Merge pull request #11 from MeierTobias/lectures/week10

Lectures/week10

src/BSB/freq_weights_example.tex

@@ -0,0 +1,13 @@
+\begin{tikzpicture}[auto, node distance=1.5cm,>=latex']
+    \node [input, name=uinput]          (uinput)        {};
+    \node [tmp, right of=uinput]        (ut)            {};
+    \node [block, right of=ut]          (P)             {$P(s)$};
+    \node [block, above of=P]           (W)             {$W(s)$};
+    \node [output, right of=P]          (y)             {};
+    \node [output, right of=W]          (z)             {};
+
+    \draw [->] (rinput) -- node[pos=0.2]{$u$} (P);
+    \draw [->] (ut) |- node{} (W);
+    \draw [->] (P) -- node{$y$} (y);
+    \draw [->] (W) -- node{$z$} (z);
+\end{tikzpicture}

src/main.tex

@@ -4,7 +4,6 @@
 
 % load the template headers
 \input{\templatePath/summary_headers.tex} % ChkTex 27
-\usepackage{cancel}
 
 \begin{document}
 \begin{multicols*}{3}
@@ -23,6 +22,7 @@
     \input{sections/LQG.tex}
     \input{sections/MIMO_intro.tex}
     \input{sections/stab_and_perf_robustness.tex}
+    \input{sections/modern_controller_synthesis.tex}
     \newpage
     \input{sections/Appendix.tex}
 

