| The PID stabilization problem involving all the controller gains(kP,kI,kD)and the delay τ for time-delay systems is of theoretical as well as practical importance,but,to the best of the authors’ knowledge,it has not been fully explored.Therefore,in this thesis,based on the frequency-sweeping mathematical framework,we implement the PIDCDTDS toolbox,which is used to solve the PID stabilization problem for time-delay systems involving totally four free parameters and no constraints are imposed on the controlled plants.This not only fills the technical gap of the PID stabilization problem for time-delay systems but also establishes the basis for the PID controller design.In the past,the frequency-sweeping method is a classic tool for analysis of time-delay systems.Because of the specifical algebraic relation between the Puiseux series and the dual Puiseux series,so the asymptotic behavior of the critical imaginary roots can be studied from a new analytic of the curve perspective.However,for the PID stabilization problem of time-delay systems,the frequency-sweeping method can solve the complete stability problem with respect to r for a fixed(kP,kI,kD)and can not deal with the case where(kP,kI,kD,τ)are simultaneously free parameters.In response to this question,we introduce a new algebraic criterion to solve the PID stabilization problem for time-delay systems.By this criterion,the critical imaginary roots(CIRs)are associated with ΔNU,which stands for the number change of unstable roots,so we can compeletely analyze the asymptotic behavior of the CIRs.Then we will implement an algebraic algorithm to solve the complete stability problem with respect to τ.In this way,the PID stabilization problem for time-delay systems involving totally four free parameters can be solved,and the stability set in the(kP,kI,kD,τ)-space is obtained.Based on the above theory,we implement the PIDCDTDS toolbox and verify the effectiveness of the toolbox by comparing examples with the existing literature.Because the results of the complete stability analysis generated by the PIDCDTDS toolbox are not conservative and multiple stable intervals can be found,so it has certain reference to the conservative conclusions given in the previous literature.Finally,the application of PIDCDTDS toolbox in DC motor control problem shows that the toolbox is suitable for the PID stabilization problem for time-delay systems involving totally four free parameters in practical applications,and will offer convenience for tuning of PID controller parameters. |
