1 - Automatic Controls

1 - Automatic Controls - INTRODUCTION TO AUTOMATIC CONTROLS...

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Tom Penick [email protected] www.teicontrols.com/notes AutomaticControls.pdf 5/10/2000 Page 1 of 10 INTRODUCTION TO AUTOMATIC CONTROLS INDEX adjoint. ............................... 6 block diagrams. .................. 4 closed loop system. ....... 5, 10 E ( s ).................................... 6 e ( t )..................................... 6 error steady state tracking. ........ 6 tracking. ............................... 6 e ss ..................................... 6 glossary. ........................... 10 impulse function. ................ 3 inputs. ................................ 5 inverse Laplace transform1, 2 lag compensation. ............... 7 Laplace transform. .............. 1 lead compensation. ............. 7 Mason's gain rule. ............... 4 open loop system. ......... 5, 10 P-D Controllers . ................. 8 phase lag compensation . ..... 7 phase lead compensation. ....7 P-I Controllers. ................... 8 PID. ................................. 10 PID Controllers . ................. 8 pole of a function. ........... 2, 3 proportional derivative. ....... 8 proportional integral. .......... 8 proportional integral derivative. ..................... 8 R ( s ).................................... 5 r ( t )..................................... 5 ramp . ................................. 5 residues. ......................... 2, 3 repeated roots. ................. 3 stability. ............................. 2 state vector model. .......... 5, 6 steady state tracking error . ..6 steady-state response. ....... 10 step. ................................... 5 tracking error. ................. 5, 6 LaPlace transform . ........... 6 transient response. ............ 10 trig identities. ..................... 9 type 0 system . .................... 5 type 1 system . .................... 5 type 2 system . .................... 5 unit ramp. ........................... 5 unit step. ............................ 5 unity feedback. ................... 4 zero of a function. ........... 2, 3 LAPLACE TRANSFORMS We use Laplace transforms because we are dealing with linear dynamic systems and it is easier than solving differential equations. We don't use Fourier transforms because we are dealing with the transient response and because a Fourier transform won't handle a system that "blows up". LAPLACE TRANSFORM The Laplace transform is used to convert a function f ( t ) in the time domain to a function F ( s ) in the s domain , where s is a complex number: ( 29 ( 29 0 st F s e ft dt - = f ( t ) is 0 for t <0 . f ( t ) can "blow up" or be piecewise. We are free to pick the value of s to make the integral converge; however, once the calculation is made you can use the result everywhere. For example if ( 29 10 t f te = , then s must be 10 or greater to do the integration. But the result is ( 29 ( 29 1 / 10 F ss =- , in which s can be less than 10. Misc: sj =σ+ϖ , 1 jx e = INVERSE LAPLACE TRANSFORM The inverse Laplace transform is used to convert a function F ( s ) in the s domain to a function f ( t ) in the time domain , where s is a complex number: ( 29 ( 29 1 2 Cj st f t Fse ds j +∞ -∞ = π In the conceptual view, c is a real number defining a line in the s-plane as shown at right. All poles of F ( s ) must lie to the left of this line. Poles are always symmetric about the real axis. c-j c+j Real axis c Imaginary axis
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Tom Penick [email protected] www.teicontrols.com/notes AutomaticControls.pdf 5/10/2000 Page 2 of 10 INVERSE LAPLACE TRANSFORM Second Order Conjugate Pair Example ( 29 ( 29( 29 100 1 1 0 1 10 Fs s s j sj = + + +- ( 29 10 1 0 1 0 cos1 0 sin10 101 tt f t e t et --  =  A second order conjugate pole pair in the left-hand side of the s-plane results in a damped sinusoid in the time domain. SYSTEM STABILITY Stable: A system is stable is there are no roots in the right- hand plane and no repeated roots on the j ϖ axis.
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This note was uploaded on 04/03/2012 for the course EE 366 taught by Professor Pore during the Spring '08 term at University of Texas at Austin.

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1 - Automatic Controls - INTRODUCTION TO AUTOMATIC CONTROLS...

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