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

612 Chapter 20 Ordinary differential equations II
More generally, the standard form ofthe stateequations for a linear system isgiven by
ẋ(t) =Ax(t) +Bu(t)
y(t) =Cx(t) +Du(t)
For a systemwithnstatevariables,rinputs and poutputs:
x(t) is ann-component column vector representing the states of thenth-order system.
It is usuallycalled thestatevector.
u(t) is an r-component column vector composed of the input functions to the
system. It is usuallycalled theinputvector.
y(t) is a p-component column vector composed of the defined outputs of the
system. It is referred toas theoutputvector.
A = (n ×n)matrix, knownas the statematrix.
B = (n ×r)matrix, knownas the inputmatrix.
C = (p ×n)matrix, knownas the outputmatrix.
D = (p ×r)matrix, knownas the directtransmission matrix.
A,B,C andDhave constant elements if the system is time invariant. When presented
in this form the equations appear to be extremely complicated. In fact, the problem is
only one of notation. The general nature of the notation allows any linear system to be
specified but in many cases the matrices are zero or have simple coefficient values. For
example,thematrixDisoftenzeroforasystemasitisunusualtohave directcoupling
betweentheinputandtheoutputofasystem.Inthisformatitwouldbestraightforward
topresent the equations toadigital computer for solution.
Engineeringapplication20.2
Anarmature-controlledd.c.motor
Deriveastate-spacemodelforanarmature-controlledd.c.motorconnectedtoamechanicalloadwithcombinedmomentofinertiaJ,andviscousfrictioncoefficientB.
The arrangement isshown inFigure 20.8.
v a
= applied armature voltage
i a
= armature current
R a
= armature resistance
L a
= armature inductance
e b
= back e.m.f.of the motor
ω m
= angular speed of the motor
T = torque generated by the motor.
Solution
Letusassumethatthesysteminputisthearmaturevoltage, v a
,andthesystemoutput
istheangularspeedofthemotor, ω m
.Thisisasecond-ordersystemandsotwostate
variablesarerequired.Wewillchoosethearmaturecurrentandtheangularspeedof
the motor. So,
x 1
=i a
x 2
=ω m