### Paper Menu >>

### Journal Menu >>

Intelligent Control and Automation, 2010, 1, 36-47 doi:10.4236/ica.201.11005 Published Online August 2010 (http://www.SciRP.org/journal/ica) Copyright © 2010 SciRes. ICA Stability Analysis and Hadamard Synergic Control for a Class of Dynamical Networks* Xinjin Liu, Yun Zou School of Automation, Nanjing University of Science and Technology, Nanjing, China E-mail: liuxinjin2006@163 .com, zouyun@vip.163.com Received February 9, 2010; revised March 21, 2010; accepted June 28, 2010 Abstract Hadamard synergic control is a new kind of control problem which is achieved via a composite strategy of the state feedback control and the direct regulation of the part of connection coefficients of system state variables. Such a control is actually used very often in the practical areas. In this paper, we discuss Ha- damard synergic stabilization problem for a class of dynamical networks. We analyze three cases: 1) Syner- gic stabilization problem for the general two-node-network. 2) Synergic stabilization problem for a special kind of networks. 3) Synergic stabilization problem for special kind of networks with communication time-delays. The mechanism of the synergic action between two control strategies: feedback control and the connection coefficients regulations are presented. Keywords: Hadamard Synergic Control, Algebraically Graph Theory, Decentralized Feedback Control, Connection Coefficient Gain Matrix 1. Introduction Complex networks of dynamic agents have attracted great interesting in recently years. This is partly due to broad applications of multiagent systems in many areas including physicists, biologists, social scientists and con- trol scientists [1-3], distributed sensor networks [4], and congestion control in communication networks [5] and so on. In fact, a complex dynamical network can be viewed as a large-scale system with special interconnec- tions among its dynamical nodes from a system-theoretic point of view and when we solved the control problems of electric power systems, socioeconomic systems, etc., large-scale interconnected systems with many state vari- ables often appear. In order to stabilize large-scale inter- connected systems via the local feedback, the traditional methods usually ignore or try to reduce the influence of interconnections under the condition that the subsystems are controllable. The interconnections among subsystems in large-scale systems are thought to be one of the most important roots to produce complexity recently [6]. To enhance the effects of stabilization, the strategy of cou- pling two decoupled subsystems via designing a suitable combined feedback are considered in [7,8], which is called the harmonic control. Along the development of society, interconnections play more and more important roles in social systems, economic systems, power systems, etc. The connections of the system states are a type of the most important structures of a system. In fact, in many fields and even in our daily life, besides the usual feedback controls, it is also very useful for us to control our business by regu- lating the connections among the subsystems directly. For examples, the damages in power and transportation ties is one of the main facts to result in the huge loss in the freeze disaster in several provinces in southern China in 2008, and reflects the effects of the connections of the subsystems for the social large-scale system; the strict and active quarantine and isolation measures among re- gions in the SARS and H1N1 is also an example. In fact, in our daily life, we always deal with the interpersonal relationship between ourselves and those around us and the inter-relations between ourselves and the collective around us. Therefore, we can say that the human world is a complex network system through these relationships, and the handling of these relationships is actually the regulation of the connection among persons. The consideration of the connection problem were mainly seen in the power system research early times, including the transient stability analysis [9] and splitting control [10-12]. The interconnection coefficients of the *This work was supported in part by the National Natural Science Foundation of P. R. China under Grant 60874007 and the Research Fund for the Doctoral Program of Higher Education 200802550024. X. J. LIU ET AL. Copyright © 2010 SciRes. ICA 37 state variables of the subsystems are considered as the control variables that are regulated directly in the split- ting control for power system. Furthermore, the isolation treatment strategy is further discussed in the emergency control [13-15]. The studies give a theoretical interpreta- tion for the practical experiences that the early quaran- tine and isolation strategies are critically important to control the outbreaks of epidemics. Finally, a new kind of concept called Hadamard synergic control is intro- duced based on the Hadamard matrix product [16]. It is achieved via a composite strategy of the state feedback control and the direct regulation of the part of connection coefficients of system state variables. Such a control im- proves the limitations of the traditional feedback control [17-19] and may be of some potential applications in the emergency treatment such as isolation and obstruction control. For clear, we give this model again here. Consider the following linear time-invariant system: x tAxtBut (1) Here, ,,,1,2,, Tm iji i nn A aBbbRin Obviously, the element ij a is the interconnection co- efficient between the i-th state and the j-th state, for con- venience, we call the system matrix A as system inter- connection matrix. In many practical cases, such as the switches and circuit breakers in the power systems, fire- wall in the Internet etc., the system interconnection ma- trix A can be directly regulated, of course, can be pre-designed in some extent. Thus, the system intercon- nection matrix A can be divided into two parts: 12 A A . 