a control methodology for constrained linear …a control methodology for constrained linear systems...

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A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse, France Co-workers: Marina VASSILAKI University of Patras, GREECE Jean-Paul BEZIAT CISI Bordeaux, FRANCE Eugˆ enio B. CASTELAN UFSC, Florian´ opolis, BRAZIL. Carlos E. T. D ´ OREA UFBA, Bahia, BRAZIL 1

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Page 1: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

A Control Methodology for Constrained

Linear Systems Based on Positive

Invariance of Polyhedra

Jean-Claude HENNET

LAAS-CNRS Toulouse, France

Co-workers:

Marina VASSILAKIUniversity of Patras, GREECE

Jean-Paul BEZIATCISI Bordeaux, FRANCE

Eugenio B. CASTELANUFSC, Florianopolis, BRAZIL.

Carlos E. T. DOREAUFBA, Bahia, BRAZIL

1

Page 2: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

INTRODUCTION

� Most real systems are subject to

- constraints on state and control,

- disturbances and uncertainties.

� The positive invariance approach is able to tacklethese features in a computationally efficient way.

� Until now, the positive invariance approach hasbeen essentially developed in theoretical worksrather than in practical applications.

� This conference presents some simple positiveinvariance concepts and methodologies usefulto treat practical control problems.

� The basic computational ingredients of the meth-ods are spectral assignment and Linear Program-ming.

2

Page 3: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

OUTLINE OF THE PRESENTATION

� Part 1 : The positive invariance approach

– Positive invariance ; Definitions and Basic Properties

– Mathematical frameworks

– Invariance and stability

– Positive invariance of polyhedral sets w.r.t. linear sys-tems : An algebraic characterization

� Part 2 : Linear Constrained Regulation Problems

– A unified model

� discrete-time systems / continuous-time systems

� state constraints / control constraints

– Positive invariance of polyhedral sets by state feed-back

– Resolution of positive invariance relations

– Examples

� Part 3 : Disturbance attenuation and constrained regula-tion

– Domain of satisfactory performance

– Positive invariance with disturbance attenuation

– Regulator design with positive invariance properties

– The feasible domain

– An application in production planning

3

Page 4: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

PART 1THE POSITIVE INVARIANCE APPROACH

� Positive invarianceDefinitions and Basic Properties

� Mathematical frameworks

� Invariance and stability

� Positive invariance of polyhedral setsw.r.t. linear systems

– Geometric characterization

– Algebraic characterization

4

Page 5: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

POSITIVE INVARIANCEDefinitions and Basic Properties

� DefinitionConsider a discrete or continuous time domain,T � ��, � � T .Let �S� be a dynamical system characterized ateach time t � T by its state vector x�t� � X .

The set � � X is positively invariant with re-spect to system �S� if and only if :

x��� � � �� x�t� � � �t � T �

� A geometric characterization of positive in-varianceLet Dt be the reachable set of states at timet � T from any initial state in �:

Dt � fx�t� jx��� � �g�

A necessary and sufficient condition for positiveinvariance of the set � with respect to system�S� is :

Dt � � �t � T �

5

Page 6: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

POSITIVE INVARIANCEA geometric characterization

Ω Ω Ω

D (t)

D (t’)

Dt � �

6

Page 7: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

SOME MATHEMATICAL FRAMEWORKS

� Differential inclusions(some links with Viability Theory (Aubin))

�x�t� � F�x�t��

� Differential equations

�x�t� � f�x�t��

� Recurrent Linear Equations

- classical :

xk�� � Axk �B�wk � B�uk

- marking evolution in Petri Nets

Mk���p� �Mk�p� � Csk

- probabilistic ( Markov chains)

�k�� � �kP

- (max,+) Algebra for Discrete Event Systems

xk�� � A� xk � B � uk

7

Page 8: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

POSITIVE INVARIANCE FOR DETERMINISTICLINEAR SYSTEMS

� Geometric Approach ”a la Wonham” :Invariance of subspaces

E A-invariant � AE � E

with AE � fx � �njx � Ay� y � Eg.

