thesis

1. Study of Sample Rate Dependent Controller and its
performance with linear and non-linear plants
Presented By
SHASHI KUMAR
Registration number : 129061 of 2014-15
Examination Roll number : M4CTL1609
Under the guidance of
Prof. (Dr.) Samar Bhattacharya
Department of Electrical Engineering
Faculty of Engineering and Technology
Jadavpur University
Kolkata – 700 032
YEAR-2016 1

2. Introduction
Selection of Sampling Period
SRD with second order system
Compression of discrete controllers
SRD with Buck converter
SRD with Slosh pendulum model
Conclusion
References
Overview
2

3. Introduction
PLANT
3

4. Continuous Discrete
T?
4

5. Sine wave
Second order system
𝑠𝑖𝑛2𝑡
10000
𝑠2 + 20𝑠 + 10000
Selection of Sampling Period
5

6. SRD with different plants
Second
order
system
Buck
Converter
Slosh
Pendulum
Model
SRD
1
2
3
0
S
6

7. Conclusion
1. Range of sampling period is
𝑡 𝑠
100
≤ 𝑇 ≤
𝑡 𝑠
5
.
2. The SRD controller works well in higher sampling rates.
3. For faster response SRD controller is preferred but for less overshoot and settling
time preferred MPC.
4. In Buck converter voltage response is observed, SRD controller gave the better
performance than PID.
7

8. 1. S.S.Saab, "A MIMO Sampling-Rate-Dependent Controller", IEEE Trans.
Ind.Electron.,vol. 62, no. 6,pp. 3662-3671, June 2015.
2. S.S.Saab, "Learn while tracking," American Control Conf., pp 3733 - 3738, 2007.
3. Liuping Wang, "Model Predictive Control System Design and Implementation
using MATLAB", Springer, 2009.
4. S.Kurode, S. Spurgeon, B. Bandyopadhyay, P. S. Gandhi, "Sliding Mode Control
for Slosh-Free Motion Using a Nonlinear Sliding Surface", IEEE/ASME
Transactions on Mechatronics, Vol.18, Issue 2, pp.714-724, Apr, 2013.
5. Rashid, M. "Power Electronics: Circuits, Devices and Applications", 3rd ed. 7th
Impression, Pearson (LPE), pp. 166-220, 621-629. 2009.
References
8

9. 6. F. Franklin,L.workman,"Digital control of dynamic system", 3rd ed.,1997.
7. A. TUSTIN,"A method of analysing the behaviour of linear systems in terms of time
series", Journal of the inst.,vol.94,pp.130-142,Jan. 2010.
8. C. Vibet, "Guide to choosing the sampling rate in computerised controllers", Electron.
Lett., Volume 23, Issue 19, pp.1002-1004, Sep.1987.
9. A. Turner, "Computer in process control: a panel discussion" , Instruments and Control
Systems, 1970.
10. Leopoldo Angrisani,"Automatic Selection of Optimal Sample Rate for Band-Pass
Signals,"IEEE instr. and Measur,pp. 1430-1435,2006.
11. Chen-Chien Hsu, Tsung-Chi Lu,"Minimum-phase criterion on sampling time for
sampled-data interval systems using genetic algorithms"1670-1679,2008.
References Cont.......
9

10. Thank you
10

11. When we consider signal is
𝑦 = 𝑠𝑖𝑛2𝑡
There are three types of sampling:-
1. Critical Sampling:
𝑤𝑠 = 2 ∗ 𝑤 𝑚
= 2 ∗ 2
𝑇 = 1.57 𝑠𝑒𝑐.
2. Under Sampling:
𝑤𝑠 < 2 ∗ 𝑤 𝑚
< 2 ∗ 2
𝑇 > 1.57 𝑠𝑒𝑐.
3. Over Sampling:
𝑤𝑠 > 2 ∗ 𝑤 𝑚
> 2 ∗ 2
𝑇 < 1.57 𝑠𝑒𝑐.
𝑁𝑜𝑡𝑒: 𝐷𝑖𝑠𝑐𝑟𝑒𝑡𝑒 𝑡𝑖𝑚𝑒 𝑠𝑝𝑒𝑐𝑡𝑟𝑢𝑚 =
1
𝑇
𝐶𝑜𝑛𝑡𝑖𝑛𝑒𝑜𝑢𝑠 𝑡𝑖𝑚𝑒 𝑠𝑝𝑒𝑐𝑡𝑟𝑢𝑚
11

12. 12

13. 5 ∗ 𝑤𝑐 < 𝑤 < 100 ∗ 𝑤𝑐
5 ∗
2𝜋
𝑡 𝑠
≤
2𝜋
𝑇
≤ 100 ∗
2𝜋
𝑡 𝑠
5
𝑡 𝑠
≤
1
𝑇
≤
100
𝑡 𝑠
𝑡 𝑠
100
≤ 𝑇 ≤
𝑡 𝑠
5
Where,
 𝑡 𝑠 = 𝑠𝑒𝑡𝑡𝑙𝑖𝑛𝑔 𝑡𝑖𝑚𝑒 𝑜𝑓 𝑐𝑙𝑜𝑠𝑒𝑙𝑜𝑜𝑝
 𝑇 = 𝑠𝑎𝑚𝑝𝑙𝑖𝑛𝑔 𝑝𝑒𝑟𝑖𝑜𝑑
13

