A Path-Complete Approach for Optimal Control of Switched Systems

TL;DR

This work develops a path-complete framework to bound the value function of discrete-time switched systems under arbitrary switching. By encoding dynamic programming inequalities on directed graphs and combining node-specific quadratic templates, it yields convex, tractable upper bounds via LMIs, with extensions to autonomous and affine-control switched linear systems. The authors prove convergence of bounds on dual De Bruijn graphs as the order grows and provide relative-accuracy guarantees that bound the true value function within a computable factor. They validate the approach numerically, showing tightening bounds and non-conservativeness in the limit, and demonstrate how the method supports controller synthesis through structured policies. Overall, the framework offers scalable, provable bounds for safety- and performance-critical switching scenarios with practical applicability to control design.

Abstract

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.

Figures (4)

• Figure 1: (a) Path-complete graph with two nodes, for a system with two switching modes. (b) Graph not path-complete.
• Figure 2: Upper bound $V(x) = \max_{\alpha\in\{1,2\}} x^\top P_\alpha x$ on the value function of the switched linear system with matrices in \ref{['eq:matrices_simple_ex']}, plotted along the unit circle $x=(\cos(\theta), \sin(\theta))$ for $\theta\in[0,\pi]$. The bound is computed using the path-complete framework with the co-complete graph in \ref{['fig:pc-graph']}, and the true value function $J$ is shown for comparison.
• Figure 3: Upper and lower bounds on the value function, $V^{l}(x)$ and $W^{l}(x)\coloneqq \frac{1}{\mu} V^l(x)$, for $x=(\cos(\theta),\,\sin(\theta))$ with $\theta\!\in\![0,\pi]$, for the autonomous switched linear system with matrices given in \ref{['eq:matrices_simple_ex']}, and using dual De Bruijn graphs of order $l\!\in\!\{1,2,3\}$. The gap between the upper and lower bound curves decreases as the order increases.
• Figure 4: Upper bound $V^{l}(x)$ on the value function, for $x = (\cos(\theta), \sin(\theta))$ with $\theta \in [0,\pi]$, computed for the controlled switched linear system defined by the matrices in \ref{['eq:ex_2D_controlled']}, using De Bruijn graphs of order $l \in \{1,2,\ldots,5\}$. The upper bound decreases as the order increases, with a more pronounced effect for some points.

Theorems & Definitions (21)

• Definition 1: Cost-to-go and value function
• Definition 2: Path-complete graph
• Definition 3: Complete and co-complete graphs
• Proposition 1
• Proposition 2: Upper bound
• Proposition 3: Lower bound
• Example 1
• Theorem 1
• Remark 1
• Remark 2