Complex‐order controller design examples and their implementation

A systematic method for approximating the complex‐order Laplacian operator by realizable integer‐order transfer functions is presented in this work. The realization is performed by a simple structure where only one active element is used. Thanks to the employment of complex‐order impedances, both integrators and differentiators can be readily implemented by the same core simply by interchanging the associated impedance locations. The validity of the presented concept is verified through simulation and experimental results, using the OrCAD PSpice suite and a Field Programmable Analog array device.

5][6] But the problem there is that the derived integer-order transfer functions are not realizable, due to the negative coefficients in their denominators.This problem will be overcome through the procedure introduced in this work, where after a detailed investigation, it was concluded that the order of approximation, as well as the frequency range, must be smaller than those considered in the aforementioned work, and this is not a problem when dealing with most controllers.A similar procedure has been followed in Elwakil et al, 12 for approximating the behavior of the complex-order capacitors.The resulting approximation functions of the complex-order integrators and differentiators are now realizable and their implementation is based on their expression as a ratio of appropriate impedances.This offers the benefit that the differentiator transfer function is readily realized just by interchanging the impedances in the active core that implements the transfer function of the integrator.
This work is organized as follows: The procedure for approximating integrators and differentiators of complex-order accompanied by the proposed implementation, where only one current feedback operational amplifier (CFOA) is utilized as an active element, is presented in Section 2. Various complex-order controller design examples are provided in Section 3, while their performance is evaluated through simulation results, in Section 4, using the OrCAD PSpice suite.In Section 5, experimental results using a Field Programmable Analog Array (FPAA) device are also provided.

| Procedure
Let us consider the transfer function of a fractional complex-order integrator described by the transfer function The approximation of (2) can be performed by writing it as HðsÞ ¼ H 1 ðsÞ þ H 2 ðsÞ, where with The next steps are as follows: • Step#1: The frequency response of the integrator is derived as the sum of the intermediate frequency responses of H 1 ðsÞ and H 2 ðsÞ.• Step#2: The frequency response H(s) derived from step#1 is approximated by an N th -order rational integer-order transfer function, using the curve-fitting based fitfrd command of the MATLAB software, which is based on the Sanathanan-Koerner (S-K) least square iterative method, having the general form with A i and B j i ¼ 0,1, …, N À 1, j ¼ 0,1, …, N ð Þ being positive and real coefficients. 13he efficiency of the fitfrd command of the MATLAB for approximating a complex-order integrator is compared with those offered by the following curve-fitting-based MATLAB commands: (a) the tfest command, which performs Sanathanan-Koerner iterations to solve the nonlinear least-squares problem (minimization of an imposed loss function), and (b) the invfreqs command of the MATLAB, which uses an equation error method to identify the best model from the provided frequency response data, by creating a system of linear equations.Considering a 3 rd -order approximation in the range ½10 À1 ,10 þ1 rad/s, the transfer functions derived using the fitfrd, tfest, and invfreqs commands for approximating a complex-order integrator with λ ¼ 0:8 and ϑ ¼ 0:05 are provided in ( 6)- (8), respectively The obtained gain and phase responses are provided in Figure 1A, where the associated error plots are given in Figure 1B, while the corresponding ideal responses are provided by dashes.Inspecting these plots, it is readily confirmed the superiority of the fitfrd command of the MATLAB in terms of accuracy, for approximating the complex-order integrator.