Continuous time: �x�t� � Ax�t�

eAtx� � E �x� � E � �t � ��

Discrete time: xk�� � Axk

Akx� � E �x� � E � �k � N �

� Positive invariance of closed domains(bounded or not), �

- continuous time:eAtx� � � �x� � � � �t � ��,

- discrete time:Akx� � �� �x� � � � �k � N �

8

Page 9: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

INVARIANCE AND STABILITY

Direct Property

Stability �� Existence of Invariant Sets

If v�x� is a Lyapunov function of system �S�, then

� � fx � �n � v�x� �g with � � �

is a positively invariant set of �S�.In general, such a set is closed and bounded.

Example :

If v�x� � xTPx is a quadratic Lyapunov functionw.r.t. �S�, then the ellipsoıds

fx � �n � v�x� �g �� � ��

are invariant sets for �S�.

Converse Property:

Existence of closed and bounded invariant sets hav-ing the ”0” state as an interior point �� Local Lya-punov Stability in a vicinity of the origin.

9

Page 10: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

PARTICULAR SETS POSITIVELY INVARIANTFOR LINEAR SYSTEMS : stability domains

� Polyhedral Lyapunov functions

v�x� � maxifj�Qx�ij

���ig

Q � �g�n � g � n , rank�Q� � n and � a posi-tive vecteur in �g .The set S�Q� �� is a positively invariant poly-tope defined by:

S�Q� �� � fx � �n �� Qx �g

� Compact polyhedral sets containing the zero state

v�X� � maxi

maxf��Qx�i�p��i

��Qx�i�p��i

g

Under rank�Q� � n and vectors p�� p� strictlypositive, the set

S�Q� p�� p�� � fx � �n �p� Qx p�g

is not empty, compact (closed and bounded),contains the zero state and positively invariant.

10

Page 11: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

GENERAL POLYHEDRAL SETS FOR LINEARSYSTEMS

� Unbounded polyhedra:

R�G� �� � fx � �n Gx �g

This polyhedron is unbounded if Ker G � f�g.Property :A necessary condition for positive invariance ofR�G� �� with respect to linear system (S), is in-variance of the subspace Ker G for system (S).

� Polyhedral cones:Representation 1: image of the positive orthant

K � fx � �njx � By� y � �m�g� K � B�m��

Representation 2: polyhedron

K � fx � �njGx �g�

A particular cone : �n� is a positively invariantset for any system xk�� � Axk with matrix Anon-negative (componentwise).

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Page 12: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

AN ALGEBRAIC CHARACTERIZATION OFINCLUSION OF POLYEDRA

Extended Farkas’Lemma

Consider two polyhedra in �n, denoted R�L�� andR�G� ��. A necessary and sufficient condition for :

R�L� � � R�G� ��

is the existence of a non-negative matrix, U , suchthat:

U�L � G

U� �

Remark:The inclusion R�L�� � R�G� �� is equivalent to:

Lx �� Gx ��

The row-vectors of matrix U can be interpreted asdual vectors.

12

Page 13: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

POSITIVE INVARIANCE OF POLYEDRA

Discrete-time linear system�SD� : xk�� � Axk.TheoremNSC for positive invariance of R�G� �� w.r.t. �SD�:�H � � (componentwise non-negative ) such that:

HG � GA

H� ��

Remark :If rank(G� � n and � � �, H � I should be a-M-matrix and �SD� is stable.

Continuous-time linear systems�SC� : �x�t� � Ax�t�.TheoremNSC for positive invariance of R�G� �� w.r.t. �SC�:�H essentially non-negative (Hij � � � �i� j � i��such that:

HG � GA

H� ��

Remark :If rank(G� � n and � � �, H should be a -M-matrice and (S) is stable.

13

Page 14: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

POSITIVE INVARIANCE OF SYMMETRICALPOLYEDRA

Symmetrical polyhedra:

S�Q� �� � fx � �n � jQxj �g � Q � �q�n

TheoremNSC for positive invariance of S�Q� �� w.r.t. �SD�:�H � �q�q such that:

HQ � QA

jHj� ��

Remark :If rank(Q� � n and � � �, positive invariance ofS�Q� �� implies stability of �SD�.

TheoremNSC for positive invariance of S�Q� �� w.r.t. �SC�:�H � �q�q such that:

HQ � QAH� ��

with

�Hii � HiiHij � jHijj

Remark :If rank(Q� � n and � � �, positive invariance ofS�Q� �� implies stability of �SD�.

