Implicit function based adaptive control of non-canonical form discrete-time nonlinear systems

This paper presents a new study on adaptive state feedback output tracking control problem for uncertain discrete-time nonlinear systems in a general non-canonical form. Time-advance operations on the output of such systems result in the output dynamics being nonlinearly dependent on the control input and unknown parameters, which leads to three technical issues: implicit relative degree; nonlinearly parameterized uncertainties; and non-affine control input. To address these issues, this paper first employs feedback linearization and implicit function theory to construct a relative degree dependent normal form; then proposes an adaptive parametric reconstruction based method to simultaneously deal with linearly and nonlinearly parameterized uncertainties in the output dynamics; and finally constructs a key implicit function equation to derive a unique adaptive control law which ensures closed-loop stability and asymptotic output tracking. An explicitly iterative solution based adaptive control law is also proposed to ensure closed-loop stability and bounded output tracking within any degree of accuracy. The simulation verifies the effectiveness of the proposed adaptive control method.