12 Digital Control System

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Nov 15, 2013 (3 years and 11 months ago)

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12 Digital Control System
G.Franklin, J.Powell, and M.Workman: Digital Control of Dynamic Systems
(2nd ed.), Addison-Wesley, 1990
Mita, Hara, and Kondo: Introduction to Digital Control, Corona, 1988
(in Japanese)
12.1 Anal
y
sis of Di
g
ital Control S
y
stem
1
y g y
Exercises 12.1, 12.2
12.2 Synthesis of Digital Control System
Exercises 12.3, 12.4
Continuous-time Control System
Plant
(continuous-time)
disturbance
control
input
controlled
output
reference
+
-
+
+
+
error
Controller
(continuous-time)
+
to be realized via digital computer
reference
+
+
e
rr
or
Digital Control System
C t ll
control
input
disturbance
Plant
controlled
output
sensor
noise
DA converter
2
inside the computer
(the world of discrete-time signals )
real world
(the world of continuous-time signals)
+
- +
+
+
e o
sampler
hold
C
on
t
ro
ll
e
r
(discrete-time)
Plant
(continuous-time)
sensor
noise
output
AD converter
2
12.1 Analysis of Digital Control System
Continuous-time World
Discrete-time World
Plant
Disturbance
Modeling error
Anal
y
sis for Discrete-time
discretization
Plant
Disturbance
Modeling error
St bilit
3
y
System
Controller
St
a
bility
Time response
Frequency Response
・・・・・
12.1.1 Discretization
Discrete-time
Controller
Continuous-time
input
Continuous-time
output
reference
+
-
error
hold
Digital Control System
Discrete-time
input
)(tu
c
)(ty
c
][iu
Sampling period
h
Continuous-time
Plant
sampler
Discrete-time
output
][iy
Discrete-time Control System
)
(
t
u
]
[
]
[
Discrete-time Plant
4
Continuous-time
Plant
Discrete-time
Controller
reference
+
-
error
sampler
hold
)
(
t
u
c
)(ty
c
]
[
iu
]
[
iy
3
Continuous-time
Plant
sampler
hold
)(tu
c
)(ty
c
][iu ][iy
Discrete-time Plant
Sampling period
h
Continuous-time Plant
)()(
)0(),()()(
0
txCty
xxtuBtxAtx
ccc
cccccc
=
=
+
=

(12.1)
t
0
h 2h 3h 4h
5h
Hold
hold
t
0
h 2h 3h 4h
5h
][iu
)(tu
c
hitihiutu
c
)1(],[)(
+
<

=
(12.2)
5
t
t
Sampler
t
0
t
0
h 2h 3h 4h
5h
sampler
)(][ ihyiy
c
=
][iy
)(ty
c
(12.3)
Continuous-time
Plant
sampler
hold
)(tu
c
)(ty
c
][iu ][iy
Discrete-time Plant
Sampling period
h
Discrete-time Plant
][][
]0[],[][]1[
0
iCxiy
xxiBuiAxix
=
=+=+
(12.4)
c
h
c
AhA
CCdBeBeA
cc
===

,,
0
τ
τ
where,
(12.5)
Discrete-time
State Space
Model
Exercise 12.1
Derive (12.4) and (12.5) from (12.1)-(12.3).
6
( ) ( )
( )
( )
( )
( )
( )
2
2 1 2
1
2 1
2 1
1 2
1
1
( ) ( ) ( ).
[ ] ( ),1,
[ 1] [ ] ( ).
From (12.1), it holds that
By defining and setting
it is easy to see that
c c
c
c
t
A t t A t
c c c c
t
c
i h
A i h
A t t
c c
ih
x
t e x t e B u d
x i x ih t ih t i h
x
i e x i e B u d
τ
τ
τ
τ
τ
τ
− −
+
+ −

= +
= = = +
+ = +


Hint:
4
z-Transform
:
( )
0
[ ] [ ]
i
x
i X z x i z





Discrete-time signal Complex Function
12.1.2 z-Transform and Pulse Transfer Function
(12.6)
0
i
=
( )
1
0
1
[ ]
1
i i i
i
z
x i X z z
z z
λ λ
λ
λ



