By Luis T. Aguilar, Igor Boiko, Leonid Fridman, Rafael Iriarte
This monograph offers an easy and effective two-relay regulate set of rules for new release of self-excited oscillations of a wanted amplitude and frequency in dynamic structures. built via the authors, the two-relay controller includes relays switched through the suggestions got from a linear or nonlinear method, and represents a brand new method of the self-generation of periodic motions in underactuated mechanical systems.
The first a part of the ebook explains the layout methods for two-relay keep an eye on utilizing 3 diverse methodologies – the describing-function procedure, Poincaré maps, and the locus-of-a perturbed-relay-system procedure – and concludes with balance research of designed periodic oscillations.
Two ways to make sure the robustness of two-relay regulate algorithms are explored within the moment half, one in accordance with the mix of the high-order sliding mode controller and backstepping, and the opposite on higher-order sliding-modes-based reconstruction of uncertainties and their reimbursement the place Lyapunov-based balance research of monitoring mistakes is used. ultimately, the 3rd half illustrates purposes of self-oscillation iteration through a two-relay regulate with a Furuta pendulum, wheel pendulum, 3-DOF underactuated robotic, 3-DOF laboratory helicopter, and fixed-phase digital circuits.
Self-Oscillations in Dynamic Systems will attract engineers, researchers, and graduate scholars engaged on the monitoring and self-generation of periodic movement of electromechanical platforms, together with non-minimum-phase platforms. it is going to even be of curiosity to mathematicians engaged on research of periodic solutions.
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Additional info for Self-Oscillations in Dynamic Systems: A New Methodology via Two-Relay Controllers
2) depends on the amplitude value only. / is the complex frequency response characteristic (Nyquist plot) of the plant. 7) is equivalent to the condition of the complex frequency response characteristic of the open-loop system intersecting the real axis in the point . 1; j0/. 8) is given in Fig. 2. The function 1=N is a straight line the slope of which depends on c2 =c1 ratio. 9) provide an approximate solution of frequency and amplitude. In this book we consider the inverse problem, that is finding the TRC gains c1 and c2 from the desired amplitude A1 and frequency ˝.
1) driven by the two-relay controller, is determined by equation kC1 D k D p . 6) and using the principle of symmetry, one can write the following: p D p . For the T-periodic (symmetric) solution, we will use the following notation: t1 D t3 D Â1 ; t2 D t4 D Â2 D T=2 Â1 . t1 I t2 /, respectively, since yP < 0 initially and y monotonically decreases from zero and cannot cross zero before yP changes sign at t D t1 . This condition can be easily verified after parameters Â1 and Â2 are determined. 42 3 Poincaré Map-Based Design We need now to formalize the condition ensuring transition from S2 to S3 without leaving R2 .
1 ; ? 26) holds. 14) has an orbitally asymptotically stable limit cycle with a desired period 2 =˝ and amplitude A1 . 29) where p1 D q1 C J1 1 J2 q2 and x1 D J1 qP 1 C J2 qP 2 C Kp1 . x01 ; x03 ; c1 ; c2 /. x01 /? x03 /? D Â where T D Tp Tsw . To complete the design, one needs to provide the set of equations to find c1 and c2 in terms of the known parameters Tp and x01 . x01 /? x03 /? , verify if the following condition holds: ˇ ˇ C 0 ? 0 ? x3 / ; c1 ; c2 / ˇ ˇ ˇ 0? 0? x / @ 2 1 @ 6 6 6 6@ 6 4 ˇ ˇ ˇ ˇ 0?