Analysis of state commonness condition of nonlinear systems with differential geometry methods, страница 2

In turn realization of (10) is possible, if in a point where the commutator is evaluated, directions of vector fields  и  coincide. Thus two vector fields will commute, and their commutator is identically equal to zero [4]. If the given requirement will be fulfilled for every ,  in relation to a vector  describing initial object, SCC matrix (7)can be represented as:

                                  .                (11)

     SC criterion for this introduction also is the rank criterion (9) for any instant.

     On the basis of the suggested approach now it is possible to review nonlinear objects of the second and third order having the specific singularities of introduced SCC matrices.

     2.2. State commonness criterion for control objects of the second order

     SCC matrix for objects of the second order is represented according to (2) as follows:

,

where

.  (12)

Using a recurrent formula (3) for  using Lie commutators, it is possible to note:

                   . (13)

     Taking into account that  and also the property stating that result of two equal vector fields commutation is equal to zero [4], expression (13) will become:

.        (14)

     From (14) it is clear that vector  does not depend on control variable  and accuracy is equal to the commutator of the first order concerning vector fields  и , i.e. initial vectors of object under consideration. Thus, for objects of the second order the SCC matrix in view of above-stated will be noted as follows:

.        (15)

     State commonness criterion for considered objects will be realization of a requirement  () for any instant.

     The obtained expression (15) allows to make the following conclusions. Elements of block SCC matrix for the objects of the second order and, accordingly, its determinant do not depend on the control attached to the object. This, in turn, specifies that cases when coordinates of object during its movement do not fulfill to state commonness criterion (so called losses of state commonness) depend exclusively on natural properties of object and follows from the equations describing it. It is possible to judge a linear dependence of columns and rows of SCC matrices on its rank. In the given case rank definition of  matrix is not complicated because control variable  is not present in the evaluation formulas. Thus, the qualitative analysis of system movements is not at a loss. Therefore, at SCC definition for introduced control objects of the second order it is not necessary to know the law of change of  and make supposition concerning what class o functions it belongs to. When solving a problem of optimal control on speed it is supposed that control variable should belong to a class of piecewise functions. It is important to note that for the objects of the second order SC criterion coincides with the controllability conditions suggested in [5].

     2.3. State commonness criterion for control objects of the third order

     In this paragraph we briefly review the main results obtained for affine nonlinear systems of the third order. State commonness matrix for the above mentioned objects in terms of Lie algebra becomes:

                              ,            (16)

where Lie commutators of the second order are defined as follows:

                  . (17)

If  (case when vector fields  and  commutate) expression (16) can be written:

                   . (18)

State commonness criterion for systems of the third order will be requirement  () for any moment of time.

     2.4. Example. For control system

it is required to carry out SC analysis. Such mathematical models are widely applied to exposition of dynamical objects and processes in different fields of industry [2]. With the purpose of simplification we assume, that does not influence on generality of analysis. Vector functions of the accepted object description look as follows:

.

Functional vectors of SCC matrix are defined, according to (3), (14), as:

Block SCC matrix (15) composed from the obtained vectors, has the following notation:

.

Using this matrix presentation, function of a determinant  is evaluated and there is a solution of the equation  concerning variable , defining special lines:

.

     On fig.1 the graph of function of a SCC matrix determinant, on fig.2  – the graph of a solution of the equation are represented.

       

                Figure 1. Graphs of                                  Figure 2. Graph of the solution

                       and zero plane                                              concerning

     3. CONCLUSION

     In this paper analytical expressions for state commonness condition matrices are derived using differential geometry method. State commonness criterion for affine nonlinear systems of arbitrary order is stated on the base of vector fields commutators (Lie brackets). Particular criterions for the systems of second and third order are defined. Cases when vectors forming state commonness condition matrices do not include control variable are revealed. These cases essentially simplify the qualitative investigation of nonlinear systems behavior.

     4. REFERENCES

     [1] Methods of Classical and Modern Automation Control Theory: Textbook in 3 volumes. Vol. 3: Methods of Modern Automation Control Theory. / Under the editorship of N.D. Egupov. Moscow.: Publishing house of Bauman MGTU, 2000 (in Russian).

     [2] V.A. Oleinikov. Optimal Process Control in Oil and Gas Industries. Leningrad: Nedra, 1982 (in Russian).

     [3] F. Warner. Fundamentals of the Smooth Manifolds Theory and Lie Groups. Moscow.: Mir, 1987 (in Russian).

     [4] Yu.N. Andreev. Differential Geometry Methods in Control Theory. // Automation and Telemechanics, 1982. № 10. PP. 5-46 (in Russian).

[5] R. Hermann, A.J. Krener. Nonlinear Controllability and Observability. // IEEE Transactions on Automatic Control, Vol. AC-22. No. 5, October 1977. PP. 728-740.