src/sections/modern_controller_synthesis.tex

@@ -0,0 +1,327 @@
+\section{Modern Controller Synthesis}
+To unify the controller synthesis the following standard form has been established.
+
+
+\setlength{\oldtabcolsep}{\tabcolsep}\setlength\tabcolsep{2pt}
+
+\begin{tabularx}{\linewidth}{@{}ll@{}}
+    \includegraphics[width=0.5\linewidth, align=c]{controller_synthesis_paradigm.png}
+     &
+    {\begin{tabularx}{\linewidth}{@{}ll@{}}
+                 $u$: & control input            \\
+                 $y$: & measured output          \\
+                 $r$: & reference signal         \\
+                 $w$: & disturbance              \\
+                 $z$: & performance output       \\
+                 $q$: & uncertainty block input  \\
+                 $v$: & uncertainty block output
+             \end{tabularx}
+        }
+\end{tabularx}
+
+\setlength{\tabcolsep}{\oldtabcolsep}
+
+
+\newpar{}
+The objective is to stabilize the system and minimize the performance outputs $z$.
+
+\subsection{Frequency Weights}
+
+To guide the optimization process the performance outputs $z_i$ are weighted with frequency dependent weighting functions $W_i$.
+
+The most common performance outputs are
+\begin{center}
+    \includegraphics[width=0.9\linewidth]{frequency_weights.png}
+\end{center}
+
+\ptitle{Tracking Error}
+
+Is given by
+\begin{equation*}
+    z_1=W_1 G_{er} = W_1 S(s)
+\end{equation*}
+with the sensitivity function $S(s)$ and the loop transfer function $L(s)$:
+\begin{gather*}
+    S(s)=G_{er}(s) = {(I+L(s))}^{-1} \\
+    L(s)=P(s)C(s)
+\end{gather*}
+\begin{itemize}
+    \item If $W_1(s)$ is chosen large at low frequencies $S(s)$ must be small in order to minimize $z_1$. In other words, the tracking error must be small at low frequencies.
+    \item This is the same as requiring $\sigma_{\min}[L(s)] \gg |W_1(s)|$, which corresponds to the low-frequency ``Bode obstacle'' in the SISO case.
+\end{itemize}
+
+\newpar{}
+
+\ptitle{Control Effort}
+
+Is given by
+\begin{equation*}
+    z_2=W_2 G_{ur}(s) = W_2 C(s) S(s)
+\end{equation*}
+\begin{itemize}
+    \item Useful to limit the maximum control effort
+\end{itemize}
+
+\newpar{}
+
+\ptitle{Noise Rejection}
+
+Is given by
+\begin{equation*}
+    z_3= W_3 G_{yn} =-W_3 T(s)
+\end{equation*}
+with the complementary sensitivity function
+\begin{equation*}
+    T(s) = G_{yr}(s)={(I+L(s))}^{-1}L(s) = I-S(s)
+\end{equation*}
+\begin{itemize}
+    \item If $W_3$ is chosen very large for high frequencies, the complementary sensitivity must be very small at high frequencies.
+    \item This is the same as requiring $\sigma_{\max}[L(s)] \ll |W_3(s)|$, which corresponds to the high-frequency ``Bode obstacle'' in the SISO case.
+\end{itemize}
+
+\newpar{}
+
+\ptitle{Stability Robustness}
+
+For stability robustness in the presence of uncertainty $\Delta$ the same weighting method can be applied.
+\begin{equation*}
+    z_\Delta = \Delta(s)W_\Delta(s)
+\end{equation*}
+with
+\begin{gather*}
+    \|\Delta(s)\|_{\mathcal{H}_\infty} < 1 \\
+    \|M(s)\|_{\mathcal{H}_\infty} < 1
+\end{gather*}
+where $M(s)$ is the TF from the output of $\delta$ ti its input.
+
+\subsubsection{First Order Weights}
+\ptitle{Lowpass}
+\begin{equation*}
+    W(s) = \frac{M}{\frac{Ms}{\omega}+1}
+\end{equation*}
+
+\ptitle{Highpass}
+\begin{equation*}
+    W(s) = \frac{Ms}{s+M\omega}
+\end{equation*}
+
+\subsection{State-Space Representation}
+To synthesize a controller an assembled state space model of the generalized system ($G$) can be constructed.
+
+\ptitle{Plant}
+\begin{gather*}
+    P : \left[
+        \begin{array}{c|c} % ChkTex -2
+            A_p & B_p \\
+            \hline % ChkTex -2
+            C_p & D_p \\
+        \end{array}
+        \right]                            \\
+    P(s) = {C_p(sI-A_p)}^{-1}B_p+D_p
+\end{gather*}
+
+\ptitle{Frequency Weights}
+
+For each weight we get
+\begin{gather*}
+    W : \left[
+        \begin{array}{c|c} % ChkTex -2
+            -p   & 1 \\
+            \hline % ChkTex -2
+            r-qp & q
+        \end{array}
+        \right]                   \\
+    W(s) = \frac{qs+r}{s+p}
+\end{gather*}
+
+\ptitle{Generalized System}
+
+The generalized system is obtained by stacking all the state-space models:
+\begin{gather*}
+    G : \left[
+        \begin{array}{c|c c} % ChkTex -2
+            A   & B_w    & B_u    \\
+            \hline % ChkTex -2
+            C_z & D_{zw} & D_{zu} \\
+            C_y & D_{yw} & D_{yu}
+        \end{array}
+        \right]                        \\
+    G(s) = \begin{bmatrix}
+        G_{zw}(s) & G_{zu}(s) \\
+        G_{yw}(s) & G_{yu}(s) \\
+    \end{bmatrix}
+\end{gather*}
+The matrix $A$ contains both the dynamics of the plant and the frequency weights.
+
+\ptitle{Controller}
+\begin{gather*}
+    K : \left[
+        \begin{array}{c|c} % ChkTex -2
+            A_c & B_c \\
+            \hline % ChkTex -2
+            C_c & D_c \\
+        \end{array}
+        \right]                            \\
