1 Observability in state-space models

1.1 Measurement outputs and their relation to state

In state-space models, a system is described by an internal state vector and a mapping from that state to measured outputs. The core idea behind observability is whether different initial states lead to distinguishable output trajectories when the system is driven by inputs. If two distinct state trajectories produce the same output for all permissible inputs, then the component of state responsible for that indistinguishability cannot be inferred from the measurements.

1.2 Observability as a property of system pairs

For linear systems written as \[ \dot x = Ax+Bu,\qquad y=Cx+Du, \] observability is not determined by a single matrix but by the interaction of the state transition and the output mapping. In the standard linear time-invariant (LTI) setting (with \(D\) often handled separately), the observability property is associated with the pair \((A,C)\): it asks whether the state is uniquely determined by the measured outputs given knowledge of the input.

1.3 Observability matrices and rank criteria

A common tool is the observability matrix, assembled from successive products of \(C\) with powers of \(A\). When this matrix has full rank, the system is observable in the sense that any nonzero initial state affects the output at some time, for every input choice. When the rank is deficient, there exists a nontrivial set of state directions that do not influence the measured output trajectory.

1.4 Time-domain viewpoint on state distinguishability

Observability can be interpreted dynamically: the output over time acts as a “signature” of the state. If the signature is insensitive to certain state components, those components are effectively hidden. This time-domain perspective is often more intuitive than purely algebraic tests, because it ties the definition directly to distinguishability of output responses over an interval.

2 Definition of the unobservable subspace

2.1 State directions invisible to outputs

The unobservable subspace is the collection of state directions whose effect on the output is identically zero for all admissible inputs, starting from those directions as initial conditions. Said differently, moving the initial state within that subspace changes the internal evolution but leaves the output unchanged, so those directions cannot be reconstructed from the measurements.

2.2 Formal characterization via reachability/observability tests

A typical formal definition uses the concept of indistinguishability: a nonzero state vector \(x_0\) is unobservable if, for every input function, the resulting output trajectory is the same as it would be from the zero state (up to the known contribution of the input through \(D\)). In practice, rank-based conditions or matrix decompositions are used to identify all such directions without enumerating inputs.

2.3 Equivalent definitions (matrix-based and geometric)

Although the unobservable subspace can be introduced through observability matrices, it also admits geometric descriptions in terms of invariant subspaces and orthogonal complements. These equivalences provide robustness: the same hidden directions can be characterized through linear-algebraic computations, through geometric constructions, or via duality relationships with controllability concepts.

2.4 Invariance under system dynamics

The unobservable subspace is invariant under the state transition \(A\) in the LTI setting. Invariance means that once the system’s state lies in the hidden subspace, its future evolution remains within that same class of states that cannot affect the output. This property supports decomposition-based analyses and simplifies observer design by separating dynamics that can be “seen” from those that cannot.

3 Unobservable subspace in linear systems

3.1 Linear time-invariant (LTI) state-space form

In LTI models, \[ \dot x = Ax+Bu,\qquad y=Cx, \] the D matrix is omitted for simplicity because it does not alter state unobservability when inputs are known and subtracted from the output. Under this convention, the unobservable subspace depends on \((A,C)\) and can be computed using repeated measurement of the state’s images under \(C\) as the dynamics evolve.

3.2 Computation from the observability matrix

Let the observability matrix be \[ \mathcal{O}=\begin{bmatrix} C\\ CA\\ CA^2\\ \vdots\\ CA^{n-1} \end{bmatrix}. \] A standard result states that the unobservable subspace is the set of initial states \(x\) such that \(\mathcal{O}x=0\). Equivalently, it is the null space of \(\mathcal{O}\). This gives a concrete computation method: build \(\mathcal{O}\), then extract a basis for its null space.

3.3 Null space interpretations

The null-space description emphasizes what is being “killed” by the measurement process. If \(x\) lies in the null space of \(\mathcal{O}\), then every output derivative (or discrete-time analogue) up to the relevant order is consistent with zero contribution from that initial state. Hence, the output provides no information to eliminate those state components.

3.4 Relationship to Kalman decomposition

3.4.1 Observable/controllable decomposition blocks

Kalman decomposition partitions the state space into subspaces associated with different combinations of observability and controllability properties. One block corresponds to dynamics that are both controllable and observable, another to parts that are controllable but not observable, and a third to parts that are observable but not controllable. The unobservable subspace corresponds to the union of blocks that lack observability.

3.4.2 Partitioning dynamics into observable and unobservable parts

In the decomposition coordinates, the system matrices become block structured so that the observable dynamics affect the output while the unobservable dynamics do not. Concretely, after a change of coordinates, the output equation depends only on the observable state block, while the unobservable block evolves independently as far as the output is concerned. This makes the unobservable subspace explicit and clarifies how hidden dynamics persist over time.

4 Geometric and algebraic tools

4.1 Orthogonal complements and duality with controllability

A geometric viewpoint uses orthogonal complements to relate unobservability to controllability in the dual system. In many linear settings, the unobservable subspace for \((A,C)\) is connected to the controllable subspace of the transpose pair \((A^\top,C^\top)\). This duality provides alternate computational routes and conceptual clarity: hidden state directions for outputs correspond to unreachable directions in the dual formulation.

4.2 Invariant subspaces and their construction

Because the unobservable subspace is invariant, it can often be constructed as the maximal invariant subspace contained in the kernel of \(C\) along with its images under \(A\). Invariant-subspace algorithms search for the largest subspace that remains compatible with the condition of producing no output, using closure under the dynamics. This approach aligns naturally with the meaning of “hidden” as persistent under evolution.

