Minimal set of generators of controllability space for singular linear dynamical systems

Minimal set of generators of controllability space for singular linear dynamical systems MARIA ISABEL GARC´ IA-PLANAS Universitat Polit` ecnica de Catalunya Departament de Matem` atiques Mineria 1, C, 1-3 Barcelona SPAIN [email protected] Abstract: Due to the significant role played by singular systems in the form E˙x(t) = Ax(t), on mathematical modeling of science and engineering problems; in the last years recent years its interest in the descriptive analysis of its structural and dynamic properties. However, much less effort has been devoted to studying the exact controllability by measuring the minimum set of controls needed to direct the entire system E˙x(t) = Ax(t)to any desired state. In this work, we focus the study on obtaining the set of all matrices Bwith a minimal number of columns, by making the singular system E˙x(t) = Ax(t) + Bu(t)controllable. Key–Words: Controllability, exact controllability, eigenvalues, eigenvectors, singular linear systems. 1 Introduction In these recent years, the study of the control of complex networks with linear dynamics has gained importance in both science and engineering; because of this kind of systems appear in applications such us electrical networks, simulation of the dynamics of multibody systems, modelling chemical reactions among others. Controllability of a dynamical system has being largely studied by several authors and under many different points of view, (see [2], [4], [5], [6], [11], [13], [14], [19], [20] and [22] for example). Between different aspects in which we can study the controllability we have the notion of structural controllability that has been proposed by Lin [15] as a framework for studying the controllability properties of directed complex networks where the dynamics of the system is governed by a linear system. Recent studies over the structural controllability can be found on [16]. Another important aspect of control is the notion of output controllability that describes the ability of an external data to move the output from any initial condition to any final in a finite time. Some results about can be found in [11]. This concept has some interest in codes theory (see [9], for example). In this article, we analyze the exact controllability concept as a generalization for singular linear dynamical systems of the concept given in [23] for standard linear dynamical systems and in [8] for `-order standard linear systems. This concept is based on the maximum multiplicity to identify the minimum set of driver nodes required to achieve full control of networks with arbitrary structures and link-weight distributions. The notion of exact controllability has interest in different topics as for example is an adequate notion in hyperbolic problems (see [12], [17]), also, can be used to explore the effect of interconnections’ correlation on the controllability of multiplex networks, S. Nie, X. Wang and B.Wang in [18] find that the minimal number of driver nodes decreases with correlation for lower density of interconnections. We were focusing the study on the obtention of the set of all matrices Bmaking the system E˙x(t) = Ax(t) + Bu(t)exact controllable. These sets are obtained from the quasi Weierstraß reduced form and from getting the transformation matrices of the system to its reduced form. We have included some examples to make the work easier readable. Finally, we introduce exact controllability for second order singular linear systems, because they are interest in application to power systems and they are also used in conjuction with the analysis and modelling of flexible beams [1]. 