Numerical solution of equations with delay. Logistic equation with time lag

Problems for equations with delay. Let us consider a variational problem in which the control determines the phase trajectory of the system using the Cauchy problem for an 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 as an independent variable) in one or more other equations. In this case, the traditional distinction between dependent and independent variables becomes meaningless. 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 studied (matrix A in the term Ay t) of the above system of equations). Secondly, predetermined variables that are supposed to influence the former, but do not experience their influence, are lagged variables, i.e. lagged (second term) and exogenous variables defined outside this system of equations.

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

Thus, when synthesizing high-speed systems of maximum stability, it is necessary to first determine the optimal values ​​of bj that ensure the fulfillment of condition (4), ng and co, (1 = 1, n), then find c/ for which (10) holds and, finally, from condition (12) for a given value of C, select dj. Comment. From the cases considered, 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 control dimension m (1.2) and with sufficient large orders n (1.1) do not depend on the value n itself. In other words, for each given m there corresponds its own well-defined number of structures of optimal solutions, which is achieved when the order value of equation (1.1) n = n and an increase in the order n > n does not lead to the appearance new optimal solutions. Therefore, when n - > QO, the possibility of synthesizing systems of 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 of how to determine the time lag value for each indicator. To determine the corresponding time lags, we use correlation analysis of time series data. The main criterion for determining the time lag is the largest value of the cross-correlation coefficient of time series of indicators with different lag periods of their influence on the inflation rate. As a result, the equation will take the following form

In addition, the SD method makes it possible to link, within the framework of one model, numerous flows (physical management and information) and the levels of capital investments and disposals of funds accumulating these flows with the level of fixed assets. capital, fertility and mortality in various age groups with the age structure of the population, etc. The SD method most clearly reflects the structure of all feedbacks taken into account and is well adapted to take into account different forms delay, leads to a system of differential equations, the solutions of which are amenable to a fairly simple experimental study for stability, depending on the parameters and structure of the model itself.

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

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

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

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

At present, it is not completely clear which factor is the main reason for the absence of diatoms in Baikal during cold periods. In [Grachev et al., 1997], the decisive factor is considered to be the increased turbidity of water caused by the work of mountain glaciers; in [Gavshin et al., 1998], the main factor is considered to be a drop in silicon concentration due to the fading of erosion in the Baikal drainage basin. Modification of model (2.6.7), where the first equation describes the dynamics of silicon concentration, and the second - the dynamics of suspension deposition, allows us to propose an approach to identify which of these two factors is the main one. It is clear that due to the enormous water mass The biota of Lake Baikal 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 should lag behind the palynological signal. If the main reason for the disappearance of diatoms during cold periods is a decrease in silicon concentration, then such delays in reactions to warming should be greater than the delays for cooling. If the main factor in the suppression of diatoms is turbidity due to glaciers, then the delay in reactions to cold snaps should be approximately the same or even greater than to warming ones.

The last equation, as the reader may have noticed, describes the behavior of the simplest self-adjusting mechanism with a proportional delay. Appendix A provides a block diagram of

The PERRON97 procedure determines in this case the break date as 1999 07, if the break date is selected based on the minimum -statistics of the unit root criterion ta=i, taken over all possible break points. In this case, ta= = - 3.341, which is above 5% of the critical level - 5.59, and the unit root hypothesis is not rejected. The largest lag of the differences included in the right side of the equations is selected equal to 12 as part of the application of the GS procedure to reduce the model with a 10% significance level.

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

For example, if a 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 first order link with delay). These types of equations are called equations with a retarded argument,

Then equation (6.31) will be written in ordinary form

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 for the dynamics of any link with delay can be divided into two:

which corresponds to the conditional breakdown of a link with 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 m is called transport delay.

To a 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 we can approximately describe this link as an aperiodic first-order link with delay (6.31), taking the values ​​of m, Г and k from the experimental curve (Fig. 6.22, b).

Note also that the same experimental curve according to the graph in Fig. 6.22c 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 relationships written in § 4.5 for a given link, from some measurements on the experimental curve or by other methods.

function (6.36) differs little from the transfer function of the 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.

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

From this we get the following rule.

