Optimal approximation of analog PID controllers of complex fractional-order

Fractional Calculus and Applied Analysis https://doi.org/10.1007/s13540-023-00168-x ORIGINAL PAPER Optimal approximation of analog PID controllers of complex fractional-order Shibendu Mahata1·Norbert Herencsar2·Guido Maione3 Received: 3 August 2022 / Revised: 3 May 2023 / Accepted: 4 May 2023 © The Author(s) 2023 Abstract Complex fractional-order (CFO) transfer functions, being more generalized versions of their real-order counterparts, lend greater flexibility to system modeling. Due to the absence of commercial complex-order fractance elements, the implementation of CFO models is challenging. To alleviate this issue, a constrained optimization approach that meets the targeted frequency responses is proposed for the rational approximation of CFO systems. The technique generates stable, minimum-phase, and real-valued coefficients based approximants, which are not always feasible for the curve-fitting approach reported in the literature. Stability and performance studies of the CFO proportional-integral-derivative (CFOPID) controllers for the Podlubny’s, the internal model control, and the El-Khazali’s forms are considered to demonstrate the feasibility of the proposed technique. Simulation results highlight that, for a practically reasonable order, all the designs achieve good agreement with the theoretical characteristics. Performance comparisons with the CFOPID controller approximants determined by the Oustaloup’s CFO differentiator based substitution method justify the proposed approach. Keywords Complex fractional-order system (primary) ·Complex fractional-order PID controller ·Approximation ·Constrained optimization ·Differential evolution BNorbert Herencsar [email protected] Shibendu Mahata [email protected] Guido Maione [email protected] 1Department of Electrical Engineering, Dr. B. C. Roy Engineering College, Durgapur, West Bengal 713206, India 2Department of Telecommunications, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technicka 12, 616 00 Brno, Czechia 3Department of Electrical and Information Engineering, Polytechnic University of Bari, Via E. Orabona, 4, 70125 Bari, Italy 123 S. Mahata et al. Mathematics Subject Classification 26A33 (primary) ·93B50 ·93C99 ·93E11 1 Introduction Applications of fractional calculus, the branch of mathematics that is regarded as a generalization of classical calculus, are ever-increasing across all domains [37, 38]. Fractional-order (FO) differentiation/integration (differintegration) generalizes the traditional derivation from an integer to any real or complex number [58]. Various definitions of FO derivatives are available in the fractional calculus literature. Caputo’s definition of fractional derivative allows a similar representation of initial conditions to that of the integer-order derivative because initial conditions are determined by integer derivatives. Hence, it provides an interpretation of initial conditions to be used in real-world problems more easily than the Riemann-Liouville definition [36]. However, many researchers dispute the ability of Caputo’s definition to take properly into account initial conditions if used to define or simulate fractional-order systems. In synthesis, there is a discussion on the respective benefits and drawbacks of the two mentioned (or other) definitions. In detail, the Caputo and the Riemann-Liouville derivatives of a function f(t)are respectively defined as C aDα tf(t)=1 (n−α) t a f(n)(τ) (t−τ)α+1−ndτ, (1.1) RL aDα tf(t)=1 (n−α) dn dtnt a f(τ) (t−τ)α+1−ndτ, (1.2) where nis an integer, n−1≤α<n,and (·)denotes the gamma function [17]. From the input–output control point of view, initial conditions are usually set to zero such that transfer functions are considered, then the problem of initial conditions is not significant. The dynamical systems can be represented by FO differential equations; hence, the linear time-invariant models can be represented by using fractional-order transfer functions (FOTFs). Due to the presence of additional tuning knobs, the FO models can more effectively capture the dynamics of real-world systems. Applicability of FO systems is reported in control systems, signal processing, circuit theory, biomedical engineering, etc. [26,56]. The majority of the literature has dealt with the design and implementation of FOTFs comprising real-valued exponent of s. The simplest transfer function