diff --git a/config.toml b/config.toml
index 1fc4913..f9705c5 100644
--- a/config.toml
+++ b/config.toml
@@ -102,7 +102,7 @@ contentCopyright = ''
# Link custom CSS and JS assets
# (relative to /static/css and /static/js respectively)
-customCSS = []
+customCSS = ["custom.css"]
customJS = []
uglyURLs = false
diff --git a/content/book/ewins00_modal.md b/content/book/ewins00_modal.md
index 9eb754a..1a5d503 100644
--- a/content/book/ewins00_modal.md
+++ b/content/book/ewins00_modal.md
@@ -424,26 +424,34 @@ which is now complex, containing both magnitude and phase information:
All structures exhibit a degree of damping due to the **hysteresis properties** of the material(s) from which they are made.
-A typical example of this effect is shown in the force displacement plot in [ 1](#org-target--fig-material-histeresis) in which the **area contained by the loop represents the energy lost in one cycle of vibration** between the extremities shown.
+A typical example of this effect is shown in the force displacement plot in [Figure 6a](#org-target--fig-material-histeresis) in which the **area contained by the loop represents the energy lost in one cycle of vibration** between the extremities shown.
The maximum energy stored corresponds to the elastic energy of the structure at the point of maximum deflection.
The damping effect of such a component can conveniently be defined by the ratio of these two:
\\[ \tcmbox{\text{damping capacity} = \frac{\text{energy lost per cycle}}{\text{maximum energy stored}}} \\]
-
-
+Figure 6: Force-deflection characteristics
+
Another common source of energy dissipation in practical structures, is the **friction** which exist in joints between components of the structure.
-It may be described very roughly by the simple **dry friction model** shown in [ 1](#org-target--fig-dry-friction).
+It may be described very roughly by the simple **dry friction model** shown in [Figure 6b](#org-target--fig-dry-friction).
-The mathematical model of the **viscous damper** which we have used can be compared with these more physical effects by plotting the corresponding force-displacement diagram for it, and this is shown in [ 1](#org-target--fig-viscous-damper).
+The mathematical model of the **viscous damper** which we have used can be compared with these more physical effects by plotting the corresponding force-displacement diagram for it, and this is shown in [Figure 6c](#org-target--fig-viscous-damper).
Because the relationship is linear between force and velocity, it is necessary to suppose harmonic motion, at frequency \\(\omega\\), in order to construct a force-displacement diagram.
The resulting diagram shows the nature of the approximation provided by the viscous damper model and the concept of the **effective or equivalent viscous damping coefficient** for any of the actual phenomena as being which provides the **same energy loss per cycle** as the real thing.
@@ -503,7 +511,7 @@ Similarly we could use the acceleration parameter so we could define a third FRF
-[Table 2](#table--tab:frf-alternatives) gives details of all six of the FRF parameters and of the names used for them.
+[Table 1](#table--tab:frf-alternatives) gives details of all six of the FRF parameters and of the names used for them.
**Inverse response** can also be defined. For instance, the **dynamic stiffness** is defined as the force over the displacement.
@@ -521,7 +529,7 @@ It should be noted that that the use of displacement as the response is greatly
- Table 2:
+ Table 1:
Definition of Frequency Response Functions
@@ -549,18 +557,26 @@ Any simple plot can only show two of the three quantities and so there are diffe
##### Bode Plot {#bode-plot}
-Bode plot are usually displayed using logarithmic scales as shown on [Table 3](#table--fig:bode-plots).
+Bode plot are usually displayed using logarithmic scales as shown on [Figure 7](#table--fig:bode-plots).
-
-
- Table 3:
- FRF plots for undamped SDOF system
+
+
+
+Figure 7: FRF plots for undamped SDOF system
+
Each plot can be divided into three regimes:
@@ -571,54 +587,66 @@ Each plot can be divided into three regimes:
##### Real part and Imaginary part of FRF {#real-part-and-imaginary-part-of-frf}
-Real and imaginary part of a receptance FRF of a damped SDOF system is shown on [Table 4](#table--fig:plot-receptance-real-imag).
+Real and imaginary part of a receptance FRF of a damped SDOF system is shown on [Figure 8](#table--fig:plot-receptance-real-imag).
This type of display is not widely used as we cannot use logarithmic axes (as we have to show positive and negative values).
-
-
- Table 4:
- Plot of real and imaginary part for the receptance of a damped SDOF
-
-|  |  |
-|--------------------------------------------------------------------------------------|-------------------------------------------------------------------------------------------|
-| Real part | Imaginary part |
-| width=\linewidth | width=\linewidth |
+
+
+
+
+
(a) Real part
+
+
+
+
(b) Imaginary part
+
+
+Figure 8: Plot of real and imaginary part for the receptance of a damped SDOF
+
##### Real part and Imaginary part of reciprocal FRF {#real-part-and-imaginary-part-of-reciprocal-frf}
It can be seen from the expression of the inverse receptance \ref{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.
-[ 5](#org-target--fig-inverse-frf-mixed) shows an example of a plot of a system with a combination of both viscous and structural damping. The imaginary part is a straight line whose slope is given by the viscous damping rate \\(c\\) and whose intercept at \\(\omega = 0\\) is provided by the structural damping coefficient \\(d\\).
+[Figure 9a](#org-target--fig-inverse-frf-mixed) shows an example of a plot of a system with a combination of both viscous and structural damping. The imaginary part is a straight line whose slope is given by the viscous damping rate \\(c\\) and whose intercept at \\(\omega = 0\\) is provided by the structural damping coefficient \\(d\\).
-
-
- Table 5:
- Inverse FRF plot for the system
+
+
+
+Figure 9: Inverse FRF plot for the system
+
##### Real part vs Imaginary part of FRF {#real-part-vs-imaginary-part-of-frf}
-[Table 6](#table--fig:nyquist-receptance) shows Nyquist type FRF plots of a viscously damped SDOF system.
+[Figure 10](#table--fig:nyquist-receptance) shows Nyquist type FRF plots of a viscously damped SDOF system.
The missing information (in this case, the frequency) must be added by identifying the values of frequency corresponding to particular points on the curve.
-
-
- Table 6:
- Nyquist FRF plots of the mobility for a SDOF system
-
+Figure 10: Nyquist FRF plots of the mobility for a SDOF system
+
The Nyquist plot has the particularity of distorting the plot so as to focus on the resonance area.
This makes the Nyquist plot very effective for modal testing applications.
@@ -1110,25 +1138,33 @@ 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 6](#figure--fig:real-complex-modes)).
+While the real mode has the appearance of a **standing wave**, the complex mode is better described as exhibiting **traveling waves** (illustrated on [Figure 11](#figure--fig:real-complex-modes)).
-{{< figure src="/ox-hugo/ewins00_real_complex_modes.png" caption="Figure 6: Real and complex mode shapes displays" >}}
+{{< figure src="/ox-hugo/ewins00_real_complex_modes.png" caption="Figure 11: Real and complex mode shapes displays" >}}
-Another method of displaying **modal complexity** is by plotting the elements of the eigenvector on an **Argand diagram**, such as the ones shown in [Table 7](#table--fig:argand-diagram).
+Another method of displaying **modal complexity** is by plotting the elements of the eigenvector on an **Argand diagram**, such as the ones shown in [Figure 12](#table--fig:argand-diagram).
Note that the almost-real mode shape does not necessarily have vector elements with near \\(\SI{0}{\degree}\\) or near \\(\SI{180}{\degree}\\) phase, what matters are the **relative phases** between the different elements.
-
-
+Figure 12: Complex mode shapes plotted on Argand diagrams
+
#### Measurement of modal complexity {#measurement-of-modal-complexity}
@@ -1137,7 +1173,7 @@ There exist few indicators of the modal complexity.
The first one, a simple and crude one, called **MCF1** consists of summing all the phase differences between every combination of two eigenvector elements:
\\[ \text{MCF1} = \sum\_{j=1}^N \sum\_{k=1 \neq j}^N (\theta\_{rj} - \theta\_{rk}) \\]
-The second measure is shown on [ 7](#org-target--fig-argand-diagram-c) where a polygon is drawn around the extremities of the individual vectors.
+The second measure is shown on [Figure 12c](#org-target--fig-argand-diagram-c) where a polygon is drawn around the extremities of the individual vectors.
The obtained area of this polygon is then compared with the area of the circle which is based on the length of the largest vector element. The resulting ratio is used as an indication of the complexity of the mode, and is defined as **MCF2**.
@@ -1177,11 +1213,11 @@ The second definition comes from the general form of the FRF expression:
Here \\(C\_r\\) may be complex whereas \\(D\_r\\) is real.
\\(\omega\_r\\) is in general different to both \\(\bar{\omega}\_r\\) and \\(\omega\_r^\prime\\).
-[Table 8](#table--tab:frf-natural-frequencies) summarizes all the different cases.
+[Table 2](#table--tab:frf-natural-frequencies) summarizes all the different cases.
@@ -1233,21 +1269,25 @@ We write \\(\alpha\_{11}\\) the point FRF and \\(\alpha\_{21}\\) the transfer FR
It can be seen that the only difference between the point and transfer receptance is in the sign of the modal constant of the second mode.
-Consider the first point mobility ([ 9](#org-target--fig-mobility-frf-mdof-point)), between the two resonances, the two components have opposite signs so that they are substractive rather than additive, and indeed, at the point where they cross, their sum is zero.
+Consider the first point mobility ([Figure 13a](#org-target--fig-mobility-frf-mdof-point)), between the two resonances, the two components have opposite signs so that they are substractive rather than additive, and indeed, at the point where they cross, their sum is zero.
On a logarithmic plot, this produces the antiresonance characteristic which reflects that of the resonance.
-
-
- Table 9:
- Mobility FRF plot for undamped 2DOF system
+
+
+
+
+
+
(a) Point FRF
+
+
+
(b) Transfer FRF
+
+
+Figure 13: Mobility FRF plot for undamped 2DOF system
+
-|  |  |
-|-----------------------------------------------------------------------------------------|-----------------------------------------------------------------------------------------------|
-| Point FRF | Transfer FRF |
-| width=\linewidth | width=\linewidth |
-
-For the plot in [ 9](#org-target--fig-mobility-frf-mdof-transfer), between the two resonances, the two components have the same sign and they add up, no antiresonance is present.
+For the plot in [Figure 13b](#org-target--fig-mobility-frf-mdof-transfer), between the two resonances, the two components have the same sign and they add up, no antiresonance is present.
##### FRF modulus plots for MDOF systems {#frf-modulus-plots-for-mdof-systems}
@@ -1263,7 +1303,7 @@ If they have apposite signs, there will not be an antiresonance.
##### Bode plots {#bode-plots}
The resonances and antiresonances are blunted by the inclusion of damping, and the phase angles are no longer exactly \\(\SI{0}{\degree}\\) or \\(\SI{180}{\degree}\\), but the general appearance of the plot is a natural extension of that for the system without damping.
-[Figure 7](#figure--fig:frf-damped-system) shows a plot for the same mobility as appears in [ 9](#org-target--fig-mobility-frf-mdof-point) but here for a system with added damping.
+[Figure 14](#figure--fig:frf-damped-system) shows a plot for the same mobility as appears in [Figure 13a](#org-target--fig-mobility-frf-mdof-point) but here for a system with added damping.
Most mobility plots have this general form as long as the modes are relatively well-separated.
@@ -1271,42 +1311,50 @@ This condition is satisfied unless the separation between adjacent natural frequ
-{{< figure src="/ox-hugo/ewins00_frf_damped_system.png" caption="Figure 7: Mobility plot of a damped system" >}}
+{{< figure src="/ox-hugo/ewins00_frf_damped_system.png" caption="Figure 14: Mobility plot of a damped system" >}}
##### Nyquist diagrams {#nyquist-diagrams}
Each of the frequency response of a MDOF system in the Nyquist plot is composed of a number of SDOF components.
-[ 10](#org-target--fig-nyquist-point) shows the result of plotting the point receptance \\(\alpha\_{11}\\) for the 2DOF system described above.
+[Figure 15a](#org-target--fig-nyquist-point) shows the result of plotting the point receptance \\(\alpha\_{11}\\) for the 2DOF system described above.
-The plot for the transfer receptance \\(\alpha\_{21}\\) is presented in [ 10](#org-target--fig-nyquist-transfer) where it may be seen that the opposing signs of the modal constants of the two modes have caused one of the modal circle to be in the upper half of the complex plane.
+The plot for the transfer receptance \\(\alpha\_{21}\\) is presented in [Figure 15b](#org-target--fig-nyquist-transfer) where it may be seen that the opposing signs of the modal constants of the two modes have caused one of the modal circle to be in the upper half of the complex plane.
-
-
- Table 10:
- Nyquist FRF plot for proportionally-damped system
+
+
+
+
+
+
(a) Point receptance
+
+
+
(b) Transfer receptance
+
+
+Figure 15: Nyquist FRF plot for proportionally-damped system
+
-|  |  |
-|--------------------------------------------------------------------------------------|--------------------------------------------------------------------------------------------|
-| Point receptance | Transfer receptance |
-| width=\linewidth | width=\linewidth |
-
-In the two [ 11](#org-target--fig-nyquist-nonpropdamp-point) and [ 11](#org-target--fig-nyquist-nonpropdamp-transfer), we show corresponding data for **non-proportional** damping.
+In the two [Figure 16a](#org-target--fig-nyquist-nonpropdamp-point) and [Figure 16b](#org-target--fig-nyquist-nonpropdamp-transfer), we show corresponding data for **non-proportional** damping.
In this case, a relative phase has been introduced between the first and second elements of the eigenvectors: of \\(\SI{30}{\degree}\\) in mode 1 and of \\(\SI{150}{\degree}\\) in mode 2.
Now we find that the individual modal circles are no longer "upright" but are **rotated by an amount dictated by the complexity of the modal constants**.
-
-
- Table 11:
- Nyquist FRF plot for non-proportionally-damped system
-
-|  |  |
-|--------------------------------------------------------------------------------------------------|--------------------------------------------------------------------------------------------------------|
-| Point receptance | Transfer receptance |
-| width=\linewidth | width=\linewidth |
+
+
+
+
+
(a) Point receptance
+
+
+
+
(b) Transfer receptance
+
+
+Figure 16: Nyquist FRF plot for non-proportionally-damped system
+
### Non-Sinusoidal vibration and FRF properties {#non-sinusoidal-vibration-and-frf-properties}
@@ -1449,18 +1497,26 @@ The resulting parameter we shall call a **Spectral Density**, in this case the *
The Spectral Density is a real and even function of frequency, and does in fact provides a description of the frequency composition of the original function \\(f(t)\\).
