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

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.

💡 Research Summary

The paper addresses the robust stability analysis and controller synthesis problem for uncertain linear positive systems, a class of systems whose states and outputs remain non‑negative for all time. Traditional approaches for general linear systems rely on quadratic Lyapunov functions and Linear Matrix Inequalities (LMIs), which often lead to high computational complexity and do not exploit the special structure of positive systems. In contrast, the authors propose a framework built on four pillars: (1) copositive linear Lyapunov functions, (2) dissipativity theory with linear supply‑rates that directly capture L1‑ and L∞‑gain performance measures, (3) Integral Linear Constraints (ILCs) to model a broad spectrum of uncertainties, and (4) Handelman’s theorem to convert the resulting infinite‑dimensional polynomial inequalities into finite‑dimensional linear programming (LP) problems.

The first contribution is the use of copositive Lyapunov functions, i.e., linear functions that are non‑negative over the non‑negative orthant. Because a positive system never leaves this orthant, such functions provide a sufficient condition for stability that is both simple to evaluate and amenable to linear programming. By pairing this Lyapunov function with a dissipativity inequality that employs a linear supply‑rate, the authors obtain explicit L1‑gain and L∞‑gain bounds. These bounds are expressed as linear inequalities in the system matrices and the Lyapunov vector, making them directly compatible with LP formulations.

To handle model uncertainties, the paper introduces Integral Linear Constraints. An ILC describes the admissible input‑output behavior of an uncertain block Δ through an integral inequality of the form ∫ wᵀ(t) Q w(t) dt ≥ 0, where w(t) aggregates signals entering and leaving Δ and Q is a constant matrix that encodes the uncertainty description. The authors show how several common uncertainty classes—interval parameters, polytopic variations, and certain stochastic perturbations—can be captured by appropriate choices of Q. Because the constraints are linear in the signals, they preserve the positivity structure and can be combined with the copositive Lyapunov condition without introducing quadratic terms.

The third technical step is the application of Handelman’s theorem. The ILC and Lyapunov conditions together generate polynomial inequalities that must hold over the unit simplex (or more generally over a compact polytope defined by the positivity constraints). Handelman’s theorem guarantees that any polynomial that is non‑negative on such a domain can be represented as a finite non‑negative linear combination of products of the defining affine functions. By truncating the degree of these products, the infinite‑dimensional condition is approximated by a finite set of linear constraints on the coefficients. Consequently, the robust stability condition becomes a standard LP: the decision variables are the entries of the copositive Lyapunov vector, the supply‑rate parameters, and the Handelman coefficients.

Having established a tractable LP for robust stability, the authors extend the framework to robust controller synthesis. They consider state‑feedback u = Kx, where K is the feedback matrix to be designed. The closed‑loop system remains positive if K respects certain sign constraints, which are again linear. By embedding the feedback matrix into the LP, the synthesis problem simultaneously guarantees robust stability for all admissible uncertainties and optimizes a chosen performance metric (e.g., minimizing the worst‑case L1‑gain or imposing an upper bound on the L∞‑gain). The resulting optimization is convex and yields a globally optimal solution.

The paper validates the methodology on several numerical examples. Small‑scale second‑ and third‑order positive systems with interval and polytopic uncertainties are used to illustrate the construction of ILCs, the Handelman‑based LP, and the synthesis of a robust stabilizing gain. A larger network‑type positive system, representing a compartmental model, demonstrates scalability: the LP size grows linearly with the number of states, and the computational time remains modest compared with an LMI‑based counterpart. In all cases, the LP approach achieves comparable or less conservative stability margins while requiring significantly less computation.

In summary, the authors deliver a novel, computationally efficient framework for robust analysis and control of uncertain linear positive systems. By exploiting positivity through copositive Lyapunov functions, linear supply‑rates, ILC modeling, and Handelman’s polynomial decomposition, they transform what would traditionally be a set of nonlinear matrix inequalities into a family of linear programs. This not only reduces the computational burden but also opens the door to large‑scale applications in biochemical reaction networks, compartmental epidemiological models, and energy distribution systems where positivity is inherent.