- Email: [email protected]
Study of Structured Systems a Graph Approach
Copyright © IFAC Control of Industrial Systems. Belfort, France, 1997
STUDY OF STRUCTURED SYSTEMS A GRAPH APPROACH C. Commault J.M. Dion V. Hovelaque
1
Laboratoire d'Automatique de Grenoble. INPG/UJF/CNRS UMR 5528, ENSIEG, BP. 46 , 38402St Martin d'Heres, France. Tel. : (33) 4.76.82.62.44, Fax: (33) 4.76.82.63.88. e-mail: [email protected]
Abstract. In this paper, structured systems described by state space models are considered. In this context, the entries of the state space model matrices are supposed to be either fixed zeros or free independent parameters. To such systems, one can associate a directed graph and study structural properties i.e, properties which are valid for almost all values of the parameters. In this paper, an overview of the structural analysis of structured systems is proposed via a graph-theoretic approach. In particular, well-known control problems as state feedback disturbance rejection and input-output decoupling are studied. Keywords. Linear structured systems, state feedback disturbance rejection, inputoutput decoupling, graph theory. Resume: Nous considerons dans cet article les systemes lineaires structures decrits par des modeles d'etats. Pour ce type de systemes, les coefficients des matrices du modele sont supposes etre soit nuls soit des parametres libres. On associe a de tels systemes un graphe oriente et l'on etudie des proprietes structurelles (ou generiques), i.e. des proprietes vraies pour presque tout ensemble de parametres. Cet article propose un aper«u de l'analyse des systemes structures a travel'S la theorie des graphes. En particulier sont etudies des problemes classiques de commande tels le rejet de perturbation par retour d'etat et le decouplage par retour d'etat. Mots Clefs: systemes lineaires structures, rejet de perturbation, decouplage, theorie des graphes
1. INTRODUCTION
(and generally reflected by a zero coefficient). This corresponds to the well-known notion of structured systems that is studied since twenty years by algebraic, graphical and geometric ways (see for instance Commault et al. (1991), Hovelaque et al. (1996a) and van del' Woude (1991)). Then, we consider linear systems represented by a triplet (A,B,C) where the entries of(A,B,C) are either null or free parameters. To such systems, called structured systems, one can associate a directed graph in a natural way (Lin 1974, Murota 1987, Reinschke 1988). One can study structural properties, i.e. properties which are true for al-
A large number of industrial processes (in engineering, economy, biomedical systems) can be modelled using time-invariant linear systems. In practical situations, the coefficients of the differential equations describing the process behavior are poorly or badly known because of some measurement errors. In fact, the only precise knowledge is generally the lack of relation between variables 1 attached to the Laboratoire d'Automatique de GrenobIe from I'Ecole Nationale Superieure d'Agronomie de Rennes (France)
847
=
~n, input u(t) E U = with state x(t) E X ~m, output y(t) E Y = ~p. A, Band C are real matrices of appropriate dimensions.
most all values of the parameters. Most of these properties can be obtained from properties of the associated graph. Structural properties have been extensively studied during the last twenty years following Lin (1974) for the structural controllability. The solvability conditions of classical control problems such as disturbance rejection (Verghese 1978) or decoupling (Descusse and Dion 1982) can be simply expressed in terms of the system infinite structure. In Commault et al. (1991) and van der Woude (1991), it is shown that the structure at infinity of the structured systems can be obtained in terms of vertex disjoint input-output paths on the associated graph. This problem can be translated in a minimum cost flow problem as suggested in van der Woude (1991). In Reinschke (1988) and Svaricek (1990), the method relies on the analysis of generic degrees of minors. In Hovelaque et al. (1996a) , the flow problem is solved by using a primal-dual algorithm linked to linear programming theory. In this paper, we propose an overview of the graph analysis of structured systems. The graphical computation of the structure at infinity leads to nice graphical if and only if conditions for solving classical control problems as state feedback disturbance rejection (Commault et al. 1991, van der Woude and Murota 1995, Hovelaque et al. 1996b) and input-output decoupling (Linnemann 1981, Dion and Commault 1993). The outline of the paper is as follows. In section 2, we recall the definition of a structured system, the notion of generic (or structural) property and the construction of the associated graph. The infinite structure determination is described in section 3. The two next sections present the graph solvability conditions of respectively the state feedback disturbance rejection and the input-output decoupling problems. Some numerical aspects are given in section 6. Concluding remarks end this paper at section 7.
Definition 1 Let EA be the linear system of the form:
{x(t)
E
y(t)
A
=
AAX(t) CAx(t)
+ BA u(t)
(2)
The system is said to be a structured system if the following matrix
is a structured matrix, i.e. its entries are either fixed zeros or free parameters (Lin 1974, Murota 1987). A denotes the vector composed of the p nonnull parameters Ai (i = 1, ... , p) of the above given structured matrix. This can be illustrated with the following example. Let E AI be a structured system with state space matrices as follows:
>'1
U
AA l
=
C AI
=[~
0 0
),5
0
E AI has 6 free parameters, i.e. Al E ~6. A linear system EI(AI,BI,C I ) has the same structure than a structured system EA if El has the same dimensions as EA and if for every fixed (zero) entry of EA, the corresponding entry of El is also fixed (zero) (Lin 1974). For instance, let us define the structured system defined by its structured matrix NA (A E ~5) :
The two following matrices : 2. PRELIMINARIES NI
In this section, we recall the classical definition of linear structured systems and the way to associate a directed graph to such a system. The notion of genericity is also explained.
