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Minimal order realizations for a class of positive linear systems
J. Franklin Inst. Vol. 333B, No. 6, pp. 893 900, 1996
~ ) Pergamon
Plh
S0016--0032(96)00051--8
Copyright ~/ 1996 The Franklin Institute Published by Elsevier Sci. . . . Ltd Printed in Great Britain 00 l 6-~032/96 $15.00 + 0.00
Minimal Order Realizationsfor a Class of Positive Linear Systems by L O R E N Z O
FARINA
Dipartimento di Informatica e Sistemistica, Universitgt degli Studi di Roma "La Sapienza", Via Eudossiana 18, 00184 Rome, Italy (Received 4 March 1996 ; accepted 10 April 1996)
ABSTRACT" The positive realization problem for linear systems is to find, for a given transfer function, all possible realizations with a state spaee of minimal dimension such that the resulting system is a positive system. In this paper, discrete-time positive linear systems having the nonnegative orthant reachable from the origin in a finite time interval with nonnegative inputs, are considered and the solution of the positive realization problem for this class of systems is given. Copyright © 1996 Published by Elsevier Science Ltd
L Introduction
The problem tackled in this paper is the following: let H(z) be a rational transfer function of a discrete-time linear system. How one can find all the realization triples (A, b, er), with a state space of minimal dimension and such that the system (A, b, e r) is a positive linear system? Positive systems are characterized by the specific property that any initial state originating from the positive orthant remains there confined whatever the positive input sequence might be. These systems are quite common in applications since state variables may represent populations, quantities of goods, masses of chemical species and so forth [see (1, 2)]. The problem of the existence of a positive realization has received many contributions in the last decades as evidence by references ( 3 ~ ) . It is well known that every proper rational function has a realization, and the minimal dimension of realizations coincides with the McMillan degree of the transfer function. When a nonnegativity constraint is imposed on the realization, the situation is totally different: a positive realizable transfer function does not in general have a jointly controllable and observable realization, so that a dimension greater than the McMillan degree may be required in order to satisfy the nonnegativity restraints. A general characterization of minimality is currently an unsolved question. In this paper, the positive realization problem for the class of discrete-time positive reachable realizations, i.e. positive realizations with the property that the entire nonnegative orthant is reachable in a finite time interval by means of nonnegative controls, will find a solution. A dual approach, which is clearly possible, leads to consider positive 893
L. Farina
894
observable realizations. F o r the sake of brevity, only positive reachable realizations will be treated. An outline of the p a p e r is as follows: In Section II some preliminary definitions and results are given and the concept of m o n o m i a l equivalence is considered in order to give an a p p r o p r i a t e definition o f independent positive realizations. Section III contains the main result o f the paper, that is a complete answer to the positive realization p r o b l e m for the class of positive reachable systems. II. Preliminary Definitions and Problem Formulation T h r o u g h o u t the paper, R+ denotes the nonnegative real orthant. A set ~ _ R" is said to be a cone provided that c~oU _ ~ for all ~ >~ 0; if oU contains an open ball of R n then ~ is said to be solid; if s ( ~ { - J ( } = {0} ~" is said to be pointed. A cone which is closed, convex, solid and pointed will be called a proper cone. Given a cone g¥, the cone Y * = {x ~ R n : xVy >~ 0, Vy e o,U} is called the dual cone of J~ff. A cone o,U is said to be polyhedral if it is expressible as the intersection o f a finite family of closed halfspaces. Cone {v~. . . . . v,} (or cone (V) where V = Ivy...v,] is a matrix) denotes the polyhedral closed convex cone consisting of all finite nonnegative linear c o m b i n a t i o n s of vectors {v~. . . . . v,}. A matrix A is said to be nonnegative if all its entries are nonnegative; such a matrix is denoted by A ~> 0, we will not consider the trivial case of an all zero matrix. A nonnegative matrix is said to be monomial if every row and column contains exactly one positive entry. In the following we will consider single input-single o u t p u t linear strictly proper discrete-time invariant systems x(t + 1) = Ax(t) + bu(t), y(t) = erx(t) where A e R n× ~, b, e = R n× 1 and the corresponding transfer function is ~1 zn
H(z)=cT(zI-A)
~b= ).M/z i-- I
/=
1Ju~2zn-2-~-'''"~-~n
zn_~_O~lZn
I .q_O~2zn- Z Tt_° ° °..~_O~n
where 3'/,. = eVA/- ~b (i >~ 1) are the Markov parameters.
