Finite-Time Trajectory Tracking of Second-Order Systems Using Acceleration Feedback Only

In this paper, an algebraic approach for the finite-time feedback control problem is provided for second-order systems where only the second-order derivative of the controlled variable is measured. In practice, it means that the acceleration is the only variable that can be used for feedback purposes. This problem appears in many mechanical systems such as positioning systems and force-position controllers in robotic systems and aerospace applications. Based on an algebraic approach, an on-line algebraic estimator is developed in order to estimate in finite time the unmeasured position and velocity variables. The obtained expressions depend solely on iterated integrals of the measured acceleration output and of the control input. The approach is shown to be robust to noisy measurements and it has the advantage to provide on-line finite-time (or non-asymptotic) state estimations. Based on these estimations, a quasi-homogeneous second-order sliding mode tracking control law including estimated position error integrals is designed illustrating the possibilities of finite-time acceleration feedback via algebraic state estimation.

ALGORITHM FOR SELF-TUNING THE PID CONTROLLER

An algorithm for self-tuning the PID controller to the second order systems is proposed in this paper. The proposed self-tuning procedure was developed according to the maximum stability degree criterion, the criterion that permits to achieve the high stability degree, good performance and robustness of the system. According to the proposed algorithm, the controller can be tuned according to the parameters that characterize the process and they can be determinate from the experimental response of the open loop system. To demonstrate the efficiency of proposed procedure of self-tuning the PID controller, the computer simulation was performed and the obtained results were compared with Haeri’s method, maximum stability degree method with iterations and parametrical optimization method. According to the developed algorithm, it was performed the control of the thermal regime in the oven.

On the splitting type of holomorphic vector bundles induced from regular systems of differential equation

We calculate the splitting type of holomorphic vector bundles on the Riemann sphere induced by a Fuchsian system of differential equations. Using this technique, we indicate the relationship between Hölder continuous matrix functions and a moduli space of vector bundles on the Riemann sphere. For second order systems with three singular points we give a complete characterization of the corresponding vector bundles by the invariants of Fuchsian system.

Robust flocking for non-identical second-order nonlinear multi-agent systems

This paper investigates the robust flocking problem for second-order nonlinear systems with a leader and external disturbances. In contrast with most of second-order systems in the literature, the intrinsic dynamics here are nonlinear and non-identical that depend not only on the velocity but also on the position, which is more realistic. Moreover, the interaction topology is undirected and switching. Provided that the leader’s velocity may be constant or time-varying, two distributed flocking control laws have been proposed for two cases to make the differences of the velocities between all followers and the leader approach to zero asymptotically. The proposed distributed flocking control laws are both model-independent which results in the effectiveness of the controllers to cope with the different intrinsic dynamics of the followers and the leader under some assumptions on boundedness of several states. An example is given to illustrate the validity of the theoretical results.

Global Bifurcation of Periodic Solutions in Symmetric Reversible Second Order Systems with Delays

Global bifurcation and spatio-temporal patterns of periodic solutions (with prescribed period) to second order reversible equivariant autonomous systems with commensurate delays are studied using the Brouwer/Leray–Schauder [Formula: see text]-equivariant degree theory. Here, [Formula: see text] is related to the reversal symmetry combined with the autonomous form of the system, [Formula: see text] reflects extra spacial symmetries of the system, and [Formula: see text] is related to the oddness of the right-hand side. Abstract results are supported by a concrete example with [Formula: see text] — the dihedral group of order 12.