State-space models and input–output maps

Write the state equation as ẋ=Ax+Bu and output equation as y=Cx+Du. The state has n components, the input m, and the output p; the matrix dimensions must follow those choices. A describes autonomous motion, B maps actuator commands into state derivatives, C selects measured combinations, and D permits an instantaneous direct input contribution. A physical position and velocity may form a useful state, but state variables need not equal sensor readings. A single sensor can reveal information about several states through their different effects over time.

This is a continuous-time model. A sampled discrete-time model instead uses x_{k+1}=A_d x_k+B_d u_k, and its stability region is the open unit disk rather than the left half-plane. Under a held input over a sampling interval h, A_d=e^{Ah} and B_d is the integral of e^{Aτ}B from zero to h. Replacing A_d by A without a discretization argument mixes units and changes the dynamics. Linearization of a nonlinear plant also has a region of validity; a local approximation does not automatically describe saturated actuators or large excursions.

With zero initial state, taking Laplace transforms gives G(s)=C(sI−A)^{-1}B+D wherever the resolvent exists and the transforms converge. The impulse response away from time zero is Ce^{At}B; direct feedthrough contributes D times the input immediately, represented by a delta term in an impulse description. A nonzero initial state adds C(sI−A)^{-1}x₀ to the transformed output. It is therefore incorrect to use G alone to predict every response unless the initial state is zero or its natural response has been separately included.

Transfer poles need not include every eigenvalue of A. A mode may be unreachable through B or invisible through C, so algebraic cancellation can remove it from the transfer description. Such a cancellation is not evidence that the internal state is harmless. An unmeasured unstable mode can grow while the measured forced response looks stable. Later lessons identify the controllable and observable part, which is the minimal realization of the transfer behavior. For design, one must decide whether internal stability or only a particular measured response is being analyzed.

For an invertible real matrix T, define z=Tx. Then the transformed matrices are TAT^{-1}, TB, CT^{-1}, and D. Substitution proves that the measured y is unchanged for the same input and appropriately transformed initial state. Eigenvalues, controllability rank and observability rank are invariant under this similarity transformation. Individual entries and numerical conditioning are not. Scaling a velocity coordinate from meters per second to millimeters per second can produce very different matrix magnitudes while describing exactly the same physical trajectories.

A useful workflow checks dimensions, specifies initial conditions, derives the response, and only then asks whether a proposed coordinate choice makes the important structure easier to inspect. A transfer description is particularly convenient for frequency response; a state description is particularly convenient for feedback, estimation and hidden-mode analysis. Neither is a universal replacement for the other. When reducing a model, preserving its zero-state transfer map does not preserve every possible initial-state response, because discarded internal variables may have carried memory that the chosen transfer experiment never excited.