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Fifth International Conference on Dynamic Systems and Applications
May 30 - June 2, 2007
Morehouse College
Atlanta, Georgia, USA |
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Organizers
M. Sambandham, Morehouse College, IFNA
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Controllability of Matrix Second Order Systems - A Trigonometric Matrix Approach
by
Raju K George
Department Of Applied Mathematics, Faculty Of Technology And Engineering , M S University Of Baroda, Baroda 390001, India
Coauthors: J P Sharma
Abstract
Many of the real life problems are modelled as Matrix Second Order
Systems, (refer Wu and Duan[]). Necessary and sufficient
conditions for controllability of Matrix Second Order Linear(MSOL)
Systems have been established by Hughes and Skelton [].
However, no scheme for computation of control was proposed. In
this paper we first obtain another necessary and sufficient
condition for the controllability of MSOL and provide a
computional algorithm for the actual computation of steering
control. We also consider a class of Matrix Second Order
Nonlinear systems (MSON) and provide sufficient conditions for its
controllability. In our analysis we make use of Sine and Cosine
matrices and employ Páde approximation for the
computation of matrix Sine and Cosine. We also invoke tools of
nonlinear analysis like fixed point theorem to obtain
controllability result for the nonlinear system. We provide
numerical example to substantiate our results.
Key Words. Controllability, Matrix Second Order Linear system, Cosine and Sine matrices, Banach contraction principle.
1 Introduction
In this paper, we investigate the controllability property of the
system governed by a matrix second order nonlinear (MSON)
differential equation:
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d2x(t)
dt2
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+ A2 x(t) = Bu(t)+f(t, x(t)) |
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ü
ï
ý
ï
þ
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| (1) |
where, the state x(t) is in Rn and the control u(t) is in
Rm, A2 is a constant matrix of order n×n and B
is a constant matrix of order n×m and f:[0, T]×Rn→ Rn is a nonlinear function satisfying Caratheodory
conditions . The initial states x0 and y0 are in
Rn. The corresponding Matrix Second Order Linear (MSOL)
system is :
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|
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d2x(t)
dt2
|
+ A2 x(t) = Bu(t) |
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|
ü
ï
ý
ï
þ
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| (2) |
The system (2) has been studied by many researchers
because it can model the dynamics of many natural phenomenon to a
significantly large extent(refer Hughes and Skelton [],
Fitzgibbon []).
The system (1) is said to be controllable on [0, T] if
for each pair x0, x1 ∈ Rn, there exists a control
u(t) ∈ L2([t0, T];Rm) such that the corresponding
solution of (1) together with x(0)=x0 also
satisfies x(T) = x1.
We note that in our controllability definition we are concerned
only in steering the states but not the velocity vector, y0
in (1). A necessary and sufficient condition for the
controllability of the MSOL system has been proved in (Hughes and
Skelton []). They converted the second order system into
first order system and obtained controllability result. However,
no computational scheme for the steering control was proposed. In
this paper we prove another controllability result and also
provide a computational algorithm for the actual computation of
controlled state and steering control. We do not reduce the system
into first order and analyse the original form itself. We use
matrix Sine and Cosine operators to find the solution of the
systems (1) and (2). We employ Páde
approximation of matrix Sine and Cosine operators. Examples are
provided to illustrate the results.
References
- []
- P.C.Hughes and R.E.Skelton, Controllability and observability of linear matrix second
order systems, ASME J.Appl. Mech., vol. 47, pp 415-420, 1980.
- []
- W.E.Fitzgibbon, Global existence and boundedness of solutions to
the extensible beam equaiton, SIAM J.Math. Anal. 13, 739-745, 1982.
- []
- Y.L.Wu and G.R.Duan, Unified Parameter Approaches for observer Design in Matrix Second
order Linear Systems, International Journal of Control, Automation, and Systems, vol.3, no.2, pp. 159-165, 2005.
Date received: January 13, 2007
Copyright © 2007 by the author(s).
The author(s) of this document and the organizers of the conference
have granted their consent to include this abstract in
Atlas Conferences Inc.
Document # catb-62.