Convex Stability Analysis Tests

• Convex stability analysis tests are formal procedures that use convex optimization to certify robustness and stability in diverse systems such as linear, nonlinear, and PDE models.
• They employ methods like LMIs, sum-of-squares, and linear programming to derive tractable, often necessary and sufficient, stability criteria.
• These tests leverage convex Lyapunov functions and dissipativity principles to ensure scalable performance analysis and robust stability under uncertainty.

A convex stability analysis test is a formal procedure, typically formulated as a convex optimization problem, that certifies stability (or robustness of stability) of a dynamical, optimization, or feasibility system whose structure or constraints are convex. These tests leverage the convexity of Lyapunov functions, dissipation/storage functionals, constraint sets, or optimization landscapes to yield tractable (often necessary and sufficient) stability criteria expressible via linear matrix inequalities (LMIs), sum-of-squares (SOS) certificates, or linear programs (LPs). Recent research has established such tests as essential tools for stability analysis across linear, nonlinear, hybrid, PDE, infinite-dimensional, and quantum systems.

1. Convex Stability Tests in Linear and Positive Systems

For linear systems, convex stability testing frequently reduces to the search for a positive definite matrix solution to an LMI. A particularly powerful result holds for positive or positively dominated systems. Consider a linear time-invariant (LTI) system with transfer matrix $M(s)$, subject to block-diagonal norm-bounded uncertainty. The key characterization is as follows:

• For a nonnegative $M$ (entrywise), the structured singular value $\mu(M, \Delta)$ exactly equals its convex diagonal-scaling upper bound:

$$\mu(M, \Delta) = \inf_{\Theta \succ 0, \, \Theta \Delta = \Delta \Theta} \| \Theta^{1/2} M \Theta^{-1/2} \|_2$$

Thus, robust stability reduces to the feasibility of a convex LMI involving only the system's static gain $G(0)$:

$$M(0)^\top \Theta M(0) - \Theta \prec 0, \quad \Theta = \operatorname{diag}(\theta_i I_{m_i}) \succ 0$$

• For positively dominated systems, the worst-case frequency is always $\omega=0$; the infinite-frequency test collapses to a single test.

This LMI is both necessary and sufficient for robust stability in the full $\mu$-sense for nonnegative $M$ and general block-structured uncertainties (Colombino et al., 2015). The approach directly yields scalable, convex tests for networked problems such as the robust stability of the Foschini-Miljanic power control algorithm in uncertain wireless networks.

2. Convex Lyapunov and Dissipativity-Based Certificates for Nonlinear Systems

Convex stability tests for nonlinear systems exploit parameterizations of Lyapunov or storage functions to reduce stability verification to a convex program. Notable subclasses include:

2.1 SOS-Convex and Handelman-Based Lyapunov Methods

• SOS-Convexity: For polynomial discrete-time switched systems, existence of an SOS-convex Lyapunov function (i.e., one for which the Hessian is an SOS-matrix polynomial) is both necessary and sufficient for global stability. The search for such a function, along with its decrease properties, is performed via SDP over the polynomial coefficients and Gram matrices (Ahmadi et al., 2018).
• Handelman Theorem-Based LPs: For polynomial vector fields on polytopic domains, piecewise polynomial Lyapunov functions can be constructed by expanding in the Handelman basis and reducing stability to an LP over their coefficients, yielding lower computational complexity in high dimension compared to SOS/Polya SDPs (Kamyar et al., 2014).

2.2 Sum-of-Squares and Spacing Functionals for Sampled-Data Systems

For hybrid sampled-data systems, global exponential stability can be tested by formulating conditions on a Lyapunov function and a sampled-data "spacing" polynomial whose endpoint-scaling and net-decrease properties are constrained via SOS multipliers. The full system of convex constraints is solved as a semidefinite feasibility problem (Peet et al., 2014).

2.3 Stability of Positive Fuzzy and Interval Systems via LP

For positive Takagi-Sugeno discrete-time fuzzy systems, the existence of a co-positive linear Lyapunov function $V(x) = p^\top x$ leads to both positivity and asymptotic stability tests formulated as LPs in $p$ and auxiliary controller parameters. Robust interval uncertainty is handled by duplicating constraints for each extreme matrix, maintaining convexity and tractability (Ahmadi et al., 2019).

3. Convex LMIs for PDE and Distributed Parameter Systems

The stability of linear PDEs with constant or spatially varying coefficients and various boundary conditions can be analyzed by parameterizing quadratic Lyapunov functionals with matrix-valued multiplier and kernel terms (SOS-polynomial expansion). The time derivative of the functional leads to an integrated quadratic form wherein negativity only needs to be enforced on the admissible solution manifold — handled via convex projection ("spacing") operators, resulting in parameter-dependent LMI or SOS constraints (Meyer et al., 2016, Meyer et al., 2016). These matrix inequalities can be implemented in SOSTOOLS/SeDuMi, providing constructive certificates for exponential stability.

