## Conditions for the Optimality of the Averaged Problem of Nonlinear Programming and Synthesis of the Philippov's Theorem on Speed of Motion of Systems with the Discontinued Right Part of Differential Equations

Мargarita A. ZAEVA*

Citation: Conditions for the Optimality of the Averaged Problem of Nonlinear Programming and Synthesis of the Philippov’s Theorem on Speed of Motion of Systems with the Discontinued Right Part of Differential Equations. American Research Journal of Mathematics. vol 3, no. 1, 2017, pp. 1-11.

Copyright This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## Abstract:

A generalization of the Filippov's velocity averaging method in the problem of determining the slip velocity of a dynamical system with discontinuous right-hand sides is proposed. It is shown that in the multidimensional case such an extension is reduced to the solution of the averaged nonlinear programming problem and the methods of averaged optimization can be used to calculate the slip velocity. Problems with various dimensions of the vector of admissible velocities and the number of surfaces of control's discontinuity are considered.

Keywords: discontinuous systems, sliding regimes, slip velocity, averaged optimization.

## Description:

INTRODUCTION

Sliding modes in systems described by ordinary differential equations (ODEs) are the subject of many and many years of research.A brief survey of the problems of which is contained in [1], an incentive to write this article.

Formulation of the problem. We consider a dynamical system

Control u and continuous function of the argument has a discontinuity on surfaces characterized by conditions
so that
Vector-valued functions  with components  are differentiable in x and t. For a fixed the solution of the equations (1) for any of them exists and is unique. If on the surface of control`s discontinuity for all and the following conditions of attraction are satisfied, for example, of the form:
then the system, being on the switching surface, moves along this surface in the control switching mode with a high frequency (sliding mode). In this case, the switching surface is called the sliding surface.
The conditions of attraction in the form (4) are sufficient, but not necessary for the appearance of a sliding mode. The sliding mode can arise on the intersection of the surfaces  and for its emergence it is required that the system, by virtue of the equations of motion, approach the origin of coordinates in space, along the axes of which the functions  with nonzero velocity. An example of such a situation is given in [2].
The task set by numerous researchers[3]–[6], was to find the equation the motion of the system along the surface (2) is subject to
The equation is called the slip equation.
In for the case of a single discontinuity surface () it is suggested to find the slip velocity for each value x and t as an element of a convex hull spanned by vectors  and and satisfying sliding conditions along the discontinuity surface:
In other words, the slip velocity vector is found as the weighted-average value of the vectors of admissible velocities  with weight coefficients chosen so that  touched the sliding surface.
Thus, the additional definition of A.F. Filippov from [3] оis based on averaging the admissible speeds of the system.
Systems with two vectors of admissible velocities and one sliding surface are considered [3]. In the general case, the number of permissible velocities can be different, as is the number of sliding surfaces. The surfaces partition the phase space of the system (1) into  subspaces  in each of which its some combination of signs  and hence, in each of which there can be it`s its velocity vector The problem arises of determining how many and which of the admissible velocities need to be averaged and how to choose the weight factors.
In [2] it is said: «`As for the vector case, the author does not know the works in which procedures would be proposed that allow in an explicit form to write out the slip equations over the intersection of the discontinuity surface.»`.
We denote the set of vectors of admissible velocities by The number of these vectors for fixed values of and t can be less than and greater than  It is only important that in each subspace  the conditions of attraction to the intersection of sliding surfaces be satisfied.
We will follow the method of averaging the admissible velocities of AF. Filippov, to determine the slip velocity as the average velocity of the system along such a direction, for which the average rates of change of the functions are zero (see [1]).
In order to obtain algorithms for determining the slip velocity, it is necessary to formalize this definition for the case when the sliding surfaces are several and the set of permissible velocities is given in various forms. With this formalization method, the problem of determining the slip velocity by the method of averaging the admissible velocities turns out to be the averaged problem of nonlinear programming. This allows us to find the slip velocity using the averaged optimization [7].
Before we formalize the problem of determining the slip velocity and expound the algorithms for solving it, we recall the optimality conditions for the averaged nonlinear programming problem.

SOLVING THE PROBLEMS OF AVERAGED OPTIMIZATION

The averaged nonlinear programming problem has the form
Here
The functions are continuous and bounded on the set  and this set is a probability measure that determines the weight of each of the ordinates of the function when its averaging
The following assertions are true: 1. The optimal solution of the problem ((7)) — the probability measure  is concentrated at discrete points whose number does not exceed  so that
where — is the Dirac delta function. — are called the base values of the vector and  — are nonnegative weighting coefficients whose sum is equal to one.
2. There exists a nonzero vector  with components  such that the base values of the vector are solutions of the minimax problem

On an optimal solution, the values of the Lagrange function  мare maximal and the same for all

We emphasize that the existence of Lagrange multipliers does not require the smoothness of the functions  It is enough that they are continuous and limited. The set  can also consist of isolated isolated points.

