Add analysis on the sensitivity function stape
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@ -1121,59 +1121,159 @@ Similarly, to reduce the effect of measurement noise $n$ on the output $y$, we w
|
|||||||
** Sensitivity Function
|
** Sensitivity Function
|
||||||
<<sec:sensitivity_transfer_functions>>
|
<<sec:sensitivity_transfer_functions>>
|
||||||
|
|
||||||
#+begin_src matlab
|
Suppose we have developed a "/reference/" controller $K_r(s)$ and made three small changes to obtained three controllers $K_1(s)$, $K_2(s)$ and $K_3(s)$.
|
||||||
K1 = 14e8 * ... % Gain
|
The obtained sensitivity functions are shown in Figure [[fig:sensitivity_shape_effect]] and the corresponding step responses are shown in Figure [[fig:sensitivity_shape_effect_step]].
|
||||||
1/(s^2) * ... % Double Integrator
|
|
||||||
(1 + s/(2*pi*10/sqrt(8)))/(1 + s/(2*pi*10*sqrt(8))); % Lead
|
|
||||||
|
|
||||||
K2 = 1e8 * ... % Gain
|
The comparison of the sensitivity functions shapes and their effect on the step response is summarized in Table [[tab:compare_sensitivity_shapes]].
|
||||||
1/(s^2) * ... % Double Integrator
|
|
||||||
(1 + s/(2*pi*1/sqrt(8)))/(1 + s/(2*pi*1*sqrt(8))); % Lead
|
|
||||||
|
|
||||||
K3 = 1e8 * ... % Gain
|
#+name: tab:compare_sensitivity_shapes
|
||||||
1/(s^2) * ... % Double Integrator
|
#+caption: Comparison of the sensitivity function shape and the corresponding step response for the three controller variations
|
||||||
(1 + s/(2*pi*1/sqrt(2)))/(1 + s/(2*pi*1*sqrt(2))); % Lead
|
| Controller | Sensitivity Function Shape | Change of the Step Response |
|
||||||
|
|------------+----------------------------------------------------+----------------------------------|
|
||||||
|
| $K_1(s)$ | Larger bandwidth $\omega_b$ | Faster rise time |
|
||||||
|
| $K_2(s)$ | Larger peak value $\Vert S\Vert_\infty$ | Large overshoot and oscillations |
|
||||||
|
| $K_3(s)$ | Larger low frequency gain $\vert S(j\cdot 0)\vert$ | Larger static error |
|
||||||
|
|
||||||
|
#+begin_src matlab :exports none
|
||||||
|
wc = 2*pi*1; L_w = 8; wi = 2*pi*0.02;
|
||||||
|
Kr = 1/((s + wi)^2) * ... % Double Integrator
|
||||||
|
(1 + s/(wc/sqrt(L_w)))/(1 + s/(wc*sqrt(L_w))); % Lead
|
||||||
|
Kr = Kr/abs(evalfr(Kr*G, j*wc));
|
||||||
|
|
||||||
|
wc = 2*pi*2; L_w = 8; wi = 2*pi*0.045;
|
||||||
|
K1 = 1/((s + wi)^2) * ... % Double Integrator
|
||||||
|
(1 + s/(wc/sqrt(L_w)))/(1 + s/(wc*sqrt(L_w))); % Lead
|
||||||
|
K1 = K1/abs(evalfr(K1*G, j*wc));
|
||||||
|
|
||||||
|
wc = 2*pi*1; L_w = 2; wi = 2*pi*0.03;
|
||||||
|
K2 = 1/((s + wi)^2) * ... % Double Integrator
|
||||||
|
(1 + s/(wc/sqrt(L_w)))/(1 + s/(wc*sqrt(L_w))); % Lead
|
||||||
|
K2 = K2/abs(evalfr(K2*G, j*wc));
|
||||||
|
|
||||||
|
wc = 2*pi*1; L_w = 8; wi = 2*pi*0.2;
|
||||||
|
K3 = 1/((s + wi)^2) * ... % Double Integrator
|
||||||
|
