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content/book/du19_multi_actuat_system_contr.md
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@@ -83,14 +83,14 @@ and the resonance \\(P\_{ri}(s)\\) can be represented as one of the following fo
#### Secondary Actuators {#secondary-actuators}
We here consider two types of secondary actuators: the PZT milliactuator (figure [1](#figure--fig:pzt-actuator)) and the microactuator.
We here consider two types of secondary actuators: the PZT milliactuator ([Figure 1](#figure--fig:pzt-actuator)) and the microactuator.
<a id="figure--fig:pzt-actuator"></a>
{{< figure src="/ox-hugo/du19_pzt_actuator.png" caption="<span class=\"figure-number\">Figure 1: </span>A PZT-actuator suspension" >}}
There are three popular types of micro-actuators: electrostatic moving-slider microactuator, PZT slider-driven microactuator and thermal microactuator.
There characteristics are shown on table [1](#table--tab:microactuator).
There characteristics are shown on [Table 1](#table--tab:microactuator).
<a id="table--tab:microactuator"></a>
<div class="table-caption">
@@ -107,7 +107,7 @@ There characteristics are shown on table [1](#table--tab:microactuator).
### Single-Stage Actuation Systems {#single-stage-actuation-systems}
A typical closed-loop control system is shown on figure [2](#figure--fig:single-stage-control), where \\(P\_v(s)\\) and \\(C(z)\\) represent the actuator system and its controller.
A typical closed-loop control system is shown on [Figure 2](#figure--fig:single-stage-control), where \\(P\_v(s)\\) and \\(C(z)\\) represent the actuator system and its controller.
<a id="figure--fig:single-stage-control"></a>
@@ -145,7 +145,7 @@ In view of this, the controller design for dual-stage actuation systems adopts a
### Control Schemes {#control-schemes}
A popular control scheme for dual-stage actuation system is the **decoupled structure** as shown in figure [4](#figure--fig:decoupled-control).
A popular control scheme for dual-stage actuation system is the **decoupled structure** as shown in [Figure 4](#figure--fig:decoupled-control).
- \\(C\_v(z)\\) and \\(C\_p(z)\\) are the controllers respectively, for the primary VCM actuator \\(P\_v(s)\\) and the secondary actuator \\(P\_p(s)\\).
- \\(\hat{P}\_p(z)\\) is an approximation of \\(P\_p\\) to estimate \\(y\_p\\).
@@ -175,7 +175,7 @@ The sensitivity functions of the VCM loop and the secondary actuator loop are
And we obtain that the dual-stage sensitivity function \\(S(z)\\) is the product of \\(S\_v(z)\\) and \\(S\_p(z)\\).
Thus, the dual-stage system control design can be decoupled into two independent controller designs.
Another type of control scheme is the **parallel structure** as shown in figure [5](#figure--fig:parallel-control-structure).
Another type of control scheme is the **parallel structure** as shown in [Figure 5](#figure--fig:parallel-control-structure).
The open-loop transfer function from \\(pes\\) to \\(y\\) is
\\[ G(z) = P\_p(z) C\_p(z) + P\_v(z) C\_v(z) \\]
@@ -192,7 +192,7 @@ Because of the limited displacement range of the secondary actuator, the control
### Controller Design Method in the Continuous-Time Domain {#controller-design-method-in-the-continuous-time-domain}
\\(\mathcal{H}\_\infty\\) loop shaping method is used to design the controllers for the primary and secondary actuators.
The structure of the \\(\mathcal{H}\_\infty\\) loop shaping method is plotted in figure [6](#figure--fig:h-inf-diagram) where \\(W(s)\\) is a weighting function relevant to the designed control system performance such as the sensitivity function.
The structure of the \\(\mathcal{H}\_\infty\\) loop shaping method is plotted in [Figure 6](#figure--fig:h-inf-diagram) where \\(W(s)\\) is a weighting function relevant to the designed control system performance such as the sensitivity function.
For a plant model \\(P(s)\\), a controller \\(C(s)\\) is to be designed such that the closed-loop system is stable and
@@ -206,7 +206,7 @@ is satisfied, where \\(T\_{zw}\\) is the transfer function from \\(w\\) to \\(z\
{{< figure src="/ox-hugo/du19_h_inf_diagram.png" caption="<span class=\"figure-number\">Figure 6: </span>Block diagram for \\(\mathcal{H}\_\infty\\) loop shaping method to design the controller \\(C(s)\\) with the weighting function \\(W(s)\\)" >}}
Equation [1](#org60aa04e) means that \\(S(s)\\) can be shaped similarly to the inverse of the chosen weighting function \\(W(s)\\).
Equation [ 1](#orgc94b8e5) means that \\(S(s)\\) can be shaped similarly to the inverse of the chosen weighting function \\(W(s)\\).
One form of \\(W(s)\\) is taken as
\begin{equation}
@@ -219,13 +219,13 @@ The controller can then be synthesis using the linear matrix inequality (LMI) ap
The primary and secondary actuator control loops are designed separately for the dual-stage control systems.
But when designing their respective controllers, certain performances are required for the two actuators, so that control efforts for the two actuators are distributed properly and the actuators don't conflict with each other's control authority.
As seen in figure [7](#figure--fig:dual-stage-loop-gain), the VCM primary actuator open loop has a higher gain at low frequencies, and the secondary actuator open loop has a higher gain in the high-frequency range.
As seen in [Figure 7](#figure--fig:dual-stage-loop-gain), the VCM primary actuator open loop has a higher gain at low frequencies, and the secondary actuator open loop has a higher gain in the high-frequency range.
<a id="figure--fig:dual-stage-loop-gain"></a>
{{< figure src="/ox-hugo/du19_dual_stage_loop_gain.png" caption="<span class=\"figure-number\">Figure 7: </span>Frequency responses of \\(G\_v(s) = C\_v(s)P\_v(s)\\) (solid line) and \\(G\_p(s) = C\_p(s) P\_p(s)\\) (dotted line)" >}}
The sensitivity functions are shown in figure [8](#figure--fig:dual-stage-sensitivity), where the hump of \\(S\_v\\) is arranged within the bandwidth of \\(S\_p\\) and the hump of \\(S\_p\\) is lowered as much as possible.
The sensitivity functions are shown in [Figure 8](#figure--fig:dual-stage-sensitivity), where the hump of \\(S\_v\\) is arranged within the bandwidth of \\(S\_p\\) and the hump of \\(S\_p\\) is lowered as much as possible.
This needs to decrease the bandwidth of the primary actuator loop and increase the bandwidth of the secondary actuator loop.
<a id="figure--fig:dual-stage-sensitivity"></a>
@@ -261,7 +261,7 @@ A VCM actuator is used as the first-stage actuator denoted by \\(P\_v(s)\\), a P
### Control Strategy and Controller Design {#control-strategy-and-controller-design}
Figure [9](#figure--fig:three-stage-control) shows the control structure for the three-stage actuation system.
[Figure 9](#figure--fig:three-stage-control) shows the control structure for the three-stage actuation system.
The control scheme is based on the decoupled master-slave dual-stage control and the third stage microactuator is added in parallel with the dual-stage control system.
The parallel format is advantageous to the overall control bandwidth enhancement, especially for the microactuator having limited stroke which restricts the bandwidth of its own loop.
@@ -296,13 +296,13 @@ The PZT actuated milliactuator \\(P\_p(s)\\) works under a reasonably high bandw
The third-stage actuator \\(P\_m(s)\\) is used to further push the bandwidth as high as possible.
The control performances of both the VCM and the PZT actuators are limited by their dominant resonance modes.
The open-loop frequency responses of the three stages are shown on figure [10](#figure--fig:open-loop-three-stage).
The open-loop frequency responses of the three stages are shown on [Figure 10](#figure--fig:open-loop-three-stage).
<a id="figure--fig:open-loop-three-stage"></a>
{{< figure src="/ox-hugo/du19_open_loop_three_stage.png" caption="<span class=\"figure-number\">Figure 10: </span>Frequency response of the open-loop transfer function" >}}
The obtained sensitivity function is shown on figure [11](#figure--fig:sensitivity-three-stage).
The obtained sensitivity function is shown on [Figure 11](#figure--fig:sensitivity-three-stage).
<a id="figure--fig:sensitivity-three-stage"></a>
@@ -319,7 +319,7 @@ Otherwise, saturation will occur in the control loop and the control system perf
Therefore, the stroke specification of the actuators, especially milliactuator and microactuators, is very important for achievable control performance.
Higher stroke actuators have stronger abilities to make sure that the control performances are not degraded in the presence of external vibrations.
For the three-stage control architecture as shown on figure [9](#figure--fig:three-stage-control), the position error is
For the three-stage control architecture as shown on [Figure 9](#figure--fig:three-stage-control), the position error is
\\[ e = -S(P\_v d\_1 + d\_2 + d\_e) + S n \\]
The control signals and positions of the actuators are given by
@@ -335,11 +335,11 @@ Higher bandwidth/higher level of disturbance generally means high stroke needed.
### Different Configurations of the Control System {#different-configurations-of-the-control-system}
A decoupled control structure can be used for the three-stage actuation system (see figure [12](#figure--fig:three-stage-decoupled)).
A decoupled control structure can be used for the three-stage actuation system (see [Figure 12](#figure--fig:three-stage-decoupled)).
The overall sensitivity function is
\\[ S(z) = \approx S\_v(z) S\_p(z) S\_m(z) \\]
with \\(S\_v(z)\\) and \\(S\_p(z)\\) are defined in equation [1](#org3237465) and
with \\(S\_v(z)\\) and \\(S\_p(z)\\) are defined in equation [ 1](#org5626095) and
\\[ S\_m(z) = \frac{1}{1 + P\_m(z) C\_m(z)} \\]
Denote the dual-stage open-loop transfer function as \\(G\_d\\)
@@ -360,7 +360,7 @@ The control signals and the positions of the three actuators are
u\_v &= C\_v(1 + \hat{P}\_p C\_p) (1 + \hat{P}\_m C\_m) e, \ y\_v = P\_v u\_v
\end{align\*}
The decoupled configuration makes the low frequency gain much higher, and consequently there is much better rejection capability at low frequency compared to the parallel architecture (see figure [13](#figure--fig:three-stage-decoupled-loop-gain)).
The decoupled configuration makes the low frequency gain much higher, and consequently there is much better rejection capability at low frequency compared to the parallel architecture (see [Figure 13](#figure--fig:three-stage-decoupled-loop-gain)).
<a id="figure--fig:three-stage-decoupled-loop-gain"></a>

260
content/book/ewins00_modal.md
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@@ -161,7 +161,7 @@ Indeed, we shall see later how these predictions can be quite detailed, to the p
The main measurement technique studied are those which will permit to make **direct measurements of the various FRF** properties of the test structure.
The type of test best suited to FRF measurement is shown in figure [1](#figure--fig:modal-analysis-schematic).
The type of test best suited to FRF measurement is shown in [Figure 1](#figure--fig:modal-analysis-schematic).
<a id="figure--fig:modal-analysis-schematic"></a>
@@ -233,7 +233,7 @@ Thus there is **no single modal analysis method**, but rater a selection, each b
One of the most widespread and useful approaches is known as the **single-degree-of-freedom curve-fit**, or often as the **circle fit** procedure.
This method uses the fact that **at frequencies close to a natural frequency**, the FRF can often be **approximated to that of a single degree-of-freedom system** plus a constant offset term (which approximately accounts for the existence of other modes).
This assumption allows us to use the circular nature of a modulus/phase polar plot of the frequency response function of a SDOF system (see figure [2](#figure--fig:sdof-modulus-phase)).
This assumption allows us to use the circular nature of a modulus/phase polar plot of the frequency response function of a SDOF system (see [Figure 2](#figure--fig:sdof-modulus-phase)).
This process can be **repeated** for each resonance individually until the whole curve has been analyzed.
At this stage, a theoretical regeneration of the FRF is possible using the set of coefficients extracted.
@@ -272,7 +272,7 @@ Even though the same overall procedure is always followed, there will be a **dif
Theoretical foundations of modal testing are of paramount importance to its successful implementation.
The three phases through a typical theoretical vibration analysis progresses are shown on figure [3](#figure--fig:vibration-analysis-procedure).
The three phases through a typical theoretical vibration analysis progresses are shown on [Figure 3](#figure--fig:vibration-analysis-procedure).
Generally, we start with a description of the structure's physical characteristics (mass, stiffness and damping properties), this is referred to as the **Spatial model**.
<a id="figure--fig:vibration-analysis-procedure"></a>
@@ -295,7 +295,7 @@ Thus our response model will consist of a set of **frequency response functions
<div class="important">
As indicated in figure [3](#figure--fig:vibration-analysis-procedure), it is also possible to do an analysis in the reverse directly: from a description of the response properties (FRFs), we can deduce modal properties and the spatial properties: this is the **experimental route** to vibration analysis.
As indicated in [Figure 3](#figure--fig:vibration-analysis-procedure), it is also possible to do an analysis in the reverse directly: from a description of the response properties (FRFs), we can deduce modal properties and the spatial properties: this is the **experimental route** to vibration analysis.
</div>
@@ -314,7 +314,7 @@ Three classes of system model will be described:
</div>
The basic model for the SDOF system is shown in figure [4](#figure--fig:sdof-model) where \\(f(t)\\) and \\(x(t)\\) are general time-varying force and displacement response quantities.
The basic model for the SDOF system is shown in [Figure 4](#figure--fig:sdof-model) where \\(f(t)\\) and \\(x(t)\\) are general time-varying force and displacement response quantities.
The spatial model consists of a **mass** \\(m\\), a **spring** \\(k\\) and (when damped) either a **viscous dashpot** \\(c\\) or **hysteretic damper** \\(d\\).
<a id="figure--fig:sdof-model"></a>
@@ -392,7 +392,7 @@ which is a single mode of vibration with a complex natural frequency having two
- **An imaginary or oscillatory part**
- **A real or decay part**
The physical significance of these two parts is illustrated in the typical free response plot shown in figure [5](#figure--fig:sdof-response)
The physical significance of these two parts is illustrated in the typical free response plot shown in [Figure 5](#figure--fig:sdof-response)
<a id="figure--fig:sdof-response"></a>
@@ -424,26 +424,26 @@ which is now complex, containing both magnitude and phase information:
All structures exhibit a degree of damping due to the **hysteresis properties** of the material(s) from which they are made.
A typical example of this effect is shown in the force displacement plot in figure [1](#org-target--fig:material_histeresis) in which the **area contained by the loop represents the energy lost in one cycle of vibration** between the extremities shown.
A typical example of this effect is shown in the force displacement plot in [ 1](#org-target--fig-material-histeresis) in which the **area contained by the loop represents the energy lost in one cycle of vibration** between the extremities shown.
The maximum energy stored corresponds to the elastic energy of the structure at the point of maximum deflection.
The damping effect of such a component can conveniently be defined by the ratio of these two:
\\[ \tcmbox{\text{damping capacity} = \frac{\text{energy lost per cycle}}{\text{maximum energy stored}}} \\]
<a id="table--fig:force-deflection-characteristics"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--fig:force-deflection-characteristics">Table 1</a></span>:
<span class="table-number"><a href="#table--fig:force-deflection-characteristics">Table 1</a>:</span>
Force-deflection characteristics
</div>
| ![](/ox-hugo/ewins00_material_histeresis.png) | ![](/ox-hugo/ewins00_dry_friction.png) | ![](/ox-hugo/ewins00_viscous_damper.png) |
|-----------------------------------------------------------------------------------------------|---------------------------------------------------------------------------------|-------------------------------------------------------------------------------------|
| <span class="org-target" id="org-target--fig:material_histeresis"></span> Material hysteresis | <span class="org-target" id="org-target--fig:dry_friction"></span> Dry friction | <span class="org-target" id="org-target--fig:viscous_damper"></span> Viscous damper |
| <span class="org-target" id="org-target--fig-material-histeresis"></span> Material hysteresis | <span class="org-target" id="org-target--fig-dry-friction"></span> Dry friction | <span class="org-target" id="org-target--fig-viscous-damper"></span> Viscous damper |
| height=2cm | height=2cm | height=2cm |
Another common source of energy dissipation in practical structures, is the **friction** which exist in joints between components of the structure.
It may be described very roughly by the simple **dry friction model** shown in figure [1](#org-target--fig:dry_friction).
It may be described very roughly by the simple **dry friction model** shown in [ 1](#org-target--fig-dry-friction).
The mathematical model of the **viscous damper** which we have used can be compared with these more physical effects by plotting the corresponding force-displacement diagram for it, and this is shown in figure [1](#org-target--fig:viscous_damper).
The mathematical model of the **viscous damper** which we have used can be compared with these more physical effects by plotting the corresponding force-displacement diagram for it, and this is shown in [ 1](#org-target--fig-viscous-damper).
Because the relationship is linear between force and velocity, it is necessary to suppose harmonic motion, at frequency \\(\omega\\), in order to construct a force-displacement diagram.
The resulting diagram shows the nature of the approximation provided by the viscous damper model and the concept of the **effective or equivalent viscous damping coefficient** for any of the actual phenomena as being which provides the **same energy loss per cycle** as the real thing.
@@ -503,7 +503,7 @@ Similarly we could use the acceleration parameter so we could define a third FRF
</div>
Table [2](#table--tab:frf-alternatives) gives details of all six of the FRF parameters and of the names used for them.
[Table 2](#table--tab:frf-alternatives) gives details of all six of the FRF parameters and of the names used for them.
**Inverse response** can also be defined. For instance, the **dynamic stiffness** is defined as the force over the displacement.
@@ -521,7 +521,7 @@ It should be noted that that the use of displacement as the response is greatly
<a id="table--tab:frf-alternatives"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--tab:frf-alternatives">Table 2</a></span>:
<span class="table-number"><a href="#table--tab:frf-alternatives">Table 2</a>:</span>
Definition of Frequency Response Functions
</div>
@@ -549,17 +549,17 @@ Any simple plot can only show two of the three quantities and so there are diffe
##### Bode Plot {#bode-plot}
Bode plot are usually displayed using logarithmic scales as shown on figure [3](#table--fig:bode-plots).
Bode plot are usually displayed using logarithmic scales as shown on [Table 3](#table--fig:bode-plots).
<a id="table--fig:bode-plots"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--fig:bode-plots">Table 3</a></span>:
<span class="table-number"><a href="#table--fig:bode-plots">Table 3</a>:</span>
FRF plots for undamped SDOF system
</div>
| ![](/ox-hugo/ewins00_bode_receptance.png) | ![](/ox-hugo/ewins00_bode_mobility.png) | ![](/ox-hugo/ewins00_bode_accelerance.png) |
|--------------------------------------------------------------------------------------|----------------------------------------------------------------------------------|----------------------------------------------------------------------------------------|
| <span class="org-target" id="org-target--fig:bode_receptance"></span> Receptance FRF | <span class="org-target" id="org-target--fig:bode_mobility"></span> Mobility FRF | <span class="org-target" id="org-target--fig:bode_accelerance"></span> Accelerance FRF |
| <span class="org-target" id="org-target--fig-bode-receptance"></span> Receptance FRF | <span class="org-target" id="org-target--fig-bode-mobility"></span> Mobility FRF | <span class="org-target" id="org-target--fig-bode-accelerance"></span> Accelerance FRF |
| width=\linewidth | width=\linewidth | width=\linewidth |
Each plot can be divided into three regimes:
@@ -571,18 +571,18 @@ Each plot can be divided into three regimes:
##### Real part and Imaginary part of FRF {#real-part-and-imaginary-part-of-frf}
Real and imaginary part of a receptance FRF of a damped SDOF system is shown on figure [4](#table--fig:plot-receptance-real-imag).
Real and imaginary part of a receptance FRF of a damped SDOF system is shown on [Table 4](#table--fig:plot-receptance-real-imag).
This type of display is not widely used as we cannot use logarithmic axes (as we have to show positive and negative values).
<a id="table--fig:plot-receptance-real-imag"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--fig:plot-receptance-real-imag">Table 4</a></span>:
<span class="table-number"><a href="#table--fig:plot-receptance-real-imag">Table 4</a>:</span>
Plot of real and imaginary part for the receptance of a damped SDOF
</div>
| ![](/ox-hugo/ewins00_plot_receptance_real.png) | ![](/ox-hugo/ewins00_plot_receptance_imag.png) |
|--------------------------------------------------------------------------------------|-------------------------------------------------------------------------------------------|
| <span class="org-target" id="org-target--fig:plot_receptance_real"></span> Real part | <span class="org-target" id="org-target--fig:plot_receptance_imag"></span> Imaginary part |
| <span class="org-target" id="org-target--fig-plot-receptance-real"></span> Real part | <span class="org-target" id="org-target--fig-plot-receptance-imag"></span> Imaginary part |
| width=\linewidth | width=\linewidth |
@@ -590,34 +590,34 @@ This type of display is not widely used as we cannot use logarithmic axes (as we
It can be seen from the expression of the inverse receptance <eq:dynamic_stiffness> that the Real part depends entirely on the mass and stiffness properties while the Imaginary part is a only function of the damping.
Figure [5](#org-target--fig:inverse_frf_mixed) shows an example of a plot of a system with a combination of both viscous and structural damping. The imaginary part is a straight line whose slope is given by the viscous damping rate \\(c\\) and whose intercept at \\(\omega = 0\\) is provided by the structural damping coefficient \\(d\\).
[ 5](#org-target--fig-inverse-frf-mixed) shows an example of a plot of a system with a combination of both viscous and structural damping. The imaginary part is a straight line whose slope is given by the viscous damping rate \\(c\\) and whose intercept at \\(\omega = 0\\) is provided by the structural damping coefficient \\(d\\).
<a id="table--fig:inverse-frf"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--fig:inverse-frf">Table 5</a></span>:
<span class="table-number"><a href="#table--fig:inverse-frf">Table 5</a>:</span>
Inverse FRF plot for the system
</div>
| ![](/ox-hugo/ewins00_inverse_frf_mixed.png) | ![](/ox-hugo/ewins00_inverse_frf_viscous.png) |
|-------------------------------------------------------------------------------|-----------------------------------------------------------------------------------|
| <span class="org-target" id="org-target--fig:inverse_frf_mixed"></span> Mixed | <span class="org-target" id="org-target--fig:inverse_frf_viscous"></span> Viscous |
| <span class="org-target" id="org-target--fig-inverse-frf-mixed"></span> Mixed | <span class="org-target" id="org-target--fig-inverse-frf-viscous"></span> Viscous |
| width=\linewidth | width=\linewidth |
##### Real part vs Imaginary part of FRF {#real-part-vs-imaginary-part-of-frf}
Figure [6](#table--fig:nyquist-receptance) shows Nyquist type FRF plots of a viscously damped SDOF system.
[Table 6](#table--fig:nyquist-receptance) shows Nyquist type FRF plots of a viscously damped SDOF system.
The missing information (in this case, the frequency) must be added by identifying the values of frequency corresponding to particular points on the curve.
<a id="table--fig:nyquist-receptance"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--fig:nyquist-receptance">Table 6</a></span>:
<span class="table-number"><a href="#table--fig:nyquist-receptance">Table 6</a>:</span>
Nyquist FRF plots of the mobility for a SDOF system
</div>
| ![](/ox-hugo/ewins00_nyquist_receptance_viscous.png) | ![](/ox-hugo/ewins00_nyquist_receptance_structural.png) |
|--------------------------------------------------------------------------------------------------|--------------------------------------------------------------------------------------------------------|
| <span class="org-target" id="org-target--fig:nyquist_receptance_viscous"></span> Viscous damping | <span class="org-target" id="org-target--fig:nyquist_receptance_structural"></span> Structural damping |
| <span class="org-target" id="org-target--fig-nyquist-receptance-viscous"></span> Viscous damping | <span class="org-target" id="org-target--fig-nyquist-receptance-structural"></span> Structural damping |
| width=\linewidth | width=\linewidth |
The Nyquist plot has the particularity of distorting the plot so as to focus on the resonance area.
@@ -1110,24 +1110,24 @@ Equally, in a real mode, all parts of the structure pass through their **zero de
</div>
While the real mode has the appearance of a **standing wave**, the complex mode is better described as exhibiting **traveling waves** (illustrated on figure [6](#figure--fig:real-complex-modes)).
While the real mode has the appearance of a **standing wave**, the complex mode is better described as exhibiting **traveling waves** (illustrated on [Figure 6](#figure--fig:real-complex-modes)).
<a id="figure--fig:real-complex-modes"></a>
{{< figure src="/ox-hugo/ewins00_real_complex_modes.png" caption="<span class=\"figure-number\">Figure 6: </span>Real and complex mode shapes displays" >}}
Another method of displaying **modal complexity** is by plotting the elements of the eigenvector on an **Argand diagram**, such as the ones shown in figure [7](#table--fig:argand-diagram).
Another method of displaying **modal complexity** is by plotting the elements of the eigenvector on an **Argand diagram**, such as the ones shown in [Table 7](#table--fig:argand-diagram).
Note that the almost-real mode shape does not necessarily have vector elements with near \\(\SI{0}{\degree}\\) or near \\(\SI{180}{\degree}\\) phase, what matters are the **relative phases** between the different elements.
<a id="table--fig:argand-diagram"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--fig:argand-diagram">Table 7</a></span>:
<span class="table-number"><a href="#table--fig:argand-diagram">Table 7</a>:</span>
Complex mode shapes plotted on Argand diagrams
</div>
| ![](/ox-hugo/ewins00_argand_diagram_a.png) | ![](/ox-hugo/ewins00_argand_diagram_b.png) | ![](/ox-hugo/ewins00_argand_diagram_c.png) |
|-----------------------------------------------------------------------------------------|-------------------------------------------------------------------------------------|----------------------------------------------------------------------------------------------|
| <span class="org-target" id="org-target--fig:argand_diagram_a"></span> Almost-real mode | <span class="org-target" id="org-target--fig:argand_diagram_b"></span> Complex Mode | <span class="org-target" id="org-target--fig:argand_diagram_c"></span> Measure of complexity |
| <span class="org-target" id="org-target--fig-argand-diagram-a"></span> Almost-real mode | <span class="org-target" id="org-target--fig-argand-diagram-b"></span> Complex Mode | <span class="org-target" id="org-target--fig-argand-diagram-c"></span> Measure of complexity |
| width=\linewidth | width=\linewidth | width=\linewidth |
@@ -1137,7 +1137,7 @@ There exist few indicators of the modal complexity.
The first one, a simple and crude one, called **MCF1** consists of summing all the phase differences between every combination of two eigenvector elements:
\\[ \text{MCF1} = \sum\_{j=1}^N \sum\_{k=1 \neq j}^N (\theta\_{rj} - \theta\_{rk}) \\]
The second measure is shown on figure [7](#org-target--fig:argand_diagram_c) where a polygon is drawn around the extremities of the individual vectors.
The second measure is shown on [ 7](#org-target--fig-argand-diagram-c) where a polygon is drawn around the extremities of the individual vectors.
The obtained area of this polygon is then compared with the area of the circle which is based on the length of the largest vector element. The resulting ratio is used as an indication of the complexity of the mode, and is defined as **MCF2**.
@@ -1177,11 +1177,11 @@ The second definition comes from the general form of the FRF expression:
Here \\(C\_r\\) may be complex whereas \\(D\_r\\) is real.
\\(\omega\_r\\) is in general different to both \\(\bar{\omega}\_r\\) and \\(\omega\_r^\prime\\).
Table [8](#table--tab:frf-natural-frequencies) summarizes all the different cases.
[Table 8](#table--tab:frf-natural-frequencies) summarizes all the different cases.
