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@@ -68,9 +68,9 @@ and the resonance \\(P\_{ri}(s)\\) can be represented as one of the following fo
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#### Secondary Actuators {#secondary-actuators}
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We here consider two types of secondary actuators: the PZT milliactuator (figure [1](#org53c168b)) and the microactuator.
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We here consider two types of secondary actuators: the PZT milliactuator (figure [1](#orgb678385)) and the microactuator.
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{{< figure src="/ox-hugo/du19_pzt_actuator.png" caption="Figure 1: A PZT-actuator suspension" >}}
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@@ -92,9 +92,9 @@ There characteristics are shown on table [1](#table--tab:microactuator).
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### Single-Stage Actuation Systems {#single-stage-actuation-systems}
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A typical closed-loop control system is shown on figure [2](#org5941a76), where \\(P\_v(s)\\) and \\(C(z)\\) represent the actuator system and its controller.
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A typical closed-loop control system is shown on figure [2](#orgcf5d697), where \\(P\_v(s)\\) and \\(C(z)\\) represent the actuator system and its controller.
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<a id="org5941a76"></a>
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{{< figure src="/ox-hugo/du19_single_stage_control.png" caption="Figure 2: Block diagram of a single-stage actuation system" >}}
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@@ -104,7 +104,7 @@ A typical closed-loop control system is shown on figure [2](#org5941a76), where
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Dual-stage actuation mechanism for the hard disk drives consists of a VCM actuator and a secondary actuator placed between the VCM and the sensor head.
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The VCM is used as the primary stage to provide long track seeking but with poor accuracy and slow response time, while the secondary stage actuator is used to provide higher positioning accuracy and faster response but with a stroke limit.
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{{< figure src="/ox-hugo/du19_dual_stage_control.png" caption="Figure 3: Block diagram of dual-stage actuation system" >}}
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@@ -130,7 +130,7 @@ In view of this, the controller design for dual-stage actuation systems adopts a
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### Control Schemes {#control-schemes}
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A popular control scheme for dual-stage actuation system is the **decoupled structure** as shown in figure [4](#org03d53e8).
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A popular control scheme for dual-stage actuation system is the **decoupled structure** as shown in figure [4](#org7def875).
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- \\(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)\\).
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- \\(\hat{P}\_p(z)\\) is an approximation of \\(P\_p\\) to estimate \\(y\_p\\).
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@@ -138,7 +138,7 @@ A popular control scheme for dual-stage actuation system is the **decoupled stru
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- \\(n\\) is the measurement noise
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- \\(d\_u\\) stands for external vibration
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{{< figure src="/ox-hugo/du19_decoupled_control.png" caption="Figure 4: Decoupled control structure for the dual-stage actuation system" >}}
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@@ -160,14 +160,14 @@ The sensitivity functions of the VCM loop and the secondary actuator loop are
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And we obtain that the dual-stage sensitivity function \\(S(z)\\) is the product of \\(S\_v(z)\\) and \\(S\_p(z)\\).
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Thus, the dual-stage system control design can be decoupled into two independent controller designs.
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Another type of control scheme is the **parallel structure** as shown in figure [5](#org37116a9).
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Another type of control scheme is the **parallel structure** as shown in figure [5](#orgbb3c494).
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The open-loop transfer function from \\(pes\\) to \\(y\\) is
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\\[ G(z) = P\_p(z) C\_p(z) + P\_v(z) C\_v(z) \\]
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The overall sensitivity function of the closed-loop system from \\(r\\) to \\(pes\\) is
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\\[ S(z) = \frac{1}{1 + G(z)} = \frac{1}{1 + P\_p(z) C\_p(z) + P\_v(z) C\_v(z)} \\]
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<a id="org37116a9"></a>
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{{< figure src="/ox-hugo/du19_parallel_control_structure.png" caption="Figure 5: Parallel control structure for the dual-stage actuator system" >}}
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@@ -177,7 +177,7 @@ Because of the limited displacement range of the secondary actuator, the control
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### Controller Design Method in the Continuous-Time Domain {#controller-design-method-in-the-continuous-time-domain}
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\\(\mathcal{H}\_\infty\\) loop shaping method is used to design the controllers for the primary and secondary actuators.
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The structure of the \\(\mathcal{H}\_\infty\\) loop shaping method is plotted in figure [6](#org299b914) where \\(W(s)\\) is a weighting function relevant to the designed control system performance such as the sensitivity function.
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The structure of the \\(\mathcal{H}\_\infty\\) loop shaping method is plotted in figure [6](#orge3f8703) where \\(W(s)\\) is a weighting function relevant to the designed control system performance such as the sensitivity function.
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For a plant model \\(P(s)\\), a controller \\(C(s)\\) is to be designed such that the closed-loop system is stable and
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@@ -187,11 +187,11 @@ For a plant model \\(P(s)\\), a controller \\(C(s)\\) is to be designed such tha
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is satisfied, where \\(T\_{zw}\\) is the transfer function from \\(w\\) to \\(z\\): \\(T\_{zw} = S(s) W(s)\\).
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{{< figure src="/ox-hugo/du19_h_inf_diagram.png" caption="Figure 6: Block diagram for \\(\mathcal{H}\_\infty\\) loop shaping method to design the controller \\(C(s)\\) with the weighting function \\(W(s)\\)" >}}
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Equation [1](#org361ec91) means that \\(S(s)\\) can be shaped similarly to the inverse of the chosen weighting function \\(W(s)\\).
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Equation [1](#orgc402b0c) means that \\(S(s)\\) can be shaped similarly to the inverse of the chosen weighting function \\(W(s)\\).
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One form of \\(W(s)\\) is taken as
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\begin{equation}
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@@ -204,16 +204,16 @@ The controller can then be synthesis using the linear matrix inequality (LMI) ap
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The primary and secondary actuator control loops are designed separately for the dual-stage control systems.
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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.
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As seen in figure [7](#org60ad057), 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.
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As seen in figure [7](#org402df06), 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.
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{{< figure src="/ox-hugo/du19_dual_stage_loop_gain.png" caption="Figure 7: 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)" >}}
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The sensitivity functions are shown in figure [8](#org1d6afb9), 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.
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The sensitivity functions are shown in figure [8](#orge904ce1), 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.
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This needs to decrease the bandwidth of the primary actuator loop and increase the bandwidth of the secondary actuator loop.
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<a id="org1d6afb9"></a>
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<a id="orge904ce1"></a>
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{{< figure src="/ox-hugo/du19_dual_stage_sensitivity.png" caption="Figure 8: Frequency response of \\(S\_v(s)\\) and \\(S\_p(s)\\)" >}}
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@@ -246,13 +246,13 @@ A VCM actuator is used as the first-stage actuator denoted by \\(P\_v(s)\\), a P
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### Control Strategy and Controller Design {#control-strategy-and-controller-design}
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Figure [9](#org0e50764) shows the control structure for the three-stage actuation system.
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Figure [9](#org8c31dd5) shows the control structure for the three-stage actuation system.
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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.
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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.
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The reason why the decoupled control structure is adopted here is that its overall sensitivity function is the product of those of the two individual loops, and the VCM and the PTZ controllers can be designed separately.
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<a id="org0e50764"></a>
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{{< figure src="/ox-hugo/du19_three_stage_control.png" caption="Figure 9: Control system for the three-stage actuation system" >}}
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@@ -281,15 +281,15 @@ The PZT actuated milliactuator \\(P\_p(s)\\) works under a reasonably high bandw
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The third-stage actuator \\(P\_m(s)\\) is used to further push the bandwidth as high as possible.
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The control performances of both the VCM and the PZT actuators are limited by their dominant resonance modes.
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The open-loop frequency responses of the three stages are shown on figure [10](#orgefe88f9).
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The open-loop frequency responses of the three stages are shown on figure [10](#orgd95bc97).
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{{< figure src="/ox-hugo/du19_open_loop_three_stage.png" caption="Figure 10: Frequency response of the open-loop transfer function" >}}
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The obtained sensitivity function is shown on figure [11](#orgd0c25f8).
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The obtained sensitivity function is shown on figure [11](#org50990f8).
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{{< figure src="/ox-hugo/du19_sensitivity_three_stage.png" caption="Figure 11: Sensitivity function of the VCM single stage, the dual-stage and the three-stage loops" >}}
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@@ -304,7 +304,7 @@ Otherwise, saturation will occur in the control loop and the control system perf
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Therefore, the stroke specification of the actuators, especially milliactuator and microactuators, is very important for achievable control performance.
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Higher stroke actuators have stronger abilities to make sure that the control performances are not degraded in the presence of external vibrations.
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For the three-stage control architecture as shown on figure [9](#org0e50764), the position error is
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For the three-stage control architecture as shown on figure [9](#org8c31dd5), the position error is
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\\[ e = -S(P\_v d\_1 + d\_2 + d\_e) + S n \\]
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The control signals and positions of the actuators are given by
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@@ -320,11 +320,11 @@ Higher bandwidth/higher level of disturbance generally means high stroke needed.
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### Different Configurations of the Control System {#different-configurations-of-the-control-system}
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A decoupled control structure can be used for the three-stage actuation system (see figure [12](#org5bb499d)).
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A decoupled control structure can be used for the three-stage actuation system (see figure [12](#org7ec3564)).
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The overall sensitivity function is
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\\[ S(z) = \approx S\_v(z) S\_p(z) S\_m(z) \\]
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with \\(S\_v(z)\\) and \\(S\_p(z)\\) are defined in equation [1](#orga34ddfe) and
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with \\(S\_v(z)\\) and \\(S\_p(z)\\) are defined in equation [1](#org6bf8240) and
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\\[ S\_m(z) = \frac{1}{1 + P\_m(z) C\_m(z)} \\]
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Denote the dual-stage open-loop transfer function as \\(G\_d\\)
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@@ -333,7 +333,7 @@ Denote the dual-stage open-loop transfer function as \\(G\_d\\)
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The open-loop transfer function of the overall system is
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\\[ G(z) = G\_d(z) + G\_m(z) + G\_d(z) G\_m(z) \\]
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{{< figure src="/ox-hugo/du19_three_stage_decoupled.png" caption="Figure 12: Decoupled control structure for the three-stage actuation system" >}}
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@@ -345,9 +345,9 @@ The control signals and the positions of the three actuators are
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u\_v &= C\_v(1 + \hat{P}\_p C\_p) (1 + \hat{P}\_m C\_m) e, \ y\_v = P\_v u\_v
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\end{align\*}
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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](#org0a46272)).
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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](#org56aeb13)).
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{{< figure src="/ox-hugo/du19_three_stage_decoupled_loop_gain.png" caption="Figure 13: Frequency responses of the open-loop transfer functions for the three-stages parallel and decoupled structure" >}}
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@@ -141,7 +141,7 @@ The main measurement technique studied are those which will permit to make **dir
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The type of test best suited to FRF measurement is shown in figure [fig:modal_analysis_schematic](#fig:modal_analysis_schematic).
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{{< figure src="/ox-hugo/ewins00_modal_analysis_schematic.png" caption="Figure 1: Basic components of FRF measurement system" >}}
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@@ -199,7 +199,7 @@ This process itself falls into two stages:
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Most of the effort goes into this second stage, which is widely referred to as "modal parameter extraction", or simply as "modal analysis".
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We have seen that we can predict the form of the FRF plots for a multi degree-of-freedom system, and that these are directly related to the modal properties of that system.
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The great majority of the modal analysis effort involves **curve-fitting** an expression such as equation [eq:frf_modal](#eq:frf_modal) to the measured FRF and thereby finding the appropriate modal parameters.
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The great majority of the modal analysis effort involves **curve-fitting** an expression such as equation \eqref{eq:frf_modal} to the measured FRF and thereby finding the appropriate modal parameters.
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A completely general curve-fitting approach is possible but generally inefficient.
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Mathematically, we can take an equation of the form
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@@ -215,7 +215,7 @@ This assumption allows us to use the circular nature of a modulus/phase polar pl
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This process can be **repeated** for each resonance individually until the whole curve has been analyzed.
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At this stage, a theoretical regeneration of the FRF is possible using the set of coefficients extracted.
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{{< figure src="/ox-hugo/ewins00_sdof_modulus_phase.png" caption="Figure 2: Curve fit to resonant FRF data" >}}
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@@ -253,7 +253,7 @@ Theoretical foundations of modal testing are of paramount importance to its succ
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The three phases through a typical theoretical vibration analysis progresses are shown on figure [fig:vibration_analysis_procedure](#fig:vibration_analysis_procedure).
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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**.
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{{< figure src="/ox-hugo/ewins00_vibration_analysis_procedure.png" caption="Figure 3: Theoretical route to vibration analysis" >}}
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@@ -298,7 +298,7 @@ Three classes of system model will be described:
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The basic model for the SDOF system is shown in figure [fig:sdof_model](#fig:sdof_model) where \\(f(t)\\) and \\(x(t)\\) are general time-varying force and displacement response quantities.
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The spatial model consists of a **mass** \\(m\\), a **spring** \\(k\\) and (when damped) either a **viscous dashpot** \\(c\\) or **hysteretic damper** \\(d\\).
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{{< figure src="/ox-hugo/ewins00_sdof_model.png" caption="Figure 4: Single degree-of-freedom system" >}}
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@@ -374,7 +374,7 @@ which is a single mode of vibration with a complex natural frequency having two
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The physical significance of these two parts is illustrated in the typical free response plot shown in figure [fig:sdof_response](#fig:sdof_response)
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{{< figure src="/ox-hugo/ewins00_sdof_response.png" caption="Figure 5: Oscillatory and decay part" >}}
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@@ -418,7 +418,7 @@ The damping effect of such a component can conveniently be defined by the ratio
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|  |  |  |
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|-----------------------------------------------|----------------------------------------|------------------------------------------|
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| <a id="orgd01ea8f"></a> Material hysteresis | <a id="org5fb7a29"></a> Dry friction | <a id="org41cf290"></a> Viscous damper |
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| <a id="org54caaf8"></a> Material hysteresis | <a id="org0fc2b44"></a> Dry friction | <a id="org0985c72"></a> Viscous damper |
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| height=2cm | height=2cm | height=2cm |
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Another common source of energy dissipation in practical structures, is the **friction** which exist in joints between components of the structure.
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@@ -458,11 +458,11 @@ where \\(\eta\\) is the **structural damping loss factor** and replaces the crit
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#### Alternative Forms of FRF {#alternative-forms-of-frf}
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So far we have defined our receptance frequency response function \\(\alpha(\omega)\\) as the ratio between a harmonic displacement response and the harmonic force [eq:receptance](#eq:receptance).
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So far we have defined our receptance frequency response function \\(\alpha(\omega)\\) as the ratio between a harmonic displacement response and the harmonic force \eqref{eq:receptance}.
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This ratio is complex: we can look at its **amplitude** ratio \\(|\alpha(\omega)|\\) and its **phase** angle \\(\theta\_\alpha(\omega)\\).
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We could have selected the response velocity \\(v(t)\\) as the output quantity and defined an alternative frequency response function [eq:mobility](#eq:mobility).
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Similarly we could use the acceleration parameter so we could define a third FRF parameter [eq:inertance](#eq:inertance).
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We could have selected the response velocity \\(v(t)\\) as the output quantity and defined an alternative frequency response function \eqref{eq:mobility}.
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Similarly we could use the acceleration parameter so we could define a third FRF parameter \eqref{eq:inertance}.
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<div class="cbox">
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<div></div>
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@@ -537,7 +537,7 @@ Bode plot are usually displayed using logarithmic scales as shown on figure [fig
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|  |  |  |
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|-------------------------------------------|-----------------------------------------|--------------------------------------------|
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| <a id="org1478c78"></a> Receptance FRF | <a id="orgd7eb060"></a> Mobility FRF | <a id="orgf95b5b3"></a> Accelerance FRF |
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| <a id="orgea747d3"></a> Receptance FRF | <a id="orgc5e3717"></a> Mobility FRF | <a id="orgcf610b2"></a> Accelerance FRF |
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| width=\linewidth | width=\linewidth | width=\linewidth |
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Each plot can be divided into three regimes:
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@@ -560,13 +560,13 @@ This type of display is not widely used as we cannot use logarithmic axes (as we
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|  |  |
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|------------------------------------------------|------------------------------------------------|
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| <a id="org35a1358"></a> Real part | <a id="org69673f5"></a> Imaginary part |
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| <a id="org695538e"></a> Real part | <a id="org95c5960"></a> Imaginary part |
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| width=\linewidth | width=\linewidth |
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##### Real part and Imaginary part of reciprocal FRF {#real-part-and-imaginary-part-of-reciprocal-frf}
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It can be seen from the expression of the inverse receptance [eq:dynamic_stiffness](#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.
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It can be seen from the expression of the inverse receptance \eqref{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.
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Figure [fig:inverse_frf_mixed](#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\\).
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@@ -578,7 +578,7 @@ Figure [fig:inverse_frf_mixed](#fig:inverse_frf_mixed) shows an example of a plo
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|  |  |
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|---------------------------------------------|-----------------------------------------------|
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| <a id="org8ba100f"></a> Mixed | <a id="org9bed1e0"></a> Viscous |
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| <a id="org9e0909f"></a> Mixed | <a id="orge2690df"></a> Viscous |
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| width=\linewidth | width=\linewidth |
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@@ -595,7 +595,7 @@ The missing information (in this case, the frequency) must be added by identifyi
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|  |  |
|
||||
|------------------------------------------------------|---------------------------------------------------------|
|
||||
| <a id="org5cbe7df"></a> Viscous damping | <a id="orgb5e61e4"></a> Structural damping |
|
||||
| <a id="org86b8a60"></a> Viscous damping | <a id="orgb0d3b09"></a> Structural damping |
|
||||
| width=\linewidth | width=\linewidth |
|
||||
|
||||
The Nyquist plot has the particularity of distorting the plot so as to focus on the resonance area.
|
||||
@@ -607,7 +607,7 @@ This makes the Nyquist plot very effective for modal testing applications.
|
||||
|
||||
#### Free Vibration Solution - The modal Properties {#free-vibration-solution-the-modal-properties}
|
||||
|
||||
For an undamped MDOF system, with \\(N\\) degrees of freedom, the governing equations of motion can be written in matrix form [eq:undamped_mdof](#eq:undamped_mdof).
|
||||
For an undamped MDOF system, with \\(N\\) degrees of freedom, the governing equations of motion can be written in matrix form \eqref{eq:undamped_mdof}.
|
||||
|
||||
<div class="cbox">
|
||||
<div></div>
|
||||
@@ -622,7 +622,7 @@ where \\([M]\\) and \\([K]\\) are \\(N\times N\\) mass and stiffness matrices, a
|
||||
|
||||
We shall consider first the free vibration solution by taking \\(f(t) = 0\\).
|
||||
In this case, we assume that a solution exists of the form \\(\\{x(t)\\} = \\{X\\} e^{i \omega t}\\) where \\(\\{X\\}\\) is an \\(N \times 1\\) vector of time-independent amplitudes.
|
||||
Substitution of this condition into [eq:undamped_mdof](#eq:undamped_mdof) leads to
|
||||
Substitution of this condition into \eqref{eq:undamped_mdof} leads to
|
||||
|
||||
\begin{equation}
|
||||
\left( [K] - \omega^2 [M] \right) \\{X\\} e^{i\omega t} = \\{0\\}
|
||||
@@ -632,7 +632,7 @@ for which the non trivial solutions are those which satisfy
|
||||
\\[ \det \left| [K] - \omega^2 [M] \right| = 0 \\]
|
||||
from which we can find \\(N\\) values of \\(\omega^2\\) corresponding to the undamped system's **natural frequencies**.
|
||||
|
||||
Substituting any of these back into [eq:free_eom_mdof](#eq:free_eom_mdof) yields a corresponding set of relative values for \\(\\{X\\}\\): \\(\\{\psi\\}\_r\\) the so-called **mode shape** corresponding to that natural frequency.
|
||||
Substituting any of these back into \eqref{eq:free_eom_mdof} yields a corresponding set of relative values for \\(\\{X\\}\\): \\(\\{\psi\\}\_r\\) the so-called **mode shape** corresponding to that natural frequency.
