Control Systems I Lecture 9: The Nyquist condition adings: Guzzella, Chapter 9.4 6 Åstrom and Murray, Chapter 9.1 4 www.cds.caltech.edu/~murray/amwiki/index.php/first_edition Emilio Frazzoli Institute for Dynamic Systems and Control D-MAVT ETH Zürich November 17, 2017 E. Frazzoli (ETH) Lecture 9: Control Systems I 17/11/2017 1 / 31
Tentative schedule # Date Topic 1 Sept. 22 Introduction, Signals and Systems 2 Sept. 29 Modeling, Linearization 3 Oct. 6 Analysis 1: Time response, Stability 4 Oct. 13 Analysis 2: Diagonalization, Modal coordinates. 5 Oct. 20 Transfer functions 1: Definition and properties 6 Oct. 27 Transfer functions 2: Poles and Zeros 7 Nov. 3 Analysis of feedback systems: internal stability, root locus 8 Nov. 10 Frequency response 9 Nov. 17 Analysis of feedback systems 2: the Nyquist condition 10 Nov. 24 Specifications for feedback systems 11 Dec. 1 Loop Shaping 12 Dec. 8 PID control 13 Dec. 15 plementation issues 14 Dec. 22 Robustness E. Frazzoli (ETH) Lecture 9: Control Systems I 17/11/2017 2 / 31
Putting it all together: Bode plots for complicated transfer functions E. Frazzoli (ETH) Lecture 9: Control Systems I 17/11/2017 3 / 31
Example Sketch the Bode plots of s + 4 G(s) = 100 s(s 2 + 10s + 100) First thing: write the transfer function in the Bode form: s/4 + 1 G(s) = 4 s(s 2 /100 + s/10 + 1) Second: draw the Bode plot for each factor in the transfer function. Third: add all of the above together to get the final Bode plot. E. Frazzoli (ETH) Lecture 9: Control Systems I 17/11/2017 4 / 31
Example (s/4 + 1) G(s) = 4 s(s 2 /100 + s/10 + 1) 60 Bode Diagram 40 Magnitude (db) 20 0-20 -40 Phase (deg) -60 90 45 0-45 -90-135 -180-225 -270 10-2 10-1 10 0 10 1 10 2 Frequency (rad/s) E. Frazzoli (ETH) Lecture 9: Control Systems I 17/11/2017 5 / 31
Example (s/4 + 1) G(s) = 4 s(s 2 /100 + s/10 + 1) 60 Bode Diagram 40 Magnitude (db) 20 0-20 -40 Phase (deg) -60 90 45 0-45 -90-135 -180-225 -270 10-2 10-1 10 0 10 1 10 2 Frequency (rad/s) E. Frazzoli (ETH) Lecture 9: Control Systems I 17/11/2017 6 / 31
Example 60 (s/4 + 1) G(s) = 4 s(s 2 /100 + s/10 + 1) Bode Diagram 40 Magnitude (db) 20 0-20 -40 Phase (deg) -60 90 45 0-45 -90-135 -180-225 -270 10-2 10-1 10 0 10 1 10 2 Frequency (rad/s) E. Frazzoli (ETH) Lecture 9: Control Systems I 17/11/2017 7 / 31
Bode s Law In the Bode plot, the magnitude slope and the phase are not independent. In particular, if the system is open-loop stable and minimum-phase, then if the slope of the Bode magnitude plot is κ db/decade over a range of more than 1 decade, the phase in that range will be approximately κ 90. E. Frazzoli (ETH) Lecture 9: Control Systems I 17/11/2017 8 / 31
The polar plot E. Frazzoli (ETH) Lecture 9: Control Systems I 17/11/2017 9 / 31
The polar plot In the polar plot, the frequency response G(jω) is plotted on the complex plane as a parametric function of ω. No special rules for drawing it, but the same principles we used in the Bode plot apply. In fact, it is convenient to sketch a Bode plot first, so that we can have a good idea of what the polar plot looks like, especially in view of the following. The only things that really matter in the polar plot are: Where the plot intersects the unit circle ( G(jω) = 1) Where the plot crosses the real axis ( G(jω) = l 180 ). E. Frazzoli (ETH) Lecture 9: Control Systems I 17/11/2017 10 / 31
Polar plot Integrator E. Frazzoli (ETH) Lecture 9: Control Systems I 17/11/2017 11 / 31
Polar plot single real, stable pole E. Frazzoli (ETH) Lecture 9: Control Systems I 17/11/2017 12 / 31
Polar plot complex-conjugate, stable poles E. Frazzoli (ETH) Lecture 9: Control Systems I 17/11/2017 13 / 31
