Generic stabilizability for sign structured systems Report 93-101 Jacob van der Woude Technische Universiteit Delft Delft University of Technology Faculteit der Technische Wiskunde en Informatica Faculty of Technical Mathematics and Informatics ISSN 0922-5641 Copyright c 1993 by the Faculty of Technical Mathematics and Informatics , Delft , The Netherlands . No part of this Journal may be reproduced in any form , by print , photoprint , microfilm , or any other means without permission from the Faculty of Technical Mathematics and Informatics , Delft University of Technology , The Netherlands . Copies of these reports may be obtained from the bureau of the Faculty of Technical Mathematics and Informatics , Julianalaan 132 , 2628 BL Delft , phone +3115784568 . A selection of these reports is available in PostScript form at the Faculty 's anonymous ftp-site . They are located in the directory / pub/publications/tech-reports at ftp.twi.tudelft.nl Generic stabilizability for sign structured systems Jacob van der Woude Abstract In this paper we study linear sign structured systems . We assume that we know which entries in the system matrices are nonzero and what their signs are . Hence , we only require the signs of the nonzeros , and not their values . Therefore , we refer to these entries as signed free parameters . We say that the system is generically stabilizable if it is stabilizable for almost all possible values of its signed free parameters . In this paper we present a graph theoretic condition for the generic stabilizability of a sign structured system . 1 Introduction In this paper we are concerned with the generic stabilizability of a linear system of the form _ x -LRB- t -RRB- = Ax -LRB- t -RRB- + Bu -LRB- t -RRB- ; -LRB- 1 -RRB- where A is a real n ? n matrix and B is a real n ? m matrix . The vector x -LRB- t -RRB- denotes the state of the system , and the vector u -LRB- t -RRB- the input or control . In the sequel we will also refer to system -LRB- 1 -RRB- as the pair -LRB- A ; B -RRB- . To illustrate the ideas of this paper , we first consider the system -LRB- 1 -RRB- given by A = > = 1 > = 2 > = 3 ! ; B = > = 4 ! : Hence , n = 2 and m = 1 . We assume that > = 1 ; > = 2 ; > = 3 and > = 4 can have any real value . Using any well-known controllability test it follows easily that the above system is controllable if and only if > = 2 > = 4 < > -LRB- cf. -LSB- 1 -RSB- -RRB- . The set f > = = -LRB- > = 1 ; > = 2 ; > = 3 ; > = 4 -RRB- > j > = 2 > = 4 = g is -LRB- contained in -RRB- a proper algebraic variety in IR4 , and consequently has Lebesgue measure zero -LRB- cf. -LSB- 1 -RSB- -RRB- . Therefore , we can say that controllability for the above system generically holds . This implies that for almost all > = 2 IR4 the eigenvalues of the system can be placed at arbitrary , but complex conjugated places in the complex plane . In particular , this implies that for almost all > = 2 IR4 there is a feedback matrix F such that the eigenvalues of A + BF are in the open left half of the complex plane . So , it follows that the system is also generically stabilizable -LRB- cf. -LSB- 1 -RSB- -RRB- . Indeed , the set of all > = 2 IR4 for which the system is not stabilizable equals f > = 2 IR4j -LRB- > = 4 = and -LRB- > = 1 > or > = 3 > 0 -RRB- -RRB- or -LRB- > = 2 = and > = 1 > 0 -RRB- g , and is contained in f > = 2 IR4j > = 2 > = 4 = g . The above illustrates the general principle that system is generically stabilizable when it is generically controllable . However , the converse need not to be true . To clarify this , we consider the system -LRB- 1 -RRB- given by A = > = 1 > = 2 > = 3 ! ; B = > = 4 ! : So , n = 2 and m = 1 . Also here we assume that > = 1 ; > = 2 ; > = 3 and > = 4 can have any real value . Now using any well-known controllability test , it follows that the system is uncontrollable -LRB- = not controllable -RRB- for all > = 2 IR4 . Or , in other words , the set of > = 2 IR4 for which the system is controllable is empty . Since this set is contained in any proper algebraic variety in IR4 , we can say that the above system is generically uncontrollable . However , only part of the system is responsible for making it generically uncontrollable . Indeed , if we decompose A = A1 A2 A3 ! ; B = B1 ! with Ai = > = i , i = 1 ; 2 ; 3 and B1 = > = 4 , then we see immediately that the subsystem described by the pair -LRB- A1 ; B1 -RRB- is generically controllable . Actually , this subsystem is controllable for all > = 2 IR4 with > = 4 < > 0 . Further , if we know that the sign of A3 = > = 3 is negative , then we can easily conclude that for almost all > = 2 IR4 