June  2022, 12(2): 309-320. doi: 10.3934/naco.2021007

Discrete-time realization of fractional-order proportional integral controller for a class of fractional-order system

Department of Electronics and Communication Engineering, North-Eastern Hill University, Shillong-793022, Meghalaya, India

* Corresponding author: Jaydeep Swarnakar

Received  June 2020 Revised  May 2021 Published  June 2022 Early access  June 2021

The approximation of the fractional-order controller (FOC) has already been recognized as a distinguished field of research in the literature of system and control. In this paper, a two-step design approach is presented to realize a fractional-order proportional integral controller (FOPI) for a class of fractional-order plant model. The design goals are based on some frequency domain specifications. The first stage of the work is focused on developing the pure continuous-time FOC, while the second stage actually realizes the FOPI controller in discrete-time representation. The presented approach is fundamentally dissimilar with respect to the conventional approaches of z -domain. In the process of realizing the FOC, the delta operator has been involved as a generating function due to its exclusive competency to unify the discrete-time system and its continuous-time counterpart at low sampling time limit. The well-known continued fraction expansion (CFE) method has been employed to approximate the FOPI controller in delta-domain. Simulation outcomes exhibit that the discrete-time FOPI controller merges to its continuous-time counterpart at the low sampling time limit. The robustness of the overall system is also investigated in delta-domain.

Citation: Jaydeep Swarnakar. Discrete-time realization of fractional-order proportional integral controller for a class of fractional-order system. Numerical Algebra, Control and Optimization, 2022, 12 (2) : 309-320. doi: 10.3934/naco.2021007
References:
[1]

Y. Q. Chen, I. Petras and D. Xue, Fractional-order control tutorial, in Proceeding of the IEEE American Control Conference (ACC), St. Louis, USA, (2009), 1397–1411.

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C. Copot, C. Muresan, C. M. Ionescu and R. De. Keyser, Fractional order control of a DC motor with load changes, in Proceedings of the international Conference on Optimization of Electrical and Electronic Equipment (OPTIM), Bran, Romania, (2014), 956–961.

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S. FoleaR. D KeyserI. R. BirsC. I. Muresan and C. Ionescu, Discrete-time implementation and experimental validation of a fractional order PD controller for vibration suppression in airplane wings, Acta Polytechnica Hungarica, 14 (2017), 191-206. 

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R. De KeyserC. I. Muresan and C. M. Ionescu, An efficient algorithm for low-order direct discrete-time implementation of fractional order transfer functions, ISA Transactions, 74 (2018), 229-238. 

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A. N. Khovanskii, The application of continued functions and their generalizations to problems in approximation theory, Noordhoff Ltd., Groningen, 1963.

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H. LiY. Luo and Y. Chen, A fractional order proportional and derivative (FOPD) motion controller: Tuning rule and experiments, IEEE Transactions on Control Systems Technology, 18 (2010), 516-520. 

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W. Li and Y. Hori, Design of fractional-order PIα controller with two modes, in Proceedings of the IEEE 12 th international Power Electronics and Motion Control Conference, Shanghai, China, (2006), 1–5.

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A. M. Lopes and and J. T. Machado, Discrete-time generalized mean fractional order controllers, IFAC-PapersOnLine, 51 (2018), 43-47. 

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Y. Luo, H. Li and Y. Q. Chen, Fractional order proportional and derivative controller synthesis for a class of fractional order systems: tuning rule and hardware-in-the-loop experiment, in Proceedings of the 48 th IEEE Conference on Decision and Control held jointly with 28 th Chinese Control Conference, Shanghai, China, (2009), 5460–5465.

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Y. Luo, C. Y. Wang and Y. Q. Chen, Tuning fractional order proportional integral controllers for fractional order systems, in Proceedings of the Chinese Control and Decision Conference, Norteastern University, Guilin, China, (2009), 307–312. doi: 10.1016/j.sysconle.2010.01.008.

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Y. LuoY. Q. ChenC. Y. Wang and Y. G. Pi, Tuning fractional order proportional integral controllers for fractional order systems, Journal of Process Control, 20 (2010), 823-831. 

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R. L. Magin, Fractional Calculus in Bioengineering, Begell house publishers inc, Redding, 2006.

