 
Summary: FiniteTime Control of Uncertain Linear Systems Using Statistical
Learning Methods
C.T. Abdallah # , F. Amato + , M. Ariola + , P. Dorato # , V. Koltchinskii §
Abstract
In this paper we show how some di#cult linear algebra problems can be ``approximately'' solved using
statistical learning methods. We illustrate our results by considering the state and output feedback,
finitetime robust stabilization problems for linear systems subject to timevarying normbounded un
certainties and to unknown disturbances. In the state feedback case, we have obtained in an earlier
paper, a su#cient condition for finitetime stabilization in the presence of timevarying disturbances;
such condition requires the solution of a Linear Matrix Inequality (LMI) feasibility problem, which is by
now a standard application of linear algebraic methods. In the output feedback case, however, we end up
with a Bilinear Matrix Inequality (BMI) problem which we attack by resorting to a statistical approach.
Keywords: FiniteTime Stability, LMIs, Disturbance Rejection, Statistical Learning Control.
1 Introduction
The interplay between linear algebra and linear control theory has been long and fruitful [4]. Until very
recently, it was actually felt that most linear control problems can be solved using linear algebraic concepts,
an opinion further reinforced with the introduction of linear matrix inequality (LMI) methods into control
engineering [12]. However, it is now known that some apparently basic linear control questions do not admit
simple solutions or any at all [5, 6, 13]. Such are the examples of fixedorder controller design, multiobjective
robust control designs, and others. While such problems remain in a linear algebraic framework, more
