Frequency methods for the analysis of feedback systems. Lecture 6. Loop analysis of feedback systems. Nyquist approach to study stability

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1 Lecture 6. Loop analysis of feedback systems 1. Motivation 2. Graphical representation of frequency response: Bode and Nyquist curves 3. Nyquist stability theorem 4. Stability margins Frequency methods for the analysis of feedback systems Revolutionary idea: focus on frequency response rather than the characteristic equation (pole/zero location). For G LTI and stable, u(t) = 1I(t)e jω 0t y(t) = y trans (t) + G(jω 0 )u(t) Investigate propagation of sinusoids around the loop. History: Black, Nyquist, Bode Syst003 lecture VI 1 Syst003 lecture VI 2 Frequency methods are well-suited for feedback analysis in the presence of uncertainty Nyquist approach to study stability y u G(s) Stability: yes-no criterion versus stability margins modeling: frequency modeling of uncertainty is often easier Robustness and performance requirements are frequency dependent. Cut the loop. Let u be a sinusoid. If y is a sinusoid with same amplitude and phase, then the loop can be closed and the oscillation will be maintained. The condition for this is L(s) = 1 where L = CG is the loop transfer function. The condition implies that the Nyquist curve of L goes through the point ( 1, 0). -1 Syst003 lecture VI 3 Syst003 lecture VI 4

2 Graphical representations of frequency response Bode plots: 20 log L versus log ω and arg L versus log ω (matlab bode ) Nyquist curve: Plot the curve L(jω), ω IR, in complex plane (matlab nyquist or plot(l(jω)) ) Draw and compare Bode plots and Nyquist curve for simple 1 transfer functions: τs+1, ωn 2 1 s 2 +ω nζs+ω, n 2 (τ 1 s+1)(τ 2 s+1)(τ 3 s+1), 1 s, 1 s(τs+1), e T s, e T s s,... Relating the Nyquist curve to stability Stability if 1 + L(s) has no zero in the RHP principle of argument: the number of zeros of F (s) in the region surrounded by a contour Γ is related to the number of encirclements of the point (0, 0) by the image contour F (Γ). Nyquist curve is the image contour by L of a curve Γ that encircles the RHP the number of encirclements of the point ( 1, 0) by the Nyquist curve is related to the number of zeros of 1 + L(s) in the RHP! Syst003 lecture VI 5 Syst003 lecture VI 6 The argument principle (For rational functions; but the proof extends to analytical functions) p i. s p i Γ Nyquist stability theorem F (s) = 1 + L(s) N RHP zeros of 1 + L(s) are RHP closed-loop poles P RHP poles of 1 + L(s) are RHP open-loop poles The feedback system is asymptotically stable if the Nyquist curve of L(s) encircles the point ( 1, 0) as many times in counterclockwise direction as there are unstable open-loop poles. N P arg F (s) = arg(s z i ) arg(s p i ) i=1 arg(s z i ) varies by 2π (only) when Γ encircles z i If F has no pole or zero on Γ, the number of encirclements of the origin by F (Γ) is N-P. i=1 Syst003 lecture VI 7 Syst003 lecture VI 8

3 Modification of the contour Γ when L(s) possesses poles on the jω-axis poles of L R Γ Illustration L(s) = e T s s Intersections of the Nyquist curve with real axis are given by the roots of cos ωt = 0 First root at ωt = π/2 yields intersection at ( 2T/π, 0). The gain margin is π/2t. (Time-delays always limit admissible feedback gain!) (with R ) Syst003 lecture VI 9 Syst003 lecture VI 10 Stability margins 1/Gm Stability margins (-1,0) d = 1 / Ms Gain margin: K < G m feedback system tolerates an uncertain gain!m Phase margin: feedback system tolerates a phase shift φ < φ m (at all frequencies). Shortest distance: stability is preserved under multiplicative uncertainty (1 + e(jω)) if e(jω) < d. Reasonable values: G m = , φ m = 45 deg deg, d = Various indicators to measure distance to instability: gain margin G m, phase margin φ m, shortest distance d to Nyquist curve. Syst003 lecture VI 11 Syst003 lecture VI 12

4 Stability margins on Bode plots G(jw) db 0 db G PM log w. GM log w Bode s relationship between phase and gain If G(s) has no pole and no zero in RHP, one has G(jω 0 ) = 1 π + d log G(jω 0 ) du log coth u du, u = log ω 2 ω 0 The phase characteristic of G can be deduced from its amplitude characteristic! If the slope of log G does not vary too much around ω 0, Bode s relationship approximates to G(jω 0 ) π 2 slope G(jω 0) In particular: the phase margin depends on the slope of log G in the vicinity of ω c Syst003 lecture VI 13 Syst003 lecture VI 14 Non minimum phase systems For systems with no RHP pole/zero and no delay, Bode s relationship between magnitude and phase implies d log G(jω) arg G(jω) π/2 d log ω Systems with RHP pole(s) and/or zero(s) and/or time-delays have a larger phase lag. They are called non-minimum phase systems Example: a time delay G nmp (s) = e st This means an additional phase lag of arg G nmp (jω) = ωt compared to the corresponding minimum phase system. Factor G(s) as G mp (s)g nmp (s) with G nmp (jω) = 1 and arg G nmp (jω) < 0 Syst003 lecture VI 15 Syst003 lecture VI 16

5 Example: a RHP zero G nmp (s) = z s z + s This means an additional phase lag of arg G nmp (jω) = 2 arctan ω z compared to the corresponding minimum phase system. Summary of lecture Analyze feedback systems by looking at (steady-state) propagation of sinusoids around the loop Stability of the feedback system can be studied from the Nyquist plot of the loop transfer function Stability margins are important performance and robustness indicators. Syst003 lecture VI 17 Syst003 lecture VI 18