Design of Complex-Order PI/PID Speed Controllers and its FPAA Realization

Complex-order controllers are a generalized version of conventional integer-order controllers and are known to offer greater flexibility, better robustness, and improved system performance. This paper discusses the design of complex-order PI/PID controllers to control the speed of an induction motor drive and an electric vehicle. The speed-tracking performance of the complex-order controllers is compared with fractional-order controllers and conventional integer-order controllers. Implementing complex-order controllers is challenging due to commercial complex-order fractance element unavailability. Hence, it is carried out by approximating the complex-order controller transfer function using an integer-order rational function with a curve-fitting approach, namely the Sanathanan Koener (SK) iterative method. This method is quite simple and can fit the required frequency range compared to the conventional Matsuda and Oustaloup approaches. The approximated controller transfer function can easily be realized by employing the AN231E04 Field Programmable Analog Array (FPAA). Simulation and experimental results highlight that the controller behaviour is in good agreement with the theoretical expectations.

interest.The fractional-order PID (FOPID) controller is an extension of the traditional integer-order PID controller with its integral and differential actions having two extra degrees of freedom λ, μ ∈ [0,1].This results in better robustness, improved system performance, and design flexibility [3], [4].The fractional-order controller provides attractive solutions in various control applications; it is well-suited for controlling dynamic systems [5] such as controlling the speed and position in motor drives [4], [6], satellite attitude system [7], robotic manipulators [8], twin-rotor aerodynamic system (TRAS) [9], and electrohydraulic servo system [10].
To further improve the freedom of tuning FOPID controllers, integral and derivative actions of complex-order have been studied by Bingi et al. [11], [12].Furthermore, Abdulwahhab [13] proposed a complex-order PID controller for a first-order plus time delay (FOPDT) and demonstrated its superiority over conventional controllers to achieve the design specifications more accurately.A power-law FO[PI] controller with reduced active element was suggested in [14], and a power-law FO[PD] controller for robust motion control was proposed in [15].It was observed that the power law FO[PI]/FO[PD] controller outperformed both conventional PI/PD and FOPI/FOPD controllers.
To further expand the reach of traditional controllers, the main contribution of the paper is the design and study of two power-law complex-order controllers (CO[PI] and CO[PID]) for speed control of induction motor (IM) drive and electric vehicle (EV), respectively.The resulting controller transfer functions are implemented using a Field Programmable Analog Array (FPAA) device.The FPAA gives a cost-effective solution to designers in realizing complex analog circuits [16].The fractional-order operators/ complex-order operators depend numerically on past values of fractionally derived functions.Hence, the digital realization of complex-order operators requires high computational resources and memory capacity.Moreover realization using FPGA can also lead to performance degradation due to discretization in synthesizing digital controllers [17].This paper is structured as follows.Section II presents the transfer functions of the complexorder controller models, followed by an analysis of two design applications in Section III.The approximation and realization of the complex-order controllers are presented in Section IV, and controller behavior and performance assessment through simulation and experiments are outlined in Section V.

II. COMPLEX-ORDER CONTROLLER A. COMPLEX -ORDER OPERATORS
The fractional-order differentiator D(s) and fractional-order integrator I (s) with order γ ∈ [0, 1] are given in the frequency domain by respectively.The complex-order differentiator and complexorder integrator with order γ + jδ (γ ∈ [0, 1], δ ∈ R) are given respectively by Writing the complex frequency s in ( 3)-( 4) as s = jω, we get D(jω) = (jω) γ +jδ = j γ ω γ j jδ ω jδ (5) The terms j jδ and ω jδ in ( 5) and ( 6) can be calculated as = e −δ jπ 2 ( 7) By substituting ( 7) and ( 8) in ( 5) and ( 6), we get, = cos(δ ln ω) − j sin(δ ln ω) The core issue with the practical realization of complexorder differentiator or integrator is the approximation of complex-order parameters.Oustaloup approximation [18], Matsuda approximation [19], and the Carlson approach [20] are the most widely used frequency-domain methods to determine the approximate integer-order transfer functions for a given fractional-order system.However, these methods are cumbersome and do not necessarily provide the best approximation, given their limitation in the required frequency range.Also, these methods are not suitable for approximating fractional-order differentiators or integrators of complex orders.Following Bingi et al. [12] curvefitting-based approach for the approximation of the FOPID controller of complex orders, here the functions ( 9) and (10) are approximated by means of the Sanathanan-Koener (SK) iterative method [21], [22].The first step in this technique is to determine the frequency response data from (9) for Next, the integer-order transfer function model D(s) is obtained from using the SK iterative method.Therefore, the approximated D(s) is defined as follows: with the coefficients E = [e 0 , e 1 , . . ., e N ] T , F = [f 0 , f 1 , . . ., f N ] T (T for transpose), and the monomial functions φ(s) = 1, s, . . .s N , θ(s) = s, s 2 , . . .s N .Finally, the coefficient of E and F are determined considering the following Levy's objective function with SK iteration: where T is the iteration step.By solving (13), unbiased fitting is achieved when F T −1 (jω p ) approaches F T (jω p ).
Here, the curve-fitting-based technique uses the built-in functions of MATLAB software frd and fitfrd for extracting and processing the frequency response data for an order N .

