Lectures/week12

src/main.tex

@@ -24,6 +24,7 @@
     \input{sections/stab_and_perf_robustness.tex}
     \input{sections/modern_controller_synthesis.tex}
     \input{sections/h2_hinf_synthesis.tex}
+    \input{sections/nonlinear.tex}
     \newpage
     \input{sections/Appendix.tex}
 

src/sections/h2_hinf_synthesis.tex

@@ -137,14 +137,20 @@ \subsection{\texorpdfstring{$\mathcal{H}_\infty$}{H-infinity} Synthesis}
           \begin{enumerate}
               \item The cost trade-offs between the state error and the control effort
               \item The expected covariances of the disturbance and sensor noise
-          \end{enumerate}
+          \end{enumerate} i.e.\ \textbf{optimal} w.r.t.\ time-domain specifications.
     \item $\mathcal{H}_2$ specifications are given in \textbf{time domain}: difficult to handle frequency-domain specifications.
     \item For $\mathcal{H}_2$ design one could use a ``mixed sensitivity'' approach, and further augment the plant $P$ with frequency-dependent weigthing functions.
-    \item $\mathcal{H}_\infty$ provides a more direct way to handle \textbf{frequency-domain specifications.}
+    \item $\mathcal{H}_\infty$ provides a more direct way to handle \textbf{frequency-domain specifications.} i.e.\ it allows to achieve a desired level of robustness to disturbances and noise (``Bode obstacles'').
 \end{itemize}
 
 \newpar{}
-\ptitle{Optimal $\mathcal{H}_\infty$ Synthesis}
+\ptitle{Remarks}
+\begin{itemize}
+    \item $\mathcal{H}_\infty$ synthesis requires (as $\mathcal{H}_2$) a detectable and stabilizable plant.
+    \item $\mathcal{H}_\infty$ minimizes the worst-case input-output gain (singular values).
+\end{itemize}
+
+\subsubsection{Optimal \texorpdfstring{$\mathcal{H}_\infty$}{H-infinity} Synthesis}
 \begin{itemize}
     \item In principle, we would like to find a controller K that minimizes the energy ($\mathcal{L}_2$) gain of the CL system i.e.
           \begin{equation*}
@@ -153,8 +159,8 @@ \subsection{\texorpdfstring{$\mathcal{H}_\infty$}{H-infinity} Synthesis}
     \item However, the optimal controller(s) are such that
           \begin{enumerate}
               \item $\sigma_{\max}(\mathbf{T}_{zw}(j\omega))$ is a constant over all frequencies, the response does not roll off at high frequencies (\textit{Bode's integral law}).
-              \item The controller is not strictly proper. (The optimal controller is not unique.)
-              \item Computing an optimal controller is numerically challenging
+              \item The controller is not strictly proper. (The optimal controller is not unique)
+              \item Computing an optimal controller is numerically challenging.
           \end{enumerate}
 \end{itemize}
 
@@ -195,3 +201,18 @@ \subsection{\texorpdfstring{$\mathcal{H}_\infty$}{H-infinity} Synthesis}
     \item $\mathbf{D}_{zu}^{\mathsf{T}}\mathbf{D}_{zu}=1,\mathbf{D}_{yw}\mathbf{D}_{yw}^{\mathsf{T}}=1$.
 \end{itemize}
 
+\subsubsection{Applying Frequency Weights}
+One can apply frequency weights to different performance outputs, similar to the ``Bode obstacles'' for SISO systems.
+\begin{itemize}
+    \item In order for $\mathcal{H}_\infty$ synthesis to be possible, frequency weights must be \textbf{stable and proper}.
+    \item Integrators (not asymptotically stable) can be approximated by very slow poles.
+    \item MATLAB: 
+    \begin{itemize}
+        \item \texttt{Pinf=augw(G,W1,0,W3)}\\
+        creates the augmented (weighted) system
+        \item \texttt{[Kinf, CL, GAM, INFO]=hinfsyn(Pinf,1,1)}\\
+        applies optimal $\mathcal{H}_\infty$ synthesis
+        \item \texttt{[K09, CL, GAM, I]=hinfsyn(Pinf,1,1,0.9)}\\
+        applies suboptimal $\mathcal{H}_\infty$ synthesis targeting $\gamma=0.9$. This can yield a more feasible controller (less control effort)
+    \end{itemize}
+\end{itemize}

