Introduction Model Model-Reference Adaptive Systems 1. The idea 2. The miT Rule Controller Plant 3. Determination of the adaptive gain 4. Lyapunov theory 5. Design of MRAs using Lyapunov Flight control in the 1950S Two ideas phil whitaker mit 6. Bounded-input, bounded-output stability The Reference mode 7. Applications to adaptive control Parameter adjustment rule 8. Output feedback · The stability problen 9. Relations between mras and STR L 10. Nonlinear systems Input-output stability 11. Conclusions Modified adjustment rules Proliferation of algorithms Unification The Mit rule Tracking error e y-ym J(0)=e2 Change parameters such that The Mit rule de The idea EXamples where de/a0 is the sensitivity derivative Error and parameter convergence Many alternatives J(e)=lel de C K.J. 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 Introduction Adjustment mechanism u Model Controller parameters Plant y Controller ym uc • Flight control in the 1950s • Two ideas Phil Whitaker MIT – The Reference Model – Parameter adjustment rule • The stability problem – Lyapunov – Input-output stability • Modified adjustment rules • Proliferation of algorithms • Unification The MIT Rule • The idea • Examples • Error and parameter convergence The MIT Rule Tracking error e y − ym Introduce J(θ ) 1 2 e2 Change parameters such that dθ dt −γ J θ −γ e e θ where e/θ is the sensitivity derivative dJ dt e e t −γ e2 e θ 2 Many alternatives J(e)tet gives dθ dt −γ J θ −γ e θ sign(e) c K. J. Åström and B. Wittenmark 1
Adjustment of Feedforward gain Process kg(s Desired response Block Diagram ym= koG(su Controller u e= y-ym = kG(p)Ouc -koG(p)uc Sensitivity derivative G(p)u k MIT rule de de time -ryme=-ryme Real examples Robotics CD player Example koG(s) 4}① A Remark Notice that G(p)(6u)≠6G(p)l Simulation Consequences for block diagram manipula C K.J. Astrom and B Wittenmark
Adjustment of Feedforward Gain Process y kG(s) Desired response ym k0G(s)uc Controller u θuc e y − ym kG(p)θuc − k0G(p)uc Sensitivity derivative e θ kG(p)uc k k0 ym MIT rule dθ dt −γ T k k0 yme −γ yme Real examples • Robotics • CD player Block Diagram Model e y Process u Σ + – uc θ y m − γ s kG(s) k0G(s) Π Π dθ dt −γ T k k0 yme −γ yme Example Model e y Process u Σ + – uc θ y m − γ s kG(s) k0G(s) Π Π Simulation 0 5 10 15 20 −2 0 0 5 10 15 20 0 2 Time Time θ ym y γ 2 γ 1 γ 0.5 A Remark Notice that G(p)(θu) 6 θ G(p)u Consequences for block diagram manipulation c K. J. Åström and B. Wittenmark 2
Derivation of Adaptive Law A First Order System The error Process y-yI dy -ay bu be d +a+b6 b dt=-am ym+bme p+a+b82 Controller b261 +a+b02)2“c=p+a+b62 u(t)=61u2(t)-62y(t) Approximate ideal parameters p+a+b62≈p+am 81=00=m 62= dt p+am Block Diagram Simulation G(s) I Input and output l s+a Parameters p+am 62 dt Example a= 1, 6=0.5, am=bm=2 C K.J. Astrom and B Wittenmark
A First Order System Process dy dt −ay + bu Model dym dt −am ym + bmuc Controller u(t) θ 1uc(t) −θ 2 y(t) ideal parameters θ 1 θ0 1 bm b θ 2 θ0 2 am − a b Derivation of Adaptive Law The error e y − ym y bθ 1 p + a + bθ 2 uc e θ 1 b p + a + bθ 2 uc e θ 2 − b2 θ 1 (p + a + bθ 2)2 uc − b p + a + bθ 2 y Approximate p + a + bθ 2 p + am Hence dθ 1 dt −γ am p + am uc e dθ 2 dt γ am p + am y e Block Diagram − Σ Π + e u y Σ Π Π Π − + uc Gm (s) G(s) θ 1 θ 2 γ s − γ s am s + am am s + am dθ 1 dt −γ am p + am uc e dθ 2 dt γ am p + am y e Example a 1, b 0.5, am bm 2. Simulation Input and output 0 20 40 60 80 100 −1 1 0 20 40 60 80 100 −5 0 5 Time Time ym y u Parameters 0 20 40 60 80 100 0 2 4 0 20 40 60 80 100 0 2 Time Time θ 1 θ 2 γ 5 γ 1 γ 0.2 γ 5 γ 1 γ 0.2 c K. J. Åström and B. Wittenmark 3
