In The Name of God The Most Compassionate, The Most Merciful Linear Control Systems 2017 Shiraz University of Technology Dr. A. Rahideh Table of Contents 1. Introduction to Control Systems 2. Mathematical Modelling of Dynamic Systems 3. Steady State and Transient Response Analysis 4. Root Locus Analysis 5. Frequency Response Analysis 2 2017 Shiraz University of Technology Dr. A. Rahideh Chapter 5 Frequency Response Analysis 5.1. Introduction 5.2. Bode Diagrams 5.3. Polar Plots (Nyquist) 5.4. Log-Magnitude vs. Phase Plots (Nichols Plots) 5.5. Nyquist Stability Criterion 5.6. Stability Analysis 3 2017 Shiraz University of Technology Dr. A. Rahideh Introduction • By the term frequency response, we mean the steady-state response of a system to a sinusoidal input. • In frequency-response methods, we vary the frequency of the input signal over a certain range and study the resulting response. • Assume the following system, if the input is sinusoidal x(t) Asinw t X (s) Y(s) G(s) The steady state output is yss (t) AG( jw) sinw t G( jw) where G(jw) is called the sinusoidal transfer function. 4 2017 Shiraz University of Technology Dr. A. Rahideh Introduction • Example 1: Find the steady-state output of the following system in response to x(t) Asinwt Y(s) X (s) K K G(s) Ts 1 Ts 1 K G( jw) 2 2 K 1T w G( jw) jTw 1 G( jw) tan 1 Tw yss (t) AG( jw) sinwt G( jw) AK y (t) sinw t tan 1 Tw ss 2 2 5 1T w 2017 Shiraz University of Technology Dr. A. Rahideh Presenting Frequency-Response Characteristics in Graphical Forms • The sinusoidal transfer function, a complex function of the frequency w, is characterized by its magnitude and phase angle, with frequency as the parameter. • There are three commonly used representations of sinusoidal transfer functions: G( jw) vs. w 1. Bode diagram or logarithmic plot G( jw) vs. w 2. Nyquist plot or polar plot ImG( jw) vs. ReG( jw) 3. Log-magnitude-versus-phase plot (Nichols plots) G( jw) vs. G( jw) 6 2017 Shiraz University of Technology Dr. A. Rahideh Bode Diagrams A Bode diagram consists of two graphs: 1. One is a plot of the logarithm of the magnitude of a sinusoidal transfer function; 2. The other is a plot of the phase angle; Both are plotted against the frequency on a logarithmic scale. The standard representation of the logarithmic magnitude of G(jw) is 20 log |G(jw)|, where the base of the logarithm is 10. The unit used in this representation of the magnitude is the decibel (dB). 7 2017 Shiraz University of Technology Dr. A. Rahideh Basic Factors The basic factors that very frequently occur in an arbitrary transfer function G(jw)H(jw) are 1. Gain K 2. Integral and derivative factors ( jw)1 3. First-order factors (1 jw)1 2 1 4. Quadratic factors 1 2 ( jw wn ) ( jw wn ) Note that adding the logarithms of the gains corresponds to multiplying them together. 8 2017 Shiraz University of Technology Dr. A. Rahideh Basic Factors K 1. Gain G( jw) K G( jw) 20log K dB G(s) K G( jw) K 0 K 0 G( jw) 180 K 0 G( jw) G( jw) G( jw) dB K 1 K 20log K K 0 w 0 K 1 K 0 180 9 2017 Shiraz University of Technology Dr. A. Rahideh Basic Factors 2. Integral ( jw)1 1 G( jw) G( jw) 20logw w dB 1 1 G(s) G( jw) s jw G( jw) 90 G( jw) G( jw) G( jw) Both axes are dB logarithmic 10 20 1 0 w 0.1 1 10 0.1 20 90 10 2017 Shiraz University of Technology Dr. A. Rahideh Basic Factors ( jw) 3. Derivative G( jw) w G( jw) 20logw dB G(s) s G( jw) jw G( jw) 90 G( jw) G( jw) G( jw) dB 10 20 90 1 0 w 0.1 1 