Garching Control Presentation
90
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Summary of Control System
Transcript of Garching Control Presentation
Control Theory Workshop
Student Manual
Texas Instruments
v12
Texas Instruments
1
Control Theory Workshop
ver. 12
2
Agenda
1. Fundamental ConceptsLinear systemsTransient response classificationFrequency domain descriptions
2. Feedback ControlEffects of feedbackSteady state errorStability & bandwidth
3. Controller DesignPhase compensationRoot locus analysisTransient tuning
4. Discrete Time SystemsSampled systemsThe z-transformz plane mapping
5. Digital Control DesignPole-zero matchingNumerical approximationInvariant methodsDirect digital design
6. Digital Control Systems ImplementationSample rate selectionSample to output delayReconstructionControl law implementationAliasing
Tutorial 1.1: Linear systemsTutorial 1.2: Second order response
Tutorial 3.1: Phase lead compensator designTutorial 3.2: Buck control exampleTutorial 3.3: Root locus design exerciseTutorial 3.4: PID controller tuning
Tutorial 5.1: Matched pole-zero exampleTutorial 5.2: Discrete conversion comparison Tutorial 5.3: Direct digital design exercise
3
1. Fundamental Concepts
• Linear systems
• Transient response classification
• Frequency domain descriptions
4
Linearity
)()( ufkukf • This is the homogenous property of a linear system
• For a linear system, if a scale factor is applied to the input, the output is scaled by the same amount.
)()()( 2121 ufufuuf • The additive property of a linear system is
5
Terminology of Linear Systems
• A system is causal if its present output depends only on past and present values of its input
• Homogenous and additive properties combine to form the principle of superposition, which all linear systems obey
ubdt
dub
dt
udb
dt
udbya
dt
dya
dt
yda
dt
yda
m
m
mm
m
mn
n
nn
n
n 011
1
1011
1
1 ......
• The ordinary differential equation with constants a0, a1,...,an and b0, b1,...,bm ...
• If none of the coefficients depend explicitly on time, the equation is said to be time invariant. This type of equation can be used to describe the dynamic behaviour of a linear time-invariant (LTI) system
...is termed a constant coefficient differential equation
)()()( 22112211 ufkufkukukf
6
Convolution
0
)()()( dtguty
• If the impulse response g(t) of a system is known, it’s output y(t) arising from any input u(t) can be computed using a convolution integral
• The impulse response of an LTI system is it’s output when subjected to an impulse function, (t)
m
ii
i
i
n
ii
i
i dt
udb
dt
yda
00
7
The Laplace Transform
...where s is an arbitrary complex variable
If f(t) is a real function of time defined for all t > 0, the Laplace transform is defined as...
dtetftfsF st
0
)()()( L
)()()()( 22112211 sFksFktfktfk L• Linearity
)()()()( 210 21 sFsFdtfft
L• Convolution
)(lim)(lim0
sFstfst
• Final value theorem
• Shifting theorem )()( sFeTtf sTL
• The Laplace transform is a convenient method for solving differential equations
8
Poles & Zeros
ubdt
dub
dt
udbya
dt
dya
dt
yda
m
m
mn
n
n 0101 ......
012
21
1 ...)( asasasasasA nn
nn
nn
The dynamic behaviour of the system is characterised by the two polynomials
012
21
1 ...)( bsbsbsbsbsB mm
mm
mm
)()(...)()()(...)( 0101 sUbssUbsUsbsYassYasYsa mm
nn
For zero initial conditions, the CCDE can be written in Laplace form...
)(...)(... 0101 sUbsbsbsYasasa mm
nn
• The roots of B(s) are called the zeros of the system
• The roots of A(s) are called the poles of the system
9
The Transfer Function
))...()((
))...()(()(
21
21
n
m
pspsps
zszszsksG
012
21
1
012
21
1
...
...)(
asasasasa
bsbsbsbsbsG
nn
nn
nn
mm
mm
mm
Numerator & denominator can be factorised to express the transfer function in terms of poles & zeros
Tutorial 1.1
)(
)(
sA
sBThe ratio is called the transfer function of the system
• The quantity n – m is called the pole excess of the system
• For a system to be physically realizable it must satisfy the constraint m ≤ n
10
Classical First & Second Order Systems
n
n
pspspssY
...)(
2
2
1
1
tpn
tptp neeety ...)( 2121
)(...
...)(
011
1
011
1 sUasasasa
bsbsbsbsY
nn
nn
mm
mm
The nth order transfer function gives rise to n roots through partial fraction expansion.
Roots may be either real or appear as complex conjugate pairs. Hence both first and second order terms can be present. This motivates the study of “classical” systems of first and second order.
The time response will be dominated by those roots with the smallest absolute real part.
11
First Order Systems
)()()( tutyty
1
1
)(
)(
ssU
sY
)()()( sUsYsYs
The dynamics of a classical first order system are given by the differential equation
Taking Laplace transforms and re-arranging to find the transfer function
The output y(t) for any input u(t) can be found using the method of Laplace transforms
1
1)()( 1
ssUty L
...where is the time constant of the system
The response following a unit step input is t
ety
1)(
12
First Order Unit Step Response
• The output reaches ~63% of the final value after seconds
• The output y(t) is within 2% of the final value for t > 4
• A tangent to the y(t) curve at time t = 0 always meets the final value line seconds later
• The output reaches 50% of the final value after ~0.7 seconds
0 1 2 3 4 5 60
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t
y(t)
0.63
0.98
0.7
y(t) = 1 – e-t
13
Time Constant
• The time taken to reach a given output is t = -ln (1 – y(t) )
• The 10% to 90% rise time is given by tr = t2 – t1
• The rise time for any first order system is approximately 2.2
14
Second Order Systems
)()()(
2)( 22
2
2
tutydt
tdy
dt
tydnnn
• Linear constant coefficient second-order differential equations of the form
are important because they can often be used to approximate high order systems
02 22 nnss
...from which we get the characteristic equation...
is called the damping ratio
n is called the un-damped natural frequency
• Performance is defined by two quantities:
22
2
2)(
)(
nn
n
sssU
sY
• The Laplace transform of this second order system is
15
Damping Ratio
t
y(t)
1
0
Unit step response of second order system with varying damping ratio
• The second order step response can exhibit simple exponential decay, or over-shoot and oscillation, depending on value of the damping ratio
• The ability to represent both types of response means higher order systems can sometimes be approximated using a second order model based on the dominant poles
21.875 1.751.625 1.51.375 1.251.125 10.875 0.750.625 0.50.375 0.25
16
Second Order Systems
• The poles of the second-order linear system are the solutions of the characteristic equation
12 nns
02 22 nnss
• Four cases of practical interest can be identified, according to the value of
1. ( > 1 ) two real, un-equal negative roots at
2. ( = 1 ) two real, equal negative roots at
3. ( 0 < < 1 ) a pair of complex conjugate roots at
12 nns
ns
21 nn js
4. ( = 0 ) two imaginary roots at njs
17
Classification of Second Order Systems
21 nn js
over-damped
10
1
1
0
ns
12 nns
njs
under-damped
critically damped
un-damped
1
2
3
4
Damping ratio Roots Classification
18
The Under-Damped Response
• In the under-damped case (0 < < 1) we have two complex conjugate roots at
21 nn js
21 nd
1
d is the damped natural frequency of the system:
is the time constant of the system:
djs • Real and imaginary parts are denoted
• The under-damped step response will be of the form
)sin(1
1)(2
te
ty d
t
1cos...where
19
Transient Response Decay Envelope
The unit step response of an under-damped second order system is shown below. The transient part consists of a phase shifted sinusoid constrained within an exponential decay envelope.
