International Journal of Computer-aided Mechanical Design and Implementation Vol. 3, No. 1 (2017), pp.1-8 http://dx.doi.org/10.21742/ijcmdi.2017.3.1.01
Delay Compensation using the Smith Predictor: A Brief Review with Numerical Examples Andri Mirzal Department of Innovation and Technology Management College of Graduate Studies, Arabian Gulf University The Kingdom of Bahrain
[email protected] Abstract This paper discusses the using of the Smith Predictor for time delay compensation in control system. Time delay is a component that always reduce system stability, so it is important to compensate the effect of the delay for improving system stability. One of the mathematical model for capturing the structure of delay component in a control system is the Smith Predictor. This model not only beautifully captures the structure of delay component, but also gives an intuitive explanation about the compensation process. Keywords: control system, delay compensation, smith predictor.
1. Introduction Delay is defined as time interval between the starting of an event at one point in a system to the corresponding output/response at different point in the system [1]. Delay is also known as transport lag, deadtime, or time lag [2].
L (m)
Thermometer
Fuel
Furnace
v (m/s)
Blower
Figure 1. Example of delay in a heat transfer system due to mass flow
Delay is a phenomenon that commonly happens in physical, chemical, biological, economic, measurement, and computation system [1]. Delay can be caused by transportation and communication lag, time for generating feedback in a sensor system that requires sampling and analysis, time to generate ISSN: 2205-8591 IJCMDI Copyright ⓒ 2017 GV School Publication
International Journal of Computer-aided Mechanical Design and Implementation Vol. 3, No. 1 (2017)
control signal, and system parameters approximation using lower order. Delays always reduce stability of minimum phase systems (systems that do not have poles or zeros in the right-hand side of s-plane or do not have other delay component) [3], so that it is important to analyze system stability under the presence of delays. Delay in a control system is usually described in the complex frequency domain. X in (s )
X out ( s) e Ts X in (s)
e Ts
Figure 2. Delay in the frequency domain
2. The Smith Predictor General Structure The Smith Predictor is a delay compensation method that is commonly used in the control system. This method can be used for both small and large delay. The main idea in the Smith Predictor is how to remove delay component from system’s closed loop. The reason is that only delay that is in the system’s closed loop can affect system stability. Thus removing the delay from the loop can improve system stability [4]. R(s)
G p ( s )e
+
p s
Cp(s)
-
Figure 3. Closed loop control system with delay component
Fig 3 shows a control system with delay inside its closed loop. To remove the delay component from the closed loop a compensator C*(s) can be added as shown in fig 4. R(s)
C*(s) +
G p ( s )e
p s
Cp(s)
-
Figure 4. A compensator is added to the system
Fig 5 shows the structure of control system after the addition of compensator C*(s).
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International Journal of Computer-aided Mechanical Design and Implementation Vol. 3, No. 1 (2017)
R(s)
Gc(s) +
e
Gp(s)
Cp(s)
p s
-
Figure 5. The updated structure of the system due to the addition of compensator C*(s)
As system in fig 4 and fig 5 is equivalent, C*(s) can be found by
C * ( s)G p ( s)e
p s
1 C * ( s)G p ( s)e
C * ( s)G p ( s)e
p s
p s
Gc ( s)G p ( s) 1 Gc ( s)G p ( s)
e
p s
(1)
(1 Gc ( s)G p ( s)) Gc ( s)G p ( s)e
p s
(1 C * ( s)G p ( s)e
p s
.
(2)
So that
C ( s) *
Gc ( s)G p ( s)e
p s
s Gc (s)Gc2 (s)e 2 s
G p ( s) 1 Gc ( s)G p ( s) e Gc ( s )
1 Gc ( s)G p ( s) 1 e
p s
p
p
(3)
By using this formulation, we can depict the structure of the system with the Smith Predictor. R(s)
E(s) +
-
+
N(s)
Gc
G pe
p s
Cp(s)
-
Gp 1 e
p s
Figure 6. The Smith Predictor general structure for delay compensation s
The Smith Predictor structure in fig 6 is the idealistic one as predictor’s transfer function G p (1 e p ) is assumed to cancel the effect of delay in the loop perfectly. This can be explained by the following equations.
Gc ( s) N ( s) E ( s) 1 G ( s)G ( s) 1 e p s c p
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(4)
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International Journal of Computer-aided Mechanical Design and Implementation Vol. 3, No. 1 (2017)
N (s) C (s) C p (s) E ( s) N ( s) N (s) C (s) R( s) 1 E ( s) N ( s)
(5)
Gc ( s)G p ( s)e C p ( s) R( s )
p s
1 Gc ( s)G p ( s) 1 e 1
Gc ( s)G p ( s)e
p s
1 Gc ( s)G p ( s) 1 e
p s
Gc ( s)G p ( s)e C p ( s) R( s)
(6)
p s
p s
1 Gc ( s)G p ( s) 1 e
p s
G (s)G (s)e 1 G ( s)G ( s)1 e
1 Gc ( s)G p ( s) 1 e
p s
c
p s
(7)
p s
(8)
p
p s
c
C p (s) R( s)
C p ( s) R( s)
p
Gc ( s)G p ( s)e
1 Gc ( s)G p ( s) 1 e
Gc ( s)G p ( s )e
p s
p s
G (s)G (s)e c
p
p s
(9)
1 Gc ( s)G p ( s)
Eq (9) describes the system in fig 5, i.e., the expected system structure after the compensator added. The delay compensation shown in fig 6 needs a perfect estimation about process transfer s function G p e p which is not possible. To implement the Smith Predictor in real world scenario, one must s s estimate Gp and e p values. If we denote the estimates to be Gm and e m , then the Smith Predictor structure can be depicted as the following. R(s) Gc +
-
+
G pe
Cp(s)
p s
-
Gm
e m s
-
+
Cm(s)
Figure 7. The Smith Predictor with process parameter estimation The Smith Predictor transfer function in fig 7 can be obtained as follows.
