Research On Positive Real And Negative Imaginary Transfer Functions

In this paper,we consider operator-valued positive real(PR)and negative imaginary(NI)transfer functions for infinite-dimensional discrete-time systems,the stability of the feedback interconnection of discrete-time negative imaginary(D-NI)systems and robust output consensus of discrete-time networked homogeneous NI systems under l2 external disturbances.In Chapter 1,backgrounds of the PR and NI theory are introduced.And the research status are also listed.We list our main work.Chapter 2 collects preliminaries about matrices,graph theory,operator theory,PR and NI systems and integral quadratic constraints(IQCs).In Chapter 3,we give operator-valued PR and NI transfer functions.The equivalence of positive realness and passivity is proved for the exponentially stable system.In addition,we give the connection between operator-valued proper real-rational PR and NI functions.In Chapter 4,we study the stability of the feedback interconnection of D-NI systems through IQCs.Applying the latest IQC-based result to the D-NI systems with poles on the unit circle,a necessary and sufficient condition for the stability of the feedback interconnection of a D-NI system without poles at ±1 and a strictly negative imaginary(D-SNI)system is developed.The stability of the feedback interconnection of a D-NI system with poles at 1 and a D-SNI system is also studied.Moreover,a sufficient condition for the internal stability of an interconnection of unstable D-NI systems is obtained.Three examples are given to illustrate that the main results are broader than some early results.In Chapter 5,we consider robust output consensus of discrete-time networked homogeneous NI systems under l2 external disturbances via IQCs.A necessary and sufficient condition for robust output consensus of discrete-time networked NI systems without poles at ± 1 is obtained by the IQC-based D-NI stability result.Moreover,the robust output consensus problem can be solved by designing an unstable controller whose poles are related to transmission zeros of D-NI systems.The case of discrete-time networked NI systems with poles at 1 is also studied.These main results are related to all eigenvalues of the Laplacian matrix of the network graph as well as the corresponding complementary IQCs at±1 frequencies.In order to simplify the main results,some corollaries are given by constructing appropriate IQC multipliers.Finally,three examples are shown to explain the validity of the main results.Chapter 6 concludes the paper,and puts forward the next steps of our work.