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IQC analysis of linear constrained MPC

IQC analysis of linear constrained MPC. W.P. Heath*, G. Li*, A.G. Wills † , B. Lennox* *University of Manchester † University of Newcastle, Australia. TLAs:. MPC: Model Predictive Control IQC: Integral Quadratic Constraint Also: KKT: Karush-Kuhn-Tucker KYP: Kalman-Yakubovich-Popov

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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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