Controllability And Observability Analysis Of Non-smooth Sandwich System

Controllability and observability are two important concepts in modern control theory and fundamental properties of a control system,which describe the ability of the input to control the state of the system and the output to observe the state of the system.And they are the basis of the optimal control and optimal estimation.The theory about controllability and observability of linear system has been very mature.But the research of nonlinear systems,which are the most common phenomena in nature and engineering,is far less mature than linear systems.Dead-zone,saturation,backlash and hysteresis are all the typical non-smooth nonlinearities and often exist in practical engineering systems such as plane elevator droved by hydraulic acturators,ultraprecise moving stage,electronic scanner microscope with piezoceramic actuators and positioning serve control system with gear and so on.When these nonlinearities are embedded in the two linear systems,we call the whole system as a non-smooth sandwich system.In fact,Non-smooth sandwich system have a wide range of engineering applications.The main contributions are briefly concluded as follows: The non-smooth state-space function for sandwich systems with dead-zone,backlash,hysteresis are given,respectively.Meanwhile,the linearized approximated state-space model are proposed to simulate these sandwich systems.The observability and controllability matrix is constructed and an method to analysis and judge the observability and controllability of sandwich systems with dead-zone,backlash,hysteresis is proposed.Moreover,both the simulation and the modeling of the micro-position stage with piezoactuator demonstrate the effectiveness of the proposed method.These studies not only improve and extend the traditional methods to analyze the controllability and observability of linear systems,but also enrich and supplement the results of the previous literature on the controllability and observability analysis of non-smooth systems.The analysis are organized as follows: First,In order to investigate the observability and controllability of the system,a non-smooth state-space function with sandwich structure is developed.Because the embedded nonlinearity is non-smooth,the state space expression is non-smooth also.Then,based on the obtained non-smooth state-space sandwich model,a linearization method based on non-smooth optimization is proposed to derive a generalized linearized state-space function to approximate the non-smooth sandwich systems within a bounded region around the equilibrium points that we are interested in.In addition,the corresponding observability and controllability matrices are constructed and the methods to analyze the observability as well as controllability of sandwich system with dead-zone,backlash,hysteresis are derived,respectively.Through the investigation of the derived observability and controllability matrices,the necessary and sufficient conditions and other related theorems of state observability and controllability around the equilibrium point are showed.Moreover,this paper gives an analysis of relationship between dimension of observable and controllable space of sandwich systems and its subsystems.Afterwards,several simulation examples and an application of the proposed method to a micro-positioning stage with piezoactuator are presented to validate the proposed method by using Matlab and Simulink.The paper consists of six chapters:In chapter 1,the research background and the present situation of the engineering power system energy control theory is introduced,and some basic concepts and theories related to the research contents are given.In chapter 2,an approach to analyze the observability and controllability of sandwich systems with dead-zone is proposed.In this method,a non-smooth state-space function is proposed to describe the sandwich systems with dead-zone which are also non-smooth nonlinear systems.Then,a linearization method based on non-smooth optimization is proposed to derive a linearized statespace function to approximate the non-smooth sandwich systems within a bounded region around the equilibrium point that we are interested in.Afterwards,both observability and controllability matrices are constructed and the methods to analyze the observability as well as controllability of sandwich system with dead-zone are derived.Finally,numerical examples are presented.In chapter 3,using the switching function,more convenient analysis and expression of the backlash operators and non-smooth characteristics are shown.Then,based on non-smooth optimization,a non-smooth state-space function and a linearized state-space function around the equilibrium point to describe the sandwich systems with backlash is proposed.Afterwards,both observability and controllability matrices are constructed and the methods to analyze the observability as well as controllability of sandwich system with backlash are derived.However,in control engineering,it often encounters the problem that the control variable is constrained.Saturation characteristics are common in all practical amplifiers which are also used,intentionally,as limiters to restrict the range of a variable.So we added saturation nonlinearity based on sandwich system with backlash and an approach to analyze the observability and controllability of sandwich systems with backlash and input saturation is proposed.Finally,two numerical examples are presented to validate the proposed method.In chapter 4,An approach to analyze the observability and controllability of micropositioning stage with piezoelectric actuator described by sandwich model with hysteresis is proposed.As hysteresis inherent in piezoelectric actuator is a non-smooth nonlinear function with multi-valued mapping,the positioning system is also a non-smooth dynamic system.In order to investigate the observability and controllability of the system,a non-smooth state-space function with sandwich structure is developed.Moreover,the Prandtl-Ishlinksii(PI)submodel is employed to describe the characteristic of hysteresis embedded in the sandwich system.Then,based on the obtained non-smooth state-space sandwich model,a linearization method based on non-smooth optimization is proposed to derive a generalized linearized state-space function to approximate the non-smooth sandwich systems within a bounded region around the equilibrium points that we are interested in.Afterwards,both observability and controllability matrices are constructed and the methods to analyze the observability as well as controllability of sandwich system with hysteresis are derived.Then,a simulation example and an application of the proposed method to a micro-positioning stage with piezoactuator are presented to validate the proposed method.In chapter 5,an method to analyze the observability and controllability of sandwich systems with hysteresis described by Duhem model is proposed and applied to real micro-positioning stage with piezoactuator.Compared with PI model to describe the hysteresis operator in chapter four,Duhem hysteresis operator can not decompose as a piecewise linear operator combination,therefore Duhem hysteresis operator is more general and complex.After analyzing on the nonsmooth characteristic of Duhem submodel,a non-smooth state-space function is given to describe the sandwich systems with hysteresis.Then,based on non-smooth optimization,we derived a linearized state-space function to approximate sandwich systems within a bounded region around the equilibrium point that we are interested in.Afterwards,both observability and controllability matrices are constructed and the methods to analyze the observability as well as controllability of sandwich systems with hysteresis are derived.Finally,experimental performance results of a micro-positioning stage with piezoactuator are persented to validate the proposed method.In chapter 6,the work of this paper is summarized,and the further research direction is proposed.