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Graph-theoretic Approach to the Structural Controllability and Observability of Large-scale Systems
Copll'ighl © I L\C Llrge SClie S""'I1l S, Berlill . (:I>R , I ' IH'I
GRAPH·THEORETIC APPROACH TO THE STRUCTURAL CONTROLLABILITY AND OBSERVABILITY OF LARGE-SCALE SYSTEMS C. Rech 1.llillJlIlt()in' (/',1/ifrlll/{/tiqu(' (/1' (;'(' II()il/1' ( c'\'RS L'R,1 228 ) /:'.\'.\11:'(; · /.\'/'(;,
HP , Hi. 38-1112 Slllllt .\llllti ll (/'//hl'l, FUIIIU'
~.
A sufficient and graph-theoretic condition is proposed within the scope of the structural controllability and observability theory in order to express these properties on a large-scale system made up of several interconnected unities.
Keywords. Structural controllability ; Observability; Large-Scale Systems ; Graph-theoretic. Example I:
INTRODUCfION A considerable number of present day industrial processes are made up of a large quantity of interconnected unities. If we wish to examine the property of structural controllability, i.e. controllability as a function of the state model parameters, it is advantageous to break down existing criteria in order to express controllability of the physical sub-systems of which it is made up . In fact this approach has the advantage not only of applying the various algorithms on smaller sized matrices, thereby reducing calculation time, but also of using the actual structure of the system to decide whether or not it can be controlled. Since it is reasonable to assume that each unity is controllable when it is uncoupled, the results are not interprdt.taccording to its structure but according to that of the various interconnections.
(A, B)
=
5 I) 362 [~ o473
(A, B) admits for structured representation : X 0 X Xl (As, Bs) = 0 X X X [ X X X X where X symbolizes the presence of a non zero parameter.
UN (1972) introduced the concept of structural controllability for systems having poorly known parameters either due to measurement errors or lack of quantitative information. The presence or absence of links is then considered to form the data of the system, hence the name "structural", A variety of criteria have been developed to characterize this approach in the fields of boolean algebra and the graph-theoretic approach.
Definition I : A system or pair (As, Bs) is said to be structurally controllable (observable) if and only if it is controllable (observable), as interpreted by KALMAN, for practically all the parameter values. Proposition I : (UN, 1972, 1974)
In a similar manner to KALMAN's controllability, the structural observability structurally describing the observability of each state, has been defined by duality.
(As, Bs) is structurally controllable if and only if there exists a numerical pair (A', B') of identical structure which is KALMAN controllable.
This paper proposes a sufficient condition of controllability (observability) for interconnected systems (RECH c., 1988) (RECH, PERRET, 1988), using a solely graph-theoretic approach based on a type of graph specific to UN and known as a cactus.
Definition 2 : Consider any structured matrix Ms = (aij) , n x n. The directed graph G(Ms) associated with Ms has vertices X and edges U such that :
We will begin by reviewing some definitions and tools required present the for understanding the rest of the paper. We will graph-theoretic results obtained both for controllability and observability.
a) there are n vertices b) there exists a directed edge (j, i) of j towards i if and only if aij ~ O. Example 2 :
STRUCTURAL CONTROLLABILITY OF AN ISOLATED SYSTEM
MS=[:
Consider an invariant, dynamic , linear system described by its state equation dx/dt = AX + BU and its observation equation Y = CX where A(n x n), B(n x m), C(r x n).
G(Ms) =
Graph G (Ms) is said to be accessible to the vertix Xo if and only if there exists a directed edge path of Xo at 11 the vertices of G (Ms).
The "structure" of the pair (A, B) is introduced by the following equivalence relation : two numerical pairs (A , B) and (A', B') have the same structure if and only if they have the same number of coefficients fixed at zero and if these latter are located in the same place. Each equivalence class charClCterizes a system structure. In this approach, a system is represented by its structure explicitly given by the stru'!lure matrix relating to the state model (A, B), A matrix is said tOlstructured if it is made , up of free parameters (algebraically independent) or of fixed zeros,
Definition 3 : The generic rank of a structured matrix is equal to the maximal value C'f the rank depending on the matrix parameters.