with > = 3 < 0 , there is a feedback matrix F such that the eigenvalues of A+BF are in the open left half of the complex plane . In other words , for almost all > = 2 IR4 with > = 3 < 0 , the system described by the above A and B is stabilizable . So , provided the sign of > = 3 is negative , we can say that the system is generically stabilizable . In both examples the generic -LRB- un -RRB- controllability is directly related to the zero-nonzero structure of the matrices A and B . This structure can be represented by means of a directed graph , and it will be of no surprise that graph theoretic conditions for the generic controllability of a system are well known -LRB- cf. -LSB- 2 -RSB- -RRB- . With these conditions we can determine whether or not a system is generically controllable . As we have seen above , if a system is generically controllable , then it is also generically stabilizable . However , if a system is generically uncontrollable , it still can be generically stabilizable provided certain nonzero entries have the appropriate signs . The purpose of this paper is to present a graph theoretic method to determine whether or not a system is generically stabilizable . It turns out that the method in principle requires two types of information and consists of three stages . To start we need the zero-nonzero structure of the matrices A and B ; the first type of information . From this structure we can construct a directed graph and we can apply a well-known graph theoretic result to determine whether the system is generically controllable or is generically uncontrollable . If the first is the case , we are already done and we can stop . If the second is the case , we have to proceed and we enter the second stage . In this stage we use the previously mentioned graph theoretic result to determine of which entries the signs are relevant for investigating the generic stabilizability of the system . Having done this we enter the third stage in which we assume the signs of these relevant entries to be known . Then we use a second well-known graph theoretic result to determine whether or not the system is generically stabilizable under the given sign information -LRB- cf. -LSB- 3 -RSB- -RRB- . The latter can be seen as the second type of information that we need . The outline of this paper is as follows . In section 2 we recall some basic facts on the stability , controllability and stabilizability of linear systems . Also we consider the zero-nonzero structure of the system matrices , and we introduce the directed graphs by which the zerononzero structure can be represented . In section 3 we review some important results on generic controllability of linear systems and in section 4 on the generic stability of linear systems . For the generic controllability only the zero-nonzero structure of the system matrices is important , whereas for the generic stability also the sign of the nonzero entries is relevant . In section 5 we combine the results of the sections 3 and 4 , and we derive a graph theoretic method for deciding whether or not a sign structured system is generically stabilizable . In section 6 we illustrate our method by means of an example . We conclude this paper with some remarks in section 7 . 2 Preliminaries As mentioned before , we consider in this paper the linear system -LRB- 1 -RRB- given by _ x -LRB- t -RRB- = Ax -LRB- t -RRB- + Bu -LRB- t -RRB- ; with state x -LRB- t -RRB- 2 IRn and input u -LRB- t -RRB- 2 IRm . The matrices A and B are real and have dimensions n ? n and n ? m , respectively . We say that the system -LRB- 1 -RRB- is asymptotically stable , or that the matrix A is asymptotically stable , if all the eigenvalues of A are located in IC ? , where IC ? is the open left half of the complex plane , i.e. IC ? = fz 2 ICj Re z < g . Since the above notion of stability is the only stability notion that we study in this paper , we omit from now on the word asymptotically . So , we just say that system -LRB- 1 -RRB- or matrix A is stable , if all the eigenvalues of A are inside IC ? . We say that the system -LRB- 1 -RRB- is controllable , or also that the pair -LRB- A ; B -RRB- is controllable , if the state of the system can be sent from any initial state x0 at time t = to any other state x1 at some time t = o > -LRB- cf. -LSB- 4 -RSB- -RRB- . It is well known that the controllability of system -LRB- 1 -RRB- is equivalent to each one of the following statements -LRB- cf. -LSB- 1 -RSB- , -LSB- 4 -RSB- , -LSB- 5 -RSB- -RRB- . 1 . rank -LSB- B ; AB ; : : : ; An ? 