Obtaining approximate region of asymptotic stability by computer algebra: A case study

Abstract

One of the classical problems in nonlinear control system analysis and design is to find a region of asymptotic stability by the Direct Method of Lyapunov. This paper tentatively shows, via a numercial example, that this problem can be easily solved using Quantifier Elimination (QE). In particular, if the governing equations are described by differential equations containing only polynomials, then the problem can be conveniently solved by a computer algebra software packages such as Qepcad or Redlog. In our case study, we use a simple Lyapunov function and Qepcad to estimate the stability region, and the results are verified by an optimization method based on Lagrange's method.

Domain-of-attraction estimation for uncertain non-polynomial systems

“…So researchers are mainly concerned with computing underestimates of the DOAs. Many well-established techniques ( [6,8,7,11,16,19,22,20,10,21,15]) are available for computing estimates of DOAs for polynomial (control) systems, i.e., autonomous systems with polynomial vector fields. However, in practice, many autonomous systems often contain non-polynomial terms in their vector fields.…”

“…Therefore, looking for underestimates of the DOA with simple shapes has been a fundamental issue in control system analysis since a long time. Among all the methods, those based on Lyapunov functions are dominant in literature [3,4,6,7,8,11,16,19,22,20,10,21,15]. These methods not only yield a Lyapunov function as a stability certificate, but also the corresponding sublevel sets as estimates of the DOA.…”

Exact Asymptotic Stability Analysis and Region-of-Attraction Estimation for Nonlinear Systems

“…However, ( 1 , 2 ) and cannot satisfy the conditions in (21) exactly, because there exists a sample point (15/32, 85/256) such that the third condition in (21) cannot be satisfied. Therefore,…”

“…Computing exact regions of attraction (ROAs) for nonlinear dynamical systems is very hard if not impossible; therefore, researchers have focused on finding estimates of the actual ROAs. There are many wellestablished techniques for computation of ROAs [5,[16][17][18][19][20][21][22]. Among all methods, those based on Lyapunov functions are dominant in the literature.…”

“…Reference [21] employed quantifier elimination (QE) method via QEPCAD to find Lyapunov functions for estimating the ROA.…”