I I ~I
l2()
C. Re eh
Theorem I : (LIN, 1972, 1974)
Reduced representation of a cactus
The structured system (As, Bs) is structurally controllable if and only if :
According to the definition of the cascade between two cacti, the only information needed to be preserved on each cactus is its origin 0, its extremity e and an imaginary edge (symbolizing the stem) which connects them; the cactus will thus be marked in abbreviated form (0, e). Each cascade therefore constitutes an alternate path of the type (01, et, 02, e2, .... ).
a) Graph G (As, Bs) is input - reachable. b) The generic rank of (As, Bs) is equal to n. Definition 4 : (LIN, 1972, 1974)
* Stem : a stem is a finite directed edge path (non cyclic) linking
The main result is as follows :
an origin to an extrimity (Fig. I).
Theorem 3 : (RECH, 1988)
* Bud : a bud is made up of an edge linking an origin to an extremity, on which is fixed a cycle passing through one or more vertices (Fig. 2).
Consider an interconnected system made up of N sub-systems (Si).
* Single origin
cactus : a cactus is obtained from a stem T by forming a graph sequence (Go = T) C G 1... .. c. (Gp = G) p ~ 0 in the following manner: a) the first graph in the sequence is stem T b) the last graph is graph G itself c) 'V k = I, 2 .. ., P Gk is obtained from Gk-I by adding a bud Bk to Gk-I such that origin ek of Bk is the only vertex common to Gk and Gk-1. (the origin of the bud introduced can never coincide with the stem extremity). (Fig. 3).
* Multi-origin cactus : this is a disjointed assembly of multi-origin cacti (Fig. 4) . The cactus represents the purely essential, i.e. minimal system structure, ensuring maintenance of the controllability property (LIN, 1977). This "minimality" aspect should be stressed. This concept is physically justified by the fact that a system nearly always possesses inputs or (and) redundant links. (RECH, 1988) proposes a cactus extraction method. The cactus correspond to a canonical form of the system from which it is extracted and must be recognized as such in the li'1far system theory. It should, nevertheless, be emphasized thatj!s not the result of a basic change as are the canonical forms in controllability in the normal understanding of the term. A cactus matrix is obtained by suppressing certain non zero coefficients in (As, Bs) initial. Theorem 3 : (LIN, 1972) (As, Bs) is structurally controllable if and only if G (As, Bs) is spanned by a cactus. SUFFICIENT CONDmON FOR THE STRUCTURAL CONTROLLABILITY OF AN INTERCONNECTED SYSTEM This approach is based on theorem 3 and on the construction of a global cactus from the cacti spanning each sub-system. It is founded on the "cacti cascade" notion introduced in (RECH, 1988) : Definition 5 : Consider two sub-systems (SI) and (S2), single-input , and structurally controllable in the minimal sense (their graph is a cactus). Their interconnection is a cascade with respect to control Ui if and only if the stem extremity of sub-system (Si) on which it is applied, is linked to the origin of the other sub-system (Sj) (Fig.
5). Definition 6 : For two sub-systems (S I) and (S2), multi-inputs , verifying the previous hypotheses, their interconnection is a cascade with respect to Ui (control vector of (Si) if and only if each elementary cactus Ckj of the second sub-system (Sj) is linked by a cascade to an elementary cactus of the first sub-system Cli of (Si) on which Ui is applied, the pairs thus formed having no common elementary cactus (Fig. 6).
