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IQC analysis of linear constrained MPC

This paper explores the integration of Integral Quadratic Constraint (IQC) theory in analyzing Model Predictive Control (MPC) frameworks under linear constraints. It includes discussions on KKT, KYP lemmas, and the utilization of Zames-Falb multipliers to derive robust stability conditions for MPC systems. We illustrate how IQC can facilitate the stability testing of various MPC structures, emphasizing important examples and computational aspects. The work seeks to establish robustness in the presence of uncertainties and nonlinearities while evaluating the efficiency of the proposed methodologies.

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IQC analysis of linear constrained MPC

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  1. IQC analysis of linear constrained MPC W.P. Heath*, G. Li*, A.G. Wills†, B. Lennox* *University of Manchester †University of Newcastle, Australia

  2. TLAs: • MPC: Model Predictive Control • IQC: Integral Quadratic Constraint Also: • KKT: Karush-Kuhn-Tucker • KYP: Kalman-Yakubovich-Popov • LMI: Linear Matrix Inequality

  3. Overview • IQC theory • Familiar examples • Quadratic programming and sector bounds • Robustness of MPC • Example • Computation • Zames-Falb multipliers

  4. Overview • IQC theory • Familiar examples • Quadratic programming and sector bounds • Robustness of MPC • Example • Computation • Zames-Falb multipliers

  5. IQC theory:

  6. IQC notation:

  7. IQC theory:

  8. Overview • IQC theory • Familiar examples • Quadratic programming and sector bounds • Robustness of MPC • Example • Computation • Zames-Falb multipliers

  9. Example: small gain theorem

  10. Example: multivariable circle criterion f

  11. Overview • IQC theory • Familiar examples • Quadratic programming and sector bounds • Robustness of MPC • Example • Computation • Zames-Falb multipliers

  12. Quadratic programmingand sector bounds

  13. Quadratic programmingand sector bounds

  14. MPC stability We can use IQC theory to test stability of many MPC structures. For example: Remark: there is no requirement for MPC internal model to match the plant

  15. Overview • IQC theory • Familiar examples • Quadratic programming and sector bounds • Robustness of MPC • Example • Computation • Zames-Falb multipliers

  16. Diagonal augmentation

  17. So we can combine uncertainty and static nonlinearities: • D represents uncertainty • f represents static nonlinearity

  18. MPC robust stability For MPC we can combine • the quadratic programming nonlinearity • the model uncertainty into a single block satisfying a single IQC. It remains to test the condition on the remaining linear element.

  19. Overview • IQC theory • Familiar examples • Quadratic programming and sector bounds • Robustness of MPC • Example • Computation • Zames-Falb multipliers

  20. Example

  21. Example in standard form

  22. Example: • 10 step horizon • 2x2 plant • IQC made up from four separate blocks (two nonlinearities and 2 uncertainties) • Weight on states is 1/k

  23. Overview • IQC theory • Familiar examples • Quadratic programming and sector bounds • Robustness of MPC • Example • Computation • Zames-Falb multipliers

  24. KYP lemma The stability condition is equivalent to an LMI • For MPC: • LMI equation dimension grows linearly with horizon • LMI solution dimension is independent of horizon

  25. Overview • IQC theory • Familiar examples • Quadratic programming and sector bounds • Robustness of MPC • Example • Computation • Zames-Falb multipliers

  26. Multipliers and IQCs • Multipliers allow more general choice of IQC • This in turn leads to less conservative stability results • Natural expression and generalisaiton of (for example): • Commutant sets for structured uncertainty • Nonlinear results such as Popov stability criterion

  27. Zames-Falb multipliers Zames and Falb introduced general class of multipliers (1968) f is - bound - monotone nondecreasing - slope restricted Safanov and Kulkarni considered their application to multivariable nonlinearities (2000) independent of path

  28. Zames-Falb multipliers for quadratic programming Result: Zames-Falb multipliers can be applied to the quadratic programme nonlinearity. Proof: via KKT conditions and convexity. Compare: - Fiacco et al: sensitivity analysis in nonlinear programming - Geometry of multiparametric quadratic programming

  29. Conclusion • IQC theory provides a robust stability test of simple MPC loops (with arbitrary horizon) • We have illustrated the test for a 2x2 system and a 10 step horizon MPC • Current work: • How should we optimise multipliers? • How conservative is the test?

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