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Chapter 14
System model representation
This chapter is about the most commonly used forms of
representing the mathematical model of dynamic systems.
While
the actual derivation of a system model is not covered by
the
present chapter, this chapter emphasizes the treatment of
the
model once the system's governing equations have been
obtained successfully. The following forms will be
introduced
and applied to systems:
1. onfiguration form
!. "tate#space representation$. %nput#output equation
&. Transfer function
Throughout most of this chapter, it will be assumed that the
systems under consideration are linear. onlinear systems, as
well as linearization procedures, will be discussed later or
in
other chapters.
14.1 Configuration form
(efinition: ) set of independent coordinates that completely
describes the motion *dynamics+ of a system is referred to as
a
set of generalized coordinates.
otation: +,$,!,1* niqi = denote the generalized coordinates for
a
system with n degrees of freedom, where the number of
degrees of freedom *(-+ is equal to the minimum number of
independent coordinates required to describe the
position*status+ of all elements of a system.
eometrically spea/ing, generalized coordinates, +,$,!,1* niqi =
,
define an n#dimensional artesian space that is referred to as
the
configuration space.
onsider an n#degree#of#freedom system whose governing
*second#order+ differential equations are given as
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+,,,,,,,,*
+,,,,,,,,*
!1!1
!1!111
tqqqqqqfq
tqqqqqqfq
nnnn
nn
=
=
*1+
where iq and +,,!,1* niq i =
denote the generalized coordinates
and generalized velocities, respectively. -unctions +,,!,1* nifi
=
, generally nonlinear, denote the generalized forces and are
algebraic functions of +timeand,, tqqii
. The system of
differential equations represented by 0quation *1+ is subectto
initial conditions
+2*,,2svelocitiedgeneralize%nitial
+2*,+,2*scoordinatedgeneralize%nitial
1
1
n
n
q)(q
qq
*1!+
0quation *1+ together with 0quation *1!+ describes the
system's configuration form.
Example 14.1
onsider a simple mechanical system, shown in -igure 1,
subect to initial conditions
22 +2*,+2*
== xxxx
where 2x and 2
x denote the prescribed initial displacement and
velocity, respectively. 03press the system's equation of
motion
in configuration form, as defined by 0quation *1+.
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-igure 1 "ingle#degree#of#freedom mechanical system.
Solution
The governing differential equation, describing the motion ofthe
system, is given by
22 +2*,+2*conditions%nitial
2
==
=++
xxxx
kxxbxm*1$+
(ividing by m and rearranging terms in 0quation *1$+, one
obtains
22 +2*,+2*,
=== xxxxxm
kx
m
bx *1&+
bserve that the only one generalized coordinate e3ists, xq =1
,
and 0quation *1&+ is in agreement with the general form
of
0quation *1+. 4oreover, the generalized function
xm
kx
m
btqqf =
+,,* 111 is e3pressed as a linear combination of x
and
x , and hence is linear.
Example 14.2
onsider the two#degree#of#freedom system shown in -igure
1!, subect to initial conditions
!2!!2!121121 +2*,+2*,+2*,+2*
==== xxxxxxxx
03press the governing equations in configuration form.
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-igure 1! Two#degree#of#freedom mechanical system.
Solution
The differential equations describing the motion of this
mechanical system are given as
2+*+* 1!!1!!111111 =++
xxbxxkxkxbxm *1a+
2+*+* 1!!1!!!! =++
xxkxxbxm *1b+
(ividing 0quation *1a and b+ by m1 and m! separately and
rearranging yields
+,,,*+6*+*71
+,,,*+*+*71
!1!1!1!!1!!
!
!
!1!111!!1!!1111
1
1
==
=++=
xxxxfxxkxxbm
x
xxxxfxxbxxkxkxbm
x
*1+
bserve from this formulation that there are two generalized
coordinates, q19x1and q!9x!, and that 0quation *1+ is in
thegeneral form of 0quation *1+. 4oreover, the generalized
forces f1 and f! are e3pressed as linear combinations of
generalized coordinates x1andx!and generalized velocities 1
x
and !
x , and hence are linear.
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Second-order matrix form
) convenient and commonly used form of representing an nn3
system of second#order differential equations is standard
second#order matri3 form, defined as
fkxxcxm =++
*1+
where m9 mass matri3 *n3n+, c9 damping matri3 *n3n+, k9
stiffness matri3 *n3n+, x9 configuration vector *n31+ 9 vector
of
generalized coordinates, and f9 vector of e3ternal forces
*n31+.
Example 14.3
onsider the single#degree#of#freedom system of 03ample
1&.1
when subected to an applied force +*tf , as shown in -igure
1
$. ;epresent the equation of motion in second#order matri3
form.
-igure 1$ 4echanical system subected to an applied force.
Solution
The equation of motion of this system is given as
+*tfkxxbxm =++
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03ample 1&.&
"uppose the mechanical system of 03ample 1&.! is subected
to
an applied force +*tf , as shown in -igure 1$#&. 03press
the
system's equations of motion in second#order matri3 form as
given by 0quation *1+.