It has units of \\(f^2/\omega\\).
-Examples of random signals, autocorrelation function and power spectral density are shown on [Table 12](#table--fig:random-signals).
+Examples of random signals, autocorrelation function and power spectral density are shown on [Figure 17](#table--fig:random-signals).
-
-
-
-|  |  |  |
-|--------------------------------------------------------------------------------|-------------------------------------------------------------------------------------------------------|-----------------------------------------------------------------------------------------|
-| Time history | Autocorrelation Function | Power Spectral Density |
-| width=\linewidth | width=\linewidth | width=\linewidth |
+
+
+
(b) Autocorrelation Function
+
+
+
+
(c) Power Spectral Density
+
+
+Figure 17: Random signals
+
A similar concept can be applied to a pair of functions such as \\(f(t)\\) and \\(x(t)\\) to produce **cross correlation** and **cross spectral density** functions.
@@ -1540,19 +1596,23 @@ The existence of two equations presents an opportunity to **check the quality**
There are difficulties to implement some of the above formulae in practice because of noise and other limitations concerned with the data acquisition and processing.
One technique involves **three quantities**, rather than two, in the definition of the output/input ratio.
-The system considered can best be described with reference to [Table 13](#table--fig:frf-determination) which shows first in [ 13](#org-target--fig-frf-siso-model) the traditional single-input single-output model upon which the previous formulae are based.
-Then in [ 13](#org-target--fig-frf-feedback-model) is given a more detailed and representative model of the system which is used in a modal test.
+The system considered can best be described with reference to [Figure 18](#table--fig:frf-determination) which shows first in [Figure 18a](#org-target--fig-frf-siso-model) the traditional single-input single-output model upon which the previous formulae are based.
+Then in [Figure 18b](#org-target--fig-frf-feedback-model) is given a more detailed and representative model of the system which is used in a modal test.
-
-
-
-|  |  |
-|---------------------------------------------------------------------------------------|---------------------------------------------------------------------------------------------------|
-| Basic SISO model | SISO model with feedback |
-| width=\linewidth | width=\linewidth |
+
+
+
(b) SISO model with feedback
+
+
+Figure 18: System for FRF determination
+
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 \ref{eq:H3}.
@@ -1570,7 +1630,7 @@ where \\(v\\) is a third signal in the system.
##### Derivation of FRF from MIMO data {#derivation-of-frf-from-mimo-data}
-A diagram for the general n-input case is shown in [Figure 8](#figure--fig:frf-mimo).
+A diagram for the general n-input case is shown in [Figure 19](#figure--fig:frf-mimo).
We obtain two alternative formulas:
@@ -1583,7 +1643,7 @@ In practical application of both of these formulae, care must be taken to ensure
-{{< figure src="/ox-hugo/ewins00_frf_mimo.png" caption="Figure 8: System for FRF determination via MIMO model" >}}
+{{< figure src="/ox-hugo/ewins00_frf_mimo.png" caption="Figure 19: System for FRF determination via MIMO model" >}}
### Complete and Incomplete models {#complete-and-incomplete-models}
@@ -1849,11 +1909,11 @@ The experimental setup used for mobility measurement contains three major items:
2. **A transduction system**. For the most part, piezoelectric transducer are used, although lasers and strain gauges are convenient because of their minimal interference with the test object. Conditioning amplifiers are used depending of the transducer used
3. **An analyzer**
-A typical layout for the measurement system is shown on [Figure 9](#figure--fig:general-frf-measurement-setup).
+A typical layout for the measurement system is shown on [Figure 20](#figure--fig:general-frf-measurement-setup).
-{{< figure src="/ox-hugo/ewins00_general_frf_measurement_setup.png" caption="Figure 9: General layout of FRF measurement system" >}}
+{{< figure src="/ox-hugo/ewins00_general_frf_measurement_setup.png" caption="Figure 20: General layout of FRF measurement system" >}}
### Structure preparation {#structure-preparation}
@@ -1905,32 +1965,40 @@ However, we need a direct measurement of the force applied to the structure (we
The shakers are usually stiff in the orthogonal directions to the excitation.
This can modify the response of the system in those directions.
-In order to avoid that, a drive rod which is stiff in one direction and flexible in the other five directions is attached between the shaker and the structure as shown on [Figure 10](#figure--fig:shaker-rod).
+In order to avoid that, a drive rod which is stiff in one direction and flexible in the other five directions is attached between the shaker and the structure as shown on [Figure 21](#figure--fig:shaker-rod).
Typical size for the rod are \\(5\\) to \\(\SI{10}{mm}\\) long and \\(\SI{1}{mm}\\) in diameter, if the rod is longer, it may introduce the effect of its own resonances.
-{{< figure src="/ox-hugo/ewins00_shaker_rod.png" caption="Figure 10: Exciter attachment and drive rod assembly" >}}
+{{< figure src="/ox-hugo/ewins00_shaker_rod.png" caption="Figure 21: Exciter attachment and drive rod assembly" >}}
The support of shaker is also of primary importance.
-The setup shown on [ 14](#org-target--fig-shaker-mount-1) presents the most satisfactory arrangement in which the shaker is fixed to ground while the test structure is supported by a soft spring.
+The setup shown on [Figure 22a](#org-target--fig-shaker-mount-1) presents the most satisfactory arrangement in which the shaker is fixed to ground while the test structure is supported by a soft spring.
-[ 14](#org-target--fig-shaker-mount-2) shows an alternative configuration in which the shaker itself is supported.
+[Figure 22b](#org-target--fig-shaker-mount-2) shows an alternative configuration in which the shaker itself is supported.
It may be necessary to add an additional inertia mass to the shaker in order to generate sufficient excitation forces at low frequencies.
-[ 14](#org-target--fig-shaker-mount-3) shows an unsatisfactory setup. Indeed, the response measured at \\(A\\) would not be due solely to force applied at \\(B\\), but would also be caused by the forces applied at \\(C\\).
+[Figure 22c](#org-target--fig-shaker-mount-3) shows an unsatisfactory setup. Indeed, the response measured at \\(A\\) would not be due solely to force applied at \\(B\\), but would also be caused by the forces applied at \\(C\\).
-
-
- Table 14:
- Various mounting arrangement of exciter
+
+
+
+Figure 22: Various mounting arrangement of exciter
+
#### Hammer or Impactor Excitation {#hammer-or-impactor-excitation}
@@ -1944,12 +2012,12 @@ The magnitude of the impact is determined by the mass of the hammer head and its
The frequency range which is effectively excited is controlled by the stiffness of the contacting surface and the mass of the impactor head: there is a resonance at a frequency given by \\(\sqrt{\frac{\text{contact stiffness}}{\text{impactor mass}}}\\) above which it is difficult to deliver energy into the test structure.
-When the hammer tip impacts the test structure, this will experience a force pulse as shown on [Figure 11](#figure--fig:hammer-impulse).
-A pulse of this type (half-sine shape) has a frequency content of the form illustrated on [Figure 11](#figure--fig:hammer-impulse).
+When the hammer tip impacts the test structure, this will experience a force pulse as shown on [Figure 23](#figure--fig:hammer-impulse).
+A pulse of this type (half-sine shape) has a frequency content of the form illustrated on [Figure 23](#figure--fig:hammer-impulse).
-{{< figure src="/ox-hugo/ewins00_hammer_impulse.png" caption="Figure 11: Typical impact force pulse and spectrum" >}}
+{{< figure src="/ox-hugo/ewins00_hammer_impulse.png" caption="Figure 23: Typical impact force pulse and spectrum" >}}
The stiffer the materials, the shorter will be the duration of the pulse and the higher will be the frequency range covered by the impact.
Similarly, the lighter the impactor mass, the higher the effective frequency range.
@@ -1976,24 +2044,24 @@ By suitable design, such a material may be incorporated into a device which **in
#### Force Transducers {#force-transducers}
The force transducer is the simplest type of piezoelectric transducer.
-The transmitter force \\(F\\) is applied directly across the crystal, which thus generates a corresponding charge \\(q\\), proportional to \\(F\\) ([Figure 12](#figure--fig:piezo-force-transducer)).
+The transmitter force \\(F\\) is applied directly across the crystal, which thus generates a corresponding charge \\(q\\), proportional to \\(F\\) ([Figure 24](#figure--fig:piezo-force-transducer)).
-{{< figure src="/ox-hugo/ewins00_piezo_force_transducer.png" caption="Figure 12: Force transducer" >}}
+{{< figure src="/ox-hugo/ewins00_piezo_force_transducer.png" caption="Figure 24: Force transducer" >}}
There exists an undesirable possibility of a cross sensitivity, i.e. an electrical output when there is zero force \\(F\\) but, say, a transverse or shear loading.
#### Accelerometers {#accelerometers}
-In an accelerometer, transduction is indirect and is achieved using a seismic mass ([Figure 13](#figure--fig:piezo-accelerometer)).
+In an accelerometer, transduction is indirect and is achieved using a seismic mass ([Figure 25](#figure--fig:piezo-accelerometer)).
In this configuration, the force exerted on the crystals is the inertia force of the seismic mass (\\(m\ddot{z}\\)).
Thus, so long as the body and the seismic mass move together, the output of the transducer will be proportional to the acceleration of its body \\(x\\).
-{{< figure src="/ox-hugo/ewins00_piezo_accelerometer.png" caption="Figure 13: Compression-type of piezoelectric accelerometer" >}}
+{{< figure src="/ox-hugo/ewins00_piezo_accelerometer.png" caption="Figure 25: Compression-type of piezoelectric accelerometer" >}}
Analysis of a simple dynamical model for this device shows that the ratio \\(\ddot{x}/\ddot{z}\\) is effectively unity over a wide range of frequency from zero upwards until the first resonant frequency of the transducer.
@@ -2027,20 +2095,24 @@ However, they cannot be used at such low frequencies as the charge amplifiers an
The correct installation of transducers, especially accelerometers is important.
There are various means of fixing the transducers to the surface of the test structure, some more convenient than others.
-Some of these methods are illustrated in [ 15](#org-target--fig-transducer-mounting-types).
+Some of these methods are illustrated in [Figure 26a](#org-target--fig-transducer-mounting-types).
-Shown on [ 15](#org-target--fig-transducer-mounting-response) are typical high frequency limits for each type of attachment.
+Shown on [Figure 26b](#org-target--fig-transducer-mounting-response) are typical high frequency limits for each type of attachment.
-
-
+Figure 26: Accelerometer attachment characteristics
+
#### Location of transducers {#location-of-transducers}
@@ -2124,27 +2196,31 @@ That however requires \\(N\\) to be an integral power of \\(2\\).
Aliasing originates from the discretisation of the originally continuous time history.
With this discretisation process, the **existence of very high frequencies in the original signal may well be misinterpreted if the sampling rate is too slow**.
-These high frequencies will be **indistinguishable** from genuine low frequency components as shown on [Figure 14](#figure--fig:aliasing).
+These high frequencies will be **indistinguishable** from genuine low frequency components as shown on [Figure 27](#figure--fig:aliasing).
-{{< 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" >}}
+{{< figure src="/ox-hugo/ewins00_aliasing.png" caption="Figure 27: The phenomenon of aliasing. On top: Low-frequency signal, On the bottom: High frequency signal" >}}
A signal of frequency \\(\omega\\) and one of frequency \\(\omega\_s-\omega\\) are indistinguishable and this causes a **distortion of the spectrum** measured via the DFT.
As a result, the part of the signal which has frequency components above \\(\omega\_s/2\\) will appear reflected or **aliased** in the range \\([0, \omega\_s/2]\\).
-This is illustrated on [Table 16](#table--fig:effect-aliasing).
+This is illustrated on [Figure 28](#table--fig:effect-aliasing).
-
-
- Table 16:
- Alias distortion of spectrum by DFT
+
+
+
+
+
+
(a) True spectrum of signal
-
-|  |  |
-|------------------------------------------------------------------------------------------------------|-------------------------------------------------------------------------------------------------------|
-| True spectrum of signal | Indicated spectrum from DFT |
-| width=\linewidth | width=\linewidth |
+
+
+
(b) Indicated spectrum from DFT
+
+
+Figure 28: Alias distortion of spectrum by DFT
+
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.
Because the filters used are inevitably less than perfect, and have a finite cut-off rate, it remains necessary to reject the spectral measurement in a frequency range approaching the Nyquist frequency \\(\omega\_s/2\\).
@@ -2156,20 +2232,24 @@ As a results, frequencies near \\(\omega\_s/2\\) may still be contaminated by th
Leakage is a problem which is a direct **consequence of the need to take only a finite length of time history coupled with the assumption of periodicity**.
-
-
- Table 17:
- Sample length and leakage of spectrum
+
+
+
+
+
+
(a) Ideal signal
+
+
+
(b) Awkward signal
+
+
+Figure 29: Sample length and leakage of spectrum
+
-|  |  |
-|-------------------------------------------------------------------------------|----------------------------------------------------------------------------------|
-| Ideal signal | Awkward signal |
-| width=\linewidth | width=\linewidth |
-
-The problem is illustrated on [Table 17](#table--fig:leakage).
-In the first case ([ 17](#org-target--fig-leakage-ok)), the signal is perfectly periodic and the resulting spectrum is just a single line at the frequency of the sine wave.
-In the second case ([ 17](#org-target--fig-leakage-nok)), the periodicity assumption is not strictly valid as there is a discontinuity at each end of the sample.
+The problem is illustrated on [Figure 29](#table--fig:leakage).
+In the first case ([Figure 29a](#org-target--fig-leakage-ok)), the signal is perfectly periodic and the resulting spectrum is just a single line at the frequency of the sine wave.
+In the second case ([Figure 29b](#org-target--fig-leakage-nok)), the periodicity assumption is not strictly valid as there is a discontinuity at each end of the sample.
As a result, the spectrum produced for this case does not indicate the single frequency which the original time signal possessed.
Energy has "leaked" into a number of the spectral lines close to the true frequency and the spectrum is spread over several lines.
@@ -2187,14 +2267,14 @@ Leakage is a serious problem in many applications, **ways of avoiding its effect
Windowing involves the imposition of a prescribed profile on the time signal prior to performing the Fourier transform.
-The profiles, or "windows" are generally depicted as a time function \\(w(t)\\) as shown in [Figure 15](#figure--fig:windowing-examples).
+The profiles, or "windows" are generally depicted as a time function \\(w(t)\\) as shown in [Figure 30](#figure--fig:windowing-examples).