in this regard is H(s)=sα, which can be obtained by taking the Laplace transform of (1.1) and (1.2) with zero initial conditions. Assuming α∈ +,we may write F(s)=spsα−p=spsγ, where pis an integer such that p=α. Therefore, γ=α−pimplies γ∈(0,1). Fitting, numerical, optimization, and continuedfraction expansion techniques help to approximate the frequency responses of sγ using a finite-order rational transfer function for a specified bandwidth [1,19,21,43, 44,48,54]. From the perspective of signal processing, FR(s)=sγrepresents a realorder FO differentiator (RFOD) for γ∈(0,1), whereas γ∈(−1,0)corresponds to 123 Optimal approximation of analog... its inverse function [33]. The transfer function of the classical proportional-integralderivative (PID) controller has been generalized as CR(s)=Kp+Kis−λ+Kdsδ, where 0 ≤λ, δ ≤1;Kp,Ki, and Kdare the proportional, integral, and derivative gains [55]. The traditional filters, including the special types, are also extended to the FO domain [6]. The FOTF can be practically implemented by replacing one or more traditional capacitor/inductor with the constant phase element (CPE), such as the FO capacitor/inductor [15]. Several works experimentally demonstrated the performances of FO circuits and systems (controllers, filters, etc.) employing the CPEs [16,28]. The CPEs are not yet available as commercial devices in the market; though, proposals to implement the FO systems for industrial applications are drawing attraction [50,66]. The impedance characteristics of the CPE can be emulated using passive or active emulator circuits [30,31]. However, the hardware complexity of the emulators is substantial compared to a single element fractor. An alternative approach that avoids using CPEs in realizing the FO systems involves determining the integer-order approximation of the FOTF. Such models may be obtained through direct substitution of the rational approximant of sγoperator in the FOTF [67]. For the non-commensurate type or more complicated FOTFs, the substitution method incurs large overhead. Hence, numerical or optimal approaches are the preferred approximation tools [11,42]. 1.1 Complex fractional-order controller The generality introduced by fractional calculus allows complex FOTF-based system modeling, i.e., the exponent of smay be a complex number. For instance, the RFOD is a subset of the complex FO differentiator (CFOD) FC(s)=sα+jβ, where α∈[0,1] and β∈[68]. The complex FOTFs provide more degrees-of-freedom in modeling as compared to their real-order counterparts. The complex-order derivative of a sine function at steady-state is expressed as Dα+jβsin(ωt)=ωα[cos{βln(ω)}+jsin{βln(ω)}] ×sin ωt+απ 2cosh βπ 2+jcos ωt+απ 2sinh βπ 2,(1.3) where j=√−1 and ωis the angular frequency variable expressed in radians per second (rad/s) [40]. The third-generation CRONE controller is a seminal work that deals with the design and application of complex-order models [35]. Other application areas of complexorder derivatives can be found in the locomotion control of robots [51,61], chaotic oscillators [52,53], system identification [5,29], filter modeling [2], viscoelasticity [8], optimization algorithms [7,39] and neural networks [32]. The effectiveness of complex fractional-order PID (CFOPID) controller was demonstrated on time-delay process [3]. The improved time-domain responses yielded by the genetic algorithm based complex FO controllers as compared to the classical and real-order FO controllers were justified [64]. Tuning rules for the CFOPID and CFOPI controllers were proposed in [23,59]. Modeling of CFODs and CFOPIDs in the discrete-time domain 123 S. Mahata et al. was also carried out [9,40,57]. Stability analysis of complex-order fractional difference equation was reported in [10], while [47] investigated the geometrical and physical interpretation of the complex-order fractional integral. 1.2 Motivations for this work The effectiveness of complex FOTF modeling has not been experimentally validated since implementing such transfer functions requires the complex-order fractance elements; realizing such elements has not yet received much attention in the literature [22, 60]. Therefore, the rational approximation of the complex FOTFs remains a feasible choice. The motivation for carrying out this research is based on the limitations of the existing methods, as discussed below. 