State-Space Methods for Controller Designs

Question 1

Stability of State-Space Model

Answer 1

Eigenvalues of the A matrix are the poles of system. System is unstable if there is RH poles

Question 2

Controllable System of LTI system

Answer 2

Determined by calculating the controllability matrix and has a full rank
C = [B AB A^2B … A^(n-1) B]
A and B are matrix of SS Models

Question 3

Observable System of LTI system

Answer 3

An initial state can be determined based on knowledge of the system input and system output over finite time interval. Reduces number of sensing requirements of states. Observability matrix must have full rank.
O = [C; CA; CA^2; CA^(n-1)]
A and B are matrix of SS Model

Question 4

Duality of Controllable System and Observable System

Answer 4

System (A,B) is controllable if and only if system (A’,B’) is observable

Question 5

Stability and Time-Domain performance of closed-loop feedback system

Answer 5

Determined by location of eigenvalues of the matrix (A-BK) where and A and B are the matrices of the SS Model and K is the state-feedback gain matrix