Sign properties of Metzler matrices with applications
Sign properties of Metzler matrices with applications Corentin Briat, Mustafa Khammash
arXiv:1512.07043v1 [math.DS] 22 Dec 2015
Department of Biosystems Science and Engineering, ETH-Z¨ urich, Basel, Switzerland
Abstract Several results on the sign properties of Metzler matrices are obtained. It is first established that checking the sign-stability of a Metzler sign-matrix can be either characterized in terms of the Hurwitz stability of the unit sign-matrix in the corresponding qualitative class, or in terms of the acyclicity of the graph associated with the sign-pattern. Similar results are obtained for the case of block-matrices and mixed-matrices, the latter containing both sign patterns and fixed real entries. The problem of assessing the sign-stability of the convex full of a finite family of Metzler matrices is also solved, and a necessary and sufficient condition for the existence of a common Lyapunov function for all the matrices in the convex hull is obtained. The notion of relative sign-stability is also introduced and a sufficient condition for the relative sign-stability of Metzler matrices is proposed. Several applications of the results are discussed in the last section. Keywords: Sign-stability; Metzler matrices; Positive systems
1. Introduction Metzler matrices [1] are often encountered in various fields such as economics [2] and biology [3, 4, 5, 6] as they may be associated with cooperative/monotone dynamical systems [7, 8], and Markov processes [9]. These matrices have also been shown to play a fundamental role in the description of linear positive systems, a class of systems that leave the nonnegative orthant invariant [10]. Due to their particular structure, such systems have been the topic of a lot of works in the dynamical systems and control communities; see e.g. also [11, 12, 13, 14, 15, 16, 17, 18, 19, 20] and references therein. In this paper, we will be interested in the sign-properties of such matrices, that is, those general properties that can be deduced from the knowledge of the sign-pattern. This problem initially emerged from economics [21], but also found applications in ecology [22, 23, 24, 25] and chemistry [26, 27]. The rationale of the overall approach stems from the fact that, in these fields, the interactions between different participants in a given system are, in general, qualitatively known but quantitatively unknown. This incomplete knowledge is very often a direct consequence of the difficulty for identifying and discriminating models because of their inherent complexity and scarce experimental data. Email addresses: [email protected], [email protected] (Corentin Briat), [email protected] (Mustafa Khammash) URL: www.briat.info (Corentin Briat), http://www.bsse.ethz.ch/ctsb (Mustafa Khammash)
Preprint submitted to Elsevier
December 23, 2015
In this context, it seems relevant to study the properties of the system solely based on the sign pattern structure or, more loosely, from the pattern of nonzero entries. This is referred to as qualitative analysis [28]. For instance, the sign-stability of general matrices (the property that all the matrices having the same sign-pattern are all Hurwitz stable) has been studied in [21, 25, 29]. An algorithm, having worst-case O(n2 ) time- and space-complexity, verifying the conditions in [29] has been proposed in [30]. Many other problems have also been addressed over the past decades; see e.g. [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41] and references therein. The first problem addressed in this paper is the sign-stability problem for which several general necessary and sufficient conditions exist [28]. By adapting these results, we show that the sign-stability of a Metzler sign-matrix can be assessed from the Hurwitz stability of a single particular matrix, referred to as the unit sign-matrix, lying inside the qualitative class. Therefore, for this class of matrices, checking the Hurwitz stability of an uncountably infinite and unbounded family of matrices is not more difficult than checking the stability of a given matrix. Lyapunov conditions, taking in this case the form of linear programs [42], can hence be used for establishing the sign-stability of a given Metzler sign-matrix. An alternative condition is formulated in terms of the acyclicity of the graph associated with the sign-pattern, a property that can be easily checked using algorithms such as the Depth-First-Search algorithm [43]. This result is then generalized to the case of block matrices for which several results, potentially enabling the use of distributed algorithms, are provided. Sign-stability results are then generalized to the problem of establishing the sign-stability of all the matrices located in the convex hull of a finite number of Metzler sign-matrices. Necessary and sufficient conditions for the existence of a common Lyapunov function for all the matrices in the convex-hull of such matrices are also provided. Relative sign-stability, a newly introduced concept, is then considered. It is shown that the sufficient conditions characterizing relative sign-stability can be brought back to a combinatorial problem known to be NP-complete [44]. Finally, mixed-matrices consisting of matrices with both sign-type and real-type entries are considered and their sign-properties clarified, again in terms of algebraic and graph theoretical conditions analogous to those obtained in the pure sign-matrix case. Several application examples are provided in the last section. Outline: Section 2 recalls some important results on the sign-stability of sign-matrices. These results are then specialized in Section 3 to the case of Metzler sign-matrices. The analysis of the sign-stability of block matrices is carried out in Section 4. Conditions for the sign-stability of the convex-hull of Metzler sign-matrices are obtained in Section 5. Section 6 is concerned with relative structural stability analysis whereas Section 7 addresses partial sign-stability. Application examples are finally discussed in Section 8. Notations: The set of real, real positive and real nonnegative numbers are denoted by R, R>0 and R≥0 , respectively. Natural, whole and integer numbers are denoted by N, N0 and Z. The n-dimensional vector of ones is denoted by 1n . The block-diagonal matrix with diagonal elements given by Mi ∈ Rni ×mi is denoted
2
by diagi {Mi }. 2. Sign-stability of sign-pattern matrices We describe here the terminology used in the paper as well as some useful existing results. 2.1. Definitions Definition 2.1. Let us define the following sets and matrices: • A Metzler matrix with entries in an arbitrary fully ordered set with zero element is a square matrix having nonnegative off-diagonal entries. The set of all n × n Metzler matrices is denoted by Mn . • A sign-matrix is a matrix taking entries in S := {⊖, 0, ⊕}. The set of all n × m sign-matrices is denoted by Sn×m . The set S has a full-order structure with the order ⊖ < 0 < ⊕. • A unit sign-matrix is a matrix taking entries in U := {−1, 0, 1}. The set of all n × m unit-sign matrices is denoted by Un×m . • We also define for simplicity MSn := Mn ∩ Sn×n , MRn := Mn ∩ Rn×n and MUn := Mn ∩ Un×n where the considered orders are the natural one and the one mentioned above for sign-matrices. Definition 2.2. The qualitative class of a matrix A ∈ (S ∪ R)n×m is the set of matrices given by ∈ R if [A] = ⊕ >0 ij Q(A) := M ∈ Rn×m [M ]ij ∈ R0 such that v T A < 0. Lemma 3.2. Let us consider a matrix M ∈ MSn with negative diagonal elements and assume that its associated directed graph DM is a directed cycle graph. Then, the Perron-Frobenius eigenvalue of sgn(M ) is equal to 0 and the corresponding eigenvector is 1n . Moreover, sgn(M ) is an irreducible matrix. Proof : Since DM is a directed cycle graph, then sgn(M ) can be decomposed as sgn(M ) = P − In where P is a permutation matrix with 0-entries on the diagonal. Hence, sgn(M ) is irreducible. Since P is a permutation matrix, then its eigenvalues are the roots of unity and has a Perron-Frobenius eigenvalue equal to 1, which is unique. Therefore, we have that the Perron-Frobenius eigenvalue of P −In is λP F (P −In ) = λP F (P )−1 = 0. It is finally easy to verify that 1 is the associated eigenvector.
5
♦
Lemma 3.3. Let A ∈ MRn . If A is Hurwitz stable, then A has negative diagonal elements. Proof : We use the contrapositive and assume, for instance, that one of the diagonal entry is nonnegative, say, a11 ≥ 0. The case of multiple nonnegative diagonal entries can be carried out similarly. Since the matrix is Metzler, then this means that the first column is nonnegative and thus that for any v ∈ Rn>0 , we have that v T A ≮ 0. Hence, by virtue of Lemma 3.1, we can conclude that the matrix is not Hurwitz stable. This proves the result.
♦
Lemma 3.4. Let us consider a matrix M ∈ MUn . Then, the following statements are equivalent: (a) M is Hurwitz stable. (b) The diagonal elements of M are negative and the directed graph DM is an acyclic directed graph. Proof : Proof of (a) ⇒ (b). We use the contrapositive. We have two cases. The first one is when at least one of the diagonal entry is nonnegative and, in this case, Lemma 3.3 proves the implication. The second one is when there exists at least a cycle in the graph DM . Let us assume that there is a single cycle of order f ∈ MUn×n k ≥ 2 in DM (the case of multiple cycles can be addressed in the same way). Define the matrix M
with negative diagonal elements and for which the graph DM f is obtained from DM by removing all the edges f is a permutation of the matrix that are not involved in the cycle. Then, M P − Ik 0
0
−In−k
(3)
where P is a permutation matrix with diagonal elements equal to 0. The left-upper block then corresponds to the nodes involved in the cycle whereas the right-lower block corresponds to those that are not. Clearly, f (in the same permutation basis). Hence, from the Perron-Frobenius theorem and the we have that M ≥ M
f). Again due to the Perrontheory of nonnegative/Metzler matrices [1], we have that λP F (M ) ≥ λP F (M
f) = λP F (P − Ik ) = 0 and hence λP F (M ) ≥ 0, Frobenius theorem and Lemma 3.2, we have that λP F (M proving then that M is not Hurwitz stable.
Proof of (b) ⇒ (a). Assume that statement (b) holds. Since the graph DM is acyclic, then there exists a permutation matrix P such that the matrix P T M P is upper-triangular with negative elements on the diagonal and nonnegative elements in the upper-triangular part. Therefore, the matrix M is Hurwitz stable since the eigenvalues are precisely the diagonal negative entries. The proof is complete.
3.2. Main results on sign-stability We can now state our main result on the sign-stability of Metzler sign-matrices: 6
♦
Theorem 3.5. Let A ∈ MSn×n . Then, the following statements are equivalent: (a) The matrix A is sign-stable. (b) The matrix sgn(A) is Hurwitz stable. (c) There exists v ∈ Rn>0 such that v T sgn(A) < 0. (d) The directed graph DA is acyclic and the diagonal elements of A are negative. (e) There exists a permutation matrix P such that P T AP is upper-triangular and has negative elements on the diagonal. (f) There is no back edge in DA and the diagonal elements of A are negative. Proof : Proof of (a)⇒ (b). Obvious. Proof of (b)⇒ (a). We show here that the fact that sgn(A) be Hurwitz stable implies that A is signstable or, equivalently, that the conditions of Theorem 2.10 are satisfied. Noting first that since sgn(A) is Hurwitz stable, then, by Lemma 3.3, all the diagonal elements of A are negative, which implies that the statement (i) of Theorem 2.10 holds (note that, from Lemma 3.3, the case aii = 0, for some i = 1, . . . , n, cannot occur). Additionally, the negative diagonal implies that the condition of statement (iii) is trivially satisfied since, in this case, RA = V and, therefore, all the vertices are black. Noting finally that, by Lemma 3.4, we have that DA is acyclic and, therefore, that the statements (ii) and (v) of Theorem 2.10 hold (note that the case aij aji < 0 cannot occur because the matrix is Metzler). Finally, the statement (iv) holds because, when there is no 2-cycles, then the graph GA has no edges and thus has a trivial (V \RA )-complete matching. Proof of (b)⇔ (c). This follows from Lemma 3.1. Proof of (d)⇔ (b). This follows from Lemma 3.4. Proof of (d)⇔ (e). This follows from the definition of an acyclic graph. Proof of (d)⇔ (f). This follows from the property that a graph is acyclic if and only if it has no back edge; see [43].
♦
We then have the following remarks regarding the above result: Remark 3.6. The statement (b) of the above result says that it is enough to check the Hurwitz stability of the unique unit sign-matrix inside the qualitative class in order to establish the sign-stability of the signmatrix. It is interesting to note that this fact does not hold true for general matrices. For instance, the matrix
⊖ ⊕ A= ⊖ ⊖ 0 ⊕ 7
⊕ ⊖ ⊖
(4)
is not sign-stable because of the presence of a cycle of order 3. On the other hand, the associated unit sign-matrix given by
1 −1 1 sgn(A) = −1 −1 −1 0 1 −1
(5)
is Hurwitz stable since sgn(A) + sgn(A)T is negative definite.
Remark 3.7. The statement (c) indicates that the Hurwitz stability can be easily checked since the problem of finding a vector v > 0 such that v T A < 0 is a linear programming problem [42] that can efficiently be solved using modern optimization algorithms. A suggested alternative way for checking the sign-stability of a sign-matrix is in terms of the acyclicity of the associated graph. Indeed, by virtue of statement (f), checking the existence of a cycle in a given graph is simple and can be performed using an algorithm such as the Depth-First-Search algorithm [43] that has a worst-case time complexity of O(n2 ) and a worst-case space complexity of O(n). To complete the above result, we provide several necessary conditions from which it is possible to quickly identify matrices that are not sign-stable. Proposition 3.8 (Necessary conditions). Let us consider a matrix A ∈ MSn and assume that one of the following statements hold: (a) aii ≥ 0 for some i = 1, . . . , n. (b) aij aji > 0 for some i, j = 1, . . . , n, i 6= j. (c) There is a cycle in the directed graph DA . (d) The matrix A is irreducible. Then, the matrix A is not sign-stable. 3.3. Potential sign-stability Definition 3.9. A matrix M ∈ Sn is potentially stable if Q(M ) contains at least one matrix that is Hurwitz stable. We then have the following result: Theorem 3.10 (Potential sign-stability). Let A ∈ MSn be given. Then, the following statements are equivalent: (a) The diagonal entries of A are negative. (b) The matrix A is potentially sign-stable. 8
Proof : Proof of 2) ⇒ 1) Since the matrix A is Metzler and is partially sign-stable, then there exists a matrix A′ ∈ Q(A) that is Hurwitz stable. From Lemma 3.3, this matrix necessarily has negative diagonal elements and, hence, A must have negative diagonal elements. Proof of 1) ⇒ 2) Assume the matrix A has negative diagonal entries. Let us then consider the matrix Mε defined as
−1 [Mε ]ij = ε 0
if [A]ij = ⊖, if [A]ij = ⊕,
(6)
if [A]ij = 0
where ε > 0. Clearly, if ε = 0, the matrix M0 is Hurwitz stable and all the eigenvalues are equal to -1. Using the fact that the eigenvalues of Mε are continuous with respect to ε, then we can conclude that there exists an ε¯ > 0 such that for all 0 ≤ ε < ε¯, the matrix Mε is Hurwitz-stable, showing then that A is potentially sign-stable. The proof is complete.
