Topic #26 Nyquist Stability Theory Reference textbook : Control - PowerPoint PPT Presentation

ME 779 Control Systems Topic #26 Nyquist Stability Theory Reference textbook : Control Systems, Dhanesh N. Manik, Cengage Publishing, 2012 1

Control Systems: Nyquist Stability Theory Feedback transfer function ( ) ( ) C s G s   ( ) 1 ( ) ( ) R s G s H s    ( )( )...( ) K s z s z s z  1 2 m ( ) ( ) G s H s    ( )( )...( ) s p s p s p 1 2 n poles and zeros of the open-loop transfer function 2

Control Systems: Nyquist Stability Theory Characteristic equation        ( s p )( s p )...( s p ) K ( s z )( s z )...( s z )   1 2 n 1 2 m 1 G ( s ) H ( s )    ( s p )( s p )...( s p ) 1 2 n the number of closed-loop poles - that is zeros of 1+GH will be equal to the number of open-loop poles 3

Control Systems: Nyquist Stability Theory Characteristic equation        ( s p )( s p )...( s p ) K ( s z )( s z )...( s z )   1 2 n 1 2 m 1 G ( s ) H ( s )    ( s p )( s p )...( s p ) 1 2 n    ( s z )( s z )( s z )   c c c 1 G ( s ) H ( s ) 1 2 n    ( )( )...( ) s p s p s p 1 2 n , ... z z z zeros of 1+G(s)H(s ), c c c 1 2 n Are also poles of the close-loop transfer function 4

Control Systems: Nyquist Stability Theory Characteristic equation    s z s z ... s z Magnitude   c c c 1 2 n 1 ( ) ( ) G s H s    ( ) ( ) ... ( ) . s p s p s p 1 2 n       Angle s z s z s z     c c c 1 ( ) ( ) G s H s 1 2 n      ( s p ) ( s p ) ( s p ) 1 2 n 5

Control Systems: Nyquist Stability Theory s-plane to 1+GH plane mapping phase angle of the 1+G(s)H(s) vector, corresponding to a point on the s-plane is the difference between the sum of the phase of all vectors drawn from zeros of 1+GH(close loop poles) and open loops on the s plane. If this point s is moved along a closed contour enclosing any or all of the above zeros and poles, only the phase of the vector of each of the enclosed zeros or open-loop poles will change by 360 o . The direction will be in the same sense of the contour enclosing zeros and in the opposite sense for the contour enclosing open-loop poles. 6

Control Systems: Nyquist Stability Theory s-plane to 1+GH plane mapping 7

Control Systems: Nyquist Stability Theory s-plane to 1+GH plane mapping 8

Control Systems: Nyquist Stability Theory THEORY When a closed contour in the s-plane encloses a certain number of poles and zeros of 1+G(s)H(s) in the clockwise direction, the number of encirclements by the corresponding contour in the G(s)H(s) plane will encircle the point (-1,0) a number of times given by the difference between the number of its zeros of 1+G(s)H(s) and the open-loop poles it enclosed on the s-plane 9

Control Systems: Nyquist Stability Theory Contour on the s-plane for checking THEORY the existence of closed-loop poles GH from the polar plot Magnitude zero since n >m GH from the mirror image of the polar plot 10

Control Systems: Nyquist Stability Theory Modified contour on the s-plane for THEORY checking the existence of closed-loop poles e    j s Magnitude of GH remains the same along the contour, phase of β changes from 270 to 90 degrees 11

Control Systems: Nyquist Stability Theory Conclusion 12

ME 779 Control Systems Topic #28 Checking Nyquist Stability Reference textbook : Control Systems, Dhanesh N. Manik, Cengage Publishing, 2012 1

Control Systems: Checking Nyquist Stability Procedure (1) Locate open-loop poles on the s-plane (2) Draw the closed contour and avoid open-loop poles on the imaginary axis (3) Count the number of open-loop poles enclosed in the above contour of step 2, say P (4) Plot G(j  )H(j  ) and its reflection on the GH plane and map part of the small semi-circle detour on the s-plane around poles (if any) on the imaginary axis. 2

Control Systems: Checking Nyquist Stability Procedure (5) Once the entire s-plane contour is mapped on to the GH plane, count the number of encirclements of the point (-1,0) and its direction. Clockwise encirclement is considered positive, say N. (6) The number of closed-loop poles in the right- half s-plane is given by Z=N+P. if Z >0, the system is unstable. (7) Determine gain margin, phase margin, and critical value of open-loop gain. 3

Control Systems: Checking Nyquist Stability Example 1 Using Nyquist criterion, determine the stability of a feedback system for K=1, whose open-loop transfer function is given by K  ( ) ( ) G s H s  s s ( 1) . Step 1 Locate open-loop poles on the s-plane Open loop poles are at s=0 and – 1. Step 2 Draw the closed contour on the s-plane to check the existence of closed-loop poles in the right-half s-plane. 4