= [~
~ ~]
420'
and N2 =
[~1 1~ 0~]
have the same structure as the structured matrix N A. We will note this structural identification by 'E', For instance: N l EN>. N 2 E N>.
2.1. Linear structured systems We consider the time-invariant linear system E described by the classical state space equations: E
{ x(t) y(t)
=
Ax(t) + Bu(t) Cx(t)
2.2. Associated graph To a structured system EA, one can associate (Lin 1974, Linnemann 1981) a directed graph G(E A ) = (Z, W) where:
(1)
848
• the vertex set is Z = U u X U Y where: U = {U 1, ... , Urn} X = {X 1, ... , Xn} Y = {Yl,""Yp} • the arc set is W = {(ui,xj),b ji =1= O} U {(Xi, Xj), aji =1= O} U {(Xi, Yj), Cji =1= O} where: bji (resp. aji , Cji) denotes the element (j, i) of the matrix BA (resp. AA, CA)'
Definition 2 Let EA be a linear structured system with A E ~p. A property IT is generic relative to V if IT is true for all values of A = (.AI,.'" .A p ) except for those such that A E V, where V C ~p be a proper algebraic variety.
For instance, consider the structured matrix N A defined in section 2.1. Let IT be the property that N A is regular. IT is true if and only if :
This can be illustrated with the above structured system E Al which associated graph G(E A ,) is depicted in figure 1. Y1
Then, the property IT : " N A is regular" is generically true except for the vector (.AI, ... , .A5) such that equation (3) is verified. In that case, the proper variety V can be easily described :
:: Y2
Fig. 1. Associated graph G(E Al )
3. STRUCTURE AT INFINITY
For the system E described by (1), let T(s) be the transfer matrix of E which is a (p x m) proper rational matrix. T(s) = C(sI _A)-l B can be factorized as follows :
A directed path in the graph G(E A ) = (Z, W) from a vertex i llo to a vertex i llq is a sequence of arcs:
such that ill, E Z for t = 0, ... , q and (ill,_" ill,) E W for t = 1, ... , q. The length of a path is the number of its arcs, each arc being counted the number of times it appears in the sequence. For the last sequence, the path has length q. Occasionally, we denote the path by the sequence of vertices it consists of, i.e. by (i llo ' i lll , ... , i llq _" ill.)' Moreover, if i llo E U and ij,lq E Y, this path is called an input-output path. A set of k input-output paths with no common vertices is called a k vertex disjoint input-output path set. the path set For instance, {U1, Xl, X2, yd, {U2, X3, Y2} of G(E Al ) is a 2 vertex disjoint input-output path set of length 5.
where: d'lag (-n, uh() S s , ... , s -n r ) ni integers with nl S n2 S ... S n r l' = rank(T(s)) B 1 (s), B 2 (s) are bicausal matrices characterized by (i = 1,2): det(lim._ oo Bi (s)) is a nonnull constant.
=
The list {nI, ... , n r } is uniquely defined and constitutes the infinite structure of T(s). The n;'s are called the infinite zero orders of the system. For a single input single output system, the infinite zero order of the system is simply the difference of degrees between denominator and numerator of the system transfer function. The infinite structure can be computed as follows:
2.3. Genericity
Suppose that IT is some property which may be verified for some linear system which has the same structure as EA. It will be of interest that IT holds true for all system E E EA except possibly some which lie on some algebraic hypersurface in the parameter space. A variety V is defined to be the locus of common zeros of a finite number of polynomials
I
bl(T(s)) k-1
bk(T(s)) -
L
nj
(k = 2, ... ,1').
j=l
where bi(T(s)) denotes the least infinite zero order of the order i minors of T( s). This structure at infinity can be characterized for a structured system EA on the associated graph G(E A ). The structural infinite zero orders of a structured system, i.e. the generic values of these infinite zero orders relative to the given structure, can also be calculated on the graph G(E A ) as stated in the next theorem (Commault et al. 1991, van der Woude 1991) :
=0 , i=l, ... ,k}
V is proper if V =1= ~P (p is the number of free parameters .AI, .. . ,.A p composing A) and V is nontrivial if V =1= 0. A generic (or structural) property is then defined as follows (Wonham 1974) :
849
if and only if
Theorem 1 Let EA be a linear structured system and G(E A ) be the associated graph, one has the following:
(4)
i) The generic rank of EA which is the number of structural infinite zeros of EA is equal to the maximum number of input-output vertex disjoint paths in G(E A ).
i=l
Indeed, the generic row-by-row infinite zero orders are equal to (Dion and Commault 1993) :
n; = L; - 1
ii) The generic infinite zero orders of E.... are characterized on G(E A ) as follows: Ll
-
-
L:
nj -
-
Lk -
= 1, ... ,p
L;
k
j=l
Lk
i
where is the minimal length of a path from the input set to the output Yi on the associated graph
1 k-l
Lk
i=l
1 -
1
k
EA· Note that (4) is equivalent to :
= 2, ... , r
where Lk is the minimal sum of k vertex disjoint input-output path lengths in G(E A ). For instance, consider the structured system EA, described in section 2 which associated graph is depicted on figure 1. One has :
For example, consider again the structured system EA, which generic infinite structure was computed in section 3, that is : n1 = 1 and n2 = 2. The generic row-by-row infinite zero orders are obtained by computing the shortest length from the input set to each output :
1. one shortest vertex disjoint input-output path is {U2, X2, yd of length L 1 = 2. 2. a shortest pair of vertex disjoint input output paths is {U1' Xl, X2, yd,{U2, X3, yz}. This pair of paths is of length L 2 = 5. 3. there are no three vertex disjoint inputoutput paths.