Definition I A system is said to be a positive linear system if the state and output variables remain nonnegative whenever the input sequence and the initial state are nonnegative (1). The positivity constraint, in the linear case, leads to nonnegativity of the elements of the system's matrices as shown in the following. Theorem I A discrete-time linear system (A, b, e r) is a positive system if and only if the matrix A and vectors b and c have nonnegative entries (1). • This theorem is a direct link with the theory of nonnegative matrices which is a very active field of research. A n exhaustive reference b o o k on this topic is (7). We state now the positive realization problem (4): PI: Given a transfer function, what are the positive realizability conditions?
Minimal Order Realizations
895
P2: What is the minimal dimension of a positive realization? P3: How can we generate all possible minimal positive realizations? Problem P1 has been recently tackled by Anderson and co-workers (3). Problem P3 needs further specifications. In fact, given a positive system, it is possible to find an equivalent positive system by means of a monomial matrix. It is known (7) that the inverse of a nonnegative matrix M is nonnegative if and only if M is monomial, therefore, given a positive system (A+, b+, e+), r the system (MA+M -1, Mb+, c+r M - 1) is also positive. We are now allowed to define, within the class of positive systems, a monomial equivalence relation and, consequently, the concept of independence:
Definition H Two positive systems are said to be independent if they are not monomially equivalent. In view of the above definition, Problem P3 can be restated as follows: P3: How can we generate all possible independent minimal positive realizations?
IlL Positive Reachable Realizations A positive linear systems is said to be positive reachable if every state x >/0 is reachable in a finite time interval from origin (2, 8). Positive reachability of discrete-time positive linear systems has been studied by several authors (4, 6, 9-12). A realization which has the property to be positive reachable will be called a positive reachable realization. In reference (8) existence conditions on the transfer function, in terms of the poles with maximum modulus, has been obtained. We give next a geometrical characterization of the positive realization problem which is a minor restatement of a result due to Nieuwenhuis (5).
Theorem H Let H(z) be an irreducible strictly proper rational function of McMillan degree n. Then H(z) has a positive realization if and only if there exists a polyhedral proper cone such that (i) ~
__ R n+
(ii) AMJg _c Jg (iii) cone (H(n, 1)) ~ Jr" where 0
1
0
...
0
0
0
1
...
0
0
0
0
...
1
--0~n
--(Xn--I
A M
and
--0~n
2
" ""
--~1
L. Farina
896
H(n,p) =
[MM. M2 M2 M3 ...
Alp Mn+p- 1
Note that the matrix H(n,p) is obviously nonnegative and that, from (ii) and (iii), follows that cone (H(n,p)) e 3if, for every p > 0. We are now able to state the main result of the paper, i.e. the solution of the positive realization problem (P1-P3) for this class of systems.
TheoremIII Let H(z) be an irreducible strictly proper rational function of McMillan degree n with nonnegative Markov parameters. (i) A positive reachable realization (A+, b+, eT+) of
H(z)exists if and only if
cone(H(n, k + 1)) _~ cone(H(n, k))
(1)
for some positive integer k. (ii) The minimal state space dimension of (A+,b+,er+) is the minimal integer N such that Eq. (1) holds. (iii) All the independent positive reachable realizations (A+,b+,e~) with a state space of minimal dimension N can be obtained by means of a convex combination of the Q nonnegative basic solutions aT, j = 1. . . . . Q, of the system
H(n,U+l)u+l= H ( n , N ) a ~ ,
j=
1. . . . . Q
(2)
where the subscript denotes the column index. All the positive reachable realizations are given by: 0 A + ( 2 1 , • • •, '~Q) =
1 .
" I " ] ~la(N1) + ' ' ' +
• .. ... EjAj=l,
b+ =
]''Q~N~(Q)
0 I 1[
2j>~0
j=l
e+ =
..... Q
M2
(3)
N
Proof'.In order to prove (i) consider first a canonical realization Xc = (A, b, e T) of H(z)with the property that the columns of H(n, n) are the reachability vectors of such a realization [see (13)]. Define s¢"i as the polyhedral cone
M i n i m a l Order Realizations
897
oYt~i= cone {b, Ab, A2b . . . . . A i lb} and the reachability cone of the pair (A, b) as the convex cone consisting of all nonnegative combinations of all the finite subset {vl . . . . . Vm}~ {b, Ab, A2b. . . . }, namely ~ / = u+aU~. i=l
It is easily seen that the reachability cone is the set of all the states reachable from the origin with nonnegative inputs. From the above definitions immediately follows that the sequence of cones J f i is non decreasing, i.e. ~'~l ~
~C2 ~
°']~3 CZ . . .