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G. Maione, High-speed digital realizations of fractional operators in the delta-domain, IEEE Transactions on Automatic Control, 56 (2011), 697-702.  doi: 10.1109/TAC.2010.2101134.

[14]

G. Maione and M. P. Lazarevi, On the symmetric distribution of interlaced zero-pole pairs approximating the discrete fractional tustin operator, in Proceedings of the IEEE International Conference on Systems, Man and Cybernetics (SMC), Bari, Italy, (2019), 2578–2583.

[15]

F. Merrikh-BayatN. Mirebrahimi and M. R. Khalili, Discrete-time fractional-order PID controller: definition, tuning, digital realization and some applications, International Journal of Control, Automation and Systems, 13 (2015), 81-90. 

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R. H. Middleton and G. C. Goodwin, Digital Control and Estimation: A Unified Approach, Prentice Hall, Englewood Cliffs, N.J., 1990.

[17]

C. A. MonjeB. M. VinagreV. Feliu and Y. Q. Chen, Tuning and auto-tuning of fractional order controllers for industry applications, Control Engineering Practice, 16 (2008), 798-812. 

[18]

C. I. MuresanS. FoleaG. Mois and E. H. Dulf, Development and implementation of an FPGA based fractional order controller for a DC motor, Mechatronics, 23 (2013), 798-804. 

[19]

A. Narang, S. L. Shah and T. Chen, Tuning of fractional PI controllers for fractional order system models with and without time delays, in Proceedings of the American Control Conference (ACC), Marriot Waterfront, Baltimore, MD, USA, (2010), 6674–6679. doi: 10.1016/j.sysconle.2010.01.008.

[20]

A. Oustaloup, CRONE Control: Robust Control of Non-integer Order, Paris, Hermes, 1991. doi: 10.1007/BFb0120098.

[21]

I. Pan and S. Das, Intelligent Fractional Order Systems and Control: An introduction, Springer, Berlin, Heidelberg, 2012.

[22]

I. Podlubny, Fractional-order systems and PIλDµ controllers, IEEE Transactions on Automatic Control, 44 (1999), 208-214.  doi: 10.1109/9.739144.

[23]

H. Sheng, Y. Q. Chen and T. S. Qiu, Fractional Processes and Fractional-Order Signal Processing, Springer, London, 2011. doi: 10.1007/978-1-4471-2233-3.

[24]

J. Sun, C. Wang and R. Xin, Design of fractional order proportional differentiation controller for second order position servo system, in Proceedings of the IEEE Chinese Control and Decision Conference (CCDC), Shenyang, China, (2018), 5939–5944.

[25]

J. SwarnakarP. Sarkar and L. J. Singh, Realization of fractional-order proportional derivative controller for a class of fractional-order system in delta-domain, intelligent Communication, Control and Devices, Advances in intelligent Systems and Computing, 989 (2020), 303-311.  doi: 10.1007/978-3-642-31549-7.

[26]

J. SwarnakarP. Sarkar and L. J. Singh, Direct discretization method for realizing a class of fractional order system in delta-domain unified approach, Automatic Control and Computer Sciences, 53 (2019), 127-139. 

[27]

A. V. Tare, M. M. Joshi and V. A. Vyawahare, Discrete approximation methods for linear fractional-order systems: a comparative study, in Proceedings of the IEEE international Conference on Circuits, Systems, Communication and information Technology Applications (CSCITA), Mumbai, India, (2014), 105–110.

[28]

C. Wang, Y. Luo and Y. Q. Chen, An analytical design of fractional order proportional integral and [proportional integral] controllers for robust velocity servo, in Proceedings of the 4 th IEEE Conference on industrial Electronics and Applications, Xian, China, 53 (2009), 3448–3453.

[29]

C. Wang, W. Fu and Y. Shi, Tuning fractional order proportional integral differentiation controller for fractional order system, in Proceedings of the $32^nd$ IEEE Chinese Control Conference (CCC), Xian, China, (2013), 552–555.

[30]

D. Xue, C. Zhao and Y. Q. Chen, Fractional order PID control of a DC-motor with elastic shaft: a case study, in Proceedings of the IEEE American Control Conference (ACC), Minnesota, USA, (2006), 3182–3187.

[31]

H. Yang, H. Y. Xia, P. Shi and L. Zhao, Analysis and synthesis of delta operator systems, Lecture Notes in Control and information Sciences, Springer, Berlin, Heidelberg, 430 (2012). doi: 10.1007/978-3-642-28774-9.

