Instead of applying voltage as the input into the piezo elements, we will assume that we have calculated an equivalent set of forces which can be applied at the ends of the element that will replicate the voltage force function.
@@ -1230,9 +1230,9 @@ Since the same forces are being applied to both piezo elements, they represent t
### Ansys Model Description {#ansys-model-description}
-In Figure [15](#orgfc320b2) are shown the principal nodes used for the model.
+In Figure [15](#orga0e8a08) are shown the principal nodes used for the model.
-
+
{{< figure src="/ox-hugo/hatch00_disk_drive_mimo_ansys.png" caption="Figure 15: Nodes used for reduced Matlab model, shown with partial mesh at coil and piezo element" >}}
@@ -1351,11 +1351,11 @@ And we note:
G = zn * Gp;
```
-
+
{{< figure src="/ox-hugo/hatch00_z13_tf.png" caption="Figure 16: Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_3\\)" >}}
-
+
{{< figure src="/ox-hugo/hatch00_z11_tf.png" caption="Figure 17: Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_1\\)" >}}
@@ -1453,13 +1453,13 @@ G_f = ss(A, B, C, D);
### Simple mode truncation {#simple-mode-truncation}
-Let's plot the frequency of the modes (Figure [18](#orgdb5ecb9)).
+Let's plot the frequency of the modes (Figure [18](#org5f90de8)).
-
+
{{< figure src="/ox-hugo/hatch00_cant_beam_modes_freq.png" caption="Figure 18: Frequency of the modes" >}}
-
+
{{< figure src="/ox-hugo/hatch00_cant_beam_unsorted_dc_gains.png" caption="Figure 19: Unsorted DC Gains" >}}
@@ -1528,7 +1528,7 @@ dc_gain = abs(xn(i_input, :).*xn(i_output, :))./(2*pi*f0).^2;
[dc_gain_sort, index_sort] = sort(dc_gain, 'descend');
```
-
+
{{< figure src="/ox-hugo/hatch00_cant_beam_sorted_dc_gains.png" caption="Figure 20: Sorted DC Gains" >}}
@@ -1872,7 +1872,7 @@ wo = gram(G_m, 'o');
And we plot the diagonal terms
-
+
{{< figure src="/ox-hugo/hatch00_gramians.png" caption="Figure 21: Observability and Controllability Gramians" >}}
@@ -1890,7 +1890,7 @@ We use `balreal` to rank oscillatory states.
[G_b, G, T, Ti] = balreal(G_m);
```
-
+
{{< figure src="/ox-hugo/hatch00_cant_beam_gramian_balanced.png" caption="Figure 22: Sorted values of the Gramian of the balanced realization" >}}
@@ -2135,6 +2135,6 @@ pos_frames = pos([1, i_input, i_output], :);
## Bibliography {#bibliography}
-
Hatch, Michael R. 2000. _Vibration Simulation Using MATLAB and ANSYS_. CRC Press.
+
Hatch, Michael R. 2000. _Vibration Simulation Using MATLAB and ANSYS_. CRC Press.
-
Miu, Denny K. 1993. _Mechatronics: Electromechanics and Contromechanics_. 1st ed. Mechanical Engineering Series. Springer-Verlag New York.
+
Miu, Denny K. 1993. _Mechatronics: Electromechanics and Contromechanics_. 1st ed. Mechanical Engineering Series. Springer-Verlag New York.