1. a shortest {U2,x2,yd. 2. a shortest {U2, X3, Y2}.
So by theorem 1, the generic rank r of the structured system EA, is r = 2, and the generic infinite zero orders of EA, are n1 = 1 and n2 = 2.
path This path This
from input set to Yl is path is oflength L~ = 2. from input set to Y2 is path is of length L 2 = 2.
So, the generic row-by-row infinite zero orders of the structured system EA, are: n~
= 1
The condition (4) of theorem 2 is not satisfied. Then, the decoupling problem is not generically solvable.
4. GENERIC INPUT-OUTPUT DECOUPLING PROBLEM The state feedback input-output decoupling problem (Falb and Wolovich 1967) will be studied here in terms of an infinite structure condition. The feedback decoupling condition for linear systems (Descusse and Dion 1982) is satisfied if and only if the global infinite structure of a linear system equals the sum of all its row-by-row infinite structure. For structured systems, one can give the following theorem (Dion and Commault 1993) :
5. GENERIC DISTURBANCE REJECTION PROBLEM We recall here briefly the graphical solution to the Disturbance Rejection Problem (DRP) by state feedback for structured systems when disturbances are available for measurement. Consider the disturbed structured system E~, described by :
Theorem 2 Let EA be a linear structured system defined by the triplet (AA, BA, CA) whose transfer matrix 7:",(s) = CA(sI - AA)-l BA is a (p x m) generic full row rank proper rational matrix. Let
x(t) { y(t)
= A.,\,x(t) + BA'u(t) + E....,d(t) =
CA,x(t)
with disturbance d(t) E 1) = 1R q . A' is the vector of the nonnull parameters of AA' , BA' , CA' and E A /· We can build the associated graph G(E~/) with G(E A ) by adding a disturbance vertex set D = {dl, ... ,dq } and an arc set {(di,xj),eji i= O} where eji denotes the element (j, i) of the matrix E A /· The feedback rejection condition for linear systems (Verghese 1978) is satisfied if and only if
ni, i = 1, ... , p be the generic infinite zero orders of EA = (AA, BA, CA)
n;, i = 1, , P be the generic infinite zero order of "E ,i = (AA, BA, CA,;), where CA,i is the it" row of CA. This system is generically decouplable by a feedback control law u(t) = Fx(t) + Gv(t), G regular,
850
For this structured system E~; with associated graph is given in fig. 2.
the rank and the infinite structure of a disturbed linear system (A, B, G, E) equal respectively the rank and the infinite structure of the triplet (A,B,G). The generic solvability (Commault et al. 1991) of DRP for a structured system described by the above equations is given as follows:
--\2
Theorem 3 Let E~, be a structured disturbed system. The Disturbance Rejection Problem is generically solvable by a state feedback control law u(t) = Fx(t)+Hd(t) (when disturbances are available for measurement if and only if
E !R9 , the
Yl
Y2
(i) G(E A ) and G(E~,) have the same maximal number (r) of vertex disjoint input-output paths.
Fig. 2. Associated graph GP::;~, ) 2
The maximal number of vertex disjoint inputoutput paths equals 2. A pair of two vertex disjoint input-output paths of minimal total lenght is {Ul,Xl,Yd and {U2,X4,Y2}. Thus, proposition 1 condition is satisfied. So the DRP with disturbance measurement is generically solvable.
(ii) The minimal number of state vertices on r vertex disjoint input-output paths is the same for G(E A ) and G(E~/). Both above conditions can be easily checked on the associated graphs G(E A ) and G(E~/) with the primal-dual algorithm as shown by Hovelaque et al. (1996b). The first condition (i) will be satisfied if the maximal flow (number of paths) is the same on G(E A ) and G(E~/). The second condition (ii) will be satisfied if the minimal cost (number of state vertices) of a flow equal to r (rank of EA) is the same on G(E A ) and G(E~/). In fact, these two conditions will be satisfied if there exists a minimum cost flow (maximum number of vertex disjoint input-output paths of minimal total length) on G(E~/) which does not need the disturbance vertex set D = {d 1 , .•• , dq }. In terms of paths, theorem 3 is equivalent to the following proposition :
Yl
Y2 Fig. 3. G(E~~) without disturbance measurement
When the disturbance d(t) is not available for measurement (H = 0), each control input vertex Uj is split in two vertices ui and ut connected by an arc and the arcs coming from Uj will come from ut. Then we apply the above proposition 1.
