Differential Equations with Constant Delay. State Equations of Dynamic Objects with Delay

LITERATURE

Linear system with delay.

Problems for Equations with Delay. Consider a variational problem in which the control determines the phase trajectory of the system by the Cauchy problem for the equation with delay

In the literature, such systems are often called systems of simultaneous equations, meaning that here the dependent variable of one equation can appear simultaneously as a variable (but already as an independent one) in one or more other equations. In this case, the traditional distinction between dependent and independent variables loses its meaning. Instead, a distinction is made between two kinds of variables. These are, firstly, jointly dependent variables (endogenous), the influence of which on each other must be investigated (matrix A in the term Ay t) of the above system of equations). Secondly, predefined variables that are supposed to influence the first ones, but are not affected by them, are lag variables, i.e. lag (second term) and exogenous variables defined outside the given system of equations.

However, for equations with general types of delays and a more or less far-reaching specification of the remainder, there are still no sufficiently reliable results regarding the properties of estimates. Thus, the estimates for a regression equation with a general polynomial form of the lag have only the consistency property, and the estimates for equations with lagging exogenous and endogenous variables obtained by the three-step least squares method (in the presence of a first-order Markov residual autocorrelation) do not even have this property (see Fig. analysis of grades in ).

Thus, when synthesizing high-speed systems with a maximum degree of stability, it is first necessary to determine the optimal values of bj that ensure the fulfillment of condition (4), ng and ω, (1=1, n), then find с/, at which (10) and, finally, from condition (12) for a given value of C, choose dj. Comment. From the considered cases, it follows that the structures of optimal solutions, i.e., the number of real and complex conjugate pairs of extreme right roots, their combination, multiplicities, and, as a consequence, the types of hodographs of optimal solutions in the X plane, depend on the dimension of the control m (1.2) and, for sufficiently higher orders n (1.1) do not depend on the value of n itself. In other words, each given m corresponds to its own well-defined number of structures of optimal solutions new optimal solutions. Therefore, for n - > QO, the possibility of synthesizing systems of the maximum degree of stability remains, the structures of optimal solutions are determined only by m, which means that for any m, the structures of optimal solutions are also known for objects with delay.

The question arises how to determine the value of the time lag for each indicator. To determine the appropriate time lags, we use the correlation analysis of time series of data. The main criterion for determining the time lag is the largest value of the cross-correlation coefficient for the time series of indicators with different lag periods of their impact on the inflation rate. As a result, the equation will take the following form

In addition, the S. d. method allows you to connect, within the framework of one model, numerous flows (physical. control and information) and the levels of capital investment and disposal of funds accumulating these flows with the level of basic. capital, birth and death rates in different age groups with the age structure of the population, etc. -rykh lend themselves to a fairly simple experimental study of stability, depending on the parameters and structure of the model itself.

Rules can also be grouped according to other criteria. For example, according to the monetary policy instrument (exchange rate, interest rate or monetary aggregate) according to the presence of foreign economic relations (open or closed economy) according to the inclusion of the forecast of economic variables in the equation of the rule (prospective and adaptive rules) according to the amount of delay (with or without lags ) etc.

The model, taking into account the flight time of the projectile and the delay in the transfer of fire, makes it possible to take into account delays in the system of early warning of an enemy missile attack and the system of space surveillance of its nuclear missile forces. This model is defined by the equations

The block of constant delay BPZ-2M is designed to reproduce functions with a delay argument in analog computing devices and can be used in electrical modeling of processes associated with the transportation of matter or energy transfer, when approximating the equations of complex multi-capacity objects by equations of the first and second order with delay.

Decision functions are a formulation of a line of conduct that determines how the available information about the levels leads to the choice of decisions related to the values of the current flow rates. The solution function can take the form of a simple equation that determines the simplest reaction of the material flow to the states of one or two levels (for example, the performance of a transport system can often be adequately expressed by the number of goods in transit, which is a level, and a constant - the average delay for the transportation time) . On the other hand, the decision function can be a long and elaborate chain of calculations performed taking into account changes in a number of additional conditions.

