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SIGN CONTROLLABILITY: SIGN PATTERNS THAT REQUIRE COMPLETE CONTROLLABILITY MICHAEL J. TSATSOMEROS SIAM J. M ATRIX NAL. PPL 1998 Society for Industrial and Applied Mathematics Vol. 19, No. 2, pp. 355–364, April 1998 004 Abstract. We apply tools from the theory of sign-solvable systems and use the directed graph of a matrix in order to obtain suﬃcient conditions for a linear control system ( A,B ) to be completely controllable solely due to the sign patterns of the coeﬃcient matrices and . We show that such conditions are necessary and suﬃcient for a particular class of linear control systems. We also consider an alternative approach to controllability, based on a reformulation of the classical condition (that the controllability matrix is of full rank) and obtain equivalent conditions for the general case. Key words. control system, sign pattern, signing, L-matrix, directed graph, aligned vertices, balancing chain AMS subject classiﬁcations. 15A99, 93B05 PII. S0895479896300346 1. Introduction. In linear control theory, the basic concepts of controllability (and observability) are intimately related to the image of a matrix of the form =[ B AB ... A n, nm where n, n and n, m . Speciﬁcally, a control system of the form dt )= Ax )+ Bu is completely controllable if and only if rank . As the matrices A,B comprise system parameters prone to measurement errors, it is desirable to determine whether rank based on combinatorial and qualitative information about and (e.g., their directed graphs and the signs of their entries). Such qualitative approaches to controllability have been undertaken, for example, by Lin [6], Mayeda and Yamada [9], Murota [8], Johnson, Mehrmann, and Olesky [4], and Olesky, Tsatsomeros, and van den Driessche [10]. In the present work, we shall consider the following: assume that the sign patterns (namely, the location of the positive, negative, and zero entries) of and are known. When can we conclude that rank , based solely on the sign patterns and regardless of the magnitudes of the nonzero entries of and The study of this question was initiated in [4], where was assumed to have nonnegative entries and was assumed to be a column vector with positive entries. Our qualitative approach to controllability will be based on extending and com- bining techniques used in the study of zero/nonzero patterns that allow or require complete controllability (see [6, 9, 10]) with notions related to the analysis of sign patterns and sign-solvable linear systems. We will ﬁnd suﬃcient conditions for com- plete controllability for general sign patterns (Theorem 3.2), and we will identify a Received by the editors March 11, 1996; accepted for publication (in revised form) by V. Mehrmann February 6, 1997. This work was supported by a grant from the Natural Sciences and Engineering Research Council of Canada. http://www.siam.org/journals/simax/19-2/30034.html Department of Mathematics and Statistics, University of Regina, Regina, Saskatchewan S4S 0A2, Canada (tsat@math.uregina.ca). 355

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356 MICHAEL J. TSATSOMEROS class of linear control systems (Deﬁnition 2.2) for which these conditions are neces- sary and suﬃcient (Theorem 3.7). In section 4, we will consider a simple technical recasting of the classical controllability condition that rank in order to provide an alternative answer to the question posed above. 2. Preliminaries. In this section, we present some of the notation, terminology, and basic facts necessary to state and prove the main results in the following sections. In the remainder we let ,...,k for any positive integer denote the cardinality of a set ; sgn( )be0,1,or 1, when is zero, positive, or negative, respectively; Re( ) denote the real part of a complex vector ; diag( ,A ,...,A )be a block diagonal matrix whose diagonal blocks are the square matrices ,A ,...,A ] denote the submatrix of s, t whose rows and columns are indexed by the sets and , respectively; ]= ]; and denote an all 1s column vector of appropriate size. We let Γ = ( V,E ) denote a directed graph with vertex set and directed edge set consisting of ordered pairs ( i,j ) of vertices. A path from to in Γ is a sequence of vertices ,r ,...,r , with ( ,r +1 for =1 ,...,t 1. The directed graph of =( ij s, t ,s , denoted by Γ = )=( V,E ), has and i,j ij =0 . Extending the terminology in [6] and [9], when s we say that )is accessible if for every there exists 〉\ such that there is a path from to in Γ. Also, for every we denote the adjacency set of by )= 〉| i,j for some Notice that if there exists an such that ⊆R ) and if Γ is accessible, then 〉\ and hence ⊂R ). In keeping with the notation and terminology of Brualdi and Shader [1] (which is our comprehensive reference on sign patterns), we deﬁne the following. The sign pattern of s, t is the (0 1)-matrix obtained from when zero, positive, and negative entries are replaced by 0, 1, and 1, respectively. The matrix determines the qualitative class ) of all matrices with the same sign pattern as . We will write ) for any matrix having the same sign pattern as signing is a nonzero square diagonal sign pattern. A real vector is called balanced if it is the zero vector, or if it has at least one negative and at least one positive entry. A real vector is referred to as unisigned if it is not balanced. If a unisigned vector has nonnegative (respectively, nonpositive) entries, we refer to it as of positive ( respectively, negative) type . We denote the signings such that all the columns of SX are balanced by ). The matrix is called an L-matrix provided that every matrix in ) has linearly independent rows. It is well known (see [1, Theorem 2.1.1]) that s, t is an L-matrix if and only if )= . Next we introduce the notion of aligned vertices in the directed graph of a matrix. EFINITION 2.1. Let s, t , and let , be two nonempty and disjoint sets. We call aligned relative to if there exists a signing ∈B ]) such that the unisigned columns of SX (if any exist) are only of one type (either only positive type or only negative type). When ]) = is by deﬁnition not aligned relative to In other words, is aligned relative to a disjoint set if there exists a signing of the rows of | ] such that the columns of ] become