♦
4. Sign-stability of block matrices Let us consider now the case of block-matrices. It may seem at first sight that considering block-matrices is irrelevant because they can be analyzed using the tools described in the previous section. The main rationale, however, is that these matrices are commonplace in fields such as control theory, where they can be used to represent, for instance, interconnections of linear dynamical systems; see e.g. [45, 46]. The idea is then to find conditions for establishing the stability of a given block matrix based on the stability of its elementary blocks. This paradigm has led to many important results such as the small-gain theorem [45] and its variants. Dealing, moreover, with the subsystems separately may allow for the development of decentralized or distributed algorithms which are applicable to large-scale problems. First of all, linear programming conditions characterizing the stability of block matrices are obtained using the (linear) S-procedure [47, 48], an essential tool of control theory. These conditions are then adapted to the analysis of the sign-stability of Metzler block-matrices. It is emphasized that distributed algorithms may be used to check the linear programming conditions in an efficient way. 4.1. Preliminary results Lemma 4.1 (S-procedure, [47, 48]). Let σi : Rn → R, i = 0, . . . , N , be linear functions. Assume that the set S≥0 := {y ∈ Rn : σi (y) ≥ 0, i = 1, . . . , N } is nonempty. Then, the following statements are equivalent: (a) We have that σ0 (x) ≥ 0 for all x ∈ S≥0 . 9
(7)
(b) There exist some τi ≥ 0, i = 1, . . . , N , such that σ0 (x) −
N X
τi σi (x) ≥ 0 for all x ∈ Rn .
i=1
We then have the following immediate corollary: Corollary 4.2. Let σi : Rn → R, i = 0, . . . , N , be linear functions. Assume that the set S0+ := y ∈ Rn≥0 : σi (y) = 0, i = 1, . . . , N
(8)
is nonempty. Then, the following statements are equivalent: (a) We have that σ0 (x) ≥ 0 for all x ∈ S0+ .
(b) There exist some ηi ∈ R, i = 1, . . . , N , such that σ0 (x) +
N X
ηi σi (x) ≥ 0 for all x ∈ Rn≥0 .
i=1
Proof : Defining S≤0 := {y ∈ Rn : −σi (y) ≥ 0, i = 1, . . . , N }
(9)
then we can see that S0+ = S≥0 ∩ S≤0 ∩ Rn≥0 . Applying then the linear S-procedure, i.e. Lemma 4.1, we get that the statement (a) is equivalent to the existence of τi ≥ 0, i = 1, . . . , 2N , such that σ0 (x) + N X [τi − τi+N ] σi (x) ≥ 0 for all x ∈ Rn≥0 . Letting then ηi := τi − τi+N ∈ R yields the result. ♦ i=1
With the above results in mind, we are in position to state the following result: n ×nij
i Theorem 4.3. Let Ai ∈ MRni and Bij ∈ R≥0
n ×nj
ij , Cij ∈ R≥0
, i, j = 1, . . . , N , i 6= j. Then, the following
statements are equivalent: (a) The Metzler matrix
is Hurwitz stable.
A1 .. . BN 1 CN 1
B12 C12
BN 2 CN 2
. . . B1N C1N .. . ... AN
(10)
i (b) There exist vectors vi ∈ Rn>0 , ℓij ∈ Rnij , i, j = 1, . . . , N , i 6= j, such that the inequalities
viT Ai
N X
ℓTji Cji
< 0
viT Bij − ℓTij
≤ 0
+
(11)
j6=i
hold for all i, j = 1, . . . , N , j 6= i.
10
Proof : The equivalence between the two statements can be shown using Corollary 4.2. Let v = col(vi ), i
i , then the stability of the matrix (10) is equivalent to saying that the following feasibility problem vi ∈ Rn>0
has a solution:
(P)
Find s.t. for all
i vi ∈ Rn>0 , ε>0 N N X X X viT xi , viT Ai xi + Bij Cij xj ≤ −ε
j6=i
i=1
i Rn≥0 ,
xi ∈
i=1
i = 1, . . . , N
Letting now wij = Cij xj , we get that the inequality condition in (P) is equivalent to N N X X X nij i viT xi , xi ∈ Rn≥0 , wij ∈ R≥0 Bij wij ≤ −ε viT Ai xi + j6=i
i=1
(12)
i=1
together with the equality constraints Cij xj − wij = 0, i, j = 1, . . . , N , j 6= i. Invoking then Corollary 4.2, we get that the inequality condition of problem (P) reformulates as N X X Bij wij viT Ai xi + j6=i
i=1
+
X
ℓTij
(Cij xj − wij ) ≤ −ε
i6=j
N X
viT xi ,
xi ∈
(13) i Rn≥0 ,
wij ∈
nij R≥0
i=1
where ℓij ∈ Rnij , i, j = 1, . . . , N , i 6= j, are the multipliers introduced by the S-procedure. Reorganizing the terms yields N X i=1
viT Ai + εI + +
X j6=i
N X j6=i
viT Bij
ℓTji Cji xi
nij i − ℓTij wij ≤ 0, xi ∈ Rn≥0 . , wij ∈ R≥0
(14)
n
ij i Using finally the facts that the above inequality must be verified for all xi ∈ Rn≥0 and that these , wij ∈ R≥0
variables are independent allow us to state the result.
♦
We can see from this result that the stability of the overall block-matrix can be broken down to the stability analysis of N subproblems (one for each of the vi ’s) involving coupled multipliers that explicitly capture the topology of the interconnection. An important advantage of this formulation lies in its convenient form allowing for the derivation of efficient parallel algorithms for establishing the Hurwitz stability of Metzler block-matrices. The methods described in [49, 50] can for instance be used. 4.2. Main results Before stating the main result, it is convenient to define their Minkowski product as the set Q(A1 )Q(A2 ) := {M1 M2 : M1 ∈ Q(A1 ), M2 ∈ Q(A2 )} 11
(15)
n1 ×m 2 where A1 ∈ S≥0 and A2 ∈ Sm×n . It is also important to define the three following product rules for ≥0
dealing with products of nonnegative sign-matrices: 1) ⊕ + ⊕ = ⊕; 2) ⊕ · ⊕ = ⊕; 3) 0 · ⊕ = 0. Remark 4.4. Note that because of these product rules, we have that Q(A1 )Q(A2 ) ⊂ Q(A1 A2 ) and that, in general, Q(A1 )Q(A2 ) 6= Q(A1 A2 ). For instance, if h i ⊕ A1 = and A2 = ⊕ ⊕ , ⊕ then Q(A1 )Q(A2 ) is the set of all 2 × 2 rank-1 positive matrices. However, since ⊕ A1 A2 = ⊕
⊕ , ⊕
then Q(A1 A2 ) is the set of all 2 × 2 positive matrices. On the other hand, if h ⊕ A1 = and A2 = ⊕ ⊕
i 0 ,
then we have that Q(A1 )Q(A2 ) = Q(A1 A2 ). A similar statement can be made for unit-sign matrices. We indeed have, in general, that sgn(A1 A2 ) 6= sgn(A1 ) sgn(A2 ). For instance, this is the case for the matrices ⊕ A1 = 0 for which we have that
⊕ ⊕ and A2 = , ⊕ ⊕
1 2 sgn(A1 A2 ) = and sgn(A1 ) sgn(A2 ) = . 1 1
The above remark emphasizes the difficulty of the possible loss of independence between the entries of a matrix resulting from the multiplication of two sign-matrices. With this warning in mind, we can state the following result: n ×nij
i Theorem 4.5. Let Ai ∈ MSni and Bij ∈ S≥0
n ×nj
ij , Cij ∈ S≥0
. Assume, moreover, that Q(Bij Cij ) =
Q(Bij )Q(Cij ) for all i, j = 1, . . . , N , i 6= j. Then, the following statements are equivalent: (a) The Metzler sign-matrix
is sign-stable.
¯ A=
A1 .. .
B12 C12
BN 1 CN 1
BN 2 CN 2
12
. . . B1N C1N .. . ... AN
(16)
¯ is Hurwitz stable. (b) The matrix sgn(A) n
j i (c) There exist vectors vi ∈ Rn>0 , ℓij ∈ R>0 , i, j = 1, . . . , N , j 6= i, such that the inequalities
viT
N X
ℓTji
< 0
viT sgn(Bij Cij ) − ℓTij
< 0
sgn(Ai ) +
(17)
j6=i
for all i, j = 1, . . . , N , j 6= i. (d) The diagonal elements of A¯ are negative and the directed graph DA¯ is acyclic. Proof : The proof of the equivalence between statements (a), (b) and (d) follows from Theorem 3.5 and the fact that Q(Bij Cij ) = Q(Bij )Q(Cij ) for all i, j = 1, . . . , N , i 6= j; i.e. independence of the entries. The equivalence with the statement (c) follows from Theorem 4.5 and the fact that the entries in the matrix products are independent. Note, however, that, by virtue of Remark 4.4, the terms Bij Cij cannot be split up since sgn(Bij Cij ) 6= sgn(Bij ) sgn(Cij ), in general.
♦
In the case where Q(Bij Cij ) 6= Q(Bij )Q(Cij ) for some i, j, then the statements (b) and (c) are sufficient for (a) to hold, but not necessary. However, when sgn(Bij Cij ) = sgn(Bij ) sgn(Cij ), we have the following result: n ×nij
i Theorem 4.6. Let Ai ∈ MSni and Bij ∈ S≥0
n ×nj
ij , Cij ∈ S≥0
. Assume, moreover, that Q(Bij Cij ) =
Q(Bij )Q(Cij ) and sgn(Bij Cij ) = sgn(Bij ) sgn(Cij ) for all i, j = 1, . . . , N , i 6= j. Then, the following statements are equivalent: 1. The matrix A¯ in (16) is sign-stable. n
ij i 2. There exist vectors vi ∈ Rn>0 , ℓij ∈ R>0 , i, j = 1, . . . , N , j 6= i, such that the inequalities
viT sgn(Ai ) +
N X
ℓTji sgn(Cji )
0 such that v T A′i < 0 for all i = 1, . . . , N , implying, in turn, that statement (c) holds.
16
♦
6. Relative sign-stability of Metzler sign-matrices 6.1. Preliminaries In these section, we will work with a larger class of sign-matrices, referred to as indefinite sign-matrices, taking entries in S⊙ := S ∪ {⊙} where ⊙ denotes a sign indefinite entry that can be obtained from the rule ⊕ + ⊖ = ⊙. To avoid confusion, sign-matrices will also be referred to as definite sign-matrices. The n×m pre-qualitative class associated with a matrix A ∈ S⊙ is defined as
PQ(A) := M ∈ Sn×m
∈ S [M ]ij = [A]
ij
if [A]ij = ⊙ . otherwise
It is important to stress that, due to the presence of sign-indefinite entries ⊙, all the matrices in PQ(A) do not have the same sign-pattern. When there is no sign-indefinite entry in A, then PQ(A) = A. We finally introduce the following set Q(PQ(A)) :=
[
Q(A′ )
(24)
A′ ∈PQ(A)
that defines the set of all real matrices sharing the same sign-pattern of at least one matrix in the prequalitative class of the matrix A. Let us define now the concept of relative sign-stability: Definition 6.1. Let A ∈ Sn×n and B ∈ Rn×ℓ , ℓ < n, full-rank. We say that A is sign-stable relative to B or that the pair (A, B) is relatively sign-stable if for all A′ ∈ Q(A), there exists a v ∈ Rn>0 such that v T A′ < 0 and v T B = 0. The main reason behind the name is that the additional constraint v T B = 0 induces a restriction to the stability condition v T A < 0, a restriction that is relative to the choice for B. Such a problem arises, for instance, in the analysis of certain stochastic jump processes representing biological processes [4, 6]. When B = 0, the usual notion of sign-stability is retrieved. We then have the following result: Theorem 6.2. Assume that the sign-matrix A ∈ MSn is sign-stable. Then, we have the following statements: (a) The inverse sign-matrix A−1 defined as • [A−1 ]ii 6= 0 for all i = 1, . . . , n; • [A−1 ]ij 6= 0 if and only if there is a path of node j to node i in the directed graph DA . is unique. (b) A−1 is nonnegative. (c) We have that Q(A)−1 := M −1 : M ∈ Q(A) ⊂ Q(A−1 ). 17
Proof : To prove statement (a), we first assume, without loss of generality, that A is in upper-triangular form with nonnegative upper-triangular elements and negative diagonal elements. Then, let us consider the linear equation y = M x where M =−
n X
mi,i ei eTi +
n X
mi,j ei eTj
(25)
i<j
i
with mi,i > 0, mi,j ≥ 0 for i < j and where {ei }ni=1 is the standard basis of Rn . Then, by inverting recursively this system, we can see that the entry [M −1 ]ij consists of a negatively weighed sum of all possible paths from j to i. Consequently, the entry is 0 if and only if there is no path from j to i in DA , otherwise it is negative. Statement (b) readily follows from Lemma 3.1. We prove finally statement (c). We know from statements (a) and (b) that for a Metzler sign-stable sign-matrix A, the matrix inversion operation is well-defined in the sense that A−1 is a uniquely defined sign-matrix, which is nonnegative with negative diagonal elements. Therefore the inclusion in statement (c) follows from the fact that all the inverses of the matrices in Q(A) have the same sign-pattern as A−1 . Equality in statement (c) does not hold because the inverse of a Hurwitz stable real nonnegative matrix with negative diagonal is not necessarily Metzler, even in the upper-triangular case. In other words, the matrix inversion operation on the set of sign-stable upper-triangular nonnegative sign-matrices is not well-defined in the sense that the sign-pattern is non-uniquely defined. This is shown below by constructing a matrix M ∈ Q(A−1 ) such that M ∈ / Q(A)−1 where A is a sign-stable Metzler sign-matrix. For instance, −1 −1 −2 −4 −1 1 1 −2 0 −1 −1 −5 0 −1 1 4 −1 −1 M = / Q(A) ∈ Q(A ), but M = ∈ 0 0 −1 −1 0 0 −1 1 0 0 0 −1 0 0 0 −1 is not a Metzler matrix.
