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Review of Frequency Response Analysis

535 APPENDIX B Review of Frequency Response Analysis Objectives The objective of this appendix is to Review the basic concepts behind Frequency re-sponse Analysis and its use in control system design. Topics discussed include Bode and Nyquist plots. Nyquist stability theorem. Closed-loop Response characteristics. Controller performance and design criteria. INTRODUCTION By Frequency Response we mean the Response characteristics of the system when subject to sinusoidal inputs. The input Frequency is varied, and the output characteristics are computed or represented as a function of the Frequency . Frequency Response Analysis provides useful insights into stability and performance characteristics of the control system. Figure shows the hypothetical experiment that is conducted.

536 Review of Frequency Response Analysis Appendix B Input System or Process Output Figure B.1 How frequency response is defined. The system is subject to an input of the form xt A t t() sin( ) 0.= ω > (B.1) After some initial transient period, the output settles down to a sine wave of the form

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Transcription of Review of Frequency Response Analysis

1 535 APPENDIX B Review of Frequency Response Analysis Objectives The objective of this appendix is to Review the basic concepts behind Frequency re-sponse Analysis and its use in control system design. Topics discussed include Bode and Nyquist plots. Nyquist stability theorem. Closed-loop Response characteristics. Controller performance and design criteria. INTRODUCTION By Frequency Response we mean the Response characteristics of the system when subject to sinusoidal inputs. The input Frequency is varied, and the output characteristics are computed or represented as a function of the Frequency . Frequency Response Analysis provides useful insights into stability and performance characteristics of the control system. Figure shows the hypothetical experiment that is conducted.

2 536 Review of Frequency Response Analysis Appendix B InputSystem or ProcessOutput Figure How Frequency Response is defined. The system is subject to an input of the form ()sin( ) => ( ) After some initial transient period, the output settles down to a sine wave of the form ()sin(), =+>> ( ) The amplitude and phase are changed by the system, but the Frequency remains the same. This is shown in Figure y(t) or u(t) , phase lag input wave output wave ABTime, t Figure Frequency Response . Note that the output wave lags behind the input . is defined as the phase lag (usually ex-pressed in degrees or radians). The output amplitude is different from the input , and we can define a ratio.

3 /)(Ratio AmplitudeABAR= ( ) Now let us examine the effect of changing the Frequency of the input . Consider the re-sponse of the level in the tank (see Figure ) to sinusoidal changes in the input flow. Introduction 537 input FlowLevel Figure Self-regulating level Response to inlet flow variations. Let us say that the tank is self-regulating: as the level changes, the outlet flow changes due to the change in hydraulic pressure until, at steady-state, the time average of the inlet flow matches the outlet flow. If the inlet flow changes sinusoidally, the level will respond likewise. At low frequen-cies, the level will have plenty of time to keep pace with the inlet flow changes. At very high frequencies, the tank will not have time to respond to the flow variations and the amplitude of level changes will be small.

4 The tank, in effect, will average out the inlet flow fluctua-tions. The peak in level will occur sometime after the inlet flow has peaked; that is, the changes in level will lag behind the changes in inlet flow. There are a number of ways to represent the Frequency Response of a process. We will use two of these representations: Bode plots and Nyquist plots. A Bode plot is a plot of the amplitude ratio (AR) and the phase lag as a function of the Frequency of the input line wave (which is the same as the Frequency of the output wave). Logarithmic scales are used for the Frequency axis. The y-axis is often plotted using the units of decibels, which is 20 log (AR). Figure shows the Bode plot for a first-order process. In this example both the AR and decrease as the Frequency increases.