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SIGN CONTROLLABILITY 357 balanced and the columns of ] become either balanced or unisigned of only one type. Consider now a linear control system of the form dt )= Ax )+ Bu ,t (2.1) where n, n and n, m , and where represents an unconstrained, piecewise continuous control input. We denote the system in (2.1) by ( A,B ). It is known that the output (viz. solution) ) of (2.1) emanating from any initial point in is controllable (by an appropriate choice of )) to any terminal point in in ﬁnite time if and only if rank n, (2.2) where =[ B AB ... A n, nm is the controllability matrix associated with ( A,B ). When (2.2) holds, we call ( A,B completely controllable . It follows easily that if n, n is nonsingular, then ( A,B ) is completely controllable if and only if ( XAX ,XB ) is completely controllable, or if and only if ( A,B ) is completely controllable. As with many questions arising in the study of sign patterns, the presence of implicit relations among the entries of the matrix in question can complicate the qualitative analysis signiﬁcantly. In the case of the controllability matrix this diﬃculty is evident because of the presence of the products of powers of with . For this reason, it is useful to consider a condition known to be equivalent to rank (see, e.g., Theorem 4.3.3 in Lancaster and Rodman [7]), namely, rank[ λI B ]= for all (2.3) The compromise in dealing with the latter condition, rather than , is the introduc- tion of the complex parameter Given a linear control system ( A,B ), we consider the qualitative class consisting of all linear control systems ( A, ) such that ). In this paper, we say that ( A,B )is sign controllable if ( A, ) is completely controllable for all ) and all ). Next we introduce a classiﬁcation of control systems ( A,B ) based on the directed graph of [ AB ] and the signs of the diagonal entries of EFINITION 2.2. Let n, n n, m =[ AB Γ= . We call A,B strict linear control system if (a) the diagonal entries of are nonzero and have the same sign, and (b) for all such that ⊂R in , either |R is an L-matrix or |R ]) contains a nonsingular signing. In the next section, we will ﬁnd suﬃcient conditions for sign controllability and we will show that these conditions are necessary and suﬃcient for sign controllability of a strict linear control system. 3. Conditions for sign controllability. First we mention a necessary condi- tion for complete controllability (that is observed in [6] as a necessary condition for structural controllability ). We caution the reader that the term sign controllability has also been used in the literature to describe a diﬀerent property of the controllability matrix (see [3, 7]).

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358 MICHAEL J. TSATSOMEROS EMMA 3.1. Let n, n n, m , and suppose that A,B is completely controllable. Then Γ= ([ AB ]) is accessible. Proof . Suppose that Γ is not accessible. Then there exists such that there is no path from to in Γ for every 〉\ . Let be the set consisting of and all vertices of ) that lie on a path emanating from . It follows that there is no path from to in Γ for every and every 〉\ Moreover, letting be the complement of in , there exists a permutation matrix such that PAP ]0 and PB | So, if =[ 0] , where is a left eigenvector of ] corresponding to an eigenvalue , then PAP λI PB ] = 0, showing that ( PAP ,PB ) (and hence A,B )) is not completely controllable. From condition (2.3) for = 0 and the above lemma, we have that [ AB ] being an L-matrix and the directed graph of [ AB ] being accessible are two necessary conditions for sign controllability of ( A,B ). We continue by showing that these two conditions, together with some additional conditions on the directed graph of [ AB ], are also suﬃcient for sign controllability of ( A,B ). HEOREM 3.2. Let n, n n, m , and Γ= ([ AB ])=( V,E . Suppose that (1) is accessible, (2) [ AB is an L-matrix, and (3) for all satisfying ⊂R in , either there exists ∈R and exactly one such that i,j ,or is not aligned relative to Then the linear control system A,B is sign controllable. Proof . Suppose ( A,B ) is not completely controllable and that (1) and (2) hold. It is enough to show that (3) is not true. By condition (2.3) and because [ AB ]isan L-matrix, there exists \{ and \{ such that AB ]=[ λx 0] (3.1) Without loss of generality, assume that =( ,x ,...,x ,... 0) ,x = 0 for ,...,k (otherwise we can work with ( PAP ,PB ) for some permutation matrix ). Also without loss of generality assume that Re( = 0 (otherwise we can replace in our arguments by ). Observe that Re( λx = 0 or else, by (3.1), Re( )[ AB ]= 0 and (2) is contradicted. Consider an invertible signing = diag( ,s ,...,s )so that Re( λx 0 (entrywise). On letting =[ SAS SB ]=( ij ), we have from (3.1) that SAS SB ]=[ λx 0] (3.2) namely, =1 ij λx =0 ( =1 ,...,k (3.3) Now take and let be its complement in . From (3.3) we can conclude that every column of ] contains at least one nonzero entry. Hence for

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SIGN CONTROLLABILITY 359 every there exists such that ( i,j . This means that ⊆R ). Since Γ is assumed accessible, we have that ⊂R ). We also have from (3.2) that =1 ij =0 ( +1 ,k +2 ,...,n (3.4) which implies that every column of ] has either no or at least two nonzero entries. Hence for every ∈R there are at least two vertices such that ( i,j . As a consequence, to show that condition (3) is violated it remains to argue that is aligned relative to . From (3.3) and (3.4) we get, respectively, that =1 Re( ij = Re( λx 0( =1 ,...,k (3.5) and =1 Re( ij =0 ( +1 ,k +2 ,...,n (3.6) Equations (3.5) and (3.6) have the following interpretation: if we consider the signing = diag(sgn(Re( )) sgn(Re( )) ,..., sgn(Re( )) then the columns of ST ], and in particular the columns of ST |R ], are balanced, while all unisigned columns of ST ] are of positive type. Hence is aligned relative to in Γ. We continue with some examples in order to illustrate the use of Theorem 3.2 and various situations that arise. Example 3.3. Let 01 1 10 10 and Notice that the directed graph of [ AB ] is accessible and that [ AB ] is an L-matrix because det A< 0 for all ) (i.e., is a sign nonsingular matrix; see [1]). Regarding condition (3) of Theorem 3.2, we ﬁnd that ⊂R ), =1 4, where , , and . We also have that and In all four cases, the ﬁrst part of condition (3) is satisﬁed with the edge from to =1 4, being (3 1), (1 4), (1 4), and (1 4), respectively. So by Theorem 3.2, ( A,B ) is sign controllable. We comment that, in the language of [10], the pair of zero/nonzero patterns ( A, B ) associated with ( A,B )isnot qualitatively controllable (see ([10, Theorem 2.2]). Example 3.4. Let 10 1 10 0 01 and 11 00

