Robust stability analysis of constrained l₁‐norm model predictive control

Abstract

Sufficient conditions for robust closed-loop stability of a class of dynamic matrix control ( D M C ) systems arepresented. The I,-norm is used in the objective function of the on-line optimization, thus resulting in a linear programming problem. The ideas of this work, however, are expandable to other DMC-type controllers. The keys to the stability conditions are: to use an end-condition in the moving horizon on-line optimization; to have coefficients of the move suppression term in the objective function of the on-line optimization satisfy certain inequalities; and to express the uncertainty as deviations in the unit pulse response coefficients of the nominal plant. These deviations and disturbances must also satisfy certain inequalities.A n off-line tuning procedure for robust stability and performance of a class of DMC controllers is also included, which determines an optimal moving horizon length and optimal values for coefficients of the move suppression term. The applicability of our approach is elucidated through numerical simulations. IntroductionIn recent years, several multivariable control techniques falling in the general category of model predictive control (MPC) have been studied and successfully implemented to industrial processes. MPC variations include model predictive heuristic control (MPHC) (Richalet et al., 1978), model algorithmic control (MAC) (Mehra et al., 1979), dynamic matrix control (DMC) (Cutler and Ramaker, 1980; Prett and Gillete, 1979), linear dynamic matrix control (LDMC) (Morshedi et al., 1985) and quadratic dynamic matrix control (QDMC) (Garcia and Morshedi, 1986).The MPHC, MAC, DMC, LDMC and QDMC algorithms share a common characteristic in using a pulse or a step response model, unlike other MPC algorithms that use a parametric model based on physical laws.Several investigators have tried to develop a theory for analyzing the stability properties of unconstrained and constrained MPC closed-loop systems. Garcia and Morari (1982) were first to study the effects of the tuning parameters (prediction and control horizons, penalty on process input changes) on the closed-loop stability of unconstrained DMC systems in the internal model control (IMC) framework. Their later workCorrespondence concerning this article should be addressed to M. Nikolaou 1954December 1993 concentrated on utilizing robust linear control theory to determine the robust stability of unconstrained DMC systems (Prett and Garcia, 1988;Morari and Zafiriou, 1989). Zafiriou and coworkers used the contraction mapping principle proposed by Economou (1985) to develop sufficient conditions for the stability of QDMC closed-loop systems with process input/output constraints (Zafiriou, 1988(Zafiriou, , 1989(Zafiriou, , 1990 Zafiriou and Marshal, 1991;Zafiriou and Chiou, 1989). They showed that the tuning rules developed for the unconstrained case may not work well for a constrained system, and in fact may cause instability in the presence of output constraints.Rawlings and Muske (1993) used a state-space framework to de...

Model predictive control of nonlinear systems: Computational burden and stability

“…They show that, under certain assumptions, predictive control can stabilise plants when a terminal equality constraint is embedded in the optimisation problem; that is, it is required that the terminal state is at a point in the state space. Similar results also have been reported by Genceli and Nikolaou (1993) and Rawlings and Muske (1993). However, from the computational point of view, solving a nonlinear dynamic optimisation problem with equality constraints is highly computationally intensive and in many cases is impossible to perform within a limited time.…”

“…This follows from modification of the standard argument; for example, see Genceli and Nikolaou (1993), Michalska and Mayne (1993), Chen and Allgöwer (1998).…”

Optimal Rules for Central Bank Interest Rates Subject to Zero Lower Bound

“…(37) and (38) Following the preceding observations, it should be noted that the widespread practice of using a discount factor β may be more problematic than realized, in the sense that it may not result in robustly stabilizing strategies. This situation, namely the need to shape weights of the terms in the MPC objective in an increasing rather than decreasing fashion in order to ensure robustness, has been rigorously analyzed in the past (Genceli and Nikolaou, 1993;Vuthandam et al, 1995) and should be explored further.…”

“…The associated Table 15 shows the resulting coefficient for the Taylor-like solution provided by MPC. Second, it has been rigorously shown that keeping m small improves the robustness of the closed loop, namely it helps maintain closed-loop stability in the presence of discrepancies between the model used by MPC and the actual system under control (Garcia and Morari, 1982;Genceli and Nikolaou, 1993;Vuthandam et al, 1995).…”

Temporal multi-level coordination techniques oriented to regional water networks: application to the Catalunya case study

“…The most widely referenced approach to guarantee stability in MPC procedures is to add an equality constraint on the final state in the prediction horizon (a so called end-state constraint) or put a weight on the final state in the objective function (De Nicolao & Scattolini, 1998;Genceli & Nikolaou, 1993;Kwon & Pearson, 1977;Kwon & Byun, 1989;Mayne & Michalska, 1990;Thomas, 1975). Another approach is to use an infinite prediction horizon with a finite control horizon (Rawlings & Muske, 1993), making it possible to apply standard linear quadratic regulator (LQR) theory to guarantee stability (Kwakernaak & Sivan, 1972;Cheng & Krogh, 2001).…”