-{{< figure src="/ox-hugo/ewins00_windowing_examples.png" caption="Figure 15: Different types of window. (a) Boxcar, (b) Hanning, (c) Cosine-taper, (d) Exponential" >}}
+{{< figure src="/ox-hugo/ewins00_windowing_examples.png" caption="Figure 30: Different types of window. (a) Boxcar, (b) Hanning, (c) Cosine-taper, (d) Exponential" >}}
The analyzed signal is then \\(x^\prime(t) = x(t) w(t)\\).
-The result of using a window is seen in the third column of [Figure 15](#figure--fig:windowing-examples).
+The result of using a window is seen in the third column of [Figure 30](#figure--fig:windowing-examples).
The **Hanning and Cosine Taper windows are typically used for continuous signals**, such as are produced by steady periodic or random vibration, while the **Exponential window is used for transient vibration** applications where much of the important information is concentrated in the initial part of the time record.
@@ -2234,24 +2314,28 @@ The common solution to the need for finer frequency resolution is to zoom on the
There are various ways of achieving this result.
The easiest way is to use a frequency shifting process coupled with a controlled aliasing device.
-Suppose the signal to be analyzed \\(x(t)\\) has a spectrum \\(X(\omega)\\) has shown on [ 18](#org-target--fig-zoom-range), and that we are interested in a detailed analysis between \\(\omega\_1\\) and \\(\omega\_2\\).
+Suppose the signal to be analyzed \\(x(t)\\) has a spectrum \\(X(\omega)\\) has shown on [Figure 31a](#org-target--fig-zoom-range), and that we are interested in a detailed analysis between \\(\omega\_1\\) and \\(\omega\_2\\).
-If we apply a band-pass filter to the signal, as shown on [ 18](#org-target--fig-zoom-bandpass), and perform a DFT between \\(0\\) and \\((\omega\_2 - \omega\_1)\\), then because of the aliasing phenomenon described earlier, the frequency components between \\(\omega\_1\\) and \\(\omega\_2\\) will appear between \\(0\\) and \\((\omega\_2 - \omega\_1)\\) with the advantage of a finer resolution (see [Figure 16](#figure--fig:zoom-result)).
+If we apply a band-pass filter to the signal, as shown on [Figure 31b](#org-target--fig-zoom-bandpass), and perform a DFT between \\(0\\) and \\((\omega\_2 - \omega\_1)\\), then because of the aliasing phenomenon described earlier, the frequency components between \\(\omega\_1\\) and \\(\omega\_2\\) will appear between \\(0\\) and \\((\omega\_2 - \omega\_1)\\) with the advantage of a finer resolution (see [Figure 32](#figure--fig:zoom-result)).
-
-
- Table 18:
- Controlled aliasing for frequency zoom
+
+
+
+
+
+
(a) Spectrum of the signal
-
-|  |  |
-|-----------------------------------------------------------------------------------------|--------------------------------------------------------------------------------------|
-| Spectrum of the signal | Band-pass filter |
-| width=\linewidth | width=\linewidth |
+
+
+
(b) Band-pass filter
+
+
+Figure 31: Controlled aliasing for frequency zoom
+
-{{< figure src="/ox-hugo/ewins00_zoom_result.png" caption="Figure 16: Effective frequency translation for zoom" >}}
+{{< figure src="/ox-hugo/ewins00_zoom_result.png" caption="Figure 32: Effective frequency translation for zoom" >}}
When using zoom the measure FRF in a narrow frequency range, it is important to ensure that there is as little vibration energy as possible outside the frequency range of interest.
@@ -2319,11 +2403,11 @@ For instance, the typical FRF curve has large region of relatively slow changes
This is the traditional method of FRF measurement and involves the use of a sweep oscillator to provide a sinusoidal command signal with a frequency that varies slowly in the range of interest.
It is necessary to check that progress through the frequency range is sufficiently slow to check that steady-state response conditions are attained.
-If excessive sweep rate is used, then distortions of the FRF plot are introduced as shown on [Figure 17](#figure--fig:sweep-distortions).
+If excessive sweep rate is used, then distortions of the FRF plot are introduced as shown on [Figure 33](#figure--fig:sweep-distortions).
-{{< figure src="/ox-hugo/ewins00_sweep_distortions.png" caption="Figure 17: FRF measurements by sine sweep test" >}}
+{{< figure src="/ox-hugo/ewins00_sweep_distortions.png" caption="Figure 33: FRF measurements by sine sweep test" >}}
One way of checking the suitability of a sweep rate is to make the measurement twice, once sweeping up and the second time sweeping down through the frequency range.
If both curves obtained are the same, the sweep rate is not excessive.
@@ -2437,11 +2521,11 @@ where \\(v(t)\\) is a third signal in the system, such as the voltage supplied t
It is known that a low coherence can arise in a measurement where the frequency resolution of the analyzer is not fine enough to describe adequately the very rapidly changing functions such as are encountered near resonance and anti-resonance on lightly-damped structures.
-This is known as a **bias** error and leakage is often the most likely source of low coherence on lightly-damped structures as shown on [Figure 18](#figure--fig:coherence-resonance).
+This is known as a **bias** error and leakage is often the most likely source of low coherence on lightly-damped structures as shown on [Figure 34](#figure--fig:coherence-resonance).
-{{< 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" >}}
+{{< figure src="/ox-hugo/ewins00_coherence_resonance.png" caption="Figure 34: Coherence \\(\gamma^2\\) and FRF estimate \\(H\_1(\omega)\\) for a lightly damped structure" >}}
It can be shown that near resonance, \\(H\_2(\omega)\\) is a much more accurate representation of the true FRF than \\(H\_1(\omega)\\).
When this situation is encountered, the best solution is usually to make a zoom measurement as explained previously.
@@ -2480,11 +2564,11 @@ For the chirp and impulse excitations, each individual sample is collected and p
##### Burst excitation signals {#burst-excitation-signals}
-Burst excitation signals consist of short sections of an underlying continuous signal (which may be a sine wave, a sine sweep or a random signal), followed by a period of zero output, resulting in a response which shows a transient build-up followed by a decay (see [Figure 19](#figure--fig:burst-excitation)).
+Burst excitation signals consist of short sections of an underlying continuous signal (which may be a sine wave, a sine sweep or a random signal), followed by a period of zero output, resulting in a response which shows a transient build-up followed by a decay (see [Figure 35](#figure--fig:burst-excitation)).
-{{< figure src="/ox-hugo/ewins00_burst_excitation.png" caption="Figure 19: Example of burst excitation and response signals" >}}
+{{< figure src="/ox-hugo/ewins00_burst_excitation.png" caption="Figure 35: Example of burst excitation and response signals" >}}
The duration of the burst is under the control of the operator and it is selected so as to provide the ideal signal processing conditions, which are essentially that the **response signal has just died away by the end of the measurement period**.
If this condition has not been attained (burst too long), then leakage error will result.
@@ -2497,34 +2581,34 @@ In the case of burst random, however, each individual burst will be different to
##### Chirp excitation {#chirp-excitation}
-The chirp consist of a short duration signal which has the form shown in [Figure 20](#figure--fig:chirp-excitation).
+The chirp consist of a short duration signal which has the form shown in [Figure 36](#figure--fig:chirp-excitation).
The frequency content of the chirp can be precisely chosen by the starting and finishing frequencies of the sweep.
-{{< figure src="/ox-hugo/ewins00_chirp_excitation.png" caption="Figure 20: Example of chirp excitation and response signals" >}}
+{{< figure src="/ox-hugo/ewins00_chirp_excitation.png" caption="Figure 36: Example of chirp excitation and response signals" >}}
##### Impulsive excitation {#impulsive-excitation}
-The hammer blow produces an input and response as shown in the [Figure 21](#figure--fig:impulsive-excitation).
+The hammer blow produces an input and response as shown in the [Figure 37](#figure--fig:impulsive-excitation).
This and the chirp excitation are very similar in the analysis point of view, the main difference is that the chirp offers the possibility of greater control of both amplitude and frequency content of the input and also permits the input of a greater amount of vibration energy.
-{{< figure src="/ox-hugo/ewins00_impulsive_excitation.png" caption="Figure 21: Example of impulsive excitation and response signals" >}}
+{{< figure src="/ox-hugo/ewins00_impulsive_excitation.png" caption="Figure 37: Example of impulsive excitation and response signals" >}}
The frequency content of the hammer blow is dictated by the **materials** involved and is rather more difficult to control.
However, it should be recorded that in the region below the first cut-off frequency induced by the elasticity of the hammer tip structure contact, the spectrum of the force signal tends to be **very flat**.
On some structures, the movement of the structure in response to the hammer blow can be such that it returns and **rebounds** on the hammer tip before the user has had time to move that out of the way.
-In such cases, the spectrum of the excitation is seen to have "holes" in it at certain frequencies ([Figure 22](#figure--fig:double-hits)).
+In such cases, the spectrum of the excitation is seen to have "holes" in it at certain frequencies ([Figure 38](#figure--fig:double-hits)).
-{{< figure src="/ox-hugo/ewins00_double_hits.png" caption="Figure 22: Double hits time domain and frequency content" >}}
+{{< figure src="/ox-hugo/ewins00_double_hits.png" caption="Figure 38: Double hits time domain and frequency content" >}}
In order to perform the required Fourier analysis of all these cases of transient signals, an assumption is made that the data obtained from a single event can be regarded as representing one period of a **quasi-periodic process**.
This means that if exactly the same input was applied \\(T\\) seconds after the first one, then exactly the same response would be observed.
@@ -2595,11 +2679,11 @@ and so **what is required is the ratio of the two sensitivities**:
The overall sensitivity can be more readily obtained by a calibration process because we can easily make an independent measurement of the quantity now being measured: the ratio of response to force.
Suppose the response parameter is acceleration, then the FRF obtained is inertance which has the units of \\(1/\text{mass}\\), a quantity which can readily be independently measured by other means.
-[Figure 23](#figure--fig:calibration-setup) shows a typical calibration setup.
+[Figure 39](#figure--fig:calibration-setup) shows a typical calibration setup.
-{{< figure src="/ox-hugo/ewins00_calibration_setup.png" caption="Figure 23: Mass calibration procedure, measurement setup" >}}
+{{< figure src="/ox-hugo/ewins00_calibration_setup.png" caption="Figure 39: Mass calibration procedure, measurement setup" >}}
A calibration procedure of this type has the distinct advantage that it is very easy to perform and can be carried out with all the measurement equipment.
Thus, frequent checks on the overall calibration factors are strongly recommended, ideally as the beginning and end of each test.
@@ -2610,11 +2694,11 @@ Thus, frequent checks on the overall calibration factors are strongly recommende
It is very important the ensure that the force is measured directly at the point at which it is applied to the structure, rather than deducing its magnitude from the current flowing in the shaker coil or other similar **indirect** processes.
This is because near resonance, the actual applied force becomes very small and is thus very prone to inaccuracy.
-This same argument applies on a lesser scale as we examine the detail around the attachment to the structure, as shown in [Figure 24](#figure--fig:mass-cancellation).
+This same argument applies on a lesser scale as we examine the detail around the attachment to the structure, as shown in [Figure 40](#figure--fig:mass-cancellation).
-{{< figure src="/ox-hugo/ewins00_mass_cancellation.png" caption="Figure 24: Added mass to be cancelled (crossed area)" >}}
+{{< figure src="/ox-hugo/ewins00_mass_cancellation.png" caption="Figure 40: Added mass to be cancelled (crossed area)" >}}
Here, we see part of the structure, an accelerometer and a force transducer.
The dashed line shows the plane at which the force is actually measured.
@@ -2667,11 +2751,11 @@ There are two problems to be tackled:
1. measurement of rotational responses
2. generation of measurement of rotation excitation
-The first of these is less difficult and techniques usually use a pair a matched conventional accelerometers placed at a short distance apart on the structure to be measured as shown on [Figure 25](#figure--fig:rotational-measurement).
+The first of these is less difficult and techniques usually use a pair a matched conventional accelerometers placed at a short distance apart on the structure to be measured as shown on [Figure 41](#figure--fig:rotational-measurement).
-{{< figure src="/ox-hugo/ewins00_rotational_measurement.png" caption="Figure 25: Measurement of rotational response" >}}
+{{< figure src="/ox-hugo/ewins00_rotational_measurement.png" caption="Figure 41: Measurement of rotational response" >}}
The principle of operation is that by measuring both accelerometer signals, the responses \\(x\_0\\) and \\(\theta\_0\\) can be deduced by taking the mean and difference of \\(x\_A\\) and \\(x\_B\\):
@@ -2685,14 +2769,14 @@ The principle of operation is that by measuring both accelerometer signals, the
This approach permits us to measure half of the possible FRFs: all those which are of the \\(X/F\\) and \\(\Theta/F\\) type.
The others can only be measured directly by applying a moment excitation.
-[Figure 26](#figure--fig:rotational-excitation) shows a device to simulate a moment excitation.
+[Figure 42](#figure--fig:rotational-excitation) shows a device to simulate a moment excitation.
First, a single applied excitation force \\(F\_1\\) corresponds to a simultaneous force \\(F\_0 = F\_1\\) and a moment \\(M\_0 = -F\_1 l\_1\\).
Then, the same excitation force is applied at the second position that gives a force \\(F\_0 = F\_2\\) and moment \\(M\_0 = F\_2 l\_2\\).
By adding and subtracting the responses produced by these two separate excitations conditions, we can deduce the translational and rotational responses to the translational force and the rotational moment separately, thus enabling the measurement of all four types of FRF: \\(X/F\\), \\(\Theta/F\\), \\(X/M\\) and \\(\Theta/M\\).
-{{< figure src="/ox-hugo/ewins00_rotational_excitation.png" caption="Figure 26: Application of moment excitation" >}}
+{{< figure src="/ox-hugo/ewins00_rotational_excitation.png" caption="Figure 42: Application of moment excitation" >}}
Then, the full \\(6 \times 6\\) mobility matrix can be measured, however this procedure is quite demanding.
@@ -3005,10 +3089,10 @@ Then, each PRF is, simply, a particular combination of the original FRFs, and th
-On example of this form of pre-processing is shown on [Table 19](#table--fig:PRF-numerical) for a numerically-simulation test data, and another in [Table 20](#table--fig:PRF-measured) for the case of real measured test data.
+On example of this form of pre-processing is shown on [Figure 43](#table--fig:PRF-numerical) for a numerically-simulation test data, and another in [Figure 44](#table--fig:PRF-measured) for the case of real measured test data.
-The second plot [ 19](#org-target--fig-PRF-numerical-svd) helps to determine the true order of the system because the number of non-zero singular values is equal to this parameter.
-The third plot [ 19](#org-target--fig-PRF-numerical-PRF) shows the genuine modes distinct from the computational modes.
+The second plot [Figure 43b](#org-target--fig-PRF-numerical-svd) helps to determine the true order of the system because the number of non-zero singular values is equal to this parameter.
