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Robust stability and stabilization of uncertain linear positive systems via Integral Linear Constraints: L1- and Linfinity-gains characterization

Corentin Briat

arXiv:1204.3554v3eess.SYmath.CAmath.DSmath.OC

TL;DR

The paper addresses stability, gain, robustness, and control analysis for uncertain linear positive systems using copositive Lyapunov functions and linear supply-rates. It develops ILC-based robust analysis and stabilization methods, expressing the results as robust linear programs. Handelman’s Theorem provides equivalent finite-dimensional formulations, with exactness for LTI positive uncertainties having fixed static-gain matrices.

  • Problem

    Stability and control of uncertain linear positive systems require methods suited to L1- and L∞-gains, linear supply-rates, and several uncertainty classes.

  • Method

    The paper combines copositive linear Lyapunov functions, dissipativity theory, ILCs, and robust linear programming for gain analysis, robust stability, stabilization, and performance optimization.

  • Results

    The resulting conditions include exact robust stability results for LTI positive uncertainties with fixed static-gain matrices and finite-dimensional formulations obtained using Handelman’s Theorem.

  • Takeaways & Limitations

    The framework supports robust stability and stabilization analysis while retaining convex linear-programming formulations for the considered positive-system settings.

Abstract

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Copositive linear Lyapunov functions are used along with dissipativity theory for stability analysis and control of uncertain linear positive systems. Unlike usual results on linear systems, linear supply-rates are employed here for robustness and performance analysis using L1- and Linfinity-gains. Robust stability analysis is performed using Integral Linear Constraints (ILCs) for which several classes of uncertainties are discussed. The approach is then extended to robust stabilization and performance optimization. The obtained results are expressed in terms of robust linear programming problems that are equivalently turned into finite dimensional ones using Handelman's Theorem. Several examples are provided for illustration.

I. INTRODUCTION

The paper develops copositive-Lyapunov and dissipativity-based methods for L1- and L∞-gain analysis, robustness, stabilization, and performance optimization in uncertain linear positive systems. Results are formulated as robust linear programs and reduced to finite-dimensional problems using Handelman’s Theorem.

  • Scope and motivation: Linear copositive Lyapunov functions yield linear-programming stability conditions and align naturally with vector 1-norm and L1-norm analysis.This contrasts with the 2-norm and L2-norm relationship associated with quadratic Lyapunov functions.
  • Scope and motivation: The paper analyzes uncertain positive systems using linear copositive Lyapunov functions, dissipativity theory, and linear supply-rates for L1- and L∞-gain characterization.The framework addresses stability, control, robustness, and performance analysis.
  • Gain characterization: L1-gains are computed directly through dissipativity theory, while L∞-gains are obtained as the L1-gains of the transposed system.The computed gains remain valid regardless of input signs and relax positivity requirements to conditions on system matrices.
  • Stabilization and performance: The methodology extends to convex necessary-and-sufficient stabilization with full, structured, and bounded state-feedback controllers, including performance constraints.The approach preserves convexity in the presence of scalings, and examples demonstrate efficiency and exactness for uncertainty classes including delays.
  • Robust analysis: Robust uncertainty analysis rewrites systems as positive interconnections and characterizes uncertain operators through Integral Linear Constraints, the linear counterpart of IQCs.For linear time-invariant uncertainties, ILCs fully characterize static-gain matrices.
  • Robust analysis: Robust stability conditions are formulated as robust linear programs, with exact conditions for LTI positive uncertainties having fixed static-gain matrices.Handelman’s Theorem converts these problems into finite-constraint linear programs and a variable-reduction procedure lowers computational complexity.

II. PRELIMINARIES

The preliminaries define positive systems, copositive Lyapunov functions, induced gains, and the relation between L1- and L∞-gains through transposition. For positive stable systems, gain calculations reduce to static-gain and impulse-response properties.

  • Positive systems: Positive systems require a Metzler state matrix and nonnegative input, output, and feedthrough matrices.These conditions preserve nonnegative states for nonnegative initial conditions and inputs.
  • Lyapunov analysis: A linear copositive Lyapunov function has the form V(x) = λ^T x and is positive with a negative derivative on nonzero states.The definition applies to the positive system ẋ(t) = Ax(t).
  • Induced gains: Induced Lσ-gains provide a framework for defining, computing, and optimizing system norms and for determining robustness and performance properties.The preliminaries introduce the gain operator framework before specializing to L1- and L∞-gains.
  • L1- and L∞-gains: The L1-gain emphasizes the most influential input column, whereas the L∞-gain emphasizes the most sensitive output row.In the SISO case, the two induced norms coincide.
  • Duality: The L∞-gain of H equals the L1-gain of its transposed system H*.The transposed transfer function exchanges the relevant state-space input and output factors.
  • Positive-system gain formulas: For asymptotically stable positive systems, nonnegative impulse responses connect induced gains to the system’s static-gain matrix.Metzler and Hurwitz dynamics imply a nonnegative impulse response, enabling the gain formulas.