1 A is the fixed part of A which is not able to be regu- lated directly and 2 A is the flexible part of A which can be regulated directly in some extent. By using the Ha- damard matrix product, this direct regulation of the in- terconnection matrix A can be written as follows: 12K A AAK (2) Here, for convenience, we call [] ijn n Kk as the connection coefficient gain matrix. It may be need to satisfy some constraints such as 01 ij k etc. Of course, the control strategy above is different from the feedback control. 2 A K is the Hadamard matrix prod- uct defined as [20]: 22 , ij ijij nn nn AK akAa Then, the general feedback control problem formula- tion can be extended as follows: find direct connection coefficient gain matrix K and feedback gain matrix F such that the generalized closed loop system 12 x AAKBFx (3) is stable, robust stable, or some other specific perform- ances. For convenience, we call this kind of control strategy as Hadamard synergic control. In order to illustrate the idea of isolation and obstruc- tion of the connections among subsystems, we give the following examples [21]. Example 1. Replacing the scalars ,, , ij ij ij abfk by matrices , ,, ijijij ij A BFkE with appropriate dimensions and the self-loops are not allowed, the general Hadamard synergic control model (3) can be rewritten as: 111 1121221111 1 222 2212112222 2 11111 1 nnn nnn nnn nnnnnnnnnn n xAxkAx kAxBu xAxkAxkAxBu x AxkAxkAxBu Where ,1,2,, i n i x Ri n is the local state of the i-th subsystem. ij kare the control variables. System model above can be rewritten as: 12 x AAKxBu (4) Here, 12 1 11 21 2 12 12 12 1211 11 21 2122 112 2 0 0 , 0 0 0 , 0 [1] ij n n nn nn nn nn nn nnn n ijn n AA AAA AA AAA kE kE BkE kE BK BkEk E E In power systems, the model (4) can be used to describe the frequency control in multi-areas loads with the bal- ance of active powers among the networks of different areas. Example 2. Consider the network model researched in [1,3], in fact, the system interconnection matrix A is divided into two parts. By the Kronecker product, this network model can be written as [22]: 1nOC n x IACAxIBu (5) Where 12 12 ,,, ,,,, TT TT TTT T nn x xxxu uuu,nn CR is called as the outer coupling matrix, C A is the inner coupling matrix describing the interconnections. Obviously, the network model (5) is a special case of the model (4). From the definition of the matrix Kronecker product we know, the connection matrix of the system has very symmetrically consistency structure if we de- scribe the system by using the corresponding Hadamard product, this is: any two subsystems have the same basic X. J. LIU ET AL. Copyright © 2010 SciRes. ICA 38 connection structure except the coupling coefficient, i.e . 11 22 1, 2,, 1 O ij C nn A ij n A A ijn BB B Network model (5) has very specific project back- ground such as in the consensus and formation control problem. This also illustrates the rationality and general- ity of the abstract model (4). Example 3. The host population consists of six sub populations: namely susceptible individuals 1() x t, as- ymptomatic individuals 2() x t, quarantined individuals 3() x t, symptomatic individuals 4() x t, isolated indi- viduals 5() x t, recovered individuals 6() x t. The total population size is 6 1 () i i Nxt . The detailed descrip- tions of other parameters see [23]. The SARS transmis- sion model with quarantine and isolation controls u and v is given by the following nonlinear system of differential equations: 4235 11 4235 212 3223 412 114 4 542322 5 61425 6 EQJ EQJ xxxx xx N xxxx x pkux N xux kx xkx dxvx xvxkxdx xxxx Obviously, the model above is a typical interconnec- tion-regulation control of a nonlinear system with the control variables u and v (see Figure 1). Hadamard product is a classical matrix product. It has many applications in some areas especially in mathe- matics and physics. It also has some applications in sig- nal processing [24]. In the existing literatures, almost all the results about the eigenvalue estimations on Ha- damard products were obtained under the presupposition that the involved matrices are special ones such as M-matrices, Hermitian (or the form of * A A), diagonal matrices, etc. See [25-29] and the other corresponding references. Also, the discussions on the mixture products like () A BC are scarcely reported. Hence, the basic properties and expressions on Hadamard product still remain to be extensively studied. Although almost all the existing control theory and applications are implemented by feedback controls, the feedback is, in a general sense, only one of the specific measures to implement the regulations of the