Proposition 1 Let E~, be a structured system and G(E~/) be its associated graph. The Disturbance
For the structured system E~, , if the disturbance 2 is not available for measurement, we apply proposition 1 on the graph depicted on figure 3. Then, the maximal number of vertex disjoint inputoutput paths equals 2 but a pair of two vertex disjoint input-output paths of minimal total length must include the disturbance vertex d. For example {u2"",ut,X4,Y2} and {d,X3,Yd. Then, DRP is not generically solvable if the disturbance is not available for measurement. By looking on the graph, we can see that the disturbance d comes quicker on Yl than '11 1 and '112' •
Rejection Problem is generically solvable by a state feedback control law u(t) = Fx(t) + H d(t) if and only if there exists a set of maximal number of vertex disjoint input-output paths of minimal total length in G(E~,) which does not include vertices of D = {d 1 , ... , dq }. Consider E~, (A, B, G, E) the following structured 2 system:
6. NUMERICAL ASPECTS
In this paper, we show that some control problem
o >'8 o o
conditions for structured systems can be easily obtained directly on the graph by studying the maximum set of vertex disjoint input-output paths of
851
Dion, J. and C. Commault (1993). 'Feedback decoupling of structured systems'. IEEE Trans. Automat. Contr. 38, 1132-1135. Falb, P. and W.A. Wolovich (1967). 'Decoupling in the design and synthesis of multivariable control systems'. IEEE Trans. Automat. Contr. 12,651-659. Hovelaque, V., C. Commault and J .M. Dion (1996 a). ;Analysis of linear structured systems using a primal-dual algorithm'. Systems and Control Letters 27, 73-85. Hovelaque, V., C. Commault and J.M. Dion (1996b). Disturbance decoupling for structured systems via a primal-dual algorithm. In 'CESA'96'. Vol. 1. Lille, France. pp. 455-459. Hovelaque, V., C. Commault, J.M. Dion, M. Bahar and J. Jantzen (1996c). 'Graph modelling approach: application to a distillation column'.
minimum length. In the graph theory litterature, there are a lot of algorithms which find a maximal set of paths of minimal total length. As suggested in van der Woude (1991), the generic infinite structure search can be transformed into a minimum cost flow problem on a modified graph (Hovelaque et al. 1996a). This approach allows to check generic conditions of control problems quickly on a computer. For example, the proposed algorithm in Hovelaque et al. (1996a) was tested on a 81 state distillation column model (Hovelaque et al. 1996b). The generic state feedback rejection condition was checked in about 3 seconds (on a PC 486 66 Mhz). A general graphical approach is proposed on a distillation column model in Hovelaque et al. (1996c).
to appear in Studies in Informatics and Control.
7. CONCLUDING REMARKS
Lin, C. (1974). 'Structural controllability'. IEEE Trans. Autom. Contr. 19, 201-208. Linnemann, A. (1981). 'Decoupling of structured systems'. Syst. Contr. Lett. 2, 71-86. Murota, K. (1987). Systems Analysis by Graphs and Matroids. Vol. 3 of Algorithms and Combinatorics. Springer-Verlag New-York, Inc. Reinschke, K. (1988). Multivariable control: A graph-theoretic approach. Vol. 108 of Lect.
Structured systems allow to represent a large class of linear systems. The graph-theoretic approach for such systems leads to appealing and simple generic solvability conditions for classical control problems. These solvability conditions can be easily checked using efficient numerical algorithms. In this paper, we focused on state feedback disturbance rejection and input-output decoupling problems. Other control problems have been tackled using similar tools e.g. disturbance rejection by output measurement (van der Woude 1993, Commault et al. 1995), combined disturbance rejection and decoupling (Commault et al. 1993). Other approaches have been proposed in the literature, in particular a modelling allowing the presence of nonzero constant parameter (Murota 1987) and a modelling by bond-graph (Sueur and Dauphin-Tanguy 1991).
Notes in Control and Information Sciences.
Springer-Verlag. Sueur, C. and G. Dauphin-Tanguy (1991). 'Bond graph approach for structural analysis of mimo linear systems'. J. Frank/in Inst. 328, 55-70. Svaricek, F. (1990). An improved graph-theoretic algorithm for computing the structure at infinth ity of linear systems. In 'Proc. of the 29 IEEE CDC'. Vol. 5. Honolulu, Hawaii. pp. 2923-2924. van der Woude, J. (1991). 'On the structure at infinity of a structured system'. Linear Algebra and its Applications 148, 145-169. van der Woude, J. (1993). Disturbance decoupling by measurement feedback for structured systems : a graph theoretic approach. In 'Proc. 2nd Europ. Cont. ConL ECC'93'. Groningen, Holland. pp. 1132-1137. van der Woude, J. and K. Murota (1995). 'Disturbance decoupling with pole placement for structured systems: a graph-theoretic approach'. SIAM J. on Matrix Anal. and Appl. 16, 922942. Verghese, G. (1978). Infinite frequency behavior in generalized dynamical systems. PhD thesis. Dept. Elect. Eng., Stanford Univ., Stanford, Calif. Wonham, \V. (1974). Linear multivariable control : a geometric approach. Springer-Verlag. N ewYork, (3rd ed., 1985).
REFERENCES Commault, C., J .M. Dion and A. Perez (1991). 'Disturbance rejection for structured systems'. IEEE Trans. Automat. Contr. 36, 884-887. Commault, C., J .M. Dion and J. Montoya (1993). Simultaneous decoupling and disturbance rejection : a structural approach. In 'Proc. 2nd Europ. Cont. ConL ECC'93'. Groningen, Holland. pp. 1138-1142. Commault, C., J.M. Dion and V. Hovelaque (1995). A geometric approach for structured systems: Application to the disturbance decoupling problem. Technical Report 96-190. Laboratoire d'Automatique de Grenoble-France (to appear in Automatica). Descusse, J. and J .M. Dion (1982). 'On the structure at infinity of linear square decouplable systems'. IEEE Trans. Automat. Contr. 27, 971974.