+The third plot [Figure 43c](#org-target--fig-PRF-numerical-PRF) shows the genuine modes distinct from the computational modes.
@@ -3026,27 +3110,43 @@ The two groups are usually separated by a clear gap (depending of the noise pres
-
-
- Table 19:
- FRF and PRF characteristics for numerical model
+
+
+
+Figure 44: FRF and PRF characteristics for measured model
+
#### Mode Indicator Functions (MIFs) {#mode-indicator-functions--mifs}
@@ -3076,7 +3176,7 @@ The **Complex mode indicator function** (CMIF) is defined as
-The actual mode indicator values are provided by the squares of the singular values and are usually plotted as a function of frequency in logarithmic form as shown in [Figure 27](#figure--fig:mifs):
+The actual mode indicator values are provided by the squares of the singular values and are usually plotted as a function of frequency in logarithmic form as shown in [Figure 45](#figure--fig:mifs):
- **Natural frequencies are indicated by large values of the first CMIF** (the highest of the singular values)
- **double or multiple modes by simultaneously large values of two or more CMIF**.
@@ -3088,7 +3188,7 @@ Associated with the CMIF values at each natural frequency \\(\omega\_r\\) are tw
-{{< figure src="/ox-hugo/ewins00_mifs.png" caption="Figure 27: Complex Mode Indicator Function (CMIF)" >}}
+{{< figure src="/ox-hugo/ewins00_mifs.png" caption="Figure 45: Complex Mode Indicator Function (CMIF)" >}}
@@ -3157,7 +3257,7 @@ In this method, it is assumed that close to one local mode, any effects due to t
This is a method which works adequately for structures whose FRF exhibit **well separated modes**.
This method is useful in obtaining initial estimates to the parameters.
-The peak-picking method is applied as follows (illustrated on [Figure 28](#figure--fig:peak-amplitude)):
+The peak-picking method is applied as follows (illustrated on [Figure 46](#figure--fig:peak-amplitude)):
1. First, **individual resonance peaks** are detected on the FRF plot and the maximum responses frequency \\(\omega\_r\\) is taken as the **natural frequency** of that mode
2. Second, the **local maximum value of the FRF** \\(|\hat{H}|\\) is noted and the **frequency bandwidth** of the function for a response level of \\(|\hat{H}|/\sqrt{2}\\) is determined.
@@ -3181,7 +3281,7 @@ Only real modal constants and thus real modes can be deduced by this method.
-{{< figure src="/ox-hugo/ewins00_peak_amplitude.png" caption="Figure 28: Peak Amplitude method of modal analysis" >}}
+{{< figure src="/ox-hugo/ewins00_peak_amplitude.png" caption="Figure 46: Peak Amplitude method of modal analysis" >}}
Alternatives of this method can be applied using the real part of the receptance FRF instead of the modulus plot.
@@ -3204,18 +3304,22 @@ In the case of a system assumed to have structural damping, the basic function w
\end{equation}
since the only effect of including the modal constant \\({}\_rA\_{jk}\\) is to scale the size of the circle by \\(|{}\_rA\_{jk}|\\) and to rotate it by \\(\angle {}\_rA\_{jk}\\).
-A plot of the quantity \\(\alpha(\omega)\\) is given in [ 21](#org-target--fig-modal-circle).
+A plot of the quantity \\(\alpha(\omega)\\) is given in [Figure 47a](#org-target--fig-modal-circle).
-
-
+Figure 47: Modal Circle
+
For any frequency \\(\omega\\), we have the following relationship:
@@ -3252,7 +3356,7 @@ It may also be seen that an **estimate of the damping** is provided by the sweep
\end{equation}
Suppose now we have two specific points on the circle, one corresponding to a frequency \\(\omega\_b\\) below the natural frequency and the other one \\(\omega\_a\\) above the natural frequency.
-Referring to [ 21](#org-target--fig-modal-circle-bis), we can write:
+Referring to [Figure 47b](#org-target--fig-modal-circle-bis), we can write:
\begin{equation}
\begin{aligned}
@@ -3318,7 +3422,7 @@ The sequence is:
3. **Locate natural frequency, obtain damping estimate**.
The rate of sweep through the region is estimated numerically and the frequency at which it reaches the maximum is deduced.
At the same time, an estimate of the damping is derived using \ref{eq:estimate\_damping\_sweep\_rate}.
- A typical example is shown on [Figure 29](#figure--fig:circle-fit-natural-frequency).
+ A typical example is shown on [Figure 48](#figure--fig:circle-fit-natural-frequency).
4. **Calculate multiple damping estimates, and scatter**.
A set of damping estimates using all possible combination of the selected data points are computed using \ref{eq:estimate\_damping}.
Then, we can choose the damping estimate to be the mean value.
@@ -3330,7 +3434,7 @@ The sequence is:
-{{< figure src="/ox-hugo/ewins00_circle_fit_natural_frequency.png" caption="Figure 29: Location of natural frequency for a Circle-fit modal analysis" >}}
+{{< figure src="/ox-hugo/ewins00_circle_fit_natural_frequency.png" caption="Figure 48: Location of natural frequency for a Circle-fit modal analysis" >}}
Then, the theoretically regenerated FRF can be plotted against the original measured data for comparison.
In order to determines the contribution of other modes on the resonance of mode \\(r\\), the distance from the top of the principal diameter to the origin has to be measured and is equal to \\({}\_rB\_{jk}\\).
@@ -3440,19 +3544,23 @@ We need to introduce the concept of **residual terms**, necessary in the modal a
The first occasion on which the residual problem is encountered is generally at the end of the analysis of a single FRF curve, such as by the repeated application of an SDOF curve-fit to each of the resonances in turn until all modes visible on the plot have been identified.
At this point, it is often desired to construct a theoretical curve (called "**regenerated**"), based on the modal parameters extracted from the measured data, and to overlay this on the original measured data to assess the success of the curve-fit process.
-Then the regenerated curve is compared with the original measurements, the result is often disappointing, as illustrated in [ 22](#org-target--fig-residual-without).
-However, by the inclusion of two simple extra terms (the "**residuals**"), the modified regenerated curve is seen to correlate very well with the original experimental data as shown on [ 22](#org-target--fig-residual-with).
+Then the regenerated curve is compared with the original measurements, the result is often disappointing, as illustrated in [Figure 49a](#org-target--fig-residual-without).
+However, by the inclusion of two simple extra terms (the "**residuals**"), the modified regenerated curve is seen to correlate very well with the original experimental data as shown on [Figure 49b](#org-target--fig-residual-with).
-
-
- Table 22:
- Effects of residual terms on FRF regeneration
+
+
+
+
+
+
(a) without residual
-
-|  |  |
-|-----------------------------------------------------------------------------------------|------------------------------------------------------------------------------------|
-| without residual | with residuals |
-| width=\linewidth | width=\linewidth |
+
+
+
(b) with residuals
+
+
+Figure 49: Effects of residual terms on FRF regeneration
+
If we regenerate an FRF curve from the modal parameters we have extracted from the measured data, we shall use a formula of the type
@@ -3480,11 +3588,11 @@ The three terms corresponds to:
2. the **high frequency modes** not identified
3. the **modes actually identified**
-These three terms are illustrated on [Figure 30](#figure--fig:low-medium-high-modes).
+These three terms are illustrated on [Figure 50](#figure--fig:low-medium-high-modes).
-{{< figure src="/ox-hugo/ewins00_low_medium_high_modes.png" caption="Figure 30: Numerical simulation of contribution of low, medium and high frequency modes" >}}
+{{< figure src="/ox-hugo/ewins00_low_medium_high_modes.png" caption="Figure 50: Numerical simulation of contribution of low, medium and high frequency modes" >}}
From the sketch, it may be seen that within the frequency range of interest:
@@ -3772,18 +3880,22 @@ with
The composite function \\(HH(\omega)\\) can provide a useful means of determining a single (average) value for the natural frequency and damping factor for each mode where the individual functions would each indicate slightly different values.
-As an example, a set of mobilities measured are shown individually in [ 23](#org-target--fig-composite-raw) and their summation shown as a single composite curve in [ 23](#org-target--fig-composite-sum).
+As an example, a set of mobilities measured are shown individually in [Figure 51a](#org-target--fig-composite-raw) and their summation shown as a single composite curve in [Figure 51b](#org-target--fig-composite-sum).
-
-
+Figure 51: Set of measured FRF
+
The global analysis methods have the disadvantages first, that the computation power required is high and second that there may be valid reasons why the various FRF curves exhibit slight differences in their characteristics and it may not always be appropriate to average them.
@@ -4331,29 +4443,29 @@ There are basically two choices for the graphical display of a modal model:
##### Deflected shapes {#deflected-shapes}
A static display is often adequate for depicting relatively simple mode shapes.
-Measured coordinates of the test structure are first linked as shown on [Figure 31](#figure--fig:static-display) (a).
-Then, the grid of measured coordinate points is redrawn on the same plot but this time displaced by an amount proportional to the corresponding element in the mode shape vector as shown on [Figure 31](#figure--fig:static-display) (b).
+Measured coordinates of the test structure are first linked as shown on [Figure 52](#figure--fig:static-display) (a).
+Then, the grid of measured coordinate points is redrawn on the same plot but this time displaced by an amount proportional to the corresponding element in the mode shape vector as shown on [Figure 52](#figure--fig:static-display) (b).
The elements in the vector are scaled according the normalization process used (usually mass-normalized), and their absolute magnitudes have no particular significance.
-{{< 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" >}}
+{{< figure src="/ox-hugo/ewins00_static_display.png" caption="Figure 52: 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" >}}
It is customary to select the largest eigenvector element and to scale the whole vector by an amount that makes that displacement on the plot a viable amount.
##### Multiple frames {#multiple-frames}
-If a series of deflection patterns that has been computed for a different instant of time are superimposed, we obtain a result as shown on [Figure 31](#figure--fig:static-display) (c).
+If a series of deflection patterns that has been computed for a different instant of time are superimposed, we obtain a result as shown on [Figure 52](#figure--fig:static-display) (c).
Some indication of the motion of the structure can be obtained, and the points of zero motion (nodes) can be clearly identified.
-It is also possible, in this format, to give some indication of the essence of complex modes, as shown in [Figure 31](#figure--fig:static-display) (d).
+It is also possible, in this format, to give some indication of the essence of complex modes, as shown in [Figure 52](#figure--fig:static-display) (d).
Complex modes do not, in general, exhibit fixed nodal points.
##### Argand diagram plots {#argand-diagram-plots}
-Another form of representation which is useful for complex modes is the representation of the individual complex elements of the eigenvectors on a polar plot, as shown in the examples of [Figure 31](#figure--fig:static-display) (e) and (f).
+Another form of representation which is useful for complex modes is the representation of the individual complex elements of the eigenvectors on a polar plot, as shown in the examples of [Figure 52](#figure--fig:static-display) (e) and (f).
Although there is no attempt to show the physical deformation of the actual structure in this format, the complexity of the mode shape is graphically displayed.
@@ -4376,13 +4488,13 @@ We then tend to interpret this as a motion which is purely in the x-direction wh
The second problem arises when the **grid of measurement points** that is chosen to display the mode shapes is **too coarse in relation to the complexity of the deformation patterns** that are to be displayed.
This can be illustrated using a very simple example: suppose that our test structure is a straight beam, and that we decide to use just three response measurements points.
-If we consider the first six modes of the beam, whose mode shapes are sketched in [Figure 32](#figure--fig:beam-modes), then we see that with this few measurement points, modes 1 and 5 look the same as do modes 2, 4 and 6.
+If we consider the first six modes of the beam, whose mode shapes are sketched in [Figure 53](#figure--fig:beam-modes), then we see that with this few measurement points, modes 1 and 5 look the same as do modes 2, 4 and 6.
All the higher modes will be indistinguishable from these first few.
This is a well known problem of **spatial aliasing**.
-{{< figure src="/ox-hugo/ewins00_beam_modes.png" caption="Figure 32: Misinterpretation of mode shapes by spatial aliasing" >}}
+{{< figure src="/ox-hugo/ewins00_beam_modes.png" caption="Figure 53: Misinterpretation of mode shapes by spatial aliasing" >}}
### Response models {#response-models}
@@ -4425,24 +4537,28 @@ However, it must be noted that there is an important **limitation to this proced
As an example, suppose that FRF data \\(H\_{11}\\) and \\(H\_{21}\\) are measured and analyzed in order to synthesize the FRF \\(H\_{22}\\) initially unmeasured.
-The predict curve is compared with the measurements on [ 24](#org-target--fig-H22-without-residual).
+The predict curve is compared with the measurements on [Figure 54a](#org-target--fig-H22-without-residual).
Clearly, the agreement is poor and would tend to indicate that the measurement/analysis process had not been successful.
However, the synthesized curve contained only those terms relating to the modes which had actually been studied from \\(H\_{11}\\) and \\(H\_{21}\\) and this set of modes did not include **all** the modes of the structure.
Thus, \\(H\_{22}\\) **omitted the influence of out-of-range modes**.
-The inclusion of these two additional terms (obtained here only after measuring and analyzing \\(H\_{22}\\) itself) resulted in the greatly improved predicted vs measured comparison shown in [ 24](#org-target--fig-H22-with-residual).
+The inclusion of these two additional terms (obtained here only after measuring and analyzing \\(H\_{22}\\) itself) resulted in the greatly improved predicted vs measured comparison shown in [Figure 54b](#org-target--fig-H22-with-residual).
-|  |  |
-|-----------------------------------------------------------------------------------------------------------|-----------------------------------------------------------------------------------------------------------|
-| Using measured modal data only | After inclusion of residual terms |
-| width=\linewidth | width=\linewidth |
+
+
+
+
+
(a) Using measured modal data only
+
+
+
+
(b) After inclusion of residual terms
+
+
+Figure 54: Synthesized FRF plot
+
The appropriate expression for a "correct" response model, derived via a set of modal properties is thus
@@ -4492,32 +4608,36 @@ If the **transmissibility** is measured during a modal test which has a single e
-In general, the transmissibility **depends significantly on the excitation point** (\\({}\_iT\_{jk}(\omega) \neq {}\_qT\_{jk}(\omega)\\) where \\(q\\) is a different DOF than \\(i\\)) and it is shown on [Figure 33](#figure--fig:transmissibility-plots).
+In general, the transmissibility **depends significantly on the excitation point** (\\({}\_iT\_{jk}(\omega) \neq {}\_qT\_{jk}(\omega)\\) where \\(q\\) is a different DOF than \\(i\\)) and it is shown on [Figure 55](#figure--fig:transmissibility-plots).
