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1、Chapter 6: Frequency Domain AnaysisInstructor: Dr. Cailian ChenOutlineConcept Graphical methodAnalysis Introduction of Frequency ResponseFrequency Response of the Typical ElementsBode Diagram of Open-loop SystemNyquist-criterion Frequency Response Based System AnalysisFrequency Response of Closed-lo

2、op SystemsKey problem of control: stability and system performanceQuestion: Why Frequency ResponseAnalysis: 1. Weakness of root locus method relies on the existence of open-loop transfer function2. Weakness of time-domain analysis method is that time response is very difficult to obtainComputational

3、 complexDifficult for higher order systemDifficult to partition into main partsNot easy to show the effects by graphical method6-1 IntroductionThree advantages: * Frequency response(mathematical modeling) can be obtained directly by experimental approaches. * Easy to analyze effects of the system wi

4、th sinusoidal signals * Convenient to measure system sensitivity to noise and parameter variationsFrequency Response AnalysisHowever, NEVER be limited to sinusoidal input * Frequency-domain performances time-domain performances Frequency domain analysis is a kind of (indirect method) engineering met

5、hod. It studies the system based on frequency response which is also a kind of mathematical model. Frequency ResponseTransfer functionExample 5.1: RC circuit By using inverse Laplase transformTransient responseSteady state responseSteady state response ofWhen the input to a linear time-invariant (LT

6、I) system is sinusoidal, the steady-state output is a sinusoid with the same frequency but possibly with different amplitude and phase. The output has same magnitude and phase with inputMagnitude will be attenuated and phase lag is increased.Magnitude response Definition：Frequency response (or chara

7、cteristic) is the ratio of the complex vector of the steady-state output versus sinusoidal input for a linear system.Phase response Magnitude of output versus input Magnitude characteristicPhase characteristicPhase error of output and inputGeneralized to linear time-invariant systemTransfer function

8、 of closed-loop systemwhere are different closed-loop poles. Given the sinusoidal inputTransient responseSteady state responseFor a stable closed-loop system, we have Furthermore, we haveThe magnitude and phase of steady state are as followsBy knowing the transfer function G(s) of a linear system, t

9、he magnitude and phase characteristic completely describe the steady-state performance when the input is sinusoid.Frequency-domain analysis can be used to predict both time-domain transient and steady-state system performance. Substitute s=j into the transfer function Relation of transfer function a

10、nd frequency characteristic of LTI system (only for LTI system)Laplace transformInverse Laplace transformFourier transformInverse Fourier transformIt is a complex vector, and has three forms:Algebraic formPolar formExponential form Vector of frequency characteristic14 We have learned following mathe

11、matical models：differential equation, transfer function and frequency responseDifferential equationFrequency responseTransfer functionImpulsive response15Example 5.2: Given the transfer functionDifferential equation：Frequency response:166-2 Frequency Characteristic of Typical Elements of systemMeasu

12、re the amplitude and phase of the steady-state output Input a sinusoid signal to the control systemGet the amplitude ratio of the output versus input Get the phase difference between the output and inputAre the measured data enough？Data processingChange frequencyyNHow to get frequency characteristic

13、?Magnitude characteristic diagram A- plotPhase characteristic diagram- plotGain-phase characteristic diagramNyquist DiagramBode diagramNyquist diagramPolar form or algebraic form: A and define a vector for a particular frequency . Nyquist Diagram of RC Circuit01/(2)1/2/3/4/5/A()10.890.7070.450.320.2

14、40.20()0-26.6-45-63.5-71.5-76-78.7-90Bode Diagram: Logarithmic plots of magnitude response and phase responseHorizontal axis: lg (logarithmic scale to the base of 10)unit: rad/sLog Magnitude In feedback-system, the unit commonly used for the logarithm of the magnitude is the decibel (dB)Bode Diagram

15、Property 1: As the frequency doubles, the decibel value increases by 6 dB.As the frequency increases by a factor of 10, the decibel value increases by 20 dB.Log scale and linear scaleone decadeoctaveoctaveone decadeone decadeNotation：Logarithmic scale use the nonlinear compression of horizontal scal

16、e. It can reflect a large region of frequency variation. Especially expand the low-frequency range.Logarithmic magnitude response simplify the plotting. Multiplication and division are changed into addition and subtraction.We cannot sketch =0 on the horizontal scale. The smallest can be determined b

17、y the region of interest.Given T=1，plot the Bode Diagram by using Matlab bode(1,1 1) 1.Proportional element It is independent on . The corresponding magnitude and phase characteristics are as follows:Frequency Characteristic of Typical ElementsSeven typical elementsFrequency characteristic2. Integra

18、tion element Frequency characteristicThe corresponding magnitude and phase characteristics are as follows:3. Derivative Element Frequency characteristicThe corresponding magnitude and phase characteristics are as follows:4.Inertial Element Rewrite it into real and imaginary partsMagnitude and phase

19、responsesNyquist diagram is half of the circle with center at (0.5,0) and radius 0.5. Frequency characteristic1/T, L()-20lgT Low-frequency region：High-frequency region： The frequency where the low- and high-frequency asymptotes meet is called the break frequency (1/T).Log magnitude and phase charact

20、eristics are as follows: AsymptoteThe true modulus has a value of Remarks：The error of true modulus and asymptoteT12510L()0.040.170.973.010.970.170.04It can be seen that the error at the break frequency is biggest.T50.512345810()-6-11-14-27-45-63-72-76-79-83-84 () is symmetrical at all rotations about the point 1/T,()=-455. First Derivative ElementFrequency characteristicNyquist DiagramInertial ElementFirst Derivative ElementFrequency characteristics are the inverse each otherLog magnitude characteristic is symmetrical about th