Optimal Control for Linear Systems:Spring 2010

F I N A L E X A M

Problem 1.

(30 pts) Let G(s) =(s− 1)

(s+ 2)(s− 3). Find a stable co-

prime factorization G(s) =n(s)

m(s)and x(·), y(·) ∈ RH∞

such that x(s)n(s) + y(s)m(s) = 1 .

[Hints: If you would like to use a state-space method with the observer

canonical form, then F = [−6, − 2] is a stabilizing feedback gain

and L = [−4, − 8]T is a stabilizing observer gain. In case you prefer

the direct approach, you might try to search for a constant x(s) and

a first-order biproper y(s) .]

Problem 2.

Consider the simple mechanical model

d2s

dt2= u+ w, s(0) = 1,

ds

dt

∣

∣

∣

∣

t=0

= −1

with an impulse-like disturbance w = 1 · δ(t) .The goal is to design a stabilizing force u, under assump-tion that both the position s and the velocity ds

dtare

measurable, that minimizes the performance index

Φ{u} =

∫ ∞

0

(

s2 + η u2)

dt, η > 0.

1

Proceed as follows.

1. (15 pts) Derive the appropriate Riccati equation as-suming that the state vector is x =

[

s, ds

dt

]

T

.

2. (15 pts) Show that P =

[√2 η

1

4 η1

2

η1

2

√2 η

3

4

]

is the

stabilizing solution of this Riccati equation.

3. (10 pts) Find the optimal feedback control law.

What happens with the poles and with the control sig-nal in the optimal closed-loop system when η is get-ting smaller?

4. (10 pts) Compute the minimal (optimal) value of theperformance index for η = 4 .

Problem 3.

Consider the system

d2y

dt2= u+ w,

1. (10 pts) Using any approach you like, find a stabilizingfeedback controller u = K(·) y that achieves y → 0as t → 0 in the case when w ≡ 0 .

2. (10 pts) Find the matrix coefficients of a Lyapunovequation, solution of which can be used for computingthe H2 -norm of the closed-loop system.

Describe how you would proceed assuming that thissolution is found.

2