src/sections/nonlinear.tex

@@ -0,0 +1,214 @@
+\section{Nonlinear Systems}
+
+\subsection{Concepts of Stability}
+A nonlinear system
+\noindent\begin{align*}
+    \dot{\mathbf{x}}(t) & = f(\mathbf{x}(t), \mathbf{u}(t)) & \mathbf{x}(0)=\mathbf{x}_0 \\
+    \mathbf{y}(t)       & = h(\mathbf{x}(t), \mathbf{u}(t))
+\end{align*}
+\textbf{does not satisfy the superposition property} and therefore
+\begin{itemize}
+    \item Effects from inputs and from initial conditions caannot be seperated.
+    \item The input cannot be seperated into elementary inputs and the output will not be a composition of elementary outputs.
+\end{itemize}
+
+\newpar{}
+\ptitle{Properties}
+\begin{itemize}
+    \item A nonlinear system can have zero, one or multiple \textbf{equillibrium points} s.t.\ $f(\mathbf{x}_e, 0)=0$.
+    \item In contrast to linear systems that require infinite time to go to infinity, nonlinear systes can \textbf{go to infinity in finite time}.
+    \item Nonlniear systems may generate permanent oscillations of fixed amplitude - \textbf{limit cycles} - independent of initial conditions.
+    \item Nonlinear systems can generate outputs at frequencies that are submultiples or multiples of the input frequency (\textbf{Subharmonic oscillations}).
+    \item Deterministic nonlinear systems can generate \textbf{chaos}.
+\end{itemize}
+
+\subsection{Lyapunov Stability Theory}
+\subsubsection{Types of Stability}
+Assuming $\mathbf{u}=0, \mathbf{x}_e=0$, the eqilibrium $\mathbf{x}_e$ of the systems
+\noindent\begin{equation*}
+    \dot{\mathbf{x}} = f(\mathbf{x}), \quad f(\mathbf{x}_e) = 0
+\end{equation*}
+is
+\begin{itemize}
+    \item \textbf{Stable in the sense of Lyapunov}
+          \noindent\begin{equation*}
+              \|\mathbf{x}(0)\| < \delta \Rightarrow \|\mathbf{x}(t)\| \leq \varepsilon,\qquad \forall t\geq 0,\; \delta > 0,\; \varepsilon\geq0
+          \end{equation*}
+    \item \textbf{Asymtotically stable}
+          \noindent\begin{equation*}
+              \|\mathbf{x}(0)\| < \delta \Rightarrow \lim_{t\to +\infty} \mathbf{x}(t)=0, \qquad \delta>0
+          \end{equation*}
+    \item \textbf{Exponentially stable}
+          \noindent\begin{equation*}
+              \|\mathbf{x}(0)\| < \delta \Rightarrow \|\mathbf{x}(t)\| < \beta e^{-\alpha t}, \qquad \forall t\geq 0,\; \alpha, \beta, \delta >0
+          \end{equation*}
+\end{itemize}
+
+\subsubsection{Lyapunov Functions}
+\textit{Lyapunov functions are, in a sense, a notion of energy: non-negative, minimized (0) only at the quilibrium point and non-increasing along all trajectories.}
+\newpar{}
+For a given compact subset $D$ of the state space containing $\mathbf{x}_e$, a Lypunov function is defined as
+\noindent\begin{equation*}
+    V:D \to \mathbb{R} \mapsto V(\mathbf{x})
+\end{equation*}
+
+If this Lyapunov function satisfies
+\noindent\begin{align*}
+    V(\mathbf{x})                 & \geq 0                                                                                                                                                          &  & \forall \mathbf{x}\in D    \\
+    V(\mathbf{x})                 & = 0 \Leftrightarrow \mathbf{x}_e = \mathbf{x}                                                                                                                                                   \\
+    \frac{d}{dt} V(\mathbf{x}(t)) & = \frac{\partial V(\mathbf{x})}{\partial t} \frac{\partial \mathbf{x}(t)}{\partial \mathbf{x}} = \frac{\partial V(\mathbf{x})}{\partial t} f(\mathbf{x}) \leq 0 &  & \forall \mathbf{x}(t)\in D
+\end{align*}
+the equillibrium point $\mathbf{x_e}$ is \textbf{stable in the sense of Lyapunov}.
+
+\newpar{}
+Futhermore, $\mathbf{x}_e$ is \textbf{asymptotically stable} if
+\noindent\begin{equation*}
+    \dot{V}(\mathbf{x}(t)) = 0 \Leftarrow \mathbf{x}(t) = \mathbf{x}_e
+\end{equation*}