<a id="table--tab:frf-natural-frequencies"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--tab:frf-natural-frequencies">Table 8</a></span>:
<span class="table-number"><a href="#table--tab:frf-natural-frequencies">Table 8</a>:</span>
FRF Formulae and Natural Frequencies
</div>
@@ -1233,21 +1233,21 @@ We write \\(\alpha\_{11}\\) the point FRF and \\(\alpha\_{21}\\) the transfer FR
It can be seen that the only difference between the point and transfer receptance is in the sign of the modal constant of the second mode.
Consider the first point mobility (figure [9](#org-target--fig:mobility_frf_mdof_point)), between the two resonances, the two components have opposite signs so that they are substractive rather than additive, and indeed, at the point where they cross, their sum is zero.
Consider the first point mobility ([ 9](#org-target--fig-mobility-frf-mdof-point)), between the two resonances, the two components have opposite signs so that they are substractive rather than additive, and indeed, at the point where they cross, their sum is zero.
On a logarithmic plot, this produces the antiresonance characteristic which reflects that of the resonance.
<a id="table--fig:mobility-frf-mdof"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--fig:mobility-frf-mdof">Table 9</a></span>:
<span class="table-number"><a href="#table--fig:mobility-frf-mdof">Table 9</a>:</span>
Mobility FRF plot for undamped 2DOF system
</div>
| ![](/ox-hugo/ewins00_mobility_frf_mdof_point.png) | ![](/ox-hugo/ewins00_mobility_frf_mdof_transfer.png) |
|-----------------------------------------------------------------------------------------|-----------------------------------------------------------------------------------------------|
| <span class="org-target" id="org-target--fig:mobility_frf_mdof_point"></span> Point FRF | <span class="org-target" id="org-target--fig:mobility_frf_mdof_transfer"></span> Transfer FRF |
| <span class="org-target" id="org-target--fig-mobility-frf-mdof-point"></span> Point FRF | <span class="org-target" id="org-target--fig-mobility-frf-mdof-transfer"></span> Transfer FRF |
| width=\linewidth | width=\linewidth |
For the plot in figure [9](#org-target--fig:mobility_frf_mdof_transfer), between the two resonances, the two components have the same sign and they add up, no antiresonance is present.
For the plot in [ 9](#org-target--fig-mobility-frf-mdof-transfer), between the two resonances, the two components have the same sign and they add up, no antiresonance is present.
##### FRF modulus plots for MDOF systems {#frf-modulus-plots-for-mdof-systems}
@@ -1263,7 +1263,7 @@ If they have apposite signs, there will not be an antiresonance.
##### Bode plots {#bode-plots}
The resonances and antiresonances are blunted by the inclusion of damping, and the phase angles are no longer exactly \\(\SI{0}{\degree}\\) or \\(\SI{180}{\degree}\\), but the general appearance of the plot is a natural extension of that for the system without damping.
Figure [7](#figure--fig:frf-damped-system) shows a plot for the same mobility as appears in figure [9](#org-target--fig:mobility_frf_mdof_point) but here for a system with added damping.
[Figure 7](#figure--fig:frf-damped-system) shows a plot for the same mobility as appears in [ 9](#org-target--fig-mobility-frf-mdof-point) but here for a system with added damping.
Most mobility plots have this general form as long as the modes are relatively well-separated.
@@ -1278,34 +1278,34 @@ This condition is satisfied unless the separation between adjacent natural frequ
Each of the frequency response of a MDOF system in the Nyquist plot is composed of a number of SDOF components.
Figure [10](#org-target--fig:nyquist_point) shows the result of plotting the point receptance \\(\alpha\_{11}\\) for the 2DOF system described above.
[ 10](#org-target--fig-nyquist-point) shows the result of plotting the point receptance \\(\alpha\_{11}\\) for the 2DOF system described above.
The plot for the transfer receptance \\(\alpha\_{21}\\) is presented in figure [10](#org-target--fig:nyquist_transfer) where it may be seen that the opposing signs of the modal constants of the two modes have caused one of the modal circle to be in the upper half of the complex plane.
The plot for the transfer receptance \\(\alpha\_{21}\\) is presented in [ 10](#org-target--fig-nyquist-transfer) where it may be seen that the opposing signs of the modal constants of the two modes have caused one of the modal circle to be in the upper half of the complex plane.
<a id="table--fig:nyquist-frf-plots"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--fig:nyquist-frf-plots">Table 10</a></span>:
<span class="table-number"><a href="#table--fig:nyquist-frf-plots">Table 10</a>:</span>
Nyquist FRF plot for proportionally-damped system
</div>
| ![](/ox-hugo/ewins00_nyquist_point.png) | ![](/ox-hugo/ewins00_nyquist_transfer.png) |
|--------------------------------------------------------------------------------------|--------------------------------------------------------------------------------------------|
| <span class="org-target" id="org-target--fig:nyquist_point"></span> Point receptance | <span class="org-target" id="org-target--fig:nyquist_transfer"></span> Transfer receptance |
| <span class="org-target" id="org-target--fig-nyquist-point"></span> Point receptance | <span class="org-target" id="org-target--fig-nyquist-transfer"></span> Transfer receptance |
| width=\linewidth | width=\linewidth |
In the two figures [11](#org-target--fig:nyquist_nonpropdamp_point) and [11](#org-target--fig:nyquist_nonpropdamp_transfer), we show corresponding data for **non-proportional** damping.
In the two [ 11](#org-target--fig-nyquist-nonpropdamp-point) and [ 11](#org-target--fig-nyquist-nonpropdamp-transfer), we show corresponding data for **non-proportional** damping.
In this case, a relative phase has been introduced between the first and second elements of the eigenvectors: of \\(\SI{30}{\degree}\\) in mode 1 and of \\(\SI{150}{\degree}\\) in mode 2.
Now we find that the individual modal circles are no longer "upright" but are **rotated by an amount dictated by the complexity of the modal constants**.
<a id="table--fig:nyquist-frf-nonpropdamp"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--fig:nyquist-frf-nonpropdamp">Table 11</a></span>:
<span class="table-number"><a href="#table--fig:nyquist-frf-nonpropdamp">Table 11</a>:</span>
Nyquist FRF plot for non-proportionally-damped system
</div>
| ![](/ox-hugo/ewins00_nyquist_nonpropdamp_point.png) | ![](/ox-hugo/ewins00_nyquist_nonpropdamp_transfer.png) |
|--------------------------------------------------------------------------------------------------|--------------------------------------------------------------------------------------------------------|
| <span class="org-target" id="org-target--fig:nyquist_nonpropdamp_point"></span> Point receptance | <span class="org-target" id="org-target--fig:nyquist_nonpropdamp_transfer"></span> Transfer receptance |
| <span class="org-target" id="org-target--fig-nyquist-nonpropdamp-point"></span> Point receptance | <span class="org-target" id="org-target--fig-nyquist-nonpropdamp-transfer"></span> Transfer receptance |
| width=\linewidth | width=\linewidth |
@@ -1449,17 +1449,17 @@ The resulting parameter we shall call a **Spectral Density**, in this case the *
The Spectral Density is a real and even function of frequency, and does in fact provides a description of the frequency composition of the original function \\(f(t)\\).
It has units of \\(f^2/\omega\\).
Examples of random signals, autocorrelation function and power spectral density are shown on figure [12](#table--fig:random-signals).
Examples of random signals, autocorrelation function and power spectral density are shown on [Table 12](#table--fig:random-signals).
<a id="table--fig:random-signals"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--fig:random-signals">Table 12</a></span>:
<span class="table-number"><a href="#table--fig:random-signals">Table 12</a>:</span>
Random signals
</div>
| ![](/ox-hugo/ewins00_random_time.png) | ![](/ox-hugo/ewins00_random_autocorrelation.png) | ![](/ox-hugo/ewins00_random_psd.png) |
|--------------------------------------------------------------------------------|-------------------------------------------------------------------------------------------------------|-----------------------------------------------------------------------------------------|
| <span class="org-target" id="org-target--fig:random_time"></span> Time history | <span class="org-target" id="org-target--fig:random_autocorrelation"></span> Autocorrelation Function | <span class="org-target" id="org-target--fig:random_psd"></span> Power Spectral Density |
| <span class="org-target" id="org-target--fig-random-time"></span> Time history | <span class="org-target" id="org-target--fig-random-autocorrelation"></span> Autocorrelation Function | <span class="org-target" id="org-target--fig-random-psd"></span> Power Spectral Density |
| width=\linewidth | width=\linewidth | width=\linewidth |
A similar concept can be applied to a pair of functions such as \\(f(t)\\) and \\(x(t)\\) to produce **cross correlation** and **cross spectral density** functions.
@@ -1540,18 +1540,18 @@ The existence of two equations presents an opportunity to **check the quality**
There are difficulties to implement some of the above formulae in practice because of noise and other limitations concerned with the data acquisition and processing.
One technique involves **three quantities**, rather than two, in the definition of the output/input ratio.
The system considered can best be described with reference to figure [13](#table--fig:frf-determination) which shows first in [13](#org-target--fig:frf_siso_model) the traditional single-input single-output model upon which the previous formulae are based.
Then in [13](#org-target--fig:frf_feedback_model) is given a more detailed and representative model of the system which is used in a modal test.
The system considered can best be described with reference to [Table 13](#table--fig:frf-determination) which shows first in [ 13](#org-target--fig-frf-siso-model) the traditional single-input single-output model upon which the previous formulae are based.
Then in [ 13](#org-target--fig-frf-feedback-model) is given a more detailed and representative model of the system which is used in a modal test.
<a id="table--fig:frf-determination"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--fig:frf-determination">Table 13</a></span>:
<span class="table-number"><a href="#table--fig:frf-determination">Table 13</a>:</span>
System for FRF determination
</div>
| ![](/ox-hugo/ewins00_frf_siso_model.png) | ![](/ox-hugo/ewins00_frf_feedback_model.png) |
|---------------------------------------------------------------------------------------|---------------------------------------------------------------------------------------------------|
| <span class="org-target" id="org-target--fig:frf_siso_model"></span> Basic SISO model | <span class="org-target" id="org-target--fig:frf_feedback_model"></span> SISO model with feedback |
| <span class="org-target" id="org-target--fig-frf-siso-model"></span> Basic SISO model | <span class="org-target" id="org-target--fig-frf-feedback-model"></span> SISO model with feedback |
| width=\linewidth | width=\linewidth |
In this configuration, it can be seen that there are two feedback mechanisms which apply.
@@ -1570,7 +1570,7 @@ where \\(v\\) is a third signal in the system.
##### Derivation of FRF from MIMO data {#derivation-of-frf-from-mimo-data}
A diagram for the general n-input case is shown in figure [8](#figure--fig:frf-mimo).
A diagram for the general n-input case is shown in [Figure 8](#figure--fig:frf-mimo).
We obtain two alternative formulas:
@@ -1849,7 +1849,7 @@ The experimental setup used for mobility measurement contains three major items:
2. **A transduction system**. For the most part, piezoelectric transducer are used, although lasers and strain gauges are convenient because of their minimal interference with the test object. Conditioning amplifiers are used depending of the transducer used
3. **An analyzer**
A typical layout for the measurement system is shown on figure [9](#figure--fig:general-frf-measurement-setup).
A typical layout for the measurement system is shown on [Figure 9](#figure--fig:general-frf-measurement-setup).
<a id="figure--fig:general-frf-measurement-setup"></a>
@@ -1905,7 +1905,7 @@ However, we need a direct measurement of the force applied to the structure (we
The shakers are usually stiff in the orthogonal directions to the excitation.
This can modify the response of the system in those directions.
In order to avoid that, a drive rod which is stiff in one direction and flexible in the other five directions is attached between the shaker and the structure as shown on figure [10](#figure--fig:shaker-rod).
In order to avoid that, a drive rod which is stiff in one direction and flexible in the other five directions is attached between the shaker and the structure as shown on [Figure 10](#figure--fig:shaker-rod).
Typical size for the rod are \\(5\\) to \\(\SI{10}{mm}\\) long and \\(\SI{1}{mm}\\) in diameter, if the rod is longer, it may introduce the effect of its own resonances.
<a id="figure--fig:shaker-rod"></a>
@@ -1914,22 +1914,22 @@ Typical size for the rod are \\(5\\) to \\(\SI{10}{mm}\\) long and \\(\SI{1}{mm}
The support of shaker is also of primary importance.
The setup shown on figure [14](#org-target--fig:shaker_mount_1) presents the most satisfactory arrangement in which the shaker is fixed to ground while the test structure is supported by a soft spring.
The setup shown on [ 14](#org-target--fig-shaker-mount-1) presents the most satisfactory arrangement in which the shaker is fixed to ground while the test structure is supported by a soft spring.
Figure [14](#org-target--fig:shaker_mount_2) shows an alternative configuration in which the shaker itself is supported.
[ 14](#org-target--fig-shaker-mount-2) shows an alternative configuration in which the shaker itself is supported.
It may be necessary to add an additional inertia mass to the shaker in order to generate sufficient excitation forces at low frequencies.
Figure [14](#org-target--fig:shaker_mount_3) shows an unsatisfactory setup. Indeed, the response measured at \\(A\\) would not be due solely to force applied at \\(B\\), but would also be caused by the forces applied at \\(C\\).
[ 14](#org-target--fig-shaker-mount-3) shows an unsatisfactory setup. Indeed, the response measured at \\(A\\) would not be due solely to force applied at \\(B\\), but would also be caused by the forces applied at \\(C\\).
<a id="table--fig:shaker-mount"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--fig:shaker-mount">Table 14</a></span>:
<span class="table-number"><a href="#table--fig:shaker-mount">Table 14</a>:</span>
Various mounting arrangement of exciter
</div>
| ![](/ox-hugo/ewins00_shaker_mount_1.png) | ![](/ox-hugo/ewins00_shaker_mount_2.png) | ![](/ox-hugo/ewins00_shaker_mount_3.png) |
|------------------------------------------------------------------------------------------|----------------------------------------------------------------------------------------------|-------------------------------------------------------------------------------------|
| <span class="org-target" id="org-target--fig:shaker_mount_1"></span> Ideal Configuration | <span class="org-target" id="org-target--fig:shaker_mount_2"></span> Suspended Configuration | <span class="org-target" id="org-target--fig:shaker_mount_3"></span> Unsatisfactory |
| <span class="org-target" id="org-target--fig-shaker-mount-1"></span> Ideal Configuration | <span class="org-target" id="org-target--fig-shaker-mount-2"></span> Suspended Configuration | <span class="org-target" id="org-target--fig-shaker-mount-3"></span> Unsatisfactory |
| width=\linewidth | width=\linewidth | width=\linewidth |
@@ -1944,8 +1944,8 @@ The magnitude of the impact is determined by the mass of the hammer head and its
The frequency range which is effectively excited is controlled by the stiffness of the contacting surface and the mass of the impactor head: there is a resonance at a frequency given by \\(\sqrt{\frac{\text{contact stiffness}}{\text{impactor mass}}}\\) above which it is difficult to deliver energy into the test structure.
When the hammer tip impacts the test structure, this will experience a force pulse as shown on figure [11](#figure--fig:hammer-impulse).
A pulse of this type (half-sine shape) has a frequency content of the form illustrated on figure [11](#figure--fig:hammer-impulse).
When the hammer tip impacts the test structure, this will experience a force pulse as shown on [Figure 11](#figure--fig:hammer-impulse).
A pulse of this type (half-sine shape) has a frequency content of the form illustrated on [Figure 11](#figure--fig:hammer-impulse).
<a id="figure--fig:hammer-impulse"></a>
@@ -1976,7 +1976,7 @@ By suitable design, such a material may be incorporated into a device which **in
#### Force Transducers {#force-transducers}
The force transducer is the simplest type of piezoelectric transducer.
The transmitter force \\(F\\) is applied directly across the crystal, which thus generates a corresponding charge \\(q\\), proportional to \\(F\\) (figure [12](#figure--fig:piezo-force-transducer)).
The transmitter force \\(F\\) is applied directly across the crystal, which thus generates a corresponding charge \\(q\\), proportional to \\(F\\) ([Figure 12](#figure--fig:piezo-force-transducer)).
<a id="figure--fig:piezo-force-transducer"></a>
@@ -1987,7 +1987,7 @@ There exists an undesirable possibility of a cross sensitivity, i.e. an electric
#### Accelerometers {#accelerometers}
In an accelerometer, transduction is indirect and is achieved using a seismic mass (figure [13](#figure--fig:piezo-accelerometer)).
In an accelerometer, transduction is indirect and is achieved using a seismic mass ([Figure 13](#figure--fig:piezo-accelerometer)).
In this configuration, the force exerted on the crystals is the inertia force of the seismic mass (\\(m\ddot{z}\\)).
Thus, so long as the body and the seismic mass move together, the output of the transducer will be proportional to the acceleration of its body \\(x\\).
@@ -2027,19 +2027,19 @@ However, they cannot be used at such low frequencies as the charge amplifiers an
The correct installation of transducers, especially accelerometers is important.
There are various means of fixing the transducers to the surface of the test structure, some more convenient than others.
Some of these methods are illustrated in figure [15](#org-target--fig:transducer_mounting_types).
Some of these methods are illustrated in [ 15](#org-target--fig-transducer-mounting-types).
Shown on figure [15](#org-target--fig:transducer_mounting_response) are typical high frequency limits for each type of attachment.
Shown on [ 15](#org-target--fig-transducer-mounting-response) are typical high frequency limits for each type of attachment.
<a id="table--fig:transducer-mounting"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--fig:transducer-mounting">Table 15</a></span>:
<span class="table-number"><a href="#table--fig:transducer-mounting">Table 15</a>:</span>
Accelerometer attachment characteristics
</div>
| ![](/ox-hugo/ewins00_transducer_mounting_types.png) | ![](/ox-hugo/ewins00_transducer_mounting_response.png) |
|----------------------------------------------------------------------------------------------------|-----------------------------------------------------------------------------------------------------------------------|
| <span class="org-target" id="org-target--fig:transducer_mounting_types"></span> Attachment methods | <span class="org-target" id="org-target--fig:transducer_mounting_response"></span> Frequency response characteristics |
| <span class="org-target" id="org-target--fig-transducer-mounting-types"></span> Attachment methods | <span class="org-target" id="org-target--fig-transducer-mounting-response"></span> Frequency response characteristics |
| width=\linewidth | width=\linewidth |
@@ -2124,7 +2124,7 @@ That however requires \\(N\\) to be an integral power of \\(2\\).
Aliasing originates from the discretisation of the originally continuous time history.
With this discretisation process, the **existence of very high frequencies in the original signal may well be misinterpreted if the sampling rate is too slow**.
These high frequencies will be **indistinguishable** from genuine low frequency components as shown on figure [14](#figure--fig:aliasing).
These high frequencies will be **indistinguishable** from genuine low frequency components as shown on [Figure 14](#figure--fig:aliasing).
<a id="figure--fig:aliasing"></a>
@@ -2133,17 +2133,17 @@ These high frequencies will be **indistinguishable** from genuine low frequency
A signal of frequency \\(\omega\\) and one of frequency \\(\omega\_s-\omega\\) are indistinguishable and this causes a **distortion of the spectrum** measured via the DFT.
As a result, the part of the signal which has frequency components above \\(\omega\_s/2\\) will appear reflected or **aliased** in the range \\([0, \omega\_s/2]\\).
This is illustrated on figure [16](#table--fig:effect-aliasing).
This is illustrated on [Table 16](#table--fig:effect-aliasing).
<a id="table--fig:effect-aliasing"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--fig:effect-aliasing">Table 16</a></span>:
<span class="table-number"><a href="#table--fig:effect-aliasing">Table 16</a>:</span>
Alias distortion of spectrum by DFT
</div>
| ![](/ox-hugo/ewins00_aliasing_no_distortion.png) | ![](/ox-hugo/ewins00_aliasing_distortion.png) |
|------------------------------------------------------------------------------------------------------|-------------------------------------------------------------------------------------------------------|
| <span class="org-target" id="org-target--fig:aliasing_no_distortion"></span> True spectrum of signal | <span class="org-target" id="org-target--fig:aliasing_distortion"></span> Indicated spectrum from DFT |
| <span class="org-target" id="org-target--fig-aliasing-no-distortion"></span> True spectrum of signal | <span class="org-target" id="org-target--fig-aliasing-distortion"></span> Indicated spectrum from DFT |
| width=\linewidth | width=\linewidth |
The solution of the problem is to use an **anti-aliasing filter** which subjects the original time signal to a low-pass, sharp cut-off filter.
@@ -2158,18 +2158,18 @@ Leakage is a problem which is a direct **consequence of the need to take only a
<a id="table--fig:leakage"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--fig:leakage">Table 17</a></span>:
<span class="table-number"><a href="#table--fig:leakage">Table 17</a>:</span>
Sample length and leakage of spectrum
</div>
| ![](/ox-hugo/ewins00_leakage_ok.png) | ![](/ox-hugo/ewins00_leakage_nok.png) |
|-------------------------------------------------------------------------------|----------------------------------------------------------------------------------|
| <span class="org-target" id="org-target--fig:leakage_ok"></span> Ideal signal | <span class="org-target" id="org-target--fig:leakage_nok"></span> Awkward signal |
| <span class="org-target" id="org-target--fig-leakage-ok"></span> Ideal signal | <span class="org-target" id="org-target--fig-leakage-nok"></span> Awkward signal |
| width=\linewidth | width=\linewidth |
The problem is illustrated on figure [17](#table--fig:leakage).
In the first case (figure [17](#org-target--fig:leakage_ok)), the signal is perfectly periodic and the resulting spectrum is just a single line at the frequency of the sine wave.
In the second case (figure [17](#org-target--fig:leakage_nok)), the periodicity assumption is not strictly valid as there is a discontinuity at each end of the sample.
The problem is illustrated on [Table 17](#table--fig:leakage).
In the first case ([ 17](#org-target--fig-leakage-ok)), the signal is perfectly periodic and the resulting spectrum is just a single line at the frequency of the sine wave.
In the second case ([ 17](#org-target--fig-leakage-nok)), the periodicity assumption is not strictly valid as there is a discontinuity at each end of the sample.
As a result, the spectrum produced for this case does not indicate the single frequency which the original time signal possessed.
Energy has "leaked" into a number of the spectral lines close to the true frequency and the spectrum is spread over several lines.
@@ -2187,14 +2187,14 @@ Leakage is a serious problem in many applications, **ways of avoiding its effect
Windowing involves the imposition of a prescribed profile on the time signal prior to performing the Fourier transform.
The profiles, or "windows" are generally depicted as a time function \\(w(t)\\) as shown in figure [15](#figure--fig:windowing-examples).
The profiles, or "windows" are generally depicted as a time function \\(w(t)\\) as shown in [Figure 15](#figure--fig:windowing-examples).
<a id="figure--fig:windowing-examples"></a>
{{< figure src="/ox-hugo/ewins00_windowing_examples.png" caption="<span class=\"figure-number\">Figure 15: </span>Different types of window. (a) Boxcar, (b) Hanning, (c) Cosine-taper, (d) Exponential" >}}
The analyzed signal is then \\(x^\prime(t) = x(t) w(t)\\).
The result of using a window is seen in the third column of figure [15](#figure--fig:windowing-examples).
The result of using a window is seen in the third column of [Figure 15](#figure--fig:windowing-examples).
The **Hanning and Cosine Taper windows are typically used for continuous signals**, such as are produced by steady periodic or random vibration, while the **Exponential window is used for transient vibration** applications where much of the important information is concentrated in the initial part of the time record.
@@ -2210,7 +2210,7 @@ Common filters are: low-pass, high-pass, band-limited, narrow-band, notch.
#### Improving Resolution {#improving-resolution}
<span class="org-target" id="org-target--sec:improving_resolution"></span>
<span class="org-target" id="org-target--sec-improving-resolution"></span>
##### Increasing transform size {#increasing-transform-size}
@@ -2234,19 +2234,19 @@ The common solution to the need for finer frequency resolution is to zoom on the
There are various ways of achieving this result.
The easiest way is to use a frequency shifting process coupled with a controlled aliasing device.
Suppose the signal to be analyzed \\(x(t)\\) has a spectrum \\(X(\omega)\\) has shown on figure [18](#org-target--fig:zoom_range), and that we are interested in a detailed analysis between \\(\omega\_1\\) and \\(\omega\_2\\).
Suppose the signal to be analyzed \\(x(t)\\) has a spectrum \\(X(\omega)\\) has shown on [ 18](#org-target--fig-zoom-range), and that we are interested in a detailed analysis between \\(\omega\_1\\) and \\(\omega\_2\\).
If we apply a band-pass filter to the signal, as shown on figure [18](#org-target--fig:zoom_bandpass), and perform a DFT between \\(0\\) and \\((\omega\_2 - \omega\_1)\\), then because of the aliasing phenomenon described earlier, the frequency components between \\(\omega\_1\\) and \\(\omega\_2\\) will appear between \\(0\\) and \\((\omega\_2 - \omega\_1)\\) with the advantage of a finer resolution (see figure [16](#figure--fig:zoom-result)).
If we apply a band-pass filter to the signal, as shown on [ 18](#org-target--fig-zoom-bandpass), and perform a DFT between \\(0\\) and \\((\omega\_2 - \omega\_1)\\), then because of the aliasing phenomenon described earlier, the frequency components between \\(\omega\_1\\) and \\(\omega\_2\\) will appear between \\(0\\) and \\((\omega\_2 - \omega\_1)\\) with the advantage of a finer resolution (see [Figure 16](#figure--fig:zoom-result)).
<a id="table--fig:frequency-zoom"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--fig:frequency-zoom">Table 18</a></span>:
<span class="table-number"><a href="#table--fig:frequency-zoom">Table 18</a>:</span>
Controlled aliasing for frequency zoom
</div>
| ![](/ox-hugo/ewins00_zoom_range.png) | ![](/ox-hugo/ewins00_zoom_bandpass.png) |
|-----------------------------------------------------------------------------------------|--------------------------------------------------------------------------------------|
| <span class="org-target" id="org-target--fig:zoom_range"></span> Spectrum of the signal | <span class="org-target" id="org-target--fig:zoom_bandpass"></span> Band-pass filter |
| <span class="org-target" id="org-target--fig-zoom-range"></span> Spectrum of the signal | <span class="org-target" id="org-target--fig-zoom-bandpass"></span> Band-pass filter |
| width=\linewidth | width=\linewidth |
<a id="figure--fig:zoom-result"></a>
@@ -2319,7 +2319,7 @@ For instance, the typical FRF curve has large region of relatively slow changes
This is the traditional method of FRF measurement and involves the use of a sweep oscillator to provide a sinusoidal command signal with a frequency that varies slowly in the range of interest.
It is necessary to check that progress through the frequency range is sufficiently slow to check that steady-state response conditions are attained.
If excessive sweep rate is used, then distortions of the FRF plot are introduced as shown on figure [17](#figure--fig:sweep-distortions).
If excessive sweep rate is used, then distortions of the FRF plot are introduced as shown on [Figure 17](#figure--fig:sweep-distortions).
<a id="figure--fig:sweep-distortions"></a>
@@ -2437,7 +2437,7 @@ where \\(v(t)\\) is a third signal in the system, such as the voltage supplied t
It is known that a low coherence can arise in a measurement where the frequency resolution of the analyzer is not fine enough to describe adequately the very rapidly changing functions such as are encountered near resonance and anti-resonance on lightly-damped structures.
This is known as a **bias** error and leakage is often the most likely source of low coherence on lightly-damped structures as shown on figure [18](#figure--fig:coherence-resonance).
