现控英文版.doc

Modern control theory What have you learned in this course? The course is mainly focused on LTI system. State space model Linear system analysis in time domain Controllability and observability Lyapunov stability theory Linear system synthesis Detail State space model 1.1 State variable State variable is a set of variables(with minimum number), which can completely describe a dynamic system’s behavior, and the variables must be linearly independent. Suppose the number of this set of variables is N, then system is an N-order system. 1.2 State space model General form of state space model is as follows: ...... In the model, A represents system matrix, it’s an n*n order matrix; B represents input matrix, it’s an n*r order matrix; C represents output matrix, it’s an m*n order matrix; D represents direct matrix, it’s an m*r order matrix. Many system have no direct matrix. Block diagram of system is as follows: ...... (pic.) 1.3 Modeling by ordinary differential equation (ODE) E.g. Equation(输出不带微分项)...... A=......B=......C=...... Modeling by T.F. E.g. T.F= (能控规范二型实现)...... A=......B=......C=...... Application: An electric system made up by LRC circuit: (pic.) Equation: ...... *1.4 Jordan canonical form Linear transformation: ...... A,B,C,P (formular) By linear transformation, we can transform a state space model into so called Jordan canonical form. Eigenvalue: ...... Eigenvector: ...... If the geometric multiplicity of all of the eigenvalue are equal to their own algebraic multiplicity, then the system matrix can be transformed into diagonal matrix. E.g G(s)=(异根)...... A=...... P=...... Else if not all of the geometric multiplicity equals to their algebraic multiplicity, the system matrix should be transformed into Jordan canonical form, which is much more complex. E.g G(s)=(重根)...... A=...... P=...... Ji represents Jordan block, Pi is characteristic vector, which belongs to eigenvalue λi. Notice: usually we use M