Control Systems I Lecture 9: The Nyquist condition Readings: Åstrom and Murray, Chapter 9.1 4 www.cds.caltech.edu/~murray/amwiki/index.php/first_edition Jacopo Tani Institute for Dynamic Systems and Control D-MAVT ETH Zürich November 16, 2018 J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/2018 1 / 26
Tentative schedule # Date Topic 1 Sept. 21 Introduction, Signals and Systems 2 Sept. 28 Modeling, Linearization 3 Oct. 5 Analysis 1: Time response, Stability 4 Oct. 12 Analysis 2: Diagonalization, Modal coordinates 5 Oct. 19 Transfer functions 1: Definition and properties 6 Oct. 26 Transfer functions 2: Poles and Zeros 7 Nov. 2 Analysis of feedback systems: internal stability, root locus 8 Nov. 9 Frequency response 9 Nov. 16 Analysis of feedback systems 2: the Nyquist condition 10 Nov. 23 Specifications for feedback systems 11 Nov. 30 Loop Shaping 12 Dec. 7 PID control 13 Dec. 14 State feedback and Luenberger observers 14 Dec. 21 On Robustness and Implementation challenges J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/2018 2 / 26
Recap A system is a function mapping input signals into output signals. An LTI system can be described by a transfer function. u L(s) y L(s) = N(s) D(s) D(s) is the characteristic polynomial of the matrix A. Poles: the roots of D(s), zeros: the roots of N(s) u(t) = sin(ωt) y ss (t) = L(jω) sin(ωt + L(jω)). For any input U(s) Y (s) = L(s)U(s) Stable system: a system that does not blow up. The system L(s) is BIBO-stable if all of its poles are on the LHP. J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/2018 3 / 26
Recap Feedback control allows one to Stabilize an unstable system; Handle uncertainties in the system; Reject external disturbances. r e u y k L(s) The closed-loop transfer function is: kl(s) 1 + kl(s). It is also called the complimentary sensitivity function. The closed-loop poles: the zeros of 1 + kl(s). The poles of 1 + kl(s) are identical to the poles of L(s). The closed-loop system is stable if all of its poles are on the LHP. J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/2018 4 / 26
The phase rule and the magnitude rule G(s) = k (s z 1)(s z 2 )... (s z m ) (s p 1 )(s p 2 )... (s p n ) Im jω Re G(s) = k s z 1 s z 2... s z m s p 1 s p 2... s p n G(s) = k + (s z 1 ) + (s z 2 ) +... + (s z m ) (s p 1 ) (s p 2 )... (s p n ) J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/2018 5 / 26
Classical methods for feedback control The main objective here is to assess/design the properties of the closed-loop system by exploiting the knowledge of the open-loop system, and avoiding complex calculations. Stability analysis of closed-loop system: Routh-Hurwitz criterion: a mathematical evaluation of the characteristic equation of the closed-loop system. There are three geometric methods to find out the stability of a systems: They are useful both in analysis and synthesis. J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/2018 6 / 26
Classical methods for feedback control Root Locus Quick assessment of control design feasibility. The insights are correct and clear. Can only be used for finite-dimensional systems (e.g. systems with a finite number of poles/zeros) Difficult to do sophisticated design. Hard to represent uncertainty. Nyquist plot The most authoritative closed-loop stability test. It can always be used (finite or infinite-dimensional systems) Easy to represent uncertainty. Difficult to draw and to use for sophisticated design. Bode plots Potentially misleading results unless the system is open-loop stable and minimum-phase. Easy to represent uncertainty. Easy to draw, this is the tool of choice for sophisticated design. J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/2018 7 / 26
Towards Nyquist s theorem J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/2018 8 / 26
The goal r e u y k L(s) Our goal is to count the number of RHP poles (if any) of the closed-loop transfer function T (s) = kl(s) 1 + kl(s) based on the frequency response of the open-loop transfer function L(s). J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/2018 9 / 26
The principle of variation of the argument If we take a complex number in the s-plane and substitute it into a function G(s), it results in another complex number which could be plotted in the G(s)-plane. Let Γ be a simple closed curve in the s-plane, which does not pass through any pole of a function G(s). As s moves along the closed curve Γ, G(s) describes another closed curve. Im G(s) Im D Re Re Remarkable 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. J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/2018 10 / 26