2 Equivalence relation It is well known that many complex networks have linear dynamics and they have a state space representation for its description: E˙x(t) = Ax(t) + Bu(t)o(1) where E, A ∈Mn(IC) and B∈Mm×n(IC). When B= 0 the system is called homogeneous. For simplicity, from now on we will write the system 1 as the triple of matrices (E, A, B)or as a pair (E, A)for the homogeneous case. Trying to understand the properties of the system they use purely algebraic techniques. The central aspect of this focus is defining an equivalence relation preserving these properties. The equivalence relation considered corresponds to standard transformation basis changes for the state space and pre multiplication for an invertible matrix. Definition 1 The systems E˙x(t) = Ax(t)and ¯ E˙ ¯x(t) = ¯ A¯x(t)are equivalent if and only if, there exist a basis change in the state space ¯x(t) = Px(t) and an invertible matrix Q∈Gl(n; IC) such that ¯ E˙ ¯x(t) = QEP ˙x(t) = QAPx(t) = ¯ A¯x(t) We can characterize equivalent systems, by associating matrix pencils to them in a natural way: The matrix pencil λE +Ais naturally associated to the pair (E, A)representing a singular linear system E˙x(t) = Ax(t) Equivalent pairs are those whose associated matrix pencils are “strictly equivalent”. Remember that two pencils λE +Aand λ¯ E+¯ Aare strictly equivalent, if and only if, there exist invertible matrices P, A ∈Gl(n; IC) such that λ¯ E+¯ A=Q(λE +A)P=λQEP +QAP. Observe that in the case where the system is standard (i.e. E=I), the equivalence relation considered, corresponds to the similarity relation of square matrices. We will consider the case of systems where the matrix pencil λE +Ais regular, as it is usual, in order to ensure that the system has a unique solution for any sufficiently differentiable input function u(t). Under this regularity assumption, there exist invertible matrices Q, P ∈Gln(IC) such that ¯ E=QEP = diag (Ir, N),¯ A=QAP =diag (J, In−r), where J a Jordan matrix and Na nilpotent matrix and we will say that the pencil is it is canonical reduced form. So, considering x(t) = P¯x(t)and premultiplying the system 1 by Qand calling ¯ B=QB, the system can be written as ¯ E˙ ¯x=¯ A¯x(t) + ¯ Bu(t), that is to say: Ir0 0N! ˙ ¯x1(t) ˙ ¯x2(t)!= J0 0In−r! ¯x1(t) ¯x2(t)!+ ¯ B1 ¯ B2!u(t) (2) In general we say that two systems (E, A, B)and (¯ E, ¯ A, ¯ B)are equivalent if and only if, there exist invertible matrices Pand Qsuch that (¯ E, ¯ A, ¯ B) = (QEP, QAP, QB). This equivalence relation corresponds with strict equivalence of the pencil sE −A B . So, the collection of invariants of the pencil are the invariants for the system. In particular, for the systems E˙x(t) = Ax(t)the generalized eigenvalues of the system are the generalized eigenvalues of the pencil. Definition 2 λ0is a generalized eigenvalue of the system, if and only if rank (λ0E−A)< n. It is easy to observe that the generalized eigenvalues of sE −Aare the eigenvalues of the matrix Jin the reduced form 2: rank (λ0E−A) = rank (λ0Q1¯ EP−1−Q−1¯ AP−1) = rank Q−1(λ0¯ E−¯ A)P−1= rank (λ0¯ E−¯ A) and rank λ0Ir−J λ0N−In−r!