17.6: Exercises

Exercise 17.1

Let $$P(s)=e^{-2 s}-1$$ be connected in a unity feedback configuration. Is this system well-posed?

Exercise 17.2

Assume that $$P_{\alpha$$ and $$K$$ in the diagram are given by:

$P_{\alpha}(s)=\left(\begin{array}{cc} \frac{s}{s+1} & \frac{-\alpha}{s+1} \\ \frac{1}{(s+1)} & \frac{1}{s+1} \end{array}\right), \quad \alpha \in \mathbb{R}, \quad K(s)=\left(\begin{array}{cc} \frac{s+1}{s(s+5)} & 0 \\ -\frac{s+1}{s(s+5)} & \frac{s+1}{s+5} \end{array}\right)\nonumber$

1. Is the closed loop system stable for all $$\alpha > 0$$?
2. Is the closed loop system stable for $$\alpha = 0$$?

Exercise 17.3

Consider the standard servo loop, with

$P(s)=\frac{1}{10 s+1}, \quad K(s)=k\nonumber$

but with no measurement noise. Find the least positive gain such that the following are all true:

• The feedback system is internally stable.
• With no disturbance at the plant output $$(d(t) \equiv 0)$$, and with a unit step on the command signal $$r(t)$$, the error $$e(t) = r(t) - y(t)$$ settles to $$|e(\infty)| \leq 0.1$$.
• Show that the $$\mathcal{L}_{2}$$ to $$\mathcal{L}_{\infty}$$ induced norm of a SISO system is given by $$\mathcal{H}_{2}$$ norm of the system.
• With zero command $$(r(t) \equiv 0),\|y\|_{\infty} \leq 0.1$$ for all $$d(t)$$ such $$\|d\|_{2} \leq 1$$. [ADD NEW Problem]

Exercise 17.4 Parametrization of Stabilizing Controllers

Consider the diagram shown below where $$P$$ is a given stable plant. We will show a simple way of parametrizing all stabilizing controllers for this plant. The plant as well as the controllers are finite dimensional.

1. Show that the feedback controller

$K=Q(I-P Q)^{-1}=(I-Q P)^{-1} Q\nonumber$

for any stable rational $$Q$$ is a stabilizing controller for the closed loop system

2. . Show that every stabilizing controller is given by $$K=Q(I-P Q)^{-1}$$ for some stable $$Q$$. (Hint: Express $$Q$$ in terms of $$P$$ and $$K$$).

3. Suppose $$P$$ is SISO, $$w_{1}$$ is a step, and $$w_{2} = 0$$. What conditions does $$Q$$ have to satisfy for the steady state value of $$u$$ to be zero. Is it always possible to satisfy this condition?

Exercise 17.5

Consider the block diagram shown in the figure below.

(a) Suppose $$P(s)=\frac{2}{s-1}$$, $$P_{0}(s)=\frac{1}{s-1}$$ and $$Q = 2$$. Calculate the transfer function from $$r$$ to $$y$$.

(b) Is the above system internally stable?

(c) Now suppose that $$P (s) = P_{0}(s) = H(s)$$ for some $$H(s)$$. Under what conditions on $$H(s)$$ is the system internally stable for any stable (but otherwise arbitrary) $$Q(s)$$?

Exercise 17.6

Consider the system shown in the figure below.

The plant transfer function is known to be given by:

$P(s)=\left[\begin{array}{cc} \frac{s-1}{s+1} & 1 \\ 0 & \frac{s+1}{s+2} \end{array}\right]\nonumber$

A control engineer designed the controller $$K(s)$$ such that the closed-loop transfer function from $$r$$ to $$y$$ is:

$H(s)=\left[\begin{array}{cc} \frac{1}{s+4} & 0 \\ 0 & \frac{1}{s+4} \end{array}\right]\nonumber$

(a) Compute $$K(s)$$.

(b) Compute the poles and zeros (with associated input zero directions) of $$P (s)$$ and $$K(s)$$.

(c) Are there pole/zero cancellations between $$P (s)$$ and $$K(s)$$ ?

An engineer wanted to estimate the peak-to-peak gain of a closed loop system $$h$$ (the input-output map). The controller was designed so that the system tracks a step input in the steady state. The designer simulated the step response of the system and computed the amount of overshoot $$(e_{1})$$ and undershoot $$(e_{2})$$ of the response. He/She immediately concluded that
$\|h\|_{1} \geq 1+2 e_{1}+2 e_{2}\nonumber$