eee4093_qft
DESCRIPTION
Fundamentals of QFT controller designTRANSCRIPT
EEE4093 2013
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UNIVERSITY OF CAPE TOWN
Department of Electrical Engineering
EEE4093 – Quantitative Feedback Theory (QFT) design
Section 1: Introduction to Quantitative feedback design
This chapter is aimed at enhancing existing knowledge of feedback systems assuming some
background in control theory (differential equations and Laplace transform at some time in the
past!). The material will to enable the student to quantify the design problem and the plant model
and then design, analyse and tune controllers for the problem. It will be interactive and will use
practical examples where possible.
Because of very limited number of lectures, we will look at single-input, single-output design only.
QFT is by far the best control design method I know for SISO systems!
Lecture 1 Introduction, Inverse Nichols Chart, Nichols Chart
Lecture 2 Sensitivity design
Lecture 3 Sensitivity design
Lecture 4 Tracking design
Lecture 5 Tracking design
Lecture 6 Tracking error design
Lecture 7 Design for digital implementation
Lecture 8 Consolidation
1.1 The real reasons for feedback control - what feedback does and does not do
A quick scan for the keyword, “specification,” in the index of undergraduate “control engineering”
textbooks will reveal one of the problems with the subject: Because of the wonderful tools and
techniques available to the teacher, one may forget that engineering design is a creative endeavour
that balances client’s requirements for performance (utility), with cost, aesthetics and other issues.
Before undertaking a control system design, there are therefore a number of items that should be
considered.
Feedback is mandatory when
a) There are unmeasured disturbances (loads) that affect the system output in an undesirable way.
Feedback is then required for regulation whether the plant is uncertain or not because of the
signal uncertainty.
b) The plant is uncertain and accurate (dynamic or static) command following is required. The
feedback and possibly pre-filtering is then required for robust tracking.
c) If the plant is unstable, feedback may be used to stabilise it. This is really a special case of (a)
or (b) because, in a hypothetical world, if the right hand plane singularities of the plant were
known with infinite precision, one might attempt to cancel them with an open loop compensator
of infinite precision. In chemical plants the usual source of instability is exothermic reactions.
In that case, the linear model of the process has a single right hand plane pole, which can easily
be stabilised by high gain feedback with the correct sign. A distributed exothermic process
which is not sufficiently well stirred can have an essential singularity in the right hand plane in
which no control effort can stabilise it - that is a run-away reaction, e.g. in a fluidised bed
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boiler, one can have a section of the bed melting if there is sufficient fuel. These exothermic
nasties are usually avoided or a self-stabilising design is used to make the plant fail-safe against
control system malfunctions (e.g. try to reverse a car blindfolded or a drunk walk). Textbooks
often devote lots of space to stability rather than to engineering design of feedback systems so
we will not spend too much time on instability in this course - except that we should not design
a controller which makes a stable process unstable! Other examples of unstable processes
include the inverted pendulum that originated from rocket launch problems (I have seen
working double and even triple inverted pendulums - academic toys), nuclear power plants
(which do not cause a nuclear explosion and other manifestations of instability if they are not
controlled), and modern fighter aircraft. Note that some in some systems instability is not of
consequence if the effects of instability appear late enough. Missiles in final approach or
population explosions (human, seal, lion) are examples.
Control engineering design may only require a contribution to the plant design if it is possible to
achieve the required performance without the expense of feedback.
Before embarking on a feedback design, we must be convinced that there is a technical justification
to use feedback, and there must be realistic (achievable) specifications both for performance and for
cost (usually measured at the plant input).
Figure 1.1 – Prototype feedback system
Stabilising an open loop unstable process
Reducing the effect of disturbances on the system output
Reducing the effect of plant variations on performance
Controller
Plant +
-
desired
output
sensed
output
actual
output error input
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Reducing the effect of unmeasured disturbances on the plant output
Example: Flow controller
Flow control example
Control system structure
Let us take SP = [0,1], f2=[0,1], f1=[0,1] (all in m3/h).
The process gain is 1 [m3/h]/[input unit] – be careful of getting the gain correct if there are lots of
unit conversions.
The plant can deliver the expected performance so we do not make fools of ourselves trying to
“tune” the controller. (No amount of feedback and tuning can make a Volkswagen Beetle
performance the same as that of a Porsche even if they look and sound similar.)
Write down the system behaviour with feedback, assuming that there are no dynamics for the time
being (incompressible fluid etc.):
f3 = f1+f2
f1 = p g (SP - f3)
or
21
1
13 f
pgSP
pg
pgf
We see the effect that the feedback loop has on reducing the effect of disturbances: We can always
modify the SP so that we get the desired output when there is no disturbance flow. This can be
done using pre-processing of the set-point (see below) and does not have to be done by hand. We
see that the larger the controller gain, g, the smaller the effect of the disturbance on the output. To
get PV=SP with f20 requires g = . This is achievable in steady state (using an integrator) but not
generally. We need realistic expectations.
wild
flow
Fv
Fi
f1
f2
f3
g
p +
-
PV
error OP=u f3
f2 +
+ f1 SP
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What gain is required to get PV to within 10% of SP if SP=1?
Interpret specifications: 0.9 y 1.1
Worst cases:
f2 = 0 f2 = 1
11
3
pg
pgf
g 9 g
pgpg
pgf
1
11
11
13
We can be smart and introduce a pre-filter to get a 2 degree of freedom controller and use the whole
range of allowed output.
Two degree of freedom control system
Calculate the gain, g, and pre-filter gain, f (be careful not to confuse notation with flows) required
to satisfy the 10% specification above.
Worst cases:
f2 = 0, 11
3
pg
fpgf f2 = 1, 1
1
11
13
pgpg
fpgf
Solving simultaneously, g 4, f=1.125. (In general, how do we know that the worst cases of f2 are
the worst cases of the final f3?)
S
P
f
g
p +
-
PV
error OP=u f3
f2
+ + f1
pre-
filter
f2
f3 1.1
1.0
0.9
g=0
g=9
1.0
wasted
f2
f3 1.1
1.0
0.9 g=4
f=1.125
1.0
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We should be aware of why we want to minimise the gain. Often a great deal of engineering effort
goes into “optimising” some variable that should not be optimised. (e.g. Maximising yield without
regard to reagent, life cycle and input costs does not necessarily optimise profit.) Various optimal
performances are shown as a function of damping factor, below for second order systems.
ref. OGATA
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Reducing the effect of plant variations on performance
In many situations it is not practicable or possible to build a perfect plant. We would still like to
maintain some input-output performance. Later on we will do more dramatic examples but
continuing the flow control example, let us assume that the valve sizing is completely wonky (a
chemical engineering term) and that the process gain varies from 1%/% to 10%/%. With no
disturbance (f2=0) to simplify the discussion, we would like |SP-PV| (i.e. the error) to be within
10% of the SP.
