QFTCT DemoExample Description
QFTCT DemoExample Description
QFTCT DemoExample Description
TM
Demo Example
for the QFT Control Toolbox
This document presents a simple demo example to run the QFT Control Toolbox,
or QFTCT, for Matlab. The toolbox includes the latest quantitative robust control
techniques within a user-friendly and interactive environment. The toolbox, devel-
oped by Prof. Mario Garcia-Sanz, has been tested in numerous courses, universi-
ties, companies and centers over the years. A complete study of the QFT robust
control methodology with the toolbox is presented in the book: Mario Garcia-
Sanz, Robust Control Engineering: practical QFT solutions, CRC Press, Taylor &
Francis, 2017.[1]
The professional or full version of the toolbox can be requested at
http://codypower.com.[2] Additional information can be found at the sites:
http://crcpress.com and http://cesc.case.edu.
The QFTCT project file for this example is: ExampleDemo.mat
To run it:
1. Open the QFT Control Toolbox by typing QFTCT in Matlab.
2. Click File, Open project, and select ExampleDemo
A. Description
Consider the second order plant with parametric uncertainty shown in Eq.(1).
y(s) k1
= P( s) = ;
u (s) s2 2 d
+
p1
s + 1 (1)
2 np1
np1
with k1 [3, 4], np1 [1, 2], d p1 = 0.4
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2 Example Demo, QFTCT
The control system block diagram is shown in Fig.1. Design a G(s) feedback
controller and a prefilter F(s) to regulate automatically the plant output y(s) by
varying the plant input u(s), and following the next three specifications:
P ( j ) G ( j )
T1 ( j ) = 1 ( ) = Ws = 1.181
1 + P ( j ) G ( j ) (2)
[0.01 0.05 0.1 0.5 1 1.5 2 3 4 5 10 20 30 40 50 100] rad/sec
which is equivalent to PM = 50.10, GM = 5.33 dB see Eqs.(2.30), (2.31) in [1].
s
a
y ( j ) 1 d
T3 ( j ) = = 3 ( ) = ; ad = 2
d o ( j ) 1 + P ( j ) G ( j ) s (3)
a +1
d
[0.01 0.05 0.1 0.5 1 1.5 2 3 4] rad/sec
P ( j ) G ( j )
6 _lo ( ) < T6 ( j ) = F ( j ) 6 _up ( )
1 + P ( j ) G ( j ) (4)
[0.01 0.05 0.1 0.5 1 1.5 2 3 4 5 10] rad/sec
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Demo example QFTCT 3
(1 lo )
6 _lo ( s ) = 2
; aL = 1; lo = 0 (5)
s
+ 1
aL
s
+ 1 (1 + up )
aU 1.25 aU
6 _up ( s ) = ; aU = 1.2 ; = 0.8 ; n = ; up = 0.02 (6)
s 2
2 s
+
n + 1
n
Bode Diagram
0
(s)
3
-5
-10
Magnitude (dB)
-15
-20
-25
-1
10 10 0 10 1 10 2
Frequency (rad/s)
1.2
- (s)
6 up
0.8
Amplitude
0.6
- (s)
0.4 6 lo
0.2
0 1 2 3 4 5 6 7 8 9
Time (seconds)
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4 Example Demo, QFTCT
B. Solution
Figures 4 to 20 show how to apply the QFT Control Toolbox to the design and
validation of a controller G(s) and prefilter F(s), to meet the specifications
Eqs.(2) to (6), for all the plants within the uncertainty Eq.(1).
Fig. 4. Plant definition window. Introducing plant model and uncertainty, Eq.(1).
Fig. 5. Templates window. Plant model with uncertainty in the Nichols Chart.
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Demo example QFTCT 5
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6 Example Demo, QFTCT
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Demo example QFTCT 7
Fig. 11. Bounds window. Reference tracking: bounds in the Nichols chart.
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8 Example Demo, QFTCT
Fig. 12. Bounds window. Intersection of all bounds in the Nichols chart.
Figures 13 to 15 show some steps of the design of the feedback controller G(s)
= PID in the Nichols chart (loop-shaping): Derivative time from Td = 0.4 to 1.4.
The right figures show the close-loop step response with nominal plant and G(s).
Fig. 13. Controller design window: design of G(s), derivative time Td = 0.4.
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Demo example QFTCT 9
Fig. 14. Controller design window: G(s) with derivative time Td from 0.4 to 1.4.
Fig. 15. Controller design window: design of G(s), derivative time Td = 1.4.
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10 Example Demo, QFTCT
Figures 16 and 17 shows some steps of the design of the prefilter F(s) in the
Bode diagram: first with F(s) = 1, and then with F(s) = first-order low-pass filter
see also Eq.(8).
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Demo example QFTCT 11
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12 Example Demo, QFTCT
.
Fig. 20. Analysis window. Reference tracking time domain, with G(s), F(s).
Figure 20 shows that the reference tracking specification of the closed-loop
system is met for all the plants within the uncertainty: the solid lines (100 plants)
are between the upper and lower specifications (dashed lines).
The expressions found for the feedback controller G(s) and prefilter F(s) are
shown in Eqs. (7 ) and (8).
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Demo example QFTCT 13
C. References
[1]. Garcia-Sanz, M. (2017). Robust Control Engineering: Practical QFT Solu-
tions. A CRC Press book, Taylor and Francis, USA.
[2]. Garcia-Sanz, M., (2008 - present). The QFT Control Toolbox for Matlab
QFTCT. http://codypower.com.
[3]. Garcia-Sanz, M. (2015). Quantitative Feedback Theory. Chapter in Encyclo-
pedia of Systems and Control. Editors: Tariq Samad, John Baillieul. Article ID:
366609, Chapter ID: 238. Springer Verlag.
[4]. Garcia-Sanz, M. and Houpis C.H. (2012). Wind Energy Systems: Control En-
gineering Design. A CRC Press book, Taylor and Francis, USA.
[5]. Houpis C.H., Rasmussen S.J., and Garcia-Sanz, M. (2006). Quantitative
Feedback Theory: Fundamentals and Applications. 2nd Edition. A CRC Press
book, Taylor and Francis, USA.
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