|
||||
|
||||
<div class="cbox">
|
||||
<div></div>
|
||||
@@ -774,7 +774,7 @@ An alternative means of deriving the FRF parameters is used which makes use of t
|
||||
\\[ [K] - \omega^2 [M] = [\alpha(\omega)]^{-1} \\]
|
||||
Pre-multiply both sides by \\([\Phi]^T\\) and post-multiply both sides by \\([\Phi]\\) to obtain
|
||||
\\[ [\Phi]^T ([K] - \omega^2 [M]) [\Phi] = [\Phi]^T [\alpha(\omega)]^{-1} [\Phi] \\]
|
||||
which leads to [eq:receptance_modal](#eq:receptance_modal).
|
||||
which leads to \eqref{eq:receptance_modal}.
|
||||
|
||||
<div class="cbox">
|
||||
<div></div>
|
||||
@@ -783,7 +783,7 @@ which leads to [eq:receptance_modal](#eq:receptance_modal).
|
||||
[\alpha(\omega)] = [\Phi] \left[ \bar{\omega}\_r^2 - \omega^2 \right]^{-1} [\Phi]^T \label{eq:receptance\_modal}
|
||||
\end{equation}
|
||||
|
||||
Equation [eq:receptance_modal](#eq:receptance_modal) permits us to compute any individual FRF parameters \\(\alpha\_{jk}(\omega)\\) using the following formula
|
||||
Equation \eqref{eq:receptance_modal} permits us to compute any individual FRF parameters \\(\alpha\_{jk}(\omega)\\) using the following formula
|
||||
|
||||
\begin{subequations}
|
||||
\begin{align}
|
||||
@@ -800,7 +800,7 @@ where \\({}\_rA\_{jk}\\) is called the **modal constant**.
|
||||
<div class="cbox">
|
||||
<div></div>
|
||||
|
||||
It is clear from equation [eq:receptance_modal](#eq:receptance_modal) that the receptance matrix \\([\alpha(\omega)]\\) is **symmetric** and this will be recognized as the **principle of reciprocity**.
|
||||
It is clear from equation \eqref{eq:receptance_modal} that the receptance matrix \\([\alpha(\omega)]\\) is **symmetric** and this will be recognized as the **principle of reciprocity**.
|
||||
|
||||
This principle of reciprocity applies to many structural characteristics.
|
||||
|
||||
@@ -938,7 +938,7 @@ From this full matrix equation, we have:
|
||||
|
||||
Having derived an expression for the general term in the frequency response function matrix \\(\alpha\_{jk}(\omega)\\), it is appropriate to consider next the analysis of a situation where the system is **excited simultaneously at several points**.
|
||||
|
||||
The general behavior for this case is governed by equation [eq:force_response_eom](#eq:force_response_eom) with solution [eq:force_response_eom_solution](#eq:force_response_eom_solution).
|
||||
The general behavior for this case is governed by equation \eqref{eq:force_response_eom} with solution \eqref{eq:force_response_eom_solution}.
|
||||
However, a more explicit form of the solution is
|
||||
|
||||
\begin{equation}
|
||||
@@ -962,7 +962,7 @@ The properties of the normal modes of the undamped system are of interest becaus
|
||||
|
||||
</div>
|
||||
|
||||
We are seeking an excitation vector \\(\\{F\\}\\) such that the **response** \\(\\{X\\}\\) **consists of a single modal component** so that all terms in [eq:ods](#eq:ods) but one is zero.
|
||||
We are seeking an excitation vector \\(\\{F\\}\\) such that the **response** \\(\\{X\\}\\) **consists of a single modal component** so that all terms in \eqref{eq:ods} but one is zero.
|
||||
This can be attained if \\(\\{F\\}\\) is chosen such that
|
||||
\\[ \\{\phi\_r\\}^T \\{F\\}\_s = 0, \ r \neq s \\]
|
||||
|
||||
@@ -1046,7 +1046,7 @@ where \\(\omega\_r\\) is the **natural frequency** and \\(\xi\_r\\) is the **cri
|
||||
When the modes \\(r\\) and \\(q\\) are a complex conjugate pair:
|
||||
\\[ s\_r = \omega\_r \left( -\xi\_r - i\sqrt{1 - \xi\_r^2} \right); \quad \\{\psi\\}\_q = \\{\psi\\}\_r^\* \\]
|
||||
|
||||
From equations [eq:viscous_damping_orthogonality](#eq:viscous_damping_orthogonality), we can obtain
|
||||
From equations \eqref{eq:viscous_damping_orthogonality}, we can obtain
|
||||
|
||||
\begin{subequations}
|
||||
\begin{align}
|
||||
@@ -1103,7 +1103,7 @@ Equally, in a real mode, all parts of the structure pass through their **zero de
|
||||
|
||||
While the real mode has the appearance of a **standing wave**, the complex mode is better described as exhibiting **traveling waves** (illustrated on figure [fig:real_complex_modes](#fig:real_complex_modes)).
|
||||
|
||||
<a id="org71662dc"></a>
|
||||
<a id="org76fb154"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/ewins00_real_complex_modes.png" caption="Figure 6: Real and complex mode shapes displays" >}}
|
||||
|
||||
@@ -1118,7 +1118,7 @@ Note that the almost-real mode shape does not necessarily have vector elements w
|
||||
|
||||
|  |  |  |
|
||||
|--------------------------------------------|--------------------------------------------|-----------------------------------------------|
|
||||
| <a id="orgdb142d3"></a> Almost-real mode | <a id="org40e30d6"></a> Complex Mode | <a id="org4c29a71"></a> Measure of complexity |
|
||||
| <a id="orgd9e3564"></a> Almost-real mode | <a id="orgeedeefa"></a> Complex Mode | <a id="org2d21384"></a> Measure of complexity |
|
||||
| width=\linewidth | width=\linewidth | width=\linewidth |
|
||||
|
||||
|
||||
@@ -1235,7 +1235,7 @@ On a logarithmic plot, this produces the antiresonance characteristic which refl
|
||||
|
||||
|  |  |
|
||||
|---------------------------------------------------|------------------------------------------------------|
|
||||
| <a id="org0d0c340"></a> Point FRF | <a id="org13ad8cd"></a> Transfer FRF |
|
||||
| <a id="org464f787"></a> Point FRF | <a id="orgd21bcd3"></a> Transfer FRF |
|
||||
| width=\linewidth | width=\linewidth |
|
||||
|
||||
For the plot in figure [fig:mobility_frf_mdof_transfer](#fig:mobility_frf_mdof_transfer), between the two resonances, the two components have the same sign and they add up, no antiresonance is present.
|
||||
@@ -1260,7 +1260,7 @@ Most mobility plots have this general form as long as the modes are relatively w
|
||||
|
||||
This condition is satisfied unless the separation between adjacent natural frequencies is of the same order as, or less than, the modal damping factors, in which case it becomes difficult to distinguish the individual modes.
|
||||
|
||||
<a id="orgb88a5bd"></a>
|
||||
<a id="orgd6edca6"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/ewins00_frf_damped_system.png" caption="Figure 7: Mobility plot of a damped system" >}}
|
||||
|
||||
@@ -1281,7 +1281,7 @@ The plot for the transfer receptance \\(\alpha\_{21}\\) is presented in figure [
|
||||
|
||||
|  |  |
|
||||
|------------------------------------------|---------------------------------------------|
|
||||
| <a id="org7bbfae7"></a> Point receptance | <a id="org0f31112"></a> Transfer receptance |
|
||||
| <a id="org5dbb609"></a> Point receptance | <a id="orgf225939"></a> Transfer receptance |
|
||||
| width=\linewidth | width=\linewidth |
|
||||
|
||||
In the two figures [fig:nyquist_nonpropdamp_point](#fig:nyquist_nonpropdamp_point) and [fig:nyquist_nonpropdamp_transfer](#fig:nyquist_nonpropdamp_transfer), we show corresponding data for **non-proportional** damping.
|
||||
@@ -1296,7 +1296,7 @@ Now we find that the individual modal circles are no longer "upright" but are **
|
||||
|
||||
|  |  |
|
||||
|-----------------------------------------------------|--------------------------------------------------------|
|
||||
| <a id="org460aa35"></a> Point receptance | <a id="org01c8e2c"></a> Transfer receptance |
|
||||
| <a id="orgae9806e"></a> Point receptance | <a id="orgb532a2f"></a> Transfer receptance |
|
||||
| width=\linewidth | width=\linewidth |
|
||||
|
||||
|
||||
@@ -1343,7 +1343,7 @@ One these two series are available, the FRF can be defined at the same set of fr
|
||||
|
||||
##### Analysis via Fourier transform {#analysis-via-fourier-transform}
|
||||
|
||||
For most transient cases, the input function \\(f(t)\\) will satisfy the **Dirichlet condition** and so its Fourier Transform \\(F(\omega)\\) can be computed from [eq:fourier_transform](#eq:fourier_transform).
|
||||
For most transient cases, the input function \\(f(t)\\) will satisfy the **Dirichlet condition** and so its Fourier Transform \\(F(\omega)\\) can be computed from \eqref{eq:fourier_transform}.
|
||||
|
||||
\begin{equation}
|
||||
F(\omega) = \frac{1}{2 \pi} \int\_{-\infty}^\infty f(t) e^{i\omega t} dt
|
||||
@@ -1450,7 +1450,7 @@ Examples of random signals, autocorrelation function and power spectral density
|
||||
|
||||
|  |  |  |
|
||||
|---------------------------------------|--------------------------------------------------|------------------------------------------------|
|
||||
| <a id="org3a8665e"></a> Time history | <a id="org5ade2bf"></a> Autocorrelation Function | <a id="org9f69a06"></a> Power Spectral Density |
|
||||
| <a id="org30bff26"></a> Time history | <a id="org7e07ced"></a> Autocorrelation Function | <a id="orgcb31329"></a> 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.
|
||||
@@ -1493,10 +1493,10 @@ However, the same equation can be transform to the frequency domain
|
||||
\tcmbox{ S\_{xx}(\omega) = \left| H(\omega) \right|^2 S\_{ff}(\omega) }
|
||||
\end{equation}
|
||||
|
||||
Although very convenient, equation [eq:psd_input_output](#eq:psd_input_output) does not provide a complete description of the random vibration conditions.
|
||||
Although very convenient, equation \eqref{eq:psd_input_output} does not provide a complete description of the random vibration conditions.
|
||||
Further, it is clear that **is could not be used to determine the FRF** from measurement of excitation and response because it **contains only the modulus** of \\(H(\omega)\\), the phase information begin omitted from this formula.
|
||||
|
||||
A second equation is required and this may be obtain by a similar analysis, two alternative formulas can be obtained [eq:cross_relation_alternatives](#eq:cross_relation_alternatives).
|
||||
A second equation is required and this may be obtain by a similar analysis, two alternative formulas can be obtained \eqref{eq:cross_relation_alternatives}.
|
||||
|
||||
<div class="cbox">
|
||||
<div></div>
|
||||
@@ -1513,8 +1513,8 @@ A second equation is required and this may be obtain by a similar analysis, two
|
||||
|
||||
##### To derive FRF from random vibration signals {#to-derive-frf-from-random-vibration-signals}
|
||||
|
||||
The pair of equations [eq:cross_relation_alternatives](#eq:cross_relation_alternatives) provides the basic of determining a system's FRF properties from the measurements and analysis of a random vibration test.
|
||||
Using either of them, we have a simple formula for determining the FRF from estimates of the relevant spectral densities [eq:frf_estimates_spectral_densities](#eq:frf_estimates_spectral_densities).
|
||||
The pair of equations \eqref{eq:cross_relation_alternatives} provides the basic of determining a system's FRF properties from the measurements and analysis of a random vibration test.
|
||||
Using either of them, we have a simple formula for determining the FRF from estimates of the relevant spectral densities \eqref{eq:frf_estimates_spectral_densities}.
|
||||
|
||||
<div class="cbox">
|
||||
<div></div>
|
||||
@@ -1547,11 +1547,11 @@ Then in [fig:frf_feedback_model](#fig:frf_feedback_model) is given a more detail
|
||||
|
||||
|  |  |
|
||||
|------------------------------------------|--------------------------------------------------|
|
||||
| <a id="org5183bee"></a> Basic SISO model | <a id="org7eda16f"></a> SISO model with feedback |
|
||||
| <a id="orgcf49de0"></a> Basic SISO model | <a id="orgad8dce0"></a> SISO model with feedback |
|
||||
| width=\linewidth | width=\linewidth |
|
||||
|
||||
In this configuration, it can be seen that there are two feedback mechanisms which apply.
|
||||
We then introduce an alternative formula which is available for the determination of the system FRF from measurements of the input and output quantities [eq:H3](#eq:H3).
|
||||
We then introduce an alternative formula which is available for the determination of the system FRF from measurements of the input and output quantities \eqref{eq:H3}.
|
||||
|
||||
<div class="cbox">
|
||||
<div></div>
|
||||
@@ -1580,7 +1580,7 @@ We obtain two alternative formulas:
|
||||
|
||||
In practical application of both of these formulae, care must be taken to ensure the non-singularity of the spectral density matrix which is to be inverted, and it is in this respect that the former version may be found to be more reliable.
|
||||
|
||||
<a id="org1066c50"></a>
|
||||
<a id="org2388f52"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/ewins00_frf_mimo.png" caption="Figure 8: System for FRF determination via MIMO model" >}}
|
||||
|
||||
@@ -1694,7 +1694,7 @@ First, if we have a **modal incompleteness** (\\(m<N\\) modes included), then we
|
||||
|
||||
However, if we have **spatial incompleteness** (only \\(n<N\\) DOFs included), then we cannot express any orthogonality properties at all because the eigenvector matrix is not commutable with the system mass and stiffness matrices.
|
||||
|
||||
In both reduced-model cases, it is not possible to use equation [eq:spatial_model_from_modal](#eq:spatial_model_from_modal) to re-construct the system mass and stiffness matrices.
|
||||
In both reduced-model cases, it is not possible to use equation \eqref{eq:spatial_model_from_modal} to re-construct the system mass and stiffness matrices.
|
||||
First of all because the eigen matrices are generally singular and even if it is not, the obtained mass and stiffness matrices produced have no physical significance and should not be used.
|
||||
|
||||
|
||||
@@ -1852,7 +1852,7 @@ The experimental setup used for mobility measurement contains three major items:
|
||||
|
||||
A typical layout for the measurement system is shown on figure [fig:general_frf_measurement_setup](#fig:general_frf_measurement_setup).
|
||||
|
||||
<a id="org257621c"></a>
|
||||
<a id="org1415164"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/ewins00_general_frf_measurement_setup.png" caption="Figure 9: General layout of FRF measurement system" >}}
|
||||
|
||||
@@ -1909,7 +1909,7 @@ 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 [fig:shaker_rod](#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="orga7d69b0"></a>
|
||||
<a id="orgbf524e6"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/ewins00_shaker_rod.png" caption="Figure 10: Exciter attachment and drive rod assembly" >}}
|
||||
|
||||
@@ -1930,7 +1930,7 @@ Figure [fig:shaker_mount_3](#fig:shaker_mount_3) shows an unsatisfactory setup.
|
||||
|
||||
|  |  |  |
|
||||
|---------------------------------------------|-------------------------------------------------|------------------------------------------|
|
||||
| <a id="orge0af414"></a> Ideal Configuration | <a id="org21ea9d5"></a> Suspended Configuration | <a id="org272e483"></a> Unsatisfactory |
|
||||
| <a id="orga9157bf"></a> Ideal Configuration | <a id="org4b90d28"></a> Suspended Configuration | <a id="org3061b55"></a> Unsatisfactory |
|
||||
| width=\linewidth | width=\linewidth | width=\linewidth |
|
||||
|
||||
|
||||
@@ -1948,7 +1948,7 @@ The frequency range which is effectively excited is controlled by the stiffness
|
||||
When the hammer tip impacts the test structure, this will experience a force pulse as shown on figure [fig:hammer_impulse](#fig:hammer_impulse).
|
||||
A pulse of this type (half-sine shape) has a frequency content of the form illustrated on figure [fig:hammer_impulse](#fig:hammer_impulse).
|
||||
|
||||
<a id="orgfb698ca"></a>
|
||||
<a id="orgdb53d89"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/ewins00_hammer_impulse.png" caption="Figure 11: Typical impact force pulse and spectrum" >}}
|
||||
|
||||
@@ -1979,7 +1979,7 @@ By suitable design, such a material may be incorporated into a device which **in
|
||||
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 [fig:piezo_force_transducer](#fig:piezo_force_transducer)).
|
||||
|
||||
<a id="org6b146da"></a>
|
||||
<a id="org93aad2e"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/ewins00_piezo_force_transducer.png" caption="Figure 12: Force transducer" >}}
|
||||
|
||||
@@ -1992,7 +1992,7 @@ In an accelerometer, transduction is indirect and is achieved using a seismic ma
|
||||
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\\).
|
||||
|
||||
<a id="org29aa78f"></a>
|
||||
<a id="org84766b5"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/ewins00_piezo_accelerometer.png" caption="Figure 13: Compression-type of piezoelectric accelerometer" >}}
|
||||
|
||||
@@ -2040,7 +2040,7 @@ Shown on figure [fig:transducer_mounting_response](#fig:transducer_mounting_resp
|
||||
|
||||
|  |  |
|
||||
|-----------------------------------------------------|------------------------------------------------------------|
|
||||
| <a id="org1a66b31"></a> Attachment methods | <a id="org4e23863"></a> Frequency response characteristics |
|
||||
| <a id="org796e903"></a> Attachment methods | <a id="org308a233"></a> Frequency response characteristics |
|
||||
| width=\linewidth | width=\linewidth |
|
||||
|
||||
|
||||
@@ -2127,7 +2127,7 @@ Aliasing originates from the discretisation of the originally continuous time hi
|
||||
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 [fig:aliasing](#fig:aliasing).
|
||||
|
||||
<a id="orgb59b07e"></a>
|
||||
<a id="orge489af5"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/ewins00_aliasing.png" caption="Figure 14: The phenomenon of aliasing. On top: Low-frequency signal, On the bottom: High frequency signal" >}}
|
||||
|
||||
@@ -2144,7 +2144,7 @@ This is illustrated on figure [fig:effect_aliasing](#fig:effect_aliasing).
|
||||
|
||||
|  |  |
|
||||
|--------------------------------------------------|-----------------------------------------------------|
|
||||
| <a id="org7e6ecc4"></a> True spectrum of signal | <a id="org3c375b8"></a> Indicated spectrum from DFT |
|
||||
| <a id="org3c7851f"></a> True spectrum of signal | <a id="orgd31d06c"></a> 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.
|
||||
@@ -2165,7 +2165,7 @@ Leakage is a problem which is a direct **consequence of the need to take only a
|
||||
|
||||
|  |  |
|
||||
|--------------------------------------|----------------------------------------|
|
||||
| <a id="org3c7efb4"></a> Ideal signal | <a id="orga61ea56"></a> Awkward signal |
|
||||
| <a id="org62f211a"></a> Ideal signal | <a id="orgd4e0fe1"></a> Awkward signal |
|
||||
| width=\linewidth | width=\linewidth |
|
||||
|
||||
The problem is illustrated on figure [fig:leakage](#fig:leakage).
|
||||
@@ -2190,7 +2190,7 @@ Windowing involves the imposition of a prescribed profile on the time signal pri
|
||||
|
||||
The profiles, or "windows" are generally depicted as a time function \\(w(t)\\) as shown in figure [fig:windowing_examples](#fig:windowing_examples).
|
||||
|
||||
<a id="org11d5f31"></a>
|
||||
<a id="orge28ad03"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/ewins00_windowing_examples.png" caption="Figure 15: Different types of window. (a) Boxcar, (b) Hanning, (c) Cosine-taper, (d) Exponential" >}}
|
||||
|
||||
@@ -2211,7 +2211,7 @@ Common filters are: low-pass, high-pass, band-limited, narrow-band, notch.
|
||||
|
||||
#### Improving Resolution {#improving-resolution}
|
||||
|
||||
<a id="orgf44c4ab"></a>
|
||||
<a id="org81b4f25"></a>
|
||||
|
||||
|
||||
##### Increasing transform size {#increasing-transform-size}
|
||||
@@ -2247,10 +2247,10 @@ If we apply a band-pass filter to the signal, as shown on figure [fig:zoom_bandp
|
||||
|
||||
|  |  |
|
||||
|------------------------------------------------|------------------------------------------|
|
||||
| <a id="orgafd809b"></a> Spectrum of the signal | <a id="org7d44157"></a> Band-pass filter |
|
||||
| <a id="org28ed6ec"></a> Spectrum of the signal | <a id="org8a7e75c"></a> Band-pass filter |
|
||||
| width=\linewidth | width=\linewidth |
|
||||
|
||||
<a id="org8e7e54a"></a>
|
||||
<a id="org60b3e9b"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/ewins00_zoom_result.png" caption="Figure 16: Effective frequency translation for zoom" >}}
|
||||
|
||||
@@ -2322,7 +2322,7 @@ This is the traditional method of FRF measurement and involves the use of a swee
|
||||
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 [fig:sweep_distortions](#fig:sweep_distortions).
