# (197d) Reduced Linear Model Predictive Control for Non-Linear Distributed Parameter Systems

Authors:
University of Manchester
University of Manchester

The most successful among the advanced control strategies is Model Predictive Control [1]. There exist many variants of the MPC algorithm, all of which follow the general concept of implementing a control action, which is computed as a result of an optimisation problem, aimed to minimize the deviation from the set point and at the same time the control energy required. In the standard MPC formulation, the model of the system is linear, so are the constraints. However, most systems of engineering interest, and especially distributed parameter systems (DPS) which we consider in this work, are nonlinear. Nonlinear MPC [2] has significantly higher computational cost, which in practice limits its applications to rather small systems. Computational time is crucial for control applications, as control decisions have to be taken within a predifined interval (i.e. before the next control move has to be implemented). Extracting a linear model for a system exhibiting nonlinear dynamics is not straightforward. Linearization at the set point may not be satisfactory, as the model may be a poor approximation for the time horizon considered in the optimisation subproblem. To this end several self-tuning control algorithms have been developed to account for the nonlinearity of the system [3]. Even if the model is both linear and accurate, the hurdle of computational cost is not easy to overcome, especially for the case of distributed parameter systems. In parametric MPC algorithms, the optimisation problem is solved offline and the corresponding control decisions are stored and retrieved when needed [4]. This approach trades computational cost with higher storage requirements.

In this contribution a novel model reduction based scheme for linear predictive control of nonlinear DPS is presented. It exploits the natural separation of scales exhibited in many engineering problems. Roughly speaking, the concept is to apply MPC on the dominant modes of the system. Those modes however change as the system progresses in time and parameter space. Hence instead of doing a one-off linearization at the set point, which is the conventional technique, we follow a successive linearization paradigm which identifies the matrices involved in the linear state space representation of the system. The resulting system is equivalent to projecting the linearized system onto the low-dimensional subspace corresponding to the slowest modes. Reduced Jacobian and sensitivity information can be calculated with a few directional perturbations to the direction of the basis for this subspace. Recent process data can be used to reduce the computational cost of the online computation [5]. The resulting linear model can be exploited in the context of an MPC algorithm, which includes solving a reduced QP subproblem in every timestep. The key features which differentiate the proposed algorithm from the standard MPC algorithm, is the identification of a low-dimensional dominant subspace in every timestep, the adaptive and effiecient linearization on this subspace and the solution of a reduced optimisation problem rather than the full one. The proposed methodology would be particularly useful for the case of multi-scale systems. The efficiency of the proposed algorithm is demonstrated through illustrative case studies.

References

1. Mayne, DQ et al, Automatica, 2000. 36(6):789-814.

2. Henson MA, Computers & Chemical Engineering, 1998. 23(2):187-202.

3. Zhu, QM et al, IEEE Proceedings-D Control Theory and Applications, 1991. 138(1):33-40.

4. Sakizlis, V et al, Industrial & Engineering Chemistry Research, 2003. 42(20):4545-4563.

5. C. Theodoropoulos and E.L. Ortiz, in Model reduction and coarse-graining approaches for multi-scale phenomena, pp 535-560, Springer 2006.