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360 MICHAEL J. TSATSOMEROS The directed graph of =[ AB ] is accessible and is an L-matrix. Regarding condition (3) of Theorem 3.2, we ﬁnd that with one exception, for all for which ⊂R ), there is exactly one edge from to some ∈R . The only exception is for which ( . Notice that every ∈B [ |R ( ]) has its ﬁrst two diagonal entries zero and the third diagonal entry nonzero. But then the last two columns of ST [ ] are unisigned of opposite type. Hence, by Theorem 3.2, A,B ) is sign controllable. We continue with a result on sign controllability, which will lead to a characteri- zation of strict sign controllable systems. ROPOSITION 3.5. Let n, n n, m =[ AB , and Γ= Assume that there exists with ⊂R in such that |R ]) contains a nonsingular signing . Also assume that the unisigned columns of ST (if any exist) are only of one type. Then A,B is not sign controllable. Proof . Let be as prescribed above and Γ = ( V,E ). Since ( A,B ) is completely controllable if and only if ( A,B ) is, we will assume, without loss of generality, that the unisigned columns of ST (if any exist) are all of positive type. Since there is no and 6∈R ) such that ( i,j | 〉\R )] = 0. Also, since ⊂R ), every column of ] contains a nonzero entry. Letting n, n be a nonsingular signing such that ]= and considering =[ SA SB ], we have that (1) every column of ] contains a positive entry, (2) every column of |R ] is balanced, and (3) every column of | 〉\R )] is zero. Therefore, by (1)–(3) above, we can assume that the nonzero entries of and have been chosen so that the entries of each column of ] add up to one, and the entries of each column of | 〉\ ] add up to zero. That is, if we let =( ,x ,...x ,x 1if 0 otherwise, we have shown that SA SB ]=[ 0] for an invertible signing . Hence, using = 1 in condition (2.3), it follows that SA S, SB ) and thus ( A,B ) is not sign controllable. OROLLARY 3.6. Let n, n n, m =[ AB Γ= , and suppose that the diagonal entries of are nonzero and have the same sign. Let with ⊂R such that |R ]) contains a nonsingular signing. Then A,B is not sign controllable. Proof . Let be as prescribed and ∈B |R ]) be nonsingular. Since all diagonal entries of ST are nonzero and have the same sign, all the assumptions of Proposition 3.5 are satisﬁed and the corollary follows. HEOREM 3.7. Let n, n n, m =[ AB , and Γ= . Suppose that A,B is a strict linear control system. Then A,B is sign controllable if and only if the following conditions hold: (1) is accessible, (2) [ AB is an L-matrix, and (3) for all satisfying ⊂R in is not aligned relative to Proof . The suﬃciency of conditions (1)–(3) follows from Theorem 3.2. We have discussed the necessity of conditions (1) and (2) after Lemma 3.1. To prove the necessity of condition (3), assume that (1) and (2) hold and that (3) does not hold.

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SIGN CONTROLLABILITY 361 Since Γ is accessible and the diagonal entries of are nonzero, we have that for all ⊂R ) and hence . So since (3) is not true, there exists ⊂R ) such that |R ]) . Because ( A,B ) is strict, |R ]) must contain a nonsingular signing. By Corollary 3.6 it follows that ( A,B ) is not sign controllable. Example 3.8. Let 101 110 011 and 11 11 The directed graph of =[ AB ] is accessible and is an L-matrix. For all we have that ⊂R ) in Γ. In fact, for all , except |R is an L-matrix. We have that [ |R ( ]= , and = diag( 1) ∈B ). So ( A,B ) is a strict linear control system. Since ST [ ] has unisigned columns of only positive type, is aligned relative to ( . By Theorem 3.7, ( A,B ) is not sign controllable. 4. The extended controllability matrix. We will now introduce some addi- tional concepts and terminology pertaining to an alternative analysis of sign control- lability. For the purposes of this section, we append to the set of signings the zero (square) matrix and refer to them as weak signings . We let ) denote ∪{ for any s, t It is clear that for every ∈B ) there exists ) such that the column sums of equal to zero (and hence equal to the column sums of the zero matrix). Based on this observation, we extend the notion of ) as follows. Given a matrix and a weak signing , we denote by X,S ) the set of all weak signings such that there exists ) with the column sums of equal to the column sums of . Notice that )= X, 0). To illustrate the deﬁnition of X,S ), let 32 101 and 100 010 000 Then X,S ) consists of all signings such that SX has a negative entry in the ﬁrst column, a positive entry in the second column, and a balanced third column. For example, = diag( 1) ∈B X,S ) and 6∈B X,S ). EFINITION 4.1. Let n, n and n, m be given. A nonzero ordered -tuple ,S ,...,S of weak signings is called an A,B balancing chain if ∈B )( =1 ,...,n and +1 ∈B A,S )( =1 ,...,n 1) If, in addition, there exist , and entrywise positive vectors such that =0 ( =1 ,...,n