connections of system states. )( 1 tx 1 x tx 2 p 2 x tx 4 21 xk 4 x 41 xd tx 6 41 x tx 3 2 ux 3 x 32 xk tx 5 5 x 52 xd 4 vx 52 x 6 x Figure 1. A schematic representation of the populations flow. Let the feedback law be,mn j uFxF fR , ,1,2,, m j f Rj n. Then the system matrix of the closed loop is of the form: T ijijnn Aabf . The ac- tual functions of the feedback are the compensations of ij a, i.e., regulating the interconnection coefficients from ij a to T ijij abf via the input information channel. Hence, in an open-loop viewpoint, the feedback control is just a special indirect regulation of interconnections of system states. The observation above show that the feedback control strategy is only one of the specific measures to imple- ment the regulations of the connections of system states via the input information channel, rather than the direct physical regulation of the system interconnection matrix A. In this paper, we mainly discuss the Hadamard synergic stabilization problem for the general two-node-network (4), and then Hadamard synergic stabilization problem for the special model (5) is studied. Matrix algebra and algebraic graph theory are proved useful tools in model- ing the communication network and relating its topology to the discussion of the network stability. The rest of this paper is organized as follows. System models and problem formulation discussed in this paper are given in Section 2. Hadamard synergic stabilization problem for the general two-node-network (4) is studied in Section 3. In Section 4, Hadamard synergic stabiliza- tion problem for the special network model (5) is dis- cussed. Furthermore, networks with communication time-delays are investigated. The last section concludes the paper. 2. System Models and Problem Formulation In this section, we give the system models and problem formulation discussed in this paper. Because of the existence of the Hadamard product, X. J. LIU ET AL. Copyright © 2010 SciRes. ICA 39 this makes that the stability analysis of the general net- work model (4) become difficult. Therefore, we mainly consider the Hadamard synergic control problem for the special model (5). Furthermore, Hadamard synergic con- trol for the two-node-network model of the general net- work system (4) is investigated simply. For convenience, the network model (5) can be rewritten as: 1nOC n x IAKAxIBu (6) ij nn Kk is the connection coefficient gain matrix. In the general case, the control variables ij k often need to satisfy some constraints. There exist the follow- ing cases being researched. Case 1: ij k is discrete. For example 0,1 ij k. When 0 ij k, it means that we cut off the connections from the j-th subsystem to the i-th subsystem; When 1 ij k , it means that we keep the corresponding connections. In this case, the control is called the isolation treatment strategy. In fact, this kind of control strategy has been researched in many literatures [15] especially in the power electrical engineering [10-12]. Case 2: ij k is continuous. It often needs to meet some constraints. For example, 01 ij k in the epi- demic control [13,14,25]; n ii ij ji kk in the consensus or formation control problem [30,31], etc. Although the control variables ij k often need to sat- isfy some constraints, as the stability research in the classical feedback control of the system (1) required to unconstrained control ()ut we also suppose that the connection coefficient ij kR in this paper. We present the formulations of the Hadamard synergic stabilization problems as follows: Hadamard synergic stabilization problem (HSSP) [21]: Given system (1), and let 12 A AA . Find con- nection coefficient gain matrix nn K R and feedback control uFx such that the corresponding Hadamard synergic closed loop (3) is stable, i.e. 12 A AKBFC Where (.) represents the set of eigenvalues of the corresponding matrix, C means the left-half complex plane. For convenience, we call the matrix pair ),( FK as the synergic control matrix pair. Remark 1. Obviously, the HSSP is equivalent to the problem that is to find connection coefficient gain matrix K such that 12 (,) A AKB is stabilizable. One of the stronger conditions of it is to find a matrix K such that 12 (,) A AKB is completely controllable. Also, for convenience, we call these two problems as Hadamard synergic stabilization and Hadamard synergic Controlla- bilization problems respectively. In this paper, we mainly consider the Hadamard syn- ergic stabilization problem for the two cases: Case 1: The two-node-network model of the general network system (4). Case 2: The special network model (6). 3. Hadamard Synergic Stabilization for THE General Two-Node-Network In this section, we consider the Hadamard synergic Sta- bilization problem for the general dynamical network model (4). We mainly consider the two-node-network de- scribed as: 111 11212211 1 22222121 1222 x AxAx Bu x AxAxBu (7) Then, the Hadamard synergic stabilization problem for the network model (7) can be presented as: find connec- tion coefficients 1221 ,R and decentralized feed- back control 1112 22 ,uFxu Fx such that the closed loop matrix 111111212 21 212222 2 (2) loop ABFA A A ABF is stable. When 12 0 or 210 , the stability of (2) loop A is equal to the stability of the two subsystems, so we do not consider this condition. In the following, we suppose that 1221 0, 0 . 