Mitigation of Derivative Kick Using Time-Varying Fractional-Order PID Control

In this paper, a novel approach for the design of a fractional order proportional integral derivative (FOPID) controller is proposed. This design introduces a new time-varying FOPID controller to mitigate a voltage spike at the controller output whenever a sudden change to the setpoint occurs. The voltage spike exists at the output of the proportional integral derivative (PID) and FOPID controllers when a derivative control element is involved. Such a voltage spike may cause a serious damage to the plant if it is left uncontrolled. The proposed new FOPID controller applies a time function to force the derivative gain to take effect gradually, leading to a time-varying derivative FOPID (TVD-FOPID) controller, which maintains a fast system response and significantly reduces the voltage spike at the controller output. The time-varying FOPID controller is optimally designed using the particle swarm optimization (PSO) or genetic algorithm (GA) to find the optimum constants and time-varying parameters. The improved control performance is validated through controlling the closed-loop DC motor speed via comparisons between the TVD-FOPID controller, traditional FOPID controller, and time-varying FOPID (TV-FOPID) controller which is created for comparison with all three PID gain constants replaced by the optimized time functions. The simulation results demonstrate that the proposed TVD-FOPID controller not only can achieve 80% reduction of voltage spike at the controller output but also is also able to keep approximately the same characteristics of the system response in comparison with the regular FOPID controller. The TVD-FOPID controller using a saturation block between the controller output and the plant still performs best according to system overshoot, rise time, and settling time.


