Nonlinear model predictive control for models in quasi‐linear parameter varying form

Abstract

This article presents a nonlinear model predictive control (NMPC) approach based on quasi-linear parameter varying (quasi-LPV) representations of the model and constraints. Stability of the proposed algorithm is ensured by the offline solution of an optimization problem with linear matrix inequality constraints in conjunction with an online terminal state constraint. Furthermore, an iterative approach is presented with which the NMPC optimization problem can be handled by solving a series of Quadratic Programs at each time step, this being highly computationally efficient. A practical and simple way of obtaining quasi-LPV representations of the system using velocity-based linearization is presented in two examples. K E Y W O R D S efficient algorithms, linear matrix inequality, nonlinear model predictive control, quasi-linear parameter varying systems 1 Int J Robust Nonlinear Control. 2020;30:3945-3959.wileyonlinelibrary.com/journal/rnc

A fast dissipative robust nonlinear model predictive control procedure via quasi‐linear parameter varying embedding and parameter extrapolation

“…Its solution comprises a CP (maximization, via fmincon) and a QP (minimization, via Gurobi); it is henceforth denoted “min‐max (Cao and Lin 2005).” The min/max LPV MPC scheme considering bounded rates of parameter variations, 13 defined with respect to the uncertainty set given in Equation (78). This approach is also resolved via fmincon and Gurobi; it is denoted “min‐max (Li and Xi 2010).” The qLPV‐embedding NMPC method, 25 which uses a scheduling sequence estimation and solves sequential QPs, solved via through iterated uses of Gurobi. This last method is henceforth marked as “qLPV MPC (Cisneros and Werner 2020).”…”

“…We note that any kind of algorithm with bounded estimation errors could be used in the place of the Taylor expansion one proposed in this article. An alternative and elegant option could be the use of the iterated mechanism, 25 which uses the state sequence computed with the minimization QP to compute the evolution of along the horizon. The proposed method is compared to two keystone min‐max LPV MPC algorithms from the literature, 11,13 which consider, respectively, that can vary arbitrarily inside and considers bounded rates of variations for . Since the variations of the scheduling parameters and its convex set are quite large for the considered application, the results obtained with these methods are quite poor.…”

“…Recent literature has shown how approximated NMPC methods (such as CaSaDi, GRAMPC, and ACADO 48 ) and qLPV‐embedding MPC algorithms 25 are able to efficiently solve such complex control problem in the range of milliseconds. For the considered case study, through IAE and RMS indexes, the reference tracking performances obtained with the proposed qLPV‐embedding min‐max MPC method are equivalent to these fast modern NMPC methods 25 . The numerical operability of the proposed method is similar to previous works 25,48 . We note that the complexity of the problem grows with the order of the system. The proposed method solves the maximization convex programming problem with respect to the error regarding the estimation of the scheduling parameters along the prediction horizon.…”

“…With respect to the obtained results, some key points are mentioned: Full‐blown nonlinear programming NMPC is not applicable for embedded applications of processes with fast sampling rates, since the average time needed to solve the NP is usually larger than the available sampling period. Recent literature has shown how approximated NMPC methods (such as CaSaDi, GRAMPC, and ACADO 48 ) and qLPV‐embedding MPC algorithms 25 are able to efficiently solve such complex control problem in the range of milliseconds. For the considered case study, through IAE and RMS indexes, the reference tracking performances obtained with the proposed qLPV‐embedding min‐max MPC method are equivalent to these fast modern NMPC methods 25 . The numerical operability of the proposed method is similar to previous works 25,48 .…”

“…Remark Some previous papers 21,25 had already conceived NMPC formulations via qLPV embedding. Nonetheless, in the prior, optimality concerns were not ensured, since the extrapolation error is not taken into account in the design procedure.…”