(26)
♦
Two important facts are stated by the above result. First of all, the sign-pattern of the inverse signmatrix A−1 is uniquely defined by the sign-pattern of A provided that A is sign-stable. Secondly, that for any sign-stable matrix A ∈ Sn×n , the nonzero entries of the inverse matrix A−1 can simply be deduced from the examination of the graph DA . For comparison, it requires at most n2 operations, which is better than usual real matrix inversion algorithms; see e.g. [53]. A drawback, however, is that by considering the matrix A−1 , we lose some information since the class of matrices that is considered is bigger. We are now ready to state the main result of this section: Proposition 6.3. Let A ∈ MSn×n be sign-stable and B ∈ Rℓ×n , ℓ < n, be full-rank. Assume that one of the following equivalent statements holds:
18
(a) All the matrices in PQ((A−1 B)T ) are L+ -matrices; i.e. for all M ∈ Q(PQ((A−1 B)T )), we have that y T M ≥ 0 with y ≥ 0 implies that y = 0. (b) All the matrices in PQ((A−1 B)T ) have no zero row and for all M ∈ Q(PQ((A−1 B)T )), there exists a v ∈ Rℓ>0 such that M v = 0. (c) For all nonzero diagonal matrices D ∈ {−1, 0, 1}n×n and all M ∈ PQ((A−1 B)T ), some column of DM is nonzero and nonnegative. (d) For all nonzero diagonal matrices D ∈ {−1, 0, 1}n×n and all M ∈ PQ((A−1 B)T ), some column of DM is nonzero and nonpositive. (e) For all M ∈ Q(PQ((A−1 B)T )), we have {M x : x ≥ 0} = Rn . Then, the pair (A, B) is relatively sign-stable. Proof : The pair (A, B) is relatively sign-stable if and only if for any A′ ∈ Q(A), there exists a v ∈ Rn>0 such that v T A′ < 0 and v T B = 0. This is equivalent to saying that for any A′ ∈ Q(A), there exist some v, q ∈ Rn>0 such that v T A′ = −q T and v T B = 0. Since, the matrix A is sign-stable, then it is invertible and the sign-pattern of A−1 is uniquely defined (see Theorem 6.2). Hence, the pair (A, B) is relatively sign-stable if for any M ∈ Q(PQ((A−1 B)T )), there exists a q ∈ Rn>0 such that M q = 0. Note that, in general, necessity is lost since the set Q(PQ(A−1 B)) contains the set Q(A)−1 B (see Theorem 6.2). Using now Theorem 2.4 of [44], we have that a sufficient condition for the relative sign-stability of the pair (A, B) is that all the matrices in PQ((A−1 B)T ) be L+ -matrices, which means that all these matrices have no zero row and that all the matrices in Q(PQ((A−1 B)T )) have a positive vector in their null-space. The latter statement is exactly statement (b). The equivalence between all the other statements also follows from Theorem 2.4 of [44]. The proof is complete.
♦
Based on the statements (c) and (d), we can clearly see that checking whether a matrix is an L+ -matrix is a combinatorial problem. This problem turns out to be NP-complete (see [32, 44]) but remains applicable for matrices having small dimensions. 7. Partial sign-stability As we have seen in the previous sections, sign-stability requires strong structural properties for the considered matrices. The idea here is to weaken the notion of sign-stability in order to enlarge the set of matrices that can be considered. To this aim, we introduce the concept of partial sign-stability which applies to matrices with mixed entries that contain both sign entries and real entries. In this case, cycles can be allowed for the nodes associated with real entries. We notably prove here that the partial sign-stability of a class of mixed matrices can be exactly characterized in terms of Hurwitz stability conditions, sign-stability conditions and the non-existence of cycles in a particular bipartite graph obtained from the mixed matrix. 19
7.1. Preliminaries Definition 7.1. A matrix M ∈ (R ∪ S)n×n is partially sign-stable if all the matrices M ′ ∈ Q(M ) are Hurwitz stable. Lemma 7.2. Let us consider a matrix M ∈ {0, 1}n×n. Then, the following statements are equivalent: (a) M is Schur stable; (b) M − I is Hurwitz stable; (c) The directed graph DM is acyclic and the diagonal elements of M are equal to 0. Proof : The proof between the two first statements follows from the fact that the nonnegative matrix M is Schur stable if and only if there exists a positive vector v such that v T (M − I) < 0, which is equivalent to the Hurwitz stability of the Metzler matrix M − I. We know from Lemma 3.4 that M − I is Hurwitz stable if and only if there is no cycle in the graph DM−I (or equivalently DM ) and the diagonal elements of M − I are negative. This turns out the be equivalent to the last statement.
♦
7.2. Main results In what follows, we consider mixed matrices of the form
Aσϕ = n ×nσ
ϕ where Aσ ∈ MSnσ , Aϕ ∈ MRnϕ , Bϕ ∈ R≥0
Aσ
Cϕ
Bϕ
Aϕ
n ×nϕ
σ and Cϕ ∈ R≥0
(27)
. We then have the following result:
Theorem 7.3. The following statements are equivalent: (1) The matrix Aσϕ defined in (27) is partially sign-stable. (2) The following statements hold: (a) Aσ is sign-stable, (b) Aϕ is Hurwitz stable, (c) There is no cycle in the directed graph DAσϕ containing nodes in both Vσ = {v1σ , . . . , vnσσ } and Vϕ = {v1ϕ , . . . , vnϕϕ } where Vσ contains the first nσ nodes associated with DAσϕ and Vϕ the nϕ last ones. (3) The following statements hold: (a) Aσ is sign-stable, (b) Aϕ is Hurwitz stable,
20
(c) There is no cycle in the directed bipartite graph B = (Vσ , Vϕ , E) where E = Eσ ∪ Eϕ with Eσ := {viϕ , vjσ ) ∈ Vϕ × Vσ : [Mσ ]ji 6= 0} and Eϕ := {(viσ , vjϕ ) ∈ Vσ × Vϕ : [Mϕ ]ji 6= 0}
(28)
where Mϕ and Mσ are nonnegative matrices defined as −1 Mϕ := −Cϕ A−1 ϕ Bϕ and Mσ := − sgn(Aσ ).
(29)
(4) The following statements hold: (a) sgn(Aσ ) is Hurwitz stable, (b) Aϕ is Hurwitz stable, (c) The matrix sgn(Mϕ Mσ ) − I is Hurwitz stable or, equivalently, sgn(Mϕ Mσ ) is Schur stable. Proof : Proof (1) ⇔ (2). Clearly, the sign-stability of Aσ and the Hurwitz stability of Aϕ are necessary conditions for the partial sign-stability of the matrix Aσϕ . Partial sign-stability then holds if and only if there is no cycle involving nodes in Vσ and Vϕ (see Theorem 3.5). Proof (2) ⇔ (3). The equivalence between (2) and (3) follows from the fact that the no-cycle condition in DAσϕ of Statement (2c) is equivalent to the no-cycle condition of Statement (3c). Proof (3) ⇔ (4).
The no-cycle condition is equivalent to the fact that the nonnegative matrix
sgn(Mϕ Mσ ) does not have any positive diagonal entry (2-cycle in the bipartite graph) and does not have any other cycles of higher-order (so no cycle in the graph DMϕ Mσ ). Combining these conditions with Lemma 7.2 yields the result.
♦
7.3. Example Let us consider the matrix
Aσϕ
⊖ ⊕ 0 ⊖ = Bϕ
1
0 0 0 . −1 2 1 −5
Clearly, Aσ is sign-stable and Aϕ is Hurwitz stable. When 0 0 , Bϕ = 0 1
(30)
(31)
the directed graph DAσϕ does not have any cycles with nodes in both Vσ and Vϕ . Alternatively, the matrices Mσ and Mϕ are given by
0 1 1 1 2 and Mϕ = Mσ = 3 0 0 0 1 21
(32)
and we have that
sgn(Mϕ Mσ ) =
0 1 0 0
(33)
which is clearly Schur stable. Therefore, the matrix Aσϕ in (30) is partially-stable. Now, if 0 Bϕ = 1
0 0 0 0 1 and sgn(Mϕ Mσ ) = , then Mϕ = 3 1 0 0 0
0 1
which is not Schur stable. Hence, the matrix Aσϕ in (30) is not partially-stable.
8. Applications 8.1. Linear positive dynamical systems 8.1.1. Systems with delays Let us consider the following linear positive system with constant delay [54, 55]: x(t) ˙
=
A0 x(t) +
N X
Ai x(t − hi )
i=1
x(s)
=
(34)
¯ 0] φ(s), s ∈ [−h,
¯ 0], Rn ), h ¯ = max{hi }, hi > 0, i = 1, . . . , N . where x ∈ Rn , φ ∈ C([−h, i
Lemma 8.1 ([56]). The linear system with delays (34) is positive if and only if the matrix A0 is Metzler and the matrices Ai , i = 1, . . . , N , are nonnegative. n×n Lemma 8.2 ([56]). Let the matrices A0 ∈ MRn and Ai ∈ R≥0 , i = 1, . . . , N , be given. Then, the following
statements are equivalent (a) The linear positive system with delay (34) is asymptotically stable. P (b) The matrix N i=0 Ai is Hurwitz stable.
Proposition 8.3. Let M0 ∈ MSn and Mi ∈ Sn≥0 , i = 1, . . . , N . Then, the following statements are equiva-
lent: (a) The matrix
PN
i=0
Mi is a definite sign-matrix (i.e.
PN
i=0
Mi ∈ Sn×n ) with negative diagonal entries.
(b) The diagonal elements of M0 are negative and the diagonal elements of Mi , i = 1, . . . , N , are equal to 0. We then have the following result: ¯ := PN Mi . Then, Proposition 8.4. Let M0 ∈ MSn and Mi ∈ Sn≥0 , i = 1, . . . , N . Define, moreover, M i=0 the following statements are equivalent:
22
(a) For all A0 ∈ Q(M0 ) and all Ai ∈ Q(Mi ), i = 1, . . . , N , the linear positive system with delays (34) is asymptotically stable; i.e. the system (34) is structurally stable. (b) The following statements hold: ¯ is a definite sign-matrix; i.e. M ¯ ∈ Sn×n , • M ¯ is sign-stable. • The matrix M (c) The following statements hold: ¯ is a definite sign-matrix; i.e. M ¯ ∈ Sn×n , • M ¯ < 0 holds. • There exists a vector v ∈ Rn>0 such that v T sgn M
Proof : The proof follows from Lemma 8.2, Theorem 3.5 and Proposition 8.3.
♦
8.1.2. Switched systems Let us consider the following positive switched linear system [57]: x(t) ˙
= Aσ(t) x(t)
x(0)
= x0
(35)
where x ∈ Rn and σ : R≥0 → {1, . . . , N } is the piecewise constant switching signal. Lemma 8.5. The linear switched system is positive if and only if Ai ∈ MRn for all i = 1, . . . , N . We then have the following result: Proposition 8.6. Let Mi ∈ MSn , i = 1, . . . , N . Then, the following statements are equivalent: (a) For all Ai ∈ Q(Mi ), i = 1, . . . , N , the linear switched system (35) is asymptotically stable under arbitrary switching i.e. the system (35) is structurally stable under arbitrary switching. (b) For all Ai ∈ Q(Mi ), i = 1, . . . , N , the linear time-varying positive system x(t) ˙ = A(t)x(t)
(36)
where A(t) ∈ co(A1 , . . . , AN ), is robustly asymptotically stable. (c) The diagonal elements of the matrices Mi , i = 1, . . . , N , are negative and the matrix
N X
Mi is sign-stable.
i=1
(d) For all Ai ∈ Q(Mi ), i = 1, . . . , N , there exists a positive definite matrix Q ∈ Rn×n such that A˜T Q + QA˜ is negative definite for all A˜ ∈ co(A1 , . . . , AN ). (e) For all Ai ∈ Q(Mi ), i = 1, . . . , N , there exists a vector v ∈ Rn>0 such that v T A˜ < 0 for all A˜ ∈ co(A1 , . . . , AN ). 23
Proof : The proof of the equivalence between the statements (1) and (2) follows from [58, Theorem 3] while the equivalence between the statements (3), (4) and (5) follows from Theorem 5.3 and Theorem 5.5. Clearly, statement (4) implies statement (2) using standard results on switched systems; see e.g. [57]. Finally, statement (2) implies (3) since if the conditions of statement (3) does not hold, then there will exist some matrices Ai ∈ Q(Mi ), i = 1, . . . , N , for which the set co(A1 , . . . , AN ) will contain unstable matrices. The proof is complete.
