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EE451_Chapter4_Notes_F09

# EE451_Chapter4_Notes_F09 - EE451/551 Digital Control...

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EE451/551: Digital Control Chapter 4: Stability of Digital Control Systems

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Definitions of Stability Asymptotic Stability (AS): A system is said to be AS if its Asymptotic Stability (AS): A system is said to be AS if its response to any IC decays to zero asymptotically in the steady state, i.e., lim ( ) 0 y k = Marginal Stability (MS): A system is said to be MS is its k →∞ Marginal Stability (MS): A system is said to be MS is its response to an IC remains bounded but does not decay to zero, e.g.,
Definitions of Stability In the absence of unstable pole zero cancellation a zero cancellation, a discrete LTI system is AS if its z TF poles lie within the open unit circle and MS if its poles lie within the closed unit circle with no repeated poles on the unit circle.

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Definitions of Stability Bounded Input Bounded Output Stability (BIBOS): A Bounded Input Bounded Output Stability (BIBOS): A system is said to be BIBOS if its response to any bounded input remains bounded, i.e., It can be shown that a discrete LTI system is BIBOS if and ( ) 0 ( ) 0 u y u k b k y k b k < < ∞∀ < < ∞∀ only if its impulse response sequence is absolutely summable, i.e., Note in the absence of unstable pole zero cancelation 0 ( ) i h i = < ∞ Note, in the absence of unstable pole zero cancelation within a system, the conditions necessary for an LTI system to be AS are equivalent to BIBOS, i.e., all system poles must lie inside the open unit circle
Internal Stability of Closed Loop Systems The stability of a closed loop system’s overall TF is not loop system s overall TF is not sufficient to ensure stability of all parts of the system, i.e., it is essential that all internal signals in the loop be bounded, e.g., e(k), u(k), and y(k) , when bounded external inputs, e.g., r(k) and d(k) , are applied to the system

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Internal Stability of Closed Loop Systems The z TFs associated with the illustrated closed loop system are given by: ( ) ( ) ( ) ZAS ZAS C z G z G z 1 ( ) ( ) 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ZAS ZAS C z G z C z G z Y z R z U z C z G z D z C z + + = The system is Internally Stable (IS) if and only if all of the 1 ( ) ( ) 1 ( ) ( ) ZAS ZAS ZAS C z G z C z G z + + The system is Internally Stable (IS), if and only if all of the z TFs that relate inputs to outputs are BIBOS; implying the characteristic poly 1 ( ) ( ) has its zeros within C z G z + the characteristic poly. 1 ( ) ( ) has its zeros within the open unit circle, and the loop gain ( ) ( ) has no t bl l ll ti i i t id ZAS ZAS C z G z unstable pole-zero cancellation, i.e., occurring on or outside the unit circle (Theorem 4.5, p. 95)
Questions?

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In class Exercise Verify that the closed loop system with T=0 1 is not Verify that the closed loop system with T=0.1 is not internally stable when: ( ) 0.5848 0.3549 1 s + ( )( ) 2 ( ) , and 0.1828 0.8627 1 10 0 8149 0 7655 G s s s = + + ( )( ) 0.8149 0.7655 C( ) 1 1.334 z z z z z = Hint: ( ) 1 ( ) First find ( ) 1 ,then ZAS G s G z z s = Z compute ( ) ( ) and verify the conditions
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EE451_Chapter4_Notes_F09 - EE451/551 Digital Control...

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