At present, it is not completely clear what factor is the main reason for the absence of diatoms in Baikal during cold periods. In [Grachev et al., 1997], the increased turbidity of water caused by the work of mountain glaciers is considered decisive, in [Gavshin et al., 1998] the main one is the drop in the concentration of silicon due to erosion fading in the Baikal drainage basin. Modification of the model (2.6.7), where the first equation describes the dynamics of silicon concentration, and the second - the dynamics of sedimentation of suspended matter, allows us to propose an approach to identify which of these two factors is the main one. It is clear that, due to the huge water mass, Baikal's biota will respond to climate change with some delay compared to the response of plant communities in the lake's drainage basin. Therefore, the diatom signal must lag behind the palynological signal. If the main reason for the disappearance of diatoms during cold periods is a decrease in the concentration of silicon, then such delays in responses to warming should be greater than the delays for cooling. If, on the other hand, the main factor in suppressing diatoms is turbidity due to glaciers, then the delay in responses to cooling should be approximately the same or even greater than to warming.

The last equation, as the reader might notice, describes the behavior of the simplest self-adjusting mechanism with a proportional delay. Annex A provides a block diagram showing

The PERRON97 procedure in this case determines the break date as 1999 07, if the choice of the break date is carried out according to the minimum - statistics of the unit root criterion ta=i, taken over all possible break points. At the same time, ta= = - 3.341, which is above 5% of the critical level - 5.59, and the unit root hypothesis is not rejected. The largest delay of the differences included in the right side of the equations is chosen to be 12 in the framework of applying the GS procedure to reduce the model with a 10% significance level.

INTRODUCTION

Ministry of Education of the Russian Federation

International Educational Consortium "Open Education"

Moscow State University of Economics, Statistics and Informatics

ANO "Eurasian Open Institute"

E.A. Gevorkyan

Delay differential equations

Textbook Guide to the study of the discipline

Collection of tasks for the discipline Curriculum for the discipline

Moscow 2004

Gevorkyan E.A. DIFFERENTIAL EQUATIONS WITH DELAYED ARGUMENT: Textbook, a guide to the study of the discipline, a collection of tasks for the discipline, a curriculum for the discipline / Moscow State University of Economics, Statistics and Informatics - M .: 2004. - 79 p.

Gevorkyan E.A., 2004

Moscow State University of Economics, Statistics and Informatics, 2004


Tutorial
Introduction ................................................ ................................................. ...............................


1.1 Classification of differential equations with
deviant argument. Statement of the initial problem .............................................................. .
1.2 Differential equations with retarded argument. Step method. ........
1.3 Differential equations with separable
variables and with a lagging argument .............................................................. .........................
1.4 Linear differential equations with retarded argument..................................
1.5 Bernoulli differential equations with retarded argument. ...............
1.6 Differential equations in total differentials
with delayed argument .................................................................. ................................................. .
CHAPTER II. Periodic solutions of linear differential equations
with delayed argument .................................................................. ................................................. .
2.1. Periodic solutions of linear homogeneous differential equations
with constant coefficients and with a lagging argument ..............................................
2.2. Periodic solutions of linear inhomogeneous differential
..................
2.3. The complex form of the Fourier series .......................................................... ......................................
2.4. Finding a Particular Periodic Solution of Linear Inhomogeneous
differential equations with constant coefficients and retarded
argument by expanding the right side of the equation in a Fourier series .................................................................. .
CHAPTER III. Approximate methods for solving differential equations
with delayed argument .................................................................. ................................................. .
3.1. Approximate expansion method for an unknown function
with delayed argument by degrees of delay.................................................................. ........
3.2. Approximate Poincaré method. ................................................. ................................
CHAPTER IV. Delay differential equations,
appearing in the solution of some economic problems
taking into account the time lag .............................................................. ................................................. ...............

4.1. The economic cycle of Koletsky. Differential equation

with trailing argument describing the change

stock of cash capital .............................................................. ................................................. .......
4.2. Characteristic equation. The case of real
roots of the characteristic equation ............................................................... ....................................
4.3. The case of complex roots of the characteristic equation..................................................
4.4. Delay differential equation,

(consumption in proportion to national income) .............................................. ..........
4.5. Delay differential equation,
describing the dynamics of national income in models with lags
(consumption grows exponentially with growth rate).................................................................. .........
Literature................................................. ................................................. ......................

