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March  2020, 10(1): 171-187. doi: 10.3934/mcrf.2019035

A direct method based on the Clenshaw-Curtis formula for fractional optimal control problems

Mohammad Hadi Noori Skandari , , Marzieh Habibli and  Alireza Nazemi

Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran

* Corresponding author: Mohammad Hadi Noori Skandari

Received  November 2017 Revised  February 2019 Published  March 2020 Early access  August 2019

In this paper, we present a new method based on the Clenshaw-Curtis formula to solve a class of fractional optimal control problems. First, we convert the fractional optimal control problem to an equivalent problem in the fractional calculus of variations. Then, by utilizing the Clenshaw-Curtis formula and the Chebyshev-Gauss-Lobatto points, we transform the problem to a discrete form. By this approach, we get a nonlinear programming problem by solving of which we can approximate the optimal solution of the main problem. We analyze the convergence of the obtained approximate solution and solve some numerical examples to show the efficiency of the method.

Keywords: Caputo fractional derivative, Chebyshev-Gauss-Lobatto points, Clenshaw-Curtis formula, fractional optimal control, nonlinear programming.
Mathematics Subject Classification: Primary: 49M37; Secondary: 26A33.
Citation: Mohammad Hadi Noori Skandari, Marzieh Habibli, Alireza Nazemi. A direct method based on the Clenshaw-Curtis formula for fractional optimal control problems. Mathematical Control and Related Fields, 2020, 10 (1) : 171-187. doi: 10.3934/mcrf.2019035
References:
[1]

M. A. Aboelela, M. F. Ahmed and H. T. Dorrah, Design of aerospace control systems using fractional PID controller, Journal of Advanced Research, 3 (2012), 225-232. 

[2]

A. Alizadeh and S. Effati, An iterative approach for solving fractional optimal control problems, Journal of Vibration and Control, 24 (2018), 18-36.  doi: 10.1177/1077546316633391.

[3]

R. Almeida and D. F. M. Torres, A discrete method to solve fractional optimal control problems, Nonlinear Dynamics, 80 (2015), 1811-1816.  doi: 10.1007/s11071-014-1378-1.

[4]

M. Beschi, F. Padula and A. Visioli, The generalised isodamping approach for robust fractional PID controllers design, International Journal of Control, 90 (2017), 1157-1164.  doi: 10.1080/00207179.2015.1099076.

[5]

A. H. Bhrawy, S. S. Ezz-Eldien, E. H. Doha, M. A. Abdelkawy and D. Baleanu, Solving fractional optimal control problems within a Chebyshev Legendre operational technique, International Journal of Control, 90 (2017), 1230-1244.  doi: 10.1080/00207179.2016.1278267.

[6]

P. J. Davis and P. Rabinowitz, Methods of Numerical Integration Academic, New York-London, 1975.

[7]

Y. Ding, Z. Wang and H. Ye, Optimal control of a fractional-order HIV-immune system with memory, IEEE Transactions on Control Systems Technology, 20 (2012), 763-769. 

[8]

D. Feliu-Talegon and V. Feliu-Batlle, Improving the position control of a two degrees of freedom robotic sensing antenna using fractional-order controllers, International Journal of Control, 90 (2017), 1256-1281.  doi: 10.1080/00207179.2017.1281440.

[9]

E. Keshavarz, Y. Ordokhani and M. Razzaghi, A numerical solution for fractional optimal control problems via Bernoulli polynomials, Journal of Vibration and Control, 22 (2016), 3889-3903.  doi: 10.1177/1077546314567181.

[10]

A. Lotfi, M. Dehghan and S. A. Yousefi, A numerical technique for solving fractional optimal control problems, Computers and Mathematics with Applications, 62 (2011), 1055-1067.  doi: 10.1016/j.camwa.2011.03.044.

[11]

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1993.

[12]

C. I. Muresan, A. Dutta, E. H. Dulf, Z. Pinar, A. Maxim and C. M. Ionescu, Tuning algorithms for fractional order internal model controllers for time delay processes, International Journal of Control, 89 (2016), 579-593.  doi: 10.1080/00207179.2015.1086027.

