Region of attraction analysis with Integral Quadratic Constraints

Abstract

A general framework is presented to estimate the Region of Attraction of attracting equilibrium points. The system is described by a feedback connection of a nonlinear (polynomial) system and a bounded operator. The input/output behavior of the operator is characterized using an Integral Quadratic Constraint. This allows to analyze generic problems including, for example, hard-nonlinearities and different classes of uncertainties, adding to the state of practice in the field which is typically limited to polynomial vector fields. The IQC description is also nonrestrictive, with the main result given for both hard and soft factorizations. Optimization algorithms based on Sum of Squares techniques are then proposed, with the aim to enlarge the inner estimates of the ROA. Numerical examples are provided to show the applicability of the approaches. These include a saturated plant where bounds on the states are exploited to refine the sector description, and a case study with parametric uncertainties for which the conservativeness of the results is reduced by using soft IQCs.

Introduction

Stability guarantees are often valid only locally for nonlinear systems, and for this reason the notion of Region of Attraction (ROA) has been proposed (Khalil, 1996). The ROA of an equilibrium point $x^{\ast }$ is the set of all the initial conditions from which the trajectories of the system converge to $x^{\ast }$ as time goes to infinity. This paper proposes a new framework for the analysis of the ROA for generic uncertain systems. In full generality, the problem considered in this article consists of the feedback interconnection of a system $G$ with polynomial vector field and a bounded causal operator $\Delta$. Motivation for this kind of description stems from the objective to compute ROA of systems which are affected by generic nonlinearities (in addition to the polynomial ones) and/or uncertainties.

The Integral Quadratic Constraint (IQC) (Megretski & Rantzer, 1997) paradigm, building on work by Yakubovich (1971), is particularly suited to address the aforementioned operator $\Delta$, because it characterizes a broad class of nonlinearities, and allows to refine the description of the uncertainties by specifying their nature. The setup provided by the feedback interconnection $G$-$\Delta$ is thus believed to be quite general and to adequately cover a large class of nonlinear systems encountered in applications.

The time domain interpretation of IQCs is instrumental to prove the main results of the paper. In particular, the connection between dissipation inequality and IQC is exploited to provide guarantees of local stability. One of the known issues is that the dissipation inequality requires the IQC to be hard in the sense that the integral constraints must hold over all finite times (Seiler, 2015). This is not immediate because the frequency domain IQC only guarantees an equivalent counter part in the time domain as an infinite-horizon integral constraint (soft IQC) (Megretski & Rantzer, 1997). Recent studies have proposed lower bounds on the finite-horizon integral constraint when only the soft property holds (Fetzer et al., 2018, Seiler, 2015, Seiler, 2018). In Fetzer et al. (2018), this was provided as a convex constraint on the IQC multiplier, and it will be employed in this work.

The exact ROA is often difficult to compute either numerically or analytically (Genesio, Tartaglia, & Vicino, 1985), therefore algorithms have been proposed to numerically calculate inner estimates of the ROA. The state of practice in the field focuses on determining Lyapunov function level sets, which are contractive and invariant and thus are subsets of the ROA (Chakraborty et al., 2011b, Valmorbida et al., 2009). Non-Lyapunov methods have also been studied to reduce the conservatism typically associated with the characterization of ROA as contractive level sets (Henrion and Korda, 2014, Valmorbida and Anderson, 2017). All the approaches above share the common feature that are either only applicable to polynomial vector fields or rely on Sum Of Squares (SOS) techniques. As a result, a limitation holds on the types of nonlinearities that can be considered. An example of a more general approach is the so-called Zubov’s method (Zubov, 1964), which is based on a converse Lyapunov theorem and requires to solve a partial differential equation, but this makes it difficult to be employed for practical cases. Relaxed versions of this result have also been proposed, for example in Vannelli and Vidyasagar (1985) where the concept of maximal Lyapunov function was introduced. Possible extensions of Lyapunov (Chesi, 2004, Topcu et al., 2010) and non-Lyapunov (Iannelli, Marcos, & Lowenberg, 2019) methods to the case of uncertain systems have also been proposed recently. In general, a major drawback of the approaches employed to deal with uncertain systems is that they do not exploit specific properties of the uncertainties (e.g. linear time invariant, real constant). This inherently leads to conservative outcomes because the results must hold for a larger set of uncertainties than the one actually affecting the system.

The contribution of this article is therefore to propose a general and flexible framework for local stability analysis of nonlinear uncertain systems. The problem is formulated by defining an augmented plant which comprises the polynomial part of the vector field $G$ as well as the Linear Time Invariant (LTI) system $\Psi$ provided by the state–space factorization of the IQC. Based on this problem setup, Section 3 establishes certificates for the domain of attraction with both hard and soft factorizations. Specifically, the ROA is formulated as the level set of a polynomial function of generic degree (which is not necessarily a Lyapunov function for the system). Since the fictitious plant and the sought function are polynomial, the problem can be solved numerically via Sum of Squares (SOS) techniques, allowing it to be recast as a set of semidefinite programs (Parrilo, 2003).

Numerical examples of polynomial systems affected by hard nonlinearities (i.e. actuator saturation) and real parametric uncertainties illustrate the application of the approaches in Section 4. In the former case, the bounds on the states inherently given by ROA are employed to provide a less conservative expression for the sector IQC. As for the case with parametric uncertainties, the favorable feature of this framework of allowing to refine the description of the uncertainties is showcased.

The work here, extending preliminary results in Iannelli, Seiler, and Marcos (2018b) which only considered the case of hard IQC, is related to recent studies (Fetzer et al., 2018, Pfifer and Seiler, 2015, Seiler, 2015, Seiler, 2018) which focused on the reconciliation between Lyapunov function methods and multiplier theory. In particular, the distinction between hard and soft IQCs plays a crucial role in the estimation of the domain of attraction. In view of this, the active area of research concerned with finding less conservative bounds for soft IQCs (Fetzer et al., 2018, Seiler, 2015, Seiler, 2018) can have a big impact on the estimation of local stability regions with the framework presented here. This paper is also connected to the work in Chakraborty, Seiler, and Balas (2010), where the time domain interpretation of IQC (hard IQCs only were considered) was exploited for performance analysis of polynomial systems subject to hard nonlinearities.