Adaptatio n o f fe e dfo rward gain Model Analy s is emura Let r be a baunded square integ rable function and let G(s)be a transfer function that is pasitive real. The systemw hase input-output reation is given b G 叫G(s is then passive Redraw b)as Example pl adjustments (-°)a G e(t)=-u()e(t)-72/ua()(x)dr Explore the advantag es of P adjustments analytically and by simulation A mo diffe d algo rithm The Augme nte d Erro r Consider the error ange G(-θ G ntroduce the augmented error H were 7=G(0-0)uc-(0-0 Guc= GOuc-0Gue Notice that n is zero under stationary condi tons Use the adaptation law Stability now folows fromthe passivity theorem ake GG SPR. Still a problem with pole The idea can be extended to the genera case excess> details are messy. C K.. Astrom and B WittenmarkAdaptation of Feedforward Gain θ Σ – Model Process + Σ – Model Process + y y e e θ uc uc kG(s) kG(s) k0G(s) k0G(s) y m y m − γ s − γ s Π Π Π Π (a) (b) Redraw b) as 0 G e Σ θ Σ − H γ s uc θ0 Π Π θ − θ0 ( ) uc − Analysis Lemma 1 Let r be a bounded square integrable function, and let G(s) be a transfer function that is positive real. The system whose input-output relation is given by y = r (G(p)ru) is then passive. Example: PI adjustments (t) = ￾ 1uc(t)e(t) ￾ 2 Z t uc( )e( ) d Explore the advantages of PI adjustments analytically and by simulation! A modied algorithm Change 0 G e Σ θ Σ − H γ s uc θ0 Π Π θ − θ0 ( ) uc − To G G + – Σ y θ uc y m − γ s Gc θ0 Π Π Make GcG SPR. Still a problem with pole excess > 1. The Augmented Error Consider the error e = G( ￾  0 )uc = G( ￾  0 )uc + ( ￾  0 )Guc ￾ ( ￾  0 )Guc Introduce the augmented error  = e +  where  = G( ￾  0 )uc ￾ ( ￾  0 )Guc = Guc ￾ Guc Notice that  is zero under stationary condi￾tions Use the adaptation law d dt = ￾ G2uc Stability now follows from the passivity theorem The idea can be extended to the general case, details are messy. c K. J. Åström and B. Wittenmark 5