Gradient Projection Anti-windup Scheme
Thesis Defense
Justin Teo (MIT Aero/Astro)
Thesis Committee:
Thesis Readers:
Department Representative:
Jonathan P. How (Chair)
Emilio Frazzoli
Steven R. Hall
Eugene Lavretsky
Luca F. Bertuccelli
Louis Breger
(MIT Aero/Astro)
(MIT Aero/Astro)
(MIT Aero/Astro)
(Boeing)
(MIT Aero/Astro)
(Draper)
Wesley L. Harris
(MIT Aero/Astro)
December 20, 2010
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
1 / 35
Outline
Outline
1
Introduction
2
GPAW Compensated Controller
3
Input Constrained Planar LTI Systems
4
An ROA Comparison Result
5
A Numerical Comparison
6
Conclusions
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
2 / 35
Introduction
Effects of Control Saturation
Effects of Control Saturation
Well Recognized Fact [Bernstein and Michel 1995]
Control saturation affects virtually all practical control systems
Effects called “windup”, affects all dynamic controllers and leads to:
performance degradation (with certainty)
instability (possibly)
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
3 / 35
Introduction
Effects of Control Saturation
Effects of Control Saturation
Well Recognized Fact [Bernstein and Michel 1995]
Control saturation affects virtually all practical control systems
Effects called “windup”, affects all dynamic controllers and leads to:
performance degradation (with certainty)
instability (possibly)
Stable Plant, Unstable Controller
Mild effects [Visioli 2006]:
large overshoots
long settling times
unconstrained
saturated
2
x
sluggish response
4
0
−2
0
2
4
6
8
10
12
14
16
18
20
18
20
2
umax
u
1
0
−1
umin
−2
−3
0
2
4
6
8
10
12
14
16
time (s)
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
3 / 35
Introduction
Effects of Control Saturation
Effects of Control Saturation
Well Recognized Fact [Bernstein and Michel 1995]
Control saturation affects virtually all practical control systems
Effects called “windup”, affects all dynamic controllers and leads to:
performance degradation (with certainty)
instability (possibly)
Stable Plant, Unstable Controller
Mild effects [Visioli 2006]:
large overshoots
long settling times
unconstrained
saturated
2
x
sluggish response
4
0
−2
0
2
4
6
8
10
12
14
18
20
18
20
umax
u
1
0
−1
Severe effects: instability
16
2
umin
−2
−3
0
2
4
6
8
10
12
14
16
time (s)
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
3 / 35
Introduction
Effects of Control Saturation
Disasters Caused Indirectly by Windup
Disasters caused indirectly by windup include:
1986 Chernobyl (nuclear reactor) disaster [Stein 2003]
1992 crash of YF-22 fighter aircraft [Dornheim 1992]
1989 and 1993 crashes of Saab Gripen JAS 39 fighter
aircraft [Butterworth-Hayes 1994, Stein 2003]
1992 crash of YF-22
Justin Teo (ACL, MIT)
1989 and 1993 crashes of Saab Gripen
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
4 / 35
Introduction
Control Design Strategies
Control Design Strategies
Control design strategies to deal with windup:
avoiding saturation - applies when control task is well-defined,
e.g. assembly lines
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
5 / 35
Introduction
Control Design Strategies
Control Design Strategies
Control design strategies to deal with windup:
avoiding saturation - applies when control task is well-defined,
e.g. assembly lines
one-step approach
accounts for saturation in design of nominal controller - complex
often conservative and hard to tune [Tarbouriech and Turner 2009,
Sofrony et al. 2006, Mulder et al. 2009]
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
5 / 35
Introduction
