## Homework 8 - Due: 03/21

This homework is slightly on the longer side to make up for Homework 5 and Homework 6 being shorter. We suggest you start early.

### 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.

### 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 *infinite* time horizon, i.e., \(M(0,\infty)\). Show that along solutions of the system we have \[
\dot V=-|y|^2.
\]

### Problem 3

- For LTI systems, show that \((A,C)\) is observable if and only if \((-A,C)\) is observable.
- Is the same statement true for LTV systems? Prove or give a counterexample.

### Problem 4

Obtain a combined controllability/observability decomposition for the LTI system \(\dot x =Ax+Bu\), \(y=Cx\) by following these steps:

- Ignoring the control for now, write down the Kalman observability decomposition.
- Now add the control, noting that the \(B\) matrix assumes no special structure in the coordinates that give the observability decomposition.
- For the observable part of the system, switch coordinates to get the Kalman controllability decomposition for it. Repeat separately for the unobservable part.

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.

### 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 \(v\): \[\begin{align*} \dot x&=(A+BK)x+Bv\\ y&=Cx \end{align*}\]

- Is the new system controllable? Prove or give a counterexample.
- Is the new system observable? Prove or give a counterexample.

### Problem 6

Consider the system

\[ \begin{aligned} \dot x&=-2x+u\\ y&=x+u \nonumber \end{aligned} \tag{1}\]

Construct a system of the form

\[ \begin{aligned} \dot z&=az+by\\ u&=cz+dy \nonumber \end{aligned} \tag{2}\]

which serves as an *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.

### Problem 7

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.

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.)

### 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 \(-2\pm i\).