Lecture 9- Analysis of Adaptive Properties of Adaptive Systems Systems 1. Introduction Theme: Adaptive systems are nonlinear. 2. Nonlinear Dynamics What can we say about their behavior? 3. Adaptation of Feedforward Gain 1. Introduction 4. Stability of dSTR 2. Nonlinear dynamics 5. Averaging 3. Time variability 6. Applicatins of Averaging 4. Stability of STR 7. Robustness 5. Summary 8. Conclusions 1 Introduction 2. Nonlinear Dynamics Investigate a given system Structure of equations · What can be done Convergence Equilibria Convergence rate Local properties Improved algorithms Global properti General principles · A simple example Understand behavior · Structural stability Can difficulties occur? Unification Structures Identifiability Excitation Achievable performance O K.J.Astrom and BWittenmark
Lecture 9 – Analysis of Adaptive Systems Theme: Adaptive systems are nonlinear. What can we say about their behavior? 1. Introduction 2. Nonlinear dynamics 3. Time variability 4. Stability of STR 5. Summary Properties of Adaptive Systems 1. Introduction 2. Nonlinear Dynamics 3. Adaptation of Feedforward Gain 4. Stability of DSTR 5. Averaging 6. Applicatins of Averaging 7. Robustness 8. Conclusions 1. Introduction • Investigate a given system – Stability – Convergence – Convergence rate • Improved algorithms • General principles – Understand behavior – Can difficulties occur? – Unification – Structures – Identifiability – Excitation – Achievable performance 2. Nonlinear Dynamics • Structure of equations • What can be done? – Equilibria – Local properties – Global properties • A simple example • Structural stability c K. J. Åström and B. Wittenmark 1
A Typical Self-tuning Regulator Structure of Equations Continuous time mras A()5+B()v =C()5+D()y 叫 Estimation dedt=y q(t,5)e(,5) +φ(t,)q(0,5) Reference Controller Alternative representation Nonlinear system dedt=y (Goyv)(Gev v) a+(Govv) What can we say apart from stability? Discrete time str Notice two loops Fast feedback loop 5(t+1)=A()5(t)+B()v(t) Slower parameter adjustment loop 7()=()=c(o(+D()v Use difference in time scales in analy e(t+1)=6(t)+P(t+1)φ(t)e(t) P(t+1)=P(t) P(to(t)p(tP(t) λ+q(t)P(t)p(t) Example Example Process model Simplification y(t+1)=6y(t)+l(t) y(t+1)= a(t)y(t)+a+yo y(t)((6o-6()y(t)+a Controller e(t+1)=6(t)+y a+y(t) u(t)=-6(1)y(t)+y Equilibrium solutions y()(y(t+1)-6()y()-u y=y e(t+1)=6(t)+y 6=60+ +y2(t) True system Local behavior y(t+1)=6y(t)+a+u(t) a+yo Characteristi 十a1z+a2 where O K.J.Astrom and BWittenmark
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 difference in time scales in analysis Structure of Equations Continuous time MRAS dξ dt A(ϑ )ξ + B(ϑ )ν η e ϕ C(ϑ )ξ + D(ϑ )ν dθ ˆdt γ ϕ (ϑ ,ξ )e(ϑ ,ξ ) α + ϕ (ϑ ,ξ )Tϕ (ϑ ,ξ ) Alternative representation dθˆdt γ (Gϕνν) (Geνν) α + (Gϕνν) T Gϕνν Discrete time STR ξ (t + 1) A(ϑ )ξ (t) + B(ϑ )ν(t) η(t) e(t) ϕ (t) C(ϑ )ξ (t) + D(ϑ )ν(t) ˆθ(t + 1) ˆθ(t) + P(t + 1)ϕ (t)e(t) P(t + 1) P(t) − P(t)ϕ (t)ϕT(t)P(t) λ + ϕT(t)P(t)ϕ (t) Example Process model y(t + 1) θ y(t) + u(t) Controller u(t)−θˆ(t)y(t) + y0 θˆ(t+1) θˆ(t)+γ y(t) y(t + 1) − θˆ(t)y(t) − u(t) α + y2(t) True system y(t + 1) θ 0 y(t) + a + u(t) Example Simplification y(t + 1) θ 0 − θ ˆ(t) y(t) + a + y0 θ ˆ(t + 1) θ ˆ(t) + γ y(t) θ 0 −θ ˆ(t) y(t) + a α + y2(t) Equilibrium solutions y y0 θ ˆ θ 0 + a y0 Local behavior A − a y0 −y0 −γ a α+y2 0 1 −γ y2 0 α+y2 0 ! Characteristic equation z2 + a1z + a2 where a1 ay0 − 1 + γ y2 0 α + y2 0 a2 − a y0 c K. J. Åström and B. Wittenmark 2
