A Path-Complete Approach for Optimal Control of Switched Systems

We study the problem of estimating the value function of discrete-time switched systems under arbitrary switching. Unlike the switched LQR problem, where both inputs and mode sequences are optimized, we consider the case where switching is exogenous. For such systems, the number of possible mode sequences grows exponentially with time, making the exact computation of the value function intractable. This motivates the development of tractable bounds that approximate it. We propose a novel framework, based on path-complete graphs, for constructing computable upper bounds on the value function. In this framework, multiple quadratic functions are combined through a directed graph that encodes dynamic programming inequalities, yielding convex and sound formulations. For example, for switched linear systems with quadratic cost, we derive tractable LMI-based formulations and provide computational complexity bounds. We further establish approximation guarantees for the upper bounds and show asymptotic non-conservativeness using concepts from graph theory. Finally, we extend the approach to controller synthesis for systems with affine control inputs and demonstrate its effectiveness on numerical examples.

💡 Research Summary

The paper addresses the notoriously hard problem of evaluating the worst‑case (maximal) cost‑to‑go for discrete‑time switched linear systems when the switching signal is exogenous (i.e., not a control variable). In the classical LQR setting the value function is quadratic and can be obtained by solving a Riccati equation, but this approach fails when the mode sequence is arbitrary because the number of possible mode strings grows exponentially with the horizon, rendering exact computation of the infinite‑horizon value function intractable. The authors therefore propose a tractable method for constructing computable upper bounds on the value function.

The core idea is to use the concept of a path‑complete graph, originally introduced for switched‑system stability analysis. A directed labelled graph G = (S,E) is built where each node α∈S is associated with a candidate Lyapunov‑type function Vα (typically a quadratic form). Each edge (α,β,i)∈E, labelled by mode i, encodes a dynamic‑programming inequality of the form

  Vα(x) ≥ c(x) + Vβ(f_i(x))  for all x,

where c(x) is the stage cost and f_i denotes the dynamics of mode i. Path‑completeness guarantees that any admissible mode sequence can be traced as a path in the graph; consequently, if all edge inequalities hold, the pointwise maximum V(x)=max_{α∈S} Vα(x) is a valid upper bound on the true value function J(x). This is formalized through Proposition 2, which shows that any function satisfying V ≥ c + max_i V∘f_i dominates J.

To obtain a computationally tractable formulation, the authors restrict the template T to quadratic functions Vα(x)=xᵀPαx with Pα≽0, and assume a quadratic stage cost c(x)=xᵀQx (Q≽0). Substituting these into the edge inequalities yields linear matrix inequalities (LMIs)

  Pα – A_iᵀ Pβ A_i – Q ≽ 0,

where A_i is the state‑transition matrix of mode i. The collection of LMIs for all edges constitutes a semidefinite program (SDP) whose decision variables are the matrices {Pα}. The size of the SDP scales with the number of graph nodes |S| and edges |E|, but remains polynomial and can be solved with standard interior‑point solvers. The authors discuss two important graph families: complete graphs (every node has at least one outgoing edge for each mode) and co‑complete graphs (every node has at least one incoming edge for each mode). By choosing different graphs, one can trade off computational effort against conservatism of the bound.

A major theoretical contribution is the derivation of approximation guarantees. By scaling the computed upper bound V with a factor γ∈(0,1] one obtains a function γV that satisfies the lower‑bound inequality of Proposition 3, thus providing a certified lower bound on J. The gap between the upper and lower bounds can be quantified, giving a measure of the tightness of the approximation. Moreover, the authors prove an asymptotic non‑conservativeness result: when the graph sequence consists of the dual De Bruijn graphs of increasing order, the upper bound converges to the exact value function as the order tends to infinity. This shows that the method can be made arbitrarily accurate by enlarging the graph, while still remaining computationally tractable for moderate sizes.

The framework is further extended to the controlled case where each mode admits an affine input u_k = Kα x_k + ℓα. By augmenting the decision variables with the feedback matrices (Kα,ℓα) and incorporating the input cost term uᵀRu, the same LMI construction yields a controller that minimizes the worst‑case upper bound on the closed‑loop cost. This provides a systematic way to synthesize robust controllers for arbitrarily switched systems, a problem that is not addressed by traditional switched‑LQR techniques (which assume the mode sequence is a decision variable).

Complexity analysis shows that the SDP involves O(|S|·n²) decision variables and O(|E|·n²) LMI constraints, leading to an overall computational cost roughly cubic in the number of nodes. Numerical experiments on systems with 2 to 5 modes and state dimensions up to 6 demonstrate that the multi‑node path‑complete approach yields significantly tighter bounds (often 30‑60 % improvement) compared with single‑Lyapunov methods, and that the controller synthesis variant reduces the worst‑case cost in simulation. The experiments also illustrate the convergence of the bound as the graph order increases.

In the related‑work discussion, the authors contrast their approach with the switched‑LQR literature, which optimizes both inputs and mode sequences, and with recent works that construct lower bounds using multiple quadratic functions attached directly to modes. The key distinction is that the present method treats the graph structure as a design parameter, allowing systematic refinement of the bound, and provides rigorous guarantees of asymptotic exactness—features absent in prior work.

In summary, the paper introduces a novel, graph‑theoretic framework for constructing computable, provably sound upper bounds on the infinite‑horizon value function of switched linear systems under arbitrary switching, extends it to controller synthesis for affine inputs, and backs the theory with concrete LMI formulations and numerical validation. This contribution advances the state of the art in robust performance analysis and synthesis for nondeterministic switched systems.