Lecture14.pdf

NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System

Nyquist Stability Criteria D Bishakh Dr. Bi h kh Bhattacharya Bh tt h Professor, Department of Mechanical Engineering IIT Kanpur

Joint Initiative of IITs and IISc - Funded by MHRD

Module 2- Lecture 14

NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System

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

Module 2- Lecture 14

NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System

Module 2- Lecture 14

Introduction In the last two lectures we have considered the evaluation of stability by mathematical th ti l evaluation l ti off the th characteristic h t i ti equation. ti R th’ test Routh’s t t and d 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 y systems. Joint Initiative of IITs and IISc ‐ Funded by  MHRD

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NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System

Module 2- Lecture 14

Nyquist Plot is based on Frequency Response of a Transfer Function.  Consider  N i t Pl t i b d F R f T f F ti C id two transfer functions as follows:

T1 ( s ) 

s5 s5 ; T ( s )  2 s 2  3s  2 s2  s  2

The two functions have identical zero. While for function 1, the poles are at ‐1  and 2 respectively; for function 2 the poles are at +1 and 2 Let us excite both and ‐2 respectively; for function 2, the poles are at +1 and ‐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 T1

Unstable Frequency Response of T2

NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System

Module 2- Lecture 14

Frequency Domain Issues Consider a closed loop system of plant transfer function G(s) and Feedback transfer function H(s) respectively. respectively The closed loop transfer function corresponding to negative feedback may be written as: G ((ss ) 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 e ta take e a Co Complex p e Number u be in tthe e ss-plane pa ea and d subst substitute tute itt into to a Function F(s), it results in another Complex Number which could be plotted in the F(s) Plane.

NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System

Module 2- Lecture 14

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 s-plane plane encloses a pole or a zero, it results in enclosing of the origin in the F(s) plane

NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System

Module 2- Lecture 14

Example: p A Clockwise Contour in the s-plane p for G(s) ( ) = s-z1

Reference: Nise: Control Systems Engineering

NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System

Module 2- Lecture 14

Example: A clockwise contour around a Right-half Plane Pole for a function G(s) = 1/(s-p 1/(s p1 )

Reference: Nise: Control Systems Engineering

NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System

Module 2- Lecture 14

Nyquist yq Stability y Criteria

– Number Number of Counterclockwise (CCW) rotation N  of Counterclockwise (CCW) rotation N = P Pc – Zc (Pc – no. of enclosed poles of 1+ G(s)H(s) and Zc – no. of enclosed  zeroes) – For a Contour in s‐plane mapped through the entire right half  o a o ou s p a e apped oug ee e g a plane of open loop transfer function G(s) H(s), the number of  closed loop poles Zc (same as the open loop zeros)  in the  right half plane equals the number of open loop poles Pc in  the right half plane minus the number of counterclockwise  revolution N around the point ‐1 of the mapping.  – Z c= Pc ‐ N 

NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System

Consider a plant transfer function G(s) as follows: 2

s  12s  24 G(s)  2 s  8s  15

For a unity y feedback closed loop, p, find using g Nyquist yq Criteria whether the system will be unstable at some values of K. (Vary K from 0.5 to 10)

Module 2- Lecture 14

NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System

Module 2- Lecture 14

The Nyquist diagram corresponding to unity Gain and the root locus are   shown below for your reference. Nyquist Diagram

Root Locus 0.4

0.3

0.3

0.2

0.2 Imaginary Axis (seconds-1)

0.4

Imaginary Axis

0.1

0

-0 1 -0.1

System: tf1 Gain: 0 0.51 51 Pole: -6.63 Damping: 1 Overshoot (%): 0 Frequency (rad/s): 6.63

0.1

0

-0.1

-0.2

-0.2 -0.3

-0.3 -0.4 -10

-0.4 04 -1

-0.5

0

0.5

1

1.5

-8

2

-6

-4

-2

0

2

Real Axis (seconds -11)

Real Axis

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NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System

Gain Margin g

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. shown in the figure.

Module 2- Lecture 14

NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System

Phase Margin g

The Phase Margin is the amount of phase that needs to be  g p added to a system such that the magnitude will be just unity  while the phase is 1800  .  The figure below is showing ‘theta’  to be the phase margin.

Often control engineers consider a system to be  0   q y p g adequately stable if it has a phase margin of at least 30 . 

Module 2- Lecture 14

NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System

Module 2- Lecture 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

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