Lecture 15 Nyquist Criterion and Diagram


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1 Lecture Notes of Control Systems I  ME 41/Analysis and Synthesis of Linear Control System  ME86 Lecture 15 Nyquist Criterion and Diagram Department of Mechanical Engineering, University Of Saskatchewan, 57 Campus Drive, Saskatoon, SK S7N 5A9, Canada 1
2 Lecture Notes of Control Systems I  ME 41/Analysis and Synthesis of Linear Control System  ME86 1. Review of System Stability and Some Concepts Related to Poles and Zeros 1. 1 System stability Consider the following closedloop system: R( + _ E( G( C( B( H( The closedloop transfer function, T(, is C T ( R where ( s H ( s ( s ( s G 1+ G ( s ( s H ( s G is the openloop transfer function, i.e., the equivalent transfer function relating the error, E(, to the feedback signal, B(. The closedloop poles are the roots of the characteristic equation, i.e., ( s H ( G s In order that the closedloop system is stable, all of the closedloop poles must be in the left half plane (LHP. Department of Mechanical Engineering, University Of Saskatchewan, 57 Campus Drive, Saskatoon, SK S7N 5A9, Canada
3 Lecture Notes of Control Systems I  ME 41/Analysis and Synthesis of Linear Control System  ME86 1. Some Concepts Related to Poles and Zeros For a function F( with a variable of s, the poles of F( are the values of s such that F ( ; and the zeros of F( are the values of s such that F ( 0. Please note the above definition is consistent with the definition that we had for the closedloop poles/zeros and the openloop poles/zeros. The closedloop poles/zeros are G( the poles/zeros of the closedloop transfer function, i.e., ; and the open 1+ G s H s ( ( loop poles/zeros are the poles/zeros of the openloop transfer function, i.e., ( s H ( s Particularly, we have following relationships: the closedloop poles the roots of characteristic equation: 1 + G ( 0 the zeros of 1 + G( s the openloop poles 1 + G s the poles of ( G. ( s + 1( s As an example, the above relationships can be verified by usingg (. s( s Department of Mechanical Engineering, University Of Saskatchewan, 57 Campus Drive, Saskatoon, SK S7N 5A9, Canada
4 Lecture Notes of Control Systems I  ME 41/Analysis and Synthesis of Linear Control System  ME86. Cauchy s Principle of Argument.1 Mapping from splane to Fplane through a function of F( For a point. Taking a complex number in the splane and substituting it into a function of F(, the result is also a complex number, which is represented in a new complexplane (called Fplane. This process is called mapping, specifically mapping a point from s plane to Fplane through F(. For a contour. Consider the collection of points in the splane (called a contour, shown in the following figure as contour A. Using the above point mapping process through F(, we can also get a contour in the Fplane, shown in the following figure contour B. Case 1: F( has one zero, i.e., F( s a Case : F( has one pole, i.e., F( 1 s a Case : F( has a number of poles and zeros, i.e, ( s z1( s z L F( ( s p ( s p L 1 Department of Mechanical Engineering, University Of Saskatchewan, 57 Campus Drive, Saskatoon, SK S7N 5A9, Canada 4
5 Lecture Notes of Control Systems I  ME 41/Analysis and Synthesis of Linear Control System  ME86. Cauchy s Principle of Argument The Cauchy s Principle of Argument states that, if taking a clockwise contour in the s plane and mapping it to the Fplane through F(, The number of clockwise rotations about the origin of the contour in the Fplane, N The number of zeros of F( inside the contour in the splane, Z  The number of poles of F( inside the contour in the splane, P or simply N Z P Example 1 Determine the number of clockwise rotations about the origin of the mapping through ( s 1( s ( s + 5 F (. If the contour in the splane includes the entire right half ( s ( s + 4 plane, as shown in following figure. Department of Mechanical Engineering, University Of Saskatchewan, 57 Campus Drive, Saskatoon, SK S7N 5A9, Canada 5