CZ ~
(4)
We prove now sufficiency of condition (i). From Eq. (1) follows ~k = ~k+ 1
(5)
for some k. Therefore, we can write k
1
A k b = ~ ~iAib,
ai~>0
fori=0,1,...,k-1
(6)
i=0
and multiplying both sides of Eq. (6) by A h with h ~> 0, yields k
I
Ak+hb = ~ O~iAi+hb,
~i >~ 0
for i = 0, 1. . . . . k - 1
i=0
namely Ygk+h = Ylk+h+~ for every h i> O, thus ~ / = 3ffk is polyhedral having a finite number of generators. Solving the system AR=RA+
e+T = e r R
b=Rb+
(7)
where = cone(R) and where, by construction, R = H(n,k) and AR = H ( n , k + 1)2,3,...,k+1 one obtains a realization of the form (3) which is positive reachable [see (10, 12)], and this concludes the sufficiency of condition (i). A positive reachable system has, by definition, a polyhedral reachable cone so that from Eq. (4) there exists an integer k for which Eq. (5) holds. Necessity of condition (i) is therefore proved in view of the fact that polyhedrality is preserved under similarity transformations. The state space dimension of a positive reachable realization coincides with the minimal number of the edges of ~ [see (4)], so that condition (ii) holds. In order to prove (iii) rewrite Eq. (7) as follows H(n, N +
H(n, N)A+
(8)
e+r = e r H ( n , N ) .
(9)
1)2,3,...,N+1
b = H(n,N)b+,
=
Equation (9) possess a unique solution while Eq. (8) may not. Since
L. Farina
898
H(3,1)~
3,2) 2
H(3,4)4
H(3,3) 3
FIG. 1. A planar section showing the Markov cone in the example.
H ( n , N + 1)2,3,...,u = H(n,N)(A+),,2,...,u_, holds only if (A+)1.2....,N-1 is a ( N - - 1)-dimensional Jordan block, it suffices to consider Eq. (2). It is well known [see, for example, reference (14)], that any solution of Eq. (2) is a convex combination of its basic solutions. Finally, note that a monomial transformation leaves a cone unchanged: if a~ff= cone(g~ . . . . . gq) then also ~ff = cone(Mgl . . . . . M g q ) for any monomial matrix M. It follows that each of the solutions of Eq. (2) is independent and this concludes the p r o o f of the theorem. • Note that all the conditions of this theorem can be easily checked and the positive reachable realizations can be obtained by means of a simple iterative procedure and by solving a set of linear equalities with nonnegativity constraints, so that one can use effective numerical tools which are easily found in the literature. More precisely, in order to find a positive reachable realization, one has to follow the evolution of the cone described by the vectors H(n,p)p for p = 1,2 . . . . . The first value N for which the vector H(n, N + 1)u+t lies inside the cone (i.e. the cone doesn't 'grow' anymore) is the minimal order of the realization. The following step is to find the basic solutions of Eq. (2) and the realizations we are looking for are given by Eq. (3).
Example: Consider the following transfer function: //(z)
=
2z+6.8 z 3 -- 0.6z 2 + 0.12z-- 0.04"
In Fig. 1 a planar section of the first four vectors of the M a r k o v cone is depicted. It is there apparent that N = 4. Figure 1 also clearly shows that system (2)
Minimal Order Realizations
899
t,856 t i!2 8 456/ 0.8864
=
0.49152
8
4.56
4.56
1.856 a40~
1.856 0.8864
has two basic solutions (Q = 2) corresponding to the triangle (1, 2, 4) f 0.008 ]
a~,,=/°°ool~] Lo.4j and the other one corresponding to the triangle (1, 3, 4)
iO.Ool,
a~42)= I 0.08 |
L o.2~
Finally, we can write the set of all independent positive reachable realizations of H(z):
A + (,~1, ,~,2) =
[i °° o°100 LI°°is1 0046J+2,008, L
b+ =
i ]21,22e[0,1] 21 +22 = 1 4.56
Note that all these realizations are minimal within the class of positive reachable systems but, in this particular case, a positive realization of a lower dimension can be found:
02 00.2)ti) (!) A+=(~ O2o4o2°'6+= ,~+= which is, obviously, not positive reachable.