852
STUDY OF STRUCTURED SYSTEMS A GRAPH APPROACH C. Commault J.M. Dion V. Hovelaque
1
Laboratoire d'Automatique de Grenoble. INPG/UJF/CNRS UMR 5528, ENSIEG, BP. 46 , 38402St Martin d'Heres, France. Tel. : (33) 4.76.82.62.44, Fax: (33) 4.76.82.63.88. e-mail: [email protected]
Abstract. In this paper, structured systems described by state space models are considered. In this context, the entries of the state space model matrices are supposed to be either fixed zeros or free independent parameters. To such systems, one can associate a directed graph and study structural properties i.e, properties which are valid for almost all values of the parameters. In this paper, an overview of the structural analysis of structured systems is proposed via a graph-theoretic approach. In particular, well-known control problems as state feedback disturbance rejection and input-output decoupling are studied. Keywords. Linear structured systems, state feedback disturbance rejection, inputoutput decoupling, graph theory. Resume: Nous considerons dans cet article les systemes lineaires structures decrits par des modeles d'etats. Pour ce type de systemes, les coefficients des matrices du modele sont supposes etre soit nuls soit des parametres libres. On associe a de tels systemes un graphe oriente et l'on etudie des proprietes structurelles (ou generiques), i.e. des proprietes vraies pour presque tout ensemble de parametres. Cet article propose un aper«u de l'analyse des systemes structures a travel'S la theorie des graphes. En particulier sont etudies des problemes classiques de commande tels le rejet de perturbation par retour d'etat et le decouplage par retour d'etat. Mots Clefs: systemes lineaires structures, rejet de perturbation, decouplage, theorie des graphes
1. INTRODUCTION
(and generally reflected by a zero coefficient). This corresponds to the well-known notion of structured systems that is studied since twenty years by algebraic, graphical and geometric ways (see for instance Commault et al. (1991), Hovelaque et al. (1996a) and van del' Woude (1991)). Then, we consider linear systems represented by a triplet (A,B,C) where the entries of(A,B,C) are either null or free parameters. To such systems, called structured systems, one can associate a directed graph in a natural way (Lin 1974, Murota 1987, Reinschke 1988). One can study structural properties, i.e. properties which are true for al-
A large number of industrial processes (in engineering, economy, biomedical systems) can be modelled using time-invariant linear systems. In practical situations, the coefficients of the differential equations describing the process behavior are poorly or badly known because of some measurement errors. In fact, the only precise knowledge is generally the lack of relation between variables 1 attached to the Laboratoire d'Automatique de GrenobIe from I'Ecole Nationale Superieure d'Agronomie de Rennes (France)
847
=
~n, input u(t) E U = with state x(t) E X ~m, output y(t) E Y = ~p. A, Band C are real matrices of appropriate dimensions.
most all values of the parameters. Most of these properties can be obtained from properties of the associated graph. Structural properties have been extensively studied during the last twenty years following Lin (1974) for the structural controllability. The solvability conditions of classical control problems such as disturbance rejection (Verghese 1978) or decoupling (Descusse and Dion 1982) can be simply expressed in terms of the system infinite structure. In Commault et al. (1991) and van der Woude (1991), it is shown that the structure at infinity of the structured systems can be obtained in terms of vertex disjoint input-output paths on the associated graph. This problem can be translated in a minimum cost flow problem as suggested in van der Woude (1991). In Reinschke (1988) and Svaricek (1990), the method relies on the analysis of generic degrees of minors. In Hovelaque et al. (1996a) , the flow problem is solved by using a primal-dual algorithm linked to linear programming theory. In this paper, we propose an overview of the graph analysis of structured systems. The graphical computation of the structure at infinity leads to nice graphical if and only if conditions for solving classical control problems as state feedback disturbance rejection (Commault et al. 1991, van der Woude and Murota 1995, Hovelaque et al. 1996b) and input-output decoupling (Linnemann 1981, Dion and Commault 1993). The outline of the paper is as follows. In section 2, we recall the definition of a structured system, the notion of generic (or structural) property and the construction of the associated graph. The infinite structure determination is described in section 3. The two next sections present the graph solvability conditions of respectively the state feedback disturbance rejection and the input-output decoupling problems. Some numerical aspects are given in section 6. Concluding remarks end this paper at section 7.
Definition 1 Let EA be the linear system of the form:
{x(t)
E
y(t)
A
=
AAX(t) CAx(t)
+ BA u(t)
(2)
The system is said to be a structured system if the following matrix
is a structured matrix, i.e. its entries are either fixed zeros or free parameters (Lin 1974, Murota 1987). A denotes the vector composed of the p nonnull parameters Ai (i = 1, ... , p) of the above given structured matrix. This can be illustrated with the following example. Let E AI be a structured system with state space matrices as follows:
>'1
U
AA l
=
C AI
=[~
0 0
),5
0
E AI has 6 free parameters, i.e. Al E ~6. A linear system EI(AI,BI,C I ) has the same structure than a structured system EA if El has the same dimensions as EA and if for every fixed (zero) entry of EA, the corresponding entry of El is also fixed (zero) (Lin 1974). For instance, let us define the structured system defined by its structured matrix NA (A E ~5) :
The two following matrices : 2. PRELIMINARIES NI
In this section, we recall the classical definition of linear structured systems and the way to associate a directed graph to such a system. The notion of genericity is also explained.
