IRJET- Analysis of Fractional PID Controller Parameters on Time Domain Specifications using Nelder-Mead Algorithm & Interior Point Algorithm
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This document analyzes the parameters of fractional PID controllers and their effects on time domain specifications using optimization algorithms. Fractional PID controllers have five parameters compared to classical PID controllers which have three. The Nelder-Mead algorithm and Interior Point algorithm are used to optimize the controller parameters. Various plant models including first order, second order, higher order, fractional order, and first order with delay are simulated. The relationships between the fractional parameters (order of integration λ and order of differentiation μ) and time domain specifications (rise time, peak time, settling time, overshoot) are observed for each plant model. Graphs are presented showing the effects of varying λ and μ on the time domain specifications. The findings provide guidelines for
![International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056
Volume: 06 Issue: 09 | Sep 2019 www.irjet.net p-ISSN: 2395-0072
© 2019, IRJET | Impact Factor value: 7.34 | ISO 9001:2008 Certified Journal | Page 610
where n is an integer number, which satisfies the condition
(n−1) ≤ α ≤ n, α is a real number, and a and t are the limits of
integration. For example, if α is 0.8, then n would be 1
because 0 ≤ 0.8 ≤ 1.
The Riemann Liouville (RL) fractional definition is given by
the following equation
=
where n is an integer number, which satisfies the condition
(n−1) ≤ α ≤ n, α is a real number, J is the integral operator,
and a and t are the limits of integration .
In the Grunwald-Letnikov, the derivative and integral have
the same definition which is depending on the multi
derivative integer calculus, so the general definition of
Grunwald-Letnikov is:
Where = represents the coefficients of the
polynomial . The coefficientscanalsobeobtained
recursively from
=1, = r =1,2…………
A fractional PID controller has five parameters for tuning,
Fig. 1 shows a block diagram of the fractional PID controller,
which has the following structure
fig1. Fractional order PID controller
E(s) is the Laplace of error signal, KP is the proportional
constant gain, KI is the integration constant gain, KD is the
derivative constant gain, λ is the order of integrationand µis
the order of differentiator. A fractional PID controller
becomes a PID controller if λ = µ = 1 as shown in the Fig. 2.
Fig 2. Fractional order pid controller
3. literature survey:
This section provides a brief survey of the state of
the art techniques in fractional order controller design. The
main difference between integer order PID controller and
fractional order PID controller is where iopid fails to
complete the task there fopid works efficiently complete
them successfully. for example in fractional order designed
for wind turbine generator fractional order PI controller
properly tracks the input command while integer order pad
controller unable to cope up with the input command due
backlash phenomena. The fractional order controller
accurately tracks the reference input under parameter
variations while integer order pad controller becomes
unstable under the same conditions.
Podlubny given a more flexible structure PIλDµ by
extended in traditional notion of PID controllers [11] with
the controller gains define the fractional differ-integrals as
design variables. And givens several intelligent techniques
for efficient tuning of such fractional order PIλDµ controllers.
Fractional order control works in controlled environment
that suffers from classical problems of overshoot and
resonance.
4.Proposed method:
Five different systems were simulated to study the
relationships between the order of fractional parameters
and the time domain specifications by varying the order of
fractional parameters in the fractional PID controller. As
mentioned earlier, the plants were of first order, second
order, higher order, fractional order systems, andfirstorder
system with delay time system (FOPDT). The higher-order
plant was described by H. Panagopoulos in 2002 [32],
whereas the fractional-order plant was described by I.
Podlubny in 1994 [24]. The general structure of the first
order delay time and second-order systems was considered
for the simulations.
The plants are](https://image.slidesharecdn.com/irjet-v6i988-191206094057/85/IRJET-Analysis-of-Fractional-PID-Controller-Parameters-on-Time-Domain-Specifications-using-Nelder-Mead-Algorithm-Interior-Point-Algorithm-2-320.jpg)
![International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056
Volume: 06 Issue: 09 | Sep 2019 www.irjet.net p-ISSN: 2395-0072
© 2019, IRJET | Impact Factor value: 7.34 | ISO 9001:2008 Certified Journal | Page 611
The optimization approach used for tuning the
plants is shown in Fig. 3. The Nelder Mead method was used
for the simulation [33] for optimizing the parameters of the
fractional PID controller. This method finds out minimum of
a function from more than one independent variables
without using derivatives. A simplex has n+1 points in n
dimensional space, which represents the number of
independent variables. For tuning of fractional PID
controller, the integrated square error (ISE) was chosen as
the performance index. This measure is moreuseful because
the range of error was large in most cases and was thus
more appropriate for designing the controller. ., and it is
given for unity feedback system considering unit step input,
fig3. Optimization approach for fopid controller
The procedure to apply optimization can be seen in fomcon
toolbox. Using manual we can derive the fopid controller
values in simulink.