Consider a set of controls Ui applied to certain (Si). The overall system is structurally controllable if there exists a cacti cascade with respect to a sub-set of Ui, spanning G R, in which G R is the graph obtained by replacing in G(A, B) the cacti by their reduced representation and by preserving only the cascades occuring among these cacti (Fig. 7). Corollary I : A minimum number of inputs sufficient to control A is equal to the number of extreme branches of a cacti cascade arborescence spanning G R i.e. A R. Their distribution is such that a cascade is obtained (FIg. 7). SUFFICIENT CONDmON FOR THE STRUCTURAL OBSERVABILITY OF AN INTERCONNECTED SYSTEM Introducing the cactus notion with respect to the outputs (Fig. 8), similar results can be stated, by duality (RECH, 1988). Definition 7 : An interconnection of two single-output cacti is a cascade with respect to output Yi if and only if the stem extremity of sub-system Sj, j 7' i, is linked to the origin of Si (Fig. 9). In multi-outputs, configurations of the same type, in parallel, are obtained (Fig. 9). Theorem 4 : Consider a set of outputs Yi observing certain cacti (Ci) (extracted from Si). The overall system is structurally observable if there exists a cacti cascade with respect to a sub-set of Yi spanning G R in which G R is the graph obtained from G (A, C) by replacing the cacti by their reduced representation (0, e) and by preserving only the cascades occuring among these cacti (Fig.
10). Corolla!), 2 : Un.1ltthese conditions, a minimum number of outputs is equal to the number of extreme branches of the extracted arborescence and their position is these very extremities (Fig. 10). CONCLUSION The sufficient condition for structural controllability (observability) developed in this paper, constitues a method of breaking down structural criteria, in the sense that overall properties are interpreted as a function of the structure of local interconnections. This characterization based on the system structure is physically justified and presents certain advantages over other algebraic methods which have difficulty considering this aspect (REINSCHKE, 1981), (TRA VE and TITLI, 1985). Moreover, given that the sub-system properties are known and that therefore cacti may be extracted from them, this graph-theoretic approach is finally reduced to an accessibility test in a condensed graph, thereby dispensing with all global consideration of generic rank.
12 1
Structural Co ntmllabilitv a nd Obse n ·ahilit,· of La rgc-s(,ti c S'Stc llls REFERENCES Lin, C.T . (1972). Structural Controllability. Ph. D thesis University of Maryland. Lin, C.T. (1974). Structural Controllability. IEEE Trans Autom Control, l2, nO3, pp. 201 -208. Lin, C.T. (1977). System structure and minimal structural controllability. IEEE Trans AUlom Control,22. nO5, pp. 855-862. Rech, C. (1988). Commandability et observability structure lies des systems interconnectes. These de doctorat, Universite de Lille. Rech , c., and R. Perret (1988). About structural controllability of interconnected dynamical systems. LA.G Internal &l2QU nO88/65, Grenoble, France. Reinschke, K. (1981) . Structurally complete systems with minimal input and output vectors. Laree-scale systems, 2, pp. 235-242. Trave, L, and A. Titli (1985) . A sequential algorithm to conclude on structural controllability of large-scale systems, Internal Report nO85312, LA.A.S, Toulouse, France.
FIGURES
,.
~.,
>
et
o
Fig. I
Fig. 2 e~
I(
..
o~
Fig. 3
Fig. 4
Fig. 5 Cascade Cl - C2 in doned lines
u~ )\
Ut.
)
r
>
>
v
p} : .'-
)
--
-~-
-
~.' (C-1) Fig. 6 Cascade Cl - C2
~.& .•
C. Rec h
122
A
G
R
R
Fig. 7 Cascade spanning G R
Fig. 8
" >
R.