|
||||
|
||||
<a id="orgb233cbb"></a>
|
||||
<a id="orgeab1f57"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/ewins00_sweep_distortions.png" caption="Figure 17: FRF measurements by sine sweep test" >}}
|
||||
|
||||
@@ -2440,7 +2440,7 @@ It is known that a low coherence can arise in a measurement where the frequency
|
||||
|
||||
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 [fig:coherence_resonance](#fig:coherence_resonance).
|
||||
|
||||
<a id="org4d83015"></a>
|
||||
<a id="orgb72faa8"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/ewins00_coherence_resonance.png" caption="Figure 18: Coherence \\(\gamma^2\\) and FRF estimate \\(H\_1(\omega)\\) for a lightly damped structure" >}}
|
||||
|
||||
@@ -2483,7 +2483,7 @@ For the chirp and impulse excitations, each individual sample is collected and p
|
||||
|
||||
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 [fig:burst_excitation](#fig:burst_excitation)).
|
||||
|
||||
<a id="org2152665"></a>
|
||||
<a id="org681a980"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/ewins00_burst_excitation.png" caption="Figure 19: Example of burst excitation and response signals" >}}
|
||||
|
||||
@@ -2502,7 +2502,7 @@ The chirp consist of a short duration signal which has the form shown in figure
|
||||
|
||||
The frequency content of the chirp can be precisely chosen by the starting and finishing frequencies of the sweep.
|
||||
|
||||
<a id="org56036c2"></a>
|
||||
<a id="org632f8cc"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/ewins00_chirp_excitation.png" caption="Figure 20: Example of chirp excitation and response signals" >}}
|
||||
|
||||
@@ -2513,7 +2513,7 @@ The hammer blow produces an input and response as shown in the figure [fig:impul
|
||||
|
||||
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.
|
||||
|
||||
<a id="orga792905"></a>
|
||||
<a id="orgdecf769"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/ewins00_impulsive_excitation.png" caption="Figure 21: Example of impulsive excitation and response signals" >}}
|
||||
|
||||
@@ -2523,7 +2523,7 @@ However, it should be recorded that in the region below the first cut-off freque
|
||||
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 [fig:double_hits](#fig:double_hits)).
|
||||
|
||||
<a id="org43e765b"></a>
|
||||
<a id="orgea279f8"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/ewins00_double_hits.png" caption="Figure 22: Double hits time domain and frequency content" >}}
|
||||
|
||||
@@ -2598,7 +2598,7 @@ Suppose the response parameter is acceleration, then the FRF obtained is inertan
|
||||
|
||||
Figure [fig:calibration_setup](#fig:calibration_setup) shows a typical calibration setup.
|
||||
|
||||
<a id="org1176320"></a>
|
||||
<a id="org3a6c052"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/ewins00_calibration_setup.png" caption="Figure 23: Mass calibration procedure, measurement setup" >}}
|
||||
|
||||
@@ -2613,7 +2613,7 @@ This is because near resonance, the actual applied force becomes very small and
|
||||
|
||||
This same argument applies on a lesser scale as we examine the detail around the attachment to the structure, as shown in figure [fig:mass_cancellation](#fig:mass_cancellation).
|
||||
|
||||
<a id="org25ad6ad"></a>
|
||||
<a id="orgf6011aa"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/ewins00_mass_cancellation.png" caption="Figure 24: Added mass to be cancelled (crossed area)" >}}
|
||||
|
||||
@@ -2670,7 +2670,7 @@ There are two problems to be tackled:
|
||||
|
||||
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 [fig:rotational_measurement](#fig:rotational_measurement).
|
||||
|
||||
<a id="org92fa8a1"></a>
|
||||
<a id="org6c1a993"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/ewins00_rotational_measurement.png" caption="Figure 25: Measurement of rotational response" >}}
|
||||
|
||||
@@ -2691,7 +2691,7 @@ First, a single applied excitation force \\(F\_1\\) corresponds to a simultaneou
|
||||
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\\).
|
||||
|
||||
<a id="org97cd93f"></a>
|
||||
<a id="org19d9418"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/ewins00_rotational_excitation.png" caption="Figure 26: Application of moment excitation" >}}
|
||||
|
||||
@@ -2700,7 +2700,7 @@ Then, the full \\(6 \times 6\\) mobility matrix can be measured, however this pr
|
||||
Other methods for measuring rotational effects include specially developed rotational accelerometers and shakers.
|
||||
|
||||
However, there is a major problem that is encountered when measuring rotational FRF: the translational components of the structure's movement tends to overshadow those due to the rotational motions.
|
||||
For example, the magnitude of the difference in equation [eq:rotational_diff](#eq:rotational_diff) is often of the order of \\(\SI{1}{\%}\\) of the two individual values which is similar to the transverse sensitivity of the accelerometers: potential errors in rotations are thus enormous.
|
||||
For example, the magnitude of the difference in equation \eqref{eq:rotational_diff} is often of the order of \\(\SI{1}{\%}\\) of the two individual values which is similar to the transverse sensitivity of the accelerometers: potential errors in rotations are thus enormous.
|
||||
|
||||
|
||||
### Multi-point excitation methods {#multi-point-excitation-methods}
|
||||
@@ -2739,7 +2739,7 @@ The two vectors are related by the system's FRF properties as:
|
||||
\\{X\\}\_{n\times 1} = [H(\omega)]\_{n\times p} \\{F\\}\_{p\times 1}
|
||||
\end{equation}
|
||||
|
||||
However, it is not possible to derive the FRF matrix from the single equation [eq:mpss_equation](#eq:mpss_equation), because there will be insufficient data in the two vectors (one of length \\(p\\), the other of length \\(n\\)) to define completely the \\(n\times p\\) FRF matrix.
|
||||
However, it is not possible to derive the FRF matrix from the single equation \eqref{eq:mpss_equation}, because there will be insufficient data in the two vectors (one of length \\(p\\), the other of length \\(n\\)) to define completely the \\(n\times p\\) FRF matrix.
|
||||
|
||||
What is required is to make a series of \\(p^\prime\\) measurements of the same basic type using different excitation vectors \\(\\{F\\}\_i\\) that should be chosen such that the forcing matrix \\([F]\_{p\times p^\prime} = [\\{F\\}\_1, \dots, \\{F\\}\_p]\\) is non-singular.
|
||||
This can be assured if:
|
||||
@@ -3031,7 +3031,7 @@ The two groups are usually separated by a clear gap (depending of the noise pres
|
||||
|
||||
|  |  |  |
|
||||
|---------------------------------------------|---------------------------------------------|---------------------------------------------|
|
||||
| <a id="org3c82345"></a> FRF | <a id="orga931e29"></a> Singular Values | <a id="orgf4a6ae7"></a> PRF |
|
||||
| <a id="org911bfc8"></a> FRF | <a id="org60f84fb"></a> Singular Values | <a id="orgdf8522b"></a> PRF |
|
||||
| width=\linewidth | width=\linewidth | width=\linewidth |
|
||||
|
||||
<a id="table--fig:PRF-measured"></a>
|
||||
@@ -3042,7 +3042,7 @@ The two groups are usually separated by a clear gap (depending of the noise pres
|
||||
|
||||
|  |  |  |
|
||||
|--------------------------------------------|--------------------------------------------|--------------------------------------------|
|
||||
| <a id="orge590e19"></a> FRF | <a id="org176a55f"></a> Singular Values | <a id="orge859b81"></a> PRF |
|
||||
| <a id="org3d1c696"></a> FRF | <a id="orgeb81dac"></a> Singular Values | <a id="orgc25aeb3"></a> PRF |
|
||||
| width=\linewidth | width=\linewidth | width=\linewidth |
|
||||
|
||||
|
||||
@@ -3084,7 +3084,7 @@ Associated with the CMIF values at each natural frequency \\(\omega\_r\\) are tw
|
||||
- the left singular vector \\(\\{U(\omega\_r)\\}\_1\\) which approximates the **mode shape** of that mode
|
||||
- the right singular vector \\(\\{V(\omega\_r)\\}\_1\\) which represents the approximate **force pattern necessary to generate a response on that mode only**
|
||||
|
||||
<a id="org52ec446"></a>
|
||||
<a id="org5f7cb1f"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/ewins00_mifs.png" caption="Figure 27: Complex Mode Indicator Function (CMIF)" >}}
|
||||
|
||||
@@ -3179,7 +3179,7 @@ The peak-picking method is applied as follows (illustrated on figure [fig:peak_a
|
||||
It must be noted that the estimates of both damping and modal constant depend heavily on the accuracy of the maximum FRF level \\(|\hat{H}|\\) which is difficult to measure with great accuracy, especially for lightly damped systems.
|
||||
Only real modal constants and thus real modes can be deduced by this method.
|
||||
|
||||
<a id="org12489c1"></a>
|
||||
<a id="org0d4b46a"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/ewins00_peak_amplitude.png" caption="Figure 28: Peak Amplitude method of modal analysis" >}}
|
||||
|
||||
@@ -3214,7 +3214,7 @@ A plot of the quantity \\(\alpha(\omega)\\) is given in figure [fig:modal_circle
|
||||
|
||||
|  |  |
|
||||
|----------------------------------------|--------------------------------------------------------------------|
|
||||
| <a id="orge6f933c"></a> Properties | <a id="org852e910"></a> \\(\omega\_b\\) and \\(\omega\_a\\) points |
|
||||
| <a id="org187efdc"></a> Properties | <a id="org0e24b72"></a> \\(\omega\_b\\) and \\(\omega\_a\\) points |
|
||||
| width=\linewidth | width=\linewidth |
|
||||
|
||||
For any frequency \\(\omega\\), we have the following relationship:
|
||||
@@ -3226,13 +3226,13 @@ For any frequency \\(\omega\\), we have the following relationship:
|
||||
\end{align}
|
||||
\end{subequations}
|
||||
|
||||
From [eq:modal_circle_tan](#eq:modal_circle_tan), we obtain:
|
||||
From \eqref{eq:modal_circle_tan}, we obtain:
|
||||
|
||||
\begin{equation}
|
||||
\omega^2 = \omega\_r^2 \left(1 - \eta\_r \tan\left(\frac{\theta}{2}\right) \right)
|
||||
\end{equation}
|
||||
|
||||
If we differentiate [eq:modal_circle_omega](#eq:modal_circle_omega) with respect to \\(\theta\\), we obtain:
|
||||
If we differentiate \eqref{eq:modal_circle_omega} with respect to \\(\theta\\), we obtain:
|
||||
|
||||
\begin{equation}
|
||||
\frac{d\omega^2}{d\theta} = \frac{-\omega\_r^2 \eta\_r}{2} \frac{\left(1 - (\omega/\omega\_r)^2\right)^2}{\eta\_r^2}
|
||||
@@ -3317,10 +3317,10 @@ The sequence is:
|
||||
Then we obtain the **center** and **radius** of the circle and the **quality factor** is the mean square deviation of the chosen points from the circle.
|
||||
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](#eq:estimate_damping_sweep_rate).
|
||||
At the same time, an estimate of the damping is derived using \eqref{eq:estimate_damping_sweep_rate}.
|
||||
A typical example is shown on figure [fig:circle_fit_natural_frequency](#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](#eq:estimate_damping).
|
||||
A set of damping estimates using all possible combination of the selected data points are computed using \eqref{eq:estimate_damping}.
|
||||
Then, we can choose the damping estimate to be the mean value.
|
||||
We also look at the distribution of the obtained damping estimates as is permits a useful diagnostic of the quality of the entire analysis:
|
||||
- Good measured data should lead to a smooth plot of these damping estimates, any roughness of the surface can be explained in terms of noise in the original data.
|
||||
@@ -3328,7 +3328,7 @@ The sequence is:
|
||||
5. **Determine modal constant modulus and argument**.
|
||||
The magnitude and argument of the modal constant is determined from the diameter of the circle and from its orientation relative to the Real and Imaginary axis.
|
||||
|
||||
<a id="orgc0336ab"></a>
|
||||
<a id="org379e1a2"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/ewins00_circle_fit_natural_frequency.png" caption="Figure 29: Location of natural frequency for a Circle-fit modal analysis" >}}
|
||||
|
||||
@@ -3427,7 +3427,7 @@ we now have sufficient information to extract estimates for the four parameters
|
||||
<ol class="org-ol">
|
||||
<li value="3">Plot graphs of \\(m\_R(\Omega)\\) vs \\(\Omega^2\\) and of \\(m\_I(\Omega)\\) vs \\(\Omega^2\\) using the results from step 1., each time using a different measurement points as the fixing frequency \\(\Omega\_j\\)</li>
|
||||
<li>Determine the slopes of the best fit straight lines through these two plots, \\(n\_R\\) and \\(n\_I\\), and their intercepts with the vertical axis \\(d\_R\\) and \\(d\_I\\)</li>
|
||||
<li>Use these four quantities, and equation [eq:modal_parameters_formula](#eq:modal_parameters_formula), to determine the **four modal parameters** required for that mode</li>
|
||||
<li>Use these four quantities, and equation \eqref{eq:modal_parameters_formula}, to determine the **four modal parameters** required for that mode</li>
|
||||
</ol>
|
||||
|
||||
This procedure which places more weight to points slightly away from the resonance region is likely to be less sensitive to measurement difficulties of measuring the resonance region.
|
||||
@@ -3453,7 +3453,7 @@ However, by the inclusion of two simple extra terms (the "**residuals**"), the m
|
||||
|
||||
|  |  |
|
||||
|--------------------------------------------|-----------------------------------------|
|
||||
| <a id="org4fd3d88"></a> without residual | <a id="orgdb69a63"></a> with residuals |
|
||||
| <a id="orge96a388"></a> without residual | <a id="org92a8b32"></a> 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
|
||||
@@ -3484,7 +3484,7 @@ The three terms corresponds to:
|
||||
|
||||
These three terms are illustrated on figure [fig:low_medium_high_modes](#fig:low_medium_high_modes).
|
||||
|
||||
<a id="org85845f3"></a>
|
||||
<a id="org745f0a4"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/ewins00_low_medium_high_modes.png" caption="Figure 30: Numerical simulation of contribution of low, medium and high frequency modes" >}}
|
||||
|
||||
@@ -3493,7 +3493,7 @@ From the sketch, it may be seen that within the frequency range of interest:
|
||||
- the first term tends to approximate to a **mass-like behavior**
|
||||
- the third term approximates to a **stiffness effect**
|
||||
|
||||
Thus, we have a basis for the residual terms and shall rewrite equation [eq:sum_modes](#eq:sum_modes):
|
||||
Thus, we have a basis for the residual terms and shall rewrite equation \eqref{eq:sum_modes}:
|
||||
|
||||
\begin{equation}
|
||||
H\_{jk}(\omega) \simeq -\frac{1}{\omega^2 M\_{jk}^R} + \sum\_{r=m\_1}^{m\_2} \left( \frac{{}\_rA\_{jk}}{\omega\_r^2 - \omega^2 + i \eta\_r \omega\_r^2} \right) + \frac{1}{K\_{jk}^R}
|
||||
@@ -3554,7 +3554,7 @@ We can write the receptance in the frequency range of interest as:
|
||||
|
||||
In the previous methods, the second term was assumed to be a constant in the curve-fit procedure for mode \\(r\\).
|
||||
However, if we have good **estimates** for the coefficients which constitutes the second term, for example by having already completed an SDOF analysis, we may remove the restriction on the analysis.
|
||||
Indeed, suppose we take a set of measured data points around the resonance at \\(\omega\_r\\), and that we can compute the magnitude of the second term in [eq:second_term_refinement](#eq:second_term_refinement), we then subtract this from the measurement and we obtain adjusted data points that are conform to a true SDOF behavior and we can use the same technique as before to obtain **improved estimated** to the modal parameters of more \\(r\\).
|
||||
Indeed, suppose we take a set of measured data points around the resonance at \\(\omega\_r\\), and that we can compute the magnitude of the second term in \eqref{eq:second_term_refinement}, we then subtract this from the measurement and we obtain adjusted data points that are conform to a true SDOF behavior and we can use the same technique as before to obtain **improved estimated** to the modal parameters of more \\(r\\).
|
||||
|
||||
This procedure can be repeated iteratively for all the modes in the range of interest and it can significantly enhance the quality of found modal parameters for system with **strong coupling**.
|
||||
|
||||
@@ -3614,7 +3614,7 @@ If we further increase the generality by attaching a **weighting factor** \\(w\_
|
||||
|
||||
is minimized.
|
||||
|
||||
This is achieved by differentiating [eq:error_weighted](#eq:error_weighted) with respect to each unknown in turn, thus generating a set of as many equations as there are unknown:
|
||||
This is achieved by differentiating \eqref{eq:error_weighted} with respect to each unknown in turn, thus generating a set of as many equations as there are unknown:
|
||||
|
||||
\begin{equation}
|
||||
\frac{d E}{d q} = 0; \quad q = {}\_1A\_{jk}, {}\_2A\_{jk}, \dots
|
||||
@@ -3663,7 +3663,7 @@ leading to the modified, but more convenient version actually used in the analys
|
||||
\end{equation}
|
||||
|
||||
In these expressions, only \\(m\\) modes are included in the theoretical FRF formula: the true number of modes, \\(N\\), is actually one of the **unknowns** to be determined during the analysis.
|
||||
Equation [eq:rpf_error](#eq:rpf_error) can be rewritten as follows:
|
||||
Equation \eqref{eq:rpf_error} can be rewritten as follows:
|
||||
|
||||
\begin{equation}
|
||||
\begin{aligned}
|
||||
@@ -3717,7 +3717,7 @@ where \\([X], [Y], [Z], \\{G\\}\\) and \\(\\{F\\}\\) are known measured quantiti
|
||||
\end{equation}
|
||||
|
||||
Once the solution has been obtained for the coefficients \\(a\_k, \dots , b\_k, \dots\\) then the second stage of the modal analysis can be performed in which the required **modal parameters are derived**.
|
||||
This is usually done by solving the two polynomial expressions which form the numerator and denominator of equations [eq:frf_clasic](#eq:frf_clasic) and [eq:frf_rational](#eq:frf_rational):
|
||||
This is usually done by solving the two polynomial expressions which form the numerator and denominator of equations \eqref{eq:frf_clasic} and \eqref{eq:frf_rational}:
|
||||
|
||||
- the denominator is used to obtain the natural frequencies \\(\omega\_r\\) and damping factors \\(\xi\_r\\)
|
||||
- the numerator is used to determine the complex modal constants \\(A\_r\\)
|
||||
@@ -3785,7 +3785,7 @@ As an example, a set of mobilities measured are shown individually in figure [fi
|
||||
|
||||
|  |  |
|
||||
|-------------------------------------------|-----------------------------------------|
|
||||
| <a id="org9ce04ed"></a> Individual curves | <a id="orgf0fa0fe"></a> Composite curve |
|
||||
| <a id="org3f9a0d6"></a> Individual curves | <a id="org9ebc973"></a> 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.
|
||||
@@ -3949,7 +3949,7 @@ First, we note that from a single FRF curve, \\(H\_{jk}(\omega)\\), it is possib
|
||||
|
||||
Now, although this gives us the natural frequency and damping properties directly, it does not explicitly yield the mode shape: only a modal constant \\({}\_rA\_{jk}\\) which is formed from the mode shape data.
|
||||
In order to extract the individual elements \\(\phi\_{jr}\\) of the mode shape matrix \\([\Phi]\\), it is necessary to make a series of measurements of specific FRFs including, especially, the point FRF at the excitation position.