Acknowledgement The author wishes to thank Prof. S. Monaco for helpful suggestions and encouragement.
900
L. Farina
References
(1) D. G. Luenberger, "Positive linear systems", Chapter 6 in "Introduction to Dynamic Systems", Wiley, New York, 1979. (2) S. Rinaldi and L. Farina, "Positive Linear Systems. Theory and Applications", Citt~, Studi, Milan, 1995 (in Italian). (3) B. D. O. Anderson, M. Deistler, L. Farina and L. Benvenuti, "Nonnegative realization of a linear system with nonnegative impulse response", IEEE Trans. Circuits Syst. I, Vol. 43, pp. 134-142, 1996. (4) H. Maeda and S. Kodama, "Positive realization of difference equation", IEEE Trans. Circuits Syst., Vol. CAS-28, pp. 39~47, 1981. (5) J. W. Nieuwenhius, "About nonnegative realizations", Systems & Control Letters, Vol. 1, pp. 283-287, 1982. (6) Y. Ohta, H. Maeda and S. Kodama, "Reachability, observability and realizability of continuous-time positive systems", S I A M J. Control Optimization, Vol. 22, pp. 171-180, 1984.
(7) H. Minc, "Nonnegative Matrices", Wiley, New York, 1987. (8) L. Farina and L. Benvenuti, "Positive realizations of linear systems", Systems & Control Letters, Vol. 26, pp. 14, 1995. (9) P. G. Coxon and H. Shapiro, "Positive input reachability and controllability of positive systems", Linear Alg. Appl., Vol. 94, pp. 35-53, 1987. (10) M. P. Fanti, B. Maione and B. Turchiano, "Controllability of linear single-input positive discrete-time systems", Int. J. Control, Vol. 50, pp. 2523-2542, 1989. (11) D . N . P . Murthy, "Controllability of linear positive dynamic systems", Int. J. Systems Sci., Vol. 17, pp. 49-54, 1986. (12) V. G. Rumchev and D. J. G. James, "Controllability of positive linear discrete-time systems", Int. J. Control, Vol. 50, pp. 845 857, 1989. (13) T. Kailath, "Linear Systems", Prentice Hall, Englewood Cliffs, N J, 1980. (14) D. G. Luenberger, "Introduction to Linear and Nonlinear Programming", AddisonWesley, Reading, Massachusetts, 1973.
~ ) Pergamon
Plh
S0016--0032(96)00051--8
Copyright ~/ 1996 The Franklin Institute Published by Elsevier Sci. . . . Ltd Printed in Great Britain 00 l 6-~032/96 $15.00 + 0.00
Minimal Order Realizationsfor a Class of Positive Linear Systems by L O R E N Z O
FARINA
Dipartimento di Informatica e Sistemistica, Universitgt degli Studi di Roma "La Sapienza", Via Eudossiana 18, 00184 Rome, Italy (Received 4 March 1996 ; accepted 10 April 1996)
ABSTRACT" The positive realization problem for linear systems is to find, for a given transfer function, all possible realizations with a state spaee of minimal dimension such that the resulting system is a positive system. In this paper, discrete-time positive linear systems having the nonnegative orthant reachable from the origin in a finite time interval with nonnegative inputs, are considered and the solution of the positive realization problem for this class of systems is given. Copyright © 1996 Published by Elsevier Science Ltd
L Introduction
The problem tackled in this paper is the following: let H(z) be a rational transfer function of a discrete-time linear system. How one can find all the realization triples (A, b, er), with a state space of minimal dimension and such that the system (A, b, e r) is a positive linear system? Positive systems are characterized by the specific property that any initial state originating from the positive orthant remains there confined whatever the positive input sequence might be. These systems are quite common in applications since state variables may represent populations, quantities of goods, masses of chemical species and so forth [see (1, 2)]. The problem of the existence of a positive realization has received many contributions in the last decades as evidence by references ( 3 ~ ) . It is well known that every proper rational function has a realization, and the minimal dimension of realizations coincides with the McMillan degree of the transfer function. When a nonnegativity constraint is imposed on the realization, the situation is totally different: a positive realizable transfer function does not in general have a jointly controllable and observable realization, so that a dimension greater than the McMillan degree may be required in order to satisfy the nonnegativity restraints. A general characterization of minimality is currently an unsolved question. In this paper, the positive realization problem for the class of discrete-time positive reachable realizations, i.e. positive realizations with the property that the entire nonnegative orthant is reachable in a finite time interval by means of nonnegative controls, will find a solution. A dual approach, which is clearly possible, leads to consider positive 893