>~
--
Cascade mono-output (Cl) - (C2)
->-- - - - "
>
\J:
h"---->-----> 'l::'t:~' Fig.9 Cascade multi-output (Cl) - (C2)
;J---~ /
-1
, '-
/
y I
/
I
~f
/ /
G
R
-
A
, R
J-:'
f: /
4
/
J:,
/
f
I
f
,
'~
--
t:,
J y: 1 p-
,
I
Y~I
Fig. 10 An example of cascade spann ing G R
/
1:5
GRAPH·THEORETIC APPROACH TO THE STRUCTURAL CONTROLLABILITY AND OBSERVABILITY OF LARGE-SCALE SYSTEMS C. Rech 1.llillJlIlt()in' (/',1/ifrlll/{/tiqu(' (/1' (;'(' II()il/1' ( c'\'RS L'R,1 228 ) /:'.\'.\11:'(; · /.\'/'(;,
HP , Hi. 38-1112 Slllllt .\llllti ll (/'//hl'l, FUIIIU'
~.
A sufficient and graph-theoretic condition is proposed within the scope of the structural controllability and observability theory in order to express these properties on a large-scale system made up of several interconnected unities.
Keywords. Structural controllability ; Observability; Large-Scale Systems ; Graph-theoretic. Example I:
INTRODUCfION A considerable number of present day industrial processes are made up of a large quantity of interconnected unities. If we wish to examine the property of structural controllability, i.e. controllability as a function of the state model parameters, it is advantageous to break down existing criteria in order to express controllability of the physical sub-systems of which it is made up . In fact this approach has the advantage not only of applying the various algorithms on smaller sized matrices, thereby reducing calculation time, but also of using the actual structure of the system to decide whether or not it can be controlled. Since it is reasonable to assume that each unity is controllable when it is uncoupled, the results are not interprdt.taccording to its structure but according to that of the various interconnections.
(A, B)
=
5 I) 362 [~ o473
(A, B) admits for structured representation : X 0 X Xl (As, Bs) = 0 X X X [ X X X X where X symbolizes the presence of a non zero parameter.
UN (1972) introduced the concept of structural controllability for systems having poorly known parameters either due to measurement errors or lack of quantitative information. The presence or absence of links is then considered to form the data of the system, hence the name "structural", A variety of criteria have been developed to characterize this approach in the fields of boolean algebra and the graph-theoretic approach.
Definition I : A system or pair (As, Bs) is said to be structurally controllable (observable) if and only if it is controllable (observable), as interpreted by KALMAN, for practically all the parameter values. Proposition I : (UN, 1972, 1974)
In a similar manner to KALMAN's controllability, the structural observability structurally describing the observability of each state, has been defined by duality.
(As, Bs) is structurally controllable if and only if there exists a numerical pair (A', B') of identical structure which is KALMAN controllable.
This paper proposes a sufficient condition of controllability (observability) for interconnected systems (RECH c., 1988) (RECH, PERRET, 1988), using a solely graph-theoretic approach based on a type of graph specific to UN and known as a cactus.
Definition 2 : Consider any structured matrix Ms = (aij) , n x n. The directed graph G(Ms) associated with Ms has vertices X and edges U such that :
We will begin by reviewing some definitions and tools required present the for understanding the rest of the paper. We will graph-theoretic results obtained both for controllability and observability.
a) there are n vertices b) there exists a directed edge (j, i) of j towards i if and only if aij ~ O. Example 2 :
STRUCTURAL CONTROLLABILITY OF AN ISOLATED SYSTEM
MS=[:
Consider an invariant, dynamic , linear system described by its state equation dx/dt = AX + BU and its observation equation Y = CX where A(n x n), B(n x m), C(r x n).
G(Ms) =
Graph G (Ms) is said to be accessible to the vertix Xo if and only if there exists a directed edge path of Xo at 11 the vertices of G (Ms).
The "structure" of the pair (A, B) is introduced by the following equivalence relation : two numerical pairs (A , B) and (A', B') have the same structure if and only if they have the same number of coefficients fixed at zero and if these latter are located in the same place. Each equivalence class charClCterizes a system structure. In this approach, a system is represented by its structure explicitly given by the stru'!lure matrix relating to the state model (A, B), A matrix is said tOlstructured if it is made , up of free parameters (algebraically independent) or of fixed zeros,
Definition 3 : The generic rank of a structured matrix is equal to the maximal value C'f the rank depending on the matrix parameters.