This is known as a **bias** error and leakage is often the most likely source of low coherence on lightly-damped structures as shown on [Figure 18](#figure--fig:coherence-resonance).
<a id="figure--fig:coherence-resonance"></a>
@@ -2480,7 +2480,7 @@ For the chirp and impulse excitations, each individual sample is collected and p
##### Burst excitation signals {#burst-excitation-signals}
Burst excitation signals consist of short sections of an underlying continuous signal (which may be a sine wave, a sine sweep or a random signal), followed by a period of zero output, resulting in a response which shows a transient build-up followed by a decay (see figure [19](#figure--fig:burst-excitation)).
Burst excitation signals consist of short sections of an underlying continuous signal (which may be a sine wave, a sine sweep or a random signal), followed by a period of zero output, resulting in a response which shows a transient build-up followed by a decay (see [Figure 19](#figure--fig:burst-excitation)).
<a id="figure--fig:burst-excitation"></a>
@@ -2497,7 +2497,7 @@ In the case of burst random, however, each individual burst will be different to
##### Chirp excitation {#chirp-excitation}
The chirp consist of a short duration signal which has the form shown in figure [20](#figure--fig:chirp-excitation).
The chirp consist of a short duration signal which has the form shown in [Figure 20](#figure--fig:chirp-excitation).
The frequency content of the chirp can be precisely chosen by the starting and finishing frequencies of the sweep.
@@ -2508,7 +2508,7 @@ The frequency content of the chirp can be precisely chosen by the starting and f
##### Impulsive excitation {#impulsive-excitation}
The hammer blow produces an input and response as shown in the figure [21](#figure--fig:impulsive-excitation).
The hammer blow produces an input and response as shown in the [Figure 21](#figure--fig:impulsive-excitation).
This and the chirp excitation are very similar in the analysis point of view, the main difference is that the chirp offers the possibility of greater control of both amplitude and frequency content of the input and also permits the input of a greater amount of vibration energy.
@@ -2520,7 +2520,7 @@ The frequency content of the hammer blow is dictated by the **materials** involv
However, it should be recorded that in the region below the first cut-off frequency induced by the elasticity of the hammer tip structure contact, the spectrum of the force signal tends to be **very flat**.
On some structures, the movement of the structure in response to the hammer blow can be such that it returns and **rebounds** on the hammer tip before the user has had time to move that out of the way.
In such cases, the spectrum of the excitation is seen to have "holes" in it at certain frequencies (figure [22](#figure--fig:double-hits)).
In such cases, the spectrum of the excitation is seen to have "holes" in it at certain frequencies ([Figure 22](#figure--fig:double-hits)).
<a id="figure--fig:double-hits"></a>
@@ -2595,7 +2595,7 @@ and so **what is required is the ratio of the two sensitivities**:
The overall sensitivity can be more readily obtained by a calibration process because we can easily make an independent measurement of the quantity now being measured: the ratio of response to force.
Suppose the response parameter is acceleration, then the FRF obtained is inertance which has the units of \\(1/\text{mass}\\), a quantity which can readily be independently measured by other means.
Figure [23](#figure--fig:calibration-setup) shows a typical calibration setup.
[Figure 23](#figure--fig:calibration-setup) shows a typical calibration setup.
<a id="figure--fig:calibration-setup"></a>
@@ -2610,7 +2610,7 @@ Thus, frequent checks on the overall calibration factors are strongly recommende
It is very important the ensure that the force is measured directly at the point at which it is applied to the structure, rather than deducing its magnitude from the current flowing in the shaker coil or other similar **indirect** processes.
This is because near resonance, the actual applied force becomes very small and is thus very prone to inaccuracy.
This same argument applies on a lesser scale as we examine the detail around the attachment to the structure, as shown in figure [24](#figure--fig:mass-cancellation).
This same argument applies on a lesser scale as we examine the detail around the attachment to the structure, as shown in [Figure 24](#figure--fig:mass-cancellation).
<a id="figure--fig:mass-cancellation"></a>
@@ -2667,7 +2667,7 @@ There are two problems to be tackled:
1. measurement of rotational responses
2. generation of measurement of rotation excitation
The first of these is less difficult and techniques usually use a pair a matched conventional accelerometers placed at a short distance apart on the structure to be measured as shown on figure [25](#figure--fig:rotational-measurement).
The first of these is less difficult and techniques usually use a pair a matched conventional accelerometers placed at a short distance apart on the structure to be measured as shown on [Figure 25](#figure--fig:rotational-measurement).
<a id="figure--fig:rotational-measurement"></a>
@@ -2685,7 +2685,7 @@ The principle of operation is that by measuring both accelerometer signals, the
This approach permits us to measure half of the possible FRFs: all those which are of the \\(X/F\\) and \\(\Theta/F\\) type.
The others can only be measured directly by applying a moment excitation.
Figure [26](#figure--fig:rotational-excitation) shows a device to simulate a moment excitation.
[Figure 26](#figure--fig:rotational-excitation) shows a device to simulate a moment excitation.
First, a single applied excitation force \\(F\_1\\) corresponds to a simultaneous force \\(F\_0 = F\_1\\) and a moment \\(M\_0 = -F\_1 l\_1\\).
Then, the same excitation force is applied at the second position that gives a force \\(F\_0 = F\_2\\) and moment \\(M\_0 = F\_2 l\_2\\).
By adding and subtracting the responses produced by these two separate excitations conditions, we can deduce the translational and rotational responses to the translational force and the rotational moment separately, thus enabling the measurement of all four types of FRF: \\(X/F\\), \\(\Theta/F\\), \\(X/M\\) and \\(\Theta/M\\).
@@ -3005,10 +3005,10 @@ Then, each PRF is, simply, a particular combination of the original FRFs, and th
</div>
On example of this form of pre-processing is shown on figure [19](#table--fig:PRF-numerical) for a numerically-simulation test data, and another in figure [20](#table--fig:PRF-measured) for the case of real measured test data.
On example of this form of pre-processing is shown on [Table 19](#table--fig:PRF-numerical) for a numerically-simulation test data, and another in [Table 20](#table--fig:PRF-measured) for the case of real measured test data.
The second plot [19](#org-target--fig:PRF_numerical_svd) helps to determine the true order of the system because the number of non-zero singular values is equal to this parameter.
The third plot [19](#org-target--fig:PRF_numerical_PRF) shows the genuine modes distinct from the computational modes.
The second plot [ 19](#org-target--fig-PRF-numerical-svd) helps to determine the true order of the system because the number of non-zero singular values is equal to this parameter.
The third plot [ 19](#org-target--fig-PRF-numerical-PRF) shows the genuine modes distinct from the computational modes.
<div class="important">
@@ -3028,24 +3028,24 @@ The two groups are usually separated by a clear gap (depending of the noise pres
<a id="table--fig:PRF-numerical"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--fig:PRF-numerical">Table 19</a></span>:
<span class="table-number"><a href="#table--fig:PRF-numerical">Table 19</a>:</span>
FRF and PRF characteristics for numerical model
</div>
| ![](/ox-hugo/ewins00_PRF_numerical_FRF.png) | ![](/ox-hugo/ewins00_PRF_numerical_svd.png) | ![](/ox-hugo/ewins00_PRF_numerical_PRF.png) |
|-----------------------------------------------------------------------------|-----------------------------------------------------------------------------------------|-----------------------------------------------------------------------------|
| <span class="org-target" id="org-target--fig:PRF_numerical_FRF"></span> FRF | <span class="org-target" id="org-target--fig:PRF_numerical_svd"></span> Singular Values | <span class="org-target" id="org-target--fig:PRF_numerical_PRF"></span> PRF |
| <span class="org-target" id="org-target--fig-PRF-numerical-FRF"></span> FRF | <span class="org-target" id="org-target--fig-PRF-numerical-svd"></span> Singular Values | <span class="org-target" id="org-target--fig-PRF-numerical-PRF"></span> PRF |
| width=\linewidth | width=\linewidth | width=\linewidth |
<a id="table--fig:PRF-measured"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--fig:PRF-measured">Table 20</a></span>:
<span class="table-number"><a href="#table--fig:PRF-measured">Table 20</a>:</span>
FRF and PRF characteristics for measured model
</div>
| ![](/ox-hugo/ewins00_PRF_measured_FRF.png) | ![](/ox-hugo/ewins00_PRF_measured_svd.png) | ![](/ox-hugo/ewins00_PRF_measured_PRF.png) |
|----------------------------------------------------------------------------|----------------------------------------------------------------------------------------|----------------------------------------------------------------------------|
| <span class="org-target" id="org-target--fig:PRF_measured_FRF"></span> FRF | <span class="org-target" id="org-target--fig:PRF_measured_svd"></span> Singular Values | <span class="org-target" id="org-target--fig:PRF_measured_PRF"></span> PRF |
| <span class="org-target" id="org-target--fig-PRF-measured-FRF"></span> FRF | <span class="org-target" id="org-target--fig-PRF-measured-svd"></span> Singular Values | <span class="org-target" id="org-target--fig-PRF-measured-PRF"></span> PRF |
| width=\linewidth | width=\linewidth | width=\linewidth |
@@ -3076,7 +3076,7 @@ The **Complex mode indicator function** (CMIF) is defined as
</div>
The actual mode indicator values are provided by the squares of the singular values and are usually plotted as a function of frequency in logarithmic form as shown in figure [27](#figure--fig:mifs):
The actual mode indicator values are provided by the squares of the singular values and are usually plotted as a function of frequency in logarithmic form as shown in [Figure 27](#figure--fig:mifs):
- **Natural frequencies are indicated by large values of the first CMIF** (the highest of the singular values)
- **double or multiple modes by simultaneously large values of two or more CMIF**.
@@ -3157,7 +3157,7 @@ In this method, it is assumed that close to one local mode, any effects due to t
This is a method which works adequately for structures whose FRF exhibit **well separated modes**.
This method is useful in obtaining initial estimates to the parameters.
The peak-picking method is applied as follows (illustrated on figure [28](#figure--fig:peak-amplitude)):
The peak-picking method is applied as follows (illustrated on [Figure 28](#figure--fig:peak-amplitude)):
1. First, **individual resonance peaks** are detected on the FRF plot and the maximum responses frequency \\(\omega\_r\\) is taken as the **natural frequency** of that mode
2. Second, the **local maximum value of the FRF** \\(|\hat{H}|\\) is noted and the **frequency bandwidth** of the function for a response level of \\(|\hat{H}|/\sqrt{2}\\) is determined.
@@ -3204,17 +3204,17 @@ In the case of a system assumed to have structural damping, the basic function w
\end{equation}
since the only effect of including the modal constant \\({}\_rA\_{jk}\\) is to scale the size of the circle by \\(|{}\_rA\_{jk}|\\) and to rotate it by \\(\angle {}\_rA\_{jk}\\).
A plot of the quantity \\(\alpha(\omega)\\) is given in figure [21](#org-target--fig:modal_circle).
A plot of the quantity \\(\alpha(\omega)\\) is given in [ 21](#org-target--fig-modal-circle).
<a id="table--fig:modal-circle-figures"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--fig:modal-circle-figures">Table 21</a></span>:
<span class="table-number"><a href="#table--fig:modal-circle-figures">Table 21</a>:</span>
Modal Circle
</div>
| ![](/ox-hugo/ewins00_modal_circle.png) | ![](/ox-hugo/ewins00_modal_circle_bis.png) |
|-------------------------------------------------------------------------------|-------------------------------------------------------------------------------------------------------------------|
| <span class="org-target" id="org-target--fig:modal_circle"></span> Properties | <span class="org-target" id="org-target--fig:modal_circle_bis"></span> \\(\omega\_b\\) and \\(\omega\_a\\) points |
| <span class="org-target" id="org-target--fig-modal-circle"></span> Properties | <span class="org-target" id="org-target--fig-modal-circle-bis"></span> \\(\omega\_b\\) and \\(\omega\_a\\) points |
| width=\linewidth | width=\linewidth |
For any frequency \\(\omega\\), we have the following relationship:
@@ -3252,7 +3252,7 @@ It may also be seen that an **estimate of the damping** is provided by the sweep
\end{equation}
Suppose now we have two specific points on the circle, one corresponding to a frequency \\(\omega\_b\\) below the natural frequency and the other one \\(\omega\_a\\) above the natural frequency.
Referring to figure [21](#org-target--fig:modal_circle_bis), we can write:
Referring to [ 21](#org-target--fig-modal-circle-bis), we can write:
\begin{equation}
\begin{aligned}
@@ -3318,7 +3318,7 @@ The sequence is:
3. **Locate natural frequency, obtain damping estimate**.
The rate of sweep through the region is estimated numerically and the frequency at which it reaches the maximum is deduced.
At the same time, an estimate of the damping is derived using <eq:estimate_damping_sweep_rate>.
A typical example is shown on figure [29](#figure--fig:circle-fit-natural-frequency).
A typical example is shown on [Figure 29](#figure--fig:circle-fit-natural-frequency).
4. **Calculate multiple damping estimates, and scatter**.
A set of damping estimates using all possible combination of the selected data points are computed using <eq:estimate_damping>.
Then, we can choose the damping estimate to be the mean value.
@@ -3440,18 +3440,18 @@ We need to introduce the concept of **residual terms**, necessary in the modal a
The first occasion on which the residual problem is encountered is generally at the end of the analysis of a single FRF curve, such as by the repeated application of an SDOF curve-fit to each of the resonances in turn until all modes visible on the plot have been identified.
At this point, it is often desired to construct a theoretical curve (called "**regenerated**"), based on the modal parameters extracted from the measured data, and to overlay this on the original measured data to assess the success of the curve-fit process.
Then the regenerated curve is compared with the original measurements, the result is often disappointing, as illustrated in figure [22](#org-target--fig:residual_without).
However, by the inclusion of two simple extra terms (the "**residuals**"), the modified regenerated curve is seen to correlate very well with the original experimental data as shown on figure [22](#org-target--fig:residual_with).
Then the regenerated curve is compared with the original measurements, the result is often disappointing, as illustrated in [ 22](#org-target--fig-residual-without).
However, by the inclusion of two simple extra terms (the "**residuals**"), the modified regenerated curve is seen to correlate very well with the original experimental data as shown on [ 22](#org-target--fig-residual-with).
<a id="table--fig:residual-modes"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--fig:residual-modes">Table 22</a></span>:
<span class="table-number"><a href="#table--fig:residual-modes">Table 22</a>:</span>
Effects of residual terms on FRF regeneration
</div>
| ![](/ox-hugo/ewins00_residual_without.png) | ![](/ox-hugo/ewins00_residual_with.png) |
|-----------------------------------------------------------------------------------------|------------------------------------------------------------------------------------|
| <span class="org-target" id="org-target--fig:residual_without"></span> without residual | <span class="org-target" id="org-target--fig:residual_with"></span> with residuals |
| <span class="org-target" id="org-target--fig-residual-without"></span> without residual | <span class="org-target" id="org-target--fig-residual-with"></span> with residuals |
| width=\linewidth | width=\linewidth |
If we regenerate an FRF curve from the modal parameters we have extracted from the measured data, we shall use a formula of the type
@@ -3480,7 +3480,7 @@ The three terms corresponds to:
2. the **high frequency modes** not identified
3. the **modes actually identified**
These three terms are illustrated on figure [30](#figure--fig:low-medium-high-modes).
These three terms are illustrated on [Figure 30](#figure--fig:low-medium-high-modes).
<a id="figure--fig:low-medium-high-modes"></a>
@@ -3772,17 +3772,17 @@ with
</div>
The composite function \\(HH(\omega)\\) can provide a useful means of determining a single (average) value for the natural frequency and damping factor for each mode where the individual functions would each indicate slightly different values.
As an example, a set of mobilities measured are shown individually in figure [23](#org-target--fig:composite_raw) and their summation shown as a single composite curve in figure [23](#org-target--fig:composite_sum).
As an example, a set of mobilities measured are shown individually in [ 23](#org-target--fig-composite-raw) and their summation shown as a single composite curve in [ 23](#org-target--fig-composite-sum).
<a id="table--fig:composite"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--fig:composite">Table 23</a></span>:
<span class="table-number"><a href="#table--fig:composite">Table 23</a>:</span>
Set of measured FRF
</div>
| ![](/ox-hugo/ewins00_composite_raw.png) | ![](/ox-hugo/ewins00_composite_sum.png) |
|---------------------------------------------------------------------------------------|-------------------------------------------------------------------------------------|
| <span class="org-target" id="org-target--fig:composite_raw"></span> Individual curves | <span class="org-target" id="org-target--fig:composite_sum"></span> Composite curve |
| <span class="org-target" id="org-target--fig-composite-raw"></span> Individual curves | <span class="org-target" id="org-target--fig-composite-sum"></span> Composite curve |
| width=\linewidth | width=\linewidth |
The global analysis methods have the disadvantages first, that the computation power required is high and second that there may be valid reasons why the various FRF curves exhibit slight differences in their characteristics and it may not always be appropriate to average them.
@@ -4331,8 +4331,8 @@ There are basically two choices for the graphical display of a modal model:
##### Deflected shapes {#deflected-shapes}
A static display is often adequate for depicting relatively simple mode shapes.
Measured coordinates of the test structure are first linked as shown on figure [31](#figure--fig:static-display) (a).
Then, the grid of measured coordinate points is redrawn on the same plot but this time displaced by an amount proportional to the corresponding element in the mode shape vector as shown on figure [31](#figure--fig:static-display) (b).
Measured coordinates of the test structure are first linked as shown on [Figure 31](#figure--fig:static-display) (a).
Then, the grid of measured coordinate points is redrawn on the same plot but this time displaced by an amount proportional to the corresponding element in the mode shape vector as shown on [Figure 31](#figure--fig:static-display) (b).
The elements in the vector are scaled according the normalization process used (usually mass-normalized), and their absolute magnitudes have no particular significance.
<a id="figure--fig:static-display"></a>
@@ -4344,16 +4344,16 @@ It is customary to select the largest eigenvector element and to scale the whole
##### Multiple frames {#multiple-frames}
If a series of deflection patterns that has been computed for a different instant of time are superimposed, we obtain a result as shown on figure [31](#figure--fig:static-display) (c).
If a series of deflection patterns that has been computed for a different instant of time are superimposed, we obtain a result as shown on [Figure 31](#figure--fig:static-display) (c).
Some indication of the motion of the structure can be obtained, and the points of zero motion (nodes) can be clearly identified.
It is also possible, in this format, to give some indication of the essence of complex modes, as shown in figure [31](#figure--fig:static-display) (d).
It is also possible, in this format, to give some indication of the essence of complex modes, as shown in [Figure 31](#figure--fig:static-display) (d).
Complex modes do not, in general, exhibit fixed nodal points.
##### Argand diagram plots {#argand-diagram-plots}
Another form of representation which is useful for complex modes is the representation of the individual complex elements of the eigenvectors on a polar plot, as shown in the examples of figure [31](#figure--fig:static-display) (e) and (f).
Another form of representation which is useful for complex modes is the representation of the individual complex elements of the eigenvectors on a polar plot, as shown in the examples of [Figure 31](#figure--fig:static-display) (e) and (f).
Although there is no attempt to show the physical deformation of the actual structure in this format, the complexity of the mode shape is graphically displayed.
@@ -4376,7 +4376,7 @@ We then tend to interpret this as a motion which is purely in the x-direction wh
The second problem arises when the **grid of measurement points** that is chosen to display the mode shapes is **too coarse in relation to the complexity of the deformation patterns** that are to be displayed.
This can be illustrated using a very simple example: suppose that our test structure is a straight beam, and that we decide to use just three response measurements points.
If we consider the first six modes of the beam, whose mode shapes are sketched in figure [32](#figure--fig:beam-modes), then we see that with this few measurement points, modes 1 and 5 look the same as do modes 2, 4 and 6.
If we consider the first six modes of the beam, whose mode shapes are sketched in [Figure 32](#figure--fig:beam-modes), then we see that with this few measurement points, modes 1 and 5 look the same as do modes 2, 4 and 6.
All the higher modes will be indistinguishable from these first few.
This is a well known problem of **spatial aliasing**.
@@ -4425,23 +4425,23 @@ However, it must be noted that there is an important **limitation to this proced
<div class="exampl">
As an example, suppose that FRF data \\(H\_{11}\\) and \\(H\_{21}\\) are measured and analyzed in order to synthesize the FRF \\(H\_{22}\\) initially unmeasured.
The predict curve is compared with the measurements on figure [24](#org-target--fig:H22_without_residual).
The predict curve is compared with the measurements on [ 24](#org-target--fig-H22-without-residual).
Clearly, the agreement is poor and would tend to indicate that the measurement/analysis process had not been successful.
However, the synthesized curve contained only those terms relating to the modes which had actually been studied from \\(H\_{11}\\) and \\(H\_{21}\\) and this set of modes did not include **all** the modes of the structure.
Thus, \\(H\_{22}\\) **omitted the influence of out-of-range modes**.
The inclusion of these two additional terms (obtained here only after measuring and analyzing \\(H\_{22}\\) itself) resulted in the greatly improved predicted vs measured comparison shown in figure [24](#org-target--fig:H22_with_residual).
The inclusion of these two additional terms (obtained here only after measuring and analyzing \\(H\_{22}\\) itself) resulted in the greatly improved predicted vs measured comparison shown in [ 24](#org-target--fig-H22-with-residual).
</div>
<a id="table--fig:H22"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--fig:H22">Table 24</a></span>:
<span class="table-number"><a href="#table--fig:H22">Table 24</a>:</span>
Synthesized FRF plot
</div>
| ![](/ox-hugo/ewins00_H22_without_residual.png) | ![](/ox-hugo/ewins00_H22_with_residual.png) |
|-----------------------------------------------------------------------------------------------------------|-----------------------------------------------------------------------------------------------------------|
| <span class="org-target" id="org-target--fig:H22_without_residual"></span> Using measured modal data only | <span class="org-target" id="org-target--fig:H22_with_residual"></span> After inclusion of residual terms |
| <span class="org-target" id="org-target--fig-H22-without-residual"></span> Using measured modal data only | <span class="org-target" id="org-target--fig-H22-with-residual"></span> After inclusion of residual terms |
| width=\linewidth | width=\linewidth |
The appropriate expression for a "correct" response model, derived via a set of modal properties is thus
@@ -4492,7 +4492,7 @@ If the **transmissibility** is measured during a modal test which has a single e
</div>
In general, the transmissibility **depends significantly on the excitation point** (\\({}\_iT\_{jk}(\omega) \neq {}\_qT\_{jk}(\omega)\\) where \\(q\\) is a different DOF than \\(i\\)) and it is shown on figure [33](#figure--fig:transmissibility-plots).
In general, the transmissibility **depends significantly on the excitation point** (\\({}\_iT\_{jk}(\omega) \neq {}\_qT\_{jk}(\omega)\\) where \\(q\\) is a different DOF than \\(i\\)) and it is shown on [Figure 33](#figure--fig:transmissibility-plots).
This may explain why transmissibilities are not widely used in modal analysis.
<a id="figure--fig:transmissibility-plots"></a>
@@ -4503,20 +4503,20 @@ This may explain why transmissibilities are not widely used in modal analysis.
#### Base excitation {#base-excitation}
The one application area where transmissibilities can be used as part of modal testing is in the case of **base excitation**.
Base excitation is a type of test where the input is measured as a response at the drive point \\(x\_0(t)\\), instead of as a force \\(f\_1(t)\\), as illustrated in figure [25](#table--fig:base-excitation-configuration).
Base excitation is a type of test where the input is measured as a response at the drive point \\(x\_0(t)\\), instead of as a force \\(f\_1(t)\\), as illustrated in [Table 25](#table--fig:base-excitation-configuration).
We can show that it is possible to determine, from measurements of \\(x\_i\\) and \\(x\_0\\), modal properties of natural frequency, damping factor and **unscaled** mode shape for each of the modes that are visible in the frequency range of measurement.
The fact that the excitation force is not measured is responsible for the lack of formal scaling of the mode shapes.
<a id="table--fig:base-excitation-configuration"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--fig:base-excitation-configuration">Table 25</a></span>:
<span class="table-number"><a href="#table--fig:base-excitation-configuration">Table 25</a>:</span>
Base excitation configuration
</div>
| ![](/ox-hugo/ewins00_conventional_modal_test_setup.png) | ![](/ox-hugo/ewins00_base_excitation_modal_setup.png) |
|-------------------------------------------------------------------------------------------------------------------|---------------------------------------------------------------------------------------------------------|
| <span class="org-target" id="org-target--fig:conventional_modal_test_setup"></span> Conventional modal test setup | <span class="org-target" id="org-target--fig:base_excitation_modal_setup"></span> Base excitation setup |
| <span class="org-target" id="org-target--fig-conventional-modal-test-setup"></span> Conventional modal test setup | <span class="org-target" id="org-target--fig-base-excitation-modal-setup"></span> Base excitation setup |
| height=4cm | height=4cm |
@@ -4560,5 +4560,5 @@ Because the rank of each pseudo matrix is less than its order, it cannot be inve
## Bibliography {#bibliography}
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Ewins, DJ. 2000. <i>Modal Testing: Theory, Practice and Application</i>. <i>Research Studies Pre, 2nd Ed., Isbn-13</i>. Baldock, Hertfordshire, England Philadelphia, PA: Wiley-Blackwell.</div>
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Ewins, DJ. 2000. <i>Modal Testing: Theory, Practice and Application</i>. <i>Research Studies Pre, 2nd Ed., ISBN-13</i>. Baldock, Hertfordshire, England Philadelphia, PA: Wiley-Blackwell.</div>
</div>

12
content/book/fleming14_desig_model_contr_nanop_system.md
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@@ -732,7 +732,7 @@ Year
{{< figure src="/ox-hugo/fleming14_amplifier_model.png" caption="<span class=\"figure-number\">Figure 1: </span>A voltage source \\(V\_s\\) driving a piezoelectric load. The actuator is modeled by a capacitance \\(C\_p\\) and strain-dependent voltage source \\(V\_p\\). The resistance \\(R\_s\\) is the output impedance and \\(L\\) the cable inductance." >}}
Consider the electrical circuit shown in Figure [1](#figure--fig:fleming14-amplifier-model) where a voltage source is connected to a piezoelectric actuator.
Consider the electrical circuit shown in [Figure 1](#figure--fig:fleming14-amplifier-model) where a voltage source is connected to a piezoelectric actuator.
The actuator is modeled as a capacitance \\(C\_p\\) in series with a strain-dependent voltage source \\(V\_p\\).
The resistance \\(R\_s\\) and inductance \\(L\\) are the source impedance and the cable inductance respectively.
@@ -768,11 +768,11 @@ If the inductance \\(L\\) is neglected, the transfer function from source voltag
This is thus highly dependent of the load.
The high capacitive impedance nature of piezoelectric loads introduces phase-lag into the feedback path.
A rule of thumb is that closed-loop bandwidth cannot exceed one-tenth the cut-off frequency of the pole formed by the amplifier output impedance \\(R\_s\\) and load capacitance \\(C\_p\\) (see Table [1](#table--tab:piezo-limitation-Rs) for values).
A rule of thumb is that closed-loop bandwidth cannot exceed one-tenth the cut-off frequency of the pole formed by the amplifier output impedance \\(R\_s\\) and load capacitance \\(C\_p\\) (see [Table 1](#table--tab:piezo-limitation-Rs) for values).
