Decomposition Problems in Dynamics of Complex Mechanical Systems

Copyright © IFAC Control of Industrial Systems. Belfort, France, 1997

DECOMPOSnnONPROBLEMffi IN DYNAMICS OF COMPLEX MECHANICAL SYSTEMffi

Lyudmila K.Kuzmina

Kazan Aviation Institute Adamuck, 4-6, Kazan-15, 420015, RUSSIA Abstract: This paper deals with the different aspects of mathematical modelling and the analysis of complex dynamic non-linear systems, primarily as a consequence of applied problems in mechanics (in particular those for gyrosystems, for stabilization and orientation systems and for control systems of movable objects, including aviation and aerospace systems). Non-linearity, multi-connectivity and the high dimensionality, that occur at the onset of dynamic problems, lead to the need of problem narrowing, and of the decomposition of the full model, but with safe keeping on the main properties and of qualitative equivalence. The generalization of reduction principle and the elaboration of regular methods for modelling problems in dynamics are the main aims of the investigations. Keywords: stability, decomposition, singular.

I.INlRODUCTION (1)

Here the initial systems (called full systems, FS) are considered as systems, that may be analyzed into subsystems in accordance with some criteria. Main aims: to replace the investigation of FS to the investigation of subsystems; to elaborate the regular methods for decomposition of FS on base of the structure analysis of the state equations; to establish the conditions of the qualitative properties decomposition. The principal problems: when the investigation of FS may be reduced to the shortened system; and how it is possible to get the correct simplified system (as the comparison system).

my be guaranteed by the corresponding properties for subsystems (a), (b). Here

X

= Ilx I' x.2

r

is state vector for FS of

Ilx

r

corresponding dimensionality; X = J' X.2 IS vector-function with the corresponding properties in considered area.

Similar problems and first statements were fonnulated and in detail investigated by Poincare (1947) and by Lyapunov (1956).

For solving the property of quasi-stability was introduced, as sufficient property.

In investigations of Lyapunov(1956) the comparison method, as general qualitative analysis method, was developed and was strictly substantiated in solving of the stability problems. This method led to the reduction principle, that is well-known one in stability theory (Malkin, 1942; Lyapunov, 1956). In generalized statement the problem of stability property decomposition was considered by Persidsky (1951). In particular, when the stability property for FS

But the problem of the separation on the subsystems by regular methods is very important (Voronov, 1985; Siljak., 1991) in case of complex systems. In this research the initial systems are treated as the systems of the singularly perturbed class (Kuzmina, 1986). The state of these systems are covered by the equations with big and small parameters. The mathematical model is represented as singular model with the small parameters in different powers. The special non-linear, non-singular, evenly-regular 665

under which conditions it is possible to reduce system (4) to the s-system (5) ? Such problems for the equations with small parameter before derivatives were considered by many authors (Tikhonov, 1952; Gradstein, 1953; Chetayev, 1957; Krasovsky, 1961; Razumikhin, 1963; Grujic, 1981; Kuzmina, 1982;... ). Here for solving this problem, following the Chetayev's method (Chetayev, 1957), it is necessary to introduce the deviations (as new variables):

transformation (L yapunov, 1956; Lefschetz, 1961) may be constructed. The decomposition problems of dynamical properties are solved.

2. INITIAL POSmONS Asymptotic approach, combining the methods of the perturbations theory and of the stability theory, is developed. Such manner allows to detour many features in the singular systems theory and to solve the main problems.

K =

Let in accordance with the applications to the mechanics problems the initial mathematical model is Lagrange's equations

dq

-=q dt

e

Here y (4);

(2)

-

= yet ,j.i ,y)

11

(3)

KII < &

(for all t > to),

Here & >0 is any small numbers, given in advance. Further these methods permit us to receive the estimate for J.i (Chetayev, 1957).

3. SOME APPLICATIONS

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3.1 Mechanical systems with big parameters (fast rotors).

