Chapter 6 Kalman’s Formalism for State Stabilization and Estimation Chúng tôi sẽ hiển thị như thế nào, dựa trên một đại diện nhà nước của một hệ thống tuyến tính liên tục thời gian hoặc thời gian rời rạc, có thể xây dựng một vòng phản hồi tiêu cực, bằng cách giả định ban đầu rằng tất cả các biến nhà nước đo lường được. Sau đó, chúng tôi sẽ giải thích như thế nào, nếu nó không phải là trường hợp, nó có thể để xây dựng nhà nước với sự giúp đỡ của người quan. | Chapter 6 Kalman s Formalism for State Stabilization and Estimation We will show how based on a state representation of a continuous-time or discrete-time linear system it is possible to elaborate a negative feedback loop by assuming initially that all state variables are measurable. Then we will explain how if it is not the case it is possible to build the state with the help of an observer. These two operations bring about similar developments which use either a pole placement or an optimization technique. These two approaches are presented successively. . The academic problem of stabilization through state feedback Let us consider a time-invariant linear system described by the following continuous-time equations of state x t A x t Bu t x 0 0 where x e Rn is the state vector and u e Rm the control vector. The problem is how to determine a control that brings x t back to 0 irrespective of the initial condition x 0 . In this chapter our interest is mainly in the state feedback controls which depend on the state vector x. A linear state feedback is written as follows Chapter written by Gilles DUC. 160 Analysis and Control of Linear Systems u t -K x t e t where K is an m X n matrix Figure and signal e t represents the input of the looped system. The equations of the looped system are written as follows x t A - BK x t Be t Figure . State feedback linear control Hence the state feedback control affects the dynamics of the system which depends on the eigenvalues of A - BK let us recall that the poles of the open loop system are eigenvalues of A similarly the poles of the closed loop system are eigenvalues of A - B K . In the case of a discrete-time system described by the equations Xk 1 F Xk G Uk X0 0 the state feedback and the equations of the looped system can be written Uk K Xk ek xk 1 F - GK xk Ge k x0 0 so that the dynamics of the system depends on the eigenvalues of F - GK . The research for matrix K can be done in various ways.
