H0K03a : Advanced Process Control Model-based Predictive Control 4 : - - PowerPoint PPT Presentation

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H0K03a : Advanced Process Control Model-based Predictive Control 4 : - - PowerPoint PPT Presentation

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H0K03a : Advanced Process Control Model-based Predictive Control 4 : Robustness Bert Pluymers Prof. Bart De Moor Katholieke Universiteit Leuven, Belgium Faculty of Engineering Sciences Department of Electrical Engineering (ESAT) Research Group

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SLIDE 1

Bert Pluymers

  • Prof. Bart De Moor

Katholieke Universiteit Leuven, Belgium Faculty of Engineering Sciences Department of Electrical Engineering (ESAT) Research Group SCD-SISTA

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

H0K03a : Advanced Process Control

Model-based Predictive Control 4 : Robustness

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1

Overview

  • Example
  • Robustness
  • Robust MPC
  • Conclusion

Lecture 4 : Robustness

  • Example
  • Robustness
  • Robust MPC
  • Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

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SLIDE 3

2

Example

  • Example
  • Robustness
  • Robust MPC
  • Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Linear state-space system of the form with bounded parametric uncertainty Aim : steer this system towards the origin from initial state without violating the constraint

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3

Example

  • Example
  • Robustness
  • Robust MPC
  • Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Results for 4 different parameter settings :

  • Recursive feasibility ?
  • Monotonicity of the cost ?
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SLIDE 5

4

Robustness

  • Example
  • Robustness
  • Robust MPC
  • Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Robust with respect to what ?

  • Disturbances
  • Model uncertainty

Cause predictions of ‘nominal’ MPC to be inaccurate

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5

Robustness

  • Example
  • Robustness
  • Robust MPC
  • Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Main aims :

  • Keep recursive feasibility properties, despite model errors,

disturbances

  • Keep asymptotic stability (in the case without disturbances)

We need to have an idea about …

  • the size of the model uncertainty
  • the size of the disturbances
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6

Uncertain Models

  • Example
  • Robustness
  • Robust MPC
  • Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Linear Parameter-Varying state space models with polytopic uncertainty description

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7

Uncertain Models

  • Example
  • Robustness
  • Robust MPC
  • Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Linear Parameter-Varying state space models with norm-bounded uncertainty description

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8

Bounded Disturbances

  • Example
  • Robustness
  • Robust MPC
  • Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

  • Typically bounded by a polytope :
  • Can be described in two ways
  • Trivial condition for well-posedness :
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SLIDE 10

9

Robust MPC

  • Example
  • Robustness
  • Robust MPC
  • Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Main aims :

  • Keep recursive feasibility properties, despite model errors,

disturbances

  • Keep asymptotic stability (in the case without disturbances)

Necessary modifications :

  • Uncertain predictions (e.g predictions with all models within

uncertainty region)

  • worst-case constraint satisfaction over all predictions
  • worst-case cost over all predictions
  • Terminal cost has to satisfy multiple Lyap. Ineq.
  • Terminal constraint has to be a robust invariant set
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SLIDE 11

10

Robust MPC

  • Example
  • Robustness
  • Robust MPC
  • Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be model uncertainty disturbances

Uncertain predictions :

N N

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11

Uncertain Predictions

  • Example
  • Robustness
  • Robust MPC
  • Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Step 1) Robust Constraint Satisfaction Result : Sufficient to impose constraint only for vert. of : Observations :

  • depends linearly on
  • is a convex polytopic set
  • is a convex set
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12

Uncertain Predictions

  • Example
  • Robustness
  • Robust MPC
  • Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

LTI

(L=1)

LPV

(L>1, e.g. 2)

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13

Uncertain Predictions

  • Example
  • Robustness
  • Robust MPC
  • Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Impose state constraints on all nodes

  • f state prediction tree

→ number of constraints increases expon. with incr. !!!

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14

Worst-Case Cost Objective

  • Example
  • Robustness
  • Robust MPC
  • Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Step 2) Worst-Case cost minimization Observations :

  • depends linearly on
  • is a convex polytopic set
  • cost function typically convex function of

→ Also for objective function sufficient to make predictions only with vertices of uncertainty polytope

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15

Worst-Case Cost Objective

  • Example
  • Robustness
  • Robust MPC
  • Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

states inputs

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16

Worst-Case Cost Objective

(1-norm)

  • Example
  • Robustness
  • Robust MPC
  • Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

LP

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17

Worst-Case Cost Objective

(2-norm)

  • Example
  • Robustness
  • Robust MPC
  • Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

CVX ?

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18

Worst-Case Cost Objective

(2-norm)

  • Example
  • Robustness
  • Robust MPC
  • Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Constraints of the form :

SOC CVX ?