+i.e.\ given $\dot{V}(\mathbf{x}(t)) = 0$ we know that the system is in the \textbf{only} equilibrium point. Finally, $\mathbf{x}_e$ is \textbf{exponentially stable} if
+\noindent\begin{equation*}
+    \dot{V}(\mathbf{x}(t)) < -\alpha V(\mathbf{x}(t)), \qquad \alpha>0
+\end{equation*}
+
+\newpar{}
+\ptitle{Remarks}
+\begin{itemize}
+    \item If there could be multiplie equilibrium points one needs to use LaSalle.
+    %TBD: must the "only equilibrium point" from above be (0,...0)?
+    %TBD: x eventually aproaches the largest pos. inv. set. in which Vdot=0. But are there also pos. inv. sets in which Vdot \neq 0?
+\end{itemize}
+
+\paragraph{Global Stability}
+To ensure global stability, i.e. $D=\mathbb{R}^n$, a Lyapunov function $V$ satisfying the aforementioned properties needs to be \textbf{radially unbounded}:
+\noindent\begin{equation*}
+    \|\mathbf{x}\| \to +\infty \Rightarrow V(\mathbf{x}) \to +\infty
+\end{equation*}
+
+\paragraph{Indirect Method}
+According to the \textbf{Hartman-Grobman Theorem}, a linearized system
+\noindent\begin{equation*}
+    A = \left.\frac{\partial f(\mathbf{x})}{\partial \mathbf{x}} \right|_{\mathbf{x} = 0}
+\end{equation*} with no unstable pole behaves qualitatively the same as the nonlinear around the equilibrium point.
+
+\newpar{}
+As a result, the Lyapunov function of the linear system also describes the nonlinear system for sufficiently small deviations from the equilibrium point.
+
+\newpar{}
+\textbf{Remark}
+
+The solution $\mathbf{P}$ of the Riccati equation can be interpreted as a Lyapunov equation:
+\noindent\begin{equation*}
+    V(\mathbf{x}) = \mathbf{x}^{\mathsf{T}} \mathbf{Px}
+\end{equation*}
+
+\paragraph{LaSalle's Invariance Principle}
+LaSalle's invariance theorem is useful if Lyapunov stability is given by the existence of a Lyapunov function but $\dot{V}=0$ not only at $\mathbf{x}_e$:
+
+\newpar{}
+\textit{Let $\Omega\in D$ be a (compact) positively invariant set i.e., $\mathbf{x}(t_0)\in \Omega \Rightarrow \mathbf{x}(t)\in\Omega$.
+    Let $V_D\to \mathbb{R}^n$, such that $\dot{V}(\mathbf{x})\leq 0\; \forall \mathbf{x}\in \Omega$ (i.e.\ there could be multiple points with $\dot{V}=0$).
+    Then, $\mathbf{x}(t)$ will eventually approach the largest positively invariant set in which $\dot{V}=0$
+}
+\newpar{}
+\textbf{Remarks}
+\begin{itemize}
+    \item A set $\Omega$ is positively invariant if, once the system's state $\mathbf{x}(t)$ enters this set, it will stay in this set for all future time $t \geq t_0$. This means if $\mathbf{x}(t_0) \in \Omega$, then $\mathbf{x}(t) \in \Omega\; \forall t \geq t_0$.
+    \item For the pendulum, one has multiple states (turning points, origin) where $\dot{\mathbf{x}}=0$ but $\mathbf{x}(t)$ only stays at the origin (but not at the turning points) once it has ``entered'' this equilibrium set (i.e.\ stopped moving).
+\end{itemize}
+
+\subsection{Control Lyapunov Functions}
+If a Control Lyapunov Function (CLF) exists, it provides a way to construct a stabilizing feedback with an appropriate control input $\tilde{\mathbf{u}}$ (\textit{Artstein's Theorem}).
+
+\newpar{}
+A CLF satisfies
+\noindent\begin{align*}
+    V(\mathbf{x})                                              & \geq 0                                                                                           &  & \text{positive definite}  \\
+    V(\mathbf{x})                                              & = 0 \Leftrightarrow \mathbf{x} = 0                                                               &  & \text{radially unbounded} \\
+    \frac{d}{dt} V(\mathbf{x}, \tilde{\mathbf{u}}(\mathbf{x})) & = \frac{\partial V(\mathbf{x})}{\partial t} f(\mathbf{x}, \tilde{\mathbf{u}}(\mathbf{x})) \leq 0 &  & \forall \mathbf{x}\neq 0
+\end{align*}
+
+\subsubsection{Gain Scheduling}
+In gain scheduling, the state space is partitioned into disjoint subspaces and for a number of design equilibria ($\mathbf{x}_i, \mathbf{u}_i$)  nominal contorl laws $\mathbf{K}_i$ are designed using the aforementioned techniques.
