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Controller Actuator ProcessFilteryue
Measure
x
Analog Control System(ACS)
Actuator ProcessA/D
adapteryue
Measure
x
Computer Control System(ACS)
A/D adapter
D/A adapter
Controller
1.1 IntroductionComparison between ACS and CCS
ACS CCS
Process
Actuator
Measure
Controller (correcting network)
Structure:
Process
Actuator
Measure
Controller (digital computer)
Adapter (A/D, D/A)
Parts: Analog Analog + Digital
Signals: Analog
Continuous analog
Discrete analog
Discrete digital
Discrete (Sampling) System1 Introduction
2 Z-transform
3 Mathematical describing of the sampling systems
4 Time-domain analysis of the sampling systems
Chapter Discrete (Sampling) System
Make a analog signal to be a discrete signal shown as in Fig.1 .
t
x(t)
0 t1 t2 t3 t4 t5 t6
Fig.1 signal sampling
x(t)x*(t)
x(t) —analog signal .x*(t) —discrete signal .
1.2 Ideal sampling switch —sampler
Sampler —the device which fulfill the sampling. Another name —the sampling switch — which works like a switch shown as in Fig.2 . T
t
x*(t)
0 t
x(t)
0
Fig.2 sampling switch
1.3 Some terms1. Sampling period T— the time interval of the signal sampling: T = ti+1 - ti .
1 Introduction1.1 Sampling
1.3 Some terms2. Sampling frequency ωs — ωs = 2π fs = 2π / T .
3. Periodic Sampling — the sampling period Ts = constant.
4. Variable period sampling — the sampling period Ts≠constant.
1.4 Sampling (or discrete) control system
There are one or more discrete signals in a control system —
the sampling (or discrete) control system. For example the
digital computer control system:
A/D D/Acomputer process
measure
r(t) c(t)e(t)
-e*(t) u*(t) u (t)
Fig.3 computer control system
1.5 Sampling analysis
Expression of the sampling signal:
1 Introduction
)()()()()()()(*
00
kTtkTxkTttxttxtx
kkT
It can be regarded as Fig.4:
0 t
x*(t)
t
x(t)
0
T
t
δT(t)
0
× =
modulating pulse(carrier)
modulated wave
Modulation signal
Fig.4 sampling process
x(t) x*(t)
1.5 Sampling analysis
If the analog signal could be whole restituted from the sampling signal, the sampling frequency must be satisfied : s
maxmax 2
Tors
. 2 ,
.
. :
s
max
Trequencysampling f
eriodsampling pT
alnalog signy of the am frequencthe maximuhere
s
1.7 zero-order hold
Usually the controlled process require the analog signals, so we need a discrete-to-analog converter shown in Fig.7.
discrete-to-analog converter
x*(t) xh(t) Fig.7D/A convert
So we have:
1.6 Sampling theorem ( Shannon’s theorem)
1 Introduction
x*(t)x(t)
xh(t)
Fig.9
The action of the zero-order hold is shown in Fig.9.
The unity pulse response of the zero-order hold is shown in Fig.10.
The mathematic expression of xh(t) :
TktkTkTxtxh )1( )()(
The transfer function of the zero-order hold can be obtained from the unity pulse response:
s
eTttLsG
Ts
1)(1)(1)( T
Fig.10
t
g(t)
ω
A(ω) )(
Fig.8
To put the ideal frequency response inpractice is difficult, the zero -order hold is usually adopted.
1.7 zero-order hold The ideal frequency response of the D/A converter is shown in Fig.8.
2 Z-transform
2.1 DefinitionExpression of the sampled signal: )()()(*
0
kTtkTxtxk
Using the Laplace transform:
0
)()(*k
kTsekTxsx
Define: Tsez
We have the Z-transform:
0
)()(*)()(k
kzkTxtxZtxZzX
.2.2 Z-transforms of some common signals
The Z-transforms of some common signals is shown in table 8.1.
.2 Z-transform
1cos2
)cos(cos
1cos2
sinsin
1)1(
1
11)(1
11)(
)()()(
222
222
22
Tzz
Tzz
s
st
Tzz
Tz
st
ez
z
se
z
Tz
st
z
zst
t
zXsXtx
Tt
Table .1
.2.3 characteristics of Z- transform
The characteristics of Z-transform is given in table .2.