3.1. Case of 1221 1rank Arank A In this section, we discuss the Hadamard synergic stabi- lization problem of the network model (7) with the spe- cial case 12 21 () ()1rank Arank A . Based on the theorem of the Linear Algebra, let 121 2212 11122 ,,,,, TT n A abAaba b abR . Then, the sys- tem (7) without local input can be rewritten as: 11111212 2 222221211 T T x Ax abx x Ax abx (8) Note that system above is equivalent to the following system: 11111212 111 2222212122 2 T T x Axayy bx x Axayybx Let 1221 ,uyu y are the inputs of the intercon- X. J. LIU ET AL. Copyright © 2010 SciRes. ICA 40 nected control; 12 ,yy are the outputs of the first and second subsystem respectively. Then 12121 2 ,aa can be viewed as the matrix of the first and second subsys- tem accepting the interconnected control respectively. Hence, let 1 1121 111 1 2122 222 T T H sbsIAa H sbsIAa Then, the matrix1112 12 21 2122 (2) A A AAA can be vie- wed as the state matrix of the closed-loop feedback sys- tem as shown in Figure 2. Theorem 1. There exist 1221 ,R such that the matrix (2)A is stable if and only if there exist 12 , 21 R such that the polynomial 12 ()()() f sdsds 1221 12 ()() f sf s is stable. In this case, the matrices 12 A , 21 A must satisfy that 12 21 () ()0tr Atr A . Here, * 11111111 * 22222222 det,( ) det,( ) T T dssIAfsb sIAa dssIAf sbsIAa * (.),()tr denote the trace and adjoins of the correspond- ing matrix respectively. Proof. From the analysis above we know that (2)A is stable if and only if the feedback system shown in Figure 2 is stable, where the closed loop transfer func- tion in Figure 2 is 1 12 * 11 2222 121221 12 1 det T Hs Hs HsHs s IAbsIA a dsd sfsf s Therefore, (2)A is stable if and only if the polyno- mial ()ds is stable. Let 112 2 11 22 , nnnn ARA R . Then, note that ()ds and 12 () () f sfs are the polynomials with degree 12 nn and 12 2nn respectively. Hence, if we let 12 12 1 0 () () nn nn dsscsd s , then we have that 12 21 () ()0ctrA trA. This completes the proof. Remark 2. When ()1rank M, there exist vectors ,n ab R such that T M ab and the different de- compositions are unique up to a constant, so the result above is independent of the decompositions of the cou- pled matrices 12 21 , A A. The results above can be gener- alized to cases of multiple subsystems simply. )( 1 sH )( 2 sH Figure 2. The closed loop of the system (8). 3.2. Case of 12 21 1, 1rank Arank A In this section, we discuss the Hadamard synergic stabi- lization problem of the network model (7) with the gen- eral case 1221 ()1, ()1rankArank A by using the small gain theorem. Decompose 1212 , A BC 212 1 A BC, then 12 12 A 12 12 ()BC , 21 212121 () A BC . Let 1 111112 1 1 111112 1 H sCsIA B H sCsIA B Similarly as in the section 3.1, the matrix (2)A can be viewed as the state matrix of the closed-loop feedback system as shown in Figure 2. In this way, (2) loop A can be viewed as an interconnected system composing of two subsystems 1112 11 ,,, A BC 2221 22 ,, A BC under the local feedback. Using the small gain theorem, we can get the follow- ing result. Proposition 1. If there exist 1221 12 ,,, F F such that 1 111111121 1 222222212 1 CsI ABFB CsI ABFB (9) then the system (7) can be stabilized by the synergic control. In the following, we suppose that 121 12122 112 2 ,, ,, 0, 0 (10) Remark 3. Based on the Proposition 1, we know that if there exist 12 21 , such that 1 111121 1 222212 1 CsI AB CsI AB (11) then (2)A is stable. Obviously, there exist 12 21 , as in (10) such that (11) holds if and only if X. J. LIU ET AL. Copyright © 2010 SciRes. ICA 41 1 1111 1 2222 12 1 CsI AB CsI AB (12) But the decompositions of 12 21 , A A are general not unique. In the following, we give the suitable decompositions and the explain (12) by LMI method by using the result in [32]. Theorem 2. For any fixed full rank decompositions 00 00 1212212 1 , A BC ABC, there exist connection coeffi- cients 12 21 , as in (10) and decompositions 121 2 A BC , 212 1 A BC such that (11) holds if and only if there exist positive matrices 12 ,,,PPXY such that 000 1111111 111 0 11 2 2 00 0 2222222 222 0 22 2 1 0 1 0 1 TTT TTT PAA PBXBPC CP Y PAAPBYBPC CP X (13) Proof. Based on the result in [32], we know for any fixed full rank decompositions 0000 12122121 , A BC ABC , there exist connection coefficients 12 21 , and decom- positions 12212112 ,CBACBA such that (11) holds if and only if there exist positive matrices 12 ,,,PPXY such that 20 00 111111121 111 0 11 20 00 222222212 222 0 22 0 0 TTT TTT PAA PBXBPC CP Y PAA PBYBPC CP X Let 2 12 , X X 2 21 YY , then the inequalities above can translate into 000 111 1111111 0 11 2 21 000 2222222 222 0 22 2 12 0 1 0 1 TTT TTT PAA PB XBPC CP Y PAAPBYBPC CP X If there exist positive matrices 12 ,,,PPXY such that (13) holds, and we can choose connection coefficient 121 212 , such that (11) holds. Conversely, if there exist connection coefficients 120 210 , as in (10) such that (11) holds, then we can get: 000 111 1111111 0 11 2 210 00 0 2222222 222 0 22 2 120 0 1 0 1 TTT TTT PAA PBXBPC CP Y PAAPBYBPC CP X By using Schur complement, we know that the ine- qualities above are equal to: 00201 0 11111 1111201111 002 010 222 222 222102222 0 0 0 0 TT T TT T Y PAA PBXBCPYPC X PAAPBYBCPXPC Since 1 120 0 , 2210 0 , thus 22 1 120 0 , 22 2 210 