With no pre-filter (f=1) the gain required is,
9
1.11
9.0
13
g
gp
gp
SPgp
gpf
or gpgp
gp
PV
SPPV
1
1
11
1.0
…
With a pre-filter, the gain is,
Worst cases:
1.11 max
max gp
fgp and 9.0
1 min
min gp
fgp
g = 3.95, f = 1.128
(better than half the gain)
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1.2 Transfer Functions
The transfer functions to the plant output and plant input can be calculated as required. For
example:
/1
Y R
F LT
H L
/
1
1Y DoT
L
/1
U R
F GT
L
/ /1
U Do U N
H GT T
L
Observe the effect that high gain has on each of these transfer functions. For L>>1,
TF
HY R/ “Ideal” tracking by design of F/H
TY Do/ 0 “Ideal” disturbance rejection
TF
PHU R/ T T
PU Do U N/ /,
1
1.3 The Nichols and inverse Nichols charts
In the frequency domain, to design for
required reference- and disturbance- to
output response will require selecting
L(s)=G(s)P(s)H(s) to achieve certain
behaviour on 1
L
L and on
1
1 L
The relationship between complex
L(j) = x+jy and 1
1M
L
, a constant
is,
(1+x)2 + y
2 = 1/ M
2
circles centre (-1,j0), radius 1/M
error
U(s)
plant
inputreference
R(s)F(s) G(s) P(s)+
-
output
Y(s)
disturbance
Do(s)
pre-filter
H(s)
+sensor noise
N(s)
++
+
controller plant
sensor
++
Di(s)
-1
M=1/2
M=1
M=2
L
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Notice that if K=1/L (KdB = -LdB, Arg(K) = -Arg(L)), 1
L
L =
1
1 K so we can show the mapping
from L to L/(1+L) by turning the inverse Nichols chart upside down to get the Nichols chart
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0
10
20
30
40
6 db
3 db
1 db
0.5 db
0.25 db
0 db
-1 db
-3 db
-6 db
-12 db
-20 db
-40 db
Open-Loop Phase (deg)
Open-L
oop G
ain
(db)
INVERSE NICHOLS CHART
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-30
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0
10
20
30
40
Open-Loop Phase (deg)
Open-L
oop G
ain
(dB
)
NICHOLS CHART
6 dB
3 dB
1 dB
0.5 dB
0.25 dB
0 dB
-1 dB
-3 dB
-6 dB
-12 dB
-20 dB
-40 dB
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Relationship between Bode plots and the Inverse Nichols Chart
MATLAB CODE l=tf(20*[-1/100 1],conv([1 1],[1/100 1])); [m,p,w]=bode(l,{0.1,100});
figure(1)
semilogx(squeeze(w),db(squeeze(m))) axis([0.1,100,-30,30])
grid
figure(2) semilogx(squeeze(w),squeeze(p))
axis([0.1,100,-270,0])
grid figure(3)
ingrid('new')
nichols(l)
axis([-270,0,-30,30])
INVERSE NICHOLS CHART
Open-Loop Phase (deg)
Open-L
oop G
ain
(dB
)
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-20
-10
0
10
20
30
-20 db
-12 db
-6 db
-3 db
-1 db
0 db
0.25 db
0.5 db
1 db
3 db
6 db
10-1
100
101
102
-30
-20
-10
0
10
20
30BODE MAGNITUDE PLOT
FREQUENCY [rad/s]
MA
GN
ITU
DE
[dB
]
10-1
100
101
102
-250
-200
-150
-100
-50
0BODE PHASE PLOT
AN
GLE
[deg]
FREQUENCY [rad/s]
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Example
1
( )1
P ss
Effect of gain of 10
The Nichols chart provides a mapping from L (rectangular grid with |L| dB and Arg(L)) to
|L/(1+L)|, to enable graphical interpretation of consequences (on the closed loop transfer
function) of design action (on the open loop transfer function).
1.3 Gain, phase and robust stability margins
We need to design engineering systems that are “far away” from the point of instability (-1,j0)
on the arithmetic complex plane (Nyquist diagram) and (0dB, -180) on the log-polar plane
(Nichols chart). We can either do this by specifying the phase reserve at the gain cross-over
point (phase margin) and the gain margin at the phase cross-over or we can specify a
maximum value of |1/(1+L)| say dB.
(You can show that |1/(1+L)| < m (arithmetic), m > 1, gives gain margin of gm = m/(m-1) and
a phase margin of m = 2 arcsin(1/(2 m)).)
Example
Shown are Nyquist plots of 2
1
1 1L
s s s
, which have a phase margin of 52 and a
gain margin of 3.2 (=10dB) the L locus misses the M = 1.9 (5.6 dB circle of |1/(1+L)|
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-20
-10
0
10
20
30
6 db
3 db
1 db
0.5 db
0.25 db
0 db
-1 db
-3 db
-6 db
-12 db
-20 db
Open-Loop Phase (deg)
Open-L
oop G
ain
(db)
INVERSE NICHOLS CHART
100
10
1
3
30
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-20
-10
0
10
20
30
6 db
3 db
1 db
0.5 db
0.25 db
0 db
-1 db
-3 db
-6 db
-12 db
-20 db
Open-Loop Phase (deg)
Open-L
oop G
ain
(db)
INVERSE NICHOLS CHART
100
10
1
3
30
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-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 -2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Real Axis
Imag A
xis
L=2.5/(s^2+2s+1)/(s+1)
m
1/gm
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 -2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Real Axis
Imag A
xis
L=2.5/(s^2+2s+1)/(s+1)
m=1.92
The same information is available on the inverse Nichols chart
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1.4 Relative Differential Sensitivity
There may be some small, unmodelled uncertainty in the plant. (We can also handle gross
uncertainty in specific parameters but in another course!) Let us investigate the effect that
this has on the closed loop system, /
1Y R
F LT
H L
, relative to the ratio of transfer functions .
1
1
T
P
T TS
P P L
Example
What is the relative differential sensitivity of T
with respect to the plant, P, if
G=1/s, P=1/(s/2+1).
Answer: 2
2
1 2
1 2 2
T
P
s sS
L s s
The sensitivity is less than 1 (0dB) for frequencies
below 1 rad/s and this means that the closed loop
is less sensitive to changes in the plant than the
open loop. This is what we want the feedback
system to do - reduce sensitivity. At frequencies
above 1, the sensitivity is more than 1 (>0dB) and
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0
10
20
6 db
3 db
1 db
0.5 db
0.25 db
0 db
-1 db
-3 db
-6 db
-12 db
-20 db
Open-Loop Phase (deg)
Open-L
oop G
ain
(db)
INVERSE NICHOLS CHART
0.1 0.3
1.0
3.0
m
gm
10-1
100
101
-20
-15
-10
-5
0
5BODE PLOT - SENSITIVITY
|1/(
1+
L)| [
dB
]
FREQUENCY [rad/s]
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this means that the closed loop sensitivity is worse than the open loop sensitivity. To
illustrate this, let the plant be perturbed to 1P =0.9 P and 2P =1.1 P
1.5 Statistical basis for feedback
So far we have only done worst-case calculations. In many situations, the interest is in
reducing the variance or standard deviation of the error. In consumer situations there is often
a hard lower limit on some variable and properly designed feedback allows us to come close
to that limit by reduction of the variance. For example, lysine is sold by weight and there is a
maximum moisture content specified by the product specification. All other things being
equal, it is very profitable to sell customers water at the price per kg of lysine! The less we
need to over-dry the product to guarantee that the maximum moisture content will not be
violated, the more water can be sold at a huge profit.
If variation comes from a number of independent sources, the distribution is Gaussian or
normal. For a mean, and standard deviation, , the density is,
2
2
1
2
1)(
x
exf
and the standard deviation then has the following meaning:
%7.9933
%5.9522
%68
Xp
Xp
Xp
-4 -2 0 2 40
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
standard deviations
Gaussian density function
Example: The wild flow in the disturbance rejection example above with fixed plant gain of 1
has a mean of 0.5 and a standard deviation of 0.25 (it may not be possible that the flow is
negative but we have not specified the distribution). How much gain is required to ensure that
the output standard deviation is less than 0.1 if the SP=0.5 with p=1?