Guide to the Study of the Discipline

2. List of main topics ............................................... ................................................. ......
2.1. Topic 1. Basic concepts and definitions. Classification
differential equations with deviating argument.
Delay differential equations. ............................................
2.2. Topic 2. Statement of the initial problem. Solution step method
differential equations with retarded argument. Examples.......................
2.3. Topic 3. Differential equations with separable
variables and with delayed arguments. Examples. ................................................. ..
2.4. Topic 4. Linear differential equations

2.5. Topic 5. Bernoulli differential equations
with a delayed argument. Examples. ................................................. ...............................
2.6. Topic 6. Differential equations in total differentials
with a delayed argument. Necessary and sufficient conditions. Examples............
2.7. Topic 7. Periodic solutions of linear homogeneous differential
equations with constant coefficients and with a retarded argument.

2.8. Topic 8. Periodic solutions of linear inhomogeneous differential
equations with constant coefficients and with a retarded argument.
Examples. ................................................. ................................................. .................................
2.9. Topic 9. Complex form of the Fourier series. Finding a private periodic
solutions of linear inhomogeneous equations with constant coefficients and with
retarded argument by expanding the right side of the equation into a Fourier series.
Examples. ................................................. ................................................. .................................
2.10. Topic 10. Approximate solution of differential equations with
delayed argument method of decomposition of a function from delay
by degrees of delay. Examples ................................................. ......................................
2.11. Topic 11. Approximate Poincare method for finding a periodic
solutions of quasilinear differential equations with a small parameter and
with a delayed argument. Examples. ................................................. ...............................

2.12. Topic 12. The economic cycle of Koletsky. Differential equation

with lagging argument for the function K(t), showing the stock of cash

fixed capital at time t .............................................. ................................................. ...
2.13. Topic 13. Analysis of the characteristic equation corresponding to
differential equation for the function K(t). ................................................. .............
2.14. Topic 14. The case of complex solutions of the characteristic equation
(ρ = α ± ιω )..................................................................................................................................
2.15. Topic 15. Differential equation for the function y(t), showing

the consumption function has the form c(t -τ ) = (1 - α ) y (t -τ ), where α is a constant rate
production accumulation .............................................................. ................................................
2.16. Topic 16. Differential equation for the function y(t), showing
national income in models with capital investment lags, provided that
the consumer function has the form c (t − τ ) = c (o ) e r (t − τ ) .............................. .................................
Collection of tasks for the discipline .............................................. ...............................................
Curriculum by discipline .................................................................. ...................................

Tutorial

INTRODUCTION

Introduction

This tutorial is devoted to the presentation of methods for integrating differential equations with retarded argument encountered in some technical and economic problems.

The above equations usually describe any processes with an aftereffect (processes with a delay, with a time delay). For example, when in the process under study the value of the quantity of interest to us at time t depends on the value x at time t-τ, where τ is the time lag (y(t)=f). Or, when the value of the quantity y at time t depends on the value of the same quantity at time

less t-τ (y(t)=f).

Processes described by differential equations with a retarded argument are found in both natural and economic sciences. In the latter, this is due both to the existence of a time lag in most links of the social production cycle, and to the presence of investment lags (the period from the start of designing objects to commissioning at full capacity), demographic lags (the period from birth to entry into working age and the beginning employment after graduation).

Taking into account the time lag in solving technical and economic problems is important, since the presence of a lag can significantly affect the nature of the solutions obtained (for example, under certain conditions it can lead to the instability of solutions).

With LAGGING ARGUMENT

CHAPTER I. Method of steps for solving differential equations

with trailing argument

1.1. Classification of differential equations with deviating argument. Statement of the initial problem

Definition 1 . Differential equations with a deviating argument are called differential equations in which the unknown function X(t) enters for different values of the argument.

X(t) = f ( t, x (t), x ) ,


X(t) = f [ t, x (t), x (t - τ 1 ), x (t − τ 2 )] ,

X(t) = f t, x (t), x (t), x [ t -τ (t )] , x [ t − τ


X(t) = f t, x (t ) , x (t) , x (t/2), x(t/2) .



