1. 1 Chapter 9 Mapping Contours in the s-plane The Nyquist Criterion Relative Stability Gain Margin and Phase Margin PID Controllers in the Frequency Domain

2. 2 Review We continue discussing the stability. We see how frequency response methods are used to investigate stability. We introduce the following concepts: gain margin; phase margin; bandwidth. Frequency response stability result-known as the Nyquist stability criterion is presented. The Routh-Hurwitz method discussed in Chapter 6, is useful for investigating the characteristic equation expressed in terms of the complex variable s =  + j . In this lecture, we will investigate the stability of a system in the real frequency domain. A frequency domain stability criterion was developed by H. Nyquist in 1932 and remains fundamental approach for the stability of linear system.

3. 3 Mapping Contours in the s-Plane A contour map is a contour or trajectory in one plane mapped or translated into another plane by a relation F(s). Since s is a complex variable, s =  + j , the function F(s) is itself complex; F(s)=u + jv jj jv  u

4. 4 Cauchy’s Theorem If a contour  s in the s-plane encircles Z zeros and P poles of F(s) and does not pass through any poles or zeros of F(s) and the traversal is in the clockwise direction along the contour, the corresponding contour  F in the F(s)-plane encircles the origin of the F(s)-plane N = Z – P times in the clockwise direction.

5. 5 Nyquist Stability Criterion The Nyquist stability criterion determines the stability of a closed-loop system from its open-loop frequency response and open-loop poles. It is important to mention that although poles and zeros of the open-loop transfer function G(s)H(s) may be in the right-half s-plane, the system is stable if all the poles of the closed-loop transfer function (the roots of the characteristic equation) are in the left-half s-plane. The Nyquist stability criterion relates the open loop frequency response to the number of zeros and poles of the characteristic equation that lie in the right-half s-plane.

6. 6 The Nyquist Criterion To investigate the stability of a control system, we consider the characteristic equation, which is F(s) = 0. For a system to be stable, all the zeros of F(s) must lie in the left-hand s-plane.

7. 7 For a system to be stable, all the zeros of F(s) must lie in the left-hand s-plane. Therefore the roots of a stable system must lie to the left of the j  -axis in the s- plane. We choose a contour  s in the s-plane that encloses the entire right-hand s- plane, and we determine whether any zeros of F(s) lie within  s by utilizing Cauchy’s theorem. A feedback system is stable if and only if the contour  L in the L(s)-plane does not encircle the (-1,0) point when the number of poles of L(s) in the right-hand s- plane is zero. A feedback system is stable if and only if the contour  L, the number of counterclockwise encirclements of the (-1,0) point is equal to the number of poles of L(s) with positive real parts.

8. 8 Gain Margin and Phase Margin The Nyquist plot yields two figures of merit that utilize Bode design! The gain margin is the increase in the system gain when phase = - 180 o that will result in a marginally stable system with intersection of the -1 + j0 point on the Nyquist diagram. The phase margin is the amount of phase shift of the GH(j  ) at unity magnitude that will result in a marginally stable system with intersection of the -1 + j0 point on the Nyquist diagram. The gain and phase margins are easily evaluated from the Bode diagram, and because it is preferable to draw the Bode diagram in contrast to the polar plot, it is worthwhile to illustrate the relative stability measures for the Bode diagram. The critical point for stability is u = -1, v = 0 in the GH (j  )-plane, which is equivalent to a logarithmic magnitude of the 0 dB and a phase angle of 180 o (or -180 o ) on the Bode diagram.

9. 9 Relative Stability Phase margin: The phase margin is that amount of additional phase lag at the gain crossover frequency required to bring the system to the verge of stability. Gain Crossover Frequency: Is the frequency at which |G(j  )|, the magnitude of the open-loop transfer function, is unity. The phase margin is 180 o plus the phase angle of the open loop transfer function at the gain crossover frequency. Gain Margin: The gain margin is the reciprocal of the magnitude |G(j  )| at the frequency at which the phase angle is -180 o. The gain margin is positive if it is more than unity and negative if less than unity. Positive gain margin (in decibels) means that the system is stable. Negative gain margin (in decibels) means that the system is unstable. For satisfactory performance, the phase margin should be between 30 o and 60 o, and the gain margin should be greater than 6 dB.