控制理论基础-自控a_ch6

1,Chapter 6 Frequency Response,6.1 Introduction6.2 Diagram of Frequency Response 6.3 Frequency Response Function of Elementary Unit6.4 Diagram of Complex Frequency Response Function6.5 Summary,2,6.1 Introduction,The steady state response of a system to a sinusoidal input test signal.,3,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 is sinusoidal in the steady state; it differs from the input waveform only in amplitude and phase angle.,4,Input signal isorFor a stable linear system, we have,5,All pi have positive real parts, so the steady-state response of the system is,The steady-state output signal depends only on the magnitude and phase of T(jw) at a specific frequency w.,6,DC motor,7,G(jw)形式：U(w)是G(jw)的实部，称为实频特性，V(w)是G(jw)的虚部，称为虚频特性思考：实际物理系统对频率越高的输入响应幅值越小，说明其传递函数分母阶次高于分子阶次，为什么？,8,The advantage of frequency response method:The experimental determination of the systems frequency response is easily accomplished, and the transfer function can be deduced from the experimental frequency response.The transfer function describing the sinusoidal steady-state behavior of a system can be obtained by replacing s with jw.,9,The disadvantage of frequency response method:The direct correlation between the frequency response and the corresponding transient response characteristics are somewhat tenuous.,10,The relationship between Fourier and Laplace transformsLaplace: Fourier: When f(t) is defined only for t=0, as is often the case, the two equations differ only in the complex variables. Thus the Fourier transform can easily be obtained from Laplace transform just by setting s=jw.,11,The Laplace transform permits us to investigate the s-plane location of the poles and zeros of a transfer G(s). However, the frequency response method allows us to consider the transfer function G(jw) and to concern ourselves with the amplitude and phase characteristics of the system.,12,6.2 Diagram of Frequency Response,Polar plot (Nyquist diagram),Polar plot: The plot of frequency response G(jw) in the complex plane as w increases from 0(-) to +.,13,Steps：1. start point: magnitude and phase (angle).,14,2. end point:,15,3. circumrotate direction:4. intersection（交点 ）on negative real axis,16,Example 6.1: Polar plot of a transfer function,17,18,1、惯性环节,19,2、积分环节,20,3、纯微分环节和一阶微分环节,21,4、二阶对象,22,5、延迟环节,23,Example 6.2: Polar plot of,24,Intersection on the real axis:,25,26,Nyquist diagram can be readily obtained by Matlab num=0 0 10;den=1 3 2 0;nyquist(num,den);v=-5 1 3 3;axis(v);,return,27,Example 6.3:,28,Notes：1.,2. The polar plot is useful for investigating system stability. 3. The limitations of polar plots: the addition of poles or zeros to an existing system requires the recalculation of the frequency response, and the calculation in this manner is tedious （繁琐的）and does not indicate the effect of the individual poles or zeros.,29,Logarithmic plot (Bode plot),Consists of two plots: magnitude plot and phase plot.The magnitude and frequency are in logarithm scale.,30,Logarithmic magnitude-frequency plot the units are decibels(dB).An interval of two frequencies with a ratio equal to 10 is called a decade.,31,32,Logarithmic phase-frequency plot,33,Example：Bode plot for first-order system:,Magnitude plot,34,At the asymptotic line:The frequency w=1/T is often called the break frequency or corner frequency.,The slope of the asymptotic line for this first-order transfer function is 20dB/decade.,35,Phase plot:,36,break frequency,37,Characteristic of Bode plot 1) Expand frequency range. 2) Easy to draw (asymptote) 3) Multiply (divide) can be transformed to plus (subtract).,38,4)the curve move up or down.(gain)5) the curve move left or right.