[13]

A. Nemati, Numerical solution of 2D fractional optimal control problems by the spectral method combined with Bernstein operational matrix, International Journal of Control, 91 (2018), 2642-2645.  doi: 10.1080/00207179.2017.1334267.

[14]

M. H. Noori Skandari, A. V. Kamyad and S. Effati, Smoothing approach for a class of nonsmooth optimal control problems, Applied Mathematical Modelling, 40 (2016), 886-903.  doi: 10.1016/j.apm.2015.05.014.

[15] K. B. Oldham and J. Spanier, The Fractional Calculus, Theory and Applications of Differentiation and Integration to Arbitrary Order, Mathematics in Science and Engineering, Academic Press, New York-London, 1974. 
[16] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999. 
[17]

S. Pooseh, R. Almeida and D. F. M. Torres, A numerical scheme to solve fractional optimal control problems, Conference Papers in Science, Hindawi Publishing Corporation, 2013 (2013), Article ID 165298, 10 pages. doi: 10.1155/2013/165298.

[18]

K. Rabiei, Y. Ordokhani and E. Babolian, The Boubaker polynomials and their application to solve fractional optimal control problems, Nonlinear Dynamics, 88 (2017), 1013-1026.  doi: 10.1007/s11071-016-3291-2.

[19]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.

[20]

N. H. Sweilam, T. M. Al-Ajami and R. H. W. Hoppe, Numerical solution of some types of fractional optimal control problems, The Scientific World Journal, 2013 (2013), 306237. doi: 10.1155/2013/306237.

[21]

L. N. Trefethen, Spectral Methods in MATLAB, Society for Industrial and Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. doi: 10.1137/1.9780898719598.

[22]

J. J. Trujillo and V. M. Ungureanu, Optimal control of discrete-time linear fractional-order systems with multiplicative noise, International Journal of Control, 91 (2018), 57-69.  doi: 10.1080/00207179.2016.1266520.

show all references

References:
[1]

M. A. Aboelela, M. F. Ahmed and H. T. Dorrah, Design of aerospace control systems using fractional PID controller, Journal of Advanced Research, 3 (2012), 225-232. 

[2]

A. Alizadeh and S. Effati, An iterative approach for solving fractional optimal control problems, Journal of Vibration and Control, 24 (2018), 18-36.  doi: 10.1177/1077546316633391.

[3]

R. Almeida and D. F. M. Torres, A discrete method to solve fractional optimal control problems, Nonlinear Dynamics, 80 (2015), 1811-1816.  doi: 10.1007/s11071-014-1378-1.

[4]

M. Beschi, F. Padula and A. Visioli, The generalised isodamping approach for robust fractional PID controllers design, International Journal of Control, 90 (2017), 1157-1164.  doi: 10.1080/00207179.2015.1099076.

[5]

A. H. Bhrawy, S. S. Ezz-Eldien, E. H. Doha, M. A. Abdelkawy and D. Baleanu, Solving fractional optimal control problems within a Chebyshev Legendre operational technique, International Journal of Control, 90 (2017), 1230-1244.  doi: 10.1080/00207179.2016.1278267.

[6]

P. J. Davis and P. Rabinowitz, Methods of Numerical Integration Academic, New York-London, 1975.

[7]

Y. Ding, Z. Wang and H. Ye, Optimal control of a fractional-order HIV-immune system with memory, IEEE Transactions on Control Systems Technology, 20 (2012), 763-769. 

[8]

D. Feliu-Talegon and V. Feliu-Batlle, Improving the position control of a two degrees of freedom robotic sensing antenna using fractional-order controllers, International Journal of Control, 90 (2017), 1256-1281.  doi: 10.1080/00207179.2017.1281440.

[9]

E. Keshavarz, Y. Ordokhani and M. Razzaghi, A numerical solution for fractional optimal control problems via Bernoulli polynomials, Journal of Vibration and Control, 22 (2016), 3889-3903.  doi: 10.1177/1077546314567181.

[10]

A. Lotfi, M. Dehghan and S. A. Yousefi, A numerical technique for solving fractional optimal control problems, Computers and Mathematics with Applications, 62 (2011), 1055-1067.  doi: 10.1016/j.camwa.2011.03.044.

