# Analog and Digital Control 1_4_ by hcj

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```									Analog and Digital
Control 1

Lecture 3

Kirsten Mølgaard Nielsen
Based on lecture notes by Jesper Sandberg Thomsen

1
Outline
Discrete equivalents
   Overview (emulation – finding a discrete equivalent to
the continuous system)
   Numerical integration
   Zero-Pole matching (mapping)
   Hold equivalents
Design by emulation
   Evaluation of design
Nyquist stability criterion
   Continuous case
   Discrete case

2
Discrete Equivalents - Overview
r(t)             e(t)    controller       u(t)       plant          y(t)
+                   D(s)                      G(s)
-
Translation to
discrete plant
Translation to discrete controller (emulation)                               Zero order hold (ZOH)
Numerical Integration
• Forward rectangular rule
• Backward rectangular rule
• Trapeziod rule (Tustin’s method, bilinear transformation)
• Bilinear with prewarping
Zero-Pole Matching             Emulation
Hold Equivalents               Purpose: Find a discrete transfer function which
• Zero order hold (ZOH)        approximately has the same characteristics over
• Triangle hold (FOH)          the frequency range of interest.
Digital implementation: Control part constant between samples. Plant is not constant between samples.

3
Numerical Integration
Fundamental concept
   Represent H(s) as a differential equation.
   Derive an approximate difference equation.
We will use the following example
   Notice, by partial expansion of a transfer function this
example covers all real poles.
Example
Transfer function                 Differential equation

U ( s)             a
 H ( s)                    u  au  ae

E ( s)            sa

4
Numerical Integration
TransferFunction
U (s)            a
 H (s) 
E (s)           sa
Differential Equation
u  au  ae 

u (t )    au( )  ae( ) d
t
(continous) 
0
kT T
 au  aed  kT T  au  aed
kT
u (kT )  
0

 u (kT  T )   area of  au  ae over kT  T , kT 
5
Numerical Integration
Now, three simple ways to
approximate the area.                  kT-T   kT
       Forward rectangle
    approx. by looking forward
from kT-T
       Backward rectangle
    approx. by looking backward
kT-T   kT
from kT
       Trapezoid
    approx. by average

kT-T   kT

6
Numerical Integration
Forward rectangular rule (Euler’s rule)
(Approximation kT-T)
Difference Equation
u1 (kT )  u1 (kT  T )  T  au1 ( kT  T )  ae( kT  T )
 (1  aT )u1 (kT  T )  aTe(kT  T )
Transfer Function
U1 ( z )  (1  aT ) z 1U1 ( z )  aTz1E ( z ) 
U1 ( z )     aTz1             aT         a
H F ( z)                                     
E ( z ) 1  (1  aT ) z 1 z  1  aT z  1  a
T

7
Numerical Integration
Backward rectangular rule (app kT)
Difference Equation
u2 ( kT )  u2 ( kT  T )  T  au2 ( kT )  ae( kT ) 
u2 ( kT )  aTu2 (kT )  u2 ( kT  T )  aTe( kT ) 
1                      aT
u2 ( kT )           u2 ( kT  T )         e( kT )
1  aT                  1  aT
Transfer F     unction
1                  aT
U2 ( z)                U2 ( z)          E ( z)
(1  aT ) z           1  aT
U2 ( z)           1           aT            aTz          a
H B ( z)                                                      
E ( z) 1           1      (1  aT ) (1  aT ) z  1 z  1  a
(1  aT ) z                             Tz
8
Numerical Integration
Trapezoid rule (Tustin’s Method, bilinear trans.)
(app ½(old value + new value))
Difference Equation

u3 ( kT )  u3 ( kT  T )   au3 ( kT  T )  ae( kT  T )  au3 ( kT )  ae( kT )
T
2
1  ( aT / 2)
                u3 ( kT  T ) 
aT / 2
e(kT  T )  e(kT )
1  ( aT / 2)                 1  ( aT / 2)
Transfer F    unction
U3 ( z)       aT ( z  1)            a
HT ( z)                              
E ( z ) ( 2  aT ) z  aT  2 2 ( z  1)  a
T ( z  1)