B. COMPLEX-ORDER PID CONTROLLER
The transfer function of a FOPID is given by where K p , K i and K d are controller gains and λ, µ (0 ≤ λ, µ ≤ 1) orders of the integrator and differentiator stages, respectively.A complex FOPID controller [13] is then obtained by allowing the order of the integrator and differentiator stages to be complex numbers rather than real numbers.Its transfer function is where 0 Similarly, the transfer function of the corresponding power-law FO[PID] controller is given by where 0 ≤ γ ≤ 1.Therefore, the transfer function of the proposed complex-order [PID] (CO[PID]) controller is defined as where 0 ≤ γ ≤ 1, δ ∈ R. By substituting K i = 0 or K d = 0, the respective transfer function of complex-order [PI] (CO[PI]) and complex-order [PD] (CO[PD]) controllers are obtained.

III. APPLICATION EXAMPLES A. DESIGN EXAMPLE 1-INDUCTION MOTOR SPEED CONTROL
The fractional-order voltage source inverter (VSI) fed induction motor (IM) model is described by [23]: The CO[PI] controller C(s) is designed to control the speed of the IM plant model in (18).The magnitude and phase of ( 18) are: respectively, where P = 279.18ωFurthermore, we have: where As for the design constraints we wish to have for a robust complex controller, first, the open loop gain H (s) of the plant model is equal to 0 dB at the gain crossover frequency (ω cg ), i.e.
Second, the expression for phase margin φ m is given by 118608 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Combining ( 22) and ( 23), we get and where e j(φ m −π− ̸ G(jω gc ) = cos φ m − π − ̸ G(jω gc ) + j sin φ m − π − ̸ G(jω gc ) .Third, the constraint to improve the controller's robustness during gain variation is given by: Now the transfer function model for the controller (CO[PI]) is given by By using Taylor's series expansion and neglecting higher order terms (28), the function can be approximated by The expressions for magnitude and phase of the CO[PI] controller C 1 (jω) are given by where The differentiation of ̸ C 1 (jω) is where ) sin(δ ln ω) ) cos(δ ln ω); ) sin(δ ln ω) The controller parameters of the CO[PI] controller for the VSI-fed IM plant are obtained by solving (32), (33), and (26).Table 1 presents the controller gains of the CO[PI] controller

B. DESIGN EXAMPLE 2: ELECTRIC VEHICLE (EV) SPEED CONTROL
The EV system dynamics include both vehicle and motor dynamics.The linearized model of EV obtained using system identification by George et al. [24] is described as The CO[PID] controller C 2 (s) is designed using the Ant Colony Optimization (ACO) algorithm to control the speed of the EV system.The ACO technique can determine the optimal solution rapidly when more pheromones are released [24], [25].The pseudocode for ACO is given in Fig. 1.

IV. SIMULATIONS AND EXPERIMENTAL RESULTS
The   3.
The experimental evaluation of the proposed controllers was performed by employing the Anadigm AN231E04 FPAA development board and the Anadigm Designer 2 EDA software.The FPAA device has a 2 × 2 matrix of fully configurable analog blocks (CABs), which offer design programmability and versatility.The AN231E04 device has seven configurable input and output structures.The detailed architecture of the AN231E04 device and its features are given in [26].First, one can see that the approximated integer-order transfer functions of the CO[PI] and CO[PID] controllers shown in (35) and (40), respectively, have the following general form: Eq. ( 41) can be implemented by applying the partial fraction expansion (PFE) technique: It is observed that (42) is a sum of integer-order low-pass filters and a gain factor, where ω 0i = |p i | , i = 1, 2, 3.The p i and r i denotes the poles and residues of (41).Table 4 summarizes the scaling factors (K i ) and time constants (τ i = 1 ω 0i ) for CO[PI] and CO[PID] controllers.By substituting K i and τ i in (42) the PFE-based approximated transfer function of the controller is obtained.It is then implemented in the Anadigm Designer 2 EDA software using the SumIntegrator, SumDiff , and GainHold configurable analog modules (CAMs).The FPAA board has four AN231E04 chips, and each chip has eight CAMS.As the implementation uses a 3rd-order approximation (Eq.( 41)), a single AN231E04 chip is utilized.For an order greater than 3, two or more chips were required.The clock frequency of the chip was chosen as f clk = 800 kHz.The time-domain performance of the controller is verified using a sinusoidal signal of 100 Hz and a peak amplitude of 5 mV.Fig. 3    Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

