Stability criteria for largescale timedelay systems: the LMI approach and the Genetic Algorithms


 William Sullivan
 3 years ago
 Views:
Transcription
1 Control and Cybernetics vol 35 (2006) No 2 Stability criteria for largescale timedelay systems: the LMI approach and the Genetic Algorithms by JenqDer Chen Department of Electronic Engineering National Kinmen Institute of Technology Jinning, Kinmen, Taiwan, 892, ROC Abstract: This paper addresses the asymptotic stability analysis problem for a class of linear largescale systems with time delay in the state of each subsystem as well as in the interconnections Based on the Lyapunov stability theory, a delaydependent criterion for stability analysis of the systems is derived in terms of a linear matri inequality (LMI) Finally, a numerical eample is given to demonstrate the validity of the proposed result Keywords: delaydependent criterion, largescale systems, linear matri inequality, genetic algorithms 1 Introduction Time delay naturally arises in the processing of information transmission between subsystems and its eistence is frequently the main source of instability and oscillation in many important systems Many practical control applications are encountered, involving the phenomena in question, including electrical power systems, nuclear reactors, chemical process control systems, transportation systems, computer communication, economic systems, etc Hence, the stability analysis problem for timedelay systems has received considerable attention (Kim, 2001; Kolmanovskii and Myshkis, 1992; Kwon and Park, 2004; Li and De Souza, 1997; Niculescu, 2001; Park, 1999; Richard, 2003; Su et al, 2001) In the recent years, the problem of stability analysis for largescale systems with or without time delay has been etensively studied by a number of authors Moreover, depending on whether the stability criterion itself contains the delay argument as a parameter, stability criteria for systems can be usually classified into two categories, namely the delayindependent criteria (Hmamed, 1986; Huang et al, 1995; Lyou et al, 1984; Michel and Miller, 1977; Schoen and Geering, 1995; Siljak, 1978; Wang et al, 1991) and the delaydependent criterion (Tsay et al, 1996) In general, the latter ones are less conservative than
2 292 JD CHEN the former ones, but the former ones are also important when the effect of time delay is small However, there are few papers to derive the delaydependent and delayindependent stability criteria for a class of largescale systems with time delays, to my knowledge This has motivated my study In this paper, delayindependent and delaydependent criteria for such systems can be derived to guarantee the asymptotic stability for largescale systems with timedelays in the state of each subsystem as well as in the interconnections Appropriate model transformation of original systems is useful for the stability analysis of such systems, and some tuning parameters, which satisfy the constraint on an LMI can be easily obtained by the GAs technique A numerical eample is given to illustrate that the proposed result is useful The notation used throughout this paper is as follows We denote the set of all nonnegative real numbers by R, the set of all continuous functions from [ H, 0 to R n by C 0, the transpose of matri A (respectively, vector ) by A T (respectively T ), the symmetric positive (respectively, negative) definite by A>0 (respectively A<0) We denote identity matri by I and the set {1, 2,, N} by N 2 Main results Consider the following largescale timedelay systems, which is composed of N interconnected subsystems S i, i N Each subsystem S i, i N is described as ẋ i (t) =A i i (t) A ij j (t h ij ), t 0, (1a) where i (t) is the state of each subsystem S i, i N, then, A i, A ij, i,j N are known constant matrices of appropriate dimensions, and h ij, i, j N, are nonnegative time delays in the state of each subsystems as well as in the interconnections The initial condition for each subsystem is given by i (t) =θ i (t), t [ H, 0, H=ma {h ij}, i,j N, (1b) i,j where θ i (t), i N is a continuous function on [ H, 0 A model transformation is constructed for the largescale timedelay systems of the form: d [ [ i (t) B ij j (s)ds = A i i (t) B ij j (t) dt (A ij B ij ) j (t h ij ), (2) where B ij R ni nj, i, j N are some matrices that can be chosen by GAs such that the matri Āi = A i B ij, i, j H is Hurwitz
3 Stability criteria for largescale timedelay systems 293 Now, we present a delaydependent criterion for asymptotic stability of system (1) Theorem 21 The system (1) is asymptotically stable for any constant time delays h ij satisfying 0 h ij h ij, i, j N under the condition that Ā i are Hurwitz and there eist symmetric positive definite matrices P i, R ij, T ij, V ij, and W ijk, i, j, k N such that the following LMI condition holds: Ξ 11 Ξ12 Ξ13 Ξ14 Ξ15 Ξ T 12 Ξ Ξ = Ξ T 13 0 Ξ < 0 (3) Ξ T Ξ 44 0 Ξ T Ξ 55 where Ψ 1 B21 T P 2 P 1 B 12 BN1 T P N P 1 B 1N P 2 B 21 B12P T 1 Ψ2 BN2 T Ξ 11 = P N P 2 B 2N P M B N1 B1N T P 1 P N B N2 B2N T P 2 ΨN Ψ i = ĀT i P i P i Ā i [ hji R ji V ij h ki (T kji W kji ), ˇΞ 12 = diag(γ 1,, Γ N ), Γ i =[ hi1 A T i P i B i1 hin A T i P i B in,, ˇΞ 22 = diag(ˆγ 1,, ˆΓ N ), ˆΓi = diag(r i1,,r in ), ˇΞ 13 = diag(π 1,, Π N ), Π i =[Π i1 Π in, Π ki =[ hi1 Bik T P ib i1 hin Bik T P ib in, ˇΞ 33 = diag(ˆπ 1,, ˆΠ N ), ˆΠi = diag(ˆπ i1 ˆΠ in ), ˆΠ ki = diag(t ik1,,t ikn ), ˇΞ 14 = diag(λ 1,, Λ N ), Λ i =[(A 1i B 1i ) T P 1 (A Ni B Ni ) T P N, ˇΞ 44 = diag(ˆλ 1,, ˆΛ N ), ˆΛi = diag(v 1i,,V Ni ), ˇΞ 15 = diag(ω 1,, Ω N ), Ω i =[Ω i1 Ω in, Ω ki =[ hi1 (A ik B ik ) T P i B i1 hin (A ik B ik ) T P i B in, ˇΞ 55 = diag(ˆω 1,, ˆΩ N ), ˆΩi = diag(ˆω i1 ˆΩ in ), ˆΩ ki = diag(w ik1,,w ikn )
4 294 JD CHEN Proof From the Schur Complements of Boyd et al (1994), the condition (3) is equivalent to the following inequality: Ξ( h, P i,r ij,t ijk,v ij,w ijk )= φ 1 ( h) B21 T P 2 P 1 B 12 BN1 T P N P 1 B 1N P 2 B 21 B12P T 1 φ 2 ( h) BN2 T P N P 2 B 2N P M B N1 B1N T P 1 P N B N2 B2N T P 2 φ N ( h) < 0 (4) where h denotes { h ij,i,j N}, and φ i ( h) =ĀT i P i P i Ā i [ hij A T i P ib ij Rij 1 BT ij P ia i h ji R ji (A ji B ji ) T P j V 1 ji P j (A ji B ji )V ij [ hjk Bjk T P jb jk T 1 jik BT jk P jb ji h ki T kji h jk (A ji B ji ) T P j B jk W 1 jik BT jk P j(a ji B ji ) h ki W kji The Lyapunov functional is given by V ( t )=V 1 ( t )V 2 ( t )V 3 ( t ), (5a) where V 1 ( t )= V 2 ( t )= L i ( t ) T P L i ( t ),L i ( t ) i=1 = i (t) B ij j (s)ds, i N (A ij B ij ) T P i (V 1 ij V 3 ( t )= T j (s)[(s t h ij) R ij (5b) h ik B ik W 1 ijk BT ik)p i (A ij B ij ) j (s)ds, (5c) t h ik (s t h ik ) T k (s)(t ijk W ijk ) k (s)ds, (5d)
5 Stability criteria for largescale timedelay systems 295 are the legitimate Lyapunov functional candidates The time derivatives of V i ( t ), i = 3, along the trajectories of the system (2) are given by V 1 ( t )= T i (t)(at i P i P i A i ) i (t) i=1 2 T i (t)a T i P i B ij j (s)ds 2 T j (t)bijp T i i (t) 2 T j (t)bt ij P ib ik k (s)ds t h ik 2 T j (t h ij )(A ij B ij ) T P i i (t) 2 T j (t h ij)(a ij B ij ) T P i B ik k (s)ds It is known fact that for any, y R n and Z R n n 2 T y T Z y T Z 1 y being true, we have > 0, the inequality V 1 ( t ) T i (t)(at i P i P i A i ) i (t) i=1 [ hij T i (t)a T i P i B ij R 1 ij BT ijp i A i i (t) [ T i (t)(bii T P i P i B ii ) i (t)2 i=1,j i T i (s)r ij j (s)ds T j (t)bt ij P i i (t) [ t hik T j (t)bijp T i B ik T 1 ijk BT ikp i B ij j (t) T k (s)t ijk k (s)ds t h ik [ T j ()(A ij B ij ) T P i V 1 P i (A ij B ij ) j ( ) T i (t)v ij i (t) ij [ hik T j ( )(A ij B ij ) T P i B ik W 1 ijk BT ikp i (A ij B ij ) j ( ) t h ik T k (s)w ijk k (s)ds
6 296 JD CHEN V 2 ( t )= V 3 ( t )= { T j (t) [ h ij R ij (A ij B ij ) T P i (V 1 ij h ik B ik W 1 ijk BT ik)p i (A ij B ij ) j (t) T j (s)r ij j (s)ds T j (t h ij)[(a ij B ij ) T P i (V 1 ij h ik B ik W 1 ijk BT ik )P i(a ij B ij ) j (t h ij ) }, [ hik T k (t)(t ijk W ijk ) k (t) t h jk T k (s)(t ijk W ijk ) k (s)ds Now, considering that T j (t) [ h ij R ij (A ij B ij ) T P i V 1 ij P i (A ij B ij ) h ik B T ij P ib ik T 1 ijk BT ik P ib ij h ik (A ij B ij ) T P i B ik W 1 ijk BT ikp i (A ij B ij ) j (t) = T j (t)[h ji R ji (A ji B ji ) T P j V 1 ji P j (A ji B ji ) h jk B T jip j B jk T 1 jik BT jkp j B ji h jk (A ji B ji ) T P j B jk W 1 jik BT jkp j (A ji B ji ) i (t) H ik T k (t)(t ijk W ijk ) k (t) = h ki T i (t)(t kji W kji ) i (t)
7 Stability criteria for largescale timedelay systems 297 Then, the derivative of V ( t ) satisfies V ( t )= V 1 ( t ) V 2 ( t ) V 3 ( t ) [ T i (t)φ i(h) i (t)2 i=1,j i T j (t)bt ij P i i (t) X T (t)ξ( h, P i,r ij,t ijk,v ij,w ijk )X(t), (6) where X(t) =[ T 1 (t) T N (t)t Hence, by Theorem 981 of Hale and Verduyn Lunel (1993) with (4)(6), we conclude that systems (1) and (2) are both asymptotically stable for any constant time delays h ij satisfying 0 h ij h ij, i, j N Remark 21 Notice that for fied h ij,andb ij, i, j N, theentriesof Ξ in (3) are affine in P i, R ij, T ijk, V ij,andw ijk, i, j, k N, so that the asymptotic stability problem for the largescale timedelay systems can be converted into a strictly feasible LMI problem Remark 22 Notice that for any chosen matri B ij =0, i, j N it means that the delay terms B ij j (s)ds, i, j N have not been converted to the left side of the system (2) By the above Theorem 21, the corresponding matrices R ij, T ijk, U ijk,andw ijk, i, j, k N could be chosen as zero matrices, and the LMI condition in (3) is reduced by it s corresponding elements Letting B ij =0, i, j N in Theorem 21, we achieve the following result that does not depend on delay arguments Corollary 21 The system (1) is asymptotically stable with h ij R, i,j N, provided that A i is Hurwitz, and there eist P i, and V ij, i, j N, such that the following LMI condition holds: ˆΞ11 ˆΞ12 < 0 (7) ˆΞ T 11 ˆΞ 22 where Ψ Ψ2 0 ˆΞ 11 =, Ψi = A T i P i P i A i V ij, 0 0 ΨN Ψ 12 = diag(λ 1,, Λ N ), Λ i =[A T 1iP 1 A T NiP N, Ψ 22 = diag(ˆλ 1,, ˆΛ N ), ˆΛi = diag(v 1i,,V Ni )
8 298 JD CHEN Remark 23 Suppose that for any chosen matri B ij = A ij, i, j N we have that the delay term A ij j (s)ds, i, j N, have been converted to the left side of the system (2) By the proof of Theorem 21, the corresponding matrices V ij,andw ijk, i, j, k N, could be chosen as zero matrices, and the LMI condition in (3) is reduced by its corresponding elements Choosing B ij = A ij, i, j N in Theorem 21 we achieve the following result that depends on delay arguments Corollary 22 The system (1) is asymptotically stable for any constant time delays h ij satisfying 0 h ij h ij, i, j N, provided that Â i = A i A ii is Hurwitz and there eist symmetric positive definite matrices P i, R ij,andt ij, i, j N satisfying the following LMI condition: Ξ 11 Ξ12 Ξ13 Ξ T 12 Ξ 22 0 Ξ T 13 0 Ξ 33 < 0 (8) where Ψ 1 A T 21 P 2 P 1 A 12 A T N1 P N P 1 A 1N P 2 A 21 A T 12 Ξ 11 = P 1 Ψ2 A T N2 P N P 2 A 2N, P N A N1 A T 1N P 1 P N A N2 A T 2N P 2 ΨN Ψ 1 = ÂT i P i P i Â i ( h ji R ji h ki T kji ), Ξ 12 = diag(γ 1,, Γ N ), Γ i =[ hi1 A T i P i A i1 hin A T i P i A in, Ξ 22 = diag(ˆγ 1,, ˆΓ N ), ˆΓi = diag(r i1,,r in ), Ξ 13 = diag(π 1,, Π N ), Π i =[Π i1 Π in, Π ki =[ hi1 A T ik P ia i1 hin A T ik P ia in, Ξ 33 = diag(ˆπ 1,, ˆΠ N ), ˆΠi = diag(ˆπ i1 ˆΠ in ), ˆΠ ki = diag(t ik1,,t ikn ) Now, we provide a procedure for testing the asymptotic stability of the system (1) Step 1 Choosing all B ij =0, i, j N the delayindependent criterion in Corollary 21 is applied to test the stability of the system (1) Step 2 If the condition in Corollary 21 cannot be satisfied, we choose all B ij = A ij, i, j N and the delaydependent criterion in Corollary 22 is applied to test the stability of the system (1)
