Chapter 5 7 Example: A linear system as follows,where a is a positive constant, impulse response is g)=ce.Determine the stability of the following system. 文=-X+W wie-d Solution: So the system is BIBO stable
• Example: • A linear system as follows, where a is a positive constant, impulse response is . Determine the stability of the following system. • Solution: • So the system is BIBO stable. 7 y cx x ax u at g( t ) c e 0 0 1 0 0 a a a c g( τ ) d τ c e d τ aτ Chapter 5
Chapter 5 Theorem:A SISO system with proper rational transfer function G(s)is BIBO,if and only if the every pole of G(s) has negative real part,or lies inside left-half s-plane. (From classical control theory) ● 。 Theorem:A multi-variable system is BIBO if and only if every pole of transfer function matrix G(s)=[g(s)]has negative real part or lies inside left-half s-plane
• Theorem: A SISO system with proper rational transfer function G(s) is BIBO, if and only if the every pole of G(s) has negative real part, or lies inside left-half s-plane. (From classical control theory) • Theorem: A multi-variable system is BIBO if and only if every pole of transfer function matrix G(s)=[gij(s)] has negative real part or lies inside left-half s-plane. • Chapter 5 8
Chapter 5 9 Discrete system: Theorem:A SISO discrete system is BIBO if and only if g(k)is absolutely summable in [0,+o): ∑g(ksM<∞ 0 Theorem:A SISO discrete system is BIBO if and only if every pole of G(z)has a magnitude less than 1. Theorem:A multi-variable discrete system is BIBO if and only if every pole of matrix G(z)=[gi(z)]has a magnitude less than 1
Discrete system: • Theorem: A SISO discrete system is BIBO if and only if g(k) is absolutely summable in [0, +): • Theorem: A SISO discrete system is BIBO if and only if every pole of G(z) has a magnitude less than 1. • Theorem: A multi-variable discrete system is BIBO if and only if every pole of matrix G(z)=[gij(z)] has a magnitude less than 1. 9 g( k ) M 0 Chapter 5