iqc analysis of linear constrained mpc
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
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?