Fopid controller values for neldermead optimization
λ μ
Plant1 991.71 248.02 0.999 1.912*10-5 0.098
Plant2 987.89 43.683 0.185 999.92 1.063
Plant3 3.56 0.0002 0.145 6.98 1.7791
Plant4 92.143 549.08 0.8479 392.2 1.176
Plant5 0.1191 0.2738 0.6750 0.8728 0.7918
Fopid controller values for interior point optimization
λ μ
Plant1 991.71 227.482 0.443 208.327 1.077
Plant 2 764.09 53.54 0.9936 996.93 1.034
Plant 3 0.782 0.1017 0.0699 0.2851 0.4876
Plant 4 157.65 181.47 1.176 137.92 1.245
Plant 5 0.328 0.0425 1.1923 0.046 0.0202
These values are used in the simulink to find the behavior of
rise time, peak time, settling time percentage of peak
overshootcan be followed in fig4.
Fig 4: simulink diagram
By using the algorithms and simulink diagram weobsevethe
various values of λ, μ for changing of rise time, peak time,
settling time, percentage of peak overshoot for each plant.
Nelder mead optimization values for
The bar graphs for changing λ values for the plant 1](https://image.slidesharecdn.com/irjet-v6i988-191206094057/85/IRJET-Analysis-of-Fractional-PID-Controller-Parameters-on-Time-Domain-Specifications-using-Nelder-Mead-Algorithm-Interior-Point-Algorithm-3-320.jpg)
![International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056
Volume: 06 Issue: 09 | Sep 2019 www.irjet.net p-ISSN: 2395-0072
© 2019, IRJET | Impact Factor value: 7.34 | ISO 9001:2008 Certified Journal | Page 615
The bar graphs for changing μ values for the plant5
5. CONCLUSIONS
In this paper, the effect of differentiation (µ) and integrator
(λ) order are investigated on various time domain
specifications. Maximum overshoot has a particular
characteristic of the order of differentiation from 0 to 2.
Other specifications (rise, peak, and settling times) showed
no particular pattern that matched the increase in the value
of the parameters independently. However,followingpoints
are observed:
–There exists a particular relationship between µ and
maximum overshoot (MP).
–By changing the values of λ and µ, the time domain
specifications can be further improved, which is advantage
of the fractional PID controller.
–For a fractional order model, the influence is almost same
for different values of λ on time domain specifications.
–The settling time goes worst as µ approaches 2 for integer
order system.
The finding will facilitate in the tuning of fractional PID
controllers, an especiallyuseful featurefortheplug-and-play
type of controllers. The effect on the fractional-order
parameters may be estimated for a given system, such as a
first-order system or a second-order system.
REFERENCES
[1] Experimental analysis of fractional order PID controller
on time domain specifications. Pritesh Shah1,∗and
Sudhir Agashe2, Progr. Fract. Differ. Appl. 3, No. 2, 141-
154 (2017)
[2] M. Patil, V. Vyawahare and M. Bhole, A new and simple
method to construct root locus of general
fractionalorder systems, ISA Transact. 53(2), 380-390
(2014).
[3] M. A. Rahimian and M. S. Tavazoei, Improving integral
square error performance with implementable
fractional-order PI controllers, Opt. Contr. Appl. Meth.
35(3), 303-323 (2014).
[4] A Flexible MATLAB Tool for Optimal Fractional order
PID controller Design Subjected to specifications
TEPLJAKOV Aleksei1, PETLENKOV Eduard1, BELIKOV
Juri1,2 (2012).