(t)]

Definition 2. A differential equation with a retarded argument is a differential equation with a deviating argument, in which the highest order derivative of the unknown function appears at the same values of the argument and this argument is not less than all the arguments of the unknown function and its derivatives included in the equation.

Note that according to Definition 2, equations (1) and (3) under the conditions τ (t) ≥ 0, t − τ (t) ≥ 0 will be equations with a retarded argument, equation (2) will be the equation

with a lagging argument, if τ 1 ≥ 0, τ 2 ≥ 0, t ≥ τ 1, t ≥ τ 2, equation (4) is an equation with a lagging argument, since t ≥ 0.

Definition 3. A differential equation with a leading argument is a differential equation with a deviating argument, in which the highest order derivative of the unknown function appears at the same values of the argument and this argument is not greater than the rest of the arguments of the unknown function and its derivatives included in the equation.

Examples of differential equations with leading argument:

X(t)=

f ( t, x(t), x[ t + τ (t) ] ) ,

f [ t , x (t ), x (t + τ 1 ), x (t + τ 2 )] ,

f t , x (t ), x . (t ), x [ t + τ (t )] , x . [ t + τ



(t)] .

I. STEP METHOD FOR SOLVING DIFFERENTIAL EQUATIONS

With LAGGING ARGUMENT

Definition 4. Differential equations with a deviating argument that are not equations with a retarded or leading argument are called differential equations of neutral type.

Examples of differential equations with deviating argument of neutral type:

X (t) = f t, x(t) , x(t − τ ) , x(t − τ )

X (t) = f t, x(t) , x[ t − τ (t) ] , x[ t − τ (t) ] , x[ t − τ (t) ] .

Note that a similar classification is also used for systems of differential equations with a deviating argument by replacing the word "function" with the word "vector function".

Consider the simplest differential equation with a deviating argument:


X (t) = f [ t, x(t) , x(t − τ ) ] ,

where τ ≥ 0 and t − τ ≥ 0 (in fact, we consider a differential equation with a retarded argument). The main initial task in solving equation (10) is as follows: to determine a continuous solution X (t) of equation (10) for t > t 0 (t 0 -

fixed time) provided that X (t ) = ϕ 0 (t ) when t 0 − τ ≤ t ≤ t 0 , where ϕ 0 (t ) is a given continuous initial function. The segment [ t 0 − τ , t 0 ] is called the initial set, t 0 is called the initial point. It is assumed that X (t 0 + 0) = ϕ 0 (t 0 ) (Fig. 1).

X (t) \u003d ϕ 0 (t)

The problem is formulated as follows: to find a solution to equation (10) for t > t 0 if the initial function X (t ) = ϕ 0 t is known for t 0 − τ (t 0 ) ≤ t ≤ t 0 .

Example. Find a solution to the equation.


X (t) = f [ t, x(t) , x(t − cos 2 t) ]
for t > t 0 = 0 if the initial function X (t ) = ϕ 0 (t ) for (t 0 − cos2 t 0 ) \|	t ≤ t0
t0 = 0

− 1 ≤ t ≤ 0).

I. STEP METHOD FOR SOLVING DIFFERENTIAL EQUATIONS

With LAGGING ARGUMENT

Example. Find a solution to the equation


X (t) = f [ t, x(t) , x(t / 2 ) ]	at (t	−t	/ 2) \|
t > t 0 = 1 if initial function X (t ) = ϕ t	at (t	−t	/ 2) \|	≤ t ≤ t

			t=1		t=1

1/ 2 ≤ t ≤ 1).

Note that the initial function is usually specified or found experimentally (mainly in technical problems).

1.2. Delay differential equations. Step Method

Consider a differential equation with a retarded argument.

It is required to find a solution to equation (13) for t ≥ t 0 .

To find a solution to equation (13) for t ≥ t 0, we will use the step method (the method of successive integration).

