On the selection of tuning methodology of FOPID controllers for the control of higher order processes.

TL;DR: A new fractional order template for reduced parameter modelling of stable minimum/non-minimum phase higher order processes is introduced and its advantage in frequency domain tuning of FOPID controllers is presented.

Abstract: In this paper, a comparative study is done on the time and frequency domain tuning strategies for fractional order (FO) PID controllers to handle higher order processes. A new fractional order template for reduced parameter modelling of stable minimum/non-minimum phase higher order processes is introduced and its advantage in frequency domain tuning of FOPID controllers is also presented. The time domain optimal tuning of FOPID controllers have also been carried out to handle these higher order processes by performing optimization with various integral performance indices. The paper highlights on the practical control system implementation issues like flexibility of online autotuning, reduced control signal and actuator size, capability of measurement noise filtration, load disturbance suppression, robustness against parameter uncertainties etc. in light of the above tuning methodologies.

Summary

1. Introduction:

• Modelling of process plants for control analysis and design often give rise to higher order models in order to capture delicate dynamic behaviours of the process, with higher accuracy [1] - [3] .
• For process control applications, FO controllers have been classified in four categories in [11] among which Podlubny's PI D λ μ or FOPID [7] and Oustaloup's CRONE controller [12] and its three generations [13] - [15] deserve special merit.
• The time domain tuning techniques, on the other hand, do not necessarily require a reduced order model and hence the higher order process model is sufficient to find out the controller parameters by an optimization technique with some time domain performance indices as the design criteria.
• Hence, a single tuning methodology can not satisfy all of the above design criteria i.e. simultaneously satisfying time and frequency domain performance specifications.
• Tuning methodologies for FOPID controllers, proposed by contemporary researchers are outlined in section 2.

2. Tuning of FOPID controllers: review of the existing methodologies:

• Several methods have been proposed for tuning PI D λ μ controllers [7] by many contemporary researchers.
• Time domain techniques of FOPID controller tuning includes dominant pole placement tuning [32] - [33] and optimal tuning [34] - [37] based on time domain integral performance index [38] minimization.
• Also, the dominant pole placement tuning [32] , [33] gives inferior closed loop performance and often unstable response for time delay systems, since the Pade approximation of delay term effectively raises the order of the overall system.
• An optimization based controller tuning by minimizing matrix norms as the cost functions has been proposed by Bouafoura & Braiek [41] .
• Tavazoei in [38] has given a brief description of the finiteness of the integral performance indices for fractional order systems for step input and load disturbance excitation, which is required to be taken into account before the optimization.

≤ . (f) Elimination of Steady-state error:

• The steady-state error of the closed loop system automatically gets cancelled with the introduction of the fractional integrator.
• Monje et al. [26] , [30] has reported the results of tuning simple FOPTD plants with FOPID controllers.
• Indeed, the above methodology can not be directly applied to tune any arbitrary higher order process model without reducing it in prespecified structure.
• Hence, the chosen reduced parameter structure should be flexibile enough to capture large variety of arbitrary higher order models with high accuracy since modeling inaccuracy with FOPTD and SOPTD structures might reduce the achievable robustness of a FOPID controller.
• In the next subsection, the new reduced parameter templates are introduced which have higher capability of retaining the domiant dynamics of higher order models than the classically used FOPTD and SOPTD structures.

3.2. New approach towards reduced parameter FO modeling of higher order processes

• In conventional process control applications higher order process models are approximated using and SOPTD structures given by: (a) First Order Plus Time Delay :.
• The noninteger reduced parameter models are defined as: (c) One Non-integer Order Plus Time Delay (NIOPTD-I): ( ) EQUATION (d) Two Non-integer Orders Plus Time Delay (NIOPTD-II): ( ) EQUATION.
• Now, the model compression of higher order processes are formulated with the help of an optimization based technique.
• It is evident from Table 1 , that optimization with the proposed NIOPTD-II structure leads to a better minimization of the modelling error than that with the other ones.
• It is also found that each of the reduced order models have a delay term, whereas the original plant transfer function was delay-free.

