For constrained nonlinear stable systems, Chen and Allgöwer (1998a) reduced on-line computation of the quasi-infinite horizon nonlinear MPC (Chen & Allgöwer, 1998b) by removing the nonlinear terminal constraints and reducing the control horizon. For constrained systems with polytopic model uncertainty, the receding horizon dual-mode paradigm was introduced in Lee and Kouvaritakis (2000) to reduce computational complexity. Zheng (1999) approximated the optimal input moves over a control horizon by the current input subject to constraints and the unconstrained solution of future inputs subject to saturation. VanAntwerp and Braatz (2000) developed the iterated ellipsoid algorithm, in which an ellipsoid is used to approximate the linear input constraint set in an off-line calculation and is rescaled during on-line calculation. Connections between nominal constrained MPC and anti-windup control schemes were investigated in Cherukuri and Nikolaou (1998), De Doná and Goodwin (2000) and Zheng (1999). For nominal constrained MPC, Bemporad, Morari, Dua, & Pistikopoulos (2002) explicitly characterized the solution of the constrained QP problem of MPC as a piecewise linear and continuous state feedback law. Realizing MPC's potentially high on-line computational demand, researchers have begun to study the possibility of fast computation of an optimal or suboptimal solution to the optimization problems associated with MPC. Moreover, when MPC incorporates explicit model uncertainty, the resulting on-line computation will likely grow significantly (Braatz, Young, Doyle, & Morari, 1994) with the number of vertices of the uncertainty set which itself grows exponentially with the number of independent uncertain process parameters (Kothare et al., 1996).
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The requirement of optimality leads to high on-line MPC computation and limits the application of MPC to relatively slow dynamics and small-scale processes. In this formulation, at each sampling time, a state feedback law is computed by semidefinite programming (SDP) involving linear matrix inequality (LMI) constraints (Kothare et al., 1996). Since the optimal solution at time k+1 must be less than or equal to the feasible solution at time k+1, the closed-loop system is asymptotically stable. If the optimal feedback law computed at time k is applied at time k+1, the feasible upper bound at time k+1 must be less than the optimal upper bound at time k (Kothare et al., 1996). At each sampling time, an optimal upper bound on the worst case performance cost over the infinite horizon is obtained by forcing a quadratic function of the state to decrease at each prediction time by at least the amount of the worst case performance cost at that prediction time. This approach generally leads to a quadratic program (QP) that is solved at each sampling time.įor an infinite input and output horizon, a closed-loop feedback law must be adopted to facilitate a finite dimensional formulation (Kothare, Balakrishnan, & Morari, 1996).
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Since models are only an approximation of the real process, it is extremely important for MPC to be robust to model uncertainty (Bemporad & Morari, 1999).įor robust constrained state feedback MPC and a finite input horizon, a standard approach to synthesize a stabilizing MPC is to use the optimal input sequence at time k as a feasible input sequence at time k+1, and force the feasible cost at time k+1 to be less than the optimal cost at time k for each model in the uncertain set (Badgwell, 1997 Primbs & Nevistić, 2000). Although more than one input move is computed, the controller implements only the first computed input (Morari & Lee, 1999) and repeats these calculations at the next sampling time. At each sampling time, MPC uses an explicit process model and information about input and output constraints to compute process inputs so as to optimize future plant behavior over the prediction horizon. Model predictive control (MPC) is an effective control algorithm for dealing with multivariable constrained control problems that are encountered in the chemical process industries.