(1 + s/(wc/sqrt(L_w)))/(1 + s/(wc*sqrt(L_w))); % Lead
|
||||||
|
K3 = K3/abs(evalfr(K3*G, j*wc));
|
||||||
|
|
||||||
|
|
||||||
|
Sr = 1/(1 + Kr*G);
|
||||||
S1 = 1/(1 + K1*G);
|
S1 = 1/(1 + K1*G);
|
||||||
S2 = 1/(1 + K2*G);
|
S2 = 1/(1 + K2*G);
|
||||||
S3 = 1/(1 + K3*G);
|
S3 = 1/(1 + K3*G);
|
||||||
|
|
||||||
|
Tr = Kr*G/(1 + Kr*G);
|
||||||
T1 = K1*G/(1 + K1*G);
|
T1 = K1*G/(1 + K1*G);
|
||||||
T2 = K2*G/(1 + K2*G);
|
T2 = K2*G/(1 + K2*G);
|
||||||
T3 = K3*G/(1 + K3*G);
|
T3 = K3*G/(1 + K3*G);
|
||||||
|
|
||||||
bodeFig({S1, S2, S3})
|
|
||||||
#+end_src
|
#+end_src
|
||||||
|
|
||||||
#+begin_src matlab
|
#+begin_src matlab :exports none
|
||||||
freqs = logspace(-1, 2, 1000);
|
freqs = logspace(-2, 2, 1000);
|
||||||
|
|
||||||
figure;
|
figure;
|
||||||
tiledlayout(1, 2, 'TileSpacing', 'None', 'Padding', 'None');
|
|
||||||
|
|
||||||
ax1 = nexttile;
|
|
||||||
hold on;
|
hold on;
|
||||||
plot(freqs, abs(squeeze(freqresp(S1, freqs, 'Hz'))), 'DisplayName', '$L(s)$');
|
plot(freqs, abs(squeeze(freqresp(S1, freqs, 'Hz'))), 'DisplayName', '$K_1(s)$');
|
||||||
plot(freqs, abs(squeeze(freqresp(S2, freqs, 'Hz'))), 'DisplayName', '$L_w(s)$');
|
plot(freqs, abs(squeeze(freqresp(S2, freqs, 'Hz'))), 'DisplayName', '$K_2(s)$');
|
||||||
plot(freqs, abs(squeeze(freqresp(S3, freqs, 'Hz'))), 'DisplayName', '$L_w(s) / \gamma$, $L_w(s) \cdot \gamma$');
|
plot(freqs, abs(squeeze(freqresp(S3, freqs, 'Hz'))), 'DisplayName', '$K_3(s)$');
|
||||||
|
plot(freqs, abs(squeeze(freqresp(Sr, freqs, 'Hz'))), 'k-', 'DisplayName', '$K_r(s)$');
|
||||||
hold off;
|
hold off;
|
||||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||||
xlabel('Frquency [Hz]'); ylabel('Sensitivity Magnitude');
|
xlabel('Frquency [Hz]'); ylabel('Sensitivity Magnitude');
|
||||||
hold off;
|
hold off;
|
||||||
|
legend('location', 'southeast');
|
||||||
|
ylim([1e-3, 3]);
|
||||||
|
#+end_src
|
||||||
|
|
||||||
ax2 = nexttile;
|
#+begin_src matlab :tangle no :exports results :results file replace
|
||||||
t = linspace(0, 1, 1000);
|
exportFig('figs/sensitivity_shape_effect.pdf', 'width', 'wide', 'height', 'normal');
|
||||||
y1 = step(T1, t);
|
#+end_src
|
||||||
y2 = step(T2, t);
|
|
||||||
y3 = step(T3, t);
|
#+name: fig:sensitivity_shape_effect
|
||||||
|
#+caption: Sensitivity function magnitude $|S(j\omega)|$ corresponding to the reference controller $K_r(s)$ and the three modified controllers $K_i(s)$
|
||||||
|
#+RESULTS:
|
||||||
|
[[file:figs/sensitivity_shape_effect.png]]
|
||||||
|
|
||||||
|
#+begin_src matlab :exports none
|
||||||
|
t = linspace(0, 5, 1000);
|
||||||
|
|
||||||
|
figure;
|
||||||
hold on;
|
hold on;
|
||||||
plot(t, y1)
|
plot(t, step(T1, t), 'DisplayName', '$K_1(s)$')
|
||||||
plot(t, y2)
|
plot(t, step(T2, t), 'DisplayName', '$K_2(s)$')
|
||||||
plot(t, y3)
|
plot(t, step(T3, t), 'DisplayName', '$K_3(s)$')