L. Farina
894
observable realizations. F o r the sake of brevity, only positive reachable realizations will be treated. An outline of the p a p e r is as follows: In Section II some preliminary definitions and results are given and the concept of m o n o m i a l equivalence is considered in order to give an a p p r o p r i a t e definition o f independent positive realizations. Section III contains the main result o f the paper, that is a complete answer to the positive realization p r o b l e m for the class of positive reachable systems. II. Preliminary Definitions and Problem Formulation T h r o u g h o u t the paper, R+ denotes the nonnegative real orthant. A set ~ _ R" is said to be a cone provided that c~oU _ ~ for all ~ >~ 0; if oU contains an open ball of R n then ~ is said to be solid; if s ( ~ { - J ( } = {0} ~" is said to be pointed. A cone which is closed, convex, solid and pointed will be called a proper cone. Given a cone g¥, the cone Y * = {x ~ R n : xVy >~ 0, Vy e o,U} is called the dual cone of J~ff. A cone o,U is said to be polyhedral if it is expressible as the intersection o f a finite family of closed halfspaces. Cone {v~. . . . . v,} (or cone (V) where V = Ivy...v,] is a matrix) denotes the polyhedral closed convex cone consisting of all finite nonnegative linear c o m b i n a t i o n s of vectors {v~. . . . . v,}. A matrix A is said to be nonnegative if all its entries are nonnegative; such a matrix is denoted by A ~> 0, we will not consider the trivial case of an all zero matrix. A nonnegative matrix is said to be monomial if every row and column contains exactly one positive entry. In the following we will consider single input-single o u t p u t linear strictly proper discrete-time invariant systems x(t + 1) = Ax(t) + bu(t), y(t) = erx(t) where A e R n× ~, b, e = R n× 1 and the corresponding transfer function is ~1 zn
H(z)=cT(zI-A)
~b= ).M/z i-- I
/=
1Ju~2zn-2-~-'''"~-~n
zn_~_O~lZn
I .q_O~2zn- Z Tt_° ° °..~_O~n
where 3'/,. = eVA/- ~b (i >~ 1) are the Markov parameters.
Definition I A system is said to be a positive linear system if the state and output variables remain nonnegative whenever the input sequence and the initial state are nonnegative (1). The positivity constraint, in the linear case, leads to nonnegativity of the elements of the system's matrices as shown in the following. Theorem I A discrete-time linear system (A, b, e r) is a positive system if and only if the matrix A and vectors b and c have nonnegative entries (1). • This theorem is a direct link with the theory of nonnegative matrices which is a very active field of research. A n exhaustive reference b o o k on this topic is (7). We state now the positive realization problem (4): PI: Given a transfer function, what are the positive realizability conditions?
Minimal Order Realizations
895
P2: What is the minimal dimension of a positive realization? P3: How can we generate all possible minimal positive realizations? Problem P1 has been recently tackled by Anderson and co-workers (3). Problem P3 needs further specifications. In fact, given a positive system, it is possible to find an equivalent positive system by means of a monomial matrix. It is known (7) that the inverse of a nonnegative matrix M is nonnegative if and only if M is monomial, therefore, given a positive system (A+, b+, e+), r the system (MA+M -1, Mb+, c+r M - 1) is also positive. We are now allowed to define, within the class of positive systems, a monomial equivalence relation and, consequently, the concept of independence:
Definition H Two positive systems are said to be independent if they are not monomially equivalent. In view of the above definition, Problem P3 can be restated as follows: P3: How can we generate all possible independent minimal positive realizations?