| Realization
A possible implementation of ( 5) is that depicted in Figure 2, where a Current Feedback Operational Amplifier (CFOA) is used as an active element.Taking into account the properties of the CFOA terminal (υ x ¼ υ y , i z ¼ i x , and υ o ¼ υ z ), the realized transfer function is Therefore, a suitable choice of values of the impedances could be Z 2 ¼ R Á H approx ðsÞ and Z 1 ¼ R, with R being an arbitrary value resistor. 14he reason for choosing a CFOA is that it offers a low-impedance output terminal instead of a high-impedance one as in the case of the second generation current conveyor (CCII) and, consequently, an extra buffering stage is not Comparison of the accuracy offered by the curve-fitting-based fitfrd, tfest, and invfreqs commands of the MATLAB software for approximating a complex-order integrator with λ ¼ 0:8 and ϑ ¼ 0:05.(A) Gain/phase responses and (B) the associated error plots.required for avoiding the loading effect of the next stages.With regards to the employment of an operational amplifier (op-amp), the implementation of (9) requires two op-amps, increasing the circuit complexity in terms of active and passive component count.
In the case of a fractional complex-order differentiator described by the approximation is performed by selecting the impedances as follows: , where H approx ðsÞ is the transfer function derived through the approximation of the transfer function 1=s μþjϕ (i.e., the associated integrator).
From the above, it is obvious that the presented scheme offers design flexibility and versatility, in the sense that both fundamental blocks can be readily approximated by the same core, just by interchanging the values of the impedances.
The approximation of the frequency dependent impedance (i.e., Z 2 in the case of integrator and Z 1 in the case of differentiator) can be performed using the Foster or Cauer type-I or type-II RC networks.Just for demonstration purposes, the Foster type-I network depicted in Figure 3 will be utilized hereinafter.The values of passive elements are calculated using the design equations summarized in (11) with r i and p i being the residues and poles of the complex impedance approximation function. 15I G U R E 2 General scheme for approximating the fractional complex-order integrator/differentiator transfer function.
F I G U R E 3 Foster type-I RC network for implementing the impedance Z 1 in the case of complex-order differentiator, and for Z 2 in the case of integrator.
In order to demonstrate the validity of the concept introduced in the previous section, it will be applied in a variety of controllers transfer functions.Let us consider some of the controller transfer functions already presented in Bingi et al. 6 The controller described by (12)   C 1a ðsÞ ¼ 0:56 þ 8:27 s 0:8þj0:05 þ 0:01s 0:3þj0:05 ð12Þ can be implemented by the topology in Figure 4A, where the realized transfer function is Comparing the coefficients of ( 12)-( 13), the design equations are readily obtained as K p ¼ 0:56, Z 2 ¼ 8:27R=s 0:8þj0:05 and Z 1 ¼ 100R=s 0:3þj0:05 .
The second provided example is the realization of a controller with the following transfer function: A topology that is suitable for realizing ( 16) is shown in Figure 4B, which implements the transfer function F I G U R E 4 CFOA implementation of the controller transfer functions given (A) by Equations ( 12) and ( 14) and (B) by Equation ( 16) and, therefore, the design equations become Z 2 ¼ R=s 0:1þj0:01 , 1=RC ex1 ¼ 8:27, and RC ex2 ¼ 0:01.The approximation of the controllers described by ( 12) and ( 16) will be performed following the steps described in the previous section, considering a 3 rd -order approximation in the range [10 À1 , 10 +1 ] rad/s and R ¼ 10 kΩ.The resulting values of the passive elements of the RC network in Figure 3, rounded to the E96 series defined in IEC 60063 standard, are summarized in Table 1.

| SIMULATION RESULTS
The behavior of the controllers, which have been presented in the previous section, will be evaluated using PSpice with the model of the AD844 discrete component utilized for emulating the behavior of the CFOA.The dynamics of the plant are described by ( 18) The frequency responses of the approximated controllers described by ( 12) and ( 16) are demonstrated in Figure 5A,B, respectively, with the ideal ones given by dashes.The observed deviation of the phase responses from the corresponding theoretical ones is caused by the effect of the nonidealities of the CFOA (r x ¼ 50Ω, r z ¼ 3 MΩ, C z ¼ 4:5 pF). 16A B L E 1 Passive elements values of the Foster type-I network for realizing the controller transfer functions in ( 12) and ( 16).