|
||||
If we measure \\(H\_{kk}\\), then by using [eq:modal_model_from_frf](#eq:modal_model_from_frf), we also obtain the specific elements in the mode shape matrix corresponding to the excitation point:
|
||||
If we measure \\(H\_{kk}\\), then by using \eqref{eq:modal_model_from_frf}, we also obtain the specific elements in the mode shape matrix corresponding to the excitation point:
|
||||
|
||||
\begin{equation}
|
||||
H\_{kk}(\omega) \longrightarrow \omega\_r, \eta\_r, {}\_rA\_{jk} \longrightarrow \phi\_{kr}; \quad r=1, m
|
||||
@@ -4332,7 +4332,7 @@ Measured coordinates of the test structure are first linked as shown on figure [
|
||||
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 [fig:static_display](#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="orgbbe2809"></a>
|
||||
<a id="orge0d2fb3"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/ewins00_static_display.png" caption="Figure 31: Static display of modes shapes. (a) basic grid (b) single-frame deflection pattern (c) multiple-frame deflection pattern (d) complex mode (e) Argand diagram - quasi-real mode (f) Argand diagram - complex mode" >}}
|
||||
|
||||
@@ -4377,7 +4377,7 @@ If we consider the first six modes of the beam, whose mode shapes are sketched i
|
||||
All the higher modes will be indistinguishable from these first few.
|
||||
This is a well known problem of **spatial aliasing**.
|
||||
|
||||
<a id="org10e3c8a"></a>
|
||||
<a id="org5c16ec7"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/ewins00_beam_modes.png" caption="Figure 32: Misinterpretation of mode shapes by spatial aliasing" >}}
|
||||
|
||||
@@ -4415,7 +4415,7 @@ In this respect, the demands of the response model are more stringent that those
|
||||
|
||||
#### Synthesis of FRF curves {#synthesis-of-frf-curves}
|
||||
|
||||
One of the implications of equation [eq:regenerate_full_frf_matrix](#eq:regenerate_full_frf_matrix) is that **it is possible to synthesize the FRF curves which were not measured**.
|
||||
One of the implications of equation \eqref{eq:regenerate_full_frf_matrix} is that **it is possible to synthesize the FRF curves which were not measured**.
|
||||
This arises because if we measured three individual FRF such as \\(H\_{ik}(\omega)\\), \\(H\_{jk}(\omega)\\) and \\(K\_{kk}(\omega)\\), then modal analysis of these yields the modal parameters from which it is possible to generate the FRF \\(H\_{ij}(\omega)\\), \\(H\_{jj}(\omega)\\), etc.
|
||||
|
||||
However, it must be noted that there is an important **limitation to this procedure** which is highlighted in the example below.
|
||||
@@ -4440,7 +4440,7 @@ The inclusion of these two additional terms (obtained here only after measuring
|
||||
|
||||
|  |  |
|
||||
|--------------------------------------------------------|-----------------------------------------------------------|
|
||||
| <a id="org0111dfe"></a> Using measured modal data only | <a id="org2abbdff"></a> After inclusion of residual terms |
|
||||
| <a id="orgee3fc43"></a> Using measured modal data only | <a id="org959e2d5"></a> 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
|
||||
@@ -4495,7 +4495,7 @@ If the transmissibility is measured during a modal test which has a single excit
|
||||
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 [fig:transmissibility_plots](#fig:transmissibility_plots).
|
||||
This may explain why transmissibilities are not widely used in modal analysis.
|
||||
|
||||
<a id="org6f53493"></a>
|
||||
<a id="orgb69dd65"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/ewins00_transmissibility_plots.png" caption="Figure 33: Transmissibility plots" >}}
|
||||
|
||||
@@ -4516,7 +4516,7 @@ The fact that the excitation force is not measured is responsible for the lack o
|
||||
|
||||
|  |  |
|
||||
|---------------------------------------------------------|-------------------------------------------------------|
|
||||
| <a id="orgfcd5181"></a> Conventional modal test setup | <a id="org544b696"></a> Base excitation setup |
|
||||
| <a id="orgfb8d62b"></a> Conventional modal test setup | <a id="orgb803ff7"></a> Base excitation setup |
|
||||
| height=4cm | height=4cm |
|
||||
|
||||
|
||||
@@ -4541,7 +4541,7 @@ from which is would appear that we can write
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
|
||||
However, equation [eq:m_k_from_modes](#eq:m_k_from_modes) is **only applicable when we have available the complete \\(N \times N\\) modal model**.
|
||||
However, equation \eqref{eq:m_k_from_modes} is **only applicable when we have available the complete \\(N \times N\\) modal model**.
|
||||
|
||||
It is much more usual to have an incomplete model in which the eigenvector matrix is rectangle and, as such, is non-invertible.
|
||||
One step which can be made using the incomplete data is the construction of "pseudo" flexibility and inverse-mass matrices.
|
||||
|
2139
content/book/hatch00_vibrat_matlab_ansys.md
Normal file
2139
content/book/hatch00_vibrat_matlab_ansys.md
Normal file
File diff suppressed because it is too large
Load Diff
@@ -61,11 +61,11 @@ There are two radically different approached to disturbance rejection: feedback
|
||||
|
||||
#### Feedback {#feedback}
|
||||
|
||||
<a id="orgcd0067e"></a>
|
||||
<a id="orgff2b033"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/preumont18_classical_feedback_small.png" caption="Figure 1: Principle of feedback control" >}}
|
||||
|
||||
The principle of feedback is represented on figure [fig:classical_feedback_small](#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](#orgff2b033). 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**.
|
||||
@@ -87,12 +87,12 @@ The objective is to control a variable \\(y\\) to a desired value \\(r\\) in spi
|
||||
|
||||
#### Feedforward {#feedforward}
|
||||
|
||||
<a id="orgf21f883"></a>
|
||||
<a id="orgc74bab2"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/preumont18_feedforward_adaptative.png" caption="Figure 2: 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](#orgf21f883).
|
||||
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](#orgc74bab2).
|
||||
|
||||
The filter coefficients are adapted in such a way that the error signal at one or several critical points is minimized.
|
||||
|
||||
@@ -123,11 +123,11 @@ The table [1](#table--tab:adv-dis-type-control) summarizes the main features of
|
||||
|
||||
### The Various Steps of the Design {#the-various-steps-of-the-design}
|
||||
|
||||
<a id="orgca19f4b"></a>
|
||||
<a id="org4c2b243"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/preumont18_design_steps.png" caption="Figure 3: The various steps of the design" >}}
|
||||
|
||||
The various steps of the design of a controlled structure are shown in figure [3](#orgca19f4b).
|
||||
The various steps of the design of a controlled structure are shown in figure [3](#org4c2b243).
|
||||
|
||||
The **starting point** is:
|
||||
|
||||
@@ -154,14 +154,14 @@ 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 [fig:general_plant](#fig:general_plant)):
|
||||
From the block diagram of the control system (figure [4](#org1c8100c)):
|
||||
|
||||
\begin{align\*}
|
||||
y &= (I - G\_{yu}H)^{-1} G\_{yw} w\\\\\\
|
||||
z &= T\_{zw} w = [G\_{zw} + G\_{zu}H(I - G\_{yu}H)^{-1} G\_{yw}] w
|
||||
\end{align\*}
|
||||
|
||||
<a id="org06b8843"></a>
|
||||
<a id="org1c8100c"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/preumont18_general_plant.png" caption="Figure 4: Block diagram of the control System" >}}
|
||||
|
||||
@@ -186,12 +186,12 @@ 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 [fig:cas_plot](#fig:cas_plot).
|
||||
A typical plot of \\(\sigma\_z(\omega)\\) is shown figure [5](#orgbde617f).
|
||||
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.
|
||||
|
||||
<a id="orgefc00fd"></a>
|
||||
<a id="orgbde617f"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/preumont18_cas_plot.png" caption="Figure 5: Error budget distribution in OL and CL for increasing gains" >}}
|
||||
|
||||
@@ -398,11 +398,11 @@ With:
|
||||
D\_i(\omega) = \frac{1}{1 - \omega^2/\omega\_i^2 + 2 j \xi\_i \omega/\omega\_i}
|
||||
\end{equation}
|
||||
|
||||
<a id="orgde5a280"></a>
|
||||
<a id="orgac9e4c8"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/preumont18_neglected_modes.png" caption="Figure 6: 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 [fig:neglected_modes](#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](#orgac9e4c8)).
|
||||
|
||||
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 \\]
|
||||
@@ -441,9 +441,9 @@ 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 [fig:collocated_control_frf](#fig:collocated_control_frf)).
|
||||
\\(G\_{kk}\\) is a monotonously increasing function of \\(\omega\\) (figure [7](#org6374ca8)).
|
||||
|
||||
<a id="orgc840265"></a>
|
||||
<a id="org6374ca8"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/preumont18_collocated_control_frf.png" caption="Figure 7: Open-Loop FRF of an undamped structure with collocated actuator/sensor pair" >}}
|
||||
|
||||
@@ -457,9 +457,9 @@ 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 [fig:collocated_zero](#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](#org55d54ab).
|
||||
|
||||
<a id="orgd860ef7"></a>
|
||||
<a id="org55d54ab"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/preumont18_collocated_zero.png" caption="Figure 8: Structure with collocated actuator and sensor" >}}
|
||||
|
||||
@@ -474,9 +474,9 @@ The open-loop poles are independant of the actuator and sensor configuration whi
|
||||
|
||||
</div>
|
||||
|
||||
By looking at figure [fig:collocated_control_frf](#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](#org6374ca8), 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="org245f75f"></a>
|
||||
<a id="org8cc5426"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/preumont18_alternating_p_z.png" caption="Figure 9: Bode plot of a lighly damped structure with collocated actuator and sensor" >}}
|
||||
|
||||
@@ -486,7 +486,7 @@ 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](#org245f75f). 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](#org8cc5426). 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.
|
||||
|
||||
|
||||
@@ -508,12 +508,12 @@ 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 [fig:voice_coil_schematic](#fig:voice_coil_schematic)):
|
||||
The system consists of (see figure [10](#orga882e0c)):
|
||||
|
||||
- A permanent magnet which produces a uniform flux density \\(B\\) normal to the gap
|
||||
- A coil which is free to move axially
|
||||
|
||||
<a id="org605d681"></a>
|
||||
<a id="orga882e0c"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/preumont18_voice_coil_schematic.png" caption="Figure 10: Physical principle of a voice coil transducer" >}}
|
||||
|
||||
@@ -551,9 +551,9 @@ 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 [fig:proof_mass_actuator](#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](#orgd7b8271)).
|
||||
|
||||
<a id="org5fa45df"></a>
|
||||
<a id="orgd7b8271"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/preumont18_proof_mass_actuator.png" caption="Figure 11: Proof-mass actuator" >}}
|
||||
|
||||
@@ -583,9 +583,9 @@ 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 [fig:proof_mass_tf](#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](#orgc8992eb)).
|
||||
|
||||
<a id="org0b93a83"></a>
|
||||
<a id="orgc8992eb"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/preumont18_proof_mass_tf.png" caption="Figure 12: Bode plot \\(F/i\\) of the proof-mass actuator" >}}
|
||||
|
||||
@@ -610,7 +610,7 @@ By using the two equations, we obtain:
|
||||
|
||||
Above the corner frequency, the gain of the geophone is equal to the transducer constant \\(T\\).
|
||||
|
||||
<a id="org45de7ce"></a>
|
||||
<a id="org2a6e175"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/preumont18_geophone.png" caption="Figure 13: Model of a geophone based on a voice coil transducer" >}}
|
||||
|
||||
@@ -619,9 +619,9 @@ 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 [fig:electro_mechanical_transducer](#fig:electro_mechanical_transducer).
|
||||
The consitutive behavior of a wide class of electromechanical transducers can be modelled as in figure [14](#org12ba88f).
|
||||
|
||||
<a id="orgdc255dc"></a>
|
||||
<a id="org12ba88f"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/preumont18_electro_mechanical_transducer.png" caption="Figure 14: Electrical analog representation of an electromechanical transducer" >}}
|
||||
|
||||
@@ -646,7 +646,7 @@ With:
|
||||
Equation \eqref{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 [fig:bridge_circuit](#fig:bridge_circuit) can be used.
|
||||
To do so, the bridge circuit as shown on figure [15](#orgdf63f4a) can be used.
|
||||
|
||||
We can show that
|
||||
|
||||
@@ -656,7 +656,7 @@ We can show that
|
||||
|
||||
which is indeed a linear function of the velocity \\(v\\) at the mechanical terminals.
|
||||
|
||||
<a id="orgab771fe"></a>
|
||||
<a id="orgdf63f4a"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/preumont18_bridge_circuit.png" caption="Figure 15: Bridge circuit for self-sensing actuation" >}}
|
||||
|
||||
@@ -664,9 +664,9 @@ 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 [fig:smart_materials](#fig:smart_materials) lists various effects that are observed in materials in response to various inputs.
|
||||
Figure [16](#orgc0d19b7) lists various effects that are observed in materials in response to various inputs.
|
||||
|
||||
<a id="orga77d2f2"></a>
|
||||
<a id="orgc0d19b7"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/preumont18_smart_materials.png" caption="Figure 16: Stimulus response relations indicating various effects in materials. The smart materials corresponds to the non-diagonal cells" >}}
|
||||
|
||||
@@ -761,7 +761,7 @@ 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 [fig:piezo_stack](#fig:piezo_stack)) of \\(n\\) disks of thickness \\(t\\) and cross section \\(A\\), the global constitutive equations of the transducer are obtained by integrating \eqref{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](#orgc13be77)) of \\(n\\) disks of thickness \\(t\\) and cross section \\(A\\), the global constitutive equations of the transducer are obtained by integrating \eqref{eq:piezo_eq_matrix_bis} over the volume of the transducer:
|
||||
|
||||
\begin{equation}
|
||||
\begin{bmatrix}Q\\\Delta\end{bmatrix}
|
||||
@@ -782,7 +782,7 @@ where
|
||||
- \\(C = \epsilon^T A n^2/l\\) is the capacitance of the transducer with no external load (\\(f = 0\\))
|
||||
- \\(K\_a = A/s^El\\) is the stiffness with short-circuited electrodes (\\(V = 0\\))
|
||||
|
||||
<a id="orge0cb54d"></a>
|
||||
<a id="orgc13be77"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/preumont18_piezo_stack.png" caption="Figure 17: Piezoelectric linear transducer" >}}
|
||||
|
||||
@@ -802,7 +802,7 @@ Equation \eqref{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 [fig:piezo_discrete](#fig:piezo_discrete).
|
||||
Let us write the total stored electromechanical energy of a discrete piezoelectric transducer as shown on figure [18](#org24eee83).
|
||||
|
||||
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
|
||||
|
||||
@@ -810,7 +810,7 @@ The total power delivered to the transducer is the sum of electric power \\(V i\
|
||||
dW = V i dt + f \dot{\Delta} dt = V dQ + f d\Delta
|
||||
\end{equation}
|
||||
|
||||
<a id="orge625816"></a>
|
||||
<a id="org24eee83"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/preumont18_piezo_discrete.png" caption="Figure 18: Discrete Piezoelectric Transducer" >}}
|
||||
|
||||
@@ -844,10 +844,10 @@ 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 [fig:piezo_stack_admittance](#fig:piezo_stack_admittance), where the piezoelectric transducer is assumed massless and is connected to a mass \\(M\\).
|
||||
Consider the system of figure [19](#org76e4915), 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="org9c326d7"></a>
|
||||
<a id="org76e4915"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/preumont18_piezo_stack_admittance.png" caption="Figure 19: Elementary dynamical model of the piezoelectric transducer" >}}
|
||||
|
||||
@@ -866,9 +866,9 @@ And one can see that
|
||||
\frac{z^2 - p^2}{z^2} = k^2
|
||||
\end{equation}
|
||||
|
||||
Equation \eqref{eq:distance_p_z} constitutes a practical way to determine the electromechanical coupling factor from the poles and zeros of the admittance measurement (figure [fig:piezo_admittance_curve](#fig:piezo_admittance_curve)).
|
||||
Equation \eqref{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](#orgba7797e)).
|
||||
|
||||
<a id="orgb11212e"></a>
|
||||
<a id="orgba7797e"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/preumont18_piezo_admittance_curve.png" caption="Figure 20: Typical admittance FRF of the transducer" >}}
|
||||
|
||||
@@ -1566,7 +1566,7 @@ 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 [fig:hac_lac_control](#fig:hac_lac_control).
|
||||
The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure [21](#org278a785).
|
||||
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:
|
||||
|
||||
@@ -1574,7 +1574,7 @@ This approach has the following advantages:
|
||||
- The active damping makes it easier to gain-stabilize the modes outside the bandwidth of the output loop (improved gain margin)
|
||||
- The larger damping of the modes within the controller bandwidth makes them more robust to the parmetric uncertainty (improved phase margin)
|
||||
|
||||
<a id="orgf36086a"></a>
|
||||
<a id="org278a785"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/preumont18_hac_lac_control.png" caption="Figure 21: Principle of the dual-loop HAC/LAC control" >}}
|
||||
|
||||
|
@@ -5,7 +5,7 @@ draft = false
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Reference Books]({{< relref "reference_books" >}})
|
||||
: [Reference Books]({{< relref "reference_books" >}}), [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting" >}})
|
||||
|
||||
Reference
|
||||
: <sup id="37468bbe5988cc7f4878fcd664d9cb7f"><a href="#schmidt14_desig_high_perfor_mechat_revis_edition" title="Schmidt, Schitter \& Rankers, The Design of High Performance Mechatronics - 2nd Revised Edition, Ios Press (2014).">(Schmidt {\it et al.}, 2014)</a></sup>
|
||||
@@ -29,7 +29,7 @@ Section 2.2.2 Force and Motion
|
||||
> One should however be aware that another non-destructive source of non-linearity is found in a tried important field of mechanics, called _kinematics_.
|
||||
> The relation between angles and positions is often non-linear in such a mechanism, because of the changing angles, and controlling these often requires special precautions to overcome the inherent non-linearities by linearisation around actual position and adapting the optimal settings of the controller to each position.
|
||||
|
||||
<a id="org462f9a1"></a>
|
||||
<a id="org8d0a076"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/schmidt14_high_low_freq_regions.png" caption="Figure 1: Stabiliby condition and robustness of a feedback controlled system. The desired shape of these curves guide the control design by optimising the lvels and sloppes of the amplitude Bode-plot at low and high frequencies for suppression of the disturbances and of the base Bode-plot in the cross-over frequency region. This is called **loop shaping design**" >}}
|
||||
|
||||
|
@@ -63,7 +63,7 @@ Year
|
||||
|
||||
## Introduction {#introduction}
|
||||
|
||||
<a id="orga7066d6"></a>
|
||||
<a id="org5d595bf"></a>
|
||||
|
||||
|
||||
### The Process of Control System Design {#the-process-of-control-system-design}
|
||||
@@ -234,7 +234,7 @@ Notations used throughout this note are summarized in tables [table:notatio
|
||||
|
||||
## Classical Feedback Control {#classical-feedback-control}
|
||||
|
||||
<a id="orgaabb1e5"></a>
|
||||
<a id="orgeb70e42"></a>
|
||||
|
||||
|
||||
### Frequency Response {#frequency-response}
|
||||
@@ -283,7 +283,7 @@ Thus, the input to the plant is \\(u = K(s) (r-y-n)\\).
|
||||
The objective of control is to manipulate \\(u\\) (design \\(K\\)) such that the control error \\(e\\) remains small in spite of disturbances \\(d\\).
|
||||
The control error is defined as \\(e = y-r\\).
|
||||
|
||||
<a id="org3b71acb"></a>
|
||||
<a id="orgf5eda9f"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_classical_feedback_alt.png" caption="Figure 1: Configuration for one degree-of-freedom control" >}}
|
||||
|
||||
@@ -595,7 +595,7 @@ 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 (Fig. [fig:classical_feedback_2dof_alt](#fig:classical_feedback_2dof_alt)), rather than operating on their difference \\(r - y\_m\\).