V. CONCLUSION
The complex-order PI/PID controllers were designed for IM drive using frequency domain specifications and for EV speed control using the ACO algorithm.The complex-order controller performed better than the fractional-order and integer-older counterparts.The approximation of complexorder controller transfer function via a curve fitting approach facilitated an efficient and compact realization.The approximation order was selected equal to three to attain an accurate level of approximation.The approximated transfer function could be easily implemented and experimentally verified using the FPAA platform, exploiting its numerous advantages in performing signal processing operations like integration, differentiation, scaling, and summation.The design procedure is versatile and could be used in a variety of applications, including biological and biomedical systems, for approximating complex-order transfer functions.
CO[PI] and CO[PID] controllers for IM drive and EV speed control are realized and simulated in MATLAB-Simulink.The open-loop magnitude and phase responses of VSI-fed IM drive using PI, FOPI, FO[PI], and CO[PI] controllers are illustrated in Fig. 2 (a).It is observed that the CO[PI] controller has a higher stability margin at ω cg = 8 rad/s than other controllers.Fig. 2 (b) shows the closed loop speed response of VSI-fed IM with the designed 118610 VOLUME 11, 2023Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

FIGURE 3 .
FIGURE 3. (a) FPAA-based realization, (b) simulated results, and (c) experimental results:input (green) and output (blue) waveforms of the approximated CO[PI] controller of order 3, using a sinusoidal signal of peak amplitude 5 mV and frequency of 100 Hz.

FIGURE 4 .
FIGURE 4. (a) Performance of the proposed controllers to track the NEDC cycle, (b) control signals, and (c) error signals.
(a) shows the FPAA-based realization of the CO[PI] controller in the Anadigm design.Fig. 3 (b) shows the simulated result for the 3rd-order approximated CO[PI] controller.Fig. 3 (c) shows the experimental results (the input (green) and output (blue) waveform) for the approximated CO[PI] controller.EV speed control using CO[PID], FO[PID], FOPID, and IOPID controller is shown in Fig. 4 (a).The New European

FIGURE 5 .
FIGURE 5. (a) FPAA-based realization, (b) simulated results, and (c) experimental results:input (green) and output (blue) waveforms of the approximated CO[PID] controller of order 3, using a sinusoidal signal of peak amplitude 5 mV and frequency of 100 Hz.

TABLE 4 .
Scaling factors and time constants for realizing CO[PI] and CO[PID] controllers described in [35] and [40], respectively using PFE.Drive Cycle (NEDC) test is widely used to test EV speed tracking in India and Europe.Fig. 4 (a) shows that the CO[PID] controller has superior speed tracking performance 118612 VOLUME 11, 2023

TABLE 5 .
Comparison of simulated and experimental results for CO[PI] and CO[PID] controller using sinusoidal signal of different amplitude.than other controllers.Fig. 4 (b) and Fig. 4 (c) illustrate the control signals and tracking error of the proposed controllers.It is observed that the CO[PI] controller provides minimum controller effort and tracking error compared to the other controllers.Finally, Fig. 5 (a) shows the FPAA-based realization of the CO[PID] controllers in the Anadigm design.Fig. 5 (b) shows the simulated results for the 3rd-order approximated CO[PID] controller.Fig. 5 (c) shows experimental results (the input (green) and output (blue) waveform) for the approximated CO[PID] controller.Table 5 shows the simulated and experimental results of CO[PI] and CO[PID] controllers for sinusoidal signals of different amplitudes.It is clearly observed that experimental results closely follow the simulated response.

TABLE 2 .
Speed controllers for EV plant using ACO.
where ITSE denotes the integral time square error, ITAE the integral time absolute error, IAE the integral absolute error, ISE the integral square error, ISCO the integral square controller output, e(t) the error signal, and u(t) is the control signal.The controller parameters of the CO[PID] controller and other conventional controllers for EV speed control obtained using ACO are presented in Table2.The third-order approximation in the range ω = [−0.01,0.01] rad/s is used for approximating the transfer function of the FO[PID] and CO[PID] controllers as given below: