Adaptive asymptotic control of multivariable systems based on a one-parameter estimation approach

Multivariable adaptive control is an important and challenging research area, where it usually requires designing adaptive laws to estimate a number of unknown parameters. Hence, this process may excessively consume limitedly available computational resources and large amount of computational time, which may degrade system performances, particularly for the case of online updating of parameter estimates. Thus reducing the number of parameters to be estimated is a promising way to solve the problem. The state-of-the-art result in the area is to update just one adaptive parameter. However, the tracking errors are only ensured bounded and thus the resulting closed-loop system is not asymptotically stable. Hence, a question that arises is whether such a result of non-zero tracking errors is the price paid for reducing the number of updating parameters. Up to now, there is still no answer to this question. In this paper, we address such an issue and realize asymptotic stability with online estimation of just one parameter. To achieve such a goal, a novel dynamic loop gain function based approach is proposed to incorporate with the backstepping control design procedure, which enables us to solve the algebraic loop problem caused by all the existing traditional Nussbaum function based approaches and thus establish system stability. A bound for the tracking error is explicitly established in terms of L2 norm, which helps improve the transient performance by selecting the control parameters. Moreover, guaranteed by the Barbalats Lemma, the tracking error is further ensured to approach zero asymptotically. Finally, simulation examples are conducted to testify the effectiveness of the proposed approach.