سیستمهای کنترل خطی پاییز 1389 بسم ا... الرحمن الرحيم 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 p 1, p 2,…,p n 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 G(s)G(s)G(s)G(s) r(t) = A sin(  t) c(t) = |G(j  )| A sin(  t +  ) the steady-state output is That is, the steady-state response depends only on the magnitude and phase of T(jω). For the system

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. Advantages of the frequency response method

Key Components of Frequency Response Analysis

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

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

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. 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 How to enlarge the lower frequency band and shrink (shorten) the higher frequency band ？ Idea:

 Consider the general form of transfer functions.  This may be written in Bode form by dividing through by all the constants.   )2))((( ))( ( )( nn k jjpjj zjzjK jG       jj    ))(21 1( ) 1 1( )(          nn k B j p j j zz K jG        

 Now consider the gain of G ( j  ) in dB.  where the Bode Gain is  zzK  n B p K                dB nn k B j p j j z j z j K jGjG )( 11 )(log20)(       

 The angle of G ( j  ) 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    1 1)( k p j j             n n j          )( / B z j z j Kj G   

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

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

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

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

The polar plot and the Bode diagram: Polar plot Re Im Bode diagram 0dB, 0 o

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

5. Differentiating element Transfer function: Polar plot Re Im Re Im 1 Re Im 1 differential1th-order differential2th-order differential 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, 0 o differential 1th-order differential 0dB, 0 o dB, 0 o th-order differential Frequency Response of The Typical Elements

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

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 = [ ]; den = conv([1 1o], [1 50]); bode(num, den); Bode plots are useful as an analytic tool. method to plot the magnitude response of the Bode diagram

Example: Comparison of Exact and Bode Plots