Contraction theory-based recursive design of stabilising controller for a class of non-linear systems

The study addresses stabilisation problem of a class of non-linear systems using contraction principle. For this purpose, contraction-based systematic design of control function is presented. Generally, Lyapunov stability-based backstepping technique is widely used to design controllers for strict feedback class of non-linear systems in a recursive manner. However, the present study highlights the usage of contraction-based concepts for stability analysis and presents a systematic procedure to select a single controller for stabilisation of systems having dynamics in strict feedback form. The approach establishes exponential stability of system states. The procedure helps in identifying coordinate transformation to establish contracting nature of the given system. Design of output tracking controller for single-link manipulator system with actuator dynamics is considered to highlight the procedure. The proposed methodology is extended to address synchronisation problem of non-linear systems. As a particular case, synchronisation problem of chaotic systems belonging to the addressed class of systems is explored. Systems to be synchronised are assumed to be connected in a chain through one way coupling. The general scalar coupling function is obtained by utilising the proposed algorithm and stability results are established using partial contraction theory. Numerical simulations are presented at appropriate places to verify the efficacy of the proposed approach.