I I ~I
l2()
C. Re eh
Theorem I : (LIN, 1972, 1974)
Reduced representation of a cactus
The structured system (As, Bs) is structurally controllable if and only if :
According to the definition of the cascade between two cacti, the only information needed to be preserved on each cactus is its origin 0, its extremity e and an imaginary edge (symbolizing the stem) which connects them; the cactus will thus be marked in abbreviated form (0, e). Each cascade therefore constitutes an alternate path of the type (01, et, 02, e2, .... ).
a) Graph G (As, Bs) is input - reachable. b) The generic rank of (As, Bs) is equal to n. Definition 4 : (LIN, 1972, 1974)
* Stem : a stem is a finite directed edge path (non cyclic) linking
The main result is as follows :
an origin to an extrimity (Fig. I).
Theorem 3 : (RECH, 1988)
* Bud : a bud is made up of an edge linking an origin to an extremity, on which is fixed a cycle passing through one or more vertices (Fig. 2).
Consider an interconnected system made up of N sub-systems (Si).
* Single origin
cactus : a cactus is obtained from a stem T by forming a graph sequence (Go = T) C G 1... .. c. (Gp = G) p ~ 0 in the following manner: a) the first graph in the sequence is stem T b) the last graph is graph G itself c) 'V k = I, 2 .. ., P Gk is obtained from Gk-I by adding a bud Bk to Gk-I such that origin ek of Bk is the only vertex common to Gk and Gk-1. (the origin of the bud introduced can never coincide with the stem extremity). (Fig. 3).
* Multi-origin cactus : this is a disjointed assembly of multi-origin cacti (Fig. 4) . The cactus represents the purely essential, i.e. minimal system structure, ensuring maintenance of the controllability property (LIN, 1977). This "minimality" aspect should be stressed. This concept is physically justified by the fact that a system nearly always possesses inputs or (and) redundant links. (RECH, 1988) proposes a cactus extraction method. The cactus correspond to a canonical form of the system from which it is extracted and must be recognized as such in the li'1far system theory. It should, nevertheless, be emphasized thatj!s not the result of a basic change as are the canonical forms in controllability in the normal understanding of the term. A cactus matrix is obtained by suppressing certain non zero coefficients in (As, Bs) initial. Theorem 3 : (LIN, 1972) (As, Bs) is structurally controllable if and only if G (As, Bs) is spanned by a cactus. SUFFICIENT CONDmON FOR THE STRUCTURAL CONTROLLABILITY OF AN INTERCONNECTED SYSTEM This approach is based on theorem 3 and on the construction of a global cactus from the cacti spanning each sub-system. It is founded on the "cacti cascade" notion introduced in (RECH, 1988) : Definition 5 : Consider two sub-systems (SI) and (S2), single-input , and structurally controllable in the minimal sense (their graph is a cactus). Their interconnection is a cascade with respect to control Ui if and only if the stem extremity of sub-system (Si) on which it is applied, is linked to the origin of the other sub-system (Sj) (Fig.
5). Definition 6 : For two sub-systems (S I) and (S2), multi-inputs , verifying the previous hypotheses, their interconnection is a cascade with respect to Ui (control vector of (Si) if and only if each elementary cactus Ckj of the second sub-system (Sj) is linked by a cascade to an elementary cactus of the first sub-system Cli of (Si) on which Ui is applied, the pairs thus formed having no common elementary cactus (Fig. 6).
