Lecture 10-Averaging Properties of Adaptive Systems Lecture 11- robust ness and 1.nt roduct ion Convergence Rate 2. Nonlinear DynamIcs 1. The idea 3. Adapt at io n of Feedfo rward Gain 2. Averag ing theorems 4. St ability of dstr 5. Averag ing pplicat ions to adapt ive cont rol 6. Applicat io ns of Averaging Perfo rmance 7. Robust ness e Co nvergence rates 8. Conclusio ns Sensitivity to ass umptions ypical Result d f(e, y, t) dy e aea eg(a, y, t) Separate Fast and Slow Motions Ass ume y is const ant and solve the fast · The Origin of the ldea de Class ical mechanics =f(x,y,) Numerical analys is Let the solutio n be S(t). the averag ed equa Nonlinear Oscillations n Krylov and Bo boliubov 1937 Mino sky 1962 d eavgg(5(t),y, t)=EG(g) . Two Time scales Consider solutions such that Slow parameters 3(0)=y(0) Fast st at es What did we really ass ume. Think. vy(t)-y(t川<KE for0<t<T/E If the averaged equation is stable then y is also sta ble and the inequality holds for0<t<∞ C K.J. Astrom and BWittenmark
Properties of Adaptive Systems 1. Introduction 2. Nonlinear Dynamics 3. Adaptation of Feedforward Gain 4. Stability of DSTR 5. Averaging 6. Applications of Averaging 7. Robustness 8. Conclusions Lecture 10 - Averaging Lecture 11 - Robustness and Convergence Rate 1. The idea 2. Averaging theorems 3. How to do it? 4. Applications to adaptive control Performance Convergence rates Sensitivity to assumptions 5. A new look at MRAS The Idea Separate Fast and Slow Motions The Origin of the Idea { Classical Mechanics { Numerical Analysis { Nonlinear Oscillations { Krylov and Boboliubov 1937 { Minorsky 1962 Two Time Scales { Slow parameters { Fast states { What did we really assume. Think! Typical Result dx dt = f (x; y; t) dy dt = g(x; y; t) Assume y is constant and solve the fast equation dx dt = f (x; y; t) Let the solution be (t). The averaged equation is then dy dt = avgg((t); y; t ) = G(y) Consider solutions such that y(0) = y(0) = y0 Then jy(t) y(t)j < K for 0 t < T =. If the averaged equation is stable then y is also stable and the inequality holds for 0 t < 1 c K. J. Åström and B. Wittenmark 1
M any ways to Compute Averages avg f(a,(6,t),t) (6:6+分}=黑吗/(( Re {(6(+)}=E((4 Consider para met ers co nst ant when analysing system Sinusoidal sig nals Average fast motion w hen analysing the v(s=G(sU(s estimat s=Gu(sU(s Many ways to take averages If u= sin wt we have Go(iw )l Ga(iw)lcos(arg Gw(iw)-arg Ga(in)) Structure of ada A Typical self-tuning reg ulator Continuo us time mras d=40+B(o)y Process parameters C()E+D(0)y Estimation dt 7(6,.)(,5) Controller parameters Reference Alternat ive at i Process dt Nonlinear syster Discrete time st What can we say apart fro m stability? ￡(+ob=A()(t)+B(0)v(t) Fast feed back loop p() C()(t)+D(0)(t) Slower parameter adjustment loo p A(t+op=0(t)+P(t+opp(te(t) Use difference in time scales in a nalysis P(t+op= P(t-P(t(ts(t)P() +p(tP(tp(t) CK.J. Astrom and BWittenmark
