Robust stability and stabilization of uncertain linear positive systems via Integral Linear Constraints: L1- and Linfinity-gains characterization (1204.3554v3)

Abstract: 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.

• The paper presents a novel robust stabilization framework for uncertain linear positive systems via copositive Lyapunov functions and integral linear constraints.
• It converts complex stability analysis into efficient linear programming problems to rigorously characterize L1- and Linf-gains.
• The methodology adapts to both constant and time-varying uncertainties using Handelman’s Theorem, offering practical insights for controller design.

Overview of Robust Stabilization in Uncertain Positive Systems

In the paper "Robust stability and stabilization of uncertain linear positive systems via Integral Linear Constraints," the author, Corentin Briat, presents an in-depth analysis of stability and performance optimization for uncertain linear positive systems. This is accomplished through the novel application of copositive linear Lyapunov functions and integral linear constraints (ILCs). The paper innovatively utilizes linear supply-rates for the characterization of $L_1$- and $L_\infty$-gains, framing the robustness and performance analysis within a linear programming context and leveraging Handelman’s Theorem to address computational complexities.

Stability Analysis and Control Synthesis

The paper highlights several key achievements in the field of positive systems stability. Unlike general linear systems which often rely on quadratic Lyapunov functions, this work employs linear copositive Lyapunov functions to analyze both stability and robust stabilization. This methodological shift enables the conversion of robust stability analysis into linear programming problems, thus admitting a more straightforward numerical resolution.

In general, positive systems present unique stability characteristics. For instance, the concept of diagonal stability greatly simplifies the structure of stability and stabilization problems, which is adeptly capitalized upon in this paper. The necessity and sufficiency of certain stability conditions are rigorously established via linear programming relaxations with detailed numerical examples illustrating their effectiveness.

Robustness via Integral Linear Constraints

Robust stability against parametric uncertainties is ensured through Integral Linear Constraints (ILCs). The use of ILCs is tactical, as it caters to the linear nature of positive systems, while also sidestepping the computationally extensive components typically associated with IQCs. The adoption of ILCs serves to broaden the problem's applicability to diverse classes of uncertainties, including, but not limited to, time-varying delays and nonlinearities.

One core contribution is the proof of equivalent robust stability results applicable to both time-invariant and time-varying uncertainties. For practitioners, this facet of the paper implies a solid foundation for extending analysis techniques beyond fixed-delay scenarios to more complex dynamic uncertainty environments.

Extensions and Numerical Examples

Further extending the theoretical contributions, the paper exploits Handelman’s Theorem to resolve the robust optimization problems, converting the infinite-dimensional feasibility problems into finite-dimensional counterparts without losing tractability. Several numerical examples poignantly demonstrate the approach’s scalability, as well as its theoretical parity with current methods in dealing with constant and time-varying delays.

Practical and Theoretical Implications

Practically, the results offer a powerful tweak in how engineers and systems theorists approach the design of controllers for positive systems under uncertainty. The theoretical implications are compelling, suggesting that static-gain matrices are pivotal to understanding dynamic interconnections for positive systems. Future research may delve into enhancing computational techniques, expanding the scope of uncertainties managed, or refining the assumptions around system positivity for broader real-world applications.

Thus, this work positions itself as a cornerstone for future explorations that aim to marry the robust simplicity of linear analysis techniques with the complex realities of non-standard system constraints and varying uncertain environments.