Element
Z 2 (Figure 4A) Z 1 (Figure 4A) Z 2 (Figure 4B) F I G U R E 5 Gain and phase responses of the controllers described (A) by Equation ( 12) and (B) by Equation ( 16) The corresponding open-loop responses are shown in Figure 6A,B, where the phase margin and crossover gain frequency are {65.04 , 4.63 rad/s} and {53.66 , 3.23 rad/s} for each one of the considered controller-plant systems, close to the corresponding theoretical values {64.1 , 4.71 rad/s}, and {52.8 , 3.31 rad/s}.
Applying an input step voltage, the derived output waveforms are demonstrated in Figure 7, where the values of the rise time and settling time are {335 ms, 1.74 s} and {421 ms, 1.92 s}, close to the theoretically predicted values {329 ms, 1.72 s} and {417 ms, 1.95 s}.
Considering a AE5% deviation of the passive elements values from their nominal ones and employing the Monte-Carlo analysis offered by the Advanced Analysis tool of the OrCAD PSpice suite, the statistical plots about the phase margin obtained for 500 runs are provided in Figure 8.The values of the standard deviation of the considered performance parameter are 0.49 and 0.14 and, taking into account that the corresponding mean values are 64.99 and 53.66 , it is concluded that both systems have reasonable sensitivity characteristics.

| EXPERIMENTAL RESULTS
The controller described by ( 19) Open-loop gain and phase responses of the systems controller-plant described (A) by Equations ( 12) and ( 18) and (B) by Equations ( 16) and ( 18) Step responses of the systems controller-plant described (A) by Equations ( 12) and ( 18) and (B) by Equations ( 16) and (18)  employed with the plant described by (18), will be experimentally verified through the utilization of the FPAA AN231E04 device from Anadigm, 17 with the clock frequency being equal to f clk ¼ 1 kHz.The approximation transfer function of the controller is given in (20) and it will be decomposed as sum of a constant term and of 1 st -order low-pass filters, as given in (21) where K 0 ¼ 0:7082, K i ¼ r i =jp i j, and τ i ¼ 1=jp i j, with r i , p i being the residues and poles of (20), respectively. 18he resulting values of the gain factors of the low-pass filters factors are K 1 ¼ 1:885, K 2 ¼ 8:069, and K 3 ¼ 172:455, while the values of time constants are τ 1 ¼ 0:425 s, τ 2 ¼ 2:87 s, and τ 3 ¼ 32:25 s.The functional block diagram for implementing the expression in Equation ( 21) is demonstrated in Figure 9.It is realized using the Fil-terLowFreqBilinear1 configured analog module (CAM) offered by theAnadigm Designer ® 2 EDA software with two external capacitors (due to the inherent differential operation) for each filter section.Each one of three sets has equal valued capacitors 0.258, 1.8, and 20.1 nF, respectively.The same is followed for the plant, with the external capacitor having a value equal to 0.154 nF.The resulting full design in open-loop configuration is depicted in Figure 10.
F I G U R E 8 Statistical plots of the phase margin of the systems controller-plant described (A) by Equations ( 12) and ( 18) and (B) by Equations ( 16) and ( 18 The time-domain behavior of the system is initially evaluated in open-loop configuration.For this purpose, the system is stimulated by a 500-mV pp sinusoidal signal with variable frequency, and the experimentally obtained values of the phase margin and gain crossover frequency are {62 , 4.78 rad/s}, close to the theoretical values {63.7 , 4.71 rad/s}.The associated screenshot from DSO6034A digital oscilloscope is demonstrated in Figure 11A.
As a next step, the system is configured in closed-loop and applying a step voltage of amplitude 1 V, the resulting output waveform is demonstrated in Figure 11B.The measured values of the rise time and settling time are {310 ms, 1.65 s}, with the theoretical ones being {327 ms, 1.72 s}.

| CONCLUSIONS
Various types of complex-order controller transfer functions have been realized in this work, for demonstrating a procedure for the complex-order Laplacian operator approximation.The provided simulation, as well as experimental results, support the efficiency of the procedure and, also, confirm the offered design flexibility and versatility.It must be mentioned at this point that the implementation of the controller structures is independent from the employed active element and, therefore, the exploitation of other types of active elements and, also, of alternative design techniques are included in the future research steps.

DATA AVAILABILITY STATEMENT
The data that support the findings of this study are available from the corresponding author upon reasonable request.
F I G U R E 1 1 Input and output waveforms of the (A) open-loop system, stimulated at the crossover gain frequency and (B) closed-loop system, stimulated by a step voltage.
) F I G U R E 9 Functional block diagram for implementing the expression in Equation (21)