Conjugate Gradient Method For Solving Vibration Control Problems Based On Extended Systems

In real life, we often encounter such a situation: in order to achieve the desired results, we should do specific things. Actually, the physical processes in engineering usually can be controlled, that means we could follow our needs to achieve the results in different ways. Then there is one problem, how to achieve the desired results? For example, how to achieve the final destination with the minimum time; or how to achieve the purpose with the least energy, etc.System-control problems exist in all fields of engineering technology, simply to say, system-control problems are to determine the control scheme or control parameters under certain conditions, so that a process commence with the initial state X(t₀)at t = t₀ ,and arrives at the expected goals at the terminal timet = t_f. In terms of mathematics, system-control problems are to determine the input function of system by such a way that the objective functional is minimized. Vibration control is a very important branch of system control. There are many phenomena involved with vibration control either in life or engineering, such as the motion control of bridges or buildings in the external interference; control of transmission of the noise and the vibration control of cars, trains and even aircrafts.Iterative method is an important numerical method for solving such problems, based on the different iteration methods of determining the direction and step size; there has been a variety of iterative algorithms, such as gradient methods, parallel tangent method and conjugate gradient methods. Among them, the conjugate gradient method is widely applied, plays an important role in solving problems related with vibration control. However solving comprehensive problems concerned with both of differential and integral, is inevitable during the course of calculation. It really deserves to take into account how to compile programs and control algorithm accuracy for such a method.In this paper, a conjugate gradient method of extended system is proposed. With the introduction of transform, the method extends the previous system, which is relevant with both of differential and integral, to a differential system with given initial conditions, during the course, there is only a slight increase in size of solution. Conjugate gradient algorithm of extended system is a simple procedure. Furthermore we can ensure calculation accuracy of previous mixed problems as long as improving the accuracy of solution about the initial value problem.The extension is described as follows.The extension of the sub-problem of state equation is given by:The extension of the sub-problem of conjugate equation is:The extension of the sub-problem of sensitivity equation is:Besides the extension of the conjugate gradient method, we also summarize the steps of proposed algorithms. And then an inspection robot and a vibration control problem with two degrees of freedom are numerically calculated to verify the validity of the proposed method. They corresponding two extended systems are both calculated by Euler predictor corrector algorithm. The results show that the proposed method is effective.