New improved fractional order integrators using PSO optimisation

This paper is concerned with the discretization of the fractional-order differentiator and integrator, which is the foundation of the digital realization of fractional order controller. Firstly, the parameterized Al-Alaoui transform is presented as a general generating function with one variable parameter, which can be adjusted to obtain the commonly used generating functions (e.g. Euler operator, Tustin operator and Al-Alaoui operator). However, the following simulation results show that the optimal variable parameters are different for different fractional orders. Then the weighted square integral index about the magtitude and phase is defined as the objective functions to achieve the optimal variable parameter for different fractional orders. Finally, the simulation results demonstrate that there are great differences on the optimal variable parameter for differential and integral operators with different fractional orders, which should be attracting more attentions in the design of digital fractional order controller.

SummaryThis paper presents a new method for designing digital recursive integrators and differentiators by using nonlinear stochastic global optimization based on particle swarm optimization (PSO). A modified PSO is used to optimize the unknown coefficients of second‐order and third‐order recursive integrators in order to obtain a better magnitude response closer to the ideal integrator. The coefficients of the state‐of‐the‐art available filters in the literature are chosen as initial points in the PSO process. By choosing a good starting point, convergence to an optimal solution is greatly facilitated. A dynamic modification of the fitness function used in the PSO process leads to the design of a set of digital integrators each optimized for a specific frequency range. Then, second‐order and third‐order recursive digital differentiators are designed by inverting and stabilizing the transfer functions of the designed recursive integrators. The obtained stabilized differentiators are further optimized using PSO to further improve their performance. The magnitude responses of the designed filters outperform the existing integrators and differentiators. Copyright © 2015 John Wiley & Sons, Ltd.

Design of wideband infinite impulse response (IIR) digital fractional order integrators (DFOIs) based on a bio-inspired metaheuristic optimization approach called the cat swarm optimization (CSO) algorithm is presented in this paper. To investigate the efficiency of the proposed approach, the CSO-based DFOIs are evaluated against those of the approximations designed using real-coded genetic algorithm (RGA), standard particle swarm optimization (PSO), and differential evolution (DE) by different magnitude and phase response error metrics. Simulation results reveal the better frequency response of the CSO-based DFOIs in comparison with the competing designs. Both parametric and nonparametric statistical hypothesis tests validate the performance consistency of CSO. Comparisons with the cited literature confirm the efficacy of the proposed models.

In order to implement fractional order transfer function (FOTF) model in digital systems, transfer function discretization methods are developed to obtain discrete filters that provide acceptable amplitude or phase response approximations to FOTFs in operating frequency ranges. This paper presents an approach to improve discrete IIR filter approximations to FOTFs according to application requirements by using Particle Swarm Optimization (PSO). For this purpose, particles of PSO are initialized around the solution of a well-known analytical approximation method and then, these particles search stable IIR filter solutions in the filter design space to improve magnitude and phase response approximation performance over a desired frequency region. Specifically, PSO algorithm is used to fine-tune results of Tustin recursive discretization method according to a weighed multi-objective cost function. This cost function is expressed as the weighted sum of magnitude and phase response matching objectives with a frequency weighting. The frequency weighting is employed for prioritization of frequency regions in optimization. The proposed method also ensures the stability of optimized IIR filter approximations by enforcing particles to move into stable filter solution regions of the search space. This design approach can contribute to the digital realization of fractional order systems for practical applications.

Pulse density modulation (PDM) can find applications in consumer electronics, power and switch control systems. PDM signals can be generated by asynchronous sigma delta modulator (ASDM) structures and improvement of ASDM performance provides significant contribution for practical applications. This study numerically investigates effects of fractional order integration (FOI) on the PDM signal, produced by ASDM. We implemented fractional order integratorin ASDM structure and present numerical analysis to discuss effects of fractional order integrator on PDM signals, obtained for bipolar digital signal transmission application. Theoretical analysis indicates that FOI provides finer pulse trains and thus improves representation of original signal. In simulations, we tested FOI for various fractional-order values. Signal to noise ratio and signal to noise and distortion ratio (SINAD) levels of demodulated signals are reported for sinusoidal and audio test signals. Simulation results demonstrate that FOI can improve SNR and SINAD performance of ASDMs.

