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Uniform observability criteria for linear time-varying systems

Soulaimane Berkane

arXiv:2610.34581v1math.OCcs.SYeess.SY

TL;DR

Uniform observability tests for LTV systems often require a state-transition matrix that is unavailable in closed form. This paper develops coefficient-based excitation criteria and derivative-based reductions for structured systems. The criteria isolate relevant state subspaces, including an exact condition for the lifted range-measured integrator-chain model.

  • Problem

    Uniform observability of LTV systems is difficult to verify because evaluating the observability Gramian generally requires a state-transition matrix unavailable in closed form.

    Source: §1 · p. 2

  • Method

    The paper characterizes excitation on eigenspaces and uses finite output-derivative combinations to eliminate measured states in cascades.

    Source: §1 · p. 3

  • Results

    Exact tests require excitation only on relevant eigenspaces or hidden eigenspace-kernel intersections; range-chain excitation is needed only orthogonal to anchor differences.

    Source: §6 · p. 13

  • Takeaways & Limitations

    The conditions identify which state directions require excitation and characterize uniform observability of the full lifted range-measurement LTV model.

    Source: §6 · p. 13

    Similarly direct tests for nonreal eigenvalues, including purely imaginary pairs, remain an open formulation problem because oscillatory dynamics and time-varying outputs can cancel.

    Source: §6 · p. 13

Abstract

from arXiv · show

Verifying uniform observability of a linear time-varying system generally requires its state-transition matrix. This paper develops necessary and sufficient conditions that can be checked directly from the coefficient matrices for two classes of systems. For constant dynamics with real spectrum and a measurable, essentially bounded output matrix, uniform observability is equivalent to persistent excitation of the output matrix on each eigenspace. An equivalent condition requires persistent excitation of the Kalman observability stack. The second result concerns cascades with a constant observable measured subsystem coupled to a hidden state. Under uniform regularity assumptions, linear combinations of output derivatives eliminate the measured state and yield an equivalent observability test for the hidden dynamics. When those dynamics have real spectrum, the test reduces to excitation on each eigenspace intersected with the kernel of the direct measurements. An application to arbitrary-order integrator chains with measurements of ranges to fixed anchors gives an exact excitation condition for the associated lifted LTV system. A second application treats direct hidden measurements and compares the condition with existing sufficient tests.

1 Introduction

Uniform observability is difficult to verify when the state-transition matrix is unavailable, motivating tests based on model signals and coefficient matrices. This section introduces exact excitation criteria for constant real-spectrum systems and cascades, then illustrates their use in applications.

1 Introduction

  • The observability Gramian gives a precise condition, but evaluating it generally requires a transition matrix unavailable in closed form; applications therefore use structural excitation tests.The introduction situates these tests among derivative-based and structure-specific observability criteria.Source: §1 · pp. 2–3
  • For constant real-spectrum dynamics, uniform observability is equivalent to persistent excitation of the output matrix on each eigenspace separately.An equivalent test uses the Kalman observability stack; the factored-output condition can be weaker than full measurement-space excitation.Source: §1 · p. 3
  • For cascades with a constant observable measured subsystem, a finite output-derivative stack eliminates the measured state and gives an equivalent hidden-state test under uniform regularity assumptions.The reduction permits multiple measured outputs and direct hidden measurements, and needs no spectral restriction; with real hidden spectrum, excitation is required on each eigenspace intersected with the direct-measurement kernel.Source: §1 · p. 3
  • Range-measured arbitrary-order integrator chains and a second-order cascade illustrate that the exact subspace requirement can be weaker than full-space persistent excitation.These conclusions concern LTV observability; nonlinear observer stability requires additional hypotheses.Source: §1 · p. 3

2 Uniform observability and standard tools

This section defines uniform observability through a uniformly positive observability Gramian and recalls derivative-based tests that can be checked from coefficients. It also distinguishes instantaneous rank requirements from window-averaged excitation conditions.