3.3. Tuning results of FOPID controllers based on NIOPTD-II models

• The robust frequency domain design of FOPID controllers was first proposed by Monje et al. [26] , [30] and Dorcak et al. [31] , based on a constrained nonlinear optimization with frequency domain specifications.
• The derivative of phase of the controller ( 16) with respect to frequency (ω ) is EQUATION EQUATION Now, having known the frequency response of the reduced NIOPTD-II models (9) and FOPID controllers (16) , by satisfying the design specifications (1)-( 5), the controller parameters can be calculated.
• Also, depending on a fixed model, a predefined graphical solution [27] - [29] restricts the application from the flexibility of online autotuning of the controller parameters.
• Numerical solution with function fsolve may diverge for simultaneous demand of large m φ for low overshoot and also demand of high gc ω to get faster time response.
• Now, with a sufficiently large flat phase curve around gc ω , system's dc gain can be increased to get faster time response by keeping the overshoot at the same level.

4. Time domain design of FOPID controllers

• The time domain optimal tuning method of FOPID controllers has been formulated for the control of higher order processes ( 12)-( 15).
• This technique searches for an optimal set of controller parameters while minimizing a suitable time domain integral performance index [6] , [38] .
• Also, the time domain optimization based tuning methodology can not be applied directly without restricting the unstable modes of the closed loop system within the search space.
• Strictly second order systems with no delay, theoretically can be controlled by dominant pole placement technique and it has been found that performance is not satisfactory for systems with large time-delay, higher order and also fractional order systems, having several dominant poles and zeros.
• So, the present study is restricted in the optimal time domain performance index based tuning only for performance study of FOPID controllers.

4.1. Choice of a suitable time domain integral performance index:

• The presence of the time multiplication term and its higher powers in the performance indices ( 25), ( 27), ( 28), ( 29), puts more penalties on the chance of oscillation at later stages in the time response curve and thus effectively helps to reduce the settling time ( s t ) of the closed loop system.
• Similarly, higher powers of error term put larger penalties for the larger values of ( ) e t and thus minimize the chance of large overshoot.
• Zamani et al. [39] proposed a customized performance index for optimization based tuning which minimizes sum of several specifications like overshoot, rise time, settling-time, steady-state error, absolute value of the error-signal, squared value of the controller outout signal and simultaneously maximizes the gain-margin and phasemargin.
• To show that a customized objective function comprising of several other performance indices like [39] indeed averages the true potential of each of them and deteriorates the performance of the closed loop system than each of the individual performance index, a new objective function has been formulated which is the sum of all the previous ones ( 24)- (29) .
• Since the controller parameters (i.e. controller gains) may take very large values while searching for the minimum value of the objective functions, thus creating problem in practical implementation.

J w IAE w ITAE w ISE w ITSE w ISTES w ISTSE

• Sometimes, MATLAB's constrained optimization function fmincon may get trapped in local minimas.
• To ensure that the global minima has been found in the optimization process, the initial guesses of the controller parameters are perturbed enough and the simulation has been run several times and only the best results are reported.
• As discussed earlier, the optimal controller parameter search are restricted with the MATLAB functions isstable and isfinite to avoid the undesirable modes, especially the unstable modes.
• A large penalty function has been included in the objective function in each occurance of the undesirable modes which strongly discourages parameter search with unstable zones, as suggested by Zamani et al. [39] .

4.2. Comparison of FOPID design with different performance indices

• The upper limit of the integral performance indices are chosen as 50 seconds.
• The corresponding closed loop responses are shown in Fig. 3-6 .
• Atherton [55] for integer order PID controllers.
• Though for plant (Fig. 3 ), IAE has been found to be the best performance index over the others.

5.1. Comparative results of parametric robustness (iso-damping property):

• In section 3.3, the iso-damping nature of frequency domain design of FOPID controllers have been shown which uses a flat-phase criterion around gc ω for controller tuning.
• On the other hand, the optimal time domain tuning presented in section 4.2 can not force the phase curve of the open loop system (comprising of the FOPID and the process plant) to be flat around gc ω .
• This fact is evident from the increase in overshoot with variation in system gain (Fig. 7 ) for time domain optimal tuning of FOPID controllers.
• The frequency domain design method, presented in section 3.3 uses an inherent robustness criterion while finding the controller parameters.
• This allows considerable variation in system gain to have a faster time response while keeping the overshoot constant (Fig. 1 ).