|
||||||
|
plot(t, step(Tr, t), 'k-', 'DisplayName', '$K_r(s)$')
|
||||||
hold off
|
hold off
|
||||||
xlabel('Time [s]'); ylabel('Step Response');
|
xlabel('Time [s]'); ylabel('Step Response');
|
||||||
|
legend('location', 'northeast');
|
||||||
|
ylim([0, 1.7])
|
||||||
#+end_src
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab :tangle no :exports results :results file replace
|
||||||
|
exportFig('figs/sensitivity_shape_effect_step.pdf', 'width', 'wide', 'height', 'normal');
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+name: fig:sensitivity_shape_effect_step
|
||||||
|
#+caption: Step response (response from $r$ to $y$) for the different controllers
|
||||||
|
#+RESULTS:
|
||||||
|
[[file:figs/sensitivity_shape_effect_step.png]]
|
||||||
|
|
||||||
|
#+begin_src latex :file h-infinity-spec-S.pdf
|
||||||
|
\begin{tikzpicture}
|
||||||
|
\begin{axis}[%
|
||||||
|
width=8cm,
|
||||||
|
height=4cm,
|
||||||
|
at={(0,0)},
|
||||||
|
xmode=log,
|
||||||
|
xmin=0.01,
|
||||||
|
xmax=10000,
|
||||||
|
ymin=-80,
|
||||||
|
ymax=40,
|
||||||
|
ylabel={Magnitude [dB]},
|
||||||
|
xlabel={Frequency [Hz]},
|
||||||
|
ytick={40, 20, 0, -20, -40, -60, -80},
|
||||||
|
xminorgrids,
|
||||||
|
yminorgrids
|
||||||
|
]
|
||||||
|
\addplot [thick, color=black, forget plot]
|
||||||
|
table[row sep=crcr]{%
|
||||||
|
0.01 -60\\
|
||||||
|
0.1 -60\\
|
||||||
|
190 6\\
|
||||||
|
10000 6\\
|
||||||
|
};
|
||||||
|
|
||||||
|
\draw[<-] (0.05, -60) -- (0.1, -70);
|
||||||
|
\draw (0.1, -70) -- (2, -70) node[right, fill=white, draw]{\footnotesize{Small static error}};
|
||||||
|
\draw[<-] (70, -3) -- (3, -3) node[left, fill=white, draw]{\footnotesize{Speed}};
|
||||||
|
|
||||||
|
\draw[<-] (300, 6) -- (200, 20);
|
||||||
|
\draw (200, 20) -- (10, 20) node[left, fill=white, draw]{\footnotesize{Robustness}};
|
||||||
|
\end{axis}
|
||||||
|
\end{tikzpicture}
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_definition
|
||||||
|
- Closed-Loop Bandwidth ::
|
||||||
|
The closed-loop bandwidth $\omega_b$ is the frequency where $|S(j\omega)|$ first crosses $1/\sqrt{2} = -3dB$ from below.
|
||||||
|
|
||||||
|
In general, a large bandwidth corresponds to a faster rise time.
|
||||||
|
#+end_definition
|
||||||
|
|
||||||
|
#+begin_important
|
||||||
|
From the simple analysis above, we can draw a first estimation of the wanted shape for the sensitivity function in Figure [[fig:h-infinity-spec-S]].
|
||||||
|
|
||||||
|
The wanted characteristics on the magnitude of the sensitivity function are then:
|
||||||
|
- A small magnitude at low frequency to make the static errors small
|
||||||
|
- A wanted minimum closed-loop bandwidth in order to have fast rise time and good rejection of perturbations
|
||||||
|
- A small peak value in order to limit large overshoot and oscillations.