9
Numerical Integration
Comparison with H(s)
H ( s)      Method           Transfer function

a                                        a
Forward rule       H F ( z) 
sa                                   z 1
a
T
a                                        a
Backward rule     H B ( z) 
sa                                   z 1
a
Tz
a                                        a
Trapezoid rule   HT ( z) 
sa                                 2 ( z  1)
a
T ( z  1)

10
Numerical Integration
Transform s ↔ z

Method                 Approximation
z 1
Forwardrule        s         or z  1  Ts
T
z 1            1
Backward rule       s         or z 
Tz          1  Ts
2 ( z  1)         1  (T / 2) s
Trapezoid rule s             or z 
T ( z  1)         1  (T / 2) s

Comparison with respect to stability
   In the s-plane, s = jw is the boundary between stability
and instability.
11
Numerical Integration - stability
Forward      z  1  Ts       for s  jw
1
Backward     z               for s  jw
1  Ts
1  (T / 2) s
Trapezoid z                  for s  jw
1  (T / 2) s
(Forward)                                         (Trapezoid)
(Backward)   Region of
Region of
Region of    s-LHP
s-LHP
s-LHP

12
Numerical Integration
Some conclusions
   Higher distortion with the forward and backward rule.
   The forward rectangular rule could cause a stable
continuous filter to be mapped into an unstable digital
filter.
   The trapezoid rule maps the stable region in the s-
plane exactly into the stable region of the z-plane.
Though, the jw-axis is compressed into the unit circle
with length 2p.
   The trapezoid rule appears superior. But, let us make
some further investigations..... let us calculate the
power |H(jw|2 at, for example, w = a – the change in
frequency response.

13
Numerical Integration
a
H ( jw ) 
jw  a

a2        1
H ( jw )  2      2 2
2

w  a 2 (w / a )  1
 (w  a )
1
H ( ja) 
2

2

   at which frequency is |HT(z)|2 = 0.5 ?

14
Numerical Integration
a                          1              HT(z) as
HT ( z)                          
2                         1 2                     before but re-
j tan(wT / 2)  a           j tan(wT / 2)  1
T                         aT                      formulated
with z  e jwT . Power wh en z1  e jw1T                    (page 194)

2                 12                  1
H T ( z1 )                           2

1 2                          2
    tan(w1T / 2)   12
a T              

2                    2        aT
a  tan(w1T / 2) or w1  arctan( )
T                    T         2                           I.e. high
2 aT                                             sampling
Notice, w1  ( )  a , aT / 2  1                            frequency
T 2
15
Numerical Integration
Trapezoid with pre-warping
    maintain the power at some specified frequency
1.   Choose a critical frequency w1
2.   Replace w1 by a´ such that new a = a’,
3.   and insert in H(s).         a’ = (2/T) tan(w1 T/2)
4.   Find the discrete filter H(z) based on the modified
continuous transfer function H(s).

s
H ( z)  H (        )
w1
s         1            z 1
w1  tan(w1T / 2) z 1

w  w1  H ( z)  H ( j1)
16
Numerical Integration   Plots for T = 0.1
ws ≈ 60 wb
Example, 3th order filter
H (s) 
1
s 3  2s 2  2s  1
with
0.1

T  1
2

Prewarpingat
w / wb  1
17
Numerical Integration

Plots for T = 1
ws ≈ 6 wb
Notice,
prewarp

18
Numerical Integration

Plots for T = 2
ws ≈ 3 wb
Notice,
prewarp

19
Zero-Pole Matching
Exact map between poles (z = esT) exists.
Basic idea, simply use same map for zeros.
Rules
1. All poles are mapped by z = esT.
   For example, s = -a maps to z = e-aT
2. All finite zeros are mapped by z = esT.
   For example, s = -b maps to z = e-bT
3. Basically, zeros at jw = infinity maps to z = ejp = -1
(representing the highest frequency)
4. Identical gain at some critical freq. (typically, s=0)
   H(s) at s=0 = H(z) at z=1

