1、The Theory of Singular Systems1. The development of the theory of singular systemsSingular system is more extensive forms of dynamical systems than a normal system, the theory of singular system is began to take shape and gradually developed an independent branch of the theory of modern control sinc
2、e 1970s. In 1974, Rosenbrock H.H. is a British scholar, in magazine delivered a speech entitled “The structure of the general nature of dynamic systems“, he had done a limited study to linear singular systems and system decoupling zero equivalence.First, he proposed the concept of singular systems.S
3、ubsequently, the American scholar Luenberger DG had study to solution of linear singular systems of the existence and uniqueness of the study and so on. From then on, it is began to the curtain of the research on the theory of singular systems.The theory of singular systems in the early stages of de
4、velopment is slow progress in research, in addition to the groundbreaking achievements,the outstanding results during this period is Luenberger DG about the non-linear singular systems. 20th century 80s, more and more theorists of the control of singular systems have had a keen interest the theory o
5、f singular systems,it has also entered a new stage of development.From the 20th century, the theory of singular system has been vigorous development.The representativeness of the results of this stage,for example: Cobb D. had studied a singular system controllability, observability and the principle
6、 of duality; further, Dai L. extensed to discrete singular systems; Yang C., make a singular system realization of the smallest; Fabmy M.M. had developed observer design; Fletcher L.R. studied the disturbance decoupling of singular systems and issues,such as eigenstructure assignment; Dai L. had dev
7、eloped on the continuous and discrete singular systems dynamic compensator design;Lin J. and Lin X., respectively, discussed the problem of the time-varying singular systems and time-invariant optimal control. Above a series of basic questions of the research results, Dai L. systematically introduce
8、d the basic theory of singular systems, which marks the singular system has formed the basis of the theory, the theoretical study of singular systems will enter a new stage of development.In the final stages of development, that is, the 20th century, early 90s until now, the theory of singular syste
9、m has been developed more than ten years, the study of singular systems to develop in depth from the foundation, covering from linear to nonlinear, from continuous to discrete,from certainty to uncertainty,from never the delay to the delay, from the linear quadratic optimal control to and control an
10、d other topics, it has achieved fruitful results.2HSingular systems exist in the social production model of the various fields, such as: power systems, economic systems, robotics systems, electronic networks and space systems and so on.It is well-known,such as Leontief dynamic input-output model, Ho
11、pfield neural network model, the coordination of multiple robots operating in the main as well as non-linear dynamic model of power system load models are singular systems. Therefore, the study of the theory of singular systems has far-reaching practical significance.2. The association and differenc
12、e between the singular systems and the normal systemsSingular systems and the corresponding normal systems both intrinsically linked to the existence and nature of the distinction.Singular system associated with the normal system,If the E of the singular systems is non-singular, the singular systems
13、 has been a normal systems.So,if the matrix E )()(txAtEfrom the significance of a wide range of values considered normal singular systems is the promotion system.Theory of Systems as a result of normal maturation of the basic research, has become a more complete theoretical system, therefore, in ord
14、er to distinguish between them, a habit for the singular matrix E as a clear sign of singular systems, The theory of singular systems as an branch of the independent research.Singular systems and the normal systems, there are still many essentially different. For example: consider the case of linear
15、 time-invariant,the difference between the singular systems and the normal systems are mainly reflected in the following areas:(1) The solution of linear singular systems contain not only the normal system of the index, and the general state of the system containing a normal response does not appear
16、 in the pulse and the pulse of the derivative, the singular system so that there is not a normal system pulse behavior.(2) The singular systems with the object of the general level, and one for the dynamic characteristics of the object; another layer to the static properties of objects,but the prope
17、rties of the normal system is not static.(3) The normal systems have n degrees of freedom, and the singular system has only rankE degrees of freedom. rankE(4) The singular systems with a polynomial transfer function matrix form, and has infinite pole.(5) The normal systes have causality, and the sin
18、gular systems typically have non-causality, that is, the solution of singular systems is not only dependent on the past and the present input, but also depend on future input.These features of the singular system reflects the singular systems have more complex and wealth to the novelty than the norm
19、al system in the structure,more difficulty and more challenging in the theoretical research. Therefore,based on the normal system, the theory of singular systems have gradually attracted a great deal of attention of many scholars at home and abroad.3.The research methods of the theory of the singula
20、r systems.So far, the theory of singular systems research ideas have been mostly with reference to the normal system theory to singular systems and the promotion of transplantation, and its research methods are mainly geometric method, frequency domain methods and state-space methods.Geometric metho
21、d is the singular systems have change to the issues of a state of the geometry of space.Its advantage is the structure of the systems has a unique portrait.For example:the structure and the control space of singular systems and the temper of the portrait of the same sub-space and so on.Moreover, the
22、 concise geometric method avoid the state-space methods derived a large number of complex matrix operation, and the resulting structure can be translated into matrix operations.Its drawback is that the systems robustness analysis seems powerless.Multivariable frequency-domain method (referred to as
23、frequency-domain method) is a state-space representation of the singular system of the system frequency-domain and frequency domain description of the calculation method for research. Frequency-domain method has strong intuitive physics for the design of the advantages of regulation.As for state-spa
24、ce methods (or time-domain method) is described in state-space systems using singular matrix calculation and the calculation of transformation matrix, direct time-domain system of research, the theory of singular systems is most commonly used method.State space method describes the way the simple in
25、tuitive,and can be designed to support the software suitable for operation on the computer, so the method most widely depth to the singular system analysis and synthesis of all aspects of control by the majority of engineers are preferred.The two basic of methods are Riccati method and the current popular LMI(Linear Matrix Inequality) method,as the system has to reveal the internal structure of computer-aided design and the advantages become a time-domain state-space.