[11]

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1993.

[12]

C. I. Muresan, A. Dutta, E. H. Dulf, Z. Pinar, A. Maxim and C. M. Ionescu, Tuning algorithms for fractional order internal model controllers for time delay processes, International Journal of Control, 89 (2016), 579-593.  doi: 10.1080/00207179.2015.1086027.

[13]

A. Nemati, Numerical solution of 2D fractional optimal control problems by the spectral method combined with Bernstein operational matrix, International Journal of Control, 91 (2018), 2642-2645.  doi: 10.1080/00207179.2017.1334267.

[14]

M. H. Noori Skandari, A. V. Kamyad and S. Effati, Smoothing approach for a class of nonsmooth optimal control problems, Applied Mathematical Modelling, 40 (2016), 886-903.  doi: 10.1016/j.apm.2015.05.014.

[15] K. B. Oldham and J. Spanier, The Fractional Calculus, Theory and Applications of Differentiation and Integration to Arbitrary Order, Mathematics in Science and Engineering, Academic Press, New York-London, 1974. 
[16] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999. 
[17]

S. Pooseh, R. Almeida and D. F. M. Torres, A numerical scheme to solve fractional optimal control problems, Conference Papers in Science, Hindawi Publishing Corporation, 2013 (2013), Article ID 165298, 10 pages. doi: 10.1155/2013/165298.

[18]

K. Rabiei, Y. Ordokhani and E. Babolian, The Boubaker polynomials and their application to solve fractional optimal control problems, Nonlinear Dynamics, 88 (2017), 1013-1026.  doi: 10.1007/s11071-016-3291-2.

[19]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.

[20]

N. H. Sweilam, T. M. Al-Ajami and R. H. W. Hoppe, Numerical solution of some types of fractional optimal control problems, The Scientific World Journal, 2013 (2013), 306237. doi: 10.1155/2013/306237.

[21]

L. N. Trefethen, Spectral Methods in MATLAB, Society for Industrial and Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. doi: 10.1137/1.9780898719598.

[22]

J. J. Trujillo and V. M. Ungureanu, Optimal control of discrete-time linear fractional-order systems with multiplicative noise, International Journal of Control, 91 (2018), 57-69.  doi: 10.1080/00207179.2016.1266520.