<a id="table--tab:piezo-limitation-Rs"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--tab:piezo-limitation-Rs">Table 1</a></span>:
<span class="table-number"><a href="#table--tab:piezo-limitation-Rs">Table 1</a>:</span>
Bandwidth limitation due to \(R_s\)
</div>
@@ -784,11 +784,11 @@ A rule of thumb is that closed-loop bandwidth cannot exceed one-tenth the cut-of
The inductance \\(L\\) does also play a role in the amplifier bandwidth as it changes the resonance frequency.
Ideally, low inductance cables should be used.
It is however usually quite high compare to \\(\omega\_c\\) as shown in Table [2](#table--tab:piezo-limitation-L).
It is however usually quite high compare to \\(\omega\_c\\) as shown in [Table 2](#table--tab:piezo-limitation-L).
<a id="table--tab:piezo-limitation-L"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--tab:piezo-limitation-L">Table 2</a></span>:
<span class="table-number"><a href="#table--tab:piezo-limitation-L">Table 2</a>:</span>
Bandwidth limitation due to \(R_s\)
</div>
@@ -827,7 +827,7 @@ For sinusoidal signals, the maximum positive and negative current is equal to:
<a id="table--tab:piezo-required-current"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--tab:piezo-required-current">Table 3</a></span>:
<span class="table-number"><a href="#table--tab:piezo-required-current">Table 3</a>:</span>
Minimum current requirements for a 10V sinusoid
</div>

60
content/book/hatch00_vibrat_matlab_ansys.md
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@@ -23,14 +23,14 @@ Matlab Code form the book is available [here](https://in.mathworks.com/matlabcen
## Introduction {#introduction}
<span class="org-target" id="org-target--sec:introduction"></span>
<span class="org-target" id="org-target--sec-introduction"></span>
The main goal of this book is to show how to take results of large dynamic finite element models and build small Matlab state space dynamic mechanical models for use in control system models.
### Modal Analysis {#modal-analysis}
The diagram in Figure [1](#figure--fig:hatch00-modal-analysis-flowchart) shows the methodology for analyzing a lightly damped structure using normal modes.
The diagram in [Figure 1](#figure--fig:hatch00-modal-analysis-flowchart) shows the methodology for analyzing a lightly damped structure using normal modes.
<div class="important">
@@ -58,7 +58,7 @@ Because finite element models usually have a very large number of states, an imp
<div class="important">
Figure [2](#figure--fig:hatch00-model-reduction-chart) shows such process, the steps are:
[Figure 2](#figure--fig:hatch00-model-reduction-chart) shows such process, the steps are:
- start with the finite element model
- compute the eigenvalues and eigenvectors (as many as dof in the model)
@@ -78,11 +78,11 @@ Figure [2](#figure--fig:hatch00-model-reduction-chart) shows such process, the s
### Notations {#notations}
Tables [3](#figure--fig:hatch00-n-dof-zeros), [2](#table--tab:notations-eigen-vectors-values) and [3](#table--tab:notations-stiffness-mass) summarize the notations of this document.
[Figure 3](#figure--fig:hatch00-n-dof-zeros), [Table 2](#table--tab:notations-eigen-vectors-values) and [Table 3](#table--tab:notations-stiffness-mass) summarize the notations of this document.
<a id="table--tab:notations-modes-nodes"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--tab:notations-modes-nodes">Table 1</a></span>:
<span class="table-number"><a href="#table--tab:notations-modes-nodes">Table 1</a>:</span>
Notation for the modes and nodes
</div>
@@ -97,7 +97,7 @@ Tables [3](#figure--fig:hatch00-n-dof-zeros), [2](#table--tab:notations-eigen-ve
<a id="table--tab:notations-eigen-vectors-values"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--tab:notations-eigen-vectors-values">Table 2</a></span>:
<span class="table-number"><a href="#table--tab:notations-eigen-vectors-values">Table 2</a>:</span>
Notation for the dofs, eigenvectors and eigenvalues
</div>
@@ -112,7 +112,7 @@ Tables [3](#figure--fig:hatch00-n-dof-zeros), [2](#table--tab:notations-eigen-ve
<a id="table--tab:notations-stiffness-mass"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--tab:notations-stiffness-mass">Table 3</a></span>:
<span class="table-number"><a href="#table--tab:notations-stiffness-mass">Table 3</a>:</span>
Notation for the mass and stiffness matrices
</div>
@@ -127,12 +127,12 @@ Tables [3](#figure--fig:hatch00-n-dof-zeros), [2](#table--tab:notations-eigen-ve
## Zeros in SISO Mechanical Systems {#zeros-in-siso-mechanical-systems}
<span class="org-target" id="org-target--sec:zeros_siso_systems"></span>
<span class="org-target" id="org-target--sec-zeros-siso-systems"></span>
The origin and influence of poles are clear: they represent the resonant frequencies of the system, and for each resonance frequency, a mode shape can be defined to describe the motion at that frequency.
We here which to give an intuitive understanding for **when to expect zeros in SISO mechanical systems** and **how to predict the frequencies at which they will occur**.
Figure [3](#figure--fig:hatch00-n-dof-zeros) shows a series arrangement of masses and springs, with a total of \\(n\\) masses and \\(n+1\\) springs.
[Figure 3](#figure--fig:hatch00-n-dof-zeros) shows a series arrangement of masses and springs, with a total of \\(n\\) masses and \\(n+1\\) springs.
The degrees of freedom are numbered from left to right, \\(z\_1\\) through \\(z\_n\\).
<a id="figure--fig:hatch00-n-dof-zeros"></a>
@@ -150,12 +150,12 @@ The resonances of the "overhanging appendages" of the constrained system create
## State Space Analysis {#state-space-analysis}
<span class="org-target" id="org-target--sec:state_space_analysis"></span>
<span class="org-target" id="org-target--sec-state-space-analysis"></span>
## Modal Analysis {#modal-analysis}
<span class="org-target" id="org-target--sec:modal_analysis"></span>
<span class="org-target" id="org-target--sec-modal-analysis"></span>
Lightly damped structures are typically analyzed with the "normal mode" method described in this section.
@@ -193,7 +193,7 @@ Summarizing the modal analysis method of analyzing linear mechanical systems and
#### Equation of Motion {#equation-of-motion}
Let's consider the model shown in Figure [4](#figure--fig:hatch00-undamped-tdof-model) with \\(k\_1 = k\_2 = k\\), \\(m\_1 = m\_2 = m\_3 = m\\) and \\(c\_1 = c\_2 = 0\\).
Let's consider the model shown in [Figure 4](#figure--fig:hatch00-undamped-tdof-model) with \\(k\_1 = k\_2 = k\\), \\(m\_1 = m\_2 = m\_3 = m\\) and \\(c\_1 = c\_2 = 0\\).
<a id="figure--fig:hatch00-undamped-tdof-model"></a>
@@ -293,7 +293,7 @@ One then find:
\end{bmatrix}
\end{equation}
Virtual interpretation of the eigenvectors are shown in Figures [5](#figure--fig:hatch00-tdof-mode-1), [6](#figure--fig:hatch00-tdof-mode-2) and [7](#figure--fig:hatch00-tdof-mode-3).
Virtual interpretation of the eigenvectors are shown in [Figure 5](#figure--fig:hatch00-tdof-mode-1), [Figure 6](#figure--fig:hatch00-tdof-mode-2) and [Figure 7](#figure--fig:hatch00-tdof-mode-3).
<a id="figure--fig:hatch00-tdof-mode-1"></a>
@@ -341,7 +341,7 @@ There are many options for change of basis, but we will show that **when eigenve
The n-uncoupled equations in the principal coordinate system can then be solved for the responses in the principal coordinate system using the well known solutions for the single dof systems.
The n-responses in the principal coordinate system can then be **transformed back** to the physical coordinate system to provide the actual response in physical coordinate.
This procedure is schematically shown in Figure [8](#figure--fig:hatch00-schematic-modal-solution).
This procedure is schematically shown in [Figure 8](#figure--fig:hatch00-schematic-modal-solution).
<a id="figure--fig:hatch00-schematic-modal-solution"></a>
@@ -687,7 +687,7 @@ Absolute damping is based on making \\(b = 0\\), in which case the percentage of
## Frequency Response: Modal Form {#frequency-response-modal-form}
<span class="org-target" id="org-target--sec:frequency_response_modal_form"></span>
<span class="org-target" id="org-target--sec-frequency-response-modal-form"></span>
The procedure to obtain the frequency response from a modal form is as follow:
@@ -695,7 +695,7 @@ The procedure to obtain the frequency response from a modal form is as follow:
- use Laplace transform to obtain the transfer functions in principal coordinates
- back-transform the transfer functions to physical coordinates where the individual mode contributions will be evident
This will be applied to the model shown in Figure [9](#figure--fig:hatch00-tdof-model).
This will be applied to the model shown in [Figure 9](#figure--fig:hatch00-tdof-model).
<a id="figure--fig:hatch00-tdof-model"></a>
@@ -877,7 +877,7 @@ Equations <eq:general_add_tf> and <eq:general_add_tf_damp> shows that in general
</div>
Figure [10](#figure--fig:hatch00-z11-tf-example) shows the separate contributions of each mode to the total response \\(z\_1/F\_1\\).
[Figure 10](#figure--fig:hatch00-z11-tf-example) shows the separate contributions of each mode to the total response \\(z\_1/F\_1\\).
<a id="figure--fig:hatch00-z11-tf-example"></a>
@@ -888,12 +888,12 @@ The zeros for SISO transfer functions are the roots of the numerator, however, f
## SISO State Space Matlab Model from ANSYS Model {#siso-state-space-matlab-model-from-ansys-model}
<span class="org-target" id="org-target--sec:siso_state_space"></span>
<span class="org-target" id="org-target--sec-siso-state-space"></span>
### Introduction {#introduction}
In this section is developed a SISO state space Matlab model from an ANSYS cantilever beam model as shown in Figure [11](#figure--fig:hatch00-cantilever-beam).
In this section is developed a SISO state space Matlab model from an ANSYS cantilever beam model as shown in [Figure 11](#figure--fig:hatch00-cantilever-beam).
A z direction force is applied at the midpoint of the beam and z displacement at the tip is the output.
The objective is to provide the smallest Matlab state space model that accurately represents the pertinent dynamics.
@@ -976,7 +976,7 @@ If sorting of DC gain values is performed prior to the `truncate` operation, the
## Ground Acceleration Matlab Model From ANSYS Model {#ground-acceleration-matlab-model-from-ansys-model}
<span class="org-target" id="org-target--sec:ground_acceleration"></span>
<span class="org-target" id="org-target--sec-ground-acceleration"></span>
### Model Description {#model-description}
@@ -990,9 +990,9 @@ If sorting of DC gain values is performed prior to the `truncate` operation, the
## SISO Disk Drive Actuator Model {#siso-disk-drive-actuator-model}
<span class="org-target" id="org-target--sec:siso_disk_drive"></span>
<span class="org-target" id="org-target--sec-siso-disk-drive"></span>
In this section we wish to extract a SISO state space model from a Finite Element model representing a Disk Drive Actuator (Figure [12](#figure--fig:hatch00-disk-drive-siso-model)).
In this section we wish to extract a SISO state space model from a Finite Element model representing a Disk Drive Actuator ([Figure 12](#figure--fig:hatch00-disk-drive-siso-model)).
### Actuator Description {#actuator-description}
@@ -1001,14 +1001,14 @@ In this section we wish to extract a SISO state space model from a Finite Elemen
{{< figure src="/ox-hugo/hatch00_disk_drive_siso_model.png" caption="<span class=\"figure-number\">Figure 12: </span>Drawing of Actuator/Suspension system" >}}
The primary motion of the actuator is rotation about the pivot bearing, therefore the final model has the coordinate system transformed from a Cartesian x,y,z coordinate system to a Cylindrical \\(r\\), \\(\theta\\) and \\(z\\) system, with the two origins coincident (Figure [13](#figure--fig:hatch00-disk-drive-nodes-reduced-model)).
The primary motion of the actuator is rotation about the pivot bearing, therefore the final model has the coordinate system transformed from a Cartesian x,y,z coordinate system to a Cylindrical \\(r\\), \\(\theta\\) and \\(z\\) system, with the two origins coincident ([Figure 13](#figure--fig:hatch00-disk-drive-nodes-reduced-model)).
<a id="figure--fig:hatch00-disk-drive-nodes-reduced-model"></a>
{{< figure src="/ox-hugo/hatch00_disk_drive_nodes_reduced_model.png" caption="<span class=\"figure-number\">Figure 13: </span>Nodes used for reduced Matlab model. Shown with partial finite element mesh at coil" >}}
For reduced models, we only require eigenvector information for dof where forces are applied and where displacements are required.
Figure [13](#figure--fig:hatch00-disk-drive-nodes-reduced-model) shows the nodes used for the reduced Matlab model.
[Figure 13](#figure--fig:hatch00-disk-drive-nodes-reduced-model) shows the nodes used for the reduced Matlab model.
The four nodes 24061, 24066, 24082 and 24087 are located in the center of the coil in the z direction and are used for simulating the VCM force.
The arrows at the nodes indicate the direction of forces.
@@ -1074,7 +1074,7 @@ From Ansys, we have the eigenvalues \\(\omega\_i\\) and eigenvectors \\(\bm{z}\\
## Balanced Reduction {#balanced-reduction}
<span class="org-target" id="org-target--sec:balanced_reduction"></span>
<span class="org-target" id="org-target--sec-balanced-reduction"></span>
In this chapter another method of reducing models, “balanced reduction”, will be introduced and compared with the DC and peak gain ranking methods.
@@ -1189,9 +1189,9 @@ The **states to be kept are the states with the largest diagonal terms**.
## MIMO Two Stage Actuator Model {#mimo-two-stage-actuator-model}
<span class="org-target" id="org-target--sec:mimo_disk_drive"></span>
<span class="org-target" id="org-target--sec-mimo-disk-drive"></span>
In this section, a MIMO two-stage actuator model is derived from a finite element model (Figure [14](#figure--fig:hatch00-disk-drive-mimo-schematic)).
In this section, a MIMO two-stage actuator model is derived from a finite element model ([Figure 14](#figure--fig:hatch00-disk-drive-mimo-schematic)).
### Actuator Description {#actuator-description}
@@ -1217,7 +1217,7 @@ Since the same forces are being applied to both piezo elements, they represent t
### Ansys Model Description {#ansys-model-description}
In Figure [15](#figure--fig:hatch00-disk-drive-mimo-ansys) are shown the principal nodes used for the model.
In [Figure 15](#figure--fig:hatch00-disk-drive-mimo-ansys) are shown the principal nodes used for the model.
<a id="figure--fig:hatch00-disk-drive-mimo-ansys"></a>
@@ -1440,7 +1440,7 @@ State Space Model
### Simple mode truncation {#simple-mode-truncation}
Let's plot the frequency of the modes (Figure [18](#figure--fig:hatch00-cant-beam-modes-freq)).
Let's plot the frequency of the modes ([Figure 18](#figure--fig:hatch00-cant-beam-modes-freq)).
<a id="figure--fig:hatch00-cant-beam-modes-freq"></a>
@@ -2123,6 +2123,6 @@ Reduced Mass and Stiffness matrices in the physical coordinates:
## Bibliography {#bibliography}
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Hatch, Michael R. 2000. <i>Vibration Simulation Using Matlab and Ansys</i>. CRC Press.</div>
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Hatch, Michael R. 2000. <i>Vibration Simulation Using MATLAB and ANSYS</i>. CRC Press.</div>
<div class="csl-entry"><a id="citeproc_bib_item_2"></a>Miu, Denny K. 1993. <i>Mechatronics: Electromechanics and Contromechanics</i>. 1st ed. Mechanical Engineering Series. Springer-Verlag New York.</div>
</div>

26
content/book/morrison16_groun_shiel.md
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@@ -148,7 +148,7 @@ In a few elements, the atomic structure is such that atoms align to generate a n
The flow of electrons is another way to generate a magnetic field.
The letter \\(H\\) is reserved for the magnetic field generated by a current.
Figure [6](#figure--fig:morrison16-H-field) shows the shape of the \\(H\\) field around a long, straight conductor carrying a direct current \\(I\\).
[Figure 6](#figure--fig:morrison16-H-field) shows the shape of the \\(H\\) field around a long, straight conductor carrying a direct current \\(I\\).
<a id="figure--fig:morrison16-H-field"></a>
@@ -167,7 +167,7 @@ Ampere's law states that the integral of the \\(H\\) field intensity in a closed
\boxed{\oint H dl = I}
\end{equation}
The simplest path to use for this integration is the one of the concentric circles in Figure [6](#figure--fig:morrison16-H-field), where \\(H\\) is constant and \\(r\\) is the distance from the conductor.
The simplest path to use for this integration is the one of the concentric circles in [Figure 6](#figure--fig:morrison16-H-field), where \\(H\\) is constant and \\(r\\) is the distance from the conductor.
Solving for \\(H\\), we obtain
\begin{equation}
@@ -179,7 +179,7 @@ And we see that \\(H\\) has units of amperes per meter.
### The solenoid {#the-solenoid}
The magnetic field of a solenoid is shown in Figure [7](#figure--fig:morrison16-solenoid).
The magnetic field of a solenoid is shown in [Figure 7](#figure--fig:morrison16-solenoid).
The field intensity inside the solenoid is nearly constant, while outside its intensity falls of rapidly.
Using Ampere's law <eq:ampere_law>:
@@ -196,7 +196,7 @@ Using Ampere's law <eq:ampere_law>:
### Faraday's law and the induction field {#faraday-s-law-and-the-induction-field}
When a conducting coil is moved through a magnetic field, a voltage appears at the open ends of the coil.
This is illustrated in Figure [8](#figure--fig:morrison16-voltage-moving-coil).
This is illustrated in [Figure 8](#figure--fig:morrison16-voltage-moving-coil).
The voltage depends on the number of turns in the coil and the rate at which the flux is changing.
<a id="figure--fig:morrison16-voltage-moving-coil"></a>
@@ -236,7 +236,7 @@ The unit of inductance if the henry.
</div>
For the coil in Figure [7](#figure--fig:morrison16-solenoid):
For the coil in [Figure 7](#figure--fig:morrison16-solenoid):
\begin{equation} \label{eq:inductance\_coil}
V = n^2 A k \mu\_0 \frac{dI}{dt} = L \frac{dI}{dt}
@@ -476,21 +476,21 @@ For example, signals that overload an input stage can produce noise that may loo
### The basic shield enclosure {#the-basic-shield-enclosure}
Consider the simple amplifier circuit shown in Figure [9](#figure--fig:morrison16-parasitic-capacitance-amp) with:
Consider the simple amplifier circuit shown in [Figure 9](#figure--fig:morrison16-parasitic-capacitance-amp) with:
- \\(V\_1\\) the input lead
- \\(V\_2\\) the output lead
- \\(V\_3\\) the conducting enclosure which is floating and taken as the reference conductor
- \\(V\_4\\) a signal common or reference conductor
Every conductor pair has a mutual capacitance, which are shown in Figure [9](#figure--fig:morrison16-parasitic-capacitance-amp) (b).
The equivalent circuit is shown in Figure [9](#figure--fig:morrison16-parasitic-capacitance-amp) (c) and it is apparent that there is some feedback from the output to the input or the amplifier.
Every conductor pair has a mutual capacitance, which are shown in [Figure 9](#figure--fig:morrison16-parasitic-capacitance-amp) (b).
The equivalent circuit is shown in [Figure 9](#figure--fig:morrison16-parasitic-capacitance-amp) (c) and it is apparent that there is some feedback from the output to the input or the amplifier.
<a id="figure--fig:morrison16-parasitic-capacitance-amp"></a>
{{< figure src="/ox-hugo/morrison16_parasitic_capacitance_amp.svg" caption="<span class=\"figure-number\">Figure 9: </span>Parasitic capacitances in a simple circuit. (a) Field lines in a circuit. (b) Mutual capacitance diagram. (b) Circuit representation" >}}
It is common practice in analog design to connect the enclosure to circuit common (Figure [10](#figure--fig:morrison16-grounding-shield-amp)).
It is common practice in analog design to connect the enclosure to circuit common ([Figure 10](#figure--fig:morrison16-grounding-shield-amp)).
When this connection is made, the feedback is removed and the enclosure no longer couples signals into the feedback structure.
The conductive enclosure is called a **shield**.
Connecting the signal common to the conductive enclosure is called "**grounding the shield**".
@@ -502,13 +502,13 @@ This "grounding" usually removed "hum" from the circuit.
Most practical circuits provide connections to external points.
To see the effect of making a _single_ external connection, open the conductive enclosure and connect the input circuit common to an external ground.
Figure [11](#figure--fig:morrison16-enclosure-shield-1-2-leads) (a) shows this grounded connection surrounded by an extension of the enclosure called the _cable shield_.
[Figure 11](#figure--fig:morrison16-enclosure-shield-1-2-leads) (a) shows this grounded connection surrounded by an extension of the enclosure called the _cable shield_.
A problem can be caused by an incorrect location of the connection between the cable shield and the enclosure.
In Figure [11](#figure--fig:morrison16-enclosure-shield-1-2-leads) (a), the electromagnetic field in the area induces a voltage in the loop and a resulting current to flow in conductor (1)-(2).
In [Figure 11](#figure--fig:morrison16-enclosure-shield-1-2-leads) (a), the electromagnetic field in the area induces a voltage in the loop and a resulting current to flow in conductor (1)-(2).
This conductor being the common ground that might have a resistance \\(R\\) or \\(1\\,\Omega\\), this current induced voltage that it added to the transmitted signal.
Our goal in this chapter is to find ways of keeping interference currents from flowing in any input signal conductor.
To remove this coupling, the shield connection to circuit common must be made at the point, where the circuit common connects to the external ground.
This connection is shown in Figure [11](#figure--fig:morrison16-enclosure-shield-1-2-leads) (b).
This connection is shown in [Figure 11](#figure--fig:morrison16-enclosure-shield-1-2-leads) (b).
This connection keeps the circulation of interference current on the outside of the shield.
There is only one point of zero signal potential external to the enclosure and that is where the signal common connects to an external hardware ground.
@@ -655,7 +655,7 @@ If the resistors are replaced by capacitors, the gain is the ratio of reactances
This feedback circuit is called a **charge converter**.
The charge on the input capacitor is transferred to the feedback capacitor.
If the feedback capacitor is smaller than the transducer capacitance by a factor of 100, then the voltage across the feedback capacitor will be 100 times greater than the open-circuit transducer voltage.
This feedback arrangement is shown in Figure [17](#figure--fig:morrison16-charge-amplifier).
This feedback arrangement is shown in [Figure 17](#figure--fig:morrison16-charge-amplifier).
The open-circuit input signal voltage is \\(Q/C\_T\\).
The output voltage is \\(Q/C\_{FB}\\).
The voltage gain is therefore \\(C\_T/C\_{FB}\\).

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content/book/preumont18_vibrat_contr_activ_struc_fourt_edition.md
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@@ -67,7 +67,7 @@ There are two radically different approached to disturbance rejection: feedback
{{< figure src="/ox-hugo/preumont18_classical_feedback_small.png" caption="<span class=\"figure-number\">Figure 1: </span>Principle of feedback control" >}}
The principle of feedback is represented on figure [1](#figure--fig:classical-feedback-small). The output \\(y\\) of the system is compared to the reference signal \\(r\\), and the error signal \\(\epsilon = r-y\\) is passed into a compensator \\(K(s)\\) and applied to the system \\(G(s)\\), \\(d\\) is the disturbance.
The principle of feedback is represented on [Figure 1](#figure--fig:classical-feedback-small). The output \\(y\\) of the system is compared to the reference signal \\(r\\), and the error signal \\(\epsilon = r-y\\) is passed into a compensator \\(K(s)\\) and applied to the system \\(G(s)\\), \\(d\\) is the disturbance.
The design problem consists of finding the appropriate compensator \\(K(s)\\) such that the closed-loop system is stable and behaves in the appropriate manner.
In the control of lightly damped structures, feedback control is used for two distinct and complementary purposes: **active damping** and **model-based feedback**.
@@ -94,14 +94,14 @@ The objective is to control a variable \\(y\\) to a desired value \\(r\\) in spi
{{< figure src="/ox-hugo/preumont18_feedforward_adaptative.png" caption="<span class=\"figure-number\">Figure 2: </span>Principle of feedforward control" >}}
The method relies on the availability of a **reference signal correlated to the primary disturbance**.
The idea is to produce a second disturbance such that is cancels the effect of the primary disturbance at the location of the sensor error. Its principle is explained in figure [2](#figure--fig:feedforward-adaptative).
The idea is to produce a second disturbance such that is cancels the effect of the primary disturbance at the location of the sensor error. Its principle is explained in [Figure 2](#figure--fig:feedforward-adaptative).
The filter coefficients are adapted in such a way that the error signal at one or several critical points is minimized.
There is no guarantee that the global response is reduced at other locations. This method is therefor considered as a local one.
Because it is less sensitive to phase lag than feedback, it can be used at higher frequencies (\\(\omega\_c \approx \omega\_s/10\\)).
The table [1](#table--tab:adv-dis-type-control) summarizes the main features of the two approaches.
The [Table 1](#table--tab:adv-dis-type-control) summarizes the main features of the two approaches.
<a id="table--tab:adv-dis-type-control"></a>
<div class="table-caption">
@@ -129,7 +129,7 @@ The table [1](#table--tab:adv-dis-type-control) summarizes the main features of
{{< figure src="/ox-hugo/preumont18_design_steps.png" caption="<span class=\"figure-number\">Figure 3: </span>The various steps of the design" >}}
The various steps of the design of a controlled structure are shown in figure [3](#figure--fig:design-steps).
The various steps of the design of a controlled structure are shown in [Figure 3](#figure--fig:design-steps).
The **starting point** is:
@@ -156,7 +156,7 @@ If the dynamics of the sensors and actuators may significantly affect the behavi
### Plant Description, Error and Control Budget {#plant-description-error-and-control-budget}
From the block diagram of the control system (figure [4](#figure--fig:general-plant)):
From the block diagram of the control system ([Figure 4](#figure--fig:general-plant)):
\begin{align\*}
y &= (I - G\_{yu}H)^{-1} G\_{yw} w\\\\
@@ -186,7 +186,7 @@ Even more interesting for the design is the **Cumulative Mean Square** response
It is a monotonously decreasing function of frequency and describes the contribution of all frequencies above \\(\omega\\) to the mean-square value of \\(z\\).
\\(\sigma\_z(0)\\) is then the global RMS response.
A typical plot of \\(\sigma\_z(\omega)\\) is shown figure [5](#figure--fig:cas-plot).
A typical plot of \\(\sigma\_z(\omega)\\) is shown [Figure 5](#figure--fig:cas-plot).
It is useful to **identify the critical modes** in a design, at which the effort should be targeted.
The diagram can also be used to **assess the control laws** and compare different actuator and sensor configuration.
@@ -398,7 +398,7 @@ With:
{{< figure src="/ox-hugo/preumont18_neglected_modes.png" caption="<span class=\"figure-number\">Figure 6: </span>Fourier spectrum of the excitation \\(F\\) and dynamic amplitification \\(D\_i\\) of mode \\(i\\) and \\(k\\) such that \\(\omega\_i < \omega\_b\\) and \\(\omega\_k \gg \omega\_b\\)" >}}
If the excitation has a limited bandwidth \\(\omega\_b\\), the contribution of the high frequency modes \\(\omega\_k \gg \omega\_b\\) can be evaluated by assuming \\(D\_k(\omega) \approx 1\\) (as shown on figure [6](#figure--fig:neglected-modes)).
If the excitation has a limited bandwidth \\(\omega\_b\\), the contribution of the high frequency modes \\(\omega\_k \gg \omega\_b\\) can be evaluated by assuming \\(D\_k(\omega) \approx 1\\) (as shown on [Figure 6](#figure--fig:neglected-modes)).
