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Redictive control smtegies which handle inputloutput constraints optimize output tracking over a horizon, and thus tend to drive the controls to the constraint limits; this can lead to infeasibility and/or instability. Here we develop necessary and sufficient conditions for feasibility and stability, and propose an algorithm which overcomes finite horizon infeasibility and gives stability and asymptotic tracking. hmdncho -I IThe aim in predictive control is to predict, over a horizon n,, the vector of future tracking e m r s and minimize its norm over a given number 4 of fi~tun control moves. This is effective but can only guarantee stability for special cases. To remedy this, ConstrainedReceding Horizon Predictive Control (CRHPC) [I] adopts the basic Generalized Predictive Control (GPC) [2] strategy but introduces some terminal constraints; a similar algorithm was also proposed in [3]. An alternative approach was taken by Stable Generalized Redictive Control (SGFC) [4] which forms a stabilizing loop around the system first and then applies GFC to a closed loop configuration with finite impulse responses (FIRs); the use of FIRS implies that " i z a t i o n of the predicted error norm yields a monotonically decreasing cost, and this guarantees stability and asymptotic tracking.These properties carry over to the case of predictive control with constraints so long as the implied opti-tion problem (a quadratic programming problem in Quadratic Programming Generalized Predictive Control, QPGPC, see Ref. [SI, or a mixed weights least squares, MWLS, problem in Constrained Stable Generalized Redictive Control, CSGFC, see Ref.[a]) is feasible. This is a strong assumption: it requires "short term" feasibility (feasibility over finite horizons); shofl term infeasibility does not imply overall infeasibility. Thus the requirement that the output reach a target value may be sensible, in that a feasible solution exists, but this does not imply that the target can be reached within n, steps without constraint violations.What is worse, in some cases both QpGFC and CSGFC, can convert a feasible problem into an infeasible one: in order to cause the predicted output IO reach its target within n, steps it may be necessary to drive the controls to their limits, but this is done without taking future stability into account. For unstable and/or non-minimum phase systems, future stability may require even harder future control moves which are not possible. The inevitable result of all this is instability.In this paper we develop necessary and sufficient conditions under which this situation can be avoided; these conditions are a posteriori conditions in that they are based on past data. Violation of a posteriori conditions will cause instability. These results provide a test for when things have gone wrong, but do not provide the mcchanism for avoiding instability. To do this one can advance the a posteriori conditions one step ahead in time and derive a priori conditions which can be used to limit the choice of future control moves. Thi...