5.1
25.01
11.0
1
1
2
2
2
2
2
2
2
3
g
gp
gpff
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Notice that when we deal with linear dynamic systems, noise signals pass through the system
with their power spectral density modified by the square of the transfer function gain. Also,
the variance at the output (y = P u) is given by Parseval’s identity from the power spectral
density (i.e. variance is related to the area under the curve with linear frequency and
amplitude axes)
djSjP
djS
uu
yyy
2
2
2
1
2
1
Summary of introduction
Feedback has the following purposes:
Stabilising unstable processes
Reducing the effects of disturbances on the output
Reducing the effect of plant variation on the behaviour
High gain feedback solves the last two problems but will have effects at the plant input to be
explored in future chapters. As in any engineering activity there is a trade-off between
performance (output) and cost (control effort). Reasonable expectations of control loop
performance will allow realistic controllers to be designed. Some rough calculations will give
good insight into the design or tuning of feedback loops.
Two degree of freedom feedback system
error
U(s)
plant
inputreference
R(s)F(s) G(s) P(s)+
-
output
Y(s)
disturbance
Do(s)
pre-filter
H(s)
+sensor noise
N(s)
++
+
controller plant
sensor
++
disturbance
Di(s)
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Section 2: Design for disturbance rejection and robust
stability with plant uncertainty
Recall that the transfer of output disturbance to the plant output with feedback is,
LGPHT DoY
1
1
1
1/ (1.1.1)
so all output disturbance rejection and sensitivity designs are based on the inverse Nichols
chart.
Example 1.1.1 (revision)
Given P=1/s, design for
i. |1/(1+L)| -20dB, 0.5
ii. |1/(1+L)| 3dB,
Confirm that G=5/(s/6+1) does
the job
Example 1.1.2
Suppose that the above plant is a 1m3 tank and that the inlet valve has a gain between 1 and 5
m3/%/hour. The gain uncertainty could be because of non-linear installed characteristic as a
result of wrong sizing; poor turndown; or upstream pressure variations. The same regulation
specifications must be met for all possible plant conditions – i.e. robust performance.
The plant model is now
P(s) = k/s, k [1, 5]
We could repeat the design for different values of process gain and try to guess at a controller
that would meet the specifications. It is better to have a systematic method.
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-5
0
5
10
15
20
25
30
Open-Loop Phase (deg)
Op
en
-Lo
op
Gain
(d
b)
6 db
3 db
1 db -1 db
-3 db
-6 db
-12 db
-20 db
L(j0.5)
L(j10)
L(j3.0)
Inverse Nichols Chart
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-20
-15
-10
-5
0
5
10
15
20
25
30
Open-Loop Phase (deg)
Op
en
-Lo
op
Gain
(d
b)
6 db
3 db
1 db -1 db
-3 db
-6 db
-12 db
-20 db
L(j0.5)
L(j10)
L(j3.0)
Inverse Nichols Chart
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2.1 Quantitative Feedback Design:
1) Using the log-polar plane (i.e. the Nichols chart), the controller elements only shift the
plant elements by the controller gain and phase at each given frequency.
dBdBdBGPL , GPL ArgArgArg
The controller does not change the relative gain between different instances of the plant.
2) At any particular design frequency (that we choose), we can draw the locus of the plant
elements as the plant parameters vary. This will define a region on the Nichols chart,
corresponding to the controller, G=1 (0dB, 0). This is called the template,
)()( ii jPj T
3) We choose an arbitrary, fixed plant called the “nominal” to use as a handle on the design
process. By moving the template on the (inverse) Nichols chart we can define the region
where the specifications (at the design frequency) are met for plant elements by plotting
the locus of the nominal plant. i.e.
nominal boundaries satisfied for the nominal plant
specification boundaries satisfied plants
4) The controller is now designed for the nominal plant to satisfy the nominal boundaries at
each design frequency. By construction, we simultaneously satisfy the specification
boundaries for all plants in the plant set.
5) Practice makes perfect!
International Journal of Robust and Nonlinear Control Special Issue: Isaac Horowitz, August 2001
http://onlinelibrary.wiley.com/doi/10.1002/rnc.v11:10/issuetoc
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Example 1.1.2 (continued)
Given, P(s) = k/s, k [1, 5], design for
i. |1/(1+L)| -20dB, 0.5
ii. |1/(1+L)| 3dB,
For pedagogic reasons, choose the nominal as P0 = 2/s. Because there is only gain
uncertainty, it is independent of frequency and all the templates are vertical lines, 14dB
(= dB(5) - dB(1)) high.
Matlab Code for Example 1.1.2
(This code requires the QFT toolbox. Failing that – by hand!) % Uncertainty example 1.1.2 w=[0.5,3,10,30]; k=[2,[1:5]]; % Nominal first k=k(:); num=k; % numerator - columns of cases den=[ones(size(k)),zeros(size(k))]; % denominator coefficients in rows, columns of cases P=freqcp(num,den,w); % frequency response
b1=sisobnds(2,w,0.5,10^(-20/20),P); % |1/(1+L)| -20dB, = 0.5rad/s
b2=sisobnds(2,w,[3,10,30],10^(3/20),P); %|1/(1+L)| 3dB bnd=grpbnds(b1,b2); lpshape(w,bnd,num(1),den(1,:)) % CAD tool for loop-shaping grid Ingrid
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0
5
10
15
20
25
30
Open-Loop Phase (deg)
Op
en
-Lo
op
Gain
(d
b)
6 db
3 db
1 db
0.5 db
-1 db
-3 db
-6 db
-12 db
-20 db
* 3
* 3
B0(30)
B0(0.5)
B(30)
B(0.5)
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-5
0
5
10
15
20
25
30
Open-Loop Phase (deg)
Op
en
-Lo
op
Gain
(d
b)
6 db
3 db
1 db
0.5 db
-1 db
-3 db
-6 db
-12 db
-20 db
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-20
-15
-10
-5
0
5
10
15
20
25
30
Open-Loop Phase (deg)
Op
en
-Lo
op
Gain
(d
b)
6 db
3 db
1 db
0.5 db
-1 db
-3 db
-6 db
-12 db
-20 db
* 3
* 3
B0(30)
B0(0.5)
B(30)
B(0.5)
* 3* 3
* 3* 3
B0(30)
B0(0.5)
B(30)
B(0.5)
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Confirm that G=5/(s/30+1) does the job. We need the same gain as before for {|1/(1+L)| –
20db, = 0.5} as the k=1 case is the worst. We can only place the controller roll-off at =30
rather than =5 in Example 1. There is a controller bandwidth penalty for the performance
robustness. This is a design trade-off – there is no free lunch.
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-20
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-5
0
5
10
15
20
25
30
Open-Loop Phase (deg)
Op
en
-Lo
op
Gain
(d
b)
2
6 db
3 db
1 db -1 db
-3 db
-6 db
-12 db
-20 dbL0(j0.5)
L0(j30)
L0(j10)
L0(j3)
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-5
0
5
10
15
20
25
30
Open-Loop Phase (deg)
Op
en
-Lo
op
Gain
(d
b)
2
6 db
3 db
1 db -1 db
-3 db
-6 db
-12 db
-20 dbL0(j0.5)
L0(j30)
L0(j10)
L0(j3)
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2.2 Template
A 'template' at the design frequency i is a collection of frequency responses (typically of the
plant P(ji) P). It can represent a set of experimentally obtained frequency responses, or a
set of frequency responses derived from a parametrically uncertain plant set.
