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373029 Output Feedback Lyapunov Model Predictive Control of Stochastic Nonlinear Systems

**Output Feedback Lyapunov Model Predictive Control of Stochastic Nonlinear Systems**

Summary

The challenges of strong nonlinearities, system constraints, disturbances, and uncertainty are ubiquitous in the modern control landscape, from chemical reactors to fighter jet airframes.

In addressing these challenges, controllers based on Nonlinear Model Predictive Control (NMPC) [13] have enjoyed huge popularity due to their explicit constraint-handling ability [14], and ease of deployment. This controller anticipates the plant’s response to control actions to devise the optimal future control plan, which is then repeatedly re-optimized as real-time information is incorporated. When NMPC is combined with Lyapunov analysis (LMPC) [15], the designer can guarantee feasibility and stability [12] prior to implementation, provided the system is within a known space of initial conditions.

However, without adequate modelling of disturbances, performance will be degraded. Since many process disturbances are inherently stochastic, control laws designed to attenuate stochastic disturbances can achieve better performance than conventional controllers based on ‘worst case’ bounded disturbance models.

In one approach, a stochastic LMPC framework is used to achieve optimal plant control. This controller accounts for system constraints as well as the distributions of probabilistic variables, but is also re-optimized as random variable sample paths are realized [8].

In [1], an LMPC formulation was developed for a class of stochastic nonlinear systems with affine input and additive vanishing noise. The paper first addressed the system’s closed-loop stability and boundedness properties subject to discrete control implementation. The resulting system was stable in the sense that it had a known probability of being contained within given bounds using sufficiently small hold times. It was then shown that the Lyapunov function decay constraints were always feasible and that receding horizon implementation of the above scheme would provide stability in probability guarantees.

In many applications, the model’s states are not measured (for technical or economic reasons), but can sometimes be inferred from the response of the system through the use of a state observer. For linear systems, the observer’s design is exact, separable, and has asymptotically stable error dynamics [7]. Even for a stochastic linear system, filtering can typically yield accurate observers. In a nonlinear setting, techniques that rely on approximate local linearization have seen the most use [6], but the need for nonlinear observers with globally stable error has led to results on observer designs for completely feedback linearizable systems [5].

In [2] it was shown that systems in stochastic output feedback canonical form admit global state estimators. Using integrator backstepping and stochastic Lyapunov techniques, the connected system was stabilized in probability. The analysis also shows that the observer’s error dynamics, which are in effect being afflicted by the random processes of the plant, can be designed to be asymptotically stable in probability as well.

In the present paper, we consider the problem of an output feedback Lyapuov-based MPC design that recognizes the stochastic nature of the system. To this end, we first define a state observer for systems diffiomorphic [4] to the specially structured stochastic output feedback form [3]. Then, we determine how the risk margins for stability and boundedness of the estimated states are affected by the observer dynamics, and how the dynamics of the controlled system under output feedback differ from a state feedback configuration. Lastly, we present a unified output feedback stochastic LMPC implementation. Simulation results are presented to illustrate the efficacy of the proposed results, using an isothermal reactor with sequential reaction kinetics.

References

[1] M. Mahmood, P. Mhaskar. Lyapunov-based model predictive control of stochastic nonlinear systems. *Automatica*, 48(9):2271-2276, 2012.

[2] H. Deng, M. Krstic. Output-feedback stochastic nonlinear stabilization. *IEEE Transactions on Automatic Control*, 44(2):328-333, 1999.

[3] S. Liu, Z. Jiang, J. Zhang. Global output-feedback stabilization for a class of stochastic non-minimum-phase nonlinear systems. *Automatica*, 44(8):1944-1957, 2008.

[4] Z. Pan. Differential geometric condition for feedback complete linearization of stochastic nonlinear system. *Automatica*, 37(1):145-149, 2001.

[5] A. Krener and W. Respondek. Nonlinear Observers with Linearizable Error Dynamics. *SIAM Journal of Control and Optimization*, 23(2):197-216, 1985.

[6] S. Julier and J. Uhlmann. Unscented Filtering and Nonlinear Estimation*. Proceedings of the IEEE*, 92(3):401-422, 2004.

[7] C. Chen, Linear Systems Theory and Design, 3^{rd} Edition. *Oxford University Press*, New York. Chapter 8, Page 253, 1999.

[8] M. Cannon, B. Kouvaritakis, and X. Wu. Probabilistic Constrained MPC for Multiplicative and Additive Stochastic Uncertainty. *IEEE Transactions on Automatic Control*, 54(7):1626–1632, 2009.

[9] R. Chabour and M. Oumoun. On a universal formula for the stabilization of control stochastic nonlinear systems. *Stochastic Analysis and Applications*, 17:359–368, 1999.

[10] H. Deng, M. Krstic, and R. J. Williams. Stabilization of stochastic nonlinear systems driven by noise of unknown covariance. *IEEE Transactions on Automatic Control*, 46(8):1237–1253, 2001.

[11] D. V. Hessem and O. Bosgra. Stochastic closed-loop model predictive control of continuous nonlinear chemical processes. *Journal of Process Control*, 16(3):225–241, 2006.

[12] M. Mahmood and P. Mhaskar. Enhanced stability regions for model predictive control of nonlinear process systems. *AIChE J.*, 54:1487–1498, 2008.

[13] D. Q. Mayne. Nonlinear model predictive control: An assessment. In *Proceedings of 5th International Conference on Chemical Process Control*, Pages 217–231, Tahoe City, CA, 1997.

[14] D. Q. Mayne, J. B. Rawlings, C. V. Rao, and P. O. M. Scokaert. Constrained model predictive control: Stability and optimality. *Automatica*, 36:789–814, 2000.

[15] P. Mhaskar, N. H. El-Farra, and P. D. Christofides. Techniques for uniting Lyapunov-based and model predictive control. In Rolf Findeisen, Frank Allgцwer, and Lorenz Biegler, Editors, *Assessment and Future Directions of Nonlinear Model Predictive Control*, Volume 358 of *Lecture Notes in Control and Information Sciences*, Pages 77–91. Springer Berlin / Heidelberg, 2007.

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