Polar plot complicated transfer function 60 40 Bode Diagram Magnitude (db) 20 0-20 -40-60 90 45 0 Phase (deg) -45-90 -135-180 -225-270 10-2 10-1 10 0 10 1 10 2 Frequency (rad/s) E. Frazzoli (ETH) Lecture 9: Control Systems I 17/11/2017 14 / 31
Towards Nyquist s theorem E. Frazzoli (ETH) Lecture 9: Control Systems I 17/11/2017 15 / 31
The principle of variation of the argument Let D C be a bounded, simply-connected region of the complex plane, and let Γ be its boundary. As s moves along the closed curve Γ, G(s) describes another closed curve. G(s) D markable fact: The number of times G(s) encircles the origin, or, equivalently, the total variation in its argument G(s), as s moves along Γ, counts the number of zeros and poles of G(s) in D. E. Frazzoli (ETH) Lecture 9: Control Systems I 17/11/2017 16 / 31
No poles/zeros in D member that if G(s) = (s z)/(s p), then G(s) = (s z) (s p). If D contains no poles/zeros, the net variation of the argument of G(s) across one complete cycle of Γ is zero. E. Frazzoli (ETH) Lecture 9: Control Systems I 17/11/2017 17 / 31
A zero in D member that if G(s) = (s z)/(s p), then G(s) = (s z) (s p). If D contains one zero, the net variation of the argument of G(s) across one complete cycle of Γ is 2π. E. Frazzoli (ETH) Lecture 9: Control Systems I 17/11/2017 18 / 31
A pole in D member that if G(s) = (s z)/(s p), then G(s) = (s z) (s p). If D contains one pole, the net variation of the argument of G(s) across one complete cycle of Γ is 2π. E. Frazzoli (ETH) Lecture 9: Control Systems I 17/11/2017 19 / 31
The general case Theorem (Variation of the argument [Proof in A&M, pp. 277 278]) N = Z - P The number N of times that G(s) encircles the origin of the complex plane as s moves along the boundary Γ of a bounded simply-connected region of the plane satisfies N = Z P, where Z and P are the numbers of zeros and poles of G(s) in D, respectively. Note that the encirclements are counted positive if in the same direction as s moves along Γ, and negative otherwise. E. Frazzoli (ETH) Lecture 9: Control Systems I 17/11/2017 20 / 31
How do we use these results for feedback control? E. Frazzoli (ETH) Lecture 9: Control Systems I 17/11/2017 21 / 31
The Nyquist or D contour For closed-loop stability, the closed-loop poles, which corresponds to the roots (i.e., zeros!) of the characteristic polynomial 1 + kl(s), must have negative real part. The poles of 1 + kl(s) are also the poles of L(s). Construct the region D as a D-shaped region containing an arbitrarily large (but finite) part of the complex right-half plane. As s moves along the boundary of this region, 1 + kl(s) encircles the origin N = Z P times, where Z is the number of unstable closed-loop poles (zeros of 1 + kl(s) in the rhp); P is the number of unstable open-loop poles (poles of 1 + kl(s) in the rhp); E. Frazzoli (ETH) Lecture 9: Control Systems I 17/11/2017 22 / 31
The Nyquist plot The previous statement can be rephrased: As s moves along the boundary of this region, L(s) encircles the 1/k point N = Z P times, where Z is the number of unstable closed-loop poles (zeros of 1 + kl(s) in the Nyquist contour); P is the number of unstable open-loop poles (poles of 1 + kl(s) in the Nyquist contour); Symmetry of poles/zeros about the real axis implies that L( jω) = L(jω), hence the plot of L(s) when s moves on the boundary of the Nyquist contour is just the polar plot + its symmetric plot about the real axis. This is what is called the Nyquist plot. The key feature of the Nyquist plot is the number of encirclements of the 1/k point. E. Frazzoli (ETH) Lecture 9: Control Systems I 17/11/2017 23 / 31