Control Design Strategies
Control Design Strategies
Control design strategies to deal with windup:
avoiding saturation - applies when control task is well-defined,
e.g. assembly lines
one-step approach
accounts for saturation in design of nominal controller - complex
often conservative and hard to tune [Tarbouriech and Turner 2009,
Sofrony et al. 2006, Mulder et al. 2009]
two-step approach or anti-windup compensation
ignores saturation in design of nominal controller (step 1)
design controller modifications to account for windup (step 2)
Anti-windup compensation preferred by practitioners due to [Tarbouriech
and Turner 2009]:
design of nominal controller greatly simplified
can be retrofitted to existing controllers
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
5 / 35
Introduction
Control Design Strategies
Anti-windup Compensation
Anti-windup compensation well studied for linear time invariant (LTI)
case [Kothare et al. 1994, Edwards and Postlethwaite 1998, Tarbouriech
and Turner 2009]
r
Σ̃c
ũ
u
sat(u)
v
ẋ = Ax + Bv
y
y = Cx + Dv
Unconstrained plant
Σ̃aw
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
6 / 35
Introduction
Control Design Strategies
Anti-windup Compensation
Anti-windup compensation well studied for linear time invariant (LTI)
case [Kothare et al. 1994, Edwards and Postlethwaite 1998, Tarbouriech
and Turner 2009]
r
Σ̃c
ũ
u
sat(u)
v
y = Cx + Dv
−
yaw2
yaw1
Σ̃aw
y
ẋ = Ax + Bv
Unconstrained plant
w
Anti-windup compensated controller
Anti-windup compensator driven by w = sat(u) − u
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
6 / 35
Introduction
Control Design Strategies
Anti-windup Compensation
Anti-windup compensation well studied for linear time invariant (LTI)
case [Kothare et al. 1994, Edwards and Postlethwaite 1998, Tarbouriech
and Turner 2009]
r
Σ̃c
ũ
u
sat(u)
y = Cx + Dv
−
yaw2
yaw1
Σ̃aw
y
ẋ = Ax + Bv
v
Unconstrained plant
w
Anti-windup compensated controller
Anti-windup compensator driven by w = sat(u) − u
Open Problem [Tarbouriech and Turner 2009]
Anti-windup compensation for saturated nonlinear systems
Most practical control systems are nonlinear - LTI are approximations
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
6 / 35
Introduction
Problem Statement
Problem Statement
Nominal system Σn : feedback
interconnection (FI) of Σp , Σc
Saturated Plant:
(
ẋ = f (x, sat(u))
Σp :
y = g(x, sat(u))
Anti-windup (AW) compensated
system Σaws : FI of Σp , Σaw
Nominal Controller:
(
ẋc = fc (xc , y, r)
Σc :
u = gc (xc , y, r)
General Anti-windup Problem
Design Σaw and determined initialization
xaw (0) such that Σaws satisfies:
AW Compensated Controller:
(
ẋaw = faw (xaw , y, r)
Σaw :
u = gaw (xaw , y, r)
Justin Teo (ACL, MIT)
when no controls saturate for Σn ,
then nominal performance
recovered, i.e. Σaws ≡ Σn
when some controls saturate,
stability and performance of Σaws is
no worse than that of Σn
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
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Introduction
Literature Review
Literature Review
Anti-windup methods (partial citations) applicable to nonlinear systems:
Conditioning Technique [Hanus et al. 1987] - computationally
prohibitive and severely limited for nonlinear systems
Feedback Linearizable Nonlinear Systems [Yoon et al. 2008] - requires
feedback linearizable plant and feedback linearizing controller
For some Particular Controllers [Hu and Rangaiah 2000, Johnson and
Calise 2001, 2003, Do et al. 2004] - not general purpose
Nonlinear Anti-windup for Euler-Lagrange Systems [Morabito et al.