3. The weighting factors  corresponding to the base values  satisfy the system of linear equations

The existence of a nonnegative solution of this system corresponds to the fact that in the m-dimensional space with coordinates  the origin of coordinates belongs to the convex hull of the setof the vectors  f the number of base values of is less then  then the origin in this space lies on the boundary of the convex hull .

Thus, the solution of the averaged problem consists of two stages. On the first the most time-consuming stage,we find the basic values  from the solution of the minimax problem (10), in the second stage, at fixed we find the weight coefficients  by the conditions (11) .

In a number of problems, the number of base values of always to be less than . In particular, if the set can be partitioned into subsets so that for each of them the problem (10) has a unique solution , for example, each subset strictly convex, then v changes from one to (see [8] ). In this case, on each subset, the optimality conditions of the convex problem are written, and the vector  is found by the condition that the values of the Lagrangian are equal in the maxima found.

We emphasize that the number of base values of is not related to the dimension of this vector, but depends only on the number of average conditions in the problem (7). In this case, the set of basic values of may not be unique (see example 2).

If in the problem (7) instead of the requirement of maximum the requirement of minimum appears, then in (10) and are interchanged.

Degenerate Case

As in the problem of non-linear programming, in the averaged problem (7) addition to regular solutions, degenerate solutions can also exist. For degenerate solutions, the factor  in the Lagrange function is zero, which corresponds to the case when the constraint system completely determines the solution, and consequently, the target function  does not affect it. Such a situation, in particular, takes place when the set  consists of discrete values of the vector and the number of these values does not exceed . In this case the problem (4) is degenerate, in the Lagrange function  the factor  and all the admissible values are basic. The solution of the problem reduces to determining the weight coefficients  from the system of linear equations (11).

When the number of admissible values of is equal to  , then the equations have no solutions in the general case. But from the number of functions  we can choose  functions whose average values are zero (there exists a nonnegative solution of the equations (11) with respect to K weighting coefficients).

If the number of admissible values of is greater than  then the possible slip speed is not unique. Each of them has its own set of basic permissible speeds and weight coefficients. The choice of the slip velocity in this case is carried out according to the conditions (10) taking into account the objective function . The form of this objective function is determined by the specific content of the task. But in any case it contains the operation of averaging the permissible velocities. In particular, the criterion of optimality can be the maximum of the average rate of change of some function  from state variables along a slip trajectory or the maximum of projection of the slip velocity on the direction of the selected vector . The last statement corresponds to the case when the function  is linear.

Formulation of the Averaged Problem

Average on the set of permissible velocities  is the rate of change of the state vector

The choice of the measure  is bounded by the condition that the mean value of the rate of change of each of the functions  equal to zero, the intersection of which forms a sliding surface, by the equations (1):

Here

When averaging in this expression, the functions  are replaced by the functions .

In addition to the conditions the required solution  satisfies the following requirements:

The problem of determining the maximum mean velocity of moving of the system in the direction of the vector , normalized so that , and tangent to the slip surface, provided that the average rates of change of the functions  are zero, takes the form

under conditions

Here averaging over s performed on the set  of admissible values of control. Since the motion occurs along the intersection of the switching surfaces, the vector  must satisfy the requirements:

If , then this vector is not unique.

The set of admissible velocities can be determined by conditions of the next type

The number of admissible velocities in this case is arbitrarily large.

The velocity of the system in the sliding mode  is the average velocity of the system on the optimal solution of the problem (16), (17).

Optimality Conditions of the Solution of the Averaged Problem

The additional determination of the slip velocity requires the calculation of the base values of control  and the corresponding velocity vectors  from the averaged solution of the non-linear programming problem (16), (17). The number of these base values does not exceed .They correspond to a minimum in of the maximum in the Lagrange function

In this problem, the value  is not zero, which means that it can be considered equal to one. On the optimal solution for the base values of the velocity vector  expression

maximal, and therefore, the same for all values of  It is clear that the base velocities , which are the solution of the averaged problem (16), (17), are different for different directions

The system in sliding mode is characterized by equations

where

The weight coefficients  are found from equations

The averaged problem (16), (17) is degenerate if the number of vectors of admissible velocities does not exceed In this case, all these vectors are basic and it is only necessary to solve the problem of calculating the weight coefficients  the conditions (23)

When the number of admissible velocity vectors  is less than  the number of sliding surfaces does not exceed . And the problem is degenerate. For example, these vectors can be two and  and switching surfaces are greater than one. In this case, the switching between the vectors  and  is determined by the sign of only one of the functions  and the conditions of attraction to other sliding surfaces are provided by the dynamic properties of the system. The surface  is a sliding surface. Slip velocity vector

where are determined from the conditions ((23)) for when summing over  from zero to one.

Let`s illustrate by examples the use of additional definition, based on averaging of permissible speeds. It coincides with the additional definition of A.F. Filippov from for the considered case of motion along the switching line on a two-dimensional phase plane with two admissible velocities.

