Automatic Control Theory School of Automation NWPU Teaching Group of Automatic Control Theory

Excecies(23) 5 — 19 ( 用坐标纸 ), 20, 21 5 — 22, 23 ( 选做 ) Automatic Control Theory

Review Unstable Stable Error Note: 1. The Nyquist path follows a circle with infinite small radius in the right- half s plane when there exist open-loop poles on the imaginary axis. Accordingly, there is a large circle with radius infinity in the G plane. Z: The number of closed-loop poles in the right-half s plane P: The number of open-loop poles in the right-half s plane N: The number of circles the Nyquist plot of GH(j  ) surrounding the point (- 1, j0). Nyquist Stability Criterion 2. The minimum unit of N is.

§5.4.3 Stability Criterion in Bode Diagrams (5) Example 2

§5.4.3 Stability Criterion in Bode Diagrams (6) Example 3 Unstable Stable Error

Automatic Control Theory （ Lecture 23) §5. Analysis and Adjustments of Linear Systems in Frequency-Domain §5.1 Concept of Frequency-Response Characteristics §5.2 Amplitude-phase Frequency Characteristics §5.3 Bode Diagrams §5.4 Nyquist Stability Criterion §5.5 Stability Margins §5.6 System Analysis by Frequency Response Characteristics of Open-Loop Systems §5.7 Nichols Chart §5.8 System Analysis by Frequency Response Characteristics of Closed-Loop Systems §5.9 Control Systems Design by Frequency Response

Automatic Control Theory §5.5 Stability Margins §5.5.1 Definitions of Stability Margins §5.5.2 Calculations of Stability Margins （ Lecture 23 ）

§5.5 Stability Margins (1) Time domain （ t ） The dynamical performance Stable boundary Frequency domain (  ) How stable the system is Imaginary axis Damped ratio  The distance to (-1,j0) (-1,j0) Stability margins (Open-loop frequency indices ) How stable it is

§5.5 Stability Margins (2) § The Definition of Stability Margins The geometry meaning of Cutoff frequency  c Phase margin Gain margin The stability depth on the Gain Phase Generally The physics meaning of Phase crossover frequency

§5.5 Stability Margins (3) §5.5.2 Calculations of Stability Margins Solution I ： Obtain  and h by the Nyquist plot Example 4 ， obtain (1) Let T. & E.

§5.5 Stability Margins (4) (2.1) Let We have

§5.5 Stability Margins (5) Rewrite G(j  ) into the real part plus the imaginary part Let We have Substitute to the RP

§5.5 Stability Margins (6) From L(  ): We have Solution II ： Determine by Bode diagram

§5.5 Stability Margins (7) Solution. Determine by L(  ) Solution I: Example 5 ， Determine Solution II:

§5.5 Stability Margins (8) Obtain  g Rewritten as We have

Summary Concept (Open-loop frequency index) Definitions Calculations Cutoff frequency  c Phase margin  Phase crossover frequency  g Amplitude margins h Meanings The geometry meaning of The physical meaning of

Excecies(23) 5 — 19 ( 用坐标纸 ), 20, 21 5 — 22, 23 ( 选做 ) Automatic Control Theory

§5.3.2 Bode Diagram For Open-loop Systems （ 11 ） Example 8 The Bode diagram of open-loop is shown in figure. Determine the G(s) Solution: According to the problem Determine K:

§5.3.2 Bode Diagram For Open-loop Systems （ 12 ） ⑴ ⑵ ⑶ ⑷

§5.3.2 Bode Diagram For Open-loop Systems （ 13 ） Non-minimum phase system ——The system which exists open-loop zeros or poles in the right half s-plane ★ Non-minimum phase system is not always unstable ★ Non-minimum phase system doesn't always sketch 0°root-locus 非最小相角系统由 L(  ) 不能惟一确定 G(s) ★ 最小相角系统由 L(  ) 可以惟一确定 G(s) ★ The absolute value of phase angle variation of the non-minimum phase system is larger than minimum phase system