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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 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 & Optimization, doi: 10.3934/naco.2021007
##### References:
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Khovanskii, The application of continued functions and their generalizations to problems in approximation theory, Noordhoff Ltd., Groningen, 1963.  Google Scholar [6] H. Li, Y. 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.   Google Scholar [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. Google Scholar [8] A. M. Lopes and and J. T. Machado, Discrete-time generalized mean fractional order controllers, IFAC-PapersOnLine, 51 (2018), 43-47.   Google Scholar [9] Y. Luo, H. Li and Y. Q. 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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.  Google Scholar [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. Google Scholar [15] F. Merrikh-Bayat, N. 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.   Google Scholar [16] R. H. Middleton and G. C. Goodwin, Digital Control and Estimation: A Unified Approach, Prentice Hall, Englewood Cliffs, N.J., 1990. Google Scholar [17] C. A. Monje, B. M. Vinagre, V. Feliu and Y. Q. Chen, Tuning and auto-tuning of fractional order controllers for industry applications, Control Engineering Practice, 16 (2008), 798-812.   Google Scholar [18] C. I. Muresan, S. Folea, G. Mois and E. H. Dulf, Development and implementation of an FPGA based fractional order controller for a DC motor, Mechatronics, 23 (2013), 798-804.   Google Scholar [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.  Google Scholar [20] A. Oustaloup, CRONE Control: Robust Control of Non-integer Order, Paris, Hermes, 1991. doi: 10.1007/BFb0120098.  Google Scholar [21] I. Pan and S. Das, Intelligent Fractional Order Systems and Control: An introduction, Springer, Berlin, Heidelberg, 2012. Google Scholar [22] I. Podlubny, Fractional-order systems and PIλDµ controllers, IEEE Transactions on Automatic Control, 44 (1999), 208-214.  doi: 10.1109/9.739144.  Google Scholar [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.  Google Scholar [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. Google Scholar [25] J. Swarnakar, P. 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.  Google Scholar [26] J. Swarnakar, P. 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.   Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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.  Google Scholar [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. Google Scholar [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. Google Scholar

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. Google Scholar [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. Google Scholar [3] S. Folea, R. D Keyser, I. R. Birs, C. 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.   Google Scholar [4] R. De Keyser, C. 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.   Google Scholar [5] A. N. Khovanskii, The application of continued functions and their generalizations to problems in approximation theory, Noordhoff Ltd., Groningen, 1963.  Google Scholar [6] H. Li, Y. 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.   Google Scholar [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. Google Scholar [8] A. M. Lopes and and J. T. Machado, Discrete-time generalized mean fractional order controllers, IFAC-PapersOnLine, 51 (2018), 43-47.   Google Scholar [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. Google Scholar [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.  Google Scholar [11] Y. Luo, Y. Q. Chen, C. Y. Wang and Y. G. Pi, Tuning fractional order proportional integral controllers for fractional order systems, Journal of Process Control, 20 (2010), 823-831.   Google Scholar [12] R. L. Magin, Fractional Calculus in Bioengineering, Begell house publishers inc, Redding, 2006. Google Scholar [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.  Google Scholar [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. Google Scholar [15] F. Merrikh-Bayat, N. 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.   Google Scholar [16] R. H. Middleton and G. C. Goodwin, Digital Control and Estimation: A Unified Approach, Prentice Hall, Englewood Cliffs, N.J., 1990. Google Scholar [17] C. A. Monje, B. M. Vinagre, V. Feliu and Y. Q. Chen, Tuning and auto-tuning of fractional order controllers for industry applications, Control Engineering Practice, 16 (2008), 798-812.   Google Scholar [18] C. I. Muresan, S. Folea, G. Mois and E. H. Dulf, Development and implementation of an FPGA based fractional order controller for a DC motor, Mechatronics, 23 (2013), 798-804.   Google Scholar [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.  