I. INTRODUCTION
A traditional proportional integral derivative (PID) controller is one of the most used type of controller in industrial applications, because it provides stability and rapid responses, for a wide range of operating conditions [1]. The traditional PID has only three controller parameters: proportional gain constant (K P ), integral gain constant (K I ), and derivative gain constant (K D ). The tuning of these parameters is classified as traditional or intelligent methods. The traditional methods like the Ziegler-Nichols [1] do not provide the best tuning. To achieve optimal turning, the use of intelligent methods such as the genetic algorithm (GA) is needed [1]- [9]. The GA is based on genetics and natural evolution, and it mimics the properties of natural selection. Elements generated by the GA are analogous to the chromosome found in deoxyribonucleic acid (DNA). The algorithm searches for the most suitable chromosomes that will successfully build the population in the required solutions space. There has to be a balance between the expansion of the search space and the best solution. Each particle from the selected population of the GA corresponds to a solution to the optimization of the PID parameters. The traditional PID controller has been used for many years now. Most literatures are focused on the controlling aspect of the plant with a reduced error management and not much on the peak values of the control signal leaving the PID controller. There is always a compromise between overshoot, settling time, and the voltage spike, called the derivative kick, coming from the derivative control from the PID controller [10]- [12]. This voltage spike can be attributed to sudden changes to the setpoint, which will give rise to an impulse signal coming from the controller output. The output signal coming from the controller is then fed to control elements such as: electric motors, electronics or control valves, which may be damaged by such sudden high voltage spikes. To mitigate this derivative kick, several researchers have applied the parallel controller structures such as the PI-PD [10], I-PD [11], [12], and the most recent fractionalorder proportional integral-derivative (FOPI-D) controller [13]. The PI-PD controller design provides a good overall control of unstable and resonant system's responses to setpoint changes. The I-PD is used to control unstable systems successfully. However, just as the PI-PD controller design, the I-PD controller does not solely target elimination of the derivative or proportional kick. Instead, the elimination of the derivative kick is considered more as a byproduct of the design. As for the FOPI-D [13], it breaks up the controller in order to tackle the derivative kick by placing the derivative block in the feedback path, leading to a change of the control system characteristics. The fractional order proportional integral derivative (FOPID) has been drawn attention with its advantages over the traditional PID controller. The integral and derivative components in the FOPID controller use fractional integral and derivative calculus [14], [15]. The traditional design methods of FOPID controllers in time-domain and frequency domain can be found in [16]. This way, the FOPID controller ends up with five controller parameters: K p , K i , K d , and the fractional components: order of fractional integration of λ and order of the fractional derivative of µ [16]- [21]. The FOPID controllers provide better energy efficiency in controls [22]. Because the fractional operator has memory, it uses past states to decide on the filtering action, making the control of the plant more efficient. Also, it does provide a more flexible control than the traditional PID controller by allowing the adjustments of the gain and phase characteristics with the help of the fractional components. Such flexibility offered by the FOPID controller is significant for drives such as permanent magnet direct current motors (PMDC). DC motors are usually used in applications that require constant speed, or adjustable speed drives. There are two major types of DC motors: the self-excited and the externally excited ones. The DC motor model used in this paper is an externally excited motor whose speed control is achieved via voltage control applied at the armature. The mathematical model of the DC motor used as the plant can be found in [23]- [29]. The optimal design with the intelligent algorithms is generally employed for FOPID design. Besides the genetic algorithm discussed previously, another popular algorithm is the particle swarm optimization. The particle swarm optimization (PSO) is a population based scholastic optimization technique that mimics swarm intelligence such as school of fish or bird flock; and it does not rely on the survival of the fittest value [9], [30]- [38]. Each individual particle from the swarm moves at certain speed in the search space; and it records its best position in its memory. The best position reached is compared to the best global position; and if it is better than the current global position, then the global position will be updated with that value, which tends to accelerate the search for the global value.
The optimization design techniques used by the PSO and GA need an objective function for optimization. There are four single variable objective functions: integral of time multiplied absolute error (ITAE), integral of absolute error (IAE), integral of time multiplied squared error (ITSE), and the integral of squared error (ISE) [23], [31], [32], [34], [39]. Among these four objective functions, the ITAE yields less settling and rise times with comparable overshoot [23], [31], [34], [39]. The DC motor speed control is one area of focus in our study. In general, the FOPID controller designed using optimization techniques offers better control performance over the generic PID controller.
However, although the FOPID controller using extra parameters does provide better control of the DC motor speed, the derivative kick exists whenever a sudden change to VOLUME 9, 2021 the setpoint occurs [5], [40], [41]. Again, placing the derivative block in the feedback path reported in [13] is a technique to tackle the derivative kick, but it changes the overall system transfer function, which is not desired. Therefore, there is room to develop a novel approach. The motivation in our research is to force the derivative gain constant in the FOPID controller to take effect gradually, by weighting it using an optimized time function, while maintaining the controlled system output as close as possible to the regular FOPID controller. Note that applying the time-varying FOPID controller does not have total control over the spike, but definitely has a positive impact.
The contributions of the paper are listed below: 1. In this research, time-varying fractional order PID controllers are developed with a purpose to mitigate the derivative kick which exhibits at the controller output in the form of a voltage spike. 2. A particle swarm optimization algorithm is modified to perform the optimal design for the proposed time-varying derivative FOPID (TVD-FOPID) controller and timevarying FOPID (TV-FOPID) controller. 3. Validation of the proposed TVD-FOPID controller is performed through the evaluation of performance indices of different configurations. As a result, the TVD-FOPID controller outperforms the regular FOPID controller. With nearly the same performance indices, the TVD-FOPID offers 80% reduction to the control voltage spike. The paper is organized as follows. Section II introduces the preliminary concepts, where the basic definitions of the fractional calculus are presented. Section III describes the design of regular time-invariant FOPID controllers. Section IV depicts the derivative kick problem, where the time invariant controllers generate a large spike in the output control voltage whenever a sudden change in the setpoint occurs. Section V presents the structure of the proposed TVD-FOPID and TV-FOPID controllers, which are achieved by optimal design using the PSO as well as GA algorithms to mitigate the voltage spike at the controller output. Section VI illustrates the design of the TVD-FOPID and TV-FOPID controllers for DC motor speed control, and the performance evaluations with different configurations. Section VII presents the conclusions.

