Time-Dependent LMI Conditions

• Time-dependent LMIs are matrix inequalities parameterized by time, extending classical static LMIs to certify the stability and performance of systems with temporal variations.
• They leverage convex optimization techniques—such as SOS programming and orthogonal basis projections—to approximate infinite-dimensional functionals and enable controller synthesis.
• Applications span delay systems, switched/hybrid systems, and LPV models, where these methods ensure robust performance and constraint satisfaction under time-varying dynamics.

Time-dependent linear matrix inequality (LMI) conditions are a central analytical and synthesis tool in modern control theory for dynamical systems whose parameters, structure, or stability requirements explicitly depend on time. These conditions are formulated as matrix inequalities parameterized—either explicitly or implicitly—by time or related variables such as scheduling parameters, switching intervals, or structured uncertainties. Time-dependent LMI conditions provide a convex or semidefinite-programming framework to certify or synthesize stability, performance, or invariance in systems that are time-varying, subject to switching, delays, or other temporal constraints. Their development has enabled rigorous treatment of continuous-time, discrete-time, and hybrid systems beyond the limitations of classical time-invariant, quadratic Lyapunov theory.

1. Foundational Concepts and Mathematical Structure

Time-dependent LMIs generalize the classical matrix inequality paradigm from constant (or static) decision variables to parameterized and, in many cases, infinite-dimensional settings.

The canonical template is the enforcement of a quadratic dissipation inequality via a Lyapunov or storage functional $V(t,x)$, whose derivative along trajectories satisfies

$$\dot V(t, x(t)) + \text{cross/coupling terms} + \text{uncertainty terms} \le 0$$

for all admissible time $t$ or parameter schedules. Instead of a constant matrix $P$, time-dependent LMIs introduce decision variables that are functions of time, such as $P(t)$, matrix-valued polynomial kernels, or nonlinear functionals.

Key mathematical forms appearing in the literature include:

• Piecewise or polynomially parameterized $P(t)$ for LTV or LPV systems (Ahmed et al., 2023, Mozelli et al., 2018)
• Occupation/measures-based (SOS polynomial) parametrization for time-dependent or time-indexed functionals (Briat et al., 2012, Peet, 2016, Aangenent et al., 2011)
• Hierarchical basis projection (Legendre, Bessel-Legendre, or other orthogonal bases) for infinite-dimensional Lyapunov-Krasovskii operators (Bajodek et al., 2022)

The structure of the LMI is typically dictated by the requirement that the dissipation inequality holds uniformly for all possible values of the time (or scheduling) argument in a domain, leading to "infinite-dimensional" or "differential" LMIs, which often require additional convexification (e.g., via the S-procedure, Positivstellensatz with SOS multipliers, or copositivity relaxations in funnel synthesis).

2. Principal Classes and Parametric Contexts

Time-dependent LMI conditions are applied to a broad span of system classes, including:

(a) Linear Time-Delay and Delay-Differential Systems

Sufficient and, under certain conditions, necessary LMI conditions for delay systems are constructed by projecting infinite-dimensional Lyapunov-Krasovskii functionals onto finite-dimensional polynomial bases, often using the Legendre expansion. For example, in single-delay LTI systems $ẋ(t) = A x(t) + A_d x(t-h)$, feasible LMIs at increasing Legendre order $n$ yield a hierarchy of sufficient (and, for $n\to\infty$, necessary) stability certificates (Bajodek et al., 2022). The key functional forms involve projections $\zeta_n(z)$ via Legendre polynomials and exploit integral inequalities such as the Bessel-Legendre bound.

(b) Switched and Hybrid Systems with Dwell-Time and Switching Constraints

Affine, time-dependent LMI conditions provide looped-functional characterizations of minimum and mode-dependent dwell-times without recourse to nonconvex matrix exponentials. These are formulated as infinite-dimensional LMIs on differentiable matrix-valued functions $Z_{ij}(t)$ subject to differential and boundary constraints, parameterized polynomially and solved via SOS programming (Briat et al., 2012). For discrete-time switched systems, convex "lifted" time-dependent LMIs replace nonconvex matrix powers by sequences of quadratic forms, indexed by elapsed time since the last switch, yielding tractable and robust stability certificates for prescribed minimum dwell-time (Briat, 2013).