=
= → = = =



Example
Properties of z-Transform
:
i

(
)
(
)
(
)
[ ],[ ],[ ]
g
i G z x i X z y i Y z→ → →
(12 7)
7
(
)
(
)
(
)
0
孝 [ ] [ ]
k
y i g i k x k Y z G z X z
=
=
− → =

(
)
(
)
[ ] [ 1] [0]y i x i Y z zX z zx= + → = −
Convolution sum
(12
.
7)
One step ahead
(12.8)
Exercise 12.2
Prove (12.7) and (12.8).
Discrete-time
Plant
(
)
U z
][iu ][iy
(
)

(
)
(
)
(
)
Y z G z U z=
Pulse Transfer Function
The ratio of the output to the input in the z-transform
1 2 3 1 2 3
[ ]:[ ]
[ 3] [ 2] [ 1] [ ] [ 2] [ 1] [ ]
u i y i
y i a y i a y i a y i bu i b u i b u i+ + + + + + = + + + +
input, : output
Difference equation:
( ) ( )
(
)
(
)
(
)
(
)
(
)
3 2 2
1 2 3 1 2 3
2
z Y z a z Y z a zY z a Y z b z U z b zU z bU z
b z b z b
+ + + = + +
+ +
Example
1)
z-transform
by assuming that all the initial conditions are zeros.
8
(
)
1 2 3
3 2
1 2 3
b z b z b
G z
z a z a z a
+ +

=
+ + +
2)
0
[ 1] [ ] [ ],[0]
[ ] [ ]
x
i Ax i Bu i x x
y i Cx i
+
= + =
=
State space equation
( ) ( )
1
n
G z C zI A B

= −
z-transform
(12.9)
5
[ 1] [ ] [ ] [0]
x i Ax i Bu i x x
+ +

12.1.3 Stability
Definition 12.1
(BIBO stability)
A pulse transfer function matrix G(z) is called BIBO stable
if G(z) generates bounded output y[i] for any bounded input u[i].
0
[ 1] [ ] [ ]
,
[0]
[ ] [ ]
x i Ax i Bu i x x
y i Cx i
+
=
+
=


=

(12.10)
Definition 12.2
(asymptotic stability)
The state space model (12.10) is called asymptotically stable
if for an identically zero input u[i]=0, the system state x[i] will
converge to zero from any initial state x[0].
Lemma 12 3
9
Lemma

12
.
3
1) G(z) is BIBO stable
All the poles of G(z) are inside the unit circle.
2) (12.10) is asymptotically stable
All the eigenvalues of A are inside the unit circle.
Discrete-time Plant
[ 1] [ ] [ ]
x
i Ax i Bu i+ = +


Continuous-time Plant
( ) ( ) ( )
c c c c c
x
t A x t B u t= +



Continuous-time
Plant
sampler
hold
)(tu
c
)(ty
c
][iu ][iy
Discrete-time Plant
( ) ( )
1
[ ] [ ] y i Cx i
G z C zI A B


=

= −
0
,
c c
h
A h A
c
c
A
e B e B d
C C
τ
τ
= =
=

( ) ( )
1
( ) ( )

c c c
c c c c
y t C x t
G s C sI A B


=

= −
:
:( )
eigenvalue of
pole of
c c
c c
A
G s
λ
λ



( )
:
:
eigenvalue of
pole of
A
G z
λ
λ



h
c
e
λ
λ=
(
12.11
)
10
x
x
x
Re
Im
Re
Im
x
x
x
1
-1
1
-1
c
λ
λ
s-plane
z-plane
( )
6
(
)
G z
(
)
θ
ω
+
=
hiriy sin][
( )
hiiu
ω
sin][ =
12.1.4 Frequency Response and Aliasing
BIBO stable
Th t d
t t t t
(
)
( )
: gain
: phase
j h
j h
r G e
G e
ω
ω
θ
=
=∠
Discrete-time signal created by sampling
the continuous-time signal
with sampling period h.
(
)
(
)
sin
u t t
ω
=
Th
e s
t
ea
dy
-s
t
a
t
e ou
t
pu
t
(12.12)
(
)
(
)
. is called a of
j h
G e frequency response G z
ω

11
Remark 12.4
( )
2
. is a periodic function of with the period of
j h
G e
h
ω
π
ω
1
)
(
G
(
)
1
h
e


samping period
h
Continuous-time
Plant
sampler
hold
)(tu
c
)(ty
c
][iu ][iy
Discrete-time Plant
Continuous-time Plant
Discrete-time Plant
1
)
(
+
=
s
s
G
c
(
)
1
h
e
Gz
ze