< n if and only if rank (λ0Ir−J)< r. In a more general form we have the following proposition. Proposition 3 Let (E, A)and (¯ E, ¯ A)be two equivalent systems, λ0is an eigenvalue of (E, A)if and only if it is is an eigenvalue of (¯ E, ¯ A)=(QEP, QAP). If λ0is a generalized eigenvalue of (E, A), then there exists a vector 06=w0such that (λ0E−A)w0= 0. Definition 4 This vector is called the generalized eigenvector associated to λ0. Proposition 5 Let (E, A)and (¯ E, ¯ A)be two equivalent systems, w0is an eigenvector of (E, A)if and only if P−1w0is an eigenvector of (¯ E, ¯ A) = (QEP, QAP). Proof: Let (E, A)and (¯ E, ¯ A)=(QEP, QAP)two equivalent systems. (λ0E−A)w0= 0 if and only if (λ0Q−1¯ EP−1− Q−1¯ AP−1)w0= 0, equivalently if and only if Q−1(λ0¯ E−¯ A)P−1w0= 0, that is yo say, if and only if (λ0¯ E−¯ A)P−1w0= 0.utIn the particular case where the equivalent system is in the reduced form P−1w0= (v1 0,0) ∈ICr×ICn−rand v1 0is an eigenvector of J: λ0Ir λ0N! v1 0 v2 0!= J In−r! v1 0 v2 0! Clearly v2 0= 0 and Jv1 0=λ0v1 0. It is important the following result. Proposition 6 Eigenvectors corresponding to different eigenvalues are independent. Proof: Let w1, . . . , w``eigenvectors corresponding to λ1, . . . , λ`with λi6=λjfor all i6=jand consider P` i=1 αiwi. Then P` i=1 αiP−1wi= 0 and ¯ AkP` i=1 αiP−1wi=P` i=1 αiλk iP−1wi= 0 Solving the system P` i=1 αiP−1wi= 0 P` i=1 αiλiP−1wi= 0 . . . P` i=1 αiλ`−1 iP−1wi= 0            , we obtain αi= 0, for i= 1,...`. Consequently, the vectors are linearly independent. ut It is important to remark the following proposition corresponding to the eigenvectors of infinity. Proposition 7 Let (E, A)and (¯ E, ¯ A)be two equivalent systems and 06=w∈ICn.w∈Ker Eif and only if P−1w∈Ker ¯ Ewith (¯ E, ¯ A)=(QEP, QAP). Proof: Let (E, A)and (¯ E, ¯ A)=(QEP, QAP)two equivalent systems. Ew = 0 if and only if Q−1¯ EP−1w= 0, equivalently if and only if ¯ EP−1w= 0.ut 2.1 Quasi-Weierstraß form The pair of matrices (E, A)corresponding to a regular pencil, can be reduced to a weaker form called “Quasi-Weierstraß form” (see [3]) in the following manner: Let P=V W and Q=EV AW −1. Matrices V∈Mn×r(C)and W∈Mn×(n−r)(C)are in such a way that V W and EV AW are invertible. (QEP, QAP) = Ir N!, Ar In−r!!= (˜ E, ˜ A), where Aris some matrix and Nis nilpotent. The vector spaces Im Vand Im Ware spanned by the generalized eigenvector at the finite and infinite eigenvalues respectively, and they are derived by the following recursive subspace iteration with a limited number of steps called Wong sequences [21]. V0=Cn, Vi+1 ={v∈Cn|Av ∈E(Vi)} W0={0}, Wi+1 ={v∈Cn|Ev ∈A(Wi)} verifying V0⊇V1⊇. . . ⊇V`=V`+1 =. . . V`+q=V∗⊇Ker A W0⊆W1⊆. . . ⊆wm=Wm+1 =. . . Wm+q=W∗ It is easy to prove that `=mand satisfy AV ∗⊆ EV ∗and EW∗⊆AW∗. Matrices Vand Ware defined in such away that V∗=Im Vand W∗=Im W. Example 8 Let (E, A)a system with E=   112 123 112  and A=   2−1−1 −1 2 −1 −1−1 2    W0={0} W1=Ker E=   1 1 −1  =W2=W V0= IR3 V1=   1 0 0 1 1 0   =V2=V then,    3 1 2 4 2 2 3 1 −4   −1    2−1−1 −1 2 −1 −1−1 2      1 0 1 0 1 1 1 0 −1  =    100 010 000  =˜ E    3 1 2 4 2 2 3 1 −4   −1    112 123 112     1 0 1 0 1 1 1 0 −1  =    2−2 0 −550 0 0 1   =˜ A Similarly to propositions 3, 5 and 7 we can prove the following results. Proposition 9 Let (E, A)be a system and (˜ E, ˜ A)its quasi-Weierstraß form. λ0is an eigenvalue of (E, A) if and only if it is is an eigenvalue of (˜ E, ˜ A). Proposition 10 Let (E, A)be a system and (˜ E, ˜ A)its quasi-Weierstraß form. w0is an eigenvector of (E, A) if and only if P−1w0is an eigenvector of (˜ E, ˜ A)and P−1w0= (v1 0,0) ∈ICr×ICn−rand v1 0is an eigenvector of Ar. It is important to remark the following proposition corresponding to the eigenvectors at the infinity. Proposition 11 Let (E, A)be a system and (˜ E, ˜ A) its quasi-Weierstraß form. Let us consider a non-zero vector w∈ICn. Then, w∈Ker Eif and only if P−1w∈Ker ˜ Ewith (˜ E, ˜ A)=(QEP, QAP). 