E.g. Draw the template for
, , at = 3 rad/s. Choose
.
E.g. Draw the template for
, , at = 3 rad/s.
Compare to
for the worst case.
2.3 Bounds
QFT translates the known frequency response (templates) and the desired performance
specifications into required conditions on the phase and magnitude of the nominal loop,
L0(ji). These requirements are referred to as 'bounds’ at the design frequencies, i
Bounds are computed at several frequencies, and are represented in the QFT Toolbox by two
line styles: a solid line designates 'stay above' and a dashed line designates 'stay below.' That
is, L0 must stay above or below its bound at the frequency of that bound.
The bounds are labelled via colours in the upper left-hand corner of the figure window in an
ascending order by frequency: r-red, g-green, b-blue, y-yellow, p-purple, c-cyan. Colours are
then repeated if there are more than 6 frequencies.
2.3 QFT Design
We add any of the following terms to the nominal loop transfer function to shape it to satisfy
the bounds with the lowest high frequency gain and controller complexity
Gain (K) - current gain
First Order (1) - add first order pole/zero element
Second Order (2) - add second order pole/zero element
Lead/Lag (L/L) - add lead/lag element
Notch (NTC) - add notch element
Second-over-Second (2/2) - add second-order over second-order element (bi-quadratic)
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2.4 General bound computation
The general computation of bounds follows directly by calculation (similar to M-circles on
the Nichols and inverse Nichols charts) but there has been some work published on bound
computation (for example, Borghesani, Chait and Yaniv, 1998; Chait and Yaniv, 1993). Some
insight into the calculations may be helpful.
Consider the general design problem given by a linear fractional mapping on design
parameter, g typically the controller,
dgc
bga (1.1.2)
Each of a, b, c, and d is a known complex number, e.g. ajaeRa
, and the specification, ,
is a given positive number. At a fixed controller angle, g, the magnitude Rg is calculated as,
2222
222
sinsincoscos2
sinsincoscos2
gdgdcgdcgdcc
gbgbagbagbaa
RRRRRR
RRRRRR (1.1.3)
Clearly this is a quadratic inequality in Rg for which a closed form solution can be found at
equality and some logic applied to discover if the acceptable solution lies above or below the
corresponding gains required for equality
A typical special case is sensitivity design where a = c = 1, b = 0, and d = p, giving,
pg1
1 (1.1.4)
and,
222 /1cos21 gpgpgp RRRR (1.1.5)
A special case of bound computation is dealt with in Section 1.3 where uncertainty reduction
calculations are undertaken.
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Section 3: SISO Tracking design
Figure 1.2.1 - Two degree of freedom feedback system
For tracking, we are interested in the reference to output behaviour, L
L
H
FTY/R
1.
We only need feedback if there is uncertainty and we have two degrees of design freedom as
we can process R and Ymeas independently.
3.1 Tracking design with magnitude specifications
Usually there are client specifications, A() and B() on TY/R in the form,
)()( BTAdBY/R (1.2.1)
In dB, )()(11
ABL
L
L
L
H
FTY/R since there is no uncertainty in F/H. (For
simplicity of exposition, H = 1 (i.e. high sensor bandwidth) is often assumed, but sensor
bandwidth determination / specification is part of the feedback loop design.) We solve the
design problem in two stages:
1) Uncertainty reduction
a) Select an arbitrary nominal P0 {P} and find boundaries using the Nichols chart (i.e.
L/(1+L) mapping) so that )()(1
ABL
L. This is done at each of a selected
set of discrete design frequencies by moving the template on the Nichols chart to
identify the boundary between the region where the inequality is satisfied and where it
is not satisfied.
b) Design L(s) = G(s) P(s) H(s) using standard synthesis, possibly using CAD tools.
c) This stage fixes G(s).
2) Pre-filter design
a) We calculate |L/(1+L)|, either by CAD tool, or by placing the nominal of the template
at i on the Nichols chart on the point L0(ji) and reading off max {|L/(1+L)|}, and
min{|L/(1+L)|}.
b) Put away the Nichols chart and design |F| using a Bode plot!
reference
R(s)
F(s)
pre-filter
+ -
H(s)
error G(s)
controller
sensor
sensor
noise
N(s)
output
Y(s)
+
+ +
+
plant
input
U(s) P(s)
plant
output disturbance
D(s)
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Design example for uncertainty reduction
Given, )1/(
)(
as
ksP , ]10,2[],5,1[ ak , design a controller, G(s), and pre-filter, F(s), to
achieve the specifications,
i) 13/)3/(
1.1
14/12/2)2/(
9.0
2/2
ssT
sssRY
ii)
dBL
31
1
Figure 1.2.2 – Open plant response, SISO tracking example
10-1
100
101
102
-35
-30
-25
-20
-15
-10
-5
0
5
10
15Open loop plant
|P| [d
B]
frequency [rad/s]
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Figure 1.2.3 – Desired closed loop tracking response – SISO tracking example
Figure 1.2.4 - Step response of plants from Bode magnitude plot extremes
10-1
100
101
102
-40
-35
-30
-25
-20
-15
-10
-5
0
5Desired closed loop Bode magnitude envelope
TY
/R [
dB
]
frequency [rad/s]
Time (sec.)