[5] Y. Luo, Y. Q. Chen, C. Y. Wang and Y. G. Pi, Tuning
fractional order proportional integral controllers for
fractional order systems, J. Proc. Contr. 20(7), 823-831
(2010)
[6] I. Podlubny, Fractional differential equations: an
introduction to fractional derivatives, fractional
differential equations, to methods of their solution and
some of their applications, vol. 198, Academic Press,
1998.
[7] C. A. Monje, Y. Q. Chen, B. M. Vinagre, D. Xue and V. Feliu,
Fractional-order systems and controls: fundamentals
and applications, Springer, 2010
[8] I. Podlubny, L. Dorcak and I. Kostial, On fractional
derivatives, fractional-order dynamic systems and PIl
Dm controllers, Proceedings of the 36th Conference on
Decision & Control, 5, 4985-4990 (1997).](https://image.slidesharecdn.com/irjet-v6i988-191206094057/85/IRJET-Analysis-of-Fractional-PID-Controller-Parameters-on-Time-Domain-Specifications-using-Nelder-Mead-Algorithm-Interior-Point-Algorithm-7-320.jpg)
IRJET- Analysis of Fractional PID Controller Parameters on Time Domain Specifications using Nelder-Mead Algorithm & Interior Point Algorithm
- 1. International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 09 | Sep 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.34 | ISO 9001:2008 Certified Journal | Page 609 Analysis of Fractional PID Controller Parameters on Time Domain Specifications using Nelder-Mead Algorithm & Interior Point Algorithm D. Shanmukha Chandra Kumar1, B.T.Krishna2 1Student, Dept. of ECE, UCEK, JNTU Kakinada, AP, India 2Professor, Dept. of ECE, UCEK, JNTU Kakinada, AP, India ---------------------------------------------------------------------***---------------------------------------------------------------------- Abstract - Fractional PID controller is an extension of classical order PID controller having five parameters rather than three guidelines for the effect of classical PIDcontroller parameters on the time domain analysisareavailablebutfor fractional order PID controller there is no guidelines available particularly for the order of integration and order of differentiation. To assist fine tuning ,the effectoftheorder of differentiation and integration parameters on the time domain specification on various order plants are investigated using nelder mead algorithm and interiorpoint algorithm. The relation between parameters (integration and differentiation) and time domain parameters (rise time, peak time, overshoot, settling time) are observed using nelder mead algorithm and interior point algorithm. The design and simulation is done by using MATLAB and fractional order modeling and control tool box. In general classical PID controller is a kind of feedback control loop mechanism that is widely used in control systems. PID has good stability, In order to improve its stability in control systems several attempts to enhance the classical PID controller, one of them is Fractional order PID controller. Key Words: fractional order PID controller, fractional order calculus, effect of parameters 1. INTRODUCTION In general for classical PID controller guidelinesare available for the effect of classical controller parameters on the time domain specifications. However, no guidelines are available for fractional PID controllers, particularly for the order of differentiation (µ) andintegration(λ).Toassistwith fine tuning, the effect of the order of differentiation and integration parametersonthetimedomainspecifications for various plants are investigated. Fractional calculus provides an excellent instrument for the description of memory and hereditary properties of various materials and processes. This is the primary advantage of fractional derivatives incomparisontoclassical integer order models, where such dynamics not taken into account. The advantages of fractional derivatives become more appealing in the modeling of mechanical, electrical and electro-mechanical properties of real materials, as well as in the description of rheological properties of rocks, and in many other fields. Recent times havewideapplicationoffield fractional integrals and derivatives also in the theory of control of dynamical systems, where the controlled system or/and the controller is described by a set of fractional differential equations 2. FRACTIONAL PID CONTROLLER A. FRACTIONAL CALCULUS: The mathematical modeling and simulation of systems and processes, based on the description of theirpropertiesin terms of fractional derivatives, naturally leads to differential equations of fractional order the necessity to solve such equations to obtain the response for a particular input. Thought in existence for more than 300 years, the idea of fractional derivatives and integrals has remained quite a strange topic, very hard to explain, due to absence of a specific tool for the solution of fractional order differential equations. Fractional order calculushasgainedacceptancein last couple of decades. J.Liouville made the first major study of fractional calculus in 1832.In 1867,A.K.Grunwald worked on the fractional operations. G. F. B. Riemann developed the theory of fractional integration in 1892. Fractional order mathematical phenomena allow us to describe and model a real object more accurately than the classical “integer” methods. Earlier due to lackofavailablemethods,afractional order system was used to be approximated as an integer order model. But at the present time, there are many available numerical techniques which are used to approximate the fractional order derivatives and integrals. In fractional calculus, the differentiation integration operator, is defined as follows Here α >0 it becomes differentiation if α<0 it becomes integration. B. DEFINITIONS: (Caputo’sdefinition of Fractional Order differentiation). Caputo’s definition is given by
