Evaluation

Given input sequence x(0), x(1), x(2), …, the state response and the output sequence are determined by

s(0) = initialState = s₀ , say and for n ³ 0

s(n + 1) = As(n) + Bx(n),

y(n) = Cs (n) + Dx(n).

In this notation

• s is a N ´ 1 column vector,
• x is a M ´ 1 column vector,
• y is a K ´ 1 column vector, and
• A, B, C, and D are all matrices with dimensions that make the equations valid.

Recursive solution of these equations leads to

s(n) = Aⁿs₀ + S _{(m = 0 to n-1)} A^n-1-m Bx(m)

y(n) = CAⁿ s₀ + S _{(m = 0 to n-1)} C A^n-1-m Bx(m) + Dx(n)

The first term is the zero-input response and the second-term is the zero-state response.