2.2 Controllability An important concept concerning structural properties is the controllability that is defined as follows Definition 12 The system 1 is called controllable if, for any t1>0,x(0) ∈ICnand w∈ICn, there exists a control inpunt u(t)sufficiently smooth such that x(t1) = w. The controllability character can be computed by means the generalized Hautus test for controllability of singular systems. Proposition 13 ([10]) The system 1 is controllable if and only if: rank E B =n rank sE −A B =n, ∀s∈IC    .(3) Remark 14 The first condition of the proposition implies that the system is standardizable under derivative feedback. Remember that, system is standardizable under derivative feedback if and only if, there exists a matrix F∈Mm×n(IC) such that E+BF is invertible and the system obtained by derivative feedback (E+BF) ˙x(t) = Ax(t)+Bu(t)can be standardized premultiplying it by (E+BF)−1. Remark 15 Controllability character can be computed by means the rank of a certain numerical matrix constructed gluing matrix blocks E B 0 A0B! in the lower right corner (see [10]). 3 Exact controllability There are many possible control matrices Bin the system 1 that satisfy the controllability condition. The goal is to find the set of all possible matrices B, having the minimum number of columns corresponding to the minimum number nB(E, A)of independent controllers required to control the whole network. Definition 16 Let (E, A)be a pair of matrices. The exact controllability nB(E, A)is the minimum of the rank of all possible matrices Bmaking the system 1 controllable. nB(E, A) = min {rank B, ∀B∈Mn×i1≤i≤n, (E, A, B)controllable}. (4) If confusion is not possible we will write simply nB. Taking into account the generalized Hautus condition 3, it is straightforward the following proposition. Proposition 17 The exact controllability nBis invariant under equivalence relation considered, that is to say: for any couple of invertible matrices (Q, P), nB(E, A) = nB(QEP, QAP). Proof: rank QEP QB = rank QE B  P I!=rank E B  rank sQEP −QAP QB = rank QsE −A B  P I!= rank sE −A B ut As a consequence, if necessary we can consider (E, A)in its canonical reduced form. Example 18 1) If E=A= 0,nB=n 2) If E=Iand A=diag(λ1, . . . , λn)with λi6= λjfor all i6=j, then nB= 1, (it suffices to take B= (1 . . . 1)t). 3) If E= 1 0!and A= 0 1!,nB= 1. It suffices to consider B= 1 1!. Remark 19 Not every matrix Bhaving nBcolumns is valid to make the system controllable. For example if E=I,A=diag(1,2,3) and B= (1,0,0)t, the system (A, B)is not controllable, (rank B AB A2B= 1 <3, or equivalently rank A−λI B = 2 for λ= 2,3. For standard systems we have the following result. Proposition 20 ([23]) nB=maxi{µ(λi)} where µ(λi) = dim Ker (A−λiI)is the geometric multiplicity of the eigenvalue λi. Example 21 ([7]) 1) If A= 0,nD=n 2) If A=diag(λ1, . . . , λn)with λi6=λjfor all i6=j, then nD= 1, (it suffices to take B= (1 . . . 1)t). 3) Not every matrix Bhaving nDcolumns is valid to make the system controllable. For example if A=diag(1,2,3) and B= (1,0,0)t, the system (A, B)is not controllable, (rank B AB A2B= 1 <3, or equivalently rank A−λI B = 2 for λ= 2,3. For singular systems it is obvious that nB≥n− rank E=nE. Theorem 22 Let (E, A)be a singular system. The exact controllability nBis computed in the following manner. nB=max {nE, µ(λi)} where µ(λi) = dim Ker (λiE−A)and λi(for each i) is the eigenvalue of pencil sE −A. Proof: Proposition 17 permit us to consider the system in its canonical reduced form rank (E, B) = rank I N! ¯ B1 ¯ B2!! = n1+rank (N, ¯ B2) N=diag (N1, . . . , NnE)with Ni=       0 1 ...... 0 1 0       Taking ¯ B2=                       0 . . . 1 ... 0 . . . 1 0 . . . 0                       =w∞ 1. . . w∞ nE, we have that rank (N, ¯ B2) = n2 rank λiE−A B = rank λi I N!