- 2. International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 09 | Sep 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.34 | ISO 9001:2008 Certified Journal | Page 610 where n is an integer number, which satisfies the condition (n−1) ≤ α ≤ n, α is a real number, and a and t are the limits of integration. For example, if α is 0.8, then n would be 1 because 0 ≤ 0.8 ≤ 1. The Riemann Liouville (RL) fractional definition is given by the following equation = where n is an integer number, which satisfies the condition (n−1) ≤ α ≤ n, α is a real number, J is the integral operator, and a and t are the limits of integration . In the Grunwald-Letnikov, the derivative and integral have the same definition which is depending on the multi derivative integer calculus, so the general definition of Grunwald-Letnikov is: Where = represents the coefficients of the polynomial . The coefficientscanalsobeobtained recursively from =1, = r =1,2………… A fractional PID controller has five parameters for tuning, Fig. 1 shows a block diagram of the fractional PID controller, which has the following structure fig1. Fractional order PID controller E(s) is the Laplace of error signal, KP is the proportional constant gain, KI is the integration constant gain, KD is the derivative constant gain, λ is the order of integrationand µis the order of differentiator. A fractional PID controller becomes a PID controller if λ = µ = 1 as shown in the Fig. 2. Fig 2. Fractional order pid controller 3. literature survey: This section provides a brief survey of the state of the art techniques in fractional order controller design. The main difference between integer order PID controller and fractional order PID controller is where iopid fails to complete the task there fopid works efficiently complete them successfully. for example in fractional order designed for wind turbine generator fractional order PI controller properly tracks the input command while integer order pad controller unable to cope up with the input command due backlash phenomena. The fractional order controller accurately tracks the reference input under parameter variations while integer order pad controller becomes unstable under the same conditions. Podlubny given a more flexible structure PIλDµ by extended in traditional notion of PID controllers [11] with the controller gains define the fractional differ-integrals as design variables. And givens several intelligent techniques for efficient tuning of such fractional order PIλDµ controllers. Fractional order control works in controlled environment that suffers from classical problems of overshoot and resonance. 4.Proposed method: Five different systems were simulated to study the relationships between the order of fractional parameters and the time domain specifications by varying the order of fractional parameters in the fractional PID controller. As mentioned earlier, the plants were of first order, second order, higher order, fractional order systems, andfirstorder system with delay time system (FOPDT). The higher-order plant was described by H. Panagopoulos in 2002 [32], whereas the fractional-order plant was described by I. Podlubny in 1994 [24]. The general structure of the first order delay time and second-order systems was considered for the simulations. The plants are
- 3. International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 09 | Sep 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.34 | ISO 9001:2008 Certified Journal | Page 611 The optimization approach used for tuning the plants is shown in Fig. 3. The Nelder Mead method was used for the simulation [33] for optimizing the parameters of the fractional PID controller. This method finds out minimum of a function from more than one independent variables without using derivatives. A simplex has n+1 points in n dimensional space, which represents the number of independent variables. For tuning of fractional PID controller, the integrated square error (ISE) was chosen as the performance index. This measure is moreuseful because the range of error was large in most cases and was thus more appropriate for designing the controller. ., and it is given for unity feedback system considering unit step input, fig3. Optimization approach for fopid controller The procedure to apply optimization can be seen in fomcon toolbox. Using manual we can derive the fopid controller values in simulink. Fopid controller values for neldermead