Google Scholar [20] A. Oustaloup, CRONE Control: Robust Control of Non-integer Order, Paris, Hermes, 1991. doi: 10.1007/BFb0120098.  Google Scholar [21] I. Pan and S. Das, Intelligent Fractional Order Systems and Control: An introduction, Springer, Berlin, Heidelberg, 2012. Google Scholar [22] I. Podlubny, Fractional-order systems and PIλDµ controllers, IEEE Transactions on Automatic Control, 44 (1999), 208-214.  doi: 10.1109/9.739144.  Google Scholar [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.  Google Scholar [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. Google Scholar [25] J. Swarnakar, P. 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.  Google Scholar [26] J. Swarnakar, P. 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.   Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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.  Google Scholar [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. Google Scholar [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. Google Scholar
The plot for $K_{i}$ versus $\nu$
Frequency responses of $C(s)$ and $C_{\delta}(\gamma)$ taking different sampling instants
Frequency responses of $G_{OL}(s)$ and $G_{OL\delta}(\gamma)$ taking different sampling instants
Comparison of step responses between the original closed loop system and its approximation taking two different sampling instants
Frequency responses of the open loop system by varying the plant gain taking $\triangle$ = 0.01 sec
Step responses of the closed loop system by varying the plant gain taking $\triangle$ = 0.01sec
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})$
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 $G_{OL\delta}(\gamma)$ $\triangle=1$ $\frac{2.921\times10^{6}\gamma^{6}+1.805\times10^{7}\gamma^{5}+4.172\times10^{7}\gamma^{4}+4.428 \times 10^{7}\gamma^{3}+2.126\times10^{7}\gamma^{2}+3.878\times10^{6}\gamma+1.61\times10^{5}} { \gamma \left(4.254\times10^{5}\gamma^{5}+1.658\times10^{6}\gamma^{4}+2.147\times10^{6}\gamma^{3}+1.066\times 10^{6}\gamma^{2}+1.757\times10^{5}\gamma+5360 \right)}$ $\triangle=0.1$ $\frac{1.291\times10^{4}\gamma^{6}+4.316\times10^{5}\gamma^{5} +4.34\times10^{6}\gamma^{4}+1.356\times10^{7}\gamma^{3} +1.303\times10^{7}\gamma^{2}+3.444\times10^{6}\gamma+1.61\times10^{5}} {\gamma \left( 2.398\times10^{4}\gamma^{5}+3.106\times10^{5}\gamma^{4}+9.161\times10^{5}\gamma^{3}+7.606\times 10^{5}\gamma^{2}+1.636\times10^{5}\gamma+5360 \right)}$ $\triangle=0.01$ $\frac{ 1094\gamma^{6}+1.188\times10^{5}\gamma^{5}+2.664\times10^{6}\gamma^{4}+1.129\times10^{7}\gamma^{3} +1.226\times10^{7}\gamma^{2}+3.4\times10^{6}\gamma+1.61\times10^{5}}{ \gamma\left(1.179\times10^{4}\gamma^{5}+2.327\times10^{5}\gamma^{4}+8.154\times10^{5}\gamma^{3}+7.312 \times10^{5}\gamma^{2}+1.624\times10^{5}\gamma+5360 \right) }$
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 $C_{\delta}(\gamma)$ $\triangle=1$ $\frac{5.698\times10^{11}\gamma^{11} +5.884\times10^{12}\gamma^{10}+2.604\times10^{13}\gamma^{9}+6.462 \times10^{8}\gamma^{8}+9.874\times10^{13}\gamma^{7}+9.615\times10^{13}\gamma^{6}+5.98\times10^{13}\gamma^{5} +2.32\times10^{13}\gamma^{4}+5.33\times10^{12}\gamma^{3}+6.624\times10^{11}\gamma^{2} +3.769 \times10^{10}\gamma+7.713\times10^{8} }{\gamma\left(1.274\times10^{11}\gamma^{10} +1.016\times10^{12}\gamma^{9}+3.38\times10^{12}\gamma^{8}+6.098 \times10^{12}\gamma^{7}+6.506\times10^{12} \gamma^{6}+4.203\times10^{12}\gamma^{5}+1.622\times10^{12}\gamma^{4}+3.565\times10^{11}\gamma^{3} +4.06\times10^{10}\gamma^{2}+1.995\times10^{9}\gamma +3.381\times10^{7}\right)}$ $\triangle=0.1$ $\frac{ 1.072\times10^{8}\gamma^{11}+5.213\times10^{9} \gamma^{10}+9.603\times10^{10}\gamma^{9}+8.515 \times10^{11}\gamma^{8}+ 3.874\times10^{12}\gamma^{7}+9.389\times10^{12}\gamma^{6} +1.22\times10^{13}\gamma^{5} +8.435\times10^{12}\gamma^{4}+2.979\times10^{12}\gamma^{3} +5.013\times10^{11}\gamma^{2} +3.387 \times 10^{10}\gamma+7.713\times10^{8}}{ \gamma\left(3.33\times10^{8}\gamma^{10}+9.124\times10^{9}\gamma^{9}+9.082\times10^{10}\gamma^{8} +4.135\times 10^{11}\gamma^{7}+9.518\times10^{11}\gamma^{6}+1.141\times10^{12}\gamma^{5} +7.148\times10^{11}\gamma^{4}+ 2.245\times10^{11}\gamma^{3}+3.283\times10^{10}\gamma^{2} +1.843\times10^{9}\gamma+3.381\times10^{7} \right)}$ $\triangle=0.01$ $\frac{ 3.798\times10^{6}\gamma^{11} +5.103\times10^{8}\gamma^{10}+2.031\times10^{10}\gamma^{9}+3.232\times 10^{11}\gamma^{8}+2.106\times10^{12}\gamma^{7}+6.439\times10^{12}\gamma^{6}+9.708\times10^{12}\gamma^{5} +7.399\times10^{12}\gamma^{4}+2.779\times10^{12}\gamma^{3}+4.862\times10^{11}\gamma^{2} +3.349\times10^{10}\gamma+7.713\times10^{8}}{ \gamma\left(7.238\times10^{7}\gamma^{10} +3.138\times10^{9}\gamma^{9}+4.556\times10^{10}\gamma^{8}+2.641 \times10^{11}\gamma^{7}+7.16\times10^{11}\gamma^{6}+9.579\times10^{11}\gamma^{5}+6.459\times10^{11}\gamma^{4} +2.128\times10^{11}\gamma^{3}+3.209\times10^{10}\gamma^{2} +1.828\times10^{9}\gamma +3.381\times10^{7} \right)}$