Analysis of the joint spectral radius via lyapunov functions on path-complete graphs

Ahmadi, Amir Ali; Jungers, Raphaël M.; Parrilo, Pablo A.; Roozbehani, Mardavij

doi:10.1145/1967701.1967706

Cited by 24 publications

(20 citation statements)

References 29 publications

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“…In this direction, we have developed two frameworks: path-complete graph Lyapunov functions (with Jungers and Roozbehani) [39], [40] and non-monotonic Lyapunov functions [22], [41]. The first approach is based on the idea of using multiple Lyapunov functions instead of one and brings in concepts from automata theory to establish how Lyapunov inequalities should be written among multiple Lyapunov functions.…”

Section: Computational Advances In Lyapunov Theory [Areas 1and2]mentioning

confidence: 99%

“…In [39], [40], we give SDP-based approximation algorithms for the JSR by applying Lyapunov analysis techniques to system (8). We show that considerable improvements in scalability are possible (especially for high dimensional systems) if instead of a common Lyapunov function of high degree for the set A, we use multiple Lyapunov functions of low degree (quadratic ones).…”

Section: A Lyapunov Inequalities and Transitions In Finite Automata mentioning

confidence: 99%

“…Definition 3: (Ahmadi, Jungers, Parrilo, Roozbehani [39]) Given a directed graph G(N, E) whose edges are labeled with words (matrices) from the set A, we say that the graph is path-complete, if for all finite words A σ k . .…”

Section: A Lyapunov Inequalities and Transitions In Finite Automata mentioning

confidence: 99%

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Towards scalable algorithms with formal guarantees for Lyapunov analysis of control systems via algebraic optimization

Ahmadi

Parrilo²

2014

53rd IEEE Conference on Decision and Control

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Abstract-Exciting recent developments at the interface of optimization and control have shown that several fundamental problems in dynamics and control, such as stability, collision avoidance, robust performance, and controller synthesis can be addressed by a synergy of classical tools from Lyapunov theory and modern computational techniques from algebraic optimization. In this paper, we give a brief overview of our recent research efforts (with various coauthors) to (i) enhance the scalability of the algorithms in this field, and (ii) understand their worst case performance guarantees as well as fundamental limitations. Our results are tersely surveyed and challenges/opportunities that lie ahead are stated. I. ALGEBRAIC METHODS IN OPTIMIZATION AND CONTROLIn recent years, a fundamental and exciting interplay between convex optimization and algorithmic algebra has allowed for the solution or approximation of a large class of nonlinear and nonconvex problems in optimization and control once thought to be out of reach. The success of this area stems from two facts: (i) Numerous fundamental problems in optimization and control (among several other disciplines in applied and computational mathematics) are semialgebraic; i.e., they involve optimization over sets defined by a finite number of possibly quantified polynomial inequalities. (ii) Semialgebraic problems can be reformulated as optimization problems over the set of nonnegative polynomials. This makes them amenable to a rich set of algebraic tools which lend themselves to semidefinite programming-a subclass of convex optimization problems for which global solution methods are available.Application areas within optimization and computational mathematics that have been impacted by advances in algebraic techniques are numerous: approximation algorithms for NP-hard combinatorial problems from classical linear control to a principled framework for design of nonlinear (polynomial) controllers that are provably safer, more agile, and more robust. As a concrete example, Figure 1 demonstrates our recent work with Majumdar and Tedrake [14] in this area applied to the field of robotics. As the caption explains, sos techniques provide contollers with much larger margins of safety along planned trajectories and can directly reason about the nonlinear dynamics of the system under consideration. These are crucial assets for more challenging robotic tasks such as walking, running, and flying. Sum of squares methods have also recently made their way to actual industry flight control problems, e.g., to explain the falling leaf mode phenomenon of the F/A-18 Hornet aircraft [15], [16] or to design controllers for hypersonic aircraft [17]. The "swing-up and balance" task via sum of squares optimization for an underactuated and severely torque limited double pendulum (the "Acrobot"). Top: projections of basins of attraction around a nominal swing-up trajectory designed by linear quadratic regulator (LQR) techniques (blue) and by SOS techniques (red). Bottom: projections of basins of...

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Section: Computational Advances In Lyapunov Theory [Areas 1and2]mentioning

confidence: 99%

Section: A Lyapunov Inequalities and Transitions In Finite Automata mentioning

confidence: 99%

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Towards scalable algorithms with formal guarantees for Lyapunov analysis of control systems via algebraic optimization

Ahmadi

Parrilo²

2014

53rd IEEE Conference on Decision and Control

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“…This problem is known to be extremely hard for general sets of matrices; see [30] and the references therein for NP-hardness and undecidability results. Nevertheless, several algorithms have been proposed, which are efficient in many practical cases (see, for instance, [1,12,26]). …”

mentioning

confidence: 99%

“…In the case p = ∞ this idea was put to good use in several works, where the extremal norm was approximated by polyhedral functions, ellipsoidal norms, sums of squares polynomials, etc. [1,4,5,12,21,23]. Recently, conic norms have been used for this purpose [26].…”

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confidence: 99%

Fast Methods for Computing thep-Radius of Matrices

Jungers¹,

Protasov²

2011

SIAM J. Sci. Comput.

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Abstract. The p-radius characterizes the average rate of growth of norms of matrices in a multiplicative semigroup. This quantity has found several applications in recent years. We raise the question of its computability. We prove that the complexity of its approximation increases exponentially with p. We then describe a series of approximations that converge to the p-radius with a priori computable accuracy. For nonnegative matrices, this gives efficient approximation schemes for the p-radius computation.

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Graph‐Based Dwell Time Computation Methods for Discrete‐Time Switched Linear Systems

Ilhan

Karabacak

2016

Asian Journal of Control

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Analytical computation methods are proposed for evaluating the minimum dwell time and the average dwell time guaranteeing the asymptotic stability of a discrete-time switched linear system whose switchings are assumed to respect a given directed graph. The minimum and average dwell time can be found using the graph that governs the switchings, and the associated weights. This approach, which is used in a previous work for continuous-time systems having non-defective subsystems, has been adapted to discrete-time switched systems and generalized to allow defective subsystems. Moreover, we present a novel method to improve the dwell time estimation in the case of bimodal switched systems. In this method, scaling algorithms to minimize the condition number have been used to give better minimum dwell time and average dwell time estimates.

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Analysis of the joint spectral radius via lyapunov functions on path-complete graphs

Cited by 24 publications

References 29 publications

Towards scalable algorithms with formal guarantees for Lyapunov analysis of control systems via algebraic optimization

Towards scalable algorithms with formal guarantees for Lyapunov analysis of control systems via algebraic optimization

Fast Methods for Computing thep-Radius of Matrices

Graph‐Based Dwell Time Computation Methods for Discrete‐Time Switched Linear Systems

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