Robust Nonlinear


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892 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 44, NO. 10, OCTOBER 1997 the chaotic slave system can be synchronized in this sense to a master system which behaves chaotically, shows limit cycle behavior or shows stable equilibrium behavior. Another example along this direction has been presented in [18], where full state error feedback has been used to synchronize two systems which are different (such as Chua’s circuit and the Lorenz attractor). In this paper, we extend the ideas from the autonomous case to the case where there exists an external input. From the example of Chua’s circuit it will follow that the allowed parameter mismatch is much smaller than for the autonomous case. By using a single transmission signal, the dynamic output feedback case leads to a simpler implementation of the synchronization than the full static state feedback scheme, but the latter has higher performance and a better flexibility for defining keys in a cryptographical scheme [25], [26]. This paper is organized as follows. In Section II, we present master–slave synchronization schemes with full static state error feedback and linear dynamic output error feedback. In Section III, we approach the synchronization problem from the viewpoint of modern control theory, by deriving standard plant representations. We take into account parameter mismatch between the Lur’e systems. In Section IV, we derive Theorems for dissipativity with finite -gain of the synchronization schemes, these conditions being expressed as matrix inequali- ties. In Section V, we formulate the robust nonlinear H syn- chronization problem, based on the Theorems of Section IV. In Section VI, we present an example on Chua’s circuit. Both static state and dynamic output feedback are applied and a comparison is made. Channel noise and parameter mismatch are taken into account in the design. II. SYNCHRONIZATION SCHEME In this section, we consider the master–slave synchro- nization schemes with vector field modulation proposed in [25] and [26], but with parameter mismatch between the systems. A. Full Static State Error Feedback Consider the master–slave synchronization scheme with full static state error feedback for nonidentical master–slave Lur’e systems: (1) with master system , slave system , full static state error feedback controller , and linear filter (Fig. 1). The index refers to the static feedback case. The subsystems have state vectors , and output vectors , , . The message signal is . At the transmitter , a linear transformation is applied to the state vector . The resulting vector is sent along the channel and is corrupted by the disturbance signal or channel noise . At the receiver, full static state error feedback between the output of and is applied with feedback matrix . The nonidentical master–slave Lur’e systems have system matrices , , and , where corresponds to the number of hidden units (if one interprets the Lur’e system as a class of recurrent neural networks [16], [23], [31]). The diagonal nonlinearity is assumed to belong to sector [16], [31]. At the master system the vector field is ...

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