Robust Model Predictive Control for Time Delayed Systems

Abstract

Model Predictive Control (MPC) is frequently implemented as one of the layers of a control structure where a Real Time Optimization (RTO) algorithm - laying in an upper layer of this structure - defines optimal targets for some of the inputs and/or outputs (Kassmann et al., 2000). The main scope is to reach the most profitable operation of the process system while preserving safety and product specification constraints. The model predictive controller is expected to drive the plant to the optimal operating point, while minimizing the dynamic error along the input and output paths. Since in the control structure considered here the model predictive controller is designed to track the optimal targets, it is expected that for nonlinear process systems, the linear model included in the controller will become uncertain as we move from the design condition to the optimal condition. The robust MPC presented in this chapter explicitly accounts for model uncertainty of open loop stable systems, where a different model corresponds to each operating point of the process system. In this way, even in the presence of model uncertainty, the controller is capable of maintaining all outputs within feasible zones, while reaching the desired optimal targets. In several other process systems, the aim of the MPC layer is not to guide all the controlled variables to optimal targets, but only to maintain them inside appropriate ranges or zones. This strategy is designated as zone control (Maciejowski, 2002). The zone control may be adopted in some systems, where there are highly correlated outputs to be controlled, and there not enough inputs to control all the outputs. Another class of zone control problems relates to using the surge capacity of tanks to smooth out the operation of a process unit. In this case, it is desired to let the level of the tank to float between limits, as necessary, to buffer disturbances between sections of a plant. The paper by Qin and Badgwell (2003), which surveys the existing industrial MPC technology, describes a variety of industrial controllers and mention that they always provide a zone control option. Other example of zone control can be found in Zanin et al, (2002), where the authors exemplify the application of this strategy in the real time optimization of a FCC system. Although this strategy shows to have an acceptable performance, stability is not usually proved, even when an infinite horizon is used, since the control system keeps switching from one controller to another throughout the continuous operation of the process. There are several research works that treat the problem of how to obtain a stable MPC with fixed output set points. Although stability of the closed loop is commonly achieved by means of an infinite prediction horizon, the problem of how to eliminate output steady state offset when a supervisory layer produces optimal economic set points, and how to explicitly incorporate the model uncertainty into the control problem formulation for this case, remain an open issue. For the nominal model case, Rawlings (2000), Pannochia and Rawlings (2003), Muske and Badgwell (2002), show how to include disturbance models in order to assure that the inputs and states are led to the desired values without offset. Muske and Badgwell (2002) and Pannochia and Rawlings (2003) develop rank conditions to assure the detectability of the augmented model. For the uncertain system, Odloak (2004) develops a robust MPC for the multi-plant uncertainty (that is, for a finite set of possible models) that uses a non-increasing cost constraint (Badgwell, 1997). In this strategy, the MPC cost function to be minimized is computed using a nominal model, but the non-increasing cost constraint is settled for each of the models belonging to the set. The stability is then achieved by means of the recursive feasibility of the optimization problem, instead of the optimality. On the other hand, there exist some recent MPC formulations that are based on the existence of a control Lyapunov function (CLF), which is independent of the control cost function. Although the construction of the CFL may not be a trivial task, these formulations also allow the explicit characterization of the stability region subject to constraints and they do not need an infinite output horizon. Mashkar et al. (2006) explore this approach for the control of nominal nonlinear systems, and Mashkar (2006) extends the approach for the case of model uncertainty and control actuator fault. More recently, González et al. (2009) extended the infinite horizon approach to stabilize the closed loop with the MPC controller for the case of multi-model uncertainty and optimizing targets. They developed a robust MPC by adapting the non-increasing cost constraint strategy to the case of zone control of the outputs and it is desirable to guide some of the manipulated inputs to the targets given by a supervisory stationary optimization stage, while maintaining the controlled output in their corresponding zones, taking into account a finite set of possible models. This problem, that seems to interchange an output tracking by an input-tracking formulation, is not trivial, since once the output lies outside the corresponding zone (because of a disturbance, or a change in the output zones), the priority of the controller is again to control the outputs, even if this implies that the input must be settled apart from its targets. Since in many process systems, mainly from the chemical and petrochemical industries, the process model shows significant time delays, the main contribution of this chapter is to extend the approach of González et al. (2009) to the case of input delayed multi-model systems by introducing minor modifications in the state space model, in such a way that the structure of the control algorithm is preserved. Simulation of a process system of the oil refining industry illustrates the performance of the proposed strategy.