Transcript Presentation 5 (ppt, 474kb)

Feasibility, uncertainty and
interpolation
J. A. Rossiter (Sheffield, UK)
Overview






2
Predictive control (MPC)
Interpolation instead of optimisation
Invariant sets
Combining invariant sets
Illustrations
Conclusions.
IEEE Colloquium, April 4th 2005
BACKGROUND
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Notation

Assume a state space model and constraints
xk 1  Axk  Buk ; yk  Cxk
u  uk  u; x  xk  x;


4
Let the control law be u k   Kxk
Define the maximal admissible set (MAS), that
is region within which constraints are met, as
S 0  {x : M 0 x  d 0 }
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1.5
Invariant set and closed-loop
trajectories
Inputs
2
1
1
0
-1
0.5
-2
0
5
10
15
0
-0.5
-1
5
-1.5
-1.5
-1
-0.5
0
0.5
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1
1.5
20
25
30
Predictive control

Minimise a performance index of the form

min J   x Qxk  u Ruk
uk ,...,uk nc1

k 0
T
k
T
k
Can write solutions as
uk   Kxk  ck , k  1,
uk   Kxk ,
6
k  nc
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u  uk  u;


s.t.  x  xk  x;
x

S
k

n
0
c


, nc
Impact on invariant sets of
adding d.o.f.
2
OMPC (nc =10)
1.5
OMPC (nc =5)
OMPC (nc =2)
1
S1
0.5
0
-0.5
-1
-1.5
-2
-3
7
-2
-1
0
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1
2
3
Observations

If terminal control is optimal, then the terminal
region may be small.
–
–

If terminal control is detuned, terminal region
may be large.
–
–
8
Need large d.o.f. to get large feasible region.
Good performance
Small d.o.f. to get large feasible region.
Suboptimal performance.
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INTERPOLATION
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Alternative strategy
Interpolation is known to:
1. Allow efficient (often trivial) optimisations.
2. Combine qualities of different strategies.
Interpolate between K1 and K2 where:
 K1 has optimal performance but possibly a
small feasible region
 K2 has large feasible region.
10
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MAS with K1 and K2
2
1.5
1
x2
0.5
0
-0.5
-1
-1.5
-2
-4
11
-3
-2
-1
0
x1
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1
2
3
4
How to interpolate
A simple summary: split the state into 2
components and predict separately through the
2 closed-loop dynamics, then recombine.
x  x1  x2




n
n
x
(
k

1
)


x
(
k
)

x
(
k

n
)


x
(
k
)


 1

1 1
1 1
2 x2 ( k )
 x (k  1)   x (k )
2 2
 2

Decomposition into x1 and x2 to ensure constraint
satisfaction.
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Feasible regions with
Interpolation
Ellipsoidal invariant sets
 Find max. volume feasible
invariant ellipsoid.
 By necessity conservative in
volume.
 Can be computed easily, even
with model uncertainty.
 Generalised interpolation
algorithm takes convex hull of
several ellipsoids.
 SDP solver required.
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Polytopic invariant sets
 Can use MAS – maximum
possible feasible regions.
 Easily computed for nominal
case only.
 Various interpolation
algorithms for certain case.
 Still limited to convex hull of
underlying sets.
 Optimisation requires QP or
LP.
Weakness of ellipsoidal sets
4
2
0
-2
-4
-6
-8
-4
14
-2
0
2
x-plane
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4
6
8
Feasible regions (figures)
1.5
GIMPC
S2
1
S1
0.5
0
-0.5
-1
15
-1.5
-3
-2
-1
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0
4th
2005
1
2
3
When to use Interpolation?
Which is more efficient:
–
–
A normal MPC algorithm with d.o.f.?
An interpolation?
ONEDOF interpolations have only one d.o.f. but
severely restricted feasibility.
General interpolation requires nx d.o.f. (nx the
state dimension).
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Feasible regions with general
interpolation, ONEDOF and nc d.o.f.
2
GIMPC
S2
1.5
S1
1
OMPC (nc =10)
OMPC (nc =5)
0.5
OMPC (nc =2)
0
-0.5
-1
-1.5
-2
-3
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-2
-1
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0
1
2
3
Weaknesses of interpolation
1.
2.
3.
Algorithms using MAS can only be applied to
the nominal case.
Easy to show that uncertainty can cause
infeasibility and instability.
Need modifications to cater for uncertainty.
Here we consider changes to cater for LPV
systems.
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POLYTOPIC INVARIANT SETS
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Polytopic invariant sets (MAS)
for nominal systems


