Proceedings of the American Control Conference San Diego, California June zyxwvutsrqponm 1999 Efficient robust predictive control zyx B. Kouvaritakis Dept. Engineering Science, Parks Road, Oxford. OX1 3PJ, UK Email: zyxwvu Basil.KouvaritakisQeng.ox.ac.uk J. A. Rossiter J. Schuurmans Department of Mathematical Sciences, Loughborough University, Leicestershire. LE11 3TU, UK hail: J.A.RossiterOlboro.ac.uk Abstract Invariant sets can be used in conjunction with known control laws to guarantee convergence of LTV or un- certain systems with constraints. However, invariant sets are often associated with fixed control laws and hence can be limited in size by constraints. Here it is shown how to utilise degrees of freedom in the tran- sient predictions in order to enlarge the invariant set and so widen the applicability of the convergence proof, for a given control law. The same degrees of freedom allow for computationally efficient and systematic per- formance optimisation. 1 Introduction Predictive control 131 is a popular strategy because it systematically accounts for performance and con- straints. Moreover, it is possible to give stability re- sults for linear systems, even during online constraint handling (e.g. [9], [8], [13]). Researchers are now con- centrating more on non-linear and uncertain systems where rigourous results and computationally efficient algorithms are more difficult to find. One early result guaranteeing convergence for non-linear and uncertain systems uses the concept of a dual mode controller [5], that is a control law which takes a different form de- pending on how close the system state is to its desired value; this paper will make use of that result. Dual mode schemes rely on a terminal condition that once the system state enters a region near the origin, a fixed control law is sufficient for convergence and constraint satisfaction. A recent study [6] followed this approach further and defined an appropriate invariant set (a set such that once the state enters, it will remain within). An algorithm was proposed for selecting a control law which gave an invariant set containing the current state consequently giving a guarantee of stability even for un- certain linear time varying (LTV) systems. However, although this control scheme allowed the control law to vary from sample to sample, it assumed a fixed linear control law in the predictions; this limits performance and requires a significant on-line computational burden which limits its applicability. The scheme is also lim- ited to those states lying inside invariants sets for fixed linear control laws. Performance can be improved considerably by allowing the controller to be nonlinear in the predictions. One way of achieving this is by allowing the first zyxw N future moves to become degrees of freedom. In [12] the con- dition that the control law becomes fixed and the state enter the invariant set is delayed until a few samples into the future. However since the system considered is uncertain the predictions over the transients are not single valued but rather are contained in polytopic sets and for anything other than small horizons the number of vertices becomes unmanageable. Here we overcome this problem by using invariant sets to capture the sys- tem prediction, even in the transient part where the control law is not fixed. It is shown that the appropri- ate invariant set can be significantly larger than that obtained with a fixed linear law (e.g. [SI) and moreover the online computational burden can be reduced dra- matically. The particular predictive control strategy of ([7], [lo], [ll]) is best suited to these developments and is used here. In this approach the predictive con- trol law optimises closed-loop predictions rather than open-loop predictions; the predictions are optimised by varying a free variable which is a loop input. The ad- vantage being that apart from giving better numerical conditionning to the computations, it also allows the nominal closed-loop to be ‘optimally designed’ in some sense, say with regard to performance and robustness. When there are no predicted constraint violations, the extra d.0.f. can be set to zero. The paper is structured as follows. Section 2 gives some backgrund and notation. In section 3 it is demon- strated how d.0.f. in transients can enlarge invariant sets. Section 4 illustrates the advantages and discusses how to augment the robustness yet further. The paper finishes with a conclusion. 0-7803-4990-6/99 $10.00 zyxwvuts 0 1999 AACC 4283