现代控制系统第六章.ppt

SUMMARY 1. The Routh-Hurwitz Stability Criterion 2. The Relative Stability of Feedback Control Systems The Routh – hurwitz criterion states that the number of roots of q(s) with positive real parts is equal to the number of changes in sign of the first column of the Routh array. The Routh – hurwitz criterion is a necessary and sufficient criterion for the stability of linear systems. 1. The Routh-Hurwitz Stability Criterion Case1 No element in the first column is zero. Four Cases: Case 2 There is a zero in the first column ,but some other elements of the row containing the zero in the first column are nonzero. Case 3. There is a zero in the first column ,and the other elements of the row containing the zero are also zero. Case 4 As in (3) with repeated roots on the Figure 6.6 Root locations in the s-plane 2. The Relative Stability of Feedback Control Systems THANKS Chapter 6 Feedback Control System characteristics 6.1 The Concept of Stability 6.2 The Routh-Hurwitz Stability Criterion 6.3 The Relative Stability of Feedback Control Systems 6.1 The Concept of Stability A stable system is a dynamic system with a bounded response to a bounded input. The concept of stability can be illustrated by following figure Figure 6.1 The stability of a cone. The location in the s-plane of the poles of a system indicates the resulting transient response, as following figure. Figure 6.2 Stability in the s-plane. Tacoma Narrows Bridge Figure 6.3 Tacoma Narrows Bridge (a) as oscillation begins (b) at catastrophic failure. For linear system, the closed-loop system transfer function is: Where is the characteristic equation whose roots are the poles of the closed-loop system. The output response for a impulse function input (when N=0) is A necessary and sufficient condition for a feedback system to be stable is that: all the poles of the system function have negative real part. Figure 6.4 The M2 robot is more energy-