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362 MICHAEL J. TSATSOMEROS and +1 +1 =1 ,...,n 1) we call ,S ,...,S compatible ( A,B -balancing chain. The notion of an ( A,B )-balancing chain depends only on the sign patterns of and . Indeed, if ( ,S ,...,S )isan( A,B )-balancing chain, then there always exist ), ), and with positive entries such that = 0 for =1 ,...,n and +1 +1 +1 for =1 ,...,n 1. In fact, we can take for all . In the deﬁnition of a compatible ( A,B )-balancing chain we require, in addition, that there are common matrices ) and ) that satisfy the above conditions. Observe that an ( A,B )-balancing chain ( ,S ,...,S ) may contain some zero weak signings, which could appear only as the leading part of the chain. Indeed, if = 0, then since +1 must have the same column sums as for some ), +1 must be nonzero. EFINITION 4.2. With the linear control system A,B we will associate (in Lemma 4.3) the extended controllability matrix deﬁned as follows: ... ... ... ... ... ... ... AI ... ... ... ... ... A I ... ... ... ... 00 00 ... AI ... ... 00 ... AB ... ... ... ,n 1) The following result is a recasting of the classical condition for controllability in (2.2); its proof can be found in Casti [2]. EMMA 4.3 (see [2, Corollary 5, section 3.5]). Let n, n n, m . The control system A,B is completely controllable if and only if rank We now have the following equivalent condition for sign controllability. HEOREM 4.4. Let n, n n, m . The linear control system A,B is sign controllable if and only if there is no compatible A,B -balancing chain. Proof . Let n, n n, m , and suppose that ( A,B ) is not sign control- lable. Then, by Lemma 4.3, there are matrices in ) and ) so that the corresponding extended controllability matrix is of deﬁcient rank, i.e., =0 for some \{ . Now let ,n be a signing such that Sx , where has positive entries. It follows that = 0. Hence if is partitioned into diagonal blocks ,S ,...,S of size , then ( ,S ,...,S ) is a compatible A,B )-balancing chain. Conversely, if ( ,S ,...,S ) is a compatible ( A,B )-balancing chain, then there exist ), ), and vectors with positive entries such that =0 ( =1 ,...,n and +1 +1 =1 ,...,n 1) It follows that for = diag( ,S ,...,S ) and for =[ ,x ,...,x Sx is a nonzero left nullvector of the extended controllability matrix of ( A, ); that is, by Lemma 4.3, ( A,B ) is not sign controllable.

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SIGN CONTROLLABILITY 363 The existence or not of a compatible ( A,B )-balancing chain can be a hard con- dition to check, but in some instances the clauses in the deﬁnition of a compatible A,B )-balancing chain can serve as useful necessary or suﬃcient conditions. This is illustrated in the following examples. Example 4.5. This example is mentioned in [4]. Let 001 101 100 and Notice that ) consists of the weak signings ,S ,...,S having their (1,1) entry equal to zero. It is easy to check the sign patterns of for =1 ,..., 9 and discover that there is no ( A,B )-balancing chain and hence, by Theorem 4.4, ( A,B is sign controllable. Example 4.6. Let 101 10 and It can be checked that 100 010 001 ,S 10 0 01 0 00 ,T 100 10 00 are in ), and that ( R,T,R T,R,T T,R,S ) are some of the ( A,B )-balancing chains. In this case the knowledge of a balancing chain leads to a straightforward search for the vectors and the matrices and in the deﬁnition of a compatible balancing chain. One ﬁnds that with and 301 30 R,T,R ) is a compatible ( A,B )-balancing chain and thus, by Theorem 4.4, ( A,B )is not sign controllable. In conclusion, we have presented an alternative approach to sign controllability of a linear control system ( A,B ) based on the existence of a balancing chain of signings. We do not know if there exists an algorithm to verify the (non)existence of a com- patible balancing chain, regardless of the complexity. We have also found suﬃcient conditions for sign controllability, based on the sign pattern and the directed graph of AB ], which are necessary and suﬃcient when the linear control system is strict. We have not addressed computational matters regarding these conditions. However, we remark that the recognition of one of these conditions, namely, that the rectangular matrix [ AB ] be an L-matrix, has been shown to be an NP-complete problem (see Klee, Ladner, and Manber [5]). REFERENCES [1] R. A. B RUALDI AND B. L. S HADER Matrices of Sign-Solvable Linear Systems , Cambridge University Press, London, 1995.

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364 MICHAEL J. TSATSOMEROS [2] J. L. C ASTI Dynamical Systems and Their Applications: Linear Theory , Academic Press, New York, 1977. [3] L. E. F AIBUSOVICH Algebraic Riccati equation and symplectic algebra , Internat. J. Control, 43 (1986), pp. 781–792. [4] C. R. J OHNSON ,V.M EHRMANN AND D. D. O LESKY Sign controllability of a nonnegative matrix and a positive vector , SIAM J. Matrix Anal. Appl., 14 (1993), pp. 398–407. [5] V. K LEE ,R.L ADNER AND R. M ANBER Signsolvability revisited , Linear Algebra Appl., 59 (1984), pp. 131–157. [6] C. T. L IN Structural Controllability , IEEE Trans. Automat. Control, AC–19 (1974), pp. 201 208. [7] P. L ANCASTER AND L. R ODMAN Algebraic Riccati Equations , Oxford University Press, New York, 1995. [8] K. M UROTA Systems Analysis by Graphs and Matroids–Structural Solvability and Controlla- bility , Algorithms and Combinatorics 3, Springer-Verlag, Berlin, Heidelberg, 1987. [9] H. M AYEDA AND T. Y AMADA Strong structural controllability , SIAM J. Control Optim., 17 (1979), pp. 123–138. [10] D. D. O LESKY ,M.T SATSOMEROS AND P. VAN DEN RIESSCHE Qualitative controllability and uncontrollability by a single entry , Linear Algebra Appl., 187 (1993), pp. 183–194.

