EE451_Chapter8_Notes_F09

# EE451_Chapter8_Notes_F09 - EE451/551 Digital Control...

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EE451/551: Digital Control Chapter 8: Properties of State Space Models

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Equilibrium State Definition 8.1: An equilibrium point or state is an initial x state from which the system nevers departs unless perturbed or the general state eqn : e () For the general state eqn.: ( 1) ( ) xk f xk += equilibriu ( ) ( ) m points satisfy the condition: ( 1 )( ) e e x f f xx + == = For LTI systems, equilibrium points satisfy: (1 ) ( ) 0 ee e x kA k A A I [ ] [] the soln. for invertible asymptotic s n e n x x AI x + = −= −⇔ tability is: 0 e x =
Asymptotic Stability efinition82:AnLTIsystemisasympto cally stable (AS) if 0 Definition 8.2: An LTI system is asymptotically stable (AS) if all of its trajectories converge to zero for any initial state ( ) xk ( lim ) 0 e k x →∞ ⇒→ = Theorem 8.1: A discrete-time LTI system is AS if and only if all eigenvalues of its state matrix lie inside the unit d A [] () 1 circle exits or rank , lled full rank dn AI n ⇔− = called full rank

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Bounded Input Bounded Output Stability efinition 8 3: An LTI system is BIBO st le if its output is Definition 8.3: An LTI system is BIBO stable if its output is bounded for any bounded input, i.e., ( ) ( ) heorem 8 2: A discrete time LTI system BIBO stable if uy uk b yk b < <∞⇒ < <∞ Theorem 8.2: A discrete-time LTI system is BIBO stable if and only if the norm of its impulse response is absolutely 0 (see matrix norm in Appendix III) summable, i.e., ( ) k hk = Theorem 8.3: If a discrete-time LTI system is AS, then it is IBO stable; furthermore in he absence of unstable pole BIBO stable; furthermore, in the absence of unstable pole- zero cancellation, the system is AS if it is BIBO stable
Stability Example

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EE451_Chapter8_Notes_F09 - EE451/551 Digital Control...

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