= [~
~ ~]
420'
and N2 =
[~1 1~ 0~]
have the same structure as the structured matrix N A. We will note this structural identification by 'E', For instance: N l EN>. N 2 E N>.
2.1. Linear structured systems We consider the time-invariant linear system E described by the classical state space equations: E
{ x(t) y(t)
=
Ax(t) + Bu(t) Cx(t)
2.2. Associated graph To a structured system EA, one can associate (Lin 1974, Linnemann 1981) a directed graph G(E A ) = (Z, W) where:
(1)
848
• the vertex set is Z = U u X U Y where: U = {U 1, ... , Urn} X = {X 1, ... , Xn} Y = {Yl,""Yp} • the arc set is W = {(ui,xj),b ji =1= O} U {(Xi, Xj), aji =1= O} U {(Xi, Yj), Cji =1= O} where: bji (resp. aji , Cji) denotes the element (j, i) of the matrix BA (resp. AA, CA)'
Definition 2 Let EA be a linear structured system with A E ~p. A property IT is generic relative to V if IT is true for all values of A = (.AI,.'" .A p ) except for those such that A E V, where V C ~p be a proper algebraic variety.
For instance, consider the structured matrix N A defined in section 2.1. Let IT be the property that N A is regular. IT is true if and only if :
This can be illustrated with the above structured system E Al which associated graph G(E A ,) is depicted in figure 1. Y1
Then, the property IT : " N A is regular" is generically true except for the vector (.AI, ... , .A5) such that equation (3) is verified. In that case, the proper variety V can be easily described :
:: Y2
Fig. 1. Associated graph G(E Al )
3. STRUCTURE AT INFINITY
For the system E described by (1), let T(s) be the transfer matrix of E which is a (p x m) proper rational matrix. T(s) = C(sI _A)-l B can be factorized as follows :
A directed path in the graph G(E A ) = (Z, W) from a vertex i llo to a vertex i llq is a sequence of arcs:
such that ill, E Z for t = 0, ... , q and (ill,_" ill,) E W for t = 1, ... , q. The length of a path is the number of its arcs, each arc being counted the number of times it appears in the sequence. For the last sequence, the path has length q. Occasionally, we denote the path by the sequence of vertices it consists of, i.e. by (i llo ' i lll , ... , i llq _" ill.)' Moreover, if i llo E U and ij,lq E Y, this path is called an input-output path. A set of k input-output paths with no common vertices is called a k vertex disjoint input-output path set. the path set For instance, {U1, Xl, X2, yd, {U2, X3, Y2} of G(E Al ) is a 2 vertex disjoint input-output path set of length 5.
where: d'lag (-n, uh() S s , ... , s -n r ) ni integers with nl S n2 S ... S n r l' = rank(T(s)) B 1 (s), B 2 (s) are bicausal matrices characterized by (i = 1,2): det(lim._ oo Bi (s)) is a nonnull constant.
=
The list {nI, ... , n r } is uniquely defined and constitutes the infinite structure of T(s). The n;'s are called the infinite zero orders of the system. For a single input single output system, the infinite zero order of the system is simply the difference of degrees between denominator and numerator of the system transfer function. The infinite structure can be computed as follows:
2.3. Genericity
Suppose that IT is some property which may be verified for some linear system which has the same structure as EA. It will be of interest that IT holds true for all system E E EA except possibly some which lie on some algebraic hypersurface in the parameter space. A variety V is defined to be the locus of common zeros of a finite number of polynomials
I
bl(T(s)) k-1
bk(T(s)) -
L
nj
(k = 2, ... ,1').
j=l
where bi(T(s)) denotes the least infinite zero order of the order i minors of T( s). This structure at infinity can be characterized for a structured system EA on the associated graph G(E A ). The structural infinite zero orders of a structured system, i.e. the generic values of these infinite zero orders relative to the given structure, can also be calculated on the graph G(E A ) as stated in the next theorem (Commault et al. 1991, van der Woude 1991) :
=0 , i=l, ... ,k}
V is proper if V =1= ~P (p is the number of free parameters .AI, .. . ,.A p composing A) and V is nontrivial if V =1= 0. A generic (or structural) property is then defined as follows (Wonham 1974) :
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if and only if
Theorem 1 Let EA be a linear structured system and G(E A ) be the associated graph, one has the following:
(4)
i) The generic rank of EA which is the number of structural infinite zeros of EA is equal to the maximum number of input-output vertex disjoint paths in G(E A ).
i=l
Indeed, the generic row-by-row infinite zero orders are equal to (Dion and Commault 1993) :
n; = L; - 1
ii) The generic infinite zero orders of E.... are characterized on G(E A ) as follows: Ll
-
-
L:
nj -
-
Lk -
= 1, ... ,p
L;
k
j=l
Lk
i
where is the minimal length of a path from the input set to the output Yi on the associated graph
1 k-l
Lk
i=l
1 -
1
k
EA· Note that (4) is equivalent to :
= 2, ... , r
where Lk is the minimal sum of k vertex disjoint input-output path lengths in G(E A ). For instance, consider the structured system EA, described in section 2 which associated graph is depicted on figure 1. One has :
For example, consider again the structured system EA, which generic infinite structure was computed in section 3, that is : n1 = 1 and n2 = 2. The generic row-by-row infinite zero orders are obtained by computing the shortest length from the input set to each output :
1. one shortest vertex disjoint input-output path is {U2, X2, yd of length L 1 = 2. 2. a shortest pair of vertex disjoint input output paths is {U1' Xl, X2, yd,{U2, X3, yz}. This pair of paths is of length L 2 = 5. 3. there are no three vertex disjoint inputoutput paths.
