The passiv ity t heo re m TI Nonlinear static Syste ms y f(a) Consider a system o bt ained by connecting two systems Hi and H2 in a feed back loo p ly jui f (a(tu(t)dt · Passive if af() Input st rictly passive(ISP)if af(a) 6|-2|2 H Out put st rictly passive if rf(m)≥6f2(m Let Hi be st rictly out put passive and H2 be passive. the closed-loop system is then BIBO stable Geo met ric Interpret at ion Passivity is an invariant under feed back .f(a=2+a input st rictly passive Use of passivity in system design .f(a=a(1+=out put strictly passive erce cont rol in ro bot ics Remote manipulator How to think abo ut the pro blem Relations between sm all gain and P ass iv ity t here ms a pplicat io ns to adaptive co ntro Struct ure of Adaptive Systems r+H时 esults Insig ht Modified Algo rit ht a→b:H1→(I+H1)-1H1,H2→I-H2 b→c:S;=(H1+1)-(H;-I) C K.. Astrom and B WittenmarkNonlinear Static Systems y = f (u) hy j ui = Z 1 0 f (u(t))u(t) dt  Passive if xf (x)  0  Input strictly passive (ISP) if xf (x)  jxj2  Output strictly passive if xf (x)  f 2(x) Geometric Interpretation Example  f (x) = x + x3 input strictly passive  f (x) = x=(1 + jxj) output strictly passive. The Passivity Theorem Theorem 3 Consider a system obtained by connecting two systems H1 and H2 in a feedback loop Σ u e y H1 − H2 Let H1 be strictly output passive and H2 be passive. The closed-loop system is then BIBO stable. Passivity is an invariant under feedback. Use of passivity in system design.  Force control in robotics  Remote manipulator  How to think about the problem Relations Between Small Gain and Passivity Theorems Σ Σ a) b) I d) 2 Σ I Σ c) H1 − I − H2 − I I − H2 S1 − S2 H1 − H2 ( ) I + H1 −1 H1 1 2 a ! b :H1 ! (I + H1)￾1H1; H2 ! I ￾ H2 b ! c :Si = (Hi + I)￾1 (Hi ￾ I ) Applications to adaptive control  Structure of Adaptive Systems  Apply passivity results  Insight  Modied Algorithms  PI adjustments c K. J. Åström and B. Wittenmark 4