control systems with actuator saturation: anti-windup · pdf file3/23/2011. ieee central...

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


Actuator saturation and integrator windup

G (s)kI ∫

kP

R Y

Actuator saturation is a common phenomenon

Integrators are commonly presence in a controller



kI=1 ∫

kP=1

R =1(t ) Y

Y(0)=0

1

-1

1

2s


kI=1 ∫

kP=1

R =1(t ) Y

Y(0)=0

1

-1

1

2s

-

Actuator saturation and integrator windupSteady state error and instability


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



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 )

http://www.boston.com/ae/theater_arts/exhibitionist/scissors.png


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


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.


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


Example:


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



Example:

0 1 0, 1

1 0 1x x u u

sign sin 2.3213u t

2RC



0 0.5 0 ,

1 1.5 1A B

Example [Hu-Lin-Qiu, SCL „02] :

12 : [0, ]Ate I A B t C


Example [Hu-Lin-Qiu, SCL „02] :


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


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


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


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


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.


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


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


Anti-windup designL2 design: Immediate activation vs anticipatory activation

Wu and Lin, CDC‟10

ha

hg


Anti-windup designL2 design: Comparison


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]


Anti-windup designDesign for a large domain of attraction: Delayed activation

[Wu and Lin, CDC ‟10]


Anti-windup designDesign for a large domain of attraction: Anticipatory activation

[Wu and Lin, CDC ‟10]


Anti-windup designDesign for a large domain of attraction: Comparison

Stable plant:

PI controller:



Unstable plant:

PI controller:

control systems with actuator saturation: anti-windup · pdf file3/23/2011. ieee central...

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