IQC analysis of linear constrained MPC W P

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 • LMI: Linear Matrix Inequality

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

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

IQC theory:

IQC notation:

IQC theory:

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

Example: small gain theorem

Example: multivariable circle criterion f

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

Quadratic programming and sector bounds

Quadratic programming and sector bounds

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

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

Diagonal augmentation

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

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.

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

Example

Example in standard form

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

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

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

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

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

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

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

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