TSATSOMEROS SIAM J M ATRIX NAL PPL 1998 Society for Industrial and Applied Mathematics Vol 19 No 2 pp 355364 April 1998 004 Abstract We apply tools from the theory of signsolvable systems and use the directed graph of a matrix in order to obtain su6 ID: 23041

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SIGN CONTROLLABILITY: SIGN PATTERNS THAT REQUIRE COMPLETE CONTROLLABILITY MICHAEL J. TSATSOMEROS SIAM J. M ATRIX NAL. PPL 1998 Society for Industrial and Applied Mathematics Vol. 19, No. 2, pp. 355–364, April 1998 004 Abstract. We apply tools from the theory of sign-solvable systems and use the directed graph of a matrix in order to obtain suﬃcient conditions for a linear control system ( A,B ) to be completely controllable solely due to the sign patterns of the coeﬃcient matrices and . We show that such conditions are necessary and suﬃcient for a particular class of linear control systems. We also consider an alternative approach to controllability, based on a reformulation of the classical condition (that the controllability matrix is of full rank) and obtain equivalent conditions for the general case. Key words. control system, sign pattern, signing, L-matrix, directed graph, aligned vertices, balancing chain AMS subject classiﬁcations. 15A99, 93B05 PII. S0895479896300346 1. Introduction. In linear control theory, the basic concepts of controllability (and observability) are intimately related to the image of a matrix of the form =[ B AB ... A n, nm where n, n and n, m . Speciﬁcally, a control system of the form dt )= Ax )+ Bu is completely controllable if and only if rank . As the matrices A,B comprise system parameters prone to measurement errors, it is desirable to determine whether rank based on combinatorial and qualitative information about and (e.g., their directed graphs and the signs of their entries). Such qualitative approaches to controllability have been undertaken, for example, by Lin [6], Mayeda and Yamada [9], Murota [8], Johnson, Mehrmann, and Olesky [4], and Olesky, Tsatsomeros, and van den Driessche [10]. In the present work, we shall consider the following: assume that the sign patterns (namely, the location of the positive, negative, and zero entries) of and are known. When can we conclude that rank , based solely on the sign patterns and regardless of the magnitudes of the nonzero entries of and The study of this question was initiated in [4], where was assumed to have nonnegative entries and was assumed to be a column vector with positive entries. Our qualitative approach to controllability will be based on extending and com- bining techniques used in the study of zero/nonzero patterns that allow or require complete controllability (see [6, 9, 10]) with notions related to the analysis of sign patterns and sign-solvable linear systems. We will ﬁnd suﬃcient conditions for com- plete controllability for general sign patterns (Theorem 3.2), and we will identify a Received by the editors March 11, 1996; accepted for publication (in revised form) by V. Mehrmann February 6, 1997. This work was supported by a grant from the Natural Sciences and Engineering Research Council of Canada. http://www.siam.org/journals/simax/19-2/30034.html Department of Mathematics and Statistics, University of Regina, Regina, Saskatchewan S4S 0A2, Canada (tsat@math.uregina.ca). 355

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356 MICHAEL J. TSATSOMEROS class of linear control systems (Deﬁnition 2.2) for which these conditions are neces- sary and suﬃcient (Theorem 3.7). In section 4, we will consider a simple technical recasting of the classical controllability condition that rank in order to provide an alternative answer to the question posed above. 2. Preliminaries. In this section, we present some of the notation, terminology, and basic facts necessary to state and prove the main results in the following sections. In the remainder we let ,...,k for any positive integer denote the cardinality of a set ; sgn( )be0,1,or 1, when is zero, positive, or negative, respectively; Re( ) denote the real part of a complex vector ; diag( ,A ,...,A )be a block diagonal matrix whose diagonal blocks are the square matrices ,A ,...,A ] denote the submatrix of s, t whose rows and columns are indexed by the sets and , respectively; ]= ]; and denote an all 1s column vector of appropriate size. We let Γ = ( V,E ) denote a directed graph with vertex set and directed edge set consisting of ordered pairs ( i,j ) of vertices. A path from to in Γ is a sequence of vertices ,r ,...,r , with ( ,r +1 for =1 ,...,t 1. The directed graph of =( ij s, t ,s , denoted by Γ = )=( V,E ), has and i,j ij =0 . Extending the terminology in [6] and [9], when s we say that )is accessible if for every there exists 〉\ such that there is a path from to in Γ. Also, for every we denote the adjacency set of by )= 〉| i,j for some Notice that if there exists an such that ⊆R ) and if Γ is accessible, then 〉\ and hence ⊂R ). In keeping with the notation and terminology of Brualdi and Shader [1] (which is our comprehensive reference on sign patterns), we deﬁne the following. The sign pattern of s, t is the (0 1)-matrix obtained from when zero, positive, and negative entries are replaced by 0, 1, and 1, respectively. The matrix determines the qualitative class ) of all matrices with the same sign pattern as . We will write ) for any matrix having the same sign pattern as signing is a nonzero square diagonal sign pattern. A real vector is called balanced if it is the zero vector, or if it has at least one negative and at least one positive entry. A real vector is referred to as unisigned if it is not balanced. If a unisigned vector has nonnegative (respectively, nonpositive) entries, we refer to it as of positive ( respectively, negative) type . We denote the signings such that all the columns of SX are balanced by ). The matrix is called an L-matrix provided that every matrix in ) has linearly independent rows. It is well known (see [1, Theorem 2.1.1]) that s, t is an L-matrix if and only if )= . Next we introduce the notion of aligned vertices in the directed graph of a matrix. EFINITION 2.1. Let s, t , and let , be two nonempty and disjoint sets. We call aligned relative to if there exists a signing ∈B ]) such that the unisigned columns of SX (if any exist) are only of one type (either only positive type or only negative type). When ]) = is by deﬁnition not aligned relative to In other words, is aligned relative to a disjoint set if there exists a signing of the rows of | ] such that the columns of ] become

