Exercise 2

2.1

Sketch the root locus diagram for the designs done in Lecture 3 on the resonant system $P(s) =\frac{100}{s^2+0.02s+100}$ with

a) $C(s) = \frac{0.01}{s}$

b) $C(s) = \frac{1}{s(s/2+1)^2}$

c) $C(s) = \frac{s^2+4.8s+144}{s(s/2+1)(s^2+1.76s+77.44)}$

d) $C(s) = 1.5\frac{e^{-0.3s}}{s}$

a-c are easily done by e.g. rlocusplot(P*C) in matlab. In subproblem d you will need to figure out some way to plot (an approximation of the most relevant part of) the root locus diagram since the exact characteristic equation has infinitely many roots. (Hint: Pade)

2.2

Let us consider different ways of sampling the resonant system $P(s) =\frac{100}{s^2+0.02s+100}$. In matlab this can be achieved with Pd = c2d(P,h,method).

Study the effects of sampling by plotting P and Pd in the same Bode-diagram. Use sampling period  $h=0.05$  and

a) method = 'zoh'

b) method = 'impulse'

c) method = 'tustin'

d) method = 'matched'

e) method = 'least-squares'

(note: faster sampling would be slightly better).

Compare the different methods regarding their ability to minimize distortion of the frequency response.

 

2.3

a) Simulate the step response of a sampled version of the resonance process P in problem 2.2 with sampled version of the the controller C in problem 2.1c. Use h=0.05, and the sampling method of your choice. You will notice that there is significant ringing in the step response

b) Design a discrete time prefilter F(z) and simulate the step response (this can be done in matlab by step(feedback(P*C,1)*F))

Try to achieve a settling time of 5 second such that $|y(t)-1| \leq 0.01$ when t>5.

2.4

Define the average residence time $T_\text{ar}$ as the first moment of the impulse response, i.e. $T_\text{ar} = \int_0^\infty th(t) dt / \int_0^\infty h(t) dt$. A possible definition of rise time $T_r$ that is sometimes used is given by

$T_r := \left(\dfrac{\int_0^\infty(t - T_\text{ar})^2 h(t) dt}{\int_0^\infty h(t) dt}\right)^{1/2}$.

1. Show that $T_\text{ar} = -\dfrac{P'(0)}{P(0)}$ and that $T_r^2 = \dfrac{P''(0)}{P(0)} - T_\text{ar}^2$
2. Calculate $T_\text{ar}$ and $T_r$ for $e^{-sT}$, $1 / (1 + sT)^n$ and $(1 - e^{-sh})/(sh)$
3. Consider a system composed of $n$ cascaded systems, each with $h_k(t) \geq 0$ (monotone step responses) and residence time $T_{\text{ar}, k}$ and rise time of $T_{r, k}$ respectively. Show that the residence time and rise time of the cascaded system equals
   $T_{\text{ar}, \text{tot}} = T_{\text{ar}, 1} + \ldots + T_{\text{ar}, n}$
   $T_{r, \text{tot}} = (T^2_{r,1} + \ldots + T^2_{r, n})^{1/2}$
   
   

2.5

There are many equivalent versions of Bode's relations valid for stable minimum phase systems. Here, we will use this variant

$v(i\omega_0) = \dfrac{2\omega_0}{\pi}\int_0^\infty \dfrac{u(i\omega) - u(i\omega_0)}{\omega^2 - \omega_0^2}d\omega$, and apply it to $u + iv = \log|G| + i\arg{G}$.

Verify by direct calculation that the system $G(s) = e^{-\sqrt{s}}$ satisfies this relation.