Control Systems: Checking Nyquist Stability Example 1 Open-loop poles and s-plane contour 5

Control Systems: Checking Nyquist Stability Example 1 1    G j ( ) H ( )    2 1          1 G j ( ) H j ( ) tan 2 6

Control Systems: Checking Nyquist Stability No. Frequency, Magnitude Phase, rad/s degrees 1 0.2 4.9029 259 2 0.4 2.3212 248 3 0.8 0.9761 231 4 1 0.7071 225 5 4 0.0606 194 6 10 0.01 186 7 50 0.0004 181 8 100 0.0001 181 9 200 0 180 7

Control Systems: Checking Nyquist Stability No. Frequency, Magnitude Phase, rad/s degrees 10 -200 0 180 11 -100 0.0001 179 12 -50 0.0004 179 13 -10 0.01 174 14 -4 0.0606 166 15 -1 0.7071 135 16 -0.8 0.9761 129 17 -0.4 2.3212 112 18 -0.2 4.9029 101 8

Control Systems: Checking Nyquist Stability  , s-plane,  , GH plane,  , s-plane,  , GH plane, deg deg deg deg 0 0 270 101 10 348 280 91 20 337 290 80 30 325 300 69 40 314 310 58 50 302 320 46 60 291 330 35 70 280 340 23 80 269 350 12 90 259 9

Control Systems: Checking Nyquist Stability The system is stable Example 1 10

Control Systems: Checking Nyquist Stability Example 2 Determine the stability of the feedback system whose open-loop transfer function given by 55  ( ) ( ) G s H s .   ( 2)( 4) s s s Locate open-loop poles on the s-plane Draw the closed loop contour on the s-plane 11

Control Systems: Checking Nyquist Stability Example 2 55  ( ) ( ) G s H s   ( 2)( 4) s s s 12

Control Systems: Checking Nyquist Stability Example 2 The number of open-loop pole enclosed, P is zero K    ( ) ( ) G j H j      2 2 4 16             1 1 G j ( ) H j ( ) tan tan 2 2 4 13

Control Systems: Checking Nyquist Stability No. Phase, Frequency Magnitude degrees 1 1.5 Positive 3.4332 213 frequencies 2 2 2.1741 198 3 2.5 1.4568 187 4 2.83 1.1446 180 5 3 1.017 177 6 3.5 0.7334 169 7 4.5 0.4122 156 8 5 0.319 150 9 5.5 0.2513 146 10 6 0.201 142 11 7 0.1339 136 12 8 0.0932 131 13 9 0.0673 126 14

Control Systems: Checking Nyquist Stability 14 -9 Negative 0.0673 234 frequencies 15 -8 0.0932 229 16 -7 0.1339 224 17 -6 0.201 218 18 -5.5 0.2513 214 19 -5 0.319 210 20 -4.5 0.4122 204 21 -3.5 0.7334 191 22 -3 1.017 183 23 -2.83 1.1446 180 24 -2.5 1.4568 173 25 -2 2.1741 162 26 -1.5 3.4332 147 15

Control Systems: Checking Nyquist Stability Example 2  , GH  , s-plane, plane,  , s-  , GH deg deg plane, plane, 0 0 deg deg 10 343 270 147 20 326 280 132 30 309 290 117 40 292 300 101 50 276 310 84 60 259 320 68 70 243 330 51 80 228 340 34 90 213 350 17 16

Control Systems: Checking Nyquist Stability Example 2 Z=N+P=2 The system is unstable 17

Control Systems: Checking Nyquist Stability Example 3 Determine the stability of the feedback system whose open-loop transfer function given by K .  ( ) ( ) G s H s  2 ( 1) s s 18

Control Systems: Checking Nyquist Stability Example 3 Locate open-loop poles on the s-plane Draw the closed loop contour on the s-plane 19

Control Systems: Checking Nyquist Stability Example 3 Count the number of open-loop poles enclosed in the closed contour of the s-plane The number of open-loop poles enclosed, P is zero. K    G j ( ) H j ( )    2 2 1          1 ( ) ( ) tan G j H j 20

Control Systems: Checking Nyquist Stability No. Phase, degree Frequency Magnitude s 1 0.49 Positive 3.74 154 frequencies 2 0.50 3.58 153 3 0.60 2.38 149 4 0.80 1.22 141 5 1.00 0.71 135 6 2.00 0.11 117 7 3.00 0.04 108 8 4.00 0.02 104 9 5.00 0.01 101 10 6.00 0.00 99 11 7.00 0.00 98 12 8.00 0.00 97 13 9.00 0.00 96 14 10.00 0.00 96 21