HYBRID APPROXIMATION METHOD FOR TIME RESPONSE IMPROVEMENT OF CFE BASED APPROXIMATE FRACTIONAL ORDER DERIVATIVE MODELS BY USING GRADIENT DESCENT ALGORITHM

Due to its high computational complexity, fractional order (FO) derivative operators have been widely implemented by using rational transfer function approximation methods. Since these methods commonly utilize frequency domain approximation techniques, their time responses may not be prominent for time-domain solutions. Therefore, time response improvements for the approximate FO derivative models can contribute to real-world performance of FO applications. Recent works address the hybrid use of popular frequency-domain approximation methods and time-domain approximation methods to deal with time response performance problems. In this context, this study presents a hybrid approach that implements Continued Fraction Expansion (CFE) method as frequency domain approximation and applies the gradient descent optimization (GDO) for step response improvement of the CFE-based approximate model of FO derivative operators. It was observed that GDO can fine-tune coefficients of CFE-based rational transfer function models, and this hybrid use can significantly improve step and impulse responses of CFE-based approximate models of derivative operators. Besides, we demonstrate analog circuit realization of this optimized transfer function model of the FO derivative element according to the sum of low pass active filters in Multisim and Matlab simulation environments. Performance improvements of hybrid CFE-GDO approximation method were demonstrated in comparison with the stand-alone CFE method

A Robust Fuzzy Fractional Order PID Design Based On Multi-Objective Optimization For Rehabilitation Device Control

In this context, Fuzzy Fractional Order Proportional Integral Derivative (FOPID-FLC) controllers are emerged as efficient approaches due to their flexibility and ability to handle nonlinearities and uncertainties. This paper proposes the use of a FOPID-FLC controller for a two-degree-of-freedom (2-DOF) lower limb exoskeleton. Our proposal is based on an enhanced control approach that combines fuzzy logic advantages and fractional calculus benefits. Contrary to popular existing methods, that use the FLC to tune the FOPID  parameters, the FLC in this work is used to generate the system torque depending on patient morphology. Indeed, our fundamental contribution is to design and implement an enhanced FOPID-FLC that achieves an adequate optimal control based on system rules composed of optimal torques and input data. The fractional calculus is approximated using successive first order filters. Next, a multi-objective optimization is established for the tuning of each FOPID parameters. Finally, the FLC is used to adjust the torque depending on the kid's age. The effectiveness of the proposed controller in various scenarios is validated based on numerical simulations. Extensive analyses prove that the FOPID-FLC outperforms the FOPID with a 90\% of improvement in terms of error performance indices and 20\% of improvement for the control action. Moreover, the controller exhibits improved robustness against uncertainties and disturbances encountered in rehabilitation environments

Frequency Response Based Curve Fitting Approximation of Fractional–Order PID Controllers

Fractional-order PID (FOPID) controllers have been used extensively in many control applications to achieve robust control performance. To implement these controllers, curve fitting approximation techniques are widely employed to obtain integer-order approximation of FOPID. The most popular and widely used approximation techniques include the Oustaloup, Matsuda and Cheraff approaches. However, these methods are unable to achieve the best approximation due to the limitation in the desired frequency range. Thus, this paper proposes a simple curve fitting based integer-order approximation method for a fractional-order integrator/differentiator using frequency response. The advantage of this technique is that it is simple and can fit the entire desired frequency range. Simulation results in the frequency domain show that the proposed approach produces better parameter approximation for the desired frequency range compared with the Oustaloup, refined Oustaloup and Matsuda techniques. Furthermore, time domain and stability analyses also validate the frequency domain results

Frequency Response Based Curve Fitting Approximation of Fractional–Order PID Controllers

Fractional-order PID (FOPID) controllers have been used extensively in many control applications to achieve robust control performance. To implement these controllers, curve fitting approximation techniques are widely employed to obtain integer-order approximation of FOPID. The most popular and widely used approximation techniques include the Oustaloup, Matsuda and Cheraff approaches. However, these methods are unable to achieve the best approximation due to the limitation in the desired frequency range. Thus, this paper proposes a simple curve fitting based integer-order approximation method for a fractional-order integrator/differentiator using frequency response. The advantage of this technique is that it is simple and can fit the entire desired frequency range. Simulation results in the frequency domain show that the proposed approach produces better parameter approximation for the desired frequency range compared with the Oustaloup, refined Oustaloup and Matsuda techniques. Furthermore, time domain and stability analyses also validate the frequency domain results