9 Stability criteria for largescale timedelay systems 299 Step 3 By employing the method of GAs (Chen, 1998; Eshelman and Schaffer, 1993; Michalewicz, 1994; Gen and Cheng, 1997), the matri parameters B ij, i, j N that maimize the fitness function h ij, i, j N and Theorem 21 will be search in this algorithm Continue this algorithm until finding optimal matri parameters B ij =0, i, j N such that the system (1) and (2) are both asymptotically stable for any constant timedelays h ij satisfying 0 h ij h ij, i, j N 3 Eample Eample 31 Largescale systems with time delays: Subsystem 1: ẋ 1 (t) = [ Subsystem 2: ẋ 2 (t) = [ (t) 2 (t) (t h 11 ) (t h 12 ), (t h 21 ) (t h 22 ) (9) Comparison of system (1) with system [ (9), we have N =2 By Theorem and GAs with h11 =21143, B 11 =,andb = B 21 = B 22 =, the LMI (3) is satisfied with P 00 1 =, P = , R =, V =, V 21 =, V =, V =, [ T 111 =, W =, W = Hence, we conclude that system (9) is asymptotically stable for 0 h , andh 12,h 21,h 22 R The delayindependent criteria in Hmamed (1986), Huang et al (1995), Lyou et al (1984), Schoen and Geering (1995), Wang et al (1991) cannot be satisfied The delaydependent stability criteria of Tsay et al (1996) cannot be applied for sufficiently large time delays, h 12,h 21,h 22 R 4 Conclusions In this paper, asymptotic stability analysis problem for a class of largescale neutral timedelay systems has been considered A model transformation and
10 300 JD CHEN the Lyapunov stability theorem have been applied to obtain some stability criteria for such systems Furthermore, the computational programming LMI with GAs has been used to improve the proposed results Finally, it has been shown using a numerical eample that the result shown in this paper is fleible and sharp Acknowledgement The author would like to thank anonymous reviewers for their helpful comments The research reported here was supported by the National Science Council of Taiwan, ROC under grant NSC E References Boyd,S,Ghaoui,LEl,Feron,Eand Balakrishnan, V (1994) Linear Matri Inequalities In System And Control Theory SIAM, Philadelphia Chen, ZM, Lin, HS and Hung, CM (1998) The Design and Application of a GeneticBased Fuzzy Gray Prediction Controller The Journal of Grey System 1, Eshelman, LJ and Schaffer, JD (1993) RealCoded Genetic Algorithms and Interval Schemata In: LD Whitley, ed, Foundation of Genetic Algorithms2, Morgan Kaufmann Publishers Gen, M and Cheng, RW (1997) Genetic Algorithms and Engineering Design John Wiley and Sons, New York Hale, JK and Verduyn Lunel, SM (1993) Introduction to functional differential equations SpringerVerlag, New York Hmamed, A (1986) Note on the stability of large scale systems with delays International Journal of Systems Science 17, Huang, SN, Shao, HH and Zhang, ZJ (1995) Stability analysis of largescale systems with delays Systems & Control Letters 25, Kim, J H (2001) Delay and its timederivative dependent robust stability of timedelayed linear systems with uncertainty IEEE Transaction Automatic Control 46, Kolmanovskii, VB, and Myshkis, A (1992) Applied Theory of Functional Differential Equations Kluwer Academic Publishers, Dordrecht, Netherlands Kwon, OM, and Park, JH (2004) On improved delaydependent robust control for uncertain timedelay systems IEEE Transaction Automatic Control 49, Li, X, and De Souza, CE (1997) Delaydependent robust stability and stabilization of uncertain linear delay systems: a linear matri inequality approach IEEE Transaction Automatic Control 42,
11 Stability criteria for largescale timedelay systems 301 Lyou, J, Kim, YS and Bien, Z (1984) A note on the stability of a class of interconnected dynamic systems International Journal of Control 39, Michalewicz, Z (1994) Genetic Algorithm Data Structures = Evolution Programs SpringerVerlag, New York Michel, AN and Miller, RK (1977) Qualitative Analysis of Large Scale Dynamic Systems Academic Press, New York Niculescu, SI (2001) Delay Effects on Stability A Robust Control Approach SpringerVerlag Park, P (1999) A delay dependent stability criterion for systems with uncertain timeinvariant delays IEEE Transactions on Automatic Control 44, Richard, JP (2003) Timedelay systems: an verview of some recent advances and open problems Automatica 39, Schoen, GM and Geering, HP (1995) A note on robustness bounds for largescale timedelay systems International Journal of Systems Science 26, Siljak, DD (1978) LargeScale Dynamic Systems: Stability and Structure Elsevier NorthHolland, New York Su, TJ, Lu, CY, and Tsai, JSH (2001) LMI approach to delay dependent robust stability for uncertain timedelay systems IEE Proceeding Control Theory Applications 148, Tsay, JT, Liu, PL and Su, TJ (1996) Robust stability for perturbed largescale timedelay systems IEE Proceedings Control Theory Applications 143, Wang, WJ, Song, CC and Kao, CC (1991) Robustness bounds for largescale timedelay systems with structured and unstructured uncertainties International Journal of Systems Science 22,