2004] - hard to generalize
Optimal Directionality Compensation [Soroush and Daoutidis 2002] plant needs to be square
Reference Governor [Gilbert and Kolmanovsky 2002] - some
conservatism introduced
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
8 / 35
Introduction
Contributions
Contributions
Contributions of this research include:
developed general purpose anti-windup scheme
motivated new paradigm for anti-windup problem
demonstrated need to consider asymmetric saturation constraints for
general saturated systems
developed region of attraction (ROA) comparison and stability results
for GPAW compensated (nonlinear) systems
demonstrated viability of GPAW scheme as a candidate anti-windup
scheme for general systems
related GPAW compensated systems to projected dynamical systems
and linear systems with partial state constraints
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
9 / 35
Introduction
Contributions
Contributions
Contributions of this research include:
developed general purpose anti-windup scheme
motivated new paradigm for anti-windup problem
demonstrated need to consider asymmetric saturation constraints for
general saturated systems
developed region of attraction (ROA) comparison and stability results
for GPAW compensated (nonlinear) systems
demonstrated viability of GPAW scheme as a candidate anti-windup
scheme for general systems
related GPAW compensated systems to projected dynamical systems
and linear systems with partial state constraints
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
9 / 35
GPAW Compensated Controller
Outline
1
Introduction
2
GPAW Compensated Controller
3
Input Constrained Planar LTI Systems
4
An ROA Comparison Result
5
A Numerical Comparison
6
Conclusions
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
10 / 35
GPAW Compensated Controller
Conditional Integration
Conditional Integration
Conditional integration (CI) for PID controllers [Fertik and Ross 1967]


0, if u ≥ umax ∧ e > 0
ėi = 0, if u ≤ umin ∧ e < 0
ėi = e
CI

−→

e, otherwise
u = Kp e + Ki ei + Kd ė
u = Kp e + Ki ei + Kd ė
Stop integration when nominal update will aggravate saturation
constraints, or stop integration when departing unsaturated region
K(e, ė) = {ēi ∈ R | sat(Kp e + Ki ēi + Kd ė) = Kp e + Ki ēi + Kd ė}
Attempts to achieve controller state-output consistency sat(u) = u
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
11 / 35
GPAW Compensated Controller
Conditional Integration
Conditional Integration
Conditional integration (CI) for PID controllers [Fertik and Ross 1967]


0, if u ≥ umax ∧ e > 0
ėi = 0, if u ≤ umin ∧ e < 0
ėi = e
CI

−→

e, otherwise
u = Kp e + Ki ei + Kd ė
u = Kp e + Ki ei + Kd ė
Stop integration when nominal update will aggravate saturation
constraints, or stop integration when departing unsaturated region
K(e, ė) = {ēi ∈ R | sat(Kp e + Ki ēi + Kd ė) = Kp e + Ki ēi + Kd ė}
Attempts to achieve controller state-output consistency sat(u) = u
ẋci = fci (xci , y, r)
Extends easily to decoupled nonlinear controllers
uci = gci (xci , y, r)
ẋc = fc (xc , y, r)
For coupled nonlinear controllers, need projection opuc = gc (xc , y, r)
erator - project onto K(e, ė) analogue
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
11 / 35
GPAW Compensated Controller
Gradient Projection Method for Nonlinear Programming
Gradient Projection Method for Nonlinear
Programming [Rosen 1960, 1961]
−∇J(x2 )
Nonlinear program:
minq J(x)
−∇J(x1 )
x∈R
h̃(x) ≤ 0
x3
−∇J(x0 )
z3
x0
Projections: z1 , z2 , z3
Justin Teo (ACL, MIT)
−∇J(x3 )
K̃
Gradient Projection Anti-windup Scheme
x 3)
H 3(
G3
Boundaries:
H1 , H2 , G3
z2
(x 3 )
Feasible region:
K̃ = {x̄ | h̃(x̄) ≤ 0}
x1 z 1
H1
zd
H2 ∇h̃2
∇h̃ 3
subject to
x2
∇h̃ 1
Dec. 20, 2010
12 / 35
GPAW Compensated Controller
Gradient Projection Method for Nonlinear Programming
Gradient Projection Method for Nonlinear
Programming [Rosen 1960, 1961]
−∇J(x2 )
Nonlinear program:
minq J(x)
−∇J(x1 )
x∈R
h̃(x) ≤ 0
x3
−∇J(x0 )
z3
−∇J(x3 )
K̃
x0
Projections: z1 , z2 , z3
x 3)
H 3(
G3
Boundaries:
H1 , H2 , G3
z2
(x 3 )
Feasible region:
K̃ = {x̄ | h̃(x̄) ≤ 0}
x1 z 1
H1
zd
H2 ∇h̃2
∇h̃ 3
subject to
x2
∇h̃ 1
Extended to continuous-time to yield projection operator - requires
solution to combinatorial optimization subproblem at each point
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
12 / 35
GPAW Compensated Controller
GPAW Compensated Controller
GPAW Compensated Controller
Gradient projection anti-windup (GPAW) compensated controller:
obtained by applying projection operator from continuous-time
gradient projection method on nominal controller
defined by online solution to a combinatorial optimization subproblem
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
13 / 35
GPAW Compensated Controller
GPAW Compensated Controller
GPAW Compensated Controller
Gradient projection anti-windup (GPAW) compensated controller:
obtained by applying projection operator from continuous-time
gradient projection method on nominal controller
defined by online solution to a combinatorial optimization subproblem
For “strictly proper” nonlinear controllers,
ẋc = fc (xc , y, r)
uc = gc (xc )
GPAW,Γ=ΓT >0
−−−−−−−−−−→
ẋg = RI ∗ (xg , y, r)fc (xg , y, r)
ug = gc (xg )
Everything rests on projection operator RI ∗ !
