Synthesis of synergetic laws of control of nonlinear dynamic plants

. Given in the work the facilities of applying the methods of the synergistic approach for the synthesis of the law of control of nonlinear dynamic plants. To give, robust properties of the control law were proposed by applying a principle of integral adaptation, which permitted compensation for the impact of the external and parametric disturbances. The realization synergistic law of control is carried out by when constructing the analytical design of aggregated controllers, which procure asymptotic sustainability of the control system for nonlinear dynamic plants. For procuring technological invariance proposed principle of the synergistic control. Formulated an extended dynamical model of the system, which included external and internal disturbances. The essence of proposing the approach is to use an extended model of the plant, taking into account disturbances due to the switching on of integrators and procuring compensation for the influence of disturbances, in distinction from known ones, where uses separate models for each disturbance. The suggested methodology has confirmed examples of digital simulation and has shown an efficient proposed approach to tasks synthesis of a nonlinear system of control with dynamical plants, procuring asymptotic sustainability of the control system and recouping unmeasured and internal disturbance.


Introduction
For really acting dynamic plants are characterized by structural and parametric indeterminacy, which significantly worsens the quality of control systems.To solve the problem of improving the quality of control systems, methods of adaptive and robust control are currently widely applied.To give the properties of adaptation to control systems, a hybrid application of adaptive control methods with modern methods of intelligent control, such as neural networks, fuzzy logic and genetic algorithms, has recently been proposed [1][2][3][4].
Another direction in the creation of highly efficient control systems for dynamic plants in the presence of various views of indeterminacies is the synergistic method of control theory (STC), the peculiarities of which are the possibility of obtaining a nonlinear control law in an analytical form, as well as the formation of a new mechanism of the control law [5][6][7].
The application of STC allows for providing insensitivity of nonlinear dynamic systems to external and parametric disturbances and to attach adaptation to systems through the use of nonlinear integrators that compensate for disturbances.
It should be noted that a synergistic control system based on the principle of adaptation does not require the use of state and disturbance observers to assess these disturbances.

Solution method
To compensate for various external and internal disturbances, which are the worst disturbances affecting the dynamic properties of the controlled system, it is most convenient to apply the adaptation principle with an integrator, which consists of the construction of guaranteeing controllers.
Consider the case, when the worst disturbance is represented as    () =  0 signμ() changing randomly, from M i0 = const on a given interval.Compensation for the measured disturbances is provided by a controller with an integrating property, which controls with astaticism and ensures the stability of the control system of complex dynamic plants.The application of the principle of adaptation with the STC integrator makes it possible to compensate for disturbances of influences by introducing the control loop of integrators and note that the analytical design of aggregated controllers (ACAR) does not allow compensating for harmonic disturbances, but to provide the equation of state with a significant weakening of the influence of a disturbance with three integrators [8][9][10].
The dynamic model of disturbance is presented as: where   -are the disturbance state variables,   () -function from of the plant of the state variables, which provides the system invariance property.The essence of the synthesis of the synergetic control law for the system under consideration is as follows.Let the dynamic model of the electric drive designed to the regulation of the movement of the executive body be described in the form of [11]: where  1 -the angle of rotation of the electric drive,  2 -speed of the rotation of the electric drive,  3 -current in the anchor chain of the electric drive,  -control,   ,   , , parameters of electric drive.The dynamical model of the electric drive (2) characterized the speed of the rotation electric drive and leads to the angle of rotation of the performing mechanism.Parametric uncertainty is the change in the reduced moment of inertia () over time.The change in the moment of resistance   is taken as external disturbances.The main task of controlling the electric drive is to maintain the desired speed of rotation of the electric drive, the above mentioned disturbances [12].To solve the delivered tasks, are suggested the following options, which are based on changes in ().
In the first version the moment of the inertia () changes linearly and is described by an equation of the form  1 =  0 + , and in the second version of the inertia is changing in the form of harmonic function  2 = sinσ 0 .Wherein resistance of moment   = const.To solve the delivered task, we are building one disturbance model with maximum execution.For each view, disturbances are using ACAR synthesize synergetic controllers, which are designed to compensate for various types of disturbances execution [13].
To compensate disturbances of the form  1 =  0 +  + sinσ 0  and bearing in mind this principle of integrated adaptation.Will be formed an extended model of the system taking into account disturbances.In an extended model of the moment, inertia () is assumed to be constant () =  0 , then  () = 0. Then the dynamic model of the electric drive has the following form with the integrator connected in series: where  1 =  ̂ -is the output variable characterizing the state of the system (3), α -constant coefficient.Combining ( 2) and ( 3), we obtain an extended system of equation: ̇1() =  2 ; (4) The resulting model will be used to solve the problem of synthesis of the synergetic control law  = (, ), which allows compensating for unmeasured disturbances  1 =  0 +  + sinσ 0  and   = const ≠ 0 and ensuring are fulfilled.To do this, using the ACAR method, we introduce a macro variable: where γ  -constant coefficients.
The functional equation of the AKAR method, taking into account the macro variable, will be obtained in the form  ̈() +  1  ̇() +  2 () = 0, (6) and solving the functional equation together with (4) and ( 5), we find the control law: Under the provided of fulfilment, the asymptotic stability   > 0,  = 1,2 of the variables γ  affects the nature of the transient process.The dynamics of the system at the intersection of manifolds Ψ = 0 and Ψ ̇() = 0 is described by a system of equations of the following form: ̇1() =  2 ; ̇2() =  3 ; To determine the value of the roots of the desired characteristic equation, we compose equations of the form: here in  0 < 0 -is the required root providing asymptotic stability.
Having calculated the characteristic equation, we determine its roots on the basis of which are determined: The asymptotic stability of the developed system (2) with synergistic control ( 7) is provided by calculations γ  according to formula (9) under the condition γ  > 0,  = 1,2.
To the reliability of the results, we conduct a computational experiment with measured disturbances of the moment of inertia of the form (Fig. 1, 2).
To determine the unknown coefficients of this equation (20), they are equated with the coefficients at the same powers of p: Choosing T>0 and defining coefficients γ  , β  -ensuring asymptotic stability of the system (15).

Results and discussion
To check the reliability of the approach, a computational experiment was carried out.The results of modelling a system with a synthesized control law were carried out with unmeasured disturbances:  = 0,5( 1 )(2 2 )(3 3 ); () = 0,2().
where λ ̂0, ̂0,  ̂0,  ̂0,  ̂0, α ̂0, β ̂0 -initial values of parameters, γ -positive constant.The parameters of the control law are found: The results of modelling a control system with a synergistic control law (23), (24) where showed that the quality of the proposed coincides with the proposed system.Unlike classical control laws, the structure of the synergetic control system has a simple form, due to the presence of only one non-linear component.In addition, unlike the known system, which contains seven dynamic components, the synergistic control system has only three dynamic components.As a result of this proposed synergistic control law, high-frequency changes in the control amplitude occur, which are significant advantages of using the synergistic law to control a nonlinear plant.

Conclusion
In this work, for the synthesis of synergetic control laws for nonlinear dynamic systems, which ensures invariance to external disturbances, the principle of integral adaptation is proposed.The essence of the proposed approach lies in the use of an extended model of the plant, taking into account disturbances by turning off the integrators that provide compensation for the influence of disturbances, in contrast to the well-known ones, where separate models are used for each disturbance.The proposed technique is confirmed by examples of digital simulation and the effectiveness of the proposed approach to the problems of synthesis of a nonlinear control system for dynamic plants is shown, which ensures the stability of the control system and compensates for unmeasured and external disturbances.