II. PRELIMINARY CONCEPTS BASIC DEFINITIONS FOR FRACTIONAL CALCULUS
In the realm of elementary calculus [19], a differentiation operator for the first derivative is defined as D = d dx and the n th derivative is defined as dx n , where n is a positive integer. The question from L'Hopital, back in 1695, towards Leibniz was about the meaning of D n f (x), when n is a fraction such as n = 1/2, instead of a positive integer. To which, Leibniz responded: ''Thus it follows that will be equal to d 0.5 f (x)/dx 0.5 , an apparent paradox, from which one day useful consequences will be drawn.'' [18], [42]. Fractional Calculus is a branch of mathematical analysis where the differentiation and integration are generalized to non-integer order fundamental operator a D α t . One of the generalized forms of the differ-integrator [5] can be represented as: where a and t are the limits of the operation, α represents the real order of the differ-integral, and α ∈ R. Since a represents the lower limit, it is considered to be zero if not stated and it will not be represented.
Euler's Gamma function of (n) is defined for all complex numbers, excluding the negative integer numbers, and it is an extension of the factorial function if its argument is represented as: where n is the positive integer number.

VOLUME 9, 2021
To find the equivalent transfer function of a system from its differential form to the frequency domain or Laplace domain, zero initial conditions are considered. A general form of the Laplace transform of the fractional order derivative, according to [8], [14], [27], [43]- [45], is given by: Equation (8) is useful in calculating the inverse Laplace of the elementary transfer functions such as s α and s −α , where α is a positive real number.

III. FRACTIONAL ORDER PROPORTIONAL INTEGRAL DERIVATIVE CONTROLLER PI λ D µ
Using the Laplace domain, the construction of a closed loop FOPID system is presented next. Fig.1 depicts a closed loop control system with a FOPID controller. The FOPID controller can be considered as an extension of the traditional PID controllers and is less sensitive to changes of its parameters. A general form of the controller in Laplace domain is given as [5]- [8], [26], [27], [37], [43], [44]: (9) where U (s) is the control signal, E(s) is the error signal, K p , K I , and K D are the proportional, integral, derivative constant gains, respectively; while λ and µ denote the fractional components: the order of fractional integration and order of the fractional derivative.
In the case of the traditional PID controller as shown in Fig. 2 in the FOPID plane, the values of the fractional components can take combinations of zero or one [6], [18], [31], [43]. In this way, the PID plane consists only of four of points: if λ = 1 and µ = 1 then it's a PID controller if λ = 1 and µ = 0 then it's a PI controller if λ = 0 and µ = 1 then it's a PD controller if λ = 0 and µ = 0 then it's a P controller In the case of the FOPID [18], again as shown in Fig. 2, the values of λ and µ can take any values in the interval λ, µ ∈ [0, 2]. Instead of jumping between points for certain fixed values like in the case of the integer order PID controller, the fractional components allow continuous movement in the FOPID plane, which is proven to be useful for improving the closed loop control performance [32].

IV. DERIVATIVE KICK PROBLEM
Although the FOPID controller [14], [17]- [21], [32] depicted in Fig. 1 is proven to be useful in improving the closed loop control performance, there exists a general problem caused by the derivative kick, that is, an instant response of the controller that generates a huge spike in the designed control signal U (s) for a step input command. Even though, many researches focus on the improvement of overshoot and settling time [10]- [12] on FOPID controller design, the spike in the control signal is often overlooked. Such a spike is harmful to the control systems and may result in damage to the system devices. To improve overall control system performance, the effect of derivative kick must be investigated. In our research, this aspect of the voltage spike mitigation in the control signal is attempted by designing a time-varying derivative FOPID (TVD-FOPID) controller. This TVD-FOPID controller initially does null the effect of the derivative portion of the controller and then gradually allows for the derivative portion to take its action. The structure of this proposed TVD-FOPID controller is presented in the following section.

V. PROPOSED TIME-VARYING FOPID CONTROLLERS
In this section, we will propose the structure of new timevarying FOPID controllers and illustrate the optimal design methods.

A. THE STRUCTURE OF TIME-VARYING FOPID CONTROLLER
A proposed framework for a closed loop control system using a time-varying FOPID controller is depicted in Fig. 3.  with the following six parameters: K P , K I , K D , α D , λ and µ 3. Time-varying FOPID (TV-FOPID) controller with the following eight parameters: K P , K I , K D , α P , α I , α D , λ and µ As it can be seen, the constants of K P , K I , and K D from the traditional FOPID controller depicted in Fig. 1 are changed to be time-varying functions, that is, where K P , K I , and K D are the gain parameters, while α P , α I , and α D are the time constants for the fractionalproportional, integral, and derivative controls, respectively. These six parameters along with λ and µ can be designed using optimization algorithms as proposed in the next section.