20
Zero-Pole Matching
Example (1)
Compute the discrete equivalent by zero-pole matching
a
H(s) 
sa
Pole (rule 1) s  a  z  e aT  
Zero (rule 3) H ( s ) s   0  z  1

z 1             a                     1  e aT
Gain (rule 4) K                                  1  K 
z  e aT   z 1 s  a   s 0                2
z 1
H ( z)  K
z 

21
Zero-Pole Matching
a    zp-match                z 1
H(s)                    H ( z)  K
sa                           z 

s-plane                           z-plane

22
Zero-Pole Matching
Including delay in computer
Notice, u(k) depends on e(k). This comes from rule 3

U ( z)               z 1        1  z 1
 H ( z)  K         K
E( z)                z        1  z 1
u(k )   u(k  1)  Ke (k )  Ke (k  1)
Solution, first zero for s=infinity is mapped to z=infinity
(as in Matlab). This gives
U ( z)        1           z 1
K          K
E( z)       z        1  z 1
u(k )   u(k  1)  Ke (k  1)

23
Hold Equivalents
Purpose, design a discrete filter H(z), such that
    input e(k) is samples of e(t)
    H(z) has an output u(k)
    u(k) approximates the output of the continuous filter
H(s) whose input is the continuous signal e(t).

Zero order hold (ZOH) (step invariant), approach below

e(t)             e(k)          eh(t)          u(t)             u(k)
Sampler          Hold           H(s)          Sampler

24
Hold Equivalents
Hold operation
q(t )  1(t )  1(t  T )
1
s

Q ( s )  1  e Ts        
E h ( s )  Q ( s ) e( k )
e( k ) is a gain
Q(s)
e(t)                e(k)              eh(t)          u(t)             u(k)
Sampler                 Hold           H(s)          Sampler

25
Hold Equivalents
ZOH transformation
    Notice, we can find H(z) by e(t )   (t )  e(k )   k
    Take inverse Laplace to find a continuous signal.
    Sample this continuous signal to find a discrete signal.

U ( s)  H ( s) Q( s) e(k )  H ( s) Q( s) (k )

u(t )  L H ( s) Q( s )  (k )  L  H ( s) 1  e Ts
1

1  1
s
         

 (k )


e(t)             e(k)           eh(t)              u(t)             u(k)
Sampler          Hold               H(s)           Sampler

26
Hold Equivalents
Continuous output signal

1     1
s

u(t )  L H ( s) 1  e Ts

 (k )  1(t)  1(t  T ) L1  Hs(s)   (k )







Discrete output signal and transfer function H(z)

u(k )  u(t )kT t
 H ( s) 
Z u(k )   U ( z )  H ( z )  (1  z 1 ) Z         
 s 
 H ( s)       1  H ( s ) 
where        Z          Z L            
 s             s 

27
Hold Equivalents
Triangle hold equivalent
      Extrapolate the samples to connect
sample to sample in a straight line.           1.0
      Impulse response for triangle hold

e(k )                            u (t )
Hold         System
T            T
eTs  2  e Ts    H (s )
Ts 2                              Result
( z  1)  H ( s ) 
H ( z)          Z 2 
Tz     s 

28
Hold Equivalents

Same third
order filter
as before

+ Zero-pole
o ZOH
x Triangle

T=1
(next T=2)

29
Hold Equivalents

Same filter
as before

+ Zero-pole
o ZOH
x Triangle

T=2

30
Hold Equivalents
Concluding remarks
       Methods, we have looked at
    Numerical integration methods
    Zero-pole matching (mapping)
    Hold equivalents
       All methods works fine for high sample frequency
       Matlab

31
Evaluation of Design
Discrete controller by emulation
   Can be calculated using one of several techniques.
Evaluation discrete equivalent (design)
   We need to analyze in the z-domain.
   Process expressed in z-domain (ZOH)
   Notice, other eval. might be necessary (simulation)

