# zbMATH — the first resource for mathematics

Guaranteeing ultimate boundedness and exponential rate of convergence for a class of nominally linear uncertain systems. (English) Zbl 0714.93013
We consider a system that can be appropriately described by a state equation of the form $$\dot x(t)=Ax(t)+Bu(t)+Ba(x(t),u(t),t),x(t_ 0)=x_ 0$$, where $$x\in {\mathbb{R}}^ n$$ is the state, $$u\in {\mathbb{R}}^ m$$ is the control variable, a: $${\mathbb{R}}^ n\times {\mathbb{R}}^ m\times {\mathbb{R}}\to {\mathbb{R}}^ m$$ is a continuous (possibly nonlinear) vector, A and B are matrices of appropriate dimensions, and rank $$(B)=m$$. This system is termed uncertain because the information available to the designer about function a($$\cdot,\cdot,\cdot)$$ is not complete; it is only known that its structure is “cone bounded”.
Generally one must design a linear controller that makes the nominal system asymptotically stable and, successively, design a nonlinear feedback capable of driving the trajectories of the system into a suitable neighborhood of the origin regardless of the realization of the uncertainties. In this paper we have focused our attention on how to combine the linear and the nonlinear part of the design so as to guarantee not only ultimate boundedness but also a convergence rate as close as desired to that of the nominal controller system. We showed that, even though the rate of convergence to the set of ultimate boundedness is generally slower than the nominal rate, there exists a final attractor of the trajectories toward which the nominal rate is assured.

##### MSC:
 93B50 Synthesis problems 93C41 Control/observation systems with incomplete information 93C05 Linear systems in control theory 93C15 Control/observation systems governed by ordinary differential equations 93C35 Multivariable systems, multidimensional control systems
##### Keywords:
cone bounded; ultimate boundedness
Full Text: