The synthesis of adaptive multi-mode regulators based on combined control of the combined maximum principle

The problem of synthesis of the optimal controls using nonlinear dynamic systems in general form is not yet analytically solved. Fundamental results of L.S. Pontryagin leads to programs of optimal control, and the solution to the synthesis problem can be obtained using complex constructions only in special cases. The implementation of such solutions implies the need to take into account the sliding modes, the regimes of frequent switching, the negative influence of interference, and so on. This was the reason for the development of methods for synthesizing systems of variable structure, which allow to increase the accuracy of control, eliminate undesirable modes, and counteract the influence of external and parametric disturbances. The results of the research showed that using the combined maximum principle leads to obtaining a general structure of quasi-optimal control for different classes of constraints and allows to provide parametric and structural adaptation to of the state and the parameters due to the possibility of complex implementation of control principles with respect to deviation and disturbance.


Introduction
The trend of development of modern controlled mechanical systems is due to the fact that these objects become essentially the multimode multipurpose systems of wide use. Therefore, the natural situation is when the control objective, the dynamic parameters of the control object and the external environment are all continuously changing. Linear control laws, as a rule, ensure the achievement of the goal of the solution for one mode of motion [1]. Meanwhile, the real controlled objects are often characterized by several modes of motion [2,3]. The theory of systems with variable structure leads to different controls obtained, each of that is constructed for the corresponding subsystem. As a result, we need to solve particular problems related to the choice of switching times, ensuring stability, reducing power consumption and increasing the performance [4,5]. Overcoming these drawbacks can be carried out by changing the switching surfaces, for example, using fuzzy control methods or artificial neural networks [5].
The basis for the procedures to obtain the equations of motion in sliding mode is the physical approach [3]. Its use for solving the problem of synthesis of regulators for the systems which dynamics satisfies the Lagrange equation of the second kind leads to a combined maximum principle [5][6][7][8]. The controls built on this base are characterized by the property to adapt to the change in the dynamic parameters of motion [7,6] and by nonlinear structure of controls. The variety of solutions of the inverse problems is related to the different behavior of dynamic measures of motion on the switching hypersurface. Combination of the controls of the combined maximum principle makes it possible to achieve the stated synthesis goal for a set of motion modes that satisfy the Hamilton-Ostrogradskii principle [8].
The purpose of the work is to perform the synthesis of adaptive multi-mode regulators based on combined control of the combined maximum principle.

Formulation of the problem
The principle of combined control includes the implementation of control processes for deviation and disturbance (see Figure 1). Where ЗУ is the driver unit; УУ is control unit; КУ is compensating unit; ОУ is object of control; q s (t) is a given trajectory; q(t) is the trajectory of the ОУ motion; N(t) is external uncontrolled influence; u(t) is the control action (control): u 1 (t) is control for deviation; u 2 (t) is control for disturbance ; ∆q(t) is deviation.
The structure of the combined management is the sum of the control actions u s defined on the sets In this case, the controls are the set of allowed impacts, following one after the other In addition the combined control can be defined on the intersection of sets In this case, the controls are the set of allowed impacts, performed simultaneously and taken with certain weights, i.e. complementing each other. Such combined controls are complementary:

The control of the pendulum oscillations in the presence of noise and disturbances
Let us consider a pendulum, to which the control and the external action are applied (see Figure 2) [7,8]. The notation in Fig. 2 is as follows: 2 ml I  is moment of inertia; g is the gravity acceleration; m is the mass; l is the pendulum length; u is the control; P is the external influence. where I is moment of inertia of the body; m is the mass; l is the length; u is the control; Pis the external influence.
The state of the pendulum is determined by the generalized coordinates The kinetic energy T and the momentum  the moving pendulum are determined by formulas The potential energy is given by expression The work of generalized forces (control and disturbance) is The target functional is as follows is stochastic process with known characteristics.
The Lagrange equation of the second kind for P = 0 has the form

The control laws for classes of measurable and piecewise-continuous controls
To obtain the conditions for a minimum of the target functional, we use the combined maximum principle, according to which we can write the extended Hamiltonian of the system as The generalized power is determined by function of the form The control from (16) can be defined as follows: where µ is synthesizing function.
The following variants to build the combined control are possible.

The control is chosen from the class of bounded measurable functions
The equation that defines the control switching line can be written as Along this line the Hamiltonian is constant The use of the operation of the Poisson bracket allows us to write: The law of the combined control with respect to deviation and disturbance has the form The motion equation has the form

The control is chosen from the class of bounded functions
Compensation for the perturbation can be performed using the forces superposition principle. Then we can put 2 1 u u u   , and here  sin The law of the combined control with respect to deviation and disturbance has the form The motion equation has the form The control is chosen from the class of discontinuous piecewise-continuous and relay controls The piecewise constant equation The generalized power function To achieve the maximum we need to satisfy the following conditions: The substitution of (25) into (23) gives the combined control for the laws (19), (21) and (26), respectively.
The results of the studies confirm the effectiveness of the synthesized combined controls for obtaining the required quality of the process to control a nonlinear system -monotonic, aperiodic or oscillatory process with requirements for accuracy, overshoot, regulation time, etc.

Conclusions
On the basis of the theorem of the combined maximum principle, the basic laws to control the dynamic objects under the influence of external uncontrollable disturbances are obtained. The use of optimal laws to build the combined controls makes it possible: to exclude the chattering regimes generated by discontinuous (relay) controls in problems with the nonsmooth Hamiltonian; to ensure the stability of the object in an unbalanced state.