• Decay rate is determined by the time constant of the system ()
• Settling time is determined by the time taken by the envelope to decay to a given error band ()
20
Under-Damped Unit Step Response
)sin(1
1)(2
te
ty d
t
nrt
5.28.0
21ln st
The unit step response for the second order case with 0 < < 1 is
• An approximation for rise time is
• Settling time is given by
d
d2
d3
21
Transient Response Peaks
)sin(1
1)(2
te
ty d
t
The unit step response of the classical second order system is given by
To find the peaks and troughs, differentiate with respect to time and equate the result to zero.
0)(cos1
)(sin1
)(22
te
te
ty dd
t
d
t
)(cos1)(sin 2 tt dd
21)(tan
td
211 1
tantan d
)tan()(tan td
The RHS of this equation is recognisable from
ntd d
nt
orSo the peaks & troughs occur when ...for integer n > 0
22
Second Order Step Response
)sin(1
1)(2
te
ty d
t
Tutorial 1.2
t
y(t)
1
0
21.875 1.751.625 1.51.375 1.251.125 10.875 0.750.625 0.50.375 0.25
Mp
2d
√
e t
d1+e
tp
Peak overshoot
Decay envelope
Damped frequency
Overshoot delay
d
23
djs
Second Order Pole Interpretation
22
2
2)(
nn
n
sssG
02 22 nnss
21 nn js
For the classical second order system
...we have the characteristic equation
For 0 < < 1, we have two roots at
or
...where the parameters are:
These have physical meaning in terms of the unit step response
)sin(1
1)(2
te
ty d
t
n decay rate:
21 nddamped natural frequency:
24
Second Order Pole Interpretation
• Decay rate and damped natural frequency are the real and imaginary components of the poles
• Un-damped natural frequency and phase delay are the modulus and argument of the poles
= cos-1
j d
j d
n
n
n
21 n
25
Constant Parameter Loci
• Concentric circles about the origin indicate constant un-damped natural frequency (n)
For the second order case, poles are located at
n8
1
2
3
4
5
67
n9
n10
n11
n12
n13
n8
n9
n10
n11
n12
n13
1
2
3
4
5
67
• Radial lines from the origin indicate constant damping ratio ()
21 js n
These are the usual grid lines drawn on a pole-zero map to aid in identifying transient response based on dominant pole location.
26
Pole Location vs. Step Response
Upper left quadrant of complex plane shown
27
Constant Parameter Loci
• Horizontal lines indicate constant damped natural frequency (d )
For the second order case, poles are located at
• Vertical lines indicate constant decay rate ()
21 nn js
djs
• Poles located further to the left have faster decay rate
• Poles with larger imaginary component are more oscillatory
28
Example Pole Specifications
Shaded regions excluded from design because system will not meet transient requirements
Poles positioned far to the left have fast response but require large control effort
Line of constant d
Line of constant defines region of acceptable overshoot
Line of constant defined by maximum acceptable settling time
29
Root Locus for Varying Damping
22
2
2)(
nn
n
sssG
Root locus for the system with fixed n as varies from 0 to infinity.
• For 0 < < 1 the locus is a semi-circle with radius n
• For = 0 both roots are imaginary
• For = 1 the roots are equal and real
• For > 1 both roots lie on the negative real axis
As →∞ one root approaches the origin while the other moves to the left along the negative real axis
These cases correspond to the transient response classifications given earlier.
30
Effect of LHP Zero
)(1
)()( tyz
tyty
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
0.5
1
1.5
2
time (s)
)(1
)(1
)( sYsz
sYs
sY
)(1)( sGz
ssG
For a unit step input, )(1
)( sGs
sY
• The effect of adding a LHP zero is to add a scaled version of the derivative of the step response of the original system
Adding a LHP zero at s = -z to the original transfer function:
Taking inverse Laplace transforms
• Rise time is decreased and overshoot increases
31
Effect of RHP Zero
1
1
stfus e
yy
RHP zeros imply inverse response behaviour in the time domain. The step response of a stable plant with n real RHP zeros will cross zero (it’s original value) at least n times
yf = final valuets = settling time = error bound = Re(z)
• For a single RHP zero, peak undershoot is bounded by the inequality
• The effect of adding an RHP zero is to increase rise time (make the response slower) and induce undershoot
32
Frequency Response
• The output y(t) is phase shifted with respect to u(t) by =
)(0
0 jGu
y
)sin()( 0 tyty
0
0
u
y
• If the steady state sinusoid u(t) = u0 sin(t + ) is applied to the linear system G(s), then the output is
• Amplitude and phase change from input to output are determined by G(s):
• The amplitude of output y(t) is modified by
)( jG
33
Polar Plots
1
1)(
ssG
221
1)(
jG
)(tan)( 1 jG
01)( jG
For the simple lag system
20)(
jG
The polar plot for a first order simple lag system stays in the fourth quadrant of the s-plane
When = 0:
As
We can plot the evolution of magnitude and phase of G(j) over frequency as a vector in the complex plane. This is known as a polar plot.
34
Second Order System Polar Plot
nn
jjG
21
1)(
2
2
01)( jG
22
1)(
jGn
When = 0
When
0)( jGAs
2
1222 21
1)(
vv
jvG
21
1
2tan)(
v
vjvG
n
v
Defining ...
The polar plot of a second order system ( n-m = 2 ) enters the third quadrant of the s-plane
2
1j
For the classical second order system
35
The Bode Plot
For simple transfer functions the gain and phase curves are related by a derivative
-40
-20
0
20
40
Mag
nitu
de (
dB)
10-2
10-1
100
101
102
-90
-45
0
45
90
Pha
se (
deg)
Frequency (rad/sec)
s
1
1
1
s
s
Bode plots show gain and phase on separate graphs against a common logarithmic frequency axis
36
Minimum Phase Systems
• A linear system is called minimum phase if all its poles and zeros lie in the left half of the s-plane and its response exhibits no time delay
• For minimum phase systems the phase curve is related to the rate of change of the gain curve and vice-versa.
)log(
)(log
2)(
d
jGdjG
• Non-minimum phase systems exhibit more phase lag and are therefore harder to control
2
n• If the gain curve has a constant slope n, the phase curve has constant value
37
Bode Asymptotes – First Order
Bode asymptotes for the first order case
2
1
4
38
Bode Asymptotes – Second Order
Bode asymptotes for the second order case – plots shown for varying damping ratio
2
39
Resonant Peak
| Y(j ) |
0
log
n
r
2
10:21 2 nr• The resonant peak occurs at frequency
212
1
rM• Resonant peak magnitude is given by
r → n as → 0The resonant peak r approaches n as damping ratio approaches zero:
40
Second Order Resonant Peak
For the classical second order system nn
n
jjG
2)(
22
2
Resonant frequency can be found by differentiating with respect to normalised frequencyn
v
0)(
dv
jvGdThe resonant peak occurs at normalised frequency
221 nr...which is at
212
1
rM
The peak value Mr can be found by substituting the normalised frequency r into the magnitude equation
...for2
10
41
2. Feedback Control
• Effects of feedback
• Steady state error
• Stability & bandwidth
42
Effects of Feedback
The objective of control is to make the output of a plant precisely follow a reference input
• Change the gain or phase of the system over some desired frequency range
• Cause an unstable system to become stable
• Reduce the effects of load disturbance and noise on system performance
• Reduce the sensitivity of the system to parameter changes
When properly applied, feedback can...