The essence of the step method is that first we find a solution to equation (13) for t 0 ≤ t ≤ t 0 + τ , then for t 0 + τ ≤ t ≤ t 0 + 2τ, etc. At the same time, we note, for example, that since in the region t 0 ≤ t ≤ t 0 + τ the argument t − τ changes within t 0 − τ ≤ t − τ ≤ t 0 , then in the equation

(13) in this region, instead of x (t − τ ), we can take the initial function ϕ 0 (t − τ ) . Then

we obtain that to find a solution to equation (13) in the region t 0 ≤ t ≤ t 0		+ τ need re-
sew an ordinary differential equation without delay in the form:
	[ t, x(t) , ϕ 0 (t − τ ) ] ,
X(t) = f	[ t, x(t) , ϕ 0 (t − τ ) ] ,
for t 0 ≤ t ≤ t 0 + τ	with the initial condition X (t 0 ) = ϕ (t 0 ) (see Fig. 1).
	finding a solution to this initial problem in the form X (t) = ϕ 1 (t) ,	we can post-

solve the problem of finding a solution on the segment t 0 + τ ≤ t ≤ t 0 + 2τ, etc.

So we have:

			0 (t − τ ) ] ,
	X (t) = f [ t, x(t) , ϕ		0 (t − τ ) ] ,
at t 0	≤ t ≤ t0 + τ , X (t0 )		= ϕ 0 (t 0 ) ,

	X (t) = f [ t, x(t) , ϕ 1 (t − τ ) ] ,
for t 0 +τ ≤ t ≤ t 0 + 2 τ ,			X (t 0 + τ ) = ϕ 1(t 0 + τ ) ,

	X (t) = f [ t, x(t) , ϕ 2 (t − τ ) ] ,
for t 0 + 2τ ≤ t ≤ t 0 + 3τ ,			X (t 0 + 2 τ ) = ϕ 2 (t 0 + 2 τ ) ,

	X (t) = f [ t, x(t) , ϕ n (t − τ ) ] ,
for t 0 + n τ ≤ t ≤ t 0 + (n +1 ) τ , X (t 0 + n τ ) = ϕ n (t 0 + n τ ) ,
	ϕ i (t ) is	solution of the considered initial		tasks on the segment
t 0 + (i −1 ) ≤ t ≤ t 0 +i τ		(I=1,2,3…n,…).

I. STEP METHOD FOR SOLVING DIFFERENTIAL EQUATIONS

With LAGGING ARGUMENT

This method of steps for solving a differential equation with a retarded argument (13) allows us to determine the solution X (t) on a certain finite interval of change in t.

Example 1. Using the method of steps, find a solution to a first-order differential equation with a retarded argument

	(t) = 6 X (t − 1 )

in the region 1 ≤ t ≤ 3 if the initial function for 0 ≤ t ≤ 1 has the form X (t ) = ϕ 0 (t ) = t .
Decision. First, let's find a solution to equation (19) in the region 1 ≤ t ≤ 2 . For this in
(19) we replace X (t − 1) by ϕ 0 (t − 1) , i.e.,
X (t − 1 ) = ϕ 0 (t − 1 ) = t\| t → t − 1 = t − 1
and take into account X (1) = ϕ 0 (1) = t \|

So, in the region 1 ≤ t ≤ 2, we obtain an ordinary differential equation of the form
	(t )= 6 (t − 1 )

or dx(t)	6 (t −1 ) .

Solving it taking into account (20), we obtain the solution of Eq. (19) for 1 ≤ t ≤ 2 in the form
X (t) = 3 t 2 − 6 t + 4 = 3 (t − 1 ) 2 + 1.
To find a solution in the region 2 ≤ t ≤ 3 in equation (19), we replace X (t − 1) by
ϕ 1 (t −1 ) = 3 (t −1 ) 2 +1 \| t → t − 1		3(t − 2) 2 + 1. Then we get the ordinary	differential
the equation:
	(t ) = 6[ 3(t − 2) 2 + 1] , X( 2) = ϕ 1 ( 2) = 4 ,

whose solution has the form (Fig. 2)
X ( t ) = 6 ( t − 2 ) 3 + 6 t − 8 .

Systems with delay differ from the systems considered earlier in that in one or more of their links they have a delay in the time of the start of the change in the output value (after the start of the change in the input) by a value t, called the delay time, and this delay time remains constant throughout the subsequent during the process.