For building amplitude-phase characteristics of any link with a delay, you need to take the characteristic of the corresponding ordinary link and move 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. 6.23, a).

the starting point remains unchanged, and the end of the characteristic asymptotically wraps 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 (time characteristics) of the form in Fig. 6.22, b 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 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 along her.

Transfer function of an open-loop system without delay.

The characteristic equation of a closed-loop system, as shown in Chap. 5, looks like

the equation can have an infinite number of roots.

The outline of the amplitude-phase characteristic of the open-loop value constructed using the frequency transfer function changes significantly

Moreover, the system is opened according to a certain rule, which is given below.

As a consequence, for stability linear systems of the first and second order with delay, it turns out that only the positivity of the coefficients is no longer enough, and for systems of the third and higher order with delay the stability criteria of Vyshnegradsky, Routh and Hurwitz are not applicable.

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

1Construction of the amplitude-phase characteristic and study of stability using the Nyquist criterion is best done if the transfer function of the open-loop system is presented in the form (6.38). To achieve this, it is necessary to open the system accordingly.

For the case shown in Fig. 6.24, a, the opening can be made anywhere in the main circuit, for example, as shown. Then the transfer function of the open-loop system will be the same in form as (6.41).

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

functions of an open-loop system, not convenient for further research:

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

The frequency transfer function (6.41) can be represented as

Therefore, presenting expression (6.41) in the form

By taking a step back you find yourself, then you move and you lose yourself.

U. Eco. Foucault pendulum

Examples of mathematical models. Basic Concepts

Preliminary terminological notes. In this chapter we will talk about models based on the use of so-called retarded differential equations. This is a special case of equations with deviating coefficients 1. Synonyms for this class are functional differential equations or differential difference equations. However, we prefer to use the term “delayed equation” or “delayed equation.”

We will encounter the term “differential-difference equations” in another context when analyzing numerical methods for solving partial differential equations and has no relation to the content of this chapter.

An example of an ecological model with lag. In the book by V. Volterra, the following class of hereditary models is given, taking into account not only the current population size of predator and prey, but also the prehistory of population development:

The general theory of equations with deviating argument is presented in the works: Bellman R., Cook K. Differential-difference equations. M.: Mir, 1967; Myshkis A. D. Linear differential equations with retarded argument. M.: Nauka, 1972; Hale J. Theory of functional differential equations. M.: Mir, 1984; ElsgoltsL. E., Norkin S. B. Introduction to the theory of differential equations with deviating argument. M.; Science, 1971.

System (7.1) belongs to the class of integral-differential models of the Volterra type, K ( , K 2 - some integral kernels.

In addition, other modifications of the “predator-prey” system are found in the literature:

Formally, there are no integral terms in system (7.2), unlike system (7.1), but the increase in predator biomass depends on the number of species not in this moment, and at the moment of time t - T(under T often refers to the lifespan of one generation of a predator, the age of sexual maturity of female predators, etc. depending on the meaningful meaning of the models). For predator-prey models, see also paragraph 7.5.

It would seem that systems (7.1) and (7.2) have significantly different properties. However, with a special form of kernels in system (7.1), namely the 8-function /?,(0 - t) = 8(0 - 7^), K 2 (d - t) = 8(0 - T 2) (we have to talk about the 8-function somewhat conditionally, since generalized functions are defined as linear functionals, and the reduced system is nonlinear), system (7.1) becomes the system

It is obvious that system (7.3) is structured as follows: the change in population size depends not only on the current size, but also on the size of the previous generation. On the other hand, system (7.3) is a special case of the integral-differential equation (7.1).

Linear equation with delay (delay type). Linear differential equation delayed type with constant coefficients we will call the equation of the form

Where a, b, t - permanent; T> 0;/ is a given (continuous) function on K. Without loss of generality in system (7.4) we can put T= 1.

Obviously, if the function is given x(t)y t e [-G; 0], then it is possible to determine x(t) at te and which is a solution to the equation (7.4) for t> 0. If f(?) has a derivative at the point t = 0, andφ(0) = atom derivative 4"(φ|,_ 0 is two-sided.