SOME EXAMPLES

The simplest way is to find the slip velocity in the case when the averaged problem turns out to be degenerate, since in this case all admissible controls are basic and the problem reduces to the calculation of the weight coefficients. These are the examples most often considered in publications. Let us show that for such cases the introduced general additional definition of the slip velocity by the method of averaging the admissible velocities leads to known results. Then we consider the more general case when the averaged problem is nondegenerate. In this case, its solution allows us to find the slip velocity without any geometric constructions.

Example1. A degenerate problem. Linear second-order system with relay control: [1], [3]

For this system, the conditions of attraction ((4)) to the slip line are fulfilled. The set consists of two, i.e. from  elements, so that the averaged problem (16), (17) is degenerate, and the two allowable velocity vectors are basic. To solve the averaged nonlinear programming problem, we need to find the weighting factors  corresponding to the basic controls. The equations of sliding have the form:

The factor  is determined by the condition that the mean rate of change is equal zero

where from

After substitution in (26) the slip equations will take the form:

which corresponds to the known result.

Example 2. Nondegenerate problems.

A.      Third order system [1]:

In this case,  and the number of vectors of admissible speeds  i.e. more than  Permissible speeds

We denote their projections on the coordinate axes by

To determine the slip velocity, we need to select the vector  and solve the averaged nonlinear programming problem (16), (17), by finding three basic vectors of admissible velocities.

As a vector we select a unit vector that coincides in direction with the axis  Let us find the maximum projection of the slip velocity

The averaged problem (16), (17) takes the form

under conditions

Here the averaging is carried out on the set of admissible velocities

The problem of choosing three basic values of the velocity vector from four reduces to a minimum in from the maximum  in of the Lagrange function

For

We choose from the condition that the maximum of the Lagrange function be equal at the base points

From the first equation

From the second equation

The value of

Similarly by condition

we get

By the condition  we have

Thus, of the four vectors  the basic ones are three:

Maximum sliding speed along

In this case, the equations (23) for calculating weight factors will take the form:

Whence After substituting the weight factors in (30) we obtain

Similarly, for the minimum speed along the which coincides with the result of considering this example in [1]n the basis of geometric considerations.

B.      A system in which the vectors of permissible velocities depend on the control. Consider a system characterized by the equations:

The line of switching . Control of  undergoes a break on the line of switching

the vector  is directed along the slip line, i.e. on the axis

In this problem, you first need to find the base values of the control vector, and then the corresponding weighting coefficients and the slip speed.

The averaged problem (16), (17) takes the form

under conditions

The Lagrange function for each of the values  has the form:

We find the base values  by the condition of the maximum of each of these expressions by  as a functions of  We get:

We substitute these expressions in  One of them grows with the growth  and the other decreases. The minimum of the maximum of the Lagrange function corresponds to the equality From this equality we find

The basic values of the control vector, whose number is

And

After determining the basic values of the control vector, we find the weight factors in the same way as in the degenerate problem. Substituting the base velocities in the expression for the rate of change  and averaging the resulting velocities with the coefficients so that the average rate of change  was zero, we get that  Finally, after substitution the base velocities and the found weights in the expression for the rate of change  we obtain the slip velocity along this axis.

C.      Tracking relay system with feedback. In the industry, widely distributed systems with electric motors, controlled by starters, the signal of which takes one of two possible values. The relay elements themselves are covered by feedback. In this case, a sliding mode occurs in the system, in which system behaves as a linear system, and the characteristics of such a «`quasilinear»` system depend on the type of feedback.

Let  be the rotation angle of the motor shaft and  be the signal at its input. The signal at the system input is  and the signal at the feedback output is  Equations of motion:

We choose the vector  so that its direction coincides with the axis  Values of the Lagrange function for different signs of

And

must be maximal by  and equal to each other  which leads to equalities for basic controls, and the factor  For weighting coefficients we have equation

whence  taking the boundary values ??when the right-hand side of the equation is greater than one or less than zero.

In sliding mode, the system obeys the equations:

After excluding  we get the connection between  so the system behaves as linear in the sliding mode.

DISCUSSION

In an equivalent control`s method is proposed to determine  according to which the slip velocity vector is found after substitution of the equivalent control in the equations of motion, so that  The equivalent control for each t,  is determined from the conditions

where φ_i is defined by the equality (14)

For systems that are affine in control, in particular for an important class of systems with variable structure, the method of equivalent control leads to the same results as the averaging of the permissible velocities according to A.F. Filippov if the system of linear equations for  has a unique solution. All the difficulties that can arise when solving this system are discussed in detail in [2].

When the right-hand sides of the equations of motion are nonlinear in control, the functions  found by the method of averaging admissible velocities and the equivalent control method are different (see the example in [1]).

In the general case, the velocities of the system can be undetermined outside the set of admissible controls. From their additional definition, the equivalent control, and hence the slip speed found from it, depends. In the method of averaging the velocities, it depends only on the admissible base velocities. In addition, the equations (36) may not have a real solution for all or for some  For example, the dimension of the vector of controls can be less than