ME5659_LecNotes_RootLocus

# ME5659_LecNotes_RootLocus - ClosedLoop Stability Dealing...

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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.

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2 Stability of Dynamical Systems Stable Unstable Neutral
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).

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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.
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

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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 .
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

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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
9 Stability Stability Special Case : Roots on the Imaginary Axis • A system that has poles on the imaginary axis is marginally stable .

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

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ME5659_LecNotes_RootLocus - ClosedLoop Stability Dealing...

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