-igure 1& 4echanical system of 03ample 1&.! subected to
an
applied force.
"olution
The equations of motion, a slight modification of 0quation
*1
5+, are
2+*+* 1!!1!!111111 =++
xxbxxkxkxbxm*1=+
+*+*+* 1!!1!!!! tfxxkxxbxm =++
The system of equations in 0quation *1=+ can then be
e3pressed using matri3 notation into the standard
second#order
matri3 form, 0quation *1+, as
fkxxcxm =++
or, equivalently,
=
++
++
fx
x
kk
kkk
x
x
bb
bbb
x
x
m
m 2
2
2
!
1
!!
!!1
!
1
!!
!!1
!
1
!
1
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Problems
1&.1 onsider the two#degree#of#freedom mechanical system
in
-igure >1&.1. (erive the governing equations of motion
and
e3press them in configuration form and identify the
generalized
forces. )ssume the following parameter values: m19 1/g,m!9
1/g, b19 !.s?m, k19 1?m, k!9 1?m.
-igure >1&.1 4echanical system.
1&.! ) mechanical system e3periencing translational and
rotational motion is governed by
2+*
+*+*
=+
=+++
xRkxbxm
tRuxRRkKBJ
where bkmRBKJ and,,,,,, denote physical parameters and are
regarded as constants, andx represent displacement and
angular displacement, respectively, and +*tu is an applied
force.
03press the equations in second#order matri3 form.
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14.2 State-space representation
The concept of state#space relies on what are /nown as state
variables.
Definition: The smallest possible set of independent
variables
that completely describes the state of a system is referred to
as
the set of state variables. These variables at some fi3ed time+*
2tt= and system inputs for all ott will provide a complete
description of system behavior at any time ott .
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Example 14.5
-or the mechanical system discussed in 03ample 1&.$,
determine the state variables.
Solution
The equation of motion is given as +*tfkxxbxm =++
. %n order to
completely solve this second#order differential equation,
/nowledge of two initial conditions is required, namely,
xx and+2*
Aence, there are two state variables *according to
@1+.-urthermore, because initial conditions correspond to x and
x ,
state variables should be chosen, according to @!, as x and
x C
i.e.,
== xxxx !1 ,
03ample 1&.8
"uppose that the governing equations for a certain dynamic
system are found to be
=+
=++
2!
2
!1!
!111
xxx
xxxx
where 1x and !x represent displacements, and 1
x and !
x denotevelocities. (etermine the state variables.
"olution
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%t is important to distinguish between state variables and
physical variables such as displacement and velocity. "tate
variables are generally mathematical quantities that are
employed to represent a system's governing equations in a
convenient form. "ometimes the conventional notation of ix ,
which is reserved for state variables, may indeed coincide
with
some of the physical variables involved in a system model.
-or
instance, in the current e3ample there are three state
variables
which, by convention, will be denoted by $!1 and,, xxx .
Aowever, we note that 1x and !x also represent displacements
of the bloc/s. This should not cause any concern because we
have determined that
>hysical quantities chosen as state variables: 1!1 ,,
xxx
4athematical quantities denoting state variables: $!1 ,, xxx
)ll that remains is to ma/e one#to#one assignments between
the
elements of the two sets. %n this process, it is customary to
use
up the physical variables in increasing order of derivatives.
%n
the current problem, this means 1x and !x , followed by 1
x . Tothat end, the assignments are made as follows:
-irst state 1x 9 -irst displacement 1x
"econd state !x 9 "econd displacement !x
Third state $x 9 -irst velocity 1
x
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)eneral formulation
nce the state variables are appropriately selected, the ne3t
tas/
is to construct the state#variable equations. %n general, let
us
consider a multi#input?multi#output *4%4+ system with n
state variables, x1, x!, D,3n, m inputs, u1, u!, D,um, and p
outputs, y1, y!, D, yp. Then, the state#variable equations are
in
the following general form:
"tate#variable equations
+C,,C,*
+,,C,*
+C,C,,*
11
11!!
1111
tuuxxfx
tuuxxfx
tuuxxfx
mnnn
mn
mn
=
=
=
*1E+
wheref1,f!, D,fnare nonlinear, in general.
"ystem outputs:
+C,,C,,*
+C,,C,,*
+C,,C,,*
11
11!!
1111
tuuxxhy
tuuxxhy
tuuxxhy
mnpp
mn
mn
=
=
=
*1+
where h1, h!, hp are nonlinear in general. %n the event that
nonlinear elements are present in the system, the algebraic
functions +,,!,1* nifi = and +,,!,1* pkhk = turn out to
nonlinear
and quite comple3 in nature, thereby complicating the
analysis.
0quations *1E+ and *1+ may be presented more
conveniently through matri3 notation. To that end, define
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13p
!
1
13p
!
1
13m
!
1
13
!
1
13
!