• Reduce or eliminate steady state error
• Linearise a non-linear component
43
Feedback System Notation
e / r = error ratio
ym / r = primary feedback ratio
r = reference input
e = error signal
u = control effort
y = output
ym = feedback
FG = forward path
H = feedback path
L = FGH = open loop
y / r = closed loop
F = controller
G = plant
44
The Loop Transfer Function
)()()( 000 rjLym
• The loop transfer function is obtained by breaking the loop at the summing point
• If a sinusoid of frequency 0 is applied at r, in the steady state the signal ym will also be a sinusoid of the same frequency 0 but it’s amplitude and phase will be modified by L(j)
• The condition necessary to sustain oscillation inside the loop is | L(j) | ≥ -1 where is the frequency at which phase lag is
L = FGH
y
r
H
GFu
ym
45
Negative Feedback
Combining error and output equations gives y = FG(r – Hy)
direct
1 + loopFGH
FG
r
y
1• The closed loop transfer function is
• The open loop transfer function is L = FGH
Error equation is: e = r - Hy Output equation is: y = FGe
46
The Closed Loop Transfer Function
1
1
GDefine the transfer functions of the forward and feedback elements as F = k, 2
2
Hand
2
2
1
1
1
1
1
k
k
r
y
2121
21
k
k
r
y
The closed loop transfer function is
2
2
1
1
47
Closed Loop Poles & Zeros
• Closed loop zeros comprise the zeros of the forward path and the poles of the feedback path
2121
21
k
k
r
y
• Zeros in the feedback path affect the poles of the closed loop system
021 k
• Closed loop poles are the roots of the characteristic equation 02121 k
• If any (s) have positive real parts the open loop will be unstable, but the closed loop can be stable
• Closed loop zeros are the roots of
• Poles in the feedback path appear as zeros in the closed loop system
48
Classification by Type
)(
)(
)(
)(1
1
11
ss
sk
pss
zskFGH
ln
ii
l
m
ii
A canonical feedback system with open-loop transfer function
...where l ≥ 0 and –zi and –pi are non-zero finite zeros and poles of FGH, is called a type l system.
• The type number reflects the number of integrators in the open-loop transfer function
• The steady state error will be either zero, finite or infinite, depending on the integer l
49
The Steady State Condition
10
1
1
2
nlif
nlifk
nlif
ess
)(lim
11
0sLs
en
s
ss
)(1
11lim
0 sLsse
nsssns
sr1
)( The steady state error following an input is given by
If L represents a type-l system: )(
)()( 1
s
s
s
ksL
l
21
0lim
1
kse
ln
s
ss
)0(
)0(12
kk ...where
• Steady state error is zero if L contains at least n-1 integrators:
50
Error Ratio
LFGHr
e
1
1
1
1The error ratio is
• The error ratio determines loop sensitivity to disturbance and is called the sensitivity function
Two ratios play an important role in closed loop performance:
LS
1
1
51
Feedback Ratio
The feedback ratio isL
L
FGH
FGH
r
ym
11
• The feedback ratio determines the reference tracking accuracy of the loop and is called the complementary sensitivity function.
L
LT
1
• Closed loop transfer function is related to T by:H
T
r
y
52
Plant Model Sensitivity
G
TS
FG
F
dG
dT
2)1(
2)1(
)1(
FG
FGFFGF
dG
dT
SGdG
TdT
/
/
For the unity feedback system, the sensitivity function can viewed in a different way.
• S is the relative sensitivity of the closed loop to relative plant model error.
If we differentiate T with respect to the plant, G...
53
S + T = 1
L
LT
1LS
1
1
11
1
L
LTS
Sensitivity function is: Complementary sensitivity function is:
• The shape of L(j) means we cannot maintain a desired S or T over the entire frequency range
10-3
10-2
10-1
100
101
102
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Mag
nitu
de (
abs)
Frequency (rad/sec)
|S||T|
54
Feedback with Control Disturbance
FGrGdFGHy 1
rH
TdGSy
HyrFdGy
Superposition allows the effects of output disturbance to be included in y...
Substituting S and T gives
++
+
_ yr
H
GF
d
ue
55
Closed Loop Performance
H
ry
rH
TdGSy
For good disturbance rejection we want S = 0
For good reference tracking we want T = 1 )( jL
• As |L(j)| rolls off both tracking performance and disturbance rejection will deteriorate
If loop gain can be kept very high, then
• Controller design involves shaping the open loop plot L(j) to achieve the best compromise between reference tracking and disturbance rejection
++
+
_ yr
H
GF
d
ue
56
Basic Feedback Equations
y = G ( d + u )
ym = H ( n + y )
u = F ( r - ym )
x1 = r – H x3
x2 = d + F x1
x3 = n + G x2
r = x1 + H x3
d = x2 – F x1
n = x3 – G x2
The equations for the exogenous signals in terms of internal signals are:
57
Basic Feedback Equations
3
2
1
10
01
01
x
x
x
G
F
H
n
d
r
nL
Hd
L
GHr
Lx
111
11
The loop equations can be written in matrix form as:
n
d
r
GFG
FHF
HGH
FGHx
x
x
1
1
1
1
1
3
2
1
If the 3x3 matrix is non-singular, the loop equation can be re-arranged to find the three internal signals
nL
FHd
Lr
L
Fx
11
1
12
nL
dL
Gr
L
FGx
1
1
113
58
Internal Stability
HSGSFSS
L
H
L
G
L
F
L 1111
1
L
FG
L
FH
L
GH
111
• External stability is a pre-requisite for internal stability
• Internal stability requires that x1, x2 & x3 do not grow without bound, so the following seven transfer functions must all be stable:
H
T
G
T
F
T
These can be written in terms of S and T :
59
The Nyquist D Contour
• The Nyquist D contour encloses the RHP with small indentations to avoid any poles of L(s) on the imaginary axis and an arc at infinity
• The Nyquist D contour encloses the RHP with small indentations to avoid any poles of L(s) on the imaginary axis and an arc at infinity
60
The Nyquist Plot
• The Nyquist plot is the image of the loop transfer function L(s) when s traverses D in the clockwise direction
• If L(s)→0 as s→∞ then the portion of the D contour at infinity maps to the origin
61
Closed Loop Stability
Closed loop stability can be defined in terms of either pole location or frequency response.
• The poles of the characteristic equation are determined by solving 1 + L(s) = 0
• For stability, all the system poles must lie in the open left half plane
• The root locus diagram provides an effective means of determining optimum loop gain
1) Pole location
2) Frequency response
• Plot the gain and phase of the open loop L(j) vs. frequency
• Time delays must first be approximated by rational transfer functions
• Evaluate gain and phase margins graphically to determine relative stability
• Information clearly presented on Bode or Nyquist plots
• Can represent non-minimum phase systems (but Bode relations do not apply)
62
Phase Margin
Phase margin is the angular difference between the point on the frequency response at the unit circle crossing and -180º. For a robust control system we want large positive m.
)( cjLPM• Phase Margin (PM) is defined as: ... where c is the gain crossover frequency
63
Gain Margin
Gain margin is the amount of extra loop gain we would have to add for the L(j) line to reach the [-1,0] point. For a robust control system we want large positive GM.
