MATH319 Exercises 7 Assessed Exercise 3 9 Assessed Exercise 4

8 Workshop Exercise 4

W4.1 Find the gain $\Gamma$ and phase change $\phi$ of the transfer function $T$ associated with the linear differential equation

y^{\prime\prime}+6y^{\prime}+y=-3u^{\prime}+u.

with $y(0)=y^{\prime}(0)=0=u(0)$ ; here $y$ is the output and $u$ is the input.

W4.2 Nyquist and Bode Plots Recall $s=i\omega$ and let

T(s)={{8s+8\imath+4}\over{(s+1)(s+2+\imath)}}.

(i) Plot the Nyquist locus of $T$ .

(ii) Let $\Gamma(\omega)$ be the gain and let $\phi(\omega)$ be the phase of $T$ . The Bode plot consists of the graphs of $\log\Gamma(\omega)$ and $\phi(\omega)$ against $\omega$ . Produce the Bode plot for $-100<\omega<100.$

W4.3 Descartes’s Rule of Signs Let $\sigma$ be the number of changes in sign in the real sequence $a_{0},\dots,a_{n}$ , ignoring $0$ . Let $r$ be the number of positive roots of

a_{0}+a_{1}x+\dots+a_{n}x^{n}=0.

Then $r\leq\sigma$ , and $\sigma-r$ is even.

Deduce the possible value of $r$ for the polynomial equations:

(i) $-2+3x+5x^{2}+x^{3}=0$ ;

(ii) $2+3x-4x^{2}+(1/2)x^{3}+x^{4}-x^{5}+6x^{2}-x^{7}=0.$

(iii) Find the roots of

-2+3x+5x^{2}+x^{3}=0

numerically; hence find $r$ .

(iv) Likewise, find the roots of

2+3x-4x^{2}+(1/2)x^{3}+x^{4}-x^{5}+6x^{6}-x^{7}=0

numerically; hence find $r$ .

W4.4 (i) Find the zeros of the polynomial

p(s)=s^{3}+10s^{2}+16s+160.

(ii) Obtain numerical approximations to the zeros of

q(s)=s^{3}+11s^{2}+16s+160,

r(s)=s^{3}+9s^{2}+16s+160.

(iii) Discuss which of these polynomials $p, q, r$ is stable.

W4.5 More Bode Plots (i) Let $T(s)=p(s)/q(s)$ , where $p(s)$ and $q(s)$ are polynomials with real coefficients; then $T(s)$ is said to be a real rational function. Show that the gain $\Gamma$ and phase $\phi$ of $T$ satisfy

\Gamma(\omega)=\Gamma(-\omega),\quad\phi(-\omega)=-\phi(\omega)\qquad(\omega% \in{\bf R}).

(ii) For $T(s)=1/(1+s)$ and $s=\imath\omega$ , plot $\log\Gamma(\omega)$ and $\phi(\omega)$ against $\omega$ for $-100<\omega<100$ .

(iii) When $T(s)$ is a transfer function as in (i), we can plot $\log\Gamma$ and $\phi$ against $\log\omega$ for $0<\omega<\infty$ . Do this for $T(s)=1/(1+s)$ .

W4.6 Let $K$ be the matrix

K=\left[\begin{pmatrix}m&n\cr-x&y\cr\end{pmatrix}\right],

where $m, n, x, y$ are all integers.

(i) By considering $\det K^{-1}$ , show that $K$ has an inverse $K^{-1}$ with integer entries, if and only if $my+xn=\pm 1$ .

(ii) Show that the condition of (i) is equivalent to $m$ and $n$ having highest common factor $1$ .

(iii) Show conversely that if $m$ and $n$ have highest common factor $1$ , then one can choose integers $x$ and $y$ such that $K$ as above is invertible and $K^{-1}$ has integer entries.

W4.7 Let $T(s)=a+(b/s)$ with complex $a, b$ .

(i) Compute $T(i\omega)\overline{T(i\omega)}$ and show that the gain satisfies

\Gamma(\omega)=\sqrt{|a|^{2}+{{2\Im a\bar{b}}\over{\omega}}+{{|b|^{2}}\over{% \omega^{2}}}}.

(ii) Derive an expression for $e^{i\phi(\omega)}.$

W4.9. For

A=B=\left[\begin{pmatrix}1&0\cr 0&0\cr\end{pmatrix}\right],

let $T:M_{2}({\bf C})\rightarrow M_{2}({\bf C})$ be the operator

T(X)=AX+XB\qquad(X\in M_{2}({\bf C})).

(i) Show that $T$ is linear.

(ii) Find ${\hbox{ker}}(T)=\{X:T(X)=0\}$ and ${\hbox{image}}(T)=\{T(X):X\in M_{2}({\bf C})\}.$