Luenberger D.G. "Dynamic Equations in Descriptor Form." IEEE Transactions on Automatic Control, Vol.AC-20, No. 3, pp. 239-246, June 1977.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....class of time evolutionary phenomena and are often the product of problem formulation in system theory, especially when the variables used are the natural describing variables of the underlying process. This topic has received a lot of attention over the last twenty years; see, in particular, [5, 6]. Within the general class of linear descriptor systems, periodic systems form an important subclass they are suitable models for many natural and man made phenomena, and are finding increasing use in control theory as well [1] In this paper, we restrict attention to solvability and ....

....responsibility rests with its authors. Partial support came from NSF grants CCR 96 19596 and ECS 96 24152. A preliminary version was presented at a special session on periodic systems organized by Dr. V. Hernandez during MTNS, St. Louis, MO, USA, June 24 28, 1996. 1 defined by Luenberger [5, 6] and present new results which extend those obtained by him for the time invariant (TI) case. We begin by briefly recalling some definitions and prior work (see [5] for details) A linear discrete time descriptor variable system defined on the time interval [0, N ] has the form E k x k 1 = A k x k ....

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D. G. Luenberger. Dynamic equations in descriptor form. IEEE Trans. Auto. Cntrl, AC-22:312--321, June 1977.

....(0) W (N ) 3 7 7 7 7 7 7 7 5 Delta 2 6 6 6 6 6 6 6 4 x(0) x(1) x(N Gamma 1) x(N ) 3 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 4 0 0 . 0 w 3 7 7 7 7 7 7 7 5 : 3) Here N simply specifies a time interval of interest. For a discussion of the general time varying TPBVP, see Luenberger [2, 3]. When studying equation (1) it is common to write V (k) E(k) W (k 1) x(k 1) V (k) A(k) W (k) x(k) 4) where V (k) V (k K) W (k) W (k K) are nonsingular for all k, and try to achieve as much simplification in form as possible for E(k) A(k) For instance, in the time invariant ....

D. G. Luenberger. Dynamic equations in descriptor form. IEEE Trans. Auto. Cntrl, AC-22:pp. 312--321, June 1977.

....form. Descriptor systems, also referred to as generalized state space systems, arise in diverse applications such as the study of large scale power systems, interconnected systems, robotics, econometrics, decentralized control and decision networks, population models, optimization problems etc. [4]. This paper is organized as follows. Section 1.1 introduces notation and some preliminary facts about linear periodic discrete time systems. For easy reference, and to avoid undue repetition, we list below some oft used acronyms: DPLE: Discrete time periodic Lyapunov equation DALE: ....

D. G. Luenberger, "Dynamic equations in descriptor form," IEEE Trans. Automat. Control, vol. AC-22, pp. 312--321, June 1977.

....with system matrices E 2 C n Thetan , A 2 C n Thetan , B 2 C n Thetam , C 2 C p Thetan , state x = x(t) 2 C n , input u = u(t) 2 C m , and output y = y(t) 2 C p . Descriptor systems arise naturally in circuit design, mechanical multi body systems and a variety of other applications [25, 32, 33]. They have recently attracted the attention of many authors to all aspects of control including pole placement, filtering, stabilization, controllability, observability, optimal control problems, invertibility, duality, realization etc. See for example [6, 14, 13, 27] and the references therein. ....

D. G. Luenberger, Dynamic equations in descriptor form, IEEE Trans. Automat. Control, AC-22 (1977), pp. 312--321.

.... Gamma1 2 Q 2 From a computational viewpoint, a most ideal situation for a simulation model is where the dynamic equations have an initial condition vector which when propagated forward serves as a state vector for every time instant. A model whose equations have that property is called regular [12]. Regularity is essentially equivalent to real time representation or causality [12] A necessary and sufficient condition for the simulation model to be regular are: 1. all the p and q variables are independent, 2. the causal graph has no strong components. The three diode circuit does not ....

....model is where the dynamic equations have an initial condition vector which when propagated forward serves as a state vector for every time instant. A model whose equations have that property is called regular [12] Regularity is essentially equivalent to real time representation or causality [12]. A necessary and sufficient condition for the simulation model to be regular are: 1. all the p and q variables are independent, 2. the causal graph has no strong components. The three diode circuit does not have a regular model since its causal graph (fig. 12) contains a strong component. An ....

