Robust mixed order backstepping control of non-linear systems

Robust backstepping control of non-linear systems with derivation orders (commensurate or non-commensurate) lying at interval (0, 2) is proposed in this study. The stability and robustness properties are proved using a linearisation procedure in contrast to the classic recursive Lyapunov approach to backstepping. Vector, adaptive and robustness extensions are then developed to solve general tracking and stabilisation control problems. The adaptive extension is proved with a non-Lyapunov approach based on inequalities on non-negative fractional integrals. The mixed-order nature enables to control integer order systems using fractional operators. Through simulations, it is shown that the performance of the controllers depends on their derivation orders. In particular, for transient behaviour and the root mean square value of the control signal, the fractional controllers proposed exhibit improved responses as compared with the integer ones.