Abstract
The aim of this work is the design of an adaptive controller based on Mamdani-type fuzzy inference systems. The input control is constructed with saturation functions’ fuzzy-equivalents, which works as the adaptive scheme of the controller. This control law is designed to stabilize the error system to synchronize a pair of chaotic nonhomogeneous piecewise systems. Finally, an illustrative example as numerical evidence is developed.
Highlights
Problem StatementWhere both the master and slave system are piecewise switching systems [33,34,35,36] of the form z_ Az + B,
Among the systems studied in synchronization, the ones that stand out are the chaotic systems; chaotic systems exhibit more complex dynamics, and they must satisfy the conditions according to Devaney’s definition of chaos [20]: (i) sensitive dependence to initial conditions, (ii) dense periodic orbits, and (iii) must be transitive
In [25], the design of a rule-based fuzzy controller for a class of master-slave chaos synchronization is presented; the whole control action is substituted by the fuzzy controller, while Xi et al propose an adaptive robust finite-time control method based on a global sliding surface for the synchronization of a class of chaotic systems in [32]
Summary
Where both the master and slave system are piecewise switching systems [33,34,35,36] of the form z_ Az + B,. That undergo chaotic behaviour, where x, y ∈ R3 are the state variables, Bj (bj, bj2, bj3)T, which are the switching laws of the piecewise chaotic systems, and u ∈ R3 is the actuator in charge of achieve the synchronization of the slave over the master. E objective of this work is to design the controller u as an adaptive controller of the form u u1 u2 ⋮. (1) e equivalence relation between the fuzzy-based controller and the saturation functions (2) Achieve adaptive synchronization between systems (3) Ensure asymptotic stability of the error system via Lyapunov’s theory
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