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3/23/2011 IEEE Central Virginia Section 1
Control Systems with Actuator Saturation:
Anti-windup Design
Zongli Lin
Electrical and Computer Engineering
University of Virginia
1
3/23/2011 IEEE Central Virginia Section 2
Actuator saturation and integrator windup
G (s)kI ∫
kP
R Y
Actuator saturation is a common phenomenon
Integrators are commonly presence in a controller
3/23/2011 IEEE Central Virginia Section 3
Actuator saturation and integrator windup
kI=1 ∫
kP=1
R =1(t ) Y
Y(0)=0
1
-1
1
2s
3/23/2011 IEEE Central Virginia Section 4
kI=1 ∫
kP=1
R =1(t ) Y
Y(0)=0
1
-1
1
2s
-
Actuator saturation and integrator windupSteady state error and instability
3/23/2011 IEEE Central Virginia Section 5
Actuator saturation and integrator windupDegradation in transience performance
kI=4 ∫
kP=2
R =1(t ) Y
Y(0)=0
1
-1
1
s
-
Without
saturation
With
saturation
3/23/2011 IEEE Central Virginia Section 6
Actuator saturation and integrator windup
Role of integrator: elimination of steady state error = reset of reference
kI=1 ∫
kP=1
R1=1(t ) Y
Y(0)=0
1
-1
1
2s
-R2 =3x1(t )
R1=1(t ) R2 =3x1(t )
3/23/2011 IEEE Central Virginia Section 7
Actuator saturation and integrator windupAnti-windup design
kI=4 ∫
kP=2
R =1(t ) Y
Y(0)=0
1
-1
1
s
-
ka -
-
Ka=0
Ka=1
Ka=10
3/23/2011 IEEE Central Virginia Section 8
Actuator saturation and integrator windupObservations and motivations for research
Actuator saturation reduces a control system‟s ability to
follow a command input;
Actuator saturation degrades transience performance;
Anti-windup mitigates the adverse effects of actuator saturation;
Traditional way for the design of anti-windup gain is ad hoc.
Direct methods to control design in the presence of
actuator saturation;
Systematic ways to design anti-windup gains, with
guarantee of stability and performance.
3/23/2011 IEEE Central Virginia Section 9
Direct approach to dealing with actuator saturationControllability with bounded controls
1
-1
u sat(u) ysat( )x Ax B u
y Cx
Example:
(0) : , 1 and 0, s.t. ( ) 0nx R u u T x T
C
Null controllable region C :
, 0, 1x ax u a u
: 1/x R x a C
– C shrinks as a increases and expands otherwise
– C is bounded and open
3/23/2011 IEEE Central Virginia Section 1010
Example:
Direct approach to dealing with actuator saturationControllability with bounded controls
0 , 1x x u u
(0) 0, 1 (0) 0x u x x
(0) 0, 1 (0) 0x u x x
Example:
0 1 0,
0 0 1
1
x x u
u
1u 1u
RC
2RC
3/23/2011 IEEE Central Virginia Section 1111
Direct approach to dealing with actuator saturationControllability with bounded controls
Example:
0 1 0, 1
1 0 1x x u u
sign sin 2.3213u t
2RC
3/23/2011 IEEE Central Virginia Section 1212
Direct approach to dealing with actuator saturationControllability with bounded controls
0 0.5 0 ,
1 1.5 1A B
Example [Hu-Lin-Qiu, SCL „02] :
12 : [0, ]Ate I A B t C
3/23/2011 IEEE Central Virginia Section 1313
Example [Hu-Lin-Qiu, SCL „02] :
Direct approach to dealing with actuator saturationControllability with bounded controls
0.2 1 0 1
0 0.2 0 , 1
0 0 0.4 1
A B
2( ) 1
22 2 :0A t tAte e I A B t t
C
3/23/2011 IEEE Central Virginia Section 1414
General characterization of [Hsu, PhD Dissertation „76]
Assume that is controllable.