5.2. Comparison of control signal and load disturbance rejection capability:

• It is well known that, the sensitivity function indicates the ability of the system to suppress load disturbances and achieve good set-point tracking.
• Whereas, the complementary sensitivity function indicates the robustness against measurement noise and other unmodelled system dynamics [19] , [22] .
• Clearly, lower value of control signal helps to reduce the size of the actuator and hence the cost involved and also the chance of actuator saturation and integral wind-up [19] .
• Clearly in frequency domain design of FOPID controller, the load disturbance response is slightly poor in comparison with that with the time domain design (Fig. 9 ) but significant reduction in controller output signal is evident from Fig. 10 .
• Thus it is evident that a single tuning technique cannot fulfill all of the contradictory controller design objectives simultaneously.

5.3. Summary of the results and few discussions

• In the previous subsections, a comparative study on the frequency and time domain design of FOPID controllers are presented.
• It is shown that the frequency domain design methodology is capable of providing high robustness against loop gain variation but it can not be applied to a higher order process model directly.
• But for online controller tuning, having the process model well known from the governing system physical laws or system identification techniques, a time domain method can be easily applied since tuning of the controllers can be done much faster.
• These intelligent optimization algorithms have been proved to give better performance over the deterministic optimization algorithms as these are able to take care of the trapping of the search at local minimas.
• But these stochastic algorithms take much computation time and also due to their randomness, satisfactory performance can not be guaranted without running the algorithms for a large number of times.

6. Conclusion

• Comparative performance study of two design methodologies of FOPID controllers is done in this paper.
• The frequency domain approach is shown to give better performance in terms of robustness (iso-damping), better capability of high frequency noise rejection, lower value of control signal and hence reduced size of the actuator.
• Rather all the philosophies of controller tuning, discussed in this paper possess some strength and also some weakness and needs an engineering decision, depending on the nature of application in process controls.
• Enhancement of robustness of a FOPID controller for frequency domain tuning technique with highly accurate (flexible order) reduced parameter templates.

Figures

Fig. 1. Bode diagram of the open loop system comprising of the NIOPTD-II model and robust FOPID controller.

Fig. 2. Iso-damped closed loop response of the test-bench plants.

Table 6: Comparison of closed loop performance of plant 3P with different performance indices.

Table 7 Comparison of closed loop performance of plant with different performance indices. 4P

Table 5: Comparison of closed loop performance of plant with different performance indices. 2P

Table 1 Choice of reduced parameter model structure based on minimum modeling error

Table 3: Frequency domain tuning results of FOPID controllers for test-bench processes

Table 4: Comparison of closed loop performance of plant with different performance indices. 1P

Fig. 7. Loss of iso-damping in time-domain performance index based optimal tuning.

Table 2 Reduced parameter models of the test-bench process plants.

Fig. 8. Comparison of ( )S s and ( )T s with frequency and time domain design.

Fig. 5. Optimal performance index based tuning of FOPID controllers for plant 3P

Fig. 6. Optimal performance index based tuning of FOPID controllers for plant . 4P

Fig. 10. Comparison of the control signals for the frequency and time domain.

Fig. 9. Comparison of responses due to unit step change in set-point and load disturbance.

Fig. 4. Optimal performance index based tuning of FOPID controllers for plant 2P

Fig. 3. Optimal performance index based tuning of FOPID controllers for plant 1P

Frequently asked questions

Q1. What are the contributions mentioned in the paper "On the selection of tuning methodology of fopid controllers for the control of higher order processes" ?

In this paper, a comparative study is done on the time and frequency domain tuning strategies for fractional order ( FO ) PID controllers to handle higher order processes. The paper highlights on the practical control system implementation issues like flexibility of online autotuning, reduced control signal and actuator size, capability of measurement noise filtration, load disturbance suppression, robustness against parameter uncertainties etc. in light of the above tuning methodologies. 

Q2. What future works have the authors mentioned in the paper "On the selection of tuning methodology of fopid controllers for the control of higher order processes" ?

Future scope of work can be directed towards fractional order modelling of open loop unstable plants ; plants with fractional differ-integrators with several minimum or nonminimum-phase zeros and desgning suitable frational order controllers for such processes. 

Q3. What is the tuning technique for PI D controllers?

Time domain techniques of FOPID controller tuning includes dominant pole placement tuning [32]-[33] and optimal tuning [34]-[37] based on time domain integral performance index [38] minimization. 