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
13 / 35
GPAW Compensated Controller
GPAW Compensated Controller
GPAW Compensated Controller
Gradient projection anti-windup (GPAW) compensated controller:
obtained by applying projection operator from continuous-time
gradient projection method on nominal controller
defined by online solution to a combinatorial optimization subproblem
For “strictly proper” nonlinear controllers,
ẋc = fc (xc , y, r)
uc = gc (xc )
GPAW,Γ=ΓT >0
−−−−−−−−−−→
ẋg =
fc (xg , y, r)
ug = gc (xg )
Everything rests on projection operator RI ∗ !
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
13 / 35
GPAW Compensated Controller
GPAW Compensated Controller
GPAW Compensated Controller
Gradient projection anti-windup (GPAW) compensated controller:
obtained by applying projection operator from continuous-time
gradient projection method on nominal controller
defined by online solution to a combinatorial optimization subproblem
For “strictly proper” nonlinear controllers,
ẋc = fc (xc , y, r)
uc = gc (xc )
GPAW,Γ=ΓT >0
−−−−−−−−−−→
ẋg = RI ∗ (xg , y, r)fc (xg , y, r)
ug = gc (xg )
Everything rests on projection operator RI ∗ !
Projection operator RI ∗ defined by Γ = ΓT > 0, online solution to
combinatorial optimization subproblem I ∗ , and projection matrix
(
I − ΓNI (NIT ΓNI )−1 NIT (xg ), if I =
6 ∅
RI (xg ) =
I,
otherwise
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
13 / 35
GPAW Compensated Controller
GPAW Compensated Controller
Other Properties
GPAW compensated controller has a single parameter Γ = ΓT > 0:
can be defined equivalently by online (unique) solution to:
convex quadratic program (with numerous efficient solvers)
projection onto convex polyhedral cone (algorithms available)
- valid regardless of nonlinearities in plant/controller
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
14 / 35
GPAW Compensated Controller
GPAW Compensated Controller
Other Properties
GPAW compensated controller has a single parameter Γ = ΓT > 0:
can be defined equivalently by online (unique) solution to:
convex quadratic program (with numerous efficient solvers)
projection onto convex polyhedral cone (algorithms available)
- valid regardless of nonlinearities in plant/controller
can be realized by closed-form expressions when uc ∈ R or uc ∈ R2
- computationally efficient
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
14 / 35
GPAW Compensated Controller
GPAW Compensated Controller
Other Properties
GPAW compensated controller has a single parameter Γ = ΓT > 0:
can be defined equivalently by online (unique) solution to:
convex quadratic program (with numerous efficient solvers)
projection onto convex polyhedral cone (algorithms available)
- valid regardless of nonlinearities in plant/controller
can be realized by closed-form expressions when uc ∈ R or uc ∈ R2
- computationally efficient
attempts to enforce control saturation constraints
h
i
g (x )−u
h(xg ) = −gcc (xgg )+umax
≤0
⇔
xg ∈ K := {x̄ | h(x̄) ≤ 0}
min
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