1.2.1 Limitations of fitting routines Curve-fitting techniques were employed for the approximation of analog CFOD [14], CFOPID [13], and complex-order filter [12]. The fitting technique is based on the Sanathanan-Koerner algorithm with Levy’s cost function [14]. The fifth-order approximation of the CFOPID controller reported in (28) of [14]isgivenby FC(s)=1+1 s0.8+0.1j+s0.8+0.1j ≈−2353s5+2.14 ×106s4+7.53 ×106s3+3.77 ×106s2+2922s−55.91 s5−1.43 ×104s4+4.58 ×106s3+1.63 ×105s2−100s−0.03 .(1.4) The approximated model in (1.4) comprises real-valued coefficients for both the numerator and denominator polynomials. However, the poles of the rational approximant are located at {13972,327.8301,8.19 ×10−4,−0.0362,−2.21 × 10−4}, whereas the zeros lie at {912.9844,−2.9027,−0.6036,−0.0043,0.0035}.The reported CFOPID approximant is an unstable and non-minimum-phase system, since three poles (13972, 327.8301, 8.19 ×10−4) and two zeros (912.9844, 0.0035) exist on the right-half s-plane. Vector Fitting (VF) [24,25] is another popular technique to fit the frequencydomain responses with rational functions. The VF method is applied for a bandwidth of [0.001, 1000] rad/s with 1000 sample points to generate the fourth and fifth-order approximations of the theoretical CFOPID controller given in (1.4). The corresponding approximants are given by FC(s)≈307s4−12272s3+1781582s2+256374s+71.9481 s4+1383s3+418560s2+1247s+0.0851 ,(1.5) FC(s)≈1389.97s5+68427s4−819142s3+3320489s2+52141s+62.24 s5+5875s4+1273888s3+109639s2+315.95s+0.2286 .(1.6) The zeros of the VF-based fourth-order approximant lie at {−0.000281,−0.1434,20.0588±73.5296 j}; the poles reside at {−0.000069, −0.002909,−447.338,−935.659}. For the fifth-order approximant, the locations 123 Optimal approximation of analog... Fig. 1 Magnitude and phase plots of the VF-based fourth and fifth-order approximations for the theoretical CFOPID controller given in (1.4) of zeros and poles are obtained as {−59.7594,5.2730±3.5107 j,−0.0143,−0.0013} and {−5649.5172,−225.3966,−0.0831,−0.0016,−0.0013}, respectively. The VF technique can generate a stable approximant since all the poles reside on the lefthalf s-plane. However, the presence of two right-half plane zeros (occurring in complex-conjugate form) for both the approximants highlight non-minimum phase behavior. Such systems may lead to stability issues in a feedback control mechanism. The magnitude and phase plots for the VF-based approximants are presented in Fig. 1. It can be seen that both the approximants fail to attain good conformity with the theoretical response. 1.2.2 Limitations of a built-in MATLAB function Using the MATLAB function invfreqs with minimization of the norm of the gradient (with parameter iter set as 30 and considering 1000 sample points), the fourth and fifth-order approximations of the theoretical CFOPID controller function given by (1.4) are respectively determined as FC(s)≈8.26 ×1017s4+2.49 ×1021s3+8.25 ×1022s2+4.15 ×1024s+3.86 ×1023 s4+1.26 ×1016s3+1.02 ×1022s2+7.02 ×1023s+1.78 ×1021 , (1.7) 123 S. Mahata et al. Fig. 2 Magnitude and phase plots of the invfreqs-based fourth and fifth-order approximations for the theoretical CFOPID controller given in (1.4) FC(s)≈ 1.84 ×1018s5+4.59 ×1021s4+5.91 ×1023s3 +3.06 ×1025s2+7.47 ×1026s+7.91 ×1025 s5+4.27 ×1017s4+1.87 ×1022s3+2.47 ×1024s2 +1.28 ×1026s+3.23 ×1023 .(1.8) Note that the approximants generated by invfreqs without the iter parameter are both unstable and non-minimum-phase. The poles of the fourth and fifth-order rational functions are located at {−1.26 ×1016,−8.09 ×105,−68.8268, −0.0025} and {−4.27×1017,−4.36×104,−66.1630±49.8776 j,−0.0025}, respectively; the zeros reside at {−16.4200±37.5811 j,−0.0931, −2981.59} and {−2361.5033, −67.4670, −32.7442±38.2736 j,−0.1063}, respectively. Although stable and minimum-phase approximations are obtained, this method gives an ill-conditioned function (coefficients with extremely large or small values), such that its implementation will lead to impractical values of passive components. The magnitude and phase responses of the invfreqs-based fourth and fifth-order approximants are presented in Fig. 2. These plots highlight a significant deviation from the theoretical responses for a wide range of the considered bandwidth. 123 Optimal approximation of analog... 