Tracking control for the output using an observer-based
H
∞
fuzzy synchronization of time-varying delayed discrete- and continuous-time chaotic systems is proposed in this paper. First, from a practical point of view, the chaotic systems here consider the influence of time-varying delays, disturbances, and immeasurable states. Then, to facilitate a uniform control design approach for both discrete- and continuous-time chaotic systems, the dynamic models along with time-varying delays and disturbances are reformulated using the T-S (Takagi–Sugeno) fuzzy representation. For control design considering immeasurable states, a fuzzy observer achieves master-slave synchronization. Third, combining both a fuzzy observer for state estimation and a controller (solved from generalized kinematic constraints) output tracking can be achieved. To make the design more practical, we also consider differences of antecedent variables between the plant, observer, and controller. Finally, using Lyapunov’s stability approach, the results are sufficient conditions represented as LMIs (linear matrix inequalities). The contributions of the method proposed are threefold: (i) systemic and unified problem formulation of master-slave synchronization and tracking control for both discrete and continuous chaotic systems; (ii) practical consideration of time-varying delay, immeasurable state, different antecedent variables (of plant, observer, and controller), and disturbance in the control problem; and (iii) sufficient conditions from Lyapunov’s stability analysis represented as LMIs which are numerically solvable observer and controller gains from LMIs. We carry out numerical simulations on a chaotic three-dimensional discrete-time system and continuous-time Chua’s circuit. Satisfactory numerical results further show the validity of the theoretical derivations.
Numerical detection of unstable periodic orbits in continuous-time dynamical systems with chaotic behaviors