14 / 35
GPAW Compensated Controller
GPAW Compensated Controller
Other Properties
GPAW compensated controller has a single parameter Γ = ΓT > 0:
can be defined equivalently by online (unique) solution to:
convex quadratic program (with numerous efficient solvers)
projection onto convex polyhedral cone (algorithms available)
- valid regardless of nonlinearities in plant/controller
can be realized by closed-form expressions when uc ∈ R or uc ∈ R2
- computationally efficient
attempts to enforce control saturation constraints
h
i
g (x )−u
h(xg ) = −gcc (xgg )+umax
≤0
⇔
xg ∈ K := {x̄ | h(x̄) ≤ 0}
min
Non-“strictly proper” nonlinear controllers can be approximated arbitrarily
well to be “strictly proper” (singular perturbation theory [Khalil 2002])
ẋc = fc (xc , y, r)
uc = gc (xc , y, r)
Justin Teo (ACL, MIT)
a∈(0,∞)
−−−−−→
≈
x̃˙ c = f˜c (x̃c , y, r)
uc = g̃c (x̃c )
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
14 / 35
GPAW Compensated Controller
Controller State-output Consistency
Controller State-output Consistency
Controller State-output Consistency
sat(gc (xg )) ≡ gc (xg )
⇔
sat(ug ) ≡ ug
⇔
xg (t) ∈ K, ∀t ∈ R
- implicit objective of anti-windup schemes (majority) driven by signal
Figure
(sat(u) − u)
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
15 / 35
GPAW Compensated Controller
Controller State-output Consistency
Controller State-output Consistency
Controller State-output Consistency
sat(gc (xg )) ≡ gc (xg )
⇔
sat(ug ) ≡ ug
⇔
xg (t) ∈ K, ∀t ∈ R
- implicit objective of anti-windup schemes (majority) driven by signal
Figure
(sat(u) − u)
Theorem (GPAW Controller State-output Consistency)
For GPAW compensated controller, if there exists a T ∈ R such that
sat(ug (T )) = ug (T ), then sat(ug (t)) = ug (t) holds for all t ≥ T
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
15 / 35
GPAW Compensated Controller
Controller State-output Consistency
Controller State-output Consistency
Controller State-output Consistency
sat(gc (xg )) ≡ gc (xg )
⇔
sat(ug ) ≡ ug
⇔
xg (t) ∈ K, ∀t ∈ R
- implicit objective of anti-windup schemes (majority) driven by signal
Figure
(sat(u) − u)
Theorem (GPAW Controller State-output Consistency)
For GPAW compensated controller, if there exists a T ∈ R such that
sat(ug (T )) = ug (T ), then sat(ug (t)) = ug (t) holds for all t ≥ T
Implications - GPAW compensated closed-loop system:
ẋ = f (x, sat(gc (xg )))
ẋg = RI ∗ fc (x, xg , sat(gc (xg )), r)
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
15 / 35
GPAW Compensated Controller
Controller State-output Consistency
Controller State-output Consistency
Controller State-output Consistency
sat(gc (xg )) ≡ gc (xg )
⇔
sat(ug ) ≡ ug
⇔
xg (t) ∈ K, ∀t ∈ R
- implicit objective of anti-windup schemes (majority) driven by signal
Figure
(sat(u) − u)
Theorem (GPAW Controller State-output Consistency)
For GPAW compensated controller, if there exists a T ∈ R such that
sat(ug (T )) = ug (T ), then sat(ug (t)) = ug (t) holds for all t ≥ T
Implications - GPAW compensated closed-loop system:
ẋ = f (x, sat(gc (xg )))
