linear system, functional differential equations, initial conditions, numerical schemes, structure, partial differential equations, functional equations, system, u2, perturbations, differential equations, stability radii, negative exponents
Content:
Dynamical Systems with Lossless Propagation and Neutral Functional Differential Equations Vladimir Rasvan Department of Automatic Control, University of Craiova, A.I.Cuza, 13, Craiova, RO1100, Romania Tel: +4051143198, Fax: +4051143198 http://
[email protected]1 Introduction By lossless propagation it is understood the phenomenon associated with long (in a definite sense) transmission lines for physical signals. In electrical and electronic engineering there are considered in various applications circuit structures consisting of multipoles connected through LC transmission lines (With respect to this a long list of references may be provided, starting with a pioneering paper of Brayton ([2]) and going up to a quite recent book of Marinov and NeittaanmЁaki ([13])). Worth to mention that even in power distribution the propagation phenomena are to be met if the distribution area is quite large  see the book of Karaev ([12]). The lossless propagation occurs also for nonelectric signals as water, steam or gas flows and pressures. With respect to this we may cite the pioneering (but almost forgotten) papers of Kabakov and Sokolov ([10]) on steam pipes for combined heat
electricity generation, the long list of papers dealing with waterhammer and many other. From the mathematical
Point of view the common model is represented by hyperbolic partial differential equations in the plane (i.e. with a single space variable besides the time variable) for which nonstandard (e.g. derivative or operatorVolterra type)
boundary conditions are formulated. Since dynamics studies are concerned with evolutions for t > 0,
initial conditions are also involved and the resulting model is a mixed initialboundary value problem for hyperbolic partial differential equations in the plane. Since the problem is not a standard one, the solving methods are also not quite standard. The pioneering papers cited above (of both electrical and nonelectrical enginnering) deal mainly with linear or linearized systems and the problem of interest is (exponential) stability. Since in usual cases such problem is solved by locating in the complex plane the roots of a certain characteristic equation the approach is that of applying the Laplace transform in a (more or less) formal way. Some echos of this approach are present in more recent papers (as the ones of Smirnova, ([17],[18])) where inversion of a formally applied Laplace transform leads to an inte
gral equation. The study of this
integral equation provides necessary information for the fundamental theory (existence, uniqueness, datadependence) as well as for stability for the initial problem. The second approach has its roots in a paper of Abolinia and Myshkis ([1]) and consists of the following steps: 1. Introduction of the Riemann invariants. 2. Integration of the invariants along the characteristics and association to the main problem of an initialvalue problem for some functional equation (operatorVolterra or other). 3. Qualitative study (fundamental theory, stability, oscillations) of the associated functional equation. 4. Establishing the properties of solutions of the mixed initialboundary value problem using the results obtained for the functional equation and a representation formula of the solutions of the mixed problems in terms of the solutions of the associated equation. This approach corresponds in fact to the method of D'Alembert but the cited paper of Abolinia and Myshkis, together with some earlier references, seem to insist for the first time on the functional differential equations generated by the method of D'Alembert: they are viewed as legitimate but selfcontained mathematical objects. Concerning this line of research we have to cite here two papers belonging to Cooke and Krumme ([4]) and Cooke ([3]) the last one remaining, unfortunately, unpublished. In this papers the boundary conditions are much simpler than in the papers of Abolinia and Myshkis; they are suggested, especially in the paper of Cooke and Krumme, by the equations of the electrical multipoles connected by LC transmission lines and they lead to differential equations with deviated arguments. Moreover it is shown, in the second paper, that, according to the type of the boundary conditions, equations of delayed, neutral or advanced type are obtained and this classification is consistent with that of Kamenskii ([11]) which is widespread in the field.
Once this technique is applied, a way is open for various types of research: stability and absolute stability (Smirnova, ([17],[18])); Rasvan, ([14])), forced oscillations (Halanay and Rasvan, ([6],[7])), control (Rasvan, ([15])), numerical schemes (Halanay and Rasvan, ([8])). 2 The method of the characteristics
where i(; t, ) are the solutions of the characteristic systems: d dt = ± () 2) Let (x(t), i(t)) a solution of (2) with the above
initial conditions. Defining:
We shall consider here a model representing a generalization of many specific models occuring in thermal, hydraulic or
electrical engineering.