=

samping

period
h
)(sG
c
(
)

01.0
=
h
1.0=h
1=h
sec][rad/6=
ω
( )
( ) sin
[ ] sin 0.1.
Time response to
and where
c
u t t
u i hi h
ω
ω
=
= =
Frequency response
12
sec][rad/60=
ω
Aliasing
7
(
)
( ) sin
c
f
t t
ω
=
0
0
,
2
For any
consider with integer,
h h
k k
h
π π
ω
π
ω ω
⎡ ⎤
∈ −
⎢ ⎥
⎣ ⎦
= +
,

sampler
sampling period
h
(
)
hiif
ω
sin][
=
Sampling Theorem

O i l f it
Nyquist frequency
N
h
π
ω = :
Aliasing
( )
( )
0 0
2
[ ] sin sin sin
then
h
f
i hi k hi hi
h
π
ω ω ω
⎧ ⎫
⎛ ⎞
= = + =
⎨ ⎬
⎜ ⎟
⎝ ⎠
⎩ ⎭
)(ωjF
c
)(tf
c
][if
][
hj
eF
ω
Aliasing
: Distortion of
frequency response
O
ne can recover a s
i
gna
l

f
rom
it
s
samples, if the highest frequency in the
signal is less than Nyquist frequency.
13
0
h
π
h
π

sec][rad/
ω
h
π
2
h
π
2

sampler
0
h
π
h
π

sec][rad/
ω
h
π
2
h
π
2

12.2 Synthesis of Digital Control System
Continuous
time
Discrete
time
Specifications
Specifications
Continuous
-
time
Discrete
-
time
Plant,
Disturbance,
Modeling error,
etc.
Plant,
Disturbance,
Modeling error,
etc.
the continuous-time
design methods
the discrete-time
design methods
discrete-time
design
Discretization
14
Controller
Controller
re-design
Discretization
8
Continuous-time
Pl t
disturbance
input
output
reference
input
+
+
error
Continuous-time
Controller
12.2.1 Re-design Method
(
)
πt数=1)†Desi杮⁡⁣→ntin×潵→-time=c→湴r→ller= ⁷桩c栠慣hiev敳=t桥⁣→湴r→l⁳peci∞icati潮→.
c
K s
(
)

Pl
an
t
noise
-
+
+
+
(
)
c
K s
Step 2) Set up the sampling period by considering the colosed-loop frequency response,
Nyquist frequency, and so on.
h
(
)
(
)
.Step 3) Transform the continuous-time controller to a discrete-time controller
c
K
s K z
(
)
Step 4) Evaluate the discrete-time controller to the digital control system by numerical
simulations and experiments
K z
15
noise
disturbance
input
output
reference
input
+
-
+
+
+
+
error
sampler
hold
Discrete-time
Controller

simulations

and

experiments
.
If the obtained control performance is not safisfied, make the sampling period smaller
and go to Step 3.

h
Continuous-time
Plant
(
)
K
z
(a) Approximate Differential
(
)
1
( )
z
K K K

⎛ ⎞
⎜ ⎟
)(sK
c
(
)
K
z
)(te
c
)(tu
c
][ie
][iu
sapmpling period h:
Transform Technique form K
c
(s) to K(z)
(12 13)
(
)
( )
c c
K
s
K
z
K
hz
⎛ ⎞
⇒ =
⎜ ⎟
⎝ ⎠
( )
1 1
( )
[ ] [ 1] 1 1 1
( )
c
c
de t
e i e i z z z
sE s E z s
dt h h h hz
− −
− − − − −
≈ ⇒ ≈ ⇒ ≈ =
(12
.
13)
(b) Step Invariant Transform

( )
c
K
s
(
)

16
( ) ( ) ( )
( ) ( ) ( )
cK cK cK cK c
c cK cK cK c
x
t A x t B e t
u t C x t D e t
= +


= +


0
[ 1] [ ] [ ]
[ ] [ ] [ ]
,,
,

cK cK
K K K K
K K K
h
A h A
K K cK
K cK K cK
x i A x i B e i
u i C x i D e i
A e B e B d
C C D D
τ
τ
+ = +