− J I! ¯ B1 ¯ B2!! = n2+rank λiI−J¯ B1. J=diag (J1(λ1), . . . , Jr(λr)),Ji(λi) = diag (Ji1(λi), . . . , Jiri(λi)) and Jij(λi) =       λi1 ...... λi1 λi       Taking ¯ B1i=                       0 . . . 1 ... 0 . . . 1 0 . . . 0                       =wλi 1. . . wλi µi(λi), we have that rank (λiI−J, ¯ Bii) = µ(λi) Consider now the following collection of vectors wλ1 1, . . . , wλ1 ` . . . wλr 1, . . . , wλr ` w∞ 1, . . . , w∞ ` where `=max (µ(λ1), . . . , µ(λr), nE)and we complete each series of vectors with the zero vectors in case its length is less than `. Finally we construct the family w1=wλ1 1+. . . +wλr 1+w∞ 1, . . . , w`= wλ1 `+. . . +wλr `+w∞ ` Clearly, rank E B =n rank λE −A B =n, for all λ∈IC Now, it suffices to remark that if we consider B= (bij)∈Mn×m(IC) with m<` i) if `=nEthen rank E B < n ii) if `=µ(λi)then rank λiE−A B < n ut Example 23 Let (E, A)be a singular system with E=                          1000000000000 0100000000000 0010000000000 0001000000000 0000100000000 0000010000000 0000001000000 0000000000000 0000000100000 0000000000000 0000000001000 0000000000000 0000000000010                          , and A=                          3000000000000 1300000000000 0130000000000 0003000000000 0001300000000 0000020000000 0000012000000 0000000100000 0000000010000 0000000001000 0000000000100 0000000000010 0000000000001                          We have: rank E= 10 rank (sE −A) =      11 for s= 3 12 for s= 2 13 for all s6= 2,3. So, nE= 3, µ(3) = 2, µ(2) = 1, then nB=max(3,2,1) = 3. In fact, taking B=                          100 000 000 001 000 100 000 100 000 010 000 001 000                          rank E B = 13 rank sE −A B = 13 for all s∈IC. Obviously, for all matrix B=                          b11 b12 b21 b22 b31 b32 b41 b42 b51 b52 b61 b62 b71 b72 b81 b82 b91 b92 b101 b102 b111 b112 b121 b122 b131 b132                          rank E B <13. 4 Generators of control space As we have discussed in the previous section, not every matrix B serves to make the system controllable, Of all possible, we want to find those with the least number of columns. To make the paper more understandable, we begin showing some particular cases. Proposition 24 Let (E, A)be a system with Einvertible. Then, the matrices Bmaking the system (E, A, B)controllable are those that make the standard system ˙z=AE−1z, controllable. Proof: rank E B =nfor all matrix B rank sE −A B = rank sI −AE−1B E I!= rank sI −AE−1B Then, rank sE −A B =n if and only if rank sI −AE−1B=n. ut The minimal sets of matrices Bmaking a standard systems controllable are described in [7]. We want to remark that the solution of the problem for standard systems is linked to the eigenstructure of the matrix AE−1. In our particular setup, the eigenstructure of the AE−1corresponds to the eigenstructure of the pair (E, A)because of det(sI − AE−1) det E= det((sI −AE−1)E) = det(sE − A). Proposition 25 Suppose that the pencil (˜ E, ˜ A)is in its quasi-Weirstraß form corresponding to a fast singular systems and let m1≥. . . msthe nilpotent indices of ˜ E. Consider 06= ¯w1∈Ker ˜ Em1\Ker ˜ Em1−1, 06= ¯w2∈Ker ˜ Em2\Ker ˜ Em2−1, linearly independent with w1. . .,06= ¯ws∈Ker ˜ Ems\Ker ˜ Ems−1, linearly independent with w1, . . . , ws. If we consider Im B= [w1, . . . , ws], then (˜ E, ˜ A, ˜ B)is controllable. Corollary 26 Let (E, A)be a singular fast system and (˜ E, ˜ A)=(QEP, QAP)its quasi-Weirestraß form. Then, taking B=Q˜ Bwith ˜ Bas in proposition 25, the system (E, A, B)is controllable. Proof: rank E Q ˜ B= rank Q−1E Q ˜ B P−1 I!= rank ˜ E˜ B rank sE −A Q ˜ B= rank Q−1E Q ˜ B P−1 I!