u1 t
+
()
u1
= 0,
u2 t

()
u2
= 0,
0 1, t > 0
u1(0, t) + 1u2(0, t) = 1(t, x(t)) u2(1, t) + 2u1(1, t) = 2(t, x(t))
(1)
x = f (t, m(t), x(t), u2(0, t), u1(1, t)) x(0) = x0 IRn, ui(, 0) = i(), 0 1
u1(, t) = 1(t1(0; , t)), u2(, t) = 2(t2(1; , t))
the collection (ui(, t), x(t)) is a solution of (1) i.e. the equations, the initial and the boundary condi tions are verified for 0 1, t 0, except on those characteristics satsfying t1(0; , t) = nh, t2(1; , t) = nh, n IN. Here ti(; , t) are solutions of the characteristic system:
dt
1
=±
d ()
where () > 0, 0 1 and m(t) is a
control signal or a forcing term. Obviously we have here linear hyperbolic partial differential eequations with usual linear boundary conditions that are "controlled" by a nonlinear system of ordinary differential equations of a rather general form. It will appear in the following that the qualitative problems are due to the nonlinear differential equations while the partial differential equations introduce only some delays (the propagation effect) that give to the differential equations a functional character. Following Abolinia and Myshkkis ([1]) or Cooke ([3]) we associate to the mixed problem above the following system of functional differential equations:
x (t) = f (t, m(t), x(t), 1(t  h), 2(t  h))
1(t) = 12(t  h) + 1(t, x(t))
(2)
2(t) = 21(t  h) + 2(t, x(t))
where
1 ds h= 0 (s)
An important result is the following:
Theorem 1 Consider the mixed problem (1), and the system of functional equations (2); assume that () and i() are continuously differentiable. Then the following statements are true:
1) Let (ui(, t), x(t)) be a solution of (1). Defining:
1(t) = u1(0, t), 2(t) = u2(1, t)
the collection (x(t), i(t)) is a solution of (2) with the following initial conditions:
x(0) = x0 10(t) = 1(1(0; t + h, 1)) 20(t) = 2(2(0; t + h, 0)) h t 0
Once this onetoone correspondence is established, the above mentioned problems on stability, forced oscillations, numerical schemes and control, while formulated for partial differential equations, may be transferred and solved for the associated system of functional differenial equations. Worth to mention that the associated system (2), may be considered as belonging to the class of neutral functional differential equations due to the following facts: a) the components of the solutions i(t) are not smoothed for increasing t > 0 but they do not loose smoothness either; b) if a discontinuity occurs at t = 0 it propagates for t = nh, n IN; this fact accounts for singularity propagation, a property that is specific for hyperbolic partial differential equations and is a consequence of boundary and initial conditions mismatching. Also if i(t) and (t, x) are differentiable then differentiating i(t) in (2), we clearly obtain a system of neutral type.
3 Stability radii
A special case of (2) is the linear sytem:
x = A0x(t) + A1y(t  h) y(t) = A2x(t) + A3y(t  h)
(3)
where x IRn, y IRp (in particular p = 2 but this is not compulsory). For this system we shall consider structured perturbations leading to:
x = (A0 + B1C1)x(t) + (A1 + B1C2)y(t  h) y(t) = (A2 + B2C1)x(t) + (A3 + B2C2)y(t  h) (4) This class of structured perturbations is interesting since it does not affect the delay structure of the system or the neutral character of the equation with deviated argument.
We define the stability radius of (3) with respect to perturbations of the structure (4) as rIK = rIK(Ai; Bi, Ci) = inf { IKmp, ()C+ = }
where () is the set of the roots of the characteristic equation of (4):
(
detH() =
)
I  A0  B1C1
(A1 + B1C2)eh
A2  B2C1
I  (A3 + B2C2)eh
=0
If the items of Ai, Bi, Ci are real we obtain two stability radii rIR and rC according to whether real or complex perturbations (items of ) are considered. We shall have:
Theorem 2 Let ( T (s) = C1
C2esh
) (H(s))1 (
B1 B2
)
exponential estimates with negative exponents (Rasvan, ([14])). Moreover the transfer function
( (s) =
c
0
) (H(s))1 (
b0 b1
)
(7)
where H(s) is that of Theorem 2, has its poles in the left hand plane Re s  < 0 hence the original of this Laplace transform, namely (t) satisfy also an exponential estimate with negative exponent hence
c = (t)dt <
(8)
0
We may state now
Theorem 3 Consider (5) under the following assumptions
i) the linear system (3) is exponentially stable;
be the transfer matrix function associated with (3) and with the structured perturbations defined by (4); here H() = H() for = 0. Then rC = [maxIR T (j)]1
ii) () is continuous and bounded i.e. () m; there exist also 0 and k (0, +) such that
()  2  2() > 0
(9)
k
4 Saturated controls
Designing feedback systems with saturated controls is quite an actual problem (see, for instance, a quite recent volume of Lecture Notes published by Springer (S.Tarbouriech (ed.), ([19])) but also the pioneering paper of Slemrod,([16]). In practice the compensators with bounded (saturated) outputs are very often of imposed structure e.g. the standard PID controllers. For the systems considered in this paper the assumption of above leads for a broad class of applications to the folowing system:
x(t) = A0x(t) + A1y(t  h)  b0(cx(t)) y(t) = A2x(t) + A3y(t  h)  b1(cx(t))
(5)
where () satisfies not only the usual sector condition 0 () Ї2 but is also bounded e.g. saturated. For this system global asymptotic stability may be studied by extending the result of Gelig ([5]) to this class of system, following the hints from author's book (Rasvan, ([14])). In order to state the Gelig type result we shall need the following development. Let Z11(t) and Z12(t) be the matrix solution of the system:
Z11 = Z11(t)A0 + Z12(t)A2 Z12(t) = Z12(t  h)A3 + Z11(t)A1
(6)
with the initial conditions Z11(t) 0, Z12(t) 0, t < 0; Z11(0) = I, Z120 = 0. If the linear system (3) is exponentially stable then Z11(t) and their derivatives satisfy
provided 0 <  mc, c being defined by (8);
iii) there exists some real parameter in order that the frequency domain inequality holds
1 + (j2 + Re(1 + j)(j) 0
(10)
k
where (s) is defined by (7). Then
lim x(t) = 0, lim y(t) = 0
t
t
Outline of the proof The output (t) = cx(t) satisfies the integral equation
t
(t) = (t)  (t  )(( ))d
(11)
0
where (t) has been defined above and (t) has the form
[
0
]
(t) = c Z11(t)x0 + Z12(t  )y0( )d
(12)
h
(x0, y0( ), h 0) being the initial state of (5). If the assumptions of Theorem 3 hold, the assumptions of Theorem 1 of Gelig ([5]) hold. Therefore
lim (t) = 0 t We may now apply the asymptoticity results of (Rasvan, ([14]); Chapter 6, §18) and get the final result.