1B -RSB- = n 2 . rank -LSB- A ? ? I ; B -RSB- = n for every eigenvalue ? of A 3 . for every monic n-th order polynomial p -LRB- s -RRB- there is a real matrix F such that the characteristic polynomial of A + BF equals p -LRB- s -RRB- The matrix F in the third statement can be interpreted as the matrix in the state feedback u -LRB- t -RRB- = Fx -LRB- t -RRB- . The statement implies that if system -LRB- 1 -RRB- is controllable the eigenvalues of the system can be shifted by a real state feedback to arbitrary places in the complex plane , provided these places are located symmetrically around the real axis . In particular , this means that if system -LRB- 1 -RRB- is controllable , the eigenvalues of A+BF all can be placed inside IC ? by a proper choice of F . The converse is not true as can be seen from A = ? I and B = 0 . System -LRB- 1 -RRB- with these matrices is not controllable , but the application of any feedback u -LRB- t -RRB- = Fx -LRB- t -RRB- shifts the eigenvalues to -1 -LRB- in fact , keeps them at -1 -RRB- . In view of the previous it makes sense to call a system -LRB- 1 -RRB- stabilizable , if there is a matrix F such that all the eigenvalues of A + BF are inside IC ? -LRB- cf. -LSB- 6 -RSB- -RRB- . In the latter reference it is also shown that system -LRB- 1 -RRB- is stabilizable if and only if rank -LSB- A ? ? I ; B -RSB- = n for all ? with Re ? > = 0 , or equivalently , if rank -LSB- A ? ? I ; B -RSB- = n for all eigenvalues ? of A with Re ? > = 0 . In particular , if A has an eigenvalue ? = 0 , the condition rank -LSB- A ; B -RSB- = n is both a part of the necessary condition and the sufficiency condition for the stabilizability of system -LRB- 1 -RRB- . In many cases the zero-nonzero structure of the matrices A and B is known . The actual value of the nonzero entries in the matrices is not known -LRB- exactly -RRB- due to various reasons . In some cases however the signs of the nonzero entries are available . For instance , if a nonzero entry corresponds to a mass or an RC-time it is often not known what its exact value is , but it is known that its sign must be positive . Two examples of ` structured systems ' , however still without the sign information , are the systems given in the introduction with A = > = 1 > = 2 > = 3 ! ; B = > = 4 ! ; and with A = > = 1 > = 2 > = 3 ! ; B = > = 4 ! : The zero-nonzero structure of general system matrices A and B can be represented by means of a directed graph -LRB- cf. -LSB- 2 -RSB- , -LSB- 7 -RSB- -RRB- . In this paper we shall in fact use two types of directed graphs . The first type will be denoted G and will consist of a vertex set V = X -LSB- U and an arc set A . Here X denotes the set of state vertices and U the set of input vertices , X = fx1 ; x2 ; : : : ; xng and U = fu1 ; u2 ; : : : ; umg . The arc set A is defined as A = f -LRB- xj ; xi -RRB- jaij < > g -LSB- f -LRB- uj ; xi -RRB- jbij < > g , where the ordered pair -LRB- xj ; xi -RRB- represents an arc from vertex xj to vertex xi , and aij < > means that the -LRB- i ; j -RRB- entry of A is a nonzero . Similarly , for -LRB- uj ; xi -RRB- and bij < > 0 . The second type of directed graph that we use in this paper will be denoted Ge -LRB- e for extended -RRB- and will consist of the vertex set V as above and an arc set Ae , which is defined as Ae = A -LSB- f -LRB- xj ; ui -RRB- ji = 1 ; 2 ; : : : ; m ; j = 1 ; 2 ; : : : ; ng . So , Ae is obtained from A by adding all the ` feedback ' arcs from state vertices to input vertices . Below we occasionally make no distinction between state and input vertices , and we then write V = fv1 ; v2 ; : : : ; vn + mg . With this we can now introduce some terminology for both graphs , especially for graph G. Given two vertices ~ v , ^ v 2 V we say that there is a path in G from ~ v to ^ v , or that ^ v can be reached from ~ v , if there are vertices vi1 ; vi2 ; : : : ; vit 2 V such that vi1 = ~ v , vit = ^ v and -LRB- vij ; vij +1 -RRB- 2 A for all j = 1 ; 2 ; : : : ; t ? 1 . The begin vertex of the path is ~ v , and the end vertex is ^ v . In a similar way a path from ~ v to ^ v in Ge can be defined . In section 3 we say a state vertex is reachable from the from the input , if there is a path from some input vertex to that state vertex . In section 5 we call a state vertex unreachable from the input if the state vertex can not be reached from any input vertex . Below we also need the notion of cycle , in G as well as in Ge . A cycle is a path such that its begin vertex and end vertex coincide , and such that each vertex on it appears only once . The length of a cycle is the number of arcs it consists of . This number is also equal to the number of distinct vertices that appear on the cycle . Cycles of length one are also referred to as self loops . We say that a collection of cycles is pairwise disjoint if every two cycles of the collection have no vertex in common . We call a collection of pairwise disjoint cycles a cycle family , and we say that it has width k if in total k distinct state vertices appear on the cycles of the cycle family . In section 4 we consider the graph G without taking into account the set of input vertices U . Hence , V = X and A = f -LRB- xj ; xi -RRB- jaij < > g . We then say that the graph G is strongly connected if for every two vertices ~ x , ^ x 2 X , there is a path from ~ x to ^ x , and there is a path from ^ x to ~ x . It is well known that if the graph G is not strongly connected , then G can be decomposed into a number of strongly connected subgraphs that themselves can be ordered in a hierarchic way -LRB- cf. -LSB- 7 -RSB- -RRB- . The statement that the graph G is not strongly connected is equivalent to the statement that there exists a permutation matrix P such that PAP ? 1 = A11 A12 A22 ! with Aij an ni ? nj matrix , -LRB- i ; j -RRB- = -LRB- 1 ; 1 -RRB- ; -LRB- 1 ; 2 -RRB- ; -LRB- 2 ; 2 -RRB- and n1 > 0 , n2 > 0 , n1 + n2 = n . If a permutation matrix P as above exists , then we call A reducible , if not we say that A is irredicible . In section 3 we extend the notion of -LRB- ir -RRB- reducibility to also include the matrix B In addition to the zero-nonzero structure of the matrices A and B sometimes also the signs of the nonzeros in A and B are available . We then say that the matrices A and B are sign structured , and together form a sign structured system . The signs of the nonzeros can also be incorporated in the graph G by assigning them to the corresponding arcs as illustrated below . Given these arc signs we say that a cycle has a positive -LRB- negative -RRB- sign if the cycle contains an even -LRB- odd -RRB- number of arcs with a negative sign . We conclude this section by depicting the graphs associated to the examples of section 1 . The graphs G and Ge for the first example are as depicted below . G : - - r r r u x2 x1 - - Ge : - - r r r u x2 x1 - - oe oe The graphs G and Ge for the second example can be depicted as follows . G : - oer r r x2 x1 u - - Ge : - oer r r x2 x1 u - - -- If it is known in the second example that > = 2 ; > = 3 are negative and > = 1 ; > = 4 are positive , then the arcs in G can labeled as follows . G : - ? oe + r r r x2 x1 u - ? - + In section 2 we have seen that the above graph corresponds to a sign structured system that is not generically controllable , but that it is generically stabilizable . The next graph corresponds to a sign structured system that is generically unstabilizable . G : - ? oe + r r r x2 x1 u - + - ? 3 Generic controllability In this section we recall some results on generic controllability -LRB- cf. -LSB- 2 -RSB- , -LSB- 7 -RSB- , -LSB- 8 -RSB- , -LSB- 9 -RSB- -RRB- . Therefore , we assume that we know the zero-nonzero structure of the matrices A and B . In other words , we assume that we are given a pair of structured matrices -LRB- A ; B -RRB- , also called a structured pair , which together form a structured system . We call a pair -LRB- ~ A ; ~ B -RRB- of numerically specified matrices admissible to the structured pair -LRB- A ; B -RRB- if -LRB- ~ A ; ~ B -RRB- can be obtained from -LRB- A ; B -RRB- by fixing the nonzeros in the matrices A and B to some specific values . For instance , the next pair is admissible to the structured pair in the first example of the introduction 4 2 ? 8 ! ; 7 ! : To obtain this pair we must fix > = 1 = 4 , > = 2 = 2 , > = 3 = ? 8 and > = 4 = 7 . We say that the structured pair -LRB- A ; B -RRB- is generically controllable if there is an admissible pair -LRB- ~ A ; ~ B -RRB- for the pair -LRB- A ; B -RRB- that is controllable in the usual sense , i.e. rank -LSB- ~ B ; ~ A ~ B ; : : : ; ~ An ? 1 ~ B -RSB- = n , or equivalently rank -LSB- ~ A ? ? I ; ~ B -RSB- = n for every eigenvalue ? of ~ A . It can be shown that if there is one admissible pair that is controllable , then almost every admissible pair is controllable -LRB- cf. -LSB- 2 -RSB- -RRB- . To present graph theoretic conditions for the generic controllability of a structured system we need two notions . First we say that the structured pair -LRB- A ; B -RRB- is reducible if there is a permutation matrix P such that PAP ? 