The small gain Theorem Examples DEFINITION 1 Linear systems with sig nals in Asystemis called bounded-input, bounded output(B/BO) sta ble if the system has lyl‖≤maG(i!川·‖u bounded gain uo= sin THEOREM 1 Linear Systems with sig nals in L Consider the system H (G) h() dr o(s)=uo sign(h(t-s) H 2 Static nonlinear system LEt y1 and y2 be the g ains cf the systems H1 and H2. The dased -loop system is BIBOstable if <1 and its gain is less than 172 A Formal statement Passivity DEFINITION 2 A systemwith input u and output y is passive ● The id dissipati Capacitors, inductors, resistances The systemis input strictly passive(IsP)if there exists e>0 such that Circuit Theory y|u)≥ul Mechatronics and output strictly passive(aSP )if there exists ●Ⅳ Mathematica formaizatio E>0 suchthat The nation of p hase y|u)≥ely‖ Postive red linear system · The passivity theoren tutte Using passivity in system design Think about u and v as voitage and arment or farce and veoaty Causality? C K.. Astrom and B WittenmarkExamples Linear systems with signals in L2e kyk  max ! jG(i!)j
kuk u0 = sin !t Linear Systems with signals in L1 (G) = Z 1 0 jh( )j d u0(s) = u0 sign(h(t ￾ s)) Static nonlinear system x f(x) f = γx f = −γx The Small Gain Theorem Definition 1 A system is called bounded-input, bounded￾output (BIBO) stable if the system has bounded gain. Theorem 1 Consider the system. Σ u e y H1 − H2 Let 1 and 2 be the gains of the systems H1 and H2. The closed-loop system is BIBO stable if 1 2 < 1 and its gain is less than = 1 1 ￾ 1 2 Passivity  The idea { Energy dissipation { Capacitors, induktors, resistances { Mass, spring, dashpot { Circuit Theory { Mechatronics  Mathematical Formalization  The Notion of Phase  Examples  Postive real linear system  The passivity theorem  Using passivity in system design A Formal Statement Definition 2 A system with input u and output y is passive if hy j ui  0 The system is input strictly passive (ISP) if there exists " > 0 such that hy j ui  "kuk 2 and output strictly passive (OSP) if there exists " > 0 such that hy j ui  "kyk 2 Intuitively  Think about u and v as voltage and current or force and velocity  Causality? c K. J. Åström and B. Wittenmark 2