14. 14

15. wn
2
wn
2
15

16. 16

17. Response SRD gain k Rise Time
(seconds)
Settling
Time
(seconds)
% peak
overshoot
Close-loop - 0.017 0.40 72
CPID - 0.21 0.80 20
Sampling period
[0.0021 -0.0022 0.0001] 1 1.7 4
1
0.1 [0.02777 -0.0315 0.0038] 0.12 0.80 39
0.01 [1.1388 -2.07 0.9324] 0.016 0.11 42
0.008 [1.7347 -3.2129 1.4782] 0.012 0.09 42
0.004 [6.5889 -12.6712 6.0823] 0.0065 0.05 41
SRD performance with second order system
Where,
CPID=continuous PID
17

18. Comparison between discrete Controllers
Response Rise Time (seconds) Settling Time (seconds) % peak overshoot
0.00571 0.008 0.010 0.00571 0.008 0.010 0.00571 0.008 0.010
SRD 0.009 0.012 0.0016 0.065 0.09 0.11 42 42 42
DPID 0.0171 0.019 0.020 0.15 0.20 0.22 66 57 47.2
MPC 0.017 0.0224 0.030 0.039 0.046 0.05 5.6 5.7 7.2
Where,
SRD=sample rate dependent controller
DPID=discrete PID
MPC=model predictive controller 18

19. 2.Buck-Converter
Where,
DCM=Discontinuous Conduction Mode
[5]
19

20. 20

21. 21

22. Performance of Buck converter at 𝑣𝑖𝑛= 20 v
Response Rise Ti me
(sec) *𝟏𝟎−𝟓
Settling Time
(sec)*𝟏𝟎−𝟓
% Peak
overshoot
Final voltage
Open-loop 24.7 100 13.2 11.1
CPID 38.5 200 24 20
SRD at T=𝟏𝟎−𝟓 1.5 9.4 28 20
MPC at T=𝟏𝟎−𝟓 16 3.2 8.3 20
Where, MPC=Model predictive controller
CPID = continuous PID 22

23. 3. Slosh Pendulum Model
23
• : Total rigid mass including mass of the tank,
mass of the base plate, and mass of liquid that
does not slosh. (kg)
• m : mass of pendulum (slosh mass) (kg)
• M : total mass ( 𝑀𝑟 + m) (kg)
• : total mass of the liquid (kg)
• l : length of pendulum (meters)
• f : force applied for translational motion (N)
• x : displacement of rigid mass (meters) with
reference to fixed reference x − z frame
• X : displacement of m in horizontal direction
(m) with reference to fixed reference x − z
frame
• Z : displacement of m in vertical direction (m),
• φ : pendulum angle or slosh angle (radians)
Fig Representation of slosh by simple pendulum [4]
rM
lM

24. 24

25. Performance of Container position
Response Rise Time (sec) Settling Time (sec) peak overshoot (m)
SMC 5 7 0
SRD at T=1.5 0.16 1.2 0.04
25

26. Sample Rate Dependent Controller
• A discrete time-varying system [1]
𝑋 𝑘 + 1 = 𝐴𝑥 𝑘 + 𝐵𝑢 𝑘
𝑦 𝑘 = 𝐶𝑥 𝑘 + 𝑣
• Augmented Error system:
𝑒 𝑘 + 1 = 𝑦 𝑑 𝑘 + 1 − 𝑦 𝑘 + 1
𝛿𝑥 𝑘 + 1 = 𝑥 𝑑 𝑘 + 1 − 𝑥 𝑘 + 1
=𝐴𝛿𝑥 𝑘 + 𝐵𝛿𝑢(𝑘) ……….(1)
𝛿𝑢 𝑘 + 1 = 𝑢 𝑑 𝑘 + 1 − 𝑢 𝑘 + 1
= 1 − 𝑀1 𝛿𝑢 𝑘 − 𝑀2 𝛿𝑥 𝑘 − 1 − 𝑀3 𝛿𝑢 𝑘 − 1 + 𝑤 (𝑘) …….(2)
Where,
𝑀1 = 𝑘0 𝐵𝐶, 𝑀2 = 𝐾0 𝐶𝐴𝐴 + 𝐾1 𝐶𝐴 + 𝑘2 𝐶, 𝑀3 = 𝐾0 𝐶𝐴𝐵 + 𝑘1 𝐶𝐵
Combine equation (1) and (2) then augmented matrix form
𝑍 𝑘 + 1 = 𝑀 𝑘 𝑍 𝑘 + 𝑤 𝑘
26

27. Where,
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𝑆𝑒𝑙𝑒𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑆𝑅𝐷 𝐺𝑎𝑖𝑛𝑠[1]
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00
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E CB
CAB
C
CA
CAACB
𝑀 = 𝐻 − 𝐼 𝑘𝐸
   21O KKKKand,000 
T
pII
1. 𝐼𝑓 𝐸 𝑖𝑠 𝑓𝑢𝑙𝑙 − 𝑐𝑜𝑙𝑜𝑢𝑚 𝑟𝑎𝑛𝑘 𝑡ℎ𝑒𝑛,
𝐾 = 𝐼 𝑇 𝐻
0
( 𝐸 𝑇
𝐸)−1
𝐸 𝑇
2. 𝐼𝑓 𝐸 𝑖𝑠 𝑓𝑢𝑙𝑙 − 𝑟𝑜𝑤 𝑟𝑎𝑛𝑘 𝑡ℎ𝑒𝑛,
𝑘 = 𝐼 𝑇 𝐻𝐸 𝑇(𝐸𝐸 𝑇)−1
27
Control law [1]: 𝑢 𝑘 + 1 = 𝑢 𝑘 + 𝑘0 𝑒 𝑘 + 1 + 𝑘1 𝑒 𝑘 + 𝑘2 𝑒 𝑘 − 1