1 = A11 A12 A22 ! ; PB = B1 ! ; with Aij an ni ? nj matrix , -LRB- i ; j -RRB- = -LRB- 1 ; 1 -RRB- ; -LRB- 1 ; 2 -RRB- ; -LRB- 2 ; 2 -RRB- , B1 an n1 ? m matrix and n1 > 0 , n2 > 0 , n1 + n2 = n . If such a permutation matrix does not exist , we call the structured pair -LRB- A ; B -RRB- irreducible . It is obvious how the present notion of -LRB- ir -RRB- reducibility generalizes the one given in the previous section . Next we define the generic rank of the structured n ? -LRB- n + m -RRB- matrix -LSB- A ; B -RSB- to be the maximum of the rank of the n ? -LRB- n + m -RRB- matrix -LSB- ~ A ; ~ B -RSB- considering all pairs -LRB- ~ A ; ~ B -RRB- that are admissible to the structured pair -LRB- A ; B -RRB- . We denote this rank by g-rank -LSB- A ; B -RSB- . With the above two notions we can formulate the next well-known results -LRB- cf. -LSB- 2 -RSB- , -LSB- 8 -RSB- , -LSB- 9 -RSB- -RRB- . Theorem 3.1 The structured pair -LRB- A ; B -RRB- is generically controllable if and only if -LRB- 1 -RRB- . g-rank -LSB- A ; B -RSB- = n , and -LRB- 2 -RRB- . the pair -LRB- A ; B -RRB- is irreducible . Theorem 3.1 can also be formulated in graph theoretic terms in the following way -LRB- cf. -LSB- 7 -RSB- -RRB- . Theorem 3.2 The structured pair -LRB- A ; B -RRB- is generically controllable if and only if -LRB- 1 -RRB- . in the graph Ge there is a cycle family of width n , and -LRB- 2 -RRB- . in the graph G -LRB- or Ge -RRB- every state vertex can be reached from the input . Theorem 3.2 immediately follows from theorem 3.1 from the facts that g-rank -LSB- A ; B -RSB- = n if and only if in Ge there is a cycle family of width n , and -LRB- A ; B -RRB- is irreducible if and only if in G every every state vertex is reachable from the input -LRB- cf. -LSB- 7 -RSB- -RRB- . Finally , theorem 3.2 can also be formulated completely in terms of the graph G using a more sophisticated graph theoretic notion . Theorem 3.3 The structured pair -LRB- A ; B -RRB- is generically controllable if and only if the graph G is spanned by means of a cacti . For the definition of a cactus or cacti we refer to -LSB- 2 -RSB- . 4 Generic stability In this section we recall some results on generic stability . For that purpose , we consider the original system -LRB- 1 -RRB- , but we disregard the input . Hence , we consider _ x -LRB- t -RRB- = Ax -LRB- t -RRB- : -LRB- 2 -RRB- As we have seen in the examples in the introduction , for the stability of system -LRB- 2 -RRB- it is not enough to know the zero-nonzero structure of A alone . Also some knowledge of the signs of the nonzeros in A is required . Therefore , we assume in this section that , in addition to the zero-nonzero structure of A , also the signs of the nonzeros are available when required . Because of this we can say that the matrix A is sign structured . In the context of the above , we say that the numerically specified matrix ~ A is sign admissible to the sign structured matrix A , if ~ A can be obtained from A by fixing its nonzero entries to some specific numerical values in accordance to the given signs . For instance , ~ A is sign admissible to A , when A = + ? + ! ; ~ A = 1 ? 8 21 ! : Inspired by -LSB- 10 -RSB- and -LSB- 11 -RSB- -LRB- see also -LSB- 3 -RSB- and -LSB- 12 -RSB- -RRB- , we say that the sign structured matrix A is generically stable , if almost every matrix ~ A that is sign admissible to A , is stable -LRB- we recall that throughout this paper stability actually means asymptotic stability -RRB- . We note that generic stability of a sign structured matrix A is a strong property , since it requires the stability of almost every sign admissible matrix ~ A . This is in contrast with generic controllablity , where controllability is required for just one admissible pair and where it can be proved that controllability for one admissible pair implies controllability for almost every admissible pair . To develop and present results on generic stability of the sign structured matrix A , we may assume without loss of generality that the matrix A is irreducible , or equivalently that its graph is strongly connected . Indeed , if A is not irreducible , then there is a permutation matrix P such that PAP ? 