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 Wittenmark
Model-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 = sup0tT 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
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 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 Wittenmark
Examples Linear systems with signals in L2e kyk max ! jG(i!)jkuk 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, boundedoutput (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 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
Linear Time-invariant Systems The Notion of p hase =0a=/()de Let th 2/ G(ia ) u(iu u (=iu)daw The phase for a given input u can then be defined as Re fGiw)](iu (iw) y u) (Hu u DEFINITION a rat io nal transfer funct io n g wit h real Passivity implies that the phase is in the rang coefficients is positive real(PR)if ≤y )≥0 for Re s≥0 a transfer funct io n g is strictly positive real (SPR)if Gs-e)is positive real for so me real C haracteriz ing p ositive real Ex ample Transfer f unctions Recall THEOREM 2 al transfer funct n G(s)with real Re)lu(iw u(ia)dw A coefficients is Pr if and only if the following co ndit io ns hold Positive real pr .(i The funct io n has no poles in the rig ht ReG(iu)≥0 .(ii If the funct io n has poles on the Input st rictly passive ISP imag inary axis or at infinity, they are ReG(u)≥E> simple poles wit h positive resid (ii) The real part of G Out put st ricky passive OSP along the lw axis, that is ReG(ia)≥e|Giu)2 Re(G(u)≥0 G(s)=s+1 SPR and ISP not OSP A transfer funct io n is sPr if conditions(i and (ii hold and if condition() is replaced by the G(s)=+1 SPR and OSP not ISP co io n that G(s)has no pol G(s)=2 OSP and ISP not OSP I maginary axis 口G(6)= OPS or SP K.. Astrom and B wittenmark
The Notion of Phase Let the signal space have an inner product The phase for a given input u can then be de ned as cos ' = hy j ui kuk kyk = hHu j ui kuk kHuk Passivity implies that the phase is in the range 2 ' 2 Linear Time-invariant Systems hy j ui = Z1 0 y(t)u(t) dt = 1 2 Z1 1 Y (i!)U (i!) d! = 1 2 Z1 1 G(i!)U (i!)U (i!) d! = 1 Z1 0 Re fG(i!)g U (i!)U (i!) d! Definition 3 A rational transfer function G with real coecients is positive real (PR) if Re G(s) 0 for Re s 0 A transfer function G is strictly positive real (SPR) if G(s ") is positive real for some real " > 0. Characterizing Positive Real Transfer Functions Theorem 2 A rational transfer function G(s) with real coecients is PR if and only if the following conditions hold. (i) The function has no poles in the right half-plane. (ii) If the function has poles on the imaginary axis or at in nity, they are simple poles with positive residues. (iii) The real part of G is nonnegative along the i! axis, that is, Re (G(i!)) 0 A transfer function is SPR if conditions (i) and (iii) hold and if condition (ii) is replaced by the condition that G(s) has no poles or zeros on the imaginary axis. Examples Recall hy j ui = 1 Z1 0 Re fG(i!)g U(i!)U(i!) d! Positive real PR Re G(i!) 0 Input strictly passive ISP Re G(i!) " > 0 Output stricly passive OSP Re G(i!) "jG(i!)j2 G(s) = s + 1 SPR and ISP not OSP G(s) = 1 s+1 SPR and OSP not ISP G(s) = s 2+1 (s+1)2 OSP and ISP not OSP G(s) = 1 s PR not SPR, OPS or ISP c K. J. Åström and B. Wittenmark 3
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 Wittenmark
Nonlinear 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
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 Wittenmark
Adaptation 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 conditions 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
a Minor extensi where the transfer function G1 is SPR The errore=y- ym can then be witten as MRAS with aug mented error e=G(8-0 )uc=(G1G2( 0-8 )uc Model G1(G2(0-6)uc+(0-")G2uc-(0-6)G2uo Introduce Process -(@-引° where n is the error augmentation defi ned by G1(6-6°)G Use adaptation law dt Compare Str and Mras W hat You should Know MRAS d e e the ideas Ypf How to make abstractions pf=-Gf(p)grade(t) The( E=GSPR(y-ym)+n=G SpRe +n Nations df g ain phase and passivity Direct str PR and sPr y(t)=pf(t-do)8 The key resuts The small gain theorem e(t)=v(t)-=y(t)-9(t-do)(t-1) The passivity theorem (t)=6(t-1)+P(t)9f(t-do)e(t) The arde criterion Abilit impute g e(t=yt-y(t) Determne passway =y(t)-ym(t)+ym()-y(t Apply to adaptive cortrd e(t)+、(t) Similarities between MRAS STIR
A Minor Extension Factor G = G1G2 where the transfer function G1 is SPR. The error e = y ym can then be written as e = G( 0 )uc = (G1G2)( 0 )uc = G1G2( 0 )uc + ( 0 )G2uc ( 0 )G2uc Introduce " = e + where is the error augmentation de ned by = G1( 0 )G2uc G( 0 )uc = G1(G2uc) Guc Use adaptation law d dt = G2uc MRAS with Augmented Error – + Model Process y – e Σ Σ θ Σ θ + η ε uc ym k0G kG − γ s G2 G1 u c Π Π Π Compare STR and MRAS MRAS d dt = 'f " 'T f = Gf (p)grad"(t) " = GSPR(y ym) + = GSPRe + Direct STR y(t) = ' T f (t d0) "(t) = y(t) =^ y(t) 'T f (t d0) ^ (t 1) ^ (t) = ^ (t 1) + P (t)' T f (t d0) "(t) Residual "(t) = y(t) y^(t) = y(t) ym(t) + ym(t) y^(t) = e(t) + (t) What You Should Know! The ideas { How to make abstractions The concepts { Notions of gain phase and passivity { PR and SPR The key results { The small gain theorem { The passivity theorem { The circle criterion Abilities { Compute gain { Determine passivity { Apply to adaptive control Similarities between MRAS STR c K. J. Åström and B. Wittenmark 6