{'fraction_non_alphanumeric': 0.013266239707227814, 'fraction_numerical': 0.0, 'mean_word_length': 5.143258426966292, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Synchronization of oscillations for coupled Van der Pol oscillators by output

A number of automatic control tasks, in particular, the synchronization of trajectories, the tracking task, control by a reference system are associated with the synthesis of control algorithms for dynamic cascade systems, which are a set of interconnected active subsystems. In this paper, the oscillation synchronization problem is considered for two Van der Pol coupled oscillators. It is assumed that the driven subsystem depends on the external control action, in addition, the phase vector is not fully known. On the first step the solution of the problem of synchronization in the form of state feedback is written. The aim of the work is to find the synchronizing control in the form of feedback on the state estimation. Such a formulation is relevant, since for many practical applications of control theory, a typical situation is when the complete state vector of the system is unknown and only some of the functions of the state variables - the outputs of the system are accessible to measurement. One can try to use the control law obtained from feedback by replacing the state with its estimate obtained by observer - a special dynamical system whose state eventually approaches (asymptotically or exponentially) to the state of the original system. In this case a question arises whether such control will be solving the synchronization problem. In mathematical control theory, in particular for the stabilization problem of dynamical systems, similar questions constitute the content of the known principle of separation. For the observation problem solving the apparatus of the method of synthesis of auxiliary invariant relations for constructing a nonlinear observer was used. In accordance with this approach a nonlinear observer is constructed for the system under consideration, which ensures the exponential estimates of the phase vector. It is further shown that the use in the control law instead of the state of the system of its evaluation under simultaneously solving the problems of observation and synchronization leads to the local solution of the problem under consideration.