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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
H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness [email protected]
2
Example
• Example
• Robustness
• Robust MPC
• Conclusion
Signal processing
Identification
System Theory Automation
H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness [email protected]
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
System Theory Automation
H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness [email protected]
Results for 4 different parameter settings :
• Recursive feasibility ?
• Monotonicity of the cost ?
4
Robustness
• Example
• Robustness
• Robust MPC
• Conclusion
Signal processing
Identification
System Theory Automation
H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness [email protected]
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
System Theory Automation
H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness [email protected]
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
System Theory Automation
H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness [email protected]
Linear Parameter-Varying state space models with
polytopic uncertainty description
7
Uncertain Models
• Example
• Robustness
• Robust MPC
• Conclusion
Signal processing
Identification
System Theory Automation
H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness [email protected]
Linear Parameter-Varying state space models with
norm-bounded uncertainty description
8
Bounded Disturbances
• Example
• Robustness
• Robust MPC
• Conclusion
Signal processing
Identification
System Theory Automation
H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness [email protected]
• 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
System Theory Automation
H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness [email protected]
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
System Theory Automation
H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness [email protected]
model uncertainty
disturbances
Uncertain predictions :
N
N
11
Uncertain Predictions
• Example
• Robustness
• Robust MPC
• Conclusion
Signal processing
Identification
System Theory Automation
H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness [email protected]
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
Uncertain Predictions
• Example
• Robustness
• Robust MPC
• Conclusion
Signal processing
Identification
System Theory Automation
H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness [email protected]
LTI (L=1)
LPV (L>1, e.g. 2)
13
Uncertain Predictions
• Example
• Robustness
• Robust MPC
• Conclusion
Signal processing
Identification
System Theory Automation
H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness [email protected]
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
System Theory Automation
H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness [email protected]
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
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 [email protected]
sta
tes
inp
uts
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 [email protected]
LP
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 [email protected]
CVX ?
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 [email protected]
Constraints of the form :
SOC
CVX ?
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 [email protected]
SOCP
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 [email protected]
SOCP
By rewriting we now get
Terminal cost
Terminal constraint
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 [email protected]
“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
System Theory Automation
H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness [email protected]
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
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 [email protected]
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
System Theory Automation
H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness [email protected]
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
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 [email protected]
Consider linear terminal controller ,
then the resulting closed loop system is :
Robust positive invariance :
Again : sufficient to satisfy inclusion
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 [email protected]
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
Robust Terminal Constraint
• Example
• Robustness
• Robust MPC
• Conclusion
Signal processing
Identification
System Theory Automation
H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness [email protected]
LTI (L=1,n=2)
LPV (L>1, e.g. 2, n=2)
28
• Example
• Robustness
• Robust MPC
• Conclusion
Signal processing
Identification
System Theory Automation
H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness [email protected]
-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
System Theory Automation
H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness [email protected]
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
System Theory Automation
H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness [email protected]
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 :
31
• Example
• Robustness
• Robust MPC
• Conclusion
Signal processing
Identification
System Theory Automation
H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness [email protected]
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
32
• Example
• Robustness
• Robust MPC
• Conclusion
Signal processing
Identification
System Theory Automation
H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness [email protected]
Polyhedral invariant sets for LPV systems
Example
Initialization
33
• Example
• Robustness
• Robust MPC
• Conclusion
Signal processing
Identification
System Theory Automation
H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness [email protected]
Polyhedral invariant sets for LPV systems
Example
Iteration 10
34
• Example
• Robustness
• Robust MPC
• Conclusion
Signal processing
Identification
System Theory Automation
H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness [email protected]
Polyhedral invariant sets for LPV systems
Example
Iteration 10 + garbage collection
35
• Example
• Robustness
• Robust MPC
• Conclusion
Signal processing
Identification
System Theory Automation
H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness [email protected]
Polyhedral invariant sets for LPV systems
Example
Iteration 20
36
• Example
• Robustness
• Robust MPC
• Conclusion
Signal processing
Identification
System Theory Automation
H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness [email protected]
Polyhedral invariant sets for LPV systems
Example
Final Result
37
• Example
• Robustness
• Robust MPC
• Conclusion
Signal processing
Identification
System Theory Automation
H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness [email protected]
Polyhedral invariant sets for LPV systems
Example
Final Result
38
• Example
• Robustness
• Robust MPC
• Conclusion
Signal processing
Identification
System Theory Automation
H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness [email protected]
Polyhedral invariant sets for LPV systems
Example
39
• Example
• Robustness
• Robust MPC
• Conclusion
Signal processing
Identification
System Theory Automation
H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness [email protected]
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
System Theory Automation
H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness [email protected]
Results for 4 different parameter settings :
• Recursive feasibility ?
• Monotonicity of the cost ?
41
• Example
• Robustness
• Robust MPC
• Conclusion
Signal processing
Identification
System Theory Automation
H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness [email protected]
Example revisited…
42
• Example
• Robustness
• Robust MPC
• Conclusion
Signal processing
Identification
System Theory Automation
H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness [email protected]
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 !