Discussion on: “Improved MPC Design based on Saturating Control Laws”

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This paper provides an interesting contribution on the integration of model predictive control (MPC) with control Lyapunov functions (CLF). Such an integration, as shown in Ref. [1], yields control strategies outperforming control design based on either MPC or CLF. In particular, the authors address the design of MPC strategies for linear discrete-time systems subject to input and/or state constraints with a large domain of attraction and a relatively low computational burden. Standard MPC with guaranteed stability either requires large control horizons or typically yields a much smaller domain of attraction. Fixed the length of the prediction horizon N, the domain of attraction can be enlarged by increasing the size of the terminal set without increasing the on-line computational burden. The authors combine the computation of an estimate of the domain of attraction, denoted as C1, for the saturated LQR regulator via linear difference inclusions (LDI) proposed in Ref. [2] and model predictive control ideas. The main contribution of the paper consists of a dual-mode algorithm that switches among two MPC algorithms with different terminal cost/set according to the feasibility. More precisely, a standard MPC algorithm with terminal cost/set related to the unsaturated LQR terminal control law and a modified MPC strategy employing as terminal set the estimate of the domain of attraction of the saturated LQR control law and as terminal cost the associated quadratic CLF, are used. The proposed terminal cost is an upper bound of the optimal one, and the resulting optimization problem amounts to the solution of a quadratic programming problem. The proposed procedure can encompass multivariable constrained control problems in the presence of state constraints. A first remark to be made is that although the authors consider only systems without uncertainty, the extension of the approach to the case of systems affected by additive unknown bounded noise and/or described by an uncertain polytopic model is rather straightforward [3]. According to the practitioners, what limits the performance and applicability of MPC are difficulties in modeling. Then it becomes fundamental to implement MPC algorithms for uncertain models with large domain of attraction and relatively low computational burden. However, the choice of presenting the results for systems without any kind of uncertainty is welcome from the point of view of simplicity. A second remark can be made in relation to the choice of the adopted LDI in order to describe the saturated system. The authors claim that the present results can be easily extended to any LDI description of the saturated system. The usage of an LDI in describing the closed-loop system clearly introduces some conservatism and then an interesting topic for future investigation would be the study of which is the best LDI description in order to reduce the conservatism.