Proof. Let's define the function x(t) =φ(?) on |-7"; 0]. Then the solution (7.4) can be written on in the form

(the formula for variation of constants is applied). Since the function x(t) is known on . This process can be continued indefinitely. Conversely, if the function x(?) satisfies formula (7.5) on ). Let's find out the question about sustainability of this decision. Substituting small deviations from the unit solution into equation (7.8) z(t) = 1 - y(t), we get

This equation has been studied in the literature, where it is shown that it satisfies a number of theorems on the existence of periodic solutions. At a = m/2, a Hopf bifurcation occurs - from fixed point a limit cycle is born. This conclusion is drawn from the results of the analysis of the linear part of equation (7.9). The characteristic equation for the linearized Hutchinson equation is

Note that the study of the stability of the linearized equation (7.8) is a study of the stability of the stationary state y(t)= 0. This gives A, = a > 0, the steady state is unstable and no Hopf bifurcation occurs.

J. Hale further shows that equation (7.9) has a nonzero periodic solution for every a > n/2. In addition, there is given without proof a theorem on the existence of a periodic solution (7.9) with any period p> 4.

A generalization of the recording of equations of state to nonlinear multidimensional dynamic objects and control systems with extension in space and elements of transport delay has been carried out. The generalization is carried out by including delay links, along with integrators, into the simplest dynamic ones, i.e. those whose output values ​​are treated as independent state variables.

1. Inertial dynamic objects

The traditional mathematical description of a dynamic object in state variables includes a vector equation of state that connects the rates of change of state variables with impacts on the object and the values ​​of the state variables themselves, as well as a vector equation that connects the values ​​of the output quantities of the object (or the results of their measurements) with its state variables and impacts on him :

• x - vector of state variables;
• u is the vector of impacts on the object;
• y - vector of object output values;
• z - interference vector;
• f(.) and g(.) are some, quite general view functions.

System (K.1.1) is a system of vector differential-algebraic equations of state variables of a multidimensional nonstationary spatially concentrated (point) nonlinear inertial-dynamic control object.

From equations (K.1.1) it is easy to see that the description of a dynamic object without delay structurally contains only three types of operators: linear differentiating (actually dynamic, inertial) and two inertia-free nonlinear ones: a coupling element and a composition element:

The linear differentiating operator describes inertia because it specifies instantaneous speed changes in the state variable, and, therefore, determines the value of the currently known variable for some, albeit small, time interval in advance. This should be interpreted as inertia, i.e. some predetermination of behavior.

Rice. K.1.1. Description of the inertial object and its structural model. Differential equation reflects the cause-and-effect relationship of the impact X and reaction (response) y simplest inertial link: impact X leads to a change in the output value y such that speed this change is directly proportional influence. The integrator is a model of the simplest, fundamental dynamic (inertial) element. The structural model reflects how the cause, the impact, is transformed into a consequence, the output value: the model of the simplest (fundamental) inertia ensures the accumulation and preservation of the impact

In a linearized model of an object, the principle of superposition is valid and therefore the composition operator of variables is their weighted sum, and the connection operator becomes linear:

The equations of a dynamic object in state variables can also be presented in an integral form, which is more visual for structural modeling:

The equation of state describes the own, internal inertia of a dynamic object. The output equation takes into account interference in the measurement of the components of the vector of output quantities.

The state and tendency of behavior, at least for an infinitesimal interval forward, of a purely inertial dynamic object is determined by the set of values ​​of all state variables of the object at some point in time and is displayed by the corresponding position of the representing point in the multidimensional state space. Since this information for an inertial object without delay is exhaustive, the coordinates of any point in the trajectory of the representing point can be considered as the initial conditions for integrating the equations of state, i.e. to determine the entire subsequent trajectory of movement of the representing point, assessing the behavior of a dynamic object under external influences or in the absence of them.

To illustrate this, we present phase portraits (trajectories of movement of the representing points of objects in a two-dimensional state space) for a model of a free oscillatory system with different initial conditions:

Rice. K.1.1. The phase portraits of a free inertial oscillatory system under different initial conditions corresponding to the same phase trajectory coincide, i.e. the coordinates of any point in the phase trajectory can be considered as initial conditions that completely determine the further free behavior of the object

Thus, the behavior of point (exclusively inertial, without delay elements) dynamic objects is completely described by the equations of state and output, as well as initial conditions, which are the values ​​of all state variables of the object at some point in time, and is displayed by a certain trajectory, and the current state of the object is characterized point in the multidimensional space of state variables.

