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
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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.
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.
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 , an incentive to write this article.
Formulation of the problem. We consider a dynamical system
SOLVING THE PROBLEMS OF AVERAGED OPTIMIZATION
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  ). 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.
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 m 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.
THE TASK OF ADDITIONAL DETERMINING THE SPEED OF SLIDING
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):
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
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
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.
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: , 
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
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 :
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
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
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
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 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
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
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
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.
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 .
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 ).
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