♦
It is interesting to note that the above result also applies to polytopic LPV systems of the form [46, 59]: ! N X αi (t)Ai x(t) x(t) ˙ = (37) i=1 x(0) =
where α : R≥0 →
(
β∈
RN ≥0
:
N X i=1
βi = 1
)
x0
is any sufficiently regular function.
8.2. Nonlinear positive systems Let us consider the following class of nonlinear positive systems that arises, among others, in reaction network theory x(t) ˙
=
x(0) =
Ax(t) + Bf (x(t)) + b
(38)
x0
where x ∈ Rn≥0 , f : Rn≥0 → Rℓ≥0 , b ∈ Rn≥0 , A ∈ Rn×n and B ∈ Zn×ℓ . We then have the following result: Proposition 8.7. Let A ∈ MRn and assume that there exists a v ∈ Rn>0 and an ε > 0 such that v T A ≤ −εv T and v T B = 0. Then, the system (38) is stable and, for all x0 ∈ / S, the trajectories of the system (38) converge to the compact set S := {x ∈ Rn≥0 : V (x) ≤ v T b/ε},
(39)
i.e. S is forward-invariant and attractive. Proof : Let us consider the function V (x) = v T x which is positive in Rn≥0 \{0}. The derivative of this function along the solutions of (38) is given by V˙ (x) = v T Ax+v T Bf (x)+v T b. Assuming the conditions of the theorem vT b hold, then we have that V˙ (x) ≤ −εV (x) + v T b and, therefore, that V (x(t)) ≤ e−εt V (x0 ) + (1 − e−εt ) . ε Hence, for all x0 ∈ / S, the trajectories of the system (38) converge to the compact set. The proof is complete. ♦ We can now derive the following structural version of the above result: Proposition 8.8. Let M ∈ MSn and assume, for simplicity, that Z := (A−1 B)T ∈ Sℓ×n . Suppose further that 24
(a) M is sign-stable; (b) Z is an L+ -matrix. Then, for all A ∈ Q(M ), there exists a v ∈ Rn>0 and an ε > 0 such that v T A ≤ −εv T and v T B = 0. Hence, the system (38) is structurally stable and its trajectories structurally converge to a compact set inside the positive orthant. 8.3. Ergodicity of stochastic reaction networks In [4], it is shown that a sufficient condition for the ergodicity of an irreducible stochastic reaction network involving bimolecular reactions with mass-action kinetics is that the drift condition xT M (v)x + v T Ax + v T b ≤ c1 − c2 v T x
(40)
holds for all x ∈ Nn0 and for some c1 , c2 > 0 where M (v) ∈ Rn×n is symmetric, A ∈ MRn and b ∈ Rn≥0 . Above, the matrices M (v), A and b are data of the problem which can be explicitly constructed from the network reactions. The following result, proved in [4], has been shown to be applicable to a broad class of reaction networks: Proposition 8.9. Let us consider an irreducible1 stochastic reaction network whose drift condition writes as (40). Assume further that there exists a v ∈ Rn>0 such that M (v) = 0 and v T A < 0. Then, the drift condition (40) holds and the reaction network is exponentially ergodic. We then have the following structural version: Proposition 8.10. Let B ∈ Zn×ℓ be such that v T B = 0 ⇒ M (v) = 0, let Z ∈ MSn be a Metzler sign-matrix and assume, for simplicity, that W := (Z −1 B)T ∈ Sℓ×n . Suppose, further, that (a) Z is sign-stable; (b) W is an L+ -matrix. Then, for all A ∈ Q(Z), there exists a v ∈ Rn>0 such that v T A < 0 and v T B = 0. Then the underlying Markov process describing the network is structurally exponentially ergodic.
Bibliography References [1] A. Berman and R. J. Plemmons. Nonnegative matrices in the mathematical sciences. SIAM, Philadelphia, USA, 1994. 1
Irreducibility of a reaction network can be checked using the method developed in [60].
25
[2] H. Nikaido. Convex structures and economic theory. Academic Press, New York, USA, 1968. [3] C. Briat and M. Khammash. Computer control of gene expression: Robust setpoint tracking of protein mean and variance using integral feedback. In 51st IEEE Conference on Decision and Control, pages 3582–3588, Maui, Hawaii, USA, 2012. [4] A. Gupta, C. Briat, and M. Khammash. A scalable computational framework for establishing long-term behavior of stochastic reaction networks. PLOS Computational Biology, 10(6):e1003669, 2014. [5] F. Parise, M. E. Valcher, and J. Lygeros. On the reachable set of the controlled gene expression system. In 53rd IEEE Conference on Decision and Control, pages 4597–4604, Los Angeles, USA, 2014. [6] C. Briat, A. Gupta, and M. Khammash. Antithetic integral feedback: A new motif for robust perfect adaptation in noisy biomolecular networks. to appear in Cell Systems, 2016. [7] H. L. Smith. Monotone Dynamical Systems - An Introduction to the Theory of Competitive and Cooperative Systems. American Mathematical Society, Providence, Rhode Island, 1995. [8] D. Angeli and E. D. Sontag. Monotone control systems. IEEE Transactions on Automatic Control, 48(10):1684–1698, 2003. [9] T. M. Liggett. Continuous-time Markov processes. An introduction, volume 112 of Graduate Studies in Mathematics. American Mathematical Society, 2010. [10] L. Farina and S. Rinaldi. Positive Linear Systems: Theory and Applications. John Wiley & Sons, 2000. [11] M. Ait Rami and F. Tadeo. Controller synthesis for positive linear systems with bounded controls. IEEE Transactions on Circuits and Systems – II. Express Briefs, 54(2):151–155, 2007. [12] W. M. Haddad, V. Chellaboina, and Q. Hui. Nonnegative and compartmental dynamical systems. Princeton University Press, 41 William Street, Princeton, New Jersey, USA, 2010. [13] C. Briat. Robust stability and stabilization of uncertain linear positive systems via integral linear constraints - L1 - and L∞ -gains characterizations. International Journal of Robust and Nonlinear Control, 23(17):1932–1954, 2013. [14] F. Mazenc and O. Bernard. Interval observers for linear time-invariant systems with disturbances. Automatica, 47:140–147, 2011. [15] Y. Ebihara, D. Peaucelle, and D. Arzelier. L1 gain analysis of linear positive systems and its applications. In 50th Conference on Decision and Control, Orlando, Florida, USA, pages 4029–4034, 2011.
26
[16] A. Rantzer. Distributed control of positive systems. In 50th IEEE Conference on Decision and Control, pages 6608–6611, 2011. [17] T. Tanaka and C. Langbort. Symmetric formulation of the S-Procedure, Kalman-Yakubovich-Popov Lemma and their exact losslessness conditions. IEEE Transactions on Automatic Control, 58(6):1486– 1496, 2013. [18] T. Tanaka, C. Langbort, and V. Ugrinovskii. DC-dominant property of cone-preserving transfer functions. Systems & Control Letters, 62(8):699–707, 2013. [19] M. Colombino and R. S. Smith. Convex characterization of robust stability analysis and control synthesis for positive linear systems. In 53rd IEEE Conference on Decision and Control, pages 4379–4384, Los Angeles, USA, 2014. [20] C. Briat. L∞ -to-L∞ interval observers for continuous-time systems with persistent inputs. 2015. URL http://arxiv.org/abs/1511.07888. [21] J. Quirk and R. Ruppert. Qualitative economics and the stability of equilibrium. The review of economic studies, 32(4):311–326, 1965. [22] R. M. May. Will a large complex system be stable? Nature, 238:413–414, 1972. [23] R. M. May. Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton, New Jersey, 1973. [24] A. Roberts. The stability of a feasible random ecosystem. Nature, 251:607–608, 1974. [25] C. Jeffries. Qualitative stability and digraphs in model ecosystems. Ecology, 55:1415–1419, 1974. [26] J. J. Tyson. Classification of instabilities in chemical reaction systems. The Journal of Chemical Physics, 62:1010–1015, 1974. [27] B. L. Clarke. Theorems on chemical network stability. The Journal of Chemical Physics, 62:773–775, 1975. [28] J. Maybee and J. Quirk. Qualitative problems in matrix theory. SIAM Reviews, 11(1):30–51, 1969. [29] C. Jeffries, V. Klee, and P. van den Driessche. When is a matrix sign stable? Canadian Journal of Mathematics, 29:315–326, 1977. [30] V. Klee and P. van den Driessche. Linear algorithms for testing the sign stability of a matrix and for finding z-maximum matchings in acyclic graphs. Numerische Mathematik, 28(3):273–285, 1977.
27
[31] C.-T. Lin. Structural controllability. IEEE Transactions on Automatic Control, 19:201–208, 1974. [32] V. Klee, R. Ladner, and R. Manber. Signsolvability revisited. Linear Algebra and Its Applications, 59: 131–157, 1984. [33] J. L. Willems. Structural controllability and observability. Systems & Control Letters, 8:5–12, 1986. [34] M. A. Berger and A. Felzenbaum. Sign patterns of matrices and their inverses. Linear Algebra and its Applications, 86:161–177, 1987. [35] C. Thomassen. When the sign pattern of a square matrix determines uniquely the sign patter of its inverse. Linear Algebra and its Applications, 119:27–34, 1989. [36] R. A. Brualdi and B. L. Shader. Matrices of sign-solvable linear systems. Cambridge University Press, Cambridge, UK, 1995. [37] E. Kaszkurewicz and A. Bhaya. Matrix diagonal stability in systems and computation. Springer Science+Business, New York, USA, 2000. [38] P. Santsesso and M. E. Valcher. On the zero pattern properties and asymptotic behavior of continuoustime positive system trajectories. Linear Algebra and its Applications, 425:283–302, 2007. [39] M. Catral, D. D. Olesky, and P. van den Driessche. Allow problems concerning spectral properties of sign pattern matrices: A survey. Linear Algebra and its Applications, 430(11-12):3080–3094, 2009. [40] C. Mendes Ara´ ujo and J. R. Torregrosa. Sign pattern matrices that admit M -, N -, P - or inverse M -matrices. Linear Algebra and its Applications, 431(5–7):724–731, 2009. [41] O. Pryporova. Qualitative convergence of matrices. Linear Algebra and its Applications, 431(1):28–38, 2009. [42] S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press, Cambridge, MA, USA, 2004. [43] D. Knuth. The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 3rd Edition. AddisonWesley, Boston, USA, 1997. [44] G.-Y. Lee and B. L. Shader. Sign-consistency and solvability of constrained linear systems. Electronic Journal of Linear Algebra, 4:1, 1998. [45] K. Zhou, J. C. Doyle, and K. Glover. Robust and Optimal Control. Prentice Hall, Upper Saddle River, New Jersey, USA, 1996.
28
[46] C. Briat. Linear Parameter-Varying and Time-Delay Systems – Analysis, Observation, Filtering & Control. Advances on Delays and Dynamics, Vol. 3. Springer-Verlag, Heidelberg, Germany, 2015. [47] V. A. Yakubovich. S-procedure in nonlinear control theory. Vestnik Leningrad Univ., pages 62–77, 1971. English translation in Vestnik Leningrad Univ. Math., vol. 4, pp. 73–93, 1977. [48] U. T. J¨onsson.
A lecture on the S-procedure.
Technical report, Division of Optimiza-
tion and Systems Theory, Royal Institute of Technology, Stockholm, Sweden, 2001.
