Transcript Presentation 6 (ppt, 788kb)

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
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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 G y (z )
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
• 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
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)
B
 f ( x)dx
A
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?