Linear Systems Theory: HW6

(Note: whenever you use Matlab, copy your Matlab work in one pdf.)

1. Show that if the time-invariant linear state equation x ̇=Ax+Bu, y=Cx+Du

with m ≥ p, where m is the dimension of the input vector and p is the dimension of the output vector, is controllable, and

 ?A B?

rankC D =n+p

then the state equation

?A 0? ?B? z ̇ = C 0 z + D u

is controllable. Also prove the converse.

2. Consider the uncontrollable state equation

2 1 0 0 0 x ̇=0 2 0 0x+1u

 00−10 1

000−1 1

Is it possible to find a gain K so that the equation with state feedback u = r − Kx has eigenvalues −2. − 2, −1, −1? Is it possible to have eigenvalues −2. − 2, −2, −1? How about −2, −2, −2, −2? Is the equation stabilizable?

3. Consider the system

−3 0 1 1 0 1

x ̇=3 −4 −3 6 x+−1 0u, y=?−2 1 2 −4?x

−5 3 3 −11  1 1

−1 1 1 −6 0 0

(a) Find the controllable subspace

(b) Decompose the system into controllable and uncontrollable parts.

(c) What are the controllable eigenvalues?

(d) Find the unobservable subspace.

(e) Decompose the system into observable and unobservable parts.

(f) What are the unobservable eigenvalues?

(g) Find a vector (or vectors) that belong to the intersection of the controllable and unobservable subspaces.

(h) What is the dimension of the intersection of the controllable and unobserv- able subspaces.

(i) Transform the system into the Kalman canonical decomposition

4. Consider the system x ̇ = Ax + Bu, y = Cx, where

0 1 0 0

A=0 0 1; B=0; C=?1 2 0?.

1 0 −1 1

Determine an appropriate linear feedback, u = F x + v, such that the transfer

function of the closed-loop system is

H(s) = 1

s2 +3s+2

Comment on your control design.