## An adaptive $$H^ \infty$$ tracking control for a class of nonlinear multiple-input-multiple-output (MIMO) systems.(English)Zbl 1048.93024

This note addresses the problem of designing robust tracking controls for a class of nonlinear MIMO systems involving parametric uncertainties, unmodelled dynamics and external disturbances, represented by the following set of differential equations $y_i^{(n_i)}=f_i(x)+\sum_{j=1}^pg_{ij}(x)u_j+d_j,\quad i=1,\dots,p,$ where $$u\in\mathbb R^p$$ is the control input, $$y\in\mathbb R^p$$ is the output, $$d \in\mathbb R^p$$ is an external disturbance, $$y_i^{(j)}\in\mathbb R^p$$ is the $$j$$th derivative of $$y_i$$ and $$n_1,\dots,n_p$$ are positive integers.
The functions $$f_i(x)$$ and $$g_{ij}(x)$$ for $$i,j=1,\dots,p$$ are unknown smooth nonlinearities which could depend on $$y_i$$, $$y_1^{(1)},\dots,y_i^{(n_i-1)}$$. Hybrid adaptive-robust $$H^\infty$$ tracking control schemes are developed to guarantee a transient and asymptotical output tracking performance in the sense that all the signals and states of the closed-loop system are bounded, the tracking error is uniformly ultimately bounded and an $$H^\infty$$ tracking performance is achieved. The $$H^\infty$$ tracking control relies only on the solution to a modified algebraic Riccati-like matrix equation and so the developed control law can easily be implemented. Consequently, compared with the existing $$H^\infty$$ tracking control scheme and the robust/adaptive control scheme, the developed adaptive-robust $$H^\infty$$ tracking control design can be extended to handle a broader class of uncertain nonlinear MIMO systems.

### MSC:

 93B36 $$H^\infty$$-control 93B51 Design techniques (robust design, computer-aided design, etc.) 93C10 Nonlinear systems in control theory 93C40 Adaptive control/observation systems
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