)()(
)()()()(nconvolutio Real
)()1lim()(lim value Final
)(lim)(lim value Initial
)()(
)()(
)()(
)()()(
)()()(
)()()()(
)()(
21
02121
1
0
1
0
1
0
22112211
zXzX
iTkTxiTxZtxtxZ
zXztx
zXtx
zXtxa
zXtxedz
zdXTzttx
ziTxzXzkTtx
zmTiTxzXzmTtx
zXkzXktxktxk
zXtx
k
i
zt
zt
a
zz
kzez
t
m
i
imm
m
i
im
T
Table .2
m
a
zz
k
T
T
zez
t
T
T
T
zmkt
z
zzT
z
Tz
dz
dTzt
az
z
z
za
ez
Tze
z
Tzte
ezz
ze
ez
z
z
ze
zXtx
T
t
)(
)1(
)1(
)1(
1
)()1(
))(1(
)1(
11
)()(
3
2
22
22
Table .3
Using the characteristics of Z-transform we can conveniently deduce the Z-transforms of some signals.
Such as the examples shown in table.3:
2.3 characteristics of Z-transform
n
iTa
i
n
n
n
iez
zKzXthen
as
K
as
K
as
K
asasas
A(s)X(s)If
1
2
2
1
1
21
)( :
)())(( :
Example .1
TT ez
z
ez
z
z
z
sssZ
sss
sZ
2
515
1
10
2
5
1
1510
)2)(1(
)4(5
2. Residues approaches
of X(s)r polesFor q-ordeez
zsXas
sqR
ResiduesRez
zsXreszX
Tsq
iq
q
asi
n
ii
as
n
iTs
i
i
)()(
lim)1(
1
)()(
1
1
11
!
.2 Z-transform2.4 Z-transform methods
1. Partial-fraction expansion approaches
22
10
22
)1(
1
10
10lim
)12(
1
)1(
10
)1(
10
T
T
Tsss
Ts
ez
Tzez
z
z
ez
z
ssez
z
ssssZ
!
2.5 Inverse Z-transform
transformInverse z-zXZkTx )()( 1
1. Partial-fraction expansion approaches
n
i
kTai
TaTaTaTaTa
i
n
eKkTXthen
es
zK
ez
zK
esezez
A(z)X(z)If
1
21
)( :
)())(( :
2121
Example .2
2.4 Z-transform methods
kTTT
T
eez
z
z
zZ
ezz
ezZkTx 2
21
2
21 1
1))(1(
)1()(
2. Power-series approaches
)3()2()()( :
)( :
321
33
22
11
TtKTtKTtKkTXthen
zKzKzKzB
A(z)X(z)If
Example .4
)3(375.6)2(75.4)(5.31
375.675.45.31
5.05.1
12)(
3211
23
231
TtTtTt
zzzZ
zzz
zzZkTx
Example.3
2.5 Inverse Z-transform
3. Residues approaches
of X(z)r polesFor q-ordezzXazzq
R
ResiduesRzzXreskTx
kqiq
q
azi
n
ii
n
i
k
i
)()(
lim)1(
1
)()(
11
1
11
1
!
2.5 Inverse Z-transform
Example 5
1
1221 )1(
)5.0)(1()5.0)(1(
z
k
zzz
zz
zz
zZ
5.0
12
)5.0()5.0)(1(
z
k
zzz
zz
T
kT
)5.0(2
.3.2 Z-transfer (pulse) function
Definition: Z-transfer (pulse) function — the ratio of the Z-transformation of the output signal versus input signal for the
linear sampling systems in the zero-initial conditions, that is:
)(
)()(
zR
zCzG
1. The Z-transfer function of the open-loop system
TG1(s)
r(t)G2(s) c(t)
c*(t)
G1(z)G2(z)R(z) C(z)
G1(s) G2(s)
T T
r(t)c(t)
c*(t)
G1G2(z)R(z) C(z)
G1G2(z) =Z [ G1(s)G2(s) ]
.3 Mathematical modeling of the sampling systems
G1(z) =Z [ G1(s)] G2(z) =Z [ G2(s)]
G(s)r c
-H(s)
rG2(s)
c-
G1(s)
H(s)
r-
G2(s)c
G1(s)
H(s)
r c-
G(s)
H(s)
r-
cG(s)
H(s)
)()()( )(
)()()( sHsGZzGH
zGH
zGzRzC
1
)()()( )(
)()( sGsRZzRG
zGH
zRGzC
1
)()(
)()()(
zHzG
zGzRzC
1
)(
)()()(
zHGG
zGzRGzC
21
21
1
)()(
)()()()(
zHGzG
zGzGzRzC
21
21
1
.3.2 Z-transfer (pulse) function
2. The z-transfer function of the closed-loop system
r-
G3(s)c
G2(s)
H(s)
G1(s)
)()()()(1
)()()()(
132
321
zHzGzGzG
zGzGzRGzC
.3.2 Z-transfer (pulse) function
Chapter Discrete (Sampling) System4 Time-domain analysis of the sampling systems4.1 The stability analysis
The characteristic equation of the sampling control systems:0)(1 zGH
0)(1 TsTs eGHez∵Suppose: TjTjTTs eeeejs )(
In s-plane, α need to be negative for a stable system, it means:
1 TTs eze
So we have:
The sufficient and necessary condition of the stability for the sampling control systems is: The roots zi of the characteristic equation 1+GH(z)=0 must all be inside the unity circle of the z-plane, that is: 1iz
1. The stability condition
1
Re
Imz-plane
Stable zone
Fig..4.1
The graphic expression of the stability
condition for the sampling control systems
is shown in Fig.4.1.
