Negative Imaginary Property Criterions Based On State Space Model

Negative imaginary systems are a class Lyapunov stable dynamical systems with dissipative properties.Over the last decade,the research on negative imaginary sys-tems has attracted extensive attention in the field of system and control.By appropri-ately choosing the systems input and output,many practical dynamic systems can be modelled as negative imaginary systems,such as atomic force microscope micro-nano positioning control.The negative imaginary internal stability plays an important role in robust performance analysis,which extends and complements existing dissipation re-sults.While we need to check the negative imaginary properties before robust controller synthesis.In this thesis,we focus on characterizing the negative imaginary properties based on state space model.Currently,although many efforts have been made on characterizing negative imag-inary properties based on state space model,there still exist many problems to be solved.Firstly,for lossless negative imaginary systems,previous results are only suitable to those transfer functions which do not have poles at the origin.Secondly,the research on checking descriptor lossless negative imaginary properties based on state space mod-el has not been reported.Thirdly,for descriptor negative imaginary systems,a lim-itation of previous results is that the transfer functions have to be proper.However,non-properness of descriptor system transfer functions is an important feature that is not found in classical systems.In view of these problems,this thesis will study negative imaginary properties based on state space model.The main contribution of this thesis is stated as follows:Aiming to lossless negative imaginary systems,this thesis is concerned with ex-tending lossless negative imaginary lemmas to the dynamical system with poles at the origin.Firstly,two versions of lossless negative imaginary lemma are es-tablished in terms of a set of linear matrix equations.They can be considered as extensions of the previous results.Meanwhile,the conjugate property of lossless negative imaginary systems is studied.Secondly,a new type of lossless negative imaginary lemma is derived based on Kalman canonical decomposition of system state space model.The second type of lossless negative imaginary lemma avoids solving linear matrix equations and is more computationally efficient.This thesis introduces the concept of descriptor lossless negative imaginary sys-tems.Then,the superposition theorem and decomposition theorem of descriptor lossless negative imaginary systems are derived based on canonical equivalent form.Finally,we derive a sufficient and necessary condition to check descriptor lossless negative imaginary property.Aiming to descriptor negative imaginary systems,this thesis extends descriptor negative imaginary lemmas to the non-proper transfer functions.Firstly,base on the relationship between the descriptor negative imaginary systems and pos-itive real systems,we analysis the controllability and observability of these two kinds of systems.Secondly,we derive a sufficient condition to check descriptor negative imaginary property in terns of solving a set of linear matrix inequations.Finally,the validity of the developed theories is illustrated by numerical examples.