In this paper we consider the problem of diagonal stability of interval systems. We investigate the existence and evaluation of a common diagonal solution to the Lyapunov and Stein matrix inequalities for third order interval systems. We show that these problems are equivalent to minimax problem with polynomial goal functions. We suggest an interesting approach to solve the corresponding game problems. This approach uses the opennes property of the set of solutions. Examples show that the proposed method is effective and sufficiently fast.

In this paper for second order interval systems we obtain necessary and sufficient conditions for the existence of a common diagonal solutions to the Lyapunov (Stein) inequality. Hurwitz and Schur cases are considered separately. One necessary and sufficient condition is given for n×n interval family of Z-matrices. The obtained results also give diagonal solution in the case of existence.

The main point of the paper is to show the close relation between the nonzero principal components and the difference subspace together with the complementary close relation between the zero principal components and the common vector. A common vector representing each word-class is obtained from the eigenvectors of the covariance matrix of its own word-class; that is, the common vector is in the direction of a linear combination of the eigenvectors corresponding to the zero eigenvalues of the covariance matrix. The methods that use the nonzero principal components for recognition purposes suggest the elimination of all the features that are in the direction of the eigenvectors corresponding to the smallest eigenvalues (including the zero eigenvalues) of the covariance matrix whereas the common vector approach suggests the elimination of all the features that are in the direction of the eigenvectors corresponding to the largest, all nonzero eigenvalues of the covariance matrix.

A voice signal contains the psychological and physiological properties of the speaker as well as dialect differences, acoustical environment effects, and phase differences. For these reasons, the same word uttered by different speakers can be very different. In this paper, two theories are developed by considering two optimization criteria applied to both the training set and the test set. The first theory is well known and uses what is called Criterion 1 here and ends up with the average of all vectors belonging to the words in the training set. The second theory is a novel approach and uses what is called Criterion 2 here, and it is used to extract the common properties of all vectors belonging to the words in the training set. It is shown that Criterion 2 is superior to Criterion 1 when the training set is of concern. In Criterion 2, the individual differences are obtained by subtracting a reference vector from other vectors, and individual difference vectors are used to obtain orthogonal vector basis by using the Gram-Schmidt orthogonalization method. The common vector is obtained by subtracting projections of any vector of the training set on the orthogonal vectors from this same vector. It is proved that this common vector is unique for any word class in the training set and independent of the chosen reference vector. This common vector is used in isolated word recognition, and it is also shown that Criterion 2 is superior to Criterion 1 for the test set. From the theoretical and experimental study, it is seen that the recognition rates increase as the number of speakers in the training set increases. This means that the common vector obtained from Criterion 2 represents the common properties of a spoken word better than the common or average vector obtained from Criterion 1.