Midterm Examination of Linear Systems

2010. 04 27

Dept./School:             ID number:               Name:

1. (15 points) A linear operator A with respect to the basis {[eqn: n _{i}]} has a generalized eigenvector v of grade k. When we choose a new basis by the chain of generalized eigenvectors, derive the new representation of the linear operator.

2. (15 points) Prove that [eqn: m TIMES n] matrix H, [eqn: m GEQ n], has rank [eqn: n] if and only if [eqn: H ^{T} H] has rank [eqn: n] or det([eqn: H ^{T} H])[eqn: != 0.]

3. (15 points) Consider a state equation of [eqn: dot{x} =- {1} over {1+t} x.]

(a) Is the equilibrium [eqn: x=0] asymptotically stable? Why?

(b) Is the equilibrium [eqn: x=0] uniformly stable? Why?

(c) Is the equilibrium [eqn: x=0] uniformly asymptotically stable? Why?

4. (20 points) Consider a linear time invariant system

[eqn: dot{x} (t)=Ax(t)].

Prove that [eqn: Re( lambda _{i} (A))<0] if and only if for any [eqn: N ^{T} =N>0], there exists

[eqn: M ^{T} =M>0`] such that [eqn: A ^{T} M+MA=-N.]

5. (20 points) Prove that [eqn: W _{c} (t)= int _{0} ^{t} {e ^{A tau } BB ^{T} e ^{A ^{T} tau } d tau } `] is nonsingular if and only if [eqn: C=[B````AB````A ^{2} B````...```A ^{n-1} B]] has full rank.

6. (15 points) A linear time invariant system

[eqn: dot{x} (t)=Ax(t)+Bu(t),````y(t)=Cx(t)]

is observable. Find the initial value of the state from the known input and output signals during [eqn: [0,`t _{1} ]] without using derivatives of input and output.

Good Luck !!