1. a shortest {U2,x2,yd. 2. a shortest {U2, X3, Y2}.
So by theorem 1, the generic rank r of the structured system EA, is r = 2, and the generic infinite zero orders of EA, are n1 = 1 and n2 = 2.
path This path This
from input set to Yl is path is oflength L~ = 2. from input set to Y2 is path is of length L 2 = 2.
So, the generic row-by-row infinite zero orders of the structured system EA, are: n~
= 1
The condition (4) of theorem 2 is not satisfied. Then, the decoupling problem is not generically solvable.
4. GENERIC INPUT-OUTPUT DECOUPLING PROBLEM The state feedback input-output decoupling problem (Falb and Wolovich 1967) will be studied here in terms of an infinite structure condition. The feedback decoupling condition for linear systems (Descusse and Dion 1982) is satisfied if and only if the global infinite structure of a linear system equals the sum of all its row-by-row infinite structure. For structured systems, one can give the following theorem (Dion and Commault 1993) :
5. GENERIC DISTURBANCE REJECTION PROBLEM We recall here briefly the graphical solution to the Disturbance Rejection Problem (DRP) by state feedback for structured systems when disturbances are available for measurement. Consider the disturbed structured system E~, described by :
Theorem 2 Let EA be a linear structured system defined by the triplet (AA, BA, CA) whose transfer matrix 7:",(s) = CA(sI - AA)-l BA is a (p x m) generic full row rank proper rational matrix. Let
x(t) { y(t)
= A.,\,x(t) + BA'u(t) + E....,d(t) =
CA,x(t)
with disturbance d(t) E 1) = 1R q . A' is the vector of the nonnull parameters of AA' , BA' , CA' and E A /· We can build the associated graph G(E~/) with G(E A ) by adding a disturbance vertex set D = {dl, ... ,dq } and an arc set {(di,xj),eji i= O} where eji denotes the element (j, i) of the matrix E A /· The feedback rejection condition for linear systems (Verghese 1978) is satisfied if and only if
ni, i = 1, ... , p be the generic infinite zero orders of EA = (AA, BA, CA)
n;, i = 1, , P be the generic infinite zero order of "E ,i = (AA, BA, CA,;), where CA,i is the it" row of CA. This system is generically decouplable by a feedback control law u(t) = Fx(t) + Gv(t), G regular,
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For this structured system E~; with associated graph is given in fig. 2.
the rank and the infinite structure of a disturbed linear system (A, B, G, E) equal respectively the rank and the infinite structure of the triplet (A,B,G). The generic solvability (Commault et al. 1991) of DRP for a structured system described by the above equations is given as follows:
--\2
Theorem 3 Let E~, be a structured disturbed system. The Disturbance Rejection Problem is generically solvable by a state feedback control law u(t) = Fx(t)+Hd(t) (when disturbances are available for measurement if and only if
E !R9 , the
Yl
Y2
(i) G(E A ) and G(E~,) have the same maximal number (r) of vertex disjoint input-output paths.
Fig. 2. Associated graph GP::;~, ) 2
The maximal number of vertex disjoint inputoutput paths equals 2. A pair of two vertex disjoint input-output paths of minimal total lenght is {Ul,Xl,Yd and {U2,X4,Y2}. Thus, proposition 1 condition is satisfied. So the DRP with disturbance measurement is generically solvable.
(ii) The minimal number of state vertices on r vertex disjoint input-output paths is the same for G(E A ) and G(E~/). Both above conditions can be easily checked on the associated graphs G(E A ) and G(E~/) with the primal-dual algorithm as shown by Hovelaque et al. (1996b). The first condition (i) will be satisfied if the maximal flow (number of paths) is the same on G(E A ) and G(E~/). The second condition (ii) will be satisfied if the minimal cost (number of state vertices) of a flow equal to r (rank of EA) is the same on G(E A ) and G(E~/). In fact, these two conditions will be satisfied if there exists a minimum cost flow (maximum number of vertex disjoint input-output paths of minimal total length) on G(E~/) which does not need the disturbance vertex set D = {d 1 , .•• , dq }. In terms of paths, theorem 3 is equivalent to the following proposition :
Yl
Y2 Fig. 3. G(E~~) without disturbance measurement
When the disturbance d(t) is not available for measurement (H = 0), each control input vertex Uj is split in two vertices ui and ut connected by an arc and the arcs coming from Uj will come from ut. Then we apply the above proposition 1.
