State-space representation

Why it matters

Transfer functions describe only what goes in and what comes out. Modern instrumentation and control — multivariable process units, digital controllers, Kalman-filter sensor fusion, observers that estimate unmeasured variables — works with the internal state of the system. State-space models are also exactly what simulation software and embedded controllers use, so being able to move between a physical model, a state-space model and a transfer function is a core skill.

Key ideas

State: the smallest set of variables x1, …, xn whose values at t0, together with the input for t ≥ t0, fix the system's future completely. The natural choice is one variable per independent energy store: capacitor voltages, inductor currents, mass positions and velocities, tank levels. The number of states n is the order of the system.

State-space model of an LTI system

• State equation: ẋ = A·x + B·u — n first-order differential equations.
• Output equation: y = C·x + D·u — algebraic.
• A (n×n) is the system matrix, B (n×m) the input matrix, C (p×n) the output matrix, D (p×m) the direct-transmission (feed-through) matrix, for m inputs and p outputs. For a strictly proper transfer function, D = 0.

State-space model is not unique. Any invertible change of variables x = P·z gives a new model (P⁻¹AP, P⁻¹B, CP, D) with the same transfer function and the same eigenvalues. Common standard forms:

• Phase-variable (controllable canonical) form: states are y, ẏ, ÿ, …; A has 1s on the super-diagonal and the negated denominator coefficients in the last row.
• Observable canonical form: the transpose (dual) of the controllable form.
• Diagonal (Jordan) canonical form: from partial fractions; each state is one mode, and A holds the poles on its diagonal (Jordan blocks for repeated poles).

From state space to transfer function: G(s) = C(sI − A)⁻¹B + D. The denominator is det(sI − A), so the eigenvalues of A are the poles — unless a pole-zero cancellation hides a mode. A cancelled mode is either uncontrollable or unobservable (next topic) and still exists inside the system.

Useful matrix facts: the characteristic equation is det(sI − A) = 0; the sum of the eigenvalues equals the trace of A; their product equals det A. For a 2×2 matrix, det(sI − A) = s² − (trace A)·s + det A.

State diagram: a block diagram built only from integrators, gains and summing points, with the output of each integrator being a state. It maps directly onto an analogue computer or a digital simulation.

Advantages over transfer functions: handles MIMO, time-varying and nonlinear systems; includes initial conditions; reveals hidden modes; and supports state-feedback and observer design.

Formulas

• ẋ(t) = A·x(t) + B·u(t), y(t) = C·x(t) + D·u(t)
• G(s) = Y(s)/U(s) = C·(sI − A)⁻¹·B + D
• (sI − A)⁻¹ = adj(sI − A)/det(sI − A)
• Characteristic equation: det(sI − A) = 0
• 2×2 inverse: [[a, b], [c, d]]⁻¹ = (1/(ad − bc))·[[d, −b], [−c, a]]
• Phase-variable form of G(s) = (b1·s + b0)/(s² + a1·s + a0): A = [[0, 1], [−a0, −a1]], B = [0; 1], C = [b0, b1], D = 0.
• Similarity transform: à = P⁻¹AP, B̃ = P⁻¹B, C̃ = CP, D̃ = D.

Units: each state carries its physical unit (V, A, m, m/s, …); entries of A have units of s⁻¹ times the ratio of state units; G(s) has the unit of output per unit input.

Worked examples

Example 1 — series RLC circuit in state space Given: R = 2 Ω, L = 1 H, C = 0.5 F in series, input u = vin, output y = vC. States x1 = vC, x2 = i.

1. Capacitor: C·dvC/dt = i → ẋ1 = (1/C)·x2 = 2·x2.
2. KVL: L·di/dt = vin − R·i − vC → ẋ2 = −x1 − 2·x2 + u.
3. A = [[0, 2], [−1, −2]], B = [0; 1], C = [1, 0], D = 0.
4. sI − A = [[s, −2], [1, s + 2]], det = s(s + 2) + 2 = s² + 2s + 2.
5. (sI − A)⁻¹·B = [2; s]/(s² + 2s + 2); taking the first row: G(s) = 2/(s² + 2s + 2).
6. Check with circuit theory: 1/(LCs² + RCs + 1) = 1/(0.5s² + s + 1) = 2/(s² + 2s + 2), which matches. Poles −1 ± j1.

Result: A = [[0, 2], [−1, −2]], B = [0; 1], C = [1, 0], and G(s) = 2/(s² + 2s + 2).

Example 2 — a hidden mode (GATE level) Given: A = [[0, 1], [−2, −3]], B = [0; 1], C = [1, 1], D = 0. Find the eigenvalues and G(s).

1. det(sI − A) = s² + 3s + 2 = (s + 1)(s + 2) → eigenvalues −1 and −2 (check: trace −3 = sum, det 2 = product).
2. sI − A = [[s, −1], [2, s + 3]]; (sI − A)⁻¹ = [[s + 3, 1], [−2, s]]/(s² + 3s + 2).
3. (sI − A)⁻¹·B = [1; s]/(s² + 3s + 2).
4. G(s) = C·[1; s]/(…) = (1 + s)/[(s + 1)(s + 2)] = 1/(s + 2).

Result: eigenvalues −1, −2 but G(s) = 1/(s + 2). The mode at −1 is cancelled by a zero: it is invisible at the output (unobservable for this C), yet it still responds to initial conditions inside the system. A transfer function alone would never reveal it.

Common mistakes

• Writing G(s) = (sI − A)⁻¹B without the C (and D).
• Inverting sI − A with the off-diagonal signs wrong.
• Taking the eigenvalues of sI − A or of −A instead of A.
• Assuming the state-space model of a system is unique, or that every eigenvalue must appear as a pole of G(s).
• Choosing dependent state variables (e.g. two capacitor voltages in parallel), which gives a singular, non-minimal model.
• Forgetting D when the transfer function has equal numerator and denominator degree.

For GATE IN

Expect questions on converting a transfer function to phase-variable form, finding G(s) from (A, B, C, D), finding eigenvalues or the characteristic equation, writing state equations for an RLC or mechanical system, and recognising pole-zero cancellation. Practise 2×2 and 3×3 (sI − A)⁻¹ calculations, and use trace and determinant as quick checks.

Quick check

1. For A = [[0, 1], [−5, −6]], what are the eigenvalues?
2. What is the sum of the poles for A = [[0, 1], [−2, −3]]?
3. Write A in phase-variable form for G(s) = 4/(s² + 5s + 6).
4. For A = [[0, 1], [−2, −3]], B = [0; 1], C = [1, 0], D = 0, what is the DC gain G(0)?

Answers: 1. s² + 6s + 5 = 0 → −1 and −5; 2. trace = −3; 3. [[0, 1], [−6, −5]] (with B = [0; 1], C = [4, 0]); 4. G(s) = 1/(s² + 3s + 2), so G(0) = 0.5.