(time constant),39,6) response inverse frequency plot is the plot mirror about the horizontal axis.,40,In comparison with the Nyiquist diagrams, the Bode plots contain additional information about the system in the frequency data. Knowing the Bode plot one can construct the corresponding Nyquist diagram, but the reverse is not possible.,41,Logarithmic Magnitude-Phase Diagram (Nichols),Nichols chartx-coordinate: phase in degree, y-coordinate: amplitude (in logarithmic scale).Nichols chart is a graphical tool for obtaining closed-loop frequency response from open-loop response. It is a popular tool before computers are widely available.,42,6.3 Frequency Response Function of Elementary Unit,1. Proportion(constant gain) Unit,43,44,2. Integrator(poles at the origin),The slope of the magnitude curve is -20dB/dec. The phase angle is 90.,45,46,In this case the slope due to the multiple poles is 20n dB/dec. The phase angle is 90n.,47,3. Differentiator (a zero at the origin),The slope of the magnitude curve is +20dB/dec. The phase angle is + 90.,48,49,4. First-order Unit (a pole on the real axis),50,Approximately, the magnitude response,51,52,The actual logarithmic gain at w=1/T is 3dB. Phase response: range from 0 to 90，center symmetry about the point（1/T,45）.,53,5. First-order Differential Unit (a zero on the real axis) The frequency response can be obtained by the reciprocal relationship (inverse frequency response),54,55,6. Oscillations Unit (complex conjugate poles),56,57,The slope is -40dB/dec.,58,59, =0.1, =0.3, =0.5, =0.7, =1, =0.1, =0.3, =0.5, =0.7, =1,60,Resonant frequency:,When the damping ratio approaches zero, them wr approaches wn.,61,And the maximum value of the magnitude is,62,7. Second-order Differential Unit (complex conjugate zeros) symmetry with respect to the frequency axis.,63,64,8. Non-minimum phase system A transfer function is called a minimum phase transfer function if all its poles and zeros lie in the left-hand s-plane. It is called a non-minimum phase transfer function if it has poles and zeros in the right-hand s-plane.The range of phase shift of a minimum phase transfer function is the least possible or minimum corresponding to a given amplitude curve.,65,1. A unit of delay: Transfer function of a delay of Td seconds and the frequency response is,66,A time delay added into the loop does not change the magnitude but increases the phase lag of the frequency response. The main effects of time delay are to reduce phase margin and reduce the closed-loop bandwidth.,67,2. Unstable unit,68,6.4 Diagram of Complex Frequency Response Function,69,There have four different kinds of factors:Constant gain KbPoles (or zeros) at the origin (jw)Poles (or zeros) on the real axis (jwt+1)Complex conjugate poles (or zeros),70,1. Decompose transfer function to the elementary unit.,71,2. Calculate the break frequencies.3. Calculate the slope and gain of the low frequency(only consider the proportion and the integrators),72,4. Obtain the asymptotic magnitude plot for the complete transfer function.*5. Furthermore, obtain the actual curves only at specific important frequencies.,73,6. Phase plot： (1). Write the phase angle equation. (2). Calculate the range and estimate the change direction. (3). List the phase angle of some points (the every corner frequency point and wc） (4). Draw the phase angle curve.,74,Example 6.1: The Bode diagram of the TF:1.2. The break frequencies3.,75,4. Asymptotic magnitude plot slopes: 0.4, -40dB/dec 2, -20dB/dec 10, -40dB/dec 20, -60dB/dec*5. revise.,76,6.,77,7. Phase plotRange from 90 to 270.,78,2,0.4,10,20,36,79,Bode plot can be readily obtained by Matlab:num=0 0 10;den=1 3 2 0;bode(num,den),80,Example 6.3 Sketching a bode plot,The factors are1. A constant gain K=52. A pole at the origin 3. A pole at =24. A zero at =105. A pair of complex poles at w=wn =50,81,Plot the magnitude characteristic:1. The constant gain is: 20lg5=14 dB;2. The magnitude of the pole has a slope of 20 dB/decade and is 0 dB at =1.3. Beyond the break frequency at = 2, the slope is - 20 dB/decade and below the break frequency 0 dB/decade.4. Beyond the break frequency at = 10, the slope is +20 dB/decade and below the break frequency 0 dB/decade.5. At = n = 50, the slope of the magnitude for the pair of complex poles is - 40 dB/decade. Because the damping ratio is = 0.3, the approximation must be corrected to the actual magnitude.,82,83,By adding the asymptotes due to each factor, we obtain the total asymptotic magnitude as,84,Plot the phase characteristic:,