\subsection{Homework 8 - Due: 03/21}
This homework is slightly on the longer side to make up for
\href{hw05.html}{Homework 5} and \href{hw06.html}{Homework 6} being
shorter. We suggest you start early.
\subsubsection{Problem 1}
Consider the discrete-time linear system with output \begin{align*}
x(k+1)&=Ax(k)\\
y(k)&=Cx(k)
\end{align*} and call it observable if different initial conditions
produce different output strings. Derive a condition for observability
in terms of \(A\) and \(C\). Show that if two initial conditions produce
outputs that coincide for the first \(n\) steps, then these outputs are
the same for all future steps.
\subsubsection{Problem 2}
Consider the LTI system \begin{align*}
\dot x&=Ax\\
y&=Cx
\end{align*} and suppose that the eigenvalues of \(A\) have negative
real parts. Consider the function \(V(x)=x^TMx\) where \(M\) denotes the
observability Gramian for the \emph{infinite} time horizon, i.e.,
\(M(0,\infty)\). Show that along solutions of the system we have \[
\dot V=-|y|^2.
\]
\subsubsection{Problem 3}
\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\tightlist
\item
For LTI systems, show that \((A,C)\) is observable if and only if
\((-A,C)\) is observable.
\item
Is the same statement true for LTV systems? Prove or give a
counterexample.
\end{enumerate}
\subsubsection{Problem 4}
Obtain a combined controllability/observability decomposition for the
LTI system \(\dot x =Ax+Bu\), \(y=Cx\) by following these steps:
\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\tightlist
\item
Ignoring the control for now, write down the Kalman observability
decomposition.
\item
Now add the control, noting that the \(B\) matrix assumes no special
structure in the coordinates that give the observability
decomposition.
\item
For the observable part of the system, switch coordinates to get the
Kalman controllability decomposition for it. Repeat separately for the
unobservable part.
\end{enumerate}
In the resulting system, make sure to specify all controllability and
observability properties of various subsystems. Identify four types of
modes: controllable and observable, uncontrollable but observable,
controllable but unobservable, and uncontrollable and unobservable.
\subsubsection{Problem 5}
Consider the system \(\dot x=Ax+Bu\), \(y=Cx\) and suppose that it is
both controllable and observable. Now consider the feedback of the form
\(u=Kx+v\), which leads to the system with new control \(v\):
\begin{align*}
\dot x&=(A+BK)x+Bv\\
y&=Cx
\end{align*}
\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\tightlist
\item
Is the new system controllable? Prove or give a counterexample.
\item
Is the new system observable? Prove or give a counterexample.
\end{enumerate}
\subsubsection{Problem 6}
Consider the system
\[
\begin{aligned}
\dot x&=-2x+u\\
y&=x+u \nonumber
\end{aligned}
\] \{\#eq-1\}
Construct a system of the form
\[
\begin{aligned}
\dot z&=az+by\\
u&=cz+dy \nonumber
\end{aligned}
\] \{\#eq-2\}
which serves as an \emph{inverse} to (1), in the sense that if we take
an input signal \(u\), feed it into the system (1), compute the output
\(y\), and feed this \(y\) into the system (2), we get the original
signal \(u\) back as the output of (2) (assuming zero initial conditions
for both \(x\) and \(z\)).
Run computer simulations to verify that the inverse you constructed in
part (a) indeed works as expected. Check what happens if you vary the
initial conditions.
\subsubsection{Problem 7}
\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
On Thursday before the break, we will discuss a lemma which says that
we can go from a controllable pair \((A,b)\) to its corresponding
controllable canonical form \((\bar A,\bar b)\) via a coordinate
transformation \(x=P\bar x\). In class we will derive
\(P=\mathcal C(A,b) \mathcal C^{-1}(\bar A,\bar b)\) and verify
\(\bar b=P^{-1}b\), but not that \(\bar A=P^{-1}AP\). Finish the proof
by verifying this last claim.
\item
Prove that any two minimal realizations of a given transfer function
\(g(s)\) can be obtained from each other by a coordinate
transformation. (Hint: use the result of the previous problem.)
\end{enumerate}
\subsubsection{Problem 8}
Consider the system
\[
\dot x=\begin{bmatrix} 0 & 1 & 0\\
-1 & 2 & 5\\
1 & -1 & -3\end{bmatrix}x+
\begin{bmatrix} 1\\-1\\1\end{bmatrix}u
\]
Find a state feedback law \(u=Kx\) such that the poles of the
closed-loop system are \(-1\) and \(-2\pm i\).