14

Page 15: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

PART 2LINEAR CONSTRAINED REGULATION

PROBLEMS

� A unified model

– discrete-time systems / continuous-time sys-tems

– state constraints / control constraints

� Positive invariance of polyhedral sets by statefeedback

� Resolution of positive invariance relations

– by Linear Programming

– by eigenstructure assignment

� Examples

15

Page 16: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

A UNIFIED MODEL OF CONSTRAINED LINEARSYSTEMS

p�xt� � Axt�But � B � �n�m� � m n� �S�

Continuous-time case : p is the derivative operator

�xt � Axt� But

Discrete-time case : p is the advance operator

xt�� � Axt� But

Case of linear constraints on the state vector:

�� Qxt � with rank Q� r n� �i � �� i� � ��� r�

Constraints generate a polyhedral domain S�Q� ��

in the state space:

S�Q� �� � fx � �n � � � Qx �g

16

Page 17: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

A UNIFIED MODEL OF CONSTRAINED LINEARSYSTEMS

Case of linear constraints on the control vector:

� Mut with rank M � c m� i � �� i � � ��� c�

Constraints generate the polyhedral domain S�M��

in the control space:

S�M�� � fu � �c � � Mu g

Under a state-feedback regulation law:

ut � Fxt

the linear control constraints define a polyhedron inthe state space, S�MF��.

S�MF�� � fx � �n � � MFx g

17

Page 18: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

POSITIVE INVARIANCE BY STATE FEEDBACK

Positive Invariance relations

A necessary and sufficient condition for S�Q� �� tobe a positively invariant set of system (S) is the ex-istence of a matrix H � �r�r and of a scalar � suchthat:

HQ � Q�A� BF�

�H� ��

with �H � jHj� � � � in the discrete-timecase,�H � H� � � in the continuous-time case.

A structural interpretation

yt � Qxt can be interpreted as an output vector.The first relation imposes (A+BF)-invariance ofKer Q.

18

Page 19: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

POSITIVE INVARIANCE BY LINEARPROGRAMMING

Basic LP formulation:

Minimize

subject to HG�GBF � GA

H� �

Hij � � ��i� j� in the discrete-time case

Hij � � ��i� j � i� in the continuous-time case

� � in the discrete-time case

Result: R�G� �� is �A� BF�-invariant if: in the discrete-time case, � in the continuous-time case.

RemarkMany additional constraints can be added to this prob-lem, to take into account:- control constraints- parametric uncertainties- regional pole placement.

19

Page 20: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

POSITIVE INVARIANCE BY EIGENSTRUCTUREASSIGNMENT

� This scheme applies to systems with linear con-straints on the state vector or on the control vec-tor or on the output vector.

� In this scheme, the domain of constraints S�Q� ��is made positively invariant by state feedback.

� The construction applies only if

rank�Q� � r m�

� The problem is decomposed into two stages:

– (A+BF)-invariance of Ker Q. This is equiva-lent to locating n�r closed-loop generalizedeigenvectors in alKerQ.

– Resolution of positive invariance conditionsin �n�Ker Q

20

Page 21: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

SPECTRAL SUFFICIENT CONDITIONS

A spectral condition - discrete time caseIf matrix H satisfying :

HQ � QA

has the real Jordan form, and its eigenvalues,i� j�i verify:

jij� j�ij �

then, �� � � such that:

jHj� ��

And polyhedron S�Q� �� is positively invariant.

������������������������������������������������������������������������������������������������������������������������������������

I

R0 1

1

Spectral domain

21

Page 22: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

SPECTRAL SUFFICIENT CONDITIONS

A spectral condition - continuous time caseIf matrix H satisfying :

HQ � QA

has the real Jordan form, and its eigenvalues,i� j�i verify:

i �j�ij�

then, �� � � such that:

H� ��

And polyhedron S�Q� �� is positively invariant.

R

I

0

Spectral domain

22

Page 23: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

POSITIVE INVARIANCE BY EIGENSTRUCTUREASSIGNMENT

Consider the system matrix (Rosenbrock 1970):

P��� �

��I �A �B

Q �r�m

(A+BF)-invariance of Ker Q is possible if and only ifthe equation

P ��i�

�viwi

��

���

has at least n � r solutions ��i� vi�, with vectors viindependent.Structural conditionIf r m and r � n, condition rank�QB� � r issufficient for (A,B)-invariance of Ker Q .

