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A Minimum-Order Functional Observer Beyond the Darouach and Luenberger Constructions

Tyrone Fernando

arXiv:2609.33615v1eess.SYcs.SY

TL;DR

Estimating prescribed linear functions of a system state need not require reconstructing the full state, but classical Darouach and reduced-order Luenberger constructions do not cover every minimum-order case. The paper characterizes the minimum algebraically permitted dimension and tests spectral feasibility across all augmentations attaining it. The framework recovers both classical cases and admits intermediate-order observers when spectral feasibility holds, even when neither classical construction is available.

  • Problem

    The Darouach and reduced-order Luenberger constructions do not cover cases where a minimum functional observer may have order strictly between the prescribed functionals and the unmeasured state directions.

    Source: §I · p. 1

  • Method

    The framework uses functional observability indices to determine the algebraically minimum dimension and characterizes the complete family of augmentations attaining it for spectral-feasibility testing.

    Source: §I · pp. 1–2, §VIII · p. 16

  • Results

    When spectral feasibility holds, the framework recovers the Darouach and reduced-order Luenberger cases and admits intermediate-order observers even when neither classical observer is available.

    Source: §VIII · p. 16

  • Takeaways & Limitations

    The minimum observer order can lie strictly between the number of prescribed functionals and the number of unmeasured state directions.

    Source: §I · p. 1

Abstract

from arXiv · show

This paper develops a general minimum-order functional observer framework that contains the classical Darouach functional observer and the reduced-order Luenberger observer as special cases. The required functional dimension is determined from the triple $(A,C,L)$ through a set of functional observability indices. The minimum dimension permitted by the algebraic observer-existence condition is characterized, and the complete family of augmentations attaining this dimension is determined. Necessary-and-sufficient spectral feasibility conditions are then established for this dimension to be the minimum achievable observer order. The resulting existence conditions reduce exactly to the Darouach conditions when no augmentation is required and to the classical observability condition in the reduced-order Luenberger case. More generally, the framework admits minimum-order functional observers of intermediate order, which may exist even when neither classical observer is available.

I. INTRODUCTION

The paper seeks the least observer order needed to estimate prescribed state functions, rather than reconstructing the full state. It derives that order from functional observability and spectral feasibility, extending the classical cases to an intermediate regime.

I. INTRODUCTION

  • The required observer order is motivated by estimating z = Lx without introducing unnecessary dynamics to reconstruct the complete state.The motivation is especially relevant when r ≪ n.Source: §I · p. 1
  • The algebraic condition sets the minimum functional dimension q0 = Σηj, where ηj are the functional observability indices, and no observer can have smaller order.The observer-existence test also includes a separate spectral condition.Source: §I · p. 1
  • Although q0 is fixed, its attaining augmentation need not be unique, so spectral feasibility must be checked across the complete minimum-dimensional family.Failure of condition (S) for one such augmentation does not rule out another of the same dimension satisfying it.Source: §I · p. 2
  • The framework recovers Darouach conditions when no augmentation is needed and the reduced-order Luenberger observability condition when all unmeasured state directions are represented.These are the q0 = r and q0 = n −p cases, respectively.Source: §I · p. 2
  • When r < q0 < n −p, spectral feasibility can yield a minimum-order observer even when the system is unobservable and neither classical construction is available.In this regime, an order-r Darouach observer does not exist, and reconstructing all n −p unmeasured state directions is unnecessary.Source: §I · p. 2

A. Functional Observability

  • Functional observability means the prescribed functional can be reconstructed from available input and output, with equivalent reconstruction and augmentation-based characterizations.The paper assumes functional observability and seeks a minimum-order observer under that assumption.Source: A. Functional Observability · p. 2, §II · p. 2

III. ORDERED FUNCTIONAL-OBSERVABILITY CONSTRUCTION

The section constructs an ordered rank-retention procedure from the rows of C and L and their A-iterates, then uses the retained rows to determine functional-observability indices. The procedure identifies independent directions while terminating after finitely many examinations.

