Assessment and Future Directions of Nonlinear Model Predictive ...

182 A. Grancharova, T.A. Johansen, and P. Tøndelstates of the system and the stability of the closed-loop system is guaranteed.An approach for NMPC design for constrained input-affine nonlinear systemshas been suggested in [9], which deploys state space partitioning and graph theoryto retain the on-line computational efficiency. In [10], [11], [12], approachesfor off-line computation of explicit sub-optimal PWL predictive controllers forgeneral nonlinear systems with state and input constraints have been developed,based on the multi-parametric Nonlinear Programming (mp-NLP) ideas [13]. Ithas been shown that for convex mp-NLP problems, it is straightforward to imposetolerances on the level of approximation such that theoretical propertieslike asymptotic stability of the sub-optimal feedback controller can be ensured[11], [14]. However, for non-convex problem there is a need to investigate practicalcomputational methods that not necessarily lead to guaranteed properties,but when combined with verification and analysis methods will give a practicaltool for development and implementation of explicit NMPC.The present paper focuses on computational and implementation aspects ofexplicit NMPC for general nonlinear systems with state and input constraintsand is structured as follows. In section 2, the formulation of the NMPC problemis given. In section 3, computational methods for approximate explicit NMPCare suggested. The application of the developed approaches to compressor surgecontrol is considered in section 4.2 Formulation of Nonlinear Model Predictive ControlProblemConsider the discrete-time nonlinear system:x(t +1)=f(x(t),u(t)) (1)y(t) =Cx(t) (2)where x(t) ∈ R n , u(t) ∈ R m ,andy(t) ∈ R p are the state, input and output variable.It is also assumed that the function f is sufficiently smooth. It is supposedthat a full measurement of the state x(t) is available at the current time t. Forthe current x(t), MPC solves the following optimization problem:V ∗ (x(t)) = minJ(U, x(t)) (3)Usubject to x t|t = x(t) and:y min ≤ y t+k|t ≤ y max , k =1, ..., N (4)u min ≤ u t+k ≤ u max , k =0, 1, ..., N − 1 (5)x T t+N|t x t+N|t ≤ δ (6)x t+k+1|t = f(x t+k|t ,u t+k ),k≥ 0 (7)y t+k|t = Cx t+k|t ,k≥ 0 (8)with U = {u t ,u t+1 , ..., u t+N−1 } and the cost function given by: