A novel fractional order fuzzy PID controller and its optimal time domain tuning based on integral performance indices

TL;DR: A novel fractional order fuzzy Proportional-Integral-Derivative (PID) controller is proposed in this paper which works on the closed loop error and its fractional derivative as the input and has a fractional integrator in its output.

About: This article is published in Engineering Applications of Artificial Intelligence.The article was published on 2012-03-01 and is currently open access. It has received 221 citations till now. The article focuses on the topics: PID controller & Open-loop controller.

Summary

Introduction

• So far the focus of the engineering community had been primarily on expressing systems with integer order differential equations and using a multitude of analytical and numerical solutions to optimize the formulation and analysis procedure.
• However recent developments in hardware implementation [1]-[3] of fractional order elements have brought a renewed wave in the modeling and analysis of new class of fractional order systems which look at natural phenomenon from a whole new perspective.
• However the controller output generated as a result of his actions may be approximated by appropriate mathematical operations which have the required compensation characteristics.
• The present study assumes fixed MFs and rule base for the FLC as in its IO counterpart [6] and then tunes the fractional rate of error, fractional order integration of FLC output along with the input-output SFs to achieve optimum performance in time domain i.e. low control signal and error index.

2. Review of the existing intelligent tuning techniques of FO controllers:

• Classical notion of PID controllers has been extended to a more flexible structure PI Dλ μ by Podlubny [8] with the fractional differ-integrals as the design variables along with the controller gains.
• Dorcak et al. [19] proposed a frequency domain robust PI Dλ μ controller tuning methodology using Self-Organizing Migrating Algorithm (SOMA), which is an extension of that proposed by Monje et al. [20] using constrained Nelder-Mead Simplex algorithm.
• Arena et al. [34]-[35] introduced a new Cellular Neural Network (CNN) with FO cells and studied existence of chaos in it.
• In the present study, the tuning of a new fuzzy FOPID controller has been attempted with GA and the closed loop performances are compared with an optimal PI Dλ μ controller.
• While [31], [32] focuses on fractional order fuzzy sliding mode controllers, the present work is concerned with the fuzzy analogue of the conventional PID controller which is widely used in the process control industry.

3.2. Formulation of the objective function for time domain optimal controller tuning:

• Various time domain integral performance indices like ITAE, ITSE, ISTES and ISTSE are considered in the problem similar to that in [43].
• The ITSE criterion penalizes the error more than the ITAE and due to the time multiplication term, the oscillation damps out faster.
• These result in faster rise time and settling time while also ensuring the minimization of the peak overshoot.
• The objective functions used for controller tuning has been taken as the weighted sum of several performance indices along with the controller outputs as shown below.
• The proposed family of time domain integral performance indices based tuning technique is especially needed for processes, governed by highly nonlinear differential equations and not mere linear systems with actuator nonlinearities, commonly encountered in process controls.

3.3. Optimization algorithm used for the tuning of optimal controllers:

• Gradient based classical optimization algorithms for minimization of the objective function often get trapped in the local minimas.
• The population is encoded as a double vector and the bit string representation is not used.
• Thus to overcome this problem, whenever the time evolution of the objective function shows instability, a large value of the objective function is assigned (10000 for this case) without simulating it for the entire time horizon.
• Each solution vector in the present population undergoes reproduction, crossover and mutation stochastically, in each generation, to produce a better population of solution vectors (in terms of fitness values) in the next generation.
• In this case rank fitness scaling is used which scales the raw scores on the basis of its position in the sorted score list.

4.1. Nonlinear process with time delay:

• The optimal tuning of the proposed FO fuzzy PID and other three controllers viz.
• Differ-integral orders greater than 1 leads to improper transfer function upon rational approximation and thus has been divided in two parts for simulation as suggested in Das et al. [47].
• Fig. 6 shows the controller output for this plant based on the ITAE tuning.
• The PID and the FOPID controllers have a larger initial controller output, while the controller outputs for the fuzzy PID and the fractional fuzzy PID are better.
• It is logical that ITSE based tuning gives better time response Fig. 7) for the most flexible controller structures but at the cost of increased control signal (Fig. 8), since the penalties on large errors increases for ITSE criteria.

4.2. Unstable process with time delay:

• The next plant considered for performance study of the optimal controllers is that of an open loop unstable process with time delay as studied by Visioli [48].
• Both the PID and FOPID have higher overshoot but have a considerably faster settling time.
• The PID and FOPID controller have a higher overshoot than the fuzzy PID and fuzzy FOPID controllers.
• The load disturbance rejection for the fuzzy PID controller is better than the fuzzy FOPID controller.
• 2P Fig. 20 shows the controller output for plant optimized with the ISTSE criterion.

4.3. Comparative performance analysis of the different controllers and few discussions

• Table 5, lists the best found controller structure from the simulations in a tabular form.
• Also since the load disturbance attenuation level is not optimized (as the maximum sensitivity specification for linear systems and controllers) by including it in the performance criterion, different controllers give better results in different cases.
• Table 6 reports the mean and standard deviation of the two processes with four controllers each with four different performance indices.
• Fuzzy controller gives better performance than conventional PID in the presence of parametric uncertainties, measurement noise and process nonlinearities.
• Thus time domain tuning is the preferred method for the tuning of such controllers which works well for a wide variety of processes.