Consider a set of controls Ui applied to certain (Si). The overall system is structurally controllable if there exists a cacti cascade with respect to a sub-set of Ui, spanning G R, in which G R is the graph obtained by replacing in G(A, B) the cacti by their reduced representation and by preserving only the cascades occuring among these cacti (Fig. 7). Corollary I : A minimum number of inputs sufficient to control A is equal to the number of extreme branches of a cacti cascade arborescence spanning G R i.e. A R. Their distribution is such that a cascade is obtained (FIg. 7). SUFFICIENT CONDmON FOR THE STRUCTURAL OBSERVABILITY OF AN INTERCONNECTED SYSTEM Introducing the cactus notion with respect to the outputs (Fig. 8), similar results can be stated, by duality (RECH, 1988). Definition 7 : An interconnection of two single-output cacti is a cascade with respect to output Yi if and only if the stem extremity of sub-system Sj, j 7' i, is linked to the origin of Si (Fig. 9). In multi-outputs, configurations of the same type, in parallel, are obtained (Fig. 9). Theorem 4 : Consider a set of outputs Yi observing certain cacti (Ci) (extracted from Si). The overall system is structurally observable if there exists a cacti cascade with respect to a sub-set of Yi spanning G R in which G R is the graph obtained from G (A, C) by replacing the cacti by their reduced representation (0, e) and by preserving only the cascades occuring among these cacti (Fig.
10). Corolla!), 2 : Un.1ltthese conditions, a minimum number of outputs is equal to the number of extreme branches of the extracted arborescence and their position is these very extremities (Fig. 10). CONCLUSION The sufficient condition for structural controllability (observability) developed in this paper, constitues a method of breaking down structural criteria, in the sense that overall properties are interpreted as a function of the structure of local interconnections. This characterization based on the system structure is physically justified and presents certain advantages over other algebraic methods which have difficulty considering this aspect (REINSCHKE, 1981), (TRA VE and TITLI, 1985). Moreover, given that the sub-system properties are known and that therefore cacti may be extracted from them, this graph-theoretic approach is finally reduced to an accessibility test in a condensed graph, thereby dispensing with all global consideration of generic rank.
12 1
Structural Co ntmllabilitv a nd Obse n ·ahilit,· of La rgc-s(,ti c S'Stc llls REFERENCES Lin, C.T . (1972). Structural Controllability. Ph. D thesis University of Maryland. Lin, C.T. (1974). Structural Controllability. IEEE Trans Autom Control, l2, nO3, pp. 201 -208. Lin, C.T. (1977). System structure and minimal structural controllability. IEEE Trans AUlom Control,22. nO5, pp. 855-862. Rech, C. (1988). Commandability et observability structure lies des systems interconnectes. These de doctorat, Universite de Lille. Rech , c., and R. Perret (1988). About structural controllability of interconnected dynamical systems. LA.G Internal &l2QU nO88/65, Grenoble, France. Reinschke, K. (1981) . Structurally complete systems with minimal input and output vectors. Laree-scale systems, 2, pp. 235-242. Trave, L, and A. Titli (1985) . A sequential algorithm to conclude on structural controllability of large-scale systems, Internal Report nO85312, LA.A.S, Toulouse, France.
FIGURES
,.
~.,
>
et
o
Fig. I
Fig. 2 e~
I(
..
o~
Fig. 3
Fig. 4
Fig. 5 Cascade Cl - C2 in doned lines
u~ )\
Ut.
)
r
>
>
v
p} : .'-
)
--
-~-
-
~.' (C-1) Fig. 6 Cascade Cl - C2
~.& .•
C. Rec h
122
A
G
R
R
Fig. 7 Cascade spanning G R
Fig. 8
" >
R.
>~
--
Cascade mono-output (Cl) - (C2)
->-- - - - "
>
\J:
h"---->-----> 'l::'t:~' Fig.9 Cascade multi-output (Cl) - (C2)
;J---~ /
-1
, '-
/
y I
/
I
~f
/ /
G
R
-
A
, R
J-:'
f: /
4
/
J:,
/
f
I
f
,
'~
--
t:,
J y: 1 p-
,
I
Y~I
Fig. 10 An example of cascade spann ing G R
/
1:5