+    K(s) = {C_c(sI-A_c)}^{-1}B_c+D_c
+\end{gather*}
+
+The closed loop TF from the exogenous inputs $w$ to the performance outputs $z$ is given by the \textit{Linear Fractional Transformation}:
+\begin{equation*}
+    F(s) = G_{zw}(s) + G_{zu}(s)K(s){(I-G_{yu}(s))}^{-1}G_{yw}(s)
+\end{equation*}
+
+\begin{examplesection}[Example Assembly]
+    For the system
+    \begin{center}
+        \input{BSB/freq_weights_example.tex}
+    \end{center}
+    the generalized system $G(s)$ has to be calculated
+    \begin{equation*}
+        \begin{bmatrix}
+            y \\
+            z
+        \end{bmatrix}
+        = \underbrace{\begin{bmatrix}
+                G_{yu}(s) \\
+                G_{zu}(s)
+            \end{bmatrix}}_{G(s)} u
+    \end{equation*}
+    The combined states evolve with
+    \begin{equation*}
+        \begin{bmatrix}
+            \dot{x}_p \\
+            \dot{x}_w
+        \end{bmatrix}
+        = \underbrace{\begin{bmatrix}
+                A_p & 0   \\
+                0   & A_w
+            \end{bmatrix}}_{A_G}
+        \begin{bmatrix}
+            x_p \\
+            x_w
+        \end{bmatrix}
+        + \underbrace{\begin{bmatrix}
+                B_p \\
+                B_w
+            \end{bmatrix}}_{B_G} \: u
+    \end{equation*}
+
+    and the combined output is given by
+
+    \begin{equation*}
+        \begin{bmatrix}
+            y \\
+            z
+        \end{bmatrix}
+        = \underbrace{\begin{bmatrix}
+                C_p & 0   \\
+                0   & C_w
+            \end{bmatrix}}_{C_G}
+        \begin{bmatrix}
+            x_p \\
+            x_w
+        \end{bmatrix}
+        +
+        \underbrace{\begin{bmatrix}
+                D_p \\
+                D_w
+            \end{bmatrix}}_{D_G}
+        \: u
+    \end{equation*}
+    The generalized system representation is then given by
+    \begin{equation*}
+        G = \left[
+            \begin{array}{c|c} % ChkTex -2
+                A_G & B_G \\
+                \hline % ChkTex -2
+                C_G & D_G
+            \end{array}
+            \right]
+        =
+        \left[
+            \begin{array}{cc|c} % ChkTex -2
+                A_p & 0   & B_p \\
+                0   & A_w & B_w \\
+                \hline % ChkTex -2
+                C_p & 0   & D_p \\
+                0   & C_w & D_w
+            \end{array}
+            \right]
+    \end{equation*}
+
+\end{examplesection}
+
+\subsection{Youla's Q Parameterization}
+One can get all stabilizing controllers for a given plant as a function of a single stable transfer function Q(s). This is called the Youla parameterization (Q-parameterization). It's the extension of a full state feedback controller (including observer) with a system \textcolor{purple}{$Q$} that takes the innovation as an input.
+\begin{center}
+    \includegraphics[width=0.8\linewidth]{youlas_Q_parameterization.png}
+\end{center}
+$Q$ needs to be \textbf{stable} and has to consist of a \textbf{proper rational} transfer function.
+\newpar{}
+The closed-loop dynamics are given by
+\begin{equation*}
+    \begin{bmatrix}
+        \dot{x}   \\
+        \dot{x}_Q \\
+        \dot{\eta}
+    \end{bmatrix}
+    =
+    \begin{bmatrix}
+        A+BK & BC_Q & -BK   \\
+        0    & A_Q  & B_Q C \\
+        0    & 0    & A+LC
+    \end{bmatrix}
+    \begin{bmatrix}
+        x   \\
+        x_Q \\
+        \eta
+    \end{bmatrix}
+    +
+    \begin{bmatrix}
+        I & 0 \\
+        0 & C \\
+        0 & C
+    \end{bmatrix}
+    \begin{bmatrix}
+        r \\
+        d
+    \end{bmatrix}
+\end{equation*}
+The closed-loop poles are the union of the full-state feedback poles $A+BK$, the observer poles $A+LC$ and the poles of $Q(s)$.
+\newpar{}
+The TF from $y$ to $u$ hence the TF of the controller $C(s)$ can be written as
+\begin{equation*}
+    C(s) = -{(X_0(s)+Q(s)N(s))}^{-1}(Y_0(s)-Q(s)D(s))
+\end{equation*}
+where
+\begin{equation*}
+    C_0(s)=-{X_0(s)}^{-1}Y_0(s)
+\end{equation*}
+is the TF when $Q(s)=0$ with controller gain $K_0$ and observer gail $L_0$
+\noindent\begin{align*}
+    X_0(s) & = -K_0{(sI-A-L_0C)}^{-1}B   \\
+    Y_0(s) & = -K_0{(sI-A-L_0C)}^{-1}L_0
+\end{align*}
+and $N$, $D$ are the nominator and denominator of the plant
+\begin{align*}
+    N(s) & =C{(sI-A-L_0C)}^{-1}B    \\
+    D(s) & = -C(sI-A-L_0-C^{-1}L_0)
+\end{align*}
+where $L_0$ is an observer gain that would stabilize the error dynamics.
+\newpar{}
+It can be show that the interconnection of the system is stable if the \textbf{Bezout identity} is fulfilled
+\begin{equation*}
+    D(s)X(s)-N(s)Y(s)=1
+\end{equation*}
+and that all feedback stabilizing controllers for $P$ are given by
+\begin{equation*}
+    C(s) = \frac{Y_0(s) -D(s)Q(s)}{X_0(s)-N(s)Q(s)}
+\end{equation*}
+The sensitivity function is given by
+\begin{equation*}
+    S(s) = D(s)(X_0(s)-N(S)Q(s))
+\end{equation*}
+and the complementary sensitivity function by
+\begin{equation*}
+    T(s) = -N(s)(Y_0(s)+D(s)Q(s))
+\end{equation*}