Study On Eigenvalue Distribution And Controller Design For Time Delay Systems

Time delay systems are also called systems with the feature that the future evolution of the state variables not only depends on their current values,but also on their past values.From the point of system theory,the current state variables of any pratical system are affected by their past values.Thus,time delay is frequently encountered in pratical engineering systems.Time delay can lead to degraded performance,even stability,in closed-loop systems.Furthermore,the existence of delay causes difficulty in analysis and controller design of systems.In the latest decades,the analysis and study on systems with time delay has attracted considerable interest in international control theory and control engineering.The distribution of eigenvalues of time delay systems plays an important role in their stability and dynamic properties.However,delay gives rise to a closed-loop characteristic equation with an infinity number of roots,which is one of the features time delay systems and makes the study of the distribution of eigenvalues more complex.On the other hand,the distribution of eigenvalues of time delay systems plays a fundamental role in the design of their controllers.This dissertation studies the problem of Eigenvalue Distribution and Controller Design for Time Delay Systems and gives several innovative achievements.The main researches and contributions of the dissertation can be briefly described as follows:(1)The study on the distribution of eigenvalues of time delay systems.The equivalent relation on the distribution of roots between a time delay characteristic equation and a quasi-polynomial is stated.By applied part of Pontryagin's conclusions on quasi-polynomial zero location,some new results applicable to the case of a quasi-polynomial which possesses some pure imaginary roots and the case of a quasi-polynomial which possesses several right-half plane roots are provided.(2)The study on stabilization of systems with time delay.This dissertation considers the problems of proportional-derivative and proportional-integral stabilization of several types of time delay systems.By employing an extended Hermite-Biehler theorem proposed by Pontryagin,this dissertation provides some straightforwardly computational approachs to determine the complete stabilizing set of the controllers.The constructive characterization of the results makes it easier be adopted in industrial practical application.Furthermore,the control problems on other performances of systems can be studied on the basis of these results.(3)The study on controller design for delay systems via eigenvalue assignment.Stability is a fundamental problem to be solved in control systems.However,the final control requirement in a system is to improve the performance under the condition of stability.In fact,many properties of a closed-loop system directly depend on the locations of its poles.In this dissertation,the extension of Pontryagin's result is applied in the problem on PI/PID control of time delay systems and an approach for controller design via eigenvalue assignment is produced.Such a result can improve the performance of systems(including overshoot,settling time and so forth)on the promise of stability.(4)The study on controller design for delay systems via dominant eigenvalue assignment.It has been indicated from theorical study that the locations of the rightmost eigenvalues of a closed-loop system in the complex plane play a critical role in its many properties.When they are isolated from the rest of the spectrum with a large distance in the horizontal direction,it is generally thought that the dynamic properties of the closed-loop system depend on these eigenvalues which are also called dominant poles.In this dissertation,based on the generalization of the Hermite-Bichler theorem for polynomials and the presented theorem on the distribustion of quais-polynomial roots,new methods on PID controller design via dominant pole assignment for high-order systems and high-order delay systems are proposed respectively.All the obtained sets of controller gains can assign the dominant poles at the desired positions and meanwhile to set all other poles to the left of them larger than a fixed distance in the horizontal direction,which is able to make the closed-loop system more close to the desired performance indexes.The methods of controller gain tuning provided in this dissertation can be achieved in a straightforwardly computational way,which results in implementing easily by a program· The given examples also illustrate the effectiveness and correctness of the proposed methods.Furthermore,all the presented conclusions play a significant role in deeply studying the property and control of time delay systems.