IlL Positive Reachable Realizations A positive linear systems is said to be positive reachable if every state x >/0 is reachable in a finite time interval from origin (2, 8). Positive reachability of discrete-time positive linear systems has been studied by several authors (4, 6, 9-12). A realization which has the property to be positive reachable will be called a positive reachable realization. In reference (8) existence conditions on the transfer function, in terms of the poles with maximum modulus, has been obtained. We give next a geometrical characterization of the positive realization problem which is a minor restatement of a result due to Nieuwenhuis (5).
Theorem H Let H(z) be an irreducible strictly proper rational function of McMillan degree n. Then H(z) has a positive realization if and only if there exists a polyhedral proper cone such that (i) ~
__ R n+
(ii) AMJg _c Jg (iii) cone (H(n, 1)) ~ Jr" where 0
1
0
...
0
0
0
1
...
0
0
0
0
...
1
--0~n
--(Xn--I
A M
and
--0~n
2
" ""
--~1
L. Farina
896
H(n,p) =
[MM. M2 M2 M3 ...
Alp Mn+p- 1
Note that the matrix H(n,p) is obviously nonnegative and that, from (ii) and (iii), follows that cone (H(n,p)) e 3if, for every p > 0. We are now able to state the main result of the paper, i.e. the solution of the positive realization problem (P1-P3) for this class of systems.
TheoremIII Let H(z) be an irreducible strictly proper rational function of McMillan degree n with nonnegative Markov parameters. (i) A positive reachable realization (A+, b+, eT+) of
H(z)exists if and only if
cone(H(n, k + 1)) _~ cone(H(n, k))
(1)
for some positive integer k. (ii) The minimal state space dimension of (A+,b+,er+) is the minimal integer N such that Eq. (1) holds. (iii) All the independent positive reachable realizations (A+,b+,e~) with a state space of minimal dimension N can be obtained by means of a convex combination of the Q nonnegative basic solutions aT, j = 1. . . . . Q, of the system
H(n,U+l)u+l= H ( n , N ) a ~ ,
j=
1. . . . . Q
(2)
where the subscript denotes the column index. All the positive reachable realizations are given by: 0 A + ( 2 1 , • • •, '~Q) =
1 .
" I " ] ~la(N1) + ' ' ' +
• .. ... EjAj=l,
b+ =
]''Q~N~(Q)
0 I 1[
2j>~0
j=l
e+ =
..... Q
M2
(3)
N
Proof'.In order to prove (i) consider first a canonical realization Xc = (A, b, e T) of H(z)with the property that the columns of H(n, n) are the reachability vectors of such a realization [see (13)]. Define s¢"i as the polyhedral cone
M i n i m a l Order Realizations
897
oYt~i= cone {b, Ab, A2b . . . . . A i lb} and the reachability cone of the pair (A, b) as the convex cone consisting of all nonnegative combinations of all the finite subset {vl . . . . . Vm}~ {b, Ab, A2b. . . . }, namely ~ / = u+aU~. i=l
It is easily seen that the reachability cone is the set of all the states reachable from the origin with nonnegative inputs. From the above definitions immediately follows that the sequence of cones J f i is non decreasing, i.e. ~'~l ~
~C2 ~
°']~3 CZ . . .
CZ ~
(4)
We prove now sufficiency of condition (i). From Eq. (1) follows ~k = ~k+ 1
(5)
for some k. Therefore, we can write k
1
A k b = ~ ~iAib,
ai~>0
fori=0,1,...,k-1
(6)
i=0
and multiplying both sides of Eq. (6) by A h with h ~> 0, yields k
I
Ak+hb = ~ O~iAi+hb,
~i >~ 0
for i = 0, 1. . . . . k - 1
i=0
namely Ygk+h = Ylk+h+~ for every h i> O, thus ~ / = 3ffk is polyhedral having a finite number of generators. Solving the system AR=RA+
e+T = e r R
b=Rb+
(7)
where = cone(R) and where, by construction, R = H(n,k) and AR = H ( n , k + 1)2,3,...,k+1 one obtains a realization of the form (3) which is positive reachable [see (10, 12)], and this concludes the sufficiency of condition (i). A positive reachable system has, by definition, a polyhedral reachable cone so that from Eq. (4) there exists an integer k for which Eq. (5) holds. Necessity of condition (i) is therefore proved in view of the fact that polyhedrality is preserved under similarity transformations. The state space dimension of a positive reachable realization coincides with the minimal number of the edges of ~ [see (4)], so that condition (ii) holds. In order to prove (iii) rewrite Eq. (7) as follows H(n, N +
H(n, N)A+
(8)
e+r = e r H ( n , N ) .
