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Complete Characterization of Minimum-Order Functional Observers from Darouach to Luenberger

Tyrone Fernando

arXiv:2609.37992v1eess.SYcs.SY

TL;DR

The paper asks how functional observers can be characterized at orders between the algebraic minimum and the upper endpoint set by functional observability. It develops fixed-order nullspace tests for the algebraic and spectral conditions, then identifies the minimum order as the first order satisfying both. The endpoint cases recover the Darouach and Luenberger orders under their respective conditions.

  • Problem

    The paper addresses how to characterize functional-observer existence at each order between the algebraic lower endpoint and the functional-observability upper endpoint, including when the lower order fails the spectral condition.

    Source: §I · p. 1

  • Method

    It represents reduced realizations through nullspaces in the original state space and uses rank and matrix-pencil conditions to test algebraic and spectral feasibility at prescribed orders.

    Source: §I · p. 2, A. Discussion · p. 14

  • Results

    The feasible-order spectrum is characterized between q0 and n0 − p, and the minimum observer order is its smallest order satisfying both conditions.

    Source: §I · p. 2, B. Conclusion · p. 15

  • Takeaways & Limitations

    The endpoint framework recovers Darouach’s order when q0 = r and Luenberger’s reduced order n − p under complete observability.

    Source: B. Conclusion · p. 15

Abstract

from arXiv · show

This paper gives an order-by-order characterization of linear functional observers between the minimum dimension permitted by the algebraic observer condition and the upper endpoint determined by functional observability. Functional observability indices determine the minimum algebraically admissible order $$ q_0=\sum_{j=1}^r\eta_j, $$ where $r$ is the number of independent functionals to be estimated and $\eta_j$ is the functional observability index associated with the $j$th functional. For an $n$-dimensional system with $p$ independent measured outputs, let $n_0$ denote the rank of the observability matrix. Under functional observability, the upper endpoint of the functional-observer order range is $n_0-p$. The algebraic and spectral observer conditions are characterized at each prescribed order between these two endpoints. A nullspace representation in the original state space yields rank and matrix-pencil characterizations of the algebraic and spectral conditions, respectively. The minimum observer order is obtained as the smallest spectrally feasible order in the resulting order spectrum. The two endpoints include the classical Darouach and Luenberger orders as special cases: the lower endpoint reduces to the Darouach order when $q_0=r$, while, under complete observability, the upper endpoint reduces to the Luenberger order $n-p$.

I. INTRODUCTION

The paper characterizes functional-observer orders between an algebraic lower endpoint and an upper endpoint set by functional observability. It asks how to characterize algebraic and spectral feasibility at each order in that range.

I. INTRODUCTION

  • Functional observers estimate prescribed state functions without reconstructing the full state, and the paper asks which orders between the two endpoints admit an observer.The problem includes whether feasibility can be characterized at each later order when order q0 fails the spectral condition, without prescribing an augmentation mechanism.Source: §I · p. 1
  • The functional observability indices set the minimum algebraically admissible order q0, while functional observability sets the upper endpoint at n0 − p.Here r counts independent functional rows modulo measured outputs, and n0 is the rank of the observability matrix.Source: §I · p. 1
  • The endpoint cases recover Darouach’s order when q0 = r and the reduced-order Luenberger order n − p under complete observability.The lower endpoint is based on functional observability indices; the upper endpoint uses the observability-matrix rank.Source: §I · p. 1

B. Problem Formulation

  • The problem is to characterize the order spectrum and the smallest order satisfying recoverability, reducedness, and both observer conditions.The analysis retains the original n-dimensional state-space representation rather than introducing an n0-dimensional realization.Source: B. Problem Formulation · pp. 2–3

III. ALGEBRAICALLY ADMISSIBLE OBSERVER ORDERS

Functional observability indices determine the minimum order at which the algebraic condition can hold, and algebraic feasibility persists at every order through the upper endpoint. Spectral feasibility is not thereby guaranteed, so the observer spectrum must be assessed order by order.

