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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 VASSILAKIUniversity of Patras, GREECE
Jean-Paul BEZIATCISI Bordeaux, FRANCE
Eugenio B. CASTELANUFSC, Florianopolis, BRAZIL.
Carlos E. T. DOREAUFBA, Bahia, BRAZIL
1
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
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
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
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
POSITIVE INVARIANCEA geometric characterization
Ω Ω Ω
D (t)
D (t’)
Dt � �
6
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
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
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
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
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).
11
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
-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
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
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
-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
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
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
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
POSITIVE INVARIANCE ;A geometric characterization
������������������������������������������������������������������������������������������������
R(G,η) P (R(G,η))
P�R�G� ��� � R�G� ��
36
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� ��.
37
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� ��
38
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 ���
39
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
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��
41
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
42
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��
43
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
� �
�����
� � � � �� � � � ���� ��� � � �� � ��� � ���� ��� ��� � �
���� �
44
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�
45
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
���� �
46
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� ���
47
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.
48
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��� ���
49
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