CONTINUOUS SYSTEM mras MRAS for first-order system with Bad parameters good control? INPUT y uc OUTPUT u The closed loop transfer function is STAtE ym th1 th2 x1 x2 deR dym dth1 dth2 dx1 dx2 Gc(s) 61G(s 61b +62G(s) +2b u=th1*uc-th2*y 62 dym=-am*ym+bm*uc dx1=-am*x1+ dth1=-gamma*e* dth2=-gamma*e*x2 am: 2 model para gamma: 2 adaptation gain END Error and Parameter Convergence Consider adaptation of feedforward gain (k-和0ue=k(6-60)u de Determination of Adaptation gain y22(0-) ult problem Solution Approximations give insight 6(t)=6°+(6(0)-6")e Leads to modified algorithms ()dr Exponential convergence with persistant excitation C K.J. Astrom and B Wittenmark
CONTINUOUS SYSTEM mras "MRAS for first-order system with " Gm=bm/(s+am) INPUT y uc OUTPUT u STATE ym th1 th2 x1 x2 DER dym dth1 dth2 dx1 dx2 u=th1*uc-th2*y dym=-am*ym+bm*uc dx1=-am*x1+am*uc dx2=-am*x2-am*y e=y-ym dth1=-gamma*e*x1 dth2=-gamma*e*x2 am:2 "model parameter bm:2 "model parameter gamma:2 "adaptation gain END Bad Parameters Good Control? The closed loop transfer function is Gcl(s) θ 1G(s) 1 + θ 2G(s) θ 1b s + a + θ 2b 01234 −1 0 1 2 θ 2 θ 1 Error and Parameter Convergence Consider adaptation of feedforward gain e (kθ − k0)uc k(θ − θ0 )uc with θ0 k0/k dθ dt −γ k2u2 c (θ − θ0) Solution θ (t) θ0 + (θ (0) −θ0)e−γ k2It where It Z t 0 u2 c (τ ) dτ Exponential convergence with persistant excitation Determination of Adaptation Gain • A difficult problem • Approximations give insight • Leads to modified algorithms c K. J. Åström and B. Wittenmark 4
An Example Example daptation of feedforward gain Process y= kG(p)u G(s) a18+ e =y==ym Approximate CE dt s3+a1s2+a2s+M=0 Parameter equation u=yymuok Stability condition dt +yym(kg(p)Ouc)=yy yym uck <a Why approximate? A thought experiment ym de d t t rymuc(G(p)a)=r(ym)2 Characteristic equation s+rymuchG(s Key parameter u=youch Example G(s)=九 Summary Modified Algorithm . The ide MIT rule Model following dt=rpe The mit rule Normalized adaptation law · The error equation e(t)=(G(p, 0)-Gm (p))uc(t) dt a+pp · a gradient procedure · Approximations normalization de dt C K.J. Astrom and B Wittenmark
An Example daptation of feedforward gain y kG(p)u ym k0G(p)uc u θuc e y − ym dθ dt −γ yme Parameter equation dθ dt + γ ym (kG(p)θuc) γ y2 m Why approximate? A thought experiment dθ dt + γ yo muo c (kG(p)θ) γ (yo m)2 Characteristic equation s + γ yo muo ckG(s) 0 Key parameter µ γ yo muo ck Example G(s) 1 s+1 Example Process G(s) 1 s2 + a1s + a2 Approximate CE s3 + a1s2 + a2s + µ 0 µ γ yo muo ck. Stability condition γ yo muo ck < a1a2 0 20 40 60 80 100 −0.1 0.1 0 20 40 60 80 100 −1 1 0 20 40 60 80 100 −10 10 Time Time Time (a) ym y (b) ym y (c) y ym Modified Algorithm MIT rule dθ dt γ ϕ e Normalized adaptation law dθ dt γ ϕ e α + ϕTϕ 0 20 40 60 80 100 −0.1 0.1 0 20 40 60 80 100 −1 1 0 20 40 60 80 100 −10 10 Time Time Time (a) ym y (b) ym y (c) ym y Summary • The idea – Model following – The MIT rule • The error equation e(t)(G(p,θ) − Gm(p))uc(t) • A gradient procedure dθ dt γ ϕ e ϕ G(p,θ ) θ uc • Approximations • Normalization dθ dt γ ϕ e α + ϕTϕ c K. J. Åström and B. Wittenmark 5