GM
11
)(
1
jLGM • Gain Margin (GM) is defined as: ... where is the phase crossover frequency
64
Gain & Phase Margins on Bode Plot
)( cjLPM• Phase Margin (PM) is defined as: ... where c is the gain crossover frequency
)(
1
jLGM • Gain Margin (GM) is defined as: ... where is the phase crossover frequency
| L( ) |
log
L( )
c
c
1
0
0
Phase Margin
Gain Margin
1
log
65
Loop Sensitivity from the Nyquist Plot
-1
L(j )
| L(j 0) |
| 1+L(j 0) |
• Estimation of S & T curves is straightforward from the Nyquist diagram
• The vectors |L(j)| and |1+L(j)| can readily be obtained from the Nyquist plot for any frequency 0
66
Conditional Stability
2
2
)1(
)6(3)(
ss
ssLBode diagram for the system with open loop transfer function
• Open loop phase lag exceeds 180 degrees at frequencies below cross-over but closed loop is stable
-10
0
10
20
30
100.1
100.3
100.5
100.7
100.9
-180
-170
-160
-150
Bode DiagramGM = -13.1 dB (at 3 rad/sec) , PM = 18.4 deg (at 5.92 rad/sec)
Frequency (rad/sec)
GM
PM
67
Conditional Stability
2
2
)1(
)6(3)(
ss
ssLNyquist diagram for the system with open loop transfer function
• The plot of L(s) as s traverses the Nyquist contour does not make a clockwise encirclement of the critical point, hence the closed loop system will be stable
-1
L(j ) L(j )
• In this case, reducing the loop gain will cause instability
68
Stability Margins - Example
)5.006.0()1(
)55.01.0(38.0)(
2
2
ssss
sssL
-100
-50
0
50
Mag
nitu
de (
dB)
10-2
10-1
100
101
102
-180
-135
-90
Pha
se (
deg)
Frequency (rad/sec)
PM
• In general, the Nyquist plot provides more complete stability information than the Bode plot
Although gain and phase margins are adequate (GM infinite, PM = 70 deg.), the system has two lightly
damped modes (1 = 0.81, 2 = 0.014) causing a highly oscillatory step response.
69
Stability Index
-1
L(j )
sn
SjLjLsn )(1
1max
)(1min
11
• A large value of || S ||∞ indicates L(j) comes close to the critical point and the feedback system is nearly unstable
• A better measure of relative stability is to gauge the closest L(j) passes to the critical point [-1,0]. This is called the stability index
• The reciprocal of the sensitivity index sn is the infinity norm of the sensitivity function
)(1min
jLsn
70
Sensitivity Norm - Example
10-2
10-1
100
101
0
0.5
1
1.5
2
2.5
3
3.5
4
Mag
nitu
de (
abs)
Frequency (rad/sec)
|| S ||∞
| S |
• The infinity norm of the sensitivity function shows considerable peaking, | S | >> 1, indicating that
L(j) passes close to the critical point and possible robustness issues
• Although the S infinity norm is a good measure of relative stability and robustness, is does not indicate stability in the binary sense: for example, a Nyquist plot which encloses the critical point might have a similar S response to that shown above
71
Maximum Peak Criteria
)(max
jSM S )(max
jTMT
• The maximum peaks of sensitivity and complementary sensitivity are
Typical design requirements are: MS < 2 (6dB) and MT < 1.25 (2dB)
0
0.5
1
1.5
10-3
10-2
10-1
100
101
102
-360
-270
-180
-90
0
90
Frequency (rad/sec)
| S(j ) || T(j ) |
Ms
72
Bandwidth
Bandwidth is defined as the frequency range [ 2] over which control is effective: B = 1 – 2
Normally we require steady state performance, so = 0 and B = 2
• Closed loop bandwidth is defined as the frequency at which | S(j) | first crosses -3dB from below
• Gain crossover frequency c is defined as the frequency at which | L(j) | first crosses 0dB from above
10-2
10-1
100
101
102
103
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Frequency (rad/sec)
cB
0.707
|S(j )|
|L(j )|
73
Regions of Control Effectiveness
Bandwidth can be defined in terms of the |S(j)| crossing of -3dB from below.
• Below performance is improved by control
• Between and BT control affects response but does not improve performance
• Above BT control has no significant effect
Mag
nitu
de (
abs)
74
Nyquist Diagram: Sensitivity Function
-1
L(j )
1
Sensitivity bandwidth reached when L(j ) first
crosses this circle
Sensitivity peaking when L(j ) lies inside this circle
1
j
-j
L(j )
√2
2
1
LS
1
1
• Peaking & bandwidth properties of the sensitivity function can be inferred from the Nyquist diagram
212
1 LS• Sensitivity bandwidth reached when first crosses
11 L
from below:
• Sensitivity peaking occurs when
75
Nyquist Diagram: Tracking Performance
2
1
L
LT
1LLT 21
2
1• Tracking bandwidth reached when first crosses from above:
• Tracking peaking occurs when Re{ L(j) } < -0.5
• Peaking & bandwidth of the tracking response can be inferred from the Nyquist diagram
12
1
76
Nyquist Diagram Grid Lines
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
-1.5
-1
-0.5
0.5
1
1.50 dB
-20 dB
-10 dB
-6 dB
-4 dB
-2 dB
20 dB
10 dB
6 dB
4 dB
2 dB
• Conventional Nyquist diagram grid displays peaking of tracking response in dB
• -3 dB line corresponds to tracking bandwidth locus
77
Effect of Time Delay
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
42.1174
36.1664
30.2154
24.2643
18.3133
12.3623
6.4113
0.460286
-5.49073
-11.4417
PM
)25.4()2(
6)(
2
sssesL s
Nyquist plot shows progressively larger time delay added to a stable third order system
• Time delay is equivalent to a frequency dependent phase lag which erodes phase margin.
78
RHP Zero: Additional Phase Lag
1
1)(1
ssL
1
1
1
1)(2
s
s
ssLThe above example shows Nyquist plots for the systems:
Both systems are strictly proper with pole excess of 1, but L2(j) has significantly higher phase lag due to its’ RHP zero, causing it to enter the unit circle centred at [-1,0].
L1(j )
L2(j )
1
-1
79
Loopshaping Limitations
N
iipdjS
10
)Re()(ln
• If L(s) is rational and has a pole excess of at least 2, then for closed-loop stability
...where there are N RHP poles at locations pi
• For a stable loop... 0)(ln0
djS
• For a given system, any increase in bandwidth (|S| < 1 over larger frequency range) must come at the expense of a larger sensitivity peak. This is termed the waterbed effect.
Equal areas
| S(j ) |
log
Mag
nitu
de (
abs)
80
3. Controller Design
• Phase compensation
• Root locus analysis
• Transient tuning
81
Loop Shaping Requirements
• |L| cannot be maintained large over the entire frequency range so compromises must be made
• Design of L(j) is most critical in the region between c and .:
– For fast response, we want large c and .
– For stability, we need | L(j) | < 1
– Phase at crossover L(jc) should be small
• For low frequency performance, L(s) must contain at least one integrator for each integrator in r
• Loop shaping means designing the shape of the L(j) gain & phase curves
• Requirements for good control are S ≈ 0 and T ≈ 1, so |L| must be large
82
Phase Compensation
• Start with gain k1 and introduce phase lead at high frequencies to achieve specified PM, GM, Mp, ...etc.