For example, if the link is described by the equation

(aperiodic link of the first order), then the equation of the corresponding link with delay will have the form

(aperiodic link of the first order with delay). This type of equation is called an equation with a retarded argument,

Then equation (6.31) will be written in the ordinary

changes abruptly from zero to one (Fig. 6.20,

standing on the right side of the link equation,

). In the general case, as for (6.31), the equation of dynamics of any link with delay can be divided into two:

which corresponds to the conditional breakdown of a link with a delay (Fig. 6.21, a) into two: an ordinary link of the same order and with the same coefficients and the delay element preceding it (Fig. 6.21.6).

means the time of movement of the metal from the rolls to the thickness gauge. In the last two examples, the value of m is called the transport delay.

In the first approximation, pipelines or long electrical lines included in the links of the system can be characterized by a certain delay value t.

shown in fig. 6.22, b, then this link can be approximately described as a first-order aperiodic link with a delay (6.31), taking the values of m, r and k from the experimental curve (Fig. 6.22, b).

Note also that the same experimental curve according to the graph in Fig. 6.22, in can also be interpreted as a time characteristic of an ordinary second-order aperiodic link with the equation

and k can be calculated from the ratios written in § 4.5 for a given link, from some measurements on the experimental curve, or by other means.

function (6.36) differs little from the transfer function of a link with delay (6.35).

The equation of any linear link with delay (6.33) will now be written in the form

The transfer function of a linear link with delay will be

the transfer function of the corresponding ordinary link without delay is indicated.

- modulus and phase of the frequency transfer function of the link without delay.

Hence we get the following rule.

To build the amplitude-phase characteristic of any link with a delay, you need to take the characteristic of the corresponding ordinary link and shift each of its points along the circle clockwise by an angle that, where w is the value of the oscillation frequency at a given point of the characteristic (Fig. 6.23, a).

the starting point remains unchanged, and the end of the characteristic winds asymptotically around the origin (if the degree of the operator polynomial B is less than that of the polynomial C).

It was said above that real transient processes (temporal characteristics) of the form in Fig. 6.22b can often be described with the same degree of approximation by both equation (6.31) and (6.34). The amplitude-phase characteristics for equations (6.31) and (6.34) are shown in fig. 6.23, a and b, respectively. The fundamental difference between the first one is that it has a point D of intersection with the axis (/. When comparing both characteristics with each other and with the experimental amplitude-phase characteristic of a real link, one must take into account not only the shape of the curve, but also the nature of the distribution of frequency marks ω along her.

Transfer function of an open system without delay.

The characteristic equation of a closed system, as shown in Chap. 5 has the form

An equation can have an infinite number of roots.

The shape of the amplitude-phase characteristic of the open circuit, constructed but the frequency transfer function, changes significantly

moreover, the opening of the system is carried out according to a certain rule, which is given below.

As a consequence, for the stability of linear systems of the first and second order with delay, it turns out that only the positiveness of the coefficients is no longer sufficient, and for systems of the third and higher order with delay, the stability criteria of Vyshnegradsky, Routh, and Hurwitz are inapplicable.

Below we will consider the determination of stability only by the Nyquist criterion, since its use for this sing turns out to be the simplest.

1 The construction of the amplitude-phase characteristic and the study of stability according to the Nyquist criterion are best done if the transfer function of an open-loop system is presented in the form (6.38). To obtain this, it is necessary to properly open the system.

For the case shown in Fig. 6.24, a, opening can be done anywhere in the main circuit, for example, as shown. Then the transfer function of the open system will be which coincides in form with (6.41).

For the case shown in Fig. 6.24, b, opening the main circuit gives the expression

open-loop functions, not convenient for further research:

Finally, in the case shown in Fig. 6.24, c, when the system is opened in the indicated place, we obtain an expression that also coincides with (6.41):

The frequency transfer function (6.41) can be represented as

Therefore, presenting the expression (6.41) in the form

Special Course

Classification of equations with deviating argument. The main initial problem for differential equations with delay.

Sequential integration method. The principle of smoothing solutions of equations with delay.

The principle of compressed mappings. Existence and uniqueness theorem for the solution of the basic initial problem for an equation with several lumped delays. Existence and uniqueness theorem for the solution of the main initial problem for a system of equations with distributed delay.

Continuous dependence of solutions of the main initial problem on parameters and initial functions.

