Final Examination of Linear Systems

2010. 06. 11

Dept./School:             ID number:               Name:

1. (20 points) Consider a transfer function

[eqn: g(s)= {s-1} over {s ^{2} +s-2}].

Realize the transfer function into a 2-nd order state equation with the state of

[eqn: x _{1} (t)=y(t),``x _{2} (t)= dot{y} (t),``]

where y(t) is output and u(t) is input.

Is this realization controllable? observable?

2. (30 points) Consider a state equation

[eqn: dot{x} (t)= LEFT [ {matrix{2&1#-1&1}} RIGHT ] ``x+ LEFT [ pile{1#2} RIGHT ] u# y=[`1```1``]x.]

(a) Calculate the transformation matrix P to transform the eq. into

controllable canonical form and write the controllable canonical

form.

(b) Design the state feedback controller by using the results in (a)

so that the closed loop equation has eigenvalues at -1 and -2.

(c) Design a reduced-dimensional state estimator with eigenvalue

at -5 by using Lyapunov equation method.

3. (15 points) The full rank Henkel matrix T(n,n) obtained from a coprime transfer function has relation of [eqn: T(n,n)=OC`,``]where O and C are observability and controllability matrix, respectively. By singular value decomposition, T(n,n) can be expressed as [eqn: T(n,n)=K LAMBDA L ^{T} `,``]where K and L are orthogonal matrices and [eqn: LAMBDA ``]is diagonal matrix with the distinct singular values of T(n,n). Determine a balanced realization {A, b, c} using the relation of [eqn: tilde{T} (n,n)=OAC`,``](Hint, balanced realization satisfies [eqn: O ^{T} O=C`C ^{T} = LAMBDA ``]).

4. (15 points) Is the following polynomial matrix column reduced? row reduced?

[eqn: M(s)= {bmatrix{3s ^{2} +2s&````2s+1#s ^{2} +s-3&s}}].

5. (20 points) Given a plant with [eqn: g(s)=(s-2)/(s ^{2} -1),] we want to design a two-parameter compensator for model matching into a closed loop transfer function of [eqn: g _{o} (s)=-(s-2)/(s ^{2} +2s+2).] Determine L(s) for  design of a minimum order compensator and write an algebraic equation to obtain M(s) and A(s) of the compensator.

Good Luck !!