|
||||
|
||||
<a id="org9265d45"></a>
|
||||
<a id="orgb16dfc9"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_classical_feedback_2dof_alt.png" caption="Figure 2: 2 degrees-of-freedom control architecture" >}}
|
||||
|
||||
@@ -604,7 +604,7 @@ The controller can be slit into two separate blocks (Fig. [fig:classical_fe
|
||||
- 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
|
||||
|
||||
<a id="org0e3d8d7"></a>
|
||||
<a id="org66943f1"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_classical_feedback_sep.png" caption="Figure 3: 2 degrees-of-freedom control architecture with two separate blocs" >}}
|
||||
|
||||
@@ -673,7 +673,7 @@ With (see Fig. [fig:performance_weigth](#fig:performance_weigth)):
|
||||
|
||||
</div>
|
||||
|
||||
<a id="org0656ee4"></a>
|
||||
<a id="orgac866a9"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_weight_first_order.png" caption="Figure 4: Inverse of performance weight" >}}
|
||||
|
||||
@@ -697,7 +697,7 @@ After selecting the form of \\(N\\) and the weights, the \\(\hinf\\) optimal con
|
||||
|
||||
## Introduction to Multivariable Control {#introduction-to-multivariable-control}
|
||||
|
||||
<a id="org25e187e"></a>
|
||||
<a id="org62230ab"></a>
|
||||
|
||||
|
||||
### Introduction {#introduction}
|
||||
@@ -740,7 +740,7 @@ For negative feedback system (Fig. [fig:classical_feedback_bis](#fig:classi
|
||||
- \\(S \triangleq (I + L)^{-1}\\) is the transfer function from \\(d\_1\\) to \\(y\\)
|
||||
- \\(T \triangleq L(I + L)^{-1}\\) is the transfer function from \\(r\\) to \\(y\\)
|
||||
|
||||
<a id="org10be303"></a>
|
||||
<a id="org409aae2"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_classical_feedback_bis.png" caption="Figure 5: Conventional negative feedback control system" >}}
|
||||
|
||||
@@ -1055,7 +1055,7 @@ The **structured singular value** \\(\mu\\) is a tool for analyzing the effects
|
||||
|
||||
The general control problem formulation is represented in Fig. [fig:general_control_names](#fig:general_control_names).
|
||||
|
||||
<a id="org410e618"></a>
|
||||
<a id="orgfca0967"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_general_control_names.png" caption="Figure 6: General control configuration" >}}
|
||||
|
||||
@@ -1085,7 +1085,7 @@ We consider:
|
||||
- The weighted or normalized exogenous inputs \\(w\\) (where \\(\tilde{w} = W\_w w\\) consists of the "physical" signals entering the system)
|
||||
- The weighted or normalized controlled outputs \\(z = W\_z \tilde{z}\\) (where \\(\tilde{z}\\) often consists of the control error \\(y-r\\) and the manipulated input \\(u\\))
|
||||
|
||||
<a id="org98354a0"></a>
|
||||
<a id="orgea1a525"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_general_plant_weights.png" caption="Figure 7: General Weighted Plant" >}}
|
||||
|
||||
@@ -1128,7 +1128,7 @@ where \\(F\_l(P, K)\\) denotes a **lower linear fractional transformation** (LFT
|
||||
|
||||
The general control configuration may be extended to include model uncertainty as shown in Fig. [fig:general_config_model_uncertainty](#fig:general_config_model_uncertainty).
|
||||
|
||||
<a id="orgaee1f77"></a>
|
||||
<a id="org9ebb3a4"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_general_control_Mdelta.png" caption="Figure 8: General control configuration for the case with model uncertainty" >}}
|
||||
|
||||
@@ -1156,7 +1156,7 @@ MIMO systems are often **more sensitive to uncertainty** than SISO systems.
|
||||
|
||||
## Elements of Linear System Theory {#elements-of-linear-system-theory}
|
||||
|
||||
<a id="orgb820714"></a>
|
||||
<a id="org0d33f26"></a>
|
||||
|
||||
|
||||
### System Descriptions {#system-descriptions}
|
||||
@@ -1442,7 +1442,7 @@ RHP-zeros therefore imply high gain instability.
|
||||
|
||||
### Internal Stability of Feedback Systems {#internal-stability-of-feedback-systems}
|
||||
|
||||
<a id="orgb1e4209"></a>
|
||||
<a id="org3b4a8f6"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_classical_feedback_stability.png" caption="Figure 9: Block diagram used to check internal stability" >}}
|
||||
|
||||
@@ -1589,7 +1589,7 @@ It may be shown that the Hankel norm is equal to \\(\left\\|G(s)\right\\|\_H = \
|
||||
|
||||
## Limitations on Performance in SISO Systems {#limitations-on-performance-in-siso-systems}
|
||||
|
||||
<a id="org76a7a2f"></a>
|
||||
<a id="orgaa2db6e"></a>
|
||||
|
||||
|
||||
### Input-Output Controllability {#input-output-controllability}
|
||||
@@ -1981,7 +1981,7 @@ Uncertainty in the crossover frequency region can result in poor performance and
|
||||
|
||||
### Summary: Controllability Analysis with Feedback Control {#summary-controllability-analysis-with-feedback-control}
|
||||
|
||||
<a id="org4ab6880"></a>
|
||||
<a id="org19dc4dc"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_classical_feedback_meas.png" caption="Figure 10: Feedback control system" >}}
|
||||
|
||||
@@ -2010,7 +2010,7 @@ In summary:
|
||||
Sometimes, the disturbances are so large that we hit input saturation or the required bandwidth is not achievable. To avoid the latter problem, we must at least require that the effect of the disturbance is less than \\(1\\) at frequencies beyond the bandwidth:
|
||||
\\[ \abs{G\_d(j\w)} < 1 \quad \forall \w \geq \w\_c \\]
|
||||
|
||||
<a id="orga143a9d"></a>
|
||||
<a id="orgbee9012"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_margin_requirements.png" caption="Figure 11: Illustration of controllability requirements" >}}
|
||||
|
||||
@@ -2032,7 +2032,7 @@ The rules may be used to **determine whether or not a given plant is controllabl
|
||||
|
||||
## Limitations on Performance in MIMO Systems {#limitations-on-performance-in-mimo-systems}
|
||||
|
||||
<a id="org6b25e5b"></a>
|
||||
<a id="org5b47882"></a>
|
||||
|
||||
|
||||
### Introduction {#introduction}
|
||||
@@ -2343,7 +2343,7 @@ We here focus on input and output uncertainty.
|
||||
In multiplicative form, the input and output uncertainties are given by (see Fig. [fig:input_output_uncertainty](#fig:input_output_uncertainty)):
|
||||
\\[ G^\prime = (I + E\_O) G (I + E\_I) \\]
|
||||
|
||||
<a id="org367c804"></a>
|
||||
<a id="orgc33ff63"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_input_output_uncertainty.png" caption="Figure 12: Plant with multiplicative input and output uncertainty" >}}
|
||||
|
||||
@@ -2479,7 +2479,7 @@ However, the situation is usually the opposite with model uncertainty because fo
|
||||
|
||||
## Uncertainty and Robustness for SISO Systems {#uncertainty-and-robustness-for-siso-systems}
|
||||
|
||||
<a id="org80d55a0"></a>
|
||||
<a id="org395bba3"></a>
|
||||
|
||||
|
||||
### Introduction to Robustness {#introduction-to-robustness}
|
||||
@@ -2553,7 +2553,7 @@ which may be represented by the diagram in Fig. [fig:input_uncertainty_set]
|
||||
|
||||
</div>
|
||||
|
||||
<a id="org865770b"></a>
|
||||
<a id="orga0f232e"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_input_uncertainty_set.png" caption="Figure 13: Plant with multiplicative uncertainty" >}}
|
||||
|
||||
@@ -2607,7 +2607,7 @@ To illustrate how parametric uncertainty translate into frequency domain uncerta
|
||||
In general, these uncertain regions have complicated shapes and complex mathematical descriptions
|
||||
- **Step 2**. We therefore approximate such complex regions as discs, resulting in a **complex additive uncertainty description**
|
||||
|
||||
<a id="org168b9ff"></a>
|
||||
<a id="org87a3682"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_uncertainty_region.png" caption="Figure 14: Uncertainty regions of the Nyquist plot at given frequencies" >}}
|
||||
|
||||
@@ -2630,7 +2630,7 @@ At each frequency, all possible \\(\Delta(j\w)\\) "generates" a disc-shaped regi
|
||||
|
||||
</div>
|
||||
|
||||
<a id="org46ced8b"></a>
|
||||
<a id="org5bedb67"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_uncertainty_disc_generated.png" caption="Figure 15: Disc-shaped uncertainty regions generated by complex additive uncertainty" >}}
|
||||
|
||||
@@ -2687,7 +2687,7 @@ To derive \\(w\_I(s)\\), we then try to find a simple weight so that \\(\abs{w\_
|
||||
|
||||
</div>
|
||||
|
||||
<a id="org797da76"></a>
|
||||
<a id="org5399ef5"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_uncertainty_weight.png" caption="Figure 16: Relative error for 27 combinations of \\(k,\ \tau\\) and \\(\theta\\). Solid and dashed lines: two weights \\(\abs{w\_I}\\)" >}}
|
||||
|
||||
@@ -2726,7 +2726,7 @@ The magnitude of the relative uncertainty caused by neglecting the dynamics in \
|
||||
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 (Fig. [fig:neglected_time_delay](#fig:neglected_time_delay)). If we consider all \\(\theta \in [0, \theta\_{\text{max}}]\\) then:
|
||||
\\[ 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} \\]
|
||||
|
||||
<a id="org8ddf130"></a>
|
||||
<a id="org3b8e0bc"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_neglected_time_delay.png" caption="Figure 17: Neglected time delay" >}}
|
||||
|
||||
@@ -2736,7 +2736,7 @@ Let \\(f(s) = e^{-\theta\_p s}\\), where \\(0 \le \theta\_p \le \theta\_{\text{m
|
||||
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)\\) (Fig. [fig:neglected_first_order_lag](#fig:neglected_first_order_lag)) can be represented by a rational transfer function with \\(\abs{w\_I(j\w)} = l\_I(\w)\\) where
|
||||
\\[ w\_I(s) = \frac{\tau\_{\text{max}} s}{\tau\_{\text{max}} s + 1} \\]
|
||||
|
||||
<a id="orge3ddb3c"></a>
|
||||
<a id="orgdce561a"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_neglected_first_order_lag.png" caption="Figure 18: Neglected first-order lag uncertainty" >}}
|
||||
|
||||
@@ -2753,7 +2753,7 @@ However, as shown in Fig. [fig:lag_delay_uncertainty](#fig:lag_delay_uncert
|
||||
|
||||
It is suggested to start with the simple weight and then if needed, to try the higher order weight.
|
||||
|
||||
<a id="orgb652b95"></a>
|
||||
<a id="org4320c38"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_lag_delay_uncertainty.png" caption="Figure 19: Multiplicative weight for gain and delay uncertainty" >}}
|
||||
|
||||
@@ -2793,7 +2793,7 @@ We use the Nyquist stability condition to test for robust stability of the close
|
||||
&\Longleftrightarrow \quad L\_p \ \text{should not encircle -1}, \ \forall L\_p
|
||||
\end{align\*}
|
||||
|
||||
<a id="org0fda45b"></a>
|
||||
<a id="orgbd21fcf"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_input_uncertainty_set_feedback.png" caption="Figure 20: Feedback system with multiplicative uncertainty" >}}
|
||||
|
||||
@@ -2809,7 +2809,7 @@ Encirclements are avoided if none of the discs cover \\(-1\\), and we get:
|
||||
&\Leftrightarrow \quad \abs{w\_I T} < 1, \ \forall\w \\\\\\
|
||||
\end{align\*}
|
||||
|
||||
<a id="org4ead586"></a>
|
||||
<a id="org68e60a3"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_nyquist_uncertainty.png" caption="Figure 21: Nyquist plot of \\(L\_p\\) for robust stability" >}}
|
||||
|
||||
@@ -2847,7 +2847,7 @@ And we obtain the same condition as before.
|
||||
We will derive a corresponding RS-condition for feedback system with inverse multiplicative uncertainty (Fig. [fig:inverse_uncertainty_set](#fig:inverse_uncertainty_set)) in which
|
||||
\\[ G\_p = G(1 + w\_{iI}(s) \Delta\_{iI})^{-1} \\]
|
||||
|
||||
<a id="orgaad9987"></a>
|
||||
<a id="orgaad840d"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_inverse_uncertainty_set.png" caption="Figure 22: Feedback system with inverse multiplicative uncertainty" >}}
|
||||
|
||||
@@ -2899,7 +2899,7 @@ The condition for nominal performance when considering performance in terms of t
|
||||
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 Fig. [fig:nyquist_performance_condition](#fig:nyquist_performance_condition).
|
||||
|
||||
<a id="org8e66342"></a>
|
||||
<a id="org2f13fac"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_nyquist_performance_condition.png" caption="Figure 23: Nyquist plot illustration of the nominal performance condition \\(\abs{w\_P} < \abs{1 + L}\\)" >}}
|
||||
|
||||
@@ -2924,7 +2924,7 @@ Let's consider the case of multiplicative uncertainty as shown on Fig. [fig
|
||||
The robust performance corresponds to requiring \\(\abs{\hat{y}/d}<1\ \forall \Delta\_I\\) and the set of possible loop transfer functions is
|
||||
\\[ L\_p = G\_p K = L (1 + w\_I \Delta\_I) = L + w\_I L \Delta\_I \\]
|
||||
|
||||
<a id="org3ca06cf"></a>
|
||||
<a id="orgbf2f9c6"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_input_uncertainty_set_feedback_weight_bis.png" caption="Figure 24: Diagram for robust performance with multiplicative uncertainty" >}}
|
||||
|
||||
@@ -3090,7 +3090,7 @@ with \\(\Phi(s) \triangleq (sI - A)^{-1}\\).
|
||||
|
||||
This is illustrated in the block diagram of Fig. [fig:uncertainty_state_a_matrix](#fig:uncertainty_state_a_matrix), which is in the form of an inverse additive perturbation.
|
||||
|
||||
<a id="orgd286b2a"></a>
|
||||
<a id="orga439362"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_uncertainty_state_a_matrix.png" caption="Figure 25: Uncertainty in state space A-matrix" >}}
|
||||
|
||||
@@ -3108,7 +3108,7 @@ We also derived a condition for robust performance with multiplicative uncertain
|
||||
|
||||
## Robust Stability and Performance Analysis {#robust-stability-and-performance-analysis}
|
||||
|
||||
<a id="orgb076a9b"></a>
|
||||
<a id="org630c4f9"></a>
|
||||
|
||||
|
||||
### General Control Configuration with Uncertainty {#general-control-configuration-with-uncertainty}
|
||||
@@ -3119,13 +3119,13 @@ where each \\(\Delta\_i\\) represents a **specific source of uncertainty**, e.g.
|
||||
|
||||
If we also pull out the controller \\(K\\), we get the generalized plant \\(P\\) as shown in Fig. [fig:general_control_delta](#fig:general_control_delta). This form is useful for controller synthesis.
|
||||
|
||||
<a id="org0853688"></a>
|
||||
<a id="orge8af5ee"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_general_control_delta.png" caption="Figure 26: 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 Fig. [fig:general_control_Ndelta](#fig:general_control_Ndelta).
|
||||
|
||||
<a id="orgc524251"></a>
|
||||
<a id="orgad34338"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_general_control_Ndelta.png" caption="Figure 27: \\(N\Delta\text{-structure}\\) for robust performance analysis" >}}
|
||||
|
||||
@@ -3145,7 +3145,7 @@ Similarly, the uncertain closed-loop transfer function from \\(w\\) to \\(z\\),
|
||||
|
||||
To analyze robust stability of \\(F\\), we can rearrange the system into the \\(M\Delta\text{-structure}\\) shown in Fig. [fig:general_control_Mdelta_bis](#fig:general_control_Mdelta_bis) where \\(M = N\_{11}\\) is the transfer function from the output to the input of the perturbations.
|
||||
|
||||
<a id="orge0e68f2"></a>
|
||||
<a id="org5ac3a4a"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_general_control_Mdelta_bis.png" caption="Figure 28: \\(M\Delta\text{-structure}\\) for robust stability analysis" >}}
|
||||
|
||||
@@ -3197,7 +3197,7 @@ Three common forms of **feedforward unstructured uncertainty** are shown Fig.&nb
|
||||
|
||||
|  |  |  |
|
||||
|----------------------------------------------------|----------------------------------------------------------|-----------------------------------------------------------|
|
||||
| <a id="org94556ee"></a> Additive uncertainty | <a id="org205e138"></a> Multiplicative input uncertainty | <a id="org884d99b"></a> Multiplicative output uncertainty |
|
||||
| <a id="org05d42ff"></a> Additive uncertainty | <a id="orgae1758d"></a> Multiplicative input uncertainty | <a id="org2db9e52"></a> Multiplicative output uncertainty |
|
||||
|
||||
In Fig. [fig:feedback_uncertainty](#fig:feedback_uncertainty), three **feedback or inverse unstructured uncertainty** forms are shown: inverse additive uncertainty, inverse multiplicative input uncertainty and inverse multiplicative output uncertainty.
|
||||
|
||||
@@ -3220,7 +3220,7 @@ In Fig. [fig:feedback_uncertainty](#fig:feedback_uncertainty), three **feed
|
||||
|
||||
|  |  |  |
|
||||
|--------------------------------------------------------|------------------------------------------------------------------|-------------------------------------------------------------------|
|
||||
| <a id="org17a4e6d"></a> Inverse additive uncertainty | <a id="org2765e1d"></a> Inverse multiplicative input uncertainty | <a id="org33356e1"></a> Inverse multiplicative output uncertainty |
|
||||
| <a id="org5ea3793"></a> Inverse additive uncertainty | <a id="org31e7d89"></a> Inverse multiplicative input uncertainty | <a id="org00c8166"></a> Inverse multiplicative output uncertainty |
|
||||
|
||||
|
||||
##### Lumping uncertainty into a single perturbation {#lumping-uncertainty-into-a-single-perturbation}
|
||||
@@ -3290,7 +3290,7 @@ where \\(r\_0\\) is the relative uncertainty at steady-state, \\(1/\tau\\) is th
|
||||
Let's consider the feedback system with multiplicative input uncertainty \\(\Delta\_I\\) shown Fig. [fig:input_uncertainty_set_feedback_weight](#fig:input_uncertainty_set_feedback_weight).
|
||||
\\(W\_I\\) is a normalization weight for the uncertainty and \\(W\_P\\) is a performance weight.
|
||||
|
||||
<a id="org2ebb26f"></a>
|
||||
<a id="orge5740dc"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_input_uncertainty_set_feedback_weight.png" caption="Figure 29: System with multiplicative input uncertainty and performance measured at the output" >}}
|
||||
|
||||
@@ -3450,7 +3450,7 @@ Where \\(G = M\_l^{-1} N\_l\\) is a left coprime factorization of the nominal pl
|
||||
|
||||
This uncertainty description is surprisingly **general**, it allows both zeros and poles to cross into the right-half plane, and has proven to be very useful in applications.
|
||||
|
||||
<a id="org71a706b"></a>
|
||||
<a id="org6fb438e"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_coprime_uncertainty.png" caption="Figure 30: Coprime Uncertainty" >}}
|
||||
|
||||
@@ -3482,7 +3482,7 @@ where \\(d\_i\\) is a scalar and \\(I\_i\\) is an identity matrix of the same di
|
||||
Now rescale the inputs and outputs of \\(M\\) and \\(\Delta\\) by inserting the matrices \\(D\\) and \\(D^{-1}\\) on both sides as shown in Fig. [fig:block_diagonal_scalings](#fig:block_diagonal_scalings).
|
||||
This clearly has no effect on stability.
|
||||
|
||||
<a id="org949fb62"></a>
|
||||
<a id="org7b2b472"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_block_diagonal_scalings.png" caption="Figure 31: Use of block-diagonal scalings, \\(\Delta D = D \Delta\\)" >}}
|
||||
|
||||
@@ -3798,7 +3798,7 @@ with the decoupling controller we have:
|
||||
\\[ \bar{\sigma}(N\_{22}) = \bar{\sigma}(w\_P S) = \left|\frac{s/2 + 0.05}{s + 0.7}\right| \\]
|
||||
and we see from Fig. [fig:mu_plots_distillation](#fig:mu_plots_distillation) that the NP-condition is satisfied.
|
||||
|
||||
<a id="orga318a7a"></a>
|
||||
<a id="org79285a0"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_mu_plots_distillation.png" caption="Figure 32: \\(\mu\text{-plots}\\) for distillation process with decoupling controller" >}}
|
||||
|
||||
@@ -3921,7 +3921,7 @@ The latter is an attempt to "flatten out" \\(\mu\\).
|
||||
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 Fig. [fig:weights_distillation](#fig:weights_distillation).