= rank s˜ E−˜ A˜ But Proposition 27 Let (˜ E, ˜ A)a singular system in its quasi-Weierstraß form. Then, (˜ E, ˜ A, ˜ B)is controllable, where ˜ B= ˜ B1 ˜ B2!, with ˜ B1and ˜ B2as in proposition and . (if both matrices do not have the same number of columns we complete the one that has less number of columns with columns of zeros). Proof: Let ˜ B1∈Mr×m(IC) and ˜ B2∈Mn−r×`(IC) be the matrix such that (N, In−r,˜ B2)are controllable constructed as proposition 24 and proposition 25 respectively . If m6=`we complete with zero columns the matrix which the number of columns is smaller, matching in this way the size. rank Ir N!, ˜ B1 ˜ B2!!= r+rank N,˜ B2 rank s Ir N!− Ar In−r!, ˜ B1 ˜ B2!!= n−r+rank sIr−Ar,˜ B1 ut Theorem 28 Let (E, A)be a singular system and (˜ E, ˜ A) = (QEP, QAP)its quasi-Weirstraß form. Then, taking B=Q˜ Bwith ˜ Bas in proposition 27, the system (E, A, B)is controllable. Example 29 Retaking example 8, we have that ˜ B1= α+ 2β α−5β! with α, β 6= 0 and ˜ B2=γwithγ6= 0. Then B=Q˜ B=    3 1 2 4 2 2 3 1 −4   −1   α+ 2β α−5β γ  =    α/3 + 25β/6 + γ/6 −α/3−67β/6−γ/6 α/6 + β/3−γ/6   And the system (E, A, B)is controllable. 5 Exact controllability of second order singular linear systems Let us consider a homogeneous second order singular linear systems in the form E¨x(t) = A1˙x(t) + A0x(t)(5) And we ask for minimum of the rank of all possible matrices Bmaking the system 5 controllable. Remember that: Definition 30 The second-order linear system E¨x(t) = A1˙x(t) + A0x(t) + Bu(t)(6) is controllable if and only if there exists a control u1(t) = u−F1˙x−F0x(0), with Fi∈Mm×n(IC) such that the equation E¨x(t) = (A1+BF1) ˙x(t)+(A0+BF0)x(t)(7) has a stable solution. For simplicity we will write the system as a quadruple of matrices (E, A1, A0, B)and as a triple (E, A1, A0)for the homogeneous case The exact controllability nB(E, A1, A0)is the minimum of the rank of all possible matrices Bmaking the system 5 controllable. Definition 31 nB(E, A1, A0) = min {rank B, ∀B∈Mn×i1≤i≤n, (E, A1, A0, B)controllable}. (8) A manner to study that is linearizing the system in the following manner: X(t) = x(t) ˙x(t)!,˙ X(t) = ˙x(t) ¨x(t)!, I E!˙ X(t) = I A0A1!X(t) + 0 B!u(t). that we can write in a simple way: E˙ X(t) = AX(t) + Bu(t)(9) Taking into account that X1(t) = x1(t) ˙x1(t)is a solution of the linear system associated ˙ X(t) = AX(t) + Bu(t), if and only if x1(t)is a solution of the equation 6, it is not difficult to prove the following proposition. Proposition 32 The second order singular linear system is controllable if and only if the singular linear system associated 9 is controllable. Proof: The controllability of X(1) =AX+Bu ensures the existence of F∈Mm×`n(IC) such that X(1) =AX(t)+Bu1(t)with u1(t) = u(t)−FX(t) has a stable solution. Partitioning the matrix Finto two blocks F=F0F1we have that the equation E¨x(t) = A1˙x(t)+A0x(t)+Bu1(t)with u1(t) = u(t)−F0x(t)−F1˙x(t)has a stable solution. Converse is analogous. ut So, controllability character of second order singular linear systems it is reflected as follows rank E B = 2n rank sE−A B = 2n, ∀s∈IC    .(10) Now we present the main result that permit us to analyze the controllability character directly from the initial equation (1.1). Theorem 33 The second order linear 6 is controllable if and only if, rank E B =n rank s2E−sA1−A0B=n. (11) for all s∈IC. Proof: Making row and column elementary transformations we obtain rank E B = I E B != n+rank E B  rank sE−A B  =rank sI −I −A0sE −A1B!= rank 0−I s2E−sA1−A0sE −A1B!= rank 0−I s2E−sA1−A00B!= n+rank s2E−sA1−A0But So, as a consequence we can enunciate the following result: Theorem 34 nB(E, A1, A0) = maxi{nE, µ(λi)} where µ(λi) = dim Ker (λiE−A). Corollary 35 nB(E, A1, A0) = maxi{nE, ν(si)} where ν(si) = dim Ker (s2E−sA1−A0) for all si∈IC such that det(s2E−sA1−A0) = 0. 