The computation of
these is generally
considered tractable.
Let constraints be
Cxk  f

Then the MAS is given
as
Mxk  d
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
Where
 C 
f
 C 
f
; d   
M 
  


 
n
C



f
for n large enough.
[Redundant rows can be
removed in general.]
Polytopic invariant sets for LPV
systems


The computation of these is
generally considered
intractable.
Consider a closed-loop LPV
system
xk 1  xk ;   Co(1 ,,  r )


21
Then computing all possible
open-loop predictions.
Clearly, there is a
combinatorial explosion in the
number of terms.
IEEE Colloquium, April 4th 2005
C 
f
M 
f
M   1 ; d   
  

 
 
f
M n 
 M i 1 
M  
M i 1   i 2 ;
  


M i r 
Polytopic invariant sets for LPV
systems



22
There is a need for an
alternative approach.
[Pluymers et al, ACC 2005]
Specifically, remove
redundant constraints from Mi
before computing Mi+1.
This will slow the rate of
growth and produce a
tractable algorithm, if, the
actual MAS is of reasonable
complexity.
IEEE Colloquium, April 4th 2005
C 
f
 Mˆ 
f
M   1 ; d   
  

 ˆ 
 
f
M n 
 Mˆ i 1 
 ˆ

M i 2 

M i 1 
;
  


ˆ
 M i  r 
Robust and nominal invariant
sets
5
Nominal
LPV
0
-5
-10
-6
23
-4
-2
0
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2
4
6
8
Polytopic invariant sets and
interpolation
MUST USE ROBUST SETS TO ENSURE FEASIBILITY!



24
We can simply use the ‘robust’ invariant sets in the
algorithm developed for the nominal case.
Proofs of recursive feasibility and convergence carry
across easily if the cost is replaced by a suitable upper
bound.
(A quadratic stabilisability condition is required.)
IEEE Colloquium, April 4th 2005
Summary
Polytopic invariant sets allow the use of interpolation with
LPV systems and hence:
1.
Large feasible regions.
2.
Robustness.
3.
Small computational load.
BUT:
General interpolation still only applicable to convex hull of
underlying regions. This could be too restrictive.
25
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EXPLICIT OR IMPLICIT
CONSTRAINT HANDLING
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Extending feasibility of
interpolation methods
General interpolation does implicit not explicit constraint
handling.
So:
1. membership of the set implies the trajectories are
feasible.
2. non-membership may not imply infeasibility.
Therefore, we know that feasibility may be extended
beyond the convex hull in general, but how ?
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Implicit constraint handling


With ellipsoidal invariant
sets this is obvious.
Constraints are
converted into an LMI,
with some conservatism
because of:
A trivial example of this
might be
 2  u  3
T
T

x
K
Kx  4

u   Kx 
or
1
0.8
0.6
0.4
Quadratic
0.2
2
2.
Asymmetry
Conversion of linear
inequalities to quadratic
inequalities.
x
1.
Linear
0
-0.2
-0.4
-0.6
-0.8
28
-1
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-1
-0.8
-0.6
-0.4
-0.2
0
x1
0.2
0.4
0.6
0.8
1
Conservatism with linear
inequalities

Define the invariant sets associated to K1,
K2,… to be
S1  x : N1 x  d , S2  x : N2 x  d ,...