Phase change and encirclements Consider a clockwise closed contour, Γ, not passing through the origin 0, traversed by s. The origin is either inside or outside Γ. What is the net change in s as s traverses Γ? The phase change as a s traverses a closed path Γ is equal to 2πN, where N is the number of clockwise encirclements of 0 by Γ. J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/2018 11 / 26
Phase change and encirclements Consider a simple closed contour, Γ, traversed clockwise by s. A fixed complex number r is either inside or outside Γ. What is the net phase change in (s r) as s traverses Γ? J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/2018 12 / 26
Principle of argument Find the image of Γ under G(s) = k (s z 1)(s z 2 )... (s z m ) (s p 1 )(s p 2 )... (s p n ) At any value of s, the angle of G(s) is: G(s) = k + (s z 1 ) +... + (s z m ) (s p 1 )... (s p n ) 2π(number of clockwise encirclements of 0 by G(Γ)) = = net change in G(s) as s traverses Γ = = net change in (s z 1 ) +... + net change in (s z m ) net change in (s p 1 )... net change in (s p n ) 2πN = 2πZ ( 2πP) J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/2018 13 / 26
The general case Theorem (Variation of the argument [Proof in A&M, pp. 277 278]) The number N of times that G(s) encircles the origin of the complex plane as s moves along the simple closed curve Γ 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. J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/2018 14 / 26
How do we use these results for feedback control? J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/2018 15 / 26
The Nyquist or D contour For closed-loop stability, the closed-loop poles, which corresponds to the roots (i.e., zeros!) of 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); J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/2018 16 / 26
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. J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/2018 17 / 26
The Nyquist contour Segment 1 corresponds to s = jω, where ω : 0. On this segment, L(Γ) is just L(jω): frequency response. Segment 2 corresponds to s = Re jθ, where R and θ : π 2 π 2. On this segment, L(Γ) collapses on a single point, since R is very large. Segment 3 corresponds to s = jω, where ω : 0. On this segment, L(Γ) is just L( jω), where ω : 0. L( jω) is complex-conjugate of L(jω), so L( jω) is the reflection of L(jω) about the real axis. J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/2018 18 / 26
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. J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/2018 19 / 26
Counting encirclements Draw a line out from the 1/k point in any arbitrary direction. Count the number of times that the Nyquist path crosses the line in the clockwise direction, and subtract the number of times it crosses in the counterclockwise direction. You get the number of clockwise encirclements of the 1/k point. J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/2018 20 / 26
Nyquist plot when L(s) has no pole on the jω axis 1 Image of Segment 1: Plot L(jω) for ω : 0. This is also called the polar plot of L(s). There is no special rules for drawing it. 2 Image of Segment 3: Reflect it about the real axis to draw L(jω) for ω : 0. 3 Image of Segment 2: This segment maps onto a point, in the case of physically realizable systems. For a strictly proper systems: if s, then L(s) maps onto origin. For a proper systems: if s, then L(s) would be a constant. 4 The points where the Nyquist plot crosses the real axis and the unit circle are of importance. J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/2018 21 / 26
Nyquist condition single real, stable pole Im L(s) = 2 s + 1 Im Re Re J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/2018 22 / 26
Nyquist condition open-loop unstable system Im L(s) = s + 2 s 2 1 Im Re Re J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/2018 23 / 26
Dealing with open-loop poles on the imaginary axis Im 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. Re J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/2018 24 / 26
Nyquist poles on the imaginary axis L(s) = 2 (s 2 + 1)(s + 1) Im Im Re Re J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/2018 25 / 26
Summary In this lecture, we learned: How to sketch a Nyquist plot. The Nyquist condition to determine closed-loop stability using a Nyquist plot. How to check the Nyquist condition.. J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/2018 26 / 26