Uniform observability

  • Uniform observability requires a positive lower bound on the normalized observability Gramian over a fixed window, uniformly for every starting time.The integral measures average squared output, and under bounded coefficients this definition is equivalent to the usual two-sided uniform complete observability condition.Source: §2 · p. 4

Weighted observability

  • A positive-semidefinite measurement weight is admissible only when its weighted Gramian has a uniform lower bound; unweighted observability alone does not guarantee this.Uniformly positive definite bounded weights preserve equivalence with unweighted observability.Source: §2 · p. 4

Output-derivative criteria

  • The derivative stack recursively combines output derivatives with the dynamics, and its rows equal successive output derivatives along system trajectories.A pointwise positive lower bound on the stack Gramian implies uniform observability under the stated regularity assumptions.Source: §2 · p. 5
  • A selected derivative stack with a uniformly positive Gramian at every instant is a sufficient observability test, but can be restrictive when measurements vary with time.The cited criterion requires the derivative stack to distinguish all state directions at every instant.Source: §2 · p. 5

Averaged determinant criterion

  • An averaged determinant condition permits temporary rank loss by requiring selected derivative rows to have positive average determinant on every observation window.For square selections, this is equivalent to the averaged absolute-determinant condition; uniform observability of selected derivative outputs transfers to the original output under the stated regularity.Source: §2 · p. 5

Averaged derivative-output criterion

  • An averaged matrix condition allows excitation to accumulate in different directions at different times, unlike determinant tests requiring full column rank on part of each window.The section presents this approach for selected derivative rows when the state matrix is constant with real spectrum.Source: §2 · p. 5

3 Constant dynamics with real spectrum

For constant dynamics with real spectrum, uniform observability is characterized by persistent excitation of the output on each eigenspace separately. Equivalent tests use an averaged Kalman stack or, for factored outputs, excitation on the images of those eigenspaces.

3 Constant dynamics with real spectrum

  • Uniform observability holds exactly when C is persistently exciting on every eigenspace, even if the eigenspaces do not span the state space.The distinct real exponential dynamics separate generalized-eigenvector directions; no additional excitation test on those directions is needed.Source: §3 · p. 6

3.1 An averaged Kalman test

  • Uniform observability is also equivalent to persistent excitation of the averaged Kalman observability stack built from C, CA, …, CA^(n−1).This test uses only C(t) and A, not output derivatives; for constant C it reduces to the usual full-rank Kalman condition.Source: §3.1 · p. 6

3.2 Factored outputs

  • For C(t)=Θ(t)H with constant observable (A,H), uniform observability holds exactly when Θ is persistently exciting on each image HEλ.This exact condition tests only the relevant eigenspace images; full measurement-space excitation is sufficient but more restrictive.Source: §3.2 · p. 7
  • For a nilpotent A, the eigenspace test reduces to excitation on ker A, or on H ker A for factored outputs.In the double-integrator example, only the position sensor must be persistently exciting; the velocity sensor may vanish identically.Source: §3.2 · p. 8

3.3 Relation to the criterion of Wang et al.

  • The selected nilpotent-power stack criterion of Wang et al. (2022, Lemma 1) is sufficient, while testing C on ker J can use a matrix of dimension dim ker J instead of n.The selected stack need not be equivalent to the full stack, though it may be easier to recognize in a particular model.Source: §3.3 · p. 8

3.2 Factored outputs

  • Separate eigenspace excitation can hold even when excitation over their sum fails, because measured components from distinct eigenspaces may cancel.Example 16 remains uniformly observable with a smooth, bounded output despite failure of full-space excitation.Source: §3.2 · p. 7, §3.3 · p. 8

4 Constant-output cascades

For cascades with an observable constant measured subsystem, a finite stack of output derivatives eliminates the measured state and yields an equivalent observability test for the hidden state under uniform regularity assumptions. With real-spectrum hidden dynamics, excitation is needed only on eigenspaces not already directly measured.

