MATH319 Exercises 8 Workshop Exercise 4 10 Workshop Exercise 5

9 Assessed Exercise 4

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A4.1 Let $p(s)=s^{2}-2s+7$ and $q(s)=s^{3}+2s^{2}+(69/4)s+65/4$ .

(i) Verify that $R(s)=p(s)/q(s)$ is stable.

(ii) By considering the Nyquist locus of $R$ , discuss whether $T=R/(1+R)$ is also stable. Supply graphs to justify your results.

(iii) Replace $p(s)$ by $r(s)=s^{2}-2s-20$ , and repeat (i) and (ii).

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A4.2 (i) Consider the system $(A,B,C,D)$ given by

A=\left[\begin{pmatrix}2&0\cr 0&-3\cr\end{pmatrix}\right],\quad B=\left[\begin% {pmatrix}0\cr 1\cr\end{pmatrix}\right],\quad C=\left[\begin{pmatrix}0&1\cr\end% {pmatrix}\right],\quad D=0.

Show that the corresponding transfer function is stable.

(ii) Show that if we replace $B$ and $C$ by

B^{\prime}=\left[\begin{pmatrix}1\cr 1\cr\end{pmatrix}\right],\quad C^{\prime}% =\left[\begin{pmatrix}1&1\cr\end{pmatrix}\right],

then the transfer function of $(A,B^{\prime},C^{\prime},D)$ is unstable.

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A4.3 Let $K$ be the matrix

K=\left[\begin{pmatrix}P(s)&Q(s)\cr-X(s)&Y(s)\cr\end{pmatrix}\right],

where $P(s),Q(s),X(s),Y(s)$ are all complex polynomials.

(i) By considering $\det K^{-1}$ , show that $K$ has an inverse with polynomial entries, if and only if $P(s)Y(s)+X(s)Q(s)=\kappa$ for some $\kappa\neq 0$ a constant.

(ii) Show that given $P(s)$ and $Q(s)$ , there exist $X(s)$ and $Y(s)$ such that $P(s)Y(s)+X(s)Q(s)=\kappa$ for some $\kappa\neq 0$ , if and only if $P(s)$ and $Q(s)$ have highest common factor $1$ .

(iii) Show conversely that if $P(s)$ and $Q(s)$ have no common zeros, then one can choose polynomials $X(s)$ and $Y(s)$ as entries of $K$ such that $K$ is invertible and $K^{-1}$ has polynomial entries.

(iv) Given $P(s)=s^{2}+3s+2$ and $Q(s)=s^{2}+2s-3$ , find a $K$ as in (i).

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A4.4 Consider the matrix

A=-\left[\begin{pmatrix}1&2&3\cr 2&5&1\cr 1&2&7\cr\end{pmatrix}\right].

(i) Show that $-A-A^{\dagger}$ is not positive definite, by considering the determinant or otherwise.

(ii) Show that there exists a positive definite $K$ such that

-AK-KA^{\dagger}=I

has a solution, and find $K$ numerically. (Use appropriate computer programs.)

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