B. OPTIMAL DESIGN WITH PARTICLE SWARM OPTIMIZATION ALGORITHM
The particle swarm optimization is a global optimization technique that mimics the behavior of schools of fish or flock of birds traveling [30], [33]. Each individual unit in the swarm moves with a dynamically adjusted velocity in the search space based on its own and the swarm group's experience [34], [35]. The memory of each unit holds the information about the best position that it traveled through and this fitness value is stored in pbest. There is another value that is kept, the gbest, which represents the best overall value and its location achieved by any of the particles from the swarm throughout the search. At each instance of time, the PSO algorithm adjusts the velocity of each individual particle towards its own pbest as well as towards the global gbest. If a new pbest is reached, it will be compared to the gbest position. Again, if this new pbest is better than the global gbest, then the global position will be updated with that value. This tends to accelerate the global value towards a global optimum solution [32]. At the beginning of the setup, a group of particles are initialized in the search space with dimension. The position of the i th particle is given by X i = [X i1 , X i2 , . . . , X iD ] and its velocity is given by The optimal position of a single particle is given by (pbest), P i = [P i1 , P i2 , . . . , P iD ] and the best global position by (gbest) P g = [P g1 , P g2 , . . . , P gD ]. The update of the positions and the velocity equations [9], [29], [31], [32], [34], [35], [38] is achieved using Equations (13) and (14), as shown below: where w− inertia weight vector t− position of current iteration c 1 , c 2 -positive acceleration constants r 1 , r 2 -random numbers in the [0,1] interval.
To overcome a poor velocity control of the particles, a time decreasing inertia [29], [31], [34], [35], [38] is introduced as: where w min and w max are the minimum and maximum of the inertia weight factor, set by the user. The parameters used in setup of PSO are: number of variables = 5 number of particles in the swarm = 50 maximum iterations = 30 inertia weight factor minimum w min = 0.05 inertia weight factor maximum w max = 0.1 positive acceleration constants c 1 , c 2 = 2 The following bounds are set for the controller parameters: To apply the PSO algorithm, an objective function using the integral of time multiplied absolute error (ITAE) is selected. Minimizing the ITAE fitness function [4], [6], [23], [24], [31], [32], [37], [39], [43], [45] through the PSO optimization is a good way to measure the quality of the system at hand [23], [32], meaning that the overshoot, settling time, and rise time are optimized for the transient response of the control system. The ITAE is expressed as where t is the time, |e(t)| is the absolute value of the error signal (the difference between the setpoint and the output of 55978 VOLUME 9, 2021 angular speed for DC motor speed control in our study), and t sim is the upper limit of the simulation time.

VI. FOPID CONTROLLER FOR DC MOTOR SPEED CONTROL
To illustrate the proposed method and validate its effectiveness, the time-varying FOPID controller is applied to control the DC motor speed. The DC motor model used in our study is described next.

A. DC MOTOR MODEL
As depicted in Fig. 4, the DC motor model used in this research [1], [9], [23] is an externally excited motor whose speed control is achieved via voltage control applied at the armature e a . At a constant flux, the rotor's angular speed ω in rad/sec is directly proportional to the induced electromotive force e b [22], [23], that is, Modeling the electric circuit around the main loop leads to the following equivalent differential equation for the armature voltage [1], [22], [23]: The electromagnetic torque produced by the armature current [1], [9], [22], [23] is given by Since the load torque T L is an external factor, it is not calculated with the electromagnetic torque. Assuming zero initial conditions, taking Laplace transform of Equations (17)- (19) leads to [23]: T em (s) = (J · s + B) · ω(s) = k T · I a (s) 0.015 0.00108 · s 2 + 0.0061 · s + 0.00163 (24) This transfer function representing the plant (DC motor) will be used in our research and implemented in MATLAB/ Simulink.