[32]

M. Zamojski, Implementation of fractional order PID controller based on recursive oustaloup'e filter, in Proceedings of the IEEE international interdisciplinary PhD Workshop, Swinoujcie, Poland, (2018), 414–417.

[33]

C. N. Zhao, D. Xue and Y. Q. Chen, A fractional order PID tuning algorithm for a class of fractional order plants, in Proceedings of the IEEE/ICMA, Niagra Falls, USA, (2005), 216–221.

show all references

References:
[1]

Y. Q. Chen, I. Petras and D. Xue, Fractional-order control tutorial, in Proceeding of the IEEE American Control Conference (ACC), St. Louis, USA, (2009), 1397–1411.

[2]

C. Copot, C. Muresan, C. M. Ionescu and R. De. Keyser, Fractional order control of a DC motor with load changes, in Proceedings of the international Conference on Optimization of Electrical and Electronic Equipment (OPTIM), Bran, Romania, (2014), 956–961.

[3]

S. FoleaR. D KeyserI. R. BirsC. I. Muresan and C. Ionescu, Discrete-time implementation and experimental validation of a fractional order PD controller for vibration suppression in airplane wings, Acta Polytechnica Hungarica, 14 (2017), 191-206. 

[4]

R. De KeyserC. I. Muresan and C. M. Ionescu, An efficient algorithm for low-order direct discrete-time implementation of fractional order transfer functions, ISA Transactions, 74 (2018), 229-238. 

[5]

A. N. Khovanskii, The application of continued functions and their generalizations to problems in approximation theory, Noordhoff Ltd., Groningen, 1963.

[6]

H. LiY. Luo and Y. Chen, A fractional order proportional and derivative (FOPD) motion controller: Tuning rule and experiments, IEEE Transactions on Control Systems Technology, 18 (2010), 516-520. 

[7]

W. Li and Y. Hori, Design of fractional-order PIα controller with two modes, in Proceedings of the IEEE 12 th international Power Electronics and Motion Control Conference, Shanghai, China, (2006), 1–5.

[8]

A. M. Lopes and and J. T. Machado, Discrete-time generalized mean fractional order controllers, IFAC-PapersOnLine, 51 (2018), 43-47. 

[9]

Y. Luo, H. Li and Y. Q. Chen, Fractional order proportional and derivative controller synthesis for a class of fractional order systems: tuning rule and hardware-in-the-loop experiment, in Proceedings of the 48 th IEEE Conference on Decision and Control held jointly with 28 th Chinese Control Conference, Shanghai, China, (2009), 5460–5465.

[10]

Y. Luo, C. Y. Wang and Y. Q. Chen, Tuning fractional order proportional integral controllers for fractional order systems, in Proceedings of the Chinese Control and Decision Conference, Norteastern University, Guilin, China, (2009), 307–312. doi: 10.1016/j.sysconle.2010.01.008.

[11]

Y. LuoY. Q. ChenC. Y. Wang and Y. G. Pi, Tuning fractional order proportional integral controllers for fractional order systems, Journal of Process Control, 20 (2010), 823-831. 

[12]

R. L. Magin, Fractional Calculus in Bioengineering, Begell house publishers inc, Redding, 2006.

[13]

G. Maione, High-speed digital realizations of fractional operators in the delta-domain, IEEE Transactions on Automatic Control, 56 (2011), 697-702.  doi: 10.1109/TAC.2010.2101134.

[14]

G. Maione and M. P. Lazarevi, On the symmetric distribution of interlaced zero-pole pairs approximating the discrete fractional tustin operator, in Proceedings of the IEEE International Conference on Systems, Man and Cybernetics (SMC), Bari, Italy, (2019), 2578–2583.

[15]

F. Merrikh-BayatN. Mirebrahimi and M. R. Khalili, Discrete-time fractional-order PID controller: definition, tuning, digital realization and some applications, International Journal of Control, Automation and Systems, 13 (2015), 81-90. 

[16]

R. H. Middleton and G. C. Goodwin, Digital Control and Estimation: A Unified Approach, Prentice Hall, Englewood Cliffs, N.J., 1990.