Example 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum. asin. k, a, and b. We study stability of the origin x
Lecture 4. LaSalle s Invariance Principle We begin with a motivating eample. Eample 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum Dynamics of a pendulum with friction can be written
More informationOn the DStability of Linear and Nonlinear Positive Switched Systems
On the DStability of Linear and Nonlinear Positive Switched Systems V. S. Bokharaie, O. Mason and F. Wirth Abstract We present a number of results on Dstability of positive switched systems. Different
More informationState Observer Design for a Class of TakagiSugeno DiscreteTime Descriptor Systems
Applied Mathematical Sciences, Vol. 9, 2015, no. 118, 58715885 HIKARI Ltd, www.mhikari.com http://dx.doi.org/10.12988/ams.2015.54288 State Observer Design for a Class of TakagiSugeno DiscreteTime Descriptor
More informationLecture 7: Finding Lyapunov Functions 1
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 7: Finding Lyapunov Functions 1
More informationCharacterizations are given for the positive and completely positive maps on n n
Special Classes of Positive and Completely Positive Maps ChiKwong Li and Hugo J. Woerdemany Department of Mathematics The College of William and Mary Williamsburg, Virginia 387 Email: ckli@cs.wm.edu
More informationLecture 13 Linear quadratic Lyapunov theory
EE363 Winter 289 Lecture 13 Linear quadratic Lyapunov theory the Lyapunov equation Lyapunov stability conditions the Lyapunov operator and integral evaluating quadratic integrals analysis of ARE discretetime
More informationSTABILITY GUARANTEED ACTIVE FAULT TOLERANT CONTROL OF NETWORKED CONTROL SYSTEMS. Shanbin Li, Dominique Sauter, Christophe Aubrun, Joseph Yamé
SABILIY GUARANEED ACIVE FAUL OLERAN CONROL OF NEWORKED CONROL SYSEMS Shanbin Li, Dominique Sauter, Christophe Aubrun, Joseph Yamé Centre de Recherche en Automatique de Nancy Université Henri Poincaré,
More informationAn interval linear programming contractor
An interval linear programming contractor Introduction Milan Hladík Abstract. We consider linear programming with interval data. One of the most challenging problems in this topic is to determine or tight
More informationA Passivity Measure Of Systems In Cascade Based On Passivity Indices
49th IEEE Conference on Decision and Control December 57, Hilton Atlanta Hotel, Atlanta, GA, USA A Passivity Measure Of Systems In Cascade Based On Passivity Indices Han Yu and Panos J Antsaklis Abstract
More informationNonlinear Systems of Ordinary Differential Equations
Differential Equations Massoud Malek Nonlinear Systems of Ordinary Differential Equations Dynamical System. A dynamical system has a state determined by a collection of real numbers, or more generally
More informationAn Adaptive Attitude Control Formulation under Angular Velocity Constraints
An Adaptive Attitude Control Formulation under Angular Velocity Constraints Puneet Singla and Tarunraj Singh An adaptive attitude control law which explicitly takes into account the constraints on individual
More informationLecture 12 Basic Lyapunov theory
EE363 Winter 200809 Lecture 12 Basic Lyapunov theory stability positive definite functions global Lyapunov stability theorems Lasalle s theorem converse Lyapunov theorems finding Lyapunov functions 12
More informationMinimizing the Number of Machines in a UnitTime Scheduling Problem
Minimizing the Number of Machines in a UnitTime Scheduling Problem Svetlana A. Kravchenko 1 United Institute of Informatics Problems, Surganova St. 6, 220012 Minsk, Belarus kravch@newman.basnet.by Frank
More informationA New Natureinspired Algorithm for Load Balancing
A New Natureinspired Algorithm for Load Balancing Xiang Feng East China University of Science and Technology Shanghai, China 200237 Email: xfeng{@ecusteducn, @cshkuhk} Francis CM Lau The University of
More informationOptimal PID Controller Design for AVR System
Tamkang Journal of Science and Engineering, Vol. 2, No. 3, pp. 259 270 (2009) 259 Optimal PID Controller Design for AVR System ChingChang Wong*, ShihAn Li and HouYi Wang Department of Electrical Engineering,
More informationPortfolio selection based on upper and lower exponential possibility distributions
European Journal of Operational Research 114 (1999) 115±126 Theory and Methodology Portfolio selection based on upper and lower exponential possibility distributions Hideo Tanaka *, Peijun Guo Department
More informationCONTROLLABILITY. Chapter 2. 2.1 Reachable Set and Controllability. Suppose we have a linear system described by the state equation
Chapter 2 CONTROLLABILITY 2 Reachable Set and Controllability Suppose we have a linear system described by the state equation ẋ Ax + Bu (2) x() x Consider the following problem For a given vector x in
More informationmax cx s.t. Ax c where the matrix A, cost vector c and right hand side b are given and x is a vector of variables. For this example we have x
Linear Programming Linear programming refers to problems stated as maximization or minimization of a linear function subject to constraints that are linear equalities and inequalities. Although the study
More informationMining changes in customer behavior in retail marketing
