(188i) Stability Analysis of Model Predictive Control Using Piecewise Affine Models Under Unstructured Uncertainty

Model predictive control (MPC) is a powerful control methodology that can deal with complex nonlinear dynamic systems. Models of system dynamics can be used in the resulting optimisation problem alongside with any physical or additional constraints (Mayne, Rawlings, Rao, & Scokaert, 2000). The dynamic model may be nonlinear and complex and additive tightening constraints can be included (Mayne, 2014)⁠. Subsequently the optimisation problem may require large computational time (Bonis et al., 2012)⁠. Therefore, in many cases, simple controllers need to be implemented. The use of linear models may solve the problem; however linearisation at the reference point may not be adequate to describe the behaviour of complex nonlinear dynamics, especially in the case of distributed parameter systems (Theodoropoulos et al., 2000). Input/output stability analysis using integral quadratic constraints (IQCs) for linear systems under uncertainty can circumvent this by providing necessary stability conditions when there are structured and unstructured uncertainties (Heath et al., 2006). Terminal and tightening constraints can be avoided as the analysis does not require any of these.

Piecewise affine (PWA) models have been widely used in MPC (Bemporad & Morari, 1999), (Rewienski & White, 2003), (Xie et al, 2011)⁠ to adequately describe the nonlinear dynamics using different linear models to span the state domain. In this work, we extend the applicability of IQCs for input/ouput stability analysis to PWA systems. To avoid the expensive computational cost of mixed integer programming while employing online optimisation, one linear model is used for each sampling time. In order to construct the model pool we collect computed trajectories and linearization is applied at selected transient points. Furthermore, even though the constructed PWA models approximate complex dynamics efficiently, uncertainties may still arise. IQCs can describe the input/output behaviour of nonlinear and uncertain components in the closed loop. Therefore all possible troublemaking parts can be considered assuming that they satisfy IQCs. Then dissipation inequalities can include IQCs to analyse input/output stability. Additionally, the constructed controller selects which model will be used every sampling time. Therefore changes from model to model may destabilise the system. Our analysis is able to guarantee that the closed loop system will remain stable. The proposed methodology is demonstrated through two chemical engineering case studies.

References

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Bonis, I., Xie, W., & Theodoropoulos, C. (2012). A Linear Model Predictive Control Algorithm for Nonlinear Large-Scale Distributed Parameter Systems. AIChE JournalAIChE Journal, 58, 801–811.

Heath, W. P., Li, G., Wills, A. G., & Lennox, B. (2006). The robustness of input constrained model predictive control to infinity-norm bound model uncertainty. IFAC Proceedings Volumes, 39(9), 495–500.

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Mayne, D. Q., Rawlings, J. B., Rao, C. V., & Scokaert, P. O. M. (2000). Constrained model predictive control: Stability and optimality. Automatica, 36(6), 789–814.

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Theodoropoulos, C., Qian, Y., & Kevrekidis, I. G. (2000). “ Coarse ” stability and bifurcation analysis using time-steppers : A reaction-diffusion example, PNAS, 97(18), 9840–9843.

Xie, W., Bonis, I., & Theodoropoulos, C. (2011). Off-line model reduction for on-line linear MPC of nonlinear large-scale distributed systems. Computers & Chemical Engineering, 35(5), 750–757.