Recipe Consider parameters constant when analysing system Average fast motion when analysing the estimate Many ways to take averages Many Ways to Compute Averages avg n f ; ^ ( ; ^ t); to = 1 T Z T 0 f ; ^ (; t ^ ); t dt avg n f ^ ; (^ ; t); to = lim T!1 Z T 0 f ^ ; (^ ; t); t dt avg n f ^ ; (^ ; t); to = E f ^ ; (^ ; t); t Sinusoidal signals V (s) = Gv(s)U (s) W(s) = Gw(s)U (s) If u = sin !t we have avg(vw) = u2 0 2 jGv(i!)j jGw(i!)j cos(arg Gv(i!) arg Gw(i!)) = u2 0 2 Re (Gv (i!)Gw(i!)) A Typical Self-tuning Regulator Process parameters Controller design Estimation Controller Process Controller parameters Reference Input Output Specification Self-tuning regulator Nonlinear system What can we say apart from stability? Notice two loops { Fast feedback loop { Slower parameter adjustment loop Use di erence in time scales in analysis Structure of Adaptive Systems Continuous time MRAS d dt = A(#) + B(#) = e ' = C(#) + D(#) d^ dt = '(#; )e(#; ) + '(#; )T '(#; ) Alternative representation d^ dt = (G' ) (Ge ) + (G' )T G' Discrete time STR (t + 1) = A(#)(t) + B(#)(t) (t) = e(t) '(t) = C(#)(t) + D(#)(t) ^ (t + 1) = ^ (t) + P (t + 1)'(t)e(t) P (t + 1) = P (t) P (t)'(t)'T (t)P (t) + 'T (t)P (t)'(t) c K. J. Åström and B. Wittenmark 2
MRAS with MIT and Lyapunov rule hat do we know? The robustness issue · What happens when G≠G0? . What is Ro bust ness? How sens it ive is the result to the 叫G(s) assumpt io ns? hat are the critical assum pt io ns? What about the assum pt io ns in the Model stability proof Ot her fields Adapt at io n of Feedfo rward gain Lyapunov versus MIT once more · A first order system yme (MIT) dt (SPR) Analysis (SPR) Analysis e=3-gn=M6(p)(0(()-4Cm(n()+6g(a)=4gcn dt +r(ho gm e)kG(Buc)=yko Gmue +y0kavgluc Guc)]=rboavg(gmu)] The equilibrium para meters are dtt mucho G(Bue)=rko Gme avg(G Averaged equations BMITF avgi(Gmue)(Guc)H G+y0kkoavgi(Gmue)(Guc))=yk3avg(Gmus)2) ko avgluc(much d+70 kavgfuel(Gue))=rk avgfuc(Gm ue)y navg(gmc)(Gu]>0 (MIT) The equilibrium para ravguc(Gu)>0(SPR G ggmu(G espr- ko avgfuc(gm ue)) CK.J. Astrom and BWittenmark
The Robustness Issue What is Robustness? { How sensitive is the result to the assumptions? { What are the critical assumptions? { What about the assumptions in the stability proof { Other elds Adaptation of Feedforward Gain { Lyapunov versus MIT once more A rst order system MRAS with MIT and Lyapunov Rule What do we know? What happens when G 6= G0? 