• Start with gain k2 and introduce phase lag at low frequencies to meet steady-state requirements
• Start with gain between k1 and k2 and introduce phase lag at low frequencies and lead at high frequencies (lag-lead compensation)
Three types of compensation are commonly used:
83
Phase Lead Compensation
• The first order phase lead compensator has one pole and one zero, with the zero frequency lower than the pole
• The simple lead compensator transfer function is ...where (z < p )p
z
s
ssF
)(
log
log
| F(j ) |
m
F(j )
m
z p
0
0
c
84
Lead Compensator Design
cm
1
1
1sin
m
m
m
sin1
sin1
Define
• If m is known, choose c to fix p, then calculate using
...where > 1
Maximum phase occurs at frequency
cs
cssF
1
11)(
Maximum phase is
Tutorials 3.1 & 3.2
85
Phase Lag Compensation
• The first order phase lag compensator has one pole and one zero, with the pole frequency lower than the zero
• The simple lag compensator transfer function is ...where (p < z )p
z
s
ssF
)(
86
Lag-Lead Compensation
))((
))(()(
21
21
pp
zz
ss
sssF
• The lag-lead compensator has two poles and two zeros arranged to induce phase lag and phase lead over different frequency ranges
• The lag-lead compensator transfer function is
• For unity gain: p1p2 = z1 z2
87
2DOF Controller
rH
TFSdy r
The 2 degrees-of-freedom structure permits disturbance rejection and tracking performance to be designed independently.
The output equation is
In practice, tuning is a two stage procedure:
• First, loop shaping is performed to design S for disturbance rejection
• Then the input filter Fr is designed to achieve desired tracking response
88
Two Degrees of Freedom Controllers
+_ yr
H
GF
Fr
+
+
FGH
FFG
r
y r
1
HFFG
GF
r
y
12
1
1
FGH
FGF
r
y r
1
Parallel Feedback Compensation
Feed-Forward Control
Feed-Forward Compensation
y+_r
H
G
F2
+_F1
89
Internal Model Principle
rGGy 1~ Then the output is
and setThe basis of the Internal Model Principle is to determine G~ 1~ GF
Suppose we have a model of G denoted G~
i.e. if the model if accurate, perfect control is achieved without feedback!
• Information about the plant may be inaccurate or incomplete
• The plant model may not be invertible or realisable
• Control is not robust, since any change in the process results in output error
In practice the approach has limited use, since...
90
Internal Model Control (IMC)
QH
QGF
11
• RHP zeros give rise to RHP poles – i.e. the controller will be unstable
FGH
FGQ
1
• An alternative to open loop shaping is to directly synthesize the closed loop transfer function
• time delay becomes time advance – i.e. the controller will be non-causal
• The approach is to specify a desirable closed-loop T.F. then find the corresponding controller
• In principle, any closed-loop response can be achieved providing knowledge of the plant is accurate and it is invertible
• if the plant is strictly proper, the inverse controller will be improper
• The plant G might be difficult to invert because...
• This method is known as Internal Model Control, or Q-parameterisation
91
IMC – Non-Minimum Phase Plant
nm GGG
HGf
GfGGF
n
nnm
111
• Step 1: factorise G into invertible and non-invertible (i.e. non-minimum phase) parts:
q
i i
isn zs
zseG
1
...where the non-invertible part Gn is given by
• Step 2: specify the desired closed loop T.F. to include Gn: Q = f Gn
• Step 3: substitute into the controller equation:
HGf
fGF
nm
11
This is an all-pass filter with delay. Any new LHP poles in Gn can be cancelled by LHP zeros in Gm
This equation does not require inversion of Gn
92
• A root locus plot is a diagram of the closed loop pole paths in the complex plane as loop gain k is varied from 0 to infinity
Root Locus Analysis
0)()()()( 2121 sskss
• Every locus begins at a pole when k = 0, and ends either at a zero or infinity (i.e. follows an asymptote)
k → ∞
k = 0
k → ∞
k → ∞
k = ∞
• The number of asymptotes is given by the pole excess, n - m
93
Angle & Magnitude Criteria
(n odd)
1FGH
kL
1
nL
Points on the root locus are given by 02121 k
i.e. all points on the root locus must satisfy the magnitude criterion:
2121 k
121
21
k
...and the angle criterion:
• When k = 0, the roots are at 12 = 0 ...i.e. at the poles of the open loop system.
• As k → ∞, the roots are at 12 = 0 ...i.e. at the zeros of the open loop system.
• For 0 < k < ∞, the roots can be found as follows...
94
• Complex conjugate poles lead to a step response which is under-damped
• If all closed loop poles are real, the step response will be over-damped
• Closed loop zeros may induce overshoot even in an over-damped system
• The response is dominated by s-plane poles closest to the imaginary axis
• When a pole and zero nearly cancel each other, the pole has little effect on overall response
Root Locus Interpretation
• Adding a pole to the forward path pushes the root loci towards the right
• Adding a LHP zero generally moves & bends the root loci towards the left
– rise time and bandwidth are inversely proportional– larger PM, GM and lower Mp improve damping
• Time domain and frequency domain specifications are loosely related:
• The further left the dominant system poles are...
– the faster the response– the greater the bandwidth
Tutorial 3.3
95
Intersection of Loci Asymptotes
mn
ii
12• For large k, the root loci are asymptotes with angles given by
• Asymptotes intersect at a point on the real axis given by
real parts of the poles of L(s) - real parts of the zeros of L(s) =n - m
Typical root locus for third order system with no zeros
3
3
96
Root Loci Asymptotes by Pole Excess
3
5
1
4
2
6
97
RHP Zero: High Gain InstabilityAs open loop gain increases, each root locus tend towards either an infinite asymptote or an open loop zero. i.e. for proper systems, each zero accommodates a closed loop pole at infinite gain.
• For each RHP zero one locus crosses into the RHP, so at sufficiently high gain the closed loop will become unstable
• Maximum achievable control gain (kcrit) occurs where the first closed loop pole crosses into the RHP
k → ∞
k = 0
Maximum value of kwhich gives stable response
k → ∞
k → ∞
98
PID Controllers
dt
dedttektu d
i
)(1
1)(
• PID (Proportional, Integral, Derivative) controllers enable intuitive loop tuning in the time domain, normally against a step response. Typical objectives are delay, overshoot, & settling time.
• Most common “parallel” form is
• General notes:– The proportional term k directly controls loop gain.– Integral action increases low frequency gain and reduces/eliminates steady state errors,
however this can have a de-stabilizing effect due to increased phase shift.– Derivative action introduces phase lead which improves stability and increases bandwidth. This
tends to stabilize the loop but can lead to large control movements.
• Many refinements and tuning methods exist (Ziegler-Nichols, Cohen-Coon, etc.)
dttei
)(1
dt
ded
99
Transient Response Tuning
A
BD
• Overshoot (A) - typically 20%• Decay ratio (D) - typically <0.3.
• Error bound () - typically 2% of step size• Settling time (ts) - small• Rise time (tr) – small• Steady state offset - small
Transient response specifications
Tutorial 3.4
100
Quality of ResponseA performance index can be used to assess transient response quality against variation of a key parameter.