Proposition 1 Let E~, be a structured system and G(E~/) be its associated graph. The Disturbance
For the structured system E~, , if the disturbance 2 is not available for measurement, we apply proposition 1 on the graph depicted on figure 3. Then, the maximal number of vertex disjoint inputoutput paths equals 2 but a pair of two vertex disjoint input-output paths of minimal total length must include the disturbance vertex d. For example {u2"",ut,X4,Y2} and {d,X3,Yd. Then, DRP is not generically solvable if the disturbance is not available for measurement. By looking on the graph, we can see that the disturbance d comes quicker on Yl than '11 1 and '112' •
Rejection Problem is generically solvable by a state feedback control law u(t) = Fx(t) + H d(t) if and only if there exists a set of maximal number of vertex disjoint input-output paths of minimal total length in G(E~,) which does not include vertices of D = {d 1 , ... , dq }. Consider E~, (A, B, G, E) the following structured 2 system:
6. NUMERICAL ASPECTS
In this paper, we show that some control problem
o >'8 o o
conditions for structured systems can be easily obtained directly on the graph by studying the maximum set of vertex disjoint input-output paths of
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Dion, J. and C. Commault (1993). 'Feedback decoupling of structured systems'. IEEE Trans. Automat. Contr. 38, 1132-1135. Falb, P. and W.A. Wolovich (1967). 'Decoupling in the design and synthesis of multivariable control systems'. IEEE Trans. Automat. Contr. 12,651-659. Hovelaque, V., C. Commault and J .M. Dion (1996 a). ;Analysis of linear structured systems using a primal-dual algorithm'. Systems and Control Letters 27, 73-85. Hovelaque, V., C. Commault and J.M. Dion (1996b). Disturbance decoupling for structured systems via a primal-dual algorithm. In 'CESA'96'. Vol. 1. Lille, France. pp. 455-459. Hovelaque, V., C. Commault, J.M. Dion, M. Bahar and J. Jantzen (1996c). 'Graph modelling approach: application to a distillation column'.
minimum length. In the graph theory litterature, there are a lot of algorithms which find a maximal set of paths of minimal total length. As suggested in van der Woude (1991), the generic infinite structure search can be transformed into a minimum cost flow problem on a modified graph (Hovelaque et al. 1996a). This approach allows to check generic conditions of control problems quickly on a computer. For example, the proposed algorithm in Hovelaque et al. (1996a) was tested on a 81 state distillation column model (Hovelaque et al. 1996b). The generic state feedback rejection condition was checked in about 3 seconds (on a PC 486 66 Mhz). A general graphical approach is proposed on a distillation column model in Hovelaque et al. (1996c).
to appear in Studies in Informatics and Control.
7. CONCLUDING REMARKS
Lin, C. (1974). 'Structural controllability'. IEEE Trans. Autom. Contr. 19, 201-208. Linnemann, A. (1981). 'Decoupling of structured systems'. Syst. Contr. Lett. 2, 71-86. Murota, K. (1987). Systems Analysis by Graphs and Matroids. Vol. 3 of Algorithms and Combinatorics. Springer-Verlag New-York, Inc. Reinschke, K. (1988). Multivariable control: A graph-theoretic approach. Vol. 108 of Lect.
Structured systems allow to represent a large class of linear systems. The graph-theoretic approach for such systems leads to appealing and simple generic solvability conditions for classical control problems. These solvability conditions can be easily checked using efficient numerical algorithms. In this paper, we focused on state feedback disturbance rejection and input-output decoupling problems. Other control problems have been tackled using similar tools e.g. disturbance rejection by output measurement (van der Woude 1993, Commault et al. 1995), combined disturbance rejection and decoupling (Commault et al. 1993). Other approaches have been proposed in the literature, in particular a modelling allowing the presence of nonzero constant parameter (Murota 1987) and a modelling by bond-graph (Sueur and Dauphin-Tanguy 1991).
Notes in Control and Information Sciences.
Springer-Verlag. Sueur, C. and G. Dauphin-Tanguy (1991). 'Bond graph approach for structural analysis of mimo linear systems'. J. Frank/in Inst. 328, 55-70. Svaricek, F. (1990). An improved graph-theoretic algorithm for computing the structure at infinth ity of linear systems. In 'Proc. of the 29 IEEE CDC'. Vol. 5. Honolulu, Hawaii. pp. 2923-2924. van der Woude, J. (1991). 'On the structure at infinity of a structured system'. Linear Algebra and its Applications 148, 145-169. van der Woude, J. (1993). Disturbance decoupling by measurement feedback for structured systems : a graph theoretic approach. In 'Proc. 2nd Europ. Cont. ConL ECC'93'. Groningen, Holland. pp. 1132-1137. van der Woude, J. and K. Murota (1995). 'Disturbance decoupling with pole placement for structured systems: a graph-theoretic approach'. SIAM J. on Matrix Anal. and Appl. 16, 922942. Verghese, G. (1978). Infinite frequency behavior in generalized dynamical systems. PhD thesis. Dept. Elect. Eng., Stanford Univ., Stanford, Calif. Wonham, \V. (1974). Linear multivariable control : a geometric approach. Springer-Verlag. N ewYork, (3rd ed., 1985).
REFERENCES Commault, C., J .M. Dion and A. Perez (1991). 'Disturbance rejection for structured systems'. IEEE Trans. Automat. Contr. 36, 884-887. Commault, C., J .M. Dion and J. Montoya (1993). Simultaneous decoupling and disturbance rejection : a structural approach. In 'Proc. 2nd Europ. Cont. ConL ECC'93'. Groningen, Holland. pp. 1138-1142. Commault, C., J.M. Dion and V. Hovelaque (1995). A geometric approach for structured systems: Application to the disturbance decoupling problem. Technical Report 96-190. Laboratoire d'Automatique de Grenoble-France (to appear in Automatica). Descusse, J. and J .M. Dion (1982). 'On the structure at infinity of linear square decouplable systems'. IEEE Trans. Automat. Contr. 27, 971974.
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