Nyquist Stability Criteria

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1 Nyquist Stability Criteria Dr. Bishakh Bhattacharya h Professor, Department of Mechanical Engineering IIT Kanpur Joint Initiative of IITs and IISc - Funded by MHRD

2 This Lecture Contains Introduction to Geometric Technique for Stability Analysis Frequency response of two second order systems Nyquist Criteria Gain and Phase Margin of a system Joint Initiative of IITs and IISc - Funded by MHRD

3 Introduction In the last two lectures we have considered the evaluation of stability by mathematical ti evaluation of the characteristic ti equation. Routh s test t and Kharitonov s polynomials are used for this purpose. There are several geometric procedures to find out the stability of a system. These are based on: Nyquist Plot Root Locus Plot and Bode plot The advantage of these geometric techniques is that they not only help in checking the stability of a system, they also help in designing controller for the systems. Joint Initiative of IITs and IISc Funded by MHRD 3

4 Nyquist tplot is based on Frequency Response of a Transfer Function. Consider two transfer functions as follows: s 5 s 5 T 1 ( s ) ; T ( s ) s 3s 2 s s 2 The two functions have identical zero. While for function 1, the poles are at 1 and 22 respectively; for function 2, the poles are at +1 and 2. 2 Let us excite both the systems by using a harmonic excitation of frequency 5 rad/sec. The responses of the two systems are plotted below: Stable Frequency Response of T 1 Unstable Frequency Response of T 2

5 Frequency Domain Issues Consider a closed loop system of plant transfer function G(s) and Feedback transfer function H(s) respectively. The closed loop transfer function corresponding to negative feedback may be written as: G(s) ( ) T ( s) 1 G( s) H ( s) Poles of 1+G(s)H(s) are identical to the poles of G(s)H(s) Zeroes of 1+G(s)H(s) are the Closed Loop Poles of the Transfer Function If we etake eaco Complex pe Number in the esp s-plane and substitute tute it into a Function F(s), it results in another Complex Number which could be plotted in the F(s) Plane.

6 Cauchy Criteria Mapping: A Clockwise contour in the s-plane results in Clockwise contour in the F(s) plane if it contains only zeros A Clockwise contour in the s-plane results in anti-clockwise contour in the F(s) plane if it contains only poles If the contour in the s-plane encloses a pole or a If the contour in the s plane encloses a pole or a zero, it results in enclosing of the origin in the F(s) plane

7 Example: A Clockwise Contour in the s-plane for G(s) = s-z 1 Reference: Nise: Control Systems Engineering

8 Example: A clockwise contour around a Right-half Plane Pole for a function G(s) = 1/(s-p 1 ) Reference: Nise: Control Systems Engineering

9 Nyquist Stability Criteria Number of Counterclockwise (CCW) rotation N = P c Z c (P c no. of enclosed poles of 1+ G(s)H(s) and Z c no. of enclosed zeroes) For a Contour ou in s plane pa mapped through the entire eright half plane of open loop transfer function G(s) H(s), the number of closed loop poles Z c (same as the open loop zeros) in the right half plane equals the number of open loop poles P c in the right half plane minus the number of counterclockwise revolution N around the point 1 of the mapping. Z c = P c N

10 Consider a plant transfer function G(s) as follows: 2 s 12s 24 G ( s ) 2 s 8s 15 For a unity feedback closed loop, find using Nyquist Criteria whether the system will be unstable at some values of K. (Vary K from 0.5 to 10)

11 The Nyquist diagram corresponding to unity Gain and the root locus are shown below for your reference. 0.4 Nyquist Diagram 0.4 Root Locus Imaginary Axis aginary Axis (seconds -1 ) Im System: tf1 Gain: 0.51 Pole: Damping: 1 Overshoot (%): 0 Frequency (rad/s): Real Axis (seconds -1 ) Real Axis Joint Initiative of IITs and IISc Funded by MHRD 11

12 Gain Margin The gain margin is the factor by which the gain can be raised such that the contour encompassed the unity point resulting in instability of the system. Following the figure below, gain margin is the inverse of the distance shown in the figure.

13 Phase Margin The Phase Margin is the amount of phase that needs to be added to a system such that the magnitude will be just unity while the phase is The figure below is showing theta to be the phase margin. Often control engineers consider a system to be adequately stable if it has a phase margin of at least 30 0.

14 Special References for this lecture Control Engineering and introductory course, Wilkie, Johnson and Katebi, PALGRAVE Control Systems Engineering Norman S Nise, John Wiley & Sons Modern Control Engineering K. Ogata, Prentice Hall Joint Initiative of IITs and IISc Funded by MHRD 14

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