Figure 1.  The exact and approximate optimal state for $ N = 4 $ in Example 6.1
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Figure 2.  The exact and approximate optimal control for $ N = 4 $ in Example 6.1
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Figure 3.  The absolute error of the approximate optimal state in Example 6.1
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Figure 4.  The absolute error of the approximate optimal control in Example 6.1
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Figure 5.  The exact and approximate optimal state for $ N = 6 $ in Example 6.2
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Figure 6.  The exact and approximate optimal control for $ N = 6 $ in Example 6.2
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Figure 7.  The absolute error of approximate optimal state in Example 6.2
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Figure 8.  The absolute error of the approximate control in Example 6.2
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Figure 9.  The exact and approximate optimal state for $ N = 6 $ in Example 6.3
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Figure 10.  The exact and approximate optimal control for $ N = 6 $ in Example 6.3
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Figure 11.  The absolute error of the approximate optimal state in Example 6.3
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Figure 12.  The absolute error of the approximate optimal control in Example 6.3
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Table 1.  The maximum absolute error for $ N = 4 $ in Example 6.1
$\alpha = 0.5$ $\alpha = 0.6$ $\alpha = 0.7$
$\underset{t}{\mathrm{Max}}|E_x(t)|$ $6.077585\times 10^{-5}$ $2.236463 \times 10^{-5} $ $1.662597\times 10^{-3}$
$\underset{t}{\mathrm{Max}}|E_u(t)|$ $2.031847\times 10^{-5} $ $6.593832 \times 10^{-5}$ $4.616561\times 10^{-4}$
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$\alpha = 0.5$ $\alpha = 0.6$ $\alpha = 0.7$
$\underset{t}{\mathrm{Max}}|E_x(t)|$ $6.077585\times 10^{-5}$ $2.236463 \times 10^{-5} $ $1.662597\times 10^{-3}$
$\underset{t}{\mathrm{Max}}|E_u(t)|$ $2.031847\times 10^{-5} $ $6.593832 \times 10^{-5}$ $4.616561\times 10^{-4}$
Table 2.  The optimal value of the objective functional for $N = 4$ in Example 6.name-style="western"
$ \alpha = 0.5 $ $ \alpha = 0.6 $ $ \alpha = 0.7 $
$ J^* $ $ 7.986571 \times 10^{-10} $ $ 1.200881\times 10^{-8} $ $ 6.080480\times 10^{-7} $
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$ \alpha = 0.5 $ $ \alpha = 0.6 $ $ \alpha = 0.7 $
$ J^* $ $ 7.986571 \times 10^{-10} $ $ 1.200881\times 10^{-8} $ $ 6.080480\times 10^{-7} $
Table 3.  The maximum absolute error for $ N = 6 $ in Example 6.2
$\alpha = 0.5$ $\alpha = 0.6$ $\alpha = 0.7$
$\underset{t}{\mathrm{Max}}|E_x(t)|$ $1.128957\times 10^{-3}$ $2.967422 \times 10^{-4} $ $1.393032\times 10^{-3}$
$\underset{t}{\mathrm{Max}}|E_u(t)|$ $1.508986\times 10^{-2} $ $5.269326 \times 10^{-3}$ $2.011737\times 10^{-3}$
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$\alpha = 0.5$ $\alpha = 0.6$ $\alpha = 0.7$
$\underset{t}{\mathrm{Max}}|E_x(t)|$ $1.128957\times 10^{-3}$ $2.967422 \times 10^{-4} $ $1.393032\times 10^{-3}$
$\underset{t}{\mathrm{Max}}|E_u(t)|$ $1.508986\times 10^{-2} $ $5.269326 \times 10^{-3}$ $2.011737\times 10^{-3}$
Table 4.  The optimal value of the objective function for $N = 6$ in Example 6.2
$ \alpha = 0.5 $ $ \alpha = 0.6 $ $ \alpha = 0.7 $
$ J^* $ $ 4.699837 \times 10^{-6} $ $ 5.516687\times 10^{-7} $ $ 8.165173\times 10^{-8} $
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$ \alpha = 0.5 $ $ \alpha = 0.6 $ $ \alpha = 0.7 $
$ J^* $ $ 4.699837 \times 10^{-6} $ $ 5.516687\times 10^{-7} $ $ 8.165173\times 10^{-8} $
Table 5.  The maximum absolute error for $ N = 6 $ in Example 6.3
$\alpha = 0.5$ $\alpha = 0.6$ $\alpha = 0.7$
$\underset{t}{\mathrm{Max}}|E_x(t)|$ $2.475186\times 10^{-3}$ $3.593088 \times 10^{-4} $ $1.005558\times 10^{-3}$
$\underset{t}{\mathrm{Max}}|E_u(t)|$ $5.873404\times 10^{-2} $ $5.118654 \times 10^{-3}$ $5.399343\times 10^{-3}$
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$\alpha = 0.5$ $\alpha = 0.6$ $\alpha = 0.7$
$\underset{t}{\mathrm{Max}}|E_x(t)|$ $2.475186\times 10^{-3}$ $3.593088 \times 10^{-4} $ $1.005558\times 10^{-3}$
$\underset{t}{\mathrm{Max}}|E_u(t)|$ $5.873404\times 10^{-2} $ $5.118654 \times 10^{-3}$ $5.399343\times 10^{-3}$
Table 6.  The optimal value of the objective functional for $N = 6$ in Example 6.3
$ \alpha = 0.5 $ $ \alpha = 0.6 $ $ \alpha = 0.7 $
$ J^* $ $ 1.080409 \times 10^{-7} $ $ 2.074337\times 10^{-9} $ $ 4.013053\times 10^{-9} $
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$ \alpha = 0.5 $ $ \alpha = 0.6 $ $ \alpha = 0.7 $
$ J^* $ $ 1.080409 \times 10^{-7} $ $ 2.074337\times 10^{-9} $ $ 4.013053\times 10^{-9} $
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