Exercise 3

Problem 1

Consider the systems $P_1(s) = 87.5 \frac{s + 1/5}{(s-5)(s + 10)}$ and $P_2(s) = 50 \frac{(1/5 - s)(s + 5)}{ (1/5 + s)(5 - s) (s + 10)}$

1. Factorize both systems into minimum phase and normalized nonminimum phase components.
2. Compute step responses for both systems. As the systems are unstable, simulate for $t\in\left\lbrack0,1\right\rbrack$. Are the systems very different?
3. Plot the nyquists, nichols and bode diagrams for the loop transfer function $L = PC$ where the controller is given by $C=1+\frac{1}{s}$. Which (if any) of the closed loops are stable? Why/Why not?
4. Where applicable, design a low-pass filtered PI controller that does not inject too much noise, gives closed-loop bandwidth of about 15 rad/s, and Ms < 1.4 and Mt < 2.

Problem 2

Show that $\left| \frac{z \bar p - 1}{z - p} \right| = 1$ for all $z$ on the unit circle.

Problem 3

The purpose of this exercise is to translate the nonminimum phase elements of lecture 4 to discrete time. Factor the following systems into minimum phase and all-pass nonminimum phase elements, $P = P_\text{mp} P_\text{nmp}$such that $P_\text{mp}(1) > 0$ and $|P_\text{nmp}(z)| = 1, \quad \forall |z| = 1$. Furthermore, for sampling time $h = 0.01$ plot the bode diagrams of $P_\text{nmp}$.

1. Unstable zero in $z_0 = 1.02$ , $P(z) = z_0 - z$
2. Unstable pole $p_0 = 1.01$, $P(z) = \frac{1}{z - p_0}$
3. unstable pole zero pair with $p_0 = 1.01$, $z_0 = 1.02$, $P(z) = \frac{z_0 - z}{z - p}$
4. Unstable pole zero pair with $p_0 = 1.02$ ,$z_0 = 1.01$, $P(z) = \frac{z_0 - z}{z - p}$

Problem 4

In continuous time, a rhp zero in $z_0$ limits the bandwidth of the sensitivity function: let $S_\text{req}(s) = \frac{s M_s}{s + aM_s}$ and require$|S(i\omega)| < |S_\text{req}(i\omega)|$ for $\omega \in [0, \infty)$, then $a < z \frac{M_s - 1}{M_s}$. Derive a discrete-time version.

Problem 5

Sketch the level curves in the complex plane of $\textrm{Re}(1/z) = c$

Problem 6

Use QFT design on the uncertain system

$P(s) = \frac{k}{(s + a)(s + b)}, \quad k \in [1, 10], \quad a \in [1, 5], \quad b \in [20, 30]$.

The specifications are:

• $\max_\omega |T(i\omega| < 1.2$
• $|S(i\omega)| \leq \left| \frac{0.02s^3 + 1.28s^2 + 14.96s + 48}{s^2 + 14.4\textcolor{red}{s} + 169}\right|$ for $\omega \in [0, 10]$ rad/s
• $|PS(i\omega)| \leq 0.01$ for$\omega \in [0, 50]$ rad/s

HINT: This is Example 1 in the manual for the QFTIT tool.

Problem 7

1. Let $H(s)$ be a strictly proper transfer function that has all its poles in the half plane $\textrm{Re}\ s \leq - \alpha$ for some $\alpha > 0$, i.e. $H(s)$ is analytic on $\textrm{Re } s > -\alpha$. Let $h(t)$ be the corresponding time domain function, i.e. $H = \mathcal L h$, where $\mathcal L$ denotes the Laplace transform. Show that for any $\textrm{Re } s_0 > -\alpha$ we have $\int_0^\infty e^{-s_0 t}h(t) dt = \lim_{s\to s_0} H(s)$. Hint: Look at the definition of the (one-sided) Laplace transform.
2. For a process $P(s)$ and controller $C(s)$. Consider our standard error feedback configuration, i.e. the 2-DOF configuration with $F = I$. Suppose that the closed loop is stable. Show that for a reference step input
   
   1. If $\lim_{s\to0}\textcolor{red}{s}P(s)C(s)=c_1$, for some $0 < |c_1| < \infty$, then $\lim_{t \to \infty} e(t) = 0$ and $\int_0^\infty e(t)dt = \frac{1}{c_1}$
   2. If $\lim_{s \to 0}s^2 P(s)C(s) = c_2$, for some $0 < |c_2| < \infty$, then $\lim_{t \to \infty} e(t) = 0$ and $\int_0^\infty e(t) dt = 0$
      Hint: Consider $P(s)C(s) = \tilde L(s) / s$ where $\lim_{s \to 0}\tilde L(s) = c_1$ and apply the final value theorem.
3. Suppose that a process $P(s)$ has an unstable pole at $s = p$ such that $\textrm{Re}\ p > 0$. Show that if the closed loop is stable, then for a reference step input
   
   1. $\int_0^\infty e^{-pt}e(t) dt = 0$
   2. $\int_0^\infty e^{-pt}y(t)dt = \frac{1}{p}$
      Hint: Interpolation constraints, use the result in A
4. Suppose instead that $P(s)$ has an unstable zero at $s = z$ such that $\textrm{Re}\ p > 0$. Show that if the closed loop is stable, then for a reference step input
   1. $\int_0^\infty e^{-qt}e(t)dt = \frac{1}{q}$
   2. $\int_0^\infty e^{-qt}y(t) = 0$
5. Discuss implications of B, C and D on overshoot and undershoot.