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StabilityLecture

# StabilityLecture - ME 279 Automatic Control of Dynamic...

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Stability ME 279 Automatic Control of Dynamic Systems Dr. Robert G. Landers

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Definitions 2 Stability Dr. Robert G. Landers A general system response is: y(t) = y n (t) + y f (t), where y n (t) is the natural response due to initial conditions and y f (t) is the forced response due to the forcing function. A Linear, Time–Invariant (LTI) system is stable if y n (t) → 0 as t → ∞. An LTI system is unstable if y n (t) is unbounded as t → ∞. An LTI system is marginally stable if y n (t) remains constant or oscillates as t → ∞. An LTI system is stable if every bounded input yields a bounded output. An LTI system is unstable if any bounded input yields an unbounded output.
Relationship to Transfer Function Poles 3 Stability Dr. Robert G. Landers Stable: all transfer function poles are strictly in the left s–plane. Unstable: at least one transfer function pole is strictly in the right s– plane or at least one transfer function pole with multiplicity n, where n > 1, is on the imaginary axis. A transfer function pole with multiplicity n on the imaginary axis leads to solutions of the form Atcos( ϖ t– φ ), At 2 cos( ϖ t– φ ), etc. Marginally Stable: at least one transfer function pole with multiplicity 1 is on the imaginary axis and all other transfer function poles are strictly in the left s–plane.

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Necessary Stability Conditions 4 Stability Dr. Robert G. Landers Assume the transfer function pole locations of a dynamic system are characterized by the characteristic polynomial d(s). The necessary conditions for stability are: If the signs of the coefficients of d(s) are the same, the system may be stable. If the signs of the coefficients of d(s) are not the same, the system is unstable. If d(s) has missing powers of s, the system is not stable; however, if the signs of the coefficients of d(s) are the same, the system may be marginally stable.
Relationship to Pole Locations 5 Stability Dr. Robert G. Landers Determine if the system is stable, marginally stable, or unstable if R(s) = 1/s. ( 29 ( 29 ( 29 ( 29 ( 29 { 3 4 forcedresponse naturalresponse 1 1 8 9 3 4 12 12 12 t t Y s s y t e e R s s s - - + = = + - + + 1 4 4 2 4 4 3 The transfer function poles are located at –3 and –4. Since the transfer function poles are in the left s–plane, the system is stable. The forced response does not affect the system stability.

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Example 1 6 Stability Dr. Robert G. Landers Determine if the system is stable, marginally stable, or unstable if R(s) = 1/s 2 . ( 29 ( 29 ( 29 ( 29 ( 29 3 4 forcedresponse naturalresponse 1 5 1 32 27 3 4 144 12 144 144 t t Y s s y t t e e R s s s - - + = = + - + + + 142 43 1 4 442 4 4 43 Determine if the system is stable, marginally stable, or unstable if R(s) = 1/s. ( 29 ( 29 ( 29 ( 29 ( 29 { 3 4 forcedresponse naturalresponse 1 1 1 1 3 4 12 21 28 t t Y s s y t e e R s s s - + = = - + + + - 1 4 4 2 4 4 3
Example 1 7 Stability Dr. Robert G. Landers The transfer function poles of the first system are located at –3 and –4. Since the transfer function poles are in the left s–plane, the system is stable. The forced response does not affect the system stability.

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StabilityLecture - ME 279 Automatic Control of Dynamic...

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