The two degree of freedom ball balancing table (BBT) is a well-known didactic tool used to evaluate the effectiveness and performances of many control algorithms for dynamic systems. The present paper proposes to control the ball position of the BBT system via a linear feedback controller based on a functional observer. The parameters of the linear functional observer are determined by applying the direct method which requires neither a Sylvester equation resolution nor canonical transformations. The use of a digital controller has motivated the elaboration of the equations in the discrete time case. In this work, the BBT is tested in real-time to evaluate the proposed controller performances when stabilizing a ball on a reference point. This paper is a continuity of the previous work [12], in which only simulation results have been carried out.

In this paper, the direct approach which was introduced for the first time by [Rotella and Zambettakis, 2011] in designing minimal functional state observers is extended to deal with discrete-time systems. One of the benefits of this approach is that it does not require solving the Sylvester matrix equation that appears in other observer design procedures. Both stable observers and arbitrary fixed poles observers problems are considered for minimality. A numerical example and simulation results explain the effectiveness and the benefits of the proposed algorithm.

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In this article, an algorithm for designing functional state observers for linear discrete time-varying systems using the direct approach is proposed. Both the single function and the multiple functions of the states are studied. The design method is simple, and it is based neither on the solution of the generalised Sylvester equation that appears in the existence conditions nor on canonical system transformations. This algorithm is explained through a detailed example.