III. STABILITY AND PERFORMANCE ANALYSIS OF UNPERTURBED SYSTEMS

For unperturbed positive systems, dissipativity with a linear supply rate yields equivalent stability and L1-performance conditions. The exact L1-gain is computable through a linear program whose complexity grows linearly with system size.

  • Scope: The section derives nonconservative stability and performance criteria for positive systems with zero control input.The analysis assumes throughout that the system is positive and u ≡ 0.
  • L1-gain characterization: Asymptotic stability and an L1-gain below γ are equivalent to static-gain inequalities and feasibility of a linear program.The equivalent conditions include transfer evaluation at zero frequency and inequalities involving C, E, and F.
  • Dissipativity proof: Dissipativity uses the supply rate s(w, z) = γ||w||1 − ||z||1 and a copositive storage functional decreasing along trajectories.Because λ > 0, the dissipativity condition also implies asymptotic stability.
  • Validity of the gain: The L1-gain computed using nonnegative inputs and states remains valid for arbitrary L1 inputs and arbitrary initial states.This follows from the alternative gain definition based only on nonnegative impulse responses.
  • Computation: The exact L1-gain of any asymptotically stable positive linear system is the optimum of a linear programming problem.The formulation uses n + 1 decision variables and 2n + p + 1 constraints.

B. L∞-gain characterization and computation

The L∞-gain characterization is obtained as the transposed counterpart of the L1 result, with exact computation by linear programming. The same framework supports positive stabilization with unconstrained, structured, and bounded controllers.

  • L∞-gain characterization: The L∞-gain characterization gives equivalent conditions involving the gain bound, the zero-frequency transfer matrix, and the static-gain matrix.The conditions are asymptotic stability plus bh(0)1p < γ1q or (F − CA^-1E)1p < γ1q.
  • L∞-gain computation: The L∞-gain is computable through a linear program with 2n + q + 1 constraints and the same number of variables as the L1 formulation.Its computational complexity grows linearly with system size.
  • Stabilization: State-feedback design is formulated so that positivity, asymptotic stability, and an L∞-gain below γ are equivalent to feasibility of linear constraints.A change of variables linearizes the controller-design problem, while Metzler and nonnegative constraints enforce closed-loop positivity.
  • Controller classes: The controller framework covers unconstrained, structured, and bounded gains, with structure and coefficient bounds imposed through supplementary linear constraints.Structured controllers enforce selected zero entries, while bounded controllers satisfy lower and upper gain limits.
  • Complexity: Unlike general LTI systems, structured and bounded controller design is not NP-hard in this positive linear setting.The tractability follows from using a diagonal Lyapunov function and the resulting necessary and sufficient conditions.
  • Exactness: Necessary and sufficient conditions are preserved for the structured and bounded controller cases, making the resulting approach nonconservative.The result also allows additional constraints such as asymmetric input bounds and bounded states.

V. ROBUST STABILITY ANALYSIS AND ROBUST PERFORMANCE

The robust analysis treats positive systems with real parametric uncertainties and extends to broader positive interconnections. Positivity makes uncertain transfer-function problems reducible to static parametric uncertainty through critical static-gain behavior.

  • Uncertain positive systems: The robust setting considers real parametric uncertainties δ ∈ [0, 1]^N in positive uncertain linear systems.The system matrices are continuous in δ, and positivity is required for every admissible uncertainty.
  • Positivity assumptions: The uncertain model assumes Metzler state matrices and nonnegative matrices for Eδ(δ), Cδ(δ), and Fδ(δ).These assumptions hold throughout the uncertainty set.
  • Scope of interconnections: The methodology applies beyond parametric uncertainty to delays, positive infinite-dimensional operators, uncertain positive operators, and static sign-preserving nonlinearities.The stated scope includes both time-invariant and time-varying uncertain positive operators.
  • Static-gain reduction: For positive systems, the static-gain matrix is critical for evaluating interconnection stability.Consequently, many uncertain positive transfer-function problems equivalently reduce to constant parametric uncertainty problems.
  • Positive LFR: An LFT representation is used to express the uncertain system, with robustness in the L1-norm relying on nonnegative loop signals.A positive LFR requires nonnegative loop signals and positive operators, although the representation is not unique.
  • LFR construction: Suitable LFR matrices can always be selected so that the loop signals are nonnegative, consistent with recurrent positive interconnections.The construction is linked to prior treatments of positive interconnections.