Expert Systems with Applications 28 (2005) 773 781 www.elsevier.com/locate/eswa Mining changes in customer behavior in retail marketing MuChen Chen a, *, AiLun Chiu b, HsuHwa Chang c a Department of
More informationR mp nq. (13.1) a m1 B a mn B
This electronic version is for personal use and may not be duplicated or distributed Chapter 13 Kronecker Products 131 Definition and Examples Definition 131 Let A R m n, B R p q Then the Kronecker product
More informationHackingproofness and Stability in a Model of Information Security Networks
Hackingproofness and Stability in a Model of Information Security Networks Sunghoon Hong Preliminary draft, not for citation. March 1, 2008 Abstract We introduce a model of information security networks.
More information4. MATRICES Matrices
4. MATRICES 170 4. Matrices 4.1. Definitions. Definition 4.1.1. A matrix is a rectangular array of numbers. A matrix with m rows and n columns is said to have dimension m n and may be represented as follows:
More informationAn improved online algorithm for scheduling on two unrestrictive parallel batch processing machines
This is the PrePublished Version. An improved online algorithm for scheduling on two unrestrictive parallel batch processing machines Q.Q. Nong, T.C.E. Cheng, C.T. Ng Department of Mathematics, Ocean
More information3.7 Nonautonomous linear systems of ODE. General theory
3.7 Nonautonomous linear systems of ODE. General theory Now I will study the ODE in the form ẋ = A(t)x + g(t), x(t) R k, A, g C(I), (3.1) where now the matrix A is time dependent and continuous on some
More informationMATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix.
MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix. Matrices Definition. An mbyn matrix is a rectangular array of numbers that has m rows and n columns: a 11
More informationOPTIMAL SELECTION BASED ON RELATIVE RANK* (the "Secretary Problem")
OPTIMAL SELECTION BASED ON RELATIVE RANK* (the "Secretary Problem") BY Y. S. CHOW, S. MORIGUTI, H. ROBBINS AND S. M. SAMUELS ABSTRACT n rankable persons appear sequentially in random order. At the ith
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationSystems with Persistent Memory: the Observation Inequality Problems and Solutions
Chapter 6 Systems with Persistent Memory: the Observation Inequality Problems and Solutions Facts that are recalled in the problems wt) = ut) + 1 c A 1 s ] R c t s)) hws) + Ks r)wr)dr ds. 6.1) w = w +
More informationNOTES ON LINEAR TRANSFORMATIONS
NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all
More informationIntroduction to Flocking {Stochastic Matrices}
Supelec EECI Graduate School in Control Introduction to Flocking {Stochastic Matrices} A. S. Morse Yale University Gif sur  Yvette May 21, 2012 CRAIG REYNOLDS  1987 BOIDS The Lion King CRAIG REYNOLDS
More informationBrief Paper. of discretetime linear systems. www.ietdl.org
Published in IET Control Theory and Applications Received on 28th August 2012 Accepted on 26th October 2012 Brief Paper ISSN 17518644 Temporal and onestep stabilisability and detectability of discretetime
More informationOn closedform solutions of a resource allocation problem in parallel funding of R&D projects
Operations Research Letters 27 (2000) 229 234 www.elsevier.com/locate/dsw On closedform solutions of a resource allocation problem in parallel funding of R&D proects Ulku Gurler, Mustafa. C. Pnar, Mohamed
More informationOPTIMAl PREMIUM CONTROl IN A NONliFE INSURANCE BUSINESS
ONDERZOEKSRAPPORT NR 8904 OPTIMAl PREMIUM CONTROl IN A NONliFE INSURANCE BUSINESS BY M. VANDEBROEK & J. DHAENE D/1989/2376/5 1 IN A OPTIMAl PREMIUM CONTROl NONliFE INSURANCE BUSINESS By Martina Vandebroek
More informationMinimizing Probing Cost and Achieving Identifiability in Probe Based Network Link Monitoring
Minimizing Probing Cost and Achieving Identifiability in Probe Based Network Link Monitoring Qiang Zheng, Student Member, IEEE, and Guohong Cao, Fellow, IEEE Department of Computer Science and Engineering
More informationSection 6.1  Inner Products and Norms
Section 6.1  Inner Products and Norms Definition. Let V be a vector space over F {R, C}. An inner product on V is a function that assigns, to every ordered pair of vectors x and y in V, a scalar in F,
More informationSome Research Problems in Uncertainty Theory
Journal of Uncertain Systems Vol.3, No.1, pp.310, 2009 Online at: www.jus.org.uk Some Research Problems in Uncertainty Theory aoding Liu Uncertainty Theory Laboratory, Department of Mathematical Sciences
More informationLecture 5: Conic Optimization: Overview
EE 227A: Conve Optimization and Applications January 31, 2012 Lecture 5: Conic Optimization: Overview Lecturer: Laurent El Ghaoui Reading assignment: Chapter 4 of BV. Sections 3.13.6 of WTB. 5.1 Linear
More information4 Lyapunov Stability Theory
4 Lyapunov Stability Theory In this section we review the tools of Lyapunov stability theory. These tools will be used in the next section to analyze the stability properties of a robot controller. We
More informationA matrix over a field F is a rectangular array of elements from F. The symbol
Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F) denotes the collection of all m n matrices over F Matrices will usually be denoted
More informationLECTURE 1 I. Inverse matrices We return now to the problem of solving linear equations. Recall that we are trying to find x such that IA = A
LECTURE I. Inverse matrices We return now to the problem of solving linear equations. Recall that we are trying to find such that A = y. Recall: there is a matri I such that for all R n. It follows that