1 = BBBB @ A11 A12 : : : A1k A22 : : : A2k ... ... ... . . : : : Akk 1 CCCCA with A11 ; A22 ; : : : ; Akk all irreducible matrices . It is clear that the generic stability of A is equivalent to the generic stability of Aii for all i = 1 ; 2 ; : : : ; k. For the graph G the above means the decomposition into k strongly connected subgraphs that are hierarchically ordered . From the previous it is clear that without loss of generality we can assume that A is irreducible , and that its graph is strongly connected . We now formulate the next theorem -LRB- cf. -LSB- 11 -RSB- -RRB- , where det stands for determinant . Theorem 4.1 The irreducible sign structured matrix A is generically stable if and only if for any matrix ~ A that is sign admissible to A the following five conditions hold : 1 . det ~ A < > 2 . ~ aii < = for all i = 1 ; 2 ; : : : ; n 3 . ~ aii < for all at least one i = 1 ; 2 ; : : : ; n 4 . ~ aij ~ aji < = for all i ; j = 1 ; 2 ; : : : ; n with i < > j 5 . ~ ai1i2 ~ ai2i3 : : : ~ aiki1 = for all i1 ; i2 ; : : : ; ik = 1 ; 2 ; : : : ; n with ij < > it for j < > t and k > = 3 A graph theoretic version of theorem 4.1 can be formulate as follows . Theorem 4.2 The irreducible sign structured matrix A is generically stable if and only if in its strongly connected graph G the following five conditions hold : 1 . there is a cycle family of width n 2 . there is no self loop with a positive sign 3 . there is at least one self loop with a negative sign 4 . the sign of any cycle of length two is negative 5 . there are no cycles of length three or more Theorem 4.2 can be proved easily from theorem 4.1 , using the fact that an n ? n -LRB- sign -RRB- structured matrix A generically has rank n if and only if in its graph G there is a cycle family of width n -LRB- cf. -LSB- 7 -RSB- -RRB- . Hence , the equivalence of the first condition in theorems 4.1 and 4.2 easily follows . The equivalence of the other conditions in both theorems is also immediate . For the case the matrix A is not irreducible we can easily prove the following corollary . Corollary 4.3 The sign structured matrix A is generically stable if and only if in its graph G the following five conditions hold : 1 . there is a cycle family of width n 2 . there is no self loop with a positive sign 3 . in each strongly connected component of G there is a self loop with a negative sign 4 . the sign of any cycle of length two is negative 5 . there are no cycles of length three or more In the next section we combine the results of the present section and the previous section . In that respect , we note that if there is no cycle family of width n in Ge , there is also no such family in G. Hence , from theorems 3.2 and 4.2 it follows that if there is no cycle family of width n in Ge , then there is no generic controllability but also no generic stability . To conclude this section we recall that generic stability is a strong notion , since it requires almost an entire class of systems to be stable . We also note that when a system is not generically stable , it is not necessarily generically unstable . To see this , we consider system -LRB- 2 -RRB- with A = + + ? ? ! ; ~ A = > = 1 > = 2 > = 3 > = 4 ! : Hence , > = 1 ; > = 2 2 -LRB- 0 ; 1 -RRB- and > = 3 ; > = 4 2 -LRB- ? 1 ; 0 -RRB- . It follows from theorem 4.1 that A is not generically stable , because the sign of a11 is always positive . With the Routh-Hurwitz criterion it follows that ~ A is stable for a given > = = -LRB- > = 1 ; > = 2 ; > = 3 ; > = 4 -RRB- > if and only if trace ~ A = > = 1 + > = 4 < and det ~ A = > = 1 > = 4 ? > = 2 > = 3 > 0 . It is not difficult to show that these two inequalities , with the fact that > = 1 ; > = 2 2 -LRB- 0 ; 1 -RRB- and > = 3 ; > = 4 2 -LRB- ? 1 ; 0 -RRB- , define a set in IR4 that has a positive Lebesgue measure . Hence , the matrix A is also not generically unstable . 5 Generic stabilizability In this section we combine the results of the previous sections and we derive a method to determine whether or not a system is generically stabilizable . Therefore , we first introduce the notion generic stabilizability for a sign structured pair -LRB- A ; B -RRB- . Such a pair consists of matrices A and B of which the zero-nonzero structure is given , together with the signs of the nonzero entries . We say that a sign structured pair -LRB- A ; B -RRB- is generically stabilizable , if almost every sign admissible pair -LRB- ~ A ; ~ B -RRB- is stabilizable in the usual sense , i.e. is such that ~ A + ~ B ~ F only has eigenvalues in IC ? for a suitably chosen ~ F . In the spirit of the examples in the introduction we can first investigate whether or not a given sign structured pair -LRB- A ; B -RRB- is generically controllable . Here we do not yet have to pay attention to the signs of the nonzeros in the matrices A and B . If -LRB- A ; B -RRB- is generically controllable , then we are done and we can conclude that the pair -LRB- A ; B -RRB- also is generically stabilizable . If -LRB- A ; B -RRB- is not generically controllable , or in other words , is generically uncontrollable , then by theorem 3.1 either g-rank -LSB- A ; B -RSB- < n , or -LRB- A ; B -RRB- is reducible , or both . The inequality g-rank -LSB- A ; B -RSB- < n is equivalent to the fact that rank -LSB- ~ A ; ~ B -RSB- < n for every pair -LRB- ~ A ; ~ B -RRB- that is admissible to -LRB- A ; B -RRB- . The latter is equivalent to the statement that ? = is an unstable and uncontrollable eigenvalue of ~ A for every admissible pair -LRB- ~ A ; ~ B -RRB- . In turn , this is equivalent to the statement that ~ A + ~ B ~ F has an eigenvalue ? = for every admissible pair -LRB- ~ A ; ~ B -RRB- and for every feedback matrix ~ F . So , if g-rank -LSB- A ; B -RSB- < n , the pair -LRB- A ; B -RRB- is not generically stabilizable , and in fact , the pair -LRB- A ; B -RRB- can said to be generically unstabilizable . Conversely , if A has an unstable eigenvalue ? = -LRB- since g-rank A < n -RRB- and this eigenvalue is generically uncontrollable , then g-rank -LSB- A ; B -RSB- < n . We note that in the previous the signs of the nonzeros are not relevant . Now assume that g-rank -LSB- A ; B -RSB- = n , but that -LRB- A ; B -RRB- is reducible . So , there exists a permutation matrix P such that ? PAP ? 1 PB ? = A11 A12 B1 A22 ! with g-rank -LSB- A11 ; B1 -RSB- = n1 , g-rank A22 = n2 , n1 > 0 , n2 > 0 , n1 + n2 = n and -LRB- A11 ; B1 -RRB- irreducible . Hence , the pair -LRB- A11 ; B1 -RRB- is generically controllable and the eigenvalues of A22 are fixed . Therefore , the original sign structured pair -LRB- A ; B -RRB- is generically stabilizable if and only if the sign structured matrix A22 is generically stable . These observations lead to the next theorems , where we refer to the subsystem _ x2 -LRB- t -RRB- = A22x2 -LRB- t -RRB- as the original system modulo its generically controllable part . Theorem 5.1 The sign structured pair -LRB- A ; B -RRB- is generically stabilizable if and only if -LRB- 1 -RRB- . g-rank -LSB- A ; B -RSB- = n , and -LRB- 2 -RRB- . the system modulo its generically controllable part is generically stable . Note that in the previous the matrix A22 still can be reducible . Hence , there may exist a permutation matrix P such that P AP ? 1 = BBBB @ A011 A012 : : : A01l A022 : : : A02l ... ... ... . . : : : A0ll 1 CCCCA with each A0ii , i = 1 ; 2 ; : : : ; l irreducible . To check whether or not A22 is generically stable , we can apply theorem 4.1 for each A0ii , i = 1 ; 2 ; : : : ; l . In order to present graph theoretic results we first decompose X = Xr -LSB- Xu , Xr \ Xu = ; with Xr the set of state vertices that are reachable from the input -LRB- r for reachable , and u for unreachable -RRB- . Next we remove from G all vertices in Xr , together with all the arcs that have a vertex in Xr as begin vertex or as end vertex . We denote the graph that remains by Gu . Now combining corollary 4.3 with theorem 5.1 we obtain the following theorem . Theorem 5.2 The pair -LRB- A ; B -RRB- is generically stabilizable if and only if the following five conditions hold : 1 . in the graph Ge there is a cycle family of width n 2 . in Gu there is no self loop with a positive sign 3 . in each strongly connected component of Gu there is a self loop with a negative sign 4 . in Gu the sign of any cycle of length two is negative 5 . in Gu there are no cycles of length three or more Let Gu consist of n2 state vertices -LRB- and no input vertices -RRB- , then a proof of theorem 5.1 easily follows from the previous results with the fact that condition 1 . of the theorem implies that in Gu there is a cycle family of width n2 . The above theorems are the main result of this paper and will be illustrated in the next section . 6 Example In this section we give an illustration of the method of this paper . To do this , we consider the following structured pair -LRB- A ; B -RRB- : A = BBBBBBBBBBBBBBBB @ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 1 CCCCCCCCCCCCCCCCA ; B = BBBBBBBBBBBBBBBB @ ? ? ? 1 CCCCCCCCCCCCCCCCA : In the above matrices ? denotes a nonzero entry , and for the time being we do not pay attention to their signs . The zero-nonzero structure in A and B can be represented by the graph G as depicted below . r r r r r r r r r r r r 6 6 6 6 6 ? ????????? 