xg (0)∈K
ẋg = RI ∗ fc (x, xg , sat(gc (xg )), r)
−−−−−→
ẋ = f (x, gc (xg ))
ẋg = RI ∗ fc (x, xg , gc (xg ), r)
saturation function sat(·) eliminated: significant simplification
all complications arising from saturation accounted for by RI ∗
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
15 / 35
GPAW Compensated Controller
Controller State-output Consistency
GPAW Scheme Visualization
Nominal controller:
ẋc = fc (xc , y, r)
fc2
fc1
uc = gc (xc )
H2 ∇h2
xg3
fc0
fc3
H 3(
K
f g3
Boundaries:
H1 , H2 , G3
xg1 f g1
fg2
)
H1
xg2
(x g3
ug = gc (xg )
∇h 1
∇h 3
GPAW controller:
ẋg = RI ∗ fc (xg , y, r)
xg0
)
Justin Teo (ACL, MIT)
x g3
Unsaturated region:
Nominal update:
Projections:
G3
Gradients:
∇hi (xg ) = ±∇gci (xg )
K := {x̄ | sat(gc (x̄)) = gc (x̄)}
fci := fc (xgi , y(ti ), r(ti )) for xgi := xg (ti )
fgi := RI ∗ fci
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
16 / 35
Input Constrained Planar LTI Systems
Outline
1
Introduction
2
GPAW Compensated Controller
3
Input Constrained Planar LTI Systems
4
An ROA Comparison Result
5
A Numerical Comparison
6
Conclusions
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
17 / 35
Input Constrained Planar LTI Systems
Projected Dynamical System
Input Constrained Planar LTI Systems
Simplest possible feedback system, 1st order LTI plant and controller:
Σplant : ẋ = ax + b sat(u),
GPAW compensated


0,
u̇ = 0,


cx + du,
Justin Teo (ACL, MIT)
u̇ = cx + du
system:
if u ≥ umax , cx + du > 0
if u ≤ umin , cx + du < 0
otherwise
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
18 / 35
Input Constrained Planar LTI Systems
Projected Dynamical System
Input Constrained Planar LTI Systems
Simplest possible feedback system, 1st order LTI plant and controller:
Σplant : ẋ = ax + b sat(u),
GPAW compensated


0,
u̇ = 0,


cx + du,
u̇ = cx + du
feedback
−−−−−→
Σn
feedback
Σg
Σplant
system:
if u ≥ umax , cx + du > 0
if u ≤ umin , cx + du < 0
otherwise
−−−−−→
Σplant
Assumption (Unconstrained Stability)
The unconstrained system Σu (umax = −umin = ∞) is globally stable
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
18 / 35
Input Constrained Planar LTI Systems
Projected Dynamical System
Input Constrained Planar LTI Systems
Simplest possible feedback system, 1st order LTI plant and controller:
Σplant : ẋ = ax + b sat(u),
GPAW compensated


0,
u̇ = 0,


cx + du,
u̇ = cx + du
feedback
−−−−−→
Σn
feedback
Σg
Σplant
system:
if u ≥ umax , cx + du > 0
if u ≤ umin , cx + du < 0
otherwise
−−−−−→
Σplant
Assumption (Unconstrained Stability)
The unconstrained system Σu (umax = −umin = ∞) is globally stable
Proposition (Relation to Projected Dynamical Systems)
The GPAW compensated system Σg is a projected dynamical
system [Dupuis and Nagurney 1993, Zhang and Nagurney 1995]
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
18 / 35
Input Constrained Planar LTI Systems
Region of Attraction Containment
Region of Attraction Containment
Region of Attraction (ROA) limits utility of systems, defined as:
Rn := {z̄ ∈ R2 | lim φn (t, z̄) = 0}
t→∞
Justin Teo (ACL, MIT)