URL
http://www.math.kth.se/ulfj/5B5746/Lecture.ps. [49] D. P. Bertsekas and J. N. Tsitsiklis. Parallel and Distributed Computation. Prentice Hall, 1989. [50] S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein. Distributed optimization and statistical learning via the alternating direction method of multipliers. Foundations and Trends in Machine Learning, 3(1): 1–122, 2011. [51] E. Fornasini and M. E. Valcher. Linear copositive Lyapunov functions for continuous-time positive switched systems. IEEE Transactions on Automatic Control, 55(8):1933–1937, 2010. [52] D. Liberzon, J. P. Hespanha, and A. S. Morse. Stability of switched systems: a Lie-algebraic condition. System and Control Letters, 37:117–122, 1999. [53] F. Le Gall. Powers of tensors and fast matrix multiplication. In 39th International Symposium on Symbolic and Algebraic Computation, pages 296–303, Kobe, Japan, 2014. [54] S. I. Niculescu. Delay effects on stability. A robust control approach, volume 269. Springer-Verlag: Heidelbeg, 2001. [55] K. Gu, V. L. Kharitonov, and J. Chen. Stability of Time-Delay Systems. Birkh¨auser, Boston, 2003. [56] W. M. Haddad and V. Chellaboina. Stability theory for nonnegative and compartmental dynamical systems with time delay. Systems & Control Letters, 51(5):355–361, 2004. [57] D. Liberzon. Switching in Systems and Control. Birkh¨auser, New York, 2003. [58] H. Lin and P. J. Antsaklis. Stability and stabilizability of switched linear systems: A survey of recent results. IEEE Transactions on Automatic Control, 54(2):308–322, 2009. [59] P. Apkarian, P. Gahinet, and G. Becker. Self-scheduled control of linear parameter varying systems: A design example. Automatica, 31(9):1251–1261, 1995. [60] A. Gupta and M. Khammash. Determining the long-term behavior of cell populations: A new procedure for detecting ergodicity in large stochastic reaction networks. Technical Report arXiv:1312.2879, ETHZ¨ urich, 2013. 29
arXiv:1512.07043v1 [math.DS] 22 Dec 2015
Department of Biosystems Science and Engineering, ETH-Z¨ urich, Basel, Switzerland
Abstract Several results on the sign properties of Metzler matrices are obtained. It is first established that checking the sign-stability of a Metzler sign-matrix can be either characterized in terms of the Hurwitz stability of the unit sign-matrix in the corresponding qualitative class, or in terms of the acyclicity of the graph associated with the sign-pattern. Similar results are obtained for the case of block-matrices and mixed-matrices, the latter containing both sign patterns and fixed real entries. The problem of assessing the sign-stability of the convex full of a finite family of Metzler matrices is also solved, and a necessary and sufficient condition for the existence of a common Lyapunov function for all the matrices in the convex hull is obtained. The notion of relative sign-stability is also introduced and a sufficient condition for the relative sign-stability of Metzler matrices is proposed. Several applications of the results are discussed in the last section. Keywords: Sign-stability; Metzler matrices; Positive systems
1. Introduction Metzler matrices [1] are often encountered in various fields such as economics [2] and biology [3, 4, 5, 6] as they may be associated with cooperative/monotone dynamical systems [7, 8], and Markov processes [9]. These matrices have also been shown to play a fundamental role in the description of linear positive systems, a class of systems that leave the nonnegative orthant invariant [10]. Due to their particular structure, such systems have been the topic of a lot of works in the dynamical systems and control communities; see e.g. also [11, 12, 13, 14, 15, 16, 17, 18, 19, 20] and references therein. In this paper, we will be interested in the sign-properties of such matrices, that is, those general properties that can be deduced from the knowledge of the sign-pattern. This problem initially emerged from economics [21], but also found applications in ecology [22, 23, 24, 25] and chemistry [26, 27]. The rationale of the overall approach stems from the fact that, in these fields, the interactions between different participants in a given system are, in general, qualitatively known but quantitatively unknown. This incomplete knowledge is very often a direct consequence of the difficulty for identifying and discriminating models because of their inherent complexity and scarce experimental data. Email addresses: [email protected], [email protected] (Corentin Briat), [email protected] (Mustafa Khammash) URL: www.briat.info (Corentin Briat), http://www.bsse.ethz.ch/ctsb (Mustafa Khammash)
Preprint submitted to Elsevier
December 23, 2015
In this context, it seems relevant to study the properties of the system solely based on the sign pattern structure or, more loosely, from the pattern of nonzero entries. This is referred to as qualitative analysis [28]. For instance, the sign-stability of general matrices (the property that all the matrices having the same sign-pattern are all Hurwitz stable) has been studied in [21, 25, 29]. An algorithm, having worst-case O(n2 ) time- and space-complexity, verifying the conditions in [29] has been proposed in [30]. Many other problems have also been addressed over the past decades; see e.g. [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41] and references therein. The first problem addressed in this paper is the sign-stability problem for which several general necessary and sufficient conditions exist [28]. By adapting these results, we show that the sign-stability of a Metzler sign-matrix can be assessed from the Hurwitz stability of a single particular matrix, referred to as the unit sign-matrix, lying inside the qualitative class. Therefore, for this class of matrices, checking the Hurwitz stability of an uncountably infinite and unbounded family of matrices is not more difficult than checking the stability of a given matrix. Lyapunov conditions, taking in this case the form of linear programs [42], can hence be used for establishing the sign-stability of a given Metzler sign-matrix. An alternative condition is formulated in terms of the acyclicity of the graph associated with the sign-pattern, a property that can be easily checked using algorithms such as the Depth-First-Search algorithm [43]. This result is then generalized to the case of block matrices for which several results, potentially enabling the use of distributed algorithms, are provided. Sign-stability results are then generalized to the problem of establishing the sign-stability of all the matrices located in the convex hull of a finite number of Metzler sign-matrices. Necessary and sufficient conditions for the existence of a common Lyapunov function for all the matrices in the convex-hull of such matrices are also provided. Relative sign-stability, a newly introduced concept, is then considered. It is shown that the sufficient conditions characterizing relative sign-stability can be brought back to a combinatorial problem known to be NP-complete [44]. Finally, mixed-matrices consisting of matrices with both sign-type and real-type entries are considered and their sign-properties clarified, again in terms of algebraic and graph theoretical conditions analogous to those obtained in the pure sign-matrix case. Several application examples are provided in the last section. Outline: Section 2 recalls some important results on the sign-stability of sign-matrices. These results are then specialized in Section 3 to the case of Metzler sign-matrices. The analysis of the sign-stability of block matrices is carried out in Section 4. Conditions for the sign-stability of the convex-hull of Metzler sign-matrices are obtained in Section 5. Section 6 is concerned with relative structural stability analysis whereas Section 7 addresses partial sign-stability. Application examples are finally discussed in Section 8. Notations: The set of real, real positive and real nonnegative numbers are denoted by R, R>0 and R≥0 , respectively. Natural, whole and integer numbers are denoted by N, N0 and Z. The n-dimensional vector of ones is denoted by 1n . The block-diagonal matrix with diagonal elements given by Mi ∈ Rni ×mi is denoted
2
by diagi {Mi }. 2. Sign-stability of sign-pattern matrices We describe here the terminology used in the paper as well as some useful existing results. 2.1. Definitions Definition 2.1. Let us define the following sets and matrices: • A Metzler matrix with entries in an arbitrary fully ordered set with zero element is a square matrix having nonnegative off-diagonal entries. The set of all n × n Metzler matrices is denoted by Mn . • A sign-matrix is a matrix taking entries in S := {⊖, 0, ⊕}. The set of all n × m sign-matrices is denoted by Sn×m . The set S has a full-order structure with the order ⊖ < 0 < ⊕. • A unit sign-matrix is a matrix taking entries in U := {−1, 0, 1}. The set of all n × m unit-sign matrices is denoted by Un×m . • We also define for simplicity MSn := Mn ∩ Sn×n , MRn := Mn ∩ Rn×n and MUn := Mn ∩ Un×n where the considered orders are the natural one and the one mentioned above for sign-matrices. Definition 2.2. The qualitative class of a matrix A ∈ (S ∪ R)n×m is the set of matrices given by ∈ R if [A] = ⊕ >0 ij Q(A) := M ∈ Rn×m [M ]ij ∈ R0 such that v T A < 0. Lemma 3.2. Let us consider a matrix M ∈ MSn with negative diagonal elements and assume that its associated directed graph DM is a directed cycle graph. Then, the Perron-Frobenius eigenvalue of sgn(M ) is equal to 0 and the corresponding eigenvector is 1n . Moreover, sgn(M ) is an irreducible matrix. Proof : Since DM is a directed cycle graph, then sgn(M ) can be decomposed as sgn(M ) = P − In where P is a permutation matrix with 0-entries on the diagonal. Hence, sgn(M ) is irreducible. Since P is a permutation matrix, then its eigenvalues are the roots of unity and has a Perron-Frobenius eigenvalue equal to 1, which is unique. Therefore, we have that the Perron-Frobenius eigenvalue of P −In is λP F (P −In ) = λP F (P )−1 = 0. It is finally easy to verify that 1 is the associated eigenvector.
5
♦
Lemma 3.3. Let A ∈ MRn . If A is Hurwitz stable, then A has negative diagonal elements. Proof : We use the contrapositive and assume, for instance, that one of the diagonal entry is nonnegative, say, a11 ≥ 0. The case of multiple nonnegative diagonal entries can be carried out similarly. Since the matrix is Metzler, then this means that the first column is nonnegative and thus that for any v ∈ Rn>0 , we have that v T A ≮ 0. Hence, by virtue of Lemma 3.1, we can conclude that the matrix is not Hurwitz stable. This proves the result.
♦
Lemma 3.4. Let us consider a matrix M ∈ MUn . Then, the following statements are equivalent: (a) M is Hurwitz stable. (b) The diagonal elements of M are negative and the directed graph DM is an acyclic directed graph. Proof : Proof of (a) ⇒ (b). We use the contrapositive. We have two cases. The first one is when at least one of the diagonal entry is nonnegative and, in this case, Lemma 3.3 proves the implication. The second one is when there exists at least a cycle in the graph DM . Let us assume that there is a single cycle of order f ∈ MUn×n k ≥ 2 in DM (the case of multiple cycles can be addressed in the same way). Define the matrix M
with negative diagonal elements and for which the graph DM f is obtained from DM by removing all the edges f is a permutation of the matrix that are not involved in the cycle. Then, M P − Ik 0
0
−In−k
(3)
where P is a permutation matrix with diagonal elements equal to 0. The left-upper block then corresponds to the nodes involved in the cycle whereas the right-lower block corresponds to those that are not. Clearly, f (in the same permutation basis). Hence, from the Perron-Frobenius theorem and the we have that M ≥ M
f). Again due to the Perrontheory of nonnegative/Metzler matrices [1], we have that λP F (M ) ≥ λP F (M
f) = λP F (P − Ik ) = 0 and hence λP F (M ) ≥ 0, Frobenius theorem and Lemma 3.2, we have that λP F (M proving then that M is not Hurwitz stable.
Proof of (b) ⇒ (a). Assume that statement (b) holds. Since the graph DM is acyclic, then there exists a permutation matrix P such that the matrix P T M P is upper-triangular with negative elements on the diagonal and nonnegative elements in the upper-triangular part. Therefore, the matrix M is Hurwitz stable since the eigenvalues are precisely the diagonal negative entries. The proof is complete.
3.2. Main results on sign-stability We can now state our main result on the sign-stability of Metzler sign-matrices: 6
♦
Theorem 3.5. Let A ∈ MSn×n . Then, the following statements are equivalent: (a) The matrix A is sign-stable. (b) The matrix sgn(A) is Hurwitz stable. (c) There exists v ∈ Rn>0 such that v T sgn(A) < 0. (d) The directed graph DA is acyclic and the diagonal elements of A are negative. (e) There exists a permutation matrix P such that P T AP is upper-triangular and has negative elements on the diagonal. (f) There is no back edge in DA and the diagonal elements of A are negative. Proof : Proof of (a)⇒ (b). Obvious. Proof of (b)⇒ (a). We show here that the fact that sgn(A) be Hurwitz stable implies that A is signstable or, equivalently, that the conditions of Theorem 2.10 are satisfied. Noting first that since sgn(A) is Hurwitz stable, then, by Lemma 3.3, all the diagonal elements of A are negative, which implies that the statement (i) of Theorem 2.10 holds (note that, from Lemma 3.3, the case aii = 0, for some i = 1, . . . , n, cannot occur). Additionally, the negative diagonal implies that the condition of statement (iii) is trivially satisfied since, in this case, RA = V and, therefore, all the vertices are black. Noting finally that, by Lemma 3.4, we have that DA is acyclic and, therefore, that the statements (ii) and (v) of Theorem 2.10 hold (note that the case aij aji < 0 cannot occur because the matrix is Metzler). Finally, the statement (iv) holds because, when there is no 2-cycles, then the graph GA has no edges and thus has a trivial (V \RA )-complete matching. Proof of (b)⇔ (c). This follows from Lemma 3.1. Proof of (d)⇔ (b). This follows from Lemma 3.4. Proof of (d)⇔ (e). This follows from the definition of an acyclic graph. Proof of (d)⇔ (f). This follows from the property that a graph is acyclic if and only if it has no back edge; see [43].
♦
We then have the following remarks regarding the above result: Remark 3.6. The statement (b) of the above result says that it is enough to check the Hurwitz stability of the unique unit sign-matrix inside the qualitative class in order to establish the sign-stability of the signmatrix. It is interesting to note that this fact does not hold true for general matrices. For instance, the matrix
⊖ ⊕ A= ⊖ ⊖ 0 ⊕ 7
⊕ ⊖ ⊖
(4)
is not sign-stable because of the presence of a cycle of order 3. On the other hand, the associated unit sign-matrix given by
1 −1 1 sgn(A) = −1 −1 −1 0 1 −1
(5)
is Hurwitz stable since sgn(A) + sgn(A)T is negative definite.
Remark 3.7. The statement (c) indicates that the Hurwitz stability can be easily checked since the problem of finding a vector v > 0 such that v T A < 0 is a linear programming problem [42] that can efficiently be solved using modern optimization algorithms. A suggested alternative way for checking the sign-stability of a sign-matrix is in terms of the acyclicity of the associated graph. Indeed, by virtue of statement (f), checking the existence of a cycle in a given graph is simple and can be performed using an algorithm such as the Depth-First-Search algorithm [43] that has a worst-case time complexity of O(n2 ) and a worst-case space complexity of O(n). To complete the above result, we provide several necessary conditions from which it is possible to quickly identify matrices that are not sign-stable. Proposition 3.8 (Necessary conditions). Let us consider a matrix A ∈ MSn and assume that one of the following statements hold: (a) aii ≥ 0 for some i = 1, . . . , n. (b) aij aji > 0 for some i, j = 1, . . . , n, i 6= j. (c) There is a cycle in the directed graph DA . (d) The matrix A is irreducible. Then, the matrix A is not sign-stable. 3.3. Potential sign-stability Definition 3.9. A matrix M ∈ Sn is potentially stable if Q(M ) contains at least one matrix that is Hurwitz stable. We then have the following result: Theorem 3.10 (Potential sign-stability). Let A ∈ MSn be given. Then, the following statements are equivalent: (a) The diagonal entries of A are negative. (b) The matrix A is potentially sign-stable. 8
Proof : Proof of 2) ⇒ 1) Since the matrix A is Metzler and is partially sign-stable, then there exists a matrix A′ ∈ Q(A) that is Hurwitz stable. From Lemma 3.3, this matrix necessarily has negative diagonal elements and, hence, A must have negative diagonal elements. Proof of 1) ⇒ 2) Assume the matrix A has negative diagonal entries. Let us then consider the matrix Mε defined as
−1 [Mε ]ij = ε 0
if [A]ij = ⊖, if [A]ij = ⊕,
(6)
if [A]ij = 0
where ε > 0. Clearly, if ε = 0, the matrix M0 is Hurwitz stable and all the eigenvalues are equal to -1. Using the fact that the eigenvalues of Mε are continuous with respect to ε, then we can conclude that there exists an ε¯ > 0 such that for all 0 ≤ ε < ε¯, the matrix Mε is Hurwitz-stable, showing then that A is potentially sign-stable. The proof is complete.
