CONTINUOUS-DETERMINED MODELS (X-SCHEMES), Basic relations. - Modeling of systems


Consider the features of a continuously deterministic approach using the example of using differential equations as mathematical models. Differential equations are those equations in which functions of one or several variables are unknown, and the equation includes not only functions, but also their derivatives of different orders. If the unknowns are functions of several variables, then the equations are called partial differential equations , otherwise, when considering a function of only one independent variable, the equations are called ordinary differential equations.

Basic relationships.

Usually, in such mathematical models, the time t is used as the independent variable on which the unknown unknown functions depend. Then the mathematical relation for deterministic systems (2.6) in general will be

where y '= dy/dt, ^ y = (y l , y 2 , y.) and f 2 , -n-dimensional

vectors;/(y, /) is a vector-valued function that is defined on some (n4 - 1) -dimensional (y, /) and continuous.

Since mathematical schemes of this kind reflect the dynamics of the system under study, that is, its behavior over time, then they are called schemes (English, dynamic) [4, 37].

In the simplest case, an ordinary differential equation has the form

The most important for system engineering is the application of d-circuits as a mathematical apparatus in the theory of automatic control. To illustrate the features of constructing and applying D-schemes, let us consider the simplest example of the formalization of the process of functioning of two elementary systems of various physical nature: mechanical 5 M (oscillations of a pendulum, Fig. 2.1a) and electric sub> t (oscillatory circuit, Figure 2.1, b).

Fig. 2.1. Elementary systems

The process of small oscillations of a pendulum is described by an ordinary differential equation

where m *, 4, is the mass and length of the pendulum suspension; % - acceleration of gravity; in (/) - angle of deviation of the pendulum at time t.

From this equation of free oscillation of the pendulum, one can find estimates of the characteristics of interest. For example, the period of oscillation of the pendulum

Similarly, processes in an electric oscillatory circuit are described by an ordinary differential equation

where C to is the inductance and capacitance of the capacitor; q (/) is the charge of the capacitor at the moment of time t.

From this equation, it is possible to obtain various estimates of the process characteristics in the oscillatory circuit. For example, the period of characteristic oscillations

Obviously, by entering the notation AND 0 = m V4 1 m - b to , = 0,

L 2 = m & pound; 4 = 1/C x , in (0 = H (0 = r (0, > we obtain an ordinary differential equation of the second order describing the behavior of this closed system:

where A 0 ,/!, And 2 are the system parameters; r (r) is the state of the system at time t.

Thus, the behavior of these two objects can be investigated on the basis of the general mathematical model (2.9). In addition, it should be noted that the behavior of one of the systems can be analyzed with the help of another. For example, the behavior of a pendulum (system 5 M ) can be studied using an electric oscillatory circuit (system 5,).

If the system in question 5, that is, a pendulum or contour, interacts with the external medium & pound; then an input effect x (/) (external force for the pendulum and energy source for the circuit) appears

and a continuously-deterministic model of such a system will have the form

From the point of view of the general scheme of the mathematical model (see § 2.1) * (/) is the input (control) action, and the state of the system H in this case can be considered as the output characteristic, ie to assume that the output variable coincides with the state of the system at a given instant of time y-r.

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