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 ≥ 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 + ∑ (m = 0 to n-1) An-1-m Bx(m)
y(n) = CAn s0 + ∑ (m = 0 to n-1) C An-1-m Bx(m) + Dx(n)
The first term is the zero-input response and the second-term is the zero-state response.