This may explain why transmissibilities are not widely used in modal analysis.
-{{< figure src="/ox-hugo/ewins00_transmissibility_plots.png" caption="Figure 33: Transmissibility plots" >}}
+{{< figure src="/ox-hugo/ewins00_transmissibility_plots.png" caption="Figure 55: Transmissibility plots" >}}
#### Base excitation {#base-excitation}
The one application area where transmissibilities can be used as part of modal testing is in the case of **base excitation**.
-Base excitation is a type of test where the input is measured as a response at the drive point \\(x\_0(t)\\), instead of as a force \\(f\_1(t)\\), as illustrated in [Table 25](#table--fig:base-excitation-configuration).
+Base excitation is a type of test where the input is measured as a response at the drive point \\(x\_0(t)\\), instead of as a force \\(f\_1(t)\\), as illustrated in [Figure 56](#table--fig:base-excitation-configuration).
We can show that it is possible to determine, from measurements of \\(x\_i\\) and \\(x\_0\\), modal properties of natural frequency, damping factor and **unscaled** mode shape for each of the modes that are visible in the frequency range of measurement.
The fact that the excitation force is not measured is responsible for the lack of formal scaling of the mode shapes.
-
-
+Figure 56: Base excitation configuration
+
### Spatial models {#spatial-models}
diff --git a/content/book/skogestad07_multiv_feedb_contr.md b/content/book/skogestad07_multiv_feedb_contr.md
index 2a9899b..19a3217 100644
--- a/content/book/skogestad07_multiv_feedb_contr.md
+++ b/content/book/skogestad07_multiv_feedb_contr.md
@@ -3627,7 +3627,7 @@ However, the inclusion of parametric uncertainty may be more significant for MIM
Unstructured perturbations are often used to get a simple uncertainty model.
We here define unstructured uncertainty as the use of a "full" complex perturbation matrix \\(\Delta\\), usually with dimensions compatible with those of the plant, where at each frequency any \\(\Delta(j\w)\\) satisfying \\(\maxsv(\Delta(j\w)) < 1\\) is allowed.
-Three common forms of **feedforward unstructured uncertainty** are shown [Table 4](#table--fig:feedforward-uncertainty): additive uncertainty, multiplicative input uncertainty and multiplicative output uncertainty.
+Three common forms of **feedforward unstructured uncertainty** are shown [Figure 29](#table--fig:feedforward-uncertainty): additive uncertainty, multiplicative input uncertainty and multiplicative output uncertainty.
@@ -3641,17 +3641,26 @@ Three common forms of **feedforward unstructured uncertainty** are shown [Table
+Figure 30: Common feedback unstructured uncertainty
+
##### Lumping uncertainty into a single perturbation {#lumping-uncertainty-into-a-single-perturbation}
@@ -3768,12 +3786,12 @@ where \\(r\_0\\) is the relative uncertainty at steady-state, \\(1/\tau\\) is th
### Obtaining \\(P\\), \\(N\\) and \\(M\\) {#obtaining-p-n-and-m}
-Let's consider the feedback system with multiplicative input uncertainty \\(\Delta\_I\\) shown [Figure 29](#figure--fig:input-uncertainty-set-feedback-weight).
+Let's consider the feedback system with multiplicative input uncertainty \\(\Delta\_I\\) shown [Figure 31](#figure--fig:input-uncertainty-set-feedback-weight).
\\(W\_I\\) is a normalization weight for the uncertainty and \\(W\_P\\) is a performance weight.
-{{< 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" >}}
+{{< figure src="/ox-hugo/skogestad07_input_uncertainty_set_feedback_weight.png" caption="Figure 31: System with multiplicative input uncertainty and performance measured at the output" >}}
We want to derive the generalized plant \\(P\\) which has inputs \\([u\_\Delta,\ w,\ u]^T\\) and outputs \\([y\_\Delta,\ z,\ v]^T\\).
@@ -3906,7 +3924,7 @@ Then the \\(M\Delta\text{-system}\\) is stable for all perturbations \\(\Delta\\
#### Application of the Unstructured RS-condition {#application-of-the-unstructured-rs-condition}
-We will now present necessary and sufficient conditions for robust stability for each of the six single unstructured perturbations in [Table 4](#table--fig:feedforward-uncertainty) and [Table 5](#table--fig:feedback-uncertainty) with
+We will now present necessary and sufficient conditions for robust stability for each of the six single unstructured perturbations in [Figure 29](#table--fig:feedforward-uncertainty) and [Figure 30](#table--fig:feedback-uncertainty) with
\begin{equation\*}
E = W\_2 \Delta W\_1, \quad \hnorm{\Delta} \le 1
@@ -3951,7 +3969,7 @@ In order to get tighter condition we must use a tighter uncertainty description
Robust stability bound in terms of the \\(\hinf\\) norm (\\(\text{RS}\Leftrightarrow\hnorm{M}<1\\)) are in general only tight when there is a single full perturbation block.
An "exception" to this is when the uncertainty blocks enter or exit from the same location in the block diagram, because they can then be stacked on top of each other or side-by-side, in an overall \\(\Delta\\) which is then full matrix.
-One important uncertainty description that falls into this category is the **coprime uncertainty description** shown in [Figure 30](#figure--fig:coprime-uncertainty), for which the set of plants is
+One important uncertainty description that falls into this category is the **coprime uncertainty description** shown in [Figure 32](#figure--fig:coprime-uncertainty), for which the set of plants is
\begin{equation\*}
G\_p = (M\_l + \Delta\_M)^{-1}(Nl + \Delta\_N), \quad \hnorm{[\Delta\_N, \ \Delta\_N]} \le \epsilon
@@ -3963,7 +3981,7 @@ This uncertainty description is surprisingly **general**, it allows both zeros a
-{{< figure src="/ox-hugo/skogestad07_coprime_uncertainty.png" caption="Figure 30: Coprime Uncertainty" >}}
+{{< figure src="/ox-hugo/skogestad07_coprime_uncertainty.png" caption="Figure 32: Coprime Uncertainty" >}}
Since we have no weights on the perturbations, it is reasonable to use a normalized coprime factorization of the nominal plant.
In any case, to test for RS we can rearrange the block diagram to match the \\(M\Delta\text{-structure}\\) with
@@ -4007,12 +4025,12 @@ To this effect, introduce the block-diagonal scaling matrix
where \\(d\_i\\) is a scalar and \\(I\_i\\) is an identity matrix of the same dimension as the \\(i\\)'th perturbation block \\(\Delta\_i\\).
-Now rescale the inputs and outputs of \\(M\\) and \\(\Delta\\) by inserting the matrices \\(D\\) and \\(D^{-1}\\) on both sides as shown in [Figure 31](#figure--fig:block-diagonal-scalings).
+Now rescale the inputs and outputs of \\(M\\) and \\(\Delta\\) by inserting the matrices \\(D\\) and \\(D^{-1}\\) on both sides as shown in [Figure 33](#figure--fig:block-diagonal-scalings).
This clearly has no effect on stability.
-{{< figure src="/ox-hugo/skogestad07_block_diagonal_scalings.png" caption="Figure 31: Use of block-diagonal scalings, \\(\Delta D = D \Delta\\)" >}}
+{{< figure src="/ox-hugo/skogestad07_block_diagonal_scalings.png" caption="Figure 33: Use of block-diagonal scalings, \\(\Delta D = D \Delta\\)" >}}
Note that with the chosen form for the scalings we have for each perturbation block \\(\Delta\_i = d\_i \Delta\_i d\_i^{-1}\\), that is we have \\(\Delta = D \Delta D^{-1}\\).
@@ -4302,7 +4320,7 @@ Note that \\(\mu\\) underestimate how bad or good the actual worst case performa
### Application: RP with Input Uncertainty {#application-rp-with-input-uncertainty}
-We will now consider in some detail the case of multiplicative input uncertainty with performance defined in terms of weighted sensitivity ([Figure 29](#figure--fig:input-uncertainty-set-feedback-weight)).
+We will now consider in some detail the case of multiplicative input uncertainty with performance defined in terms of weighted sensitivity ([Figure 31](#figure--fig:input-uncertainty-set-feedback-weight)).
The performance requirement is then
@@ -4416,11 +4434,11 @@ with the decoupling controller we have:
\overline{\sigma}(N\_{22}) = \overline{\sigma}(w\_P S) = \left|\frac{s/2 + 0.05}{s + 0.7}\right|
\end{equation\*}
-and we see from [Figure 32](#figure--fig:mu-plots-distillation) that the NP-condition is satisfied.
+and we see from [Figure 34](#figure--fig:mu-plots-distillation) that the NP-condition is satisfied.
-{{< figure src="/ox-hugo/skogestad07_mu_plots_distillation.png" caption="Figure 32: \\(\mu\text{-plots}\\) for distillation process with decoupling controller" >}}
+{{< figure src="/ox-hugo/skogestad07_mu_plots_distillation.png" caption="Figure 34: \\(\mu\text{-plots}\\) for distillation process with decoupling controller" >}}
##### RS {#rs}
@@ -4431,7 +4449,7 @@ In this case \\(w\_I T\_I = w\_I T\\) is a scalar times the identity matrix:
\mu\_{\Delta\_I}(w\_I T\_I) = |w\_I t| = \left|0.2 \frac{5s + 1}{(0.5s + 1)(1.43s + 1)}\right|
\end{equation\*}
-and we see from [Figure 32](#figure--fig:mu-plots-distillation) that RS is satisfied.
+and we see from [Figure 34](#figure--fig:mu-plots-distillation) that RS is satisfied.
The peak value of \\(\mu\_{\Delta\_I}(M)\\) is \\(0.53\\) meaning that we may increase the uncertainty by a factor of \\(1/0.53 = 1.89\\) before the worst case uncertainty yields instability.
@@ -4439,7 +4457,7 @@ The peak value of \\(\mu\_{\Delta\_I}(M)\\) is \\(0.53\\) meaning that we may in
##### RP {#rp}
Although the system has good robustness margins and excellent nominal performance, the robust performance is poor.
-This is shown in [Figure 32](#figure--fig:mu-plots-distillation) where the \\(\mu\text{-curve}\\) for RP was computed numerically using \\(\mu\_{\hat{\Delta}}(N)\\), with \\(\hat{\Delta} = \text{diag}\\{\Delta\_I, \Delta\_P\\}\\) and \\(\Delta\_I = \text{diag}\\{\delta\_1, \delta\_2\\}\\).
+This is shown in [Figure 34](#figure--fig:mu-plots-distillation) where the \\(\mu\text{-curve}\\) for RP was computed numerically using \\(\mu\_{\hat{\Delta}}(N)\\), with \\(\hat{\Delta} = \text{diag}\\{\Delta\_I, \Delta\_P\\}\\) and \\(\Delta\_I = \text{diag}\\{\delta\_1, \delta\_2\\}\\).
The peak value is close to 6, meaning that even with 6 times less uncertainty, the weighted sensitivity will be about 6 times larger than what we require.
@@ -4576,11 +4594,11 @@ The latter is an attempt to "flatten out" \\(\mu\\).
#### Example: \\(\mu\text{-synthesis}\\) with DK-iteration {#example-mu-text-synthesis-with-dk-iteration}
For simplicity, we will consider again the case of multiplicative uncertainty and performance defined in terms of weighted sensitivity.
-The uncertainty weight \\(w\_I I\\) and performance weight \\(w\_P I\\) are shown graphically in [Figure 33](#figure--fig:weights-distillation).
+The uncertainty weight \\(w\_I I\\) and performance weight \\(w\_P I\\) are shown graphically in [Figure 35](#figure--fig:weights-distillation).
-{{< figure src="/ox-hugo/skogestad07_weights_distillation.png" caption="Figure 33: Uncertainty and performance weights" >}}
+{{< figure src="/ox-hugo/skogestad07_weights_distillation.png" caption="Figure 35: Uncertainty and performance weights" >}}
The objective is to minimize the peak value of \\(\mu\_{\tilde{\Delta}}(N)\\), \\(\tilde{\Delta} = \text{diag}\\{\Delta\_I, \Delta\_P\\}\\).
\\(\Delta\_I\\) is a diagonal \\(2 \times 2\\) matrix representing the diagonal input uncertainty and \\(\Delta\_P\\) is a full \\(2 \times 2\\) matrix representing the performance specifications.
@@ -4592,8 +4610,8 @@ The scaling matrix \\(D\\) for \\(DND^{-1}\\) then has the structure \\(D = \tex
- Iteration No. 1.
Step 1: with the initial scalings, the \\(\mathcal{H}\_\infty\\) synthesis produced a 6 state controller (2 states from the plant model and 2 from each of the weights).
- Step 2: the upper \\(\mu\text{-bound}\\) is shown in [Figure 34](#figure--fig:dk-iter-mu).
- Step 3: the frequency dependent \\(d\_1(\omega)\\) and \\(d\_2(\omega)\\) from step 2 are fitted using a 4th order transfer function shown in [Figure 35](#figure--fig:dk-iter-d-scale)
+ Step 2: the upper \\(\mu\text{-bound}\\) is shown in [Figure 36](#figure--fig:dk-iter-mu).
+ Step 3: the frequency dependent \\(d\_1(\omega)\\) and \\(d\_2(\omega)\\) from step 2 are fitted using a 4th order transfer function shown in [Figure 37](#figure--fig:dk-iter-d-scale)
- Iteration No. 2.
Step 1: with the 8 state scalings \\(D^1(s)\\), the \\(\mathcal{H}\_\infty\\) synthesis gives a 22 state controller.
Step 2: This controller gives a peak value of \\(\mu\\) of \\(1.02\\).
@@ -4603,25 +4621,25 @@ The scaling matrix \\(D\\) for \\(DND^{-1}\\) then has the structure \\(D = \tex
-{{< figure src="/ox-hugo/skogestad07_dk_iter_mu.png" caption="Figure 34: Change in \\(\mu\\) during DK-iteration" >}}
+{{< figure src="/ox-hugo/skogestad07_dk_iter_mu.png" caption="Figure 36: Change in \\(\mu\\) during DK-iteration" >}}
-{{< figure src="/ox-hugo/skogestad07_dk_iter_d_scale.png" caption="Figure 35: Change in D-scale \\(d\_1\\) during DK-iteration" >}}
+{{< figure src="/ox-hugo/skogestad07_dk_iter_d_scale.png" caption="Figure 37: Change in D-scale \\(d\_1\\) during DK-iteration" >}}
-The final \\(\mu\text{-curves}\\) for NP, RS and RP with the controller \\(K\_3\\) are shown in [Figure 36](#figure--fig:mu-plot-optimal-k3).
