دکتر حسين بلندي- دکتر سید مجید اسما عیل زاده بسم ا... الرحمن الرحيم سیستمهای کنترل خطی پاییز 1389 دکتر حسين بلندي- دکتر سید مجید اسما عیل زاده

What is frequency response So far we have described the response and performance of a system in terms of complex frequency variable s=σ+jω and the location of poles and zeros in the s-plane. An important alternative approach to system analysis and design is the frequency response method. The frequency response of a system is defined as the steady-state response of the system to a sinusoidal input signal. We will investigate the steady-state response of the system to the sinusoidal input as the frequency varies.

When the input signal is a sinusoid, the resulting output signal for LTI systems is sinusoidal in the steady state, it differs from the input only in amplitude and phase. where p1, p2,…,pn are distinctive poles, then in partial fraction expansion form, we have Taking the inverse Laplace transform yields Suppose the system is stable, then all the poles are located in the left half plane and thus the exponential terms decay to zero as t→∞. Hence, the steady-state response of the system is

Process Exposed to a Sinusoidal Input For the system the steady-state output is That is, the steady-state response depends only on the magnitude and phase of T(jω). G(s) r(t) = A sin(w t) c(t) = |G(jw)| A sin(w t + q )

Advantages of the frequency response method The sinusoidal input signal for various ranges of frequency and amplitude is readily available. It is the most reliable and uncomplicated method for the experimental analysis of a system. Control of system bandwidth as well as some measure of the response of the system to undesired noise and disturbances. The TF describing the sinusoidal steady-state behavior of the system is easily obtained by replacing s with jω in the system TF.

Key Components of Frequency Response Analysis

Graphic expression of the frequency response 1. Rectangular coordinates plot Example

2. Polar plot The polar plot is easily useful for investigating system stability. Example The magnitude and phase response: Calculate A(ω) and 　　 for different ω: Re Im -117o -135o

Bode diagram(logarithmic plots) The shortage of the polar plot and the rectangular coordinates plot: to synchronously investigate the cases of the lower and higher frequency band is difficult. Idea: How to enlarge the lower frequency band and shrink (shorten) the higher frequency band？ Bode diagram(logarithmic plots) Plot the frequency characteristic in a semilog coordinate: Magnitude response — Y-coordinate in decibels: X-coordinate in logarithm of ω: logω Phase response — Y-coordinate in radian: X-coordinate in logarithm of ω: logω

Constructing Bode Plots: Asymptotic Approximations Consider the general form of transfer functions. This may be written in Bode form by dividing through by all the constants. L ) 2 )(( ( )( 1 n k j p z K G w Vw + = L ) )( 2 1 ( ÷ ø ö ç è æ - + = n k B j p z K G w V w w j j

Constructing Bode Plots: Asymptotic Approximations z K 2 1 n B p w = where the Bode Gain is Now consider the gain of G(jw) in dB. L - ÷ ø ö ç è æ + = dB n k B j p z K G 2 1 10 ) ( log 20 w V

Constructing Bode Plots: Asymptotic Approximations The angle of G(jw) may be written as Thus it is clear that for both magnitude in dB and the angle, the total transfer function may be written in terms of the sum of its components L + = 2 1 ) ( / B z j K G w + - 1 ) ( k p j w L ÷ ø ö ç è æ 2 n V

Frequency Response of The Typical Elements The typical elements of the linear control systems 1. Proportional element Transfer function: Frequency response: Re Im K 0dB, 0o 100 10 1 0.1 Polar plot Bode diagram

Frequency Response of The Typical Elements 2. Integrating element Transfer function: Frequency response: 0dB, 0o 100 10 1 0.1 Re Im Polar plot Bode diagram

Frequency Response of The Typical Elements 3. Inertial element Transfer function: 1/T: break frequency 0dB, 0o 100 10 1 0.1 Re Im 1 Polar plot Bode diagram

Frequency Response of The Typical Elements 4. Oscillating element Transfer function: maximum value of ： Make:

The polar plot and the Bode diagram: Re Im 0dB, 0o 100 10 1 0.1 1 Polar plot Bode diagram

The transfer function of a 2nd-order system: Second-Order System The transfer function of a 2nd-order system: The frequency response of this system can be modeled as: When : 40 dB/decade Changes by  ]i=k,

Frequency Response of The Typical Elements 5. Differentiating element Transfer function: Re Im Re Im Re Im 1 1 differential 1th-order differential 2th-order differential Polar plot

Frequency Response of The Typical Elements Because of the transfer functions of the differentiating elements are the reciprocal of the transfer functions of Integrating element, Inertial element and Oscillating element respectively, that is: the Bode curves of the differentiating elements are symmetrical to the logω-axis with the Bode curves of the Integrating element, Inertial element and Oscillating element respectively. Then we have the Bode diagram of the differentiating elements:

Frequency Response of The Typical Elements 0dB, 0o 100 10 1 0.1 0dB, 0o 100 10 1 0.1 differential 0dB, 0o 100 10 1 0.1 2th-order differential 1th-order differential

Frequency Response of The Typical Elements 6. Delay element Transfer function: Re Im 0dB, 0o 100 10 1 0.1 R=1 Polar plot Bode diagram

method to plot the magnitude response of the Bode diagram Transfer function: The critical frequencies are  = 2 (zero), 10 (pole), and 50 (pole). MATLAB (exact resp.): w = logspace(-1,3,300); s = j*w; H = 1000*(s+2)./(s+10)./(s+50); magdB = 20*log10(abs(H)); phase = angle(H)*180/pi; MATLAB (Bode): num = [1000 2000]; den = conv([1 1o], [1 50]); bode(num, den); Bode plots are useful as an analytic tool. ]i=k,

Example: Comparison of Exact and Bode Plots ]i=k,

Then add them together to get the system asymptotic approximation. method to plot the magnitude response of the Bode diagram ) 2 )( 5 ( 10 + = s G Bode Form: Plot the asymptotic approximations for each term separately, for both magnitude and angle. Then add them together to get the system asymptotic approximation. Sketch in the Bode plot curve. ) 2 / 1 )( 5 ( 10 w j G + =

) 2 / 1 )( 5 ( 10 w j + 5 / 1 w j + 2 Frequency (rad/sec) Phase (deg) Magnitude (dB) -60 -40 -20 20 40 60 10 -2 -1 -180 -160 -140 -120 -100 .2 .5 -90

method to plot the magnitude response of the Bode diagram

( ) 5 100 G ( s ) = × s + 1 s + 2 s + 100 Bode Form: method to plot the magnitude response of the Bode diagram 5 100 G ( s ) = × s + 1 s 2 + 2 s + 100 Bode Form: The damping ratio for the second-order term is z = 0.1 and the natural frequency is 10 rad./s . ( ) 2 10 / . 1 5 100 w - + × = j G

( ) 10 / . 1 5 100 w - + × = j G 2 -20dB/dec. 14dB -60dB/dec. 1 5 100 w - + × = j G Frequency (rad/sec) Phase (deg) Magnitude (dB) -60 -40 -20 10 -1 1 2 -250 -200 -150 -100 -50 14 -60dB/dec. -20dB/dec. -270 p = 1 wn = 10 -45/dec. -135/dec. -90/dec. 14dB

method to plot the magnitude response of the Bode diagram 0dB, 0o 100 10 1 0.1 －20dB/dec －20dB/dec 20dB, 45o -20dB, -45o -40dB, -90o 40dB, 90o -80dB,-180o -60dB.-135o -100dB,-225o -120dB,-270o 1.25dB －60dB/dec There is a resonant peak Mr at: