feasibility, uncertainty and interpolation

Feasibility, uncertainty and interpolation

J. A. Rossiter (Sheffield, UK)

IEEE Colloquium, April 4th 20052

Overview

Predictive control (MPC) Interpolation instead of optimisation Invariant sets Combining invariant sets Illustrations Conclusions.


BACKGROUND


Notation

Assume a state space model and constraints

Let the control law be Define the maximal admissible set (MAS), that

is region within which constraints are met, as

1 ;

; ;

k k k k k

k k

x Ax Bu y Cx

u u u x x x

k ku Kx

0 0 0{ : }S x M x d


Invariant set and closed-loop trajectories

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

0 5 10 15 20 25 30-2

-1

0

1

2Inputs


Minimise a performance index of the form

Can write solutions as

1,..., 0

0

;

min . . ;k k nc

c

k

T Tk k k k k

u u k

k n

u u u

J x Qx u Ru s t x x x

x S

Predictive control

, 1, ,

,k k k c

k k c

u Kx c k n

u Kx k n


Impact on invariant sets of adding d.o.f.

-3 -2 -1 0 1 2 3-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

OMPC (nc=10)

OMPC (nc=5)

OMPC (nc=2)

S1


Observations

If terminal control is optimal, then the terminal region may be small. – Need large d.o.f. to get large feasible region.– Good performance

If terminal control is detuned, terminal region may be large.– Small d.o.f. to get large feasible region.– Suboptimal performance.


INTERPOLATION


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.


MAS with K1 and K2

-4 -3 -2 -1 0 1 2 3 4-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

x1

x 2


How to interpolate

A simple summary: split the state into 2 components and predict separately through the 2 closed-loop dynamics, then recombine.

Decomposition into x1 and x2 to ensure constraint satisfaction.

)()()(

)()1(

)()1( 2211

222

111

21

kxkxnkx

kxkx

kxkx

xxxnn


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.

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 8-8

-6

-4

-2

0

2

4

x-plane


Feasible regions (figures)

-3 -2 -1 0 1 2 3-1.5

-1

-0.5

0

0.5

1

1.5

GIMPCS

2

S1


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).


Feasible regions with general interpolation, ONEDOF and nc d.o.f.

-3 -2 -1 0 1 2 3-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

GIMPCS

2

S1

OMPC (nc=10)

OMPC (nc=5)

OMPC (nc=2)


Weaknesses of interpolation

1. Algorithms using MAS can only be applied to the nominal case.

2. Easy to show that uncertainty can cause infeasibility and instability.

3. Need modifications to cater for uncertainty.

Here we consider changes to cater for LPV systems.


POLYTOPIC INVARIANT SETS


Polytopic invariant sets (MAS) for nominal systems

The computation of these is generally considered tractable.

Let constraints be

Then the MAS is given as

Where

for n large enough.[Redundant rows can be

removed in general.]

fCxk

dMxk

f

f

f

d

C

C

C

M

n

;


Polytopic invariant sets for LPV systems

The computation of these is generally considered intractable.

Consider a closed-loop LPV system

Then computing all possible open-loop predictions.

Clearly, there is a combinatorial explosion in the number of terms.

),,(; 11 rkk Coxx

f

f

f

d

M

M

C

M

n

;1

;2

1

1

ri

i

i

i

M

M

M

M


Polytopic invariant sets for LPV systems

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.

f

f

f

d

M

M

C

M

n

;

ˆ

ˆ1

;

ˆ

ˆ

ˆ

2

1

1

ri

i

i

i

M

M

M

M


Robust and nominal invariant sets

-6 -4 -2 0 2 4 6 8-10

-5

0

5

Nominal

LPV


Polytopic invariant sets and interpolation

MUST USE ROBUST SETS TO ENSURE FEASIBILITY!

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.)


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.


EXPLICIT OR IMPLICIT CONSTRAINT HANDLING


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 ?


Implicit constraint handling

With ellipsoidal invariant sets this is obvious.

Constraints are converted into an LMI, with some conservatism because of:1. Asymmetry2. Conversion of linear

inequalities to quadratic inequalities.

A trivial example of this might be

or

432

KxKxKxu

u TT

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x1

x 2

Quadratic Linear


Conservatism with linear inequalities

Define the invariant sets associated to K1, K2,… to be

Then, general interpolation first splits x into several components and uses the constraints

,...:,: 2211 dxNxSdxNxS

ii

i

i

iii

xKu

Sx

xxx ;

1

0;...21


Conservatism with linear inequalities (b)

The constraint enforces feasibility.

However, consider the following hypothetical illustration:

This implies that

iii Sx

xnkxdxN

xmkxdxN

22222

11111

)(

)(

xxxx

xxxx

)min()min(

)max()max(

21

21


Remarks

The constraint

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

1;0; iiiii Sx

0; iiii Sx


Relaxed constraints

General interpolation can be composed as

We propose to replace this as a single inequality:

NOTE: No longer any variables!

1

0

...

;21

222

111

i

i

xxx

dxN

dxN

...

;

21

2211

xxx

dxNxN


Structure of inequalities (nominal case)

Consider the predictions

And hence the explicit constraints are

...)()()();(

)(

)2(

)1(

21

2

rkxrkxrkxkx

nkx

kx

kx

i

ni

i

i

i

i

i

f

f

f

kx

C

C

C

kx

C

C

C

nn

...)()( 2

2

22

2

1

1

21

1


ILLUSTRATIONS


Illustrations

1. There can be surprisingly large increases in feasibility.

2. Probably because the directionality of trajectories for each controller are different.

-4 -3 -2 -1 0 1 2 3 4-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

GIMPC2

GIMPCS

2

S1


Extensions to the LPV case

Unfortunately, explicit constraint handling requires a direct link between the prediction equations and the inequalities.

However, the algorithm for finding polytopic invariant sets in the LPV case, relied, for efficiency, on removing redundant constraints from the predictions.

dxNxN

f

f

f

kx

C

C

C

kx

C

C

C

nn

22112

2

22

2

1

1

21

1

...)()(


Extensions to the LPV case (b)

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!


Constraints for general interpolation with LPV systems

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.


Illustration of inequalities

N1 N2 other Total d.o.f.

GIMPC 30 12 2 44 3

GIMPC2 412 412 0 412 2

RMPC

(nc=5)

448 5


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:– 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).

feasibility, uncertainty and interpolation

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