More informationTwo classes of ternary codes and their weight distributions
Two classes of ternary codes and their weight distributions Cunsheng Ding, Torleiv Kløve, and Francesco Sica Abstract In this paper we describe two classes of ternary codes, determine their minimum weight
More informationDATA ANALYSIS II. Matrix Algorithms
DATA ANALYSIS II Matrix Algorithms Similarity Matrix Given a dataset D = {x i }, i=1,..,n consisting of n points in R d, let A denote the n n symmetric similarity matrix between the points, given as where
More informationOptimization under fuzzy ifthen rules
Optimization under fuzzy ifthen rules Christer Carlsson christer.carlsson@abo.fi Robert Fullér rfuller@abo.fi Abstract The aim of this paper is to introduce a novel statement of fuzzy mathematical programming
More informationDeterminants. Dr. Doreen De Leon Math 152, Fall 2015
Determinants Dr. Doreen De Leon Math 52, Fall 205 Determinant of a Matrix Elementary Matrices We will first discuss matrices that can be used to produce an elementary row operation on a given matrix A.
More informationAN EVALUATION METHOD FOR ENTERPRISE RESOURCE PLANNING SYSTEMS
Journal of the Operations Research Society of Japan 2008, Vol. 51, No. 4, 299309 AN EVALUATION METHOD FOR ENTERPRISE RESOURCE PLANNING SYSTEMS ShinGuang Chen Tungnan University YiKuei Lin National Taiwan
More informationDirect Methods for Solving Linear Systems. Matrix Factorization
Direct Methods for Solving Linear Systems Matrix Factorization Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University c 2011
More informationApplication of Fuzzy Logic in Operation Management Research
International Journal of Scientific and Research Publications, Volume 4, Issue 10, October 2014 1 Application of Fuzzy Logic in Operation Management Research Preeti Kaushik Assistant Professor, Inderprastha
More informationGenerating Valid 4 4 Correlation Matrices
Applied Mathematics ENotes, 7(2007), 5359 c ISSN 16072510 Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/ Generating Valid 4 4 Correlation Matrices Mark Budden, Paul Hadavas, Lorrie
More informationHierarchical Facility Location for the Reverse Logistics Network Design under Uncertainty
Journal of Uncertain Systems Vol.8, No.4, pp.5570, 04 Online at: www.jus.org.uk Hierarchical Facility Location for the Reverse Logistics Network Design under Uncertainty Ke Wang, Quan Yang School of Management,
More information7 Gaussian Elimination and LU Factorization
7 Gaussian Elimination and LU Factorization In this final section on matrix factorization methods for solving Ax = b we want to take a closer look at Gaussian elimination (probably the best known method
More informationLyapunov Stability Analysis of Energy Constraint for Intelligent Home Energy Management System
JAIST Reposi https://dspace.j Title Lyapunov stability analysis for intelligent home energy of energ manageme Author(s)Umer, Saher; Tan, Yasuo; Lim, Azman Citation IEICE Technical Report on Ubiquitous
More informationDetermining Feasible Solution in Imprecise Linear Inequality Systems
Applied Mathematical Sciences, Vol. 2, 2008, no. 36, 17891797 Determining Feasible Solution in Imprecise Linear Inequality Systems A. A. Noura and F. H. Saljooghi Department of Mathematics Sistan and
More informationAN ITERATIVE METHOD FOR THE HERMITIANGENERALIZED HAMILTONIAN SOLUTIONS TO THE INVERSE PROBLEM AX = B WITH A SUBMATRIX CONSTRAINT
Bulletin of the Iranian Mathematical Society Vol. 39 No. 6 (2013), pp 1291260. AN ITERATIVE METHOD FOR THE HERMITIANGENERALIZED HAMILTONIAN SOLUTIONS TO THE INVERSE PROBLEM AX = B WITH A SUBMATRIX CONSTRAINT
More informationPredictive Control Algorithms: Stability despite Shortened Optimization Horizons
Predictive Control Algorithms: Stability despite Shortened Optimization Horizons Philipp Braun Jürgen Pannek Karl Worthmann University of Bayreuth, 9544 Bayreuth, Germany University of the Federal Armed
More informationNetwork Traffic Modelling
University of York Dissertation submitted for the MSc in Mathematics with Modern Applications, Department of Mathematics, University of York, UK. August 009 Network Traffic Modelling Author: David Slade
More informationNON SINGULAR MATRICES. DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that
NON SINGULAR MATRICES DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that AB = I n = BA. Any matrix B with the above property is called
More informationEquilibria and Dynamics of. Supply Chain Network Competition with Information Asymmetry. and Minimum Quality Standards
Equilibria and Dynamics of Supply Chain Network Competition with Information Asymmetry in Quality and Minimum Quality Standards Anna Nagurney John F. Smith Memorial Professor and Dong (Michelle) Li Doctoral
More informationReal Roots of Quadratic Interval Polynomials 1
Int. Journal of Math. Analysis, Vol. 1, 2007, no. 21, 10411050 Real Roots of Quadratic Interval Polynomials 1 Ibraheem Alolyan Mathematics Department College of Science, King Saud University P.O. Box:
More informationLecture 5  Triangular Factorizations & Operation Counts
LU Factorization Lecture 5  Triangular Factorizations & Operation Counts We have seen that the process of GE essentially factors a matrix A into LU Now we want to see how this factorization allows us
More informationCONTROL SYSTEMS, ROBOTICS AND AUTOMATION Vol. XVI  Fault Accomodation Using Model Predictive Methods  Jovan D. Bošković and Raman K.