B. FOPID DESIGN
The fractional order controllers are built in MATLAB/ Simulink with the fractional order blocks from the fractional-order modeling and control (FOMCON) package [4], [18], [26], [43]- [46]. The FOMCON was developed by Aleksei Tepljakov for the purpose of modeling and controlling of the fractional order dynamic systems. The FOPID design is implemented with the help of MATLAB/Simulink, as depicted in Fig. 6. The R(s) reference is a step input that kicks in at t = 0. The response of the closed loop system to the step input is Y(s). The error E(s) is tapped to a set of blocks to obtain the ITAE output, which is then minimized. The controller configurations and parameter bounds were set below: Case 1: Regular time-invariant FOPID controller K p (t) = K P , K i (t) = K I , K d (t) = K D with the following bounds: with the following bounds:   Running the PSO and GA optimization algorithms on the above setups, the convergence curves are obtained for both linear and non-linear configurations. Fig. 8 shows ITAE values obtained by the PSO, while Fig.9 depicts the ITAE values obtained by GA.
As shown in Figs. 8 and 9, it is observed that the PSO optimization offers lower errors as well as the lower computational complexity. The optimization parameters from both PSO and GA algorithms are listed in Table 2 for the linear configuration without using a saturation block and Table 3 for the nonlinear configuration with a saturation block. Table 2 details the parameter values found by PSO and GA, including the time constants and the fractional order of operators for the case of the linear controllers. Table 3 details the parameter values found by PSO and GA, including the time constants and the fractional order   operators, for the case of the nonlinear controllers with a saturation block. After obtaining the optimized parameters, the step responses and the control voltage spikes are compared for evaluation.  Step response for PSO using ITAE. Fig. 10 shows the step responses of the linear systems, where it is observed that there is no overshoot in any of the presented cases. There is very little difference in the rise time and the settling time between case 1 and case 2, that is, the timeinvariant FOPID, and the time-varying derivative FOPID (TVD-FOPID). In fact, the 6-element TVD-FOPID controller achieves almost the same performance as the 5-element timeinvariant FOPID controller.

C. CONTROL PERFORMANCE EVALUATION OF FOPID CONTROLLERS
The control voltages of the linear systems, U(s), with the afferent spikes that this research tries to mitigate, are presented in Fig. 11. The most preeminent spike comes from the time-invariant FOPID with maximum at 1300V, which occurs at 1ms. The TVD-FOPID controller only has a maximum value of 260V occurring much later at 3.8ms. Clearly there is an 80% reduction in the derivative kick, with minimal loss in the rise time and the settling time. In addition, the 8-element TV-FOPID has the maximum value of 45V, which occurs at 5.4ms. 96.5% of reduction in the derivative kick is evidenced with a trade-off that comes with longer rise time and settling time.  Step response for PSO using ITAE with saturation block. Fig. 12 shows the step responses of the nonlinear systems (saturation unit). It is observed that the TVD-FOPID configuration offers the best control performance in terms of the overshoot, rise time, and settling time.   13 shows the control voltages of the nonlinear systems, U(s). As shown in Fig. 13, the TVD-FOPID controller with VOLUME 9, 2021 Step response for PSO using ITAE.
the saturation unit takes a little bit more time at saturation level of 12 volts; and it has the largest negative peak. It is clear that all the control signals are well within the practical range |u(t)| < 12 V .
Additional simulations and comparisons were conducted using the developed TVD-FOPID, TV-FOPID, standard FOPID, FOPI (no derivative component), TVD-PID, TV-PID, and genetic PID controllers. Further comparison with the other type of control methods such as the sliding mode control (SMC) method [47], [48] will be our future research work.
The step responses from the developed TVD-FOPID, FOPID, FOPI [49], and PID controllers are displayed in Fig. 14(a). As shown in Fig. 14(a), the responses from the TVD-FOPID, FOPID, and generic PID controllers are similar. However, the response of the FOPI controller has unacceptable overshoot and settling time. Fig 14(b) presents a more comprehensive comparison of step responses from time-varying, non-time varying, fractional-order, and integer-order PID controllers. We can see that besides the TV-FOPID and TV-PID controllers, the TVD-FOPID and FOPID controllers have faster response than the TVD-PID and PID controllers, respectively. Fig. 15(a) shows the controller voltage outputs from TVD-FOPID, FOPID, FOPI [49], and generic PID controllers. This figure indicates that the voltage spikes from the FOPID and generic PID controllers are similarly high, while the voltage spike from our developed TVD-FOPID controller is reduced by 80% when compared to the FOPID and generic PID controllers. The voltage spike using the FOPI controller is minimal due to the absence of the derivative component. Fig 15(b) shows the controller outputs for all different PID controllers. From Fig. 15 (b), the TVD-PID controller offers more spike reduction in comparison with the TVD-FOPID controller but it is less favorable to the TVD-FOPID controller due its slow response.
The step responses of the nonlinear systems from the developed TVD-FOPID, FOPID, FOPI [49] and PID controllers are presented Fig. 16 (a). As shown in Fig. 16 (a), the responses from our developed TVD-FOPID controller and regular FOPID controller are similarly fast, while the responses from the FOPI and generic PID controllers Step response for PSO using ITAE with saturation block.
are sluggish. As shown in Fig. 16 (b) for a comparison of all the PID controllers, the TVD-FOPID, TVD-PID, and FOPID controllers have a high performance. Although TVD-PID and FOPID have quicker rise times, they suffer from the over shoot and longer settling time. Instead, the proposed TVD-FOPID controller offers the best control quality in terms of no overshoot and faster settling time. Fig. 17(a) depicts the control voltage outputs of the nonlinear systems from the developed TVD-FOPID, FOPID, FOPI [49], and PID controllers. It can be observed that the generic PID controller has the shortest saturation time, followed by the FOPID, TVD-FOPID, and lastly, by the FOPI controllers. For the comparison of all the various PID controllers depicted in Fig. 17 (b), it can be seen that besides the generic PID controller, all the controllers spend time in the saturation region. The generic PID controller spends the least amount of time in the saturation region and has no negative spike, but it accommodates an unacceptable settling time, as shown in Fig. 16 (b).
Clearly, our developed TVD-FOPID controller retains the same transient characteristics as the standard FOPID controller but has a significant reduction of voltage spike at the controller output. Furthermore, the TVD-FOPID controller offers faster settling time. For the rest of this paper, the focus is on the FOPID and time-varying FOPID controllers.