D.G. Luenberger. Dynamic equations in descriptor form. IEEE Trans. Automatic Control, 22:312--321, 1977.

....= Cx(t) Du(t) 1.3) for which numerous analysis and design methods exist. In recent years, there has been considerable interest in the study of modified state space systems where now T ( E Gamma A) and E is a general matrix that may be singular (see e.g. Dervi soglu and Desoer (1975) Luenberger (1977), Verghese, Van Dooren and Kailath (1979) Campbell (1980) Verghese, Levy and Kailath (1981) Van Dooren (1981) Cobb (1984) Lewis (1985a, 1985b, 1986) Bender and Laub (1987) Misra and Patel (1989a, 1989b) Miminis (1993) and the references therein) The equations corresponding to this case ....

.... arises from their applications in representing and resolving problems concerning differential equations with perturbed coefficients, singular perturbations (Saxena, O Reilly and Kokotovic, 1984) noncausal systems (Bernhard, 1982) identification (Adams, Levy and Willsky, 1984) economic systems (Luenberger, 1977), interconnected systems (Rosenbrock and Pugh, 1974) and modeling of electronic circuits (Chua and P. M. Lin, 1975) A system described by (1.4) is said to be non singular if E has full rank and singular otherwise. The zeros of a non singular GSS system are identical to those of the corresponding ....

Luenberger, D.G. (1977). Dynamic equations in descriptor form. IEEE Trans. Automat. Contr., AC-22, 312-321.

....= Cx; 2) where E; A 2 R n Thetan , B 2 R n Thetam , C 2 R p Thetan , x 2 R n , u 2 R m , y 2 R p and x = dx=dt. For ease of notation, a descriptor system of the form (1) 2) is denoted here by (E; A; B; C) Descriptor systems arise naturally in a variety of practical circumstances [23, 32] and have recently been investigated in [8, 10, 11, 14, 19, 20, 21, 25, 26, 28, 29, 30, 33, 34, 36, 37, 38, 40, 41] We consider only square systems, since they arise naturally from realizations [11] and also since the nonsquare case can be reduced to the square case [6] With a little more ....

D. G. Luenberger. Dynamic equations in descriptor form. IEEE Trans. Automat. Control, AC-22:312--321, 1977.

.... This kind of restriction does not result in a loss of modeling power since any negative vector can be represented as the difference between two nonnegative vectors (Dantzig, 1963) In addition, when V 1 is a zero matrix, equation 1 reduces to the traditional Luenberger s descriptor system (Luenberger, 1977). Equation 1 becomes a conventional linear state equation when V 1 is a zero matrix and E 1 is an invertible square matrix. In general, E 1 is singular in equation 1 and slack variables v are introduced to represent uncertain system dynamics. We illustrate the system modeling with a simple ....

Luenberger, David G. (1977). Dynamic equations in descriptor form. IEEE Trans. on Automatic Control.

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Luenberger D.G. "Dynamic Equations in Descriptor Form." IEEE Transactions on Automatic Control, Vol.AC-20, No. 3, pp. 239-246, June 1977.

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Luenberger D.G.,Dynamic Equations in Descriptor Form, IEEE Trans .Autom. Contr., Vol-22, No 3, June 1977, pp. 312-321.

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D.G. Luenberger. Dynamic equations in descriptor form. IEEE Trans. Automat. Control, AC-22(3):312-321, 1977.

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Luenberger D.G.,'Dynamic Equations in Descriptor Form', IEEE Trans .Autom. Contr., Vol-22, No 3, June 1977, pp. 312-321.

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Luenberger D.G.,1977, Dynamic Equations in Descriptor Form, IEEE Trans .Autom. Contr., Vol-22, No 3, 312-321

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Luenberger D. G., Dynamic equations in descriptor form, IEEE Trans. Auto. Control, Vol. AC-22, 312-321, 1977.

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Luenberger, David, "Dynamic Equations in Descriptor Form," IEEE Transactions on Automatic Control, vol. AC-22, no. 3 (June 1977), 312-321.

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