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19

Worst-Case Cost Objective

(2-norm)

  • Example
  • Robustness
  • Robust MPC
  • Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

SOCP

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20

Robust MPC

(2-norm)

  • Example
  • Robustness
  • Robust MPC
  • Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

SOCP

By rewriting we now get Terminal cost Terminal constraint

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21

Robust Terminal Cost

  • Example
  • Robustness
  • Robust MPC
  • Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

“non-robust” stability condition for terminal cost: In case of…

  • LPV system with polytopic uncertainty
  • linear feedback controller
  • quadratic cost criterion
  • quadratic terminal cost

… this becomes :

  • r equivalent :
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22

Robust Terminal Cost

  • Example
  • Robustness
  • Robust MPC
  • Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Robust stability condition for terminal cost: Observations :

  • inequality is convex and linear in and (i.e. LMI in )
  • is a convex polytopic set

Hence, inequality satisfied iff

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23

Robust Terminal Cost : Design

  • Example
  • Robustness
  • Robust MPC
  • Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

  • 1. Find a robustly stabilizing controller
  • 2. Find a terminal cost satisfying

by solving the following optimization problem :

SDP

  • ptimization variables

Minimization of eigenvalues of

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24

Robust Terminal Constraint

  • Example
  • Robustness
  • Robust MPC
  • Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Recursive feasibility is guaranteed if 1) 2) 3)

Terminal constraint is feasible w.r.t state constraints Terminal constraint is feasible w.r.t input constraints Terminal constraint is a positive invariant set w.r.t

Reminder : nominal case

remain unchanged Has to be modified in order to Model uncertainty into account Robust positive invariance

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25

Robust Terminal Constraint

  • Example
  • Robustness
  • Robust MPC
  • Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Consider linear terminal controller , then the resulting closed loop system is : Robust positive invariance : Again : sufficient to satisfy inclusion

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26

Robust Terminal Constraint

  • Example
  • Robustness
  • Robust MPC
  • Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Reminder : invariant sets for LTI systems Given an LTI system subject to linear constraints then the largest size feasible invariant set can be found as with a finite integer. Given an LTI system subject to linear constraints then the largest size feasible invariant set can be found as with a finite integer. Comes down to making forward predictions using

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27

Robust Terminal Constraint

  • Example
  • Robustness
  • Robust MPC
  • Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

LTI

(L=1,n=2)

LPV

(L>1, e.g. 2, n=2)

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28

  • Example
  • Robustness
  • Robust MPC
  • Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

  • 0.8
  • 0.6
  • 0.4
  • 0.2

0.2 0.4 0.6 0.8

  • 1
  • 0.8
  • 0.6
  • 0.4
  • 0.2

0.2 0.4 0.6 0.8 1

S X

  • Constructed by solving semi-definite program (SDP)
  • Conservative with respect to constraints

Ellipsoidal invariant sets for LPV systems

(Kothare et al.,1996, Automatica)

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29

  • Example
  • Robustness
  • Robust MPC
  • Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Polyhedral invariant sets for LPV systems

A set is invariant with respect to a system defined by iff with Reformulate invariance condition : Sufficient condition : Also necessary condition

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30

  • Example
  • Robustness
  • Robust MPC
  • Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Polyhedral invariant sets for LPV systems

Advantages :

  • in step 2 only ‘significant’ constraints are added to :

significant insignificant

  • Initialize
  • iteratively add constraints from to until

Algorithm :

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31

  • Example
  • Robustness
  • Robust MPC
  • Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Polyhedral invariant sets for LPV systems

Algorithm : Advantages :

  • prediction tree never explicitly constructed
  • given a polyhedral set , it is straightforward

to calculate :

  • Initialize
  • iteratively add constraints from to until
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32

  • Example
  • Robustness
  • Robust MPC
  • Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Polyhedral invariant sets for LPV systems

Example

Initialization

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33

  • Example
  • Robustness
  • Robust MPC
  • Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Polyhedral invariant sets for LPV systems

Example

Iteration 10

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34

  • Example
  • Robustness
  • Robust MPC
  • Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Polyhedral invariant sets for LPV systems

Example

Iteration 10 + garbage collection

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35

  • Example
  • Robustness
  • Robust MPC
  • Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Polyhedral invariant sets for LPV systems

Example

Iteration 20

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SLIDE 37

36

  • Example
  • Robustness
  • Robust MPC
  • Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Polyhedral invariant sets for LPV systems

Example

Final Result

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SLIDE 38

37

  • Example
  • Robustness
  • Robust MPC
  • Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Polyhedral invariant sets for LPV systems

Example

Final Result

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SLIDE 39

38

  • Example
  • Robustness
  • Robust MPC
  • Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Polyhedral invariant sets for LPV systems

Example

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39

  • Example
  • Robustness
  • Robust MPC
  • Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Recursive feasibility, stability guarantee ?

Open loop

  • ptimal input sequence

Closed loop

  • ptimal input sequence

NO recursive feasibility !!! Recursive feasibility

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40

Example revisited…

  • Example
  • Robustness
  • Robust MPC
  • Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Results for 4 different parameter settings :

  • Recursive feasibility ?
  • Monotonicity of the cost ?
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SLIDE 42

41

  • Example
  • Robustness
  • Robust MPC
  • Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Example revisited…

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42

  • Example
  • Robustness
  • Robust MPC
  • Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Conclusion

  • Robustness w.r.t

a) bounded model uncertainty b) bounded disturbances

  • necessary modifications :
  • worst-case constraints satisfaction
  • worst-case objective function
  • terminal cost
  • terminal constraint
  • “open-loop” vs. “closed-loop” predictions

→ currently hot research topic !

  • convex optimization but problem size impractical

→ currently hot research topic !