A. Order Endpoints

  • The functional observability indices determine q0, the minimum order at which condition (A) can hold.At q0, the complete algebraically admissible family is available from prior work [25].Source: A. Order Endpoints · p. 3

B. No-Gap Property

  • For q > q0, the extension result ensures an admissible higher-order realization but does not parameterize every realization at that order.The complete fixed-order characterization is instead developed using nullspace basis matrices.Source: B. No-Gap Property · pp. 4–5
  • Under functional observability, condition (A) is feasible at every order from q0 through n0 − p, and at no order below q0.The one-row admissible extension supports the no-gap result for algebraic feasibility.Source: B. No-Gap Property · pp. 4–5
  • At q0, a functional observer exists if and only if at least one member of the complete algebraically admissible family satisfies condition (S).If none does, the order must increase; the fixed-order tests determine where both conditions hold.Source: B. No-Gap Property · p. 5
  • Algebraic feasibility at every order does not imply joint algebraic and spectral feasibility at those orders.The lower bound q0 need not itself be feasible as a functional observer, nor must feasibility first occur at q0 + 1.Source: B. No-Gap Property · p. 5

IV. NULLSPACE MATRIX CHARACTERIZATION OF CONDITION (A)

The section represents a reduced realization through its nullspace, yielding a matrix characterization of condition (A) at a prescribed order.

IV. NULLSPACE MATRIX CHARACTERIZATION OF CONDITION (A)

  • A full-column-rank nullspace basis Z turns the order-q realization requirement and condition (A) into matrix forms.The characterization uses d = n − p − q and a basis RZ for ker(CAZ).Source: §IV · p. 5

Z AZRZ

The nullspace basis representation provides fixed-order tests for algebraic and spectral observer conditions. The spectral condition is expressed through a matrix pencil.

Z AZRZ

  • For a prescribed order q, the complete family satisfying the realization requirement and condition (A) is parameterized through full-column-rank matrices Z.The characterization uses a basis RZ for ker(CAZ); RZ is determined by Z rather than being an independent design variable.Source: Z AZRZ · pp. 5–6

V. NULLSPACE MATRIX CHARACTERIZATION OF CONDITION (S)

  • The spectral condition (S) is equivalent to a rank condition on a matrix pencil using the same nullspace basis Z.The tests depend on the column space of Z, so replacing Z by ZQ for nonsingular Q leaves them unchanged.Source: §V · pp. 6–7

VI. COMPLETE FIXED- AND ALL-ORDER CHARACTERIZATION

The fixed-order characterization tests observer existence through a nullspace matrix at each prescribed order. At the minimum order, it recovers the complete characterization and existence test from [25].

A. Fixed-Order Characterization

  • For every order in the admissible range, Corollary 1 parameterizes the complete condition-(A) family through Z, and Theorem 4 supplies the spectral test.The resulting test is necessary and sufficient for fixed-order existence.Source: A. Fixed-Order Characterization · p. 7
  • At each prescribed order, observer existence is equivalent to finding a full-column-rank nullspace matrix Z satisfying the stated conditions.The characterization combines algebraic and spectral conditions.Source: A. Fixed-Order Characterization · p. 7
  • At q = q0, the nullspace characterization is equivalent to the complete minimum-order characterization of [25].The corresponding nullspace representation is independent of the parameter K.Source: A. Fixed-Order Characterization · pp. 7–8
  • The order-q0 nullspace representation recovers the complete family of algebraically admissible realizations and its minimum-order spectral existence test.The correspondence is up to a change of nullspace basis and row equivalence.Source: A. Fixed-Order Characterization · p. 8

B. Matrix Parameterization at a Prescribed Order

At a prescribed order, the paper parameterizes candidate nullspaces with finitely many matrix charts and tests the algebraic and spectral conditions on those charts.

