Cont roller Windup it? The feed back loop is bro ken Pol ynomial controllers Trouble if there are unst able systems in Let the cont roller be How to avoid it? R()u(t=t(uc(t)-s(a)y(t nt ro duce the observer dy namics ex plicitel Estima 叫 Actuate A(q)u(t)=T(q)u2(t)-S(q)y(t)+(A(q)-R(q)u(t) Let the sat urating act uator be described by the Estima feedback 叫 Actuator Process function f the co nt roller wit h anti-windup then becomes A(q)v(t)=T(q)u2(t)-S(q)y(t)+(Aa(q)-R(q)u(t) Notice u(t)=f(u(t) Level sat urat ion . Rate sat urat io n Ot her no nlineariti Even worse for adaptive syst ems. Why? An Example Process model G(s y Singularities in Diophantine uation C K.J. Astrom and BWittenmarkController Windup What is it?  The feedback loop is broken  Trouble if there are unstable systems in the loop How to avoid it? (b) (a) Estimator Actuator Process State feedback x ˆ u y Estimator Actuator Process State feedback x ˆ u ua y Notice  Level saturation  Rate saturation  Other nonlinearities  Even worse for adaptive systems. Why? Polynomial Controllers Let the controller be R(q)u(t) = T (q)uc(t) ￾ S(q)y(t Introduce the observer dynamics explicitely Ao (q)u(t) = T (q)uc(t) ￾ S(q)y(t)+(Ao (q) ￾ R(q))u(t) Let the saturating actuator be described by the function f the controller with anti-windup then becomes Ao (q)v(t) = T (q)uc(t) ￾ S(q)y(t)+(Ao (q) ￾ R(q))u(t) u(t) = f (v(t)) An Example Process model G(s) = 1 s(s+1) 0 20 40 60 80 100 −1 0 1 2 0 20 40 60 80 100 −50 0 50 Time Time uc y u 0 20 40 60 80 100 −1 0 1 0 20 40 60 80 100 −0.5 0.0 0.5 Time Time uc y u Singularities in Diophantine equation c K. J. Åström and B. Wittenmark 4