And \\(G(\omega)\\) can be rewritten on terms of the **low frequency modes only**:
\\[ G(\omega) \approx \sum\_{i=1}^m \frac{\phi\_i \phi\_i^T}{\mu\_i \omega\_i^2} D\_i(\omega) + R \\]
@@ -436,7 +436,7 @@ The open-loop FRF of a collocated system corresponds to a diagonal component of
If we assumes that the collocated system is undamped and is attached to the DoF \\(k\\), the open-loop FRF is purely real:
\\[ G\_{kk}(\omega) = \sum\_{i=1}^m \frac{\phi\_i^2(k)}{\mu\_i (\omega\_i^2 - \omega^2)} + R\_{kk} \\]
\\(G\_{kk}\\) is a monotonously increasing function of \\(\omega\\) (figure [7](#figure--fig:collocated-control-frf)).
\\(G\_{kk}\\) is a monotonously increasing function of \\(\omega\\) ([Figure 7](#figure--fig:collocated-control-frf)).
<a id="figure--fig:collocated-control-frf"></a>
@@ -451,7 +451,7 @@ For lightly damped structure, the poles and zeros are just moved a little bit in
</div>
If the undamped structure is excited harmonically by the actuator at the frequency of the transmission zero \\(z\_i\\), the amplitude of the response of the collocated sensor vanishes. That means that the structure oscillates at the frequency \\(z\_i\\) according to the mode shape shown in dotted line figure [8](#figure--fig:collocated-zero).
If the undamped structure is excited harmonically by the actuator at the frequency of the transmission zero \\(z\_i\\), the amplitude of the response of the collocated sensor vanishes. That means that the structure oscillates at the frequency \\(z\_i\\) according to the mode shape shown in dotted line [Figure 8](#figure--fig:collocated-zero).
<a id="figure--fig:collocated-zero"></a>
@@ -467,7 +467,7 @@ The open-loop poles are independant of the actuator and sensor configuration whi
</div>
By looking at figure [7](#figure--fig:collocated-control-frf), we see that neglecting the residual mode in the modelling amounts to translating the FRF diagram vertically. That produces a shift in the location of the transmission zeros to the right.
By looking at [Figure 7](#figure--fig:collocated-control-frf), we see that neglecting the residual mode in the modelling amounts to translating the FRF diagram vertically. That produces a shift in the location of the transmission zeros to the right.
<a id="figure--fig:alternating-p-z"></a>
@@ -479,7 +479,7 @@ The open-loop transfer function of a lighly damped structure with a collocated a
G(s) = G\_0 \frac{\Pi\_i(s^2/z\_i^2 + 2 \xi\_i s/z\_i + 1)}{\Pi\_j(s^2/\omega\_j^2 + 2 \xi\_j s /\omega\_j + 1)}
\end{equation}
The corresponding Bode plot is represented in figure [9](#figure--fig:alternating-p-z). Every imaginary pole at \\(\pm j\omega\_i\\) introduces a \\(\SI{180}{\degree}\\) phase lag and every imaginary zero at \\(\pm jz\_i\\) introduces a phase lead of \\(\SI{180}{\degree}\\).
The corresponding Bode plot is represented in [Figure 9](#figure--fig:alternating-p-z). Every imaginary pole at \\(\pm j\omega\_i\\) introduces a \\(\SI{180}{\degree}\\) phase lag and every imaginary zero at \\(\pm jz\_i\\) introduces a phase lead of \\(\SI{180}{\degree}\\).
In this way, the phase diagram is always contained between \\(\SI{0}{\degree}\\) and \\(\SI{-180}{\degree}\\) as a consequence of the interlacing property.
@@ -501,7 +501,7 @@ Two broad categories of actuators can be distinguish:
A voice coil transducer is an energy transformer which converts electrical power into mechanical power and vice versa.
The system consists of (see figure [10](#figure--fig:voice-coil-schematic)):
The system consists of (see [Figure 10](#figure--fig:voice-coil-schematic)):
- A permanent magnet which produces a uniform flux density \\(B\\) normal to the gap
- A coil which is free to move axially
@@ -543,7 +543,7 @@ Thus, at any time, there is an equilibrium between the electrical power absorbed
#### Proof-Mass Actuator {#proof-mass-actuator}
A reaction mass \\(m\\) is conected to the support structure by a spring \\(k\\) , and damper \\(c\\) and a force actuator \\(f = T i\\) (figure [11](#figure--fig:proof-mass-actuator)).
A reaction mass \\(m\\) is conected to the support structure by a spring \\(k\\) , and damper \\(c\\) and a force actuator \\(f = T i\\) ([Figure 11](#figure--fig:proof-mass-actuator)).
<a id="figure--fig:proof-mass-actuator"></a>
@@ -574,7 +574,7 @@ with:
</div>
Above some critical frequency \\(\omega\_c \approx 2\omega\_p\\), **the proof-mass actuator can be regarded as an ideal force generator** (figure [12](#figure--fig:proof-mass-tf)).
Above some critical frequency \\(\omega\_c \approx 2\omega\_p\\), **the proof-mass actuator can be regarded as an ideal force generator** ([Figure 12](#figure--fig:proof-mass-tf)).
<a id="figure--fig:proof-mass-tf"></a>
@@ -610,7 +610,7 @@ Designing geophones with very low corner frequency is in general difficult. Acti
### General Electromechanical Transducer {#general-electromechanical-transducer}
The consitutive behavior of a wide class of electromechanical transducers can be modelled as in figure [14](#figure--fig:electro-mechanical-transducer).
The consitutive behavior of a wide class of electromechanical transducers can be modelled as in [Figure 14](#figure--fig:electro-mechanical-transducer).
<a id="figure--fig:electro-mechanical-transducer"></a>
@@ -637,7 +637,7 @@ With:
Equation <eq:gen_trans_e> shows that the voltage across the electrical terminals of any electromechanical transducer is the sum of a contribution proportional to the current applied and a contribution proportional to the velocity of the mechanical terminals.
Thus, if \\(Z\_ei\\) can be measured and substracted from \\(e\\), a signal proportional to the velocity is obtained.
To do so, the bridge circuit as shown on figure [15](#figure--fig:bridge-circuit) can be used.
To do so, the bridge circuit as shown on [Figure 15](#figure--fig:bridge-circuit) can be used.
We can show that
@@ -655,7 +655,7 @@ which is indeed a linear function of the velocity \\(v\\) at the mechanical term
### Smart Materials {#smart-materials}
Smart materials have the ability to respond significantly to stimuli of different physical nature.
Figure [16](#figure--fig:smart-materials) lists various effects that are observed in materials in response to various inputs.
[Figure 16](#figure--fig:smart-materials) lists various effects that are observed in materials in response to various inputs.
<a id="figure--fig:smart-materials"></a>
@@ -748,7 +748,7 @@ It measures the efficiency of the conversion of the mechanical energy into elect
</div>
If one assumes that all the electrical and mechanical quantities are uniformly distributed in a linear transducer formed by a **stack** (see figure [17](#figure--fig:piezo-stack)) of \\(n\\) disks of thickness \\(t\\) and cross section \\(A\\), the global constitutive equations of the transducer are obtained by integrating <eq:piezo_eq_matrix_bis> over the volume of the transducer:
If one assumes that all the electrical and mechanical quantities are uniformly distributed in a linear transducer formed by a **stack** (see [Figure 17](#figure--fig:piezo-stack)) of \\(n\\) disks of thickness \\(t\\) and cross section \\(A\\), the global constitutive equations of the transducer are obtained by integrating <eq:piezo_eq_matrix_bis> over the volume of the transducer:
\begin{equation}
\begin{bmatrix}Q\\\\Delta\end{bmatrix}
@@ -789,7 +789,7 @@ Equation <eq:piezo_stack_eq> can be inverted to obtain
#### Energy Stored in the Piezoelectric Transducer {#energy-stored-in-the-piezoelectric-transducer}
Let us write the total stored electromechanical energy of a discrete piezoelectric transducer as shown on figure [18](#figure--fig:piezo-discrete).
Let us write the total stored electromechanical energy of a discrete piezoelectric transducer as shown on [Figure 18](#figure--fig:piezo-discrete).
The total power delivered to the transducer is the sum of electric power \\(V i\\) and the mechanical power \\(f \dot{\Delta}\\). The net work of the transducer is
@@ -831,7 +831,7 @@ The ratio between the remaining stored energy and the initial stored energy is
#### Admittance of the Piezoelectric Transducer {#admittance-of-the-piezoelectric-transducer}
Consider the system of figure [19](#figure--fig:piezo-stack-admittance), where the piezoelectric transducer is assumed massless and is connected to a mass \\(M\\).
Consider the system of [Figure 19](#figure--fig:piezo-stack-admittance), where the piezoelectric transducer is assumed massless and is connected to a mass \\(M\\).
The force acting on the mass is negative of that acting on the transducer, \\(f = -M \ddot{x}\\).
<a id="figure--fig:piezo-stack-admittance"></a>
@@ -853,7 +853,7 @@ And one can see that
\frac{z^2 - p^2}{z^2} = k^2
\end{equation}
Equation <eq:distance_p_z> constitutes a practical way to determine the electromechanical coupling factor from the poles and zeros of the admittance measurement (figure [20](#figure--fig:piezo-admittance-curve)).
Equation <eq:distance_p_z> constitutes a practical way to determine the electromechanical coupling factor from the poles and zeros of the admittance measurement ([Figure 20](#figure--fig:piezo-admittance-curve)).
<a id="figure--fig:piezo-admittance-curve"></a>
@@ -1552,7 +1552,7 @@ Their design requires a model of the structure, and there is usually a trade-off
When collocated actuator/sensor pairs can be used, stability can be achieved using positivity concepts, but in many situations, collocated pairs are not feasible for HAC.
The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure [21](#figure--fig:hac-lac-control).
The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in [Figure 21](#figure--fig:hac-lac-control).
The inner loop uses a set of collocated actuator/sensor pairs for decentralized active damping with guaranteed stability ; the outer loop consists of a non-collocated HAC based on a model of the actively damped structure.
This approach has the following advantages:

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content/book/skogestad07_multiv_feedb_contr.md
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@@ -183,7 +183,7 @@ In order to obtain a linear model from the "first-principle", the following appr
### Notation {#notation}
Notations used throughout this note are summarized in [1](#table--tab:notation-conventional), [2](#table--tab:notation-general) and [3](#table--tab:notation-tf).
Notations used throughout this note are summarized in [Table 1](#table--tab:notation-conventional), [Table 2](#table--tab:notation-general) and [Table 3](#table--tab:notation-tf).
<a id="table--tab:notation-conventional"></a>
<div class="table-caption">
@@ -272,7 +272,7 @@ We note \\(N(\w\_0) = \left( \frac{d\ln{|G(j\w)|}}{d\ln{\w}} \right)\_{\w=\w\_0}
#### One Degree-of-Freedom Controller {#one-degree-of-freedom-controller}
The simple one degree-of-freedom controller negative feedback structure is represented in [1](#figure--fig:classical-feedback-alt).
The simple one degree-of-freedom controller negative feedback structure is represented in [Figure 1](#figure--fig:classical-feedback-alt).
The input to the controller \\(K(s)\\) is \\(r-y\_m\\) where \\(y\_m = y+n\\) is the measured output and \\(n\\) is the measurement noise.
Thus, the input to the plant is \\(u = K(s) (r-y-n)\\).
@@ -592,13 +592,13 @@ For reference tracking, we typically want the controller to look like \\(\frac{1
We cannot achieve both of these simultaneously with a single feedback controller.
The solution is to use a **two degrees of freedom controller** where the reference signal \\(r\\) and output measurement \\(y\_m\\) are independently treated by the controller ([2](#figure--fig:classical-feedback-2dof-alt)), rather than operating on their difference \\(r - y\_m\\).
The solution is to use a **two degrees of freedom controller** where the reference signal \\(r\\) and output measurement \\(y\_m\\) are independently treated by the controller ([Figure 2](#figure--fig:classical-feedback-2dof-alt)), rather than operating on their difference \\(r - y\_m\\).
<a id="figure--fig:classical-feedback-2dof-alt"></a>
{{< figure src="/ox-hugo/skogestad07_classical_feedback_2dof_alt.png" caption="<span class=\"figure-number\">Figure 2: </span>2 degrees-of-freedom control architecture" >}}
The controller can be slit into two separate blocks ([3](#figure--fig:classical-feedback-sep)):
The controller can be slit into two separate blocks ([Figure 3](#figure--fig:classical-feedback-sep)):
- the **feedback controller** \\(K\_y\\) that is used to **reduce the effect of uncertainty** (disturbances and model errors)
- the **prefilter** \\(K\_r\\) that **shapes the commands** \\(r\\) to improve tracking performance
@@ -672,7 +672,7 @@ Which can be expressed as an \\(\mathcal{H}\_\infty\\):
W\_P(s) = \frac{s/M + \w\_B^\*}{s + \w\_B^\* A}
\end{equation\*}
With (see [4](#figure--fig:performance-weigth)):
With (see [Figure 4](#figure--fig:performance-weigth)):
- \\(M\\): maximum magnitude of \\(\abs{S}\\)
- \\(\w\_B\\): crossover frequency
@@ -750,7 +750,7 @@ The main rule for evaluating transfer functions is the **MIMO Rule**: Start from
#### Negative Feedback Control Systems {#negative-feedback-control-systems}
For negative feedback system ([5](#figure--fig:classical-feedback-bis)), we define \\(L\\) to be the loop transfer function as seen when breaking the loop at the **output** of the plant:
For negative feedback system ([Figure 5](#figure--fig:classical-feedback-bis)), we define \\(L\\) to be the loop transfer function as seen when breaking the loop at the **output** of the plant:
- \\(L = G K\\)
- \\(S \triangleq (I + L)^{-1}\\) is the transfer function from \\(d\_1\\) to \\(y\\)
@@ -1109,7 +1109,7 @@ The **structured singular value** \\(\mu\\) is a tool for analyzing the effects
### General Control Problem Formulation {#general-control-problem-formulation}
The general control problem formulation is represented in [6](#figure--fig:general-control-names) (introduced in (<a href="#citeproc_bib_item_1">Doyle 1983</a>)).
The general control problem formulation is represented in [Figure 6](#figure--fig:general-control-names) (introduced in (<a href="#citeproc_bib_item_1">Doyle 1983</a>)).
<a id="figure--fig:general-control-names"></a>
@@ -1141,7 +1141,7 @@ Then we have to break all the "loops" entering and exiting the controller \\(K\\
#### Controller Design: Including Weights in \\(P\\) {#controller-design-including-weights-in-p}
In order to get a meaningful controller synthesis problem, for example in terms of the \\(\hinf\\) norms, we generally have to include the weights \\(W\_z\\) and \\(W\_w\\) in the generalized plant \\(P\\) ([7](#figure--fig:general-plant-weights)).
In order to get a meaningful controller synthesis problem, for example in terms of the \\(\hinf\\) norms, we generally have to include the weights \\(W\_z\\) and \\(W\_w\\) in the generalized plant \\(P\\) ([Figure 7](#figure--fig:general-plant-weights)).
We consider:
- The weighted or normalized exogenous inputs \\(w\\) (where \\(\tilde{w} = W\_w w\\) consists of the "physical" signals entering the system)
@@ -1199,7 +1199,7 @@ where \\(F\_l(P, K)\\) denotes a **lower linear fractional transformation** (LFT
#### A General Control Configuration Including Model Uncertainty {#a-general-control-configuration-including-model-uncertainty}
The general control configuration may be extended to include model uncertainty as shown in [8](#figure--fig:general-config-model-uncertainty).
The general control configuration may be extended to include model uncertainty as shown in [Figure 8](#figure--fig:general-config-model-uncertainty).
<a id="figure--fig:general-config-model-uncertainty"></a>
@@ -1595,14 +1595,14 @@ RHP-zeros therefore imply high gain instability.
{{< figure src="/ox-hugo/skogestad07_classical_feedback_stability.png" caption="<span class=\"figure-number\">Figure 9: </span>Block diagram used to check internal stability" >}}
Assume that the components \\(G\\) and \\(K\\) contain no unstable hidden modes. Then the feedback system in [9](#figure--fig:block-diagram-for-stability) is **internally stable** if and only if all four closed-loop transfer matrices are stable.
Assume that the components \\(G\\) and \\(K\\) contain no unstable hidden modes. Then the feedback system in [Figure 9](#figure--fig:block-diagram-for-stability) is **internally stable** if and only if all four closed-loop transfer matrices are stable.
\begin{align\*}
&(I+KG)^{-1} & -K&(I+GK)^{-1} \\\\
G&(I+KG)^{-1} & &(I+GK)^{-1}
\end{align\*}
Assume there are no RHP pole-zero cancellations between \\(G(s)\\) and \\(K(s)\\), the feedback system in [9](#figure--fig:block-diagram-for-stability) is internally stable if and only if **one** of the four closed-loop transfer function matrices is stable.
Assume there are no RHP pole-zero cancellations between \\(G(s)\\) and \\(K(s)\\), the feedback system in [Figure 9](#figure--fig:block-diagram-for-stability) is internally stable if and only if **one** of the four closed-loop transfer function matrices is stable.
### Stabilizing Controllers {#stabilizing-controllers}
@@ -2234,7 +2234,7 @@ Uncertainty in the crossover frequency region can result in poor performance and
{{< figure src="/ox-hugo/skogestad07_classical_feedback_meas.png" caption="<span class=\"figure-number\">Figure 10: </span>Feedback control system" >}}
Consider the control system in [10](#figure--fig:classical-feedback-meas).
Consider the control system in [Figure 10](#figure--fig:classical-feedback-meas).
Here \\(G\_m(s)\\) denotes the measurement transfer function and we assume \\(G\_m(0) = 1\\) (perfect steady-state measurement).
<div class="important">
@@ -2654,7 +2654,7 @@ The issues are the same for SISO and MIMO systems, however, with MIMO systems th
In practice, the difference between the true perturbed plant \\(G^\prime\\) and the plant model \\(G\\) is caused by a number of different sources.
We here focus on input and output uncertainty.
In multiplicative form, the input and output uncertainties are given by (see [12](#figure--fig:input-output-uncertainty)):
In multiplicative form, the input and output uncertainties are given by (see [Figure 12](#figure--fig:input-output-uncertainty)):
\begin{equation\*}
G^\prime = (I + E\_O) G (I + E\_I)
@@ -2873,7 +2873,7 @@ In most cases, we prefer to lump the uncertainty into a **multiplicative uncerta
G\_p(s) = G(s)(1 + w\_I(s)\Delta\_I(s)); \quad \abs{\Delta\_I(j\w)} \le 1 \\, \forall\w
\end{equation\*}
which may be represented by the diagram in [13](#figure--fig:input-uncertainty-set).
which may be represented by the diagram in [Figure 13](#figure--fig:input-uncertainty-set).
</div>
@@ -2940,7 +2940,7 @@ This is of course conservative as it introduces possible plants that are not pre
#### Uncertain Regions {#uncertain-regions}
To illustrate how parametric uncertainty translate into frequency domain uncertainty, consider in [14](#figure--fig:uncertainty-region) the Nyquist plots generated by the following set of plants
To illustrate how parametric uncertainty translate into frequency domain uncertainty, consider in [Figure 14](#figure--fig:uncertainty-region) the Nyquist plots generated by the following set of plants
\begin{equation\*}
G\_p(s) = \frac{k}{\tau s + 1} e^{-\theta s}, \quad 2 \le k, \theta, \tau \le 3
@@ -2968,7 +2968,7 @@ The disc-shaped regions may be generated by **additive** complex norm-bounded pe
\end{aligned}
\end{equation}
At each frequency, all possible \\(\Delta(j\w)\\) "generates" a disc-shaped region with radius 1 centered at 0, so \\(G(j\w) + w\_A(j\w)\Delta\_A(j\w)\\) generates at each frequency a disc-shapes region of radius \\(\abs{w\_A(j\w)}\\) centered at \\(G(j\w)\\) as shown in [15](#figure--fig:uncertainty-disc-generated).
At each frequency, all possible \\(\Delta(j\w)\\) "generates" a disc-shaped region with radius 1 centered at 0, so \\(G(j\w) + w\_A(j\w)\Delta\_A(j\w)\\) generates at each frequency a disc-shapes region of radius \\(\abs{w\_A(j\w)}\\) centered at \\(G(j\w)\\) as shown in [Figure 15](#figure--fig:uncertainty-disc-generated).
</div>
@@ -3044,7 +3044,7 @@ To simplify subsequent controller design, we select a delay-free nominal model
\end{equation\*}
To obtain \\(l\_I(\w)\\), we consider three values (2, 2.5 and 3) for each of the three parameters (\\(k, \theta, \tau\\)).
The corresponding relative errors \\(\abs{\frac{G\_p-G}{G}}\\) are shown as functions of frequency for the \\(3^3 = 27\\) resulting \\(G\_p\\) ([16](#figure--fig:uncertainty-weight)).
The corresponding relative errors \\(\abs{\frac{G\_p-G}{G}}\\) are shown as functions of frequency for the \\(3^3 = 27\\) resulting \\(G\_p\\) ([Figure 16](#figure--fig:uncertainty-weight)).
To derive \\(w\_I(s)\\), we then try to find a simple weight so that \\(\abs{w\_I(j\w)}\\) lies above all the dotted lines.
</div>
@@ -3092,7 +3092,7 @@ The magnitude of the relative uncertainty caused by neglecting the dynamics in \
##### Neglected delay {#neglected-delay}
Let \\(f(s) = e^{-\theta\_p s}\\), where \\(0 \le \theta\_p \le \theta\_{\text{max}}\\). We want to represent \\(G\_p(s) = G\_0(s)e^{-\theta\_p s}\\) by a delay-free plant \\(G\_0(s)\\) and multiplicative uncertainty. Let first consider the maximum delay, for which the relative error \\(\abs{1 - e^{-j \w \theta\_{\text{max}}}}\\) is shown as a function of frequency ([17](#figure--fig:neglected-time-delay)). If we consider all \\(\theta \in [0, \theta\_{\text{max}}]\\) then:
Let \\(f(s) = e^{-\theta\_p s}\\), where \\(0 \le \theta\_p \le \theta\_{\text{max}}\\). We want to represent \\(G\_p(s) = G\_0(s)e^{-\theta\_p s}\\) by a delay-free plant \\(G\_0(s)\\) and multiplicative uncertainty. Let first consider the maximum delay, for which the relative error \\(\abs{1 - e^{-j \w \theta\_{\text{max}}}}\\) is shown as a function of frequency ([Figure 17](#figure--fig:neglected-time-delay)). If we consider all \\(\theta \in [0, \theta\_{\text{max}}]\\) then:
\begin{equation\*}
l\_I(\w) = \begin{cases} \abs{1 - e^{-j\w\theta\_{\text{max}}}} & \w < \pi/\theta\_{\text{max}} \\\ 2 & \w \ge \pi/\theta\_{\text{max}} \end{cases}
@@ -3105,7 +3105,7 @@ Let \\(f(s) = e^{-\theta\_p s}\\), where \\(0 \le \theta\_p \le \theta\_{\text{m
##### Neglected lag {#neglected-lag}
Let \\(f(s) = 1/(\tau\_p s + 1)\\), where \\(0 \le \tau\_p \le \tau\_{\text{max}}\\). In this case the resulting \\(l\_I(\w)\\) ([18](#figure--fig:neglected-first-order-lag)) can be represented by a rational transfer function with \\(\abs{w\_I(j\w)} = l\_I(\w)\\) where
Let \\(f(s) = 1/(\tau\_p s + 1)\\), where \\(0 \le \tau\_p \le \tau\_{\text{max}}\\). In this case the resulting \\(l\_I(\w)\\) ([Figure 18](#figure--fig:neglected-first-order-lag)) can be represented by a rational transfer function with \\(\abs{w\_I(j\w)} = l\_I(\w)\\) where
\begin{equation\*}
w\_I(s) = \frac{\tau\_{\text{max}} s}{\tau\_{\text{max}} s + 1}
@@ -3131,7 +3131,7 @@ There is an exact expression, its first order approximation is
w\_I(s) = \frac{(1+\frac{r\_k}{2})\theta\_{\text{max}} s + r\_k}{\frac{\theta\_{\text{max}}}{2} s + 1}
\end{equation\*}
However, as shown in [19](#figure--fig:lag-delay-uncertainty), the weight \\(w\_I\\) is optimistic, especially around frequencies \\(1/\theta\_{\text{max}}\\). To make sure that \\(\abs{w\_I(j\w)} \le l\_I(\w)\\), we can apply a correction factor:
However, as shown in [Figure 19](#figure--fig:lag-delay-uncertainty), the weight \\(w\_I\\) is optimistic, especially around frequencies \\(1/\theta\_{\text{max}}\\). To make sure that \\(\abs{w\_I(j\w)} \le l\_I(\w)\\), we can apply a correction factor:
\begin{equation\*}
w\_I^\prime(s) = w\_I \cdot \frac{(\frac{\theta\_{\text{max}}}{2.363})^2 s^2 + 2\cdot 0.838 \cdot \frac{\theta\_{\text{max}}}{2.363} s + 1}{(\frac{\theta\_{\text{max}}}{2.363})^2 s^2 + 2\cdot 0.685 \cdot \frac{\theta\_{\text{max}}}{2.363} s + 1}
@@ -3167,7 +3167,7 @@ where \\(r\_0\\) is the relative uncertainty at steady-state, \\(1/\tau\\) is th
#### RS with Multiplicative Uncertainty {#rs-with-multiplicative-uncertainty}
We want to determine the stability of the uncertain feedback system in [20](#figure--fig:feedback-multiplicative-uncertainty) where there is multiplicative uncertainty of magnitude \\(\abs{w\_I(j\w)}\\).
We want to determine the stability of the uncertain feedback system in [Figure 20](#figure--fig:feedback-multiplicative-uncertainty) where there is multiplicative uncertainty of magnitude \\(\abs{w\_I(j\w)}\\).
The loop transfer function becomes
\begin{equation\*}
@@ -3189,7 +3189,7 @@ We use the Nyquist stability condition to test for robust stability of the close
##### Graphical derivation of RS-condition {#graphical-derivation-of-rs-condition}
Consider the Nyquist plot of \\(L\_p\\) as shown in [21](#figure--fig:nyquist-uncertainty). \\(\abs{1+L}\\) is the distance from the point \\(-1\\) to the center of the disc representing \\(L\_p\\) and \\(\abs{w\_I L}\\) is the radius of the disc.
Consider the Nyquist plot of \\(L\_p\\) as shown in [Figure 21](#figure--fig:nyquist-uncertainty). \\(\abs{1+L}\\) is the distance from the point \\(-1\\) to the center of the disc representing \\(L\_p\\) and \\(\abs{w\_I L}\\) is the radius of the disc.
Encirclements are avoided if none of the discs cover \\(-1\\), and we get:
\begin{align\*}
@@ -3236,7 +3236,7 @@ And we obtain the same condition as before.
#### RS with Inverse Multiplicative Uncertainty {#rs-with-inverse-multiplicative-uncertainty}
We will derive a corresponding RS-condition for feedback system with inverse multiplicative uncertainty ([22](#figure--fig:inverse-uncertainty-set)) in which
We will derive a corresponding RS-condition for feedback system with inverse multiplicative uncertainty ([Figure 22](#figure--fig:inverse-uncertainty-set)) in which
\begin{equation\*}
G\_p = G(1 + w\_{iI}(s) \Delta\_{iI})^{-1}
@@ -3290,7 +3290,7 @@ The condition for **nominal performance** when considering performance in terms
</div>
Now \\(\abs{1 + L}\\) represents at each frequency the distance of \\(L(j\omega)\\) from the point \\(-1\\) in the Nyquist plot, so \\(L(j\omega)\\) must be at least a distance of \\(\abs{w\_P(j\omega)}\\) from \\(-1\\).