For second order systems, a damping ratio of ~0.7 gives a minimum value for IAE & ITAE, and ~0.5 for IES.
dtte
0
2)(
IES = Integral of the Error Squared
dtte
0
|)(|
IAE = Integral of the Absolute Error
ITAE = Integral of Time x Absolute Error
dttet
0
|)(|
101
4. Discrete Time Systems
• Sampled systems
• The z-transform
• z plane mapping
102
The Digital Control System
+_ y(t)r(k)
H(s)
G(s)F(z)e(k) u(k)
DACu(t)
ADCb(t)b(k)
Continuous timeDiscrete time
+_
r(k)
F(z)e(k) u(k)
DAC u(t)ADCb(t)b(k)
t
b(t)
k
b(k)
k
u(k)
t
u(t)
103
Sampled Systems
Tfs
1
• The sampler converts a continuous function of time y(t) into a discrete time function y(kT)
• Almost all samplers operate at a fixed rate
• The dynamic properties of the signal are changed as it passes through the sampler
• The T is implicit in notation, so for example y(k) is equivalent to y(kT)
y(t) y(k)
t
y(t)
k
y(k)
Sampler
T
1 2 3 4 5 6 700
104
Discrete Convolution
Once the impulse response f (nT) is known, the controller output u(nT) arising from any arbitrary input e(nT) can be computed using the summation
n
k
TknfkTenTu0
)]([)()(
The impulse response of a discrete system is its’ response to a single input pulse of unit amplitude at time t = 0.
• The design task is to find the f (nT) coefficients which deliver a desired output u(nT) for some e(nT)
105
Discrete Convolution
T = 0: u(0) = f (0)e(0)
T = 1: u(1) = f (0)e(1) + f (1)e(0)
T = 2: u(2) = f (0)e(2) + f (1)e(1) + f (2)e(0)
T = 3: u(3) = f (0)e(3) + f (1)e(2) + f (2)e(1) + f (3)e(0)
T = n: u(3) = f (0)e(n) + f (1)e(n-1) + ... + f (n)e(0)
• Discrete convolution consists of sequence reversal, cross-multiplication, & summation
• The digital controller implements this n-term sum-of-products at each sample instant, T
k
e(k) u(k)
k
f (k)
k0 1 2 3 4 5 6 7 8 9 10 11 0 1 2 3 40 1 2 3 ... ...
Input e(k) Unit pulse response f (k) Output u(k)
F(z)e(k) u(k)
106
The Delta Function
)()()( afdttfat
ta
(t - a)
0
f (t)
• If a delta impulse is combined with a continuous signal the result is given by the screening property
• The delta function, denoted (t), represents an impulse of infinite amplitude, zero width, and unit area.
107
Impulse Modulation
0
)()()(*n
nTttftf
0
)()(n
T nTtt
108
The z Transform
Applying the screening property of the delta function at each sample instant, we find
)()()(*0
nTtnTftfn
0
)()(n
nznTfzF
...)2()2()()()()0()(* TtTfTtTftfsF L
The shifting theorem allows us to take the Laplace transform of this series term-by-term...
0
)()(*n
snTenTfsF
The z transform of f(t) is found from the above series after making the substitution z = esT
109
z Transforms
)(t nz
)(0
nTtn
1z
z
)(1 t 1z
z
nT 2)1( z
Tz
(unit step)
(unit ramp)
2)(nT 3
2
)1(2
)1(
z
zzT
anTe aTez
z
anTenT 2)( aT
aT
ez
Tze
nTa1 ))(1()1(
azzaz
ate1 ))(1(
)1(aT
aT
ezz
ez
nTsin 1cos2
sin2 Tzz
Tz
nTe anT sin aTaT
aT
eTezzTez
22 cos2sin
nTcos 1cos2
)cos(2
Tzz
Tzz
nTe anT cos aTaT
aT
eTezzTezz
22
2
cos2cos
f (nT) F (z) f (nT) F (z)
110
z Transform Theorems
)()()]()([ 22112211 zFazFanTfanTfa Z• Linearity:
)()()]([)( 2120
1 zFzFTknfkTfn
k
Z• Convolution of time sequences:
)()1(lim)(lim1
zFznTfzn
• Final value theorem:
)()( zFzknf kZ• Time shift:
111
z Plane Poles & Zeros
))...()((
))...()(()(
21
21
n
m
pspsps
zszszsksF
The z-transform of an equivalent sampled data system can be found using a discrete transformation, which yields a transfer function in the complex variable z:
Assume we have a continuous system with the Laplace transfer function:
))...()((
))...()(()(
21
21
n
m
pzpzpz
zzzzzzkzF
• Poles & zeros in the discrete domain are in different positions compared with those in the Laplace domain
• The pole excess of the continuous and discrete representations may not be the same
• Time and frequency domain performance of the two systems will be different
112
Stability
j
-j
-1 1
a
1
az
b
ze
zu
)(
)(
1
1
)(n
i
i ineab
k u(k)
1
2 be(1)
be(2) + abe(1)3
be(3) + abe(2) + a2be(1)
be(4) + abe(3) + a2be(2) + a3be(1)
4
5
• For stability we have the constraint on the pole: | a | ≤ 1
The first order discrete time transfer function
...corresponds to the difference equation u(k + 1) = be(k) + au(k)
The evolution of the time sequence is:
n
0
113
Simple z Plane Poles
Recall for continuous-time systems:
If p = + j, u(t) = et ejt, and for stability we want et to remain finite.
i.e. the pole at (s = p) must have a negative real part, and
• stable poles must lie in the left half of the s-plane
G(s) = 1
s – phas an impulse response of the form y(t) = ept
Now, in discrete systems:
Therefore, for stability |p| must be less than 1, and
• stable poles must lie within the unit circle in the z-plane
G(z) = z
z – p has a unit pulse response y(nT) involving the term pn
114
Complex z Plane Poles
))(()(
2
jj pezpez
zzG
As for continuous time systems, discrete time complex poles always exist as conjugate pairs.
