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State-space representation describes a dynamical system using first-order matrix differential equations: ẋ = Ax + Bu, y = Cx + Du, where x is the state vector, u is the input, y is the output, and A, B, C, D are constant system matrices. Unlike transfer functions, state-space models naturally represent MIMO (multiple-input, multiple-output) systems and capture all internal dynamics including unobservable or uncontrollable modes. The eigenvalues of the A matrix are the system's natural frequencies (closed-loop poles). State variables are not unique — any invertible linear transformation yields an equivalent representation with different A, B, C matrices but identical input-output behavior.
Practice converting second-order differential equations into companion-form state space, then verify by computing H(s) = C(sI−A)⁻¹B + D and confirming it matches the original transfer function. Implement state-space simulations using scipy.signal.StateSpace.
You already know how to describe a linear system using a transfer function H(s) = Y(s)/U(s) — a ratio of polynomials in the Laplace variable s. Transfer functions work well for single-input, single-output (SISO) systems, but they hit two walls: they cannot naturally represent systems with multiple inputs and outputs (MIMO), and they can hide internal dynamics that cancel in the ratio. State-space representation solves both problems by working directly in the time domain with first-order matrix differential equations.
The key move is variable selection. Any differential equation of order n can be rewritten as n first-order equations by introducing state variables. For a second-order mechanical system mẍ + bẋ + kx = u, define x₁ = x (position) and x₂ = ẋ (velocity). Then ẋ₁ = x₂ and ẋ₂ = (u − bx₂ − kx₁)/m, which you write compactly as ẋ = Ax + Bu. The state vector [x₁, x₂] is the system's complete "memory" — given the current state and all future inputs, you can compute all future outputs. This is the fundamental role of the state: it separates the past from the future.
The A matrix is the heart of the representation. Its eigenvalues are the system's natural frequencies — the poles. An eigenvalue with negative real part corresponds to a stable, decaying mode. A positive real part means that mode grows without bound: instability. Complex conjugate eigenvalues with nonzero imaginary parts produce oscillation. These are exactly the poles you know from transfer functions, which is not a coincidence: the poles of H(s) = C(sI−A)⁻¹B + D are precisely the eigenvalues of A. The two representations are equivalent for what they predict about input-output behavior.
Where they diverge is in what they hide. A transfer function is formed by a ratio, and ratios can cancel. If a mode is uncontrollable — no input signal can excite it — and unobservable — it never appears in the output — then it cancels as a pole-zero pair in H(s) and becomes invisible. That mode still exists physically, and if it is unstable, the real system will diverge even while your transfer-function model predicts stability. State-space models retain all eigenvalues of A regardless of cancellations, making every internal mode visible. This is why stability analysis, controllability, and observability are always done in state-space.
One liberating insight is that state variables are not unique. Applying any invertible matrix T to the state vector gives new state variables x̃ = Tx, with different system matrices à = TAT⁻¹, B̃ = TB, C̃ = CT⁻¹, but the identical transfer function. Different choices of state variables are called different realizations of the same system. Companion form is good for reading off coefficients. Modal form (diagonalizing A) decouples the system into independent modes. Balanced realization is useful for model reduction. The freedom to choose is a tool, not an ambiguity.