Local Analysis Global Properties- Stable Characteristic equation Equilibrium z2+a1z+a2=0 Explore the properties at the boundaries Stability conditions(Schur-Cohn) ()a20 quilibrium (i)a2+a1+1>0 6 gIves ()y 2(a+y) Consider e and y equations separately (t+1)=(6-6)y(t)+a+ e(t+1) g(t) a+ a+y(t) Intuitive discussion Global Properties-Unstable y(t+1)=(6-6)y(t)+a+y0 Equilibrium and 6(t+1)=(1 y2()-)6(t Choose 0o= 1, a =0.1 a+y2(t) a+ y2(t) a=0.9. Equilibrium is y 1 and 0=1.9 Stable if y <0.22. Choose y=0.5 The frozen motion fix e y(t+1)=(60-6)y(t)+a+y Equilibrium a+ y 1+b-60 What is the character of the motion of y for different 6? When isis stable? Bursts When does it converge fast? When is is oscillatory Explain intuitively e When is it monotone? O K.J.Astrom and BWittenmark
Local Analysis Characteristic equation z2 + a1z + a2 0 Stability conditions (Schur-Cohn) (i) a2 0 (iii) a2 + a1 + 1> 0 gives (i) a y0 > −1 (ii) γ 0 γ − y0 y 0 a 2(α + y 0 2 ) y0 2 Global Properties - Stable Equilibrium Explore the properties at the boundaries (home work)! Simulation α 0.1, γ 0.1, θ 0 1.5, y0 1, and a 0.9. Unique stable equilibrium −10 0 10 20 0 1 2 θˆ y Consider θˆ and y equations separately y(t + 1)(θ 0 −θˆ)y(t) + a + y0 ˜θ(t+1) 1 −γ y2(t) α + y2(t) ˜θ(t)+γ ay(t) α + y2(t) Intuitive Discussion y(t + 1)(θ 0 −θˆ)y(t) + a + y0 θ˜(t+1) 1 −γ y2(t) α + y2(t) θ˜(t)+γ ay(t) α + y2(t) The frozen motion fix θˆ. y(t + 1)(θ 0 −θˆ)y(t) + a + y0 Equilibrium y a + y0 1 + θˆ − θ 0 • What is the character of the motion of y for different θˆ? • When is is stable? • When does it converge fast? • When is is oscillatory? • When is it monotone? Global Properties - Unstable Equilibrium Choose θ 0 1, α 0.1, y0 1, and a 0.9. Equilibrium is y 1 and θ 1.9. Stable if γ < 0.22. Choose γ 0.5. 0 50 100 150 200 −1 0 1 2 3 0 50 100 150 200 1 2 3 Time Time y θˆ Bursts Explain intuitively! c K. J. Åström and B. Wittenmark 3
Intuitive discussion Phase plane y=0.5 a=0.9 y(t+1)=(60-6)y(t)+a+y (t+1) a+y2(t) e(t) a+y2(t) =6-60 Equilibrium Chaos !! Show the double pendulum! Phase plane y=0.5 a=-1.1 3. Adaptation of Feedforward Gain dy d=k6(t)(t)-y(t) Parameter adjustment MIT rule de dt =-yym(t)e(t)=-rym(t)(y(t)-ym(t) Complete system d 0-ym()/6,/ry2(t) dt 0 O K.J.Astrom and BWittenmark
Intuitive Discussion y(t + 1)(θ 0 −θˆ)y(t) + a + y0 θ˜(t+1) 1 −γ y2(t) α + y2(t) θ˜(t)+γ ay(t) α + y2(t) θ˜ θˆ −θ 0 Equilibrium θ˜ 1 y Phase plane γ 0.5 a 0.9 −1 0 1 2 1 2 3 θˆ y Chaos!! Show the double pendulum! Phase plane γ 0.5 a −1.1 −2 −1 0 1 2 0 1 2 3 4 3. Adaptation of Feedforward Gain dy dt kθˆ(t)uc(t) − y(t) Parameter adjustment MIT rule dθˆ dt −γ ym(t)e(t)−γ ym(t) (y(t) − ym(t)) Complete system d dt ˆ θ y 0 −γ ym(t) kuc(t) −1 ˆ θ y + γ y2 m(t) 0 0 5 10 15 20 0 1 0 5 10 15 20 0 1 0 5 10 15 20 0 1 Time Time Time (a) θˆ (b) θˆ (c) θˆ c K. J. Åström and B. Wittenmark 4
Linear Periodic Systems Typical Stability Boundary Consider the system Adjustment of feedforward gain MIT rule Process model d=A(t)中 G(s) s+1 A(t+t)=A(t) Command signal Solution has the form u(t)=sin t Φ(t)=D(t)e where D(t)=D(t+r) Stability condition All eigenvalues of c less than one in magnitude How to compute C? 0 Summary 4. Stability of dstR Adaptive systems are nonlinear Review of Algorithm An Properties of Estimator Equilibria Main Result Local analysis- linearization · Discuss Assumptions e Global behavio Disturbances Difficult Behavior can be very complicated Simulation roximations · Use special structure e more of this to follow e K.J. Astrom and BWittenmark