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International Journal of Computer-aided Mechanical Design and Implementation Vol. 3, No. 1 (2017)
Gc G p e C p ( s) R( s)
C p ( s) R( s)
1 Gc Gm (1 e m s ) 1
p s
Gc G p e
(10)
p s
1 Gc Gm (1 e m s ) Gc G p e
p s
1 Gc G m Gc G p e
p s
Gm e m s
(11)
Eq (11) shows that if process parameter can be modeled without error then system transfer function in fig 7 is equivalent to the system transfer function in fig 5.
3. Simulation results Modeling inaccuracy can be caused by the difference between estimated process parameter and the real process parameter. The inaccuracy is more sensitive with respect to the delay modeling m than to the plant modeling Gm because the delay is modeled by using an exponential function. This implies that delay modeling is more crucial than plant modeling. The following presents simulation results of the system response before and after compensation to demonstrate the effect of delay compensation for system in fig 6 with assumptions that Gp(s) and p are known. There are three cases considered, first order, second order, and third order. 3.1. First case: Gc ( s) 1 , G p ( s)
(a) Without delay
1 , and p 0.5,1, dan 3 second. s 1
(b) With delay, no compensation
(c) Delay is compensated
Figure 8. First order system response with 0.5-second delay
(a) With delay, no compensation
(b) Delay is compensated
Figure 9. First order system response with 1-second delay
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International Journal of Computer-aided Mechanical Design and Implementation Vol. 3, No. 1 (2017)
(a) With delay, no compensation
(b) Delay is compensated
Figure 10. First order system response with 3-second delay
3.2. Second case: Gc ( s) 1 , G p ( s)
(a) Without delay
1 , and p 0.5,1, dan 3 second. s 1.4s 4 2
(b) With delay, no compensation
(c) With compensation
Figure 11. Second order system response with 0.5-second delay
(a) With delay, no compensation
(b) Delay is compensated
Figure 12. Second order system response with 1-second delay
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International Journal of Computer-aided Mechanical Design and Implementation Vol. 3, No. 1 (2017)
(a) With delay, no compensation
(b) Delay is compensated
Figure 13. Second order system response with 3-second delay
3.3. Third case: Gc ( s) 1 , G p ( s)
(d) Without delay
1 , and p 0.5,1, dan 3 seconds. s 3s 4s 2 3
2
(e) With delay, no compensation
(f)
With compensation
Figure 14. Third order system response with 0.5-second delay
(c) With delay, no compensation
(d) Delay is compensated
Figure 15. Third order system response with 1-second delay
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International Journal of Computer-aided Mechanical Design and Implementation Vol. 3, No. 1 (2017)
(c) With delay, no compensation
(d) Delay is compensated
Figure 16. Third order system response with 3-second delay
4. Conclusion The Smith Predictor is a powerful mathematical model for delay compensation to improve system stability. The Smith Predictor works by removing out the delay component from system’s closed loop so that the system can be analyzed and compensated similarly as system without delay component. Simulation has been conducted to numerically show the Smith Predictor capability in stabilizing the system response due to the delay. As shown in fig 8 to fig 16, the fluctuations in the system responses have been reduced due to the Smith Predictor. Thus it can be concluded that the Smith Predictor can improve stability of system with time delay.
References [1]. O’ Dwyer, “The Estimation and Compensation of Processes with Time Delays”, Ph.D. Thesis, School of Electronic Engineering, Dublin City University, (1996). [2]. G.F. Franklin, J.D. Powell and A. Emami-Naeni, “Feedback Control of Dynamic Systems”, 3rd edition, Addison-Wesley Publishing Company, (1994). [3]. O’Dwyer, “Performance and robustness issues in the compensation of FOLPD processes with PI and PID controllers”, School of Control Systems and Electrical Engineering, Dublin Institute of Technology, Kevin St, Dublin, Vol. 8, (1998). [4]. G. Stephanopoulus, “Chemical Process Control: An Introduction to Theory and Practice”, Prentice-Hall Int. Ed., (1984).
Authors Andri Mirzal Andri Mirzal received PhD and MSc in Information Science and Technology from Hokkaido University (Japan), and BEng in Electrical Engineering from Institut Teknologi Bandung (Indonesia). His research interests include machine learning, bioinformatics, optimization methods, web search engine, and linear inverse problems. Currently, he is an associate professor in Department of Innovation & Technology Management, College of Graduate Studies, Arabian Gulf University, The Kingdom of Bahrain. He was the recipient of Monbukagakusho Scholarship from the Japanese Government for both MSc and PhD programs (2006 - 2011). He is also the winner of National Research Award 2015 in Information and Communication Technology sector from The Research Council of Oman.
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