The frequency method of analysis of stability of the multidimensional automatic control system.

Kramar, Vadim

1. INTRODUCTION

The linear multidimensional control systems are the part of the linear stationary systems class. That is why their theory is based, by facts, on the same basis that the theory of one-dimensional systems. At the same time the multidimensional systems class in comparison with the one-dimensional systems class requires further evolution of the linear systems theory because of the particular mathematical research problems (Doyle, 1992). There are no such problems in one-dimensional systems class. Such general specific problem is the one of the system characteristic polynomial, which determines, with combination of the other description elements, the dynamics of the great number of processes on its outputs. The core of a problem is in that polynomial is to be formed so that it is possible to determine a quantity of the significant image poles of the system outputs (Gasparjan, 2008).

This article presents an approach to building a characteristic polynomial of the multidimensional control system in case of unstructured model.

2. INFORMATION

2.1 System model

The mathematical model of multidimensional automatic control system is to be viewed. This system can be presented as structured model. There is an input/output description of the system is presented:

y(s) = [PHI](s)u(s), (1)

where y [member of] [R.sup.n], u [member of] [R.sup.n] are the output and the input vectors correspondingly, and [PHI](s) is a rational transfer function (the transfer function matrix) of the system, which is given in the form of:

[PHI](s) = [[I + W (s)].sup.-1] W(s) = W(s)[[I + W(s)].sup.-1], (2)

i.e. in the form of the system, which is shown on Figure 1.

W (s) is the matrix of the forward path transfer functions of n x n dimension.

W(s) = {[w.sub.ij](s)}, [w.sub.ij](s) = [p.sub.ij] (s) / [q.sub.ij] (s), i, j = 1,2, ...n. (3)

[FIGURE 1 OMITTED]

Let's present the W(s) matrix in the form of:

W(s) = [Q.sup.-1](s)[PI](s). (4)

Q(s) is the matrix presented as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5)

where [q.sub.i](s), i = 1, ..., n is the least common multiple of the characteristic polynomial [q.sub.ij] (s), j = 1, ..., n from equation (3).

[q.sub.i](s) = [c.sub.ij] (s)[q.sub.ij] (s), i, j = 1, ..., n. (6)

The matrix [PI](s) is given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (7)

where

[[pi].sub.ij] (s) = [c.sub.ij] (s) [p.sub.ij] (s), i, j = 1, ..., n. (8)

Then for matrix [PHI](s) there is a right presentation in the form of:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (9)

It is need to get the requisite and sufficient conditions of stability. For that the presentation (10) is introduced (Gantmacher, 1977):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (10)

where C(s) is a left greatest common divisor of matrixes, and [??](s) and [??](s) are mutually distinct.

2.2 Stability analysis

Let's present:

q(s) = det Q(s), (11)

the characteristic polynomial of open-loop system.

By denotation of det C(s) = [D.sub.n] (s) it is possible to write characteristic polynomial of the system in the form of:

[delta](s) = det[Q(s) + [PHI](s) / [D.sub.n](s)]. (12)

The r is a number of all right roots of polynomial q(s), d is a number of all right roots of polynomial [D.sub.n](s), N = deg q(s), v = deg [D.sub.n](s) (Coffey T.C. & Williams I.J., 1968). In view of equation (11) the equation (31) can be written in the form of:

[delta](s) = -q(s) / [D.sub.n] (s) det[I + W(s)]. (13)

If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] then:

deg [delta](s) = N - v. (14)

Therefore the frequency stability criterion for the polynomial [delta](s) is given by:

1. [delta](j[omega]) [not equal to] 0, [for all] [omega]; 2. [DELTA] arg [delta](j [omega]) [|.sup.+[infinity].sub.-[infinity]] = [pi](N - v). (15)

But at the same time

[DELTA] arg [delta](s) = [DELTA] arg q(s) - [DELTA] arg [D.sub.n] (s) + [DELTA] arg det[I + W (s)], (16)

In which connection

[DELTA] arg q (s) [|.sup.+[infinity].sub.-[infinity]] = [pi]N - 2[pi]r, (17)

[DELTA] arg [D.sub.n] (s)[|.sup.+[infinity].sub.-[infinity]] = [pi]v - 2[pi]d. (18)

By computation of argument increment of second condition in equation (15) with help of equations (16), (17) and (18) it can be found that:

[DELTA] arg det[I + W(s)][|.sup.+[infinity].sub.-[infinity]] = 2[pi](r - d). (19)

Thus the equation (38) is equal to the second condition of equation (15).

At the same time the number of all right roots of polynomial [delta](s) with condition of [delta](j[omega]) [not equal to] 0 is equal to:

k = r - 1 / 2[pi] [DELTA] arg det[I + W(j [omega])][|.sup.+[infinity].sub.-[infinity]]. (20)

By writing of characteristic polynomial of system in the form of

[delta](s) = g(s)det[I + W(s)], (21)

where

g(s) = q(s) / [D.sub.n] (s) (22)

is the "reduced" characteristic polynomial of the open-loop system.

If taking into account the response ratio, r is a number of all definitely right roots of the polynomial q(s), d is a number of all definitely right roots of the polynomial [D.sub.n] (s), p is a number of all definitely right roots of the polynomial g(s), then:

[rho] = r - d, (23)

since the multitude of the roots of the polynomial [D.sub.n] (s) is included in the multitude of the roots of the polynomial q(s) (or r = [rho] + d).

Thus it comes in that with q(j [omega]) [not equal to] 0, [for all] [omega], det[I + W([infinity])] [not equal to] 0, W([infinity]) [not equal to] [infinity], for the concerned system to be steady it is necessary and sufficient the fulfillment of the two conditions:

1. det[I + W(j [omega])] [not equal to] 0, -[infinity] < [omega] < + [infinity], (24)

2. [DELTA]arg det[I + W(j [omega])][|.sup.+[infinity].sub.-[infinity]] = 2[pi][rho]. (25)

3. CONCLUSION

The method is presented in this article allows to get a mathematical model of the structured multidimensional automatic control system presented in the form of the system with single degenerative feedback. The frequency approach to analysis of the system stability based on analysis of frequency locus function is offered for the obtained model. The use of offered approach allows carrying out the stability analysis, including the discrete-continuous systems, whose mathematical model is presented with the special approach to the type species (Kramar & Kirilova, 2009).

The specified approach allows carrying out the analysis and synthesis of discrete-continuous automatic control systems after finding their equivalent single-cycle model (Kramar, 2008).

4. REFERENCES

Doyle, J.C. (1992). Feedback Control Theory, Macmillan Publishing Company, New York

Gantmacher, F.R. (1977). Teorija matrits, Nauka, Moscow

Gasparjan O.N. (2008). Linear and Nonlinear Multivariable Feedback Control: A Classical Approach, John Wiley

Kramar V.A. (2008) Design of Equivalent One-Time Model of Multivariable Multi-Stage Control System. Applied Computer Science Vol.4, No1, 2008, pp: 1-17

Coffey T.C. & Williams I.J. Stability Analysis of Multiloop, Multirate, Sampled Systems. AIAA Journal, Vol.-4, December 1968, pp: 129-144.

Kramar V & Kirilova O. (2009) Matematicheskaya model mnogomernoy sistemy avtomaticheskogo upravleniya v forme sistemy s edinichnoy otricatelnoy obratnoy svyazy. Zagadnienia Teoretyczno--Eksperymentalne w Przemysle Maszynowym 2009, pp: 158-161