Am
plit
ude
Desired closed loop step response envelope
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.2
0.4
0.6
0.8
1
1.2
1.4
From: U(1)
To: Y
(1)
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-100 -90 -80 -70 -60 -50 -40 -30 -20 -10 0 10 -35
-30
-25
-20
-15
-10
-5
0
5
10
15
0.1 1
10
100
Phase (degrees)
Ma
gn
itud
e (
dB
)
Plant templates, w =[0.1,1,10,100]
Figure 1.2.5 – Plant templates, SISO tracking example
-250 -200 -150 -100 -50 0 -40
-30
-20
-10
0
10
20
X: Phase (degrees) Y: Magnitude (dB)
30
10
10 3
3
1
1
0.3, 0.1
L0(j)
Design of G
Figure 1.2.6 - Controller design:
145/1
15/5.3
ss
sG , SISO tracking example
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Figure 1.2.7 – Pre-filter design, SISO tracking example with magnitude specifications
Figure 1.2.8 – Pre-filter, 15/12/
16.1
ssF design – achieved performance
10-1
100
101
102
-40
-35
-30
-25
-20
-15
-10
-5
0
5
|L/(1+L)| and specifications on |TY/R
|
frequency [rad/s]
magnitude [
dB
]
10-1
100
101
102
-40
-35
-30
-25
-20
-15
-10
-5
0
5
Achieved and specifications on |TY/R
|
frequency [rad/s]
magnitude [
dB
]
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Matlab code for above example: % SisoTrackingG.m % SISO tracking design - Feedback controller w=[0.1,0.3,1,3,10,30,100]; % Plant: k/(s/a+1), k=[1,5], a=[2,10] a=[linspace(2,10,4),linspace(2,10,4),ones([1,4])*2,ones([1,4])*10]'; k=[ones([1,4])*1,ones([1,4])*5,linspace(1,5,4),linspace(1,5,4)]'; num=k; den=[1./a,ones(size(a))]; P=freqcp(num,den,w); figure plottmpl(w,[0.1,1,10,100],P) title('Plant templates, w =[0.1,1,10,100]') %Specifications Tup=abs(freqcp(1.1,[1/9,1/3,1],w)); % upper bound Tlo=abs(freqcp(0.9,conv([1/4,1,1],[1/4,1]),w)); % lower bound figure step(tf(1.1,[1/9,1/3,1]),'k',tf(0.9,conv([1/4,1,1],[1/4,1])),'k',5) title('Desired closed loop step response envelope') ww=logspace(-1,2); TTup=abs(freqcp(1.1,[1/9,1/3,1],ww)); % upper bound TTlo=abs(freqcp(0.9,conv([1/4,1,1],[1/4,1]),ww)); % lower bound figure semilogx(ww,db([TTup;TTlo]),'k') axis([0.1,100,-40,5]) grid title('Desired closed loop Bode magnitude envelope') ylabel('T_Y_/_R [dB]') xlabel('frequency [rad/s]') % Design b1=sisobnds(7,w,w,[Tup;Tlo],P); b2=sisobnds(2,w,[10,30,100],10^(3/20),P); bnd=grpbnds(b1,b2); plotbnds(bnd) grid bnd=sectbnds(bnd); % Controller numg=3.5*[1/5,1]; deng=conv([1,1],[1/45,1]); lpshape(w,bnd,num(1),den(1,:),0,numg,deng) grid
% SisoTrackingF.m % SISO tracking example - Pre-filter design s=logspace(-1,2)*i; w=abs(s); %Plant a=[linspace(2,10,4),linspace(2,10,4),ones([1,4])*2,ones([1,4])*10]'; k=[ones([1,4])*1,ones([1,4])*5,linspace(1,5,4),linspace(1,5,4)]'; num=k; den=[1./a,ones(size(a))]; P=freqcp(num,den,w); % Controller numg=3.5*[1/5,1]; deng=conv([1,1],[1/45,1]); G=freqcp(numg,deng,w); for kk=1:16, L(kk,1:50)=G.*P(kk,:); end T=L./(1+L); Tup=abs(freqcp(1.1,[1/9,1/3,1],w)); Tlo=abs(freqcp(0.9,conv([1/4,1,1],[1/4,1]),w)); pfshape(7,w,w,[Tup;Tlo],P,[],G,[],1.16,conv([1/2 1],[1/5 1]))
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3.2 Tracking design with reference model tracking error specifications
Eitelberg (2000) investigated quantitative feedback theory (QFT) design for single-input,
single-output systems with tracking error specifications and Boje (2001) showed how the pre-
filter is designed for this approach.
The two-degree-of-freedom control system shown in Fig. 1.3.1 above and as usual in
quantitative control system design, it is assumed that the linear(ised) plant transfer function is
an element a set, }{)s(P P , including structured and unstructured uncertainty. In order to
undertake engineering design, there must also be some closed loop specifications available. In
most QFT work (see Horowitz (1991, or 1993) for general reference), the tracking (and other)
specifications are assumed to be only on the magnitude of the closed loop transfer functions.
As Eitelberg (2000) argued, bounding the closed loop transfer function within a disk around a
nominal (model) performance often makes engineering sense. Such performance
specifications result in sensitivity designs for tracking performance. They are compatible with
norm-based methods, but the exact (structured and/or unstructured) plant uncertainty
description is retained. QFT design with norm bounded specifications has previously been
discussed for example by Horowitz (1979, 1991), and Nwokah, Jayasuriya & Chait (1991).
Given the system in Fig. 1.3.1 and client-specified model, M(s), with model output,
R(s)M(s)(s)Ym , the design task is to specify feedback controller, G(s) and pre-filter, F(s) to
meet relative tracking error specifications. The relative tracking error transfer function, rk
E ,
of the kth
plant (with Lk = PkG ) is defined as:
RL
FLMYYRE
k
kkm
rk
1 (1.2.2)
Eitelberg (2000) does not use a model reference (i.e. M(s) = 1) which means that the relative
tracking error must become large at frequencies where the closed loop transfer function
(reference to output) has low gain. A model reference extends the frequency range over which
the relative tracking error specification is useful as it captures knowledge of the expected
closed loop roll-off. Eitelberg (2000) accounts for unstructured uncertainty in the
measurement system and pre-filter. This important consideration will not be pursued here. Let
H=1 (no significant measurement dynamics) and D=0 (no disturbance as tracking behaviour
is our focus). Define the complimentary sensitivity, kkk LLIT1
. Simple specifications
on the relative tracking error would be to contain the relative error (element-wise) within a
disk of client-specified, frequency-dependant radius, A(),
}{P),(A)j(F)j(T)j(M)j(E kkrk
P (1.2.3)
A two-degree-of-freedom design is required to properly solve the design problem with
minimum feedback bandwidth and to take care of unstructured and structured uncertainty in
the plant. The design is performed at a set at of discrete design frequencies, i, i = 1, 2…
EEE4093 2013
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3.2.1 Feedback design
Following Eitelberg's (2000) approach, take the difference between the tracking error in two
plant cases, i and k.
MELI
PP
FLILLILEE
ri
k
ik
iikkrk
ri
1 (1.2.4)
In the usual implementation of QFT designs, a nominal (or handle) plant is chosen arbitrarily.
It is plausible that a special nominal plant, Pi = P0, exists for which it is required that the
tracking error be zero, 00 rE , or at least small with respect to the worst tracking error (i.e.
AE r 0 ). Eitelberg (2000) has shown that at worst, the gain is over-designed by 6dB if the
nominal is chosen “badly”. As discussed below, such over-design may ease the pre-filter
design. With a nominal chosen to give zero tracking error in eq(1.2.4), the feedback controller
design can be separated from the pre-filter design as,
MLI
PPE
k
krk
01
(1.2.5)
Eq(1.2.5) is a linear fractional mapping on G of the form,
kkkkkk
rk
PdcbMPPa
dGc
bGaE
,1,0,1 0
(1.2.6)
Eq(1.2.6) along with the specification results in quadratic inequalities for each plant case. The
nominal design boundary is the intersection of all feasible regions over the plant uncertainty
set.
3.2.2 Pre-filter design
Once the controller, G, has been designed to meet or exceed the feedback design constraints,
the pre-filter, F, can be designed to satisfy the tracking error specification, eq(1.2.3). A simple
approach to the design of F is to make use of the assumption that the nominal relative tracking
error is zero,
00 T/MFF (1.2.7)
By correct (realistic) specification of the model, M, the designer can ensure that F is rational,
(strictly) proper and stable (even if L0 is transcendental). The model would be required to
exactly anticipate and replicate right hand plane transmission zeros and other singularities in
the nominal loop transfer function. If, as would be usual, the model is approximate, direct
application of eq(1.3.7) may yield nuisance right hand plane singularities with small residuals
that should obviously be discarded. Direct use of eq(1.2.7) may also result in a pre-filter of
unnecessarily high order for practical problems.
To avoid high order feedback controllers, practical QFT designs usually have some over-
design of the feedback controller. As opposed to “standard” QFT designs where the choice of
nominal plant is arbitrary, there may also be over-design as a result of a poor or constrained
EEE4093 2013
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choice of the nominal plant. At each design frequency, i, any over-design of the feedback
controller can be exploited to reduce the complexity of the pre-filter by finding the exact
region (around F0) within which F may lie. For any particular L, M, and A, eq(1.2.3) is also a
linear fractional mapping in F and therefore results in simple quadratic inequalities on a plant
by plant basis.
