082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

20.4 State-space models 611
statevariableisrequiredasthesystemissecondorder.Inthiscasethevelocityofthe
masswillbechosen.Thevelocityofthemassisnotdirectlydependentonitsposition
and so the two variables are independent. Another possible choice would have been
dw
−w.Thismayseemaclumsychoicebutforcertainproblemssuchchoicesmay
dt
lead tosimplifications inthe statevariable equations.
Itiscustomarytousethesymbolsx 1
,x 2
,x 3
,...torepresentvariablesforreasons
that will become clear shortly. So,
x 1
=w
(20.9a)
x 2
= dw
(20.9b)
dt
Because of the particular choice of state variables, it is easy to obtain the first of
thefirst-orderdifferentialequations--thusillustratingtheneedforexperiencewhen
choosing statevariables.
Differentiating Equation (20.9a) gives
dx 1
= dw =x
dt dt 2
This isthe first stateequation although itisusually written as
ẋ 1
=x 2
where ẋ 1
denotes dx 1
. The second of the first-order equations is obtained by rearranging
Equation
dt
(20.8):
d 2 w
= − K dt 2 M w − B dw
+ 1 f (t) (20.10)
M dt M
However, differentiating Equation (20.9b) we get
d 2 w
dt 2 =ẋ 2
Then, usingEquation (20.10) weobtain
ẋ 2
=− K M x 1 − B M x 2 + 1 M f(t)
Finally, itisusual toarrange the stateequations inaparticular way:
ẋ 1
= + x 2
ẋ 2
=− K M x 1 − B M x 2 + 1 M f(t)
w=x 1
Note that it is conventional to relate the output of the system to the state variables.
Assume that for this system the required output variable is the position of the mass,
that is w.
Itisstraightforward torewrite these equations inmatrix form:
) ( )( ) ( )
(ẋ1 0 1 x1 0
=
+ f(t)
ẋ2 −K/M −B/M x 2
1/M
w = ( 1 0 ) ( )
x 1
x 2