1.2.3 Applicability of the Oustaloup’s recursive algorithm An alternative technique to approximate the theoretical CFOD characteristics was reported by Oustaloup et al. [48]. According to the Oustaloup’s method, the rational approximant of a CFOD comprises of complex poles and zeros. The CFOD approximant for sv, where v=a+jb, based on this technique is given by Dv M(s)=μ−v/2 M  k=−M 1+s ω k 1+s ωk ,(1.9) where Mis a positive integer, μ=ωh ωb , and the bandwidth of approximation is in the interval [ωb,ω h]rad/s. The values of ωkand ω kare respectively given by ωk=ρkejθ,(1.10) ω k=ρ ke−jθ,(1.11) where ρk=ωbωh ωbk+M+1/2+a/2 2M+1,ρ k=ωbωh ωbk+M+1/2−a/2 2M+1, and θ=b 2(2M+1)log(μ). The Oustaloup’s CFOD approximant may be re-written as a ratio of complex functions as given by DM(s)=P(s)+jQ(s) P(s)+jQ(s),(1.12) where P(s), P(s), Q(s), and Q(s)are real coefficients based polynomials. Alternatively, (1.12) may be written as DM(s)=PM(s) QM(s)+jP M(s) QM(s),(1.13) where PM(s)=P(s)P(s)+Q(s)Q(s),P M(s)=P(s)Q(s)−Q(s)P(s), and QM(s)=P2(s)+Q2(s). A French patent [49] employing operational amplifiers for the circuit implementation of (1.13) was reported three decades ago. However, it may be more straightforward to practically implement a real-valued function as compared to (1.13), which is a complex function. Moreover, as shown in Sect.3, the Oustaloup’s approximation even with M= 1 may lead to a CFOPID controller of high order, with coefficients having large values. Therefore, it is worth investigating the following question: Within the bandwidth approximation limits where six (or less) decades may be sufficient (in general) for process control applications, is it possible to achieve reasonable accuracy using a realvalued rational transfer function with smaller orders (even and odd) to approximate the characteristics of the CFOPID controller? 123 S. Mahata et al. 1.3 Contributions of this work In this paper, the approximation of the CFOPID controller is presented wherein the approximant is strictly a real function. The real-valued coefficients of the rational approximant are determined optimally, while enforcing the poles and zeros to lie on the left-half s-plane. Thus, the proposed work helps in alleviating the issues associated with the existing approaches. The primary contributions of this work are the following: 1. A constrained optimization approach based on a state-of-the-art evolutionary method is presented that satisfies the conditions of stability, minimum-phase, and achieves real-valued coefficients real rational approximants of both even and odd orders for the CFOPID controller. 2. The feasibility and effectiveness of the proposed approach is validated on three different variants of the CFOPID controller, namely the Podlubny’s form [55], the internal model control (IMC) form [65], and the El-Khazali’s form [20]. Effects of the approximation order on the approximation performance are evaluated using various frequency response error metrics. 3. The accuracy of the proposed technique is compared with the Oustaloup’s method. Results demonstrate that the proposed approach provides a competitive performance when compared against the much higher order of the competing designs. While the Oustaloup’s CFOD approximants are substituted in the theoretical CFOPID transfer function to obtain the corresponding approximation, the proposed method directly determines the CFOPID controller coefficients. In the rest of the paper, Sect. 2presents the proposed optimization problem formulation and the solution methodology. In Sect.3, design performance studies are carried out, and the simulation results are discussed. The paper concludes in Sect.4. 2 Proposed technique 2.1 Problem formulation The generalized transfer function of a real FOTF is defined as FR(s)=Bmsλm+Bm−1sλm−1+...+B0sλ0 Ansμn+An−1sμn−1+...+A0sμ0,(2.1) where Bi(i=0,1,...,m),Ak(k=0,1,...,n),λi(i=0,1,...,m), and μk(k=0,1,...,n)are real numbers; λm>λ m−1>··· >λ 0≥0; and μn>μ n−1>···>μ 0≥0. Replacing the real-valued exponents of sin (2.1) with complex numbers, such as λi=ri+jy i(i=0,1,...,m)and μk=wk+jzk(k= 0,1,...,n), leads to the complex FOTF, as defined below, which provides further generalization and more degrees-of-freedom than (2.1): FC(s)=Bmsrm+jym+Bm−1srm−1+jym−1+...+B0sr0+jy0 Answn+jzn+An−1swn−1+jzn−1+...+A0sw0+jz0.(2.2) 123 Optimal approximation of analog... Any classical continuous-time system can be represented as a ratio of polynomials in s, as defined by HN P(s)=PN(s) QN(s)=aNsN+aN−1sN−1+...+a0 sN+bN−1sN−1+bN−2sN−2+...