Projection Operator and Integral with Anti-windup :Theory and
Practical Implications in Adaptive Control
Author1, Author2, Author3
AbstractIn this paper theoretical and practical implica-tions of
Projection operator and Integral with Anti-windup inadaptive
control are discussed.
I. INTRODUCTION
In adaptive control projection operator based adaption lawsare
used, which ensures the bounds on parameters at thesame time it
prevents integrators against the undesirablewindup phenomenon. In
PID based control, integrator withanti-windup are widely used to
saturate the control signaland avoid windup phenomenon. In this
paper, we study thetheoretical and practical implications of
projection operator(PO) and integral with anti-windup (IWAW) in
adaptivecontrol. Section II presents the basics of projection
operatorand integral with anti-windup.
II. PROJECTION OPERATOR AND INTEGRAL WITHANTI-WINDUP
A. Projection Operator
Definition 1: Consider a convex compact set with asmooth
boundary given by
c , { Rn|f() c}, 0 c 1where f : R R, is the following smooth
convex function:
f() , ( + 1)> 2max
2max
with max being the norm bound imposed on the vector ,and > 0
is the projection tolerance bound of our choice.The projection
operator is defined as
Proj(, y) =
y if f() < 0,y if f() 0 Of>y 0,y gf() if f() 0 Of>y
> 0.
where g is OfOf
OfOf , y
Property 1: The projection operator Proj(, y) does not
alter y if belongs to the set 0 , { Rn|f() 0}. Inthe set 0 , {
Rn|0 f() 1}, if Of>y > 0, theProj(, y) operator subtracts a
vector normal to the boundaryf() = { Rn|f() = f()}, so that we get
a smoothtransformation from the original vector field y to an
inwardor tangent vector field for 1. Figure 1 gives the
pictorialrepresentation of projection operator for the cases when 0
0) ( max e < 0),0 if max e > 0,0 if min e < 0.
Figure 2 shows the block diagram representation ofintegrator
with anti-windup.
C. Comparison of Projection Operator and Integrator
withanti-windup
To show the difference between projection operator andintegrator
with anti-windup, we have simulated a simplecase of constant input
both for positive and negative, to theprojection operator and
integrator with anti-windup. Figure3 shows the plot of output for
both the schemes with varyingtolerance on bound. It is evident that
increase in the toleranceleads to smooth trajectory near the bound
for projectionoperator.
REFERENCES[1] C. Cao and N. Hovakimyan, Design and analysis of a
novel L1 adaptive
controller, Part II: Guaranteed transient performance, American
ControlConference, pages 3403-3408, 2006.