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) d^ dt = yme (MIT) d^ dt = uce (SPR) Analysis d^ dt = yme (MIT) d^ dt = uce (SPR) e = y ym = kG(p) ^ (t)uc(t)k0Gm(p)uc(t) Hence d^ dt + (k0Gmuc)kG(^ uc) = k0Gmuc d^ dt + uckk0G(^ uc) = k0Gmuc Averaged equations d dt + kk0avgf(Gmuc)(Guc)g = k2 0 avgf(Gmuc)2g d dt + kavgfuc(Guc)g = k0avgfuc(Gmuc)g The equilibrium parameters are MIT = k0 k avgf(Gmuc)2g avgf(Gmuc)(Guc)g SPR = k0 k avgfuc(Gmuc)g avgfuc(Guc)g Analysis d dt + kk0avgf(Gmuc)(Guc)g = k2 0 avgf(Gmuc)2g d dt + kavgfuc(Guc)g = k0avgfuc(Gmuc)g The equilibrium parameters are MIT = k0 k avgf(Gmuc)2g avgf(Gmuc)(Guc)g SPR = k0 k avgfuc(Gmuc)g avgfuc(Guc)g avgf(Gmuc)(Guc)g > 0 (MIT) avgfuc(Guc)g > 0 (SPR) c K. J. Åström and B. Wittenmark 3
An Example G(s +a)(s+ Stability co ndit io ns ko b2 Analysis of MRAs for First Order MIT- b2 System AspR k b(ab-w2) Equilibrium conditions Local st ability (a) ce rate ● Ro bust ness The system Design Mod Desired Respo nse Gm(s)=bm Anal - yuce Cont roller u(t)=6 dt rye Lyapunov design 叫G(s) y= G(pju Gm(puc u=61ue-62y C K.J. Astrom and BWittenmark
An Example Gm(s) = a s + a G(s) = ab (s + a)(s + b) Stability conditions MIT = k0 k b 2 + !2 b2 SPR = k0 k a(b 2 + !2 ) b(ab !2 ) ! < p ab 0 100 300 500 0.0 0.5 1.0 0 100 300 500 0.0 0.5 1.0 0 100 300 500 0 5 10 0 100 300 500 0 5 10 Time Time Time Time (a) (b) (c) (d) ^ ^ ^ ^ Analysis of MRAS for First Order System Equilibrium conditions Local stability Convergence rate Robustness The System Design Model G(s) = b s + a Desired Response Gm(s) = bm s + am Controller u(t) = 1uc 2y Lyapunov design − Σ Π + e u y Σ Π Π Π − + uc Gm (s) G(s) θ 1 θ 2 γ s − γ s Analysis d^ 1 dt = uce d^ 2 dt = ye e = y ym y = G(p)u ym = Gm(p)uc u = ^ 1uc ^ 2y c K. J. Åström and B. Wittenmark 4
Equilibrium values trans fer function A vera ag ing A nalysis c 61G 62G Control error e(t)=y(t-ynht)=(Gdp)-Gmp))udt) dt Yuo Re f(o, 8, 2)) d62 F(a,61,62) nhiw)G(iw)+gri B.-Imt1/G(iw)l FO Gmi) IGm(iw/G(iw) Disc uss hig h and lo Lo cal stabi accuracy of averaging Linearize the a verged equations Consider t he b= 2, am= bm= 3 G(s A 2 GmI -Gm2cos gm w here arctan(w/am GM(s A4+a(1 Dis c uss convergence rate as a function of CK.J. Astrom and BWittenmark
Equilibrium Values Closed loop transfer function Gc = ^ 1G 1 + ^ 2G Control error e(t) = y(t) ym(t)=(Gc(p) Gm(p)) uc(t) Sinusoidal signals ^ 0 1G(i!) = ^ 0 2Gm(i!)G(i!) + Gm(i!) Hence ^ 0 1 = Imf1=G(i!)g Imf1=Gm(i!)g ^ 0 2 = ImfGm(i!)=G(i!)g ImGm(i!) Discuss high and low !, signs etc. Averaging Analysis Ge = ^ 1G 1 + ^ 2G Gm GT ' = 1 ^ 1G 1 + ^ 2G d 1 dt = u2 0 2 Re F (!; 1; 2) d 2 dt = u2 0 2 Re F (!; 1; 2) 1G(i!) 1 + 2G(i!) F (!; 1; 2) = 1G(i!) (1 + 2G(i!) Gm(i!) Accuracy of Averaging Consider the case a = 1, b = 2, am = bm = 3, = 1, uc = sin !t G(s) = 2 s + 1 GM (s) = 2 s + 2 0 20 40 60 80 100 0 1 Time Local Stability Linearize the averaged equations A = u2 0jGmj 2 0 1 cos m jGmj cos 2m jGmj jGmj2 cos m where m = arctan(!=am) 2 + (1 + cos2 m) + 2 sin2 m = 0 where = u2 0amb 2 (a2m + !2) Discuss convergence rate as a function of frequency. c K. J. Åström and B. Wittenmark 5