The Analysis Of Controllability, Observability And Stability Of Discrete Non-square Singular Systems

The concept of singular system was set up in 1970s. In 1974, Rosenbrock H.H. first came up with the concept of the singular system when he studied the complex circuit network system. Singular systems exist in a lot of fields, such as engineering domain, society domain, economy domain and so on. So singular systems have been concerned so many scientists in the fields of control and mathematics. They studied the singular systems in different facet and obtained some valuable results.Controllability, observability and stability are very important attribute of singular systems and play a vital role. The coefficient matrix for the singular systems being square, many scholars have conducted in-depth research, and have achieved a lot of more useful results. However, the coefficient matrix for the singular systems being non-square, research was relatively rare; particularly the systems for the rectangular coefficient matrix of discrete singular systems is less.Coefficient matrix for the rectangular systems takes square one in singular systems as special case, because when the rectangular coefficient matrix of row vectors and column vectors are equal, then systems will naturally become the singular systems. Therefore, research of the non-square singular systems, not only has a high theoretical meaning, but also has a very important application.Regularity is a basic condition for the research of the singular systems, but non-square descriptor systems can not meet the requirements of regularity, that is non-square singular systems are irregular singular systems. This thesis provides a practical example, in which it is a non-square singular system by modeling.Based on the analysis and research scholars in the past for non-square singular systems, for discrete non-square singular systems we introduce into a generalized inverse multiplier (γE-A)+ and get a generalized inverse equivalence form of the discrete non-square singular systems. After the conversion, the coefficient matrix of the generalized inverse equivalence form of the discrete non-square singular systems is no longer a non-square matrix, which becomes a square. It makes the systems more convenient to study. On this basis, the thesis focuses on the C-controllable, R-controllable, C-observability and R-observability of the discrete non-square singular systems, and gets some basic results. For the generalized inverse equivalent form of the discrete non-square singular systems, Lyapunov function is constructed and its stability is discussed. A basic conclusion of the generalized inverse equivalent form of the discrete non-square singular systems is obtained.