We propose in this article a procedure for testing unit and fractional orders of integration, with the roots simultaneously occurring in the trend, the seasonal and the cyclical component of the time series. The tests have standard null and local limit distributions. However, finite sample critical values are computed, and several Monte Carlo experiments conducted across the paper show that the rejection frequencies against unit (and fractional) orders of integration are relatively high in all cases. The tests are applied to the UK consumption and income series, the results showing the importance of the roots corresponding to the trend and the seasonal components and, though the unit roots are found to be fairly suitable models, we show that fractional processes (including one for the cyclical component) may also be plausible alternatives in some cases.

Digital filters play a major role in signal processing that are employed in many applications such as in control systems, audio or video processing systems, noise reduction applications and different systems for communication. In this regard, FIR filters are employed because of frequency stability and linearity in their phase response. FIR filter design requires multi-modal optimization problems. Therefore, PSO (Particle Swarm Optimization) algorithm and its variants are more adaptable techniques based upon particles’ population in the search space and a great option for designing FIR filter. PSO and its different variants improve the solution characteristic by providing a unique approach for updating the velocity and position of the swarm. An optimized set of filter coefficient is produced by PSO and its variant algorithms which gives the optimized results in passband and stopband. In this research paper, Digital FIR filter is effectively designed by using PSO Algorithm and its two variants ARPSO and CRPSO in MATLAB. The outcomes prove that the filter design technique using CRPSO is better than filter design by PM algorithm. PSO and ARPSO Algorithms in the context of frequency spectrum and RMS error.

Complex system design problems tend to be high dimensional and nonlinear, and also often involve multiple objectives and mixed-integer variables. Heuristic optimization algorithms have the potential to address the typical (if not most) characteristics of such complex problems. Among them, the Particle Swarm Optimization (PSO) algorithm has gained significant popularity due to its maturity and fast convergence abilities. This paper seeks to translate the unique benefits of PSO from solving typical continuous single-objective optimization problems to solving multi-objective mixed-discrete problems, which is a relatively new ground for PSO application. The previously developed Mixed-Discrete Particle Swarm Optimization (MDPSO) algorithm, which includes an exclusive diversity preservation technique to prevent premature particle clustering, has been shown to be a powerful single-objective solver for highly constrained MINLP problems. In this paper, we make fundamental advancements to the MDPSO algorithm, enabling it to solve challenging multi-objective problems with mixed-discrete design variables. In the velocity update equation, the explorative term is modified to point towards the non-dominated solution that is the closest to the corresponding particle (at any iteration). The fractional domain in the diversity preservation technique, which was previously defined in terms of a single global leader, is now applied to multiple global leaders in the intermediate Pareto front. The multi-objective MDPSO (MO-MDPSO) algorithm is tested using a suite of diverse benchmark problems and a disc-brake design problem. To illustrate the advantages of the new MO-MDPSO algorithm, the results are compared with those given by the popular Elitist Non-dominated Sorting Genetic Algorithm-II (NSGA-II).

International audience

At a macroscopic level derivatives and integrals are the usual mathematical tools to model real time processes and to perform the basic control and signal processing actions. However, the analysis, design, synthesis and implementation of fractional order differentiator and integrator is a difficult task because of its irrational behaviour. Therefore, for mathematical evaluation of any fractional order system, conversion to its approximate integer order equivalent is essential. In this paper approximated integer order models of fractional differentiator and integrator are developed using the continued fraction expansion technique. A continued fraction is an expression obtained through an iterative process. For any iteration to terminate, a finite numerical value is assigned, which in this paper is equal to the number of frequency points within the desired frequency band. It includes both the lower and upper limit values. A set of coefficients are obtained by finding the gains of the fractional term at respective frequencies and thereby applying the recursive formula. The coefficients thus obtained are substituted in the expression of continued fraction which results in a polynomial function of finite order. The developed models can be directly applied for analysis and realization of fractional order systems. The models are developed for fractional terms 0.1 to 0.9 in steps of 0.1, and also for 0.25 and 0.75. A detailed discussion on the sensitivity analysis is presented, which includes the influence of variable parameters on the accuracy and length of the order. Simulations have been performed in MATLAB. A comparison with both, the ideal values and also with existing methods is performed and tabulated to validate the correctness of the developed models both in terms of accuracy and integer order of the model. It shows that the Matsuda method yield very good results both in terms of magnitude and phase. And, is most suitable for linear phase circuits. Also, the proposed models can be directly used for the realization of customized fractional order Proportional Integral (PI), Proportional-Derivative (PD) and PID controllers. To establish the correctness of CFE based technique for hardware realization, the integer order approximated model of one-tenth and seven-tenth differentiator is decomposed to obtain the circuit parameters resistor (R) and capacitor (C). Then its implementation in OrCAD Capture CIS is performed. It can be seen that the results of realization closely match the actual response.