= +

= =
= =

(12.14)
9
(c) Bilinear ( or Tustin’s ) Transform
( )
(
)
(
)
2 1
( )
1
c c
z
K s K z K
h z
⎛ ⎞

⇒ =
⎜ ⎟
⎜ ⎟
+
⎝⎠
)(sK
c
(
)
K
z
)(te
c
)(tu
c
][ie
][iu
sapmpling period
h

(12.15)
(
)
1
h z
+
⎝ ⎠
cKcKKcKcKK
BAI
h
h
BAI
h
AI
h
A
11
,
22
,
22
−−






−=






+






−=
( ) ( ) ( )
( ) ( ) ( )
cK cK cK cK c
c cK cK cK c
x
t A x t B e t
u t C x t D e t
= +


= +


[ 1] [ ] [ ]
[ ] [ ] [ ]
K K K K
K K K
x
i A x i B e i
u i C x i D e i
+ = +


= +


(12 16)
17
cKcKcKcKKcKcKK
BAI
h
CDDAI
h
C
h
C
h
h
h
h
11
2
,
22
−−






−+=






−=






(12
.
16)
( )
( )
2 1
2
,
1 2
bilinear transform between and
z
hs
s
z s z
h z hs

+
= =
+ −

-planes
-planez
Im
Im
Bilinear transform maps
the left-half s-plane into the unit disk in the z-plane
in one-to-one correspondence.
Bilinear transform maps
the imaginary axis in the s-plane into the unit circle
in the z-plane.
j
hj
e
Ω
Re
Re
x x xx
h
2

1
ω
j
h
Ω
2
tan
2 h
h
Ω
=ω
( )
( )
( )
( )
2
)
2
/
i (
2
2
1
12
,
2/2/
2/2/
h
h
j
ee
ee
heh
e
j
ezjs
hjhj
hjhj
hj
hj
hj
Ω
Ω
+

=
+

=
==
Ω−Ω
Ω−Ω
Ω
Ω
Ω
ω
ω
とすると
(12.17)
R l ti b t
K
(
)d
K
(
)
18
2
tan
2
)2/cos(
)
2
/
s
i
n
(
2
h
h
j
h
h
j
h
Ω
=
Ω
Ω
=
(
)
( )
j
h
c
K j K eω
Ω
=
)( ωjK
c
ω
Ω
h
π
( )
j
h
K e
Ω
R
e
l
a
ti
on
b
e
t
ween
K
c
(
s
)
an
d

K
(
z
)
In frequency response
(12.18)
10
)(ty
c
K(s)
+
-
sampler
hold
G
c
(s)
)(tu
c
][ir
( )( )
sss
sG
c
5.011.01
3
)(
++
=
Example
Continuous-time Plant
s
s
sK
c
532.02.01
532.01
4.1)(
×+
+
=
Continuous-time Controller
(Time-lead Compensator)
Step Response ⇒
19
G
c
(s)
K
c
(s)
+
-
)(tu
c
)(tr
c
)(ty
c
Transform K
c
(s) to K(z) by three transforms
s
s
sK
c
532.02.01
532.01
4.1)(
×+
+
=
(
)
K
z
(a) Approximate Differential
(b) Step Invariant Transform
(c) Bilinear Transform
Sampling period
h
=0.01[sec]
blue=(a), read=(b), green=(c), black=Continuous-time System)
20
11
Sampling period
h
=0.05[sec]
blue=(a), read=(b), green=(c), black=Continuous-time System)
E i
12 3
21
E
xerc
i
se
12
.
3
Continue the above example to carry out numerical simulations in the cases of
the sampling periods h=0.1[sec] and h=0.2[sec].
( )
0
Continuous-time Plant
c c
c
c
A B
G s
C
⎡ ⎤
=
⎢ ⎥
⎣ ⎦
12.2.2 Discrete-time Design Method
Digital control system
disturbance
f
Discrete-time
C t ll
( )
0
Discrete-time Plant
A B
G z
C
⎡ ⎤
=
⎢ ⎥
⎣ ⎦
noise
input
output
re
f
erence
input
+
-
+
+
+
+
error
sampler
hold
C
on
t
ro
ll
e
r
Continuous-time
Plant
( )
K
z
22
0
0
,,
c c
h
A h A
c c
C
A
e B e B d C C
τ
τ
⎢ ⎥
⎣ ⎦
=
= =