+
+\newpar{}
+The dynamics of the system around a equilibrium point $\mathbf{x}_i$ can be approximated by
+\noindent\begin{equation*}
+    \dot{\mathbf{x}} = \mathbf{A}_i(\mathbf{x}-\mathbf{x}_i) + \mathbf{B}_i(\mathbf{u}-\mathbf{u}_i)
+\end{equation*}
+
+The stability of
+\noindent\begin{equation*}
+    \mathbf{u} = \mathbf{u}_i+\mathbf{K}_i(\mathbf{x}-\mathbf{x}_i)
+\end{equation*}
+can be proven with the Lyapunov function ($\forall i$)
+\noindent\begin{equation*}
+    V(\mathbf{x}) = \min_{i} {(\mathbf{x}-\mathbf{x}_i)}^{\mathsf{T}}\mathbf{P}(\mathbf{x}-\mathbf{x}_i)
+\end{equation*}
+
+\subsubsection{Multiple Lyapunov Functions}
+Another option is to consider multiple Lyapunov functions $V_i$ with corresponding sets $X_i$ that satisfy
+\begin{itemize}
+    \item positive definite in $X_i$
+    \item $\dot{V_i}(\mathbf{x})\leq 0$ when $\mathbf{x}\in X_i$
+\end{itemize}
+
+and to check if the system satisfies the \textbf{sequence non-increasing condition}:
+\noindent\begin{equation*}
+    V_i[k+1] < V_i[k] \qquad \forall k \in \mathbb{N}
+\end{equation*}
+
+\subsubsection{Linear Parameter-Varying Systems}
+A third option is to rewrite the nonlinear system in a ``linear'' fashion with slow varying parameter $\sigma(\mathbf{x})$
+\noindent\begin{equation*}
+    \dot{\mathbf{x}}_\delta = \mathbf{A} (\sigma(\mathbf{x}))\mathbf{x}_\delta + \mathbf{B}(\sigma(\mathbf{x}))\mathbf{u}_\delta\quad
+    \begin{cases}
+        \mathbf{x}_\delta = \mathbf{x}-\mathbf{x}_e(\sigma(\mathbf{x})) \\
+        \mathbf{u}_\delta = \mathbf{u}-\mathbf{u}_e(\sigma(\mathbf{x}))
+    \end{cases}
+\end{equation*}
+
+This system is \textbf{exponentially stable} if
+\noindent\begin{equation*}
+    {\bar{\mathbf{A}}(\sigma)}^{\mathsf{T}} \mathbf{P} + \mathbf{P}\bar{\mathbf{A}}(\sigma) + \sum \rho_i \frac{\partial \mathbf{P}(\sigma)}{\partial \sigma_i} < 0,\qquad
+    \begin{cases}
+        \sigma(t)\in S       \\
+        \dot{\sigma(t)}\in R \\
+        \forall \sigma\in S, \rho\in R
+    \end{cases}
+\end{equation*}
+In other words, the existence of a family of positive matrices $\mathbf{P}(\sigma)$ (common Lyapunov functions) prove the systems exponential stability.
+
+\subsubsection{Backstepping Control}
+Backstepping is a form of \textit{cascaded control} where a control input $\mathbf{u}$ is chosen s.t.\ the output $\mathbf{z}$ of the outer loop stabilizes the inner loop:
+
+\noindent\begin{align*}
+    \dot{\mathbf{x}} & = f_0(\mathbf{x}) + g_0(\mathbf{x})\mathbf{z}                         &  & \text{inner loop} \\
+    \dot{\mathbf{z}} & = f_1(\mathbf{x},\mathbf{z}) + g_1(\mathbf{x}, \mathbf{z}) \mathbf{u} &  & \text{outer loop}
+\end{align*}
+where the inner system has an equilibrium point in $\mathbf{x}=0, \mathbf{z}=0$.
+
+\newpar{}
+If $\mathbf{z} = \mathbf{u}_0(\mathbf{x})$ then 
+\noindent\begin{equation*}
+    \frac{d}{dt} V_0(\mathbf{x}) = -W(\mathbf{x}) < 0
+\end{equation*}
+proving stability in the sense of Lyapounov.
+
+\newpar{}
+\ptitle{Controlling the Error}
+
+In order to ensure stability, Lyapunov stability of the error $\mathbf{e} = \mathbf{z}-\mathbf{u}_0$ needs to be established.
+
+The Lypunov candidate
+\noindent\begin{equation*}
+    V_1(\mathbf{x}, \mathbf{e}) = V_0(\mathbf{x})+\frac{1}{2} \mathbf{e}^2
+\end{equation*}
+satisfies
+\noindent\begin{equation*}
+    \frac{d}{dt}V_1(\mathbf{x}, \mathbf{e}) = -W(\mathbf{x}) - k_1 \mathbf{e}^2 <0 ,\qquad \forall(\mathbf{x},\mathbf{e}) \neq 0
+\end{equation*}
+and thus proves the stability of the ``error'' system
+\noindent\begin{align*}
+    \dot{\mathbf{x}} &= f_0(\mathbf{x})+g_0(\mathbf{x})\mathbf{u}_0(\mathbf{x}) + g_0(\mathbf{x})\mathbf{e}\\
+    \dot{\mathbf{e}} &= \dot{\mathbf{z}} - \underbrace{\frac{\partial \mathbf{u}_0(\mathbf{x})}{\partial \mathbf{x}}(\dot{\mathbf{x}})}_{\dot{\mathbf{u}}_0} = \mathbf{v}
+\end{align*}