0,1,,
dcTbMa
dFc
bFaFTME
k
krk (1.2.8)
Eq(1.2.8) can be solved for r, given any [-360, 0], for example using the Matlab QFT
toolbox (Borgesani, Chait & Yaniv, 1998). The solution of eq(1.2.8) for a particular plant
case and design frequency will divide the complex plane of F into acceptable and
unacceptable regions. If the feedback controller design was successful, by construction, there
is a non-empty intersection (over the plant set) of the acceptable regions for F at each
frequency (or at worst the point, MT/F 01 ). (F(s) must satisfy Bode gain-phase
relationships and, as discussed above, correctly specifying the model will ensure that a proper,
rational and stable F(s) exists.) Usually, in QFT the pre-filter is designed by magnitude only
but the design outlined here is conveniently undertaken on the log-complex plane (arg{F} vs.
dB{F}). The intersection of the regions given by eq(1.3.8) at frequency i provides an exact
bound for the design of F(i).
3.2.3 Example
The example is modified from the example above, with plant: 1/
)(
as
ksP , ],5,1[k
]10,2[a .
Original tracking specifications: 13/)3/(
1.1
14/12/2)2/(
9.0
2/2
ssT
sssRY
Robust stability specification:
dBL
31
1
New specifications:
Nominal (specifically chosen): 15/
5.20
ssP .
Approximately equivalent tracking error specifications, (illustrated at s=j3 in Fig. 1.3.9):
Model response (for specific P0) 14/15.3/4.15.3/
1
2
sss
sM
Tracking error tolerance:
jssss
sA
16/14.2/2.14.2/
14.0/1.0
2
Design
As illustrated in Figure 1.2.10, the feedback controller,
140/13.1/
110/8.4
ss
sG , satisfies the
tracking error bounds. The constraints on the pre-filter design are illustrated in Figure 1.2.11
EEE4093 2013
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along with the pre-filter design, 13/4.1)3/(
1.1
2
ssF , that satisfies the constraints with
low order and low bandwidth.
-1.5 -1 -0.5 0 0.5 1 1.5 -1.5
-1
-0.5
0
0.5
1
1.5
M(j1))
A(j1)
min{TY/R(j1)}
max{TY/R(j1)}
real
imag
Figure 1.2.9 – SISO tracking error specifications compared to magnitude specifications
EEE4093 2013
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-250 -200 -150 -100 -50 0 -30
-20
-10
0
10
20
30
X: Phase (degrees) Y: Magnitude (dB)
=30
=3 B0(1)
L0(j1)
=10
0.3, 0.1
Feedback design, G, Tracking error specification
Figure 1.2.10 – Feedback design SISO tracking error design,
140/13.1/
110/8.4
ss
sG
-180 -160 -140 -120 -100 -80 -60 -40 -20 0 -60
-50
-40
-30
-20
-10
0
X: Phase (degrees) Y: Magnitude (dB)
Design of pre-filter, F- Tracking error
=1
=10
=3
=30
=100
Figure 1.2.11 – Pre-filter design for tracking error specifications 13/4.1)3/(
1.1
2
ssF
(dashed lines = “below” bounds, solid lines = “above” bounds)
EEE4093 2013
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Eitelberg E, “Quantitative feedback design for tracking error tolerance” Automatica, vol. 36, no. 2, 319-326,
http://dx.doi.org/10.1016/S0005-1098(99)00149-1
Boje E, “Multivariable quantitative feedback design for tracking error specifications”, Automatica, vol. 38 no. 1,
2002, 131-138. http://dx.doi.org/10.1016/S0005-1098(01)00177-7 Boje E, “Pre-filter Design for Tracking Error Specifications in QFT”, International Journal of Robust and
Nonlinear Control, vol. 13, no 7, 637-642, 2003. http://dx.doi.org/10.1002/rnc.829
10-1
100
101
102
0
0.2
0.4
0.6
0.8
1
1.2
1.4
rad/s
Arith
metic m
agnitude
Example 1.3.2.3
Tlow
Thigh
Model
Model Error|T
high|-|T
low|
Probably don't care
what happens here!
EEE4093 2013
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Section 4: Discrete controller design
The vast majority of digital designs are for sampled data systems shown in Figure 2.1.1 where
the plant is continuous but the controller is discrete. There are many possible ways to go
about digital controller design. I prefer using the w-domain as continuous time design
philosophy can be used (Eitelberg, 1988; Boje, 1990; Eitelberg and Boje, 1991). The results
of this section can also be applied for systems with pulse width modulators (PWM) as part of
the actuator as this is also a sampling process (Boje, 1999).
pre-filter
F(w) G(w)
controller
-
+P(s)
plant+
+
d(t)
disturbance
r(t) u(t) y(t)A/D
outputreference
A/D
D/A
Computer
input
Figure 2.1.1 - Sampled data control system
4.1 – Approximate effect of discretisation
It is possible to show (see Eitelberg, 1988) that the effect of sampling a continuous time
process for implementing a digital controller is approximately to add an additional right hand
plane zero, (1-wT/2), where T is the sampling period to the loop transfer function. To see this,
consider a single-input, single-output, strictly proper, continuous time system,
bAIc1
)(
ssPs , described by the state space equations,
(1.3.1)
At the sampling instances,
idid
iT
iT
i
u
udee
bxA
bxxAA
01 (1.3.2)
( bbAAA dee
Td
Td
0 and are respectively the discrete time state transition and input
matrices corresponding to the given sampling rate.) Taking z- transforms and solving for the
transfer function,
ddz zz/UzYzP bAIc1
)()()(
(1.3.3)
Now take the w-transform (Tustin or bi-linear transform) of the z-domain transfer function,
1
12,
2/1
2/1
z
z
Tw
wT
wTz (1.3.4)
This gives
2/12/12/1)(1
wTwTwTwP ddz
bAIc (1.3.5)
cx
bAxx
y
u
EEE4093 2013
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Writing out the Taylor series expansion for ...!222 TTd AAIA and
bAb ...!22 TTd , we can express Pz(w) approximately in terms of the parameters of
Ps(s),
2/1)(2/1)(1
wTwPwTwwP sz
bAIc (1.3.6)
The approximation in eq(1.3.6) is accurate to order, O(T2) (Boje, 1990). The result can be
generalised to systems with input and measurement delay, PWM inputs, sampling jitter, etc.
Another view of the result is obtained by writing Ps(s) as partial fraction expansion and then
taking the z-transform (transfer function with “zero-order hold”, not signal) and w-transform.
Assuming no repeated roots (for convenience),
n
i i
is
ps
rsP
1
)( , and
n
iTp
iTp
iz
i
i
ez
perzP
1
1)( , giving,
2
1
21
2
2
2
2
21
22
1
1tanh
tanh
111
1)(
Tn
i i
i
Tn
iT
iT
iT
iTi
Tn
iTpTT
iTp
iz
wpw
r
wpw
ppr
weww
perwP
i
i
(1.3.7)
The approximations require that piT << 1, meaning that the sampling rate, 1/T must be more
than the highest plant corner frequency that has significant residual, ri. Eq. (1.3.7) illustrates
the effect of frequency warping when translating between the s- and w- domains.
