Page 736 - The Mechatronics Handbook

0066_Frame_C23 Page 44 Friday, January 18, 2002 6:02 PM

where the number of states is two (n = 2). If the position z(t) and velocity (t) of the mass are known
z ˙
at time t 0 , along with the applied voltage V z (t) defined for t ≥ t 0 , then the future behavior of the system
(i.e., the state x(t)) can be determined by solving the differential Eq. (23.100).
The Linear State-Space Equation and Its Solution
For a linear system, the evolution of the states of a system over time can be described by a set of linear
first order differential equations of the form:
dx 1 t()
x ˙ 1 t() = --------------- = a 11 t()x 1 t() + … + a 1n t()x n t() + b 11 t()u 1 t() + … b 1p t()u p t()
dt
dx 2 t()
x ˙ 2 t() = --------------- = a 21 t()x 1 t() + … + a 2n t()x n t() + b 21 t()u 1 t() + … b 2p t()u p t()
dt (23.103)
M
dx n t()
x ˙ n t() = --------------- = a n1 t()x 1 t() + … + a nn t()x n t() + b n1 t()u 1 t() + … b np t()u p t()
dt

2
where n is the number of states (or the order of the system) and p is the number of inputs. Defining
the input vector as

u 1 t()

ut() = u 2 t() (23.104)
M
u p t()

and the state vector x(t) as defined in Eq. (23.101), the set of first order differential equations given by
Eq. (23.103) can be rewritten in compact matrix form as [8, Chapter 2, section 2]

a 11 t() a 12 t() … a 1n t() b 11 t() b 12 t() … b 1p t()

x ˙ t() = a 21 t() a 22 t() … a 2n t() xt() + b 21 t() b 22 t() … b 2p t() ut()
M M O M M M O M (23.105)
a n1 t() a n2 t() … a nn t() b n1 t() b n2 t() … b np t()
= At()xt() + Bt()ut()

where A(t) is an n × n matrix and B(t) is an n × p matrix. For a system defined with q outputs y(t), which
are assumed to be a linear combination of the state x(t) and input u(t), we can write the output equation as

c 11 t() c 12 t() … c 1n t() d 11 t() d 12 t() … d 1p t()

yt() = c 21 t() c 22 t() … c 2n t() xt() + d 21 t() d 22 t() … d 2p t() ut()
M M O M M M O M (23.106)
c q1 t() c q2 t() … c qn t() d q1 t() d q2 t() … d qp t()
= Ct()xt() + Dt()ut()

2
Given a higher order differential equation, a set of first order differential equations can be obtained by a procedure
known as reduction to first order as presented in [9].

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