+The final \\(\mu\text{-curves}\\) for NP, RS and RP with the controller \\(K\_3\\) are shown in [Figure 38](#figure--fig:mu-plot-optimal-k3).
The objectives of RS and NP are easily satisfied.
The peak value of \\(\mu\\) is just slightly over 1, so the performance specification \\(\overline{\sigma}(w\_P S\_p) < 1\\) is almost satisfied for all possible plants.
-{{< figure src="/ox-hugo/skogestad07_mu_plot_optimal_k3.png" caption="Figure 36: \\(mu\text{-plots}\\) with \\(\mu\\) "optimal" controller \\(K\_3\\)" >}}
+{{< figure src="/ox-hugo/skogestad07_mu_plot_optimal_k3.png" caption="Figure 38: \\(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 [Figure 37](#figure--fig:perturb-s-k3).
+To confirm that, 6 perturbed plants are used to compute the perturbed sensitivity functions shown in [Figure 39](#figure--fig:perturb-s-k3).
-{{< figure src="/ox-hugo/skogestad07_perturb_s_k3.png" caption="Figure 37: Perturbed sensitivity functions \\(\overline{\sigma}(S^\prime)\\) using \\(\mu\\) "optimal" controller \\(K\_3\\). Lower solid line: nominal plant. Upper solid line: worst-case plant" >}}
+{{< figure src="/ox-hugo/skogestad07_perturb_s_k3.png" caption="Figure 39: Perturbed sensitivity functions \\(\overline{\sigma}(S^\prime)\\) using \\(\mu\\) "optimal" controller \\(K\_3\\). Lower solid line: nominal plant. Upper solid line: worst-case plant" >}}
### Further Remarks on \\(\mu\\) {#further-remarks-on-mu}
@@ -4696,7 +4714,7 @@ By multivariable transfer function shaping, therefore, we mean the shaping of th
The classical loop-shaping ideas can be further generalized to MIMO systems by considering the singular values.
-Consider the one degree-of-freedom system as shown in [Figure 38](#figure--fig:classical-feedback-small).
+Consider the one degree-of-freedom system as shown in [Figure 40](#figure--fig:classical-feedback-small).
We have the following important relationships:
\begin{align}
@@ -4706,7 +4724,7 @@ We have the following important relationships:
-{{< figure src="/ox-hugo/skogestad07_classical_feedback_small.png" caption="Figure 38: One degree-of-freedom feedback configuration" >}}
+{{< figure src="/ox-hugo/skogestad07_classical_feedback_small.png" caption="Figure 40: One degree-of-freedom feedback configuration" >}}
@@ -4750,11 +4768,11 @@ 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 [Figure 39](#figure--fig:design-trade-off-mimo-gk).
+Typically, the open-loop requirements 1 and 3 are valid and important at low frequencies \\(0 \le \omega \le \omega\_l \le \omega\_B\\), while conditions 2, 4, 5 and 6 are conditions which are valid and important at high frequencies \\(\omega\_B \le \omega\_h \le \omega \le \infty\\), as illustrated in [Figure 41](#figure--fig:design-trade-off-mimo-gk).
-{{< figure src="/ox-hugo/skogestad07_design_trade_off_mimo_gk.png" caption="Figure 39: Design trade-offs for the multivariable loop transfer function \\(GK\\)" >}}
+{{< figure src="/ox-hugo/skogestad07_design_trade_off_mimo_gk.png" caption="Figure 41: Design trade-offs for the multivariable loop transfer function \\(GK\\)" >}}
The control engineer must design \\(K\\) such that \\(\minsv(GK)\\) lies above a performance boundary for all \\(\omega\\) up to \\(\omega\_l\\), and such that \\(\maxsv(GK)\\) lies below a robustness boundary for all \\(\omega\\) above \\(\omega\_h\\).
@@ -4810,11 +4828,11 @@ The optimal state estimate is given by a **Kalman filter**.
The solution to the LQG problem is then found by replacing \\(x\\) by \\(\hat{x}\\) to give \\(u(t) = -K\_r \hat{x}\\).
-We therefore see that the LQG problem and its solution can be separated into two distinct parts as illustrated in [Figure 40](#figure--fig:lqg-separation): the optimal state feedback and the optimal state estimator (the Kalman filter).
+We therefore see that the LQG problem and its solution can be separated into two distinct parts as illustrated in [Figure 42](#figure--fig:lqg-separation): the optimal state feedback and the optimal state estimator (the Kalman filter).
-{{< figure src="/ox-hugo/skogestad07_lqg_separation.png" caption="Figure 40: The separation theorem" >}}
+{{< figure src="/ox-hugo/skogestad07_lqg_separation.png" caption="Figure 42: The separation theorem" >}}
@@ -4842,7 +4860,7 @@ and \\(X\\) is the unique positive-semi definite solution of the algebraic Ricca
-The **Kalman filter** has the structure of an ordinary state-estimator, as shown on [Figure 41](#figure--fig:lqg-kalman-filter), with:
+The **Kalman filter** has the structure of an ordinary state-estimator, as shown on [Figure 43](#figure--fig:lqg-kalman-filter), with:
\begin{equation} \label{eq:kalman\_filter\_structure}
\dot{\hat{x}} = A\hat{x} + Bu + K\_f(y-C\hat{x})
@@ -4864,9 +4882,9 @@ Where \\(Y\\) is the unique positive-semi definite solution of the algebraic Ric
-{{< figure src="/ox-hugo/skogestad07_lqg_kalman_filter.png" caption="Figure 41: The LQG controller and noisy plant" >}}
+{{< figure src="/ox-hugo/skogestad07_lqg_kalman_filter.png" caption="Figure 43: The LQG controller and noisy plant" >}}
-The structure of the LQG controller is illustrated in [Figure 41](#figure--fig:lqg-kalman-filter), its transfer function from \\(y\\) to \\(u\\) is given by
+The structure of the LQG controller is illustrated in [Figure 43](#figure--fig:lqg-kalman-filter), its transfer function from \\(y\\) to \\(u\\) is given by
\begin{align\*}
L\_{\text{LQG}}(s) &= \left[ \begin{array}{c|c}
@@ -4881,11 +4899,11 @@ The structure of the LQG controller is illustrated in [Figure 41](#figure--fig:l
It has the same degree (number of poles) as the plant.
-For the LQG-controller, as shown on [Figure 41](#figure--fig:lqg-kalman-filter), it is not easy to see where to position the reference input \\(r\\) and how integral action may be included, if desired. Indeed, the standard LQG design procedure does not give a controller with integral action. One strategy is illustrated in [Figure 42](#figure--fig:lqg-integral). Here, the control error \\(r-y\\) is integrated and the regulator \\(K\_r\\) is designed for the plant augmented with these integral states.
+For the LQG-controller, as shown on [Figure 43](#figure--fig:lqg-kalman-filter), it is not easy to see where to position the reference input \\(r\\) and how integral action may be included, if desired. Indeed, the standard LQG design procedure does not give a controller with integral action. One strategy is illustrated in [Figure 44](#figure--fig:lqg-integral). Here, the control error \\(r-y\\) is integrated and the regulator \\(K\_r\\) is designed for the plant augmented with these integral states.
-{{< figure src="/ox-hugo/skogestad07_lqg_integral.png" caption="Figure 42: LQG controller with integral action and reference input" >}}
+{{< figure src="/ox-hugo/skogestad07_lqg_integral.png" caption="Figure 44: LQG controller with integral action and reference input" >}}
For an LQG-controller system with a combined Kalman filter and LQR control law, there are **no guaranteed stability margins**, and there exist LQG combinations with arbitrary small gain margins.
However, there are procedures for improving robustness properties of LQG control such as **Loop Transfer Recovery** (LTR).
@@ -4905,11 +4923,11 @@ Their main limitation is that they can only be applied to minimum phase plants.
There are many ways in which feedback design problems can be cast as \\(\htwo\\) and \\(\hinf\\) optimization problems.
It is very useful therefore to have a **standard problem formulation** into which any particular problem may be manipulated.
-Such a general formulation is afforded by the general configuration shown in [Figure 43](#figure--fig:general-control).
+Such a general formulation is afforded by the general configuration shown in [Figure 45](#figure--fig:general-control).
-{{< figure src="/ox-hugo/skogestad07_general_control.png" caption="Figure 43: General control configuration" >}}
+{{< figure src="/ox-hugo/skogestad07_general_control.png" caption="Figure 45: General control configuration" >}}
The system is described by
@@ -5085,7 +5103,7 @@ Then the LQG cost function is
#### \\(\hinf\\) Optimal Control {#hinf-optimal-control}
-With reference to the general control configuration on [Figure 43](#figure--fig:general-control), the standard \\(\hinf\\) optimal control problem is to find all stabilizing controllers \\(K\\) which minimize
+With reference to the general control configuration on [Figure 45](#figure--fig:general-control), the standard \\(\hinf\\) optimal control problem is to find all stabilizing controllers \\(K\\) which minimize
\begin{equation\*}
\hnorm{F\_l(P, K)} = \max\_{\omega} \maxsv\big(F\_l(P, K)(j\omega)\big)
@@ -5196,7 +5214,7 @@ In general, the scalar weighting functions \\(w\_1(s)\\) and \\(w\_2(s)\\) can b
This can be useful for **systems with channels of quite different bandwidths**.
In that case, **diagonal weights are recommended** as anything more complicated is usually not worth the effort.
-To see how this mixed sensitivity problem can be formulated in the general setting, we can imagine the disturbance \\(d\\) as a single exogenous input and define and error signal \\(z = [z\_1^T\ z\_2^T]^T\\), where \\(z\_1 = W\_1 y\\) and \\(z\_2 = -W\_2 u\\) as illustrated in [Figure 44](#figure--fig:mixed-sensitivity-dist-rejection).
+To see how this mixed sensitivity problem can be formulated in the general setting, we can imagine the disturbance \\(d\\) as a single exogenous input and define and error signal \\(z = [z\_1^T\ z\_2^T]^T\\), where \\(z\_1 = W\_1 y\\) and \\(z\_2 = -W\_2 u\\) as illustrated in [Figure 46](#figure--fig:mixed-sensitivity-dist-rejection).
We can then see that \\(z\_1 = W\_1 S w\\) and \\(z\_2 = W\_2 KS w\\) as required.
The elements of the generalized plant are
@@ -5215,16 +5233,16 @@ The elements of the generalized plant are
-{{< figure src="/ox-hugo/skogestad07_mixed_sensitivity_dist_rejection.png" caption="Figure 44: \\(S/KS\\) mixed-sensitivity optimization in standard form (regulation)" >}}
+{{< figure src="/ox-hugo/skogestad07_mixed_sensitivity_dist_rejection.png" caption="Figure 46: \\(S/KS\\) mixed-sensitivity optimization in standard form (regulation)" >}}
-Another interpretation can be put on the \\(S/KS\\) mixed-sensitivity optimization as shown in the standard control configuration of [Figure 45](#figure--fig:mixed-sensitivity-ref-tracking).
+Another interpretation can be put on the \\(S/KS\\) mixed-sensitivity optimization as shown in the standard control configuration of [Figure 47](#figure--fig:mixed-sensitivity-ref-tracking).
Here we consider a tracking problem.
The exogenous input is a reference command \\(r\\), and the error signals are \\(z\_1 = -W\_1 e = W\_1 (r-y)\\) and \\(z\_2 = W\_2 u\\).
-As the regulation problem of [Figure 44](#figure--fig:mixed-sensitivity-dist-rejection), we have that \\(z\_1 = W\_1 S w\\) and \\(z\_2 = W\_2 KS w\\).
+As the regulation problem of [Figure 46](#figure--fig:mixed-sensitivity-dist-rejection), we have that \\(z\_1 = W\_1 S w\\) and \\(z\_2 = W\_2 KS w\\).
-{{< figure src="/ox-hugo/skogestad07_mixed_sensitivity_ref_tracking.png" caption="Figure 45: \\(S/KS\\) mixed-sensitivity optimization in standard form (tracking)" >}}
+{{< figure src="/ox-hugo/skogestad07_mixed_sensitivity_ref_tracking.png" caption="Figure 47: \\(S/KS\\) mixed-sensitivity optimization in standard form (tracking)" >}}
Another useful mixed sensitivity optimization problem, is to find a stabilizing controller which minimizes
@@ -5235,7 +5253,7 @@ Another useful mixed sensitivity optimization problem, is to find a stabilizing
The ability to shape \\(T\\) is desirable for tracking problems and noise attenuation.
It is also important for robust stability with respect to multiplicative perturbations at the plant output.
-The \\(S/T\\) mixed-sensitivity minimization problem can be put into the standard control configuration as shown in [Figure 46](#figure--fig:mixed-sensitivity-s-t).
+The \\(S/T\\) mixed-sensitivity minimization problem can be put into the standard control configuration as shown in [Figure 48](#figure--fig:mixed-sensitivity-s-t).
The elements of the generalized plant are
@@ -5254,7 +5272,7 @@ The elements of the generalized plant are
-{{< figure src="/ox-hugo/skogestad07_mixed_sensitivity_s_t.png" caption="Figure 46: \\(S/T\\) mixed-sensitivity optimization in standard form" >}}
+{{< figure src="/ox-hugo/skogestad07_mixed_sensitivity_s_t.png" caption="Figure 48: \\(S/T\\) mixed-sensitivity optimization in standard form" >}}
The shaping of closed-loop transfer functions as described above with the stacked cost functions becomes difficult with more than two functions whereas with two, the process is relatively easy.
The bandwidth requirements on each are usually complementary and simple, stable low-pass and high-pass filters are sufficient to carry out the required shaping and trade-offs.
@@ -5277,20 +5295,20 @@ 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 [Figure 47](#figure--fig:input-uncertainty-hinf), where \\(G\\) represents the nominal model, \\(W\\) is a weighting function that captures the relative model fidelity over frequency, and \\(\Delta\\) represents unmodelled dynamics usually normalized such that \\(\hnorm{\Delta} < 1\\).
+Weights are also used if a perturbation is used to model uncertainty, as in [Figure 49](#figure--fig:input-uncertainty-hinf), where \\(G\\) represents the nominal model, \\(W\\) is a weighting function that captures the relative model fidelity over frequency, and \\(\Delta\\) represents unmodelled dynamics usually normalized such that \\(\hnorm{\Delta} < 1\\).
-{{< figure src="/ox-hugo/skogestad07_input_uncertainty_hinf.png" caption="Figure 47: Multiplicative dynamic uncertainty model" >}}
+{{< figure src="/ox-hugo/skogestad07_input_uncertainty_hinf.png" caption="Figure 49: Multiplicative dynamic uncertainty model" >}}
LQG control is a simple example of the signal based approach, in which the exogenous signals are assumed to be stochastic and the error signals are measured in terms of the 2-norm.