+b0 ,(2.3) where ak(k=0,1,...,N)and bk(k=0,1,...,N−1)are the real-valued coefficients of the numerator and denominator polynomials, respectively, of HN P(s);and N is the order of the system. HN P(s)is bounded-input bounded-output (BIBO) stable if all the poles reside in the left-half s-plane; if all the zeros lie on the left-half s-plane, then the system attains minimum-phase behavior. In the case of complex conjugate poles or zeros, their real part must be negative. A necessary condition for BIBO stability and minimum-phase response is that all the coefficients of QN(s)and PN(s), respectively, must possess the same sign with no missing terms. The frequency-domain transfer functions of (2.2) and (2.3) can be obtained by substituting swith jω, as respectively defined below FC(jω) =Bm(jω)rm+jym+Bm−1(jω)rm−1+jym−1+...+B0(jω)r0+jy0 An(jω)wn+jzn+An−1(jω)wn−1+jzn−1+...+A0(jω)w0+jz0 ,(2.4) HN P(jω) =aN(jω)N+aN−1(jω)N−1+...+a0 (jω)N+bN−1(jω)N−1+bN−2(jω)N−2+...+b0 .(2.5) Table 1shows the transfer function and frequency domain expressions for the different CFOPID controllers considered for approximation in this work. The design problem can be solved in the optimal sense, where the purpose of the search routine is to minimize the frequency response error between the theoretical (targeted) model FC(s)and the rational approximant HN P(s), subject to satisfying the stability and minimum-phase criteria. The objective function fof the proposed constrained optimization (minimization) problem can be formulated as f=1 L L  i=1|FC(jωi)|−HN P(jωi,X)+ ∠FC(jωi)−∠HN P(jωi,X), (2.6) Subject to :σz<0 and σp<0(p, z =1,...,N), where Lis the total number of log-spaced data points in the bandwidth of approximation [ωmin,ω max]rad/s; Xrepresents the decision variables vector, i.e., X=[aN aN−1…a0bN−1bN−2…b0]; σzand σpdenote the real part of the roots of PN(s) and QN(s), respectively. The total number of decision variables for this optimization problem is 2N+1, which represents the dimension (D). 2.2 Solution methodology A metaheuristic algorithm facilitates the integration of various constraint handling strategies to generate a feasible solution [34]. Although (2.6) may be minimized using 123 S. Mahata et al. Fig. 3 Flowchart of the proposed design method as {105,10 5}, {104,10 4}, {104,10 4}, and {105,10 6}, respectively, based on the trial-and-error method mentioned previously. Note that increasing Nshould improve the approximation by reducing the errors in the Bode gain and phase plots. A higher order Nis needed to represent the long memory of the system very accurately, but the computational burden increases. On the contrary, a lower Nproduces more ripples in gain and phase plots, but is advantageous, because the changes in coefficients due to tolerances of components (in an analog implementation) or due to the limitations of microprocessor words and the quantization effects (for a digital implementation) can be contained [45]. Hence a low sensitivity to parameter variations is obtained. Moreover, an accurate implementation over a wide frequency range is difficult for a conventional hardware realization due to computational complexity [27] and hardware costs can be limited if the accuracy is required in a restricted frequency range. Finally, from the practical point of view, approximations are not required over a very large frequency range. Namely, in systems and control engineering applications, approximations should work within a bounded frequency range, where a few decades are usually satisfactory. 123 Optimal approximation of analog... Fig. 4 The magnitude and phase-frequency comparison plots of the proposed optimal approximants of the CFOPID controllers with the theoretical models for (a) case I, (b) case II, (c) case III, (d) case IV The stability and minimum-phase of the designs can be confirmed from Table 4, which shows that all the poles and zeros reside on the left-half s-plane. This is an important property since the occurrence of non-minimum-phase zeros in a controller may lead to stability issues in a closed-loop control system, including approximants of such controller. The frequency responses of the designed controllers are compared with the theoretical models for the four cases, as shown in Figs. 4(a)–(d). It may be observed that: (i) for the case I, the sixth-order design achieves lower magnitude error than the fifthorder model in the frequency range [0.001, 0.028] rad/s. The fifth-order approximant outperforms the sixth-order one in terms of phase response accuracy in the interval [0.001, 0.145] rad/s. Frequency responses