(c) Time-Varying Systems, LPV, and Markovian Jump Systems

LPV systems with parameter trajectory and rate bounds require enforcing negativity of a Lyapunov derivative over all admissible parameter and rate trajectories, leading to LMI constraints indexed over the vertices of a polytopic representation of the rate-polytope—often requiring combinatorial computation, with linear-complexity relaxations available via simplex approximation (Mozelli et al., 2018). For LTV models, piecewise-linear time dependence permits parametrization of $P(t)$ on a time grid, with interpolation-induced LMIs ensuring exponential decay or robust stability over bounded gain uncertainty (Ahmed et al., 2023). In periodically time-varying Markov-jump systems, time-dependent Lyapunov functions $P_i(k)$ indexed by both mode and time step lead to coupled LMIs enforcing mean-square stability, quadratic cost minimization, or region of attraction maximization (Shrivastava et al., 17 Sep 2024).

(d) Funnel and Controlled Invariant Set Synthesis

Time-dependent LMI conditions for funnel synthesis impose continuous-time differential LMI (DLMI) constraints ensuring the invariance of a set defined by the sublevel of a time-varying quadratic form $V(t,\eta) = \eta^\top Q(t)^{-1}\eta$. These DLMIs are reduced to a finite set of copositivity-based static LMIs via affine parametrization and convex (slack-variable or diagonal-dominance) tests on each time interval (Kim et al., 23 Feb 2024).

(e) Control Synthesis under Temporal Constraints

The enforcement of time-domain constraints on closed-loop system signals translates into enforcing polynomial nonnegativity on compact time intervals via SOS LMI relaxations. This allows simultaneous satisfaction of performance specifications such as overshoot, settling time, and steady-state accuracy, while maintaining tractable LMI feasibility problems (Aangenent et al., 2011).

3. Infinite-Dimensionality, Convexification, and Hierarchical Structure

A critical technical challenge in time-dependent LMI methods is the convexification of inherently infinite- or high-dimensional dissipation constraints. Several frameworks address this:

• SOS polynomial parameterizations: Matrix-valued functionals ($Z_{ij}(t), P(t)$) are restricted to low-degree polynomial forms; positivity on $[0,T]$ is enforced by the existence of SOS multipliers, which, via Putinar's theorem, guarantee non-negativity on a compact domain (Briat et al., 2012, Peet, 2016, Aangenent et al., 2011).
• Projection onto orthogonal bases: Approximating infinite-dimensional functionals with finite bases (e.g., Legendre polynomials), admissible error is bounded via integral inequalities, and an explicit hierarchy or convergence result is obtained (Bajodek et al., 2022).
• Lifted/Indexed quadratic forms: Variables such as $R_i(k)$ in discrete-time switched systems or $P_k$ on time intervals in LTV permit reduction of nonconvex (or infinite) conditions to a finite set of convex LMIs, sometimes with a monotonic decrease in conservatism as dimension increases (Briat, 2013, Ahmed et al., 2023).

Convergence and necessity are key outstanding questions in infinite-dimensional schemes. For instance, (Bajodek et al., 2022) proves that feasibility of the sequence of finite-dimensional LMIs is both necessary and sufficient for global exponential stability, provided the Legendre order $n$ exceeds an analytically computable $N^*$.