Using developed method, here the gyrostabilization systems are considered. Assuming the gyroscopes rotors are fast the simplified models are obtained. Let the differential equations for such systems are represented (Kuzmina, 1982) in the form

a, - nonnegative members, 0 ~ a, ~ r; E are identity matrices. Taking into consideration in (4) only the members containing J.i in power no greater than s (0 ~ s < r), it will be possible to get the shortened systems

= Y,(t,j.i,y)

(6)

and to determine the conditions, with which

The equations (4) are studied, where

dy MsCj.i) dt

dK

M(j.i)"'d( = K(t,j.i,K)

the initial system (2) is reduced to the standard form of the singular equations

M(j.i) dy dt

= y(t, J.i) is the solution of the initial system yS = yS(t, J.i) is the solution of the simplified

system (5). Using the Lyapunov's methods it is possible to investigate the differential equations for these variables

Here q is k-dimensional vector, k is number of the freedom degrees of the initial system. In accordance with the using manner, supposing that (2) is the model of the singularly perturbed object, the small parameter J.i >0 is introduced in (2). Further with the help of the corresponding transformation of the variables (q, qe) ~ y

y_yS

o e e -d aq M + (h + g 0) q M = Q'M + Q"M dt

d dt

(5)

(5=0,1,2,..., r-1)

Lq£e + B OqJe + RO q£e

=Q'£ + Q"£

dqM =qM' e Q'M =110, A Oq£, e ---;j["

The equations (5) are called the simplified systems of s-level (s-systems). For the singularly perturbed systems, that are considering here, the order of system (5) is lower than order of initial system (4). In applications to mechanics these shortened systems lead to the simplified models (as the asymptotic models of s-level). It is possible to obtain the sequence of asymptotic models (as set models) in mechanics, corresponding the sequence of these s-systems (s=0,l.2...,r-1). The problem (important for the theory and applications): in which cases and

Q£' - - 11 (j) 0 ql'

n

(7)

-

0 C q4

liT '

q£, 0liT

0 e

Here in accordance with applied theory it is accepted (Merkin, 1956) that g=g"H, H=l/J.i J.i>0 is small parameter. Here the necessary transformation of the variables is

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was marked). This result gives the strict substantiation of a simplified model use, corresponding to precessional system in critical case. With the transition to this model the stability properties are kept and there is a proximity of solutions in an infinite time interval. and the system (7) in new variables is

dz

dT

= Z(T,Ji,Z,X)

dx M(Ji) dT

Theorem 2. If

(8)

half-plane

I-level

leads

and

equations

lao A + bO +II =0

x=llx j ,x.?,X3 r,a/=2, a2=1, a3=O, of

and all

roots (except m zero roots) of the characteristic equation for limit system (10) are placed in the left

= P(Ji)x + X(T,Ji,Z,X)

The shortened system precessional model

Igo 1* 0, Igo [:.: * 0

IU+ It +!:il =0,

satisfy the Hurwitz conditions

then with sufficiently big H-values from zerosolution stability of the limit system (10) the zerosolution stability of the full system (7) follows and for preassigned positive numbers ~ , TJ , y (~ and r can be taken as small as one wishes), there is such a H..-value, that with all H> H•• in disturbed motion for all t ~ to+ Y

to

(9)

The shortened system of G-levelleads to limit model (it's new model for gyrosystems):

(by "**,, index the solution of limit system (10) was marked).

The possibility of use of simplified system more simple than precessional model was pointed before (Merkin, 1956). This approach allows to obtain these models by strict means as asymptotic models of corresponding levels. These shortened models get obvious "physical" interpretation in application to mechanics. The conditions of the models correctness in critical case may be found by the stability theory methods.

Theorem 1.

If

IgOI * 0,

Ig,~I:::: * 0

These conditions are determined, in which a transition to a limit model in analysis of mechanical systems with fast gyroscopes is permissible; the decomposition of stability property is shown. Remark. These results supplement and generalize already known in the gyroscopes theory. Shortened models are asymptotic models corresponding of levels, which have (u+n/2) and (n+u)/2 of freedom degrees.

and all

roots (except m zero roots) of the characteristic equation for precessional system (9) have negative real

parts

and

equation

System (8) belongs to the specific critical case (Lyapunov, 1956), when all eigenvalues of matrix, corresponding to the fast variables x I , are imaginary. Well-known results of theory of singularly perturbed systems (Tikhonov, 1952; Grujic, 1981) are non-suitable here.

lao A + bO + gOI = 0

satisfies the Hurwitz conditions then with sufficiently big H-values from the zero-solution stability of precessional system (9) the zero-solution stability of full system (7) is followed; and for preassigned numbers; > 0, TJ> 0, y>O (; and r can be taken as small as one wishes), there is such a Hr-value that with all H> H. in disturbed motion for all t ~ to+ Y

3.2 Mechanical systems with the small parameters (small constants of time).

Let us consider the gyrostabilization system of type (7) without writing and denoting it (11). Let the transition processes in electric follow-up systems are quick-response. Therefore let assume, that

is small parameter. In this case the corresponding of variables transformation is constructed:

(by index "*" the solution of precessional system (7)

667

boundary of stability domains. In these cases the direct use of known results of the singular perturbations theory are non-suitable. Here initial systems are non-Tikhonov's systems: eigenvalues of corresponding matrices are zero and imaginary. These results summarize the available ones for the theory of singular perturbations in critical cases and solve the mathematical modelling problems in mechanics of singular systems (in the theory of oscillations, the gyrosystems, the stabilization systems).

