IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 56, NO. 12, DECEMBER 2011 2997 Design of a Nonlinear Anti-Windup Gain by Using a Composite Quadratic Lyapunov Function Liang Lu, Member, IEEE, and Zongli Lin, Fellow, IEEE Abstract—This technical note revisits the problem of designing a static anti-windup gain for enlarging the domain of attraction of the resulting closed-loop system. By utilizing a composite quadratic Lyapunov func- tion, which was originally proposed to study the stabilization problem for linear systems under actuator saturation, an existing LMI based de- sign algorithm is enhanced to result in a nonlinear, possibly continuous, anti-windup gain. This nonlinear anti-windup gain enables us to obtain an estimate of the domain of attraction in the form of the convex hull of a group of ellipsoids, instead of a single ellipsoid that would result from a single Lyapunov function based design. Simulation results demonstrate the features of the proposed design the composite quadratic Lyapunov function brings about. Index Terms—Actuator saturation, anti-windup, composite quadratic Lyapunov functions, domain of attraction. I. INTRODUCTION A large portion of the modern control theory does not take into ac- count the magnitude constraints on control input. As a result, when its control input saturates the actuators, the control system suffers a more severe performance degradation than one whose control design takes saturation into account would. This has motivated the concept of anti-windup compensation, the augmentation of a closed-loop system that was designed without taking actuator saturation into consideration so that the adverse effect of such saturation can be weakened. The design of an anti-windup compensator can be carried out in ei- ther a direct way or an indirect way. A direct way to anti-windup design would try to minimize the difference between the input and output of the actuator. Much of the earlier work adopts this approach (see, for example, [1], [4]). In an indirect way to anti-windup design, the objec- tive is not to reduce the difference between the input and output of the actuator but to improve the stability and performance of the resulting compensated closed-loop system (see, for example, [2], [3], [5]–[8], [11]–[15], [17]–[21] for a small sample of the vast literature on the topic). A particular stability and performance objective of enlarging the do- main of attraction has been adopted in the recent literature ([2], [8], [15]). In particular, in [2], by expressing a saturated linear feedback law on the convex hull of some auxiliary linear feedbacks [10] and by using a single quadratic Lyapunov function, Cao et al. formulate the de- sign of the anti-windup gain that maximizes the contractively invariant Manuscript received March 18, 2010; revised September 24, 2010, May 25, 2011, and June 19, 2011; accepted July 05, 2011. Date of publication July 14, 2011; date of current version December 07, 2011. This work was supported in part by the National Natural Science Foundation of China under grant 60928008 and 60934007. Recommended by Associate Editor F. Wu. L. Lu was with Shanghai Jiao Tong University, Shanghai, China and Eindhoven University of Technology, Eindhoven, The Netherlands and is now with the State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang 110004, China (e-mail: liangup@gmail.com). Z. Lin is with the Charles L. Brown Department of Electrical and Com- puter Engineering, University of Virginia, Charlottesville, VA 22904-4743 USA (e-mail: zl5y@virginia.edu). Color versions of one or more of the figures in this technical note are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAC.2011.2161832 ellipsoid, an estimate of the domain of attraction, as a constrained op- timization problem with bilinear matrix inequality constraints. An iter- ative LMI algorithm is developed to solve this optimization problem. Later in [8], by observing the relationship between the saturation func- tion and the deadzone function and by using a sector type condition that results, Gomes da Silva and Tarbouriech formulate an alternative constrained optimization problem. A nice feature of this alternative op- timization problem is that it can be solved as an LMI problem. In an effort to further enlarge the domain of attraction, we recently used multiple Lyapunov functions to design a switching anti-windup compensator [15]. This switching anti-windup compensator involves switching among a group of anti-windup gains, but results in an esti- mate of domain of attraction in the form of the union of a group of ellipsoids. Simulation results have indicated that such a switching anti- windup compensator has the ability to enlarge the domain of attraction significantly beyond what a single anti-windup gain is able to achieve. However, the design in [15] involves the solution of a non-convex op- timization problem. In this technical note, we will show how the LMI optimization based design of [8] can be further enhanced to arrive at a nonlinear, but pos- sibly continuous, anti-windup gain that will result in an estimate of the domain of attraction in the form of the convex hull of a group of el- lipsoids. This enhancement is made possible by the use of a composite quadratic Lyapunov function, which was originally proposed to study the stabilization problem for linear systems under actuator saturation [9]. The composite Lyapunov quadratic functions take the form of a quadratic Lyapunov function, , with a state dependent .A different state dependent has been used in the context of global stabilization of linear systems with bounded inputs (see, e.g., [16]). The remainder of our technical note is organized as follows. In Sec- tion II, we state the problem to be studied and recall some preliminary materials that will be needed to develop the results of this technical note. Section III recalls the definition and some key properties of com- posite quadratic Lyapunov functions from [9]. Section IV develops the algorithm for designing the nonlinear anti-windup gain. A numerical example is presented in Section V. Section VI concludes the technical note. II. PROBLEM FORMULATION AND PRELIMINARIES Consider the following linear system subject to actuator saturation: (1) where is the state, is the control input, is the measured output, the function is the vector valued standard saturation function defined as , . Here we have slightly abused the notation by using sat to denote both the scalar valued and the vector valued saturation functions. Also note that it is without loss of generality to assume unity saturation level. A non-unity saturation level can be absorbed into matrix . We assume that a linear dynamic controller of the form has been designed that stabilizes system (1) in the absence of actuator saturation. 0018-9286/$26.00 © 2011 IEEE