Homework 9 - Due: 04/02

Tip

We plan to release the solutions the day the homework is due so that you may prepare for the midterm; therefore (a) seek help early if you are stuck and (b) absolutely no extensions.

Problem 1

Construct minimal realizations of the following transfer functions: $$G_1(s)={\displaystyle \frac{s-3}{s^2-5s+6}},G_2(s)={\displaystyle \frac{s^2+1}{s^3-2s^2+s}}$$

Problem 2

Consider the system $$\dot{x}=\left[\begin{array}{ccc}0& 1& 2\\ 0& 0& 3\\ 0& 0& -1\end{array}\right]x+\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]u$$

1. Is this system asymptotically stabilizable by a state feedback $u=Kx$? If yes, find such a $K$; if not, explain why not.

2. Consider the output $$y=\left[\begin{array}{ccc}1& 0& 0\end{array}\right]x$$ Determine the transfer function of the resulting (open-loop) system . Explain your answer.

Problem 3

Consider the system $$\dot{x}=\left[\begin{array}{ccc}0& -1& 1\\ 1& 2& -1\\ 0& 5& -3\end{array}\right]x,y=\left[\begin{array}{ccc}1& -1& 1\end{array}\right]x$$

1. Design an asymptotic observer for this system such that the eigenvalues of the error dynamics are $-1$ and $-2\pm i$. (Hint: You can carefully use the result of Problem 8 of previous homework).

2. What would happen to the entries of the output injection matrix $L$ if the desired observer eigenvalues were moved to $-10$ and $-20\pm i$, or to $-100$ and $-200\pm i$? Explain why.

Problem 4

Consider the system $$\dot{x}=\left[\begin{array}{ccc}-1& 0& 4\\ 2& -2& 3\\ 0& 1& 1\end{array}\right]x,y=\left[\begin{array}{ccc}0& 0& 1\end{array}\right]x$$

Design an Luenberger (i.e. a reduced order) observer for $x_1,x_2$ in the form of a dynamical system with state dimension 2.

Problem 5

Consider the harmonic oscillator with position measurements: $$\ddot{x}+x=u,y=x$$

1. Show that it cannot be asymptotically stabilized by static output feedback of the form $u=ky$.

2. Find a dynamic output feedback law that asymptotically stabilizes this system.

Problem 6

Consider the system $$\dot{x}=Ax+Bu+d_{1}y=Cx+d_2$$

Suppose the objective is to make $y(t)$ asymptotically approach a reference signal $r(t)$, in spite of the disturbances. In this problem, take $r(t)$ to a sinusoidal signal: $$r(t)=\sin (t),t\ge 0$$ Solve the problem by following these steps, along the lines of the construction given in class for tracking constants & ramps.

1. Reduce the problem to asymptotic stabilization of an auxiliary system whose state contains the tracking error $e:=y-r$. You can assume here that the disturbances belong to the same class as $r$, but make this assumption precise and explain how it helps your analysis.

2. Assuming controllability of this auxiliary system, state what type of control law can be used to stabilize it (you don’t need to investigate conditions for controllability in terms of the original data $A$, $B$, $C$ like we did in class). Your final answer for the controller should be a state-space dynamical system (don’t leave it in the form containing integrators).

3. Comment in what sense your results reflect the internal model principle.