of both the approximants are similar at high frequency regions. Increasing the order of the proposed approximant further slightly reduces the error in phase approximation but does not significantly affect the error in magnitude approximation; (ii) for case II with both the approximation orders, a similar accuracy in magnitude is obtained in the frequency range [0.181, 6.884] rad/s, whereas the same accuracy for phase characteristic is obtained between 0.774 and 3.4 rad/s. Overall, an improved accuracy in magnitude and phase is respectively attained using 123 S. Mahata et al. Table 4 Locations of zeros and poles of the proposed optimal approximants of the CFOPID controllers Case NZeros Poles I5−2.1488 −0.9432, −0.1116, −0.0074, –59290, −0.1733, −0.0202, −0.0037, −0.0015 −0.0007 6−2.5457, −0.0865±1.0970j, −0.8593, –39646, −0.0941±1. 0932j, −0.1997, −0.1136, −0.0103 −0.0172, −0.0009 II 4 −43.22, −0.7714, −0.0756, −0.0048 −9991.7, −0.2740, −0.0220, −0.0019 5−192.73, −0.6720, −0.0877, −0.0138, −9496.7, −0.2209, −0.0248, −0.0045, −0.0026 −0.0006 III 5 −1236.5, −2.8835, −0.5352, −0.0279, −1238.4, −3.5570, −0.0707, −0.0088, −0.0029 −0.0010 6−2217.0, −0.8230±0.6719j, −0.3221, −2328.1, −0.8829±0.2645j, −0.0749, −0.0372, −0.0036 −0.0114, −0.0011 IV 4 −7.0858, −0.4112, −0.0327, −0.0019 –19642, −0.0671, −0.0035, −0.00015 5−6.4104, −0.4287, −0.0728, −0.0117, –183440, −0.1225, −0.0180, −0.0018, −0.0003 −0.00015 123 Optimal approximation of analog... Table 5 Comparison of error metrics for the proposed optimal approximants of the CFOPID controllers (best performance is highlighted in boldface) Case NMax Absolute Error Mean Absolute Error Magnitude (dB) Phase (◦) Magnitude (dB) Phase (◦) I 5 49.841 22.469 29.939 6.029 649.153 28.412 25.023 12.405 II 4 34.570 43.250 14.499 4.668 5−0.795 51.978 −21.040 20.903 III 5 46.443 5.696 23.371 0.913 645.982 3.265 22.728 1.580 IV 4 42.075 36.109 17.247 15.584 5 45.052 21.782 21.639 11.798 the designs with N= 5 and N= 4; (iii) for case III, the frequency characteristic plots of both the approximants exhibit similarity. While the phase response exhibits proximity with the theory throughout the bandwidth, the magnitude plot starts deviating at the lower frequency end below 0.04 rad/s; (iv) for case IV, the magnitude-frequency profiles for the proposed fourth and fifth-order aproximants are similar, whereas the fifth-order design markedly outperforms the fourth-order model in the range [0.001, 0.026] rad/s regarding the phase response accuracy. Table 5presents the performance metrics for all the designed CFOPIDs, which show that: (i) for case I, maximum APE for N=5 (49.841 dB) and N=6 (49.153 dB) is similar. The fifth-order design outperforms the sixth-order controller about the maximum and mean APE, whereas the sixth-order design achieves smaller mean AME; (ii) for case II, the fifth-order design is inferior to its lower-order counterpart regarding the maximum and mean APE metrics, but significantly outperforms the fourth-order approximant about both the magnitude response indices; (iii) for case III, the performances of both the designed controllers are similar; (iv) substantial improvement in maximum APE is achieved by the fifth-order approximant as compared to the fourth-order one for case IV. However, it is remarked that the high values of maximum absolute errors are typically obtained at the extremum low and high frequencies of the considered intervals, where the approximations are obviously worst. As a consequence, mean absolute errors are affected by such values. On the contrary, the approximations give suitable results inside the frequency interval [ωmin,ωmax]. 3.2 Comparison with the literature The CFOPID controller can be approximated using Oustaloup’s CFOD approximant by a simple substitution method. For instance, the normalized CFOD approximant for s0.8+0.1jbased on the Oustaloup’s method with M=1, ωb=0.001 rad/s, and 123 S. Mahata et al. ωh=1000 rad/s, is given by s0.8+0.1j≈D0.8+0.1j 1=1 1000(0.8+0.1j)⎛ ⎜ ⎜ ⎝ 1+s ω −1 1+s ω−1 ⎞ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎝ 1+s ω 0 1+s ω0 ⎞ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎝ 1+s ω 1 1+s ω1 ⎞ ⎟ ⎟ ⎠ , (3.3) where ω−1=0.0614+0.0144 j,ω −1=0.0015−0.0004 j,ω0=6.1430+1.4400 j, ω 0=0.1543 −0.0362 j,ω1=614.3+144 j, and ω 1=15.4306 −3.6172 j. Thus, the theoretical CFOPID controller given by (1.4) can be approximated using the Oustaloup’s technique as FC(s)=1+s0.8+0.1j+1 s0.8+0.1j≈1+D0.8+0.1j 1(s)+1 D0.8+0.1j 1(s) .