5 Concluding remarks The idea of associating a functional
differential system to the mixed initialboundary value problem for hyperbolic PDE is stimulating for research for both classes of systems (described by FDE and PDE). A natural extension would be to consider lossy transmission lines: this could complicate the FDE leading, for instance, to distributed delays. Remark also that, from the point of view of the general theory of PDE, the integration along the characteristics remains in the area of classical solutions (except the propagation of singularities). References [1] Abolinia,V.E., Myshkis, A.D., Mixed problem for an almost linear hyperbolic system in the plane, Matem.sbornik 1960 vol.12, pp.423442, no.4 (in Russian) [2] Brayton,R.K., Smallsignal stability criterion for electrical networks containing lossless transmission lines, IBM Journ.Res.Develop. 1968 vol. 12, pp.431440, no.6 [3] Cooke,K.L., A linear mixed problem with derivative boundary conditions (unpublished), Pomona College (U.S.A.) Res.Dept., 1970 [4] Cooke,K.L., Krumme,D.W., DifferentialDifference Equations and Nonlinear PartialBoundary Value Problems for Linear Hyperbolic Partial Differential Equations, J.Math.Anal.Appl. 1968, vol.24, pp.372387, no.2 [5] Gelig,A.Kh., Absolute stability of nonlinear
control systems with bounded nonlinearities, Automat.i telemekhanika 1969, no.4 (in Russian)
[11] Kamenskii, G.A., On general theory of the equations with deviated argument, Dokl.A.N.SSSR, 1958 vol.120, no.4, pp.697700 (in Russian) [12] Karaev,R.I., Transient processes in long distance transmission lines, Moscow, Energia Publ.House, 1978 (in Russian) [13] Marinov, C., NeittaanmЁaki, P.,
mathematical models in Electrical Circuits:Theory and Applications, Kluwer Academic, 1991 [14] Rasvan, Vl., Absolute stability of time lag
Control Systems, Ed.Academiei, Bucharest, 1975 (in Romanian, Russian
revised edition by Nauka, Moscow, 1983) [15] Rasvan, Vl., Stability of bilinear control systems occuring in combined heatelectricity generation. II: Stabilization of the reduced models, Rev.Roumaine Sci.Techn. Sґerie Electrotechn. et Energ., 1984 vol.29, no.4, pp.423432 [16] Slemrod, M. Stabilization of Bilinear Control Systems with Applications to Nonconservative Problems in Elasticity, SIAM J.Control Optim., 1978 vol.16, no.1, pp.131141 [17] Smirnova, V.B., Asymptotics of solutions for a problem with discontinuous nonlinearity, Diff.uravnenia, 1973 vol.9, no.1, pp.149157 (in Russian) [18] Smirnova, V.B., On asymptotic behaviour of a class of control systems with distributed parameters, Autom.i telemekhanika, 1973 no.10, pp.512 (in Russian) [19] Tarbouriech, S. (Editor) Control of systems with bounded
control inputs, Lect.Notes in Control, 1997, Springer Verlag
[6] Halanay, A., Rasvan, Vl., Almost periodic solutions for a class of systems described by delaydifferential and difference equations, Nonlinear Analysis. Theory, Methods & Applications, 1977 vol.1, no.1
[7] Halanay, A., Rasvan, Vl., Forced oscillations in difference systems, Rev.Roum.Sci.Techn.Sґerie Electrotechn.et Energ. 1979 vol.24, no.1
[8] Halanay, A., Rasvan, Vl., Approximation of delays by differential equations, in Recent Advances in Differential Equations (R.Conti, Ed.), Acad.Press, 1981
[9] Kabakov, I.P., Processes for
Steam Pressure control, Inzh.sbornik, 1946 vol.2, no.2, pp.2760 (in Russian)
[10] Kabakov, I.P., Sokolov, A.A., Influence of the hydraulic shock on the process of steam turbine
speed control, Inzh.sbornik., 1946 vol.2, no.2, pp.6176 (in Russian)
WE Dixon, DM Dawson