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 BWittenmarkThe 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