Certifying the robustness of model predictive controllers W. P. Heath and B. Lennox Control Systems Centre The University of Manchester

Overview Motivating examples: cross-directional control edible oil refining active 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 control edible oil refining active vibration control Robustness of MPC (i) Robust linear control IQC framework Geometry of quadratic programs Robustness of MPC (ii) Cross-directional control example Challenges

Generic web forming process Paper making, plastic film extrusion, steel rolling etc.

Plastic film extrusion Process variable: - thickness Manipulated variables: - bolts at slice lip - machine speed

Profile response Slice actuators (paper) Modeled deflection (paper) Observed step response (plastic film)

Typical control improvement

Actuator Picketing Actuator constraints of the form:

Overview Motivating examples: cross-directional control edible oil refining active 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): , March 2005.

Separator operation

Quality/yield trade-off open loop

Quality/yield trade-off closed loop

MPC algorithm Muske and Rawlings (1993):

Before Flow Pressure 1 Pressure 2

After Pressure 1 Pressure 2 Flow

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

Start-up Manipulated variables Process variables

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

Self-cleaning Manipulated variables Process variables

Overview Motivating examples: cross-directional control edible oil refining active 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, KHz 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 control edible oil refining active 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 Scattolini NMPC 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 issues loss 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 horizons Remark: length of horizon and terminal weight depend on both current state and projected steady state position Detune your controller Remark: no theory!

Zafiriou Computers chem. Engng “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 control edible oil refining active vibration control Robustness of MPC (i) Robust linear control IQC framework Geometry of quadratic programs Robustness of MPC (ii) Cross-directional control example Challenges

Gain margin

Phase margin

Gain margin

How do we generalise ideas to multivariable? Rosenbrock Manchester/Cambridge School H ∞ theory, μ synthesis etc.

Sensitivity analysis

Plant model

Closed loop system

Small gain theorem If and then the loop is stable

Overview Motivating examples: cross-directional control edible oil refining active vibration control Robustness of MPC (i) Robust linear control IQC framework Geometry of quadratic programs Robustness of MPC (ii) Cross-directional control example Challenges

IQC theory:

IQC notation:

IQC theory:

Example: small gain theorem

Example: multivariable circle criterion 

Overview Motivating examples: cross-directional control edible oil refining active 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

Multi-parametric quadratic programming

Sector bound

MPC stability We can use IQC theory to test stability of many MPC structures. For example:

Equivalent loop  represents static nonlinearity (quadratic program)

Overview Motivating examples: cross-directional control edible oil refining active 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 stability For 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

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 Motivating examples: cross-directional control edible oil refining active 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. Heath Numerical design of robust cross-directional control with saturating actuators. Control Systems 06, Finland.

Robustness analysis

Corresponding range in the modes.

Tuning parameter at each mode

Final profile value (with and without control) in both profile and mode space.

Corresponding actuator position, and second moment of actuators.

Overview Motivating examples: cross-directional control edible oil refining active vibration control Robustness of MPC (i) Robust linear control IQC framework Geometry of quadratic programs Robustness of MPC (ii) Cross-directional control example Challenges

From analysis to control design Robust performance Output (and state) constraints Open loop unstable plant (e.g. integrating plants)

Thank you! W. P. Heath and B. Lennox Control Systems Centre The University of Manchester