4.3 Jordan/real canonical forms and intuition

Canonical forms (such as Jordan or real canonical forms) provide intuition by exposing how different eigenstructure components influence measured outputs. Components associated with certain eigenmodes may have zero projection through the output mapping across the required chains, resulting in hidden behavior. While the raw canonical form may be sensitive to basis choice, the observable/unobservable partition is invariantly defined, so the intuition persists even if the representation changes.

4.4 Coordinate transformations that isolate hidden dynamics

A key technique in practice is to apply a similarity transformation that separates the state into observable and unobservable coordinates. In the transformed coordinates, the output matrix has zeros in the columns corresponding to unobservable states, and the state transition matrix is block upper-triangular or block diagonal in the relevant idealized setting. This separation reduces estimation problems by restricting attention to the observable part.

5 Consequences and interpretation

5.1 Non-identifiability of initial conditions

If a state component lies in the unobservable subspace, then different initial conditions that differ by that component are indistinguishable in the output. As a result, the initial state cannot be uniquely identified from measurement data, even with perfect knowledge of inputs and noise-free sensing. The unobservable subspace thus quantifies a structural limitation rather than a practical one.

5.2 Limits on observer performance

Observers (such as Luenberger observers or Kalman filters) can track the observable portion of the state, but they cannot correct errors in directions that do not influence outputs. In error-dynamics terms, the estimation error restricted to the unobservable subspace evolves without correction from the measurement residual. Therefore, observer gains cannot eliminate uncertainty in hidden state directions; they can at best ensure desired convergence behavior in the observable subspace.

5.3 Distinguishability over finite horizons

Full observability is a property that typically considers how outputs behave up to a maximal necessary time order (often tied to system dimension). Over shorter horizons, additional ambiguities may appear: even if the system is observable, limited data may not yet expose all distinguishable state components. Conversely, unobservable subspace directions are ambiguous over every horizon length because they never affect the output at all.

5.4 Detectability vs. full observability distinctions

Detectability weakens observability by allowing unobservable modes, provided they are not harmful for estimation over time. In many observer designs, what matters is whether the unobservable part does not produce growing estimation errors (e.g., unstable unobservable modes are absent). The unobservable subspace underlies this distinction: even if full state reconstruction is impossible, stable unobservable behavior may still permit bounded estimation and convergence of the observable component.

6 Computation and practical workflow

6.1 Building the observability matrix

The standard workflow begins by forming the observability matrix \(\mathcal{O}\) from \(C\) and powers of \(A\). For systems of order \(n\), one typically uses up to \(CA^{n-1}\). In software implementations, care is taken to construct \(\mathcal{O}\) reliably, often using stable matrix-multiplication routines rather than explicit repeated powers where possible.

6.2 Extracting a basis for the unobservable subspace

Once \(\mathcal{O}\) is formed, an estimate of its null space yields a basis for the unobservable subspace. Numerically this is usually done via singular value decomposition (SVD) or related rank-revealing factorizations. The resulting basis vectors span exactly those state directions that satisfy \(\mathcal{O}x\approx 0\) within numerical tolerance.

6.3 Numerical considerations and rank tolerance

In floating-point arithmetic, theoretical rank conditions become approximate. Practical computation requires a tolerance to decide which singular values are effectively zero. This choice can affect the estimated dimension of the unobservable subspace. Robust workflows report sensitivity to tolerance, and they may cross-check results using invariant-subspace computations or decomposition methods to reduce misclassification.

6.4 Validating invariance and correctness

After computing a candidate basis, validation checks typically confirm that the subspace is invariant under \(A\) and that applying \(C\) to the evolved states yields no output contribution. In the exact arithmetic setting, invariance and the null-space condition are equivalent ways to verify the definition. In numerical settings, validation helps catch issues arising from poor conditioning, scaling, or tolerance misestimation.

7.1 Detectable subspaces and minimum required observability

Instead of focusing on whether the entire state is observable, one may seek the smallest subspace that must be observable to achieve a desired estimation objective. Detectable subspaces capture which portions can be inferred adequately for stability or convergence requirements. The unobservable subspace then appears as a complement to what remains identifiable under those weaker goals.

7.2 Output injection and observer error dynamics

Observer design often introduces an output error term multiplied by a gain matrix. The effect of this injection can be analyzed by projecting the error dynamics onto observable and unobservable coordinates. While the injection term directly corrects estimation errors in directions that affect outputs, it has no leverage over the unobservable part, which continues according to the system’s internal evolution projected onto the hidden subspace.

7.3 Reduction to minimal realizations

Minimal realizations remove redundant dynamics: portions of the state that are either uncontrollable or unobservable do not contribute to the input-output behavior. The unobservable subspace therefore plays a direct role in model reduction: states spanning this subspace can be eliminated without changing the system’s transfer characteristics. This yields a smaller model that is both controllable and observable.

7.4 Invariant zeros vs. unobservable state components

Invariant zeros are properties related to input-output transmission, describing values for which certain output behavior can be forced to vanish. Although invariant zeros and unobservable state components both relate to limitations of inference, they are distinct: unobservable state components concern initial state effects that never appear in outputs, whereas invariant zeros concern the system’s ability (through inputs) to produce particular output responses. Nonetheless, both concepts reflect structure in how internal dynamics manifest at the measurement level.