Abstract
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.
Highlights
Chaotic behavior exists in many systems, in some cases being problematic where control is needed
E results [1, 2] represent the synchronization as a T-S (Takagi–Sugeno) fuzzy representation. e result [3] achieves robust state estimation influenced by time delay. e result [4] represents that synchronization is achieved by solving LMIs
We summarize the overall design procedure for output tracking control as follows: Step G1: construct T-S fuzzy model for the nonlinear time-delay system as (2)
Summary
Chaotic behavior exists in many systems, in some cases being problematic where control is needed. From a practical point of view, a unified approach to synchronization and control of both discrete-time and continuous chaotic systems considering immeasurable state, time-varying delay, different antecedent variables, and disturbance is needed. Note that feedback without delay methods do not need any information of the delay where approaches are more suitable for practical applications Both stability criteria are represented into linear matrix inequality problems (LMIPs) [7]. E contributions of the proposed approach are (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, mismatched antecedent variable, disturbance in control problem; and (iii) numerically solvable observer and controller gains from LMIs of sufficient conditions derived using Lyapunov’s stability analysis.
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