2. Equations of state of extended objects with delay elements

Taking into account delay links in object models as a second, independent type of the simplest dynamic elements, along with inertial ones (integrators), allows us to uniformly describe dynamic objects of almost any complexity in state variables and, on this basis, carry out their analysis and optimization.

2.1. Equations and structure of models of extended dynamic objects

Differential form of equations of state of an extended object

The presence of delay elements in some branches of the dynamic object model significantly, and often fundamentally, changes the dynamic properties of the object compared to an object without delay elements. Therefore, the space of states corresponding only to the output values ​​of inertial elements (integrators) does not fully define the state and behavior of an object that has delay links.

The delay element of a dynamic object, just like an inertial one, should be considered as dynamic, and its output value as a separate state variable.

The basis for classifying the link that delays the signal for a finite time interval as elementary dynamic is based on the similarities and differences between two types of simplest dynamic elements of models of real objects and is as follows.

The external difference is that the inertial element is described by an elementary differential equation, while the retarded element is described by an algebraic one.

The term “dynamic” refers to objects whose behavior under external influence can be predicted at least over an infinitesimal interval. The inertial element, the integrator, traditionally considered the only dynamic one, meets this requirement. But the delay link also meets this requirement if the background of the impact on it is known. In this case, the delay link makes it possible to rigidly determine the behavior of its output value for a finite time interval ahead. That. The delay link can be classified as dynamic.

On the other hand, the delay link corresponds in real objects to either the transfer of materials (“transport delay”), or the delay in the arrival of a signal (impact model) at the input of some element of the object, associated with its propagation in space. Thus, the delay link can also be attributed to the communication elements.

A non-stationary delay element with dispersion, and its special case, a pure delay element, as well as the simplest inertial element, is dynamic because its output signal is unique and cannot be obtained by an inertia-free composition of other, only inertial state variables. This is the result of a time delay in such a composition.

To generalize the equations of state of point objects, presented in Cauchy form, to extended objects and objects with transport delay, we formally introduce the prediction operator Fwd(τ) :

This operator in the general case, of course, is not physically realizable, since it must absolutely accurately predict the value of the variable on which it acts for a finite time interval τ in advance. But this operator is needed only for a formal “beautiful” initial representation of the equations of state, and their structural solution is possible using the implemented delay operator. On the other hand, the prediction operator in the state equations acts only on a state variable whose values ​​are determined by the history of the behavior of all state variables of the object with a delay and input influences, i.e. some composition of such, and therefore, in this particular case, it is realizable, since the forecast is strictly determined by the background.

So, let's write the vector equations of the state variables of an extended dynamic object in the form:

In (K.2.1.2), for ease of writing and reading, state variables are divided into two groups. Variables x (1) of the first group are state variables of the simplest inertial elements of the object, their output values. Variables x (2) are state variables corresponding to the outputs of the delay links of the object. It is obvious that, in principle, “inertial” and “delayed” state variables can be written and numbered in any order and combined in one vector equation.

Note that the generalized system of equations of state of a dynamic object has only one independent variable - time t. The spatial characteristics of the object in (K.2.1.2) are described indirectly, by taking into account the vector of delay times τ, caused by the propagation of impacts in space with a finite (not infinite) speed or transport delay.

Consideration of dynamic objects with delay based on their description by equations of state was carried out by some authors earlier.

In paragraph 2.1, (2.1.2), the description is limited to indicating delays only on the right sides of the equations and does not include delay links in the structure of the model as functioning elements determined by their own state variables. A similar initial representation of the equations of state is used in “1.5. Optimal control of systems with transport delay”, p. 188 ff, as well as in .

The form of equations (K.2.1.2) differs from that proposed in the introduction of special state variables corresponding to the output values ​​of the delay links. In this way, the delay links are related to the simplest dynamic ones and the description of dynamic objects becomes universal.

