Polyhedral Lyapunov Functions with Fixed Complexity

Abstract

Polyhedral Lyapunov functions can approximate any norm arbitrarily well. Because of this, they are used to study the stability of linear time varying and linear parameter varying systems without being conservative. However, the computational cost associated with using them grows unbounded as the size of their representation increases. Finding them is also a hard computational problem.Here we present an algorithm that attempts to find polyhedral functions while keeping the size of the representation fixed, to limit computational costs. We do this by measuring the gap from contraction for a given polyhedral set. The solution is then used to find perturbations on the polyhedral set that reduce the contraction gap. The process is repeated until a valid polyhedral Lyapunov function is obtained.The approach is rooted in linear programming. This leads to a flexible method capable of handling additional linear constraints and objectives, and enables the use of the algorithm for control synthesis. D. Kousoulidis is supported by the Engineering and Physical Sciences Research Council (EPSRC) of the United Kingdom. D. Kousoulidis and F. Forni are with the

Learning fixed-complexity polyhedral Lyapunov functions from counterexamples

“…i=1 and η. Bilinear optimization programs are in general NP-hard to solve [16]. An alternating descent procedure to solve a program similar to (3) is proposed in [13]. An important drawback of this approach is that the convergence to a feasible point is not guaranteed.…”

“…An important drawback of these methods is that they are restricted to discrete-time systems and often lack complexity bounds. Optimization-based methods [11]- [13] aim to solve the nonlinear, nonconvex optimization problem accounting for the existence of a polyhedral Lyapunov function. This is achieved, for instance, by considering convex relaxations of this problem [12], arbitrarily fixing some variables [11], or using an alternating descent procedure to solve locally the optimization problem [13].…”

“…Optimization-based methods [11]- [13] aim to solve the nonlinear, nonconvex optimization problem accounting for the existence of a polyhedral Lyapunov function. This is achieved, for instance, by considering convex relaxations of this problem [12], arbitrarily fixing some variables [11], or using an alternating descent procedure to solve locally the optimization problem [13]. These methods are not guaranteed to converge to a global optimum of the optimization problem, so that they may fail to find a polyhedral Lyapunov function for the system, even if one exists.…”

Polyhedral Estimation of L-1 and L-Infinity Incremental Gains of Nonlinear Systems

“…The optimization problems in (21) and ( 22) are adjoint. We mean this in the sense that each solution of (21) for system matrices {A, B, C} with variables (η w , η z , V, P, M) is also a solution of (22) for system matrices {A T , C T , B T } (the adjoint system) with variables (η z , η w , V T , P T , M T ) and vice versa.…”

“…When the polyhedral function/set is fixed the conditions are reduced to linear programming problems. This allows us to adapt our previous work on finding polyhedral Lyapunov functions of fixed complexity [21] to this problem for both analysis and synthesis. We provide definitions and the conditions at an abstract level in Section 2, specialize them to polyhedral functions and provide our novel conditions in Section 3, provide an overview of our adapted analysis and synthesis algorithms in Section 4, and apply the algorithms on some numerical examples in Section 5.…”