Adaptive suboptimal tracking for the first-order plant with Lipschitz uncertainty

A tracking problem for the first-order discrete-time plant with time-invariant uncertainty of the Lipschitz type and bounded exogenous disturbance is considered in the adaptive setting and two different adaptive laws are proposed. The first law is based on a natural estimation of the unknown linear part of the plant and introduces some conservatism in stability margin and tracking performance. The second adaptive law is suboptimal and includes additionally the estimation of the unknown Lipshitz constant and the unknown norm of exogenous disturbance. The suboptimality is achieved by exploiting a cone estimation algorithm and the linearity of the control criterion with respect to unknown parameters. Both adaptive laws utilize the nonlinear feedback by Xie and Guo based on nonparametric estimation of the uncertainty.