III. ORDERED FUNCTIONAL-OBSERVABILITY CONSTRUCTION

  • The ordered construction preserves the joint structure generated by measured-output rows C and functional rows L, providing independent directions for later augmentation.The resulting finite set of directions is generated by those rows and their successive A-iterates.Source: §III · p. 3

A. Global examination order and rank-retention rule

  • The procedure examines C and L generators and their successive A-iterates by increasing powers of A, retaining a candidate row only when it increases rank.Retained rows are linearly independent, and the global examination order fixes which rows are considered first.Source: A. Global examination order and… · p. 3
  • Once a generator's A-generated sequence reaches a dependent row, all subsequent rows in that sequence are dependent, so the search terminates after finitely many examinations.Cayley–Hamilton ensures each sequence encounters a dependent row no later than power n.Source: A. Global examination order and… · pp. 3–4

B. Functional-observability indices

  • The functional-observability indices are read from each generator's highest retained power, with one added to identify its first dependent row.The indices can be determined without regrouping the rows of the retained matrix.Source: B. Functional-observability indices · p. 4

C. First-dependence relations

The first-dependence relations express each generator's first dependent A-iterate using previously retained rows. These relations preserve the ordered structure and supply the ingredients for constructing admissible functional augmentations.

C. First-dependence relations

  • Each generator's first dependent A-iterate is expressed as a linear combination of rows retained before that iterate in the prescribed global order.The preceding rows may include directions generated by both C and L.Source: C. First-dependence relations · p. 4
  • At a generator's terminal power, only retained rows from preceding generators can appear in its first-dependence relation.At lower powers, retained rows from any generator may occur.Source: C. First-dependence relations · pp. 4–5

IV. CONSTRUCTION OF ADMISSIBLE FUNCTIONAL AUGMENTATIONS

  • The ordered rank-retention construction supplies a finite set of independent directions and first-dependence relations for building augmentations that satisfy conditions (A) and (S).Condition (A) requires closure under A modulo measured directions supplied by C and CA; condition (S) imposes a spectral rank requirement.Source: §IV · p. 5

A. Augmentation required by condition (A)

  • Condition (A) requires each augmented functional row's A-image to lie in the row space generated by the augmented functional rows, C, and CA.If the prescribed L does not meet this inclusion, additional functional directions are required.Source: A. Augmentation required by condition… · p. 5

B. Construction of the augmentation for condition (A)

  • For each functional generator, the construction retains rows from its A-generated sequence up to the first dependent row and forms added rows by successively reducing the positive powers of A.The resulting augmentation satisfies condition (A), and the following result establishes that its number of added rows is minimum.Source: B. Construction of the augmentation… · pp. 5–6

C. Verification of condition (A)

The constructed augmented functional matrix satisfies condition (A). This construction establishes feasibility, while the minimum-dimension result that follows leaves open which minimum augmentation will satisfy the additional condition (S).

C. Verification of condition (A)

  • Theorem 1 establishes that the constructed augmented functional matrix satisfies condition (A).The proof checks that the A-image of its prescribed and augmented rows lies in the required row space.Source: C. Verification of condition (A) · pp. 6–7
  • The construction verifies condition (A) for both terminal rows with ηj = 1 and augmented sequences with ηj > 1.When ηj = 1, the prescribed row is already terminal; otherwise the proof also checks nonterminal and terminal augmented rows.Source: C. Verification of condition (A) · p. 7
  • Minimum augmentations need not be unique, and one satisfying condition (A) may fail condition (S) while another of the same dimension satisfies it.Thus testing condition (S) requires considering the family of minimum augmentations, not just one construction.Source: C. Verification of condition (A) · p. 7

D. Minimum augmentation for condition (A)

The section establishes the minimum number of additional rows needed for condition (A) and distinguishes this fixed minimum dimension from the potentially nonunique augmentations that attain it.