This is illustrated graphically in [23](#figure--fig:nyquist-performance-condition).
This is illustrated graphically in [Figure 23](#figure--fig:nyquist-performance-condition).
<a id="figure--fig:nyquist-performance-condition"></a>
@@ -3312,7 +3312,7 @@ For robust performance, we require the performance condition to be satisfied for
</div>
Let's consider the case of multiplicative uncertainty as shown on [24](#figure--fig:input-uncertainty-set-feedback-weight-bis).
Let's consider the case of multiplicative uncertainty as shown on [Figure 24](#figure--fig:input-uncertainty-set-feedback-weight-bis).
The robust performance corresponds to requiring \\(\abs{\hat{y}/d}<1\ \forall \Delta\_I\\) and the set of possible loop transfer functions is
\begin{equation\*}
@@ -3326,7 +3326,7 @@ The robust performance corresponds to requiring \\(\abs{\hat{y}/d}<1\ \forall \D
##### Graphical derivation of RP-condition {#graphical-derivation-of-rp-condition}
As illustrated on [23](#figure--fig:nyquist-performance-condition), we must required that all possible \\(L\_p(j\omega)\\) stay outside a disk of radius \\(\abs{w\_P(j\omega)}\\) centered on \\(-1\\).
As illustrated on [Figure 23](#figure--fig:nyquist-performance-condition), we must required that all possible \\(L\_p(j\omega)\\) stay outside a disk of radius \\(\abs{w\_P(j\omega)}\\) centered on \\(-1\\).
Since \\(L\_p\\) at each frequency stays within a disk of radius \\(|w\_I(j\omega) L(j\omega)|\\) centered on \\(L(j\omega)\\), the condition for RP becomes:
\begin{align\*}
@@ -3524,7 +3524,7 @@ In the transfer function form:
with \\(\Phi(s) \triangleq (sI - A)^{-1}\\).
This is illustrated in the block diagram of [25](#figure--fig:uncertainty-state-a-matrix), which is in the form of an inverse additive perturbation.
This is illustrated in the block diagram of [Figure 25](#figure--fig:uncertainty-state-a-matrix), which is in the form of an inverse additive perturbation.
<a id="figure--fig:uncertainty-state-a-matrix"></a>
@@ -3562,13 +3562,13 @@ The starting point for our robustness analysis is a system representation in whi
where each \\(\Delta\_i\\) represents a **specific source of uncertainty**, e.g. input uncertainty \\(\Delta\_I\\) or parametric uncertainty \\(\delta\_i\\).
If we also pull out the controller \\(K\\), we get the generalized plant \\(P\\) as shown in [26](#figure--fig:general-control-delta). This form is useful for controller synthesis.
If we also pull out the controller \\(K\\), we get the generalized plant \\(P\\) as shown in [Figure 26](#figure--fig:general-control-delta). This form is useful for controller synthesis.
<a id="figure--fig:general-control-delta"></a>
{{< figure src="/ox-hugo/skogestad07_general_control_delta.png" caption="<span class=\"figure-number\">Figure 26: </span>General control configuration used for controller synthesis" >}}
If the controller is given and we want to analyze the uncertain system, we use the \\(N\Delta\text{-structure}\\) in [27](#figure--fig:general-control-Ndelta).
If the controller is given and we want to analyze the uncertain system, we use the \\(N\Delta\text{-structure}\\) in [Figure 27](#figure--fig:general-control-Ndelta).
<a id="figure--fig:general-control-Ndelta"></a>
@@ -3588,7 +3588,7 @@ Similarly, the uncertain closed-loop transfer function from \\(w\\) to \\(z\\),
&\triangleq N\_{22} + N\_{21} \Delta (I - N\_{11} \Delta)^{-1} N\_{12}
\end{align\*}
To analyze robust stability of \\(F\\), we can rearrange the system into the \\(M\Delta\text{-structure}\\) shown in [28](#figure--fig:general-control-Mdelta-bis) where \\(M = N\_{11}\\) is the transfer function from the output to the input of the perturbations.
To analyze robust stability of \\(F\\), we can rearrange the system into the \\(M\Delta\text{-structure}\\) shown in [Figure 28](#figure--fig:general-control-Mdelta-bis) where \\(M = N\_{11}\\) is the transfer function from the output to the input of the perturbations.
<a id="figure--fig:general-control-Mdelta-bis"></a>
@@ -3627,7 +3627,7 @@ However, the inclusion of parametric uncertainty may be more significant for MIM
Unstructured perturbations are often used to get a simple uncertainty model.
We here define unstructured uncertainty as the use of a "full" complex perturbation matrix \\(\Delta\\), usually with dimensions compatible with those of the plant, where at each frequency any \\(\Delta(j\w)\\) satisfying \\(\maxsv(\Delta(j\w)) < 1\\) is allowed.
Three common forms of **feedforward unstructured uncertainty** are shown [4](#table--fig:feedforward-uncertainty): additive uncertainty, multiplicative input uncertainty and multiplicative output uncertainty.
Three common forms of **feedforward unstructured uncertainty** are shown [Table 4](#table--fig:feedforward-uncertainty): additive uncertainty, multiplicative input uncertainty and multiplicative output uncertainty.
<div class="important">
@@ -3651,7 +3651,7 @@ Three common forms of **feedforward unstructured uncertainty** are shown [4](#ta
|-------------------------------------------------------------------------------------------------|----------------------------------------------------------------------------------------------------------|------------------------------------------------------------------------------------------------------------|
| <span class="org-target" id="org-target--fig-additive-uncertainty"></span> Additive uncertainty | <span class="org-target" id="org-target--fig-input-uncertainty"></span> Multiplicative input uncertainty | <span class="org-target" id="org-target--fig-output-uncertainty"></span> Multiplicative output uncertainty |
In [5](#table--fig:feedback-uncertainty), three **feedback or inverse unstructured uncertainty** forms are shown: inverse additive uncertainty, inverse multiplicative input uncertainty and inverse multiplicative output uncertainty.
In [Table 5](#table--fig:feedback-uncertainty), three **feedback or inverse unstructured uncertainty** forms are shown: inverse additive uncertainty, inverse multiplicative input uncertainty and inverse multiplicative output uncertainty.
<div class="important">
@@ -3768,7 +3768,7 @@ where \\(r\_0\\) is the relative uncertainty at steady-state, \\(1/\tau\\) is th
### Obtaining \\(P\\), \\(N\\) and \\(M\\) {#obtaining-p-n-and-m}
Let's consider the feedback system with multiplicative input uncertainty \\(\Delta\_I\\) shown [29](#figure--fig:input-uncertainty-set-feedback-weight).
Let's consider the feedback system with multiplicative input uncertainty \\(\Delta\_I\\) shown [Figure 29](#figure--fig:input-uncertainty-set-feedback-weight).
\\(W\_I\\) is a normalization weight for the uncertainty and \\(W\_P\\) is a performance weight.
<a id="figure--fig:input-uncertainty-set-feedback-weight"></a>
@@ -3906,7 +3906,7 @@ Then the \\(M\Delta\text{-system}\\) is stable for all perturbations \\(\Delta\\
#### Application of the Unstructured RS-condition {#application-of-the-unstructured-rs-condition}
We will now present necessary and sufficient conditions for robust stability for each of the six single unstructured perturbations in [4](#table--fig:feedforward-uncertainty) and [5](#table--fig:feedback-uncertainty) with
We will now present necessary and sufficient conditions for robust stability for each of the six single unstructured perturbations in [Table 4](#table--fig:feedforward-uncertainty) and [Table 5](#table--fig:feedback-uncertainty) with
\begin{equation\*}
E = W\_2 \Delta W\_1, \quad \hnorm{\Delta} \le 1
@@ -3951,7 +3951,7 @@ In order to get tighter condition we must use a tighter uncertainty description
Robust stability bound in terms of the \\(\hinf\\) norm (\\(\text{RS}\Leftrightarrow\hnorm{M}<1\\)) are in general only tight when there is a single full perturbation block.
An "exception" to this is when the uncertainty blocks enter or exit from the same location in the block diagram, because they can then be stacked on top of each other or side-by-side, in an overall \\(\Delta\\) which is then full matrix.
One important uncertainty description that falls into this category is the **coprime uncertainty description** shown in [30](#figure--fig:coprime-uncertainty), for which the set of plants is
One important uncertainty description that falls into this category is the **coprime uncertainty description** shown in [Figure 30](#figure--fig:coprime-uncertainty), for which the set of plants is
\begin{equation\*}
G\_p = (M\_l + \Delta\_M)^{-1}(Nl + \Delta\_N), \quad \hnorm{[\Delta\_N, \ \Delta\_N]} \le \epsilon
@@ -4007,7 +4007,7 @@ To this effect, introduce the block-diagonal scaling matrix
where \\(d\_i\\) is a scalar and \\(I\_i\\) is an identity matrix of the same dimension as the \\(i\\)'th perturbation block \\(\Delta\_i\\).
Now rescale the inputs and outputs of \\(M\\) and \\(\Delta\\) by inserting the matrices \\(D\\) and \\(D^{-1}\\) on both sides as shown in [31](#figure--fig:block-diagonal-scalings).
Now rescale the inputs and outputs of \\(M\\) and \\(\Delta\\) by inserting the matrices \\(D\\) and \\(D^{-1}\\) on both sides as shown in [Figure 31](#figure--fig:block-diagonal-scalings).
This clearly has no effect on stability.
<a id="figure--fig:block-diagonal-scalings"></a>
@@ -4302,7 +4302,7 @@ Note that \\(\mu\\) underestimate how bad or good the actual worst case performa
### Application: RP with Input Uncertainty {#application-rp-with-input-uncertainty}
We will now consider in some detail the case of multiplicative input uncertainty with performance defined in terms of weighted sensitivity ([29](#figure--fig:input-uncertainty-set-feedback-weight)).
We will now consider in some detail the case of multiplicative input uncertainty with performance defined in terms of weighted sensitivity ([Figure 29](#figure--fig:input-uncertainty-set-feedback-weight)).
The performance requirement is then
@@ -4416,7 +4416,7 @@ with the decoupling controller we have:
\overline{\sigma}(N\_{22}) = \overline{\sigma}(w\_P S) = \left|\frac{s/2 + 0.05}{s + 0.7}\right|
\end{equation\*}
and we see from [32](#figure--fig:mu-plots-distillation) that the NP-condition is satisfied.
and we see from [Figure 32](#figure--fig:mu-plots-distillation) that the NP-condition is satisfied.
<a id="figure--fig:mu-plots-distillation"></a>
@@ -4431,7 +4431,7 @@ In this case \\(w\_I T\_I = w\_I T\\) is a scalar times the identity matrix:
\mu\_{\Delta\_I}(w\_I T\_I) = |w\_I t| = \left|0.2 \frac{5s + 1}{(0.5s + 1)(1.43s + 1)}\right|
\end{equation\*}
and we see from [32](#figure--fig:mu-plots-distillation) that RS is satisfied.
and we see from [Figure 32](#figure--fig:mu-plots-distillation) that RS is satisfied.
The peak value of \\(\mu\_{\Delta\_I}(M)\\) is \\(0.53\\) meaning that we may increase the uncertainty by a factor of \\(1/0.53 = 1.89\\) before the worst case uncertainty yields instability.
@@ -4439,7 +4439,7 @@ The peak value of \\(\mu\_{\Delta\_I}(M)\\) is \\(0.53\\) meaning that we may in
##### RP {#rp}
Although the system has good robustness margins and excellent nominal performance, the robust performance is poor.
This is shown in [32](#figure--fig:mu-plots-distillation) where the \\(\mu\text{-curve}\\) for RP was computed numerically using \\(\mu\_{\hat{\Delta}}(N)\\), with \\(\hat{\Delta} = \text{diag}\\{\Delta\_I, \Delta\_P\\}\\) and \\(\Delta\_I = \text{diag}\\{\delta\_1, \delta\_2\\}\\).
This is shown in [Figure 32](#figure--fig:mu-plots-distillation) where the \\(\mu\text{-curve}\\) for RP was computed numerically using \\(\mu\_{\hat{\Delta}}(N)\\), with \\(\hat{\Delta} = \text{diag}\\{\Delta\_I, \Delta\_P\\}\\) and \\(\Delta\_I = \text{diag}\\{\delta\_1, \delta\_2\\}\\).
The peak value is close to 6, meaning that even with 6 times less uncertainty, the weighted sensitivity will be about 6 times larger than what we require.
@@ -4576,7 +4576,7 @@ The latter is an attempt to "flatten out" \\(\mu\\).
#### Example: \\(\mu\text{-synthesis}\\) with DK-iteration {#example-mu-text-synthesis-with-dk-iteration}
For simplicity, we will consider again the case of multiplicative uncertainty and performance defined in terms of weighted sensitivity.
The uncertainty weight \\(w\_I I\\) and performance weight \\(w\_P I\\) are shown graphically in [33](#figure--fig:weights-distillation).
The uncertainty weight \\(w\_I I\\) and performance weight \\(w\_P I\\) are shown graphically in [Figure 33](#figure--fig:weights-distillation).
<a id="figure--fig:weights-distillation"></a>
@@ -4592,8 +4592,8 @@ The scaling matrix \\(D\\) for \\(DND^{-1}\\) then has the structure \\(D = \tex
- Iteration No. 1.
Step 1: with the initial scalings, the \\(\mathcal{H}\_\infty\\) synthesis produced a 6 state controller (2 states from the plant model and 2 from each of the weights).
Step 2: the upper \\(\mu\text{-bound}\\) is shown in [34](#figure--fig:dk-iter-mu).
Step 3: the frequency dependent \\(d\_1(\omega)\\) and \\(d\_2(\omega)\\) from step 2 are fitted using a 4th order transfer function shown in [35](#figure--fig:dk-iter-d-scale)
Step 2: the upper \\(\mu\text{-bound}\\) is shown in [Figure 34](#figure--fig:dk-iter-mu).
Step 3: the frequency dependent \\(d\_1(\omega)\\) and \\(d\_2(\omega)\\) from step 2 are fitted using a 4th order transfer function shown in [Figure 35](#figure--fig:dk-iter-d-scale)
- Iteration No. 2.
Step 1: with the 8 state scalings \\(D^1(s)\\), the \\(\mathcal{H}\_\infty\\) synthesis gives a 22 state controller.
Step 2: This controller gives a peak value of \\(\mu\\) of \\(1.02\\).
@@ -4609,7 +4609,7 @@ The scaling matrix \\(D\\) for \\(DND^{-1}\\) then has the structure \\(D = \tex
{{< figure src="/ox-hugo/skogestad07_dk_iter_d_scale.png" caption="<span class=\"figure-number\">Figure 35: </span>Change in D-scale \\(d\_1\\) during DK-iteration" >}}
The final \\(\mu\text{-curves}\\) for NP, RS and RP with the controller \\(K\_3\\) are shown in [36](#figure--fig:mu-plot-optimal-k3).
The final \\(\mu\text{-curves}\\) for NP, RS and RP with the controller \\(K\_3\\) are shown in [Figure 36](#figure--fig:mu-plot-optimal-k3).
The objectives of RS and NP are easily satisfied.
The peak value of \\(\mu\\) is just slightly over 1, so the performance specification \\(\overline{\sigma}(w\_P S\_p) < 1\\) is almost satisfied for all possible plants.
@@ -4617,7 +4617,7 @@ The peak value of \\(\mu\\) is just slightly over 1, so the performance specific
{{< figure src="/ox-hugo/skogestad07_mu_plot_optimal_k3.png" caption="<span class=\"figure-number\">Figure 36: </span>\\(mu\text{-plots}\\) with \\(\mu\\) \"optimal\" controller \\(K\_3\\)" >}}
To confirm that, 6 perturbed plants are used to compute the perturbed sensitivity functions shown in [37](#figure--fig:perturb-s-k3).
To confirm that, 6 perturbed plants are used to compute the perturbed sensitivity functions shown in [Figure 37](#figure--fig:perturb-s-k3).
<a id="figure--fig:perturb-s-k3"></a>
@@ -4696,7 +4696,7 @@ By multivariable transfer function shaping, therefore, we mean the shaping of th
The classical loop-shaping ideas can be further generalized to MIMO systems by considering the singular values.
Consider the one degree-of-freedom system as shown in [38](#figure--fig:classical-feedback-small).
Consider the one degree-of-freedom system as shown in [Figure 38](#figure--fig:classical-feedback-small).
We have the following important relationships:
\begin{align}
@@ -4750,7 +4750,7 @@ Thus, over specified frequency ranges, it is relatively easy to approximate the
</div>
Typically, the open-loop requirements 1 and 3 are valid and important at low frequencies \\(0 \le \omega \le \omega\_l \le \omega\_B\\), while conditions 2, 4, 5 and 6 are conditions which are valid and important at high frequencies \\(\omega\_B \le \omega\_h \le \omega \le \infty\\), as illustrated in [39](#figure--fig:design-trade-off-mimo-gk).
Typically, the open-loop requirements 1 and 3 are valid and important at low frequencies \\(0 \le \omega \le \omega\_l \le \omega\_B\\), while conditions 2, 4, 5 and 6 are conditions which are valid and important at high frequencies \\(\omega\_B \le \omega\_h \le \omega \le \infty\\), as illustrated in [Figure 39](#figure--fig:design-trade-off-mimo-gk).
<a id="figure--fig:design-trade-off-mimo-gk"></a>
@@ -4810,7 +4810,7 @@ The optimal state estimate is given by a **Kalman filter**.
The solution to the LQG problem is then found by replacing \\(x\\) by \\(\hat{x}\\) to give \\(u(t) = -K\_r \hat{x}\\).
We therefore see that the LQG problem and its solution can be separated into two distinct parts as illustrated in [40](#figure--fig:lqg-separation): the optimal state feedback and the optimal state estimator (the Kalman filter).
We therefore see that the LQG problem and its solution can be separated into two distinct parts as illustrated in [Figure 40](#figure--fig:lqg-separation): the optimal state feedback and the optimal state estimator (the Kalman filter).
<a id="figure--fig:lqg-separation"></a>
@@ -4842,7 +4842,7 @@ and \\(X\\) is the unique positive-semi definite solution of the algebraic Ricca
<div class="important">
The **Kalman filter** has the structure of an ordinary state-estimator, as shown on [41](#figure--fig:lqg-kalman-filter), with:
The **Kalman filter** has the structure of an ordinary state-estimator, as shown on [Figure 41](#figure--fig:lqg-kalman-filter), with:
\begin{equation} \label{eq:kalman\_filter\_structure}
\dot{\hat{x}} = A\hat{x} + Bu + K\_f(y-C\hat{x})
@@ -4866,7 +4866,7 @@ Where \\(Y\\) is the unique positive-semi definite solution of the algebraic Ric
{{< figure src="/ox-hugo/skogestad07_lqg_kalman_filter.png" caption="<span class=\"figure-number\">Figure 41: </span>The LQG controller and noisy plant" >}}
The structure of the LQG controller is illustrated in [41](#figure--fig:lqg-kalman-filter), its transfer function from \\(y\\) to \\(u\\) is given by
The structure of the LQG controller is illustrated in [Figure 41](#figure--fig:lqg-kalman-filter), its transfer function from \\(y\\) to \\(u\\) is given by
\begin{align\*}
L\_{\text{LQG}}(s) &= \left[ \begin{array}{c|c}
@@ -4881,7 +4881,7 @@ The structure of the LQG controller is illustrated in [41](#figure--fig:lqg-kalm
It has the same degree (number of poles) as the plant.<br />
For the LQG-controller, as shown on [41](#figure--fig:lqg-kalman-filter), it is not easy to see where to position the reference input \\(r\\) and how integral action may be included, if desired. Indeed, the standard LQG design procedure does not give a controller with integral action. One strategy is illustrated in [42](#figure--fig:lqg-integral). Here, the control error \\(r-y\\) is integrated and the regulator \\(K\_r\\) is designed for the plant augmented with these integral states.
For the LQG-controller, as shown on [Figure 41](#figure--fig:lqg-kalman-filter), it is not easy to see where to position the reference input \\(r\\) and how integral action may be included, if desired. Indeed, the standard LQG design procedure does not give a controller with integral action. One strategy is illustrated in [Figure 42](#figure--fig:lqg-integral). Here, the control error \\(r-y\\) is integrated and the regulator \\(K\_r\\) is designed for the plant augmented with these integral states.
<a id="figure--fig:lqg-integral"></a>
@@ -4905,7 +4905,7 @@ Their main limitation is that they can only be applied to minimum phase plants.
There are many ways in which feedback design problems can be cast as \\(\htwo\\) and \\(\hinf\\) optimization problems.
It is very useful therefore to have a **standard problem formulation** into which any particular problem may be manipulated.
Such a general formulation is afforded by the general configuration shown in [43](#figure--fig:general-control).
Such a general formulation is afforded by the general configuration shown in [Figure 43](#figure--fig:general-control).
<a id="figure--fig:general-control"></a>
@@ -5085,7 +5085,7 @@ Then the LQG cost function is
#### \\(\hinf\\) Optimal Control {#hinf-optimal-control}
With reference to the general control configuration on [43](#figure--fig:general-control), the standard \\(\hinf\\) optimal control problem is to find all stabilizing controllers \\(K\\) which minimize
With reference to the general control configuration on [Figure 43](#figure--fig:general-control), the standard \\(\hinf\\) optimal control problem is to find all stabilizing controllers \\(K\\) which minimize
\begin{equation\*}
\hnorm{F\_l(P, K)} = \max\_{\omega} \maxsv\big(F\_l(P, K)(j\omega)\big)
@@ -5196,7 +5196,7 @@ In general, the scalar weighting functions \\(w\_1(s)\\) and \\(w\_2(s)\\) can b
This can be useful for **systems with channels of quite different bandwidths**.
In that case, **diagonal weights are recommended** as anything more complicated is usually not worth the effort.<br />
To see how this mixed sensitivity problem can be formulated in the general setting, we can imagine the disturbance \\(d\\) as a single exogenous input and define and error signal \\(z = [z\_1^T\ z\_2^T]^T\\), where \\(z\_1 = W\_1 y\\) and \\(z\_2 = -W\_2 u\\) as illustrated in [44](#figure--fig:mixed-sensitivity-dist-rejection).
To see how this mixed sensitivity problem can be formulated in the general setting, we can imagine the disturbance \\(d\\) as a single exogenous input and define and error signal \\(z = [z\_1^T\ z\_2^T]^T\\), where \\(z\_1 = W\_1 y\\) and \\(z\_2 = -W\_2 u\\) as illustrated in [Figure 44](#figure--fig:mixed-sensitivity-dist-rejection).
We can then see that \\(z\_1 = W\_1 S w\\) and \\(z\_2 = W\_2 KS w\\) as required.
The elements of the generalized plant are
@@ -5217,10 +5217,10 @@ The elements of the generalized plant are
{{< figure src="/ox-hugo/skogestad07_mixed_sensitivity_dist_rejection.png" caption="<span class=\"figure-number\">Figure 44: </span>\\(S/KS\\) mixed-sensitivity optimization in standard form (regulation)" >}}
Another interpretation can be put on the \\(S/KS\\) mixed-sensitivity optimization as shown in the standard control configuration of [45](#figure--fig:mixed-sensitivity-ref-tracking).
Another interpretation can be put on the \\(S/KS\\) mixed-sensitivity optimization as shown in the standard control configuration of [Figure 45](#figure--fig:mixed-sensitivity-ref-tracking).
Here we consider a tracking problem.
The exogenous input is a reference command \\(r\\), and the error signals are \\(z\_1 = -W\_1 e = W\_1 (r-y)\\) and \\(z\_2 = W\_2 u\\).
As the regulation problem of [44](#figure--fig:mixed-sensitivity-dist-rejection), we have that \\(z\_1 = W\_1 S w\\) and \\(z\_2 = W\_2 KS w\\).
As the regulation problem of [Figure 44](#figure--fig:mixed-sensitivity-dist-rejection), we have that \\(z\_1 = W\_1 S w\\) and \\(z\_2 = W\_2 KS w\\).
<a id="figure--fig:mixed-sensitivity-ref-tracking"></a>
@@ -5235,7 +5235,7 @@ Another useful mixed sensitivity optimization problem, is to find a stabilizing
The ability to shape \\(T\\) is desirable for tracking problems and noise attenuation.
It is also important for robust stability with respect to multiplicative perturbations at the plant output.
The \\(S/T\\) mixed-sensitivity minimization problem can be put into the standard control configuration as shown in [46](#figure--fig:mixed-sensitivity-s-t).
The \\(S/T\\) mixed-sensitivity minimization problem can be put into the standard control configuration as shown in [Figure 46](#figure--fig:mixed-sensitivity-s-t).
The elements of the generalized plant are
@@ -5277,7 +5277,7 @@ The focus of attention has moved to the size of signals and away from the size a
</div>
Weights are used to describe the expected or known frequency content of exogenous signals and the desired frequency content of error signals.
Weights are also used if a perturbation is used to model uncertainty, as in [47](#figure--fig:input-uncertainty-hinf), where \\(G\\) represents the nominal model, \\(W\\) is a weighting function that captures the relative model fidelity over frequency, and \\(\Delta\\) represents unmodelled dynamics usually normalized such that \\(\hnorm{\Delta} < 1\\).
Weights are also used if a perturbation is used to model uncertainty, as in [Figure 47](#figure--fig:input-uncertainty-hinf), where \\(G\\) represents the nominal model, \\(W\\) is a weighting function that captures the relative model fidelity over frequency, and \\(\Delta\\) represents unmodelled dynamics usually normalized such that \\(\hnorm{\Delta} < 1\\).
<a id="figure--fig:input-uncertainty-hinf"></a>
@@ -5288,9 +5288,9 @@ As we have seen, the weights \\(Q\\) and \\(R\\) are constant, but LQG can be ge
When we consider a system's response to persistent sinusoidal signals of varying frequency, or when we consider the induced 2-norm between the exogenous input signals and the error signals, we are required to minimize the \\(\hinf\\) norm.
In the absence of model uncertainty, there does not appear to be an overwhelming case for using the \\(\hinf\\) norm rather than the more traditional \\(\htwo\\) norm.
However, when uncertainty is addressed, as it always should be, \\(\hinf\\) is clearly the more **natural approach** using component uncertainty models as in [47](#figure--fig:input-uncertainty-hinf).<br />
However, when uncertainty is addressed, as it always should be, \\(\hinf\\) is clearly the more **natural approach** using component uncertainty models as in [Figure 47](#figure--fig:input-uncertainty-hinf).<br />
A typical problem using the signal-based approach to \\(\hinf\\) control is illustrated in the interconnection diagram of [48](#figure--fig:hinf-signal-based).
A typical problem using the signal-based approach to \\(\hinf\\) control is illustrated in the interconnection diagram of [Figure 48](#figure--fig:hinf-signal-based).
\\(G\\) and \\(G\_d\\) are nominal models of the plant and disturbance dynamics, and \\(K\\) is the controller to be designed.
The weights \\(W\_d\\), \\(W\_r\\), and \\(W\_n\\) may be constant or dynamic and describe the relative importance and/or the frequency content of the disturbance, set points and noise signals.
