自动控制理论 I_ch4.pptVIP

特征多项式/辅助多项式-余下的多项式重新做劳思表 辅助多项式对s求导后系数代替全0行系数-继续劳思表 * * Chapter 4 The Stability of Linear Feedback Systems 4.1 The Concept of Stability4.2 Routh-Hurwitz Stability Criterion 4.1 The Concept of Stability A stable system is dynamic system with a bounded response to a bounded input. Stability is the fundamental requirement of a control system. Absolute stability —— stable / not stable Relative stability —— the degree of stability The response of a linear system to a stimulus has two component: (1). Steady state terms which are directly related to the input. (2). Transient terms which are either exponential or oscillatory with an envelope at exponential form. 　 If the exponential terms decay as time increases then the system is said to be stable; otherwise the system is said to be unstable. Stability means that with no input, the output will decay to zero eventually.　 When the transfer function of system is T(s), the output Y(s) is: According to inverse Laplace transform, there is: the system is stable. then total response , so the system is unstable. The conclusion will be gotten as following: The system is stable only when all the closed-loop poles are located in the left-hand position (LHP) of the s-plane. The system becomes unstable as soon as one closed-loop pole is located in the right-hand position (RHP) of the s-plane. If a system has simple roots on the imaginary axis with all other roots in the LHP, it is called marginally stable. 4.2 Routh-Hurwitz Stability Criterion Consider the following system: A necessary but not sufficient criterion: For a stable system, all the coefficients of the polynomial D(s) must have the same sign and be nonzero. The Routh-Hurwitz criterion is a necessary and sufficient criterion. ? Col 1 Col 2 Col 3 Row 0 s n a n a n-2 a n-4 Row 1 s n-1 a n-1 a n-3 a n-5 Row 2 s n-2 b 1 b 2 b 3 Row 3 s n-3 c 1 ... ... . ... ... ... ... . ... ... ... ... Row n-1 s 1 y 1 y 2 Row n s 0 z 1 Routh