3 ?????? ? SSSSSSSSSSSS w S S S S S S S S S S S S o ?????? > = Q Q Q Q Q Q Q Q Q k oe 6 ? - oe - - - 6 6 u1 x8 x5 x1 u2 x9 x6 x10 x7 x4 x2 x3 G : As indicated in section 3 we will also be dealing with an extension of the graph G which is denoted Ge , and which can be obtained from G by including all possible ` feedback ' arcs -LRB- xj ; ui -RRB- , i = 1 ; 2 ; j = 1 ; 2 ; : : : ; 10 . We shall not depict the graph Ge , but it will be clear that in the following a cycle family in Ge of width 10 is depicted . r r r r r r r r r r r r 6 6 6 6 6 6 ? - oe ? ? - u1 x8 x5 x1 u2 x9 x6 x10 x7 x4 x2 x3 From the graph G it follows that not every state vertex is reachable from the input . So , by theorem 3.2 , the structured pair -LRB- A ; B -RRB- is not generically controllable . It follows from G that the set of state vertices that are reachable from the input is given by fx1 ; x2 ; x3 ; x5 ; x6 ; x8 ; x9g , and consequently fx4 ; x7 ; x10g is the set of unreachable state vertices . To investigate whether or not the pair -LRB- A ; B -RRB- is generically stabilizable , we need to know the signs of -LRB- some of -RRB- the arcs between the unreachable state vertices . Or , in other words , we need to know the signs of -LRB- some of -RRB- the nonzero entries in A that have row and column index in f4 ; 7 ; 10g . We assume that sign a44 = ? , sign a47 = + , sign a74 = ? , sign a107 = + , sign a1010 = ? , and we consider the subgraph Gu obtained from G by removing in the latter all the reachable state vertices and arcs that have these vertices as begin vertex or end vertex . The graph Gu can be depicted as follows . r r r - oe - - x10 x7 x4 Gu : Note that Gu is not strongly connected and that it can be decomposed into two strongly connected components as is depicted in the next , where we have labeled the arcs in Gu in accordance to their signs . r r r - + oe ? - ? - ? x10 x7 x4 The two associated irreducible matrices have dimensions 2 ? 2 and 1 ? 1 , respectively , and have the following sign structure . ? ? + ! ; ? ? ? : According to theorem 4.1 both matrices are generically stable . Note that not all signs of arcs between unreachable state vertices are relevant . Indeed , in the above only the signs of a44 , a47 , a74 and a1010 are relevant for the generic stabilizability . In summary , if are given A and B as in the beginning of this section and we know that the signs of a44 , a1010 and a47a74 are negative , we can conclude that the structured pair -LRB- A ; B -RRB- is stabilizable for almost every value of its nonzero entries . 7 Remarks In this paper we introduced the notion of generic stabilizability for structured pairs -LRB- A ; B -RRB- of which the signs of the nonzeros may be used also . We developed a method to determine whether or not such a pair is generically stabilizable . To do this , we used graph theoretic representations of the zero-nonzero structure of the matrices A and B. With the method it is possible to indicate which signs are relevant for generic stabilizability . We believe that the method can be used to establish stabilizability in situations in which the values , but not the signs , of nonzero entries are missing . Also we see our conditions as easy sufficient condition for the stabilizability of a system . References -LSB- 1 -RSB- W.M. Wonham , Linear Multivariable Control : a geometric approach , 3rd ed. , SpringerVerlag , New York , 1985 . -LSB- 2 -RSB- C.T. Lin , `` Structural controllability '' , IEEE Trans . Aut . Contr. , Vol . AC-19 , 1974 , pp. 201-208 . -LSB- 3 -RSB- C. Jeffries , `` Qualitative stability and digraphs in model ecosystems '' , Ecology , Vol . 55 , 1974 , pp 1415-1419 . -LSB- 4 -RSB- E.D. Sontag , Mathematical Control Theory : Deterministic Finite Dimensional Systems , Springer-Verlag , New York , 1990 . -LSB- 5 -RSB- M.L.J. Hautus , `` Controllability and observability conditions of linear autonomous systems '' , Proc . Kon . Ned . Akad . Wetensch. , Ser . A , Vol . 72 , 1969 , pp 443-448 . -LSB- 6 -RSB- M.L.J. Hautus , `` Stabilizability , controllability and observability of linear autonomous systems '' , Proc . Kon . Ned . Akad . Wetensch. , Ser . A , Vol . 73 , 1970 , pp 448-455 . -LSB- 7 -RSB- K.J. Reinschke , Multivariable Control : A Graph-theoretic Approach , Springer-Verlag , New York , 1988 . -LSB- 8 -RSB- R.W. Shields , J.B. Pearson , `` Structural controllability of multi-input linear systems '' , IEEE Trans . Aut . Contr. , Vol . AC-21 , 1976 , pp. 203-212 . -LSB- 9 -RSB- K. Glover , L.M. Silverman , `` Characterization of structural controllability '' , IEEE Trans . Aut . Contr. , Vol . AC-21 , 1976 , pp . 354-537 . -LSB- 10 -RSB- J. Quirk , R. Ruppert , `` Qualitative economics and stability of equilibrium '' , Rev. Economic Studies , Vol . 32 , 1965 , pp 311-326 . -LSB- 11 -RSB- T. Yamada , `` Generic matrix sign-stability '' , Canad . Math . Bull. , Vol . 30 , 1987 , pp 370 - 376 . -LSB- 12 -RSB- C. Jeffries , V. Klee , P. van den Driessche , `` When is a matrix sign stable ? '' , Can . J. Math. , Vol . XXIX , 1977 , pp 315-326 .

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O B I O O O O O O B I O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O B I O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O

directed graph
generic controllability
generic stability