The identification techniques developed in [1] can be used to monitor the state of dynamic objects operating in a natural environment in order to take appropriate measures when critical limits are in danger of being approached. One such measure is an external control interaction for returning an object to a stable state. For example, in geophysics a long elastic wave created by powerful vibrator is an action of this type that inhibits the blocking of natural impediments in boundary structures of geological media in order to prevent earthquakes [2, 3].

At present, there is no theory for the types of control actions or for evaluating their effectiveness as applied to arbitrary objects. Here a practical method that is universal for any object is proposed to deal with this problem. It is based on a new mathematical description of a dynamic object in the form of a difference equation for the evolution of its state [1]. All that is needed to construct this equation is the measured parameters, without a need for specific information on the object. This means that the mathematical description has a universality, as does the method based on it for evaluating external effects on the state of a dynamic object.

Analytic Technique. In the absence of an external interaction, we represent the object by a homogeneous difference equation for the evolution of its state. For practical purposes, we can stop at the second order, second degree equation [1]:

$$ {x_{k+2 }}=a{x_{k+1 }}+b{x_k}+cx_{k+1}^2+dx_k^2, $$
(1)

where x k is a measured parameter of the object; k = 0, 1, 2, … correspond to discrete measurement times; a, b, c, and d are coefficients specific to different objects which are evaluated from the sequence of values x k of the measured parameter [4]; and x 0 and x 1 are the recorded initial values.

The monitored indicator is the state of the object. Numerically it is characterized by coefficients a and b which determine a point on the field of solutions of Eq. (1). Depending on its location, the state can be classified as stable (the point lies in the region of convergent solutions), unstable (in the region of undamped solutions), or catastrophic (in the region of diverging solutions).

An object subjected to an interaction y k+1(k = 0, 1, 2, …) is mathematically represented in the form of an inhomogeneous quadratic, second order difference equation derived from Eq. (1) by including the interaction on its right hand side:

$$ {x_{k+2 }}=a{x_{k+1 }}+b{x_k}+cx_{k+1}^2+dx_k^2+g{y_{k+1 }}, $$
(2)

where g is an external interaction coefficient set experimentally.

This was analyzed by computer simulation with the aid of a new version of the AnDynSys program with an additional subprogram for generating external interactions [5]. Numerical values of the coefficients and interaction y k+1(k = 0, 1, 2, …) are introduced in Eq. (2). A sequence is generated which can be regarded as measurements on the object subjected to the external interaction. The coefficients in Eq. (1) are recalculated in this sequence, along with the field of solutions and the point corresponding to the actual state; all these now characterize the object exposed to the external interaction.

The result of the control action should be to achieve the set goals. One of these is to shift the object from an unstable or, especially, catastrophic state into a stable state. Experiments must establish whether changes in the critical state of the object (1) can be achieved by means of artificial external interactions and what kind of actions are most effective with the lowest value of g.

Numerical Simulations. Six forms of interaction were used:

  • harmonic sin(kT + w); with chosen T, w; k ≥ 0;

  • chaotic 4y k (1 – y k ); y 0 = 0.14;

  • random from a random number generator with a uniform distribution from 0 to 1 with y 1 = 0;

  • almost periodic 0.9y k – y k–1 + y k 2; y 0 = –0.001; y 1 = 0.5;

  • fixed y 0, k = 0, 1, 2, …; and

  • pulsed y ki = y kf, with the start k i and end k ƒ of the pulse and its duration k i – k f specified.

By varying just the coefficient g, since all the interactions were reduced to the same amplitude, we find the minimum of g in terms of its modulus at which the point corresponding to the state is shifted from the catastrophic region to the region of converging solutions. We repeat the simulation in analogous fashion for the other forms of interaction. More than 100 simulations were carried out with variations in the parameters of Eq. (1).

Results of the Simulations. The possibility, in principle, of shifting an object from a catastrophic state to a stable state by using an external control interaction of appropriate form has been established. Figure 1 shows the field of solutions of the equation for an object in a catastrophic state: the point circumscribed by a circle is in the region of diverging solutions (white field). The grey color indicates the region of converging (stable) solutions and the black, the region of undamped oscillations (pseudo-noise and cyclical). In the black region, the states are unstable. Both regions have been calculated for Eq. (1) with parameters a = –0.8, b = 3.5 (the coordinates of the point corresponding to the state of the object), c = 0, d = 1, x0 = –2, and x 1 = –1.

Fig. 1
figure 1

A catastrophic state of an object: the point (circled) corresponding to the state is in the region of diverging solutions (white field). The triangle is the field of solutions for a linear model of the object (c = d = 0).

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Figure 2 shows the field of solutions of the equation for the evolution of the state for the same object when a stabilizing interaction is present. Now the point corresponding to the state of the object lies in the region of converging solutions (grey region) and the object is in a stable state. The external interaction is fixed with y 0 = 1 and g = –1. The state of the object subjected to this interaction is described by Eq. (1) with the new parameters a = –0.3596, b = 3.9170, c = 0.0942, d = 1.0883, x 0 = –2, x 1 = –1.

Fig. 2
figure 2

The result of a fixed, stabilizing external interaction. The object is shifted from a catastrophic state into a stable state. The point corresponding to the state lies in the grey region.

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The most efficient of the external interactions considered here was the fixed interaction: stability was achieved in these simulations with the lowest modulus of g. (All the interactions were reduced to unit amplitude.)

It was found that the object cannot be stabilized in all catastrophic states (at least for the types of interactions considered here). This possibility depends on the values of x 0 and x 1 for the process at the onset of the interaction. Figure 3 shows the field of solutions of the equation for the evolution of the state of a real object in a catastrophic state [3]. For the actual initial values x 0 = 1.736, x 1 = 1.724 and coefficients a = 1.509602, b = –1.279645, c = 0.065652, d = 0.378055 in Eq. (1), stabilization could not be achieved with any of the forms of interaction considered here. However, when the initial values were replaced by x 0 = 0.736, x 1 = 0.724, stabilization did set in for a fixed interaction (y 0 = 1, g = 0.3). Under real conditions of constant monitoring of the measured values for a process, this kind of result could be attained by choosing the time of onset of the interaction, since at any step k two neighboring values of the sequence x k , x k+1 can be treated as the initial values for the subsequent steps.

Fig. 3
figure 3

An example of a catastrophic state that cannot be stabilized with any of the forms of external interaction considered here.

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In concluding, we note that, as these simulations show, the same forms of interaction can lead to stabilization or destabiliztion, depending on the state of the object and the value of the interaction coefficient. The method proposed here can be used for the mathematical analysis of dynamical objects which are described by inhomogeneous, nonlinear equations and for which analytic solution techniques are not currently available.