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 

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

12. Quadratic programming and 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:  represents uncertainty  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 is equivalent to an LMI For MPC: LMI equation dimension grows linearly with horizon LMI solution dimension is independent of horizon The stability condition

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)  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?