The impulse response is given by
• complex poles give rise to an oscillatory response and have the same stability constraints as simple poles
njnj peApeAnTy *11)(
)()()(
*11
jj pez
A
pez
Azy
...where the residual A1 has the form aej
npnTy n cos2)(The time sequence can be written
115
z Transforms of Common Signals
2)1( z
z
az
z
)1)((
)1(
zaz
azna1
na
nT
1z
z1
1][T
Waveform f (nT) z-planeF(z)
116
z Transforms of Common Signals
Waveform f (nT) z-planeF(z)
1cos2
)cos(2
aTzz
aTzz
anTsin
anTcos
1cos2
sin2 aTzz
aTz
bnTan sin 22 cos2
sin
abTazz
bTaz
117
Frequency Response
*))(()(
azaz
bzzG
For the first order system magnitude and phase are found from
*00
0
0 )(aeae
beeG
TjTj
TjTj
321*)( 0000 aeaebeeG TjTjTjTj
j
-j
-1 1
0
3
1
1
b
a
a*
• The response of G(j) at frequency = 0 is evaluated by TjezzG 0)(
118
z Plane Mapping
Equivalent regions shown cross-hatched
-j
j
AB C
DE F
A
BC
DE
F
2sj
2sj
s plane z plane
119
Complex Plane Mapping
1 = T
2 = ’T
r1 = e-T
r2 = eT
r3 = e-T
z = esT = e(a+jb)T = eaTejbT = rej• Points in the s-plane are mapped according to
120
The Nyquist Frequency
• The Nyquist frequency represents the highest unique frequency in the discrete time system
• Uniqueness is lost for higher continuous time frequencies after sampling
z-planes-plane
2sj
2sj
121
Aliasing
• Loss of uniqueness means an infinite number of congruent strips are mapped into the unit circle
122
Pole Location vs. Step Response
123
Complex Plane Grid
Lines of constant decay rate ( ) and damped natural frequency (d )
d8
123456
d9
d10
d11
d12
d13
d8
d9
d10
d11
d12
d13
d
s plane z plane
0.1 /T
0.2 /T
0.3 /T
0.4 /T0.5 /T
0.6 /T
0.7 /T
0.8 /T
0.9 /T
/T
-0.1 /T
-0.2 /T
-0.3 /T
-0.4 /T-0.5 /T
-0.6 /T
-0.7 /T
-0.8 /T
-0.9 /T
0- /T
124
Complex Plane Grid
Lines of constant damping ratio ( ) and un-damped natural frequency (n)
125
Root Locus Design Constraints
• Equivalent second order loci allow regions of the complex plane to be marked out which correspond to closed loop root locations yielding acceptable transient response
126
5. Digital Control Design
• Pole-zero matching
• Numerical approximation
• Invariant methods
• Direct digital design
127
Pole – Zero Matching
))()((
)()(
221
1
pspsps
zsAsF
))()((
)()1()(
321
1
1 TpTpTp
Tz
ezezez
ezzAzF
1. Transform the poles & zeros of the transfer function using z = esT
2. Map any infinite zeros to z = -1 (but maintain a pole excess of 1)
3. Match the gain of the transformed system at z = 1 to that of the original at s = 0
Tutorials 5.1 & 5.2
128
Numerical Approximation
)()()( taetautu as
asF
)(Starting with the simple controller we get the differential equation
t
dttuteatu0
)()()(The solution to the continuous equation is
An equivalent discrete time controller performs this integration in discrete time:
t
k k+1
e(t)-u(t)
1
)()()()1(k
k
dttuteakuku
129
Forward Approximation Method
tk k+1
)()()()1( kukeaTkuku
aT
za
ze
zuzF
1)(
)()(
T
zs
1
The forward approximation method implies we can find the z-transform directly from the Laplace transform by making the substitution:
Approximating the unknown area using a rectangle of height a{e(k) - u(k)}...
Application of the shifting theorem and simple algebra leads to...
The forward approximation rule maps the ROC of the s plane into the region shown. The unit circle is a subset of the mapped region, so stability is not necessarily preserved under this mapping.
130
Backward Approximation Method
)1()1()()1( kukeaTkuku
aTz
za
zF
1
)(
Tz
zs
1
The backward approximation method implies we can find the z-transform directly from the Laplace transform by making the substitution:
Approximating the unknown area using a rectangle of height a{e(k+1) - u(k+1)}...
Application of the shifting theorem and simple algebra leads to...
The backward approximation rule maps the ROC of the s plane into a circle of radius 0.5 within the z plane unit circle. Pole-zero locations are very distorted under this mapping.
131
Trapezoidal Approximation Method
tk k+1
)1()1()()(2
)()1( kukekukeaT
kuku
azz
T
azF
112
)(
The trapezoidal approximation method implies we can find the z-transform directly from the Laplace transform by making the substitution:
Approximating the unknown area using a trapezoid...
Application of the shifting theorem and simple algebra leads to...
Trapezoidal approximation maps the ROC of the s plane exactly into the unit circle.
1
12
z
z
Ts
This method is also known as Tustin’s method or the bi-linear transform.
132
Numerical Approximation Methods
Forward approximation
Backward approximation
Trapezoidal approximation
)()( kukeaTI
)1()1( kukeaTI
)()()1()1(2
kukekukeT
aI
T
zs
1
Tz
zs
1
1
12
z
z
Ts
-j
j
-1 1
j
-1 1
-j
j
-1 1
-j
133
Step Invariant Method
2. Find the corresponding z-transform of the response
• Invariant methods emulate the response of the continuous system to a specific input
1. Determine the output of the output of the continuous time system for the selected hold input
3. Divide by the z-transform of the selected input
• Step and ramp invariant methods are also known as ZOH and FOH equivalent methods respectively
• Invariant methods capture the gain & phase characteristics of the respective hold unit
134
Direct Digital Design
• In direct design we begin by transforming the plant model into discrete form using the step invariant method. This captures the action of the ZOH element which precedes the plant.
• Standard design techniques can then be used to synthesize the controller.
• The design iterates as many times as necessary until a satisfactory controller is reached.
• Control performance with the direct method is usually significantly better than with emulation methods.
Tutorial 5.3
135
6. Digital Control System Implementation
• Sample rate selection
• Sample to output delay
• Reconstruction
• Control law implementation
• Aliasing
136
Sample Rate Selection
Slow sample rate
• Reduced CPU load
• Cheaper processor
• Cheaper data converters
• Performance trade-offs required
• Larger passives components required
• Increased output ripple
• More sensitive to disturbances
• More complex & expensive anti-aliasing filters
Fast sample rate
• Increased CPU load
• More expensive processor
• More expensive data converters
• Faster transient response
• Smaller & cheaper passive components
• Reduced output ripple
• Better disturbance rejection
• Simpler & cheaper anti-aliasing filters
137
Sample Rate Selection
• Acceptable results will normally be obtained for the range 4 ≤ N ≤ 10
T
tN rDefine N as the ratio of rise time to sample period:
N
tT ror
• For control systems, choice of sample rate is based on the 10% to 90% rise time of the plant
138
Sample to Output Delay
td = sample to output delay
139
Phase Delay
• Delay in the time domain translates into frequency dependent phase lag in the frequency domain
• Phase lag is indistinguishable from delay in the time domain.
)()( sFtf LFrom the shifting property of the Laplace transform, if )()( sFetf s Lthen
Therefore, if F(j) = re j then e-jD F(j) = re j(-D)
140
Reconstruction
)()( kutu
Zero Order Hold (ZOH)
kTtT
kukukutu
)1()()()(
First Order Hold (FOH)
Hold functions attempt to construct a smooth continuous time signal from a discrete time sequence.
TktkT )1( To do so, the function must approximate over the interval
141
Zero Order Hold
j
ejF
Tj
ZOH
1)(
The frequency response of the Zero Order Hold function can be modelled using a unit pulse
sZOH
TjF
2)(
222
2
2sin
22
2)(
Tj
Tj
TjT
j
ZOH eT
j
j
j
eeejF
2/
2sinc)( Tj
ZOH eT
TjF
This can be simplified using the exponential form of the sine function
Further simplification using the sinc function yields
This is a complex number expressed in polar form, where the angle is given by
• The Zero Order Hold contributes a frequency dependent phase lag to the loop response
142
ZOH Phase Approximation
Pha
se e
rror
(de
g)
2
T
• Comparison of phase error vs. normalised frequency for various sample rates shows the
4s
2
T holds reasonably well for approximation
143
Data Converter Gain
22maxN
bb
12maxN
uu
bK ADC
1 uKDAC
• Converter gains are related to quantization limits by:
• End-to-end controller gain is KADC | F(z) | KDAC
• Quantization takes place at each continuous/discrete boundary. Resolution limits can be referred to the continuous domain by:
144
Resolution Based Limit Cycles
Dis
plac
emen
t (m
m)
Dis
plac
emen
t (m
m)
• Steady-state limit cycles may appear if the sense resolution (b) is worse than the control resolution (u)
• Limit cycle oscillation will be low amplitude & high frequency, and may not cause an issue in low bandwidth systems, or in systems with high inertia or friction. In some cases, the oscillation may even be beneficial as it introduces a “dither-like” effect.