Remark:If any invariant zero is unstable, (A+BF)-invarianceof Ker Q and closed-loop stability will not be simul-taneously obtained.

23

Page 24: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

EIGENSTRUCTURE ASSIGNMENT IN Ker Q

If rank�QB� � r m, the following technique canbe applied:

1. Select n� r stable closed-loop poles:

- The p invariant zeros of �A�B�Q� have to be se-lected as closed-loop poles. They must be stable.

- The n� p� r remaining closed-loop poles are se-lected in the stable region as desired.

- J� is the Jordan form of the restriction of �A�BF �to Ker Q .

2. Eigenvectors spanning Ker Q

- Define V� � �v�� ���� vn�r� such that:

GV� � �

�A� BF�V� � V�J�

- For any �i, solve

P ��i�

�viwi

��

���

24

Page 25: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

EIGENSTRUCTURE ASSIGNMENT IN ACOMPLEMENTARY SUBSPACE OF Ker Q

1. Selection of r appropriate closed-loop poles:

The eigenvalues ��i of �A� BF�j��n�Ker Q� areselected so as to satisfy:

jij� j�ij � in the discrete-time case

i � �j�ij in the continuous-time case

J� : real Jordan form of �A� BF�j��n�Ker Q�.

2. Eigenvectors spanning R

Vectors v�i and w�i are computed to satisfy:

P���i�

�v�iw�i

��

��ei

�, with ei �

��������

� �

�������� i�

25

Page 26: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

CONSTRUCTION OF THE GAIN MATRIX

Let V � �V� j V�� be the matrix of the desired realgeneralized eigenvectors, and W � �W� j W�� theassociated input directions.

The selected real Jordan form of �A�BF� is:

J �

�J� �� J�

The feedback gain matrix providing the desired eigen-structure assignment is:

F �WV ��

By construction, it satisfies for some positive vector�:

J�Q� Q�A� BF�

�J�� ��

26

Page 27: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

EXAMPLE 1 : State Constraints

Consider the following data:

A �

��� ������ ������ ������ ���� �� ������ �������������� ������ ������

��

B �

��� ������� ������

������ ����� �������� ������

��

The open-loop system has unstable eigenvalues:

��A� �

��� ����� � j����������� � j������

����

��

The state constraints are defined by

�� Qx �

with:

Q�

�������� ����� ������������� ������� �������

�� ��

�������

System �A�B�Q� has an stable invariant zero at������.

27

Page 28: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

Positive invariance of S�Q� �� and global asymptoticstability are obtained when selecting:

��� �� � ������ ( stable zero)��� � �� � j������� � �� � j���

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

Invariance of the domain of constraints

28

Page 29: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

-2

-1

0

1

2

3

4

5

0 5 10 15 20 25 30

Convergence in ���Ker Q

-6

-5

-4

-3

-2

-1

0

0 5 10 15 20 25 30

Convergence in Ker Q

29

Page 30: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

EXAMPLE 2 : Control Constraints

Third order system:

A �

�� ����� ������ ����������� ������ ��������� ����� ������

� �

B �

�� ���� ��������� �������� ����

The open-loop system has two unstable eigenvalues, ���� � ��� �,and one stable, ���.

��� � Suk � �� with S �

����� ��������� ����

��

In this example, r � m � �. The stable pole, �� � ��� is leftunchanged.

We select

��� � ���� ���j�� � ��� � ���j

, and S�� as the matrix of real

input vectors associated with these last two eigenvalues.

30

Page 31: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

EXAMPLE 2 (contd.)

The matrix of closed-loop generalized real eigenvectors be-comes:

V �

�� ����� �� �� ������

��� ���� ���������� ���� ������

under the feedback gain matrix:

F �

������ ���� ��������� ����� ����

��

to obtain:

J �

�� ��� � �

� ��� ����� ���� ���

� �

A better eigenstructure assignment is obtained by simply in-verting the order of �� and ��.

Under the new feedback gain matrix,

F �

������ ����� ��������� ����� ����

�the size of the invariant domain is increased by more than40�.

