State Space Analysis

State-space model

An LTI system can be represented by xdot = Ax + Bu and y = Cx + Du. States and inputs, together with an initial condition, determine the evolution. The zero-state transfer matrix is G(s) = C(sI − A)^−1B + D; a transfer function alone need not reveal hidden internal modes. The complete solution is x(t) = exp(At)x(0) + ∫0ᵗ exp(A(t − τ))Bu(τ)dτ.

Controllability and observability

For an n-state system, the controllability matrix is [B AB … A^(n−1)B]. Full row rank n means every state can be reached by a suitable unconstrained input in the linear model. The observability matrix stacks C, CA, …, CA^(n−1). Rank n means the initial state can be distinguished from output data with known input. Numerical rank should be assessed with an appropriate tolerance.

Worked example

For y'' + 3y' + 2y = u, take x1 = y and x2 = y'. Then A = [[0,1],[-2,-3]], B = [[0],[1]], C = [1,0], D = 0. Controllability matrix = [[0,1],[1,-3]], determinant −1, so rank is 2. Observability matrix = [[1,0],[0,1]], determinant 1, so rank is 2. G(s) = 1/(s² + 3s + 2), with poles −1 and −2. Both state modes decay in the unforced system.

Design connection

With state feedback u = −Kx + r, the closed-loop state matrix is A − BK. Arbitrary pole placement requires controllability. An observer uses measured output to estimate state; arbitrary observer-error pole placement requires observability. Actuator limits and noise remain practical constraints.

Quick check

1. Are state coordinates unique? No; invertible coordinate changes preserve input-output behavior and eigenvalues.
2. Does D represent memory? No, it represents direct feedthrough.
3. Does a stable transfer function always prove internal stability? No, hidden modes may remain.