References
Eitelberg E (1988): “Sampling rate design based on (1-sT/2)”, International Journal of Control, 48:4, 1423-1432
http://dx.doi.org/10.1080/00207178808906259
Boje E, “Further results on Eitelberg’s sampling rate design based on (1-sT/2)”, International Journal of
Control, Vol. 51, No 5, 1990, pp1155-1158. http://dx.doi.org/10.1080/00207179008934123
Eitelberg E and Boje E, “Feedback controller design for plants with modes and disturbances above the sampling
frequency”, International Journal of System Science, Vol. 22, No 9, pp1553-1562, 1991.
http://dx.doi.org/10.1080/00207729108910730
EEE4093 2013
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Figure 1.3.2 Warping of poles from s- to w-domains, T=1
4.2 – Effect of discretisation on design
For design, the performance specifications (sensitivity or tracking) are at low frequency,
within the system bandwidth where |Ts(j)| |Tz(j)| for any transfer function, T.
The main effect of sampling is to add (non-minimum) phase lag to the loop transfer function.
For example, with a sampling rate of 1 sample per second, we get approximately 45 of extra
phase lag and +3dB of extra gain at 2 rad/s. (Think what that does on the inverse Nichols
chart.) For sampling rate, Ts = 1s, the table below gives other values of lag and gain.
Table 1.3.1 – Normalised effect of sampling (i.e. T=1s)
frequency (rad/s) 0.1 0.2 0.4 1.0
lag -2.8 -5.7 -12.1 -26.6
gain +0.0 dB +0.0 dB +0.2 dB +1.0dB
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1Warping of poles from s- to w-domain
real
imag
EEE4093 2013
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Figure 1.3.2 - Inverse Nichols chart showing effect of sampling at T=0.1s on L=1/s – an
integrator
Design
There are two possible design scenarios.
1) The sampling rate is fixed and we must design the discrete controller. The accurate
approach is to find the exact Pz(w). We can obtain the exact w-domain transfer function
using Matlab or tables, for example,
» Ps=tf(2,[1 3])
3
2)(
ssPs
» Pz=c2d(Ps,0.1,'zoh') % T=0.1
7408.0
1728.0)(
zzPz
» Pw=d2c(Pz,'tustin')
3
05012
9782
9851099260
w
w.
.w
.w.(w)Ps
The controller, G(w) is designed to meet specifications on T/(w) and then implemented
as G(z). We may save time by simply taking Pz(w) Ps(s)(1-sT/2)|s=w.
2) The sampling rate must be determined as part of the design.
Here, we design with P(s) but include the term (1-sT/2) with T a design variable to be
specified as an output of the controller design. Of course G(w) does not include the term,
it is part of L(w) to be designed but is the sampling effect and the phase lag comes from
sampling the plant.
-250 -200 -150 -100 -50 0-40
-30
-20
-10
0
10
20
30
40
6 db
3 db
1 db
0.5 db
0.25 db
0 db
-1 db
-3 db
-6 db
-12 db
-20 db
Open-Loop Phase (deg)
Open-L
oop G
ain
(db)
INVERSE NICHOLS CHART
0.03
0.1
0.3
1.0
3.0
10 10
30 100
L1=1/s
L2=(1-sT/2)/s
T=0.1
EEE4093 2013
E Boje 38
Z{y(iT)} y(t) L{y(t)} or
Ps(s)
iTy
sz
s
sP
z
zzP
)(1)( 1LZ
1z
z
(t)
s
1
1z
T
21z
Tz
t
2
1
s
2
2
)1(
)1(2/
z
zT
aTez
z
e-at
as
1
aT
aT
ez
ae
/)1(
2aT
aT
ez
Tze
te
-at
21
as
22
1)1(1
aez
aTeezaTe
aT
ataTaT
aTaT
at
ebTezz
bTez
22 cos2
sin
bte at sin
22)( bas
b
aTaT ebTezz
babnzn
22
2201
cos2
bTb
abTeen
bTb
abTen
aTaT
aT
sincos
sincos1
0
1
aTaT
at
ebTezz
bTezz
22
2
cos2
cos
bte at cos
22)( bas
as
aTaT ebTezz
baanzn
22
2201
cos2
bTa
bbTeen
bTa
bbTen
aTaT
aT
sincos
sincos1
0
1
Table 2.1.2 Quick table of Laplace and Z-transforms. (After Eitelberg, 1999)
4.3 – A/D Conversion and Calculation delays
We must often allow for the A/D conversion time and calculation delay between the sampling
of the input and the control action at the output. It is unlikely that this will be more than one
sample on a single processor system unless they are badly designed. For one sample delay,
allocate an extra 1
2/1
2/1
z
wT
wT. For sub-sample delay, see the literature (e.g. Eitelberg,
1988).
Example: As before, ]5,1[],5,1[,1/
kaas
kP , with specifications,
i) 0.5 dB,201
1
L
ii) 1 dB,101
1
L
EEE4093 2013
E Boje 39
iii)
dB,31
1
L
iv) Control action follows measurement with a one sample delay. Initially allow 30
for sampling effects at 30 rad/s.
Choose 1
10
sP . (Previous design was,
130/
15/
11.0/
60)(
s
s
ssG )
Specification (iv) tells us that the approximate effect of sampling is
delayn calculatio sampling
2/1
2/12/1)()(
wT
wTwTwPwP sz
and 30/3=10 is allocated to each term at 30
rad/s.
Design: As before except the high frequency pole must be at higher frequency to leave
sampling effect: 1120/
15/
11.0/
60)(
s
s
ssG
-250 -200 -150 -100 -50 0 -40
-30
-20
-10
0
10
20
30
Open-Loop Phase (deg)
Open-L
oop G
ain
(db)
2
2
6 db
3 db
1 db
0 db
-1 db
-3 db
-6 db
-12 db
-20 db 0.5
1.0
10
30
B0(10) B0(30) B0(100
) 30
Figure 1.3.3 – Design with 30 margin at 30 rad/s, 1120/
15/
11.0/
60)(
s
s
ssG
The margin allows arg(1-jT/2) = -10 at =30, T=11.8ms, 2/T=170 rad/s
EEE4093 2013
E Boje 40
-250 -200 -150 -100 -50 0 -40
-30
-20
-10
0
10
20
30
Open-Loop Phase (deg)
Open
-Loop G
ain
(db)
2
2
6 db
3 db
1 db
0 db
-1 db
-3 db
-6 db
-12 db
-20 db
Specs. violated
0.5
1.0
10
30
Figure 1.3.3 – Design with 2/1
2/12/1)()()(
wT
wTwTwGwPwL s
, T=11.8ms
To prevent violating high frequency specifications, we must reduce the sampling and
calculation delay or increase the sampling rate. With T=8ms, extra lag around 15 rad/s, and
some tuning, we get,
-250 -200 -150 -100 -50 0 -40
-30
-20
-10
0
10
20
30
Open-Loop Phase (deg)
Open-L
oop G
ain
(db)
2
2
6 db
3 db
1 db
0 db
-1 db
-3 db
-6 db
-12 db
-20 db
30
1.0
10
Figure 1.3.3 – Design with 2/1
2/12/1)()()(
wT
wTwTwGwPwL s
, T=8ms,
110/
120/
1120/
14/
11.0/
60)(
w
w
w
w
wwG
EEE4093 2013
E Boje 41
Notice that as a result of the phase lag introduced by sampling and delay, this G(w) has
considerably more bandwidth than the G(s). A digital controller can never have lower
bandwidth than an equivalent continuous one.