B. Handling positive uncertainties via Integral Linear Constraints

The paper introduces ILCs to analyze positive uncertain interconnections using linear Lyapunov functions and linear supply-rates. For positive operators, the ILC framework supports frequency-domain reasoning while addressing the absence of an L1 Plancherel theorem.

  • Motivation: ILCs provide a linear-constraint framework for stability analysis of interconnected uncertain systems, complementing small-gain, quadratic-separation, and IQC approaches.The paper motivates ILCs because IQCs do not fit the current framework, while ILCs use linear rather than quadratic forms.
  • Scaling choices: Scalings are selected according to the uncertainty set, and positivity makes componentwise absolute values disappear for nonnegative inputs and positive operators.The vectors ϕ1,i and ϕ2,i are called scalings and must be chosen according to Σ.
  • Frequency-domain interpretation: In the L1 setting, frequency-domain analysis remains possible, but the absence of the Plancherel Theorem prevents the direct L2-style conversion to tractable LMI problems.The paper notes that frequency-domain conditions can nevertheless be considered for positive systems.
  • ILC formulation: For positive operators, the ILC applies to every pair of nonnegative signals satisfying z = Σw, linking time-domain inequalities with algebraic inequalities.The algebraic formulation uses Laplace transforms of the input and output signals.

2) Generic robustness results and remarks:

The generic robustness results reduce ILC verification for positive LTI uncertainties to conditions on static-gain matrices. The paper then derives equivalent L1- and L∞-performance conditions and robust linear feasibility formulations.

  • Static-gain reduction: Because only the static-gain matrix matters, the uncertainty analysis reduces to constant parametric uncertainties rather than full frequency-dependent dynamics.The paper states that higher-order dynamics have no impact on stability in these positive interconnections.
  • Generic robustness: Theorem 1 establishes equivalence between an LTI positive operator satisfying an ILC and its static-gain matrix satisfying the corresponding algebraic condition.The equivalence also holds after adding sΘ(s), where Θ(s) is positive, asymptotically stable, and proper.
  • Uncertainty classes: The admissible uncertainty set can be very large because Θ(s) may have essentially unrestricted coefficient magnitudes acting on powers of s.The paper describes this as a consequence of the static-gain characterization.
  • Optimization formulation: The robust conditions are expressed as linear optimization or feasibility problems, but some resulting problems are difficult to solve directly.Handelman’s Theorem is proposed as an exact solving scheme for these robust optimization problems.
  • L1-gain results: For constant nonnegative uncertainty Δ0, asymptotic stability and an L1-gain below γ are equivalent to the corresponding positive LTI uncertainty with the same static gain.Theorem 3 explicitly equates the constant and dynamic cases when bΔ(0) = Δ0 ≥ 0.
  • L∞-gain results: The analogous L∞ result guarantees asymptotic stability and an L∞-gain smaller than γ when the robust feasibility condition holds for every uncertain parameter.The paper states that the L∞ extension follows from the transposed system.

VI. ROBUST STABILIZATION

The robust stabilization section extends the positive-system framework to controller synthesis with L∞ performance. Feasibility conditions preserve convexity and yield a controller satisfying a prescribed closed-loop gain bound.

  • Problem formulation: Robust stabilization with L∞ performance is formulated through robust linear feasibility conditions involving controller and scaling variables.The conditions include inequalities indexed by uncertain parameters and system dimensions.
  • Convexity and parameter dependence: The presence of scalings does not destroy convexity, even though rational parameter dependence initially produces rational constraints.A common denominator converts these constraints into polynomial constraints on numerators when its sign is fixed.
  • Positivity assumptions: The approach only requires nonnegative disturbance input matrices, so the open-loop system itself need not be positive.The requirement is E(δ), F(δ) nonnegative for all admissible parameters.
  • Stability synthesis: Theorem 5 gives a sufficient feasibility-based condition for asymptotic stability of the controlled uncertain system.The result searches for vectors and auxiliary variables satisfying the robust constraints.
  • Performance guarantee: A feasible solution produces a controller K for which the closed-loop system satisfies ||z1||L∞≤γ||w1||L∞.The performance guarantee is stated directly for the closed-loop transfer.
  • Computational solution: Handelman’s Theorem is proposed to solve the robust linear optimization problems arising in the stabilization results.The theorem is used to replace parameter-dependent robust problems with finite-dimensional formulations.

A. Handelman’s Theorem

Handelman’s Theorem represents polynomials positive on compact polytopes as nonnegative combinations of products of defining linear forms. This converts robust polynomially parameterized linear programs into finite-dimensional linear programs.