Discrete-time control system
K(z)
noise
output
reference
input
+
-
+
+
+
error
G(z)
disturbance
input
+
12



=
=+=+
][][
]0[],[][]1[
0
iCxiy
xxiBuiAxix
(a) State Feedback Control
Discrete-time Plant
(12.19)
( )
0
[ 1] [ ],[0]
[ ] [ ]
x
i A BF x i x x
y i Cx i

+ = − =

=

State Feedback Control
[ ] [ ]u i Fx i= −
Closed-loop System
( )
( )
0
0
[ ]
[ ]
i
i
x
i A BF x
y i C A BF x
= −
= −
(12.20)
(12.21)
23
Pole Assignment
When (A,B) is controllable, there exists a feedback gain matrix F such that
all the eigenvalues of A-BF can be assigned as you like.
Observer
Discrete-time
Plant
u
y
ˆ
(b) Observer



=
=+=+
][][
]0[],[][]1[
0
iCxiy
xxiBuiAxix
Discrete-time Plant
(12.22)
x
ˆ
(
)
( )
[ ]






+−=
=−++=+
][
][
][
ˆ
ˆ
)0(
ˆ
,][
ˆ
][][][
ˆ
]1[
ˆ
0
iy
iu
GBixGCA
xxixCiyGiBuixAix
If all the eigenvalues of are inside the unit circle,
it holds that
A GC

observer
(12.23)
24
( )
ˆ
lim [ ] [ ] 0.
it

holds

that

t
x i x i
→∞
− =
(
)
( )
ˆ
[ ] [ ] [ ],
[ 1] [ ].

By defining the estimation error
it follows that
e i x i x i
e i A GC e i

+ = −
∵ 
When (A,C) is observable, there exists a matrix G such that
all the eigenvalues of A-GC can be assigned as you like, and
thus you can design a observer.
13
:
p
ositive semi-definite
Q
(c) Optimal Regulator



=
=+=+
][][
]0[],[][]1[
0
iCxiy
xxiBuiAxix
Discrete-time Plant
(12.24)
{ }


=
+=
0
0
][][][][
2
1
)(
i
TT
iRuiuiQxixxJ
:
p
positive definite
Q
R
The term for the state → 0
as soon as possible
The term for input be
as small as possible
Performance Index
(12.25)
Problem:Find the optimal input u[i] by which the performance index (12.25)
is minimum.
25
( )
( )
1
1
,
.
where by using the positive definite solutiuon to the following
Riccati equation

the feedback gain matrix is defined as follows.

T T T T
T T
P
P Q A PA A PB R B PB B PA
F
F R B PB B PA


= + − +
= +
][][ iFxiu

=
Optimal Control
(12.26)
state feedback
(12.27)
(12.28)
{ }
{ }
{ }
{
}
{
}
0 0 0
E,[ ],[ ]
E [ ] 0,E [ ] [ ],:
E [ ] 0 E [ ] [ ]
where
and are mutually independent
positive semi-definite
iti d fi it
T
ij
T
x x x w i v i
w i w i w j W W
i i j V V
δ
δ
=
= =
(d) Kalman Filter



=
=+=+
][][
]0[],[][]1[
0
iCxiy
xxiBuiAxix
Discrete-time Plant (12.29)
{
}
{
}
E [ ] 0
,
E [ ] [ ]
,: pos
iti
ve
d
e
fi
n
it
e
T
ij
v
i
v
i
v
j V V
δ
= =
( ) ( )
{
}
ˆ ˆ
E [ ] [ ] [ ] [ ]
T
e
J
x i x i x i x i= − −
(
)
(
)
[
]
[ ]
ˆ
ˆ
ˆ
ˆ
[ 1] [ ] [ ] [ ] [ ] [ ]
u i
A B G C A GC B AG
⎡ ⎤
Performance Index: (12.30)
Optimal estimate
(12 31)
Problem:Find the optimal state estimate which minimizes (12.30)
26
(
)
(
)
[
]
0

