MATH319 Exercises 9 Assessed Exercise 4 11 Projects for MATH319 Linear Systems: Instructions

10 Workshop Exercise 5

W5.1 Express the rational function

G(s)={{s^{2}+s+1}\over{s^{2}-2}}

as the quotient $G=P/Q$ of stable rational functions $P$ and $Q$ that are coprime in ${\cal S}$ .

W5.2 Let $p(s)$ be a complex polynomial with leading term $s^{n}$ , and let $r(s)=p(s)-(s+1)^{n}$ .

(i) Show that

R(s)={{r(s)}\over{(s+1)^{n}}}

is stable.

(ii) Show that $p(s)=0$ has no roots in the right half plane if and only if the Nyquist contour of $R(s)$ does not pass through or wind around $-1$ . .

(ii) Hence show that

p(s)=s^{4}+3s^{3}+2s^{2}+s+1

has roots in the right half plane.

W5.3 Consider a MIMO system $(A,B,I,0)$ . By adding a feedback controller $K$ , we replace $A$ by $A-BK$ . Let $T(s)$ be the transfer function of $(A,B,K,I).$ Show that

\det T(s)={{\det(sI-A+BK)}\over{\det(sI-A)}}.

Note the result of W3.9, which is helpful here.

W5.4 Consider the matrices

A=\left[\begin{pmatrix}-1&-3&-2&-4\cr 1/2&-5&-1&-4\cr-2&-3&-5&-5\cr-1&0&-2&-7% \cr\end{pmatrix}\right],\quad P=\left[\begin{pmatrix}11&0&0&0\cr 0&1&0&0\cr 0&% 0&3&1\cr 0&0&1&3\end{pmatrix}\right].

(i) Show that $P$ is positive definite.

(ii) Give numerical approximations to the eigenvalues of $A$ .

(iii) Obtain numerically a positive definite $K$ such that

-AK-KA^{\dagger}=P.

W5.5 Suppose that $X,Y,M,N\in{\cal S}$ satisfy $XM+YN=1$ , and let

K={{NQ+X}\over{-MQ+Y}}.

Solve this for $Q$ .

W5.6 An amplifier and its controller have transfer functions

G(s)={{\alpha}\over{1+\beta s}},\quad K(s)=b+{{c}\over{s}},

where $\alpha,\beta,b,c$ are real constants with $\alpha,\beta\neq 0$ .

(i) State conditions under which $G(s)$ is stable.

(ii) Compute the entries of

\Psi={{1}\over{1+GK}}\left[\begin{pmatrix}1&G\cr K&GK\cr\end{pmatrix}\right],

and state conditions for all the entries to be stable.

(iii) Deduce that for all $G$ there exists $K$ such that $\Psi$ is stable.