In these variables the system (11) has the form of type (8) where a,=l, a2=2, a;=O. . In accordance with the above adduced results it is possible to obtain here (for the electromechanical system with the quick-response, small-inertial electrical circuits) two types of simplified models:

~ aq~ + (b o + gO)q~ = Q~ + Q~

For all considered systems there exists the necessary variables transformation, that allows to lead the initial mathematical model (2) to the standard form of (4); to obtain the shortened models on elaborated scheme (as set models, correct in dynamic problems); to get the acceptability conditions; to decompose the system and the initial problem.

(12)

Rq~ = Q~ + Qi', d~; = q~ and

(b + g)q~

= Q~ + Q~ Rq ~ = Q~ + Q~ , dq =q ~

REFERENCES

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Chetayev, N.G. (1957). On estimations of approximate integrations. Appl.Math. and Mech., vo1.21, No.3, 419-421. Gradstein, I.M. (1953). On solutions on time line for differential equations with small parameters. Mathem.Rev., vo1.32, No.3, 533-544. Grujic, Lj.T. (1981). Uniform asymptotic stability of non-linear singularly perturbed general and large-scale systems. Int. Journ. Cont., 33, No.3, 481-504. Klimushev, A.I., Krasovsky, N.N. (1961). Uniform asymptotic stability of systems of differential equations with small parameters. Appl. Math. and Mech., vo1.25, No.4, 680-690. Kuzmina, L.K. (1982). On some properties ofsingularly perturbed systems in critical cases. Appl. Math.and Mech., vo1.46, No.3, 382-388. Kuzmina, L.K. (1986). Stability theory methods and singularly perturbed systems. Papers abstract of 6-th Union Congress on Theoret.and Appl Mech.. 398, FAN, Tashkent. Lyapunov, AM. (1956). The general problem of motion stability. ColI. of papers, vol.2, 7-264, USSR AS, Moscow. Malkin, I.G. (1942). Some theorems of motion stability theory in critical cases. Appl Math and Mech., vo1.6, No.6, 411-448. Merkin, D.R. (1956). Gyroscopic systems. Gostechizdat, Moscow. Persidsky, K.P. (1951). Some specific cases in systems. News of Kaz .AS, Math. and Mech., No.S, 3-24. Poincare, H. (1947 On curves, that are determined by differential equations. Mir, Moscow. Razumikhin, B.S. (1963). On solutions stability of differential equations with small multipliers under derivatives. Sib. Math. Journal, voI.4, No.l, 206-211. Siljak, D.D. (1991). Decentralized Control of Complex Systems .Ac.Press, New York-Boston. Tikhonov, A.N. (1952). Systems of differential equations with small parameters under derivatives Math. Rev.. vo1.31, No.3, 575-586. Voronov, AA (1985). Introduction in dynamics of complex control systems. Nauka, Moscow.

M

dt

The conditions of the admissibility of these simplified models may be to get by same method based on ideas of Chetayev (with the estimate of 11" values).

Remark. The system (12) and the system (10) are basicly the different simplified models, with the different conditions of their acceptability: (10) is SM of O-level on l1-parameter, (12) is SM of O-level on I1rparameter. 3.3 System with the small gyroscopes. Here let us suppose, that mass and the inertia moments of gyroscopes and their mountings are smaller in comparison with ones of stabilized platforms. In accordance with this in (11) 112 small IS introduced; the necessary parameter transformation of variables is constructed;' and the simplified model (limit model on 112 parameter) is obtained. It is new model. Also the conditions of acceptability are new, different from known ones (Merkin, 1956).

4. CONCLUSION Used method allows to organize the decomposition schemes of the initial systems and to obtain the admissible shortened systems as comparison ones. For systems with the peculiarities (bad stipulated matrices; the critical spectrums) the conditions are determined, when the qualitative investigation of singularly perturbed systems is reduced to the investigation of shortened systems. With the Lyapunov's methods the research deals with the special cases, when unperturbed subsystems are on

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