Adaptive control with guaranteed transient and steady state tracking error bounds for strict feedback systems

Two robust adaptive control schemes for single-input single-output (SISO) strict feedback nonlinear systems possessing unknown nonlinearities, capable of guaranteeing prescribed performance bounds are presented in this paper. The first assumes knowledge of only the signs of the virtual control coefficients, while in the second we relax this assumption by incorporating Nussbaum-type gains, decoupled backstepping and non-integral-type Lyapunov functions. By prescribed performance bounds we mean that the tracking error should converge to an arbitrarily predefined small residual set, with convergence rate no less than a prespecified value, exhibiting a maximum overshoot less than a sufficiently small prespecified constant. A novel output error transformation is introduced to transform the original ''constrained'' (in the sense of the output error restrictions) system into an equivalent ''unconstrained''one. It is proven that the stabilization of the ''unconstrained'' system is sufficient to solve the problem. Both controllers are smooth and successfully overcome the loss of controllability issue. The fact that we are only concerned with the stabilization of the ''unconstrained'' system, severely reduces the complexity of selecting both the control parameters and the regressors in the neural approximators. Simulation studies clarify and verify the approach.