As we have seen, the weights \\(Q\\) and \\(R\\) are constant, but LQG can be generalized to include frequency dependent weights on the signals leading to what is called Wiener-Hopf design or \\(\htwo\\) control.
When we consider a system's response to persistent sinusoidal signals of varying frequency, or when we consider the induced 2-norm between the exogenous input signals and the error signals, we are required to minimize the \\(\hinf\\) norm.
In the absence of model uncertainty, there does not appear to be an overwhelming case for using the \\(\hinf\\) norm rather than the more traditional \\(\htwo\\) norm.
-However, when uncertainty is addressed, as it always should be, \\(\hinf\\) is clearly the more **natural approach** using component uncertainty models as in [Figure 47](#figure--fig:input-uncertainty-hinf).
+However, when uncertainty is addressed, as it always should be, \\(\hinf\\) is clearly the more **natural approach** using component uncertainty models as in [Figure 49](#figure--fig:input-uncertainty-hinf).
-A typical problem using the signal-based approach to \\(\hinf\\) control is illustrated in the interconnection diagram of [Figure 48](#figure--fig:hinf-signal-based).
+A typical problem using the signal-based approach to \\(\hinf\\) control is illustrated in the interconnection diagram of [Figure 50](#figure--fig:hinf-signal-based).
\\(G\\) and \\(G\_d\\) are nominal models of the plant and disturbance dynamics, and \\(K\\) is the controller to be designed.
The weights \\(W\_d\\), \\(W\_r\\), and \\(W\_n\\) may be constant or dynamic and describe the relative importance and/or the frequency content of the disturbance, set points and noise signals.
The weight \\(W\_\text{ref}\\) is a desired closed-loop transfer function between the weighted set point \\(r\_s\\) and the actual output \\(y\\).
@@ -5313,9 +5331,9 @@ The problem can be cast as a standard \\(\hinf\\) optimization in the general co
-{{< figure src="/ox-hugo/skogestad07_hinf_signal_based.png" caption="Figure 48: A signal-based \\(\hinf\\) control problem" >}}
+{{< figure src="/ox-hugo/skogestad07_hinf_signal_based.png" caption="Figure 50: A signal-based \\(\hinf\\) control problem" >}}
-Suppose we now introduce a multiplicative dynamic uncertainty model at the input to the plant as shown in [Figure 49](#figure--fig:hinf-signal-based-uncertainty).
+Suppose we now introduce a multiplicative dynamic uncertainty model at the input to the plant as shown in [Figure 51](#figure--fig:hinf-signal-based-uncertainty).
The problem we now want to solve is: find a stabilizing controller \\(K\\) such that the \\(\hinf\\) norm of the transfer function between \\(w\\) and \\(z\\) is less that 1 for all \\(\Delta\\) where \\(\hnorm{\Delta} < 1\\).
We have assumed in this statement that the **signal weights have normalized the 2-norm of the exogenous input signals to unity**.
This problem is a non-standard \\(\hinf\\) optimization.
@@ -5327,7 +5345,7 @@ It is a robust performance problem for which the \\(\mu\text{-synthesis}\\) proc
-{{< figure src="/ox-hugo/skogestad07_hinf_signal_based_uncertainty.png" caption="Figure 49: A signal-based \\(\hinf\\) control problem with input multiplicative uncertainty" >}}
+{{< figure src="/ox-hugo/skogestad07_hinf_signal_based_uncertainty.png" caption="Figure 51: A signal-based \\(\hinf\\) control problem with input multiplicative uncertainty" >}}
However, whilst the structured singular value is a useful analysis tool for assessing designs, \\(\mu\text{-synthesis}\\) is sometimes difficult to use and often too complex for the practical problems.
@@ -5378,7 +5396,7 @@ The objective of robust stabilization is to stabilize not only the nominal model
where \\(\epsilon > 0\\) is then the **stability margin**.
-For the perturbed feedback system of [Figure 50](#figure--fig:coprime-uncertainty-bis), the stability property is robust if and only if the nominal feedback system is stable and
+For the perturbed feedback system of [Figure 52](#figure--fig:coprime-uncertainty-bis), the stability property is robust if and only if the nominal feedback system is stable and
\begin{equation\*}
\gamma \triangleq \hnorm{\begin{bmatrix}
@@ -5391,7 +5409,7 @@ Notice that \\(\gamma\\) is the \\(\hinf\\) norm from \\(\phi\\) to \\(\begin{bm
-{{< figure src="/ox-hugo/skogestad07_coprime_uncertainty_bis.png" caption="Figure 50: \\(\hinf\\) robust stabilization problem" >}}
+{{< figure src="/ox-hugo/skogestad07_coprime_uncertainty_bis.png" caption="Figure 52: \\(\hinf\\) robust stabilization problem" >}}
The lowest achievable value of \\(\gamma\\) and the corresponding maximum stability margin \\(\epsilon\\) are given as
@@ -5456,11 +5474,11 @@ If \\(W\_1\\) and \\(W\_2\\) are the pre and post compensators respectively, the
G\_s = W\_2 G W\_1
\end{equation}
-as shown in [Figure 51](#figure--fig:shaped-plant).
+as shown in [Figure 53](#figure--fig:shaped-plant).
-{{< figure src="/ox-hugo/skogestad07_shaped_plant.png" caption="Figure 51: The shaped plant and controller" >}}
+{{< figure src="/ox-hugo/skogestad07_shaped_plant.png" caption="Figure 53: The shaped plant and controller" >}}
The controller \\(K\_s\\) is synthesized by solving the robust stabilization problem for the shaped plant \\(G\_s\\) with a normalized left coprime factorization \\(G\_s = M\_s^{-1}N\_s\\).
The feedback controller for the plant \\(G\\) is then \\(K = W\_1 K\_s W\_2\\).
@@ -5491,13 +5509,13 @@ Systematic procedure for \\(\hinf\\) loop-shaping design:
- A small value of \\(\epsilon\_{\text{max}}\\) indicates that the chosen singular value loop-shapes are incompatible with robust stability requirements
7. **Analyze the design** and if not all the specification are met, make further modifications to the weights
8. **Implement the controller**.
- The configuration shown in [Figure 52](#figure--fig:shapping-practical-implementation) has been found useful when compared with the conventional setup in [Figure 38](#figure--fig:classical-feedback-small).
+ The configuration shown in [Figure 54](#figure--fig:shapping-practical-implementation) has been found useful when compared with the conventional setup in [Figure 40](#figure--fig:classical-feedback-small).
This is because the references do not directly excite the dynamics of \\(K\_s\\), which can result in large amounts of overshoot.
The constant prefilter ensure a steady-state gain of \\(1\\) between \\(r\\) and \\(y\\), assuming integral action in \\(W\_1\\) or \\(G\\)
-{{< figure src="/ox-hugo/skogestad07_shapping_practical_implementation.png" caption="Figure 52: A practical implementation of the loop-shaping controller" >}}
+{{< figure src="/ox-hugo/skogestad07_shapping_practical_implementation.png" caption="Figure 54: A practical implementation of the loop-shaping controller" >}}
We will conclude this section with a summary of the **advantages** offered by the above \\(\hinf\\) loop-shaping design procedure:
@@ -5518,25 +5536,25 @@ Many control design problems possess two degrees-of-freedom:
Sometimes, one degree-of-freedom is left out of the design, and the controller is driven by an error signal i.e. the difference between a command and the output.
But in cases where stringent time-domain specifications are set on the output response, a one degree-of-freedom structure may not be sufficient.
-A general two degrees-of-freedom feedback control scheme is depicted in [Figure 53](#figure--fig:classical-feedback-2dof-simple).
+A general two degrees-of-freedom feedback control scheme is depicted in [Figure 55](#figure--fig:classical-feedback-2dof-simple).
The commands and feedbacks enter the controller separately and are independently processed.
-{{< figure src="/ox-hugo/skogestad07_classical_feedback_2dof_simple.png" caption="Figure 53: General two degrees-of-freedom feedback control scheme" >}}
+{{< figure src="/ox-hugo/skogestad07_classical_feedback_2dof_simple.png" caption="Figure 55: General two degrees-of-freedom feedback control scheme" >}}
The presented \\(\mathcal{H}\_\infty\\) loop-shaping design procedure in section is a one-degree-of-freedom design, although a **constant** pre-filter can be easily implemented for steady-state accuracy.
However, this may not be sufficient and a dynamic two degrees-of-freedom design is required.
-The design problem is illustrated in [Figure 54](#figure--fig:coprime-uncertainty-hinf).
+The design problem is illustrated in [Figure 56](#figure--fig:coprime-uncertainty-hinf).
The feedback part of the controller \\(K\_2\\) is designed to meet robust stability and disturbance rejection requirements.
A prefilter is introduced to force the response of the closed-loop system to follow that of a specified model \\(T\_{\text{ref}}\\), often called the **reference model**.
-{{< figure src="/ox-hugo/skogestad07_coprime_uncertainty_hinf.png" caption="Figure 54: Two degrees-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping design problem" >}}
+{{< figure src="/ox-hugo/skogestad07_coprime_uncertainty_hinf.png" caption="Figure 56: Two degrees-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping design problem" >}}
-The design problem is to find the stabilizing controller \\(K = [K\_1,\ K\_2]\\) for the shaped plant \\(G\_s = G W\_1\\), with a normalized coprime factorization \\(G\_s = M\_s^{-1} N\_s\\), which minimizes the \\(\mathcal{H}\_\infty\\) norm of the transfer function between the signals \\([r^T\ \phi^T]^T\\) and \\([u\_s^T\ y^T\ e^T]^T\\) as defined in [Figure 54](#figure--fig:coprime-uncertainty-hinf).
+The design problem is to find the stabilizing controller \\(K = [K\_1,\ K\_2]\\) for the shaped plant \\(G\_s = G W\_1\\), with a normalized coprime factorization \\(G\_s = M\_s^{-1} N\_s\\), which minimizes the \\(\mathcal{H}\_\infty\\) norm of the transfer function between the signals \\([r^T\ \phi^T]^T\\) and \\([u\_s^T\ y^T\ e^T]^T\\) as defined in [Figure 56](#figure--fig:coprime-uncertainty-hinf).
This problem is easily cast into the general configuration.
The control signal to the shaped plant \\(u\_s\\) is given by:
@@ -5566,11 +5584,11 @@ The main steps required to synthesize a two degrees-of-freedom \\(\mathcal{H}\_\
5. Replace the prefilter \\(K\_1\\) by \\(K\_1 W\_i\\) to give exact model-matching at steady-state.
6. Analyze and, if required, redesign making adjustments to \\(\rho\\) and possibly \\(W\_1\\) and \\(T\_{\text{ref}}\\)
-The final two degrees-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping controller is illustrated in [Figure 55](#figure--fig:hinf-synthesis-2dof).
+The final two degrees-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping controller is illustrated in [Figure 57](#figure--fig:hinf-synthesis-2dof).
-{{< figure src="/ox-hugo/skogestad07_hinf_synthesis_2dof.png" caption="Figure 55: Two degrees-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping controller" >}}
+{{< figure src="/ox-hugo/skogestad07_hinf_synthesis_2dof.png" caption="Figure 57: Two degrees-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping controller" >}}
#### Observer-Based Structure for \\(\hinf\\) Loop-Shaping Controllers {#observer-based-structure-for-hinf-loop-shaping-controllers}
@@ -5650,11 +5668,11 @@ When implemented in Hanus form, the expression for \\(u\\) becomes
where \\(u\_a\\) is the **actual plant input**, that is the measurement at the **output of the actuators** which therefore contains information about possible actuator saturation.
-The situation is illustrated in [Figure 56](#figure--fig:weight-anti-windup), where the actuators are each modeled by a unit gain and a saturation.
+The situation is illustrated in [Figure 58](#figure--fig:weight-anti-windup), where the actuators are each modeled by a unit gain and a saturation.
-{{< figure src="/ox-hugo/skogestad07_weight_anti_windup.png" caption="Figure 56: Self-conditioned weight \\(W\_1\\)" >}}
+{{< figure src="/ox-hugo/skogestad07_weight_anti_windup.png" caption="Figure 58: Self-conditioned weight \\(W\_1\\)" >}}
The Hanus form prevents windup by keeping the states of \\(W\_1\\) consistent with the actual plant input at all times.
When there is no saturation, \\(u\_a=u\\), the dynamics of \\(W\_1\\) remains unaffected.
@@ -5713,11 +5731,11 @@ Moreover, one should be careful about combining controller synthesis and analysi
### Introduction {#introduction}
-In previous sections, we considered the general problem formulation in [Figure 57](#figure--fig:general-control-names-bis) and stated that the controller design problem is to find a controller \\(K\\) which based on the information in \\(v\\), generates a control signal \\(u\\) which counteracts the influence of \\(w\\) on \\(z\\), thereby minimizing the closed loop norm from \\(w\\) to \\(z\\).
+In previous sections, we considered the general problem formulation in [Figure 59](#figure--fig:general-control-names-bis) and stated that the controller design problem is to find a controller \\(K\\) which based on the information in \\(v\\), generates a control signal \\(u\\) which counteracts the influence of \\(w\\) on \\(z\\), thereby minimizing the closed loop norm from \\(w\\) to \\(z\\).
-{{< figure src="/ox-hugo/skogestad07_general_control_names_bis.png" caption="Figure 57: General Control Configuration" >}}
+{{< figure src="/ox-hugo/skogestad07_general_control_names_bis.png" caption="Figure 59: General Control Configuration" >}}
In this chapter we are concerned with the **structural decisions** associated with the following selection tasks of control structure design:
@@ -5748,31 +5766,40 @@ The reference value \\(r\\) is usually set at some higher layer in the control h
- **Optimization layer**: computes the desired reference commands \\(r\\)
- **Control layer**: implements these commands to achieve \\(y \approx r\\)
-Additional layers are possible, as is illustrated in [Figure 58](#figure--fig:control-system-hierarchy) which shows a typical control hierarchy for a chemical plant.
+Additional layers are possible, as is illustrated in [Figure 60](#figure--fig:control-system-hierarchy) which shows a typical control hierarchy for a chemical plant.
-{{< figure src="/ox-hugo/skogestad07_system_hierarchy.png" caption="Figure 58: Typical control system hierarchy in a chemical plant" >}}
+{{< figure src="/ox-hugo/skogestad07_system_hierarchy.png" caption="Figure 60: Typical control system hierarchy in a chemical plant" >}}
-In general, the information flow in such a control hierarchy is based on the higher layer sending reference values (setpoints) to the layer below reporting back any problems achieving this (see [ 6](#org-target--fig-optimize-control-b)).
