Review Automatic Control Instructor: Cailian Chen, Associate Professor Department of Automation 27 December 2012

Structure of the course Linear Time-invariant System (LTI) Analysis Time Domain Complex Domain Frequency Domain System Model Concepts Performance Compensation Design 各章概念融会贯通 解题方法灵活运用 2017/4/21

Time Constant Canonical Form (Bode Form) ； Root Locus Canonical Form (Evan’s Form) ； 2017/4/21

Stability Analysis Method (1) Routh Formula (2) Root Locus Method (3) Nyquist Stability Criterion Z＝N＋P j  Stable Region Unstable Region [S plane] Stability: All of the roots of characteristic equation of the closed-loop system locate on the left-hand half side of s plane. 2017/4/21

Summary of Chapter 5 Understand and remember of root locus equation Rules for sketching of root locus Calculate Kg and K by using magnitude equation 2017/4/21

当k’ 从 变化时，S平面上系统特征根的变化形成轨迹。每一个k’ 值，按幅值条件对应于根轨迹上的n个点。 R(s) G(s) H(s) + - C(s) E(S) Characteristic equation of the closed-loop system 当k’ 从 变化时，S平面上系统特征根的变化形成轨迹。每一个k’ 值，按幅值条件对应于根轨迹上的n个点。 根轨迹 上的点符合相角条件，且 符合相角条件的点一定在根轨迹上。故利用相角条件就可以绘制根轨迹，无需考虑幅值条件。 Magnitude equation Phase equation Argument equation 2017/4/21 2017/4/21 6 6

Rules for Plotting Root Locus Content Rules 1 Continuity and Symmetry Symmetry Rule 2 Starting and end points Number of segments n segments start from n open-loop poles, and end at m open-loop zeros and (n-m) zeros at infinity. 3 Segments on real axis On the left of an odd number of poles or zeros 4 Asymptote n-m segments： 5

Breakaway and break-in points 6 Breakaway and break-in points 7 Angle of emergence and entry Angle of emergence Angle of entry 8 Cross on the imaginary axis Substitute s = j to characteristic equation and solve Routh’s formula

Important rules Rule 4: Segments of the real axis Rule 5: Asymptotes of locus as s Approaches infinity Rule 6: Breakaway and Break-in Points on the Real Axis 根据根轨迹的定义,当系统增益K值由0→∞ 时,根轨迹的起点必为K=0时的闭环特征根,而终点则为K→∞时的闭环特征根。 Rule 7: The point where the locus crosses the imaginary axis substituting s=jω into the characteristic equation and solving for ω； Use Routh Formula 2017/4/21 2017/4/21 9 9

Summary of Chapter 6 Complete Nyquist Diagram Bode Diagram Nyquist Stability Criterion Relative Stability 2017/4/21

Open-loop transfer function with integration elements Type I system（ν＝1） Type II system (ν = 2)

Type I open-loop system only with inertial elements Type II open-loop system with inertial elements Intercept with real axis is most important, and can be determined by the following method: A. Solve Im[G(jω)]=0 to get ω and then get Re[G(jω)]； B. Solve ∠G(jω) = k·180°(k is an integer)

Draw the complete Nyquist Diagram (ω) ：＋90ν°→ 0°→－90ν°

Bode Diagram: Always Use Asymptotes Change the open-loop transfer function into the Bode Canonical form The slope of lower frequency line is －20νdB/dec，where ν is the type of open-loop system. For ω＝1, L(1)=201gK If there exist any break frequency less than 1, the point with ω＝1 and L(1)=201gK is on the extending line of lower frequency line.

Nyquist stability criterion If N≠－P，the closed-loop system is unstable. The number of poles in the right-hand half s plane of closed-loop system is Z＝N＋P. If the open-loop system is stable，i.e. P=0，then the condition for the stability of closed-loop system is: the complete Nyquist diagram does not encircle the point (－1, j0), i.e. N=0.

Relative Stability Phase margin γ 2. Gain margin γ

相角裕度和增益裕度

Summary of Chapter 7 Phase Lead Compensation Multiplying the transfer function by α 2017/4/21 2017/4/21 18

Rules to design phase lead compensation (1) Determine K to satisfy steady-state error constraint (2) Determine the uncompensated phase margin γ0 (3) estimate the phase margin in order to satisfy the transient response performance constraint (4) Determine Extra margin: 5o~10o (5) Calculate ωm (6) Determine T (7) Confirmation