Stability Of Neutral Multi-delay Systems With Applications

Time delay is an inevitable factor in real systems,like robot,internet,metal cutting,drilling,digital control,man-machine interaction,and so on.Time delay affects the systems' behaviors greatly,to study time delay systems provides both scientific significance and engineering requirement.Usually time delay systems are categorized into two classes: Retarded type and neutral type,this dissertation focuses on the stability of neutral time delay systems mainly.Compared to the retarded time delay systems,the neutral ones have peculiar dynamical behaviors:Strong stability and weak stability.For a weak stable system,even if it is asymptotic stable,the system can lose stability when applied to an arbitrarily small perturbation of time delay.Weak stable systems has no robustness against delay variations hence has no practical meaning.Usually we only concern about the strong stable cases,where the systems' stabilities do not change if the perturbations of time delays are small enough.This dissertation mainly considers the strong stable systems,while discusses a few about weak stability too.Among the current stability methods for time delay systems,the ones analyzing the characteristic roots are often more delicate than the others,and can be used to judge the system frequencies,to construct the stability regions with respect to certain parameters.The Argument-Principle-based Mikhailov criterion calculates the number of the unstable characteristic roots of retarded time delay systems.The equivalent improper integral form of Mikhailov criterion,simple in form,works effectively,it judges the system stability with a rough estimation of the improper integral.However,there are no general rules to choose a suitable upper limit of the definite integral.In this dissertation,the definite integral method is extended to neutral time delay systems,and is reduced to a simplified form under the strong stability condition,the simplified form improves the computational efficiency greatly.Moreover,the definite integral method is further improved by proposing two ways of choosing the upper limit of the integral,which is suitable for judging the stability of systems with fixed or variable parameters,respectively.The definite integral method is efficient,and convenient for computer coding,and suits for multi-delay systems.Furthermore,based on the integral method,a new scheme to calculate the characteristic roots is proposed under strong stability condition.This scheme can calculate the relatively important roots including the rightmost one on the complex plain,at the same time.Nyquist method is an intuitive way to judge the system's stability,mainly for systems with fixed parameter.However,Nyquist method may fail for certain cases,this dissertation improves it inthis regard.For the applications of neutral time delay systems,this dissertation first studies the Wheeled Inverted Pendulum with delayed acceleration feedback.A Wheeled Inverted Pendulum is a two-wheeled robot,it is of simple form and unique performances such as zero-turning radius and high energy efficiency.To measure the inclination of the body relative to the vertical direction of the Wheeled Inverted Pendulum,a gyroscope or a combination of gyroscope and accelerometer is usually used.This dissertation studies the possibility of using only one single-axis accelerometer as the sensor of the Wheeled Inverted Pendulum.The output of an accelerometer is a signal that is the combination of the angular displacement and acceleration.With such an output being fed back,the controlled Wheeled Inverted Pendulum is a neutral time delay system.By introducing the fly-wheel damper,the balancing of Wheeled Inverted Pendulum is achieved,and it is of strong stability.How to choose the location of the accelerometer,time delay,and control gains is also discussed.This dissertation also studies the fractional ordered mass-damping-spring system with delayed acceleration feedback,where the Scott-Blair fractional model is used to model the damping force generated by the viscoelastic material.The oscillator's forced vibration is controlled via acceleration feedback,which results in a fractional-delay system.We extend the definite integral method to the fractional-delay systems.By introducing a factor to the characteristic function,the calculation of the unstable characteristic roots can be carried out uniformly.Moreover,the strong stability and weak stability of the fractional-delay systems are discussed,and an example of a neutral fractional-delay system is given to show the peculiar properties of weak stability.