The weight \\(W\_\text{ref}\\) is a desired closed-loop transfer function between the weighted set point \\(r\_s\\) and the actual output \\(y\\).
@@ -5315,7 +5315,7 @@ The problem can be cast as a standard \\(\hinf\\) optimization in the general co
{{< figure src="/ox-hugo/skogestad07_hinf_signal_based.png" caption="<span class=\"figure-number\">Figure 48: </span>A signal-based \\(\hinf\\) control problem" >}}
Suppose we now introduce a multiplicative dynamic uncertainty model at the input to the plant as shown in [49](#figure--fig:hinf-signal-based-uncertainty).
Suppose we now introduce a multiplicative dynamic uncertainty model at the input to the plant as shown in [Figure 49](#figure--fig:hinf-signal-based-uncertainty).
The problem we now want to solve is: find a stabilizing controller \\(K\\) such that the \\(\hinf\\) norm of the transfer function between \\(w\\) and \\(z\\) is less that 1 for all \\(\Delta\\) where \\(\hnorm{\Delta} < 1\\).
We have assumed in this statement that the **signal weights have normalized the 2-norm of the exogenous input signals to unity**.
This problem is a non-standard \\(\hinf\\) optimization.
@@ -5378,7 +5378,7 @@ The objective of robust stabilization is to stabilize not only the nominal model
where \\(\epsilon > 0\\) is then the **stability margin**.<br />
For the perturbed feedback system of [50](#figure--fig:coprime-uncertainty-bis), the stability property is robust if and only if the nominal feedback system is stable and
For the perturbed feedback system of [Figure 50](#figure--fig:coprime-uncertainty-bis), the stability property is robust if and only if the nominal feedback system is stable and
\begin{equation\*}
\gamma \triangleq \hnorm{\begin{bmatrix}
@@ -5456,7 +5456,7 @@ If \\(W\_1\\) and \\(W\_2\\) are the pre and post compensators respectively, the
G\_s = W\_2 G W\_1
\end{equation}
as shown in [51](#figure--fig:shaped-plant).
as shown in [Figure 51](#figure--fig:shaped-plant).
<a id="figure--fig:shaped-plant"></a>
@@ -5491,7 +5491,7 @@ Systematic procedure for \\(\hinf\\) loop-shaping design:
- A small value of \\(\epsilon\_{\text{max}}\\) indicates that the chosen singular value loop-shapes are incompatible with robust stability requirements
7. **Analyze the design** and if not all the specification are met, make further modifications to the weights
8. **Implement the controller**.
The configuration shown in [52](#figure--fig:shapping-practical-implementation) has been found useful when compared with the conventional setup in [38](#figure--fig:classical-feedback-small).
The configuration shown in [Figure 52](#figure--fig:shapping-practical-implementation) has been found useful when compared with the conventional setup in [Figure 38](#figure--fig:classical-feedback-small).
This is because the references do not directly excite the dynamics of \\(K\_s\\), which can result in large amounts of overshoot.
The constant prefilter ensure a steady-state gain of \\(1\\) between \\(r\\) and \\(y\\), assuming integral action in \\(W\_1\\) or \\(G\\)
@@ -5518,7 +5518,7 @@ Many control design problems possess two degrees-of-freedom:
Sometimes, one degree-of-freedom is left out of the design, and the controller is driven by an error signal i.e. the difference between a command and the output.
But in cases where stringent time-domain specifications are set on the output response, a one degree-of-freedom structure may not be sufficient.<br />
A general two degrees-of-freedom feedback control scheme is depicted in [53](#figure--fig:classical-feedback-2dof-simple).
A general two degrees-of-freedom feedback control scheme is depicted in [Figure 53](#figure--fig:classical-feedback-2dof-simple).
The commands and feedbacks enter the controller separately and are independently processed.
<a id="figure--fig:classical-feedback-2dof-simple"></a>
@@ -5528,7 +5528,7 @@ The commands and feedbacks enter the controller separately and are independently
The presented \\(\mathcal{H}\_\infty\\) loop-shaping design procedure in section is a one-degree-of-freedom design, although a **constant** pre-filter can be easily implemented for steady-state accuracy.
However, this may not be sufficient and a dynamic two degrees-of-freedom design is required.<br />
The design problem is illustrated in [54](#figure--fig:coprime-uncertainty-hinf).
The design problem is illustrated in [Figure 54](#figure--fig:coprime-uncertainty-hinf).
The feedback part of the controller \\(K\_2\\) is designed to meet robust stability and disturbance rejection requirements.
A prefilter is introduced to force the response of the closed-loop system to follow that of a specified model \\(T\_{\text{ref}}\\), often called the **reference model**.
@@ -5536,7 +5536,7 @@ A prefilter is introduced to force the response of the closed-loop system to fol
{{< figure src="/ox-hugo/skogestad07_coprime_uncertainty_hinf.png" caption="<span class=\"figure-number\">Figure 54: </span>Two degrees-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping design problem" >}}
The design problem is to find the stabilizing controller \\(K = [K\_1,\ K\_2]\\) for the shaped plant \\(G\_s = G W\_1\\), with a normalized coprime factorization \\(G\_s = M\_s^{-1} N\_s\\), which minimizes the \\(\mathcal{H}\_\infty\\) norm of the transfer function between the signals \\([r^T\ \phi^T]^T\\) and \\([u\_s^T\ y^T\ e^T]^T\\) as defined in [54](#figure--fig:coprime-uncertainty-hinf).
The design problem is to find the stabilizing controller \\(K = [K\_1,\ K\_2]\\) for the shaped plant \\(G\_s = G W\_1\\), with a normalized coprime factorization \\(G\_s = M\_s^{-1} N\_s\\), which minimizes the \\(\mathcal{H}\_\infty\\) norm of the transfer function between the signals \\([r^T\ \phi^T]^T\\) and \\([u\_s^T\ y^T\ e^T]^T\\) as defined in [Figure 54](#figure--fig:coprime-uncertainty-hinf).
This problem is easily cast into the general configuration.
The control signal to the shaped plant \\(u\_s\\) is given by:
@@ -5566,7 +5566,7 @@ The main steps required to synthesize a two degrees-of-freedom \\(\mathcal{H}\_\
5. Replace the prefilter \\(K\_1\\) by \\(K\_1 W\_i\\) to give exact model-matching at steady-state.
6. Analyze and, if required, redesign making adjustments to \\(\rho\\) and possibly \\(W\_1\\) and \\(T\_{\text{ref}}\\)
The final two degrees-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping controller is illustrated in [55](#figure--fig:hinf-synthesis-2dof).
The final two degrees-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping controller is illustrated in [Figure 55](#figure--fig:hinf-synthesis-2dof).
<a id="figure--fig:hinf-synthesis-2dof"></a>
@@ -5650,7 +5650,7 @@ When implemented in Hanus form, the expression for \\(u\\) becomes
where \\(u\_a\\) is the **actual plant input**, that is the measurement at the **output of the actuators** which therefore contains information about possible actuator saturation.
The situation is illustrated in [56](#figure--fig:weight-anti-windup), where the actuators are each modeled by a unit gain and a saturation.
The situation is illustrated in [Figure 56](#figure--fig:weight-anti-windup), where the actuators are each modeled by a unit gain and a saturation.
<a id="figure--fig:weight-anti-windup"></a>
@@ -5713,7 +5713,7 @@ Moreover, one should be careful about combining controller synthesis and analysi
### Introduction {#introduction}
In previous sections, we considered the general problem formulation in [57](#figure--fig:general-control-names-bis) and stated that the controller design problem is to find a controller \\(K\\) which based on the information in \\(v\\), generates a control signal \\(u\\) which counteracts the influence of \\(w\\) on \\(z\\), thereby minimizing the closed loop norm from \\(w\\) to \\(z\\).
In previous sections, we considered the general problem formulation in [Figure 57](#figure--fig:general-control-names-bis) and stated that the controller design problem is to find a controller \\(K\\) which based on the information in \\(v\\), generates a control signal \\(u\\) which counteracts the influence of \\(w\\) on \\(z\\), thereby minimizing the closed loop norm from \\(w\\) to \\(z\\).
<a id="figure--fig:general-control-names-bis"></a>
@@ -5748,19 +5748,19 @@ The reference value \\(r\\) is usually set at some higher layer in the control h
- **Optimization layer**: computes the desired reference commands \\(r\\)
- **Control layer**: implements these commands to achieve \\(y \approx r\\)
Additional layers are possible, as is illustrated in [58](#figure--fig:control-system-hierarchy) which shows a typical control hierarchy for a chemical plant.
Additional layers are possible, as is illustrated in [Figure 58](#figure--fig:control-system-hierarchy) which shows a typical control hierarchy for a chemical plant.
<a id="figure--fig:control-system-hierarchy"></a>
{{< figure src="/ox-hugo/skogestad07_system_hierarchy.png" caption="<span class=\"figure-number\">Figure 58: </span>Typical control system hierarchy in a chemical plant" >}}
In general, the information flow in such a control hierarchy is based on the higher layer sending reference values (setpoints) to the layer below reporting back any problems achieving this (see [6](#org-target--fig-optimize-control-b)).
In general, the information flow in such a control hierarchy is based on the higher layer sending reference values (setpoints) to the layer below reporting back any problems achieving this (see [ 6](#org-target--fig-optimize-control-b)).
There is usually a time scale separation between the layers which means that the **setpoints**, as viewed from a given layer, are **updated only periodically**.<br />
The optimization tends to be performed open-loop with limited use of feedback. On the other hand, the control layer is mainly based on feedback information.
The **optimization is often based on nonlinear steady-state models**, whereas we often use **linear dynamic models in the control layer**.<br />
From a theoretical point of view, the optimal performance is obtained with a **centralized optimizing controller**, which combines the two layers of optimizing and control (see [6](#org-target--fig-optimize-control-c)).
From a theoretical point of view, the optimal performance is obtained with a **centralized optimizing controller**, which combines the two layers of optimizing and control (see [ 6](#org-target--fig-optimize-control-c)).
All control actions in such an ideal control system would be perfectly coordinated and the control system would use on-line dynamic optimization based on nonlinear dynamic model of the complete plant.
However, this solution is normally not used for a number a reasons, included the cost of modeling, the difficulty of controller design, maintenance, robustness problems and the lack of computing power.
@@ -5885,7 +5885,7 @@ Thus, the selection of controlled and measured outputs are two separate issues.
### Selection of Manipulations and Measurements {#selection-of-manipulations-and-measurements}
We are here concerned with the variable sets \\(u\\) and \\(v\\) in [57](#figure--fig:general-control-names-bis).
We are here concerned with the variable sets \\(u\\) and \\(v\\) in [Figure 57](#figure--fig:general-control-names-bis).
Note that **the measurements** \\(v\\) used by the controller **are in general different from the controlled variables** \\(z\\) because we may not be able to measure all the controlled variables and we may want to measure and control additional variables in order to:
- Stabilize the plant, or more generally change its dynamics
@@ -5977,9 +5977,9 @@ Then when a SISO control loop is closed, we lose the input \\(u\_i\\) as a degre
A cascade control structure results when either of the following two situations arise:
- The reference \\(r\_i\\) is an output from another controller.
This is the **conventional cascade control** ([7](#org-target--fig-cascade-extra-meas))
This is the **conventional cascade control** ([ 7](#org-target--fig-cascade-extra-meas))
- The "measurement" \\(y\_i\\) is an output from another controller.
This is referred to as **input resetting** ([7](#org-target--fig-cascade-extra-input))
This is referred to as **input resetting** ([ 7](#org-target--fig-cascade-extra-input))
<a id="table--fig:cascade-implementation"></a>
<div class="table-caption">
@@ -6013,7 +6013,7 @@ where in most cases \\(r\_2 = 0\\) since we do not have a degree-of-freedom to c
##### Cascade implementation {#cascade-implementation}
To obtain an implementation with two SISO controllers, we may cascade the controllers as illustrated in [7](#org-target--fig-cascade-extra-meas):
To obtain an implementation with two SISO controllers, we may cascade the controllers as illustrated in [ 7](#org-target--fig-cascade-extra-meas):
\begin{align\*}
r\_2 &= K\_1(s)(r\_1 - y\_1) \\\\
@@ -6023,12 +6023,12 @@ To obtain an implementation with two SISO controllers, we may cascade the contro
Note that the output \\(r\_2\\) from the slower primary controller \\(K\_1\\) is not a manipulated plant input, but rather the reference input to the faster secondary controller \\(K\_2\\).
Cascades based on measuring the actual manipulated variable (\\(y\_2 = u\_m\\)) are commonly used to **reduce uncertainty and non-linearity at the plant input**.
In the general case ([7](#org-target--fig-cascade-extra-meas)) \\(y\_1\\) and \\(y\_2\\) are not directly related to each other, and this is sometimes referred to as _parallel cascade control_.
However, it is common to encounter the situation in [59](#figure--fig:cascade-control) where the primary output \\(y\_1\\) depends directly on \\(y\_2\\) which is a special case of [7](#org-target--fig-cascade-extra-meas).
In the general case ([ 7](#org-target--fig-cascade-extra-meas)) \\(y\_1\\) and \\(y\_2\\) are not directly related to each other, and this is sometimes referred to as _parallel cascade control_.
However, it is common to encounter the situation in [Figure 59](#figure--fig:cascade-control) where the primary output \\(y\_1\\) depends directly on \\(y\_2\\) which is a special case of [ 7](#org-target--fig-cascade-extra-meas).
<div class="important">
With reference to the special (but common) case of cascade control shown in [59](#figure--fig:cascade-control), the use of **extra measurements** is useful under the following circumstances:
With reference to the special (but common) case of cascade control shown in [Figure 59](#figure--fig:cascade-control), the use of **extra measurements** is useful under the following circumstances:
- The disturbance \\(d\_2\\) is significant and \\(G\_1\\) is non-minimum phase.
If \\(G\_1\\) is minimum phase, the input-output controllability of \\(G\_2\\) and \\(G\_1 G\_2\\) are the same and there is no fundamental advantage in measuring \\(y\_2\\)
@@ -6065,7 +6065,7 @@ Then \\(u\_2(t)\\) will only be used for **transient control** and will return t
##### Cascade implementation {#cascade-implementation}
To obtain an implementation with two SISO controllers we may cascade the controllers as shown in [7](#org-target--fig-cascade-extra-input).
To obtain an implementation with two SISO controllers we may cascade the controllers as shown in [ 7](#org-target--fig-cascade-extra-input).
We again let input \\(u\_2\\) take care of the **fast control** and \\(u\_1\\) of the **long-term control**.
The fast control loop is then
@@ -6086,7 +6086,7 @@ It also shows more clearly that \\(r\_{u\_2}\\), the reference for \\(u\_2\\), m
<div class="exampl">
Consider the system in [60](#figure--fig:cascade-control-two-layers) with two manipulated inputs (\\(u\_2\\) and \\(u\_3\\)), one controlled output (\\(y\_1\\) which should be close to \\(r\_1\\)) and two measured variables (\\(y\_1\\) and \\(y\_2\\)).
Consider the system in [Figure 60](#figure--fig:cascade-control-two-layers) with two manipulated inputs (\\(u\_2\\) and \\(u\_3\\)), one controlled output (\\(y\_1\\) which should be close to \\(r\_1\\)) and two measured variables (\\(y\_1\\) and \\(y\_2\\)).
Input \\(u\_2\\) has a more direct effect on \\(y\_1\\) than does input \\(u\_3\\) (there is a large delay in \\(G\_3(s)\\)).
Input \\(u\_2\\) should only be used for transient control as it is desirable that it remains close to \\(r\_3 = r\_{u\_2}\\).
The extra measurement \\(y\_2\\) is closer than \\(y\_1\\) to the input \\(u\_2\\) and may be useful for detecting disturbances affecting \\(G\_1\\).
@@ -6173,7 +6173,7 @@ Four applications of partial control are:
The outputs \\(y\_1\\) have an associated control objective but are not measured.
Instead, we aim at indirectly controlling \\(y\_1\\) by controlling the secondary measured variables \\(y\_2\\).
The table [8](#table--tab:partial-control) shows clearly the differences between the four applications of partial control.
The table [Table 8](#table--tab:partial-control) shows clearly the differences between the four applications of partial control.
In all cases, there is a control objective associated with \\(y\_1\\) and a feedback involving measurement and control of \\(y\_2\\) and we want:
- The effect of disturbances on \\(y\_1\\) to be small (when \\(y\_2\\) is controlled)
@@ -6201,7 +6201,7 @@ By partitioning the inputs and outputs, the overall model \\(y = G u\\) can be w
\end{aligned}
\end{equation}
Assume now that feedback control \\(u\_2 = K\_2(r\_2 - y\_2 - n\_2)\\) is used for the "secondary" subsystem involving \\(u\_2\\) and \\(y\_2\\) ([61](#figure--fig:partial-control)).
Assume now that feedback control \\(u\_2 = K\_2(r\_2 - y\_2 - n\_2)\\) is used for the "secondary" subsystem involving \\(u\_2\\) and \\(y\_2\\) ([Figure 61](#figure--fig:partial-control)).
We get:
\begin{equation} \label{eq:partial\_control}
@@ -6270,7 +6270,7 @@ The selection of \\(u\_2\\) and \\(y\_2\\) for use in the lower-layer control sy
##### Sequential design of cascade control systems {#sequential-design-of-cascade-control-systems}
Consider the conventional cascade control system in [7](#org-target--fig-cascade-extra-meas) where we have additional "secondary" measurements \\(y\_2\\) with no associated control objective, and the objective is to improve the control of \\(y\_1\\) by locally controlling \\(y\_2\\).
Consider the conventional cascade control system in [ 7](#org-target--fig-cascade-extra-meas) where we have additional "secondary" measurements \\(y\_2\\) with no associated control objective, and the objective is to improve the control of \\(y\_1\\) by locally controlling \\(y\_2\\).
The idea is that this should reduce the effect of disturbances and uncertainty on \\(y\_1\\).
From <eq:partial_control>, it follows that we should select \\(y\_2\\) and \\(u\_2\\) such that \\(\\|P\_d\\|\\) is small and at least smaller than \\(\\|G\_{d1}\\|\\).
@@ -6338,7 +6338,7 @@ Then to minimize the control error for the primary output, \\(J = \\|y\_1 - r\_1
### Decentralized Feedback Control {#decentralized-feedback-control}
In this section, \\(G(s)\\) is a square plant which is to be controlled using a diagonal controller ([62](#figure--fig:decentralized-diagonal-control)).
In this section, \\(G(s)\\) is a square plant which is to be controlled using a diagonal controller ([Figure 62](#figure--fig:decentralized-diagonal-control)).
<a id="figure--fig:decentralized-diagonal-control"></a>

90
content/book/taghirad13_paral.md
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@@ -56,7 +56,7 @@ The control of parallel robots is elaborated in the last two chapters, in which
Six independent parameters are sufficient to fully describe the spatial location of a rigid body.
Consider a rigid body in a spatial motion as represented in [1](#figure--fig:rigid-body-motion).
Consider a rigid body in a spatial motion as represented in [Figure 1](#figure--fig:rigid-body-motion).
Let us define:
- A **fixed reference coordinate system** \\((x, y, z)\\) denoted by frame \\(\\{\bm{A}\\}\\) whose origin is located at point \\(O\_A\\)
@@ -89,7 +89,7 @@ It can be **represented in several different ways**: the rotation matrix, the sc
##### Rotation Matrix {#rotation-matrix}
We consider a rigid body that has been exposed to a pure rotation.
Its orientation has changed from a state represented by frame \\(\\{\bm{A}\\}\\) to its current orientation represented by frame \\(\\{\bm{B}\\}\\) ([2](#figure--fig:rotation-matrix)).
Its orientation has changed from a state represented by frame \\(\\{\bm{A}\\}\\) to its current orientation represented by frame \\(\\{\bm{B}\\}\\) ([Figure 2](#figure--fig:rotation-matrix)).
A \\(3 \times 3\\) rotation matrix \\({}^A\bm{R}\_B\\) is defined by
@@ -363,7 +363,7 @@ There exist transformations to from screw displacement notation to the transform
##### Consecutive transformations {#consecutive-transformations}
Let us consider the motion of a rigid body described at three locations ([5](#figure--fig:consecutive-transformations)).
Let us consider the motion of a rigid body described at three locations ([Figure 5](#figure--fig:consecutive-transformations)).
Frame \\(\\{\bm{A}\\}\\) represents the initial location, frame \\(\\{\bm{B}\\}\\) is an intermediate location, and frame \\(\\{\bm{C}\\}\\) represents the rigid body at its final location.
<a id="figure--fig:consecutive-transformations"></a>
@@ -536,7 +536,7 @@ The position of the point \\(O\_B\\) of the moving platform is described by the
{{< figure src="/ox-hugo/taghirad13_stewart_schematic.png" caption="<span class=\"figure-number\">Figure 6: </span>Geometry of a Stewart-Gough platform" >}}
The geometry of the manipulator is shown [6](#figure--fig:stewart-schematic).
The geometry of the manipulator is shown [Figure 6](#figure--fig:stewart-schematic).
#### Inverse Kinematics {#inverse-kinematics}
@@ -676,7 +676,7 @@ The matrix \\(\bm{\Omega}^\times\\) denotes a **skew-symmetric matrix** defined
\end{bmatrix}}
\end{equation}
Now consider the general motion of a rigid body shown in [7](#figure--fig:general-motion), in which a moving frame \\(\\{\bm{B}\\}\\) is attached to the rigid body and **the problem is to find the absolute velocity** of point \\(P\\) with respect to a fixed frame \\(\\{\bm{A}\\}\\).
Now consider the general motion of a rigid body shown in [Figure 7](#figure--fig:general-motion), in which a moving frame \\(\\{\bm{B}\\}\\) is attached to the rigid body and **the problem is to find the absolute velocity** of point \\(P\\) with respect to a fixed frame \\(\\{\bm{A}\\}\\).
<a id="figure--fig:general-motion"></a>
@@ -935,7 +935,7 @@ We obtain that the **Jacobian matrix** constructs the **transformation needed to
#### Static Forces of the Stewart-Gough Platform {#static-forces-of-the-stewart-gough-platform}
As shown in [8](#figure--fig:stewart-static-forces), the twist of moving platform is described by a 6D vector \\(\dot{\bm{\mathcal{X}}} = \left[ {}^A\bm{v}\_P \ {}^A\bm{\omega} \right]^T\\), in which \\({}^A\bm{v}\_P\\) is the velocity of point \\(O\_B\\), and \\({}^A\bm{\omega}\\) is the angular velocity of moving platform.<br />
As shown in [Figure 8](#figure--fig:stewart-static-forces), the twist of moving platform is described by a 6D vector \\(\dot{\bm{\mathcal{X}}} = \left[ {}^A\bm{v}\_P \ {}^A\bm{\omega} \right]^T\\), in which \\({}^A\bm{v}\_P\\) is the velocity of point \\(O\_B\\), and \\({}^A\bm{\omega}\\) is the angular velocity of moving platform.<br />
<a id="figure--fig:stewart-static-forces"></a>
@@ -1093,7 +1093,7 @@ in which \\(\sigma\_{\text{min}}\\) and \\(\sigma\_{\text{max}}\\) are the small
#### Stiffness Analysis of the Stewart-Gough Platform {#stiffness-analysis-of-the-stewart-gough-platform}
In this section, we restrict our analysis to a 3-6 structure ([9](#figure--fig:stewart36)) in which there exist six distinct attachment points \\(A\_i\\) on the fixed base and three moving attachment point \\(B\_i\\).
In this section, we restrict our analysis to a 3-6 structure ([Figure 9](#figure--fig:stewart36)) in which there exist six distinct attachment points \\(A\_i\\) on the fixed base and three moving attachment point \\(B\_i\\).
<a id="figure--fig:stewart36"></a>
@@ -1222,7 +1222,7 @@ Linear acceleration of a point \\(P\\) can be easily determined by time derivati
Note that this is correct only if the derivative is taken with respect to a **fixed** frame.<br />
Now consider the general motion of a rigid body, in which a moving frame \\(\\{\bm{B}\\}\\) is attached to the rigid body and the problem is to find the absolute acceleration of point \\(P\\) with respect to the fixed frame \\(\\{\bm{A}\\}\\).
The rigid body performs a general motion, which is a combination of a translation, denoted by the velocity vector \\({}^A\bm{v}\_{O\_B}\\), and an instantaneous angular rotation denoted by \\(\bm{\Omega}\\) (see [7](#figure--fig:general-motion)).
The rigid body performs a general motion, which is a combination of a translation, denoted by the velocity vector \\({}^A\bm{v}\_{O\_B}\\), and an instantaneous angular rotation denoted by \\(\bm{\Omega}\\) (see [Figure 7](#figure--fig:general-motion)).
To determine acceleration of point \\(P\\), we start with the relation between absolute and relative velocities of point \\(P\\):
\begin{equation}
@@ -1347,7 +1347,7 @@ On the other hand, if the reference frame \\(\\{B\\}\\) has **pure rotation** wi
##### Linear Momentum {#linear-momentum}
Linear momentum of a material body, shown in [11](#figure--fig:angular-momentum-rigid-body), with respect to a reference frame \\(\\{\bm{A}\\}\\) is defined as
Linear momentum of a material body, shown in [Figure 11](#figure--fig:angular-momentum-rigid-body), with respect to a reference frame \\(\\{\bm{A}\\}\\) is defined as
\begin{equation}
{}^A\bm{G} = \int\_V \frac{d\bm{p}}{dt} \rho dV
@@ -1376,7 +1376,7 @@ This highlights the important of the center of mass in dynamic formulation of ri
##### Angular Momentum {#angular-momentum}
Consider the solid body represented in [11](#figure--fig:angular-momentum-rigid-body).
Consider the solid body represented in [Figure 11](#figure--fig:angular-momentum-rigid-body).
Angular momentum of the differential masses \\(\rho dV\\) about a reference point \\(A\\), expressed in the reference frame \\(\\{\bm{A}\\}\\) is defined as
\\[ {}^A\bm{H} = \int\_V \left(\bm{p} \times \frac{d\bm{p}}{dt} \right) \rho dV \\]
in which \\(d\bm{p}/dt\\) denotes the velocity of differential mass with respect to the reference frame \\(\\{\bm{A}\\}\\).
@@ -1504,7 +1504,7 @@ With \\(\bm{v}\_{b\_{i}}\\) an **intermediate variable** corresponding to the ve
\bm{v}\_{b\_{i}} = \bm{v}\_{p} + \bm{\omega} \times \bm{b}\_{i}
\end{equation}
As illustrated in [12](#figure--fig:free-body-diagram-stewart), the piston-cylinder structure of the limbs is decomposed into two separate parts, the masses of which are denoted by \\(m\_{i\_1}\\) and \\(m\_{i\_2}\\).
As illustrated in [Figure 12](#figure--fig:free-body-diagram-stewart), the piston-cylinder structure of the limbs is decomposed into two separate parts, the masses of which are denoted by \\(m\_{i\_1}\\) and \\(m\_{i\_2}\\).
The position vector of these two center of masses can be determined by the following equations:
\begin{align}
@@ -1539,7 +1539,7 @@ We assume that each limb consists of two parts, the cylinder and the piston, whe
We also assume that the centers of masses of the cylinder and the piston are located at a distance of \\(c\_{i\_1}\\) and \\(c\_{i\_2}\\) above their foot points, and their masses are denoted by \\(m\_{i\_1}\\) and \\(m\_{i\_2}\\).
Moreover, consider that the pistons are symmetric about their axes, and their centers of masses lie at their midlengths.
