Studies On The Problem Of The Zero-free Spectral Factors Of Descriptor Systems

Factor analysis is used for example in multivariate data analysis, by definition less latent variables to explain more variables in the scalar matrix. The associated transfer function matrix has more rows than columns, and when the factors variables are independent zero mean white noise sequences, then the output spectrum is singular. The descriptor system with complexity is studied. When output spectrum is singular and transfer function matrix has no zero, the research and applying of the properties of zero-free spectral matrices are aroused by more and more scholars’attentions.In this paper, the conclusions about the discrete-time descriptor systems are considered and discussed the spectral factors of the spectral factorization of descriptor systems with the inner-outer factorization. In addition, output of the system is more than input that excited by independent zero mean white noise; the associated transfer function matrix has more rows comparing with columns. We assume that the output power spectrum is known, in minimal state-variable form. And matrices defining the system are unknown. Our purpose is to solve the minimum phase spectral factors of associated system as well as achieve the system by using Riccati difference equation and Kalman filtering ideas. The main results in this dissertation are shown below:(1) The research is about the problem of the definition and the conclusion of zero-free of the rectangular discrete-time descriptor systems. In this part, we make an expansion with the definition of the no finite zeros of the rectangular discrete-time descriptor systems. And we use the invariant zero, Markov parameter and the invertibility of descriptor systems to discuss the finite zeros and prove conditions are necessary and enough with the no finite zeros of this system. Moreover, we get the related definitions and conclusions of the finite zeros and no finite zeros of the rectangular discrete-time descriptor systems as we studied before.(2) We use state-space realization to solve the inner-outer and spectral factorization problems for a discrete-time descriptor systems by the knowledge of pole separated realization of G(z) The algorithm is based on orthogonal transformations and standard reliable procedures to solve algebraic Riccati equation which order is usually much smaller than the degree of McMillan of the transfer matrix of the system. Through this condition above, we get the inner-outer factor of the system by analyzing the relation of inner-outer and spectral factorization for the given system. This inner-outer factor is the spectral factor of the system when the system has a spectral factorization. Finally, a simple numerical example has demonstrated the theoretical method mentioned previously.(3) We also discuss the condition under the number of latent variables exceeds the number of factor variables. When associated transfer function matrix has more rows than columns and the factor variables are independent zero mean white noise sequences, the transfer function matrix is stable and the output spectrum is singular. When we do not know the matrices defining the system and it has no zeros. And if the output power spectrum is known, we use the spectral factorization to calculate the spectral factor of the system and then we can achieve the system though this method. In fact, the problem is solved by the thought of Riccati difference equation and Kalman filtering ideas indeed.We also have one more method to resolve this problem. Descriptor systems are changed into normal subsystems by using Jordan decomposition of matrixes. The observability of these two systems is consistent. We know the method to solve the stable minimum phase spectral factor of linear systems. By this method, we can solve a zero free minimum phase spectral factor of the normal linear subsystem. We can find out a zero free minimum phase spectral factor from this spectral factor. In the end, there is a numerical method that can support the method.