D. SPEED CONTROL OF DC MOTOR USING VARIOUS APPROACHES 1) OVERSHOOT, RISE TIME AND SETTLING TIME
To compare the performances from the FOPID, novel TVD-FOPID, and TV-FOPID controllers, the overshoot, rise time, and settling time were measured and gathered in Table 4. From the overshoot prospective there is no significant difference between the compared controllers. When comparing the settling times, the TVD-FOPID controller offers far better settling times than the TV-FOPID controllers. Table 5 contains the data gathered at the controller output. It provides a comparison between the voltage spike for the case of the linear controller, and the duration of saturation for the case of the nonlinear controller with a saturation block present. For the linear controllers, once again, designed by VOLUME 9, 2021   the PSO does provide roughly 50% lower spikes for both the FOPID and TVD-FOPID, than the ones designed by the GA. When comparing the FOPID to the TVD-FOPID, the latter does offer a reduction of 80% in the voltage spike. When comparing the nonlinear controllers, the best scenario is offered by the controller that will stay the least time in the saturation zone. There is no overall significant difference observed between the various FOPID controllers. The average time is 58.79 ms. It can be concluded that, the PSO TVD-FOPID offers a far superior performance in mitigating the voltage spike, due to the derivative kick, when compared to the other designed controllers.

2) COMPARISON OF PERFORMANCE INDICES
In addition, the integral of time multiplied squared error (ITSE) is also evaluated to compare with the ITAE. The ITSE is also widely used in industry [23], [29], [31], [34], [35], [38], [39], [43], [45] and it is defined below: where t is the time, e(t) is the error signal (the difference between the setpoint and the angular speed output for DC motor speed control in our study), and t sim is the upper limit of the simulation time. From Table 6, it can be seen that the errors from the PSO design are lower in ITSE index, when compared to the ITAE.