Specific features of solutions of equations with delay. The possibility of continuing the solution. Move starting point. Theorems on sufficient conditions for sticking intervals. Theorem on sufficient conditions for nonlocal extendability of solutions.

Derivation of the general solution formula for a linear system with linear delays.

Investigation of equations with delay for stability. Method of D-partitions.

Application of the method of functionals for the study of stability. Theorems of N. N. Krasovskii on necessary and sufficient conditions for stability. Examples of constructing functionals.

Application of the method of Lyapunov functions for the study of stability. Razumikhin's theorems on stability and asymptotic stability of solutions of equations with delay. Examples of constructing Lyapunov functions.

Construction of program controls with delay in systems with complete and incomplete information. Theorems of V. I. Zubov. The problem of distribution of capital investments by industries.

Construction of optimal program controls in linear and non-linear cases. Pontryagin's maximum principle.

Stabilization of the system of equations by control with constant delays. Influence of variable delay on uniaxial stabilization of a rigid body.

Zhabko A.P., Zubov N.V., Prasolov A.V. Methods for studying systems with aftereffect. L., 1984. Dep. VINITI, No. 2103-84.
Zubov V.I. On the theory of linear stationary systems with retarded argument // Izv. universities. Ser. mathematics. 1958. No. 6.
Zubov V.I. Lectures on control theory. Moscow: Nauka, 1975.
Krasovsky N. N. Some problems of the theory of motion stability. M., 1959
Malkin I. G. Theory of motion stability.
Myshkis A. D. General theory of differential equations with retarded argument // Uspekhi Mat. Sciences. 1949. V.4, No. 5.
Prasolov A.V. Analytical and numerical studies of dynamic processes. St. Petersburg: Publishing house of St. Petersburg State University, 1995.
Prasolov A.V. Mathematical models of dynamics in the economy. St. Petersburg: Publishing House of St. Petersburg. University of Economics and Finance, 2000.
Chizhova O. N. Construction of solutions and stability of systems of differential equations with retarded argument. L., 1988. Dep. in VINITI, No. 8896-B88.
Chizhova O. N. Stabilization of a rigid body taking into account linear delay // Bulletin of St. Petersburg State University. Ser.1. 1995. Issue 4, No. 22.
Chizhova O. N. On the non-local extension of equations with variable delay // Questions of mechanics and control processes. Issue. 18. - St. Petersburg: Publishing House of St. Petersburg State University, 2000.
Elsgolts L. E., Norkin S. B. Introduction to the theory of differential equations with deviating argument. M., 1971.

Linear systems with delay are such automatic systems that, having in general the same structure as ordinary linear systems (Section II), differ from the latter in that in one or more of their links they have a delay in the start time of changing the output quantity (after the beginning of the input change) by a value called the delay time, and this delay time remains constant throughout the subsequent course of the process.

For example, if an ordinary linear link is described by the equation

(aperiodic link of the first order), then the equation of the corresponding linear link with delay will have the form

(aperiodic link of the first order with delay). Equations of this type are called equations with a retarded argument or differential-difference equations.

Denote Then the equation (14.2) will be written in the ordinary form:

So, if the input value changes abruptly from zero to one (Fig. 14.1, a), then the change in the value standing on the right side of the link equation will be depicted by the graph in Fig. 14.1b (jump one second later). Using now the transient response of an ordinary aperiodic link as applied to equation (14.3), we obtain a change in the output value in the form of a graph in Fig. 14.1, c. This will be the transient response of the aperiodic link of the first order with a delay (its aperiodic "inertial" property is determined by the time constant T, and the delay is determined by the value

Linear link with delay. In the general case, as for (14.2), the equation of dynamics of any linear link with delay can be

split into two:

which corresponds to the conditional breakdown of a linear link with a delay (Fig. 14.2, a) into two: an ordinary linear link of the same order and with the same coefficients and the delay element preceding it (Fig. 14.2, b).

The time characteristic of any link with a delay will therefore be the same as that of the corresponding ordinary link, but only shifted along the time axis to the right by .

An example of a “pure” delay link is an acoustic communication line - the sound transit time). Other examples are a system for automatic dosing of a substance moving with a belt conveyor - the time the belt moves in a certain area), as well as a system for regulating the thickness of the rolled metal, where it means the time the metal moves from the rolls to the thickness gauge

In the last two examples, the quantity is called the transport delay.