optimization λ μ Plant1 991.71 248.02 0.999 1.912*10-5 0.098 Plant2 987.89 43.683 0.185 999.92 1.063 Plant3 3.56 0.0002 0.145 6.98 1.7791 Plant4 92.143 549.08 0.8479 392.2 1.176 Plant5 0.1191 0.2738 0.6750 0.8728 0.7918 Fopid controller values for interior point optimization λ μ Plant1 991.71 227.482 0.443 208.327 1.077 Plant 2 764.09 53.54 0.9936 996.93 1.034 Plant 3 0.782 0.1017 0.0699 0.2851 0.4876 Plant 4 157.65 181.47 1.176 137.92 1.245 Plant 5 0.328 0.0425 1.1923 0.046 0.0202 These values are used in the simulink to find the behavior of rise time, peak time, settling time percentage of peak overshootcan be followed in fig4. Fig 4: simulink diagram By using the algorithms and simulink diagram weobsevethe various values of λ, μ for changing of rise time, peak time, settling time, percentage of peak overshoot for each plant. Nelder mead optimization values for The bar graphs for changing λ values for the plant 1
- 4. International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 09 | Sep 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.34 | ISO 9001:2008 Certified Journal | Page 612 The bar graphs for changing μ values for the plant 1 The bar graphs for changing λ values for the plant2 The bar graphs for changing μ values for the plant2 The bar graphs for changing λ values for the plant3 The bar graphs for changing μ values for the plant3 The bar graphs for changing λ values for the plant4
- 5. International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 09 | Sep 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.34 | ISO 9001:2008 Certified Journal | Page 613 The bar graphs for changing μ values for the plant4 The bar graphs for changing λ values for the plant5 The bar graphs for changing μ values for the plant5 Interior point optimization graphs The bar graphs for changing λ values for the plant1 The bar graphs for changing μ values for the plant1 The bar graphs for changing λ values for the plant2
- 6. International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 09 | Sep 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.34 | ISO 9001:2008 Certified Journal | Page 614 The bar graphs for changing μ values for the plant2 The bar graphs for changing λ values for the plant3 The bar graphs for changing μ values for the plant3 The bar graphs for changing λ values for the plant4 The bar graphs for changing μ values for the plant4 The bar graphs for changing λ values for the plant5
- 7. International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 09 | Sep 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.34 | ISO 9001:2008 Certified Journal | Page 615 The bar graphs for changing μ values for the plant5 5. CONCLUSIONS In this paper, the effect of differentiation (µ) and integrator (λ) order are investigated on various time domain specifications. Maximum overshoot has a particular characteristic of the order of differentiation from 0 to 2. Other specifications (rise, peak, and settling times) showed no particular pattern that matched the increase in the value of the parameters independently. However,followingpoints are observed: –There exists a particular relationship between µ and maximum overshoot (MP). –By changing the values of λ and µ, the time domain specifications can be further improved, which is advantage of the fractional PID controller. –For a fractional order model, the influence is almost same for different values of λ on time domain specifications. –The settling time goes worst as µ approaches 2 for integer order system. The finding will facilitate in the tuning of fractional PID controllers, an especiallyuseful featurefortheplug-and-play type of controllers. The effect on the fractional-order parameters may be estimated for a given system, such as a first-order system or a second-order system. REFERENCES [1] Experimental analysis of fractional order PID controller on time domain specifications. Pritesh Shah1,∗and Sudhir Agashe2, Progr. Fract. Differ. Appl. 3, No. 2, 141- 154 (2017) [2] M. Patil, V. Vyawahare and M. Bhole, A new and simple method to construct root locus of general fractionalorder systems, ISA Transact. 53(2), 380-390 (2014). [3] M. A. Rahimian and M. S. Tavazoei, Improving integral square error performance with implementable fractional-order PI controllers, Opt. Contr. Appl. Meth. 35(3), 303-323 (2014). [4] A Flexible MATLAB Tool for Optimal Fractional order PID controller Design Subjected to specifications TEPLJAKOV Aleksei1, PETLENKOV Eduard1, BELIKOV Juri1,2 (2012). [5] Y. Luo, Y. Q. Chen, C. Y. Wang and Y. G. Pi, Tuning fractional order proportional integral controllers for fractional order systems, J. Proc. Contr. 20(7), 823-831 (2010) [6] I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, vol. 198, Academic Press, 1998. [7] C. A. Monje, Y. Q. Chen, B. M. Vinagre, D. Xue and V. Feliu, Fractional-order systems and controls: fundamentals and applications, Springer, 2010 [8] I. Podlubny, L. Dorcak and I. Kostial, On fractional derivatives, fractional-order dynamic systems and PIl Dm controllers, Proceedings of the 36th Conference on Decision & Control, 5, 4985-4990 (1997).