Linear Periodic Systems Consider the system dΦ dt A(t)Φ with A(t + τ ) A(t) Solution has the form Φ(t) D(t)eCt where D(t) D(t + τ ). Stability condition: All eigenvalues of C less than one in magnitude. How to compute C? Typical Stability Boundary Adjustment of feedforward gain MIT rule Process model G(s) 1 s + 1 Command signal uc(t) sinωt 100 0 0 1 2 ω γ Stable Summary Adaptive systems are nonlinear! Analysis • Equilibria • Local analysis - Linearization • Global behavior – Difficult – Behavior can be very complicated • Simulation • Approximations • Use special structure • More of this to follow 4. Stability of DSTR • Review of Algorithm • Properties of Estimator • Main Result • Discuss Assumptions • Disturbances c K. J. Åström and B. Wittenmark 5
The Algorithm Main result Process model Theorem 1 A' (q y(t)=B(q )u(t-d) Assume that A1: The time delay d is known Desired response A2: Upper bounds on the degrees of the Am(q y(t=toue(t-d) polynomials a* and b* are known A3: The polynomial b has all its zeros Notice all process zeros cancelled Estimate inside the unit disc parameters of the model A4: The sign of bo= ro is known ACAmy(t+d)=Ru(t)+S'y(t)=o(t)8Then e(t)=6(t-1)+ +p(t-d)(t-dre(t) ( The sequences u(t)) and y(t)) are bounded e(t)=y(t)-(t-d)(t-1) ()limt-oo Am( q-)(t)touc(t-d)=0 Control law Ru(t)+S'y(t)=toAue(t) Idea of proof Disturbances Process model A( y(t=B( )ut-d)+u(t) Assume Upper bounds on degrees Stability results(Egardt) plating R C Update only wh max(bo/b0,1) Projection Parameters bounded apriori. Modify stimator to give estimates insi O K.J.Astrom and BWittenmark
The Algorithm Process model A∗(q−1 )y(t) B∗(q−1 )u(t − d) Desired response A∗ m(q−1 )y(t) t0uc(t − d) Notice all process zeros cancelled Estimate parameters of the model A∗ oA∗ m y(t + d) R∗u(t) + S∗ y(t) ϕT(t)θ ˆθ(t) ˆθ(t − 1) + γ ϕ (t − d) α + ϕT(t − d)ϕ (t − d) e(t) e(t) y(t) − ϕT(t − d) ˆθ(t − 1) Control law Rˆ ∗u(t) + Sˆ ∗ y(t) t0A∗ ouc(t) Main Result Theorem 1 Assume that A1: The time delay d is known. A2: Upper bounds on the degrees of the polynomials A∗ and B∗ are known. A3: The polynomial B has all its zeros inside the unit disc. A4: The sign of b0 r0 is known. Then (i) The sequences {u(t)} and {y(t)} are bounded (ii) limt→∞ A∗ m(q−1)y(t) − t0uc(t − d) 0 Idea of Proof Disturbances Process model A∗ (q−1 )y(t) B∗(q−1 )u(t − d) + v(t) Assume • Upper bounds on degrees • Minimum phase • Sign of b0 known Stability results (Egardt) • Conditional Updating sup t t R1 AoAm vt ≤ c Update only when tet ≥ 2c 2 − max(b0/ˆ b0, 1) • Projection Parameters bounded apriori. Modify estimator to give estimates inside prior bounds. c K. J. Åström and B. Wittenmark 6
Summan 1. Nonlinear Nature of Adaptive Systems Local properties Global properties Complicated behavior 2. Adaptive Feedforward Much simpler Linear but time-varying Periodic case tractable Also complicated, not as bad as 1 3. Direct STR Use special properties · Assumptions required 4. Next we will exploit difference in time scale O K.J.Astrom and BWittenmark
Summary 1. Nonlinear Nature of Adaptive Systems • Equilibria • Local properties • Global properties • Complicated behavior 2. Adaptive Feedforward • Much simpler • Linear but time-varying • Periodic case tractable • Also complicated, not as bad as 1 3. Direct STR • Use special properties • Assumptions required 4. Next we will exploit difference in time scales c K. J. Åström and B. Wittenmark 7
