Synthesis of multivariate control systems by objects with inexact data

Yunicheva N.R.

e-mail: naduni@ mail.ru

Institute  of Informatics and Control Problems,

Almaty, Kazakhstan

 

   For the decision of tasks of the analysis and synthesis, research of quality indicators of intellectual control systems interval and indistinct methods [1] are used.

 Application of the given methods is caused by that discrepancies of the data in parameters of object of management or, otherwise, uncertainty have a statistical property.

 In given article the decision of task of parametrical synthesis for multivariate intellectual control systems of objects with the inexact data is carried out with the help of a method [2] and methods of the interval analysis. The computing algorithm in C ++ is developed.

 

1 Statement of task

Let's assume, that the mathematical model of multivariate uncertain object of control looks as follows:

,                                         (1)

 where - a n-dimensional vector of conditions,  - a m-dimensional vector of control actions,  – interval matrixes of control object.

The purpose of synthesis task we shall count definition of an interval matrix  in the equation of  feedback

,                                                  (2)

providing reception of desirable dynamic properties in the closed control system it will be presented in the following kind:

,    (3)

.where – an interval matrix of the closed control system.

The interval characteristic polynomial of the closed system will look like:

 

,          (4)

where  - interval numbers of a characteristic polynomial.

The algorithm of the task decision in view of synthesis will consist of the following steps:

Step 1:

Decision of parametrical synthesis task for multivariate stationary object of the management belonging to family (1).

Before directly to proceed to the decision of a task in view of synthesis, from family of mathematical models (1) we shall allocate mathematical model with the fixed values of elements of matrixes , i.e. mathematical model of the following kind:

 

                                        (5)

where and – constant matrixes. For model of a kind (5) equation of a feedback will be presented as follows:

 

                                                 (6)

 where K – the matrix of adjusted parameters, which definition will answer the decision of a problem(task) of synthesis for stationary  control object, carried out on a technique suggested in [3].

 Step 2:

Decision of parametrical synthesis task for multivariate object of management on the basis of a method of common parameter and the matrix found above K  as initial approximation the decision of synthesis task of control we shall increase the first column of a matrix K  on the common parameter β:

 

  or

Then instead of the equation (2) we shall write down:

                                                 (7)

The characteristic polynomial of the closed control system with the account (7) will be written down in the following kind:

                        (8)

 β. also it should be equal to a desirable characteristic polynomial (4). Equating factors at identical degrees λ expressions (9) and (4), we shall receive system of the linear algebraic interval equations for definition of parameter β.

Let's consider the equation of the received system which looks like:

 

     ,                                     (9)

where   - the interval sizes dependent on elements of matrixes  and K .

By formal disclosing brackets and reduction similar composed we shall receive expression:

                                           (10)

According to the subdistributive law of interval mathematics for set β we have:

 

,                                              (11)

The parameter  will be found from expression:

 

,                                     (12)

The found value  is substituted in:

 

,                                           (13)

Thus, received in view of found  factors of the characteristic equation of the closed system will be included in factors of the desirable characteristic equation. The system of the equations (13) according to a synthesis algorithm will be transformed to a kind:

,                                                 (14)

         The opportunity of such transformation is determined with the help of operation of accommodation by summation [3].

The decision of the received system (15) is given with the following theorem: the valid parameter  is the decision íåâûðîæäåííîé systems of the linear algebraic equations in only case when  it represents the decision of system of the equations:

,                                                (15)

also satisfies to restrictions:

,                                                  (16)

 where  the centers and  lengths of interval numbers .

         From (16) sequences of parameters  found on a condition it is chosen the minimal value of the adjusted parameter.

 

References

1.      Jolen  L., Kiefer Ì., Didri Î. Walter E.. The applied interval analysis. M.: Institute of computer researches. 2007. – 467p.

2.     Ashimov A.A., Syzdykov D.Zh. Identification by the common parameter method: Reference book on the theory of automatic control / Edited by Krasovsky A.A. – M.:Science, 1987. –P.263-271.

3.     Khlebalin N.A. Modal Control of Plants with Uncertain Interval Parameters, in: Proc. Intern. Workshop «Control System Syntesis: Theory and Application», Novosibirsk, 1991. -P. 168-173.