Rg := {z̄ ∈ R2 | lim φg (t, z̄) = 0}
Gradient Projection Anti-windup Scheme
t→∞
Dec. 20, 2010
19 / 35
Input Constrained Planar LTI Systems
Region of Attraction Containment
Region of Attraction Containment
Region of Attraction (ROA) limits utility of systems, defined as:
Rn := {z̄ ∈ R2 | lim φn (t, z̄) = 0}
t→∞
Rg := {z̄ ∈ R2 | lim φg (t, z̄) = 0}
t→∞
Anti-windup schemes aim to improve performance only when saturated
Require ROA to be maintained/enlarged to be valid anti-windup
scheme, i.e. Rn ⊂ Raw
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
19 / 35
Input Constrained Planar LTI Systems
Region of Attraction Containment
Region of Attraction Containment
Region of Attraction (ROA) limits utility of systems, defined as:
Rn := {z̄ ∈ R2 | lim φn (t, z̄) = 0}
t→∞
Rg := {z̄ ∈ R2 | lim φg (t, z̄) = 0}
t→∞
Anti-windup schemes aim to improve performance only when saturated
Require ROA to be maintained/enlarged to be valid anti-windup
scheme, i.e. Rn ⊂ Raw
Proposition (ROA Containment)
The ROA of the origin of system Σn is contained within the ROA of the
origin of system Σg , i.e. Rn ⊂ Rg
ROA containment is a strong result:
valid for all system parameters and saturation limits
independent of any Lyapunov function
implies for every Lyapunov function Vn ⇒ Rn , then ∃Vg ⇒ Rg (⊃ Rn )
stark departure from existing stability results
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
19 / 35
Input Constrained Planar LTI Systems
Region of Attraction Containment
Numerical Results I, Rn = Rg
Unstable plant, stable controller, umax = −umin = 1
5
4
3
2
u
1
0
−1
−2
−3
−4
−5
−2.5
Justin Teo (ACL, MIT)
−2
−1.5
−1
−0.5
0
x
0.5
1
Gradient Projection Anti-windup Scheme
1.5
2
2.5
Dec. 20, 2010
20 / 35
Input Constrained Planar LTI Systems
Region of Attraction Containment
Numerical Results I, Rn = Rg
Unstable plant, stable controller, umax = −umin = 1
5
4
3
2
1
u
Rn
0
−1
−2
−3
−4
−5
−2.5
Justin Teo (ACL, MIT)
−2
−1.5
−1
−0.5
0
x
0.5
1
Gradient Projection Anti-windup Scheme
1.5
2
2.5
Dec. 20, 2010
20 / 35
Input Constrained Planar LTI Systems
Region of Attraction Containment
Numerical Results I, Rn = Rg
Unstable plant, stable controller, umax = −umin = 1
5
4
3
2
1
u
Rg
0
−1
−2
−3
−4
−5
−2.5
Justin Teo (ACL, MIT)
−2
−1.5
−1
−0.5
0
x
0.5
1
Gradient Projection Anti-windup Scheme
1.5
2
2.5
Dec. 20, 2010
20 / 35
Input Constrained Planar LTI Systems
Region of Attraction Containment
Numerical Results I, Rn = Rg
Unstable plant, stable controller, umax = −umin = 1
5
4
3
2
u
1
Rn = Rg
0
−1
−2
−3
−4
−5
−2.5
Justin Teo (ACL, MIT)
−2
−1.5
−1
−0.5
0
x
0.5
1
Gradient Projection Anti-windup Scheme
1.5
2
2.5
Dec. 20, 2010
20 / 35
Input Constrained Planar LTI Systems
Region of Attraction Containment
Numerical Results I, Rn = Rg
Unstable plant, stable controller, umax = −umin = 1
5
φn (t, z0 )
4
φg (t, z0 )
3
2
u
1
Rn = Rg
0
−1
−2
−3
−4
−5
−2.5
Justin Teo (ACL, MIT)
−2
−1.5
−1
−0.5
0
x
0.5
1
Gradient Projection Anti-windup Scheme
1.5
2
2.5
Dec. 20, 2010
20 / 35
Input Constrained Planar LTI Systems
Region of Attraction Containment
