Certifying the robustness of model predictive controllers W. P. Heath and B. Lennox Control Systems Centre The University of Manchester
Dec 21, 2015
Certifying the robustness of
model predictive controllers
W. P. Heath and B. LennoxControl Systems CentreThe University of Manchester
Overview
• Motivating examples: cross-directional controledible oil refiningactive vibration control
• Robustness of MPC (i)• Robust linear control• IQC framework• Geometry of quadratic programs• Robustness of MPC (ii)• Cross-directional control example• Challenges
Two success stories for control engineering
• Model predictive control- industry led- wide interest in academia
• Robust linear control- developed in academia- industrial applications e.g. automotive, aerospace…
Overview
• Motivating examples: cross-directional controledible oil refiningactive vibration control
• Robustness of MPC (i)• Robust linear control• IQC framework• Geometry of quadratic programs• Robustness of MPC (ii)• Cross-directional control example• Challenges
Plastic film extrusion
Process variable: - thickness
Manipulated variables: - bolts at slice lip - machine speed
Profile response
Slice actuators(paper)
Modeled deflection(paper)
Observed stepresponse(plastic film)
Overview
• Motivating examples: cross-directional controledible oil refiningactive vibration control
• Robustness of MPC (i)• Robust linear control• IQC framework• Geometry of quadratic programs• Robustness of MPC (ii)• Cross-directional control example• Challenges
A. G. Wills and W. P. Heath. Application of barrier function model predictive control to an edible oil refining process. Journal of Process Control, 15(2):183-200, March 2005.
Results
• 76% reduction in separator 1 variation (sd)
• 78% reduction in separator 2 variation (sd)
• 10% increase in input flow variation (sd)
Start-up:manual to automatic operation
• Final set-up has both manual and automatic valves
• Best solution would be to have clutched handwheels with position sensors
• Implemented solution brings in one loop at a time via MPC’s constraint handling
Self-cleaning
• Periodically separator bowl opens to atmospheric pressure:
• Circa 40% volume lost during self-clean• During operation inputs frozen and
setpoints track measured variables• Fast recovery exploits
– observer– MPC constraint handling
Overview
• Motivating examples: cross-directional controledible oil refiningactive vibration control
• Robustness of MPC (i)• Robust linear control• IQC framework• Geometry of quadratic programs• Robustness of MPC (ii)• Cross-directional control example• Challenges
A. G. Wills et al. Model Predictive Control Applied to Constraint Handling in Active Noise and Vibration Control. IEEE Transactions on Control Systems Technology, 2007.
5KHz sampling
• MPC beats LQG with antiwindup at 5kHz
• Implemented on standard DSP
• On-line active set algorithm
• 12 step horizon
• Linear state space formulation with terminal weight and observer
• Boxed input constraints only
Common to all three examples:
• Low level control application
• Multivariable interactions
• Input constraints only
Overview
• Motivating examples: cross-directional controledible oil refiningactive vibration control
• Robustness of MPC (i)• Robust linear control• IQC framework• Geometry of quadratic programs• Robustness of MPC (ii)• Cross-directional control example• Challenges
Mayne et al., Automatica 2000
“While the problem has been studied and is now well understood, the outcome of the research is conceptual controllers that work well in principle but are too complex to employ.”
Magni and ScattoliniNMPC 2005
“Despite the large number of results available, it is believed that significant process [has] still to be done towards the development of algorithms guaranteeing satisfactory performances with an acceptable computational effort”
Why is it hard?
• Satisfying constraints renders the controller inherently nonlinear.
• State constraints introduce:feasibility issuesloss of sparseness and symmetry
• Remark: standard stability approaches impose state constraints
• Approaches such as min-max make matters even worse
What can we advise practitioners?
• Rewrite your code
• Extend your horizonsRemark: length of horizon and terminal weight depend on both current state and projected steady state position
• Detune your controllerRemark: no theory!
ZafiriouComputers chem. Engng. 1990
“One should study the problem in its nonlinear nature, obtain conditions that guarantee nominal and robust stability and performance and tune the parameters of the original optimization problems to satisfy them.”
Overview
• Motivating examples: cross-directional controledible oil refiningactive vibration control
• Robustness of MPC (i)• Robust linear control• IQC framework• Geometry of quadratic programs• Robustness of MPC (ii)• Cross-directional control example• Challenges
How do we generalise ideas to multivariable?
Rosenbrock
Manchester/Cambridge School
H∞ theory, μ synthesis etc.
Overview
• Motivating examples: cross-directional controledible oil refiningactive vibration control
• Robustness of MPC (i)• Robust linear control• IQC framework• Geometry of quadratic programs• Robustness of MPC (ii)• Cross-directional control example• Challenges
Overview
• Motivating examples: cross-directional controledible oil refiningactive vibration control
• Robustness of MPC (i)• Robust linear control• IQC framework• Geometry of quadratic programs• Robustness of MPC (ii)• Cross-directional control example• Challenges
We only consider
• Open-loop stable plant
• Linear plant model
• Input constraints
• Robust stability
Overview
• Motivating examples: cross-directional controledible oil refiningactive vibration control
• Robustness of MPC (i)• Robust linear control• IQC framework• Geometry of quadratic programs• Robustness of MPC (ii)• Cross-directional control example• Challenges
MPC robust stabilityFor MPC we can combine
– the quadratic programming nonlinearity – the model uncertainty
into a single block satisfying a single IQC.
• represents uncertainty• represents static nonlinearity (quadratic program)
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
• Motivating examples: cross-directional controledible oil refiningactive vibration control
• Robustness of MPC (i)• Robust linear control• IQC framework• Geometry of quadratic programs• Robustness of MPC (ii)• Cross-directional control example• Challenges
Cross-directional control with unit prediction horizon
R. M. Morales and W. P. HeathNumerical design of robust cross-directional control with saturating actuators.Control Systems 06, Finland.
Overview
• Motivating examples: cross-directional controledible oil refiningactive vibration control
• Robustness of MPC (i)• Robust linear control• IQC framework• Geometry of quadratic programs• Robustness of MPC (ii)• Cross-directional control example• Challenges
Challenges
• From analysis to control design
• Robust performance
• Output (and state) constraints
• Open loop unstable plant (e.g. integrating plants)