Automatica 43 (2007) 158 – 163 www.elsevier.com/locate/automatica Technical communique Improving off-line approach to robust MPC based-on nominal performance cost  BaoCang Ding a , ∗ , YuGeng Xi b , Marcin T. Cychowski c , Thomas O’Mahony c a School of Electricity and Automation, Hebei University of Technology, Tianjin 300130, PR China b Institute of Automation, Shanghai Jiaotong University, Shanghai 200240, PR China c Department of Electronic Engineering, Cork Institute of Technology, Rossa Avenue, Cork, Ireland Received 17 July 2005; received in revised form 3 July 2006; accepted 21 July 2006 Available online 26 September 2006 Abstract This paper gives two alternative off-line synthesis approaches to robust model predictive control (RMPC) for systems with polytopic description. In each approach, a sequence of explicit control laws that correspond to a sequence of nested asymptotically invariant ellipsoids are constructed off-line. In order to accommodate a wider class of systems, nominal performance cost is chosen to substitute the “worst-case” one in an existing technique. In the design of control law for a larger ellipsoid, the second approach further incorporates the knowledge of control laws associated with all smaller ellipsoids, so as to further improve feasibility and optimality. The effectiveness of the alternative approaches is illustrated by a simulation example.  2006 Elsevier Ltd. All rights reserved. Keywords: Robust model predictive control; Off-line method; Nominal performance cost; Asymptotically invariant ellipsoid; Linear matrix inequality 1. Introduction Synthesis approaches for robust model predictive control (RMPC), which have been widely investigated, can be classified by on-line RMPC (see e.g. Angeli, Casavola, & Mosca, 2002; Kothare, Balakrishnan, & Morari, 1996; Kouvaritakis, Rossiter, & Schuurmans, 2000; Wan & Kothare, 2003a) and off-line RMPC (see e.g. Cychowski, Ding, Tang, & O’Mahony, 2004; Wan & Kothare, 2003b). The latter never involves on-line opti- mization. Kothare et al. (1996) on-line optimized a single state feedback gain F(k) such that a “worst-case” infinite-horizon quadratic optimization problem is solved. This method usually imposes heavy on-line computational burden and limitations on the achievable optimality and feasibility. There are many techniques towards improvements. Kouvaritakis et al. (2000)  This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form under the direction of Editor A.L. Tits. ∗ Corresponding author. E-mail address: dingbc@jsmail.hebut.edu.cn (B.C. Ding). 0005-1098/$ - see front matter  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2006.07.022 off-line designed a state feedback gain F to generate “closed- loop” state predictions and, on-line, only a summation of the 2-norm of the perturbations on F is minimized. Angeli et al. (2002) off-line constructed a series of nested ellipsoids, the smallest one corresponding to F, such that the state in one ellipsoid can be steered into the neighboring smaller one in one step. On-line, by properly choosing one of the ellipsoids as the terminal constraint set, a standard MPC (see Mayne, Rawlings, Rao, & Scokaert, 2000) with control horizon M = 1 will converge the state towards the smallest ellipsoid. More recently, Wan and Kothare (2003b) directly solved the optimization in Kothare et al. (1996) off-line, such that a number of N state feedback gains F i s with corresponding nested ellipsoidal domains of attraction  i s are constructed. On-line, when the state lies between two adjacent ellipsoids, the real-time control law is chosen as a linear interpolation of the two corresponding off-line laws. This off-line RMPC has considerably low on-line computational burden. However, with respect to optimality and feasibility it is apparently worse than Kothare et al. (1996). In the present paper, we propose two improved alternatives to Wan and Kothare (2003b). The first alternative adopts nominal performance cost in substitution of