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In1932, American physicist Nyquist (H. Nyquist)
proposed the frequency response method. (频率 响应法)
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The development of classical control theory
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The root locus method
Definition
The root locus is the path of the roots of the
characteristic equation traced out in the s-plane as a system parameter (参数)is changed. Explanation The root locus is a curve of the location of the poles of a transfer function as some parameter (generally the gain K) is varied. The number of zeros does not exceed the number of poles.
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The root locus procedure
• Step 1:Write the characteristic equation as： • Step 2:Factor(分解) P(s),write the polynomial （多项式） in the form of poles and zeros.
k 0,1,2,n m 1
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Bode Plot
A Bode plot is a graph of the transfer function of a linear, time-invariant system versus frequency, plotted with a logfrequency axis, to show the system's frequency response.
• Step 6 :Asymptotes（渐 近线） of the root loci. • 渐近线与实轴相交点的坐 m 标为： n
j
A
p z
i 1 i i 1
i
180o p3 -2 p2 -1 60o -60o p1 0
nm
• 渐近线与实轴正方向的夹 角为：
( 2k 1) A nm
• The root locus on the real axis always lies in a section of the real axis to the left of an odd number of poles and zeros.
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The root locus procedure
Bode(伯德) and Nichols（尼科尔斯） proposed
the frequency- domain analysis in the field of
classical control theory.
Wiener （维纳）published “ cybernetics（控制论） on the animal and the machine control and communication science . "
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j
p3
p2
z1
p1 0
p4 z3
The root locus procedure
• Step 4:Locate the segments of the real axis that are root loci. • Step 5: The root loci must be symmetrical（对 称的） with respect to the horizontal real axis.
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Classical control theory
• Classical control theory is mainly used to solve the analysis and design problemsstems.
In 1868, British physicist J.C. Maxwell（麦斯威尔）
proposed algebraic stability criterion.（稳定性代 数判据）
Rolls (EJ Routh) and Hurwitz (A. Hurwitz) proposed two famous stability criterion - Routh criterion and Hurwitz criterion.(劳斯判据和胡尔维茨判据)
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Bode plot
Bode magnitude plot (幅频图) • expresses the magnitude of the frequency response gain Bode phase plot (相频图) • expresses the frequency response phase shift
The famous Chinese scientist Qian Xuesen（钱学森）
applied control theory to engineering, publishing “ engineering cybernetics .“(工程控制论)
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Classical control theory
The root locus procedure
• Step 3:Locate the poles and zeros of P(s) on splane with selected symbols.
z
2
(s p ) K (s z ) 0
i i i 1 i 1
n
m
• NOTICE:
m
1 KP( s) 0
K is the parameter that is variable. Example:
1
K ( s zi )
s( s 4) P( s) 2 s ( s 4) s 5
(s p )
i i 1
i 1 n
0
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Translation The transfer function, root locus method and
welcome to use these PowerPoint templates, New
frequency-domain analysis are Content design, 10 years experience the primary
Introduction of classical control theory
Content
Development Root locus plot Frequency-domain analysis
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The development of classical control theory
• The locus of the roots of the characteristic equation 1+KP(s)=0 begins at the poles of P(s) and end at the zeros of P(s) as K increases form 0 to infinity(无穷).
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Exercise
• 系统的开环传递函数 G(s)=k/[s(T1*s+1)*(T2*s+1)] (T1>T2)，画 出伯德图。
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Nyquist stability criterion
• The Nyquist stability criterion is a graphical technique for determining the stability of a system. Because it only looks at the Nyquist plot of the open loop systems, it can be applied without explicitly computing the poles and zeros of either the closed-loop or openloop system.
analysis and design tools in classical control
theory. They constitute the basic framework of classical control theory.
传递函数，根轨迹和频域分析是经典控制论中重要的分析设 计工具。他们组成了经典控制论的基础框架。