Utilization of fractional order differentiation in nonlinear control methods and signal processing techniques

This thesis basically consists of two main parts which include fractional order nonlinear control applications and signal processing techniques with fractional integro-differantial operators. In the fractional control applications, a fractional version of the sliding mode controller, which is one of the robust controller structures was developed for real time and simulation studies have been done on this sructures. The robustness and stability of systems against various disturbing inputs and noises have been examined. In this context, a fractional sliding mode controller design for unstable time delay systems is discussed. Sliding surface and equivalent control law are created by using fractional derivative operator. System coefficients have been changed and unit step response has been examined. It has been proved by the simulation studies that the designed controller has a robust structure. The proposed controller structure is also applied in real time to control the motion of the twin-rotor multi-input multi-output model helicopter horizontally and vertically. It has been observed that fractional version has better preformence for suppressing the distortions and noises according to the traditional one. Reference model based systems from nonlinear control structures are presented as a new controller by enriching with fractional integro-differential expressions. Both simulations and real-time studies show that the proposed controller provides the desired performance. The position control of the DC servo motor has been investigated using a conventional model reference adaptive controller structure with a fractional integrator. Modifications of the controller structure are provided by using fractional operators in the adaptation rule. The reference position datas of the real time running system have been updated according to the values obtained from the camera. The system output is compared by integer and fractional adaptation rule by changing the learning coefficient. The results showed that the fractional approach yielded better responses. Controlling horizontal and vertical movement of the twin-rotor multi-input multi-output model helicopter has been studied to demonstrate the applicability of the proposed controller structure in different experimental systems. In addition, frequency response methods used for estimating limit cycle parameters using fractional order integro-differential expressions with nonlinear relay feedback systems have been investigated. Various program parts have been developed and implemented for the true analysis. In the section of fractional order signal applications, a fractional version of the integer one in first order Sigma-Delta Modulators was developed and the performance of the signal at the output of the modulator was examined. Asynchronous pulse strings are formed in the system where the hysteresis relay type is used. This system, which exhibits pulse density modulation, provides bipolar digital signal transmission. Theoretical analyzes have shown that the fractional integrator produces adjustable pulse trains. This indicates that the digital signal obtained at the system output is closer to the real signal. Performance measures such as ratios of noise to noise and rugged error and decay rate were measured in the simulation studies and it was observed that the fractional integrator increased the performance of Asynchronous Sigma-Delta Modulator. In addition, fractional digital filters are implemented using embedded programming techniques on low cost microcontroller hardware. First of all, the IIR obtained from the s domain is modeled by using the CFE and Oustaloup methods. It has been converted into discrete-time form using appropriate discretization methods. The filter in the z-domain is then expressed in first-order canonical form using the difference equations to fit the hardware structure and embedded in the ARM-based microcontroller card using the buffer structure. The work was supported by simulation and application results. Finally, a single-bit threshold structure was used along with a fractional low-pass filter for the diagnosis of sleep apnea on ECG signals. By setting the fraction level of the filtration, the slope value of the transition band is determined. Thus, the shift in the transmission band and the change in the frequency of the energy signal of the ECG are found. This increases the sensitivity of the filter output to better observe the change in R-R range. It has been observed that the desired performance has been achieved on actual data.