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SIGN CONTROLLABILITY 357 balanced and the columns of ] become either balanced or unisigned of only one type. Consider now a linear control system of the form dt )= Ax )+ Bu ,t (2.1) where n, n and n, m , and where represents an unconstrained, piecewise continuous control input. We denote the system in (2.1) by ( A,B ). It is known that the output (viz. solution) ) of (2.1) emanating from any initial point in is controllable (by an appropriate choice of )) to any terminal point in in ﬁnite time if and only if rank n, (2.2) where =[ B AB ... A n, nm is the controllability matrix associated with ( A,B ). When (2.2) holds, we call ( A,B completely controllable . It follows easily that if n, n is nonsingular, then ( A,B ) is completely controllable if and only if ( XAX ,XB ) is completely controllable, or if and only if ( A,B ) is completely controllable. As with many questions arising in the study of sign patterns, the presence of implicit relations among the entries of the matrix in question can complicate the qualitative analysis signiﬁcantly. In the case of the controllability matrix this diﬃculty is evident because of the presence of the products of powers of with . For this reason, it is useful to consider a condition known to be equivalent to rank (see, e.g., Theorem 4.3.3 in Lancaster and Rodman [7]), namely, rank[ λI B ]= for all (2.3) The compromise in dealing with the latter condition, rather than , is the introduc- tion of the complex parameter Given a linear control system ( A,B ), we consider the qualitative class consisting of all linear control systems ( A, ) such that ). In this paper, we say that ( A,B )is sign controllable if ( A, ) is completely controllable for all ) and all ). Next we introduce a classiﬁcation of control systems ( A,B ) based on the directed graph of [ AB ] and the signs of the diagonal entries of EFINITION 2.2. Let n, n n, m =[ AB Γ= . We call A,B strict linear control system if (a) the diagonal entries of are nonzero and have the same sign, and (b) for all such that ⊂R in , either |R is an L-matrix or |R ]) contains a nonsingular signing. In the next section, we will ﬁnd suﬃcient conditions for sign controllability and we will show that these conditions are necessary and suﬃcient for sign controllability of a strict linear control system. 3. Conditions for sign controllability. First we mention a necessary condi- tion for complete controllability (that is observed in [6] as a necessary condition for structural controllability ). We caution the reader that the term sign controllability has also been used in the literature to describe a diﬀerent property of the controllability matrix (see [3, 7]).

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358 MICHAEL J. TSATSOMEROS EMMA 3.1. Let n, n n, m , and suppose that A,B is completely controllable. Then Γ= ([ AB ]) is accessible. Proof . Suppose that Γ is not accessible. Then there exists such that there is no path from to in Γ for every 〉\ . Let be the set consisting of and all vertices of ) that lie on a path emanating from . It follows that there is no path from to in Γ for every and every 〉\ Moreover, letting be the complement of in , there exists a permutation matrix such that PAP ]0 and PB | So, if =[ 0] , where is a left eigenvector of ] corresponding to an eigenvalue , then PAP λI PB ] = 0, showing that ( PAP ,PB ) (and hence A,B )) is not completely controllable. From condition (2.3) for = 0 and the above lemma, we have that [ AB ] being an L-matrix and the directed graph of [ AB ] being accessible are two necessary conditions for sign controllability of ( A,B ). We continue by showing that these two conditions, together with some additional conditions on the directed graph of [ AB ], are also suﬃcient for sign controllability of ( A,B ). HEOREM 3.2. Let n, n n, m , and Γ= ([ AB ])=( V,E . Suppose that (1) is accessible, (2) [ AB is an L-matrix, and (3) for all satisfying ⊂R in , either there exists ∈R and exactly one such that i,j ,or is not aligned relative to Then the linear control system A,B is sign controllable. Proof . Suppose ( A,B ) is not completely controllable and that (1) and (2) hold. It is enough to show that (3) is not true. By condition (2.3) and because [ AB ]isan L-matrix, there exists \{ and \{ such that AB ]=[ λx 0] (3.1) Without loss of generality, assume that =( ,x ,...,x ,... 0) ,x = 0 for ,...,k (otherwise we can work with ( PAP ,PB ) for some permutation matrix ). Also without loss of generality assume that Re( = 0 (otherwise we can replace in our arguments by ). Observe that Re( λx = 0 or else, by (3.1), Re( )[ AB ]= 0 and (2) is contradicted. Consider an invertible signing = diag( ,s ,...,s )so that Re( λx 0 (entrywise). On letting =[ SAS SB ]=( ij ), we have from (3.1) that SAS SB ]=[ λx 0] (3.2) namely, =1 ij λx =0 ( =1 ,...,k (3.3) Now take and let be its complement in . From (3.3) we can conclude that every column of ] contains at least one nonzero entry. Hence for

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SIGN CONTROLLABILITY 359 every there exists such that ( i,j . This means that ⊆R ). Since Γ is assumed accessible, we have that ⊂R ). We also have from (3.2) that =1 ij =0 ( +1 ,k +2 ,...,n (3.4) which implies that every column of ] has either no or at least two nonzero entries. Hence for every ∈R there are at least two vertices such that ( i,j . As a consequence, to show that condition (3) is violated it remains to argue that is aligned relative to . From (3.3) and (3.4) we get, respectively, that =1 Re( ij = Re( λx 0( =1 ,...,k (3.5) and =1 Re( ij =0 ( +1 ,k +2 ,...,n (3.6) Equations (3.5) and (3.6) have the following interpretation: if we consider the signing = diag(sgn(Re( )) sgn(Re( )) ,..., sgn(Re( )) then the columns of ST ], and in particular the columns of ST |R ], are balanced, while all unisigned columns of ST ] are of positive type. Hence is aligned relative to in Γ. We continue with some examples in order to illustrate the use of Theorem 3.2 and various situations that arise. Example 3.3. Let 01 1 10 10 and Notice that the directed graph of [ AB ] is accessible and that [ AB ] is an L-matrix because det A< 0 for all ) (i.e., is a sign nonsingular matrix; see [1]). Regarding condition (3) of Theorem 3.2, we ﬁnd that ⊂R ), =1 4, where , , and . We also have that and In all four cases, the ﬁrst part of condition (3) is satisﬁed with the edge from to =1 4, being (3 1), (1 4), (1 4), and (1 4), respectively. So by Theorem 3.2, ( A,B ) is sign controllable. We comment that, in the language of [10], the pair of zero/nonzero patterns ( A, B ) associated with ( A,B )isnot qualitatively controllable (see ([10, Theorem 2.2]). Example 3.4. Let 10 1 10 0 01 and 11 00

