h0k03a : advanced process controlmaapc/static/files/... · [email protected] h0k03a :...

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 [email protected]

H0K03a : Advanced Process Control Model-based Predictive Control 4 : Robustness

1

Overview

• Example

• Robustness

• Robust MPC

• Conclusion

Lecture 4 : Robustness

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification

System Theory Automation


2

Example

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



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

3

Example

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



Results for 4 different parameter settings :

• Recursive feasibility ?

• Monotonicity of the cost ?

4

Robustness

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



Robust with respect to what ?

• Disturbances

• Model uncertainty

Cause predictions of

‘nominal’ MPC to be inaccurate

5

Robustness

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



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

6

Uncertain Models

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



Linear Parameter-Varying state space models with

polytopic uncertainty description

7

Uncertain Models

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



Linear Parameter-Varying state space models with

norm-bounded uncertainty description

8

Bounded Disturbances

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



• Typically bounded by a polytope :

• Can be described in two ways

•

•

• Trivial condition for well-posedness :

9

Robust MPC

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



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

10

Robust MPC

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



model uncertainty

disturbances

Uncertain predictions :

N

N

11

Uncertain Predictions

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



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

12


• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



LTI (L=1)

LPV (L>1, e.g. 2)

13


• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



Impose state constraints on all nodes

of state prediction tree

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

14

Worst-Case Cost Objective

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



Step 2) Worst-Case cost minimization

Observations :

• depends linearly on


• cost function typically convex function of

→ Also for objective function sufficient to make

predictions only with vertices of uncertainty polytope

15

Worst-Case Cost Objective

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



sta

tes

inp

uts

16

Worst-Case Cost Objective (1-norm)

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



LP

17


• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



CVX ?

18


• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



Constraints of the form :

SOC

CVX ?

19


• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



SOCP

20

Robust MPC (2-norm)

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



SOCP

By rewriting we now get

Terminal cost

Terminal constraint

21

Robust Terminal Cost

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



“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 :

or equivalent :

22

Robust Terminal Cost

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



Robust stability condition for terminal cost:

Observations :

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


Hence, inequality satisfied iff

23

Robust Terminal Cost : Design

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



1. Find a robustly stabilizing controller

2. Find a terminal cost satisfying

by solving the following optimization problem :

SDP

optimization variables

Minimization of

eigenvalues of

24

Robust Terminal Constraint

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



Recursive feasibility is guaranteed if

1)

2)

3)

Terminal constraint is

feasible w.r.t state constraints


feasible w.r.t input constraints


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

25


• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



Consider linear terminal controller ,

then the resulting closed loop system is :

Robust positive invariance :

Again : sufficient to satisfy inclusion

26


• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



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

27


• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



LTI (L=1,n=2)

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

28

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

-1

-0.8

-0.6

-0.4

-0.2

0

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)

29

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



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

30

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification




Advantages :

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

significant insignificant

• Initialize

• iteratively add constraints from to until

Algorithm :

31

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification




Algorithm :

Advantages :

• prediction tree never explicitly constructed

• given a polyhedral set , it is straightforward

to calculate :

• Initialize

• iteratively add constraints from to until

32

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification




Example

Initialization

33

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification




Example

Iteration 10

34

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification




Example

Iteration 10 + garbage collection

35

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification




Example

Iteration 20

36

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification




Example

Final Result

37

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification




Example

Final Result

38

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification




Example

39

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



Recursive feasibility, stability guarantee ?

Open loop

optimal input sequence

Closed loop

optimal input sequence

NO recursive feasibility !!! Recursive feasibility

40

Example revisited…

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



Results for 4 different parameter settings :

• Recursive feasibility ?

• Monotonicity of the cost ?

41

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



Example revisited…

42

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



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 !

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