Lecture 8- The Input-Output View Int roduct io e White boxes and black boxes Model-Reference Adaptive systems Input-o ut put descript ions 1. The idea How to generalize from linear to 2. The Mit rule 3. Det ermination of the adaptive gain 2. The Small Gain Theo rem (SGT) · The notio n of gain · Examples 5. Desig n of MRAS using Lyapunov theory The main result 6. Bounded-input, bo unded-out put st ability The Passivity Theorem(PT) 7. Applicat io ns to adaptive cont rol Passivity and phase 8. Out put feed back · Examples 9. Relations between mras and str · The Passivity Theorem 10. Nonlinear systems Rel at ions between sgt and pt 11. Conclusio ns 4. Applicat io ns to Adaptive Cont rol mRas and STR 5. Conclusi ntroduction The notion of gain White boxes Sig nal space Input-Out put Descriptions The Table Hall=(5-o u2(t)dt Linear Systems Loo: J l= suport<oo u(t) Duality between Sig nals and Systems xtended spaces Formalization of the input-out put view Si ∫at)0≤t≤r ls ig nals an sig nal s pace (t) 0 The notio ns of gain and phase The notio n of passivity ∈ Xe if aT∈X The notion of gain(o perator no rm) 7(S= sup Stability criteria Extensio ns of Nyquists theorem Gain smallest value y such th . the Mat hematical Framework Funct io nal analys SuL‖≤(S)‖ ul for all u∈Xe C K.. Astrom and B WittenmarkModel-Reference Adaptive Systems 1. The idea 2. The MIT Rule 3. Determination of the adaptive gain 4. Lyapunov theory 5. Design of MRAS using Lyapunov theory 6. Bounded-input, bounded-output stability 7. Applications to adaptive control 8. Output feedback 9. Relations between MRAS and STR 10. Nonlinear systems 11. Conclusions Lecture 8 { The Input-Output View 1. Introduction  White Boxes and Black Boxes  Input-output descriptions  How to generalize from linear to nonlinear? 2. The Small Gain Theorem (SGT)  The notion of gain  Examples  The main result 3. The Passivity Theorem (PT)  Passivity and phase  Examples  The Passivity Theorem  Relations between SGT and PT 4. Applications to Adaptive Control  The augmented error  MRAS and STR 5. Conclusions Introduction  Black Boxes and White Boxes  Input-Output Descriptions { The Table { Linear Systems { Duality between Signals and Systems  Formalization of the input-output view { Signals and signal spaces { The notions of gain and phase { The notion of passivity  Stability { Stability Concept BIBO { Stability of a system! { Stability criteria { Extensions of Nyquist's theorem  The Mathematical Framework { Functional analysis The Notion of Gain Signal spaces L2: kuk = R 1 ￾1 u2(t) dt 1 2 L1: kuk = sup0t<1 ju(t)j Extended spaces xT (t) =  x(t) 0  t  T 0 t>T u 2 Xe if xT 2 X The notion of gain (=operator norm) (S) = sup u2Xe kSuk kuk Gain smallest value such that kSuk  (S)kuk for all u 2 Xe c K. J. Åström and B. Wittenmark 1