5. Conclusion:

• Genetic algorithm based optimal time domain tuning of a novel fractional order fuzzy PID controller is attempted in this paper while minimizing a weighted sum of various integral performance indices and the control signal.
• Baogang Hu, George K. I. Mann and Raymond G. Gosine, “New methodology for analytical and optimal design of fuzzy PID controllers”, IEEE Transactions on Fuzzy Systems, Volume 7, Issue 5, pp. 521-539, October 1999. [6].
• Jun-Yi Cao, Jin Liang and Bing-Gang Cao, “Optimization of fractional order PID controllers based on genetic algorithm”, Proceedings of 2005 International Conference on Machine Learning and Cybernetics, 2005, Volume 9, pp. 5686-5689, 18-21 Aug.

Figures

Table 6: Statistical analysis of the GA based controller tuning results

Fig. 6. Control signal of 1P for step input and load disturbance with ITAE based tuning.

Fig. 7. Step input and load disturbance response for plant with ITSE based tuning. 1P

Fig. 15. Step input and load disturbance response for plant 2P with ITSE based tuning.

Fig. 2. Rule base for error, fractional rate of error and FLC output.

Fig. 16. Control signal of 2P for step input and load disturbance with ITSE based tuning.

Fig. 17. Step input and load disturbance response for plant with ISTES based tuning. 2P

Table 2: Optimal parameters for FOPID and PID controller for plant 1P

Fig. 5. Step input and load disturbance response for plant 1P with ITAE based tuning.

Fig. 12. Control signal of 1P for step input and load disturbance with ISTSE based tuning.

Table 3 and 4 gives the optimal values of the controller parameters for the four different controllers, with four set of integral performance indices. Fig. 13 shows the unit step response of the plant along with the load disturbance for ITAE criterion. Both the

Fig. 19. Step input and load disturbance response for plant with ISTSE based tuning. 2P

Fig. 18. Control signal of 2P for step input and load disturbance with ISTES based tuning.

Table 1: Optimal parameters for fuzzy FOPID and fuzzy PID controller for plant 1P

Fig. 4. Surface plot for rule base.

Fig. 3. Membership functions for error, fractional rate of error and FLC output.

Table 4: Optimal parameters for FOPID and PID controller for plant 2P

Fig. 11. Step input and load disturbance response for plant with ISTSE based tuning. 1P

Fig. 10. Control signal of 1P for step input and load disturbance with ISTES based tuning.

Table 5, lists the best found controller structure from the simulations in a tabular

Fig. 20. Control signal of 2P for step input and load disturbance with ISTSE based tuning.

Frequently asked questions

Q1. What contributions have the authors mentioned in the paper "A novel fractional order fuzzy pid controller and its optimal time domain tuning based on integral performance indices" ?

A novel fractional order ( FO ) fuzzy Proportional-Integral-Derivative ( PID ) controller has been proposed in this paper which works on the closed loop error and its fractional derivative as the input and has a fractional integrator in its output. 

Q2. What future works have the authors mentioned in the paper "A novel fractional order fuzzy pid controller and its optimal time domain tuning based on integral performance indices" ?

More stringent multi-objective optimization criteria may be imposed on the controller tuning algorithm to achieve effective results under different circumstances as a scope of future work. 

Q3. Why is it important to restrict the input scaling factors to unity?

Restricting the input scaling factors to unity is to ensure that the fuzzy inference is always between the designed universe of discourse. 

Q4. What is the process of generating a population of solution vectors?

Each solution vector in the present population undergoes reproduction, crossover and mutation stochastically, in each generation, to produce a better population of solution vectors (in terms of fitness values) in the next generation. 

Q5. What is the preferred method for the tuning of controllers?

Thus time domain tuning is the preferred method for the tuning of such controllers which works well for a wide variety of processes. 

Q6. How many independent runs were carried out to show the consistency of the GA based controller tuning algorithm?

30 independent runs (with different seeds for random number generation) were carried out to show the consistency of the GA based controller tuning algorithm. 

Q7. How did Cai, Pan & Du tune a PI D controller?

Pan & Du [15] tuned a PI Dλ μ controller by minimizing the ITAE criteria using multi-parent crossover evolutionary algorithm. 

Q8. What is the effect of change in output scaling factor on the controller performance?

Also for fuzzy enhanced PID controllers it is well known [4] that change in output scaling factor for example has more effect on the controller performance than changes in the membership functions or fuzzification-inferencing-defuzzification mechanism. 

Q9. What is the proposed family of time domain integral performance indices based tuning technique?

The proposed family of time domain integral performance indices based tuning technique is especially needed for processes, governed by highly nonlinear differential equations and not mere linear systems with actuator nonlinearities, commonly encountered in process controls. 

Q10. What are the different time domain integral performance indices?

Various time domain integral performance indices like ITAE, ITSE, ISTES and ISTSE are considered in the problem similar to that in [43]. 

Q11. What is the definition of the elite count?

The number of fittest individuals (solution vectors) that will definitely be self replicated to the next generation is denoted in the algorithm by a parameter called the elite count. 

Q12. What is the effect of different levels of positive control signal on the process output?

Hence different levels of positive control signal isrequired for different combinations of and e d e dtμμ , to reverse the course of the processoutput and make it tend towards the set point. 

Q13. What is the structure of the fuzzy PID used here?

The structure of the fuzzy PID used here is inherited from a combination of fuzzy PI and fuzzy PD controllers [4] with and as the input SFs and eK dK α and β as output SFs as described by Woo, Chung & Lin [6] and Yesil, Guzelkaya & Eksin [37].