| Negative imaginary system theory widely exists in various engineering applications.It has attracted the attention of scholars in the field of control.The current research on the theory of negative imaginary systems is limited to the extension of the definition and properties of negative imaginary systems and the theoretical part of linear systems.In actual scenarios,there are often various types of uncertainty.The stability-related problems of negative virtual systems with uncertainties are studied,and the application of negative imaginary system theory in actual control under uncertainty is generalized.Researches on the theoretical stability of negative virtual systems have achieved certain results,but there are still many shortcomings.On the one hand,at present,there is insufficient research on the absolute stability of interconnections of negative virtual systems with nonlinear uncertainties such as slope boundedness.On the other hand,there is a lack of research on the stability of discrete-time negative virtual systems with uncertainty.The existing internal stability conditions of the negative imaginary system interconnection based on the IQC method reduce the conservativeness of the DC gain condition,and the research on the stability of the discrete-time negative imaginary system interconnection is insufficient.The main contributions of this article can be summarized as the following three aspects:·For the interconnected system with negative virtual system as the controlled object,the uncertainty of the slope bounded nonlinearity and the controller with strictly negative imaginary property,we study the absolute stability related issues.First,based on the characteristics of passiveness and the absolute stability theorem,the method of ring transformation is used to transform the uncertain part into a passive operator,and then the original interconnected system is converted into a new form to facilitate stability analysis.Secondly,construct Lur'e-Postnikov-type lyapunov function,and then obtain sufficient conditions for the absolute stability of the interconnected system.Finally,based on the conclusion of absolute stability,the condi-tions to be met by a strict negative imaginary controller that keeps the system absolutely stable are obtained,and the absolute stability of the system that meets the conditions is verified with an example of a light damping model and numerical simulation.·For discrete-time negative imaginary systems with uncertainties,we study the conditions for their absolute stability.First,the inference of the discrete-time negative imaginary lemma is given.By constructing the lyapunov function,the conditions for absolute stability of the discrete-time negative virtual system with fan-shaped bounded uncertainty are obtained.Second,consider the interconnected system of any linear controlled object and fanshaped bounded uncertainty,and analyze the design conditions that the absolute ballast needs to satisfy,making the entire system absolutely stable.Furthermore,according to the conclusions obtained above,the design conditions of the ballast are obtained,which makes any controlled object have a negative virtual property,which makes the entire system absolutely stable,and the correctness and validity of the main conclusions are verified by numerical examples.·For the interconnection of discrete time negative imaginary systems,the stable IQC conditions in the positive feedback interconnection system are given.First,based on the definition of-stability,the internal stability of the interconnection of a discrete-time negative imaginary system with no pole at is studied using the IQC method.Secondly,the nature of the discrete-time negative virtual system and the bilinear transformation are used to transform the studied system into a discrete-time negative virtual system interconnection with no poles,and then use the previous internal stability conclusions to get the poles at the place.The internal stability conditions of discrete negative imaginary systems without poles,and numerical examples are used to verify the correctness and validity of the main conclusions. |
