Homework 2 Solution. The following Matlab code was used to define the system and plot its poles and zeros in the s-plane:

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Gregory Logan
6 years ago
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1 Homework Soluton Problem a Fn cloe loo tranfer functon T Y / R. Plot ole an zero. Y H T R H k Y 5 T. 9. R k For k, the tranfer functon become: T k. 9. k The followng Matlab coe wa ue to efne the ytem an lot t ole an zero n the -lane: %efnng the arcraft ytem num[]; en[. 9.]; Htfnum,en; %efnng the comenator numc*[ ]; enc[ 5]; tfnumc,enc; %efnng the cloe loo tranfer functon Tfeeback*H,; % ole an zero ma of the cloe loo ytem n the -lane zmat; The ytem ha one ar of comlex conjugate ole an one real ole. All ole an zero are n the left half of the -lane. The -lane lot of the ole an zero reente n Fgure.

2 Fgure. S-lane lot of the ole an zero of the ytem b Comute te reone by han. The Lalace tranform of the te nut /. Y Calculatng the root of the enomnator ole of the ytem we get: Y j j Ung the artal fracton exanon the equaton become: Y A B C D j.44.5 j where A Y / B 4.7 Y C jy D.44.5 jy j j j j After ome calculaton we obtan: Y

3 *.9 Y Takng the nvere Lalace tranform of th equaton we obtan: y t e e. co.5t.9 n.5t c Ue Matlab to lot the te reone for k,,. The Matlab coe that wa ue to lot the te reone of the ytem : clear all; cloe all; clc; k; tme[:.:4]; for :, %efnng the arcraft ytem num[]; en[. 9.]; Htfnum,en; %efnng the comenator numck*[ ]; enc[ 5]; tfnumc,enc; kk*; %cloe loo tranfer functon Tfeeback*H,; fgure; retet,tme; lottme,re; xlabel'tme []';ttle'ste reone of the ytem'; en The te reone for the fferent value of k lotte n the fgure, an 4. Notce from the fgure that the hgher the value of k the horter become the tme untl the ytem reache teay tate. For k the teay tate reache after 4 econ, for k the ettlng tme 5 econ an for k the ettlng tme become econ.

4 Fgure. Ste reone of the cloe loo ytem for k Fgure. Ste reone of the cloe loo ytem for k 4

5 5 Fgure 4. Ste reone of the cloe loo ytem for k Comute the teay tate error e n reone to a unt te comman. The tranfer functon from rt to et the followng: H R E H e Then T R H H R H R E The Lalace tranform of the te nut R /. Then ung the fnal value theorem we get that the teay tate error : lm lm lm T T e e For k, fn turbance tranfer functon Y/D. The turbance tranfer functon H H D Y H For k the tranfer functon :

6 H f Fn teay tate outut y n reone to a unt te turbance t. Ung the fnal value theorem we get: y lm H lm H g For k, ue Matlab to lot mule reone from t to yt. The Matlab coe that wa ue to obtan the lot of the mule reone : clear all; cloe all; clc; tme[:.:5]; %efnng the arcraft ytem num[]; en[. 9.]; Htfnum,en; %efnng the comenator numc*[ ]; enc[ 5]; tfnumc,enc; %efnng the tranfer functon TfeebackH,; fgure; remulet,tme; lottme,re; xlabel'tme []';ttle'imule reone from t to yt'; In Fgure 5 lotte the outut of the ytem wth the mule turbance nut. 6

7 Fgure 5. Imule reone for the turbance cloe loo tranfer functon Problem : T Y R T The Routh : k k k k 4 k k 4 k 9 k 4 k 9 k k 8k Ung the conton, we get < k <.5 7

8 Problem : Oen Loo ytem H.5 H Cloe Loo ytem T H Charactertc Equaton gven a H a For Proortonal Control Charactertc Equaton gven a.5 Snce all the Pole of the ytem are n rght half lane, therefore the ytem untable an roortonal control unable to make the ytem table. Root Locu lot obtane ung MATLAB k; numk; en[ ]; rlocunum,en; ttle'root locu for Proortonal Control'; xlabel'real Ax'; ylabel'imag Ax'; Root locu for Proortonal Control Imag Ax Real Ax Fgure 6. Root locu for Proortonal Control 8