32
Evaluation of Design
Example, using Matlab
Cont. sys.(antenna)          Cont. controller
1                        (10s  1)
G( s)                       D( s) 
s(10 s  1)                    s 1

sysD_c = tf([10 1],[1 1]);
T = 0.2; % around 30*wn. Also use, T = 1
sysD_d = c2d(sysD_c,T,’matched’); % zero-pole matched
sysG_c = tf([1],[10 1 0]);
sysG_d = c2d(sysG_c,T,’zoh’); % zero order hold
sysCL_c = feedback(sysD_c*sysG_c,1);
sysCL_d = feedback(sysD_d*sysG_d,1);
step(sysCL_c); hold on;
step(sysCL_d);

33
Evaluation of Design

T = 0.2
(ws ≈ 30 wn)

34
Evaluation of Design
T=1
(ws ≈ 6 wn)

The sample
delay gives
phase lag.
Thus, less PM
which gives
higher
overshoot

35
Nyquist Stability Criterion
Check closed loop stability
   with controller KD(s) and plant G(s)
Y ( s)    K D( s ) G ( s )                       b( s)
                       , D( s) G( s) 
R ( s ) 1  K D( s ) G ( s )                     a( s)

Characteristic equation (CE)
a( s)  Kb( s)
1  K D( s) G( s)  0                 0
a( s)
Notice, number of poles in CE equals number of open-
loop poles in D(s)G(S). (known)
Notice, zeros in CE are closed-loop poles! (unknown)

36
Nyquist Stability Criterion
Approach, check characteristic equation
   Let us follow a contour encircling the RHP.
   Basic idea: Evaluate CE = 1+KD(s)G(s) at the
contour, and draw image. (We can just as well
evaluate KD(s)G(s) and look at -1 )

K D( s) G( s)

Notice, poles or
zeros gives
encirclements

37
Nyquist Stability Criterion
Criterion (continuous case)

   Z=P+N

   Z = number of unstable closed-loop poles. Thus, for
stability we want Z=0. (unknown)
   P = number of unstable open-loop poles of KDG
(known).
   N = total number of encirclements of the point -1 by
evaluation of KDG (taken in same direction as s.
Thus, N might be negative).

38
Nyquist Stability Criterion
Interpretations and explanations
   If the s-plane contour encircles a zero of 1+KDG in a
certain direction, the image contour will encircle the
origin in the same direction. (related to Z)
   If the s-plane contour encircles a pole of 1+KDG in a
certain direction, the image contour will encircle the
origin in the opposite direction. (related to P)
   The net number of same-direction encirclements, N,
equals the difference N = Z- P.
   Actually, we should investigate 1+KDG and
encirclements around the origin, but easier to
investigate KDG and encirclements around -1.

39
Nyquist Stability Criterion
Discrete case
   Same approach,
almost...
   Unstable region is
outside the unit
circle.
   Easier to evaluate
using the stable
region.

40
Nyquist Stability Criterion
Discrete case
   CE: 0 = 1+KD(z)G(z)
   Total number of poles = n
   Number of stable zeros of 1+KD(z)G(z) is n-Z
   Number of stable poles of 1+KD(z)G(z) is n-P

Criterion (discrete case)
   Z = (n-Z) – (n-P)   or     Z=P–N

41
Nyquist Stability Criterion
Discrete case
   Determin the number P of unstable poles of KDG
   Plot KD(z)G(z) for the unit circle. This is the counter-
clockwise path around the unit circle.
   Set N equal to the number of Counter-clockwise
encirclements of the point -1 on the plot
   Compute Z=P-N.
   The system is stable if and only if Z=0

42
Nyquist Stability Criterion
Example (7.9)
Evaluate the stability of the
unity feedback discrete system
with the plant G(s) with
sampling time T=2 and ZOH.
K=1
Use controller KD(z) = K.

1
G( s ) 
s( s  1)
Solution (matlab)
sysC = tf([1],[1 1 0]);
sysD = c2d(sysC,2);
nyquist(sysD);

The system is openloop stable and there are no encirkelments of -1
=> the system is closed loop stable for K=1
43

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