[17]

C. A. MonjeB. M. VinagreV. Feliu and Y. Q. Chen, Tuning and auto-tuning of fractional order controllers for industry applications, Control Engineering Practice, 16 (2008), 798-812. 

[18]

C. I. MuresanS. FoleaG. Mois and E. H. Dulf, Development and implementation of an FPGA based fractional order controller for a DC motor, Mechatronics, 23 (2013), 798-804. 

[19]

A. Narang, S. L. Shah and T. Chen, Tuning of fractional PI controllers for fractional order system models with and without time delays, in Proceedings of the American Control Conference (ACC), Marriot Waterfront, Baltimore, MD, USA, (2010), 6674–6679. doi: 10.1016/j.sysconle.2010.01.008.

[20]

A. Oustaloup, CRONE Control: Robust Control of Non-integer Order, Paris, Hermes, 1991. doi: 10.1007/BFb0120098.

[21]

I. Pan and S. Das, Intelligent Fractional Order Systems and Control: An introduction, Springer, Berlin, Heidelberg, 2012.

[22]

I. Podlubny, Fractional-order systems and PIλDµ controllers, IEEE Transactions on Automatic Control, 44 (1999), 208-214.  doi: 10.1109/9.739144.

[23]

H. Sheng, Y. Q. Chen and T. S. Qiu, Fractional Processes and Fractional-Order Signal Processing, Springer, London, 2011. doi: 10.1007/978-1-4471-2233-3.

[24]

J. Sun, C. Wang and R. Xin, Design of fractional order proportional differentiation controller for second order position servo system, in Proceedings of the IEEE Chinese Control and Decision Conference (CCDC), Shenyang, China, (2018), 5939–5944.

[25]

J. SwarnakarP. Sarkar and L. J. Singh, Realization of fractional-order proportional derivative controller for a class of fractional-order system in delta-domain, intelligent Communication, Control and Devices, Advances in intelligent Systems and Computing, 989 (2020), 303-311.  doi: 10.1007/978-3-642-31549-7.

[26]

J. SwarnakarP. Sarkar and L. J. Singh, Direct discretization method for realizing a class of fractional order system in delta-domain unified approach, Automatic Control and Computer Sciences, 53 (2019), 127-139. 

[27]

A. V. Tare, M. M. Joshi and V. A. Vyawahare, Discrete approximation methods for linear fractional-order systems: a comparative study, in Proceedings of the IEEE international Conference on Circuits, Systems, Communication and information Technology Applications (CSCITA), Mumbai, India, (2014), 105–110.

[28]

C. Wang, Y. Luo and Y. Q. Chen, An analytical design of fractional order proportional integral and [proportional integral] controllers for robust velocity servo, in Proceedings of the 4 th IEEE Conference on industrial Electronics and Applications, Xian, China, 53 (2009), 3448–3453.

[29]

C. Wang, W. Fu and Y. Shi, Tuning fractional order proportional integral differentiation controller for fractional order system, in Proceedings of the $32^nd$ IEEE Chinese Control Conference (CCC), Xian, China, (2013), 552–555.

[30]

D. Xue, C. Zhao and Y. Q. Chen, Fractional order PID control of a DC-motor with elastic shaft: a case study, in Proceedings of the IEEE American Control Conference (ACC), Minnesota, USA, (2006), 3182–3187.

[31]

H. Yang, H. Y. Xia, P. Shi and L. Zhao, Analysis and synthesis of delta operator systems, Lecture Notes in Control and information Sciences, Springer, Berlin, Heidelberg, 430 (2012). doi: 10.1007/978-3-642-28774-9.

[32]

M. Zamojski, Implementation of fractional order PID controller based on recursive oustaloup'e filter, in Proceedings of the IEEE international interdisciplinary PhD Workshop, Swinoujcie, Poland, (2018), 414–417.

[33]

C. N. Zhao, D. Xue and Y. Q. Chen, A fractional order PID tuning algorithm for a class of fractional order plants, in Proceedings of the IEEE/ICMA, Niagra Falls, USA, (2005), 216–221.