ˆ
ˆ
ˆ
ˆ
嬱] [ ] [ ] [ ] [ ] [ ]
,

ˆ
嬰]
x
i
A
x i
B
u i
G
y i
C
x i
A GC
x i
B AG
y i
x x
⎡ ⎤
+
= + + − = − +




=
( )
( )
1
1
,
.
where by using the positive definite solution to the following
Riccati equation

the observer gain matrix is defined as follows

T T T T
T T
S
S W ASA ASC V CSC CSA
G
G SC V CSC


= + − +
= +
(12
.
31)
(12.32)
(12.33)
14
(e) Kalman Filter + Optimal Regulator
{ }
{ }
{ }
{
}
{
}
0 0 0
E,[ ],[ ]
E [ ] 0,E [ ] [ ],:
E [ ] 0 E [ ] [ ]
where
and are mutually independent
positive semi-definite
iti d fi it
T
ij
T
x x x w i v i
w i w i w j W W
i i j V V
δ
δ
=
= =



=
=+=+
][][
]0[],[][]1[
0
iCxiy
xxiBuiAxix
Discrete-time Plant (12.34)
( )






+=

=
∞→
f
f
N
i
TT
f
N
iRuiuiQxix
N
J
0
][][][][
2
1
E
1
lim
Di t
ti
u
y
Optimal Controller
{
}
{
}
E [ ] 0
,
E [ ] [ ]
,: pos
iti
ve
d
e
fi
n
it
e
T
ij
v
i
v
i
v
j V V
δ
= =
Performance Index
(12.35)
LQG problem:Find the optimal input u[i] by which (12.35) is minimum.
27
Kalman Filter
Di
scre
t
e-
ti
me
Plant
u
y
x
ˆ
F

Optimal Regulator
Optimal Regulator:
FRQBA ⇒,,,
Kalman Filter:
GVWCA ⇒,,,
1c
θ
2c
θ
Numerical Example:
Design the (optimal regulator + Kalman Filter) by two methods
(re-design and discrete-time) and then compare those control performances.
1
J
2
J
K
1
D
2
D
c
τ
[
]
)()(),()(
,)()()()()(
1
2121
ttyttu
tttttx
cccc
T
ccccc
θτ
θθθθ
==
=

)()(
1
tty
cc
θ
=
)()( ttu
cc
τ
=
10
01.0,05.0
,2,1
21
21
=
==
=
=
K
DD
JJ
28
][iu
Discrete-time
Controller
sampler
hold
h
:sampling period
][iy
15
(
)





−=
+−−=
)(
ˆ
)(
)()(
ˆ
)(
ˆ
txFtu
tyGtxFBCGAtx
Kccc
ccKccccccKc

Re-design Method
ccccc
ccccc
GVIWCA
FRIQBA
⇒==

=
=
1,100,,
1,100,,
4
4
1) Design a continuous-time controller
2) Transform the controller by using
bilinear transform
Discrete-time Design Method





+=
+=+
][
ˆ
][
ˆ
ˆ
][
][
ˆ
][
ˆ
ˆ
]1[
ˆ
iyDixCiu
iyBixAix
K
KK
CBACBA
ccc
,,,,⇒
2)

Transform

the

controller

by

using

bilinear

transform
1) Transform the continuous-time plant to the discrete-time one.
2) T f th i hti t i t th di t
ti
29
,,
,,
c c
c c
Q R hQ hR
W V hW hV


,,,
,,,
A
B Q R F
A
C W V G


(
)



−=
+−−=+
][
ˆ
][
][][
ˆ
]1[
ˆ
ixFiu
iGyixBFGCAix
K
KK
2)

T
rans
f
orm
th
e we
i
g
hti
ng ma
t
r
i
ces
t
o
th
e
di
scre
t
e-
ti
me ones.
3) Design a discrete-time controller.
Sampling period:h
=0.01[sec]
Re-design method
)
(
K
Both control performances are almost same.
Discrete-time Design Method
)
(
s
K
c
][zK
)(
2
d
θ
)(
2
c
θ
)(
1

)(
1
c
θ
)(du
c
)(cu
c
30
)(sK
c
][zK
)(
2
d
θ
)(
2
c
θ
)(
1
d
θ
)(
1
c
θ
)(du
c
)(cu
c
16
Re-design method
Sampling period:h
=0.2[sec]
The control performances degenerate, in particular the re-design
method is getting worse.
Discrete-time Design Method
31
Exercise 12.4
Continue the above example to carry out numerical simulations in the cases of
the sampling periods h=0.3[sec] and h=1.0[sec].
32