(9)
1)2,3,...,N+1
b = H(n,N)b+,
=
Equation (9) possess a unique solution while Eq. (8) may not. Since
L. Farina
898
H(3,1)~
3,2) 2
H(3,4)4
H(3,3) 3
FIG. 1. A planar section showing the Markov cone in the example.
H ( n , N + 1)2,3,...,u = H(n,N)(A+),,2,...,u_, holds only if (A+)1.2....,N-1 is a ( N - - 1)-dimensional Jordan block, it suffices to consider Eq. (2). It is well known [see, for example, reference (14)], that any solution of Eq. (2) is a convex combination of its basic solutions. Finally, note that a monomial transformation leaves a cone unchanged: if a~ff= cone(g~ . . . . . gq) then also ~ff = cone(Mgl . . . . . M g q ) for any monomial matrix M. It follows that each of the solutions of Eq. (2) is independent and this concludes the p r o o f of the theorem. • Note that all the conditions of this theorem can be easily checked and the positive reachable realizations can be obtained by means of a simple iterative procedure and by solving a set of linear equalities with nonnegativity constraints, so that one can use effective numerical tools which are easily found in the literature. More precisely, in order to find a positive reachable realization, one has to follow the evolution of the cone described by the vectors H(n,p)p for p = 1,2 . . . . . The first value N for which the vector H(n, N + 1)u+t lies inside the cone (i.e. the cone doesn't 'grow' anymore) is the minimal order of the realization. The following step is to find the basic solutions of Eq. (2) and the realizations we are looking for are given by Eq. (3).
Example: Consider the following transfer function: //(z)
=
2z+6.8 z 3 -- 0.6z 2 + 0.12z-- 0.04"
In Fig. 1 a planar section of the first four vectors of the M a r k o v cone is depicted. It is there apparent that N = 4. Figure 1 also clearly shows that system (2)
Minimal Order Realizations
899
t,856 t i!2 8 456/ 0.8864
=
0.49152
8
4.56
4.56
1.856 a40~
1.856 0.8864
has two basic solutions (Q = 2) corresponding to the triangle (1, 2, 4) f 0.008 ]
a~,,=/°°ool~] Lo.4j and the other one corresponding to the triangle (1, 3, 4)
iO.Ool,
a~42)= I 0.08 |
L o.2~
Finally, we can write the set of all independent positive reachable realizations of H(z):
A + (,~1, ,~,2) =
[i °° o°100 LI°°is1 0046J+2,008, L
b+ =
i ]21,22e[0,1] 21 +22 = 1 4.56
Note that all these realizations are minimal within the class of positive reachable systems but, in this particular case, a positive realization of a lower dimension can be found:
02 00.2)ti) (!) A+=(~ O2o4o2°'6+= ,~+= which is, obviously, not positive reachable.
Acknowledgement The author wishes to thank Prof. S. Monaco for helpful suggestions and encouragement.
900
L. Farina
References
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(7) H. Minc, "Nonnegative Matrices", Wiley, New York, 1987. (8) L. Farina and L. Benvenuti, "Positive realizations of linear systems", Systems & Control Letters, Vol. 26, pp. 14, 1995. (9) P. G. Coxon and H. Shapiro, "Positive input reachability and controllability of positive systems", Linear Alg. Appl., Vol. 94, pp. 35-53, 1987. (10) M. P. Fanti, B. Maione and B. Turchiano, "Controllability of linear single-input positive discrete-time systems", Int. J. Control, Vol. 50, pp. 2523-2542, 1989. (11) D . N . P . Murthy, "Controllability of linear positive dynamic systems", Int. J. Systems Sci., Vol. 17, pp. 49-54, 1986. (12) V. G. Rumchev and D. J. G. James, "Controllability of positive linear discrete-time systems", Int. J. Control, Vol. 50, pp. 845 857, 1989. (13) T. Kailath, "Linear Systems", Prentice Hall, Englewood Cliffs, N J, 1980. (14) D. G. Luenberger, "Introduction to Linear and Nonlinear Programming", AddisonWesley, Reading, Massachusetts, 1973.