Figure 1.  The plot for $ K_{i} $ versus $ \nu $
Figure 2.  Frequency responses of $ C(s) $ and $ C_{\delta}(\gamma) $ taking different sampling instants
Figure 3.  Frequency responses of $ G_{OL}(s) $ and $ G_{OL\delta}(\gamma) $ taking different sampling instants
Figure 4.  Comparison of step responses between the original closed loop system and its approximation taking two different sampling instants
Figure 5.  Frequency responses of the open loop system by varying the plant gain taking $ \triangle $ = 0.01 sec
Figure 6.  Step responses of the closed loop system by varying the plant gain taking $ \triangle $ = 0.01sec
Table 1.  Numerator and denominator coefficients obtained from fifth-order CFE approximation of $ s^{\nu}\approx\bigg(L\times\frac{\gamma}{\gamma+L}\bigg)^{\nu}. $
$ J_{0} = \alpha{_{5}}+L\alpha{_{4}}+L^{2}\alpha{_{3}}+L^{3}\alpha{_{2}}+L^{4} \alpha{_{1}}+L^{5}\alpha{_{0}} $
$ J_{1} = 5L\alpha{_{5}}+4L^{2}\alpha{_{4}}+3L^{3}\alpha{_{3}}+2L^{4}\alpha{_{2}}+L^{5}\alpha{_{1}} $
$ J_{2} = 10L^{2}\alpha{_{5}}+6L^{3}\alpha{_{4}}+3L^{4}\alpha{_{3}}+L^{5}\alpha{_{2}} $
$ J_{3} = 10L^{3}\alpha{_{5}}+4L^{4}\alpha{_{4}}+L^{5}\alpha{_{3}} $
$ J_{4} = 5L^{4}\alpha{_{5}}+L^{5}\alpha{_{4}} $
$ J_{5} = L^{5}\alpha{_{5}} $
$ K_{0} = \alpha_{0}+L\alpha_{1}+L^{2}\alpha_{2}+L^{3}\alpha_{3}+L^{4}\alpha_{4}+L^{5}\alpha_{5} $
$ K_{1} = 5L\alpha_{0}+4L^{2}\alpha_{1}+3L^{3}\alpha_{2}+2L^{4}\alpha_{3}+L^{5}\alpha_{4} $
$ K_{2} = 10L^{2}\alpha_{0}+6L^{3}\alpha_{1}+3L^{4}\alpha_{2}+L^{5}\alpha_{3} $
$ K_{3} = 10L^{3}\alpha_{0}+4L^{4}\alpha_{1}+L^{5}\alpha_{2} $
$ K_{4} = 5L^{4}\alpha_{0}+L^{5}\alpha_{1} $
$ K_{5} = L^{5}\alpha_{5} $
$ L = \frac{2}{\triangle} $
$ \alpha_{0} = -(120+274\nu+225\nu^{2}+85\nu^{3}+15\nu^{4}+\nu^{5}) $
$ \alpha_{1} = -(3000+3250\nu+1005\nu^{2}-5\nu^{3}-45\nu^{4}-5\nu^{5}) $
$ \alpha_{2} = -(12000+4000\nu-1230\nu^{2}-410\nu^{3}+30\nu^{4}+10\nu^{5}) $
$ \alpha_{3} = -(12000-4000\nu-1230\nu^{2}+410\nu^{3}+305\nu^{4}-10\nu^{5}) $
$ \alpha_{4} = -(3000-3250\nu+100\nu^{2}+5\nu^{3}-45\nu^{4}+5\nu^{5}) $
$ \alpha_{5} = -(120-274\nu+225\nu^{2}-85\nu^{3}+15\nu^{4}-\nu^{5}) $
$ J_{0} = \alpha{_{5}}+L\alpha{_{4}}+L^{2}\alpha{_{3}}+L^{3}\alpha{_{2}}+L^{4} \alpha{_{1}}+L^{5}\alpha{_{0}} $
$ J_{1} = 5L\alpha{_{5}}+4L^{2}\alpha{_{4}}+3L^{3}\alpha{_{3}}+2L^{4}\alpha{_{2}}+L^{5}\alpha{_{1}} $
$ J_{2} = 10L^{2}\alpha{_{5}}+6L^{3}\alpha{_{4}}+3L^{4}\alpha{_{3}}+L^{5}\alpha{_{2}} $
$ J_{3} = 10L^{3}\alpha{_{5}}+4L^{4}\alpha{_{4}}+L^{5}\alpha{_{3}} $
$ J_{4} = 5L^{4}\alpha{_{5}}+L^{5}\alpha{_{4}} $
$ J_{5} = L^{5}\alpha{_{5}} $