For audio testing, a pink noise generator is a very useful and widely used tool for equalization of loudspeaker systems, room acoustics and crossovers. It can be built as a digital, analog or mixed circuit. The pink noise generator consists of a white noise source followed by a fractional order integrator (FOI) with a roll-off of 10 dB/decade. In this way, the obtained output power is constant in each decade, which is essential for testing audio devices within the entire audio frequency band. In this paper, an active realization of an FOI with an order 1/2 is presented, using an active-RC circuit that uses an operational amplifier as the active element. To build an FOI, we can use different approximation methods. We use our recently published Minimax method (MMX), with unequally (optimally) spaced singularities and compare it with a method proposed by other authors. The latter intuitively considers optimal singularities equally spaced on a logarithmic scale. We demonstrate the superiority of our proposed MMX approach.

Parameter identification of fractional order linear system based on Haar wavelet operational matrix

In this paper, a fractional order LQG benchmark is proposed for the control performance assessment of fractional order control systems. Similar to the conventional LQG benchmark, the fractional order LQG performance benchmark curve is determined by the numerical calculation method, which avoids the calculation of the complex interaction matrix. The fractional order process model is discretized via fractional order calculus. Meanwhile, the fractional order integral is introduced into the conventional LQG cost function. Then solving the linear quadratic Gaussian problem under the fractional order model and fractional control, the optimal input and output variances are determined for different weighting factors and the performance curves can be achieved. The comparison between fractional order LQG and the conventional LQG shows the improvement of the proposed benchmark under the same condition. The proposed benchmark can provide a more direct and superior reference standard to evaluate the performance of fractional order control system. Finally, a case study of fractional order PID(FO-PID) controller in industrial heating furnace temperature control experiment with model matching and model mismatch conditions is used to verify the effectiveness of the proposed benchmark.

In this study, we present an effective R-wave detection method in the QRS complex of the electrocardiogram (ECG) based on digital differentiation and integration of fractional order. The detection algorithm is performed in two steps. The pre-processing step is based on a fractional order digital band-pass filter whose fractional order is obtained by maximising the signal to noise ratio of the ECG signal, followed by a five points differentiator of fractional order 1.5 then the squaring transformation and the smoothing are used to generate peaks corresponding to the ECG parts with high slopes. The detection step is a new and simple strategy which is also based on fractional order operators for the localisation of the R waves. The MIT/BIH arrhythmia database is used to test the effectiveness of the proposed method. The algorithm has provided very good performance and has achieved about 99.86% of the detection rate for the standard database. The results obtained are presented, discussed and compared to the most recent and efficient R-wave detection algorithms.

Inflation targeting is a common monetary policy regime. Inflation targets are often flexible in the sense that the central bank allows inflation to temporarily deviate from the target to avoid causing unnecessary volatility in the real economy. In this paper, we propose modeling the degree of flexibility using an autoregressive fractionally integrated moving average (ARFIMA) model. Assuming that the central bank controls the long-run inflation rate, the fractional integration order becomes a measure of how flexible the inflation target is. A higher integration order implies that inflation deviates from the target for longer periods of time and consequently, that the target is flexible. Several estimators of the fractional integration order have been proposed in the literature. Grassi and Magistris (2014) show that a state-based maximum likelihood estimator is superior to other estimators, but our simulations show that their finding is over-biased for a nearly non-stationary time series. To resolve this issue, we first proposed a Bayesian Monte Carlo Markov Chain (MCMC) estimator for fractional integration parameters. This estimator resolves the problem of over-bias. We estimate the fractional integration order for 6 countries for the period 1993M1 to 2017M9. We found that inflation was integrated to an order of 0.8 to 0.9 indicating that the inflation targets are implemented with a high degree of flexibility.