Discovering Multiple Lyapunov Functions for Switched Hybrid Systems

et al. 2018

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Summary Domain of attraction plays an important role in stability analysis and safety verification of nonlinear control systems. In this paper, based on the concept of multiple Lyapunov‐like functions, we propose iteration algorithms for computing inner estimates of domains of attraction for a class of switched hybrid systems, where the state space is composed of several regions and each region is described by polyhedral sets. Starting with an initial inner estimate of domain of attraction, we firstly present a theoretical framework for obtaining a larger inner estimate by iteratively computing multiple Lyapunov‐like functions. Successively, the theoretical framework is underapproximatively realized by using S‐procedure and sums of squares programming, associated with the coordinatewise iteration method. Afterwards, for obtaining a required initial inner estimate of domain of attraction, we propose an alternative higher‐order truncation and linear semidefinite programming based method for computing a common Lyapunov function. Especially, a bisection method based improvement is proposed for obtaining better estimates in each iteration step. Finally, we implement proposed algorithms and test them on numerical examples with comparisons. These computation and comparison results show that the advantages of our multiple Lyapunov‐like functions based algorithm. Especially, we provide alternative underapproximations for avoiding the possible numerical problem.

“…In this paper, we study a class of switched hybrid systems of the form

\overset{x}{true} false(t false) = f_{i} false(boldx false), i \in normalΓ = false{1, 2, ⋯, N false}, boldx \in D_{i} \subset R^{n},

where

f_{i} false(boldx false) : R^{n} \mapsto R^{n}

is a polynomial vector field describing the dynamics of mode i and satisfying f i ( 0 ) = 0 ,

D_{i}

is a union of polyhedral sets defined as the form

false{boldx \in R^{n} : E_{i, m} boldx \geq 0, E_{i, m} is an n \times n matrix false}

such that

{scriptD}_{i} = \cup_{m = 1}^{l_{i}} {x \in {double-struckR}^{n} : E_{i, m} x \geq 0, E_{i, m} is an n \times n matrix} and \cup_{i \in Γ} {scriptD}_{i} = {double-struckR}^{n} .

Note that

int false(D_{i} false) \cap int false(D_{j} false)

Section: Preliminariesmentioning

confidence: 99%

Computing multiple Lyapunov‐like functions for inner estimates of domains of attraction of switched hybrid systems

Zheng

et al. 2018

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“…This is similar to finding Lyapunov functions for nonlinear dynamical systems, which is usually a difficult problem. However, for polynomial dynamical systems, SOS technique provide a powerful way to search feasible SOS polynomial Lyapunov functions . On account of this, we will restrict ourselves to network with polynomial vector field, and attempt to discover an SOS polynomial V .…”

Section: Search For Feasible Polynomial V For Polynomial F and H By Smentioning

confidence: 99%

Sum‐of‐squares–based consensus verification for directed networks with nonlinear protocols

Liang

Ong

2019

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Summary In this paper, we investigate the consensus verification problem of nonlinear agents in a fixed directed network with a nonlinear protocol. Inspired by the classical Lipschitz‐like condition, we introduce a more relax condition for the dynamics of the nonlinear agents. This condition is motivated via the construction of general Lyapunov functions for achieving asymptotic consensus. Especially, for agents where dynamics are described by polynomial function of the states, our consensus criterion can be converted to a sum of squares (SOS) programming problem, solvable via semidefinite programming tools. Of interest is that the scale of the resulting SOS programming problem does not increase as the size of the network increases, and thus, the applicability to analyze consensus of large‐scale networks is promising. Finally, an example is given to illustrate the effectiveness of our theoretical results.

“…On the other hand, the semialgebraic set

\cup_{L_{1}, L_{2}} Δ_{1, L_{1}, L_{2}}

can be automatically solved to discover a sample point as a PID controller parameter via numerical or symbolic methods. And how to use the adaptive cylindrical algebraic decomposition (CAD) technique to efficiently solve the semialgebraic set will be our future work.…”

Section: Pid Controlled Time‐varying Switched Systemsmentioning

confidence: 99%

“…Switched systems consist of several subsystems along with a switching rule that orchestrates the switching among these subsystems . Recently, numerous researchers have focused on switched systems since they can appropriately model many complex systems such as constrained robotics, chemical processes, manufacturing systems, multiagent systems, aircraft control, automotive engine control, and so on …”

Section: Introductionmentioning

confidence: 99%

Design proportional‐integral‐derivative/proportional‐derivative controls for second‐order time‐varying switched nonlinear systems

Zhang

et al. 2019

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Summary Based on proportional‐integral‐derivative (PID)/PD controls, we in the article investigate the tracking problem of a class of second‐order time‐varying switched nonlinear systems. To start with, for tracking a given point under arbitrary switching signals, we propose a sufficient condition about PID controller parameters, which can be implicitly described as semialgebraic sets. Successively, we consider the tracking problem under average dwell time (ADT)‐based switching signals and propose an alternative sufficient condition about PID controller parameters. Especially, for tracking an equilibrium point of the system without controls, we can further simply utilize the proportional‐derivative control and similarly construct corresponding semialgebraic conditions about proportional‐derivative controller parameters under arbitrary switching signals and ADT‐based switching signals. Finally, two examples are given to show the applicability of our theoretical results.