《自动控制原理》课程教学资源（PPT课件讲稿）Module12

Module 12 Frequency Response and Nyquist diagram (2 hours)

Module 12 Frequency Response and Nyquist Diagram (2 hours)

12. 1 Introduction of the Frequency response 12.1.1 Why do we should study the Frequency Response (P223: Paragraph 2) 12. 1.2 What is the Frequency Response? The frequency response of a system is defined as the steady-state response of the system to a sinusoidal input signal. The sinusoid is a unique input signal, and the resulting output signal for a linear system, as well as signals throughout the system, is sinusoidal in the steady state, it differs from the input waveform only in amplitude and phase angle

12.1 Introduction of the Frequency Response 12.1.1 Why do we should study the Frequency Response (P223: Paragraph 2) 12.1.2 What is the Frequency Response? —The frequency response of a system is defined as the steady-state response of the system to a sinusoidal input signal. The sinusoid is a unique input signal, and the resulting output signal for a linear system, as well as signals throughout the system, is sinusoidal in the steady￾state; it differs from the input waveform only in amplitude and phase angle

For example, consider the system Y(s)=T(SR(S with r(o)=A sin wt. We have Ao R(S)= s2+d2 ane T!) q(s) ∏(s+P) where p i are assumed to be distinct poles. Then in partial fraction form we have kn, as+B +…+ stpr s+pn st Taking the inverse laplace transform yields y()=k1e-4+…+ke4+9 aS+β s2+

where a and B are constants which are problem dependent. If the system is stable, then lPi have negative nonzero real parts, and +β im y(t (=lim g f→0 s2+a2 since each exponential term;e 'i'decays to zero as t→的 In the limit for (t), we obtain for t-)oo(the steady state) W()=$-1 as+B AoT(ja)lsin(ot +9) =AT(ja )sin(ot +o), where

Consider the following system R CKHG(S Can we determine a value of K, for which the system is stable, without knowing the transfer function G() exactly? Yes, if we can measure the frequency response of the open loop system Recall that if we subject a linear system to a harmonic input, the output is also harmonic with a phase shift r()=he0-“G()-+c()=4m1=dee=cgl

Recall from Module 5, that the magnitude and phase of the harmonic response of a system can be obtained from its transfer function by substituting sjo Mo Glo ∠G(O There are two major representations of the frequency response: bode plot and Nyquist diagram M m▲ O=0 0=∞ q e G(O) Bode plot Nyquist diagram

The frequency response is a kind of model of system G()=G(s)= d Differential Equation Transfer Function System dt S=JO Frequency Response

• –– The frequency response is a kind of model of system:  s j G j G s = = ( ) ( ) Differential Equation Transfer Function Frequency Response System dt d s = s = j dt d j =

12.2 Different forms of the Frequency Response Nyquist Diagram Section 8.2 Frequency Response Plots X(a) Negative w R( 0 Positive a

• Nyquist Diagram 12.2 Different forms of the Frequency Response

Bode Diagram M Fig. 12.3 Bode representation of frequency response

• Bode Diagram

Nichols Diagram 10 e 10 0 「o 30 40 270 225 180 135 Phase, degrees FIGURE Log-magnitude-phase curve for GH,(w)

• Nichols Diagram