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5-10-1 Conceptof State Variable
The choice of a set of variables to be designated as state variables for a given
systemis somewhat arbitrar}r.ln other words, there is no unique method ofdefining
what should be a set of state variables for a given system. However, the state
variables have'to meet some requirements, which can be stated as follows.
1) The variables selected as state variables must be linearly independent, i.e., it
should not be possible to express any one or more of the state variables in terms of
the remaining state variables. Mathematically, ifx is an n dimensional vector with
components xt, r -. 1, n, then the components x, are said to be linearly independent
if atxi 7/0, for all a/ 7/- O and all x, +0. Here, cy/, A, e - 1, and n are arbitrary
constants.
2) Given all the initial conditions, the input for t > 0, and the solution of the
governing differential equation in terms of the selected state variables, one must be
able to describe uniquely any physical parameter (state and output) of the system
for all t > 0. In other words, if any of the physical parameters of the system
cannot be described in this manner, then the selected variables do not qualify to
be designated as state variables.
5.10.2 State-Space Representation
A state vector is a vector whose elements are the state variables satisfying the
above requirements. If n is the dimension of a state vector, then Lhe state-space
is an n dimensional space whose axes are the state variables. For example, if
XI, X2, and X3 are the elements of the state vector x, then n -. 3, and the state-space
is a three-dimensional space with xi, X2, and X3 as three axes.
508 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
The state equation is a set of n simultaneous first-order differential equations
involving n state variables and m inputs. Usually, m < n. The output equation is a
set of algebraic equations that relate the outputs of the system to the state variables.
For example,
where
A-
x - Ax + Bu
y = Cx + Du
a12
a22
a32
.
an2
a13
a23
033
.
an3
(5.171)
(5.172)
(5.173)
(5.174)
(5.175)
is a state-space representation of an nth order system. Here, the order of the
system is equal to the number of simultaneous first-order differential equations.
In Eqs. (5.171) and (5.172), x is the state vector of dimension n, A is the system
matrix of dimension n x n, B is an n x m input coupling matrix, u is an m x 1
input vector, y is the output vector of dimension q, C is a q .x n out)ut matrix,
and D is a q x m feed forward matrix. The term feed forward is used when a part
of the input directly appears at the output. A schematic diagram of the state-space
representation is shown in Fig. 5.40."v'
5.70.3 State Transition Matrix
Consider the homogeneous part (u =0) of state
given by
x - Ax
Eqs. (5.171) and (5.172) as
(5.176)
y = Cx (5.177)
The state transition matrix q>(t) is defined as a matrix that satisfies the equation
x(t) -. <l>(t)x(0) (5.178)
LINEAR SYSTEMS, THEORY, AND DESIGN: A BRIEF REVIEW 509
Fig. 5.40 Schematic diagram of state-space representation.
(t)
In other words, given the initial conditions x(0), the state transition matrix enables
us to predict the state vector at t > O.
Substituting for x(t) from Eq. (5.178) into state Eq. (5.176), we obtain
<P(t)x(0) - AQ(t)x(0) (5.179)
[<i>(t) - Aq>(t)]x(0) = 0 (5.180)
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