tailieunhanh - Advanced Control Engineering - Chapter 5

Classical design in the s-plane Tính ổn định của hệ thống năng động Đáp ứng của một hệ thống tuyến tính kích thích có hai thành phần: (a) trạng thái ổn định thuật ngữ liên quan trực tiếp đến đầu vào (b) điều kiện thoáng qua hoặc theo cấp số nhân, hoặc dao động với một phong bì hình thức theo cấp số nhân. Nếu về phân rã theo cấp số nhân làm tăng thời gian, sau đó hệ thống được cho là ổn định. Nếu các điều khoản theo cấp số nhân tăng theo thời gian ngày càng tăng, hệ. | 5 Classical design in the s-plane Stability of dynamic systems The response of a linear system to a stimulus has two components a steady-state terms which are directly related to the input b transient terms which are either exponential or oscillatory with an envelope of exponential form. If the exponential terms decay as time increases then the system is said to be stable. If the exponential terms increase with increasing time the system is considered unstable. Examples of stable and unstable systems are shown in Figure . The motions shown in Figure are given graphically in Figure . Note that b in Figure does not represent b in Figure . The time responses shown in Figure can be expressed mathematically as For a Stable xo t Ae sinU For b Unstable vo f Aeat sinU-7 For c Stable xo t Ae n For d Unstable xo t Aeơt From equations - it can be seen that the stability of a dynamic system depends upon the sign of the exponential index in the time response function which is in fact a real root of the characteristic equation as explained in section . Classical design in the s-plane 111 a Stable t Fig. Stable and unstable systems. c d Fig. Graphical representation of stable and unstable time responses. 112 Advanced Control Engineering Stability and roots of the characteristic equation The characteristic equation was defined in section for a second-order system as as1 bs c 0 The roots of the characteristic equation given in equation were shown in section . to be b ựồ2 4ức r 51 52 ------ --------- 2a These roots determine the transient response of the system and for a second-order system can be written as a Overdamping 51 -ơq 52 -Ơ2 b Critical damping 51 52 Ơ c Underdamping 51 52 Ơ jw If the coefficient b in equation were to be negative then the roots would be 51 52 jw The roots given in equation provide a stable response of the form given in Figure a and .