Study On Control Methods In Singular Jumping Systems

The singular Markov jumping system (SMJS) is a new kind of dynamical system,which has two particular characteristics. First one is structural parameters changerandomly, the other one is the system states are not known completely, or the changingrate of the internal state is inconsistency. Because of these two features, in modernscientific research, SMJS is widely concerned by more and more researchers. On theone hand, the characteristic of random jumping, which satisfies one step Markovproperty, can generate a greater impact to the performance of the system. On the otherhand, the singular perturbation is better to meet the specific requirements of manyactual engineering applications. In view of above, the investigation of SMJS is verymeaningful and helpful. With these problems in mind, this article mainly focus onstochastic stability, filter design, and the optimal control method. Main innovations areas follows:First of all, based on the stability of SMJS, new criterions of stochasticstabilization and robust H∞performance have been researched. According to thedefinitions of stochastic stability and robust H∞performance, a new discriminationmethod with the structure of linear matrix inequalities (LMI) has been proposed. Firstof all, the characteristics of regular and no-pulse are considered to ensure the systemhas a unique solution at any time. Secondly, the global stability is guaranteed when thestructural parameters are switching in different subsystems. Based on the new criterion,the design of stabilizing controller has been researched and the analytical expressionsof feedback gain are given.Secondly, based on the phenomenon of singular perturbation during the workprocess of SMJS, the relationship between singular perturbation and systemperformance has been researched. Because the time scales of state change are notsynchronous, the stability of SMJS is not equivalent to the stability of the originalmodel: singular perturbed jumping system. The equivalent conditions between originalmodel and degradation model are given. Based on this conclusion, the calculationmethod of small time allowable upper limit is researched and the algebraic basis isgiven for the establishment of SMJS model, which is equivalent to the original singularperturbed jumping system.After that, based on the problem of optimal control, the control methods of SMJS have been researched. The algebraic relations between rapidly varying sub-statevectors and slowly varying sub-state vectors are established, and the analyticalexpressions of SMJS optimal control law are given. Then, the influence ofsingular perturbation is considered. A new method is given to solve the optimalcontrol law when the small time parameters are unknown.At the last, based on the bilinear SMJS model, the stochastic stability andstabilization have been researched. The interactive product terms between statevector and control vector are introduced in the state equations of system model.The new criterion of stochastic stability has been researched and a new methodof feedback controller gain calculation has been given.