In the first approximation, pipelines or long electrical lines included in the links of the system can be characterized by a certain amount of delay (for more details on them, see § 14.2).

The value of the delay in the link can be determined experimentally by removing the time characteristic. For example, if, when a certain value, taken as unity, is applied to the input of a link, the experimental curve for shown in Fig. 1 is obtained at the output. 14.3, b, then this link can be approximately described as a first-order aperiodic link with a delay (14.2), taking the values from the experimental curve (Fig. 14.3, b).

Note also that the same experimental curve according to the graph in Fig. 14.3, c can also be interpreted as a time characteristic of an ordinary second-order aperiodic link with the equation

moreover, and k can be calculated from the relations written in § 4.5 for a given link, according to some measurements on the experimental curve, or in other ways.

So, from the point of view of the time characteristic, a real link, approximately described by a first-order equation with a retarded argument (14.2), can often be described with the same degree of approximation by a second-order ordinary differential equation (14.5). To decide which of these equations best fits a given

real link, one can also compare their amplitude-phase characteristics with the experimentally taken amplitude-phase characteristic of the link, which expresses its dynamic properties during forced vibrations. The construction of the amplitude-phase characteristics of links with a delay will be considered below.

For the sake of unity in writing the equations, we represent the second of relations (14.4) for the delay element in operator form. Expanding its right side in a Taylor series, we get

or, in the previously accepted symbolic operator notation,

This expression coincides with the formula of the delay theorem for function images (Table 7.2). Thus, for the pure delay link, we obtain the transfer function in the form

Note that in some cases the presence of a large number of small time constants in the control system can be taken into account in the form of a constant delay equal to the sum of these time constants. Indeed, let the system contain series-connected aperiodic links of the first order with a transfer coefficient equal to unity and the value of each time constant. Then the resulting transfer function will be

If then in the limit we get . Already at the transfer function (14.8) differs little from the transfer function of the link with delay (14.6).

The equation of any linear link with delay (14.4) will now be written in the form

The transfer function of a linear link with delay will be

where denotes the transfer function of the corresponding ordinary linear link without delay.

The frequency transfer function is obtained from (14.10) by substituting

where are the modulus and phase of the frequency transfer function of the link without delay. Hence we get the following rule.

To build the amplitude-phase characteristic of any linear link with a delay, you need to take the characteristic of the corresponding ordinary linear link and shift each of its points along the circle clockwise by an angle , where is the value of the oscillation frequency at a given point of the characteristic (Fig. 14.4, a).

Since at the beginning of the amplitude-phase characteristic and at the end, then the initial point remains unchanged, and the end of the characteristic winds asymptotically to the origin (if the degree of the operator polynomial is less than the polynomial

It was said above that real transient processes (temporal characteristics) of the form in Fig. 14.3, b can often be described with the same degree of approximation by both equation (14.2) and (14.5). The amplitude-phase characteristics for equations (14.2) and (14.5) are shown in fig. 14.4, a and respectively. The fundamental difference of the first one is that it has a point D of intersection with the axis

When comparing both characteristics with each other and with the experimental amplitude-phase characteristic of a real link, it is necessary to take into account not only the shape of the curve, but also the nature of the distribution of frequency marks o along it.

If the delay τ

in equation (10) depends on time t

(τ = τ (t )) , then the initial

Let a single-circuit or multi-circuit automatic system have one link with a delay among its links. Then the equation of this link has the form (14.9). If there are several such links, then they can have different delay values. All the general formulas for equations and transfer functions of automatic control systems derived in Chapter 5 remain valid for any linear systems with delay, if only the values of transfer functions are substituted into these formulas in the form ( 14.10).

For example, for an open circuit of series-connected links, among which there are two links with a delay, respectively, the transfer function of an open system will have the form

where is the transfer function of an open circuit without taking into account the delay, equal to the product of the transfer functions of the links connected in series.

Thus, when studying the dynamics of an open circuit of series-connected links, it is immaterial whether the entire delay will be concentrated in one link or spread over different links. For multi-loop circuits, more complex relationships will be obtained.

If there is a link with negative feedback, which has a delay, then it will be described by the equations;