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

  • * Corresponding author: Jaydeep Swarnakar

    * Corresponding author: Jaydeep Swarnakar
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  • 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.

    Mathematics Subject Classification: 37N35, 47N70.


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  • 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}) $
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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) }$
     | Show Table
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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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