Generalization On Linear Negative Imaginary Systems Concepts And Its Properties Research

Negative imaginary systems theory,which can be considered as a complement to the positive real systems theory,is attracting more and more interests among control theory researchers.This type of systems arises in many practical systems.For exam-ple,the transfer function characterizing the input of force actuators and the output of collocated position sensors in a damped or undamped flexible structure subjects to the negative imaginary property;the transfer function of direct current machine systems from input voltages to shaft rotational velocities,and the transfer function of electrical active filter circuits from input voltage to output voltage are also negative imaginary.One of the main contributions of negative imaginary systems theory is the internal sta-bility of positive feedback interconnected negative imaginary systems,which has been widely applied to various areas,such as the robust vibration control of flexible struc-tures.Although the study on negative imaginary systems has achieved a great success,the study on negative imaginary systems theory is far from completion.Specifically,the current research on negative imaginary systems theory mainly lies in continuous-time real-rational proper systems,while few studies were provided for non-proper neg-ative imaginary systems and discrete-time negative imaginary systems.The existing definitions of negative imaginary systems require the transfer function matrix to be real-rational proper or symmetric,that are not applicable to non-symmetric and non-proper real-rational case.To this end,the thesis proposes the definitions of continuous-time non-symmetric,non-proper negative imaginary systems and discrete-time non-symmetric negative imaginary systems,and aims at studying the properties and stability of such systems.The main works of the thesis are summarized as follows:1.Considering the pole placement problem of negative imaginary systems,the ?-and D-negative imaginary systems are studied.The definition of the ?-negative imaginary system is introduced.Then,the relationship between negative imag-inary and ?-negative imaginary transfer systems is studied.By means of the gen-eralized inverse,an ?-negative imaginary lemma is proposed to test the ?-negative imaginary property of transfer function matrices based on linear matrix inequality.Also,a necessary and sufficient condition is provided for the ?-stability of the interconnected negative imaginary systems.A state-feedback controller design condition is established such that the resulting system is ?-negative imaginary.In addition,the concept of ?-negative imaginary systems is extended to that of D-negative imaginary systems.This study is detailed in Chapter 2.2.In view of the limitation of the existing definitions and shortcomings of negative imaginary systems,continuous-time non-proper negative imaginary systems are considered.Firstly,we extend the definition of negative imaginary transfer func-tion matrices to non-proper case,which allow poles at infinity.Then,a sufficient condition is provided for a system to be real-rational non-proper negative imagi-nary in terms of s-domain in the quarter domain of analyticity.Also,by using a minor decomposition,a new relationship is established between non-proper(loss-less)positive real and non-proper(lossless)negative imaginary transfer function matrices,and some important properties of non-proper negative imaginary sys-tems are studied.The results in this thesis give us the possibility to address non-symmetric and non-proper descriptor systems with negative imaginary frequency response.This study is detailed in Chapter 3.3.In view of the current works of negative imaginary systems focusing on continuous-time system or symmetric discrete-time systems,discrete-time negative imagi-nary systems are considered without symmetric restriction.A new definition for discrete-time negative imaginary transfer function matrices without the symmet-ric restriction is proposed in the thesis,and some important properties of such sys-tems are studied.Under the new definition of discrete-time negative imaginary systems,the relations between discrete-time negative imaginary and discrete-time positive real transfer function matrices are characterized in two different method-s,and two different discrete-time negative imaginary lemma are derived in terms of the minimal state-space realization to test the discrete-time negative imaginary properties.Then,a necessary and sufficient condition is proposed for the internal stability of the interconnected systems in terms of the dc-loop gain at 1.Mean-while,the connections between discrete-time negative imaginary and continuous-time negative imaginary transfer function matrices are characterized by means of the bilinear transformation.Moreover,the definition of discrete-time lossless negative imaginary transfer function matrices is proposed,and some properties of such systems are studied.For this study,one can refer to Chapters 4-6.