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EE C128 Chapter 10 Lecture abstract EE C128 / ME C134 – Feedback Control Systems Topics covered in this presentation Lecture – Chapter 10 – Frequency Response Techniques I Advantages of FR techniques over RL I Define FR Alexandre Bayen I Define Bode & Nyquist plots I Relation between poles & zeros to Bode plots (slope, etc.) Department of Electrical Engineering & Computer Science st nd University of California Berkeley I Features of 1 -&2 -order system Bode plots I Define Nyquist criterion I Method of dealing with OL poles & zeros on imaginary axis I Simple method of dealing with OL stable & unstable systems I Determining gain & phase margins from Bode & Nyquist plots I Define static error constants September 10, 2013 I Determining static error constants from Bode & Nyquist plots I Determining TF from experimental FR data Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 1 / 64 Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 2 / 64 10 FR techniques 10.1 Intro Chapter outline 1 10 Frequency response techniques 1 10 Frequency response techniques 10.1 Introduction 10.1 Introduction 10.2 Asymptotic approximations: Bode plots 10.2 Asymptotic approximations: Bode plots 10.3 Introduction to Nyquist criterion 10.3 Introduction to Nyquist criterion 10.4 Sketching the Nyquist diagram 10.4 Sketching the Nyquist diagram 10.5 Stability via the Nyquist diagram 10.5 Stability via the Nyquist diagram 10.6 Gain margin and phase margin via the Nyquist diagram 10.6 Gain margin and phase margin via the Nyquist diagram 10.7 Stability, gain margin, and phase margin via Bode plots 10.7 Stability, gain margin, and phase margin via Bode plots 10.8 Relation between closed-loop transient and closed-loop 10.8 Relation between closed-loop transient and closed-loop frequency responses frequency responses 10.9 Relation between closed- and open-loop frequency responses 10.9 Relation between closed- and open-loop frequency responses 10.10 Relation between closed-loop transient and open-loop 10.10 Relation between closed-loop transient and open-loop frequency responses frequency responses 10.11 Steady-state error characteristics from frequency response 10.11 Steady-state error characteristics from frequency response 10.12 System with time delay 10.12 System with time delay 10.13 Obtaining transfer functions experimentally 10.13 Obtaining transfer functions experimentally Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 3 / 64 Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 4 / 64 10 FR techniques 10.1 Intro 10 FR techniques 10.1 Intro Advantages of frequency response (FR) methods, [1, p. The concept of FR, [1, p. 535] 534] I At steady-state, sinusoidal inputs to a linear system generate sinusoidal responses In the following situations of the same frequency with I When modeling TFs from physical data di↵erent amplitudes and phase I When designing lead compensators to meet a steady-state error angle from the input, each of requirements which are a function of frequency. I When finding the stability of NL systems I Phasor – complex I In settling ambiguities when sketching a root locus representation of a sinusoid I G(!) – amplitude || || I \G(!) – phase angle Figure: Sinusoidal FR: a. system; b. I M cos(!t + φ) ... M\φ TF; c. IO waveforms Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 5 / 64 Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 6 / 64 10 FR techniques 10.1 Intro 10 FR techniques 10.1 Intro The concept of FR, [1, p. 535] Analytical expressions for FR, [1, p. 536] I Steady-state output sinusoid I General input sinusoid Mo(!)\φo(!) r(t)=A cos(!t)+B sin(!t) = Mi(!)M(!)\(φi(!)+φ(!)) 2 2 1 B = A + B cos !t tan− A I Magnitude FR − p I Input phasor forms Mo(!) M(!)= I Polar, Mi\φi Mi(!) Figure: System with 2 2 Mi = A + B sinusoidal input I Phase FR p 1 B φ = tan− i − A φ(!)=φo(!) φi(!) − I Rectangular, A jB jφ − Figure: Sinusoidal FR: a. system; b. I Euler’s, M e i I FR i TF; c. IO waveforms M(!)\φ(!) Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 7 / 64 Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 8 / 64 10 FR techniques 10.1 Intro 10 FR techniques 10.2 Asymptotic approximations: Bode plots Analytical expressions for FR, [1, p. 536] 1 10 Frequency response techniques 10.1 Introduction I Forced response 10.2 Asymptotic approximations: Bode plots As + B! C(s)= G(s) 10.3 Introduction to Nyquist criterion s2 + !2 10.4 Sketching the Nyquist diagram I Steady-state forced response after partial fraction expansion 10.5 Stability via the Nyquist diagram MiMG j(φ φ ) MiMG j(φ φ ) e− i− G e i− G 10.6 Gain margin and phase margin via the Nyquist diagram C (s)= 2 + 2 ss s + j! s j! 10.7 Stability, gain margin, and phase margin via Bode plots − 10.8 Relation between closed-loop transient and closed-loop where MG = G(j!) and φG = \G(j!) || || frequency responses I Time-domain response 10.9 Relation between closed- and open-loop frequency responses c(t)=MiMG cos(!t + φi + φG) 10.10 Relation between closed-loop transient and open-loop I Time-domain response in phasor form frequency responses 10.11 Steady-state error characteristics from frequency response Mo\φo =(Mi\φi)(MG\φG) 10.12 System with time delay I FR of system 10.13 Obtaining transfer functions experimentally G(j!)=G(s) s j! | ! Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 9 / 64 Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 10 / 64 10 FR techniques 10.2 Asymptotic approximations: Bode plots 10 FR techniques 10.2 Asymptotic approximations: Bode plots History interlude General Bode plots, [1, p. 540] History (Hendrik Wade Bode) I 1905 – 1982 I American engineer I 1930s – Inventor of Bode plots, gain margin, & phase margin G(j!)=MG(!)\φG(!) I 1944 – WWII anti-aircraft (including V-1 flying bombs) I Separate magnitude and phase plots as a function of frequency systems I Magnitude – decibels (dB)vs.log(!),wheredB = 20 log(M) I Phase – phase angle vs. log(!) I 1947 – Cold War anti-ballistic missiles I 1957 – Served on NACA (now NASA) with Wernher von Braun (inventor of V-1 flying Figure: Hendrik Wade Bode bombs & V-2 rockets) Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 11 / 64 Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 12 / 64 10 FR techniques 10.2 Asymptotic approximations: Bode plots 10 FR techniques 10.2 Asymptotic approximations: Bode plots Bode plots approximations, [1, p. 542] Simple Bode plots, [1, p. 542] I TF G(s)=s + a 40 0 I Low frequencies 30 −10 20 −20 G(j!) a 0◦ Magnitude (dB) ⇡ \ Magnitude (dB) 10 −30 0 −40 I High frequencies 90 0 60 −30 G(j!) ! 90◦ \ 30 −60 Phase (deg) ⇡ Phase (deg) 0 −90 I Asymptotes – straight-line −2 −1 0 1 2 −2 −1 0 1 2 10 10 10 10 10 10 10 10 10 10 approximations Frequency (Hz) Frequency (Hz) I Low-frequency Figure: Bode plots of s + a:a. Figure: Bode plot of s+a Figure: Bode plot of a I Break frequency magnitude plot; b. phase plot a s+a I High-frequency Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 13 / 64 Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 14 / 64 10 FR techniques 10.2 Asymptotic approximations: Bode plots 10 FR techniques 10.2 Asymptotic approximations: Bode plots Simple Bode plots, [1, p. 545] Simple Bode plots, [1, p. 549] 80 0 30 0 60 −20 20 −10 40 −40 10 −20 Magnitude (dB) Magnitude (dB) Magnitude (dB) 20 −60 Magnitude (dB) 0 −30 0 −80 91 −89 180 0 90.8 −89.2 90.6 −89.4 135 −45 90.4 −89.6 90.2 −89.8 90 −90 90 −90 89.8 −90.2 −90.4 Phase (deg) Phase (deg) 89.6 Phase (deg) Phase (deg) 45 −135 89.4 −90.6 89.2 −90.8 89 −91 0 −180 −1 0 1 −1 0 1 −2 −1 0 1 2 −2 −1 0 1 2 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) Frequency (Hz) Frequency (Hz) 2 2 2 1 s +2⇣!ns+!n !n Figure: Bode plot of s Figure: Bode plot of Figure: Bode plot of 2 Figure: Bode plot of 2 2 s !n s +2⇣!ns+!n Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 15 / 64 Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 16 / 64 10 FR techniques 10.2 Asymptotic approximations: Bode plots 10 FR techniques 10.3 Intro to Nyquist criterion nd Detailed 2 -order Bode plots, [1, p. 550] 1 10 Frequency response techniques 10.1 Introduction 10.2 Asymptotic approximations: Bode plots 10.3 Introduction to Nyquist criterion 10.4 Sketching the Nyquist diagram 10.5 Stability via the Nyquist diagram 10.6 Gain margin and phase margin via the Nyquist diagram 10.7 Stability, gain margin, and phase margin via Bode plots 10.8 Relation between closed-loop transient and closed-loop frequency responses 10.9 Relation between closed- and open-loop frequency responses 10.10 Relation between closed-loop transient and open-loop frequency responses 10.11 Steady-state error characteristics from frequency response 10.12 System with time delay 2 2 !2 s +2⇣!ns+!n n Figure: Bode plot of 2 Figure: Bode plot of s2+2⇣! s+!2 10.13 Obtaining transfer functions experimentally !n n n Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 17 / 64 Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 18 / 64 10 FR techniques 10.3 Intro to Nyquist criterion 10 FR techniques 10.3 Intro to Nyquist criterion History interlude Introduction, [1, p.
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