Optimal control for elastic space constructions

Section 14-23 ZAKRZHEVSKII,A. S1135 zyxwvut Optimal control for elastic space constructions The problem of optimal control is considered for elastic constructions the s a m e as great Space constructions and rocket systems. Proper solution of the problem permits t o lover o r t o eliminate the negative inluence of elastacity o n behaviour of the construction as object of the control. Current methods used t o solve the problem are based o n creation of vibroisolation systems for elastic elements or of the optimal control that extinguishes one-two of first elastic modes, that do not guarantees nonperturbation of the rest elastic modes. Adequate accounting of elastic properties of the constructions in the criterion of optimization, recommended by the author, permits t o create the optimal control which damps all elastic modes of the controlled system. Modern space vehicles (SV) contain constructive elements (solar batteries, antenna etc) the elasticity of which at various maneuvers around of centre of masses should be taken into account in their mathematical models. For description of a motion of the SV main module it is enough to deal with mechanical model, containing a rigid body simulating a main module and attached t o it equivalent cores or plates simulating elastic elements of a construction. The determination of the eigen-modes and fundamental frequencies of equivalent elements is a separate problem, and we suppose it is solvable. Here optimization of control of the regime of reorientation of one or three axes of connected with main module basis in inertial space and carrying them into prescribed terminal conditions is one of practically important programm regimes of a motion. The terminal conditions may be as one- and biaxial orientation, the rotation etc. A typical example of such regime is EULERIAN turn of connected basis of a main module. Many of works devote to problems of control of the orientation for an absolutely rigid body. As a rule minimum of the flow of a working body, flow of energies or speed are used as criterion of an optimality. The solution of such problems appears usually in a class of disconnected functions. Such laws are unacceptable for control of an elastic SV, as they can excite the whole spectrum of elastic oscillations of system. It can appear unacceptable from the point of view of accuracy of fulfilment of the program. Following to the methods of analytical mechanics [l]we shall present the equations of a SV motion simulated by a rigid body with attached elastic elements as equation of motion around of pole, the equation of a motion of a pole about of centre of masses of system as a whole, the equations of a relative motion of elastic elements and kinematic equations. The most essential element at statement of a problem of optimal control is a choice of an optimizing functional. For considered class of problems the optimizing functional should be chosen according to the requirement of minimum of handicapes of orientation of a main module from oscillations of elastic elements. Strictly speaking, it is necessary to minimize in the right parts of the equation of SV motion around of a pole all terms, dependent on elastic displacements and their derivatives with respect t o time. In general case it is in A . LURIE'Snotations zyxw 00 C{qa[2(Aa . $ + 3 x Aa .L3) + a" x (G+ 3 x I&)] + &(23 Aa + 3 x 6")+ qaGa}. (1) a=l It is obviously that the problem of optimal control with such functional will be awkward. As the purpose of optimization of control consists in decrease of handicapes of orientation of a main module to allowable limits, it is possible to try to solve this problem for a more simple optimizing functional which nevertheless reflects the basic requirement to a motion of system. Mathematical modeling of behaviour of real system with found control will allow to give the answer on a question whether it is necessary to complicate criterion of an optimality. For a practically important case of slow angular motions (wi < 1) linear term C;='=, qaGa is the most essential term here. Starting from this we shall choose criterion of quality close t o the initial requirement on a physical essense and at the same time rather simple for the decision of a boundary problem, namely as - We shall consider further some typical problems of this class, the optimal turn of an elastic object about EULER axis located arbitrary with respect to body axes. The object contains two elastic elements which can bend S1136 ZAMM . Z. Angew. Math. Mech. 78 (1998) S3 and twist. The rotation occurs from state of rest to state of rest through an angle of the order of a radian. In assumption that N first modes of elastic oscillations must be considered correctly and N 1,. . . , 00 modes may be considered in quasistatic formulation the mathematical model described above reduces the optimal control problem for EULERIAN turn t o the form + Here z 1 is the angle of rotation of the main module around the EULERIAN axes, 22k+3 are the normal coordinates of the elastic displacements, R k are the normal frequencies of the partial elastic vibrations, wk are the weight coefficierits in the optimization criterion, ak, b k , dk are coefficients in the equations of motion determined by elastic modes, v is the controlling function, and 4J is the full rotation angle. In general case the problem is solved numerically. We next consider cases when the problem of optimal reorientation of one of the principal axes of an elastic object has analitical solution.It takes place for slow motion, when the quasistatic condition is satisfied beginning with the lowest frequency. The optimal control problem in this case takes the rather simple form zyxwvutsr /d zyxwvu zyxwvu T XI = 2 2 ; x2 = x 3 ; x, = 24; X I ( 0 ) = 4J;xi(O) = xj(T) = 0; x4 = v; J* = v2dT; (i = 2 , 3 , 4 ; j = 1,.. . ,4). In the absence of constraints the solution is 7 k=i-1 7 k=4 The analytical solutions of the problem for cases with a constraint on the angular velocity of the rotation S1 =I x 2 I -1 5 0 and with a constraint on the angular acceleration of the rotation S2 =( 2 3 1 -U 5 0 were constructed. For the optimal control problem with constraints on the phase variables the boundary-value problem was reduced to a nonlinear programming problem based on the parametrization of the solutions on multypoirit, boundary-value problems. In first case amount of parameters is 3, the parameters in our case are t l , t 2 and x 1 ( t l ) and are given by formulas In second case solution have 8 parameters which are found analitically also. The regions of accessibility of the state includes regions where the solution does not fall outside the constraints or is outside the constraints over a finite time interval, and also regions where the solutions does not exist. Aftcr analitical determining of the parameters solutions for all variables of problem are building easily. The significance of the constructed analytical solution consists not only in an opportunity easily to reveal various mechanical effects in behaviour of researched system but also that the given rather complex problem of optimal control can be test at definition of competitiveness of various numerical methods of the decision of problcrn of optimal control and also will allow simply t o construct successful initial approximation for a similar problem in which the conditions of a quasistaticness are not carried out for the lowest forms of elastic oscillations. This approach lightly can be applied for all cases of rotational and transfer motion of rigid body with inner degrees of freedom which are caused by elasticity, mobile liquid or carried mobile bodies. 1. References 1 LURIE,A.I.: Analitical Mechanics; Fizmatgiz, Moscow 1961. Address: DR. ALEKSANDR ZAKRZHEVSKII, Timoshenko Institute of Mechanics NAS of Ukraine, P. Nesterova st,r. 3 , Kiev, 252057, Ukraine