SOME NEW GENERALIZATIONS OF HADAMARD–TYPE MIDPOINT INEQUALITIES INVOLVING FRACTIONAL INTEGRALS

In this study, we formulate the identity and obtain some generalized inequalities of the Hermite–Hadamard type by using fractional Riemann–Liouville integrals for functions whose absolute values of the second derivatives are convex. The results are obtained by uniformly dividing a segment [a,b] into n equal sub-intervals. Using this approach, the absolute error of a Midpoint inequality is shown to decrease approximately n^2 times. A dependency between accuracy of the absolute error (ε) of the upper limit of the Hadamard inequality and the number (n) of lower intervals is obtained

Further results on fractional order control of a mechatronic system

Ovaj rad predstavlja jedan novi algoritam PID upravljanja necelobrojnog reda zasnovani na genetskim algoritmima (GA) u zadatku pozicioniranja robotskog sistema sa tri stepena slobode pogonjen jednosmernim motorima. Urađena su optimalna podešavanja parametara FOPID kontrolera kao i IOPID kontrolera, primenom GA pristupa za date FOPID/IOPID kontrolere na uporedni način. Efektivnost predloženog optimalnog FOPID upravljanja je demonstrirano na datom robotskom sistemu kao jednim ilustrativnim primerom. Takođe, u preostalom delu rada prezentovano je projektovanje naprednog algoritma FOPID upravljanja podešavanog primenom GA i primena u upravljanju proizvodnjom tehničkih gasova, tj. kriogenog procesa separacije vazduha. Zatim je izvedeni model linearizovan i raspregnut i gde su zatim primenjeni IOPID i FOPID kontroleri. Na sličan način, skup optimalnih parametara datih kontrolera su dobijeni primenom GA optimizacione procedure minimizujući predloženi kriterijum optimalnosti. Konačno, koristeći rezultate simulacije u vremenskom domenu pokazano je da FOPID kontroler poboljšava odgovor sistema u prelaznom režimu i obezbeđuje više robusnosti u poređenju sa klasičnim IOPID kontrolerom.This paper presents a new algorithm of the fractional order PID (FOPID) control based on genetic algorithms (GA) in the position control of a 3 DOF's robotic system driven by DC motors. The optimal settings for a FOPID controller as well as an integer order PID controller (IOPID) are done, applying the GA tuning approach and their extension for FOPID-IOPID controllers in a comparative manner. The effectiveness of the suggested optimal FOPID control is demonstrated with a given robotic system as an illustrative example. The rest of the paper presents the design of an advanced algorithm of the FOPID control tuned by GA and the application in the control of the production of technical gases, i.e. in the cryogenic air separation process. Then, the obtained model is linearized and decoupled and consequently IOPID and FOPID controllers are applied. In the same manner, a set of optimal parameters of these controllers is achieved through the GA optimization procedure through minimizing the proposed cost function. Finally, the use of the simulation results in the time domain has shown that the FOPID controller improves a transient response and provides more robustness than a conventional IOPID