ME5659_LecNotes_RootLocus

ME5659_LecNotes_RootLocus - ClosedLoop Stability Dealing...

Info iconThis preview shows pages 1–10. Sign up to view the full content.

View Full Document Right Arrow Icon
1 Closed–Loop Stability • Dealing with the issue of stability for closed– loop systems is an important part in analyzing a control system. • As mentioned, a closed–loop control system is stable if it has a bounded response to a bounded input . This is called external stability viewpoint. • A system has internal stability if the zero- input response is bounded. A necessary condition for this stability is to have all the characteristic roots (poles) on the LHP of the complex s-plane.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
2 Stability of Dynamical Systems Stable Unstable Neutral
Background image of page 2
3 Stability of Closed–Loop Systems Stable System: All poles of the closed–loop transfer function are on LHP of complex s-plane. Unstable System: At least one pole of the closed–loop transfer function is on RHP of s-plane. Marginally Stable: Poles of the closed–loop transfer function are on the j ω –axis (imaginary).
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
4 Closed–Loop as a Linear System • The necessary and sufficient conditions for a feedback system to be stable is that all the poles of the transfer function of the system have negative real parts . • This means that all the poles must be located on the left–half of the s –plane . • Determining the stability by examining the characteristic equation of the closed–loop transfer function.
Background image of page 4
5 Stability Stability • Consider the following control system _ + E ( s ) Y ( s ) R ( s ) G ( s ) H ( s ) () ( ) () () ( ) s D s N s H s G s G s R s Y s T = + = = 1 ) ( The closed-loop transfer function, T ( s ), is Numerator Denominator
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
6 Stability Stability • The system outputs is: () ( ) ( ) ( ) s R P s P s P s s N s R s D s N s Y n = = " 2 1 n P P P ..., , , 2 1 are the closed loop poles .
Background image of page 6
7 Stability Stability • Using partial fraction expansion, one has () ) ( from erms fraction t partial the ... 2 2 1 1 s R P s k P s k P s k s Y n n + + + + = Taking the inverse Laplace transform, one has: ( ) t y e k e k e k t y r t P n t P t P n + + + + = " 2 1 2 1
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
8 Stability Stability • For a control system to be stable, the poles, i.e., the roots of the characteristic equation, must lie on the left half of the S-Plane. If any one pole lies on the right half plane, the system is unstable. Re Im Unstable Region Stable Region
Background image of page 8
9 Stability Stability Special Case : Roots on the Imaginary Axis • A system that has poles on the imaginary axis is marginally stable .
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 10
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 05/26/2010 for the course ME 5659 taught by Professor Jalili during the Spring '10 term at Northeastern.

Page1 / 38

ME5659_LecNotes_RootLocus - ClosedLoop Stability Dealing...

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online