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360 MICHAEL J. TSATSOMEROS The directed graph of =[ AB ] is accessible and is an L-matrix. Regarding condition (3) of Theorem 3.2, we ﬁnd that with one exception, for all for which ⊂R ), there is exactly one edge from to some ∈R . The only exception is for which ( . Notice that every ∈B [ |R ( ]) has its ﬁrst two diagonal entries zero and the third diagonal entry nonzero. But then the last two columns of ST [ ] are unisigned of opposite type. Hence, by Theorem 3.2, A,B ) is sign controllable. We continue with a result on sign controllability, which will lead to a characteri- zation of strict sign controllable systems. ROPOSITION 3.5. Let n, n n, m =[ AB , and Γ= Assume that there exists with ⊂R in such that |R ]) contains a nonsingular signing . Also assume that the unisigned columns of ST (if any exist) are only of one type. Then A,B is not sign controllable. Proof . Let be as prescribed above and Γ = ( V,E ). Since ( A,B ) is completely controllable if and only if ( A,B ) is, we will assume, without loss of generality, that the unisigned columns of ST (if any exist) are all of positive type. Since there is no and 6∈R ) such that ( i,j | 〉\R )] = 0. Also, since ⊂R ), every column of ] contains a nonzero entry. Letting n, n be a nonsingular signing such that ]= and considering =[ SA SB ], we have that (1) every column of ] contains a positive entry, (2) every column of |R ] is balanced, and (3) every column of | 〉\R )] is zero. Therefore, by (1)–(3) above, we can assume that the nonzero entries of and have been chosen so that the entries of each column of ] add up to one, and the entries of each column of | 〉\ ] add up to zero. That is, if we let =( ,x ,...x ,x 1if 0 otherwise, we have shown that SA SB ]=[ 0] for an invertible signing . Hence, using = 1 in condition (2.3), it follows that SA S, SB ) and thus ( A,B ) is not sign controllable. OROLLARY 3.6. Let n, n n, m =[ AB Γ= , and suppose that the diagonal entries of are nonzero and have the same sign. Let with ⊂R such that |R ]) contains a nonsingular signing. Then A,B is not sign controllable. Proof . Let be as prescribed and ∈B |R ]) be nonsingular. Since all diagonal entries of ST are nonzero and have the same sign, all the assumptions of Proposition 3.5 are satisﬁed and the corollary follows. HEOREM 3.7. Let n, n n, m =[ AB , and Γ= . Suppose that A,B is a strict linear control system. Then A,B is sign controllable if and only if the following conditions hold: (1) is accessible, (2) [ AB is an L-matrix, and (3) for all satisfying ⊂R in is not aligned relative to Proof . The suﬃciency of conditions (1)–(3) follows from Theorem 3.2. We have discussed the necessity of conditions (1) and (2) after Lemma 3.1. To prove the necessity of condition (3), assume that (1) and (2) hold and that (3) does not hold.

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SIGN CONTROLLABILITY 361 Since Γ is accessible and the diagonal entries of are nonzero, we have that for all ⊂R ) and hence . So since (3) is not true, there exists ⊂R ) such that |R ]) . Because ( A,B ) is strict, |R ]) must contain a nonsingular signing. By Corollary 3.6 it follows that ( A,B ) is not sign controllable. Example 3.8. Let 101 110 011 and 11 11 The directed graph of =[ AB ] is accessible and is an L-matrix. For all we have that ⊂R ) in Γ. In fact, for all , except |R is an L-matrix. We have that [ |R ( ]= , and = diag( 1) ∈B ). So ( A,B ) is a strict linear control system. Since ST [ ] has unisigned columns of only positive type, is aligned relative to ( . By Theorem 3.7, ( A,B ) is not sign controllable. 4. The extended controllability matrix. We will now introduce some addi- tional concepts and terminology pertaining to an alternative analysis of sign control- lability. For the purposes of this section, we append to the set of signings the zero (square) matrix and refer to them as weak signings . We let ) denote ∪{ for any s, t It is clear that for every ∈B ) there exists ) such that the column sums of equal to zero (and hence equal to the column sums of the zero matrix). Based on this observation, we extend the notion of ) as follows. Given a matrix and a weak signing , we denote by X,S ) the set of all weak signings such that there exists ) with the column sums of equal to the column sums of . Notice that )= X, 0). To illustrate the deﬁnition of X,S ), let 32 101 and 100 010 000 Then X,S ) consists of all signings such that SX has a negative entry in the ﬁrst column, a positive entry in the second column, and a balanced third column. For example, = diag( 1) ∈B X,S ) and 6∈B X,S ). EFINITION 4.1. Let n, n and n, m be given. A nonzero ordered -tuple ,S ,...,S of weak signings is called an A,B balancing chain if ∈B )( =1 ,...,n and +1 ∈B A,S )( =1 ,...,n 1) If, in addition, there exist , and entrywise positive vectors such that =0 ( =1 ,...,n