|
||||
|
||||
<a id="orgd273607"></a>
|
||||
<a id="org8fb66bd"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_weights_distillation.png" caption="Figure 33: Uncertainty and performance weights" >}}
|
||||
|
||||
@@ -3944,11 +3944,11 @@ The scaling matrix \\(D\\) for \\(DND^{-1}\\) then has the structure \\(D = \tex
|
||||
- Iteration No. 3.
|
||||
Step 1: The \\(\mathcal{H}\_\infty\\) norm is only slightly reduced. We thus decide the stop the iterations.
|
||||
|
||||
<a id="org10a3970"></a>
|
||||
<a id="org5f826e8"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_dk_iter_mu.png" caption="Figure 34: Change in \\(\mu\\) during DK-iteration" >}}
|
||||
|
||||
<a id="org400285f"></a>
|
||||
<a id="orga6732c2"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_dk_iter_d_scale.png" caption="Figure 35: Change in D-scale \\(d\_1\\) during DK-iteration" >}}
|
||||
|
||||
@@ -3956,13 +3956,13 @@ The final \\(\mu\text{-curves}\\) for NP, RS and RP with the controller \\(K\_3\
|
||||
The objectives of RS and NP are easily satisfied.
|
||||
The peak value of \\(\mu\\) is just slightly over 1, so the performance specification \\(\bar{\sigma}(w\_P S\_p) < 1\\) is almost satisfied for all possible plants.
|
||||
|
||||
<a id="org519a9ca"></a>
|
||||
<a id="org7cbd772"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_mu_plot_optimal_k3.png" caption="Figure 36: \\(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 Fig. [fig:perturb_s_k3](#fig:perturb_s_k3).
|
||||
|
||||
<a id="orgfcb21f2"></a>
|
||||
<a id="orge989e28"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_perturb_s_k3.png" caption="Figure 37: Perturbed sensitivity functions \\(\bar{\sigma}(S^\prime)\\) using \\(\mu\\) \"optimal\" controller \\(K\_3\\). Lower solid line: nominal plant. Upper solid line: worst-case plant" >}}
|
||||
|
||||
@@ -4017,7 +4017,7 @@ If resulting control performance is not satisfactory, one may switch to the seco
|
||||
|
||||
## Controller Design {#controller-design}
|
||||
|
||||
<a id="orga616dec"></a>
|
||||
<a id="orgcb8e2f1"></a>
|
||||
|
||||
|
||||
### Trade-offs in MIMO Feedback Design {#trade-offs-in-mimo-feedback-design}
|
||||
@@ -4037,7 +4037,7 @@ We have the following important relationships:
|
||||
\end{align}
|
||||
\end{subequations}
|
||||
|
||||
<a id="org8d4f22a"></a>
|
||||
<a id="org68d662f"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_classical_feedback_small.png" caption="Figure 38: One degree-of-freedom feedback configuration" >}}
|
||||
|
||||
@@ -4079,7 +4079,7 @@ Thus, over specified frequency ranges, it is relatively easy to approximate the
|
||||
|
||||
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 Fig. [fig:design_trade_off_mimo_gk](#fig:design_trade_off_mimo_gk).
|
||||
|
||||
<a id="org6e3f117"></a>
|
||||
<a id="org058fc8c"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_design_trade_off_mimo_gk.png" caption="Figure 39: Design trade-offs for the multivariable loop transfer function \\(GK\\)" >}}
|
||||
|
||||
@@ -4136,7 +4136,7 @@ The solution to the LQG problem is then found by replacing \\(x\\) by \\(\hat{x}
|
||||
|
||||
We therefore see that the LQG problem and its solution can be separated into two distinct parts as illustrated in Fig. [fig:lqg_separation](#fig:lqg_separation): the optimal state feedback and the optimal state estimator (the Kalman filter).
|
||||
|
||||
<a id="org6f521b9"></a>
|
||||
<a id="org7162554"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_lqg_separation.png" caption="Figure 40: The separation theorem" >}}
|
||||
|
||||
@@ -4186,7 +4186,7 @@ Where \\(Y\\) is the unique positive-semi definite solution of the algebraic Ric
|
||||
|
||||
</div>
|
||||
|
||||
<a id="orgf0f14d9"></a>
|
||||
<a id="orgaa1f9d2"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_lqg_kalman_filter.png" caption="Figure 41: The LQG controller and noisy plant" >}}
|
||||
|
||||
@@ -4207,7 +4207,7 @@ It has the same degree (number of poles) as the plant.<br />
|
||||
|
||||
For the LQG-controller, as shown on Fig. [fig:lqg_kalman_filter](#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 Fig. [fig:lqg_integral](#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="orgb7cfb99"></a>
|
||||
<a id="orga77c0d2"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_lqg_integral.png" caption="Figure 42: LQG controller with integral action and reference input" >}}
|
||||
|
||||
@@ -4220,18 +4220,18 @@ Their main limitation is that they can only be applied to minimum phase plants.
|
||||
|
||||
### \\(\htwo\\) and \\(\hinf\\) Control {#htwo--and--hinf--control}
|
||||
|
||||
<a id="org6da7635"></a>
|
||||
<a id="org2f8c60b"></a>
|
||||
|
||||
|
||||
#### General Control Problem Formulation {#general-control-problem-formulation}
|
||||
|
||||
<a id="org1448cec"></a>
|
||||
<a id="orgbe54b98"></a>
|
||||
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 Fig. [fig:general_control](#fig:general_control).
|
||||
|
||||
<a id="org3b91a51"></a>
|
||||
<a id="orgf28c426"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_general_control.png" caption="Figure 43: General control configuration" >}}
|
||||
|
||||
@@ -4482,7 +4482,7 @@ The elements of the generalized plant are
|
||||
\end{array}
|
||||
\end{equation\*}
|
||||
|
||||
<a id="org35551f8"></a>
|
||||
<a id="org3f11e63"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_mixed_sensitivity_dist_rejection.png" caption="Figure 44: \\(S/KS\\) mixed-sensitivity optimization in standard form (regulation)" >}}
|
||||
|
||||
@@ -4491,7 +4491,7 @@ 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 Fig. [fig:mixed_sensitivity_dist_rejection](#fig:mixed_sensitivity_dist_rejection), we have that \\(z\_1 = W\_1 S w\\) and \\(z\_2 = W\_2 KS w\\).
|
||||
|
||||
<a id="org55460a0"></a>
|
||||
<a id="org758eceb"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_mixed_sensitivity_ref_tracking.png" caption="Figure 45: \\(S/KS\\) mixed-sensitivity optimization in standard form (tracking)" >}}
|
||||
|
||||
@@ -4515,7 +4515,7 @@ The elements of the generalized plant are
|
||||
\end{array}
|
||||
\end{equation\*}
|
||||
|
||||
<a id="org007c976"></a>
|
||||
<a id="org68fca49"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_mixed_sensitivity_s_t.png" caption="Figure 46: \\(S/T\\) mixed-sensitivity optimization in standard form" >}}
|
||||
|
||||
@@ -4543,7 +4543,7 @@ The focus of attention has moved to the size of signals and away from the size a
|
||||
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 Fig. [fig:input_uncertainty_hinf](#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="orgabff04a"></a>
|
||||
<a id="orge5eb5e5"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_input_uncertainty_hinf.png" caption="Figure 47: Multiplicative dynamic uncertainty model" >}}
|
||||
|
||||
@@ -4565,7 +4565,7 @@ The problem can be cast as a standard \\(\hinf\\) optimization in the general co
|
||||
w = \begin{bmatrix}d\\r\\n\end{bmatrix},\ z = \begin{bmatrix}z\_1\\z\_2\end{bmatrix}, \ v = \begin{bmatrix}r\_s\\y\_m\end{bmatrix},\ u = u
|
||||
\end{equation\*}
|
||||
|
||||
<a id="org5056f35"></a>
|
||||
<a id="org896492e"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_hinf_signal_based.png" caption="Figure 48: A signal-based \\(\hinf\\) control problem" >}}
|
||||
|
||||
@@ -4576,7 +4576,7 @@ This problem is a non-standard \\(\hinf\\) optimization.
|
||||
It is a robust performance problem for which the \\(\mu\text{-synthesis}\\) procedure can be applied where we require the structured singular value:
|
||||
\\[ \mu(M(j\omega)) < 1, \quad \forall\omega \\]
|
||||
|
||||
<a id="org7befd92"></a>
|
||||
<a id="org01deb15"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_hinf_signal_based_uncertainty.png" caption="Figure 49: A signal-based \\(\hinf\\) control problem with input multiplicative uncertainty" >}}
|
||||
|
||||
@@ -4634,7 +4634,7 @@ For the perturbed feedback system of Fig. [fig:coprime_uncertainty_bis](#fi
|
||||
|
||||
Notice that \\(\gamma\\) is the \\(\hinf\\) norm from \\(\phi\\) to \\(\begin{bmatrix}u\\y\end{bmatrix}\\) and \\((I-GK)^{-1}\\) is the sensitivity function for this positive feedback arrangement.
|
||||
|
||||
<a id="org4f3b2f4"></a>
|
||||
<a id="org5d52156"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_coprime_uncertainty_bis.png" caption="Figure 50: \\(\hinf\\) robust stabilization problem" >}}
|
||||
|
||||
@@ -4681,7 +4681,7 @@ It is important to emphasize that since we can compute \\(\gamma\_\text{min}\\)
|
||||
|
||||
#### A Systematic \\(\hinf\\) Loop-Shaping Design Procedure {#a-systematic--hinf--loop-shaping-design-procedure}
|
||||
|
||||
<a id="org929fa3b"></a>
|
||||
<a id="orge4e2f7e"></a>
|
||||
Robust stabilization alone is not much used in practice because the designer is not able to specify any performance requirements.
|
||||
|
||||
To do so, **pre and post compensators** are used to **shape the open-loop singular values** prior to robust stabilization of the "shaped" plant.
|
||||
@@ -4694,7 +4694,7 @@ If \\(W\_1\\) and \\(W\_2\\) are the pre and post compensators respectively, the
|
||||
|
||||
as shown in Fig. [fig:shaped_plant](#fig:shaped_plant).
|
||||
|
||||
<a id="orgbc1e59e"></a>
|
||||
<a id="org1f55ae2"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_shaped_plant.png" caption="Figure 51: The shaped plant and controller" >}}
|
||||
|
||||
@@ -4731,7 +4731,7 @@ Systematic procedure for \\(\hinf\\) loop-shaping design:
|
||||
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\\)
|
||||
|
||||
<a id="org83cf8d8"></a>
|
||||
<a id="org50d434b"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_shapping_practical_implementation.png" caption="Figure 52: A practical implementation of the loop-shaping controller" >}}
|
||||
|
||||
@@ -4757,7 +4757,7 @@ But in cases where stringent time-domain specifications are set on the output re
|
||||
A general two degrees-of-freedom feedback control scheme is depicted in Fig. [fig:classical_feedback_2dof_simple](#fig:classical_feedback_2dof_simple).
|
||||
The commands and feedbacks enter the controller separately and are independently processed.
|
||||
|
||||
<a id="org8f1d974"></a>
|
||||
<a id="orgbb5e97f"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_classical_feedback_2dof_simple.png" caption="Figure 53: General two degrees-of-freedom feedback control scheme" >}}
|
||||
|
||||
@@ -4768,7 +4768,7 @@ The design problem is illustrated in Fig. [fig:coprime_uncertainty_hinf](#f
|
||||
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**.
|
||||
|
||||
<a id="orgd00d786"></a>
|
||||
<a id="orgce4d7f1"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_coprime_uncertainty_hinf.png" caption="Figure 54: Two degrees-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping design problem" >}}
|
||||
|
||||
@@ -4793,7 +4793,7 @@ The main steps required to synthesize a two degrees-of-freedom \\(\mathcal{H}\_\
|
||||
|
||||
The final two degrees-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping controller is illustrated in Fig. [fig:hinf_synthesis_2dof](#fig:hinf_synthesis_2dof).
|
||||
|
||||
<a id="org3d681ec"></a>
|
||||
<a id="orgac8907e"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_hinf_synthesis_2dof.png" caption="Figure 55: Two degrees-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping controller" >}}
|
||||
|
||||
@@ -4865,7 +4865,7 @@ where \\(u\_a\\) is the **actual plant input**, that is the measurement at the *
|
||||
|
||||
The situation is illustrated in Fig. [fig:weight_anti_windup](#fig:weight_anti_windup), where the actuators are each modeled by a unit gain and a saturation.
|
||||
|
||||
<a id="org3867b27"></a>
|
||||
<a id="orgef9fa56"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_weight_anti_windup.png" caption="Figure 56: Self-conditioned weight \\(W\_1\\)" >}}
|
||||
|
||||
@@ -4913,14 +4913,14 @@ Moreover, one should be careful about combining controller synthesis and analysi
|
||||
|
||||
## Controller Structure Design {#controller-structure-design}
|
||||
|
||||
<a id="org6fc0469"></a>
|
||||
<a id="orgfc18612"></a>
|
||||
|
||||
|
||||
### Introduction {#introduction}
|
||||
|
||||
In previous sections, we considered the general problem formulation in Fig. [fig:general_control_names_bis](#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="org366605e"></a>
|
||||
<a id="orga10d2bd"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_general_control_names_bis.png" caption="Figure 57: General Control Configuration" >}}
|
||||
|
||||
@@ -4955,7 +4955,7 @@ The reference value \\(r\\) is usually set at some higher layer in the control h
|
||||
|
||||
Additional layers are possible, as is illustrated in Fig. [fig:control_system_hierarchy](#fig:control_system_hierarchy) which shows a typical control hierarchy for a chemical plant.
|
||||
|
||||
<a id="org42e952b"></a>
|
||||
<a id="org73996e3"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_system_hierarchy.png" caption="Figure 58: Typical control system hierarchy in a chemical plant" >}}
|
||||
|
||||
@@ -4977,7 +4977,7 @@ However, this solution is normally not used for a number a reasons, included the
|
||||
|
||||
|  |  |  |
|
||||
|--------------------------------------------------|--------------------------------------------------------------------------------|-------------------------------------------------------------|
|
||||
| <a id="org6986695"></a> Open loop optimization | <a id="orgaae7402"></a> Closed-loop implementation with separate control layer | <a id="orge8ee4d7"></a> Integrated optimization and control |
|
||||
| <a id="orgcd78f08"></a> Open loop optimization | <a id="orgd4993ff"></a> Closed-loop implementation with separate control layer | <a id="org0552911"></a> Integrated optimization and control |
|
||||
|
||||
|
||||
### Selection of Controlled Outputs {#selection-of-controlled-outputs}
|
||||
@@ -5184,7 +5184,7 @@ A cascade control structure results when either of the following two situations
|
||||
|
||||
|  |  |
|
||||
|-------------------------------------------------------|---------------------------------------------------|
|
||||
| <a id="org4e7be08"></a> Extra measurements \\(y\_2\\) | <a id="org1a947e7"></a> Extra inputs \\(u\_2\\) |
|
||||
| <a id="org7eda032"></a> Extra measurements \\(y\_2\\) | <a id="org765934a"></a> Extra inputs \\(u\_2\\) |
|
||||
|
||||
|
||||
#### Cascade Control: Extra Measurements {#cascade-control-extra-measurements}
|
||||
@@ -5233,7 +5233,7 @@ With reference to the special (but common) case of cascade control shown in Fig.
|
||||
|
||||
</div>
|
||||
|
||||
<a id="org664489f"></a>
|
||||
<a id="orgcccf6fb"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_cascade_control.png" caption="Figure 59: Common case of cascade control where the primary output \\(y\_1\\) depends directly on the extra measurement \\(y\_2\\)" >}}
|
||||
|
||||
@@ -5283,7 +5283,7 @@ We would probably tune the three controllers in the order \\(K\_2\\), \\(K\_3\\)
|
||||
|
||||
</div>
|
||||
|
||||
<a id="org12e1e27"></a>
|
||||
<a id="org87719b3"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_cascade_control_two_layers.png" caption="Figure 60: Control configuration with two layers of cascade control" >}}
|
||||
|
||||
@@ -5398,7 +5398,7 @@ We get:
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
|
||||
<a id="orgffa343f"></a>
|
||||
<a id="orga0f78b2"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_partial_control.png" caption="Figure 61: Partial Control" >}}
|
||||
|
||||
@@ -5518,7 +5518,7 @@ Then to minimize the control error for the primary output, \\(J = \\|y\_1 - r\_1
|
||||
|
||||
In this section, \\(G(s)\\) is a square plant which is to be controlled using a diagonal controller (Fig. [fig:decentralized_diagonal_control](#fig:decentralized_diagonal_control)).
|
||||
|
||||
<a id="org301990a"></a>
|
||||
<a id="org7292a61"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_decentralized_diagonal_control.png" caption="Figure 62: Decentralized diagonal control of a \\(2 \times 2\\) plant" >}}
|
||||
|
||||
@@ -5905,7 +5905,7 @@ The conditions are also useful in an **input-output controllability analysis** f
|
||||
|
||||
## Model Reduction {#model-reduction}
|
||||
|
||||
<a id="org7648c32"></a>
|
||||
<a id="org98665b1"></a>
|
||||
|
||||
|
||||
### Introduction {#introduction}
|
||||
|
@@ -19,7 +19,7 @@ Year
|
||||
|
||||
## Introduction {#introduction}
|
||||
|
||||
<a id="org2669f89"></a>
|
||||
<a id="orgb4a81bf"></a>
|
||||
|
||||
This book is intended to give some analysis and design tools for the increase number of engineers and researchers who are interested in the design and implementation of parallel robots.
|
||||
A systematic approach is presented to analyze the kinematics, dynamics and control of parallel robots.
|
||||
@@ -44,7 +44,7 @@ The control of parallel robots is elaborated in the last two chapters, in which
|
||||
|
||||
## Motion Representation {#motion-representation}
|
||||
|
||||
<a id="orge862be5"></a>
|
||||
<a id="org9be8358"></a>
|
||||
|
||||
|
||||
### Spatial Motion Representation {#spatial-motion-representation}
|
||||
@@ -59,7 +59,7 @@ Let us define:
|
||||
|
||||
The absolute position of point \\(P\\) of the rigid body can be constructed from the relative position of that point with respect to the moving frame \\(\\{\bm{B}\\}\\), and the **position and orientation** of the moving frame \\(\\{\bm{B}\\}\\) with respect to the fixed frame \\(\\{\bm{A}\\}\\).
|
||||
|
||||
<a id="orgf94f362"></a>
|
||||
<a id="org221cb93"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/taghirad13_rigid_body_motion.png" caption="Figure 1: Representation of a rigid body spatial motion" >}}
|
||||
|
||||
@@ -84,7 +84,7 @@ 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}\\}\\) (Figure [2](#org42e4742)).
|
||||
Its orientation has changed from a state represented by frame \\(\\{\bm{A}\\}\\) to its current orientation represented by frame \\(\\{\bm{B}\\}\\) (Figure [2](#org1448d5b)).
|
||||
|
||||
A \\(3 \times 3\\) rotation matrix \\({}^A\bm{R}\_B\\) is defined by
|
||||
|
||||
@@ -106,7 +106,7 @@ in which \\({}^A\hat{\bm{x}}\_B, {}^A\hat{\bm{y}}\_B\\) and \\({}^A\hat{\bm{z}}\
|
||||
|
||||
The nine elements of the rotation matrix can be simply represented as the projections of the Cartesian unit vectors of frame \\(\\{\bm{B}\\}\\) on the unit vectors of frame \\(\\{\bm{A}\\}\\).
|
||||
|
||||
<a id="org42e4742"></a>
|
||||
<a id="org1448d5b"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/taghirad13_rotation_matrix.png" caption="Figure 2: Pure rotation of a rigid body" >}}
|
||||
|
||||
@@ -132,7 +132,7 @@ The term screw axis for this axis of rotation has the benefit that a general mot
|
||||
The screw axis representation has the benefit of **using only four parameters** to describe a pure rotation.
|
||||
These parameters are the angle of rotation \\(\theta\\) and the axis of rotation which is a unit vector \\({}^A\hat{\bm{s}} = [s\_x, s\_y, s\_z]^T\\).
|
||||
|
||||
<a id="org4b4688a"></a>
|
||||
<a id="org778c755"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/taghirad13_screw_axis_representation.png" caption="Figure 3: Pure rotation about a screw axis" >}}
|
||||
|
||||
@@ -158,7 +158,7 @@ Three other types of Euler angles are consider with respect to a moving frame: t
|
||||
|
||||
The pitch, roll and yaw angles are defined for a moving object in space as the rotations along the lateral, longitudinal and vertical axes attached to the moving object.
