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

20.4 State-space models 609
Solutions
1 (a)
(b)
(c)
(d)
(e)
(f)
dy 1
dx =y dy 2
2
dx = −3y 1 −2y d 2 y
2 2 (a)
dx 2 + dy
dx −4y=0
dy 1
dx =y dy 2
2
dx = −9y 1 −8y 2
d 2 y
(b)
dx 2 −3dy dx +6y=0
dx 1 dx
=x 2
dt 2 = −6x
dt 1 −4x 2
d 2 x
(c)
dy 1 dy dt 2 −11x=0
=y 2
dt 2 = −7y
dt 1 −6y 2
d 2 y
(d)
dy 1
dx =y dy 2
2
dx =y dt 2 +5dy dt −38y=0
3
3 y ′′ +3y ′ +2y=0
dy 3
dx = −y 1 −2y 2 −6y 3
y 1 (t) =Ae −2t +Be −t
dx 1 dx
=x 2
y
dt 2 =x 2 (t) = − 2
dt 3
3 Ae−2t − 1 2 Be−t
( ) ( )
y ′ (y1 ) 2 6
dx 1
3
= −2x
dt 1 −4x 2 −2x 3 y 2
′ =
−2 −5 y 2
20.4 STATE-SPACEMODELS
Thereareseveralwaystomodellineartime-invariantsystemsmathematically.Oneway,
which we have already examined, is to use linear differential equations with constant
coefficients. A second method is to use transfer functions which will be discussed in
Chapter 21. A third type of model is the state-space model. The state-space technique
is particularly useful for modelling complex engineering systems in which there are
several inputs and outputs. It also has a convenient form for solution by means of a
digital computer.
Thebasisofthestate-spacetechniqueistherepresentationofasystembymeansofa
setoffirst-ordercoupleddifferentialequations,knownasstateequations.Thenumber
of first-order differential equations required to model a system defines the order of the
system. For example, if three differential equations are required then the system is a
third-ordersystem.Associatedwiththefirst-orderdifferentialequationsareasetofstate
variables, the samenumber astherearedifferential equations.
The concept of a state variable lies at the heart of the state-space technique. A system
is defined by means of its state variables. Provided the initial values of these state
variables are known, it is possible to predict the behaviour of the system with time by
means of the first-order differential equations. One complication is that the choice of
statevariablestocharacterizeasystemisnotunique.Manydifferentchoicesofasetof
state variables for a particular system are often possible. However, a system of ordern
only requires n state variables to specify it. Introducing more state variables than this
only introducesredundancy.
Thechoiceofstatevariablesforasystemis,tosomeextent,dependentonexperience
buttherearecertainguidelinesthatcanbefollowedtoobtainavalidchoiceofvariables.
One thing that is particularly important is that the state variables are independent of