145
The Limit Cycle Condition
21 22maxmaxNN
uk
u
22maxN
bb
12maxN
uu
The “limit cycle condition” is: control resolution > sense resolution: u > b
In the steady state, control and sense quantities are related by
Control resolution is:
Sense resolution is:
This gives the inequality
kNN 221 log
maxmaxmax |)0()0(| ukuHGb
21 21
2 NN
k
• To avoid a steady-state limit cycle, control effort resolution must satisfy the inequality
146
Finding the Difference Equation
nnnn
mmmm
zzz
zzz
ze
zu
11
10
11
10
...
...
)(
)(
nn
nn
nm
nm
nmnm
zazaza
zbzbzbzb
ze
zu
1
11
1
11
110
...1
...
)(
)(
nn
nn
nm
nm
nmnm zazazazuzbzbzbzbzezu
11
11
11
110 ...)(...)()(
)()2(...)1()()1(...)1()()( 11110 nkuakuakuankebnkebmnkebmnkebku nnmm
Normalizing for the term involving the highest denominator power (a0) gives
Applying the shifting property of the z-transform term-by-term yields the difference equation
The n-pole m-zero transfer function is written
147
Control Law Diagram
)()1(...)1()(...)2()1()()( 11210 nkuankuakuankebkebkebkebku nnn
The general n-pole n-zero control law can be represented in diagrammatic form as shown below...
148
2P2Z Control Law
z-1 z-1
+
z-1 z-1
+ +
-a1-a2
b0 b1 b2
e(k-2)e(k-1)e(k)
u(k-2) u(k-1)
u(k)
)2()1()2()1()()( 21210 kuakuakebkebkebku
The 2-pole 2-zero (2P2Z) control law can be represented graphically as shown below.
149
Pre-Computation
)2()1()2()1()()( 21210 kuakuakebkebkebku
)1()()( 0 kvkebku )2()1()1()()( 2121 kuakuakebkebkv
Sample-to-output delay can be reduced by pre-computing that part of the control law which is already available.
Unknownat t = k-1
Can be pre-computed at t = k-1
150
Series of Constant Samples
TjTee Tj
js
sT
sincos
The sampler modifies the frequency response of a signal according to:
ss
j
je s
2sin2cos
2
Since s
T2
, this can be written
12sin2cos2
njne
s
s
n
j
When = ns ... (n integer)
• Whenever the input frequency is a multiple of the sample rate (ns), a series of constant sample values will be produced. i.e. DC is indistinguishable from any signal which is a multiple of the sample rate.
151
Discrete Frequency Ambiguity
• Both f1 & f2 give rise to exactly the same set of samples. After sampling it is impossible to determine which frequency was sampled. In fact, any of an infinite number of possible sine waves could have produced these samples. This effect is known as aliasing.
152
Frequency Response of a Sampled System
k
kTttyty )()()(*
n
tjnn
k
seCkTt )(
dtekTtT
CT
T k
tjnn
s
2/
2/
)(1
The sampler is periodic so can be represented by the Fourier series
...where the Fourier coefficients are given by
dtetT
CT
T
tjnn
s
2/
2/
)(1 Only one term is within range of the integration, so
)()()( afdtattf
T
eT
CT
Tn
11 2/
2/0
We can integrate this easily using the “screening” property of the delta function
n
tjn
k
seT
kTt 1)(So, the Fourier series representing the sampler is given by
The sampled signal is given by
153
Frequency Response of a Sampled System
)()()(0
sFdtetftf st
L
0
1)()(*)(* dtee
TtytysY st
n
tjn sL
n
tjns dtetyT
sY s
0
)()(1
)(*
We can now find the Laplace transform of the sampled system
n
sjnsYT
sY )(1
)(*
The integral term is the same as the Laplace transform of y(t), but with a change of complex variable
n
snjYT
jY )(1
)(*
The frequency response of the samples signal is:
Each term in the infinite summation corresponds to the response of the continuous system, shifted along the frequency axis by ±ns
n
tjn
k
seT
kTt 1
)(
154
Frequency Response of a Sampled System
Continuous Spectrum
Sampled Spectrum
155
Anti-Aliasing
(dB)
c
-20 log10(2N)
s
2
0
0
2s• To prevent aliasing, we need to attenuate the input signal to less than 1 converter bit at before sampling.
Filter constraints can be relaxed if a faster sample rate is selected.
156
Suggested Design Checklist
Design controller using emulation or direct method
Select a suitable processor
Design anti-aliasing filter
Carefully select a sample rate
Account for ZOH effects
Simulate & iterate!
Evaluate sample-to-output delay
Account for data converter gains
157
Review
1. Fundamental ConceptsLinear systemsTransient response classificationFrequency domain descriptions
2. Feedback ControlEffects of feedbackSteady state errorStability & bandwidth
3. Controller DesignPhase compensationRoot locus analysisTransient tuning
4. Discrete Time SystemsSampled systemsThe z-transformz plane mapping
5. Digital Control DesignPole-zero matchingNumerical approximationInvariant methodsDirect digital design
6. Digital Control Systems ImplementationSample rate selectionSample to output delayReconstructionControl law implementationAliasing
158
Suggested Reading
• J.J.DiStefano, A.R.Stubberud & I.J.Williams, Feedback & Control Systems, Schaum, 1995
• J.Doyle, B.Francis & A.Tannenbaum, Feedback Control Theory, Macmillan, 1990
• S. Skogestad & I. Postlethwaite, Multivariable Feedback Control, Wiley, 2005
• Gene F. Franklin, J. David Powell & Michael L. Workman, Digital Control of Dynamic Systems,Addison-Wesley, 1998
• K.J.Astrom & B.Wittenmark, Computer Controlled Systems, Prentice Hall, 1997
Table of selected Laplace transforms
f (t) F(s)
)(t 1
)( Tt sTe
)(t1 (unit step)
s
1
)( Tt 1 (delayed step) sTe
s1
)()( Ttt 11 (rectangular pulse) )1(
1 sTes
t (unit ramp)
2
1
s
)!1(
1
n
t n
ns
1
ate
as 1
atn et
n
1
)!1(
1 nas )(
1
tsin
22 s
tcos 22 s
s
te at
sin
1 22)(
1
as
te at cos
22)(
as
as
f (t) F(s)
)(1 btat ee
ab
))((
1
bsas
)(
1 btat beaeba
))(( bsas
s
)1(
1 atea
)(
1
ass
)1(
12
atat ateea
2)(
1
ass
)1(
12
atat eeata
)(
12 ass
ab
ae
ab
be
ab
btat
11
))((
1
bsass
)cos1(
12
t
)(
122 ss
te d
t
d
n
sin1 22 2
1
nnss
)(sin
112
te dt
dnn
n )2(
122
nnsss
)(sin t
22
cossin
s
s
Table of selected z-transforms
f (nT) F(z)
)(t nz
)(
0
nTtn
1z
z
)(1 t (unit step)
1z
z
nT (unit ramp)
2)1( z
Tz
2)(nT 3
2
)1(2
)1(
z
zzT
anTe
aTez
z
anTenT 2)( aT
aT
ez
Tze
nTa1
))(1(
)1(
azz
az
ate1
))(1(
)1(aT
aT
ezz
ez
nTsin
1cos2
sin2 Tzz
Tz
nTe anT sin
aTaT
aT
eTezz
Tez22 cos2
sin
nTcos
1cos2
)cos(2
Tzz
Tzz
nTe anT cos
aTaT
aT
eTezz
Tezz22
2
cos2
cos
Notes
Notes
Notes
Notes
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