0 , so we have 002010 111 111 1111111 00201 0 222222 2222222 0 0 TT T TT T PAA PB XBC PYPC PAAPBYBCPXPC use Schur complement, then (13) holds. This completes the proof. Remark 4. From the Theorem above, we know that we only need to consider full rank decompositions among the different decompositions of 12 21 , A A under the minimal connection coefficients. From the proof of the Theorem 1 in [32], we know that 1 1111 ()CsI AB 1 2222 ()CsI AB can be minimized among different decompositions of 12 21 , A A by LMI method, if we let 11 11112 2220 min CsI ABCsI AB we can get 1 12 210 max . From the proof above, we can give the following algo- rithm to get the estimation of 12 21 max,max for any fixed full rank decompositions. Step 1. For any fixed full rank decompositions 00 00 1212212 1 , A BC ABC, solve the LMI (13) if it holds, go to step 2; otherwise, stop. Step 2. Solving the following LMIs: 2 2 2 1 000 1111111 111 01 11 00 0 222222 2222 01 22 0 0 TTT TTT PAA PB XBPC CP Y PAA PBYBPC CP X X. J. LIU ET AL. Copyright © 2010 SciRes. ICA 42 If it holds, then, we can get: 12 1212 max,max Otherwise, go to step 3. Step 3. Choose the appropriate step size 12 , and move one step size for the LIM (13), i.e., solve the following inequalities: 00 0 1111111 111 0 11 2 22 00 0 2222222 222 0 22 2 11 0 1 () 0 1 () TTT TTT PAA PB XBPC CP Y PAAPBYBPC CP X (14) If it does not hold, stop and get 1211121222 max,,max, Otherwise, keep on moving one step size for the LIM (14) and solve the corresponding inequalities and con- tinue the following process in step 3. If it moves the n step size, we can get: 1211 11 2122 22 max1 , max1 , nn nn Obviously, (13) is only a sufficient condition, but it is easy to establish an LMI algorithm for designing decen- tralized control 12 , F F. Theorem 3. For any fixed full rank decompositions 00 00 1212212 1 , A BC ABC, there exist 12 , F Fand 12 21 , as in (10) and decomposition 121 2212 1 , A BCABC such that (9) holds, if and only if there exist positive 12 1 2 ,, ,PP XX and any matrices 12 ,YY such that 2 2 2 1 00 0 1 1111111111111 11 1 0 11 2 000 222 222222222 22222 0 22 1 0 1 0 1 TTTTT TTTTT PA APBYYBBXBPC CP X PAAPBYYBB X BPC CP X and decentralized controllers gain are given by 1 F 1 11 YP , 1 222 F YP . Remark 5. LMIs can be solved easily by using the toolbox [33]. Compare to the result in the Subsection 3.1, result in this section is only sufficient condition, but it is easier to establish an LMI algorithm for designing de- centralized control and more simple to compute. 4. Synergic Stabilization for the Special Dynamical Network In this section, we discuss the HSSP for the special model (6). 4.1. Nyquist Criterion Method For stability analysis of network (6), we show the fol- lowing to be true. Theorem 4. There exist connection coefficient gain matrix [], ij ij K kkR such that nO C I AKA is stable if and only if there exist iR such that OiC A A are stable simultaneously for 1, 2,,in. Proof. Let nn PR be a nonsingular matrix such that 1 PKP J and J is the Jordan standard form of K.Then, based on the Properties of the matrix Kro- necker product, we can get: 11 1 nn OCn nOC PIIAKAPI IAJA Since the Jordan form matrix J is block upper-triangular, the stability of this system is equivalent to the stability of the n systems defined in the diagonal blocks. ForC J A , the diagonal blocks are each iC A , and then we can get the conclusion. This completes the proof. Remark 6. If ij k need to meet some constraints, then, i also should satisfy some constraints correspondingly. For example in the consensus or formation control prob- lem: 1, n iq qki ij ij kji k kR ji (15) Then, we need that there must exist 00 i such that 0 OiC A A is stable, i.e., O A is stable, and the associ- ated eigenvector of 0 i is 11 T . In the following discussion, we suppose that C A 11 BC , 1 mn m CR . We can get the following result. Theorem 5. There exists Hadamard synergic control matrix pair [], , ij ij K kk RF such that the network (6) is stable if and only if there exist ,1,2,, iRi n such that the controller iii uFyz simultaneously stabilizes the set of the following n systems: 1 1 iOi i ii iii x Ax Bu yx zCx (16) Proof. Using the same transform method as in the Theorem 4, we can get that the network (6) is stable if and only if the following n systems is stable simulta- X. J. LIU ET AL. Copyright © 2010 SciRes. ICA 43 neously if 11C A BC. 11 1iOi i x ABICBIFx Where 1 1 n x PIx . This is equivalent to the controller iii uFyz sta- bilizes the set of the n systems as in (16). This com- pletes the proof. Remark 7. Theorem 5 reveals that the special network (6) can be analyzed for stability by analyzing the stability of a single system with the same dynamics, modified by only a scalar, representing the interconnection, that take values according to the eigenvalues of the connection coefficient gain matrix K . Hereafter, we refer to the transfer function from i u to i y as ()Gs; the closed loop system can be shown as Figure 3 in this case. If ()Gs is single-input-single-output (SISO), we can state a second version of Theorem 5 which is useful for stability analysis. Theorem 6. Suppose ()Gs is SISO and p is the number of right-half plant poles of ()Ps. Then, the closed loop system as in Figure 3 is stable if and only if 1) If 0,1,2,, iin , then, the counterclockwise net encirclement of 1 (,0) ij by the Nyquist plot of 1()CPs is equal to p for 1, 2,,in . 