diff --git a/content/zettels/norms.md b/content/zettels/norms.md
index a721e99..7905232 100644
--- a/content/zettels/norms.md
+++ b/content/zettels/norms.md
@@ -13,13 +13,16 @@ Tags
Resources:
-- ([Skogestad and Postlethwaite 2007](#orgf3c8b69))
-- ([Toivonen 2002](#orgb2755d2))
-- ([Zhang 2011](#org3e1b2ef))
+- ([Skogestad and Postlethwaite 2007](#org352385f))
+- ([Toivonen 2002](#orga84ee63))
+- ([Zhang 2011](#org74fd92c))
## Definition {#definition}
+
+
+
A norm of \\(e\\) (which may be a vector, matrix, signal of system) is a real number, denoted \\(\\|e\\|\\), that satisfies the following properties:
1. Non-negative: \\(\\|e\\| \ge 0\\)
@@ -27,6 +30,8 @@ A norm of \\(e\\) (which may be a vector, matrix, signal of system) is a real nu
3. Homogeneous: \\(\\|\alpha \cdot e\\| = |\alpha| \cdot \\|e\\|\\) for all complex scalars \\(\alpha\\)
4. Triangle inequality: \\(\\|e\_1 + e\_2\\| \le \\|e\_1\\| + \\|e\_2\\|\\)
+
+
## Vector Norms {#vector-norms}
@@ -42,7 +47,7 @@ A norm of \\(e\\) (which may be a vector, matrix, signal of system) is a real nu
## Matrix Norms {#matrix-norms}
-
+
A norm on a matrix \\(\\|A\\|\\) is a matrix norm if, in addition to the four norm properties, it also satisfies the multiplicative property:
@@ -96,37 +101,94 @@ We normally use the same p-norm both for the vector and the signal.
## Signal Interpretation of Various System Norms {#signal-interpretation-of-various-system-norms}
+Consider a system \\(G\\) with input \\(d\\) and output \\(e\\), such that:
+\\[ e = G d \\]
+
+For performance, we may want the output signal \\(e\\) to be "small" for any allowed input signals \\(d\\).
+We therefore need to specify:
+
+1. What \\(d\\) are allowed. (Which set does \\(d\\) belong to?)
+ Some possible inputs signal sets are:
+ - \\(d(t)\\) consists of impulses \\(\delta(t)\\).
+ - These generate step changes in the states.
+ - \\(d(t) = \sin(\omega t)\\) with fixed frequency
+ - \\(d(t)\\) is bounded in energy \\(\\|d(t)\\|\_2 \le 1\\)
+ - \\(d(t)\\) is bounded in power \\(\\|d(t)\\|\_\text{pow} \le 1\\)
+ - \\(d(t)\\) is bounded in magnitude \\(\\|d(t)\\|\_\infty \le 1\\)
+2. What we mean by "small". (Which norm should be use for \\(e\\)?)
+ To measure the output signal, we may consider the following norms:
+ - 2-norm (energy): \\(\\|e(t)\\|\_2\\)
+ - \\(\infty\text{-norm}\\) (peak magnitude): \\(\\|e(t)\\|\_\infty\\)
+ - Power: \\(\\|e(t)\\|\_\text{pow}\\)
+
+We now consider which system norms result from the definition of input classes and output norms (Table [1](#table--tab:system-norms)).
+
+
+
+
Table 1:
+ System norms for sets of inputs signals and three different output norms
+
+
+| | \\(\delta(t)\\) | \\(\sin(\omega t)\\) | \\(\vert\vert d \vert\vert\_2\\) | \\(\vert\vert d \vert\vert\_\infty\\) | \\(\vert\vert d \vert\vert\_\text{pow}\\) |
+|-------------------------------------------|------------------------------------------|--------------------------------------------------------|------------------------------------------|----------------------------------------------|-------------------------------------------|
+| \\(\vert\vert e \vert\vert\_2\\) | \\(\vert\vert G(s) \vert\vert\_2\\) | \\(\infty\\) | \\(\vert\vert G(s) \vert\vert\_\infty\\) | \\(\infty\\) | \\(\infty\\) |
+| \\(\vert\vert e \vert\vert\_\infty\\) | \\(\vert\vert g(t) \vert\vert\_\infty\\) | \\(\overline{\sigma}(G(j\omega))\\) | \\(\vert\vert G(s) \vert\vert\_2\\) | \\(\vert\vert g(t) \vert\vert\_1\\) | \\(\infty\\) |
+| \\(\vert\vert e \vert\vert\_\text{pow}\\) | 0 | \\(\frac{1}{\sqrt{2}} \overline{\sigma}(G(j\omega))\\) | 0 | \\(\le \vert\vert G(s) \vert\vert\_\infty\\) | \\(\vert\vert G(s) \vert\vert\_\infty\\) |
+
## System Norms {#system-norms}
### \\(\mathcal{H}\_\infty\\) Norm {#mathcal-h-infty--norm}
-SISO Systems => absolute value => bode plot
-MIMO Systems => singular value
-Signal => maximum value
+
+
+
+Consider a proper linear stable system \\(G(s)\\).