4. Practical Implementation and Scalability

Convex stability analysis tests are highly amenable to modern convex optimization solvers:

• LMI and SDP: Most tests reduce to feasibility of semidefinite programs. Diagonal or block-diagonal Lyapunov variables yield small-scale or highly structured SDPs (e.g., $\mathcal{O}(N)$ variables for positive systems).
• LP and Handelman Expansions: Handelman-based and positive-fuzzy system methods result in large but highly sparse LPs, advantageous as state dimension and polynomial degrees grow.
• SOS Approaches: For polynomial and PDE systems, sum-of-squares relaxations can be solved reliably for moderate state dimensions and polynomial degrees.
• Efficiency: These convex relaxations are generally scalable, with complexity polynomial in the number of decision variables. For large-scale systems, sparsity, block-decomposition, and first-order solvers may be exploited.

5. Theoretical Extensions and Connections

Convex stability criteria have been extended and refined in several directions:

• Convex Lyapunov Functions and Path-Connectedness: Existence of a convex Lyapunov function ensures not only GAS for the nominal vector field but also for straight-line homotopies and convex combinations, including robust and switched settings. This leads to a "continuation" theory for convex Lyapunov certificates (Jongeneel et al., 2023).
• Robustness to Uncertainty and Switching: Convex liftings allow for necessary and sufficient dwell-time and performance analysis for switched and uncertain systems, preserving equivalence with nonconvex tests but granting algorithmic tractability (Briat, 2013).
• Convex Equilibrium-Free and Incremental Criteria: Universal and incremental stability/performance of nonlinear systems can be checked via LMIs or infinite-family LMIs (converted via parameterization to finite SDPs), using velocity/differential forms and LPV embeddings (Koelewijn et al., 2024).
• Infinite-Dimensional and Optimization Problem Stability: Stability tests for convex feasibility and infinite convex systems utilize coderivative formulas, directional-derivative characterizations, and error bound stability, deduced from convexity and epigraphical properties (Bernardi et al., 2018, Wei et al., 2021, Cánovas et al., 2011).

6. Characteristic Features and Test Construction

The essential structure of convex stability tests is summarized as follows:

System Class	Lyapunov/Functional Form	Convex Test (Program)
Linear positive/dominated LTI	Diagonal quadratic Lyapunov	Single LMI on $M(0)$
Nonlinear polynomial ODE	Polynomial (SOS-/Handelman-)	SDP/SOS or LP over coefficients
Sampled-data/hybrid	Lyapunov + spacing, polynomial	SDP (SOS constraints, multipliers)
Positive fuzzy/interval	Copositive linear	Linear program with auxiliary vars
PDEs (linear, coupled)	Quadratic multiplier/kernel	SDP over matrix polynomials, spacing

This universality, combined with necessity and sufficiency in many settings, makes convex stability analysis tests a unified and algorithmically powerful framework for robust stability certification in systems and control theory.

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Principal References:

• "A Convex Characterization of Robust Stability for Positive and Positively Dominated Linear Systems" (Colombino et al., 2015)
• "Global Stability Analysis of Nonlinear Sampled-Data Systems using Convex Methods" (Peet et al., 2014)
• "Convex lifted conditions for robust stability analysis and stabilization of linear discrete-time switched systems" (Briat, 2013)
• "A Convex Approach for Stability Analysis of Coupled PDEs using Lyapunov Functionals" (Meyer et al., 2016)
• "Constructing Piecewise Polynomial Lyapunov Functions for Local Stability of Nonlinear Systems Using Handelman's Theorem" (Kamyar et al., 2014)
• "SOS-Convex Lyapunov Functions and Stability of Difference Inclusions" (Ahmadi et al., 2018)
• "Stability and Stabilization Analysis of Interval Positive Takagi-Sugeno Fuzzy Systems by using Convex Optimization" (Ahmadi et al., 2019)
• "Convex Equilibrium-Free Stability and Performance Analysis of Discrete-Time Nonlinear Systems" (Koelewijn et al., 2024)
• "On continuation and convex Lyapunov functions" (Jongeneel et al., 2023)
• "Stability of a convex feasibility problem" (Bernardi et al., 2018)
• "Characterizations of Stability of Error Bounds for Convex Inequality Constraint Systems" (Wei et al., 2021)
• "Quantitative Stability of Linear Infinite Inequality Systems under Block Perturbations with Applications to Convex Systems" (Cánovas et al., 2011)