$ K_{0} = \alpha_{0}+L\alpha_{1}+L^{2}\alpha_{2}+L^{3}\alpha_{3}+L^{4}\alpha_{4}+L^{5}\alpha_{5} $
$ K_{1} = 5L\alpha_{0}+4L^{2}\alpha_{1}+3L^{3}\alpha_{2}+2L^{4}\alpha_{3}+L^{5}\alpha_{4} $
$ K_{2} = 10L^{2}\alpha_{0}+6L^{3}\alpha_{1}+3L^{4}\alpha_{2}+L^{5}\alpha_{3} $
$ K_{3} = 10L^{3}\alpha_{0}+4L^{4}\alpha_{1}+L^{5}\alpha_{2} $
$ K_{4} = 5L^{4}\alpha_{0}+L^{5}\alpha_{1} $
$ K_{5} = L^{5}\alpha_{5} $
$ L = \frac{2}{\triangle} $
$ \alpha_{0} = -(120+274\nu+225\nu^{2}+85\nu^{3}+15\nu^{4}+\nu^{5}) $
$ \alpha_{1} = -(3000+3250\nu+1005\nu^{2}-5\nu^{3}-45\nu^{4}-5\nu^{5}) $
$ \alpha_{2} = -(12000+4000\nu-1230\nu^{2}-410\nu^{3}+30\nu^{4}+10\nu^{5}) $
$ \alpha_{3} = -(12000-4000\nu-1230\nu^{2}+410\nu^{3}+305\nu^{4}-10\nu^{5}) $
$ \alpha_{4} = -(3000-3250\nu+100\nu^{2}+5\nu^{3}-45\nu^{4}+5\nu^{5}) $
$ \alpha_{5} = -(120-274\nu+225\nu^{2}-85\nu^{3}+15\nu^{4}-\nu^{5}) $
Table 2.  Approximation of FOPI controller at different sampling instants
Sampling time (sec) Approximation of $ G_{OL\delta}(\gamma) $
$ \triangle=1 $ $ \frac{2.921\times10^{6}\gamma^{6}+1.805\times10^{7}\gamma^{5}+4.172\times10^{7}\gamma^{4}+4.428 \times 10^{7}\gamma^{3}+2.126\times10^{7}\gamma^{2}+3.878\times10^{6}\gamma+1.61\times10^{5}} { \gamma \left(4.254\times10^{5}\gamma^{5}+1.658\times10^{6}\gamma^{4}+2.147\times10^{6}\gamma^{3}+1.066\times 10^{6}\gamma^{2}+1.757\times10^{5}\gamma+5360 \right)}$
$ \triangle=0.1 $ $ \frac{1.291\times10^{4}\gamma^{6}+4.316\times10^{5}\gamma^{5} +4.34\times10^{6}\gamma^{4}+1.356\times10^{7}\gamma^{3} +1.303\times10^{7}\gamma^{2}+3.444\times10^{6}\gamma+1.61\times10^{5}} {\gamma \left( 2.398\times10^{4}\gamma^{5}+3.106\times10^{5}\gamma^{4}+9.161\times10^{5}\gamma^{3}+7.606\times 10^{5}\gamma^{2}+1.636\times10^{5}\gamma+5360 \right)} $
$ \triangle=0.01 $ $ \frac{ 1094\gamma^{6}+1.188\times10^{5}\gamma^{5}+2.664\times10^{6}\gamma^{4}+1.129\times10^{7}\gamma^{3} +1.226\times10^{7}\gamma^{2}+3.4\times10^{6}\gamma+1.61\times10^{5}}{ \gamma\left(1.179\times10^{4}\gamma^{5}+2.327\times10^{5}\gamma^{4}+8.154\times10^{5}\gamma^{3}+7.312 \times10^{5}\gamma^{2}+1.624\times10^{5}\gamma+5360 \right) }$
Sampling time (sec) Approximation of $ G_{OL\delta}(\gamma) $
$ \triangle=1 $ $ \frac{2.921\times10^{6}\gamma^{6}+1.805\times10^{7}\gamma^{5}+4.172\times10^{7}\gamma^{4}+4.428 \times 10^{7}\gamma^{3}+2.126\times10^{7}\gamma^{2}+3.878\times10^{6}\gamma+1.61\times10^{5}} { \gamma \left(4.254\times10^{5}\gamma^{5}+1.658\times10^{6}\gamma^{4}+2.147\times10^{6}\gamma^{3}+1.066\times 10^{6}\gamma^{2}+1.757\times10^{5}\gamma+5360 \right)}$
