On The Second Order Time-Varying Systems

On The Second Order Time-Varying Systems

Vanita Jain B. K. Lande

Professor, Principal,

Bharati Vidyapeeths College of Engineering, Shah and Anchor Kutchhi Engineering College Paschim Vihar, New Delhi-110063. Chembur, Mumbai 400 088

Abstract: This paper presents a method to find the solution of second order time-varying autonomous systems. The methodology uses well defined Cayley-Hamilton theorem widely used to find the solution of linear time-invariant systems. To demonstrate the methodology a general form of second-order time-varying system with periodic coefficients is considered.

Key Words: Periodic, Autonomous Systems, Time-Varying

1. INTRODUCTION

   Most human activities involve vibration in one form or another. For example, we hear because our eardrums vibrate and see because light waves undergo vibrations. Breathing is associated with the vibration of lungs and walking involves (periodic) oscillatory motion of legs and hands. We speak due to the oscillatory motion of larynges (and tongues) [1, 2].Early scholars in the field of vibration concentrated their efforts on understanding the natural phenomena and developing mathematical

   theories to describe the vibration of physical systems.

   In this paper, we are extending the well-defined Cayley Hamilton theorem used for linear time-invariant systems [11] to calculate the solution of autonomous systems whose parameters are periodic in nature. To illustrate the methodology, a general second order periodic system is considered however, the presented method is applicable to any order of periodic system.

2. METHODOLOGY

   Consider the autonomous system

   In [3], the phenomenons, the revolution of Moon around the Earth, the rotation of Earth around its axis,

   x t At x t

   (1)

   the swinging movement of the pendulum of the clock, the wheel of a moving car, an engine in the working order, the effect of alternating current etc are described as almost periodic motions and Floquet theorem is used to find the fundamental matrix of homogeneous linear systems with periodic coefficients. In [4], many general classes of methods have been described to compute the exponential of a matrix but the systems considered are defined by linear constant coefficient ordinary differential equations. Some formations to find the state

   where At is a continuous periodic 2 2 matrix

   function of t; i.e., when there is a constant T 0 such that At T At for every t . When this

   condition is satisfied, we say, more precisely, that

   At is T periodic. If T is the smallest positive number

   for which this condition holds, we say that T is the minimal period of At

   transition matrix of continuous time-varying systems are given in [5,6,7,8]. In [9], Peano Baker series method

   Let

   At a1 cost a2 cost

   a cos t a cos t

   is given to define the transition matrix of second order time-varying system but it says that computation of

   3 4

   (2)

   solution via Peano Baker series is a frightening prospect, though calm calculations is profitable in the simplest cases. For an overview of recently developed numerical algorithms for the analysis and design of periodic systems see [10 and references therein].

   where a1, a2 , a3 and a4 are constants and cost is time-varying factor. For the simplicity in calculations let a2 is equal to zero.

   The solution of the time-varying varying system (1) under the zero initial conditions is

   x(t) (t, 0)x(0)

   (3)

   The solution of (1) under zero initial conditions is

   where

   (t, 0)

   is the state transition matrix of (1).

   3

   3

   exp(a1 sin t) 0

   According to Cayley-Hamilton technique

   x(t) a

   exp((a1 a4 ) sin t) exp(a1 sin t) 1 exp(a4 sin t)

   (t, 0) eB where

   tt aa ssinin tt aa sin t

   (a1 a4 )

3. EXAMPLE

   Example 1. Consider the system (1) with [12]

   (4)

   B A( )d 1 2

   0 a3 sin t a4 sin t

   6t2

   3t5 ,

   t 0

   with zero initial

   Using Cayley-Hamilton technique, we say

   A(t) 0

   3t2

   1 2

   1 2

   eB I B and

   ei for i 1, 2

   conditions.

   Thus (t, 0) eB where

   1 2 i

   t6

   t 2t3

   Solving for 1 and 2

   B A( )d 2

   0 0 t3

   det(I B) det a1 sin t

   a sin t

   a2 sin t

   a sin t

   Using the Cayley-Hamilton technique where we assume

   3 4

   that

   ( a1 sint)( a4 sint) 0

   eB B

   1 2

   1 a1 sin t

   a sint

   ei

   for i 1 and 2

   2 4 1 2 i

   Solving for 1 and 2

   Solving for and

   1 2

   a1 exp(a1 sin t) (1 exp(a sin t)) a4 exp(a1 sin t)

   2t3 and t3

   1 a a 1 a a 1 2

   1 4 1 4

   2

   2

   exp((a1 a4 ) sin t)

   Solving for and yields

   (a1 a4 ) sin t

   1

   t3 2t3

   2

   3 3

   3 3

   et e2t

   r s

   r s

   (t, 0) I B p q

   1 2e e

   , 2 3

   where

   1 2

   Then

   (t, 0) e

   2t3

   t

   (e

   (e

   t3 t3

   2

   e2t3

   )

   )

   3

   3

   p exp(a sin

   0

   et

   1 t)

   q 0

   a3

   Using (3), the solution of the given system under zero initial conditions is

   3

   r exp((a a ) sin t)

   (a a ) 1 4

   e2t3 t (et3 e2t3 )

   1 4 x(t) 2

   3

   3

   s exp(a1 sin t) 1exp(a4 sin t)

   0

   et

   Example 2: Consider the system (1) with

   0 2t

   0 2t

   A(t) 1 0 ,

   conditions.

   B

   B

   Thus (t,t0 ) e

   t 0

   and non-zero initial

4. CONCLUSION

   The paper presents a method to find the solution of autonomous second order time-varying systems. The method has been described for general second-order systems though the approach in general can be applied to system of any order. The application of the method has been demonstrated through examples.

   t t t 0

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   0

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   where

   B A( )d

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   t 0

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