B. Matrix Parameterization at a Prescribed Order

  • Each chart has d(mL − d) free scalar parameters, and the finite chart choices cover the candidate subspaces.These are coordinate charts of the Grassmannian Gr(d, mL).Source: B. Matrix Parameterization at a… · p. 9
  • The A-compatibility and spectral requirements determine which d-dimensional subspaces qualify among the chart coordinates.The chart matrices provide finite coordinates for these candidate subspaces.Source: B. Matrix Parameterization at a… · p. 9
  • At prescribed order q, an observer exists if and only if some matrix chart and parameter Θ satisfy the stated conditions.The chart parameterizes the fixed-order characterization and reduces its spectral condition to a finite polynomial test.Source: B. Matrix Parameterization at a… · p. 9
  • The chart family covers every candidate nullspace, with Z = KLW and rank W = d.Varying the row-index chart covers all candidate nullspaces in the fixed-order characterization.Source: B. Matrix Parameterization at a… · pp. 8–9
  • For pole assignability, the spectral test uses a maximal-minor gcd condition rather than a pointwise search over λ ∈ C.For detectability, the gcd must have no zeros in the closed right half-plane.Source: B. Matrix Parameterization at a… · pp. 9–10
  • The complete search uses finitely many charts and two rank tests, while CZ = 0 and LZ = 0 hold by construction.The tests are conditions (14a)–(14b).Source: B. Matrix Parameterization at a… · p. 10

C. All-Order Characterization and Procedure

The paper organizes candidate observer orders from q0 through n0 − p and characterizes feasibility order by order, without assuming feasible orders form an interval.

C. All-Order Characterization and Procedure

1) Terminal Functional-Observability Endpoint:

Under functional observability, the upper endpoint n0 − p is spectrally feasible and becomes the reduced-order Luenberger order under complete observability. The full order spectrum is determined by testing feasibility at each admissible order.

1) Terminal Functional-Observability Endpoint:

  • The upper endpoint n0 − p is spectrally feasible under functional observability.The endpoint has a nullspace basis Z0 satisfying the stated conditions.Source: 1) Terminal Functional-Observability… · pp. 10–11
  • If (A, C) is observable, the functional-observability endpoint reduces to the classical reduced-order Luenberger order.In this case n0 = n and Z0 has no columns.Source: 1) Terminal Functional-Observability… · p. 11
  • Functional observability is equivalent to existence of an observer at some order from q0 through n0 − p.The paper also characterizes it by existence of a full-column-rank matrix Z satisfying the fixed-order conditions.Source: 1) Terminal Functional-Observability… · p. 11
  • The complete feasible-order spectrum consists of the orders whose condition-(A) nullspace family also satisfies the spectral condition.Theorem 6 provides the chart test at each dimension, and the terminal set is nonempty.Source: 1) Terminal Functional-Observability… · p. 11
  • Failure of both tests over every chart proves nonexistence at that order, rather than failure of only one selected realization.The minimum feasible order is the smallest order in the spectrum.Source: 1) Terminal Functional-Observability… · p. 11

VII. ILLUSTRATIVE EXAMPLE

The example’s admissible order range is q = 3 through 6, but the algebraic lower bound q = 3 is not itself feasible. Exhaustive chart tests establish that no order-3 observer satisfies the spectral condition.

VII. ILLUSTRATIVE EXAMPLE

  • Condition (A) first becomes possible at q = q0, but condition (S) excludes every admissible realization at q = q0 and q = q0 + 1.The example’s first feasible order is q = q0 + 2, below the functional-observability endpoint.Source: §VII · p. 11
  • For n = 8, p = 2, and r = 2, the example is completely observable and considers the four orders q = 3, 4, 5, and 6.The functional-observability endpoint reduces to the classical reduced-order Luenberger order.Source: §VII · pp. 11–12

A. Failure at the lower bound q = q0 = 3

  • At q = q0 = 3, the algebraic lower bound is not a functional-observer order.The Grassmann charts cover all candidate three-dimensional subspaces, and no condition-(A) candidate across the charts satisfies condition (S).Source: A. Failure at the lower bound q = q0 = 3 · p. 12

B. Failure at q = q0 + 1 = 4

The order-4 candidates all fail the spectral condition, while the example identifies a feasible candidate at order 5. At the endpoint q = 6, the terminal nullspace is trivial.

