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 

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

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:  represents uncertainty  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 2x2 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 is equivalent to an LMI For MPC: LMI equation dimension grows linearly with horizon LMI solution dimension is independent of horizon The stability condition

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)  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 2x2 system and a 10 step horizon MPC Current work: –How should we optimise multipliers? –How conservative is the test?