Interval Frequency Negative Imaginary Properties Research Based On Generalized KYP Lemma

The research of negative imaginary systems theory has attracted many interests of many scholars in control field,and is a research direction closely related to the positive real systems and the dissipative systems.Many practical engineering systems satisfy negative imaginary properties,such as flexible structure systems with position sensors and collected forced actuators,the transfer functions of DC motors from input voltage to output rotating speed,the transfer functions of RLC network systems from input charge to output voltage,the transfer functions of hard disk drive servo systems,robotic manipulator arms and large vehicles platoons.The study of negative imaginary systems has attracted much attention,and has made significant progress.For example,giving the necessary and sufficient conditions for testing the negative imaginary properties,providing robust stability conditions for positive feedback connected negative imaginary systems.However,there are fewer re-search achievements for the systems satisfying interval frequency negative imaginary properties.The existing negative imaginary systems theory mainly focuses on whole positive frequency set.However,many practical systems satisfy negative imaginary property only in interval frequency set.For example,the transfer functions of piezo-electric ceramic tubes from input voltage to output position and the transfer functions of RLC network systems.In view of the limitation of study for the existing negative imaginary systems theory,this thesis provides the concept of interval frequency nega-tive imaginary,and generalizes this concept to discrete systems and descriptor systems.The main works as follows:1.Considering the problems of the limitation of the research for the negative imag-inary systems theory on interval frequency set,we study the interval frequency negative imaginary systems.Firstly,this thesis introduces the concept of the in-terval frequency negative imaginary systems allowing the poles at the origin,and establishes the relation between interval frequency negative imaginary systems and interval frequency positive real systems.Secondly,in view of the general-ized KYP lemma,we obtain the necessary and sufficient conditions for interval frequency negative imaginary transfer functions with minimal state-space real-ization.When the interval frequency set becomes the positive frequency set,the necessary and sufficient in this thesis are equivalent to the necessary and suffi-cient for normal negative imaginary transfer functions.In view of the systems input,output and state,a time domain interpretation of interval frequency nega-tive imaginary properties is provided in this thesis.2.In view of the limitation of the research for the discrete time negative imaginary systems,this thesis studies discrete time interval frequency negative imaginary systems.Firstly,we generalize the concept of continuous time interval frequency negative imaginary to the discrete time systems,and provide the discrete time in-terval frequency negative imaginary systems.This concept allows at most double pole at the infinity and the same to the origin.Secondly,the relationship between discrete time interval imaginary negative imaginary transfer functions and contin-uous time interval frequency negative imaginary systems.For the cases of interval frequency(or,low frequency,middle frequency,high frequency),we introduce two forms of generalized KYP lemma.Finally,this thesis provides discrete time interval frequency negative imaginary lemma based on bilinear transformation or generalized KYP lemma.3.In view of the limitation of the research of the descriptor negative imaginary sys-tems,this thesis studies the interval frequency negative imaginary properties of descriptor systems.Firstly,we generalize the concept of interval frequency nega-tive imaginary transfer functions to the descriptor systems allowing at most dou-ble pole at the infinity.Secondly,this thesis introduces strict interval frequency negative imaginary lemma based on non-Weierstrass form,and finite frequency negative imaginary lemma for descriptor systems.Moreover,in view of Weier-strass form,we provide a time interpretation of interval frequency negative imag-inary properties,and a time interpretation of finite frequency negative imaginary properties.