Question

Consider the system x˙= Ax+Bu,y=Cx+Du.

show that only modes that are both controllable and observable appear in ceᵃᵗb, and therefore, in the impulse response and the transfer function matrix of the system.

Answer 1

Controllability and observability are necessary for modes to appear in the matrix exponential, impulse response, and transfer function matrix of a system.

Step-by-step explanation:

In order for a mode to appear in the matrix exponential, impulse response, and transfer function matrix of a system, it must be both controllable and observable. Controllability means that the state of the system can be manipulated by applying appropriate inputs, while observability means that the state of the system can be determined by measuring the outputs.

Mathematically, a mode is controllable if the corresponding column of the controllability matrix is non-zero and observable if the corresponding row of the observability matrix is non-zero. Modes that are both controllable and observable will have non-zero elements in both the column and row.

Therefore, only modes that are both controllable and observable will appear in the matrix exponential (ce^At), impulse response, and transfer function matrix of the system.