Stability Analysis And Design Of A Class Of Linear Delay Systems With Stochastic Delays

With the rapid development of science and technology,great changes have taken place in the way of human life.The existence of high technology brings a lot of convenience to people's life.However,everything has two sides of development.With the development of science and technology,the demand for technology is increasing day by day.High and new technologies such as spaceships,the Internet,intelligent robots and industrial control put forward higher requirements for automatic control.Therefore,the development of control theory still faces great challenges.In the field of control,time-delay systems are ubiquitous.The existing research methods and achievements promote the research progress of time-delay systems,but there is still a lot of research space.For example,there is no unified and effective method to construct a(Lyapunov-Krasovskii)L-K functional for getting better results;There is still no good solution to the problem of defining the quadratic integral term after the derivation of L-K functional;The variation range and derivative of time delay have been considered by many scholars,but the random property of time delay is often ignored.Therefore,it is still a hot topic that how to reduce the conservatism of time-delay systems in the field of control research.Based on Lyapunov stability theory and linear matrix inequality(LMI),this paper combines mathematical derivation,numerical example and Simulink simulation to study stability analysis and controller design of linear systems with random delay.The following parts are the research content of this paper:(1)The stability of interval delay system with random delay is analyzed.Stochastic variables conforming with the Bernoulli distribution are considered to depict random time delays.Assuming that the probability distribution of the time-varying delays in the two intervals is known in advance,the system is transformed into a type of probability interval time-varying delay system after introducing the random variable.Secondly,in order to take full advantage of the time-delay information,an augmented matrix and a triple integral term are introduced into the L-K functional.And a combination of Wirtinger integral inequality and convex combination is used to handle the quadratic integral term which is appearing in derivation of the L-K functional,the stability criteria of the system are established.Meanwhile,the results are obtained by solving a set of linear matrix inequalities.Finally,two examples are given to demonstrate the validity of the proposed method.The distribution of time delay is given to verify that the random time delay conforms to Bernoulli distribution.(2)Considering that the error of external interference and internal parameters will affect stability of the system,on the basis of the previous chapter,the robust stability analysis of the random time-delay system is carried out.Considering the existence of delay derivative and uncertain factors,and combining with the probability interval distribution of delay,a suitable functional is constructed for the system,and then the derivative of the functional is derived.When dealing with functional derivatives,the condition of system stability is obtained by reserving this term.The numerical example is given at the end to verify that the results are less conservative,and the system state response diagram is given to judge that the final stability criterion is effective.(3)The stability problem of linear systems with random delays has been studied previously,and the robust problem of stochastic delay systems with parametric uncertainty and delay derivative dependence is also discussed.But the problem of controller design was not taken into account,Therefore,the problem of designing robust controllers for stochastic delay systems with parametric uncertainties will be considered next.By applying the Lyapunov stability theory,and the reciprocally convex combination lemma and other valid inequalities mathematical techniques,a stability criterion for a random time-varying delay system with parameter uncertainty is developed that guarantees the designed controller can satisfy the requirements of stability.Finally,the feasibility and effectiveness of the designed controller are verified by the numerical examples and the state response diagrams given at the end.