  • Theorem: A polynomial positive on a compact polytope can be expressed as a linear combination with nonnegative coefficients of products of the polytope’s defining linear functions.This is the central representation used to certify positivity over uncertain-parameter domains.
  • Illustration: For univariate degree-2 polynomials on [−1, 1], the basis functions are g1(x) = x+1 and g2(x) = 1−x, and all nonnegative polynomials use their products.The representation introduces nonnegative coefficients τi.
  • Robust optimization: The theorem converts robust linear programs with polynomial dependence over compact polytopes into more complex finite-dimensional linear programs.The construction generalizes from one uncertain parameter to multiple parameters.
  • Problem size: Before reduction, the resulting formulation has NP(b + 1) + η variables, NP(d+1) equality constraints, and NP(b+1) inequality constraints.After eliminating variables, it has NP(b−d)+η decision variables and NP(b+2) inequality constraints.
  • Computational limitation: The basis-product selection is difficult for multivariate polynomials because the number of basis functions b can become very large.A brute-force strategy considers all products up to a chosen degree, which is straightforward for univariate cases but more problematic in multiple variables.

VIII. EXAMPLES

The examples apply ILC-based gain and stability analysis to uncertain positive systems, including infinite-dimensional, delay, drug-distribution, and computational cases.

  • Uncertainty operators: The ILC framework characterizes common uncertainty operators, including multiplication, heat-equation, constant-delay, and time-varying-delay operators.The examples construct ILCs from operator gain or static-gain information.
  • Uncertain infinite-dimensional system: The heat-equation interconnection is certified stable for every ω ∈[0, +∞), because its static gain is independent of ω.The construction uses ϕ1 = −ϕ2(α + β).
  • Constant delay: Constant-delay positive systems receive a necessary and sufficient stability condition through saturated ILC scalings.The result recovers earlier constant-delay stability results.
  • Numerical computation: For randomly generated positive systems, gain computation uses n + 1 variables, with more constraints for L∞-gain than L1-gain.The text therefore expects L∞-gain computations to take longer; Table I reports mean times and standard deviations.
  • Drug distribution: In the drug-distribution model, SISO L1- and L∞-gains coincide with the static gain, while different output matrices yield different gains.The model is positive and asymptotically stable under the stated sign pattern.

D. Theoretical Robustness analysis - Time-delay systems

The time-delay examples compare L1- and L∞-based robustness conditions for constant and time-varying delays, showing distinct sensitivity to delay-rate bounds.

  • Constant time-delay: For constant delays, saturated ILC scalings yield a necessary and sufficient stability condition for positive time-delay systems.The condition is equivalent to the established inequality and is dual to a prior condition.
  • L∞ analysis: The L∞-based condition is equivalent to either (A + Ah)λ < 0 or −A^−1Ahϕ < ϕ.The latter is identified as the L∞-gain counterpart of the constant-delay condition.
  • Comparison of gain characterizations: The L∞ stability condition is identical for constant and time-varying delays, so the delay-derivative bound has no negative impact on stability.The text contrasts this with the L1-based result.
  • Lyapunov interpretation: The analysis connects L1-based results with Lyapunov-Krasovskii functionals and L∞-based results with Lyapunov-Razumikhin functions.The paper notes that Lyapunov-Razumikhin results for positive systems had not been reported.

F. Numerical robustness analysis - A polynomial system example

The polynomial uncertainty example evaluates L1- and L∞-gain estimates under parameter-dependent scalings and shows that higher-degree scalings reduce conservatism.

  • Polynomial uncertainty model: The example studies the uncertain transfer w1 →z1 for a system whose matrices depend polynomially on δ through δ and δ^2 terms.The state and output equations include A0, A1, A2, E0, E1, E2, C0, C1, C2, F0, F1, and F2.
  • Finite-dimensional computation: The robust gain and stabilization conditions are formulated as linear programming problems and converted to finite-dimensional feasibility problems.The conversion uses the complexity-reduction technique and the proposed exact relaxation scheme based on Handelman’s Theorem.
  • Gain estimation: Parameter-independent scalings estimate the L1-gain less accurately than the L∞-gain.The comparison is reported for the gain computations summarized in Tables IV and V.
  • Scaling choice: Degree-two scalings considerably reduce conservatism, accurately estimating the L∞-gain while retaining some conservatism for the L1-gain.The numerical conclusions also hold for time-varying parameters.
  • Scope of the results: The approach is nonconservative for LTI positive uncertainties with fixed static-gain matrix.The conclusion also reports validity for inputs and states without a definite sign.
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