Assessment and Future Directions of Nonlinear Model Predictive ...

6 E.F. Camacho and C. Bordons2.3 State EstimationWhen a state space model is being used, the system state is necessary for predictionand therefore has to be known. In some cases it is accessible throughmeasurements, but in general this is not the case and a state observer must beimplicitly or explicitly included in the control loop. The choice of an appropriateobserver may have influence on the closed-loop performance and stability.The most extended approach for output-feedback nmpc is based on the CertaintyEquivalence Principle. The estimate state ˆx is computed via a state observerand used in the model predictive controller. Even assuming that the observererror is exponentially stable, often only local stability of the closed-loopis achieved [24], i.e. the observer error must be small to guarantee stability ofthe closed-loop and in general nothing can be said about the necessary degreeof smallness. This is a consequence of the fact that no general valid separationprinciple for nonlinear systems exists. Nevertheless this approach is applied successfullyin many applications. A straightforward extension of the optimal linearfilter (Kalman filter) is the Extended Kalman filter ekf. The basic idea of ekfis to perform linearization at each time step to approximate the nonlinear systemas a time-varying system affine in the variables to be estimated, and toapply the linear filtering theory to it [26]. Although its theoretical propertiesremain largely unproven the ekf is popular in industry and usually performswell. Neither the Kalman Filter nor the extended Kalman Filter rely on on-lineoptimization, and neither handle constraints.There exists a dual of the nmpc approach for control for the state estimationproblem. It is formulated as an on-line optimization similar to nmpc and isnamed moving horizon estimation (mhe), see for example [1],[29],[41],[47]. Itis dual in the sense that a moving window of old measurement data is usedto obtain an optimization based estimate of the system state. Moving horizonestimation was first presented for unconstrained linear systems by Kwon et al.[22]. The first use of mhe for nonlinear systems was published by Jang et al.[15]. In their work, however, the model does not account for disturbances orconstraints. The stability of constrained linear mhe was developed by Muskeand Rawlings [30] and Rao et al. [37]. The groundwork for constrained nonlinearmhe was developed by Rao et al. [38]. Some applications in the chemical industryhave been reported [42].3 Solution of the NMPC ProblemIn spite of the difficulties associated with nonlinear modelling, the choice of appropriatemodel is not the only important issue. Using a nonlinear model changesthe control problem from a convex quadratic program to a nonconvex nonlinearproblem, which is much more difficult to solve and provides no guarantee thatthe global optimum can be found. Since in real-time control the optimum has tobe obtained in a prescribed interval, the time needed to find the optimum (oran acceptable approximation) is an important issue.