Then, general interpolation first splits x into
several components and uses the constraints
x  x1  x2  ... ;
29
xi  i Si
i  0 ; u   K i xi
 i  1
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Conservatism with linear
inequalities (b)



30
The constraint xi  i Si
enforces
feasibility.
However, consider the following hypothetical
illustration:
N1 x1  1d  x1 (k  m)  1 x
N 2 x2  2 d  x2 (k  n)  2 x
This implies that
max( x)  max( x1  x2 )  x
min( x)  min( x1  x2 )  x
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Remarks




The constraint xi  i Si ; i  0;  i  1
is necessary with ellipsoidal invariant sets as one can
not check predictions explicitly against constraints.
This is not the case with polytopic invariant sets.
Hence we propose to relax this condition and hence
increase feasible regions.
Remove the two conditions
xi  i Si ; i  0
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Relaxed constraints

General interpolation
can be composed as
We propose to replace
this as a single
inequality:
NOTE: No longer any 
variables!
N1 x1  1d
N 2 x2  2 d

;
x  x1  x2  ...
i  0
 i  1

32
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N1 x1  N 2 x2    d ;
x  x1  x2  ...
Structure of inequalities
(nominal case)
33

Consider the predictions

And hence the explicit constraints are
 xi (k  1)    i 
 x (k  2)   2 
 i
   i  x (k ); x(k  r )  x (k  r )  x (k  r )  ...
1
2

    i


  n
x
(
k

n
)
 i
  i 
 C1 
C 2 
f
C 2 
C 2 
f
1
2

 x (k )  
 x (k )  ...   
1
  
   2

 n
 n
 
C

C

f
 1
 2
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ILLUSTRATIONS
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Illustrations
2
GIMPC2
GIMPC
S2
1.5
1.
2.
There can be
surprisingly large
increases in feasibility.
Probably because the
directionality of
trajectories for each
controller are different.
S1
1
0.5
0
-0.5
-1
-1.5
-2
-4
35
-3
-2
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-1
0
1
2
3
4
Extensions to the LPV case

Unfortunately, explicit constraint handling requires a
direct link between the prediction equations and the
inequalities.
 C1 
C 2 
f
C 2 
C 2 
f
 1  x (k )   2  x (k )  ...     N x  N x  d
1 1
2 2
   1
   2

 n
 n
 
f
C1 
C 2 

36
However, the algorithm for finding polytopic invariant
sets in the LPV case, relied, for efficiency, on removing
redundant constraints from the predictions.
IEEE Colloquium, April 4th 2005
Extensions to the LPV case (b)



37
For the original GIMPC, sets S1, S2,.. could be
described as efficiently as possible. There was
no need for mutual consistency because
constraint handling was implicit.
Notably, all redundant inequalities could be
eliminated.
When doing explicit constraint handling,
redundant constraints cannot be eliminated
from Si, just in case the overall x(k+j) for that
row is against a constraint!
IEEE Colloquium, April 4th 2005
Constraints for general
interpolation with LPV systems


38
Algorithms can be written to formulate the
inequalities, but suffer more from the
combinatorial growth problems outlined earlier.
Assuming the resulting sets are not too large,
proofs of convergence and feasibility are
straightforward.
IEEE Colloquium, April 4th 2005
Illustration of inequalities
N1
N2
other
Total
d.o.f.
GIMPC
30
12
2
44
3
GIMPC2
412
412
0
412
2
448
5
RMPC
(nc=5)
39
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Conclusions



Interpolation is known to facilitate reductions in
complexity at times, particular for low dimensional
systems. However most work has focussed on the
nominal case.
Some earlier interpolation algorithms used implicit
constraint handling to cater for uncertainty. This could
lead to considerable conservatism.
We have illustrated:
–
–
–
40
How interpolation can be modified to overcome this
conservatism and the associated issues (recently submitted).
how polytopic robust MAS might be computed and used in
MPC (to be published IFAC and ACC, 2005).
how to use polytopic robust MAS with interpolation (recently
submitted).
IEEE Colloquium, April 4th 2005