Robust Stability Analysis of Nonlinear Hybrid Systems

Abstract

as (19), and compute the standard deviation of the values of the corresponding perturbed polynomial at the point ( 3 1 ; 3 2 ). We perform the test at ( (0:6; 0:6) and compare the results of both approaches in Table IV.There are several reasons showing that our approach is promising. First, we can see from Tables I and III that the number of variables in the conventional approach grows rapidly when the degrees of the monomial bases increase, while the number of variables in our approach grows linearly with respect to the number of subregions. Moreover, our approach attains the exact optimal value with less computational cost than the conventional SOS approach. Secondly, Table IV shows that the standard deviations of the perturbed polynomial from the proposed approach are less than that from the conventional approach, for all selected points. This implies that the optimal value of the proposed approach is less numerically sensitive than that of the conventional approach. Strictly speaking, the asymptotic exactness of our scheme is not guaranteed in the case of lowest-degree monomial bases. However, it is achieved apparently in this example. We expect that the asymptotic exactness can also be proved with the monomial bases of lowest degrees, and this is the direction of our further research. REFERENCES[1] A. Ben-Tal and A. Nemirovski, "Robust convex optimization," Math.Oper. Res., vol. 23, no. 4, pp. 769-805, 1998 Robust Stability Analysis of Nonlinear Hybrid Systems Antonis Papachristodoulou and Stephen PrajnaAbstract-We present a methodology for robust stability analysis of nonlinear hybrid systems, through the algorithmic construction of polynomial and piecewise polynomial Lyapunov-like functions using convex optimization and in particular the sum of squares decomposition of multivariate polynomials. Several improvements compared to previous approaches are discussed, such as treating in a unified way polynomial switching surfaces and robust stability analysis for nonlinear hybrid systems.Index Terms-Hybrid systems, linear matrix inequality, sum of squares, switched systems.

Stability analysis of networked control systems: A sum of squares approach

“…We will use the above notation to expand Theorem 1 in the spirit of [23] by applying a technique similar to the Sprocedure, called the positivstellensatz [15], [24], in order to encode the information that the inequalities (11) and (12) only have to be satisfied on the sets C\A and D\A. 17) then the set A is GAS.…”

“…These models will be converted into a hybrid systems formulations as in [12]. Assuming piecewise polynomial plant dynamics (including piecewise affine systems) and a piecewise polynomial controller Lyapunov functions can be constructed using sum of squares (SOS) tools [15], [23], [24]. As a result, this will lead to LMIbased tests for stability given bounds on the delays and transmission intervals.…”

Balancing and Step Recovery Capturability via Sums-of-Squares Optimization

“…See [27] and [18] for an overview of SOS programming. Recent applications have included verification and control design of robotic systems, [22,34], hybrid systems [25], and of mechanical systems undergoing contact [28].…”

“…Here, we present a unified approach to capturability computations, applicable to a significantly broader class of models and controllers. To enable computation for more complex models, we base our approach on recent developments in sums-of-squares (SOS) programming [27,18], which has been effectively used to compute control policies and regions of attraction [34,25,35,22,28] and to approximate reachable sets [7].…”

A Sums-of-Squares extension of policy iterations

“…Their method relies on a so-called iterative advection algorithm based on Sums-of-Squares (SOS) and semidefinite programming (SDP) to compute either inner of outer approximation of the backward reachable set, also named domain of attraction (DoA). The stability analysis of continuous-time hybrid systems with SOS certificates was investigated in [PP09]. [PTT16] have recently applied analogous techniques to perform stability analysis and controller synthesis in the context of robotics.…”