♦
4. Sign-stability of block matrices Let us consider now the case of block-matrices. It may seem at first sight that considering block-matrices is irrelevant because they can be analyzed using the tools described in the previous section. The main rationale, however, is that these matrices are commonplace in fields such as control theory, where they can be used to represent, for instance, interconnections of linear dynamical systems; see e.g. [45, 46]. The idea is then to find conditions for establishing the stability of a given block matrix based on the stability of its elementary blocks. This paradigm has led to many important results such as the small-gain theorem [45] and its variants. Dealing, moreover, with the subsystems separately may allow for the development of decentralized or distributed algorithms which are applicable to large-scale problems. First of all, linear programming conditions characterizing the stability of block matrices are obtained using the (linear) S-procedure [47, 48], an essential tool of control theory. These conditions are then adapted to the analysis of the sign-stability of Metzler block-matrices. It is emphasized that distributed algorithms may be used to check the linear programming conditions in an efficient way. 4.1. Preliminary results Lemma 4.1 (S-procedure, [47, 48]). Let σi : Rn → R, i = 0, . . . , N , be linear functions. Assume that the set S≥0 := {y ∈ Rn : σi (y) ≥ 0, i = 1, . . . , N } is nonempty. Then, the following statements are equivalent: (a) We have that σ0 (x) ≥ 0 for all x ∈ S≥0 . 9
(7)
(b) There exist some τi ≥ 0, i = 1, . . . , N , such that σ0 (x) −
N X
τi σi (x) ≥ 0 for all x ∈ Rn .
i=1
We then have the following immediate corollary: Corollary 4.2. Let σi : Rn → R, i = 0, . . . , N , be linear functions. Assume that the set S0+ := y ∈ Rn≥0 : σi (y) = 0, i = 1, . . . , N
(8)
is nonempty. Then, the following statements are equivalent: (a) We have that σ0 (x) ≥ 0 for all x ∈ S0+ .
(b) There exist some ηi ∈ R, i = 1, . . . , N , such that σ0 (x) +
N X
ηi σi (x) ≥ 0 for all x ∈ Rn≥0 .
i=1
Proof : Defining S≤0 := {y ∈ Rn : −σi (y) ≥ 0, i = 1, . . . , N }
(9)
then we can see that S0+ = S≥0 ∩ S≤0 ∩ Rn≥0 . Applying then the linear S-procedure, i.e. Lemma 4.1, we get that the statement (a) is equivalent to the existence of τi ≥ 0, i = 1, . . . , 2N , such that σ0 (x) + N X [τi − τi+N ] σi (x) ≥ 0 for all x ∈ Rn≥0 . Letting then ηi := τi − τi+N ∈ R yields the result. ♦ i=1
With the above results in mind, we are in position to state the following result: n ×nij
i Theorem 4.3. Let Ai ∈ MRni and Bij ∈ R≥0
n ×nj
ij , Cij ∈ R≥0
, i, j = 1, . . . , N , i 6= j. Then, the following
statements are equivalent: (a) The Metzler matrix
is Hurwitz stable.
A1 .. . BN 1 CN 1
B12 C12
BN 2 CN 2
. . . B1N C1N .. . ... AN
(10)
i (b) There exist vectors vi ∈ Rn>0 , ℓij ∈ Rnij , i, j = 1, . . . , N , i 6= j, such that the inequalities
viT Ai
N X
ℓTji Cji
< 0
viT Bij − ℓTij
≤ 0
+
(11)
j6=i
hold for all i, j = 1, . . . , N , j 6= i.
10
Proof : The equivalence between the two statements can be shown using Corollary 4.2. Let v = col(vi ), i
i , then the stability of the matrix (10) is equivalent to saying that the following feasibility problem vi ∈ Rn>0
has a solution:
(P)
Find s.t. for all
i vi ∈ Rn>0 , ε>0 N N X X X viT xi , viT Ai xi + Bij Cij xj ≤ −ε
j6=i
i=1
i Rn≥0 ,
xi ∈
i=1
i = 1, . . . , N
Letting now wij = Cij xj , we get that the inequality condition in (P) is equivalent to N N X X X nij i viT xi , xi ∈ Rn≥0 , wij ∈ R≥0 Bij wij ≤ −ε viT Ai xi + j6=i
i=1
(12)
i=1
together with the equality constraints Cij xj − wij = 0, i, j = 1, . . . , N , j 6= i. Invoking then Corollary 4.2, we get that the inequality condition of problem (P) reformulates as N X X Bij wij viT Ai xi + j6=i
i=1
+
X
ℓTij
(Cij xj − wij ) ≤ −ε
i6=j
N X
viT xi ,
xi ∈
(13) i Rn≥0 ,
wij ∈
nij R≥0
i=1
where ℓij ∈ Rnij , i, j = 1, . . . , N , i 6= j, are the multipliers introduced by the S-procedure. Reorganizing the terms yields N X i=1
viT Ai + εI + +
X j6=i
N X j6=i
viT Bij
ℓTji Cji xi
nij i − ℓTij wij ≤ 0, xi ∈ Rn≥0 . , wij ∈ R≥0
(14)
n
ij i Using finally the facts that the above inequality must be verified for all xi ∈ Rn≥0 and that these , wij ∈ R≥0
variables are independent allow us to state the result.
♦
We can see from this result that the stability of the overall block-matrix can be broken down to the stability analysis of N subproblems (one for each of the vi ’s) involving coupled multipliers that explicitly capture the topology of the interconnection. An important advantage of this formulation lies in its convenient form allowing for the derivation of efficient parallel algorithms for establishing the Hurwitz stability of Metzler block-matrices. The methods described in [49, 50] can for instance be used. 4.2. Main results Before stating the main result, it is convenient to define their Minkowski product as the set Q(A1 )Q(A2 ) := {M1 M2 : M1 ∈ Q(A1 ), M2 ∈ Q(A2 )} 11
(15)
n1 ×m 2 where A1 ∈ S≥0 and A2 ∈ Sm×n . It is also important to define the three following product rules for ≥0
dealing with products of nonnegative sign-matrices: 1) ⊕ + ⊕ = ⊕; 2) ⊕ · ⊕ = ⊕; 3) 0 · ⊕ = 0. Remark 4.4. Note that because of these product rules, we have that Q(A1 )Q(A2 ) ⊂ Q(A1 A2 ) and that, in general, Q(A1 )Q(A2 ) 6= Q(A1 A2 ). For instance, if h i ⊕ A1 = and A2 = ⊕ ⊕ , ⊕ then Q(A1 )Q(A2 ) is the set of all 2 × 2 rank-1 positive matrices. However, since ⊕ A1 A2 = ⊕
⊕ , ⊕
then Q(A1 A2 ) is the set of all 2 × 2 positive matrices. On the other hand, if h ⊕ A1 = and A2 = ⊕ ⊕
i 0 ,
then we have that Q(A1 )Q(A2 ) = Q(A1 A2 ). A similar statement can be made for unit-sign matrices. We indeed have, in general, that sgn(A1 A2 ) 6= sgn(A1 ) sgn(A2 ). For instance, this is the case for the matrices ⊕ A1 = 0 for which we have that
⊕ ⊕ and A2 = , ⊕ ⊕
1 2 sgn(A1 A2 ) = and sgn(A1 ) sgn(A2 ) = . 1 1
The above remark emphasizes the difficulty of the possible loss of independence between the entries of a matrix resulting from the multiplication of two sign-matrices. With this warning in mind, we can state the following result: n ×nij
i Theorem 4.5. Let Ai ∈ MSni and Bij ∈ S≥0
n ×nj
ij , Cij ∈ S≥0
. Assume, moreover, that Q(Bij Cij ) =
Q(Bij )Q(Cij ) for all i, j = 1, . . . , N , i 6= j. Then, the following statements are equivalent: (a) The Metzler sign-matrix
is sign-stable.
¯ A=
A1 .. .
B12 C12
BN 1 CN 1
BN 2 CN 2
12
. . . B1N C1N .. . ... AN
(16)
¯ is Hurwitz stable. (b) The matrix sgn(A) n
j i (c) There exist vectors vi ∈ Rn>0 , ℓij ∈ R>0 , i, j = 1, . . . , N , j 6= i, such that the inequalities
viT
N X
ℓTji
< 0
viT sgn(Bij Cij ) − ℓTij
< 0
sgn(Ai ) +
(17)
j6=i
for all i, j = 1, . . . , N , j 6= i. (d) The diagonal elements of A¯ are negative and the directed graph DA¯ is acyclic. Proof : The proof of the equivalence between statements (a), (b) and (d) follows from Theorem 3.5 and the fact that Q(Bij Cij ) = Q(Bij )Q(Cij ) for all i, j = 1, . . . , N , i 6= j; i.e. independence of the entries. The equivalence with the statement (c) follows from Theorem 4.5 and the fact that the entries in the matrix products are independent. Note, however, that, by virtue of Remark 4.4, the terms Bij Cij cannot be split up since sgn(Bij Cij ) 6= sgn(Bij ) sgn(Cij ), in general.
♦
In the case where Q(Bij Cij ) 6= Q(Bij )Q(Cij ) for some i, j, then the statements (b) and (c) are sufficient for (a) to hold, but not necessary. However, when sgn(Bij Cij ) = sgn(Bij ) sgn(Cij ), we have the following result: n ×nij
i Theorem 4.6. Let Ai ∈ MSni and Bij ∈ S≥0
n ×nj
ij , Cij ∈ S≥0
. Assume, moreover, that Q(Bij Cij ) =
Q(Bij )Q(Cij ) and sgn(Bij Cij ) = sgn(Bij ) sgn(Cij ) for all i, j = 1, . . . , N , i 6= j. Then, the following statements are equivalent: 1. The matrix A¯ in (16) is sign-stable. n
ij i 2. There exist vectors vi ∈ Rn>0 , ℓij ∈ R>0 , i, j = 1, . . . , N , j 6= i, such that the inequalities
viT sgn(Ai ) +
N X
ℓTji sgn(Cji )
0 such that v T A′i < 0 for all i = 1, . . . , N , implying, in turn, that statement (c) holds.
16
♦
6. Relative sign-stability of Metzler sign-matrices 6.1. Preliminaries In these section, we will work with a larger class of sign-matrices, referred to as indefinite sign-matrices, taking entries in S⊙ := S ∪ {⊙} where ⊙ denotes a sign indefinite entry that can be obtained from the rule ⊕ + ⊖ = ⊙. To avoid confusion, sign-matrices will also be referred to as definite sign-matrices. The n×m pre-qualitative class associated with a matrix A ∈ S⊙ is defined as
PQ(A) := M ∈ Sn×m
∈ S [M ]ij = [A]
ij
if [A]ij = ⊙ . otherwise
It is important to stress that, due to the presence of sign-indefinite entries ⊙, all the matrices in PQ(A) do not have the same sign-pattern. When there is no sign-indefinite entry in A, then PQ(A) = A. We finally introduce the following set Q(PQ(A)) :=
[
Q(A′ )
(24)
A′ ∈PQ(A)
that defines the set of all real matrices sharing the same sign-pattern of at least one matrix in the prequalitative class of the matrix A. Let us define now the concept of relative sign-stability: Definition 6.1. Let A ∈ Sn×n and B ∈ Rn×ℓ , ℓ < n, full-rank. We say that A is sign-stable relative to B or that the pair (A, B) is relatively sign-stable if for all A′ ∈ Q(A), there exists a v ∈ Rn>0 such that v T A′ < 0 and v T B = 0. The main reason behind the name is that the additional constraint v T B = 0 induces a restriction to the stability condition v T A < 0, a restriction that is relative to the choice for B. Such a problem arises, for instance, in the analysis of certain stochastic jump processes representing biological processes [4, 6]. When B = 0, the usual notion of sign-stability is retrieved. We then have the following result: Theorem 6.2. Assume that the sign-matrix A ∈ MSn is sign-stable. Then, we have the following statements: (a) The inverse sign-matrix A−1 defined as • [A−1 ]ii 6= 0 for all i = 1, . . . , n; • [A−1 ]ij 6= 0 if and only if there is a path of node j to node i in the directed graph DA . is unique. (b) A−1 is nonnegative. (c) We have that Q(A)−1 := M −1 : M ∈ Q(A) ⊂ Q(A−1 ). 17
Proof : To prove statement (a), we first assume, without loss of generality, that A is in upper-triangular form with nonnegative upper-triangular elements and negative diagonal elements. Then, let us consider the linear equation y = M x where M =−
n X
mi,i ei eTi +
n X
mi,j ei eTj
(25)
i<j
i
with mi,i > 0, mi,j ≥ 0 for i < j and where {ei }ni=1 is the standard basis of Rn . Then, by inverting recursively this system, we can see that the entry [M −1 ]ij consists of a negatively weighed sum of all possible paths from j to i. Consequently, the entry is 0 if and only if there is no path from j to i in DA , otherwise it is negative. Statement (b) readily follows from Lemma 3.1. We prove finally statement (c). We know from statements (a) and (b) that for a Metzler sign-stable sign-matrix A, the matrix inversion operation is well-defined in the sense that A−1 is a uniquely defined sign-matrix, which is nonnegative with negative diagonal elements. Therefore the inclusion in statement (c) follows from the fact that all the inverses of the matrices in Q(A) have the same sign-pattern as A−1 . Equality in statement (c) does not hold because the inverse of a Hurwitz stable real nonnegative matrix with negative diagonal is not necessarily Metzler, even in the upper-triangular case. In other words, the matrix inversion operation on the set of sign-stable upper-triangular nonnegative sign-matrices is not well-defined in the sense that the sign-pattern is non-uniquely defined. This is shown below by constructing a matrix M ∈ Q(A−1 ) such that M ∈ / Q(A)−1 where A is a sign-stable Metzler sign-matrix. For instance, −1 −1 −2 −4 −1 1 1 −2 0 −1 −1 −5 0 −1 1 4 −1 −1 M = / Q(A) ∈ Q(A ), but M = ∈ 0 0 −1 −1 0 0 −1 1 0 0 0 −1 0 0 0 −1 is not a Metzler matrix.
