Lecture 11: Canonical Forms and Minimal Realizations

Question 1

Relation of Controllability and Controllable Canonical Form

Answer 1

If system is in controllable canonical form then the system has complete controllability.
Rank(P) -> full rank

Question 2

Relation of Observability and Observable Canonical Form

Answer 2

If system is in observable canonical form then the system has complete observability
rank(Q) -> full rank

Question 3

Method to determine Controllable / Observable Canonical Form

Answer 3

H(s) -> All integrator block -> canonical form

Question 4

Property of Controllable Canonical form System Matrix

Answer 4

Via Cayley Hamilton Theorem, A’ satisfies its own characteristic equation

Question 5

Equivalence Transform Theorem to Controllable Canonical Form (A’)

Answer 5

T = [t1,..., tn] 
where 
tn = B
tn-1 = A*tn + an-1 * B
...
t1 = At2 + a1B = A^(n-1)*B + ... + a2*A*B + a1*B

where aj are the coefficients of the characteristic equation of A’

Question 6

System Matrices of Transform Controllable Canonical Form

Answer 6

A' = inv(T)*A*T
B' = inv(T)*B
C' = C*T

Question 7

Kalman Decomposition

Answer 7

Transofrm of system where all controllable and observable outcomes are displayed
T = (Tco’, Tco, Tc’o’, Tc’o)
Tco’ spans intersection of controllable and uncontrollabe subspace
Tco spans controllabe supbspace
Tc’o’ spans uncontrollabe subspace
Tc’o chosen so T is non singular

Question 8

Minimal Realization of transfer function H(s)

Answer 8

The modes of the system from Sco are minimal