Output Feedback Stabilization Of Cascaded Infinite Dimensional Systems With Disturbance

Distributed parameter system mainly described by partial differential equation,integral equation,and abstract differential equation in Banach space or Hilbert space.The specific content includes the controller design and stability analysis of the system.In recent years,the research of a class of infinite dimensional system with uncertain external disturbance has become a widespread concern in the field of distributed parameter system control.At the same time,it has become a hot and difficult issue for experts and scholars all over the world.In daily life,the motion of objects can be seen everywhere,accompanied by the occurrence of vibration phenomenon.However,some vibration plays a positive role in the motion of objects,but some vibration is not what we hope to happen.For example,hand vibration,wind vibration of the pipeline at the inlet of the turbine,the vibration of the turbine,the friction vibration and so on.All of them will cause great harm to us.For example,the famous Broughton suspension bridge,due to the resonance,caused the collapse of the bridge.We call this vibration phenomenon uncertain disturbance.Also,the vibration similar to the above mentioned needs to be suppressed.Therefore,it is particularly important to reduce and solve the harm caused by this kind of disturbance.There are many examples of partial differential equations in nature.Many physical phenomena can also be described by partial differential equations,which provides a strong practical background for distributed parameter system control.In fact,we often extract corresponding models in chemical engineering,thermal energy equation and missile control.There are many similar examples,of course,in ecological environment system and social system.In this paper,the problem of feedback stabilization for coupled systems of ordinary differential equations and partial differential equations with uncertain external disturbances is studied.There are many examples of coupling system,among which the typical coupling systems include spacecraft launch vehicle coupling system,multi-channel coupling of spacecraft,satellite attitude and trajectory coupling,and the coupling system of aero-engine double rotor rolling bearing gearbox,etc.All of these provide convenience for our life.But there are also some mechanical movements in which the coupling treatment is not good,which brings us great harm.For example,the hypersonic vehicle HTV-II of American,which was once poorly handled due to the inertia coupling problem,failed to fly.There also has a history of low positioning accuracy due to the unreasonable coupling of satellite attitude and trajectory in China.Looking at the examples of this kind of coupling system,we know that it is necessary to study this kind of coupling infinite dimensional system with disturbance.Specifically,in this paper,several kinds of stabilization problems(including the wellposedness of solutions and the stability of systems)of coupled systems of ordinary differential equation and partial differential equation are considered.The main methods applied are backstepping transformation and Lyapunov analysis.In Chapter 2 and 3,the feedback stabilization problems of coupled systems of ordinary differential equations and parabolic equations are studied.In Chapter 4,the coupling system of ordinary differential equation and Schrodinger equation is studied.In Chapter 5,the problem of feedback stabilization for the coupled system of ordinary differential equation and Korteweg de Vries(KdV)equation is analyzed.The output feedback method used in this paper provides a cost-effective method for the stabilization of coupled systems with disturbance.In Chapter 2,the problem of feedback exponential stabilization of coupled ordinary differential equation and heat equation system connected by Dirichlet on boundary is studied,in which the disturbance and control are both at the right end of the region.A coupled state observer with unknown input is designed for the original system to stabilize the original system and compensate for the external disturbance.The stable state feedback controller of the observer system is designed through three invertible transformations,that is,the output feedback stability controller based on the observer in the original system.Finally,we prove that the closed-loop system is exponentially stable.Since the boundary disturbance of the coupled system only appears on the boundary of the partial differential system.Therefore,we try to design a single observer to estimate the disturbance first and then to stabilize the coupled system.So,in Chapter 3,we consider the feedback stabilization of the cascaded system of ordinary differential equation and heat equation with both control and external unknown disturbances.Taking advantage of the stability of the auxiliary system,a state observer is designed,which can not only to stabilize the coupling system and compensate for external interference at the same time.The state feedback controller is designed for the observer system by using the classical inversion transformation.In other words,the corresponding observer based output feedback controller is designed for the original system.The results show that the closed-loop system is exponentially stable.Finally,the numerical simulation is used to verify the effectiveness of the method.The biggest difference between this Chapter and the second Chapter is that in this chapter,an extended state observer for a single partial differential equation is constructed to estimate the original system,which is also a highlight of this chapter.Considering that the regularity of Schrodinger equation is weaker than that of heat equation,so some conclusions of heat equation do not hold in Schrodinger equation.Therefore,in Chapter 4,we discuss the problem of output feedback exponential stabilization for a class of coupled systems of ordinary differential equation and Schrodinger equation.Based on three measurable signals,a new extended state observer(ESO)is proposed,which can estimate the states and disturbances of the original system simultaneously.Then,a stable controller is designed by using the inverstible transformation method.And the closed-loop system obtained is proved to be exponentially stable.At the same time,all internal systems involved are uniformly bounded.In Chapter 5,the output feedback stabilization problem of the coupled system of ordinary differential equation and linear Korteweg-de Vries equation connected by Dirichlet boundary is considered.Firstly,the unknown input extended state observer of the coupled system is constructed.And then the feedback controller is designed by backstepping transformation.Finally,the exponential stability of the closed-loop system is proved by Lyapunov analysis method.