$ \triangle=0.1 $ $ \frac{1.291\times10^{4}\gamma^{6}+4.316\times10^{5}\gamma^{5} +4.34\times10^{6}\gamma^{4}+1.356\times10^{7}\gamma^{3} +1.303\times10^{7}\gamma^{2}+3.444\times10^{6}\gamma+1.61\times10^{5}} {\gamma \left( 2.398\times10^{4}\gamma^{5}+3.106\times10^{5}\gamma^{4}+9.161\times10^{5}\gamma^{3}+7.606\times 10^{5}\gamma^{2}+1.636\times10^{5}\gamma+5360 \right)} $
$ \triangle=0.01 $ $ \frac{ 1094\gamma^{6}+1.188\times10^{5}\gamma^{5}+2.664\times10^{6}\gamma^{4}+1.129\times10^{7}\gamma^{3} +1.226\times10^{7}\gamma^{2}+3.4\times10^{6}\gamma+1.61\times10^{5}}{ \gamma\left(1.179\times10^{4}\gamma^{5}+2.327\times10^{5}\gamma^{4}+8.154\times10^{5}\gamma^{3}+7.312 \times10^{5}\gamma^{2}+1.624\times10^{5}\gamma+5360 \right) }$
Table 3.  Approximation of open loop system at different sampling instants
Sampling time (sec) Approximation of $ C_{\delta}(\gamma) $
$ \triangle=1 $ $ \frac{5.698\times10^{11}\gamma^{11} +5.884\times10^{12}\gamma^{10}+2.604\times10^{13}\gamma^{9}+6.462 \times10^{8}\gamma^{8}+9.874\times10^{13}\gamma^{7}+9.615\times10^{13}\gamma^{6}+5.98\times10^{13}\gamma^{5} +2.32\times10^{13}\gamma^{4}+5.33\times10^{12}\gamma^{3}+6.624\times10^{11}\gamma^{2} +3.769 \times10^{10}\gamma+7.713\times10^{8} }{\gamma\left(1.274\times10^{11}\gamma^{10} +1.016\times10^{12}\gamma^{9}+3.38\times10^{12}\gamma^{8}+6.098 \times10^{12}\gamma^{7}+6.506\times10^{12} \gamma^{6}+4.203\times10^{12}\gamma^{5}+1.622\times10^{12}\gamma^{4}+3.565\times10^{11}\gamma^{3} +4.06\times10^{10}\gamma^{2}+1.995\times10^{9}\gamma +3.381\times10^{7}\right)} $
$ \triangle=0.1 $ $ \frac{ 1.072\times10^{8}\gamma^{11}+5.213\times10^{9} \gamma^{10}+9.603\times10^{10}\gamma^{9}+8.515 \times10^{11}\gamma^{8}+ 3.874\times10^{12}\gamma^{7}+9.389\times10^{12}\gamma^{6} +1.22\times10^{13}\gamma^{5} +8.435\times10^{12}\gamma^{4}+2.979\times10^{12}\gamma^{3} +5.013\times10^{11}\gamma^{2} +3.387 \times 10^{10}\gamma+7.713\times10^{8}}{ \gamma\left(3.33\times10^{8}\gamma^{10}+9.124\times10^{9}\gamma^{9}+9.082\times10^{10}\gamma^{8} +4.135\times 10^{11}\gamma^{7}+9.518\times10^{11}\gamma^{6}+1.141\times10^{12}\gamma^{5} +7.148\times10^{11}\gamma^{4}+ 2.245\times10^{11}\gamma^{3}+3.283\times10^{10}\gamma^{2} +1.843\times10^{9}\gamma+3.381\times10^{7} \right)} $