2) Otherwise, 0p and this net encirclement is equal to zero. Proof. The Nyquist criterion states that the stability of the closed loop system in Figure 3 is equivalent to the number of counterclockwise encirclements of (1,0)j by the forward loop 1() iCP j being equal to the num- ber of the right-half plant poles of ()Ps, which is as- sumed to be p. This criterion is equivalent to the num- ber of encirclements of 1 (,0) ij by the Nyquist plot of 1()CPs being p. This completes the proof. Similarly, if ()Gs is MIMO, we can give the fol- lowing result. )(sG F )(sP 1 C i u y Figure 3. The closed loop of the system (16). Corollary 1. Suppose ()Gs is MIMO and p is the number of right-half plant poles of ()Ps . Then, the closed loop system as in Figure 3 is stable if and only if 1) If 0,1,2,, iin , then, the counterclockwise net encirclement of the origin by the Nyquist plot of 1 det i I CP s is equal to p for 1, 2,,in. 2) Otherwise, 0p and this net encirclement is equal to zero. Remark 8. The zero eigenvalue of K can be inter- preted as the unobservability of absolute motion in the measurements i z. The design strategy in the Theorem 6 can be interpreted as follows: firstly, close the inner loop around i y such that the internal closed loop system ()Ps has p right-half plant poles which is equal to the number of uncontrollable poles of the system (16); sec- ondly, close the outer loop around i z such that the whole network system is stable. This can be seen as the synergic action between the feedback control and the connection gain regulation. 4.2 Algebraic condition In this section, we consider the Hadamard synergic sta- bility problem by using the algebraic method. First, we give the following Lemma. Lemma 1. For any matrix K , nO CCnO I AKAKAI A if and only if OC CO A AAA . Proof. Based on the fact nO COC Cn OCO I AKAK AA K AI AKAA We can get the conclusion directly. This completes the proof. Lemma 2. [20] Let ,nn ST C and ST TS , 1,, n , 1,, n are their eigenvalues respectively. Then, there exists a permutation 1,, n ii of 1, 2,, n such that 1 1,, n ini are eigenvalues of ST . Remark 9. Lemma 2 implies that if ST TS , then ()()()STS T , i.e. max RemaxRemax Re min Remin Remin Re STS T STS T Here, () X denotes the eigenvalue of matrix X . Theorem 7. Suppose OC CO A AAA and satisfied 1) If max Re0 C A , then, X. J. LIU ET AL. Copyright © 2010 SciRes. ICA 44 max Re Re max,0 max Re O C A KA or max ReRe 0 min Re O C AK A 2) If min Re0 C A , then, max Re Re min,0 min Re O C A KA or max Re 0Re max Re O C A KA 3) If max Re0 C A , then, max Re minRe0, 0Remin Re O C C A AK A max Re min Re0,0Remin Re O C C A AK A Proof. Based on the Lemma 1 and Lemma 2, we have that max Re max Remax Re max RemaxReRe nO C nO C OC IAKA IA KA AKA then, we can get the conclusion directly. This completes the proof. Corollary 2. Suppose that the connection gain matrix [] ij K k meet constraint (15), OC CO A AAA and O A is stable. 1) If max Re0 C A , then, nO C I AKAis stable for any K satisfied 0 ij k. 2) If min Re0 C A , then, nO C I AKA is stable for any K satisfied 0 ij k. Proof. Based on the Gerschgorin disk theorem, we know that all the eigenvalues of [] ij K k are located in the union of the n disk: 1 n iiiij j GzRzk k thus, we can get that all the eigenvalues of [] ij K k are positive except zero when 0 ij k and are negative ex- cept zero when 0 ij k. Then based on the Theorem 7, we can get the conclusion directly. This completes the proof. When we consider the common decentralized control- ler ii uFx , if we want to use the conclusions above, we must require that 11 ()() OCCO A BF AAABF, this is difficult to solve. Thus, we consider the special case that ,0 Cn AaIa and can get the following re- sult. Corollary 3. If 1 (,) O A B is controllable, then for any K there must exist common decentralized controller ii uFx such that 1 ()() nO n I ABFK aI is sta- ble; otherwise, suppose 12 1 3 0 O A A TAT A , then, there exist common decentralized controller ii uFx such that 1 ()() nO n I ABFK aI is stable if and only if 3 max(Re ()) () A Ka for0a or 3 max(Re ()) () A Ka for 0a . Proof. From the fact that for any matrix, K F 1 1 nO n nn O IABFKaI K aIIAB F and based on the Lemma 2 we can get the conclusion directly. Remark 10. Based on the analysis in the Corollary 3 for this special case, synergic action between the decen- tralized feedback control and the connection gain regula- tions can be interpret as follows, that is: designing the common decentralized controller ii uFx to stabilize the controllable part firstly, and designing connection coefficient gain matrix K to stabilize the uncontrolla- ble part secondly. 