4.1 Reduction to the hidden state

  • A finite derivative stack and a left-nullspace basis cancel the measured state, producing a hidden-state pair whose uniform observability is equivalent to that of the full cascade.The equivalence requires bounded continuous coupling and derivative terms, with the highest derivative term uniformly continuous; no spectral restriction is needed for this reduction.Source: §4.1 · p. 9, Figure 2 · p. 9
  • Boundedness and smoothness alone do not ensure the reduction: a smooth bounded, non-uniformly-continuous coupling can be persistently exciting while no fixed window has a uniform Gramian bound.The example uses b(t) = sin(t^2), whose effect after integration can become arbitrarily small.Source: §4.1 · p. 10

4.2 Excitation on unmeasured eigenspaces

  • For real-spectrum hidden dynamics, the cascade is uniformly observable exactly when Γ is persistently exciting on each eigenspace intersected with ker D.If that intersection is zero for every eigenspace, the direct measurements alone suffice under the theorem’s regularity assumptions.Source: §4.2 · p. 10

4.3 Two illustrations of the reduction

  • In the two-channel illustration, both measured channels are needed for the measured subsystem to be observable because every scalar output row vanishes on some nonzero vector in ker Ao.The example’s observability conclusion follows from Corollary 22.Source: §4.3 · pp. 10–11
  • With direct hidden measurements y = col(h, Dz), the second-order example is uniformly observable exactly when b(t)⊤ is persistently exciting on ker D.The example assumes b is bounded and uniformly continuous.Source: §4.3 · p. 11
  • For the cited ideal LTV model, the paper’s condition can be weaker than the sufficient test in Tchonkeu et al. (2026a, Lemma 2): with mI = e1 and aI(t) = sin(t)e2, (14) holds while the averaged comparison matrix has rank one.The comparison assumes the paper’s uniform regularity conditions and concerns the LTV model, not nonlinear observer stability.Source: §4.3 · p. 11

5 Case study: integrator chains with range measurements

The case study lifts range measurements from an arbitrary-order position chain into an LTV cascade, then applies the cascade and eigenspace reductions. The resulting exact condition requires excitation of the highest input derivative only in directions orthogonal to the anchor differences.

5 Case study: integrator chains with range measurements

  • The lift adds 2r − 1 scalar states, independently of the number of anchors, and the two reductions yield the exact excitation test.Quadratic range measurements are lifted into a cascade before eliminating the measured subsystem.Source: §5 · pp. 11–12
  • For the full lifted LTV system, uniform observability is equivalent to persistent excitation of u(r−1)(t)⊤ on ker H, under the stated regularity assumptions.H is the anchor-difference matrix; the condition applies to the homogeneous error system on its full lifted state space.Source: §5 · p. 12
  • Excitation is required only in directions not measured by anchor differences; the condition is vacuous when anchors affinely span Rd and becomes full-space excitation with one anchor.In three dimensions with rank H = 2, it requires a uniform positive lower bound on the mean square of u(r−1) along a unit normal to the anchor plane.Source: §5 · p. 12
  • For r = 1, the anchor-geometry condition recovers the known-velocity condition of Hamel and Samson (2017, Lemmas 4.1–4.2), under the boundedness assumed here.The paper states that the anchor matrix and H have the same kernel in this comparison.Source: §5 · pp. 12–13
  • The exact equivalence applies to the full lifted LTV model, not necessarily to the original nonlinear model, whose lifted states satisfy additional constraints.For r = 2 and one anchor, the example satisfies (20) although full-space PE of R3 fails in dimension four.Source: §5 · p. 13

6 Conclusion

The paper gives direct observability tests for constant real-spectrum systems and for cascades with an observable constant measured subsystem. Its range-measurement case study yields an exact condition for the full lifted model, while direct tests for nonreal spectra remain an open issue.

6 Conclusion

  • For constant real-spectrum dynamics, uniform observability is equivalent to persistent excitation on each eigenspace, without output differentiation or instantaneous full rank.The paper also gives an equivalent averaged Kalman-stack test and explains why excitation on larger subspaces can impose unnecessary restrictions.Source: §6 · p. 13
  • For cascades, finite output-derivative combinations eliminate an observable constant measured subsystem under uniform regularity, yielding an exact hidden-state observability test.When the hidden spectrum is real, excitation is needed only on eigenspaces intersected with the kernel of direct measurements.Source: §6 · p. 13
  • For arbitrary-order position chains measured through ranges to fixed anchors, the exact test requires excitation of a known input derivative only orthogonal to the anchor differences.This characterizes uniform observability of the full lifted LTV model.Source: §6 · p. 13
  • Direct tests for state matrices with nonreal eigenvalues, including purely imaginary pairs, remain an open question; such tests must account for cancellations between oscillatory dynamics and time-varying outputs.The paper’s rotation example illustrates this cancellation issue.Source: §6 · p. 13

A Proof of Theorem 11

The proof establishes both directions of the eigenspace characterization: persistent excitation on each eigenspace implies a uniform observability bound, and uniform observability implies excitation there. It uses asymptotic dominance of a top-eigenvalue component and compactness to obtain uniform constants.