(3.4) Then, (3.4) can be represented in the form of (1.13), where the expressions for PM(s),P M(s), and QM(s)are obtained as PM(s)=194.5053s12 +1.4910 ×105s11 +8.2897 ×106s10 +1.7707 ×108s9 +1.7046 ×109s8+6.7896 ×109s7+8.8736 ×109s6+6.6288 ×109s5 +1.6625 ×109s4+1.7320 ×108s3+8.1395 ×106s2+1.4701 ×105s +187.2878,(3.5) P M(s)=159.9975s12 +8.0683 ×104s11 +3.3008 ×106s10 +4.7865 ×107s9 +2.4461 ×108s8+8.9531 ×107s7−3.9970 ×108s6−1.6790 ×108s5 +1.5350 ×108s4+3.6652 ×107s3+2.7079 ×106s2+6.8829 ×104s +144.7765,(3.6) QM(s)=s12 +1.2548 ×103s11 +4.3522 ×105s10 +1.7476 ×107s9 +2.6860 ×108s8+1.7739 ×109s7+4.5571 ×109s6+1.7733 ×109s5 +2.6834 ×108s4+1.7441 ×107s3+4.3339 ×105s2+1.2199 ×103s +0.9594.(3.7) Eqns. (3.5), (3.6), and (3.7) reveal that even with M=1, the substitution of Oustaloup’s CFOD model yields a CFOPID controller of twelfth order (N= 12) for both the real and imaginary parts of the approximant. The coefficients also attain large values as observed from the above three expressions. Note that the Oustaloup’s method yields only an even order of rational approximant for the CFOPID controller. For comparison purpose, the CFOPID controllers for all the four cases are obtained by substituting the Oustaloup’s CFOD approximant (with M=1in(1.9)) in the theoretical controller functions. 123 Optimal approximation of analog... Fig. 5 The magnitude and phase-frequency comparison plots of the proposed CFOPID controllers with the published literature for (a) case I, (b) case II, (c) case III, (d) case IV The magnitude and phase responses comparison plots of the controllers obtained by using the Oustaloup’s CFOD-based substitution and the proposed optimization with larger Nare illustrated in Figs. 5(a)-(d) for cases I-IV, respectively. It is observed that: 1. for case I, the proposed and Oustaloup’s controller achieve superior accuracy in the magnitude and phase responses, respectively, at the lower frequency values, whereas their performance is similar in the mid-band region. At the higher frequencies, the proposed controller attains better phase accuracy, whereas the Oustaloup’s approximant yields superior accuracy in magnitude; 2. for case II, the proposed controller’s phase behavior is inferior to Oustaloup’s. However, the proposed approximant attains better proximity to the theoretical magnitude plot over the entire bandwidth; 3. for case III, the magnitude response accuracy of the Oustaloup’s CFOPID approximant is superior to that of the designed one for frequencies in [0.001, 0.019] rad/s. The phase response of the proposed approximation suffers lower deviation in the interval [0.001, 0.64] rad/s, while the Oustaloup’s model achieves better conformity with the theoretical magnitude-frequency behavior. At the higher frequencies, the behavior is similar; 123 S. Mahata et al. Table 6 Comparison of error metrics for the proposed CFOPID controllers with the reported literature (best performance is highlighted in boldface) Case Model Max Absolute Error Mean Absolute Error Magnitude (dB) Phase (◦) Magnitude (dB) Phase (◦) I Oustaloup 34.665 44.338 12.262 8.517 Proposed 49.153 28.412 25.023 12.405 II Oustaloup 20.814 39.216 −0.419 8.143 Proposed −0.795 51.978 −21.040 20.903 III Oustaloup 34.665 36.745 5.992 3.817 Proposed 45.982 3.265 22.728 1.580 IV Oustaloup 34.641 44.357 12.309 7.990 Proposed 45.052 21.782 21.639 11.798 4. for case IV, in the low frequency region, the proposed model yields an inferior phase response. However, the proposed approximant achieves a smaller value for the maximum deviation in phase response beyond 335 rad/s. In Table 6, comparisons about the AME and APE metrics between the proposed and Oustaloup’s CFOPID approximants are presented. Results reveal that: (i) for case I, the Oustaloup’s model yields better performance for maximum and mean AME and mean APE, whereas the proposed design achieves significantly lower maximum APE; (ii) for case II, the magnitude errors are significantly smaller for the proposed design, although the Oustaloup’s model outperforms the proposed one about the phase response errors; (iii) for case III, significant reduction in maximum APE (3.265◦) is obtained by the proposed approximant as compared to the Oustaloup’s approximant (36.745◦), although the Oustaloup’s design yields a better magnitude response accuracy; (iv) for case IV, the proposed model provides markedly improved accuracy over Oustaloup’s for the maximum