In the representation of a dynamic object proposed in this article, the current internal state of the object is completely determined by the vector of values ​​of state variables corresponding to the output values ​​of the integrators and delay links, and the history of their behavior.

Integral form of equations of state of an extended object

Equations of the state variables of a dynamic object with delay can be presented in an integral “delayed” form, which is perhaps more visual for drawing up a structural model of the object:

where the delay operators are:

carry out the opposite action in relation to the forecast operator Fwd(.).

So, (K.2.1.3) are integral “retarded” equations of vector state variables of a multidimensional extended nonlinear nonstationary dynamic object. The part of the variables corresponding to the output signals of the simplest inertial elements and designated by the vector x (1) is the result of accumulation (integration) of some combination of all variables, which, like the variables themselves, as well as input influences, can change over time. The second part of the state variables, denoted x (2), represents a delay of some combination of all state variables, as well as the input actions of the object, for some time τ (vector), which can generally change over time. In accordance with these equations, structural equations can be constructed, incl. virtual-analogue, dynamic object models.

Initial conditions of the equations of state of an extended object

In equations (K.2.1.3), the initial conditions for the delay links (operators) are not simply the values ​​of combinations of state variables and input actions at the zero point in time, as is the case for integrators. To uniquely solve equations (K.2.1.3), it is necessary to set the initial conditions for the delay links in the form of functions that determine the history of the behavior of the input values ​​of these links for the time interval back for which they implement the delay.

That. delay links, having “memory”, require more information to unambiguously resolve the issue of the behavior of the object: not just a vector of values ​​of state variables at some, conventionally zero point in time, as is sufficient for integrators, but a vector of functions (combinations of state variables and input influences on object) specified at the time intervals corresponding to the delay links preceding the start of integration.

In other words, the state and behavior of a dynamic object, as a point and trajectory in state space for systems with delay, is determined not only by the position of the point in this space, but also by its previous trajectory both in the “delay” subspace x(2) and in the subspace x (1) “inertial” variables, as well as the history of the behavior of external influences during those time intervals for which there is a delay in the corresponding delay links.

A similar statement for the traditional form of representing the equations of state of objects with delay is given in Section 2.1:

“The state of a continuous object with delay at an arbitrary moment of time is characterized not only by a certain finite number of parameters (meaning state variables - F.B.T.) (as in the case of objects without delay), but also by certain functions defined respectively on the interval . This significantly complicates the management of such objects.”

Generally speaking, the problem of specifying initial conditions for delay links is characteristic not only of describing a dynamic object in state variables, but also of other description methods. Often, when digitally modeling dynamic objects with a delay, the initial trajectory of “delayed” variables is taken, i.e. the output values ​​of the delay units are constant. To do this, the link buffer is initially filled with zeros or a constant.

The input signal of the delay link, which is part of a dynamic object, is a composition of state variables related to other links and impacts on the object, therefore, setting a rigid forecast of changes in the output signal of the delay link is equivalent to specifying the prehistory of the behavior of the named state variables and impacts for the same time interval .

Rice. 2.1.1. The state of a dynamic object with a delay at some point in time is characterized by the position of its representing point in the state space, the coordinates of which are the values ​​of the state variables at this point in time, as well as the trajectory of this point at points in time preceding the current one. The multidimensional state space can be represented as a set of subspace of inertial state variables and a subspace of “delayed” state variables

Thus, for point objects, the position of the representing point in state space at some point in time completely determines the state of the dynamic object and the trend of its behavior in the near future. For objects extended in space, having transport delay links in their structure, their state and subsequent behavior are determined not only by the current position of the representing point, but also by the trajectory of its movement in state space in the previous, which can be quite large, time interval.

