State-Space Analysis

Why it matters

State-space models describe internal dynamics as well as input–output behavior, and naturally handle multiple inputs and outputs.

Key ideas

For a continuous-time finite-dimensional LTI model, ẋ = Ax + Bu and y = Cx + Du. The state x contains the information required, together with future input, to predict future behavior. A has size n×n, B n×m, C p×n and D p×m.

For discrete time, x[k+1] = Ax[k] + Bu[k]. Initial state must be specified; a transfer function describes the zero-state response and can hide uncontrollable or unobservable internal modes.

Formulas

• x(t) = exp(At)x(0) + integral from 0 to t of exp(A(t−τ))Bu(τ)dτ.
• With zero initial state, H(s) = C(sI−A)^−1 B + D where the inverse exists.
• Continuous-time internal asymptotic stability requires every eigenvalue of A to have negative real part. Discrete-time requires magnitude less than one.
• Controllability matrix [B AB … A^(n−1)B] must have rank n for complete controllability.
• Observability matrix formed by stacking C, CA, …, CA^(n−1) must have rank n for complete observability.

Worked example

An RC low-pass circuit has R = 1 kΩ, C_cap = 1 μF, input voltage u and state/output x = y = capacitor voltage. KCL gives C_cap ẋ = (u−x)/R. Thus ẋ = −1000x + 1000u and y = x. Here A = −1000 s^−1, B = 1000 s^−1, output coefficient C = 1 and D = 0. H(s) = 1000/(s+1000). The only eigenvalue is −1000 s^−1, so the model is internally asymptotically stable. For x(0) = 0 and a 5 V step, y(t) = 5(1−exp(−1000t)) V; y(1 ms) = 3.161 V. Both rank tests equal one, so this one-state model is controllable and observable.

Common mistakes

Do not confuse capacitor symbol C_cap with the output matrix C. Do not discard initial conditions when asked for total response. Transfer-function pole cancellation does not prove that every internal mode is stable.

Quick check

1. What does x(0) represent? Initial stored state.
2. What is the RC time constant here? 1 ms.
3. Is an eigenvalue of +2 internally stable in continuous time? No; its natural response grows exponentially.