4. Computational and Complexity Considerations

The transition from infinite- or high-dimensional parameterizations to finite LMIs is central for practical application:

• The number of LMIs in LPV settings with rate constraints grows combinatorially in the number of scheduling variables ("vertex explosion"). Approximate convex hulls or simplex-based outer approximations reduce this burden substantially, trading computational efficiency against mild conservatism (Mozelli et al., 2018).
• The required basis expansion order (e.g., Legendre order $N^*$) can be sizeable, with analytic upper bounds provided but moderate orders (typically $n\leq5$) sufficient in most practice (Bajodek et al., 2022).
• Copositivity-based relaxations for time-dependent DLMIs allow trading conservatism against computational overhead via choice of diagonal- or slack-based LMI tests (Kim et al., 23 Feb 2024).
• Modern SDP solvers (MOSEK, SeDuMi, Clarabel) can handle problem sizes arising in piecewise-LTV and periodic MJLS settings for $T\leq50$, $N\leq10$ in near real-time (Shrivastava et al., 17 Sep 2024, Ahmed et al., 2023).

5. Robustness, Uncertainty, and Mode-Dependent Extensions

Time-dependent LMI frameworks are naturally extensible to uncertain, robust, and hybrid scenarios:

• Uncertain Plant Matrices: Convexification of dependence on matrix uncertainty (e.g., polytopic or full-block S-procedure using Petersen's lemma) allows extension to robust dwell-time and delay systems (Briat et al., 2012, Briat, 2013).
• Time-Varying/Uncertain Delays: Sufficient LMI conditions for robustness of predictor-based feedback under time-varying input delays are formulated via Lyapunov-Krasovskii functionals whose cross/derivative terms depend explicitly on the delay interval and uncertainty bound; feasible LMIs directly quantify admissible uncertainty ranges (Lhachemi et al., 2019).
• Mode-Dependent and Piecewise Conditions: Dwell-time and Lyapunov functionals parameterized by mode or switching interval length treated via polynomial ansatz with respect to both time and interval length permit simultaneous characterization of mode-dependent switching regimes (Briat et al., 2012).

6. Applications, Performance Guarantees, and Synthesis Perspectives

Time-dependent LMIs underpin a variety of synthesis problems and practical applications:

• Stability Certification: Hierarchical or infinite-dimensional LMI family feasibility exactly characterizes exponential stability for classes such as single-delay systems and piecewise-linear LTV models (Bajodek et al., 2022, Ahmed et al., 2023).
• Controller Synthesis: The LMI machinery provides direct parameterizations for state-feedback laws, optimal quadratic cost minimization, region of attraction maximization, and robust dwell-time and switching constraint handling in both deterministic and Markovian settings (Briat, 2013, Shrivastava et al., 17 Sep 2024).
• Funnel and Invariant Set Synthesis: Time-dependent copositivity conditions directly yield invariant funnels for nonlinear or LPV disparate systems over finite horizons, as in trajectory planning or safety verification (Kim et al., 23 Feb 2024).
• Performance and Constraint Satisfaction: Time-domain LMI design enables simultaneous satisfaction of transient response specifications, hard state/input bounds, and set-invariance constraints (Aangenent et al., 2011).

In these applications, the trade-off between conservatism and computational tractability is managed via the choice of expansion basis, the complexity of parameterization (SOS degree, time grid size), and the convexification technique. Direct numerical experiments, as reported, demonstrate the practical efficacy of the approach, with feasible solutions typically achieved at moderate computational cost if the optimization problem is constructed with care.

7. Future Directions and Open Challenges

Ongoing research in time-dependent LMI conditions focuses on reducing the computational burden in large-scale or high-dimensional settings, improving the expressiveness of function parameterizations (e.g., higher-order, mesh-adaptive bases, non-polynomial structures), and establishing sharp necessity/sufficiency theorems for broader classes of hybrid and nonlinear systems. Robustness to structured nonlinearities, scalability for multi-rate or multi-mode systems, and integration with modern sample-based and data-driven control approaches are active topics. The convergence and tightness of hierarchical LMI hierarchies, especially in relation to control synthesis rather than mere analysis, remain central technical questions (Bajodek et al., 2022). The extension to distributed, networked, or infinite-dimensional settings is another major theme, particularly as computational solvers and symbolic algebraic tools mature to address the attendant complexities.