31

Page 32: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

-1

0

1

-1 0 1

-1

0

1

-1 0 1

x2

x1

-1

0

1

-1 0 1

-1

0

1

-1 0 1

-1

0

1

-1 0 1

(1.a)

-1

0

1

-1 0 1

-1

0

1

-1 0 1

x2

x1

-1

0

1

-1 0 1

-1

0

1

-1 0 1

-1

0

1

-1 0 1

(1.b)

The invariant domains in projection

32

Page 33: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

PART 3DISTURBANCE ATTENUATION AND

CONSTRAINED REGULATION

� Domain of satisfactory performance

� Positive invariance with disturbance attenuation

� Regulator design with positive invariance prop-erties

� The feasible domain

� An application in production planning

– The dynamical model

– A closed-loop production policy

33

Page 34: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

DOMAIN OF SATISFACTORY PERFORMANCE

Discrete-time linear system:

xk�� � Axk �B�uk � B�wk

wk � �q is the disturbance input vector.

It is random and takes its value in a closed and boundedpolyhedral set in �q :

wk � R�L� �� � fw � �qjLw �g

Constraints on the state vector:

Sxxk �x �k � N

Constraints on the control input vector:

Suuk �u �k � N

Combined performance requirements:

Zsxk � Zuuk �� �k � N �

Target state: �x�� u��.The problem is formulated as a regulation problem.Under a stabilizing linear state feedback, uk � Fxk,all the constraints and requirements define a polyhe-dral performance domain:

R�Q� �� � fx � �n j Qx ��g

with � � �q, � � �.

34

Page 35: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

POSITIVE INVARIANCE WITH DISTURBANCEATTENUATION

Autonomous linear system (S):

xk�� � A�xk � B�wk

with wk � R�L� ��.

Polyhedral set in the state space �n:

R�G� �� � fx � �n j Gx ��g

Positive invariance of R�G� ��:

x� � R�G� �� �� xn � R�G� ��

�n � N � �fwkg� wk � R�L� ���

A geometric characterization:

P�R�G� ��� � R�G� �� with

P�R�G� ��� � fy � A�x�B�w j

�x � R�G� ���w � R�L� ���

35

Page 36: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

POSITIVE INVARIANCE ;A geometric characterization

������������������������������������������������������������������������������������������������

R(G,η) P (R(G,η))

P�R�G� ��� � R�G� ��

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Page 37: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

POSITIVE INVARIANCE WITH DISTURBANCEATTENUATION

Positive invariance theorem:

A NSC for positive invariance of R�G� �� w.r.t. (S) forany disturbance vector wk in R�L� ��, is theexistence of two nonnegative matrices H and M

such that :

HG � GA�

ML � GB�

H��M� ��

This theorem is obtained by application of Farkas’Lemma to the inclusion conditionP�R�G� ��� � R�G� ��.

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Page 38: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

POSITIVE INVARIANCE OF SYMMETRICALPOLYHEDRA

Positive invariance theorem:

A NSC for positive invariance of

S��� �� � fx � �nj � � �x �g�

w.r.t. (S) for any disturbance vector wk in

S��� �� � fw � �qj � � �w �g

is the existence of two matrices H and M such that:

H� � �A�

M� � �B�

jHj� � jM j� ��

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Page 39: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

REGULATOR DESIGN

Spectral assignment methodology:

� Selection of the closed-loop spectrum :

��A� B�F�

� Closed-loop real Jordan form : J

� Matrix of generalized real eigenvectors:V such that :

JV �� � V ���A� BF��

V � �V� j V�� and W � �W� jW��

with for �i real in ��A� B�F�,

��iI � A � B�

�viwi

�� �

and F �WV ���

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Page 40: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

A SPECTRAL SUFFICIENT CONDITION FORPOSITIVE INVARIANCE WITHOUT

DISTURBANCES

If matrix J satisfying :

J� � �A�

has the real Jordan form,and its eigenvalues, i� j�i verify:

jij� j�ij �

then, �� � � such that:

jJ j� ��

And polyhedron S��� �� is positively invariant (in theundisturbed case).

������������������������������������������������������������������������������������������������������������������������������������

I

R0 1

1

Spectral domain jij� j�ij

40

Page 41: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

SUFFICIENT CONDITIONS FOR POSITIVEINVARIANCE UNDER BOUNDED

DISTURBANCES

1. Positive invariance with contractivity.Set � � V �� and A� � A� B�F .If we impose the tighter conditions :

j�ij� j�ij � � �i

with � �i ,then, row by row :

�iA� � Ji�

jJij� � � �i���

2. Disturbance attenuationCondition P�S��� ��� � S��� �� is obtainedthrough the algebraic conditions :

M� � �B�

jM j� diag��l��

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Page 42: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

THE CONSTRAINED CONTROL SCHEME

Construction of a domain � such that :

1. The zero state lies in the interior of �,

2. � is D�invariant with respect to thecontrolled system, for any disturbancevector wk in R�L� ��.