+The \\(\mathcal{H}\_\infty\\) norm is the peak value of its maximum singular value:
+\\[ \\|G(s)\\|\_\infty \triangleq \max\_{\omega} \overline{\sigma}(G(j\omega)) \\]
+
+
+
+In terms of signals, the \\(\mathcal{H}\_\infty\\) norm can be interpreted as follows:
+
+- it is the worst case steady-state gain for sinusoidal inputs at any frequency
+- it is equal to the 2-norm in the time domain:
+ \\[ \\|G(s)\\|\_\infty = \max\_{d(t)} \frac{\\|e(t)\\|\_2 \neq 0}{\\|d(t)\\|\_2} = \max\_{\\|d(t)\\|\_2 = 1} \\|e(t)\\|\_2 \\]
### \\(\mathcal{H}\_2\\) Norm {#mathcal-h-2--norm}
+
+
+
+Consider a strictly proper system \\(G(s)\\).
+The \\(\mathcal{H}\_2\\) norm is:
+
+\begin{align\*}
+\\|G(s)\\|\_2 &\triangleq \sqrt{\frac{1}{2\pi} \int\_{-\infty}^{\infty} \text{tr}\left(G(j\omega)^HG(j\omega)\right) d\omega} \\\\\\
+ &= \sqrt{\frac{1}{2\pi} \int\_{-\infty}^{\infty} \sum\_i {\sigma\_i}^2(G(j\omega)) d\omega}
+\end{align\*}
+
+
+
+In terms of signals, the \\(\mathcal{H}\_\infty\\) norm can be interpreted as follows:
+
+- it is a measure of the expected RMS value of the output to white noise excitation
+
The \\(\mathcal{H}\_2\\) is very useful when combined to [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting" >}}).
-As explained in ([Monkhorst 2004](#orgd47c42b)), the \\(\mathcal{H}\_2\\) norm has a stochastic interpretation:
+As explained in ([Monkhorst 2004](#org50a92a2)), the \\(\mathcal{H}\_2\\) norm has a stochastic interpretation:
> The squared \\(\mathcal{H}\_2\\) norm can be interpreted as the output variance of a system with zero mean white noise input.
-Minimizing the \\(\mathcal{H}\_2\\) norm can be equivalent as minimizing the RMS value of some signals in the system.
-
-
-## Link between signal and system norms {#link-between-signal-and-system-norms}
-
## Bibliography {#bibliography}
-
Monkhorst, Wouter. 2004. “Dynamic Error Budgeting, a Design Approach.” Delft University.
+
Monkhorst, Wouter. 2004. “Dynamic Error Budgeting, a Design Approach.” Delft University.
-
Skogestad, Sigurd, and Ian Postlethwaite. 2007. _Multivariable Feedback Control: Analysis and Design_. John Wiley.
+
Skogestad, Sigurd, and Ian Postlethwaite. 2007. _Multivariable Feedback Control: Analysis and Design_. John Wiley.
-
Toivonen, Hannu T. 2002. “Robust Control Methods.” Abo Akademi University.
+
Toivonen, Hannu T. 2002. “Robust Control Methods.” Abo Akademi University.
-
Zhang, Weidong. 2011. _Quantitative Process Control Theory_. CRC Press.
+
Zhang, Weidong. 2011. _Quantitative Process Control Theory_. CRC Press.