FAULT ACCOMMODATION USING MODEL PREDICTIVE METHODS Scientific Systems Company, Inc., Woburn, Massachusetts, USA. Keywords: Fault accommodation, Model Predictive Control (MPC), Failure Detection, Identification
More information1.5 Elementary Matrices and a Method for Finding the Inverse
.5 Elementary Matrices and a Method for Finding the Inverse Definition A n n matrix is called an elementary matrix if it can be obtained from I n by performing a single elementary row operation Reminder:
More information15 Limit sets. Lyapunov functions
15 Limit sets. Lyapunov functions At this point, considering the solutions to ẋ = f(x), x U R 2, (1) we were most interested in the behavior of solutions when t (sometimes, this is called asymptotic behavior
More informationDemand Forecasting Optimization in Supply Chain
2011 International Conference on Information Management and Engineering (ICIME 2011) IPCSIT vol. 52 (2012) (2012) IACSIT Press, Singapore DOI: 10.7763/IPCSIT.2012.V52.12 Demand Forecasting Optimization
More informationFurther Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1
Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1 J. Zhang Institute of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing
More informationThe Conference Call Search Problem in Wireless Networks
The Conference Call Search Problem in Wireless Networks Leah Epstein 1, and Asaf Levin 2 1 Department of Mathematics, University of Haifa, 31905 Haifa, Israel. lea@math.haifa.ac.il 2 Department of Statistics,
More informationUsing Congestion Graphs to Analyze the Stability of Network Congestion Control
INTERNATIONAL JOURNAL OF INTELLIGENT CONTROL AND SYSTEMS VOL. 14, NO. 2, JUNE 29, 125133 Using Congestion Graphs to Analyze the Stability of Network Congestion Control Damian HobsonGarcia and Tomohisa
More informationA New Method for MultiObjective Linear Programming Models with Fuzzy Random Variables
383838383813 Journal of Uncertain Systems Vol.6, No.1, pp.3850, 2012 Online at: www.jus.org.uk A New Method for MultiObjective Linear Programming Models with Fuzzy Random Variables Javad Nematian * Department
More informationLINEAR ALGEBRA OF PASCAL MATRICES
LINEAR ALGEBRA OF PASCAL MATRICES LINDSAY YATES Abstract. The famous Pascal s triangle appears in many areas of mathematics, such as number theory, combinatorics and algebra. Pascal matrices are derived
More informationMULTIPHASE FUZZY CONTROL OF SINGLE INTERSECTION IN TRAFFIC SYSTEM BASED ON GENETIC ALGORITHM. Received February 2011; revised June 2011
International Journal of Innovative Computing, Information and Control ICIC International c 2012 ISSN 13494198 Volume 8, Number 5(A), May 2012 pp. 3387 3397 MULTIPHASE FUZZY CONTROL OF SINGLE INTERSECTION
More informationA simple criterion on degree sequences of graphs
Discrete Applied Mathematics 156 (2008) 3513 3517 Contents lists available at ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam Note A simple criterion on degree
More informationDecentralized Control of Multimachine Power Systems
29 American Control Conference Hyatt Regency Riverfront, St Louis, MO, USA June 112, 29 ThA33 Decentralized Control of Multimachine Power Systems Karanjit Kalsi, Jianming Lian and Stanislaw H Żak Abstract
More informationFuzzy regression model with fuzzy input and output data for manpower forecasting
Fuzzy Sets and Systems 9 (200) 205 23 www.elsevier.com/locate/fss Fuzzy regression model with fuzzy input and output data for manpower forecasting Hong Tau Lee, Sheu Hua Chen Department of Industrial Engineering
More informationMatrix Differentiation
1 Introduction Matrix Differentiation ( and some other stuff ) Randal J. Barnes Department of Civil Engineering, University of Minnesota Minneapolis, Minnesota, USA Throughout this presentation I have
More informationOPTIMAL CONTROL OF A COMMERCIAL LOAN REPAYMENT PLAN. E.V. Grigorieva. E.N. Khailov
DISCRETE AND CONTINUOUS Website: http://aimsciences.org DYNAMICAL SYSTEMS Supplement Volume 2005 pp. 345 354 OPTIMAL CONTROL OF A COMMERCIAL LOAN REPAYMENT PLAN E.V. Grigorieva Department of Mathematics
More informationBackground: State Estimation
State Estimation Cyber Security of the Smart Grid Dr. Deepa Kundur Background: State Estimation University of Toronto Dr. Deepa Kundur (University of Toronto) Cyber Security of the Smart Grid 1 / 81 Dr.
More informationBy choosing to view this document, you agree to all provisions of the copyright laws protecting it.
