The present paper is devoted to the study of a robust adaptive control problem of some uncertain nonlinear systems of the form \[ \begin{aligned} \dot z & =q(z,x,u),\\ \dot x_i & =x_{i+1}+ \theta^T \varphi_i(x_1, \dots,x_i)+ \Delta_i (x,z,u,t),\;1\leq i\leq n-1,\\ \dot x_n &=u+\theta^T \varphi_n(x)+ \Delta_n (x,z,u,t),\\ y & =x_1,\end{aligned} \tag{1} \] where \(x=(x_1, \dots,x_n)^T \in\mathbb{R}^n\) and \(z\in\mathbb{R}^{n_0}\) are comprised of the measured states and the remaining unmeasured states, respectively, \(\theta\in \mathbb{R}^l\) is a vector of unknown constant parameters, the \(\varphi_i\)’s are known smooth functions, and the \(\Delta_i\)’s and \(q\) are unknown Lipschitz continuous functions.
The main result embodied in Theorem 4.1 is proved under the following conditions:
Assumption 1.1. \[ \forall i\in\{1,\dots,n\},\exists p^*_i\in [0,\infty) \ni\bigl |\Delta_i(x,z,u,t) \bigr|\leq p^*_i\Bigl[ \psi_{i1} \biggl(\bigl|(x_1, \dots,x_i) \bigr|\biggr) +\psi_{i2} \bigl(|z|\bigr) \Bigr], \] where \((z,x,u,t) \in\mathbb{R}^{n_0} \times\mathbb{R}^n \times\mathbb{R} \times[0, \infty)\), \(\psi_{i1}, \psi_{i2}\) are some known nonnegative smooth functions.
Assumption 4.1. The \(z\)-system in (1) has an exp-ISpS Lyapunov function in the sense of Definition 3.4 (which is too long to be cited here).
Then Theorem 4.1 says that, under the above-mentioned assumptions, all the solutions of (1) are globally uniformly ultimately bounded. A modified adaptive backstepping design procedure is used in the paper. The derived adaptive controller guarantees the global boundedness property for all signals and states, and it steers the output to a small neighborhood of the origin.
This paper is an extension from the unbiased case to the biased case of an earlier work of the same authors [Proc. IFAC 96 World Congress, San Francisco, Vol. K, pp. 73-78 (1996)].