6 Lecture Notes of Control Systems I  ME 41/Analysis and Synthesis of Linear Control System  ME86. Nyquist Criterion Nyquist Contour: is the contour in the splane that includes the entire right half plane, as shown in the proceeding page. Now, let s consider F ( 1+ G( and the contour in the splane is the Nyquist contour (i.e., the entire right half plane. Applying the Cauchy s Principle of Argument, we should have The number of rotations about the origin of the mapping through 1+ G(, N The number of zeros of 1+ G( in the right half plane, Z  The number of poles of 1+ G( in the right half plane, P Please note The mapping through G( is virtually the same as the one through 1+ G(, except that the contour is shifted one unit to the left. Thus, we can count rotations about 1 instead of rotations about the origin in the above statement. The zeros of 1+ G( the closedloop poles. The poles of 1 + G( s H ( s the openloop poles or the poles of ( s H ( s G. Therefore The number of rotations about 1 of the mapping through G (, N The number of closedloop poles in the right half plane, Z  The number of openloop poles in the right half plane, P or The number of closedloop poles in the right half plane, Z The number of rotations about 1 of the mapping through G (, N + The number of openloop poles in the right half plane, P The above relationship is called the Nyquist Criterion; and the mapping through G ( is called the Nyquist Diagram of G ( For a system to be stable, Z must be zero. Department of Mechanical Engineering, University Of Saskatchewan, 57 Campus Drive, Saskatoon, SK S7N 5A9, Canada 6
7 Lecture Notes of Control Systems I  ME 41/Analysis and Synthesis of Linear Control System  ME86 Example The following figures (a and (b show, respectively, the Nyquist contour (in which denotes the location of an open loop pole and the Nyquist diagram for a control system. Determine the system stability using the Nyquist criterion. (a Nyquist contour (b Nyquist diagram Department of Mechanical Engineering, University Of Saskatchewan, 57 Campus Drive, Saskatoon, SK S7N 5A9, Canada 7
8 Lecture Notes of Control Systems I  ME 41/Analysis and Synthesis of Linear Control System  ME86 4. Sketching the Nyquist Diagram Suppose the openloop transfer function 1 G(, sketch its Nyquist diagram. 1 + s B A C D Nyquist contour Key Points of the polar plot: GH GH ω ω 090 o Cross Re: ω 0 See above See above ω Cross Re: ω 0 See above See above ω ω 1 rad/s o Sketching the Nyquist diagram includes two steps: (1 Sketch the mapping of Point A to Point B, which is the same as the polar plot of frequency response for G (. Note that the semicircle with a infinite radius, i.e., BCD, is mapped to the origin if the order the denominator of G ( is greater than the order the numerator of G (. ( Sketch the mapping of Point A to Point D, which is the mirror image about the real axis of the mapping of Point A to Point B. Department of Mechanical Engineering, University Of Saskatchewan, 57 Campus Drive, Saskatoon, SK S7N 5A9, Canada 8
9 Lecture Notes of Control Systems I  ME 41/Analysis and Synthesis of Linear Control System  ME86 Example Sketch the Nyquist diagram for the system shown in the following figure, and then determine the system stability using the Nyquist criterion. R( + _ E( 500 ( s + 1( s + ( s + 10 C( Solution: (1 Sketch the Nyquist diagram. The openloop transfer function: 500 G (. ( s + 1( s + ( s + 10 Replacing s with jω yields the frequency response of G (, i.e., G( jω H ( jω ( jω + 1( jω + ( jω + 10 ( 14ω j(4ω ( 14ω + 0 j(4ω 500 ( 14ω (4ω Magnitude response: G ( jω H ( jω Re + Im 500 ( 14ω (4ω Phase response: G jω H ( jω tan Im tan Re ( 4ω ω 14ω ( Cross Re: Im 0 (4ω ( 14ω (4ω 4ω 0 0 ω ω 0 and ω 6.56 rad/s Department of Mechanical Engineering, University Of Saskatchewan, 57 Campus Drive, Saskatoon, SK S7N 5A9, Canada 9
10 Lecture Notes of Control Systems I  ME 41/Analysis and Synthesis of Linear Control System  ME86 Cross Im: Re 0 14ω + 0 ( 14ω (4ω 14ω ω ω 1.46 rad/s Key Points of the polar plot: Nyquist diagram GH GH ω ω 070 o Cross Re: ω 0 ω ω 6.56 rad/s See above See above 180 o Cross Re: ω 0 ω ω 1.46 rad/s See above 8.6 See above 90 o Im 8.6j 8.6j Re ( Determine the system stability using the Nyquist criterion. Department of Mechanical Engineering, University Of Saskatchewan, 57 Campus Drive, Saskatoon, SK S7N 5A9, Canada 10
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