Robust Ada ptive control Stochastic self-tuner w hat do the diffic ulties de pend on? How can the diffic ulties be avoided? A'(g)y(t)=B'(q-)u(t-d)+C'(q-)e(t) Excitation Model Dead y(t=R"(q u(t-d)+s(a y(t-d ● Leakage east squares parameter estimation dt 仟F )=6(t-1)+(+)(+)-p(t-d)e(t) e(t) +f;le(6° R()=(t-1)+7(t)(y(t-d)y2(t-d)-R(t-1) Control ormallza tion Cr(t=max (u(t)I, ly(tD) u(t)= (a ormallze ed signal y y P(t)=r(tr(t w here y( t=1/t Stochastic Averaging An example Approximate To c 1)+bu(t-2)+e(t)+ The equations then becomes B(t)=b(t-1)+?(t)R(t)f(6 Parameters: a=-099 b=0.5. and c==0.7 R(t)=R(t-1)+(t)(G(6)-Rt-1) Estim ation mod f(0)=E{y(t-d,0)(y(t)-2(-d,6)同} ()=E{y(t-d,6)92(t-d,b)} y(t=u(t Define△r=∑kt1(k),then C ntro A(t)=a(t)+ArR(t)f(a(t) R(t)=R(t)+A(G(e(t)-R(t) Change time scale t=T and t'=t+AT R(T)f((-) dT (6(7)-R(r) C K.J. Astrom and B WittenmarkRobust Adaptive Control  What do the diculties depend on?  How can the diculties be avoided?  Excitation  Dead zones  Leakage d^  dt = 'e  + 'T ' + 1( 0 ￾ ^ ) d^  dt = 'e  + 'T ' + 1jej( 0 ￾ ^ )  Normalization C r(t) = max (ju(t)j; jy(t)j) Normalized signals y~ = y r ; u~ = ur; v~ = v r Stochastic Self-tuner Process A (q￾1 )y(t) = B (q￾1 )u(t ￾ d) + C (q￾1 )e(t) Model y(t) = R (q￾1 )u(t ￾ d) + S (q￾1 )y(t ￾ d) Least squares parameter estimation ^ (t) = ^ (t ￾ 1) + (t)R(t)￾1 '(t ￾ d)e(t) e(t) =  y(t) ￾ 'T (t ￾ d)^ (t ￾ 1) R(t) = R(t ￾ 1) + (t) ￾'(t ￾ d)'T (t ￾ d) ￾ R(t ￾ 1)
Control u(t) = ￾ S^ (q￾1 ) R^ (q￾1 ) y(t) This implies '(t)T ^ (t)=0 Notice P (t) = (t)R(t)￾1 where (t)=1=t. Stochastic Averaging Approximate y(t) = 'T (t ￾ d)  'T (t ￾ d; ) The equations then becomes  (t) =  (t ￾ 1) + (t)R (t)￾1f ( ) R (t) = R (t ￾ 1) + (t) ￾G( ) ￾ R (t ￾ 1)
f ( ) = E '(t ￾ d;  ) ￾y(t) ￾ 'T (t ￾ d;  ) 
G( ) = E '(t ￾ d;  )' T (t ￾ d;  ) Dene
 = Pt 0 k=t (k), then  (t 0) =  (t)+
R (t)￾1f ￾  (t)
R(t 0) = R(t)+
 ￾G ￾ (t)
￾ R(t)
Change time scale t =  and t 0 = t +
 d  d = R( )￾1f ￾ ( )
dR d = G ￾ ( )
￾ R ( ) An example Process y(t) + ay(t ￾ 1) = u(t ￾ 1) + bu(t ￾ 2) + e(t) + ce(t ￾ 1) Parameters: a = ￾0:99, b = 0:5, and c = ￾0:7 Estimation model y(t) = u(t ￾ 1) + r1u(t ￾ 2) + s0y(t ￾ 1) Controller u(t) = ￾s0y(t) ￾ r1u(t ￾ 1) 0123 −1 0 1 0123 −1 0 1 (a) r1 s0 (b) r^1 s^0 c K. J. Åström and B. Wittenmark 8