|
||||
|
||||
<a id="orgfdb1f6b"></a>
|
||||
<a id="orga52af18"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/taghirad13_pitch-roll-yaw.png" caption="Figure 4: Definition of pitch, roll and yaw angles on an air plain" >}}
|
||||
|
||||
@@ -260,7 +260,7 @@ If the pose of a rigid body \\(\\{{}^A\bm{R}\_B, {}^A\bm{P}\_{O\_B}\\}\\) is giv
|
||||
|
||||
### Homogeneous Transformations {#homogeneous-transformations}
|
||||
|
||||
To describe general transformations, we introduce the \\(4\times1\\) **homogeneous coordinates**, and Eq. [eq:chasles_therorem](#eq:chasles_therorem) is generalized to
|
||||
To describe general transformations, we introduce the \\(4\times1\\) **homogeneous coordinates**, and Eq. \eqref{eq:chasles_therorem} is generalized to
|
||||
|
||||
\begin{equation}
|
||||
\tcmbox{{}^A\bm{P} = {}^A\bm{T}\_B {}^B\bm{P}}
|
||||
@@ -363,7 +363,7 @@ There exist transformations to from screw displacement notation to the transform
|
||||
Let us consider the motion of a rigid body described at three locations (Figure [fig:consecutive_transformations](#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="orgce1a4bb"></a>
|
||||
<a id="org49a98a7"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/taghirad13_consecutive_transformations.png" caption="Figure 5: Motion of a rigid body represented at three locations by frame \\(\\{\bm{A}\\}\\), \\(\\{\bm{B}\\}\\) and \\(\\{\bm{C}\\}\\)" >}}
|
||||
|
||||
@@ -426,7 +426,7 @@ Hence, the **inverse of the transformation matrix** can be obtain by
|
||||
|
||||
## Kinematics {#kinematics}
|
||||
|
||||
<a id="org4d32ea5"></a>
|
||||
<a id="org56e6a14"></a>
|
||||
|
||||
|
||||
### Introduction {#introduction}
|
||||
@@ -533,7 +533,7 @@ The position of the point \\(O\_B\\) of the moving platform is described by the
|
||||
\end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
<a id="org6f4d53b"></a>
|
||||
<a id="org10ade7e"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/taghirad13_stewart_schematic.png" caption="Figure 6: Geometry of a Stewart-Gough platform" >}}
|
||||
|
||||
@@ -586,7 +586,7 @@ The complexity of the problem depends widely on the manipulator architecture and
|
||||
|
||||
## Jacobian: Velocities and Static Forces {#jacobian-velocities-and-static-forces}
|
||||
|
||||
<a id="org58aa31a"></a>
|
||||
<a id="org150c88f"></a>
|
||||
|
||||
|
||||
### Introduction {#introduction}
|
||||
@@ -625,7 +625,7 @@ The direction of \\(\bm{\Omega}\\) indicates the instantaneous axis of rotation
|
||||
|
||||
</div>
|
||||
|
||||
The angular velocity vector is related to the screw formalism by equation [eq:angular_velocity_vector](#eq:angular_velocity_vector).
|
||||
The angular velocity vector is related to the screw formalism by equation \eqref{eq:angular_velocity_vector}.
|
||||
|
||||
\begin{equation}
|
||||
\tcmbox{\bm{\Omega} \triangleq \dot{\theta} \hat{\bm{s}}}
|
||||
@@ -683,7 +683,7 @@ The matrix \\(\bm{\Omega}^\times\\) denotes a **skew-symmetric matrix** defined
|
||||
|
||||
Now consider the general motion of a rigid body shown in Figure [fig:general_motion](#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="orgbdb62a8"></a>
|
||||
<a id="org29fa342"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/taghirad13_general_motion.png" caption="Figure 7: Instantaneous velocity of a point \\(P\\) with respect to a moving frame \\(\\{\bm{B}\\}\\)" >}}
|
||||
|
||||
@@ -700,7 +700,7 @@ The time derivative of the rotation matrix \\({}^A\dot{\bm{R}}\_B\\) is:
|
||||
\tcmbox{{}^A\dot{\bm{R}}\_B = {}^A\bm{\Omega}^\times \ {}^A\bm{R}\_B}
|
||||
\end{equation}
|
||||
|
||||
And we finally obtain equation [eq:absolute_velocity_formula](#eq:absolute_velocity_formula).
|
||||
And we finally obtain equation \eqref{eq:absolute_velocity_formula}.
|
||||
|
||||
<div class="cbox">
|
||||
<div></div>
|
||||
@@ -760,7 +760,7 @@ The **general Jacobian matrix** is defined as:
|
||||
\dot{\bm{q}} = \bm{J} \dot{\bm{\mathcal{X}}}
|
||||
\end{equation}
|
||||
|
||||
From equation [eq:jacobians](#eq:jacobians), we have:
|
||||
From equation \eqref{eq:jacobians}, we have:
|
||||
|
||||
\begin{equation}
|
||||
\bm{J} = {\bm{J}\_q}^{-1} \bm{J}\_x
|
||||
@@ -847,7 +847,7 @@ Moreover, we have:
|
||||
- \\({}^A\dot{\bm{R}}\_B {}^B\bm{b}\_i = {}^A\bm{\omega} \times {}^A\bm{R}\_B {}^B\bm{b}\_i = {}^A\bm{\omega} \times {}^A\bm{b}\_i\\) in which \\({}^A\bm{\omega}\\) denotes the angular velocity of the moving platform expressed in the fixed frame \\(\\{\bm{A}\\}\\).
|
||||
- \\(l\_i {}^A\dot{\hat{\bm{s}}}\_i = l\_i \left( {}^A\bm{\omega}\_i \times \hat{\bm{s}}\_i \right)\\) in which \\({}^A\bm{\omega}\_i\\) is the angular velocity of limb \\(i\\) express in fixed frame \\(\\{\bm{A}\\}\\).
|
||||
|
||||
Then, the velocity loop closure [eq:loop_closure_limb_diff](#eq:loop_closure_limb_diff) simplifies to
|
||||
Then, the velocity loop closure \eqref{eq:loop_closure_limb_diff} simplifies to
|
||||
\\[ {}^A\bm{v}\_p + {}^A\bm{\omega} \times {}^A\bm{b}\_i = \dot{l}\_i {}^A\hat{\bm{s}}\_i + l\_i ({}^A\bm{\omega}\_i \times \hat{\bm{s}}\_i) \\]
|
||||
|
||||
By dot multiply both side of the equation by \\(\hat{\bm{s}}\_i\\):
|
||||
@@ -884,9 +884,9 @@ We then omit the superscript \\(A\\) and we can rearrange the 6 equations into a
|
||||
#### Singularity Analysis {#singularity-analysis}
|
||||
|
||||
It is of primary importance to avoid singularities in a given workspace.
|
||||
To study the singularity configurations of the Stewart-Gough platform, we consider the Jacobian matrix determined with the equation [eq:jacobian_formula_stewart](#eq:jacobian_formula_stewart).<br />
|
||||
To study the singularity configurations of the Stewart-Gough platform, we consider the Jacobian matrix determined with the equation \eqref{eq:jacobian_formula_stewart}.<br />
|
||||
|
||||
From equation [eq:jacobians](#eq:jacobians), it is clear that for the Stewart-Gough platform, \\(\bm{J}\_q = \bm{I}\\) and \\(\bm{J}\_x = \bm{J}\\).
|
||||
From equation \eqref{eq:jacobians}, it is clear that for the Stewart-Gough platform, \\(\bm{J}\_q = \bm{I}\\) and \\(\bm{J}\_x = \bm{J}\\).
|
||||
Hence the manipulator has **no inverse kinematic singularities** within the manipulator workspace, but **may possess forward kinematic singularity** when \\(\bm{J}\\) becomes rank deficient. This may occur when
|
||||
\\[ \det \bm{J} = 0 \\]
|
||||
|
||||
@@ -942,7 +942,7 @@ We obtain that the **Jacobian matrix** constructs the **transformation needed to
|
||||
|
||||
As shown in Figure [fig:stewart_static_forces](#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="org6eae91e"></a>
|
||||
<a id="org55b47c8"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/taghirad13_stewart_static_forces.png" caption="Figure 8: Free-body diagram of forces and moments action on the moving platform and each limb of the Stewart-Gough platform" >}}
|
||||
|
||||
@@ -1006,7 +1006,7 @@ The relation between the applied actuator force \\(\tau\_i\\) and the correspond
|
||||
|
||||
in which \\(k\_i\\) denotes the **stiffness constant of the actuator**.<br />
|
||||
|
||||
Re-writing the equation [eq:stiffness_actuator](#eq:stiffness_actuator) for all limbs in a matrix form result in
|
||||
Re-writing the equation \eqref{eq:stiffness_actuator} for all limbs in a matrix form result in
|
||||
|
||||
\begin{equation}
|
||||
\tcmbox{\bm{\tau} = \mathcal{K} \cdot \Delta \bm{q}}
|
||||
@@ -1015,7 +1015,7 @@ Re-writing the equation [eq:stiffness_actuator](#eq:stiffness_actuator) for all
|
||||
in which \\(\bm{\tau}\\) is the vector of actuator forces, and \\(\Delta \bm{q}\\) corresponds to the actuator deflections.
|
||||
\\(\mathcal{K} = \text{diag}\left[ k\_1 \ k\_2 \dots k\_m \right]\\) is an \\(m \times m\\) diagonal matrix composed of the actuator stiffness constants.<br />
|
||||
|
||||
Writing the Jacobian relation given in equation [eq:jacobian_disp](#eq:jacobian_disp) for infinitesimal deflection read
|
||||
Writing the Jacobian relation given in equation \eqref{eq:jacobian_disp} for infinitesimal deflection read
|
||||
|
||||
\begin{equation}
|
||||
\Delta \bm{q} = \bm{J} \cdot \Delta \bm{\mathcal{X}}
|
||||
@@ -1023,19 +1023,19 @@ Writing the Jacobian relation given in equation [eq:jacobian_disp](#eq:jacobian_
|
||||
|
||||
in which \\(\Delta \bm{\mathcal{X}} = [\Delta x\ \Delta y\ \Delta z\ \Delta\theta x\ \Delta\theta y\ \Delta\theta z]\\) is the infinitesimal linear and angular deflection of the moving platform.
|
||||
|
||||
Furthermore, rewriting the Jacobian as the projection of actuator forces to the moving platform [eq:jacobian_forces](#eq:jacobian_forces) gives
|
||||
Furthermore, rewriting the Jacobian as the projection of actuator forces to the moving platform \eqref{eq:jacobian_forces} gives
|
||||
|
||||
\begin{equation}
|
||||
\bm{\mathcal{F}} = \bm{J}^T \bm{\tau}
|
||||
\end{equation}
|
||||
|
||||
Hence, by substituting [eq:stiffness_matrix_relation](#eq:stiffness_matrix_relation) and [eq:jacobian_disp_inf](#eq:jacobian_disp_inf) in [eq:jacobian_force_inf](#eq:jacobian_force_inf), we obtain:
|
||||
Hence, by substituting \eqref{eq:stiffness_matrix_relation} and \eqref{eq:jacobian_disp_inf} in \eqref{eq:jacobian_force_inf}, we obtain:
|
||||
|
||||
\begin{equation}
|
||||
\tcmbox{\bm{\mathcal{F}} = \underbrace{\bm{J}^T \mathcal{K} \bm{J}}\_{\bm{K}} \cdot \Delta \bm{\mathcal{X}}}
|
||||
\end{equation}
|
||||
|
||||
Equation [eq:stiffness_jacobian](#eq:stiffness_jacobian) implies that the moving platform output wrench is related to its deflection by the **stiffness matrix** \\(K\\).
|
||||
Equation \eqref{eq:stiffness_jacobian} implies that the moving platform output wrench is related to its deflection by the **stiffness matrix** \\(K\\).
|
||||
|
||||
<div class="cbox">
|
||||
<div></div>
|
||||
@@ -1099,7 +1099,7 @@ in which \\(\sigma\_{\text{min}}\\) and \\(\sigma\_{\text{max}}\\) are the small
|
||||
|
||||
In this section, we restrict our analysis to a 3-6 structure (Figure [fig:stewart36](#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="org65471b6"></a>
|
||||
<a id="org2921078"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/taghirad13_stewart36.png" caption="Figure 9: Schematic of a 3-6 Stewart-Gough platform" >}}
|
||||
|
||||
@@ -1129,7 +1129,7 @@ The largest axis of the stiffness transformation hyper-ellipsoid is given by thi
|
||||
|
||||
## Dynamics {#dynamics}
|
||||
|
||||
<a id="org5b3c9d5"></a>
|
||||
<a id="orgd3cd6ba"></a>
|
||||
|
||||
|
||||
### Introduction {#introduction}
|
||||
@@ -1213,7 +1213,7 @@ where \\(\\{\theta, \hat{\bm{s}}\\}\\) are the screw parameters representing the
|
||||
|
||||
</div>
|
||||
|
||||
As shown by [eq:angular_acceleration](#eq:angular_acceleration), the angular acceleration of the rigid body is also along the screw axis \\(\hat{\bm{s}}\\) with a magnitude equal to \\(\ddot{\theta}\\).
|
||||
As shown by \eqref{eq:angular_acceleration}, the angular acceleration of the rigid body is also along the screw axis \\(\hat{\bm{s}}\\) with a magnitude equal to \\(\ddot{\theta}\\).
|
||||
|
||||
|
||||
##### Linear Acceleration of a Point {#linear-acceleration-of-a-point}
|
||||
@@ -1260,7 +1260,7 @@ For the case where \\(P\\) is a point embedded in the rigid body, \\({}^B\bm{v}\
|
||||
|
||||
In this section, the properties of mass, namely **center of mass**, **moments of inertia** and its characteristics and the required transformations are described.
|
||||
|
||||
<a id="orgf3666fb"></a>
|
||||
<a id="orgdfab15a"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/taghirad13_mass_property_rigid_body.png" caption="Figure 10: Mass properties of a rigid body" >}}
|
||||
|
||||
@@ -1310,7 +1310,7 @@ in which
|
||||
|
||||
##### Principal Axes {#principal-axes}
|
||||
|
||||
As seen in equation [eq:moment_inertia](#eq:moment_inertia), the inertia matrix elements are a function of mass distribution of the rigid body with respect to the frame \\(\\{\bm{A}\\}\\).
|
||||
As seen in equation \eqref{eq:moment_inertia}, the inertia matrix elements are a function of mass distribution of the rigid body with respect to the frame \\(\\{\bm{A}\\}\\).
|
||||
Hence, it is possible to find **orientations of frame** \\(\\{\bm{A}\\}\\) in which the product of inertia terms vanish and inertia matrix becomes **diagonal**:
|
||||
|
||||
\begin{equation}
|
||||
@@ -1374,7 +1374,7 @@ in which \\({}^A\bm{v}\_C\\) denotes the velocity of the center of mass with res
|
||||
This result implies that the **total linear momentum** of differential masses is equal to the linear momentum of a **point mass** \\(m\\) located at the **center of mass**.
|
||||
This highlights the important of the center of mass in dynamic formulation of rigid bodies.
|
||||
|
||||
<a id="org0925a29"></a>
|
||||
<a id="org0c8049e"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/taghirad13_angular_momentum_rigid_body.png" caption="Figure 11: The components of the angular momentum of a rigid body about \\(A\\)" >}}
|
||||
|
||||
@@ -1398,10 +1398,10 @@ Therefore, angular momentum of the rigid body about point \\(A\\) is reduced to
|
||||
in which
|
||||
\\[ {}^C\bm{H} = \int\_V \bm{r} \times (\bm{\Omega} \times \bm{r}) \rho dV = {}^C\bm{I} \cdot \bm{\Omega} \\]
|
||||
|
||||
Equation [eq:angular_momentum](#eq:angular_momentum) reveals that angular momentum of a rigid body about a point \\(A\\) can be written as \\(\bm{p}\_c \times \bm{G}\_c\\), which is the contribution of linear momentum of the rigid body about point \\(A\\), and \\({}^C\bm{H}\\) which is the angular momentum of the rigid body about the center of mass.
|
||||
Equation \eqref{eq:angular_momentum} reveals that angular momentum of a rigid body about a point \\(A\\) can be written as \\(\bm{p}\_c \times \bm{G}\_c\\), which is the contribution of linear momentum of the rigid body about point \\(A\\), and \\({}^C\bm{H}\\) which is the angular momentum of the rigid body about the center of mass.
|
||||
|
||||
This also highlights the important of the center of mass in the dynamic analysis of rigid bodies.
|
||||
If the center of mass is taken as the reference point, the relation describing angular momentum [eq:angular_momentum](#eq:angular_momentum) is very analogous to that of linear momentum [eq:linear_momentum](#eq:linear_momentum).
|
||||
If the center of mass is taken as the reference point, the relation describing angular momentum \eqref{eq:angular_momentum} is very analogous to that of linear momentum \eqref{eq:linear_momentum}.
|
||||
|
||||
|
||||
##### Kinetic Energy {#kinetic-energy}
|
||||
@@ -1519,7 +1519,7 @@ The position vector of these two center of masses can be determined by the follo
|
||||
\bm{p}\_{i\_2} &= \bm{a}\_{i} + ( l\_i - c\_{i\_2}) \hat{\bm{s}}\_{i}
|
||||
\end{align}
|
||||
|
||||
<a id="org8ab78ec"></a>
|
||||
<a id="orgc1e3ded"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/taghirad13_free_body_diagram_stewart.png" caption="Figure 12: Free-body diagram of the limbs and the moving platform of a general Stewart-Gough manipulator" >}}
|
||||
|
||||
@@ -1615,7 +1615,7 @@ in which \\(\bm{\mathcal{X}} = [\bm{x}\_P, \bm{\theta}]^T\\) is the motion varia
|
||||
|
||||
#### Closed-Form Dynamics {#closed-form-dynamics}
|
||||
|
||||
While dynamic formulation in the form of Equation [eq:dynamic_formulation_implicit](#eq:dynamic_formulation_implicit) can be used to simulate inverse dynamics of the Stewart-Gough platform, its implicit nature makes it unpleasant for the dynamic analysis and control.
|
||||
While dynamic formulation in the form of Equation \eqref{eq:dynamic_formulation_implicit} can be used to simulate inverse dynamics of the Stewart-Gough platform, its implicit nature makes it unpleasant for the dynamic analysis and control.
|
||||
|
||||
|
||||
##### Closed-Form Dynamics of the Limbs {#closed-form-dynamics-of-the-limbs}
|
||||
@@ -1673,7 +1673,7 @@ It is preferable to use the **screw coordinates** for representing the angular m
|
||||
\ddot{\bm{\mathcal{X}}} = \begin{bmatrix}\bm{a}\_p \\ \dot{\bm{\omega}}\end{bmatrix}}
|
||||
\end{equation}
|
||||
|
||||
Equations [eq:dyn_form_implicit_trans](#eq:dyn_form_implicit_trans) and [eq:dyn_form_implicit_rot](#eq:dyn_form_implicit_rot) can be simply converted into a closed form of Equation [eq:close_form_dynamics_platform](#eq:close_form_dynamics_platform) with the following terms:
|
||||
Equations \eqref{eq:dyn_form_implicit_trans} and \eqref{eq:dyn_form_implicit_rot} can be simply converted into a closed form of Equation \eqref{eq:close_form_dynamics_platform} with the following terms:
|
||||
|
||||
\begin{equation}
|
||||
\begin{aligned}
|
||||
@@ -1733,11 +1733,11 @@ in which
|
||||
|
||||
As shown in Figure [fig:stewart_forward_dynamics](#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="orgd407041"></a>
|
||||
<a id="orgff76393"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/taghirad13_stewart_forward_dynamics.png" caption="Figure 13: Flowchart of forward dynamics implementation sequence" >}}
|
||||
|
||||
The closed-form dynamic formulation of the Stewart-Gough platform corresponds to the set of equations given in [eq:closed_form_dynamic_stewart_wanted](#eq:closed_form_dynamic_stewart_wanted), whose terms are given in [eq:close_form_dynamics_stewart_terms](#eq:close_form_dynamics_stewart_terms).
|
||||
The closed-form dynamic formulation of the Stewart-Gough platform corresponds to the set of equations given in \eqref{eq:closed_form_dynamic_stewart_wanted}, whose terms are given in \eqref{eq:close_form_dynamics_stewart_terms}.