A final observation from Figure 2.1.2 is that one should avoid calculation delays if at all
possible. Some bad hardware designs may have latching of the A/D output in one cycle and
latching of the controller output to the D/A in the next sample of even later. (For example,
Yaniv, Fried and Furst-Yust, (2002) report 4 samples of delay on a Loughborough PC31
board.) This is no problem in signal processing but unnecessarily reduces performance in
feedback systems. Parts of the control algorithm not requiring the current A/D result should
be pre-calculated and the D/A value latched as soon as it is available. There is a possible
trade-off here as one could introduce sampling jitter by not latching the D/A value
periodically and this introduces noise effects discussed below.
4.4 – PID Controller discretisation
The PID controller itself is discretised, often in a pretty obscure way. If the sampling is very
fast with respect to the process dynamics, it does not matter but the actual algorithm may
influence the behaviour in very fast loops. For example, the discretisation of the series PID,
1
111)(
sT
sT
sTp
d
d
i
ksG
, with Ei = SPi - PVi - the error, all terms in the forward path,
backward difference approximation to derivative and the velocity algorithm is,
increment D) (and alProportion
increment Integral
increment Output
output Next
1
1
iipi
ii
pi
iii
iii
XXkP
ET
hkI
IPOP
OPOPOP
(1.3.8)
We calculate Xi depending on whether there is derivative action or not:
end
else
small)(or 0 if
11
ii
didididi
d
EX
hTαETEh)(TXTαX
T
(1.3.9)
EEE4093 2013
E Boje 42
References Boje E, "Further results on Eitelberg's sampling rate design based on (1-sT/2)", International Journal of Control,
Vol. 51, No 5, pp 1155-1158, 1990.
Boje E, “Feedback design of pulse width modulators”, IEEE ACC 1999, San Diego, California, WP15-4, 2-4
June 1999, 1420-1424.
Eitelberg E, "Sampling rate design based on (1-sT/2)", International Journal of Control, Vol. 48, No 4, pp 1423-
1432, 1988.
Eitelberg E and Boje E, "Feedback Controller Design for Plants with modes and Disturbances above the
Sampling Frequency", International Journal of System Science, Vol. 22, No. 9, pp 1553-1562, 1991.
Matlab QFT Toolbox help % Quantitative Feedback Theory (QFT) Toolbox (OLD VERSION)
% Version 1.0 01-Feb-95 % Copyright (c) 1995 by C. Borghesani, Y. Chait, O. Yaniv. % % Conversions. % cp2mp - Complex to magnitude and phase. % mp2cp - Magnitude and phase to complex. % % Integrated Development Environments. % lpshape - Continuous-time controller design. % dlpshape - Discrete-time controller design. % pfshape - Continuous-time pre-filter design. % dpfshape - Discrete-time pre-filter design. % % Specialised X-Y Graphs. % plotbnds - Nichols plot of bounds. % plottmpl - Nichols plot of templates. % % Arithmetic. % addcp - Addition of frequency response sets. % addnd - Addition of transfer function num/den sets. % mulcp - Multiplication of frequency % putqft - Interactively set IDE MAT-file. % getqft - Interactively retrieve IDE MAT-file. % % General Utility. % freqcp - Continuous-time frequency response of num/den matrix. % dfreqcp - Discrete-time frequency response of num/den matrix. % qftdefs - User-defined defaults. function bdb = sisobnd2(w,wbd,W,uP,vP,R,nom,uC,vC,ctype,ph_r,info);
% Compute QFT bounds for the following closed-loop configuration % % | 1 | % |------------ | <= WS % | 1 + PGH | % % SISOBNDS(2,W,WBD,WS,P,R,NOM,C,LOC,PHS) computes bounds at frequencies % designated by WBD. WS is the performance specification, P is % the frequency response data of the plant (complex), R is the % disk radius for non-parametric uncertainty, NOM designates the % nominal plant and controller, C. LOC specifies location of unknown % controller in the loop: 1 for G, 2 for H. PHS specifies at which % phases (degrees) to compute bounds. % % SISOBNDS(2,W,[],WS,P,[],NOM,[],[],PHS) computes bounds using default % values for WBD (all frequencies in W), R (0), C (1), CTYPE (1).
EEE4093 2013
E Boje 43
function plotbnds(bdb,ptype,phase,pos)
% PLOTBNDS Plot QFT bounds. % PLOTBNDS(BDS,PTYPE,PHS) plots the QFT bounds in BDS that are % associated with PTYPE and PHS. PHS is the same phase vector used % for computing BDB with SISOBNDS(PTYPE,...) and GENBNDS(PTYPE,...). % % PLOTBNDS(BDS) plots all the QFT bounds in BDS that were computed % using the default PHS. % % PLOTBNDS(BDS,[],PHS) plots the QFT bounds in BDS that were computed % using PHS and uses default value for PTYPE. function plottmp(w,wbd,P,nom,pos)
% PLOTTMPL Plot frequency response templates. % PLOTTMPL(W,WBD,P,NOM) plots the frequency response data computed % at the frequencies in WBD in P. NOM designates the location of % the nominal plant within P. % % PLOTTMPL(W,[],P) uses the default values for WBD (all frequencies % in W) and NOM (the first row) function lpshape(wl,bdb,uL0,vL0,delay,numC0,denC0,phase)
% LPSHAPE Controller design environment for continuous-time linear systems. % LPSHAPE produces a continuous-time Nichols plot using default % settings. % % LPSHAPE(W,BDB,NUMP0,DENP0,DELAY,NUMC0,DENC0,PHS) produces % the Nichols plot design environment with the nominal plant transfer % function P0(s)=NUMP0(s)/DENP0(s) where NUMP0 and DENP0 contain % the polynomial coefficients in descending powers of s using the % user-supplied frequency vector W. BDB contains the QFT bounds, % NUMC0 and DENC0 contain the coefficients for the initial % controller, and PHS is the user-defined phase vector. % % LPSHAPE(W,[],P0,[]) produces the nichols plot as above, % yet without the bounds and P0 is now in complex number format. % W corresponds to the frequency vector used to compute P0. The % rest of the inputs are set to their defaults. % % See also DLPSHAPE, PFSHAPE, DPFSHAPE, XTRQFT function bds = grpbnds(bd1,bd2,bd3,bd4,bd5,bd6,bd7,bd8,bd9)
% GRPBNDS Group QFT bounds. % GRPBNDS(BD1,BD2,...,BD9) creates a single bound matrix containing % all the passed bounds. % % See Also: PLOTBNDS, SECTBNDS, SISOBNDS, GENBNDS function cp = freqcp(num,den,w,delay)
% FREQCP Compute continuous bode of numerator/denominator matrices. % CP=FREQCP(NUM,DEN,W,DELAY) returns the continuous bode of the % numerators and denominators in NUM and DEN, respectively. W is % the frequency vector in radians/second. DELAY is the pure % time delay and it must have as many rows as NUM or DEN or be % just one element. % % CP=FREQCP(NUM,DEN,W) returns the continuous bode as above, % except DELAY defaults to zero (0). % % Rows and columns of NUM and DEN correspond to cases and % coefficients, respectively. % % Rows and columns of CP correspond to cases and frequencies, % respectively.