10 0.1 Both axes are 20 logarithmic 11 2017 Shiraz University of Technology Dr. A. Rahideh Basic Factors 4. First-order (1 jwT )1 1 G( jw) (1) 2 2 1 1 1T w G(s) G( jw) 1Ts 1 jTw G( jw) tan 1(Tw) (2) 1 w 1 20logTw w 1 (1) G( jw) T G( jw) T Tw dB 1 1 0 w T 1 w T 1 G( jw) 1 20log 2 3 dB G( jw) 1 dB w w T T 2 12 2017 Shiraz University of Technology Dr. A. Rahideh Basic Factors 1 1 4. First-order (1 jwT) w 1 G( jw) T 1 Tw 1 G(s) 1 w T 1Ts 20logTw w 1 1 G( jw) T G( jw) tan (Tw) dB 1 0 w T G( jw) G( jw) G( jw) dB 1 10 1 T T 0 1 w 3 0.1 2 1 10 45 0.1 20 90 13 2017 Shiraz University of Technology Dr. A. Rahideh Basic Factors 5. First-order (1 jwT ) 2 2 (1) G( jw) 1T w G(s) 1Ts G( jw) 1 jTw G( jw) tan 1(Tw) (2) Tw w 1 1 T 20logTw w T (1) G( jw) G( jw) 1 w 1 dB 1 T 0 w T G( jw) 1 2 G( jw) 1 20log 2 3 dB w dB w T T 14 2017 Shiraz University of Technology Dr. A. Rahideh Basic Factors 5. First-order (1 jwT ) Tw w 1 G( jw) T 1 G(s) 1Ts 1 w T 20logTw w 1 G( jw) G(jw) tan 1(Tw) T dB 1 0 w T G( jw) G( jw) G( jw) dB 10 20 90 1 10 12 T T 3 w 0 0 .1 1 10 45 0 15 2017 Shiraz University of Technology Dr. A. Rahideh Basic Factors 6. First-order (1 jwT ) 2 2 (1) G( jw) 1T w G(s) 1Ts G( jw) 1 jTw 1 Tw G( jw) tan ( 1 ) (2) Tw w 1 1 T 20logTw w T (1) G( jw) G( jw) 1 w 1 dB 1 T 0 w T G( jw) 1 2 G( jw) 1 20log 2 3 dB w dB w T T 16 2017 Shiraz University of Technology Dr. A. Rahideh Basic Factors 6. First-order (1 jwT ) Tw w 1 G( jw) T 1 G(s) 1Ts 1 w T 20logTw w 1 G( jw) tan 1( Tw ) G( jw) T 1 dB 1 0 w T G( jw) G( jw) G( jw) dB 10 20 180 1 10 12 T T 3 135 w 0 0 .1 1 10 90 17 2017 Shiraz University of Technology Dr. A. Rahideh Basic Factors 7. Second-order w 2 1 n G(s) G(s) 2 2 2 s 2wns wn s 2 2 s 1 wn wn 1 G( jw) 2 2 w 2 w 1 2 1 w 2 w G( jw) n n w 2 w 1 2 j2 w wn wn 2 G( jw) tan 1 wn w 2 1 2 wn 18 2017 Shiraz University of Technology Dr. A. Rahideh Basic Factors 7. Second-order 2 wn G(s) 2 2 s 2wns wn 1 G( jw) 2 2 w 2 w w 2 w w 1 2 2 wn n G( jw) wn wn 1 w wn w w 40log w w wn 2 G( jw) n G( jw) tan 1 wn dB w 2 0 w wn 1 2 wn 19 2017 Shiraz University of Technology Dr. A. Rahideh Basic Factors 7. Second-order 2 wn G(s) 2 2 s 2w ns wn 2 w G( jw) tan1 wn w 2 1 2 wn w 40log w wn G( jw) wn dB 0 w wn 20 2017 Shiraz University of Technology Dr. A. Rahideh Basic Factors 7. Second-order The Resonant Frequency wr and the Resonant Peak Value Mr The peak value of G ( j w ) occurs when the denominator, g ( w ) , minimizes 2 2 w 2 w g(w) 1 2 2 wn wn dg(w) 2 0 1 0 wr wn 1 2 2 dw 1 1 M r G( jw) G( jwr ) 0 max 2 1 2 2 21 2017 Shiraz University of Technology Dr. A. Rahideh Basic Factors Corner frequency • In the first-order system of the following form, K G(s) Ts 1 1 the corner frequency is wc T • In the second-order system of the following form K G(s) s2 2w s w 2 n n the corner frequency is wc wn 22 2017 Shiraz University of Technology Dr. A. Rahideh Basic Factors Corner frequency • Example 1: 2 4 G(s) G(s) 3 3s 2 2 s 1 1 2 the corner frequency is wc T 3 • Example 2: 6 3 G(s) G(s) 2s2 2s 4 2 s s 2 the corner frequency is wc wn 2 23 2017 Shiraz University of Technology Dr. A. Rahideh Bode Diagrams Example: Plot the bode diagrams of the following system 1000 1000 G(s) G( jw) s(s 5)(s 50) jw ( jw 5)( jw 50) 4 G( jw) w w jw ( j 5 1)( j 50 1) 1 1 w 5 wc2 50 The corner frequencies are c 1 T 1 and T2 1 1 1 G( jw) 20log 4 20log jw 20log w 20log w dB j 5 1 j 50 1 1 1 1 G( jw) 0 jw j w 1 j w 1 24 5 50 2017 Shiraz University of Technology Dr.