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

Abstract

In this article, a robust model predictive control (MPC) procedure for quasi-linear parameter varying (qLPV) systems is proposed. The novelty resides in considering a recursive extrapolation algorithm to estimate the values of the scheduling parameters along the prediction horizon N p , which fastens the sluggish performances achieved with the robust qLPV MPCs from the literature.The bounds on the estimation errors of the scheduling parameters through N p are taken into account by the robust MPC, which solves an online min-max problem: first, a constrained convex program is resolved in order to determine the worst-case bound on the cost function and, subsequently, a second constrained quadratic program is solved to minimize this worst-case cost function with respect to a control sequence vector. Since the bounds on the estimation error for the scheduling parameters are usually much smaller than the bounds on the actual scheduling parameter, the conservativeness of the solution is quite reduced. Recursive feasibility and stability of the proposed algorithm are demonstrated with dissipativity arguments given in the form of a linear matrix inequality remedy, which determines the zone of attraction for which input-to-state stability is ensured. The nonlinear temperature regulation problem of a flat solar collector is considered as a case study. Using a realistic simulation benchmark, the proposed technique is compared to other robust min-max LPV MPC algorithms from the literature, proving itself efficient while achieving good performances. K E Y W O R D Sdissipativity, linear parameter varying systems, quadratic programming, robust model predictive control, solar collectors INTRODUCTIONModel predictive control (MPC) is one of the most widespread control techniques, with many industrial applications. 1 Its implementation is relatively simple, requiring the solution of an online optimization problem, written in terms of a prediction model and operational constraints. MPC is a sliding-horizon paradigm, which means that the algorithm Abbreviations: LPV, linear parameter varying; LTI, linear time-invariant; MPC, model predictive control.

NMPC via qLPV models and Taylor-based Scheduling Parameter Extrapolation: A Cartesian Robot Case Study

“…( 4) is solved l times per sample, with the scheduling parameter trajectory estimate being given by P l k = f ρ (X l k ). • The highlighted Taylor-based extrapolation single QP qLPV MPC from [13], [14], where the scheduling trajectory is estimated by the means of Eq. ( 15).…”

“…By definition, an ellipsoid ensures (C1). (C4) and (C5) are respectively satisfied by applying Schur complements to LMI (12) and (13). In turn, the terminal feedback κ t (x) = Kx ensures that the qLPV system is asymptotically stable.…”

“…Accordingly, we detail, in this Section, the recent framework from [13], [14], which uses a first-order Taylor expansion of the scheduling proxy f ρ (x(k)) to construct the future scheduling sequence. First and foremost, we stress that the main advantage of this extrapolation is that predicted sequence P k is generated through a single recursive linear law, numerically much cheaper than the state-of-the-art procedure from [7], which requires the iterative re-evaluation of…”

“…Through the sequel, we neglect the bias term ξ ρ , since it is bounded and small, as argued in [14]. In any case, robustified MPC schemes with respect to such estimation error are discussed in [8], [13]. Accordingly, we can write the vectorwise extrapolation in a recursive fashion:…”

“…Recent works have developed an alternative approach to the prior, where the future scheduling trajectories are estimated through linear recursive Taylor-based extrapolation laws [12], [13]. The novelty of this alternative is that the resulting MPC algorithm does not require to evaluate many QPs per sample, but rather just a single one.…”