We study the dynamics of a ring of unidirectionally coupled autonomous Duffing oscillators. Starting from a situation where the individual oscillator without coupling has only trivial equilibrium dynamics, the coupling induces complicated transitions to periodic, quasiperiodic, chaotic, and hyperchaotic behavior. We study these transitions in detail for small and large numbers of oscillators. Particular attention is paid to the role of unstable periodic solutions for the appearance of chaotic rotating waves, spatiotemporal structures, and the Eckhaus effect for a large number of oscillators. Our analytical and numerical results are confirmed by a simple experiment based on the electronic implementation of coupled Duffing oscillators.

In numerous electric drive applications, the mechanical phenomena in the velocity or position control loop determine real difficulties and challenges for the control system. So-called two-mass drive systems with a flexible shaft are the most important example of this situation. The problem becomes even more difficult if the characteristics of torque transmission along the shaft are nonlinear, nonlinear friction is present, and the plant parameters are unknown, as it happens in numerous robotic systems. A novel adaptive controller is derived for such a system. The recurrent design procedure is based on proper modifications of the adaptive backstepping scheme, including non-strict-feedback plant application, tuning functions to exclude controller overparameterization, robust adaptive laws, proper means to avoid controller complexity explosion, and a nonlinear PI controller in the initial loop to minimize quasi-steady-state tracking error. The closed-loop system uniform ultimate boundedness is proven using Lyapunov techniques and the design and tuning procedures are described. The attractive features of the obtained drive, including the robustness against the violation of assumptions, are presented using several examples.

In this study, the problem of master–slave synchronization of two different chaotic systems is considered and solved under a novel set of assumptions. The mathematical model of each of them contains unknown, constant parameters. Only a single output of the master system is available, and only a single input of the slave system is a control input. The proposed, novel approach is based on the active cooperation of the adaptive observer of the master system and adaptive controller of the slave. The tuning function technique is included in the observer–controller design to avoid overparameterization. Complexity explosion and unacceptable increases in adaptive parameters are prevented by proper adaptive techniques application. Due to the selected observer type, the derivation is restricted to the defined class of master systems—output-nonlinear parametric (ONP) systems. Linear transformation of several popular chaotic systems (e.g., Arneodo, Arneodo–Coullet, Genesio–Tesi, Lur’e) into the ONP form is discussed. The stability of the whole, closed-loop system is derived using Lyapunov techniques and examples of implementation (synchronization of Arneodo and 3D jerk systems) are provided.

Precise motion control remains one of the most important problems in modern technology. It is especially difficult in the case of two-mass systems with flexible coupling if only the motor position and velocity are measured. We propose a new methodology of control system design in this situation. The concept is founded on a robust observer design, based on a linear matrix inequality (LMI) solution. The observer cooperates with the original nonlinear controller. The presented approach allows us to solve the position tracking problem for a two-mass drive, with unknown parameters, in the presence of disturbances (for instance, nonlinear friction-like torques) acting on both ends of the flexible shaft. Under this set of assumptions, the problem was never solved previously. The closed-loop system stability is investigated, and the uniform ultimate boundedness of state estimation errors and tracking errors is proven using Lyapunov techniques. Numerical properties of the design procedure and characteristic features of the observer, controller, and closed-loop system are demonstrated by several examples.

We consider a two-mass drive with a flexible joint with a nonlinear characteristic of the transmitted torque as a function of the torsion angle. We propose a new, nonlinear, adaptive position-tracking controller, taking this nonlinearity of stiffness into account. The derivation of the controller is based on nonlinear adaptive control theory, incorporates several non-standard mathematical techniques and provides a proof of the uniform ultimate boundedness of tracking errors. As the result, we present a controller that solves the position tracking problem, attenuates dangerous tortional oscillations in the shaft and operates correctly in the presence of unknown torques acting on both sides of the joint, even if all plant parameters are unknown. We demonstrate experimentally that using some materials indeed introduces a nonlinear characteristic of the joint. We prove via real plant experiments that the proposed control algorithm is easily implementable with a DSP controller in real-world applications.