(26)
♦
Two important facts are stated by the above result. First of all, the sign-pattern of the inverse signmatrix A−1 is uniquely defined by the sign-pattern of A provided that A is sign-stable. Secondly, that for any sign-stable matrix A ∈ Sn×n , the nonzero entries of the inverse matrix A−1 can simply be deduced from the examination of the graph DA . For comparison, it requires at most n2 operations, which is better than usual real matrix inversion algorithms; see e.g. [53]. A drawback, however, is that by considering the matrix A−1 , we lose some information since the class of matrices that is considered is bigger. We are now ready to state the main result of this section: Proposition 6.3. Let A ∈ MSn×n be sign-stable and B ∈ Rℓ×n , ℓ < n, be full-rank. Assume that one of the following equivalent statements holds:
18
(a) All the matrices in PQ((A−1 B)T ) are L+ -matrices; i.e. for all M ∈ Q(PQ((A−1 B)T )), we have that y T M ≥ 0 with y ≥ 0 implies that y = 0. (b) All the matrices in PQ((A−1 B)T ) have no zero row and for all M ∈ Q(PQ((A−1 B)T )), there exists a v ∈ Rℓ>0 such that M v = 0. (c) For all nonzero diagonal matrices D ∈ {−1, 0, 1}n×n and all M ∈ PQ((A−1 B)T ), some column of DM is nonzero and nonnegative. (d) For all nonzero diagonal matrices D ∈ {−1, 0, 1}n×n and all M ∈ PQ((A−1 B)T ), some column of DM is nonzero and nonpositive. (e) For all M ∈ Q(PQ((A−1 B)T )), we have {M x : x ≥ 0} = Rn . Then, the pair (A, B) is relatively sign-stable. Proof : The pair (A, B) is relatively sign-stable if and only if for any A′ ∈ Q(A), there exists a v ∈ Rn>0 such that v T A′ < 0 and v T B = 0. This is equivalent to saying that for any A′ ∈ Q(A), there exist some v, q ∈ Rn>0 such that v T A′ = −q T and v T B = 0. Since, the matrix A is sign-stable, then it is invertible and the sign-pattern of A−1 is uniquely defined (see Theorem 6.2). Hence, the pair (A, B) is relatively sign-stable if for any M ∈ Q(PQ((A−1 B)T )), there exists a q ∈ Rn>0 such that M q = 0. Note that, in general, necessity is lost since the set Q(PQ(A−1 B)) contains the set Q(A)−1 B (see Theorem 6.2). Using now Theorem 2.4 of [44], we have that a sufficient condition for the relative sign-stability of the pair (A, B) is that all the matrices in PQ((A−1 B)T ) be L+ -matrices, which means that all these matrices have no zero row and that all the matrices in Q(PQ((A−1 B)T )) have a positive vector in their null-space. The latter statement is exactly statement (b). The equivalence between all the other statements also follows from Theorem 2.4 of [44]. The proof is complete.
♦
Based on the statements (c) and (d), we can clearly see that checking whether a matrix is an L+ -matrix is a combinatorial problem. This problem turns out to be NP-complete (see [32, 44]) but remains applicable for matrices having small dimensions. 7. Partial sign-stability As we have seen in the previous sections, sign-stability requires strong structural properties for the considered matrices. The idea here is to weaken the notion of sign-stability in order to enlarge the set of matrices that can be considered. To this aim, we introduce the concept of partial sign-stability which applies to matrices with mixed entries that contain both sign entries and real entries. In this case, cycles can be allowed for the nodes associated with real entries. We notably prove here that the partial sign-stability of a class of mixed matrices can be exactly characterized in terms of Hurwitz stability conditions, sign-stability conditions and the non-existence of cycles in a particular bipartite graph obtained from the mixed matrix. 19
7.1. Preliminaries Definition 7.1. A matrix M ∈ (R ∪ S)n×n is partially sign-stable if all the matrices M ′ ∈ Q(M ) are Hurwitz stable. Lemma 7.2. Let us consider a matrix M ∈ {0, 1}n×n. Then, the following statements are equivalent: (a) M is Schur stable; (b) M − I is Hurwitz stable; (c) The directed graph DM is acyclic and the diagonal elements of M are equal to 0. Proof : The proof between the two first statements follows from the fact that the nonnegative matrix M is Schur stable if and only if there exists a positive vector v such that v T (M − I) < 0, which is equivalent to the Hurwitz stability of the Metzler matrix M − I. We know from Lemma 3.4 that M − I is Hurwitz stable if and only if there is no cycle in the graph DM−I (or equivalently DM ) and the diagonal elements of M − I are negative. This turns out the be equivalent to the last statement.
♦
7.2. Main results In what follows, we consider mixed matrices of the form
Aσϕ = n ×nσ
ϕ where Aσ ∈ MSnσ , Aϕ ∈ MRnϕ , Bϕ ∈ R≥0
Aσ
Cϕ
Bϕ
Aϕ
n ×nϕ
σ and Cϕ ∈ R≥0
(27)
. We then have the following result:
Theorem 7.3. The following statements are equivalent: (1) The matrix Aσϕ defined in (27) is partially sign-stable. (2) The following statements hold: (a) Aσ is sign-stable, (b) Aϕ is Hurwitz stable, (c) There is no cycle in the directed graph DAσϕ containing nodes in both Vσ = {v1σ , . . . , vnσσ } and Vϕ = {v1ϕ , . . . , vnϕϕ } where Vσ contains the first nσ nodes associated with DAσϕ and Vϕ the nϕ last ones. (3) The following statements hold: (a) Aσ is sign-stable, (b) Aϕ is Hurwitz stable,
20
(c) There is no cycle in the directed bipartite graph B = (Vσ , Vϕ , E) where E = Eσ ∪ Eϕ with Eσ := {viϕ , vjσ ) ∈ Vϕ × Vσ : [Mσ ]ji 6= 0} and Eϕ := {(viσ , vjϕ ) ∈ Vσ × Vϕ : [Mϕ ]ji 6= 0}
(28)
where Mϕ and Mσ are nonnegative matrices defined as −1 Mϕ := −Cϕ A−1 ϕ Bϕ and Mσ := − sgn(Aσ ).
(29)
(4) The following statements hold: (a) sgn(Aσ ) is Hurwitz stable, (b) Aϕ is Hurwitz stable, (c) The matrix sgn(Mϕ Mσ ) − I is Hurwitz stable or, equivalently, sgn(Mϕ Mσ ) is Schur stable. Proof : Proof (1) ⇔ (2). Clearly, the sign-stability of Aσ and the Hurwitz stability of Aϕ are necessary conditions for the partial sign-stability of the matrix Aσϕ . Partial sign-stability then holds if and only if there is no cycle involving nodes in Vσ and Vϕ (see Theorem 3.5). Proof (2) ⇔ (3). The equivalence between (2) and (3) follows from the fact that the no-cycle condition in DAσϕ of Statement (2c) is equivalent to the no-cycle condition of Statement (3c). Proof (3) ⇔ (4).
The no-cycle condition is equivalent to the fact that the nonnegative matrix
sgn(Mϕ Mσ ) does not have any positive diagonal entry (2-cycle in the bipartite graph) and does not have any other cycles of higher-order (so no cycle in the graph DMϕ Mσ ). Combining these conditions with Lemma 7.2 yields the result.
♦
7.3. Example Let us consider the matrix
Aσϕ
⊖ ⊕ 0 ⊖ = Bϕ
1
0 0 0 . −1 2 1 −5
Clearly, Aσ is sign-stable and Aϕ is Hurwitz stable. When 0 0 , Bϕ = 0 1
(30)
(31)
the directed graph DAσϕ does not have any cycles with nodes in both Vσ and Vϕ . Alternatively, the matrices Mσ and Mϕ are given by
0 1 1 1 2 and Mϕ = Mσ = 3 0 0 0 1 21
(32)
and we have that
sgn(Mϕ Mσ ) =
0 1 0 0
(33)
which is clearly Schur stable. Therefore, the matrix Aσϕ in (30) is partially-stable. Now, if 0 Bϕ = 1
0 0 0 0 1 and sgn(Mϕ Mσ ) = , then Mϕ = 3 1 0 0 0
0 1
which is not Schur stable. Hence, the matrix Aσϕ in (30) is not partially-stable.
8. Applications 8.1. Linear positive dynamical systems 8.1.1. Systems with delays Let us consider the following linear positive system with constant delay [54, 55]: x(t) ˙
=
A0 x(t) +
N X
Ai x(t − hi )
i=1
x(s)
=
(34)
¯ 0] φ(s), s ∈ [−h,
¯ 0], Rn ), h ¯ = max{hi }, hi > 0, i = 1, . . . , N . where x ∈ Rn , φ ∈ C([−h, i
Lemma 8.1 ([56]). The linear system with delays (34) is positive if and only if the matrix A0 is Metzler and the matrices Ai , i = 1, . . . , N , are nonnegative. n×n Lemma 8.2 ([56]). Let the matrices A0 ∈ MRn and Ai ∈ R≥0 , i = 1, . . . , N , be given. Then, the following
statements are equivalent (a) The linear positive system with delay (34) is asymptotically stable. P (b) The matrix N i=0 Ai is Hurwitz stable.
Proposition 8.3. Let M0 ∈ MSn and Mi ∈ Sn≥0 , i = 1, . . . , N . Then, the following statements are equiva-
lent: (a) The matrix
PN
i=0
Mi is a definite sign-matrix (i.e.
PN
i=0
Mi ∈ Sn×n ) with negative diagonal entries.
(b) The diagonal elements of M0 are negative and the diagonal elements of Mi , i = 1, . . . , N , are equal to 0. We then have the following result: ¯ := PN Mi . Then, Proposition 8.4. Let M0 ∈ MSn and Mi ∈ Sn≥0 , i = 1, . . . , N . Define, moreover, M i=0 the following statements are equivalent:
22
(a) For all A0 ∈ Q(M0 ) and all Ai ∈ Q(Mi ), i = 1, . . . , N , the linear positive system with delays (34) is asymptotically stable; i.e. the system (34) is structurally stable. (b) The following statements hold: ¯ is a definite sign-matrix; i.e. M ¯ ∈ Sn×n , • M ¯ is sign-stable. • The matrix M (c) The following statements hold: ¯ is a definite sign-matrix; i.e. M ¯ ∈ Sn×n , • M ¯ < 0 holds. • There exists a vector v ∈ Rn>0 such that v T sgn M
Proof : The proof follows from Lemma 8.2, Theorem 3.5 and Proposition 8.3.
♦
8.1.2. Switched systems Let us consider the following positive switched linear system [57]: x(t) ˙
= Aσ(t) x(t)
x(0)
= x0
(35)
where x ∈ Rn and σ : R≥0 → {1, . . . , N } is the piecewise constant switching signal. Lemma 8.5. The linear switched system is positive if and only if Ai ∈ MRn for all i = 1, . . . , N . We then have the following result: Proposition 8.6. Let Mi ∈ MSn , i = 1, . . . , N . Then, the following statements are equivalent: (a) For all Ai ∈ Q(Mi ), i = 1, . . . , N , the linear switched system (35) is asymptotically stable under arbitrary switching i.e. the system (35) is structurally stable under arbitrary switching. (b) For all Ai ∈ Q(Mi ), i = 1, . . . , N , the linear time-varying positive system x(t) ˙ = A(t)x(t)
(36)
where A(t) ∈ co(A1 , . . . , AN ), is robustly asymptotically stable. (c) The diagonal elements of the matrices Mi , i = 1, . . . , N , are negative and the matrix
N X
Mi is sign-stable.
i=1
(d) For all Ai ∈ Q(Mi ), i = 1, . . . , N , there exists a positive definite matrix Q ∈ Rn×n such that A˜T Q + QA˜ is negative definite for all A˜ ∈ co(A1 , . . . , AN ). (e) For all Ai ∈ Q(Mi ), i = 1, . . . , N , there exists a vector v ∈ Rn>0 such that v T A˜ < 0 for all A˜ ∈ co(A1 , . . . , AN ). 23
Proof : The proof of the equivalence between the statements (1) and (2) follows from [58, Theorem 3] while the equivalence between the statements (3), (4) and (5) follows from Theorem 5.3 and Theorem 5.5. Clearly, statement (4) implies statement (2) using standard results on switched systems; see e.g. [57]. Finally, statement (2) implies (3) since if the conditions of statement (3) does not hold, then there will exist some matrices Ai ∈ Q(Mi ), i = 1, . . . , N , for which the set co(A1 , . . . , AN ) will contain unstable matrices. The proof is complete.