The Nyquist condition Theorem Consider a closed-loop system with loop transfer function kl(s), which has P poles in the region enclosed by the Nyquist contour. Let N be the net number of clockwise encirclements of 1/k by L(s) when s moves along the Nyquist contour in the clockwise direction. The closed loop system has Z = N + P poles in the Nyquist contour. In particular: If the open-loop system is stable, the closed-loop system is stable as long as the Nyquist plot of L(s) does NOT encircle the 1/k point. If the open-loop system has P poles, the closed-loop system is stable as long as the Nyquist plot of L(s) encircles the 1/k point P times in the negative (counter-clockwise) direction. E. Frazzoli (ETH) Lecture 9: Control Systems I 17/11/2017 24 / 31
Nyquist condition single real, stable pole L(s) = 2 s + 1 E. Frazzoli (ETH) Lecture 9: Control Systems I 17/11/2017 25 / 31
Nyquist condition open-loop unstable system L(s) = s + 2 s 2 1 = s + 2 s 2 + 1 E. Frazzoli (ETH) Lecture 9: Control Systems I 17/11/2017 26 / 31
Dealing with open-loop poles on the imaginary axis If there are open-loop poles on the imaginary axis, make small indentations in the Nyquist contour, e.g., leaving the imaginary poles on the left. Be careful on how you close the Nyquist plot at infinity: If moving CCW around the poles, then close the plot CW. E. Frazzoli (ETH) Lecture 9: Control Systems I 17/11/2017 27 / 31
Nyquist poles on the imaginary axis L(s) = 2 (s 2 + 1)(s + 1) E. Frazzoli (ETH) Lecture 9: Control Systems I 17/11/2017 28 / 31
The Nyquist condition on Bode plots If the open-loop is stable, then we know that in order for the closed-loop to be stable the Nyquist plot of L(s) should NOT encircle the 1 point. In other words, L(jω) < 1 whenever L(jω) = 180. On the Bode plot, this means that the magnitude plot should be below the 0 db line if/when the phase plot crosses the 180 line. member that this condition is valid only if the open loop is stable. In all other cases (including non-minimum phase zeros) it is strongly recommended to double check any conclusion on closed-loop stability using other methods (Nyquist, root locus). E. Frazzoli (ETH) Lecture 9: Control Systems I 17/11/2017 29 / 31
Gain and Phase Margin The distance from the Nyquist plot to the 1 point is a measure of robustness. On the bode plot, it is easy to measure this distance in terms of gain and phase margin. 40 20 0 Bode Diagram Magnitude (db) -20-40 -60-80 -100-120 -90-135 Phase (deg) -180-225 -270 E. Frazzoli 10 (ETH) -1 10 0 Lecture 9: Control 10 1 Systems I 10 2 17/11/2017 10 3 30 / 31
Summary In this lecture, we learned: How to sketch a polar plot (and hence a Nyquist plot), based on Bode plots The Nyquist condition to determine closed-loop stability using a Nyquist plot. How to check the Nyquist condition on a Bode plot. How to quickly assess the robustness of a feedback control system. Now we have three graphical methods to study closed-loop stability given the (open-)loop transfer function. 1 Root locus: always correct if applicable (assumes finite-dimensional system) 2 Nyquist: always correct, always appplicable; 3 Bode: very useful for control system design, however may be misleading in determining closed-loop stability (e.g., for open-loop unstable systems). So far we have only looked at analysis issues, i.e., how to determine closed-loop stability; from now on we will concentrate on control synthesis, i.e., how to design a feedback control system that makes a system behave as desired. E. Frazzoli (ETH) Lecture 9: Control Systems I 17/11/2017 31 / 31