Structure of a dynamic object model with delays

The structure of the model of a dynamic object with delays corresponding to the system (K.2.1.3) is presented in enlarged form in the figure:

Rice. K.2.1.2. Enlarged schematic representation of the main structural elements of the model of the observed multidimensional nonstationary nonlinear dynamic control object extended in space. The object's own dynamic properties are determined by the structure, characteristics and parameters of the left block; the converter block converts state variables into quantities that can be measured (or directly into measurement results)

Rice. K.2.1.3. The structure of the model of the dynamic object itself, reflecting its internal “metabolism”, i.e. directions for transferring values ​​of influences and variables, as well as operations performed on them. The behavior of an object with a delay is determined not only by the vector of initial conditions of “inertial” state variables, but also by the history of all state variables, as well as by the history of impacts on the object

A complex dynamic object with functional elements of delay is structurally represented by two parallel contours, inertial and “delayed”. State variables of the entire object are a combination of inertial and “delayed” state variables (output values ​​of the simplest inertial elements in the structure of the object, and “delayed” ones, i.e. output values ​​of delay links) into one vector.

As noted above, in the general case, the input signal of a certain delay link is determined both by all the state variables of the object and by all the influences on it. Therefore, in order to unambiguously determine the state and then the behavior of an object, it is necessary to know the values ​​and prediction of the behavior of “lagging” state variables, or, equivalently, the history of the behavior of all state variables and input actions of the object.

2.2. The simplest structural elements of extended objects

As can be seen from equations (K.2.1.2) and (K.2.1.3) of the state and output of dynamic objects with delay, only four operators are sufficient to describe them. The mathematical description of all four simplest elements (virtual analogues of these operators) of dynamic systems and objects that have spatial extension and (or transport delay), indirectly based on the physical laws that describe them, is reduced to simple equations, one of which is linear differential, and the other three are algebraic:

• x - impact on the element,
• y is his reaction,
• t - time,
• τ is some time delay.

Rice. K.2.2.1. The integrator and the stationary delay link are an exhaustive set of types of elementary dynamic objects. These simplest dynamic elements of object models with delay require setting initial conditions for a complete and unambiguous description of the state and behavior of the object. For an integrator, this is simply the value of the output quantity at a conditional zero moment of time; for the delay link, the “initial” condition is the behavior of the input quantity at previous times in the interval [-τ, 0], or, what is the same, a forecast of the behavior of the output value of the delay link (“delayed” state variable) for an interval equal to the delay time in the link

Rice. K.2.2.2. The simplest (fundamental) elements of the general form of the structural diagram of a dynamic object as its mathematical model count only four different types of elements. Elements of these types are sufficient to model an arbitrarily complex dynamic object (process plant, its control system, etc.)

By combining the simplest elements, you can build a consistent model of an arbitrarily complex dynamic object. Compiling a system of differential algebraic equations of a dynamic object in the form of equations of state is an implicit, indirect way, a kind of “sacrament,” of representing a model of a dynamic object in the form of a set of unidirectional simplest dynamic elements interacting with each other.

2.3. Observability and controllability of objects with delay

From the above consideration it follows that the unambiguous state of a dynamic object with a delay is determined not only by the current values ​​of the state variables, but also by the history of their changes at previous moments in time, over a finite and sufficiently extended interval. Therefore, for such objects it is necessary to clarify the concepts of observability and controllability.

Controllability of a dynamic object with delay elements is that it is possible, in a finite time, by a finite change in the vector of influences, to transfer the object from the current state, which was preceded by some specific behavior, to a new, required state, which is preceded by a given trajectory of the representing point in the state space.

Observability object with delay we define as the possibility of finding the current vector of state variables at any time and final part of the trajectory in the state space along which the representing point falls into the current position, according to measurements of the output quantities of the object and their behavior during some previous time interval.

More rigorous definitions of the concepts of observability and controllability of dynamic objects in the representation of delays on the right sides of the equations of state can be found in: “2.6. Controllability and observability of systems with delay.”

2.4. State and initial conditions of a dynamic object with delay

The current state of a dynamic object with a delay must unambiguously determine its behavior at subsequent moments of time, at least for a very short interval. In the absence of external influences on the object (free movement), or with known external influences, this time extends to infinity.

The state of a dynamic object with a delay is determined instantaneous value all state variables, “inertial” and “delayed”, as well as their history and the history of impacts on the object.

