Observability and minimal realizations

For the unforced continuous-time LTI system, y(t)=Ce^{At}x₀. Its derivatives at zero involve Cx₀, CAx₀, CA²x₀ and higher powers. Cayley–Hamilton again truncates the structural test at n terms. Stacking these matrices gives the observability matrix, whose full column rank means no nonzero initial state produces identically zero output. If the input is known, subtract its forced response before applying this argument. Unknown disturbances cannot generally be removed the same way and require a different estimation model.

The derivative picture is a proof tool, not a recommendation to numerically differentiate noisy sensor data many times. The finite-horizon observation Gramian integrates e^{Aᵀt}CᵀCe^{At}; its quadratic form equals the output signal’s squared energy for a chosen initial state. It is positive definite exactly when the pair is observable. Small eigenvalues identify state combinations with weak output signatures. Such directions may be structurally observable but difficult to estimate under noise, much as a small reachability Gramian eigenvalue makes actuation expensive.

The unobservable subspace is the kernel of the stacked observation matrix. It is A-invariant, so an initial ambiguity in that subspace remains invisible as it evolves. Detectability requires every mode confined to this invisible subspace to be strictly stable. Equivalently, there exists an observer injection L making A−LC Hurwitz. Observability permits arbitrary observer pole assignment, whereas detectability only asks whether some stable observer exists. An invisible stable state cannot be exactly reconstructed from output history, but its initial-error effect decays naturally.

The continuous-time PBH test says detectability holds when the stacked matrix with rows λI−A and C has full column rank for every complex λ with nonnegative real part. For observability, require the test at every eigenvalue, including stable ones. The boundary matters: an invisible eigenvalue at zero does not decay and therefore prevents detectability. In discrete time the corresponding forbidden region is |λ|≥1. This distinction mirrors controllability versus stabilizability and prevents unnecessarily rejecting a model that has harmless hidden stable modes.

A finite-dimensional proper rational transfer matrix has a minimal state-space realization when no realization with fewer states produces the same zero-state input–output map. Such a realization is exactly one that is both controllable and observable. A Kalman decomposition separates reachable and unreachable directions and then the visible and invisible parts. Only the reachable-observable block is needed for the transfer map. Removing other blocks can preserve G while changing possible initial-state responses, so minimality is not a claim that discarded physical states never matter.

Any two minimal realizations of the same transfer matrix are related by an invertible similarity transformation. Their internal coordinates may differ, but their state dimension and pole structure agree. An unstable cancellation in a nonminimal realization can leave G stable while the internal system is unstable; checking the reduced transfer poles alone misses that issue. Conversely, a well-conditioned minimal realization can be more useful for estimation and numerical computation than a larger redundant model. The right reduction depends on the experiment and the requirement one intends to preserve.