Application of regression in algorithm of nonlinear stochastic adaptation of unstable multidimensional objects

The results of a study of applicability of kernel estimation in the synergetic control systems for the objects unstable in an open-loop state (without a stabilizing control) have been presented. The effectiveness of kernel estimates has been shown for four nonlinear objects with unstable limiting states. The estimate the effectiveness of embedding the kernel predictive estimate of the state variables of a nonlinear object, subjected to disturbances of an unknown nature, into the system of synergetic control is demonstrated.


Introduction
At present time, in the methods of nonlinear control the focus is shifted towards synthesizing the regulators for the objects with predetermined target manifolds (see, e.g., [1][2][3][4]), which, based on the principles of the synergetic control theory (SCT), allow for an analytical regulator synthesis without a preliminary manipulations with a complex object. The conditions for application of these methods are natural [1,4]: • analytical description of the aim of control (target manifold) and its consistency with the physical properties of a controlled object; • boundedness of solutions and existence of a stable target system for the 'object-target manifold' pair. The merit of SCT consists in implementation of the principle of self-organization, resulting in minimization of the efforts to achieve the required target qualities including the intrinsic reactions (physically common for the object) to external impacts [1]. This peculiarity of SCT guarantees the development of an energy-saving control (if any).
On the other hand, by setting the desirable target manifolds, testing their physical attributes and achievability one can design new objects with unexpected properties (new materials, in particular, if the analytical formula of their initial description is known).
The purpose of this work is to demonstrate a possibility and to estimate the effectiveness of embedding the kernel predictive estimate [5] of the state variables of a nonlinear object, subjected to disturbances of an unknown nature, into the system of synergetic control. It is clear that an interpretation of the disturbances can be quite diverse: from the inaccuracy of a nonlinear model (which is nearly evident due to the object complexity) to the noise in the measurement of state variables.
To begin with, let us briefly describe the essence of two algorithms based on SCT, whose outputs are the control systems correcting systematic and random disturbances, respectively. We assume that the objects under study are set by the systems of ordinary differential equations or difference stochastic  A positive feature of the two algorithms for synthesizing adaptive control systems, which are implemented below, is their analytical justification [6].

Algorithm of nonlinear adaptation
The main points of the algorithm of nonlinear adaptation are the following: • perform extension of the phase space via simulating derivative models of disturbances as a linear function of the target macrovariables; • design control on the basis of the analytical design of aggregated regulators (ADAR) as the principal method of SCT as applies to the closed system obtained in Step 1; • select auxiliary macrovariables in a special form, whose attainment automatically results in the compensation of disturbances (in the final synthesis stage). Remark 1. The purpose of any control system is to maintain the standard condition. In an ideal case, an error as a deviation from this condition would be zero. If the system is subjected to random disturbances, even under the standard condition it is impossible to ensure a zero error.
Since any detailed study invariably requires a transition to a discrete description [7], the significance of the algorithms for discrete synthesis of stochastic nonlinear regulators becomes evident.
The main points of the algorithm of nonlinear stochastic adaptation (NAS) are the following. 1. Perform a classical discrete ADAR-synthesis at fixed random functions. 2. Determine the conditional mathematical expectation from the control found in Stage 1. The satisfiability of this operation is ensured by the peculiarity of the ADAR-synthesis.

Remark 2.
Not any initial data smoothing can result in a 'good' form of a controlled process. The point is that "one can throw the baby with the bathwater", since in the systems of deterministic chaos it is not always clear what we are cutting out (smoothing): a useful signal or its measurement noise.

Application example of the NAS-synthesis of a scalar regulator for the Lorentz model
Let the aim of control be set in the form of sustaining a balance between the variables Following the steps of the NAS-algorithm, obtain, respectively the following: 1) ADAR-control at the fixed random functions [ ], 0,1,... (1) and (2) 3) Dependence for the random functions as the functions for measuring the object's states for the found control (3 Let us present the results of numerical simulation of the system (4

Example 1. Immunological model.
Let us look at a couple of simulation results (figures 3 and 4) for an immunological model with the hepatitis data [8,9]. The mathematical description of this model suggests that the first two equations 'resemble' those of the Verhulst model of deterministic chaos. x k x k ax k bx k y k y k y k dy k mx k y k u k k c k x x y y The problem of stabilization of variable [ ] is solved positively using the NAS algorithm, and the regulator acquires the following form: , , τ ω ω are the discretization parameters of the object and the regulator, respectively. a) normal noise b) uniform noise Figure 5. Dynamics of RMS of the controlled index in the predator-prey model using kernel estimation (green lines) and without it (ADAR -red and NAS -blue) Shown in figures 4 and 5 are the results of simulation of control algorithms designed in accordance with two methods -classical ADAR [4] and stochastic adaptation (NAS [6]) with and without filtering, from which it follows that, firstly, starting from a certain level of noise both methods can have commensurable indices, while the use of kernel smoothing maintains the steady-state behavior of the controlled variable (judging by the output variable dispersion); secondly, the algorithm of stochastic adaptation NAS itself possesses a filtering property missing in the classical ADAR under the conditions of random disturbances, and the object with homogeneous disturbance is less prone to control than that with a normal disturbance.
Remark 3. Such parameters 1 2 , , T ω ω as a duration of reaching the vicinity of the target state require careful selection for the given individual characteristics of the model (4) and (5).
Here the variables are: х 1 -load pressure; х 2 -supporting pressure; х 3 -control valve spool velocity; х 4 -control valve spool movement; u -input solenoid voltage of the sliding spool valve (control valve input) realizing the goal achievement * * 2 2 0, , x is the target value;  According to the NAS-algorithm, we obtain a regulator of the form (figure 6) [ ] (