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 programmingand sector bounds

Quadratic programmingand 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 )(zGy

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 pathB

A

dxx)(

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?