2. The stability criterionIn the characteristic equation 1+GH(z)=0, substitute z with
1
1
w
wz —— W (bilinear) transformation.
We can analyze the stability of the sampling control systems the same as (Routh criterion in the w-plane) .
)( ) (
101 0
)1(
2
)1(
1
1
1
1
1
1
1
: , , :
2222
2222
22
z-planele of the unit circinside theethe w-planofft halffor the le
yxyx
yx
yj
yx
yx
jyx
jyx
jyx
jyx
z
zjw
thenjyxzjwsupposeProof
4.1 The stability analysis
unstable zone
critical stability
0368.0368.1
632.01)(1
2
zz
KzzG
Determine K for the stable system.Solution :
0)632.0736.2(264.16320 0368.0368.1
632.01
2
KwKw.
zz
Kz
K
KK.
nh criteriof the RoutIn terms o
632.0736.2
264.1
632.0736.26320
:
We have: 0 < K < 4.33.
1
1
w
wzmake
4.2 The steady state error analysis
The same as the calculation of the steady state , we can use the final value theorem of the z-transform:
)()1(lim1
zEzez
ss
4.1 The stability analysis
Example .7
4.2 The steady state error analysis
)(1
)(
)(1
)()()()()()(
zG
zR
zG
zGzRzRzczRzE
G(s)r c
-e
Fig.8.4.2
For the stable system shown in Fig.8.4.2
*
2
*
*
11
1
1
)(1
)()1(lim)()1(lim
a
v
p
zzss
K
T
K
TK
zG
zRzzEze
)()1(lim ;)1(
)1()( )(
)()1(lim ;)1(
)( )(
)(lim ;1
)()(1)(
2
1
*3
22
1
*2
1
*
zGzKz
zzTzRttr
zGzKz
TzzRttr
zGKz
zzRttr
za
zv
zp
Z.o.h —Zero-order hold.)5(
)( 1
ss
KsGsT
2) If r(t) = 1+t, determine ess= ?1) Determine K for the stable system.
Solution
Example .8
4.2 The steady state error analysis
52555)1(
)5()1(
)5(
1)(
2
2
s
K
s
K
s
KZe
ss
KZe
ss
K
s
eZzG
Ts
Ts
Ts1)
)0067.0)(1(
2135.02067.2
5
2515
)1(5)1(
2
1
521
zz
zzK
ez
Kz
z
Kz
z
KTzz
T
T
r- G (s)
cZ.o.hT
e
0)4202.2067.10()1.5739.993(0.9932 1
1
0)2135.00335.0()0335.52067.2()5(
0)0067.0)(1(
2135.02067.2
51)(1
:
2
2
KwKww
wz
KzKz-K
zz
zzKzG
m the systequation ofteristic eThe charec
4.2 The steady state error analysis
16.40 K
2)
KK
T
K
T
Ke
Kz
zzKzGzK
zz
zzKzGK
vpss
zzv
zzp
5
2.00
1
1
2.0)0067.0(
2135.02067.2
5lim)()1(lim
)0067.0)(1(
2135.02067.2
5lim)(lim
1T**
2
11
*
2
11
*
4 Time-domain analysis of the sampling systems4.3 The unit-step response analysis
n
i
T
kT
ii
n
i i
i
n
pAAkTcthen
pz
zA
z
zA
z
z
pspsps
zAzRzzCsuppose
10
1
0
21
)()( :
1
1)())((
)()()()( :
Fig.4.3
Analyzing c(kT) we have the graphic expression of C(kT) is shown in Fig.4.3.
Im
Re1
Chapter Discrete (Sampling) System
5 The root locus of the sampling control systems
The plotting procedure of the root loci of the sampling systems are the same as that we introduced in continuous system.
But the analysis of the root loci of the sampling systems is different from that we discussed in continuous system. (imaginary axis of the s-plane ←→ the unit circle of the z-plane)..6 The frequency response of the sampling control systems
The analysis and design methods of the frequency response of the sampling systems are the same as that we discussed continuous system, only making:
jvww
wz
and 1
1
Here: v :frequency
Stability is a basic requirement for digital and analog control systems
Asymptotic Stability
Bounded Input Bounded Output
Routh-Hurwitz Criterion
Bilinear transformation transforms the inside of the unit circle to the LHP (Left Half Plane).
Bilinear Transformation
Example:
Substituting the bilinear transformation
stability conditions
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