Static output feedback problem for Lipschitz nonlinear systems

Abstract

The output feedback stabilization of Lipschitz nonlinear systems is addressed. The synthesis of reduced-order controller is formulated as static output feedback problem. Based on coupled algebraic Riccati inequalities, the stability analysis of closed loop dynamic is presented. By utilizing some structural knowledge of Lipschitz nonlinearity, the sufficient conditions to obtain static as well as dynamic output feedback gains are given. For the class of Lipschitz nonlinearity, it is shown that the proposed condition is a necessary and sufficient condition to achieve static gain. The cone complementary linearization method is then applied to satisfy the proposed stability condition and to obtain an output feedback regulator. The effectiveness of proposed method is finally demonstrated through simulation results on some practical systems.

Introduction

The static output feedback (SOF) stabilization is one of the most known and significant problems in control theory. This problem is motivated in many practical applications due to the fact that, state variables are usually not fully accessible in real plants. Furthermore, an output feedback regulator is less expensive to be implemented and is more reliable in practice [1]. It is well-known that, many problems in dealing with synthesis of reduced-order controllers can be reformulated as a SOF controller design involving augmented plants [2], [3].

Unlike the state feedback case, the SOF synthesis for linear systems can be represented as a bilinear matrix inequality (BMI) problem. Due to this nonconvex formulation, the approaches based on necessary and sufficient conditions lead to some computational complexity. Accordingly, various numerical iterative algorithms based on sufficient conditions have been applied to design SOF regulators [4], [5], [6], [7]. The Cone Complementary Linearization (CCL) approach in [5] is well-known for solving bilinear problems and has been widely used to handle coupling constraints.

Many investigations concerning SOF problem in linear systems are presented to achieve certain desirable characteristics [8]. In this case, the SOF problem with H_∞ and robust performance are studied in [9], [10], [11], [12]. Also, the output-feedback stabilization for positive linear systems is addressed in [13], [14], [15]. Despite well-known results of SOF problem in linear systems and available synthesis methods, unfortunately the SOF problem for nonlinear systems has not been widely studied as its linear counterpart. (see [1], [16]). Based on solvability of a Hamilton – Jacobi equation, a synthesis approach to obtain output feedback controller for nonlinear systems is suggested in [16]. Also, the output-feedback problem for polynomial nonlinear systems is presented in [17], [18].

The motivation of present work is to extend the SOF problem and related synthesis method in linear systems to the Lipschitz nonlinear systems, in the sense that, many available stability results in literature can be extended and applied to this class of nonlinear systems and further some similar ideas as CCL method in [5] can be utilized to synthesis SOF for nonlinear systems. The Lipschitz nonlinear systems usually arise in practical applications [19], [20] due to increasing the domain of attraction as well as due to some perturbed nonlinearities in dynamic equation.

By considering the differentiability of dynamic system, one can achieve a linearization of dynamic model without utilizing any transformation, and further the Lipschitz condition is satisfied at least locally. Also, the considered model can be obtained via employing some coordinate transformations. Specifically, the concept of uniform observability allows to transform the affine nonlinear systems into some canonical forms in which the linear and nonlinear parts of dynamic model have some triangular structures (see direct and indirect methods in [19]). The observer synthesis for Lipschitz nonlinear systems has received considerable attention during decades (see [19], [20], [21], [22] and references therein) where some methods utilize the structural knowledge of nonlinearities to achieve less conservative results in synthesis problem [19], [21], [22]. Also the observer-based controller for Lipschitz nonlinear systems is suggested in [23], [24].

Related to the contribution of this paper, the stabilization problem via a reduced-order controller for Lipschitz nonlinear systems is first formulated as a general SOF problem, and some challenges such as equilibrium point of closed loop nonlinear dynamic and regulation problem are discussed. The main contribution of paper is presented where the stability of closed loop dynamic via both static and dynamic output feedback controllers are established for Lipschitz nonlinear systems. In this case, the sufficient conditions to obtain rth-order output feedback controller are presented. In order to achieve less conservative results, the stability criterion is derived to encompass some structural knowledge of Lipschitz nonlinearity. Accordingly, compared with linear systems, the Lyapunov-based inequalities concerning stability problem have been extended to the Riccati-based inequalities in Lipschitz nonlinear systems (see Theorems 1 and 4). Moreover, it is shown that, for the class of Lipschitz nonlinear systems, the proposed condition is a necessary and sufficient condition to ensure stability via static gain. Finally, a practical synthesis approach is introduced based on CCL method in [5]. In fact, applying CCL method allows to satisfy the proposed Riccati-based matrix inequalities in stability problem with minimum order of output feedback controller for Lipschitz nonlinear systems. Note that, the proposed method does not involve complicated approach based on solving a constrained Hamilton – Jacobi equation in [16].

The paper is organized as follows. In Section 2, the system description and some preliminarily results on SOF problem are presented. Section 3 is devoted to stability analysis of Lipschitz nonlinear systems via an output-feedback regulator. The synthesis approach by adopting CCL method is further studied. The simulation results on two practical systems are presented in Section 4 to demonstrate the effectiveness of proposed synthesis method. The concluding remarks are finally given in Section 5.

Notation: For real symmetric matrices X and Y, the notation X > Y (X ≥ Y) means that the matrix $X-Y$ is positive (semi)definite. The prime ′ represents complex conjugate transpose and star ⋆ denotes the complex conjugate transpose of corresponding element in LMI’s. The matrix I is an identity matrix and C_⊥ (respectively, B_⊥) represents a full rank orthogonal null space such that $C{C}_{\perp }=0$ (respectively, $B_{\perp }B=0$). Also, the function ρ(.) is used to express the rank of matrices and the matrices if not explicitly stated, are assumed to have compatible dimensions.