♦
It is interesting to note that the above result also applies to polytopic LPV systems of the form [46, 59]: ! N X αi (t)Ai x(t) x(t) ˙ = (37) i=1 x(0) =
where α : R≥0 →
(
β∈
RN ≥0
:
N X i=1
βi = 1
)
x0
is any sufficiently regular function.
8.2. Nonlinear positive systems Let us consider the following class of nonlinear positive systems that arises, among others, in reaction network theory x(t) ˙
=
x(0) =
Ax(t) + Bf (x(t)) + b
(38)
x0
where x ∈ Rn≥0 , f : Rn≥0 → Rℓ≥0 , b ∈ Rn≥0 , A ∈ Rn×n and B ∈ Zn×ℓ . We then have the following result: Proposition 8.7. Let A ∈ MRn and assume that there exists a v ∈ Rn>0 and an ε > 0 such that v T A ≤ −εv T and v T B = 0. Then, the system (38) is stable and, for all x0 ∈ / S, the trajectories of the system (38) converge to the compact set S := {x ∈ Rn≥0 : V (x) ≤ v T b/ε},
(39)
i.e. S is forward-invariant and attractive. Proof : Let us consider the function V (x) = v T x which is positive in Rn≥0 \{0}. The derivative of this function along the solutions of (38) is given by V˙ (x) = v T Ax+v T Bf (x)+v T b. Assuming the conditions of the theorem vT b hold, then we have that V˙ (x) ≤ −εV (x) + v T b and, therefore, that V (x(t)) ≤ e−εt V (x0 ) + (1 − e−εt ) . ε Hence, for all x0 ∈ / S, the trajectories of the system (38) converge to the compact set. The proof is complete. ♦ We can now derive the following structural version of the above result: Proposition 8.8. Let M ∈ MSn and assume, for simplicity, that Z := (A−1 B)T ∈ Sℓ×n . Suppose further that 24
(a) M is sign-stable; (b) Z is an L+ -matrix. Then, for all A ∈ Q(M ), there exists a v ∈ Rn>0 and an ε > 0 such that v T A ≤ −εv T and v T B = 0. Hence, the system (38) is structurally stable and its trajectories structurally converge to a compact set inside the positive orthant. 8.3. Ergodicity of stochastic reaction networks In [4], it is shown that a sufficient condition for the ergodicity of an irreducible stochastic reaction network involving bimolecular reactions with mass-action kinetics is that the drift condition xT M (v)x + v T Ax + v T b ≤ c1 − c2 v T x
(40)
holds for all x ∈ Nn0 and for some c1 , c2 > 0 where M (v) ∈ Rn×n is symmetric, A ∈ MRn and b ∈ Rn≥0 . Above, the matrices M (v), A and b are data of the problem which can be explicitly constructed from the network reactions. The following result, proved in [4], has been shown to be applicable to a broad class of reaction networks: Proposition 8.9. Let us consider an irreducible1 stochastic reaction network whose drift condition writes as (40). Assume further that there exists a v ∈ Rn>0 such that M (v) = 0 and v T A < 0. Then, the drift condition (40) holds and the reaction network is exponentially ergodic. We then have the following structural version: Proposition 8.10. Let B ∈ Zn×ℓ be such that v T B = 0 ⇒ M (v) = 0, let Z ∈ MSn be a Metzler sign-matrix and assume, for simplicity, that W := (Z −1 B)T ∈ Sℓ×n . Suppose, further, that (a) Z is sign-stable; (b) W is an L+ -matrix. Then, for all A ∈ Q(Z), there exists a v ∈ Rn>0 such that v T A < 0 and v T B = 0. Then the underlying Markov process describing the network is structurally exponentially ergodic.
Bibliography References [1] A. Berman and R. J. Plemmons. Nonnegative matrices in the mathematical sciences. SIAM, Philadelphia, USA, 1994. 1
Irreducibility of a reaction network can be checked using the method developed in [60].
25
[2] H. Nikaido. Convex structures and economic theory. Academic Press, New York, USA, 1968. [3] C. Briat and M. Khammash. Computer control of gene expression: Robust setpoint tracking of protein mean and variance using integral feedback. In 51st IEEE Conference on Decision and Control, pages 3582–3588, Maui, Hawaii, USA, 2012. [4] A. Gupta, C. Briat, and M. Khammash. A scalable computational framework for establishing long-term behavior of stochastic reaction networks. PLOS Computational Biology, 10(6):e1003669, 2014. [5] F. Parise, M. E. Valcher, and J. Lygeros. On the reachable set of the controlled gene expression system. In 53rd IEEE Conference on Decision and Control, pages 4597–4604, Los Angeles, USA, 2014. [6] C. Briat, A. Gupta, and M. Khammash. Antithetic integral feedback: A new motif for robust perfect adaptation in noisy biomolecular networks. to appear in Cell Systems, 2016. [7] H. L. Smith. Monotone Dynamical Systems - An Introduction to the Theory of Competitive and Cooperative Systems. American Mathematical Society, Providence, Rhode Island, 1995. [8] D. Angeli and E. D. Sontag. Monotone control systems. IEEE Transactions on Automatic Control, 48(10):1684–1698, 2003. [9] T. M. Liggett. Continuous-time Markov processes. An introduction, volume 112 of Graduate Studies in Mathematics. American Mathematical Society, 2010. [10] L. Farina and S. Rinaldi. Positive Linear Systems: Theory and Applications. John Wiley & Sons, 2000. [11] M. Ait Rami and F. Tadeo. Controller synthesis for positive linear systems with bounded controls. IEEE Transactions on Circuits and Systems – II. Express Briefs, 54(2):151–155, 2007. [12] W. M. Haddad, V. Chellaboina, and Q. Hui. Nonnegative and compartmental dynamical systems. Princeton University Press, 41 William Street, Princeton, New Jersey, USA, 2010. [13] C. Briat. Robust stability and stabilization of uncertain linear positive systems via integral linear constraints - L1 - and L∞ -gains characterizations. International Journal of Robust and Nonlinear Control, 23(17):1932–1954, 2013. [14] F. Mazenc and O. Bernard. Interval observers for linear time-invariant systems with disturbances. Automatica, 47:140–147, 2011. [15] Y. Ebihara, D. Peaucelle, and D. Arzelier. L1 gain analysis of linear positive systems and its applications. In 50th Conference on Decision and Control, Orlando, Florida, USA, pages 4029–4034, 2011.
26
[16] A. Rantzer. Distributed control of positive systems. In 50th IEEE Conference on Decision and Control, pages 6608–6611, 2011. [17] T. Tanaka and C. Langbort. Symmetric formulation of the S-Procedure, Kalman-Yakubovich-Popov Lemma and their exact losslessness conditions. IEEE Transactions on Automatic Control, 58(6):1486– 1496, 2013. [18] T. Tanaka, C. Langbort, and V. Ugrinovskii. DC-dominant property of cone-preserving transfer functions. Systems & Control Letters, 62(8):699–707, 2013. [19] M. Colombino and R. S. Smith. Convex characterization of robust stability analysis and control synthesis for positive linear systems. In 53rd IEEE Conference on Decision and Control, pages 4379–4384, Los Angeles, USA, 2014. [20] C. Briat. L∞ -to-L∞ interval observers for continuous-time systems with persistent inputs. 2015. URL http://arxiv.org/abs/1511.07888. [21] J. Quirk and R. Ruppert. Qualitative economics and the stability of equilibrium. The review of economic studies, 32(4):311–326, 1965. [22] R. M. May. Will a large complex system be stable? Nature, 238:413–414, 1972. [23] R. M. May. Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton, New Jersey, 1973. [24] A. Roberts. The stability of a feasible random ecosystem. Nature, 251:607–608, 1974. [25] C. Jeffries. Qualitative stability and digraphs in model ecosystems. Ecology, 55:1415–1419, 1974. [26] J. J. Tyson. Classification of instabilities in chemical reaction systems. The Journal of Chemical Physics, 62:1010–1015, 1974. [27] B. L. Clarke. Theorems on chemical network stability. The Journal of Chemical Physics, 62:773–775, 1975. [28] J. Maybee and J. Quirk. Qualitative problems in matrix theory. SIAM Reviews, 11(1):30–51, 1969. [29] C. Jeffries, V. Klee, and P. van den Driessche. When is a matrix sign stable? Canadian Journal of Mathematics, 29:315–326, 1977. [30] V. Klee and P. van den Driessche. Linear algorithms for testing the sign stability of a matrix and for finding z-maximum matchings in acyclic graphs. Numerische Mathematik, 28(3):273–285, 1977.
27
[31] C.-T. Lin. Structural controllability. IEEE Transactions on Automatic Control, 19:201–208, 1974. [32] V. Klee, R. Ladner, and R. Manber. Signsolvability revisited. Linear Algebra and Its Applications, 59: 131–157, 1984. [33] J. L. Willems. Structural controllability and observability. Systems & Control Letters, 8:5–12, 1986. [34] M. A. Berger and A. Felzenbaum. Sign patterns of matrices and their inverses. Linear Algebra and its Applications, 86:161–177, 1987. [35] C. Thomassen. When the sign pattern of a square matrix determines uniquely the sign patter of its inverse. Linear Algebra and its Applications, 119:27–34, 1989. [36] R. A. Brualdi and B. L. Shader. Matrices of sign-solvable linear systems. Cambridge University Press, Cambridge, UK, 1995. [37] E. Kaszkurewicz and A. Bhaya. Matrix diagonal stability in systems and computation. Springer Science+Business, New York, USA, 2000. [38] P. Santsesso and M. E. Valcher. On the zero pattern properties and asymptotic behavior of continuoustime positive system trajectories. Linear Algebra and its Applications, 425:283–302, 2007. [39] M. Catral, D. D. Olesky, and P. van den Driessche. Allow problems concerning spectral properties of sign pattern matrices: A survey. Linear Algebra and its Applications, 430(11-12):3080–3094, 2009. [40] C. Mendes Ara´ ujo and J. R. Torregrosa. Sign pattern matrices that admit M -, N -, P - or inverse M -matrices. Linear Algebra and its Applications, 431(5–7):724–731, 2009. [41] O. Pryporova. Qualitative convergence of matrices. Linear Algebra and its Applications, 431(1):28–38, 2009. [42] S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press, Cambridge, MA, USA, 2004. [43] D. Knuth. The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 3rd Edition. AddisonWesley, Boston, USA, 1997. [44] G.-Y. Lee and B. L. Shader. Sign-consistency and solvability of constrained linear systems. Electronic Journal of Linear Algebra, 4:1, 1998. [45] K. Zhou, J. C. Doyle, and K. Glover. Robust and Optimal Control. Prentice Hall, Upper Saddle River, New Jersey, USA, 1996.
28
[46] C. Briat. Linear Parameter-Varying and Time-Delay Systems – Analysis, Observation, Filtering & Control. Advances on Delays and Dynamics, Vol. 3. Springer-Verlag, Heidelberg, Germany, 2015. [47] V. A. Yakubovich. S-procedure in nonlinear control theory. Vestnik Leningrad Univ., pages 62–77, 1971. English translation in Vestnik Leningrad Univ. Math., vol. 4, pp. 73–93, 1977. [48] U. T. J¨onsson.
A lecture on the S-procedure.
Technical report, Division of Optimiza-
tion and Systems Theory, Royal Institute of Technology, Stockholm, Sweden, 2001.
URL
http://www.math.kth.se/ulfj/5B5746/Lecture.ps. [49] D. P. Bertsekas and J. N. Tsitsiklis. Parallel and Distributed Computation. Prentice Hall, 1989. [50] S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein. Distributed optimization and statistical learning via the alternating direction method of multipliers. Foundations and Trends in Machine Learning, 3(1): 1–122, 2011. [51] E. Fornasini and M. E. Valcher. Linear copositive Lyapunov functions for continuous-time positive switched systems. IEEE Transactions on Automatic Control, 55(8):1933–1937, 2010. [52] D. Liberzon, J. P. Hespanha, and A. S. Morse. Stability of switched systems: a Lie-algebraic condition. System and Control Letters, 37:117–122, 1999. [53] F. Le Gall. Powers of tensors and fast matrix multiplication. In 39th International Symposium on Symbolic and Algebraic Computation, pages 296–303, Kobe, Japan, 2014. [54] S. I. Niculescu. Delay effects on stability. A robust control approach, volume 269. Springer-Verlag: Heidelbeg, 2001. [55] K. Gu, V. L. Kharitonov, and J. Chen. Stability of Time-Delay Systems. Birkh¨auser, Boston, 2003. [56] W. M. Haddad and V. Chellaboina. Stability theory for nonnegative and compartmental dynamical systems with time delay. Systems & Control Letters, 51(5):355–361, 2004. [57] D. Liberzon. Switching in Systems and Control. Birkh¨auser, New York, 2003. [58] H. Lin and P. J. Antsaklis. Stability and stabilizability of switched linear systems: A survey of recent results. IEEE Transactions on Automatic Control, 54(2):308–322, 2009. [59] P. Apkarian, P. Gahinet, and G. Becker. Self-scheduled control of linear parameter varying systems: A design example. Automatica, 31(9):1251–1261, 1995. [60] A. Gupta and M. Khammash. Determining the long-term behavior of cell populations: A new procedure for detecting ergodicity in large stochastic reaction networks. Technical Report arXiv:1312.2879, ETHZ¨ urich, 2013. 29