A Proof of Theorem 11

  • Common excitation windows and bounds are obtained by taking extrema over the finitely many eigenvalues; boundedness of C and compactness then make the observability bound uniform over initial states.The proof first bounds the output uniformly in initial time for each fixed unit initial state.Source: A Proof of Theorem 11 · pp. 14–15
  • For sufficiency, each initial state’s dominant generalized-eigenspace component yields a nonzero vector in an eigenspace that can be tested by persistent excitation.Lower eigenvalue components decay relative to that component, while lower polynomial-degree terms from the same eigenvalue also become negligible.Source: A Proof of Theorem 11 · p. 14
  • To prove necessity, the argument constructs a common observation window and lower bound from eigenspace excitation, using a finite cover of the unit sphere.The resulting Gramian satisfies Wδo(t) ⪰ μoIn for every t ≥ 0.Source: A Proof of Theorem 11 · pp. 14–15

B Proof of Corollary 13

The proof establishes that persistent excitation of the observability stack is equivalent to persistent excitation of the output on every eigenspace. Compactness and the real spectrum of the dynamics supply the uniform lower bound needed for one direction.

B Proof of Corollary 13

  • Uniform eigenspace excitation bounds Qt on each eigenspace, placing the matrices Qt in a compact set KA.The proof uses this set to seek a uniform lower bound for FA(Qt).Source: B Proof of Corollary 13 · p. 15
  • If FA(Q) were singular for Q in KA, Cayley–Hamilton would produce a real eigenvector in ker Q^1/2, contradicting the eigenspace lower bound.Compactness of KA and continuity of the minimum eigenvalue then give a positive uniform lower bound, so OA is persistently excited.Source: B Proof of Corollary 13 · p. 15
  • Persistent excitation of OA implies that C is persistently excited on every Eλ, and Theorem 11 then gives uniform observability.The proof obtains the eigenspace bound by evaluating the stack condition on each eigenvector.Source: B Proof of Corollary 13 · p. 15

C Proof of Theorem 20

The proof shows that uniform observability of the cascade is equivalent to uniform observability of its reduced hidden-state pair. It uses regularity of the output-derivative relations for sufficiency and a state transformation built from a left inverse for necessity.

C Proof of Theorem 20

  • For sufficiency, a sequence with vanishing output energy has a convergent subsequence whose output derivatives through order ν vanish on the observation window.Uniform boundedness and equicontinuity enable the Arzelà–Ascoli limit argument.Source: C Proof of Theorem 20 · pp. 15–16
  • The vanishing derivative limits, together with uniform observability of the reduced pair, force the hidden-state component to vanish.The observable measured subsystem then forces the measured-state component to vanish as well, contradicting the normalized initial state.Source: C Proof of Theorem 20 · pp. 16–17
  • For necessity, a constant left inverse and the full left nullspace of O+ construct a transformation that separates the hidden-state dynamics from the measured state.The proof notes that using the full left nullspace is required for this step.Source: C Proof of Theorem 20 · p. 17
  • The transformed state η satisfies the reduced dynamics, with its forcing and output expressed through Γz.The identities give η̇(t) = Aoη(t) + M̄f(t) and q(t) = Coη(t) − J̄f(t).Source: C Proof of Theorem 20 · p. 17
  • Uniform observability of the cascade implies uniform observability of the reduced pair with the same window and a scaled lower bound.The proof chooses an initial measured state that cancels the transformed hidden state, then bounds the output energy by the hidden-state measurement energy.Source: C Proof of Theorem 20 · p. 17
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