Poly-Quadratic Lyapunov Functions

• Poly-quadratic Lyapunov functions are parameter-dependent quadratic forms that generalize classical certificates for assessing stability in systems with varying parameters.
• They leverage convex combinations and LMI/SDP frameworks to yield less conservative stability tests for LPV, switching, and hybrid dynamical systems.
• Applications include robust controller/observer synthesis, region-of-attraction estimation, and performance certification in nonlinear and hybrid system analysis.

A poly-quadratic Lyapunov function is a composite or parameter-dependent quadratic function that generalizes classical quadratic Lyapunov certificates to capture the stability of systems with inherent variability—such as polytopic linear parameter-varying (LPV) systems or piecewise-affine/switching models. These functions arise as convex combinations, maxima, minima, or regionally-defined collections of quadratic forms, often yielding significantly less conservative stability tests than single quadratic forms. Poly-quadratic Lyapunov methods are central in modern analysis and synthesis of LPV, switched, and hybrid dynamical systems, providing tractable linear matrix inequality (LMI) or semidefinite program (SDP) formulations. The following sections detail definitions, characterization results, computational aspects, limitations, and applications, with emphasis on rigor and explicit connections to the research literature.

1. Mathematical Formulation and Types

The prototypical discrete-time polytopic LPV system is described by

$$x_{k+1} = A(p_k)\,x_k + B\,u_k, \qquad y_k = C\,x_k,$$

where $p_k \in \mathcal{P} \subset \mathbb{R}^{n_p}$ indexes a time-varying parameter inside a convex polytope whose $N$ vertices $\{v_i\}_{i=1}^N$ define extremal system matrices $\{A_i\}$. The affine dependence $A(p) = \sum_{i=1}^N \xi_i(p) A_i$ with nonnegative barycentric weights $\xi_i(p),\ \sum_{i} \xi_i(p) = 1$ ensures a strictly polytopic structure when for every vertex $v_i$ one can choose $\xi(v_i) = e_i$ (Meijer et al., 1 Feb 2026).

A poly-quadratic Lyapunov function is then any parameter-dependent quadratic of the form

$$V(p,x) = x^\top P(p) x, \qquad P(p) = \sum_{i=1}^N \xi_i(p)\, \bar P_i, \quad \bar P_i \succ 0.$$

On the vertices, $P(v_i) = \bar P_i$, so $V$ interpolates $N$ quadratic forms over the simplex $\mathcal{P}$.

Related variants include:

• Piecewise-quadratic Lyapunov functions: Defined as $V(x) = x^\top P_j x$ on a partitioned region $R_j$ of state-space, or as envelops $V(x) = \min_{i} x^\top P_i x$, $V(x) = \max_{i} x^\top Q_i x$ (Ahmadi et al., 2015, Angeli et al., 2016).
• Max/min-of-quadratics: Common in switching systems, yielding certificates for uniform stability under arbitrary mode transitions.
• Path-complete Lyapunov functions: A generalization using a labeled directed graph associating quadratic pieces to nodes and Lyapunov inequalities to edges (Angeli et al., 2016).

2. Stabilizability and Detectability via Poly-Quadratic LMIs

For strictly polytopic LPV systems, the core theorems guarantee that stabilization and detectability can be verified via a finite family of LMIs parameterized by the polytope vertices (Meijer et al., 1 Feb 2026):

• Poly-quadratic detectability: There exist $\{\bar P_i\succ0\}$ s.t.

$$\bar P_i - A_i^\top\,\bar P_j\,A_i + C^\top C \succ 0,\quad \forall i,j\in \{1,\dots,N\}.$$

This ensures a parameter-dependent observer gain of the form

$$L(p) = -\sum_{i=1}^N \xi_i(p)\,A_i [\bar P_i + C^\top C]^{-1} C^\top.$$

• Poly-quadratic stabilizability: There exist $\{\bar S_i\succ0\}$ (set $\bar P_i = \bar S_i^{-1}$) satisfying

$$\bar S_j - A_i\,\bar S_i\,A_i^\top + B B^\top \succ 0,\quad \forall i,j,$$

and corresponding parameter-dependent state feedback law

$K(p,p^+) = -B^\top [S(p^+)+BB^\top]^{-1}A(p).$

The LMIs exhibit a $N^2$ scaling with polytope size due to cross-mode constraints, but greatly reduce conservatism relative to single quadratic Lyapunov tests, which require $P - A_i^\top P A_i \succ 0$ for all $i$ (Meijer et al., 1 Feb 2026).

3. Generalizations: Piecewise-Quadratic and Path-Complete Approaches

Piecewise-quadratic functions extend poly-QLFs to arbitrary partitions or to models with chaining of polytopes, such as Takagi-Sugeno (TS) models (Sel et al., 17 Jul 2025) and hybrid gene network models (Pasquini et al., 2019). In this framework, the system is approximated by a convex hull of LTI subsystems or regions: $\dot x = \sum_{j=1}^r w_j(x) A_j x,$ and the Lyapunov candidate adopts either a maximum, minimum, or region-wise assignment over $M$ regions/pieces: $V(x) = \max_{m=1,\dots,M} x^\top P_m x.$ LMIs are enforced blockwise, possibly with cross-region continuity constraints to guarantee global decrease.