3. � � R�Q� ��

x1

x2

R(Q,ρ)

Ω

���������������������������������������������������������������������������������������������������

����������������������������������������������������������������������������������������������������������

����������������

An admissible domain

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Page 43: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

THE DOMAIN OF ADMISSIBLE INITIAL STATES

The size of the invariant domain is maximized throughmaximization of the components of �.

The scheme is completed by solving the followingLinear Program:

Maximize C �nX

i��

ci�i with all ci � �

subject to :

jQV j� �

�iA� � Ji�

jJij� � � �i��

M� � �B�

jM j� diag��l��

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Page 44: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

APPLICATION TO PRODUCTION PLANNING

π43

θ5

θ4

θ3

θ1θ2

5

4

3

2 1

π53

1

1

11

1

π32 π31

π51

π52

A Petri net representation of a product structure

� �

�����

� � � � �� � � � ���� ��� � � �� � ��� � ���� ��� ��� � �

���� �

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Page 45: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

THE PRODUCTION MODEL

Stock equation for product i :

yik � yi�k��� ui�k��i �NXj��

�ijvjk � dik�

��� sik � yik � s�i if yik � �s�isik � � if yik �s�i

with a backorder of � yik � s�i �

Stock equation in vector form :

� � q���yk � �diag�q��i����vk � dk

Decomposition of the demand vector

dk � d� wk

with E�dk� � d� E�wk� � �.

Boundedness assumption :

�w wk w with w d�

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Page 46: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

THE PRODUCTION MODEL

Steady-state nominal control

v� � �I �����d

Property : v� � �.

Change of variable uk � vk � v� to obtain :

� � q���yk � ���� T�q��� � � �T�q

���uk�wk

State vector :

xk � �yTk�� uTk�� � � �u

Tk���

T

State equation :

xk�� � Axk � B�wk � B�uk

with

A �

�����

I T� � � � � � � T�O � � � � � � OO I O � � � O... . . . . . . ...O � � � O I O

���� � B� �

�����

I

O...

O

���� � B� �

�����

T�I

O...O

���� �

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Page 47: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

AN INVARIANT CONTROLLER

vk � v�� �F G� � � � G��xk

D-Invariance of a domain S��� �� is obtainedby choosing :

F � ��I ������

Gj � ��I �����Ej for j � � ���� �

with,

Ej �� diag�eji� with

�eji � if j �ieji � � if j � �i

S��� �� defines a set of admissible initial statesunder:

S��� �� � R�Q� ���

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Page 48: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

THE ADMISSIBLE SET OF INITIAL STATES

The positively invariant domain for the closed-loopsystem, S��� �� is defined by:

� �

��������

I E� � � � � � � E�

O ��I ��� O � � � OO ��I ��� . . . ...

... . . . OO � � � O ��I ���

��������

� � �� � � � ��T with � � ��

D�invariance of S��� �� is obtained if

�i � max�wi� wi� �i� � ���� N�

Inclusion S��� �� � R�Q� �� with maximization ofthe components �i is obtained by Linear Program-ming.

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Page 49: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

CONCLUSIONS

� The proposed methodology has been mainly basedon spectral assignment.

This approach also offers degrees of freedomin the choice of the eigenstructure. They can beused to improve:

– the robustness of the control schememinimization of condition number

k�V � � kV k�kV��k��

– the size of the set of admissible initial statesS��� ��.

� Positively invariant controls generally provide lo-cal solutions valid only if

x� � S��� ���

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Page 50: A Control Methodology for Constrained Linear …A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse,

CONCLUSIONS (contd.)

� A dual-mode control scheme can be constructedif x� �� S��� ��:

– The state is first attracted to S��� �� (in open-loop).

– Then the closed-loop scheme can be applied.

� The concepts of (A,B)-invariance and D-(A,B)-invariance have been studied to generalize thepositive invariance approach to non-linear con-trol laws and resolution of optimization problemssuch as:

– maximization of the set of admissible initialstates

– optimal attenuation of bounded persistent dis-turbances (�� problem).

50