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

Jaydeep Swarnakar

doi:10.3934/naco.2021007

Article Contents

2022, Volume 12, Issue 2: 309-320. Doi: 10.3934/naco.2021007

This issue Previous Article Convergence of interval AOR method for linear interval equations Next Article Applying particle swarm optimization based on Padé approximant to solve ordinary differential equation

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

Jaydeep Swarnakar^,

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

^* Corresponding author: Jaydeep Swarnakar
^* Corresponding author: Jaydeep Swarnakar

Received: June 2020

Revised: May 2021

Early access: June 2021

Published: June 2022

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Abstract

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.

Keywords:

Fractional-order system (FOS),
Fractional-order proportional integral (FOPI) controller,
Continued fraction expansion (CFE),
Delta operator,
Robustness.

Mathematics Subject Classification: 37N35, 47N70.

Citation:

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Figure 1. The plot for $ K_{i} $ versus $ \nu $

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Figure 2. Frequency responses of $ C(s) $ and $ C_{\delta}(\gamma) $ taking different sampling instants

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Figure 3. Frequency responses of $ G_{OL}(s) $ and $ G_{OL\delta}(\gamma) $ taking different sampling instants

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Figure 4. Comparison of step responses between the original closed loop system and its approximation taking two different sampling instants

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Figure 5. Frequency responses of the open loop system by varying the plant gain taking $ \triangle $ = 0.01 sec

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Figure 6. Step responses of the closed loop system by varying the plant gain taking $ \triangle $ = 0.01sec

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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}) $

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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) }$

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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)} $

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