This material is posted here with permission of the IEEE Such permission of the IEEE does not in any way imply IEEE endorsement of any of Helsinki University of Technology's products or services Internal
More informationLecture 11. Shuanglin Shao. October 2nd and 7th, 2013
Lecture 11 Shuanglin Shao October 2nd and 7th, 2013 Matrix determinants: addition. Determinants: multiplication. Adjoint of a matrix. Cramer s rule to solve a linear system. Recall that from the previous
More informationNotes V General Equilibrium: Positive Theory. 1 Walrasian Equilibrium and Excess Demand
Notes V General Equilibrium: Positive Theory In this lecture we go on considering a general equilibrium model of a private ownership economy. In contrast to the Notes IV, we focus on positive issues such
More informationLecture 3: Linear Programming Relaxations and Rounding
Lecture 3: Linear Programming Relaxations and Rounding 1 Approximation Algorithms and Linear Relaxations For the time being, suppose we have a minimization problem. Many times, the problem at hand can
More informationClosedloop Load Balancing: Comparison of a Discrete Event Simulation with Experiments
25 American Control Conference June 8, 25. Portland, OR, USA ThB4.2 Closedloop Load Balancing: Comparison of a Discrete Event Simulation with Experiments Zhong Tang, John White, John Chiasson, J. Douglas
More informationAN UPPER BOUND ON THE LAPLACIAN SPECTRAL RADIUS OF THE SIGNED GRAPHS
Discussiones Mathematicae Graph Theory 28 (2008 ) 345 359 AN UPPER BOUND ON THE LAPLACIAN SPECTRAL RADIUS OF THE SIGNED GRAPHS HongHai Li College of Mathematic and Information Science Jiangxi Normal University
More informationA goal programming approach for solving random interval linear programming problem
A goal programming approach for solving random interval linear programming problem Ziba ARJMANDZADEH, Mohammadreza SAFI 1 1,2 Department of Mathematics, Semnan University Email: z.arjmand@students.semnan.ac.ir,
More informationMATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix.
MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix. Inverse matrix Definition. Let A be an n n matrix. The inverse of A is an n n matrix, denoted
More informationDissipative Analysis and Control of StateSpace Symmetric Systems
28 American Control Conference Westin Seattle Hotel, Seattle, Washington, USA June 1113, 28 WeA12.3 Dissipative Analysis and Control of StateSpace Symmetric Systems Mona MeisamiAzad, Javad Mohammadpour,
More informationBasics Inversion and related concepts Random vectors Matrix calculus. Matrix algebra. Patrick Breheny. January 20
Matrix algebra January 20 Introduction Basics The mathematics of multiple regression revolves around ordering and keeping track of large arrays of numbers and solving systems of equations The mathematical
More informationDynamic Eigenvalues for Scalar Linear TimeVarying Systems
Dynamic Eigenvalues for Scalar Linear TimeVarying Systems P. van der Kloet and F.L. Neerhoff Department of Electrical Engineering Delft University of Technology Mekelweg 4 2628 CD Delft The Netherlands
More informationInner Product Spaces and Orthogonality
Inner Product Spaces and Orthogonality week 34 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,
More informationApproximability of TwoMachine NoWait Flowshop Scheduling with Availability Constraints
Approximability of TwoMachine NoWait Flowshop Scheduling with Availability Constraints T.C. Edwin Cheng 1, and Zhaohui Liu 1,2 1 Department of Management, The Hong Kong Polytechnic University Kowloon,
More informationUNIT 2 MATRICES  I 2.0 INTRODUCTION. Structure
UNIT 2 MATRICES  I Matrices  I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress
More informationSimilarity and Diagonalization. Similar Matrices
MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that
More informationThe HenstockKurzweilStieltjes type integral for real functions on a fractal subset of the real line
The HenstockKurzweilStieltjes type integral for real functions on a fractal subset of the real line D. Bongiorno, G. Corrao Dipartimento di Ingegneria lettrica, lettronica e delle Telecomunicazioni,
More informationFactorization Theorems
Chapter 7 Factorization Theorems This chapter highlights a few of the many factorization theorems for matrices While some factorization results are relatively direct, others are iterative While some factorization
More informationLinear Maps. Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007)
MAT067 University of California, Davis Winter 2007 Linear Maps Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007) As we have discussed in the lecture on What is Linear Algebra? one of
More informationScalar Valued Functions of Several Variables; the Gradient Vector
Scalar Valued Functions of Several Variables; the Gradient Vector Scalar Valued Functions vector valued function of n variables: Let us consider a scalar (i.e., numerical, rather than y = φ(x = φ(x 1,
More informationThe Australian Journal of Mathematical Analysis and Applications
The Australian Journal of Mathematical Analysis and Applications Volume 7, Issue, Article 11, pp. 114, 011 SOME HOMOGENEOUS CYCLIC INEQUALITIES OF THREE VARIABLES OF DEGREE THREE AND FOUR TETSUYA ANDO
More informationPÓLYA URN MODELS UNDER GENERAL REPLACEMENT SCHEMES
J. Japan Statist. Soc. Vol. 31 No. 2 2001 193 205 PÓLYA URN MODELS UNDER GENERAL REPLACEMENT SCHEMES Kiyoshi Inoue* and Sigeo Aki* In this paper, we consider a Pólya urn model containing balls of m different
More informationApplications of improved grey prediction model for power demand forecasting
Energy Conversion and Management 44 (2003) 2241 2249 www.elsevier.com/locate/enconman Applications of improved grey prediction model for power demand forecasting CheChiang Hsu a, *, ChiaYon Chen b a
More informationMATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix.
MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix. Nullspace Let A = (a ij ) be an m n matrix. Definition. The nullspace of the matrix A, denoted N(A), is the set of all ndimensional column
More informationVictims Compensation Claim Status of All Pending Claims and Claims Decided Within the Last Three Years
Claim#:021914174 Initials: J.T. Last4SSN: 6996 DOB: 5/3/1970 Crime Date: 4/30/2013 Status: Claim is currently under review. Decision expected within 7 days Claim#:041715334 Initials: M.S. Last4SSN: 2957
More informationAnalysis of Model and Key Technology for P2P Network Route Security Evaluation with 2tuple Linguistic Information
Journal of Computational Information Systems 9: 14 2013 5529 5534 Available at http://www.jofcis.com Analysis of Model and Key Technology for P2P Network Route Security Evaluation with 2tuple Linguistic
More information