|
||||
|
||||
|
||||
##### Inverse Dynamics Simulation {#inverse-dynamics-simulation}
|
||||
@@ -1752,11 +1752,11 @@ For such a trajectory, \\(\bm{\mathcal{X}}\_{d}(t)\\) and the time derivatives \
|
||||
The next step is to solve the inverse kinematics of the manipulator and to find the limbs' linear and angular positions, velocity and acceleration as a function of the manipulator trajectory.
|
||||
The manipulator Jacobian matrix \\(\bm{J}\\) is also calculated in this step.
|
||||
|
||||
Next, the dynamic matrices given in the closed-form formulations of the limbs and the moving platform are calculated using equations [eq:closed_form_intermediate_parameters](#eq:closed_form_intermediate_parameters) and [eq:close_form_dynamics_stewart_terms](#eq:close_form_dynamics_stewart_terms), respectively.<br />
|
||||
Next, the dynamic matrices given in the closed-form formulations of the limbs and the moving platform are calculated using equations \eqref{eq:closed_form_intermediate_parameters} and \eqref{eq:close_form_dynamics_stewart_terms}, respectively.<br />
|
||||
|
||||
To combine the corresponding matrices, an to generate the whole manipulator dynamics, it is necessary to find intermediate Jacobian matrices \\(\bm{J}\_i\\), given in [eq:jacobian_intermediate](#eq:jacobian_intermediate), and then compute compatible matrices for the limbs given in [eq:closed_form_stewart_manipulator](#eq:closed_form_stewart_manipulator).
|
||||
To combine the corresponding matrices, an to generate the whole manipulator dynamics, it is necessary to find intermediate Jacobian matrices \\(\bm{J}\_i\\), given in \eqref{eq:jacobian_intermediate}, and then compute compatible matrices for the limbs given in \eqref{eq:closed_form_stewart_manipulator}.
|
||||
Now that all the terms required to **computed to actuator forces required to generate such a trajectory** is computed, let us define \\(\bm{\mathcal{F}}\\) as the resulting Cartesian wrench applied to the moving platform.
|
||||
This wrench can be calculated from the summation of all inertial and external forces **excluding the actuator torques** \\(\bm{\tau}\\) in the closed-form dynamic formulation [eq:closed_form_dynamic_stewart_wanted](#eq:closed_form_dynamic_stewart_wanted).
|
||||
This wrench can be calculated from the summation of all inertial and external forces **excluding the actuator torques** \\(\bm{\tau}\\) in the closed-form dynamic formulation \eqref{eq:closed_form_dynamic_stewart_wanted}.
|
||||
|
||||
By this definition, \\(\bm{\mathcal{F}}\\) can be viewed as the projector of the actuator forces acting on the manipulator, mapped to the Cartesian space.
|
||||
Since there is no redundancy in actuation in the Stewart-Gough manipulator, the Jacobian matrix \\(\bm{J}\\), squared and actuator forces can be uniquely determined from this wrench, by \\(\bm{\tau} = \bm{J}^{-T} \bm{\mathcal{F}}\\), provided \\(\bm{J}\\) is non-singular.
|
||||
@@ -1766,7 +1766,7 @@ Therefore, actuator forces \\(\bm{\tau}\\) are computed in the simulation from
|
||||
\bm{\tau} = \bm{J}^{-T} \left( \bm{M}(\bm{\mathcal{X}})\ddot{\bm{\mathcal{X}}} + \bm{C}(\bm{\mathcal{X}}, \dot{\bm{\mathcal{X}}})\dot{\bm{\mathcal{X}}} + \bm{G}(\bm{\mathcal{X}}) - \bm{\mathcal{F}}\_d \right)
|
||||
\end{equation}
|
||||
|
||||
<a id="org07cbdc9"></a>
|
||||
<a id="org077ac35"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/taghirad13_stewart_inverse_dynamics.png" caption="Figure 14: Flowchart of inverse dynamics implementation sequence" >}}
|
||||
|
||||
@@ -1791,7 +1791,7 @@ Therefore, actuator forces \\(\bm{\tau}\\) are computed in the simulation from
|
||||
|
||||
## Motion Control {#motion-control}
|
||||
|
||||
<a id="org7a85646"></a>
|
||||
<a id="orgdd65dc9"></a>
|
||||
|
||||
|
||||
### Introduction {#introduction}
|
||||
@@ -1812,7 +1812,7 @@ However, using advanced techniques in nonlinear and MIMO control permits to over
|
||||
|
||||
### Controller Topology {#controller-topology}
|
||||
|
||||
<a id="org5a89036"></a>
|
||||
<a id="org5d92117"></a>
|
||||
|
||||
<div class="cbox">
|
||||
<div></div>
|
||||
@@ -1861,7 +1861,7 @@ Figure [fig:general_topology_motion_feedback](#fig:general_topology_motion_feedb
|
||||
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 />
|
||||
|
||||
<a id="orgab9400f"></a>
|
||||
<a id="orga16758c"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/taghirad13_general_topology_motion_feedback.png" caption="Figure 15: The general topology of motion feedback control: motion variable \\(\bm{\mathcal{X}}\\) is measured" >}}
|
||||
|
||||
@@ -1871,7 +1871,7 @@ The relation between the **differential motion variables** \\(\dot{\bm{q}}\\) an
|
||||
|
||||
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 [fig:general_topology_motion_feedback_bis](#fig:general_topology_motion_feedback_bis) to implement such a controller.
|
||||
|
||||
<a id="org212e259"></a>
|
||||
<a id="org72c22c0"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/taghirad13_general_topology_motion_feedback_bis.png" caption="Figure 16: The general topology of motion feedback control: the active joint variable \\(\bm{q}\\) is measured" >}}
|
||||
|
||||
@@ -1885,7 +1885,7 @@ To overcome the implementation problem of the control topology in Figure [fig:ge
|
||||
In this topology, depicted in Figure [fig:general_topology_motion_feedback_ter](#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="orgc845c97"></a>
|
||||
<a id="org65fcdb0"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/taghirad13_general_topology_motion_feedback_ter.png" caption="Figure 17: The general topology of motion feedback control: the active joint variable \\(\bm{q}\\) is measured, and the inverse kinematic analysis is used" >}}
|
||||
|
||||
@@ -1899,7 +1899,7 @@ For the topology in Figure [fig:general_topology_motion_feedback_ter](#fig:gener
|
||||
To generate a **direct input to output relation in the task space**, consider the topology depicted in Figure [fig:general_topology_motion_feedback_quater](#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="org566f432"></a>
|
||||
<a id="orgf53b839"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/taghirad13_general_topology_motion_feedback_quater.png" caption="Figure 18: The general topology of motion feedback control in task space: the motion variable \\(\bm{\mathcal{X}}\\) is measured, and the controller output generates wrench in task space" >}}
|
||||
|
||||
@@ -1909,7 +1909,7 @@ For a fully parallel manipulator such as the Stewart-Gough platform, this mappin
|
||||
|
||||
### Motion Control in Task Space {#motion-control-in-task-space}
|
||||
|
||||
<a id="orgf6bab13"></a>
|
||||
<a id="org7befb5a"></a>
|
||||
|
||||
|
||||
#### Decentralized PD Control {#decentralized-pd-control}
|
||||
@@ -1918,7 +1918,7 @@ In the control structure in Figure [fig:decentralized_pd_control_task_space](#fi
|
||||
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.
|
||||
|
||||
<a id="org80664c6"></a>
|
||||
<a id="org84d7f6d"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/taghirad13_decentralized_pd_control_task_space.png" caption="Figure 19: Decentralized PD controller implemented in task space" >}}
|
||||
|
||||
@@ -1941,7 +1941,7 @@ A feedforward wrench denoted by \\(\bm{\mathcal{F}}\_{ff}\\) may be added to the
|
||||
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) \\]
|
||||
|
||||
<a id="org5a2762c"></a>
|
||||
<a id="org11304dd"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/taghirad13_feedforward_control_task_space.png" caption="Figure 20: Feed forward wrench added to the decentralized PD controller in task space" >}}
|
||||
|
||||
@@ -2004,7 +2004,7 @@ Furthermore, mass matrix is added in the forward path in addition to the desired
|
||||
|
||||
As for the feedforward control, the **dynamics and kinematic parameters of the robot are needed**, and in practice estimates of these matrices are used.<br />
|
||||
|
||||
<a id="orgbab4b31"></a>
|
||||
<a id="org7bcc842"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/taghirad13_inverse_dynamics_control_task_space.png" caption="Figure 21: General configuration of inverse dynamics control implemented in task space" >}}
|
||||
|
||||
@@ -2056,9 +2056,9 @@ These are the reasons why, in practice, IDC control is extended to different for
|
||||
|
||||
To develop the simplest possible implementable IDC, let us recall dynamic formulation complexities:
|
||||
|
||||
- the manipulator mass matrix \\(\bm{M}(\bm{\mathcal{X}})\\) is derived from kinetic energy of the manipulator (Eq. [eq:kinetic_energy](#eq:kinetic_energy))
|
||||
- the gravity vector \\(\bm{G}(\bm{\mathcal{X}})\\) is derived from potential energy (Eq. [eq:gravity_vectory](#eq:gravity_vectory))
|
||||
- the Coriolis and centrifugal matrix \\(\bm{C}(\bm{\mathcal{X}}, \dot{\bm{\mathcal{X}}})\\) is derived from Eq. [eq:gravity_vectory](#eq:gravity_vectory)
|
||||
- the manipulator mass matrix \\(\bm{M}(\bm{\mathcal{X}})\\) is derived from kinetic energy of the manipulator (Eq. \eqref{eq:kinetic_energy})
|
||||
- the gravity vector \\(\bm{G}(\bm{\mathcal{X}})\\) is derived from potential energy (Eq. \eqref{eq:gravity_vectory})
|
||||
- the Coriolis and centrifugal matrix \\(\bm{C}(\bm{\mathcal{X}}, \dot{\bm{\mathcal{X}}})\\) is derived from Eq. \eqref{eq:gravity_vectory}
|
||||
|
||||
The computation of the Coriolis and centrifugal matrix is more intensive than that of the mass matrix.
|
||||
Gravity vector is more easily computable.
|
||||
@@ -2066,7 +2066,7 @@ Gravity vector is more easily computable.
|
||||
However, it is shown that certain properties hold for mass matrix, gravity vector and Coriolis and centrifugal matrix, which might be directly used in the control techniques developed for parallel manipulators.
|
||||
One of the most important properties of dynamic matrices is the skew-symmetric property of the matrix \\(\dot{\bm{M}} - 2 \bm{C}\\) .<br />
|
||||
|
||||
Consider dynamic formulation of parallel robot given in Eq. [eq:closed_form_dynamic_formulation](#eq:closed_form_dynamic_formulation), in which the skew-symmetric property of dynamic matrices is satisfied.
|
||||
Consider dynamic formulation of parallel robot given in Eq. \eqref{eq:closed_form_dynamic_formulation}, in which the skew-symmetric property of dynamic matrices is satisfied.
|
||||
The simplest form of IDC control effort \\(\bm{\mathcal{F}}\\) consists of:
|
||||
\\[ \bm{\mathcal{F}} = \bm{\mathcal{F}}\_{pd} + \bm{\mathcal{F}}\_{fl} \\]
|
||||
in which the first term \\(\bm{\mathcal{F}}\_{pd}\\) is generated by the simplified PD form on the motion error:
|
||||
@@ -2104,7 +2104,7 @@ A global understanding of the trade-offs involved in each method is needed to em
|
||||
|
||||
Various sources of uncertainties such as unmodelled dynamics, unknown parameters, calibration error, unknown disturbance wrenches, and varying payloads may exist, and are not seen in dynamic model of the manipulator.
|
||||
|
||||
To consider these modeling uncertainty in the closed-loop performance of the manipulator, recall the general closed-form dynamic formulation of the manipulator given in Eq. [eq:closed_form_dynamic_formulation](#eq:closed_form_dynamic_formulation), and modify the inverse dynamics control input \\(\bm{\mathcal{F}}\\) as
|
||||
To consider these modeling uncertainty in the closed-loop performance of the manipulator, recall the general closed-form dynamic formulation of the manipulator given in Eq. \eqref{eq:closed_form_dynamic_formulation}, and modify the inverse dynamics control input \\(\bm{\mathcal{F}}\\) as
|
||||
|
||||
\begin{align\*}
|
||||
\bm{\mathcal{F}} &= \hat{\bm{M}}(\bm{\mathcal{X}}) \bm{a}\_r + \hat{\bm{C}}(\bm{\mathcal{X}}, \dot{\bm{\mathcal{X}}}) \dot{\bm{\mathcal{X}}} + \hat{\bm{G}}(\bm{\mathcal{X}})\\\\\\
|
||||
@@ -2126,14 +2126,14 @@ in which
|
||||
\\[ \bm{\eta} = \bm{M}^{-1} \left( \tilde{\bm{M}} \bm{a}\_r + \tilde{\bm{C}} \dot{\bm{\mathcal{X}}} + \tilde{\bm{G}} \right) \\]
|
||||
is a measure of modeling uncertainty.
|
||||
|
||||
<a id="orgabfe014"></a>
|
||||
<a id="org3cb985a"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/taghirad13_robust_inverse_dynamics_task_space.png" caption="Figure 22: General configuration of robust inverse dynamics control implemented in the task space" >}}
|
||||
|
||||
|
||||
#### Adaptive Inverse Dynamics Control {#adaptive-inverse-dynamics-control}
|
||||
|
||||
<a id="org27f4777"></a>
|
||||
<a id="org2d5c44d"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/taghirad13_adaptative_inverse_control_task_space.png" caption="Figure 23: General configuration of adaptative inverse dynamics control implemented in task space" >}}
|
||||
|
||||
@@ -2218,7 +2218,7 @@ In this control structure, depicted in Figure [fig:decentralized_pd_control_join
|
||||
|
||||
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 />
|
||||
|
||||
<a id="org8778164"></a>
|
||||
<a id="orgef87ac5"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/taghirad13_decentralized_pd_control_joint_space.png" caption="Figure 24: Decentralized PD controller implemented in joint space" >}}
|
||||
|
||||
@@ -2240,7 +2240,7 @@ To remedy these shortcomings, some modifications have been proposed to this stru
|
||||
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 Figure [fig:feedforward_pd_control_joint_space](#fig:feedforward_pd_control_joint_space).
|
||||
|
||||
<a id="org82bffda"></a>
|
||||
<a id="org7dd1247"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/taghirad13_feedforward_pd_control_joint_space.png" caption="Figure 25: Feed forward actuator force added to the decentralized PD controller in joint space" >}}
|
||||
|
||||
@@ -2288,7 +2288,7 @@ Furthermore, the mass matrix is acting in the **forward path**, in addition to t
|
||||
Note that to generate this term, the **dynamic formulation** of the robot, and its **kinematic and dynamic parameters are needed**.
|
||||
In practice, exact knowledge of dynamic matrices are not available, and there estimates are used.<br />
|
||||
|
||||
<a id="orgb3e85c7"></a>
|
||||
<a id="orgd592b03"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/taghirad13_inverse_dynamics_control_joint_space.png" caption="Figure 26: General configuration of inverse dynamics control implemented in joint space" >}}
|
||||
|
||||
@@ -2564,7 +2564,7 @@ Hence, it is recommended to design and implement controllers in the task space,
|
||||
|
||||
## Force Control {#force-control}
|
||||
|
||||
<a id="org6d7e26a"></a>
|
||||
<a id="org7035f2e"></a>
|
||||
|
||||
|
||||
### Introduction {#introduction}
|
||||
@@ -2620,7 +2620,7 @@ The output control loop is called the **primary loop**, while the inner loop is
|
||||
|
||||
</div>
|
||||
|
||||
<a id="orgd4ccfa2"></a>
|
||||
<a id="orgc775ba2"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/taghirad13_cascade_control.png" caption="Figure 27: Block diagram of a closed-loop system with cascade control" >}}
|
||||
|
||||
@@ -2654,7 +2654,7 @@ As seen in Figure [fig:taghira13_cascade_force_outer_loop](#fig:taghira13_cascad
|
||||
|
||||
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 />
|
||||
|
||||
<a id="orgd1518f8"></a>
|
||||
<a id="orgc0138a8"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/taghira13_cascade_force_outer_loop.png" caption="Figure 28: 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" >}}
|
||||
|
||||
@@ -2662,7 +2662,7 @@ Other alternatives for force control topology may be suggested based on the vari
|
||||
If the force is measured in the joint space, the topology suggested in Figure [fig:taghira13_cascade_force_outer_loop_tau](#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="org56763bc"></a>
|
||||
<a id="orgf060936"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/taghira13_cascade_force_outer_loop_tau.png" caption="Figure 29: 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" >}}
|
||||
|
||||
@@ -2673,7 +2673,7 @@ However, as the inner loop is constructed in the joint space, the desired motion
|
||||
|
||||
Therefore, the structure and characteristics of the position controller in this topology is totally different from that given in the first two topologies.<br />
|
||||
|
||||
<a id="orgd855e74"></a>
|
||||
<a id="orgfe2ecb4"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/taghira13_cascade_force_outer_loop_tau_q.png" caption="Figure 30: Cascade topology of force feedback control: position in inner loop and force in outer loop. Actuator forces \\(\bm{\tau}\\) and joint motion variable \\(\bm{q}\\) are measured in the joint space" >}}
|
||||
|
||||
@@ -2691,7 +2691,7 @@ By this means, when the manipulator is not in contact with a stiff environment,
|
||||
However, when there is interacting wrench \\(\bm{\mathcal{F}}\_e\\) applied to the moving platform, this structure controls the force-motion relation.
|
||||
This configuration may be seen as if the **outer loop generates a desired force trajectory for the inner loop**.<br />
|
||||
|
||||
<a id="org1b62063"></a>
|
||||
<a id="org1509ffc"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/taghira13_cascade_force_inner_loop_F.png" caption="Figure 31: 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" >}}
|
||||
|
||||
@@ -2699,7 +2699,7 @@ Other alternatives for control topology may be suggested based on the variations
|
||||
If the force is measured in the joint space, control topology shown in Figure [fig:taghira13_cascade_force_inner_loop_tau](#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="org9e07f41"></a>
|
||||
<a id="org4e33f80"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/taghira13_cascade_force_inner_loop_tau.png" caption="Figure 32: 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" >}}
|
||||
|
||||
@@ -2708,7 +2708,7 @@ The inner loop is based on the measured actuator force vector in the joint space
|
||||
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.
|
||||
|
||||
<a id="org688e959"></a>
|
||||
<a id="org159af61"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/taghira13_cascade_force_inner_loop_tau_q.png" caption="Figure 33: Cascade topology of force feedback control: force in inner loop and position in outer loop. Actuator forces \\(\bm{\tau}\\) and joint motion variable \\(\bm{q}\\) are measured in the joint space" >}}
|
||||
|
||||
@@ -2727,7 +2727,7 @@ Thus, independent controllers for each joint may be suitable for this topology.
|
||||
|
||||
### Direct Force Control {#direct-force-control}
|
||||
|
||||
<a id="org7d75d2a"></a>
|
||||
<a id="orgb779c33"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/taghira13_direct_force_control.png" caption="Figure 34: Direct force control scheme, force feedback in the outer loop and motion feedback in the inner loop" >}}
|
||||
|
||||
@@ -2818,7 +2818,7 @@ The impedance of the system may be found from the Laplace transform of the above
|
||||
|
||||
</div>
|
||||
|
||||
<a id="org649bf2b"></a>
|
||||
<a id="orge490c94"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/taghirad13_impedance_control_rlc.png" caption="Figure 35: Analogy of electrical impedance in (a) an electrical RLC circuit to (b) a mechanical mass-spring-damper system" >}}
|
||||
|
||||
@@ -2877,7 +2877,7 @@ Moreover, direct force-tracking objective is not assigned in this control 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.
|
||||
By this means, no prescribed force trajectory is tracked, while the **motion control scheme would advise a force trajectory for the robot to ensure the desired impedance regulation**.<br />
|
||||
|
||||
<a id="org93dc62d"></a>
|
||||
<a id="org9902c61"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/taghira13_impedance_control.png" caption="Figure 36: Impedance control scheme; motion feedback in the outer loop and force feedback in the inner loop" >}}
|
||||
|
||||
|
Reference in New Issue
Block a user