SISO Systems

For a single-input, single-output system, M = 1, K = 1, so Inputs = Outputs = Reals. (The state, however, is N-dimensional.) So

• the N ´ 1 matrix B can be written as the column vector b = [b₁, … , b_N ]^T ,
• the 1 ´ N matrix C can be written as a 1 ´ N row vector c^T = [c₁ , … , c_N ], and
• the 1 ´ 1 matrix D can be written as a real number d.

With this notation s, b, c are all N ´ 1 column vectors.

The response of the system to an input sequence x(0), x(1), x(2), … is

s(n) = Aⁿ s₀ + S _{(m = 0 to n-1)} A^n-1-m b x(m)

y(n) = c^T Aⁿ s₀ + S _{(m = 0 to n-1)} c^T A^n-1-m b x(m) + dx(n).