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 BWittenmarkA 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 analy￾sis 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