Rice. K.2.4.1. Phase portraits and behavior of state variables of a dynamic object with delay in the absence of external influences. If we consider the delay link as an elementary dynamic one, i.e. consider its output value as an independent state variable, then for full description state and behavior trends of a dynamic object with a delay, it is necessary to specify not only the values ​​of the state variables at some point in time, but also the history of their change, placed in this case in the buffer of the delay link. Different histories lead to different trajectories of the phase portrait, i.e. to different object behavior. The forecast of the behavior of the output variable of the delay link (state variable x3) is equivalent to the prehistory of the behavior of its input value, since it represents this very prehistory delayed by the delay time, in this case τ = 1 sec. The interval at which the history should be known is determined by the amount of delay in the delay link

As you can see, to set the initial conditions of the equations of state, and also, which is equivalent, to unambiguously describe the current state of a dynamic object with a delay, it is necessary to know not only the values ​​of the state variables, but also their history.

Rice. K.2.4.2. Initial conditions, or what is equivalent, the state of inertial-dynamic objects and inertial-dynamic objects with delay. For a purely inertial object, for a comprehensive description of its properties, it is enough to know the values ​​of all state variables at some point in time, as well as the values ​​of input actions on the object, if any. An object with delays requires not only knowledge of the values ​​of all state variables, both inertial (output signals of the model integrators) and “delayed” (output signals of the model delay links), but also to have a forecast of the behavior of the “delayed” ones.

Thus, to describe objects with a delay, much more information is required than for simply inertial objects, which complicates their analysis and optimization.

2.5. On the complete state space of a digital model of a dynamic object with delay and its consistent subspace

Models of real continuous inertial dynamic objects without delay can be constructed both using exclusively integrators (W(p)-model) and using only elementary delay units (W(z)-model):

Rice. 2.5.1. (animation, 14 frames) Models of an inertial oscillatory system, built on the basis of integrators and on the basis of elementary delay units that delay by one clock cycle, are equivalent, as can be seen from the transition functions of the output quantities, x1 and z1, respectively. Naturally, the state variables of these models corresponding to the output values ​​of the integrators and signal delay units per simulation cycle are different. Therefore, the trajectories of the representing points of different pairs of variables are different. Of course, in the model at the elementary delay links the trajectory of the representing point is rather “boring”, it goes along the diagonal, since both variables differ by an insignificant amount, which is fundamentally important for ensuring the consistency of the model

Note that integrators (aperiodic links) cannot even approximately model links with a sufficiently large delay, while any delay can be modeled without problems with any accuracy by delay links per clock cycle, you just need to select a sufficient number of them.

Rice. 2.5.2. Continuous delay link and its digital models. A state variable that carries meaningful, comprehensive information is the output value of the delay link, taking into account the history of the behavior of its input effect. The output signals of the intermediate elements of the discrete model of the delay link can formally be attributed to state variables, however, since the information in them is repeated with a shift, it is sufficient to limit ourselves only to the output value of the entire link and consider it as an elementary unitary dynamic state, the state of which is determined not only by the value of the output value, but and its forecast (prehistory of the input quantity). The unitary discrete model buffer is filled with the history of the input quantity, so the prediction of the state variable is strictly determined by this history

Determining the state variable assigned to the delay link, which is actually equal to the last value of the micro-links of the delay buffer, allows us to use as a consistent subspace of states one that includes only the output values ​​of the elementary links that make up the digital model of the delay link. The relatively small number of effective state variables is especially important in the analytical study of a dynamic object and the graphical presentation of its results.

Conclusion

The finite value delay link can be considered, in addition to the integrator, as the simplest dynamic element, the output value of which is an independent state variable, and for a complete and unambiguous description of the state of the object, it is necessary to know both the position of the representing point in the state space and part of its previous trajectory, i.e. This is the history of the object's behavior.

An optimal control system, if it has already been implemented, exists objectively and its characteristics do not depend on the mathematical apparatus by which it was described and with the help of which mathematical methods and tools it was optimized. Therefore, the simplicity of the mathematical description of the control system, in particular the ACS, should be determined by the complexity of the system and correspond to it.

Literature and the Internet

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Acknowledgments

The logistic equation with a time lag can be applied to the study of predator-prey interactions. - Stable limit cycles in accordance with the logistic equation.
The existence of a time lag makes it possible to use another method of modeling a simple system of predator-prey relations.