The path-complete graph formalism (Angeli et al., 2016) systematizes such approaches by associating Lyapunov pieces to graph nodes, edge-labels to system modes, and requiring for each edge $(i,j,\sigma)$: $V_j(A_\sigma x) \leq V_i(x) \quad \forall x.$ Complete coverage of all switching sequences is encoded as path-completeness of the graph, and the main result shows that a nested min-max of the $V_i$ yields a common Lyapunov function.

A critical observation is that such flexible constructions drastically relax the conservatism of standard quadratic certificates; however, not every min-/max-of-quadratics CLF can be realized as a path-complete Lyapunov function with the minimal number of pieces (Angeli et al., 2016).

4. Hierarchies, Polynomial Lyapunov Lifting, and Computational Aspects

Higher-degree polynomial Lyapunov functions for switched and parameter-dependent systems can be generated via hierarchies of quadratic Lyapunov functions over lifted or stacked state spaces (Abdelraouf et al., 2024, Abate et al., 2019, Abate et al., 2020). In the "poly-quadratic" lifting setting, one forms variables $$\tilde\xi_i = [x, x^{\otimes 2}, \dots, x^{\otimes i}]$$ and block-diagonal $P$ to yield

$$W_i(x) = \sum_{k=1}^{i} (x^{\otimes k})^\top P_k (x^{\otimes k}).$$

This is a non-homogeneous polynomial of degree $2i$, recovering higher-order invariants from quadratic LMIs in larger spaces.

Such lifted hierarchies can yield tight outer approximations to reachable sets, pointwise explicit $L_\infty$ bounds, and facilitate convex optimization via standard SDP solvers. The methodology produces strictly improving certificates as the hierarchy level $i$ increases, trading off matrix size for reduction in conservatism (Abdelraouf et al., 2024, Abate et al., 2019, Abate et al., 2020).

5. Limitations, Lower Bounds, and Complexity Barriers

There exists no universal bound on the number of quadratic pieces (or regions) required in poly-quadratic Lyapunov functions to guarantee stability certification for all switched linear systems of a fixed dimension (Ahmadi et al., 2015). Constructive proofs elucidate families of stable two-matrix systems in $2 \times 2$ that do not admit any poly-quadratic Lyapunov function with fewer than $d$ pieces for arbitrary $d$, directly implying that the size of search problems (e.g., SDPs over $N$) must grow unboundedly as the systems approach marginal stability.

These impossibility results are linked to the non-semi-algebraicity of the absolute stability set for the joint spectral radius (JSR), the finiteness property, and undecidability phenomena. Consequently, computational searches for poly-quadratic certificates must be incremental in both the number of pieces and dimension, and can never offer a finite "master theorem" for guaranteed feasibility at bounded complexity (Ahmadi et al., 2015).

Furthermore, even for certain globally asymptotically stable polynomial or hybrid systems, global polynomial Lyapunov functions (including poly-quadratic) may fail to exist, and the degree or number of pieces required can be arbitrarily high (Ahmadi et al., 2013).

6. Applications and Algorithmic Implementations

Poly-quadratic Lyapunov functions and their variants underpin numerous modern algorithmic strategies:

• LPV controller/observer synthesis: via poly-quadratic LMI conditions, efficiently yielding parameter-dependent gains certifying robust stability and performance without rate constraints (Meijer et al., 1 Feb 2026).
• Region-of-attraction (ROA) estimation: for nonlinear systems using Takagi-Sugeno poly-quadratic Lyapunov forms, often with coordinate charts to maximize covered regions (Sel et al., 17 Jul 2025).
• Hybrid and switched system stability: through piecewise-quadratic and path-complete Lyapunov techniques. These approaches enable tractable convex optimization setups for verifying absolute asymptotic stability under arbitrary switching (Angeli et al., 2016, Pasquini et al., 2019).
• Data-driven/stochastic controller analysis: via analytic-center cutting-plane, mixed-integer, and robust SDP formulations leveraging the convexity properties of poly-quadratic LFs (Chen et al., 2020).
• Performance certification: especially for LTV and switching systems where pointwise-in-time guarantees and reachable set approximation require non-conservative certificates beyond classical quadratic methods (Abdelraouf et al., 2024, Abate et al., 2020).

Algorithmic primitives core to poly-quadratic Lyapunov analysis include construction of region partitions or parameter simplices, cross-vertex or cross-region LMI enforcement, and—where applicable—convexity preservation and invariance under system symmetries. The computational complexity is dominated by SDP dimensionality, scaling with the number of pieces and order of polynomial lifting.

7. Significance and Outlook

Poly-quadratic Lyapunov functions encapsulate a unifying framework at the intersection of LMIs, convex geometry, and algebraic systems theory for robust stability theory in non-autonomous, switched, hybrid, and nonlinear systems. Their use leads to reduced conservatism, direct interpretation in terms of physical system partitions or parameter sets, and practical tractability for controller and observer synthesis. Nonetheless, inherent complexity-theoretic limitations and the lack of finite generic bounds on piece number or degree motivate continued investigation of trade-offs between computational resources, certificate strength, and system structural features (Ahmadi et al., 2015, Ahmadi et al., 2013). The landscape continues to be shaped by advances in convex optimization, semialgebraic geometry, and graph-based Lyapunov abstractions.