APE (21.782◦versus 44.357◦) but is outperformed for the other three error indices. Overall, it may be inferred that the proposed approach does not outperform the Oustaloup’s method for both the magnitude and phase responses simultaneously. However, the proposed technique results in real coefficients based real rational transfer functions, which is in contrast to the real coefficients based complex transfer function yielded using the Oustaloup’s method. Therefore, hardware implementation of the proposed controllers can be realized using standard circuit design techniques readily available in the literature. Furthermore, unlike the Oustaloup’s method, there is no restriction on the approximation order (N: even or odd) of the proposed controllers. Among the various techniques available for continuous-time rational approximation of FOTFs, it is not possible to establish which one is the best [46]. While some methods provide better accuracy regarding the frequency response, others may achieve improved time response performance. The approximation performances can also depend on the non-integer order. The real and imaginary part of the complexvalued impulse response of the proposed sixth-order approximant and the Oustaloup’s 123 Optimal approximation of analog... 0 0.005 0.01 0.015 0.02 0.025 0.03 −2.5 −2 −1.5 −1 −0.5 0 0.5 x 106 Time (s) Impulse Response (Real Part) Theoretical Oustaloup’s approximant Proposed approximant Fig. 6 Impulse response (real part) comparison plots between the proposed (N= 6) and Oustaloup’s (N= 12) approximants for the CFOPID controller of case I approximant of twelfth order for the CFOPID controller pertaining to case I are compared in Figs. 6and 7, respectively. It is evident that the Oustaloup’s approximant attains superior impulse responses matching with the theoretical ones as compared to the proposed one. However, the deviation of the impulse response for the proposed approximant diminishes with respect to the theoretical after a short time (about 0.005s). 4 Conclusions An optimal technique that guarantees stable poles, minimum-phase zeros, and realvalued coefficients of rational approximants for the complex fractional-order PID controllers is presented in this paper. The drawbacks of the reported curve-fitting techniques are eliminated through (i) incorporation of constraints, (ii) appropriate selection of the lower bound of decision variables pertaining to the coefficients of the numerator and denominator polynomials of the proposed model, and (iii) utilizing a real-parameter constrained optimization algorithm. The effectiveness of the suggested technique is verified on the Podlubny’s, IMC, and El-Khazali’s forms of CFOPID controllers. Performance comparisons for different orders (odd and even) of approximation are demonstrated for the design examples. Comparisons with the Oustaloup’s CFOPID approximant shows that the proposed approach may achieve better magnitude or phase response fitting, though simultaneous improvements in both are 123 S. Mahata et al. Fig. 7 Impulse response (imaginary part) comparison plots between the proposed (N= 6) and Oustaloup’s (N= 12) approximants for the CFOPID controller of case I not possible. The proposed approach may be considered as an effective alternative to the Oustaloup’s method that results in complex rational approximants with real coefficients; especially, when the bandwidth considerations are limited to 3∼4 decades. For the considered non-linear optimization problem, two limitations of the proposed approach are (i) an inability to guarantee the generation of a global optimal solution, which is in general true when employing a metaheuristic method, and (ii) large dispersion of coefficients (especially for the denominator polynomial) when Nis increased. While this work solely concentrated on the approximation of CFOPID controllers, it will be interesting to consider functions such as the unstable, non-minimum phase, and conjugated-order CFO systems [4,68] in the future. Solving such problems may necessitate incorporation of additional design constraints that may involve employing the multior many-objective constrained evolutionary methods. Funding Open access publishing supported by the National Technical Library in Prague. Conflict of interest The authors declare that they have no conflict of interest. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. 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