This method is based on the logistic equation (Section 6.9):

Table 10.1. The fundamental similarity of the population dynamics obtained in the Lotka-Volterra model (and in general in models of the predator-prey type), on the one hand, and in the logistic model with a time delay, on the other. In both cases, there is a four-phase cycle with maxima (and minima) in predator abundance following the maxima (and minima) in prey abundance

The growth rate of the predator population in this equation depends on the initial size (C) and the specific growth rate, r-(K-C) I Kf where K is the maximum saturation density of the predator population. The relative rate, in turn, depends on the degree of underutilization of the environment (K-S), which in the case of a predator population can be considered as the degree to which the predator's needs exceed the availability of prey. However, prey availability and hence the relative rate of predator population growth often reflects the predator's population density at some previous time period (Sect. 6.8.4). In other words, there may be a time lag in the response of a predator population to its own density:
dC „ l ( K Cnow-Iag \
- - G. Gnow j.
If this delay is small or the predator reproduces too slowly (i.e., the value of r is small), then the dynamics of such a population will not differ noticeably from those described by a simple logistic equation (see May, 1981a). However, at moderate or high values ​​of the lag time and reproduction rate, the population oscillates with stable limit cycles. Moreover, if these stable limit cycles occur according to a logistic equation with a time lag, then their duration (or "period") is approximately four times that of the

victims in order to understand the mechanism of fluctuations in their numbers.
There are a number of examples obtained from natural populations in which regular fluctuations in the numbers of predators and prey can be detected. They are discussed in Sect. 15.4; Just one example will be useful here (see Keith, 1983). Fluctuations in hare populations have been discussed by ecologists since the twenties of our century, and hunters discovered them 100 years earlier. For example, the mountain hare (Lepus americanus) in the boreal forests of North America has a “10-year population cycle” (although in reality its duration varies from 8 to 11 years; Fig. B). The mountain hare predominates among herbivores in the area; it feeds on the tips of the shoots of numerous shrubs and small trees. Fluctuations in its numbers correspond to fluctuations in the numbers of a number of predators, including the lynx (Lynx canadensis). 10-year population cycles are also characteristic of some other herbivorous animals, namely the collared grouse and American grouse. In hare populations, 10-30-fold changes in numbers often occur, and under favorable conditions, 100-fold changes can be observed. These fluctuations are especially impressive when they occur almost simultaneously over a vast area from Alaska to Newfoundland.
The decline in the mountain hare population is accompanied by low birth rates, low survival rates of juveniles, weight loss and low growth rates; all these phenomena can be reproduced in experiment by worsening nutritional conditions. In addition, direct observations do confirm a decrease in food availability during periods of maximum hare abundance. Although, perhaps more importantly, plants respond to severe overeating by producing shoots with a high content of toxic substances, which makes them inedible for hares. And what is especially important is that the plants remain protected in this way for 2-3 years after severe nibbling. This leads to a delay of approximately 2.5 years between the start of hare population declines and the restoration of its food reserves. Two and a half years is the same time lag, amounting to a quarter of the duration of one cycle, which exactly corresponds to predictions from simple models. So, there appears to be an interaction between the hare population and plant populations that reduces the number of hares and occurs with a time delay, which causes cyclical fluctuations.
Predators most likely follow fluctuations in hare numbers, rather than cause them. Nevertheless, the fluctuations are probably more pronounced due to the high ratio of the number of predators to the number of prey during the period of decline in the number of hares, as well as due to their low ratio in the period following the minimum number of hares, when they, ahead of the predator, restore their numbers (Fig. 10.5). In addition, when high regard When the ratio of lynx to the number of hare is large, the predator eats a large amount of upland game, and if the ratio is low, it eats a small amount. This appears to be the cause of population fluctuations in these minor herbivores (Fig. 10.5). Thus, hare-plant interactions cause fluctuations in hare abundance, predators repeat fluctuations in their abundance, and population cycles in herbivorous birds are caused by changes in predator pressure. It is clear that simple models are useful for understanding the mechanisms of population fluctuations in natural conditions, but these models do not fully explain the occurrence of these oscillations.