The free-body diagrams of the limbs and the moving platforms is given in [12](#figure--fig:free-body-diagram-stewart).
The free-body diagrams of the limbs and the moving platforms is given in [Figure 12](#figure--fig:free-body-diagram-stewart).
The reaction forces at fixed points \\(A\_i\\) are denoted by \\(\bm{f}\_{a\_i}\\), the internal force at moving points \\(B\_i\\) are dentoed by \\(\bm{f}\_{b\_i}\\), and the internal forces and moments between cylinders and pistons are denoted by \\(\bm{f}\_{c\_i}\\) and \\(\bm{M\_{c\_i}}\\) respectively.
Assume that the only existing external disturbance wrench is applied on the moving platform and is denoted by \\(\bm{\mathcal{F}}\_d = [\bm{F}\_d, \bm{n}\_d]^T\\).
@@ -1566,7 +1566,7 @@ in which \\(m\_{c\_e}\\) is defined as
##### Dynamic Formulation of the Moving Platform {#dynamic-formulation-of-the-moving-platform}
Assume that the **moving platform center of mass is located at the center point** \\(P\\) and it has a mass \\(m\\) and moment of inertia \\({}^A\bm{I}\_{P}\\).
Furthermore, consider that gravitational force and external disturbance wrench are applied on the moving platform, \\(\bm{\mathcal{F}}\_d = [\bm{F}\_d, \bm{n}\_d]^T\\) as depicted in [12](#figure--fig:free-body-diagram-stewart).
Furthermore, consider that gravitational force and external disturbance wrench are applied on the moving platform, \\(\bm{\mathcal{F}}\_d = [\bm{F}\_d, \bm{n}\_d]^T\\) as depicted in [Figure 12](#figure--fig:free-body-diagram-stewart).
The Newton-Euler formulation of the moving platform is as follows:
@@ -1723,7 +1723,7 @@ in which
##### Forward Dynamics Simulations {#forward-dynamics-simulations}
As shown in [13](#figure--fig:stewart-forward-dynamics), it is **assumed that actuator forces and external disturbance wrench applied to the manipulator are given and the resulting trajectory of the moving platform is to be determined**.
As shown in [Figure 13](#figure--fig:stewart-forward-dynamics), it is **assumed that actuator forces and external disturbance wrench applied to the manipulator are given and the resulting trajectory of the moving platform is to be determined**.
<a id="figure--fig:stewart-forward-dynamics"></a>
@@ -1736,7 +1736,7 @@ The closed-form dynamic formulation of the Stewart-Gough platform corresponds to
In inverse dynamics simulations, it is assumed that the **trajectory of the manipulator is given**, and the **actuator forces required to generate such trajectories are to be determined**.
As illustrated in [14](#figure--fig:stewart-inverse-dynamics), inverse dynamic formulation is implemented in the following sequence.
As illustrated in [Figure 14](#figure--fig:stewart-inverse-dynamics), inverse dynamic formulation is implemented in the following sequence.
The first step is trajectory generation for the manipulator moving platform.
Many different algorithms are developed for a smooth trajectory generation.
For such a trajectory, \\(\bm{\mathcal{X}}\_{d}(t)\\) and the time derivatives \\(\dot{\bm{\mathcal{X}}}\_{d}(t)\\), \\(\ddot{\bm{\mathcal{X}}}\_{d}(t)\\) are known.
@@ -1847,7 +1847,7 @@ In general, the desired motion of the moving platform may be represented by the
To perform such motion in closed loop, it is necessary to **measure the output motion** \\(\bm{\mathcal{X}}\\) of the manipulator by an instrumentation system.
Such instrumentation usually consists of two subsystems: the first subsystem may use accurate accelerometers, or global positioning systems to calculate the position of a point on the moving platform; and a second subsystem may use inertial or laser gyros to determine orientation of the moving platform.<br />
[15](#figure--fig:general-topology-motion-feedback) shows the general topology of a motion controller using direct measurement of the motion variable \\(\bm{\mathcal{X}}\\), as feedback in the closed-loop system.
[Figure 15](#figure--fig:general-topology-motion-feedback) shows the general topology of a motion controller using direct measurement of the motion variable \\(\bm{\mathcal{X}}\\), as feedback in the closed-loop system.
In such a structure, the measured position and orientation of the manipulator is compared to its desired value to generate the **motion error vector** \\(\bm{e}\_\mathcal{X}\\).
The controller uses this error information to generate suitable commands for the actuators to minimize the tracking error.<br />
@@ -1859,7 +1859,7 @@ However, it is usually much **easier to measure the active joint variable** rath
The relation between the **joint variable** \\(\bm{q}\\) and **motion variable** of the moving platform \\(\bm{\mathcal{X}}\\) is dealt with the **forward and inverse kinematics**.
The relation between the **differential motion variables** \\(\dot{\bm{q}}\\) and \\(\dot{\bm{\mathcal{X}}}\\) is studied through the **Jacobian analysis**.<br />
It is then possible to use the forward kinematic analysis to calculate \\(\bm{\mathcal{X}}\\) from the measured joint variables \\(\bm{q}\\), and one may use the control topology depicted in [16](#figure--fig:general-topology-motion-feedback-bis) to implement such a controller.
It is then possible to use the forward kinematic analysis to calculate \\(\bm{\mathcal{X}}\\) from the measured joint variables \\(\bm{q}\\), and one may use the control topology depicted in [Figure 16](#figure--fig:general-topology-motion-feedback-bis) to implement such a controller.
<a id="figure--fig:general-topology-motion-feedback-bis"></a>
@@ -1870,9 +1870,9 @@ As described earlier, this is a **complex task** for parallel manipulators.
It is even more complex when a solution has to be found in real time.<br />
However, as shown herein before, the inverse kinematic analysis of parallel manipulators is much easier to carry out.
To overcome the implementation problem of the control topology in [16](#figure--fig:general-topology-motion-feedback-bis), another control topology is usually implemented for parallel manipulators.
To overcome the implementation problem of the control topology in [Figure 16](#figure--fig:general-topology-motion-feedback-bis), another control topology is usually implemented for parallel manipulators.
In this topology, depicted in [17](#figure--fig:general-topology-motion-feedback-ter), the desired motion trajectory of the robot \\(\bm{\mathcal{X}}\_d\\) is used in an **inverse kinematic analysis** to find the corresponding desired values for joint variable \\(\bm{q}\_d\\).
In this topology, depicted in [Figure 17](#figure--fig:general-topology-motion-feedback-ter), the desired motion trajectory of the robot \\(\bm{\mathcal{X}}\_d\\) is used in an **inverse kinematic analysis** to find the corresponding desired values for joint variable \\(\bm{q}\_d\\).
Hence, the controller is designed based on the **joint space error** \\(\bm{e}\_q\\).
<a id="figure--fig:general-topology-motion-feedback-ter"></a>
@@ -1881,12 +1881,12 @@ Hence, the controller is designed based on the **joint space error** \\(\bm{e}\_
Therefore, the **structure and characteristics** of the controller in this topology is totally **different** from that given in the first two topologies.
The **input and output** of the controller depicted in [17](#figure--fig:general-topology-motion-feedback-ter) are **both in the joint space**.
The **input and output** of the controller depicted in [Figure 17](#figure--fig:general-topology-motion-feedback-ter) are **both in the joint space**.
However, this is not the case in the previous topologies where the input to the controller is the motion error in task space, while its output is in the joint space.
For the topology in [17](#figure--fig:general-topology-motion-feedback-ter), **independent controllers** for each joint may be suitable.<br />
For the topology in [Figure 17](#figure--fig:general-topology-motion-feedback-ter), **independent controllers** for each joint may be suitable.<br />
To generate a **direct input to output relation in the task space**, consider the topology depicted in [18](#figure--fig:general-topology-motion-feedback-quater).
To generate a **direct input to output relation in the task space**, consider the topology depicted in [Figure 18](#figure--fig:general-topology-motion-feedback-quater).
A force distribution block is added which maps the generated wrench in the task space \\(\bm{\mathcal{F}}\\), to its corresponding actuator forces/torque \\(\bm{\tau}\\).
<a id="figure--fig:general-topology-motion-feedback-quater"></a>
@@ -1904,7 +1904,7 @@ For a fully parallel manipulator such as the Stewart-Gough platform, this mappin
#### Decentralized PD Control {#decentralized-pd-control}
In the control structure in [19](#figure--fig:decentralized-pd-control-task-space), a number of linear PD controllers are used in a feedback structure on each error component.
In the control structure in [Figure 19](#figure--fig:decentralized-pd-control-task-space), a number of linear PD controllers are used in a feedback structure on each error component.
The decentralized controller consists of **six disjoint linear controllers** acting on each error component \\(\bm{e}\_x = [e\_x,\ e\_y,\ e\_z,\ e\_{\theta\_x},\ e\_{\theta\_y},\ e\_{\theta\_z}]\\).
The PD controller is denoted by \\(\bm{K}\_d s + \bm{K}\_p\\), in which \\(\bm{K}\_d\\) and \\(\bm{K}\_p\\) are \\(6 \times 6\\) **diagonal matrices** denoting the derivative and proportional controller gains for each error term.
@@ -1927,7 +1927,7 @@ The controller gains are generally tuned experimentally based on physical realiz
#### Feed Forward Control {#feed-forward-control}
A feedforward wrench denoted by \\(\bm{\mathcal{F}}\_{ff}\\) may be added to the decentralized PD controller structure as depicted in [20](#figure--fig:feedforward-control-task-space).
A feedforward wrench denoted by \\(\bm{\mathcal{F}}\_{ff}\\) may be added to the decentralized PD controller structure as depicted in [Figure 20](#figure--fig:feedforward-control-task-space).
This term is generated from the dynamic model of the manipulator in the task space, represented in a closed form by the following equation:
\\[ \bm{\mathcal{F}}\_{ff} = \bm{\hat{M}}(\bm{\mathcal{X}}\_d)\ddot{\bm{\mathcal{X}}}\_d + \bm{\hat{C}}(\bm{\mathcal{X}}\_d, \dot{\bm{\mathcal{X}}}\_d)\dot{\bm{\mathcal{X}}}\_d + \bm{\hat{G}}(\bm{\mathcal{X}}\_d) \\]
@@ -1986,7 +1986,7 @@ By this means, **nonlinear and coupling behavior of the robotic manipulator is s
</div>
General structure of IDC applied to a parallel manipulator is depicted in [21](#figure--fig:inverse-dynamics-control-task-space).
General structure of IDC applied to a parallel manipulator is depicted in [Figure 21](#figure--fig:inverse-dynamics-control-task-space).
A corrective wrench \\(\bm{\mathcal{F}}\_{fl}\\) is added in a **feedback structure** to the closed-loop system, which is calculated from the Coriolis and centrifugal matrix and gravity vector of the manipulator dynamic formulation.
Furthermore, mass matrix is added in the forward path in addition to the desired trajectory acceleration \\(\ddot{\bm{\mathcal{X}}}\_d\\).
@@ -2133,7 +2133,7 @@ If this measurement is available without any doubt, such topologies are among th
However, as explained in Section , in many practical situations measurement of the motion variable \\(\bm{\mathcal{X}}\\) is difficult or expensive, and usually just the active joint variables \\(\bm{q}\\) are measured.
In such cases, the controllers developed in the joint space may be recommended for practical implementation.<br />
To generate a direct input to output relation in the joint space, consider the topology depicted in [16](#figure--fig:general-topology-motion-feedback-bis).
To generate a direct input to output relation in the joint space, consider the topology depicted in [Figure 16](#figure--fig:general-topology-motion-feedback-bis).
In this topology, the controller input is the joint variable error vector \\(\bm{e}\_q = \bm{q}\_d - \bm{q}\\), and the controller output is directly the actuator force vector \\(\bm{\tau}\\), and hence there exists a **one-to-one correspondence between the controller input to its output**.<br />
The general form of dynamic formulation of parallel robot is usually given in the task space.
@@ -2191,7 +2191,7 @@ Furthermore, the main dynamic matrices are all functions of the motion variable
Hence, in practice, to find the dynamic matrices represented in the joint space, **forward kinematics** should be solved to find the motion variable \\(\bm{\mathcal{X}}\\) for any given joint motion vector \\(\bm{q}\\).<br />
Since in parallel robots the forward kinematic analysis is computationally intensive, there exist inherent difficulties in finding the dynamic matrices in the joint space as an explicit function of \\(\bm{q}\\).
In this case it is possible to solve forward kinematics in an online manner, it is recommended to use the control topology depicted in [16](#figure--fig:general-topology-motion-feedback-bis), and implement control law design in the task space.<br />
In this case it is possible to solve forward kinematics in an online manner, it is recommended to use the control topology depicted in [Figure 16](#figure--fig:general-topology-motion-feedback-bis), and implement control law design in the task space.<br />
However, one implementable alternative to calculate the dynamic matrices represented in the joint space is to use the **desired motion trajectory** \\(\bm{\mathcal{X}}\_d\\) instead of the true value of motion vector \\(\bm{\mathcal{X}}\\) in the calculations.
This approximation significantly reduces the computational cost, with the penalty of having mismatch between the estimated values of these matrices to their true values.
@@ -2200,7 +2200,7 @@ This approximation significantly reduces the computational cost, with the penalt
#### Decentralized PD Control {#decentralized-pd-control}
The first control strategy introduced in the joint space consists of the simplest form of feedback control in such manipulators.
In this control structure, depicted in [24](#figure--fig:decentralized-pd-control-joint-space), a number of PD controllers are used in a feedback structure on each error component.
In this control structure, depicted in [Figure 24](#figure--fig:decentralized-pd-control-joint-space), a number of PD controllers are used in a feedback structure on each error component.
The PD controller is denoted by \\(\bm{K}\_d s + \bm{K}\_p\\), where \\(\bm{K}\_d\\) and \\(\bm{K}\_p\\) are \\(n \times n\\) **diagonal** matrices denoting the derivative and proportional controller gains, respectively.<br />
@@ -2224,7 +2224,7 @@ To remedy these shortcomings, some modifications have been proposed to this stru
#### Feedforward Control {#feedforward-control}
The tracking performance of the simple PD controller implemented in the joint space is usually not sufficient at different configurations.
To improve the tracking performance, a feedforward actuator force denoted by \\(\bm{\tau}\_{ff}\\) may be added to the structure of the controller as depicted in [25](#figure--fig:feedforward-pd-control-joint-space).
To improve the tracking performance, a feedforward actuator force denoted by \\(\bm{\tau}\_{ff}\\) may be added to the structure of the controller as depicted in [Figure 25](#figure--fig:feedforward-pd-control-joint-space).
<a id="figure--fig:feedforward-pd-control-joint-space"></a>
@@ -2266,7 +2266,7 @@ By this means, the **nonlinear and coupling characteristics** of robotic manipul
</div>
The general structure of inverse dynamics control applied to a parallel manipulator in the joint space is depicted in [26](#figure--fig:inverse-dynamics-control-joint-space).
The general structure of inverse dynamics control applied to a parallel manipulator in the joint space is depicted in [Figure 26](#figure--fig:inverse-dynamics-control-joint-space).
A corrective torque \\(\bm{\tau}\_{fl}\\) is added in a **feedback** structure to the closed-loop system, which is calculated from the Coriolis and Centrifugal matrix, and the gravity vector of the manipulator dynamic formulation in the joint space.
Furthermore, the mass matrix is acting in the **forward path**, in addition to the desired trajectory acceleration \\(\ddot{\bm{q}}\_q\\).
@@ -2596,7 +2596,7 @@ However, note that the motion control of the robot when the robot is in interact
To follow **two objectives** with different properties in one control system, usually a **hierarchy** of two feedback loops is used in practice.
This kind of control topology is called **cascade control**, which is used when there are **several measurements and one prime control variable**.
Cascade control is implemented by **nesting** the control loops, as shown in [27](#figure--fig:cascade-control).
Cascade control is implemented by **nesting** the control loops, as shown in [Figure 27](#figure--fig:cascade-control).
The output control loop is called the **primary loop**, while the inner loop is called the secondary loop and is used to fulfill a secondary objective in the closed-loop system.
</div>
@@ -2628,9 +2628,9 @@ Consider the force control schemes, in which **force tracking is the prime objec
In such a case, it is advised that the outer loop of cascade control structure is constructed by wrench feedback, while the inner loop is based on position feedback.
Since different types of measurement units may be used in parallel robots, different control topologies may be constructed to implement such a cascade structure.<br />
Consider first the cascade control topology shown in [28](#figure--fig:taghira13-cascade-force-outer-loop) in which the measured variables are both in the **task space**.
Consider first the cascade control topology shown in [Figure 28](#figure--fig:taghira13-cascade-force-outer-loop) in which the measured variables are both in the **task space**.
The inner loop is constructed by position feedback while the outer loop is based on force feedback.
As seen in [28](#figure--fig:taghira13-cascade-force-outer-loop), the force controller block is fed to the motion controller, and this might be seen as the **generated desired motion trajectory for the inner loop**.
As seen in [Figure 28](#figure--fig:taghira13-cascade-force-outer-loop), the force controller block is fed to the motion controller, and this might be seen as the **generated desired motion trajectory for the inner loop**.
The output of motion controller is also designed in the task space, and to convert it to implementable actuator force \\(\bm{\tau}\\), the force distribution block is considered in this topology.<br />
@@ -2639,7 +2639,7 @@ The output of motion controller is also designed in the task space, and to conve
{{< figure src="/ox-hugo/taghira13_cascade_force_outer_loop.png" caption="<span class=\"figure-number\">Figure 28: </span>Cascade topology of force feedback control: position in inner loop and force in outer loop. Moving platform wrench \\(\bm{\mathcal{F}}\\) and motion variable \\(\bm{\mathcal{X}}\\) are measured in the task space" >}}
Other alternatives for force control topology may be suggested based on the variations of position and force measurements.
If the force is measured in the joint space, the topology suggested in [29](#figure--fig:taghira13-cascade-force-outer-loop-tau) can be used.
If the force is measured in the joint space, the topology suggested in [Figure 29](#figure--fig:taghira13-cascade-force-outer-loop-tau) can be used.
In this topology, the measured actuator force vector \\(\bm{\tau}\\) is mapped into its corresponding wrench in the task space by the Jacobian transpose mapping \\(\bm{\mathcal{F}} = \bm{J}^T \bm{\tau}\\).<br />
<a id="figure--fig:taghira13-cascade-force-outer-loop-tau"></a>
@@ -2647,7 +2647,7 @@ In this topology, the measured actuator force vector \\(\bm{\tau}\\) is mapped i
{{< figure src="/ox-hugo/taghira13_cascade_force_outer_loop_tau.png" caption="<span class=\"figure-number\">Figure 29: </span>Cascade topology of force feedback control: position in inner loop and force in outer loop. Actuator forces \\(\bm{\tau}\\) and motion variable \\(\bm{\mathcal{X}}\\) are measured" >}}
Consider the case where the force and motion variables are both measured in the **joint space**.
[30](#figure--fig:taghira13-cascade-force-outer-loop-tau-q) suggests the force control topology in the joint space, in which the inner loop is based on measured motion variable in the joint space, and the outer loop uses the measured actuator force vector.
[Figure 30](#figure--fig:taghira13-cascade-force-outer-loop-tau-q) suggests the force control topology in the joint space, in which the inner loop is based on measured motion variable in the joint space, and the outer loop uses the measured actuator force vector.
In this topology, it is advised that the force controller is designed in the **task** space, and the Jacobian transpose mapping is used to project the measured actuator force vector into its corresponding wrench in the task space.
However, as the inner loop is constructed in the joint space, the desired motion variable \\(\bm{\mathcal{X}}\_d\\) is mapped into joint space using **inverse kinematic** solution.
@@ -2665,7 +2665,7 @@ In such a case, force tracking is not the primary objective, and it is advised t
Since different type of measurement units may be used in parallel robots, different control topologies may be constructed to implement such cascade controllers.<br />
[31](#figure--fig:taghira13-cascade-force-inner-loop-F) illustrates the cascade control topology for the system in which the measured variables are both in the task space (\\(\bm{\mathcal{F}}\\) and \\(\bm{\mathcal{X}}\\)).
[Figure 31](#figure--fig:taghira13-cascade-force-inner-loop-F) illustrates the cascade control topology for the system in which the measured variables are both in the task space (\\(\bm{\mathcal{F}}\\) and \\(\bm{\mathcal{X}}\\)).
The inner loop is loop is constructed by force feedback while the outer loop is based on position feedback.
By this means, when the manipulator is not in contact with a stiff environment, position tracking is guaranteed through the primary controller.
However, when there is interacting wrench \\(\bm{\mathcal{F}}\_e\\) applied to the moving platform, this structure controls the force-motion relation.
@@ -2676,14 +2676,14 @@ This configuration may be seen as if the **outer loop generates a desired force
{{< figure src="/ox-hugo/taghira13_cascade_force_inner_loop_F.png" caption="<span class=\"figure-number\">Figure 31: </span>Cascade topology of force feedback control: force in inner loop and position in outer loop. Moving platform wrench \\(\bm{\mathcal{F}}\\) and motion variable \\(\bm{\mathcal{X}}\\) are measured in the task space" >}}
Other alternatives for control topology may be suggested based on the variations of position and force measurements.
If the force is measured in the joint space, control topology shown in [32](#figure--fig:taghira13-cascade-force-inner-loop-tau) can be used.
If the force is measured in the joint space, control topology shown in [Figure 32](#figure--fig:taghira13-cascade-force-inner-loop-tau) can be used.
In such case, the Jacobian transpose is used to map the actuator force to its corresponding wrench in the task space.<br />
<a id="figure--fig:taghira13-cascade-force-inner-loop-tau"></a>
{{< figure src="/ox-hugo/taghira13_cascade_force_inner_loop_tau.png" caption="<span class=\"figure-number\">Figure 32: </span>Cascade topology of force feedback control: force in inner loop and position in outer loop. Actuator forces \\(\bm{\tau}\\) and motion variable \\(\bm{\mathcal{X}}\\) are measured" >}}
If the force and motion variables are both measured in the **joint** space, the control topology shown in [33](#figure--fig:taghira13-cascade-force-inner-loop-tau-q) is suggested.
If the force and motion variables are both measured in the **joint** space, the control topology shown in [Figure 33](#figure--fig:taghira13-cascade-force-inner-loop-tau-q) is suggested.
The inner loop is based on the measured actuator force vector in the joint space \\(\bm{\tau}\\), and the outer loop is based on the measured actuated joint position vector \\(\bm{q}\\).
In this topology, the desired motion in the task space is mapped into the joint space using **inverse kinematic** solution, and **both the position and force feedback controllers are designed in the joint space**.
Thus, independent controllers for each joint may be suitable for this topology.
@@ -2766,7 +2766,7 @@ Nevertheless, note that Laplace transform is only applicable for **linear time i
<div class="exampl">
Consider an RLC circuit depicted in [35](#figure--fig:taghirad13-impedance-control-rlc).
Consider an RLC circuit depicted in [Figure 35](#figure--fig:taghirad13-impedance-control-rlc).
The differential equation relating voltage \\(v\\) to the current \\(i\\) is given by
\\[ v = L\frac{di}{dt} + Ri + \int\_0^t \frac{1}{C} i(\tau)d\tau \\]
in which \\(L\\) denote the inductance, \\(R\\) the resistance and \\(C\\) the capacitance.
@@ -2781,7 +2781,7 @@ The impedance of the system may be found from the Laplace transform of the above
<div class="exampl">
Consider the mass-spring-damper system depicted in [35](#figure--fig:taghirad13-impedance-control-rlc).
Consider the mass-spring-damper system depicted in [Figure 35](#figure--fig:taghirad13-impedance-control-rlc).
The governing dynamic formulation for this system is given by
\\[ m \ddot{x} + c \dot{x} + k x = f \\]
in which \\(m\\) denote the body mass, \\(c\\) the damper viscous coefficient and \\(k\\) the spring stiffness.
@@ -2811,7 +2811,7 @@ An impedance \\(\bm{Z}(s)\\) is called
</div>
Hence, for the mechanical system represented in [35](#figure--fig:taghirad13-impedance-control-rlc):
Hence, for the mechanical system represented in [Figure 35](#figure--fig:taghirad13-impedance-control-rlc):
- mass represents inductive impedance
- viscous friction represents resistive impedance
@@ -2844,9 +2844,9 @@ In the impedance control scheme, **regulation of the motion-force dynamic relati
Therefore, when the manipulator is not in contact with a stiff environment, position tracking is guaranteed by a primary controller.
However, when there is an interacting wrench \\(\bm{\mathcal{F}}\_e\\) applied to the moving platform, this structure may be designed to control the force-motion dynamic relation.<br />
As a possible impedance control scheme, consider the closed-loop system depicted in [36](#figure--fig:taghira13-impedance-control), in which the position feedback is considered in the outer loop, while force feedback is used in the inner loop.
As a possible impedance control scheme, consider the closed-loop system depicted in [Figure 36](#figure--fig:taghira13-impedance-control), in which the position feedback is considered in the outer loop, while force feedback is used in the inner loop.
This structure is advised when a desired impedance relation between the force and motion variables is required that consists of desired inductive, resistive, and capacitive impedances.
As shown in [36](#figure--fig:taghira13-impedance-control), the motion-tracking error is directly determined from motion measurement by \\(\bm{e}\_x = \bm{\mathcal{X}}\_d - \bm{\mathcal{X}}\\) in the outer loop and the motion controller is designed to satisfy the required impedance.
As shown in [Figure 36](#figure--fig:taghira13-impedance-control), the motion-tracking error is directly determined from motion measurement by \\(\bm{e}\_x = \bm{\mathcal{X}}\_d - \bm{\mathcal{X}}\\) in the outer loop and the motion controller is designed to satisfy the required impedance.
Moreover, direct force-tracking objective is not assigned in this control scheme, and therefore the desired force trajectory \\(\bm{\mathcal{F}}\_d\\) is absent in this scheme.
However, an auxiliary force trajectory \\(\bm{\mathcal{F}}\_a\\) is generated from the motion control law and is used as the reference for the force tracking.
@@ -2859,9 +2859,9 @@ By this means, no prescribed force trajectory is tracked, while the **motion con
The required wrench \\(\bm{\mathcal{F}}\\) in the impedance control scheme, is based on inverse dynamics control and consists of three main parts.
In the inner loop, the force control scheme is based on a feedback linearization part in addition to a mass matrix adjustment, while in the outer loop usually a linear motion controller is considered based on the desired impedance requirements.
Although many different impedance structures may be considered as the basis of the control law, in [36](#figure--fig:taghira13-impedance-control), a linear impedance relation between the force and motion variables is generated that consists of desired inductive \\(\bm{M}\_d\\), resistive \\(\bm{C}\_d\\) and capacitive impedances \\(\bm{K}\_d\\).<br />
Although many different impedance structures may be considered as the basis of the control law, in [Figure 36](#figure--fig:taghira13-impedance-control), a linear impedance relation between the force and motion variables is generated that consists of desired inductive \\(\bm{M}\_d\\), resistive \\(\bm{C}\_d\\) and capacitive impedances \\(\bm{K}\_d\\).<br />
According to [36](#figure--fig:taghira13-impedance-control), the controller output wrench \\(\bm{\mathcal{F}}\\), applied to the manipulator may be formulated as
According to [Figure 36](#figure--fig:taghira13-impedance-control), the controller output wrench \\(\bm{\mathcal{F}}\\), applied to the manipulator may be formulated as
\\[ \bm{\mathcal{F}} = \hat{\bm{M}} \bm{M}\_d^{-1} \bm{e}\_F + \bm{\mathcal{F}}\_{fl} \\]
with: