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State-space models (ẋ = Ax + Bu, y = Cx + Du) and transfer functions G(s) = C(sI − A)⁻¹B + D are two representations of the same linear time-invariant system, and converting between them reveals important structural properties. The transfer function is obtained from the state-space model by G(s) = C·adj(sI − A)·B/det(sI − A) + D, where det(sI − A) gives the characteristic polynomial. Canonical forms provide standardized state-space representations: controllable canonical form places the characteristic polynomial coefficients directly in the last row of A with a specific B and C structure, making controllability transparent; observable canonical form is its dual, making observability transparent. A state-space realization is minimal if and only if it is both controllable and observable, meaning no pole-zero cancellations occur and the state dimension equals the transfer function's McMillan degree. Non-minimal realizations have a higher state dimension than necessary because hidden modes (uncontrollable or unobservable states) create pole-zero cancellations that disappear from the transfer function. Converting from transfer function to state-space always yields a minimal realization when using standard canonical forms, but converting a non-minimal state-space model to a transfer function and back loses the hidden modes permanently.
Start with a third-order state-space model and compute the transfer function by hand using the formula G(s) = C(sI − A)⁻¹B + D, verifying with MATLAB's tf(ss(A,B,C,D)) or Python's control.ss2tf(). Then construct the controllable and observable canonical forms for the same transfer function and confirm they have identical input-output behavior but different internal state variables. Finally, take a fourth-order state-space model with an uncontrollable mode, convert to transfer function, and observe the pole-zero cancellation that reduces the order — illustrating why minimality matters.
You already know two ways to describe a linear time-invariant system: the state-space model (ẋ = Ax + Bu, y = Cx + Du), which tracks the internal state evolving moment to moment, and the transfer function G(s), which compresses everything down to an input-output ratio in the s-domain. These are two windows onto the same underlying dynamics, and understanding how to convert between them reveals structure that neither view makes obvious on its own.
The conversion formula G(s) = C(sI − A)⁻¹B + D is more informative than it first appears. The matrix (sI − A)⁻¹ is the resolvent of A — it captures how the system responds at each frequency s. Its denominator is det(sI − A), the characteristic polynomial, whose roots are the system's poles. This is the same characteristic equation you solve to find eigenvalues of A, so the poles of the transfer function are exactly the eigenvalues of A. The numerator picks out how the input B drives the states and how the output matrix C observes them. The D term is the direct feedthrough that bypasses the dynamics entirely.
Canonical forms are standardized state-space structures designed to make a specific property obvious by construction. In controllable canonical form, the characteristic polynomial coefficients appear directly in the last row of A, and B has a simple structure (zeros except for a 1 in the last entry). This form guarantees controllability — every state can be driven from the input — and is the natural starting point for state feedback pole placement because the feedback gains directly modify those last-row coefficients. Observable canonical form is its transpose dual: it guarantees observability and is the natural basis for observer design. The insight is that these two forms share the same transfer function but have completely different internal representations; any invertible transformation T relating two realizations via x̃ = Tx gives another valid but structurally different model.
The deepest concept here is minimality. A realization is minimal if and only if it is both controllable and observable, meaning its state dimension equals the McMillan degree of the transfer function. When a system has an uncontrollable or unobservable mode, that mode cancels as a pole-zero pair when you compute the transfer function — it simply disappears from the input-output description. This is dangerous: a cancelled mode is not gone from the physical system, it is merely hidden. If that hidden mode is unstable, the internal state will diverge even while the output looks perfectly well-behaved. A system with a right-half-plane pole hidden by a zero cancellation can fail catastrophically while every output measurement appears nominal.
One practical implication to remember: the round-trip conversion loses information. Converting a state-space model to a transfer function and back always returns a minimal realization — the hidden modes are permanently discarded. Infinitely many state-space realizations share the same transfer function, all related by similarity transformations. This means that when you design a controller using a transfer function, you are implicitly assuming minimality. When you implement that controller on the actual system, any non-minimal modes in the physical plant remain present and must be accounted for in the stability analysis.