B. Failure at q = q0 + 1 = 4

  • At q = q0 + 1 = 4, none of the condition-(A) branches across the six Grassmann charts satisfies condition (S).The charts cover the candidate two-dimensional subspaces, proving that no order-4 functional observer exists.Source: B. Failure at q = q0 + 1 = 4 · pp. 12–13

C. Feasibility at q = q0 + 2 = 5

  • At q = 5, a feasible one-dimensional subspace is identified, and condition (S) is satisfied.The example gives a generator Z5 with rank Z5 = 1, CZ5 = 0, and LZ5 = 0.Source: C. Feasibility at q = q0 + 2 = 5 · p. 13

D. Existence at the functional-observability endpoint

  • At the functional-observability endpoint, d = 0 makes the terminal nullspace trivial, so Theorem 7 applies in this endpoint case.The passage does not state the resulting endpoint condition explicitly.Source: D. Existence at the… · p. 13

E. Explicit Observer Realization at q = 5

The example constructs a numerically verified order-5 functional observer at the true minimum, which lies strictly inside the admissible order range. The discussion and conclusion distinguish algebraic feasibility from spectral feasibility and characterize observer existence order by order.

E. Explicit Observer Realization at q = 5

  • At q = 5, the example realizes the true minimum-order observer, with eigenvalues −1 through −5 and a Sylvester-equation residual norm of approximately 1.4 × 10−13.The construction applies the classical Sylvester-equation method to an augmented functional.Source: E. Explicit Observer Realization at q… · p. 14

F. Interpretation

  • The example has r = 2 < q0 = 3 < νmin = 5 < n0 −p = 6, leaving order 6 feasible above the minimum.Thus the four quantities are distinct, and the minimum is strictly interior to the admissible range.Source: F. Interpretation · p. 14
  • Orders q0 and q0 + 1 are both excluded by condition (S), because condition (A)’s complete family lies where pencil matrix H is provably full-rank; the algebraic lower bound guarantees neither order.This two-order gap can occur independently of whether νmin coincides with the functional-observability endpoint.Source: F. Interpretation · p. 14
  • The Darouach and Fernando algorithm generically reaches q = 6, whereas the order-by-order characterization identifies the true minimum as νmin = 5.At q = 6 the cited solvability condition holds for arbitrary N; at q = 3, 4, and 5 it restricts N.Source: F. Interpretation · p. 14

A. Discussion

  • Functional-observability indices set the smallest algebraically feasible order, and algebraic feasibility then persists through n0−p; spectral feasibility remains a separate fixed-order test.The nullspace pencil supplies that spectral test at each predetermined order.Source: A. Discussion · p. 14
  • Higher-order realizations may be replacements rather than extensions of a particular minimum-order solution, so one-row extensions alone do not cover every case.The representation Z = KLW with rank conditions (14a)–(14b) parameterizes the order-q problem directly and includes replacement realizations.Source: A. Discussion · p. 14
  • The functional-observer order spectrum need not inherit the interval structure of the algebraic spectrum, so existence must be checked independently at each admissible order.Source: A. Discussion · p. 14

B. Conclusion

  • Under functional observability, condition (A) is feasible at every order from q0 through n0−p, and fixed-order rank, nullspace, and column-space tests characterize the algebraic and spectral conditions.The minimum is the first order in the resulting spectrum satisfying both conditions; when n0 = n the upper endpoint is n−p, and when q0 = r the lower endpoint is Darouach’s order.Source: B. Conclusion · p. 15
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