5 At low frequencies, the output is able to respond to the slow varying input disturbances with only a small at-tenuation (AR close to 1). However, at higher frequencies, the AR decreases rapidly, ap-proaching an asymptote with a slope of 1 in the log-log graph shown for the first-order self-regulating process of the tank. Note that this system acts as a low pass filter, that removes the high- Frequency inputs. A first-order system has decreasing phase angle, which ap-proaches 90 asymptotically at higher frequencies. This implies that the output will lag be-hind the input (hence the name first-order lag). 538 Review of Frequency Response Analysis Appendix B Frequency (rad/sec)-20-15-10-50 Amplitude Ratio, Decibels ( 20 log(AR))AR)10-1100101-100-80-60-40-200 Phase AngleFrequency (rad/sec)-20-15-10-50 Amplitude Ratio, Decibels ( 20 log(AR))AR)10-1100101-100-80-60-40-200 Phase Angle Figure Bode plot of a first-order process with )1/(1)(+=ssG.

6 The phase angle is usually negative. In this text we use the convention that phase lag is the negative of the phase angle. As increases, becomes more negative ( , the phase lag increases). This again represents the fact that at higher frequencies, the output will peak later than the input . Typically, most processes exhibit a low AR at high frequencies. Hence, any low fre-quencies present in the input signal is passed through the process, whereas high- Frequency components of the input signal are reduced significantly in amplitude as they pass through the process. We can view such a process as a low pass filter, which allows low frequencies to pass through without attenuation. Any periodic signal can be viewed as a composite sum of various Frequency compo-nents (obtained via a Fourier transform of the signal).

7 Likewise, a system can be viewed as a filter that attenuates the input signal according to the Frequency contained in the signal. A Nyquist plot is another way to show the Frequency Response . Here G(i ) is plotted in the complex plane. An example of a Nyquist plot is shown in Figure Note that there are two parts to the curve: one that shows the plot for varying from 0 to and another that shows the curve for varying from to 0. (This is true only for transfer functions with the degree of the denominator polynomial higher than the degree of the numerator.) Frequency Response from Transfer Functions 539 Real AxisImaginary AxisImaginary Figure Nyquist plot of G(s)=1/(s+1)3. Frequency Response FROM TRANSFER FUNCTIONS The Frequency Response can be derived from the transfer function using the following theo-rem: Consider a process with transfer function )(sG.

8 Then the Frequency Response is given by (),().ARG iGi == ( ) The proof of the theorem can be found in many undergraduate texts (see Luyben 1989, for example). Example Consider ,1)(+=sKsG .1)()(+= iKiG 540 Review of Frequency Response Analysis Appendix B Applying the theorem, we get ,) 1( )( 2122 +==KiGAR ).( tan )( 1 == iG This is plotted in Figure for 1 = and 1 =K. As increases, the AR gets smaller and smaller. At , =b the 21 =AR and 4 rad 45 . = = This is called the break Frequency . (Break Frequency is the Frequency at which the low Frequency and high Frequency asymptotes intersect.)

9 The maximum phase lag of 90 is reached as . At low frequencies, , KAR which means that the output amplitude is the input amplitude multiplied by the process gain. The following corollary to the theorem in Example is useful in computing the fre-quency Response of transfer functions in series. Corollary: The Frequency Response of two transfer functions in series is given by ),( s)( )(21sGGsG= ( ) ),( )( G)( 21 GARGARAR= ( ) ()).

10 ( )( 21 GGG += ( ) Thus ARs are multiplied together, whereas the phase angles are additive. For example, we can obtain the Frequency Response of a third-order transfer function using the result of equa-tion series ( ). ,) 1(1,)1 1( )1 1( )1 1( G)( ,)1 ( )1 ( )1 (1)(232 +=+ + +=+++=sARsARsARAR ssssG ).( tan 3 G)( 1 = The result is plotted in Figure Frequency Response from Transfer Functions 541 Frequency (rad/sec)Phase(deg) Magnitude (dB)-80-60-40-20010-1100101-300-200-1000 Frequency (rad/sec)Phase(deg) Magnitude (dB)-80-60-40-20010-1100101-300-200-1000 Frequency (rad/sec)Phase(deg) Magnitude (dB)-80-60-40-20010-1100101-300-200-1000 Frequency (rad/sec)Phase(deg) Magnitude (dB)-80-60-40-20010-1100101-300-200-1000 Figure Bode plot of 1/(s+1)3.)


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