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362 MICHAEL J. TSATSOMEROS and +1 +1 =1 ,...,n 1) we call ,S ,...,S compatible ( A,B -balancing chain. The notion of an ( A,B )-balancing chain depends only on the sign patterns of and . Indeed, if ( ,S ,...,S )isan( A,B )-balancing chain, then there always exist ), ), and with positive entries such that = 0 for =1 ,...,n and +1 +1 +1 for =1 ,...,n 1. In fact, we can take for all . In the deﬁnition of a compatible ( A,B )-balancing chain we require, in addition, that there are common matrices ) and ) that satisfy the above conditions. Observe that an ( A,B )-balancing chain ( ,S ,...,S ) may contain some zero weak signings, which could appear only as the leading part of the chain. Indeed, if = 0, then since +1 must have the same column sums as for some ), +1 must be nonzero. EFINITION 4.2. With the linear control system A,B we will associate (in Lemma 4.3) the extended controllability matrix deﬁned as follows: ... ... ... ... ... ... ... AI ... ... ... ... ... A I ... ... ... ... 00 00 ... AI ... ... 00 ... AB ... ... ... ,n 1) The following result is a recasting of the classical condition for controllability in (2.2); its proof can be found in Casti [2]. EMMA 4.3 (see [2, Corollary 5, section 3.5]). Let n, n n, m . The control system A,B is completely controllable if and only if rank We now have the following equivalent condition for sign controllability. HEOREM 4.4. Let n, n n, m . The linear control system A,B is sign controllable if and only if there is no compatible A,B -balancing chain. Proof . Let n, n n, m , and suppose that ( A,B ) is not sign control- lable. Then, by Lemma 4.3, there are matrices in ) and ) so that the corresponding extended controllability matrix is of deﬁcient rank, i.e., =0 for some \{ . Now let ,n be a signing such that Sx , where has positive entries. It follows that = 0. Hence if is partitioned into diagonal blocks ,S ,...,S of size , then ( ,S ,...,S ) is a compatible A,B )-balancing chain. Conversely, if ( ,S ,...,S ) is a compatible ( A,B )-balancing chain, then there exist ), ), and vectors with positive entries such that =0 ( =1 ,...,n and +1 +1 =1 ,...,n 1) It follows that for = diag( ,S ,...,S ) and for =[ ,x ,...,x Sx is a nonzero left nullvector of the extended controllability matrix of ( A, ); that is, by Lemma 4.3, ( A,B ) is not sign controllable.

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SIGN CONTROLLABILITY 363 The existence or not of a compatible ( A,B )-balancing chain can be a hard con- dition to check, but in some instances the clauses in the deﬁnition of a compatible A,B )-balancing chain can serve as useful necessary or suﬃcient conditions. This is illustrated in the following examples. Example 4.5. This example is mentioned in [4]. Let 001 101 100 and Notice that ) consists of the weak signings ,S ,...,S having their (1,1) entry equal to zero. It is easy to check the sign patterns of for =1 ,..., 9 and discover that there is no ( A,B )-balancing chain and hence, by Theorem 4.4, ( A,B is sign controllable. Example 4.6. Let 101 10 and It can be checked that 100 010 001 ,S 10 0 01 0 00 ,T 100 10 00 are in ), and that ( R,T,R T,R,T T,R,S ) are some of the ( A,B )-balancing chains. In this case the knowledge of a balancing chain leads to a straightforward search for the vectors and the matrices and in the deﬁnition of a compatible balancing chain. One ﬁnds that with and 301 30 R,T,R ) is a compatible ( A,B )-balancing chain and thus, by Theorem 4.4, ( A,B )is not sign controllable. In conclusion, we have presented an alternative approach to sign controllability of a linear control system ( A,B ) based on the existence of a balancing chain of signings. We do not know if there exists an algorithm to verify the (non)existence of a com- patible balancing chain, regardless of the complexity. We have also found suﬃcient conditions for sign controllability, based on the sign pattern and the directed graph of AB ], which are necessary and suﬃcient when the linear control system is strict. We have not addressed computational matters regarding these conditions. However, we remark that the recognition of one of these conditions, namely, that the rectangular matrix [ AB ] be an L-matrix, has been shown to be an NP-complete problem (see Klee, Ladner, and Manber [5]). REFERENCES [1] R. A. B RUALDI AND B. L. S HADER Matrices of Sign-Solvable Linear Systems , Cambridge University Press, London, 1995.

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364 MICHAEL J. TSATSOMEROS [2] J. L. C ASTI Dynamical Systems and Their Applications: Linear Theory , Academic Press, New York, 1977. [3] L. E. F AIBUSOVICH Algebraic Riccati equation and symplectic algebra , Internat. J. Control, 43 (1986), pp. 781–792. [4] C. R. J OHNSON ,V.M EHRMANN AND D. D. O LESKY Sign controllability of a nonnegative matrix and a positive vector , SIAM J. Matrix Anal. Appl., 14 (1993), pp. 398–407. [5] V. K LEE ,R.L ADNER AND R. M ANBER Signsolvability revisited , Linear Algebra Appl., 59 (1984), pp. 131–157. [6] C. T. L IN Structural Controllability , IEEE Trans. Automat. Control, AC–19 (1974), pp. 201 208. [7] P. L ANCASTER AND L. R ODMAN Algebraic Riccati Equations , Oxford University Press, New York, 1995. [8] K. M UROTA Systems Analysis by Graphs and Matroids–Structural Solvability and Controlla- bility , Algorithms and Combinatorics 3, Springer-Verlag, Berlin, Heidelberg, 1987. [9] H. M AYEDA AND T. Y AMADA Strong structural controllability , SIAM J. Control Optim., 17 (1979), pp. 123–138. [10] D. D. O LESKY ,M.T SATSOMEROS AND P. VAN DEN RIESSCHE Qualitative controllability and uncontrollability by a single entry , Linear Algebra Appl., 187 (1993), pp. 183–194.

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