3) FREQUENCY RESPONSE ANALYSIS AND STABILITY
The time-varying system stability and working frequency bandwidth are examined next. As a typical illustration, the frequency response of the PSO TVD-FOIPID is displayed in Fig. 18 since this is the one which offers the most promising  behavior in voltage spike mitigation at the controller output.
As it can be seen from Fig. 18, the PSO TVD-FOPID has a gain margin of 23.1 dB, a phase margin of 87.6 degrees, and a frequency bandwidth of 212 rad/sec, which is measured in open-loop gain at −3 dB. Since the gain margin value is much larger than 0 dB and the phase margin is larger than 15 degrees, the final settled TVD-FOPID control system has high relative stability.  Table 7 lists the gain margin, phase, and bandwidth obtained from the frequency response for each controller, designed by the PSO and GA algorithms. It can be seen that although the gain margins from the optimized TVD-FOPID controllers are relatively low (but much higher than stability requirement of 0 dB), the PSO and GA optimized TVD-FOPID controllers offer three times more bandwidth in comparison with the regular FOPID controllers, implying faster and better transient responses of the developed controller.  To validate the stability of the TVD-FOPID and TV-FOPID controllers during the time varying period, the frequency response sweep was done for different values of the time instants. Since the FOPID controller is time-invariant, it does remain constants in terms of gain margin, phase margin, and bandwidth, as the green lines shown in Fig. 19 for the PSO optimized controllers and Fig. 20 for the GA optimized controllers. The TVD-FOPID responses for gain margin, phase margin, and bandwidth (blue) were measured at the following time instants (sec): 0, 0.1, 0.2, 0.4, 0.8, 1, 2, 3, 4, 5. As for the PSO TV-FOPID controller, the responses of gain margin, phase margin, and bandwidth (red) were measured at the following time instants(sec): 0.01, 0.1, 0.2, 0.4, 0.8, 1, 2, 3, 4, 5. Note that the first time instant for measurement starts at 0.01 seconds for the PSO TV-FOPID to eliminate the condition that all controller gains are set to zero to begin with.
Based on Table 7, Figs. 19 and 20, it can be concluded that both the linear and nonlinear, and time-invariant and time-varying controllers designed by the PSO or GA exhibit a stable behavior. More specifically, during the time-varying period, the gain margins for the TVD-FOPID and TV-FOPID systems are significantly high initially and then decrease gradually, but are settled to high dB values while their phase margins and bandwidths are gradually improved. Furthermore, the TVD-FOPID controller offers a remarkable improvement on bandwidth.

VII. CONCLUSION
The time-varying derivative fractional order proportional integral derivative (TVD-FOPID) controller has been developed and proposed. The TVD-FOPID controller has a timevarying derivative gain which is achieved via an optimized time function. For a comparison purpose, the time-varying (TV-FOPID) controller with the proportional, integral, and derivative gains which are replaced by the corresponding optimized time functions, respectively, are investigated. Both TVD-FOPID and TV-FOPID controllers have been optimally designed using PSO or GA, based on the regular FOPID controller parameters and time function parameters. The developed TVD-FOPID controller initially suppresses only the effect of the derivative action from the controller, then it takes effect gradually to its full capacity, while the TV-FOPID controller initially suppresses the proportional, integral, and derivative actions, and then the three gains gradually reach their full capacity, in order to ameliorate the effectiveness of the derivative kick. The proposed TVD-FOPID controller has the ability to mitigate the derivative kick by reducing voltage spike at the controller output by a stunning value of 80% while keeping the system overshoot, settling time and rise time on par with the regular FOPID controller. The TV-FOPID controller offers maximum reduction of derivative kick, but it comes with a trade-off in control performance degradation, in terms of the delayed rise time and settling time. The TVD-FOPID controller is validated as the best choice when dealing with mitigation of the derivative kick. The proposed TVD-FOPID controller with a saturation block between the controller output and the plant process is still validated to have the best performance in terms of system overshoot, rise time and settling time. In addition, the timevarying gain margin, phase margin, and bandwidth are also simulated to demonstrate that all controllers designed by the PSO and GA exhibit stable behaviors and feasible working frequency bandwidths. Since our proposed TVD-FOPID controller with optimal design approach is developed as a generic case, it can be applied to any control system in which the derivative kick exists. Our future work will also apply the developed TVD-FOPID controller with optimal design to various practical systems, including hardware validations.