455b

Míriam R. García^{1}, Carlos Vilas^{1}, Lino O. Santos^{2}, and **Antonio A. Alonso**^{1}. (1) Process Engineering, IIM-CSIC, Eduardo Cabello 6, Vigo, Spain, (2) Departamento de Engenharia Química, Universidade de Coimbra, Largo Marquês de Pombal, 3000, Coimbra, Portugal

Model Predictive Control (MPC) has been proved over the years to be an efficient control tool for a wide class of multivariable nonlinear dynamic systems. Its appeal must be found essentially in its conceptual simplicity which can be formulated as follows: For a given system, suppose we have a good representation of its dynamics in the form of a set of algebraic and difference or differential equations (DAES) describing the evolution of the states, as well as the constraints the process is subject to. In addition, let us assume that we have access to the states of the system by appropriate measurements. Then, we should be able to "foresee" the effect of future manipulations on the system on the evolution of the states. Moreover, we could select the set of actions or manipulations that will force the system to evolve in accordance with a given criterion which usually would appear in the form of maximizing or minimizing a certain functional of the states. Finally, if we are able to process this logic fast enough as compared with the dynamics of the process, we end up with a feed-back mechanism which despite possible plant disturbances is able to drive the operation in an optimal way.

The work we present here makes use of the MPC framework to develop a stabilizing and robust control scheme for a wide class of diffusion-convection-reaction systems representative of chemical and fluid dynamic processes. These processes are good examples of extremely high dimensional (in fact infinite dimensional) nonlinear dynamic systems where the "recursive" logic we mentioned above is very likely to fail due to a number of reasons directly or indirectly related with:

1. The high level of spatial discretization required to approximate the original set of partial differential equations by DAES either using finite differences or finite elements.

2. The dependence of the degree of discretization and discretization scheme on the dynamic properties of the resulting plant description.

2. The stiff nature of the resulting nonlinear DAE system.

3. The usually large times of convergence associated with the subsequent optimization problems.

These issues, and in particular points 1 and 2, are inherent to the distributed nature of these processes. They have been usually overcome in control and optimization applications by the so-called projection techniques which founded on the dissipative nature of the parabolic (diffusion based) PDE set and the time scale separation of dynamic modes, transform the original PDE into a low dimensional dynamic system capturing the most representative (slow) dynamics [1],[2]. The so-called POD (Proper orthogonal Decomposition) technique lies into this category where, essentially, we extract by means of repetitive measurements the set of space dependent basis functions (PODs) which define the low dimensional subspace where the dynamics are more likely to live. Projection of the original PDE set onto the set of PODs leads to a reduced order model constituted by a low dimensional set of ordinary differential equations which gives account of such dynamics, at least in the region where measurements where taken. In this way, the approach allows us to derive a set of representative low dimensional nonlinear dynamic models which, on one hand capture the slow and usually conflictive (as it can be unstable) dynamics. On the other hand, the resulting plant model could be efficiently integrated in real time applications such as MPC. Unfortunately these models are only valid locally.

Therefore and in order to provide a reliable description of the plant guaranteeing convergence of the MPC control scheme, a criterion is needed to either accept or to reject a given model. In the same way, an in-line automatic procedure must be devised which based on field measurements, would be able to build up the appropriate model when required. Our work concentrates on this two aspects of model identification and model construction so to ensure a stabilizing control action despite uncertainties and disturbances. The criterion we suggest is based on previous results developed by one of the authors to analyze process-model mismatch in the context on nonlinear MPC [3]. The automatic model generation tool exploits the underlying algebraic structure of the finite element method to efficiently produce and update suitable POD sets and to project on that set the original PDE-based model of the plant.

The resulting MPC scheme can be shown to be stable and therefore converge despite model uncertainty and disturbances. Since model is built in-line, the approach we propose can be interpreted as an MPC controller making use of multiple models (thus the name multi-model predictive control MMPC), the accuracy of each one being guaranteed on given regions of the state space. Nevertheless it can be anticipated that the necessary arguments are easily obtained from the dissipative nature of the class of systems and the criterion for model selection. Dissipation which always ensure the existence of a reduced order process representation of the plant for any operation region. The selection criterion, being based on the maximum allowed mismatch for a given model, will ensure continuous decreasing of the objective function and therefore asymptotic convergence.

[1] P.D. Christofides. Nonlinear and Robust Control of PDE Systems: Methods and Applications to Transport-Reaction Processes. Birkhäuser, Boston, 2001.

[2] A.A. Alonso, C.V. Fernández, and J.R. Banga. Dissipative systems: from physics to robust nonlinear control. Int. J. Robust Nonlinear Control, 14: 157-179, 2004.

[3] L.O. Santos and L.T. Biegler. A tool to analyze robust stability for model predictive controllers. J Process Contr., 9(3): 233-246, 1999.

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See more of The 2006 Annual Meeting