D. Minimum augmentation for condition (A)

  • The minimum augmentation for condition (A) adds ηj −1 rows for each prescribed functional row Lj.The construction supplies an upper bound, while the proof establishes a matching lower bound for every augmentation satisfying condition (A).Source: D. Minimum augmentation for condition… · pp. 8–9
  • The minimum augmentation dimension is fixed, but distinct augmented functional spaces of that dimension may satisfy condition (A).The ordered rank-retention construction selects one particular minimum augmentation based on its generator and examination orders.Source: D. Minimum augmentation for condition… · pp. 8–9
  • Because minimum augmentations can differ, a particular one may fail condition (S) even when another of the same dimension satisfies it.The complete family of minimum augmentations is therefore relevant when testing condition (S).Source: D. Minimum augmentation for condition… · p. 9

V. COMPLETE FAMILY OF MINIMUM CONDITION-(A) AUGMENTATIONS

The section identifies additional independent directions for a minimum augmentation and explains how measured directions are excluded from the count.

V. COMPLETE FAMILY OF MINIMUM CONDITION-(A) AUGMENTATIONS

  • The section constructs a minimum augmentation satisfying condition (A), then describes the choices of additional directions before imposing that condition.The supplied passage also states that the minimum augmentation has additional independent directions beyond those represented by L.Source: §V · p. 9
  • Directions belonging to row(C) add nothing because the measured quantity Cx is already available.Multiplication by NC removes precisely the measured directions; rank(LNC) counts functional directions modulo row(C).Source: §V · p. 9
  • The number of additional independent directions is considered beyond those already represented by L.The section seeks minimum augmentations and then imposes condition (A).Source: §V · p. 9

A. Parameterization of the additional directions

Minimum candidates are parameterized by pivot sets and free parameters; the family is then restricted to members satisfying condition (A).

A. Parameterization of the additional directions

  • In transformed coordinates, the prescribed functional directions occupy the first r coordinates, and minimum candidates add k independent directions.Subtracting combinations of the prescribed rows yields a rank-k matrix Y for the additional directions.Source: A. Parameterization of the additional… · pp. 9–10
  • Every k-dimensional row space in R^s is represented by a pivot set J and a free parameter Θ in the chart YJ(Θ).The chart normalizes k independent columns to Ik; the remaining s − k columns form Θ.Source: A. Parameterization of the additional… · p. 10

B. Imposing condition (A)

  • The parameterized minimum candidates are restricted to those satisfying condition (A), expressed in reduced coordinates.The section introduces a reduced form of condition (A) for this restriction.Source: B. Imposing condition (A) · p. 10

(LNC) ¯A

The section reduces condition (A) to a rank test and establishes that its parameterized family captures every minimum reduced augmentation satisfying that condition.

(LNC) ¯A

  • Condition (A) is equivalent to LANC = FLNC + GBA for some matrices F and G.The reduced-coordinate condition is expressed through the measured null-space basis NC and the input term BA.Source: (LNC) ¯A · p. 11
  • For each pivot set J, the admissible parameter values Θ are those whose candidates satisfy the transformed rank condition.Adding the rows of WJ(Θ) ¯AT must not increase the rank.Source: (LNC) ¯A · p. 11

C. Completeness of the parameterization

  • The WJ(Θ) family represents, up to row equivalence, every minimum reduced augmentation satisfying condition (A).The text states both that family members satisfy condition (A) and that every such minimum augmentation has a representative in the family.Source: C. Completeness of the parameterization · pp. 11–12

D. Recovery of the augmentation matrix

Admissible reduced-coordinate candidates are recovered as augmented functional matrices in the original coordinates, with measured-row additions leaving the reduced augmentation unchanged.

