A new low-and-high gain feedback design using MPC for global stabilization of linear systems subject to input saturation Xu Wang 1 Håvard Fjær Grip 1;2 Ali Saberi 1 Tor Arne Johansen 2 Abstract—In this paper, we develop a new low-and-high gain feedback design methodology using a ultra-short-horizon Model Predictive Controller (MPC) for global asymptotic stabilization of discrete-time linear system subject to input saturation. The proposed method yields improved performance over classical low-and-high gain design and has reduced computational com- plexity and guaranteed global asymptotic stability of closed-loop system compared to MPC with a long prediction horizon. I. I NTRODUCTION Stabilization of linear systems subject to actuator sat- uration has been extensively studied during the past two decades and is still drawing renewed attention, largely because saturation is widely recognized as ubiquitous in engineering applications and inherent constraints in control system designs. Many significant results have already been obtained in the literature. Some early works in this area are summarized in [3], [21], [26], [22], [7], [10] and references therein. The low-gain method, proposed in [14], [15], [13], was originally developed as a linear feedback design methodol- ogy in the context of semi-global stabilization under actuator saturation and later on extended to the global framework with a gain scheduling [18], [6]. The low-gain feedback is parameterized by a so-called low-gain parameter, which is determined a priori in the semi-global setting according to a pre-selected compact set or adaptively with respect to states in the global setting. By properly selecting this parameter, we are able to limit the input magnitude to a sufficiently small level and avoid saturation for all time so as to stabilize the system. On one hand the low-gain proves to be successful in solving stabilization problems, on the other hand it does not utilize the full actuation level and hence is conservative and less capable regarding performance. Low-and-high gain feedback designs are conceived to rectify the drawbacks of low-gain design methods, and can make better use of available control capacity. As such, they have been used for control problems beyond stabilization, to enhance transient performance and to achieve robust stability and disturbance rejection (see [15], [16], [20], [6], [19], [27]). However, 1 School of Electrical Engineering and Computer Science, Wash- ington State University, Pullman, WA 99164-2752, U.S.A. E-mail: fxwang,saberig@eecs.wsu.edu, grip@ieee.org. The work of Ali Saberi and Xu Wang is partially supported by NAVY grants ONR KKK777SB001 and ONR KKK760SB0012. The work of Håvard Fjær Grip is supported by the Research Council of Norway 2 Department of Engineering Cybernetics, Norwegian University of Sci- ence and Technology, O.S. Bragstads plass 2D N-7491 Trondheim, Norway. E-mail: Tor.Arne.Johansen@itk.ntnu.no as will be shown in this paper, there is still a room for improvement. Model predictive controllers (MPC) have a reputation of dealing with constraints and achieving good closed-loop per- formance. It numerically solves a finite-horizon constrained optimal control problem at each sample. Hence, a MPC may choose to operate exactly at the constraints, while a low-gain strategy would be to avoid the constraints. The more aggressive approach of the MPC complements the more conservative low-gain strategy, and is an interesting approach to include in a low-and-high gain feedback design in order to improve performance. A drawback of MPC is the computational complexity of solving online numerically a constrained optimization prob- lem (usually a quadratic program) at each sample. Guaran- tees of MPC stability requires particular formulations of the finite-horizon optimal control problem, such as sufficiently long prediction horizon and the use of a terminal cost, [24], or terminal constraints, [11]. Reduction of computational complexity typically requires that the prediction horizon is made shorter, which comes at the cost of more complex terminal costs and constraints, as well as sub-optimality compared to an infinite horizon constrained optimal control formulation, see e.g. [23], [25] for examples of such refor- mulations. Explicit MPC of constrained linear systems admits a piece- wise affine state feedback solution to be pre-computed using multi-parametric quadratic programming, [2]. Although on- line computational complexity can be reduced by orders of magnitude, the approach is still limited by available computer memory and the cost of off-line pre-computations, [29], [1]. Consequently, low-complexity sub-optimal strategies are also of interest in explicit MPC in order to manage complexity due to long prediction horizon, high system order, or many constraints, while preserving stability (see e.g. [9], [12], [5], [8]). The key idea pursued in this paper is to use an ultra-short- horizon MPC as the high gain strategy in a low-high-gain feedback design, where simple constraints resulting from the low-gain design are imposed on the MPC in order to guarantee stability. II. CLASSICAL LOW- GAIN DESIGN AND MPC Consider a discrete-time system x kC1 D Ax k C B  .u k / (1) where  ./ is standard saturation, i.e. for u 2 R m ,  .u/ D Œ 0 .u 1 /I  I  0 .u m /,  0 .u i / D sign.u i / minf1; ju i jg and