Control synthesis for non-polynomial systems: A domain of attraction perspective

Abstract

Abstract-This paper studies a control synthesis problem to enlarge the domain of attraction (DA) for non-polynomial systems by using polynomial Lyapunov functions. The basic idea is to formulate an uncertain polynomial system with parameter ranges obtained from the truncated Taylor expansion and the parameterizable remainder of the non-polynomial system. A strategy for searching a polynomial output feedback controller and estimating the lower bound of the largest DA is proposed via an optimization of linear matrix inequalities (LMIs). Furthermore, in order to check the tightness of the lower bound of the largest estimated DA, a necessary and sufficient condition is given for the proposed controller. Lastly, several methods are provided to show how the proposed strategy can be extended to the case of variable Lyapunov functions. The effectiveness of this approach is demonstrated by numerical examples.

On estimating the Robust Domain of Attraction for uncertain non-polynomial systems: An LMI approach

“…Motivated from the above, this work extends the result of [22], [23] to the case of uncertain rational polynomial and non-polynomial systems, and contributes in the following three aspects:…”

“…Future effort will be devoted to designing a less conservative convex approach to further enlarge the lower bound of µ * by using variable rational Lyapunov functions, e.g., searching a Lyapunov function with the largest volume of the sublevel set V(c) (see [4], [23] for the case of polynomial Lyapunov functions). Moreover, we are interested to develop this approach combining with other robust verification methods, such as reachability analysis methods and contraction theory methods [29], [30].…”

“…It is useful to mention that the search for an optimal rational Lyapunov function to enlarge the RDA may provide a less conservative result, but it is out of the scope of this paper due to limited space. The readers may refer to [5], [23] for the case of polynomial Lyapunov functions.…”

Estimating the domain of attraction based on the invariance principle

“…Inspired by the work in [26], and based on the work in [3], [27] in the category of Lyapunov method, this paper uses invariant principles to inner-approximate the DA without searching for a Lyapunov function. The main contributions of this paper are listed as follows:…”

On enlarging backward reachable sets via Zonotopic set membership

“…Motivated by the work in [20], and by adopting the control purpose of our previous work [21], this paper proposes a Lagrangian method based on a zonotopic set representation. Different from the existing literature, this paper uses the backward reachable sets to estimate and enlarge the RA by designing an optimal controller.…”