Q4. What is the way to achieve a satisfactory time response under these disturbed conditions?

To obtain a satisfactory time response under these disturbed conditions, the sensitivity function should have small values at lower frequencies and complementary sensitivity function should have small values at higher frequencies [26], [30]. 

Q5. What is the novelty of the work with respect to the available techniques?

The novelty of the work with respect to the available techniques is to formulate a FOPID tuning stategy for the control of higher order processes in two different ways i.e. frequency domain and time domain approach and also highlighting the inadequacies inherent in these tuning philosophies. 

Q6. What is the objective of iso-damped frequency domain tuning?

The objective of iso-damped frequency domain tuning for the family of FOPID controllers, presented in this section, is to achieve gain independent overshoot in some specific robust control applications like Saha et al. [4] and Chao et al. [54]. 

Q7. What is the tuning methodology for PI D controllers?

From specified phase margin ( mφ ), gain crossover frequency ( gcω ) [23] and iso-damping/robustness criteria (i.e. flat phase curve around gcω ) [24], [25] a tuning methodology for FOPI/FOPD controllers for controlling integer order systems have been discussed in [26], [27]. 

Q8. What is the method for tuning of a FOPID controller?

for offline tuning of FO-controllers a frequency domain method is always preferred where increased computational cost due to an extra model reduction technique involved, is not of a major concern. 

Q9. What is the optimization function for controller parameters?

In this specific application the unconstrained optimization function fminsearch() should not be used, since the controller parameters (i.e. controller gains) may take very large values while searching for the minimum value of the objective functions, thus creating problem in practical implementation. 

Q10. What is the method for tuning PI D controllers?

An optimization based controller tuning by minimizing matrix norms as the cost functions has been proposed by Bouafoura & Braiek [41]. 

Q11. What is the criterion used for the controller parameters?

The frequency domain design method, presented in section 3.3 uses an inherent robustness criterion while finding the controller parameters. 

Q12. What is the controller parameter search function in MATLAB?

In the present work, the performance indices (24)-(29) are evaluated using Trapezoidal rule fornumerical integration and then minimized with the constrained Nelder-Mead Simplex algorithm [49] implemented in MATLAB’s optimization toolbox [50] function fmincon() to obtain an optimal set of FOPID controller parameters. 

Q13. what is the structure of the NIOPTD-II controller?

The structure of the FOPID controller considered here is in the parallel/noninteracting form( ) ip KC s K K s s d μ λ= + + (16) The frequency domain tuning with the specifications (1)-(5) basically uses the gain, phase and phase derivative which is now derived for the reduced parameter NIOPTD-II model and FOPID controller. 

Q14. What is the method for achieving the optimal performance of FOPID controllers?

Practical implementation of FOPID controllers can be done by fractance and analog electronic circuit realization [24], [56]-[59], FPGA based digital realization [60] or electrochemical realization by lossy capacitors [24], [61], [62]. 

Q15. Why is NIOPTD-II the accurate controller?

Hence it needs a reduced order modelling in some standard structures, among which NIOPTD-II has been found to be the most accurate one due to its superb flexibility to lower the modeling error (Table 1). 

Q16. what is the structure of the NIOPTD-II model?

In Table 1, it has been shown that compared to other reduced order structures, the NIOPTD-II can capture the higher order dynamics of a process model much efficiently and hence in the present study only the accurate NIOPTD-II model structure is used for the frequency domain tuning of FOPID controllers. 

Q17. What is the way to determine the optimal performance index for a particular process?

optimal tuning parameters with the most suitable performance index for a specific process may not produce optimal performance for other processess and hence the choice of performance index greatly depends on the process model itself for FOPID tuning and should not be chosen a priori. 

Q18. What is the description of the FOPID controller tuning technique?

Padula & Visioli [48] proposed empirical tuning rules for FOPID controllers using IAE minimization criteria with constraints on maximum sensitivity for the FOPTD processes, which is rather a simplified approximation for higher order processes with large modeling error. 

Q19. What is the purpose of the Bode diagram?

The corresponding Bode diagram (Fig. 1) shows wide flatness in the phase curves around the gain cross-over frequencies which ensures iso-damped time responses (Fig. 2). 

Q20. What is the current approach for tuning the FOPID controller?

The present approach automatically takes care of the stability of the closed loop system while tuning the FOPID controller in time domain.