Local stability analysis for large polynomial spline systems

Abstract

Polynomial switching systems such as multivariate splines provide accurate fitting while retaining an algebraic representation and offering arbitrary degrees of smoothness; yet, application of sum-of-squares techniques for local stability analysis is computationally demanding for a large number of subdomains. This communiqué presents an algorithm for region of attraction estimation that is confined to those subdomains actually covered by the estimate, thereby significantly reducing computation time. Correctness of the results is subsequently proven and the run time is approximated in terms of the number of total and covered subdomains. Application to longitudinal aircraft motion concludes the study.

Sum-of-Squares Flight Control Synthesis for Deep-Stall Recovery

“…(10)- (11). In [32], we proposed an efficient algorithm for multi-variate splines that detects the active boundaries in line 7 of Algorithm 1 and thus reduces the total number of decision variables. On the other hand, the number of constraints in line 4 of the same algorithm, that is, the number of conditions (12)-(13), increases with the number of active boundaries, too.…”

Sequential sum-of-squares programming for analysis of nonlinear systems⋆

“…Despite nonlinear sum-of-squares problems being computationally hard, local analysis of stability and other properties of polynomial dynamics with sum-of-squares programming has been extensively studied [18][19][20][21][22][23][24][25][26][27][28]. Here, the (mostly bilinear) nonconvex constraints have been mitigated by bisections [29], coordinate descent [30], and combinations of both.…”

Nonlinear Analysis for Wing-Rock System with Adaptive Control

“…However, it is hard to determine the exact ROA if not impossible [7][8][9][10]. Various kinds of methods dedicated to estimating ROA, which are roughly divided into analytical and computational methods [11,12]; or into Lyapunovbased methods and non-Lyapunov-based methods [13] regarding the theory behind the methods.…”

“…SOS polynomial optimization has been used previously to analyze the performance and robustness of systems described by polynomial dynamics. Computational algorithms have been developed not only for estimating ROA but also reachability sets, input-output gains, robustness with respect to uncertainty, and time-delay margin [10,16]. Applying the SOS technique for ROA estimation, this paper aims to show the ability of the techniques to analyze the stability and robustness verification for a nonlinear adaptive control system.…”