Optimal control of uncertain nonlinear quadratic systems

Abstract

This paper addresses the problem of robust and optimal control for the class of nonlinear quadratic systems subject to norm-bounded parametric uncertainties and disturbances. By using an approach based on the guaranteed cost control theory, a technique is proposed to design a state feedback controller ensuring for the closed-loop system: (i) the local exponential stability of the zero equilibrium point; (ii) the inclusion of a given region into the domain of exponential stability of the equilibrium point; (iii) the satisfaction of a guaranteed level of performance, in terms of boundedness of some optimality indexes. In particular, a sufficient condition for the existence of a state feedback controller satisfying a prescribed integral–quadratic index is provided, followed by a sufficient condition for the existence of a state feedback controller satisfying a given ${\mathcal{L}}_2$-gain disturbance rejection constraint. By the proposed design procedures, the optimal control problems dealt with here can be efficiently solved as Linear Matrix Inequality (LMI) optimization problems.

Introduction

The goal of this paper is to investigate the extension of the linear quadratic regulator (LQR) and ${\mathcal{H}}_{\infty }$ optimal control techniques to the class of nonlinear quadratic systems (NQSs).

The stability analysis and design of nonlinear quadratic systems has been performed in Amato, Ambrosino, Ariola, Cosentino, and Merola (2007) and Amato et al. (2007b), Amato et al. (2010); these papers provide conditions ensuring the existence of state feedback controllers, which stabilize the given quadratic system and guarantee that an assigned polytopic region belongs to the domain of attraction of the zero equilibrium point; applications of such approach are reported in Merola, Cosentino, and Amato (2008), to study the interaction dynamics between tumor and immune system, and in Cosentino, Salerno, Passanti, Merola, Bates, and Amato (2012), to investigate the bistable behavior of gene regulatory network.

The extension of the above-mentioned optimal control methodologies to NQSs will be pursued through an approach that is reminiscent of the Guaranteed Cost Control (GCC) theory (Chang & Peng, 1972). GCC-based methodologies guarantee that the control performance is bounded by a specified performance level for all admissible uncertainties of the closed loop system Costa & Oliveira (2002), Petersen & McFarlane (1994).

In the GCC literature, few works have dealt with nonlinear systems; for instance, in Aliyu (2000), a minimax optimization methodology has been developed for designing a robust GCC law for a class of uncertain nonlinear systems, whereas some LMI-based conditions have been formulated in Coutinho, Trofino, and Fu (2002) to solve a robust GCC problem for a class of input-affine nonlinear systems. A preliminary work concerning guaranteed-cost optimal control of NQSs can be found in Amato, Colacino, Cosentino, and Merola (2014).

As ${\mathcal{H}}_{\infty }$ optimal control theory for nonlinear systems is concerned, the design of state feedback controllers is tackled in Kim, Kim, and Lim (2005), where bilinear systems are considered, whereas ${\mathcal{H}}_{\infty }$ filtering for a class of Lipschitz nonlinear systems with time-varying uncertainties is proposed in Abbaszadeh and Marquez (2012), in order to attain both the exponential stability of the estimation error dynamics and robustness against uncertainties. In Qiu, Feng, & Gao, (2011), the ${\mathcal{H}}_{\infty }$ control theory has been extended to the class of discrete-time piecewise-affine systems with norm-bounded uncertainties; the basic aim of the contribution is to design a piecewise-linear static output feedback controller guaranteeing the asymptotic stability of the resulting closed-loop system with a prescribed ${\mathcal{H}}_{\infty }$ disturbance attenuation level.

Since the achievement of global stabilization and/or the determination of the optimal cost is a difficult or even impossible task when NQSs are dealt with, following the guidelines of Amato and Ambrosino, et al. (2007) and Amato et al. (2007b), Amato et al. (2010), we look for sub-optimal controllers with guaranteed performance into a certain compact region containing the origin of the state space (such region can be interpreted as the operating domain of the system). More precisely, given an uncertain NQS, possibly subject to exogenous disturbances, the main results of this paper consist of some sufficient conditions for the existence of a linear state feedback controller which will ensure for the closed-loop system: (i) the local exponential stability of the zero equilibrium point; (ii) the inclusion of a given region into the domain of exponential stability of the equilibrium point itself; (iii) the satisfaction of a guaranteed level of performance, in terms of the boundedness of a quadratic cost function in the form

$$J_2≔{\int }_0^{\infty }\left(x^T\left(t\right)Qx\left(t\right)+u^T\left(t\right)Ru\left(t\right)\right)dt,$$

where $x$ and $u$ are the system input and state, respectively (when the extension of the LQR approach is considered), or in terms of the negativeness of a quadratic cost function in the form

$$J_{\infty }≔{\int }_0^{\infty }\left(z^T\left(t\right)z\left(t\right)-w^T\left(t\right)w\left(t\right)\right)dt,$$

where $z$ and $w$ are the system controlled variable and the disturbance, respectively (when the ${\mathcal{H}}_{\infty }$ case is considered).

The remainder of the paper is organized as follows. Section 2 provides the problems statement and some preliminary results. The main results of the paper, namely some sufficient conditions for the existence of linear state feedback controllers guaranteeing optimal quadratic regulator and ${\mathcal{H}}_{\infty }$ performance, are presented in Section 3. Eventually, some concluding remarks are given in Section 4.

Notation

The symbol ${\mathcal{L}}_2^{n_w}$ denotes the subspace of vector-valued functions in ${\mathbb{R}}^{n_w}$ which are square-integrable over $[0,+\infty )$ with Euclidean vector norm $\Vert \cdot {\Vert }_2={({\int }_0^{\infty }\Vert \cdot {\Vert }^{2}dt)}^{1∕2}$. The matrix operation $A\otimes B$ denotes the Kronecker product of matrices $A$ and $B$, while $I_n$ denotes the identity matrix of order $n$. Given a square matrix $M$, $\mathrm{symm}\left(M\right)≔M+M^T$.