4.3. Network with Communication Time-Delays In this section, we consider a network of continuous-time integrators in which the i-th subsystem state i x passes through a communication channel ij e with time-delay 0 ij before getting to j-th subsystem. The transfer function associated with the edge ij e can be expressed as: () ij s ij hs e in the Laplace domain. As the discus- sion in [34], to gain further insight in the relation be- tween the connection gain matrix K and the maximum time-delays, we focus on the simplest possible case where the time-delays in all channels are equal to 0 and () () s ij hs hse . Then the network system can be written as: 1 1 n iOi ijCji j x tAxtkAxt Bu After taking the Laplace transform of both sides, we X. J. LIU ET AL. Copyright © 2010 SciRes. ICA 45 can get 1 0 n ii OiijCj j s XsxhsAXshskA Xs The set of equations above can be rewritten in a com- pact form as: 10 nO C X ssIhsIAKA x (17) The convergence analysis for a network of integrator nodes with communication time-delays reduces to stabil- ity analysis for a multiple-input-multiple-output (MIMO) transfer function: 1 NOC GssIhs IAKA In the following, we give the stability result of the model (17). Theorem 8. Consider a network of integrator nodes with equal communication time-delay 0 in all links. Assume the matrix nO C I AKA has no eigenvalue of zero or the algebraic multiplicity of the zero eigen- value is 1. Then the model (17) is stable if and only if either of the following equivalent conditions are satis- fied: 1) 0 (0,) with 0min( ) i , where 22 22 22 22 Re arcsin Re ImRe 0 Re Im Re 2arcsin Re ImRe 0 Re Im i ii i ii i i ii i ii (18) i is the eigenvalue of the matrix nO C I AKA. 2) The Nyquist plot of () s e s s has a zero encir- clement around 1 i for 0 i . Proof. To establish the stability of (17), we use fre- quency domain analysis. We have () ()(0) X sGsx. Define 1 nO C H sGssIhsI AKA . Then, we require that all the zeros of det H s are on the Left Hand Plane (LHP) or 0s. Let i be the normalized eigenvector of nO C I AKA associated with the eigenvalue i . If the matrix nO C I AKA has zero eigenvalue and suppose 10 , then 0s in the direction 1 is a zero of det H s since 11 00 nOC HIAKA ; otherwise 0s is not a zero of det H s. Furthermore, we can get that any eigenvector of () H s is an eigenvector of nO C I AKA and vice verse. Then, we can get that for any s of the zero of det H s, we must have ()0 i Hs for some one i, i.e., 0 inOCi s ii HssIhs IAKA se But 0 i , thus, 0s satisfies the following equa- tion: 10 s i e s (19) Thus, if the net encirclement of the Nyquist plot of () s e s s around 1 i for 0 i is zero, then all the poles of )(sG except 0s are stable. We calculate the upper bound on time-delay as follows. We want to find the smallest value of the time-delay 0 such that det H s has a zero on the imaginary axis. Set s j in (19), we can get 0 0 j i j i je je multiplying both sides of the two equations above, we get 22 2sin 0 ii . Let Re() Im() iii j , then, we have Re2 Re1sinIm 2Re ImImsin0 ii i ii i j Assume 0 (due to 0s), then from the equa- tion above, we can get: 22 ReIm, Resin iii This implies 22 22 Re ReIm2arcsin Re Im i ii ii k , i.e. 22 22 Re 2arcsin Re Im Re Im i ii ii k X. J. LIU ET AL. Copyright © 2010 SciRes. ICA 46 thus, the smallest 0 satisfied that 0min( ) i and i are given in (18). Due to the continuous dependence of the roots of (19) in and the fact that all the zeros of this equation ex- cept 0s for 0 are located on the open LHP, for all 0 (0, ) , the roots of (19) are on the open LHP, and therefore the poles of ()Gs are all stable except 0s, but the algebraic multiplicity of the zero eigen- value is 1. We can repeat a similar argument for the as- sumption that 0 . This completes the proof. Remark 11. From the condition (1) of the Theorem above, we can see that the upper bound on time-delay 0 is determined by the eigenvalues nO C I AKA. Thus, if OC CO A AAA, then based on the Lemma 2, we can design the desired connection gain matrix K in order to obtain expected upper bound on time-delay. 5. Conclusions In this paper, Hadamard synergic stabilization problem is investigated. Synergic stabilization problem for a special kind of networks are studied by using the Nyquist crite- rion. The mechanism of the synergic action between two control strategies: feedback control and the connection coefficients regulations are presented. Networks with communication time-delays are also discussed. Further- more, synergic stabilization problem for the general dy- namical network composed of two subsystems are inves- tigated. The regulations of the interconnections can be exploited to improve the stability of the closed-loop sys- tem. It should be noted that only some special network models have been investigated in this paper, many more general network models remain to be challenging sub- jects for future research. Although Hadamard synergic control problem has not received much attention, we suggest that it will probably turn out to be widespread in power electrical engineering and the epidemic control system. We hope that our work will stimulate further studies of this new kind of control problem. 