The Decomposition And Stability Analysis For Nonmonotone Control Systems

During the last decade, considerable attention has been devoted to the monotone dynamic systems which own extensive ecology and economics backgrounds. Strong stability and convergence are possessed by monotone dynamic systems, and the general nature of the result in abstract Banach space is the greatest advantage of the dynamic systems, that is, the methodology and the overall consistency of the result in the different branches of the study. However, the monotone control systems are obtained from the combination of the control theory and the monotone dynamical systems theory. Through the study of the monotone control systems, we can easily comprehend and investigate the complex molecular biology system model, particularly, the complex interconnected systems with feedback loop.Monotony plays an important role in biological systems which have a positive or negative feedback loop. The result of its global convergence and stability of the balance points can be obtained by the stability of the intersection of the I/O characteristics curve and feedback curve through a large number of experimental data in some appropriate assumptions in monotone control systems. In practice, the original system models are often non-monotone and contain many variables, how to analyze the dynamical behavior becomes more difficult for us. In order to make use of the monotone control systems theory, non-monotone systems will be decomposed to the connection of the monotone systems, and then discusses the system's stability. The thesis is organized as follows:In the first chapter, the born and development of the monotone dynamic systems as well as the background and development of the monotone control systems are carried on the simple description. And the development condition of the systems biology and biocybernetics are discussed.In the chapter 2, the definition of the system monotonous and a few of the pertinent concepts are explained. And the criterion of the system monotonous is given; Then the I/S characteristic and the I/O characteristic of the monotone control systems are explained; Finally, the nature of graph theory of the monotone control systems, and the function of the incidence graph theory in the analysis the concrete biology system mathematical model is explained through example.In the chapter 3, non-monotone systems are decomposed to the positive feedback monotone closed-loop system with input and output through the decomposition method, and then monotone control theory is used to study the stability of the monotone closed-loop system. The greatest difficulty in the application of monotone control systems theory is how to decide the location and quantity of the balance points. The method of the decision of the location and quantity of the balance points are given in this chapter, as well as how to analyze the stability of the balance points.In the chapter 4, the method of decomposition of the nonmonotone control systems with negative feedback is given, that is, the least edges can be removed to make the rest of the incidence graph is consistent by incidence graph theory. Then the new monotone control systems are composed by the connection of the consistent system and a negative feedback loop through the negative feedback. Then the original system's dynamic characteristic can be obtained through the interconnection monotone subsystem with the suitable interconnection criterion, which the subsystem's dynamic characteristic is simpler and is easier understood.Some concluding remarks are given in Chapter 5, and future research works are pointed out.