9 b For PI control Charactertc Equaton gven a.5 Charactertc Equaton Ue Routh Array to etermne whether the ytem table an f t table then ue t to fn the range of the ytem From the above Routh Array, we conclue that the ytem untable: Root Locu lot obtane ung MATLAB k;k num[k k]; en[ ]; rlocunum,en; ttle'root locu for PIControl'; xlabel'real Ax'; ylabel'imag Ax'; 8 Root locu for PIControl 6 4 Imag Ax Real Ax Fgure 7. Root locu for PI Control 9

10 c For PD control Charactertc Equaton gven a.5 Charactertc Equaton Ue Routh Array to etermne whether the ytem table an f t table then ue t to fn the range of the ytem From the above Routh Array, 5 & > 6. 5 for the ytem to be table > Root Locu lot obtane ung MATLAB k; k; num[k k]; en[ ]; rlocu num,en; ttle'root locu for PD Control'; xlabel'real Ax'; ylabel'imag Ax'; Root locu for PD Control Imag Ax Real Ax Fgure 8. Root locu for PD Control

11 For Realzable PD control η Charactertc Equaton gven a 6.5 η Charactertc Equaton η η 6.5η Ue Routh Array to etermne whether the ytem table an f t table then ue t to fn the range of the ytem η η η 5 η 5 6.5η 6.5η 6.5 5η 6.5η From the above Routh Array, η > 5; 6.5 5η η 5 > 6.5η & > 6. 5η for the ytem to be table, therefore for η we get > 4.75 & > 4 Root Locu lot obtane ung MATLAB k6; k-; num[k k]; en[ ]; rlocu num,en; ttle'root locu for realzable PD Control'; xlabel'real Ax'; ylabel'imag Ax'; Root locu for realzable PD Control 5 5 Imag Ax Real Ax Fgure 9. Root locu for realzaton PD Control

12 e For PID control Charactertc Equaton gven a.5 Charactertc Equaton Ue Routh Array to etermne whether the ytem table an f t table then ue t to fn the range of the ytem From the above Routh Array, > ; > 5 & > for the ytem to be table. Root Locu lot obtane ung MATLAB k5;k; k5; num[k k k]; en[ ]; rlocu num,en; ttle'root locu for PID Control'; xlabel'real Ax'; ylabel'imag Ax'; 8 Root locu for PID Control 6 4 Imag Ax Real Ax Fgure. Root locu for PD Control

13 f For Lea Comenator / Charactertc Equaton gven a.5 Charactertc Equaton Ue Routh Array to etermne whether the ytem table an f t table then ue t to fn the range of the ytem From the above Routh Array, > 7. 5 for the ytem to be table Root Locu lot obtane ung MATLAB k; num[k k]; en[ ]; rlocu num,en; ttle'root locu for Lea Comenator'; xlabel'real Ax'; ylabel'imag Ax'; Root locu for Lea Comenator Imag Ax Real Ax Fgure. Root locu for Lea Comenator

14 g For Lag Comenator / Charactertc Equaton gven a.5 Charactertc Equaton Ue Routh Array to etermne whether the ytem table an f t table then ue t to fn the range of the ytem From the above Routh Array, the ytem untable for all value of Root Locu lot obtane ung MATLAB k; num[k *k]; en[ ]; rlocu num,en; ttle'root locu for Lag Comenator'; xlabel'real Ax'; ylabel'imag Ax'; 4 Root locu for Lag Comenator Imag Ax Real Ax Fgure. Root locu for Lag Comenator 4

ELECTRONICS & COMMUNICATIONS DEP. 3rd YEAR, 2010/2011 CONTROL ENGINEERING SHEET 4 PID Controller

CAIRO UNIVERSITY FACULTY OF ENGINEERING ELECTRONICS & COMMUNICATIONS DEP. 3rd YEAR, 00/0 CONTROL ENGINEERING SHEET 4 PID Controller [] The block dagram of a tye ytem wth a cacade controller G c () hown