1
h
h
h
y
y
y
u
u
u
ppmnnnn f
f
f
x
x
x
=
=
=
=
=
hyufx
so that the state#variable equations are e3pressed as
+,,* tuxfx =
and the system outputs are
+,* tux*hy =
The comple3ity associated with the general formulation
reduces
considerably for the case of linear systems. %n the event that
all
elements in the model of a dynamic system are linear, the
algebraic 0quations *1E+ and *1+ will ta/e the following
special forms:
Finear state#variable equations:
mnmnnnmnn
mmnn
mmnn
ububxaxax
ububxaxax
ububxaxax
+++++=
+++++=
++++=
1111
!1!1!1!1!
111111111
*1+
Finear system outputs:
mpmpnpnpp
mmnn
mmnn
ududxcxcy
ududxcxcy
ududxcxcy
+++++=
+++++=
+++++=
1111
!1!1!1!1!
111111111
*1!+
%n order to represent 0quations *1+ and *1!+ in matri3
form, define the following quantities:
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toroutput vecor,input vectvector,state
13
!
1
13
!
1
13
!
1
=
==
==
=
ppmmnn
y
y
y
u
u
u
x
x
x
yux
matri3
nsmissiondirect tra
ddd
ddd
ddd
matri3,output
cpncc
ccc
ccc
matri3input
!
matri3,state
p3mpmp!p1
!m!!!1
1m1!11
p3np!p1
!n!!!1
1n1!11
31
!!!!1
11!11
3!1
!!!!1
11!11
=
==
=
=
==
=
DC
+,
mnnmnn
m
m
nnnnnn
n
n
bbb
bbb
bbb
aaa
aaa
aaa
)s a result, equations *1+ and *1!+ are e3pressed in
matri3 forms below:
"tate equation +u,xx +=
*1$a+
utput equation DuCxy += *1$b+
0quation *1$+ is /nown as the state#space representation
orstate#space form of the system model.
03ample 1&.
onsider the simple mechanical system of 03ample 1&.$.
)ssume that the displacement of the bloc/, +*tx , is the output
of
the system. btain the state#space form of the system model.
"olutionThe governing differential equation is given by
+*tfkxxbxm =++
*1&+
)s discussed in e3ample 1&.5, the state variables are xx =1
and
=xx! .
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!1 xx =
*1+
which does not depend on the dynamics of the system. The
second equation, however, is obtained directly from the
equationof motion, 0quation *1&+, by substitution of state
variables,
as follows:
+*1!! tfkxbxxm =++
"olving for !
x , one obtains
[ ]+*1
1!! tfkxbxm
x +=
*1+
0quations *1+ and *1+ form the state#variable equations
consisting of the time derivatives of the state variables and
the
algebraic functions of the state variables and system inputsC
that
is,
"tate#variable equations
!1 xx = *1+
[ ]+*1
1!! tfkxbx
mx +=
*1+
)lso, assuming linearity of the elements has caused these
equations to be in the general form of 0quation *1+. "ystem
output is, by assumption, the displacement of the bloc/C
hence
1xyxy == *1+
which agrees with the general form of 0quation *1!+.
03pressing the state#variable equations in matri3 form yields
the
state equation
fmx
x
mbmkx
x
+
=
?1
2
??
12
!
1
!
1*1=+
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so that in relation to 0quation *1$a+ we have
+u,xx +=
where
+*,?1
2,
??
12,
!
1tfu
mmbmkx
x=
=
=
= +,x
;ewriting 0quation *1+ using vector notation yields the
output equation
[ ] ux
xy .221
!
1+
=
and in relation to 0quation *1$b+ we have
Duy +=Cx where [ ] 2,21 == DC *1E+
Gltimately, a combination of 0quations *1=+ and *1E+
defines the system's state#space form
+u,xx +=
DuCxy +=
in which all vectors and matrices have been properly
defined.
bserve also that the two state variables are independent,
because it is impossible to e3press displacement as an
algebraic
function of velocity.
Example 14.8
onsider the electrical circuit in -igure 1!, in which the
voltage +*te is the input, and 1q and !q denote electric
charges.
The constant parametersR,L1,L!, C1, and C!denote resistance,
inductances, and capacitances, respectively.
*1+
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*!+ )ssuming that the system output is q1, find the output
equation. ;epeat for the case in which the outputs are 1q
and 1
q
-igure 1! Two#loop electrical circuit."olution
The system's governing equations are:
eqC
qqRqL =++
1
1
!111
1+* *1!2a+
21
+* !!
1!!! =++
qC
qqRqL *1!2b+
*a+
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The state equation is obtained by e3pressing 0quation *1!1+
in
matri3 form, u+,xx +=
, as
+*
2
?1
22
??+?*12
??2+?*1
12222122
1
&
$
!
1
!!!!
1111
&
$
!
1
teL
x
x
xx
LRLRCL
LRLRCL
x
x
xx
+
=
*b+
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-igure 1$ ;otational mechanical system.
*1+ (etermine the state#space form if the system output is 1.*!+
;epeat for the case in which both 1and ! are outputs.