6. References [1] X. F. Wang, “Complex Networks: Topology, Dynamics, and Synchronization,” International Journal of Bifurca- tion and Chaos, Vol. 5, No. 12, 2002, pp. 885-916. [2] D. J. Watts and S. H. Strogatz, “Collective Dynamics of ‘Small-World’ Networks,” Nature, Vol. 393, No. 6684, 1998, pp. 440-442. [3] X. Li, X. F. Wang and G. R. Chen., “Pinning a Complex Dynamical Network to its Equilibrium,” IEEE Transac- tions on Circuits and Systems-I, Vol. 51, No. 10, 2004, pp. 2074-2085. [4] V. Lesser, C. L. Ortiz and M. Tamb, “Distributed Sensor Networks: A Multiagent Perspective,” Kluwer Academic Publishers, Boston, 2003. [5] S. S. Mascolo, “Congestion Control in High-Speed Com- munication Networks Using the Smith Principle,” Auto- matica, Vol. 35, No. 11, 1999, pp. 1921-1935. [6] L. Huang and Z. S. Duan, “Complexity in Control Sci- ence,” Acta Automatica Sinica, Vol. 29, No. 5, 2003, pp. 748-753. [7] Z. S. Duan, L. Huang, J. Z. Wang and L. Wang, “Har- monic Control between Two Systems,” Acta Automatica Sinica, Vol. l29, No. 1, 2003, pp. 14-21. [8] Z. S. Duan, J. Z. Wang and L.Huang, “Special Decen- tralized Control Problems and Effectiveness of Parame- ter-Depend Lyaounov Function Method,” Proceedings of the American Control Conference, Boston, 2005, pp. 1697-1702. [9] Y. Zou, M. H. Yin and H. D. Chiang, “Theoretical Foun- dation of Controlling U.E.P Method in Network Structure Preserving Power System Model,” IEEE Transactions on Circuits and Systems I, Vol. 50, No. 10, 2003, pp. 1324- 1336. [10] K. Sun, D. Z. Zheng and Q. Lu, “A Simulation Study of OBDD-Based Proper Splitting Strategies for Power Sys- tems, under Consideration of Transient Stability,” IEEE Transaction on Power Systems, Vol. l20, No. 1, 2005, pp. 389-399. [11] K. Sun, D. Z. Zheng and Q. Lu, “Splitting Strategies for Islanding Operation of Large-Scale Power Systems Using OBDD-Based Methods,” IEEE Transactions on Power Systems, Vol. 18, No. 2, 2003, pp. 912-923. [12] Q. C. Zhao, K. Sun, D. Z. Zheng, J. Ma and Q. Lu, “A Study of System Splitting Strategies for Island Operation of Power System: A Two-Phase Method Based on OBDDs,” IEEE Transactions on Power Systems, Vol. 18, No. 4, 2003, pp. 1556-1565. [13] X. F. Yan, Y. Zou and J. Li, “Optimal Quarantine and Isolation Strategies in Epidemics Control,” World Jour- nal of Modeling and Simulation, Vol. 3, No. 3, 2007, pp. 202-211. [14] X. F. Yan and Y. Zou, “Optimal Control for Internet Worm,” ETRI Journal, Vol. 30, No. 1, 2008, pp. 81-88. [15] X. F. Yan and Y. Zou, “Isolation Treatment Strategy for Emergency Control,” Journal of systems Engineering, Vol. 24, No. 2, 2009, pp. 129-135. [16] Y. Zou, “Investigation on Mechanism and Extension of Feedback: An Extended Closed-loop Coordinate Control of Interconnection-regulation and Feedback,” Journal of Nanjing University of Science and Technology, Vol. 34, No. 1, 2010, pp. 1-7. [17] K. J. Astrom, “Limitations on Control System Perform- ance,” European Journal of Control, Vol. 4, No. 1, 2000, pp. 2-20. [18] R. M. Murray, K. J. Astrom, S. P. Boyd, et al., “Future Directions in Control in an Information-Rich World,” IEEE Control Systems Magazine, Vol. 23, No. 2, 2003, pp. 20-33. X. J. LIU ET AL. Copyright © 2010 SciRes. ICA 47 [19] L. Guo, “Exploring the Maximum Capability of Adaptive Feedback,” International Journal of Adaptive Control and Signal Processing, Vol. 16, No. 1, 2002, pp. 341- 354. [20] J. L. Chen and X. H. Chen, “Special Matrices,” Tsinghua Publisher, Beijing, 2001. [21] X. J. Liu and Y. Zou, “On the Hadamard Synergic Stabi- lization Problem: The Stabilization Via a Composite Strategy of Connection-Regulation and Feedback Control: Matrix Inequality Approach,” International Journal of Innovative Computing, Information and Control, Vol. 6, No. 5, 2010, pp. 2383-2392. [22] Z. S. Duan, J. Z. Wang, G. R. Chen and L. Huang, “Sta- bility Analysis and Decentralized Control of a Class of Complex Dynamical Networks,” Automatica, Vol. 44, No. 4, 2008, pp. 1028-1035. [23] A. B. Gumel, S. Ruan, T. Day, et al., “Modeling Strate- gies for Controlling SARS Outbreaks,” Proceedings of the Royal Society, Vol. 49, No. 9, 2004, pp. 1465-1476. [24] X. Zhang, X. Hu and Z. Bao, “Blind Source Separation Based on Grading Learning,” Science in China-Series E, Vol. 32, No. 5, 2002, pp. 693-793. [25] S. Chen, “A Lower Bound for the Minimum Eigenvalue of the Hadamard Product of Matrices,” Linear Algebra and its Applications, Vol. 37, No. 8, 2004, pp. 159-166. [26] H. Li, T. Huang, S. Shen, et al., “Lower Bounds for the Minimum Eigenvalue of Hadamard Product of an M- matrix and its Inverse,” Linear Algebra and its Applica- tions, Vol. 4, No. 20, 2007, pp. 235-247. [27] G. Visick, “A Quantitative Version of the Observation that the Hadamard Product is a Principal Submatrix of the Kronecker Product,” Linear Algebra and its Applications, Vol. 30, No. 4, 2000, pp. 45-68. [28] M. Neumann, “A Conjecture Concerning the Hadamard Product of Inverses of M-matrices,” Linear Algebra and its Applications, Vol. 28, No. 5, 1998, pp. 277-290. [29] L. Qiu and X. Zhan, “On the Span of Hadamard Products of Vectors,” Linear Algebra and its Applications, Vol. 422, No. 1, 2006, pp. 304-307. [30] J. A. Fax and R. M. Murray, “Information Flow and Co- operative Control of Vehicle Formations,” IEEE Trans- actions on Automatic Control, Vol. 49, No. 9, 2004, pp. 1465-1476. [31] R. O. Saber and M. Murray, “Consensus Protocols for Networks of Dynamic Agents,” Proceedings of the American Control Conference, Denver, 2003, pp. 951-956. [32] Z. S. Duan, L. Huang, L. Wang and J. Z. Wang, “Some Applications of Small Gain Theorem to Interconnected Systems,” Systems and Control Letters, Vol. 52, No. 3-4, 2004, pp. 263-273. [33] P. Gahinet, A. Nemirovski and A. J. Laub, “LMI Control Toolbox,” TheMath Works, Inc., Natick, 1995. [34] R. O. Saber and M. Murray, “Consensus Problems in Networks of Agents with Switching Topology and Time- Delays,” IEEE Transactions on Automatic Control, Vol. 49, No. 9, 2004, pp. 1520-1533. |