D. Recovery of the augmentation matrix

  • Recovery is nonunique: adding rows from the measured row space does not change the reduced augmentation.The solution of the recovery equation is not unique.Source: D. Recovery of the augmentation matrix · p. 12
  • The complete family of minimum augmented functional matrices satisfying condition (A) is represented after returning to the original coordinates.Representatives correspond to admissible pairs (J, Θ).Source: D. Recovery of the augmentation matrix · p. 12
  • The minimum condition-(A) family is nonempty because the chain construction produces a minimum augmentation satisfying condition (A).The complete parameterization therefore has at least one admissible pivot set and parameter value.Source: D. Recovery of the augmentation matrix · p. 12

VI. SPECTRAL FEASIBILITY AT THE MINIMUM CONDITION-(A) ORDER

The paper reduces spectral feasibility for minimum condition-(A) augmentations to a rank test, then gives a finite algebraic criterion for whether any member yields an observer of order q0.

VI. SPECTRAL FEASIBILITY AT THE MINIMUM CONDITION-(A) ORDER

  • For a minimum condition-(A) augmentation, condition (S) holds exactly when rank PL(λ) = rank H for every λ ∈ C.Here H = LNCA and PL(λ) = λH − LANCA.Source: §VI · p. 13

A. Finite algebraic spectral test

  • The spectral test is finite and algebraic: for h > 0, the maximal minors must have greatest common divisor 1 with respect to λ, up to a nonzero constant.This is equivalent to the minors having no common zero in C.Source: A. Finite algebraic spectral test · pp. 13–14
  • An order-q0 observer exists if and only if some member of the complete minimum condition-(A) family passes the spectral test.When h = rank H is positive, the test is that the maximal minors of PL(λ) have no nonconstant common factor; when h = 0, it holds automatically.Source: A. Finite algebraic spectral test · p. 14
  • For asymptotic convergence alone, the rank test need only hold for λ with ℜ(λ) ≥ 0.An asymptotic order-q0 observer exists if and only if at least one minimum condition-(A) augmentation satisfies this restricted test.Source: A. Finite algebraic spectral test · p. 14

B. Recovery of the Darouach and Luenberger Conditions

The general existence criterion recovers both classical observer cases at their respective functional dimensions: Darouach when no augmentation is needed, and reduced-order Luenberger at the unmeasured-state endpoint.

B. Recovery of the Darouach and Luenberger Conditions

  • When q0 = r, the minimum-order criterion reduces to the Darouach functional-observer existence conditions.At this endpoint, no augmentation is required.Source: B. Recovery of the Darouach and… · p. 14
  • When L spans the n−p state directions unavailable from the measured output, q0 = n−p and the criterion reduces to the PBH observability condition for (A, C).This is the reduced-order Luenberger case.Source: B. Recovery of the Darouach and… · p. 14

VII. EXAMPLE

The example realizes a minimum-order observer of intermediate dimension even though neither classical construction is available; the conclusion places this result within a framework that recovers both classical cases.

VII. EXAMPLE

  • The example has r = 2, q0 = 4, n0−p = 5, and n−p = 6, so its minimum dimension lies strictly between the functional dimension and both observability endpoints.The order-r Darouach conditions fail, and (A, C) is unobservable.Source: §VII · p. 15
  • Among the minimum condition-(A) augmentations, La,b satisfies condition (S) exactly when b ≠ 0.L0,0 fails (S) at λ = −5, whereas L0,1 satisfies both (A) and (S).Source: §VII · p. 15

VIII. CONCLUSION

  • The example confirms that failure of one minimum condition-(A) augmentation does not rule out another augmentation of the same dimension.The conclusion states that the general criterion recovers Darouach at q0 = r and Luenberger at q0 = n−p, while permitting intermediate orders when spectral feasibility holds.Source: §VII · p. 15, §VIII · p. 16

VII. EXAMPLE

  • A spectrally feasible fourth-order realization exists even though the cited Darouach–Fernando procedure advances to a higher admissible order.The paper constructs an observer using L0,1 and assigns N the Hurwitz eigenvalues {−2, −3, −4, −6}.Source: §VII · pp. 15–16
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