a) If is semi-stable ( ), then,
( ),A B
A 0( )A C C
b) If is anti-stable ( ), then, is a bounded convex open set.A ( )A C
Direct approach to dealing with actuator saturationControllability with bounded controls
nRC
c) If then,
where is the null controllable region of
1 1 0
1 2
2 2
0, , ( ) , ( ) ,
0
A BA B A C A C C
A B
1 1 1 1 ( )x A x B u
2
1
nR C C
1C
C
3/23/2011 IEEE Central Virginia Section 15
Direct approach to dealing with actuator saturationGlobal, semi-global, regional control designs
Stabilization
Robust stabilization
Output regulation
Disturbance attenuation
Linear feedback laws
Nonlinear feedback laws
Discontinuous feedback laws
LMI based designs
3/23/2011 IEEE Central Virginia Section 16
Direct approach to dealing with actuator saturationMany references
[1] D.S. Bernstein and A.N. Michel, “A chronological bibliography
on saturating actuators,” International Journal of Robust and
Nonlinear Control, Vol. 5, pp. 375-380, 1995.
[2] Z. Lin, Low Gain Feedback, Springer, London, 1998.
[3] T. Hu and Z. Lin, Control Systems with Actuator Saturation:
Analysis and Design, Birkhauser, Boston, 2001.
[4] V. Kapila and K.M. Grioriadis, Actuator Saturation Control, Marcel
Dekker, 2002.
[5] S. Tarbouriech, G. Garcia and A.H. Glattfelder, Advanced
Strategies in Control Systems with Input and Output Constraints,
Springer, London, 2007.
3/23/2011 IEEE Central Virginia Section 17
Anti-windup design
R Y1
-1-
Ec -
-G (s)C (s)
a large size of stability region (domain of attraction)
good closed-loop performances such as a small L2
gain from the disturbance to the output
Systematic design of Ec that guarantees
3/23/2011 IEEE Central Virginia Section 18
Anti-windup designL2 design
3/23/2011 IEEE Central Virginia Section 19
Anti-windup designL2 design
3/23/2011 IEEE Central Virginia Section20
Anti-windup designL2 design: Immediate activation vs delayed activation
Sajjadi-Kia and Jabbari,
IEEE TAC‟ 09
hd
hg
3/23/2011 IEEE Central Virginia Section 21
Anti-windup designL2 design: Immediate activation vs anticipatory activation
Wu and Lin, CDC‟10
ha
hg
3/23/2011 IEEE Central Virginia Section 22
Anti-windup designL2 design: Comparison
3/23/2011 IEEE Central Virginia Section 23
Anti-windup designL2 design: Comparison
3/23/2011 IEEE Central Virginia Section 24
Anti-windup designL2 design: Comparison
3/23/2011 IEEE Central Virginia Section 25
Anti-windup designDesign for a large domain of attraction: Immediate activation
R Y1
-1-
Ec -
-G (s)C (s)
[Cao, Lin and Ward, IEEE TAC „02]
3/23/2011 IEEE Central Virginia Section 26
Anti-windup designDesign for a large domain of attraction: Delayed activation
[Wu and Lin, CDC ‟10]
3/23/2011 IEEE Central Virginia Section 27
Anti-windup designDesign for a large domain of attraction: Anticipatory activation
[Wu and Lin, CDC ‟10]
3/23/2011 IEEE Central Virginia Section 28
Anti-windup designDesign for a large domain of attraction: Comparison
Stable plant:
PI controller:
3/23/2011 IEEE Central Virginia Section 29
Anti-windup designDesign for a large domain of attraction: Comparison
3/23/2011 IEEE Central Virginia Section 30
Anti-windup designDesign for a large domain of attraction: Comparison
Unstable plant:
PI controller:
3/23/2011 IEEE Central Virginia Section 31
Anti-windup designDesign for a large domain of attraction: Comparison
3/23/2011 IEEE Central Virginia Section 32