Control Systems with Actuator Saturation:
Anti-windup Design
Zongli Lin
Electrical and Computer Engineering
University of Virginia
3/23/2011
IEEE Central Virginia Section
1
Actuator saturation and integrator windup
kP
R
kI
∫
G (s)
Y
 Actuator saturation is a common phenomenon
 Integrators are commonly presence in a controller
3/23/2011
IEEE Central Virginia Section
2
Actuator saturation and integrator windup
kP=1
R =1(t )
3/23/2011
kI=1
∫
1
-1
IEEE Central Virginia Section
Y
1
s2
Y(0)=0
3
Actuator saturation and integrator windup
Steady state error and instability
kP=1
R =1(t )
-
3/23/2011
kI=1
∫
1
-1
IEEE Central Virginia Section
Y
1
s2
Y(0)=0
4
Actuator saturation and integrator windup
Degradation in transience performance
kP=2
R =1(t )
-
kI=4
1
∫
-1
Y
1
s
Y(0)=0
With
saturation
Without
saturation
3/23/2011
IEEE Central Virginia Section
5
Actuator saturation and integrator windup
 Role of integrator: elimination of steady state error = reset of reference
kP=1
R1=1(t )
R2 =3x1(t ) -
R1=1(t )
3/23/2011
kI=1
1
∫
-1
Y
1
s2
Y(0)=0
R2 =3x1(t )
IEEE Central Virginia Section
6
Actuator saturation and integrator windup
Anti-windup design
kP=2
R =1(t )
-
-
1
∫
kI=4
ka
-1
Y
1
s
Y(0)=0
-
Ka=0
Ka=1
Ka=10
3/23/2011
IEEE Central Virginia Section
7
Actuator saturation and integrator windup
Observations 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
8
Direct approach to dealing with actuator saturation
Controllability with bounded controls
u
1
sat(u)
 x  Ax  Bsat(u )

 y  Cx
-1
y
Null controllable region C :
C   x(0)  R n : u, u

 1 and T  0, s.t. x(T )  0
Example:
x  ax  u, a  0, u  1
C   x  R : x  1/ a
– C shrinks as a increases and expands otherwise
– C is bounded and open
3/23/2011
IEEE Central Virginia Section
9
Direct approach to dealing with actuator saturation
Controllability with bounded controls
Example:
x  0 x  u,
u 1
 x(0)  0, u  1  x  x(0)   0
C R
 x(0)  0, u  1  x  x(0)   0
Example:
0 1 
0
x
x    u,

0 0
1 
u 1
u 1
u  1
C  R2
3/23/2011
IEEE Central Virginia Section
10
Direct approach to dealing with actuator saturation
Controllability with bounded controls
Example:
 0 1
0 
x
x    u,

 1 0
1 
u 1
u  sign  sin  t  2.3213 
C  R2
3/23/2011
IEEE Central Virginia Section
11
Direct approach to dealing with actuator saturation
Controllability with bounded controls
Example [Hu-Lin-Qiu, SCL „02] :
0
A
1

0.5
0
, B 

1.5 
 1

C    2e At  I  A1B : t [0, ]
3/23/2011
IEEE Central Virginia Section
12
Direct approach to dealing with actuator saturation
Controllability with bounded controls
Example [Hu-Lin-Qiu, SCL „02] :
0 
0.2 1
1
A   0 0.2 0  , B  1
 0
1
0 0.4


C    2e At  2e A(t t2 )  I  A1B : 0  t2  t  
3/23/2011
IEEE Central Virginia Section
13
Direct approach to dealing with actuator saturation
Controllability with bounded controls
General characterization of C
[Hsu, PhD Dissertation „76]
Assume that (A, B ) is controllable.
a) If A is semi-stable ( ( A)  C   C 0 ), then, C  R n
b) If A is anti-stable ( ( A)  C  ), then,  is a bounded convex open set.
 A1 0 
 B1 
, B    ,  ( A1 )  C  ,  ( A2 )  C   C 0 , then, C  C1  Rn2
c) If A  

 0 A2 
 B2 
where C1 is the null controllable region of
x1  A1 x1  B1 (u)
3/23/2011
IEEE Central Virginia Section
14
Direct approach to dealing with actuator saturation
Global, 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
15
Direct approach to dealing with actuator saturation
Many 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
16
Anti-windup design
R
-
-
1
C (s)
G (s)
Y
-1
Ec
-
Systematic design of Ec that guarantees
 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
3/23/2011
IEEE Central Virginia Section
17
Anti-windup design
L2 design
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IEEE Central Virginia Section
18
Anti-windup design
L2 design
3/23/2011
IEEE Central Virginia Section
19
Anti-windup design
L2 design: Immediate activation vs delayed activation
Sajjadi-Kia and Jabbari,
IEEE TAC‟ 09
h
3/23/2011
IEEE Central Virginia Section
gd
h
20
Anti-windup design
L2 design: Immediate activation vs anticipatory activation
Wu and Lin, CDC‟10
h
3/23/2011
IEEE Central Virginia Section
h
ga
21
Anti-windup design
L2 design: Comparison
3/23/2011
IEEE Central Virginia Section
22
Anti-windup design
L2 design: Comparison
3/23/2011
IEEE Central Virginia Section
23
Anti-windup design
L2 design: Comparison
3/23/2011
IEEE Central Virginia Section
24
Anti-windup design
Design for a large domain of attraction: Immediate activation
[Cao, Lin and Ward, IEEE TAC „02]
R
-
-
C (s)
G (s)
Y
-1
Ec
3/23/2011
1
-
IEEE Central Virginia Section
25
Anti-windup design
Design for a large domain of attraction: Delayed activation
[Wu and Lin, CDC ‟10]
3/23/2011
IEEE Central Virginia Section
26
Anti-windup design
Design for a large domain of attraction: Anticipatory activation
[Wu and Lin, CDC ‟10]
3/23/2011
IEEE Central Virginia Section
27
Anti-windup design
Design for a large domain of attraction: Comparison
Stable plant:
PI controller:
3/23/2011
IEEE Central Virginia Section
28
Anti-windup design
Design for a large domain of attraction: Comparison
3/23/2011
IEEE Central Virginia Section
29
Anti-windup design
Design for a large domain of attraction: Comparison
Unstable plant:
PI controller:
3/23/2011
IEEE Central Virginia Section
30
Anti-windup design
Design for a large domain of attraction: Comparison
3/23/2011
IEEE Central Virginia Section
31
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IEEE Central Virginia Section
32