$ \triangle=0.01 $ $ \frac{ 3.798\times10^{6}\gamma^{11} +5.103\times10^{8}\gamma^{10}+2.031\times10^{10}\gamma^{9}+3.232\times 10^{11}\gamma^{8}+2.106\times10^{12}\gamma^{7}+6.439\times10^{12}\gamma^{6}+9.708\times10^{12}\gamma^{5} +7.399\times10^{12}\gamma^{4}+2.779\times10^{12}\gamma^{3}+4.862\times10^{11}\gamma^{2} +3.349\times10^{10}\gamma+7.713\times10^{8}}{ \gamma\left(7.238\times10^{7}\gamma^{10} +3.138\times10^{9}\gamma^{9}+4.556\times10^{10}\gamma^{8}+2.641 \times10^{11}\gamma^{7}+7.16\times10^{11}\gamma^{6}+9.579\times10^{11}\gamma^{5}+6.459\times10^{11}\gamma^{4} +2.128\times10^{11}\gamma^{3}+3.209\times10^{10}\gamma^{2} +1.828\times10^{9}\gamma +3.381\times10^{7} \right)} $
Sampling time (sec) Approximation of $ C_{\delta}(\gamma) $
$ \triangle=1 $ $ \frac{5.698\times10^{11}\gamma^{11} +5.884\times10^{12}\gamma^{10}+2.604\times10^{13}\gamma^{9}+6.462 \times10^{8}\gamma^{8}+9.874\times10^{13}\gamma^{7}+9.615\times10^{13}\gamma^{6}+5.98\times10^{13}\gamma^{5} +2.32\times10^{13}\gamma^{4}+5.33\times10^{12}\gamma^{3}+6.624\times10^{11}\gamma^{2} +3.769 \times10^{10}\gamma+7.713\times10^{8} }{\gamma\left(1.274\times10^{11}\gamma^{10} +1.016\times10^{12}\gamma^{9}+3.38\times10^{12}\gamma^{8}+6.098 \times10^{12}\gamma^{7}+6.506\times10^{12} \gamma^{6}+4.203\times10^{12}\gamma^{5}+1.622\times10^{12}\gamma^{4}+3.565\times10^{11}\gamma^{3} +4.06\times10^{10}\gamma^{2}+1.995\times10^{9}\gamma +3.381\times10^{7}\right)} $
$ \triangle=0.1 $ $ \frac{ 1.072\times10^{8}\gamma^{11}+5.213\times10^{9} \gamma^{10}+9.603\times10^{10}\gamma^{9}+8.515 \times10^{11}\gamma^{8}+ 3.874\times10^{12}\gamma^{7}+9.389\times10^{12}\gamma^{6} +1.22\times10^{13}\gamma^{5} +8.435\times10^{12}\gamma^{4}+2.979\times10^{12}\gamma^{3} +5.013\times10^{11}\gamma^{2} +3.387 \times 10^{10}\gamma+7.713\times10^{8}}{ \gamma\left(3.33\times10^{8}\gamma^{10}+9.124\times10^{9}\gamma^{9}+9.082\times10^{10}\gamma^{8} +4.135\times 10^{11}\gamma^{7}+9.518\times10^{11}\gamma^{6}+1.141\times10^{12}\gamma^{5} +7.148\times10^{11}\gamma^{4}+ 2.245\times10^{11}\gamma^{3}+3.283\times10^{10}\gamma^{2} +1.843\times10^{9}\gamma+3.381\times10^{7} \right)} $
$ \triangle=0.01 $ $ \frac{ 3.798\times10^{6}\gamma^{11} +5.103\times10^{8}\gamma^{10}+2.031\times10^{10}\gamma^{9}+3.232\times 10^{11}\gamma^{8}+2.106\times10^{12}\gamma^{7}+6.439\times10^{12}\gamma^{6}+9.708\times10^{12}\gamma^{5} +7.399\times10^{12}\gamma^{4}+2.779\times10^{12}\gamma^{3}+4.862\times10^{11}\gamma^{2} +3.349\times10^{10}\gamma+7.713\times10^{8}}{ \gamma\left(7.238\times10^{7}\gamma^{10} +3.138\times10^{9}\gamma^{9}+4.556\times10^{10}\gamma^{8}+2.641 \times10^{11}\gamma^{7}+7.16\times10^{11}\gamma^{6}+9.579\times10^{11}\gamma^{5}+6.459\times10^{11}\gamma^{4} +2.128\times10^{11}\gamma^{3}+3.209\times10^{10}\gamma^{2} +1.828\times10^{9}\gamma +3.381\times10^{7} \right)} $
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