上交自控讲义第九讲.ppt

Module 9 Routh’s Method, Root Locus: Magnitude and Phase Equations Requirements on the Module Grasp the relationship of System Stability with Roots of System Characteristic Equation Grasp the necessary and sufficient requirements for Routh’s Criterion Understand some special situations Grasp the principle of Root Locus Definition of Lyapunov stability The fundamental condition for a normally working control system is that it is stable incline to an equilibrium state in a finite time under no input but initial conditions (or disturbances) Marginally stable Asymptotically stable Linear system stability and its necessary and sufficient requirements The system will became unstable as soon as one closed-loop pole is located in the right-hand half of the complex plane System responses corresponding to closed-loop poles’ location Diverse methods on system stability determination for linear systems Methods directly depended on root: Solving Differential equation Root locus methods by determining the root scope Routh’s method Nyquist criterion Nyquist criterion on Bode diagram Lyaponuv Stability Criterion Introduction of Routh’s Method The closed-loop poles are the roots of the system characteristic equation Why the method solving equation is not taken into use? Solving a high-order characteristic equation is tedious or even impossible without using numerical methods It is unnecessary to calculate exact value of every root for determining if there is anyone on the right-hand half of the complex plane Principle of Routh’s Stability Criterion Relationship of roots with coefficients for high-order algebra equations Routh’s criterion:  Necessary requirement The necessary requirements for a stable system are that, No zero exists among the coefficients of D(s) All coefficients of D(s) possess the same sign Routh’s criterion:  Sufficient requirement Establish the Routh array in the following manner: The reminder of the row elements shoul