6 Conclusion In this work, given two n-order square matrices E, A defining regular generalized systems E˙x=Ax. We ask for minimal number of columns that must have a matrix Bin order to make the system (E, A, B)controllable. The sets of minimal generators of controllability spaces are obtained using the quasi Weierstraß reduced form. Examples have been included to make the work easier readable. References: [1] G. Antoniou, Second-order Generalized Systems: The DFT Algorithm for Computing the Transfer Function, WSEAS Transactions on Circuits. 2002, pp. 151–153. [2] J. Assan, A. Lafay and A. Perdon, Computation of maximal pre-controllability submodules over a Noetherian ring, Systems & Control Letters. 37(3), 1999, pp. 153–161. [3] T. Berger, A. Ilchmann and S. Trenn, The quasi- Weierstraß form for regular matrix pencils, Linear Algebra and its Applications. 436, 2012, pp. 4052–4069. [4] F. Cardetti and M. Gordina, A note on local controllability on lie groups, Systems & Control Letters. 57, 2008, pp. 978–979. [5] C.T. Chen, Introduction to Linear System Theory, Holt, Rinehart and Winston Inc, New York 1970 [6] M.I. Garc´ ıa-Planas, Generalized Controllability Subspcaes for Time-invariant Singular Linear Systems, IPACS Electronic Library, 2011. [7] M.I. Garc´ ıa-Planas, Exact Controllability of Linear Dynamical Systems: a Geometrical Approach, Applications of Mathematics. 1, 2017, pp. 37–47, DOI:10.21136/AM.2016.0427-15 [8] M.I. Garc´ ıa-Planas, Analizing exact controllability of l-order linear ystems. WSEAS transactions on systems and control. 12, pp. 232–239. [9] M.I. Garc´ ıa-Planas, El M. Souidi and L.E. Um, Convolutional codes under linear systems point of view. Analysis of output-controllability. WSEAS transactions on mathematics. 11(4), 2012, pp. 324–333. [10] M.I. Garcia-Planas, S. Tarragona and A. Diaz, Controllability of time-invariant singular linear systems From physics to control through an emergent view, World Scientific, 2010, pp. 112– 117. [11] M.I. Garcia-Planas and J.L. Dom´ ınguez, Alternative tests for functional and pointwise outputcontrollability of linear time-invariant systems, Systems & control letters, 62(5), 2013, pp. 382– 387. [12] M. Gugat, Exact Controllability In: Optimal Boundary Control and Boundary Stabilization of Hyperbolic Systems. Springer-Briefs in Electrical and Computer Engineering. Birkhuser, Cham, 2015. [13] A. Heniche and I. Kamwa, Using measures of controllability and observability for input and output selection, Proceedings of the 2002 IEEE International Conference on Control Applications, 2, 2002, pp. 1248–1251. [14] P. Kundur, Power System Stability and Control, McGraw-Hill, New York, USA 1994. [15] C. Lin, Structural controllability, IEEE Trans. Automat. Contr. 19, 1974, pp. 201–208. [16] Y. Liu, J. Slotine and A. Barab´ asi, Controllability of complex networks, Nature, 473(7346), 2011, pp. 167–173. [17] A.K. Nandakumaran, Exact controllability of Linear Wave Equations and Hilbert Uniqueness Method, Proceedings, Computational Mathematics, Narosa, 2005. [18] S. Nie, X. Wang, B. Wang Effect of degree correlation on exact controllability of multiplex networks Physica A: Statistical Mechanics and its Applications, 436(15), 2015, pp. 98–102. [19] M. P´ osfai, J. Gao, S.P. Cornelius, A.L. Barab´ asi and R.M. D’Souza, Controllability of multiplex, multi-time-scale networks, Physical Review E, 94(3), 2016. [20] R. Shields and J. Pearson, Structural controllability of multi-input linear systems. IEEE Trans. Automat. Contr, 21, 1976, pp.203–212.