+In general, the information flow in such a control hierarchy is based on the higher layer sending reference values (setpoints) to the layer below reporting back any problems achieving this (see [Figure 61b](#org-target--fig-optimize-control-b)).
There is usually a time scale separation between the layers which means that the **setpoints**, as viewed from a given layer, are **updated only periodically**.
The optimization tends to be performed open-loop with limited use of feedback. On the other hand, the control layer is mainly based on feedback information.
The **optimization is often based on nonlinear steady-state models**, whereas we often use **linear dynamic models in the control layer**.
-From a theoretical point of view, the optimal performance is obtained with a **centralized optimizing controller**, which combines the two layers of optimizing and control (see [ 6](#org-target--fig-optimize-control-c)).
+From a theoretical point of view, the optimal performance is obtained with a **centralized optimizing controller**, which combines the two layers of optimizing and control (see [Figure 61c](#org-target--fig-optimize-control-c)).
All control actions in such an ideal control system would be perfectly coordinated and the control system would use on-line dynamic optimization based on nonlinear dynamic model of the complete plant.
However, this solution is normally not used for a number a reasons, included the cost of modeling, the difficulty of controller design, maintenance, robustness problems and the lack of computing power.
-
-
- Table 6:
- Alternative structures for optimization and control
-
-|  |  |  |
-|-------------------------------------------------------------------------------------------------|---------------------------------------------------------------------------------------------------------------------------------|--------------------------------------------------------------------------------------------------------------|
-| Open loop optimization | Closed-loop implementation with separate control layer | Integrated optimization and control |
+
+
+
+
+
(a) Open loop optimization
+
+
+
+
(b) Closed-loop implementation with separate control layer
+
+
+
+
(c) Integrated optimization and control
+
+
+Figure 61: Alternative structures for optimization and control
+
### Selection of Controlled Outputs {#selection-of-controlled-outputs}
@@ -5885,7 +5912,7 @@ Thus, the selection of controlled and measured outputs are two separate issues.
### Selection of Manipulations and Measurements {#selection-of-manipulations-and-measurements}
-We are here concerned with the variable sets \\(u\\) and \\(v\\) in [Figure 57](#figure--fig:general-control-names-bis).
+We are here concerned with the variable sets \\(u\\) and \\(v\\) in [Figure 59](#figure--fig:general-control-names-bis).
Note that **the measurements** \\(v\\) used by the controller **are in general different from the controlled variables** \\(z\\) because we may not be able to measure all the controlled variables and we may want to measure and control additional variables in order to:
- Stabilize the plant, or more generally change its dynamics
@@ -5977,19 +6004,24 @@ Then when a SISO control loop is closed, we lose the input \\(u\_i\\) as a degre
A cascade control structure results when either of the following two situations arise:
- The reference \\(r\_i\\) is an output from another controller.
- This is the **conventional cascade control** ([ 7](#org-target--fig-cascade-extra-meas))
+ This is the **conventional cascade control** ([Figure 62a](#org-target--fig-cascade-extra-meas))
- The "measurement" \\(y\_i\\) is an output from another controller.
- This is referred to as **input resetting** ([ 7](#org-target--fig-cascade-extra-input))
+ This is referred to as **input resetting** ([Figure 62b](#org-target--fig-cascade-extra-input))
-
-
-
-|  |  |
-|--------------------------------------------------------------------------------------------------------|---------------------------------------------------------------------------------------------------|
-| Extra measurements \\(y\_2\\) | Extra inputs \\(u\_2\\) |
+
+
+
(b) Extra inputs \(u_2\)
+
+
+Figure 62: Cascade Implementations
+
#### Cascade Control: Extra Measurements {#cascade-control-extra-measurements}
@@ -6013,7 +6045,7 @@ where in most cases \\(r\_2 = 0\\) since we do not have a degree-of-freedom to c
##### Cascade implementation {#cascade-implementation}
-To obtain an implementation with two SISO controllers, we may cascade the controllers as illustrated in [ 7](#org-target--fig-cascade-extra-meas):
+To obtain an implementation with two SISO controllers, we may cascade the controllers as illustrated in [Figure 62a](#org-target--fig-cascade-extra-meas):
\begin{align\*}
r\_2 &= K\_1(s)(r\_1 - y\_1) \\\\
@@ -6023,12 +6055,12 @@ To obtain an implementation with two SISO controllers, we may cascade the contro
Note that the output \\(r\_2\\) from the slower primary controller \\(K\_1\\) is not a manipulated plant input, but rather the reference input to the faster secondary controller \\(K\_2\\).
Cascades based on measuring the actual manipulated variable (\\(y\_2 = u\_m\\)) are commonly used to **reduce uncertainty and non-linearity at the plant input**.
-In the general case ([ 7](#org-target--fig-cascade-extra-meas)) \\(y\_1\\) and \\(y\_2\\) are not directly related to each other, and this is sometimes referred to as _parallel cascade control_.
-However, it is common to encounter the situation in [Figure 59](#figure--fig:cascade-control) where the primary output \\(y\_1\\) depends directly on \\(y\_2\\) which is a special case of [ 7](#org-target--fig-cascade-extra-meas).
+In the general case ([Figure 62a](#org-target--fig-cascade-extra-meas)) \\(y\_1\\) and \\(y\_2\\) are not directly related to each other, and this is sometimes referred to as _parallel cascade control_.
+However, it is common to encounter the situation in [Figure 63](#figure--fig:cascade-control) where the primary output \\(y\_1\\) depends directly on \\(y\_2\\) which is a special case of [Figure 62a](#org-target--fig-cascade-extra-meas).
-With reference to the special (but common) case of cascade control shown in [Figure 59](#figure--fig:cascade-control), the use of **extra measurements** is useful under the following circumstances:
+With reference to the special (but common) case of cascade control shown in [Figure 63](#figure--fig:cascade-control), the use of **extra measurements** is useful under the following circumstances:
- The disturbance \\(d\_2\\) is significant and \\(G\_1\\) is non-minimum phase.
If \\(G\_1\\) is minimum phase, the input-output controllability of \\(G\_2\\) and \\(G\_1 G\_2\\) are the same and there is no fundamental advantage in measuring \\(y\_2\\)
@@ -6039,7 +6071,7 @@ With reference to the special (but common) case of cascade control shown in [Fig
-{{< 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\\)" >}}
+{{< figure src="/ox-hugo/skogestad07_cascade_control.png" caption="Figure 63: Common case of cascade control where the primary output \\(y\_1\\) depends directly on the extra measurement \\(y\_2\\)" >}}
In terms of design, it is recommended to first design \\(K\_2\\) to minimize the effect of \\(d\_2\\) on \\(y\_1\\) and then to design \\(K\_1\\) to minimize the effect of \\(d\_1\\) on \\(y\_1\\).
@@ -6065,7 +6097,7 @@ Then \\(u\_2(t)\\) will only be used for **transient control** and will return t
##### Cascade implementation {#cascade-implementation}
-To obtain an implementation with two SISO controllers we may cascade the controllers as shown in [ 7](#org-target--fig-cascade-extra-input).
+To obtain an implementation with two SISO controllers we may cascade the controllers as shown in [Figure 62b](#org-target--fig-cascade-extra-input).
We again let input \\(u\_2\\) take care of the **fast control** and \\(u\_1\\) of the **long-term control**.
The fast control loop is then
@@ -6086,7 +6118,7 @@ It also shows more clearly that \\(r\_{u\_2}\\), the reference for \\(u\_2\\), m
-Consider the system in [Figure 60](#figure--fig:cascade-control-two-layers) with two manipulated inputs (\\(u\_2\\) and \\(u\_3\\)), one controlled output (\\(y\_1\\) which should be close to \\(r\_1\\)) and two measured variables (\\(y\_1\\) and \\(y\_2\\)).
+Consider the system in [Figure 64](#figure--fig:cascade-control-two-layers) with two manipulated inputs (\\(u\_2\\) and \\(u\_3\\)), one controlled output (\\(y\_1\\) which should be close to \\(r\_1\\)) and two measured variables (\\(y\_1\\) and \\(y\_2\\)).
Input \\(u\_2\\) has a more direct effect on \\(y\_1\\) than does input \\(u\_3\\) (there is a large delay in \\(G\_3(s)\\)).
Input \\(u\_2\\) should only be used for transient control as it is desirable that it remains close to \\(r\_3 = r\_{u\_2}\\).
The extra measurement \\(y\_2\\) is closer than \\(y\_1\\) to the input \\(u\_2\\) and may be useful for detecting disturbances affecting \\(G\_1\\).
@@ -6100,7 +6132,7 @@ We would probably tune the three controllers in the order \\(K\_2\\), \\(K\_3\\)
-{{< figure src="/ox-hugo/skogestad07_cascade_control_two_layers.png" caption="Figure 60: Control configuration with two layers of cascade control" >}}
+{{< figure src="/ox-hugo/skogestad07_cascade_control_two_layers.png" caption="Figure 64: Control configuration with two layers of cascade control" >}}
#### Selectors {#selectors}
@@ -6173,7 +6205,7 @@ Four applications of partial control are:
The outputs \\(y\_1\\) have an associated control objective but are not measured.
Instead, we aim at indirectly controlling \\(y\_1\\) by controlling the secondary measured variables \\(y\_2\\).
-The table [Table 8](#table--tab:partial-control) shows clearly the differences between the four applications of partial control.
+The table [Table 4](#table--tab:partial-control) shows clearly the differences between the four applications of partial control.
In all cases, there is a control objective associated with \\(y\_1\\) and a feedback involving measurement and control of \\(y\_2\\) and we want:
- The effect of disturbances on \\(y\_1\\) to be small (when \\(y\_2\\) is controlled)
@@ -6181,7 +6213,7 @@ In all cases, there is a control objective associated with \\(y\_1\\) and a feed
@@ -6201,7 +6233,7 @@ By partitioning the inputs and outputs, the overall model \\(y = G u\\) can be w
\end{aligned}
\end{equation}
-Assume now that feedback control \\(u\_2 = K\_2(r\_2 - y\_2 - n\_2)\\) is used for the "secondary" subsystem involving \\(u\_2\\) and \\(y\_2\\) ([Figure 61](#figure--fig:partial-control)).
+Assume now that feedback control \\(u\_2 = K\_2(r\_2 - y\_2 - n\_2)\\) is used for the "secondary" subsystem involving \\(u\_2\\) and \\(y\_2\\) ([Figure 65](#figure--fig:partial-control)).
We get:
\begin{equation} \label{eq:partial\_control}
@@ -6214,7 +6246,7 @@ We get:
-{{< figure src="/ox-hugo/skogestad07_partial_control.png" caption="Figure 61: Partial Control" >}}
+{{< figure src="/ox-hugo/skogestad07_partial_control.png" caption="Figure 65: Partial Control" >}}
##### Tight control of \\(y\_2\\) {#tight-control-of-y-2}
@@ -6270,7 +6302,7 @@ The selection of \\(u\_2\\) and \\(y\_2\\) for use in the lower-layer control sy
##### Sequential design of cascade control systems {#sequential-design-of-cascade-control-systems}
-Consider the conventional cascade control system in [ 7](#org-target--fig-cascade-extra-meas) where we have additional "secondary" measurements \\(y\_2\\) with no associated control objective, and the objective is to improve the control of \\(y\_1\\) by locally controlling \\(y\_2\\).
+Consider the conventional cascade control system in [Figure 62a](#org-target--fig-cascade-extra-meas) where we have additional "secondary" measurements \\(y\_2\\) with no associated control objective, and the objective is to improve the control of \\(y\_1\\) by locally controlling \\(y\_2\\).
The idea is that this should reduce the effect of disturbances and uncertainty on \\(y\_1\\).
From \ref{eq:partial\_control}, it follows that we should select \\(y\_2\\) and \\(u\_2\\) such that \\(\\|P\_d\\|\\) is small and at least smaller than \\(\\|G\_{d1}\\|\\).
@@ -6338,11 +6370,11 @@ Then to minimize the control error for the primary output, \\(J = \\|y\_1 - r\_1
### Decentralized Feedback Control {#decentralized-feedback-control}
-In this section, \\(G(s)\\) is a square plant which is to be controlled using a diagonal controller ([Figure 62](#figure--fig:decentralized-diagonal-control)).
+In this section, \\(G(s)\\) is a square plant which is to be controlled using a diagonal controller ([Figure 66](#figure--fig:decentralized-diagonal-control)).
-{{< figure src="/ox-hugo/skogestad07_decentralized_diagonal_control.png" caption="Figure 62: Decentralized diagonal control of a \\(2 \times 2\\) plant" >}}
+{{< figure src="/ox-hugo/skogestad07_decentralized_diagonal_control.png" caption="Figure 66: Decentralized diagonal control of a \\(2 \times 2\\) plant" >}}
The design of **decentralized diagonal control systems** involves two steps:
diff --git a/content/zettels/test.md b/content/zettels/test.md
index c22a41c..b2b1009 100644
--- a/content/zettels/test.md
+++ b/content/zettels/test.md
@@ -79,7 +79,7 @@ Sed aliquam
Here is a list of links to:
- [Figure 5](#figure--fig:general-control-names)
-- [Table 3](#table--tab:table-with-equations)
+- [Table 2](#table--tab:table-with-equations)
- Listing [Code Snippet 1](#code-snippet--lst:matlab-figure)
- Specific [line of code](#org-coderef--967846-4)
- Equation \ref{eq:numbered}
@@ -465,7 +465,7 @@ Numbering can be continued by using `+n` option as shown below.
```
Code Snippet 4:
- Tikz code that is used to generate 5
+ Tikz code that is used to generate 5
@@ -506,7 +506,7 @@ Fusce blandit mauris dui, sed lobortis sapien tincidunt ac. Maecenas vitae moles
### Sub Images {#sub-images}
-Link to sub[ 2](#org-target--fig-general-control-names-1).
+Link to sub[Figure 6a](#org-target--fig-general-control-names-1).
```md
#+name: fig:subfigure
@@ -516,24 +516,29 @@ Link to sub[ 2](#org-target--fig-general-control-names-1).
| <> sub figure caption | <> sub figure caption |
```
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-
-|  |  |
-|--------------------------------------------------------------------------------------------------|--------------------------------------------------------------------------------------------------|
-| sub figure caption | sub figure caption |
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(a) sub figure caption
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(b) sub figure caption
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+Figure 6: Subfigure Caption
+
## Tables {#tables}
-[Table 3](#table--tab:table-with-equations) shows a table with some mathematics inside.
+[Table 2](#table--tab:table-with-equations) shows a table with some mathematics inside.
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