|
||||||
|
This generally means higher robustness.
|
||||||
|
This will become clear in the next section about the *module margin*.
|
||||||
|
|
||||||
|
#+name: fig:h-infinity-spec-S
|
||||||
|
#+caption: Typical wanted shape of the Sensitivity transfer function
|
||||||
|
[[file:figs/h-infinity-spec-S.png]]
|
||||||
|
#+end_important
|
||||||
|
|
||||||
** Robustness: Module Margin
|
** Robustness: Module Margin
|
||||||
<<sec:module_margin>>
|
<<sec:module_margin>>
|
||||||
|
|
||||||
@ -1655,46 +1755,6 @@ The obtained shapes are shown in Figure [[fig:high_order_weight]].
|
|||||||
** Sensitivity Function Shaping - Example
|
** Sensitivity Function Shaping - Example
|
||||||
<<sec:sensitivity_shaping_example>>
|
<<sec:sensitivity_shaping_example>>
|
||||||
|
|
||||||
#+begin_src latex :file h-infinity-spec-S.pdf
|
|
||||||
\begin{tikzpicture}
|
|
||||||
\begin{axis}[%
|
|
||||||
width=8cm,
|
|
||||||
height=4cm,
|
|
||||||
at={(0,0)},
|
|
||||||
xmode=log,
|
|
||||||
xmin=0.01,
|
|
||||||
xmax=10000,
|
|
||||||
ymin=-80,
|
|
||||||
ymax=40,
|
|
||||||
ylabel={Magnitude [dB]},
|
|
||||||
xlabel={Frequency [Hz]},
|
|
||||||
ytick={40, 20, 0, -20, -40, -60, -80},
|
|
||||||
xminorgrids,
|
|
||||||
yminorgrids
|
|
||||||
]
|
|
||||||
\addplot [thick, color=black, forget plot]
|
|
||||||
table[row sep=crcr]{%
|
|
||||||
0.01 -60\\
|
|
||||||
0.1 -60\\
|
|
||||||
190 6\\
|
|
||||||
10000 6\\
|
|
||||||
};
|
|
||||||
\draw[<-] (0.05, -60) -- (0.1, -70);
|
|
||||||
\draw (0.1, -70) -- (2, -70) node[right, fill=white, draw]{$\SI{-60}{\decibel} \rightarrow$ \footnotesize{Static error}};
|
|
||||||
\draw[<-] (1, -40) -- (10, -40) node[right, fill=white, draw]{$\SI{20}{\decibel/dec} \rightarrow$ \footnotesize{Ref. track.}};
|
|
||||||
\draw[<-] (100, 0) -- (3, 0) node[left, fill=white, draw]{$\omega_c \rightarrow$ \footnotesize{Speed}};
|
|
||||||
|
|
||||||
\draw[<-] (300, 6) -- (200, 20);
|
|
||||||
\draw (200, 20) -- (10, 20) node[left, fill=white, draw]{$\SI{6}{\decibel} \rightarrow$ \footnotesize{Module margin}};
|
|
||||||
\end{axis}
|
|
||||||
\end{tikzpicture}
|
|
||||||
#+end_src
|
|
||||||
|
|
||||||
#+name: fig:h-infinity-spec-S
|
|
||||||
#+caption: Typical wanted shape of the Sensitivity transfer function
|
|
||||||
#+RESULTS:
|
|
||||||
[[file:figs/h-infinity-spec-S.png]]
|
|
||||||
|
|
||||||
|
|
||||||
- Robustness: Module margin > 2 ($\Rightarrow \text{GM} > 2 \text{ and } \text{PM} > 29^o$)
|
- Robustness: Module margin > 2 ($\Rightarrow \text{GM} > 2 \text{ and } \text{PM} > 29^o$)
|
||||||
- Bandwidth:
|
- Bandwidth:
|
||||||
|
Loading…
Reference in New Issue
Block a user