Numerical Results I, Rn = Rg
Unstable plant, stable controller, umax = −umin = 1
5
φn (t, z0 )
4
φg (t, z0 )
3
2
u
1
Rn = Rg
0
−1
−2
−3
−4
−5
−2.5
Justin Teo (ACL, MIT)
−2
−1.5
−1
−0.5
0
x
0.5
1
Gradient Projection Anti-windup Scheme
1.5
2
2.5
Dec. 20, 2010
20 / 35
Input Constrained Planar LTI Systems
Region of Attraction Containment
Numerical Results I, Rn = Rg
Unstable plant, stable controller, umax = −umin = 1
5
φn (t, z0 )
4
φg (t, z0 )
3
2
u
1
Rn = Rg
0
−1
−2
−3
−4
−5
−2.5
Justin Teo (ACL, MIT)
−2
−1.5
−1
−0.5
0
x
0.5
1
Gradient Projection Anti-windup Scheme
1.5
2
2.5
Dec. 20, 2010
20 / 35
Input Constrained Planar LTI Systems
Region of Attraction Containment
Numerical Results I, Rn = Rg
Unstable plant, stable controller, umax = −umin = 1
5
φn (t, z0 )
4
φg (t, z0 )
3
2
u
1
Rn = Rg
0
−1
−2
−3
−4
−5
−2.5
Justin Teo (ACL, MIT)
−2
−1.5
−1
−0.5
0
x
0.5
1
Gradient Projection Anti-windup Scheme
1.5
2
2.5
Dec. 20, 2010
20 / 35
Input Constrained Planar LTI Systems
Region of Attraction Containment
Numerical Results II, Rn ⊂ Rg
Unstable plant, stable controller, 1.5 = umax > −umin = 1
5
4
3
2
u
1
0
−1
−2
−3
−4
−5
−2.5
Justin Teo (ACL, MIT)
−2
−1.5
−1
−0.5
0
x
0.5
1
Gradient Projection Anti-windup Scheme
1.5
2
2.5
Dec. 20, 2010
21 / 35
Input Constrained Planar LTI Systems
Region of Attraction Containment
Numerical Results II, Rn ⊂ Rg
Unstable plant, stable controller, 1.5 = umax > −umin = 1
5
4
3
2
u
1
Rn
0
−1
−2
−3
−4
−5
−2.5
Justin Teo (ACL, MIT)
−2
−1.5
−1
−0.5
0
x
0.5
1
Gradient Projection Anti-windup Scheme
1.5
2
2.5
Dec. 20, 2010
21 / 35
Input Constrained Planar LTI Systems
Region of Attraction Containment
Numerical Results II, Rn ⊂ Rg
Unstable plant, stable controller, 1.5 = umax > −umin = 1
5
4
3
2
u
1
Rg
0
−1
−2
−3
−4
−5
−2.5
Justin Teo (ACL, MIT)
−2
−1.5
−1
−0.5
0
x
0.5
1
Gradient Projection Anti-windup Scheme
1.5
2
2.5
Dec. 20, 2010
21 / 35
Input Constrained Planar LTI Systems
Region of Attraction Containment
Numerical Results II, Rn ⊂ Rg
Unstable plant, stable controller, 1.5 = umax > −umin = 1
5
4
3
2
u
1
Rg
0
−1
−2
Rn
−3
−4
−5
−2.5
Justin Teo (ACL, MIT)
−2
−1.5
−1
−0.5
0
x
0.5
1
Gradient Projection Anti-windup Scheme
1.5
2
2.5
Dec. 20, 2010
21 / 35
Input Constrained Planar LTI Systems
Region of Attraction Containment
Numerical Results II, Rn ⊂ Rg
Unstable plant, stable controller, 1.5 = umax > −umin = 1
5
φn (t, z0 )
4
φg (t, z0 )
3
2
u
1
Rg
0
−1
−2
Rn
−3
−4
−5
−2.5
Justin Teo (ACL, MIT)
−2
−1.5
−1
−0.5
0
x
0.5
1
Gradient Projection Anti-windup Scheme
1.5
2
2.5
Dec. 20, 2010
21 / 35
Input Constrained Planar LTI Systems
Region of Attraction Containment
Numerical Results II, Rn ⊂ Rg
Unstable plant, stable controller, 1.5 = umax > −umin = 1
5
φn (t, z0 )
4
φg (t, z0 )
3
2
u
1
Rg
0
−1
−2
Rn
−3
−4
−5
−2.5
Justin Teo (ACL, MIT)
−2
−1.5
−1
−0.5
0
x
0.5
1
Gradient Projection Anti-windup Scheme
1.5
2
2.5
Dec. 20, 2010
21 / 35
Input Constrained Planar LTI Systems
Region of Attraction Containment
Numerical Results II, Rn ⊂ Rg
Unstable plant, stable controller, 1.5 = umax > −umin = 1
5
φn (t, z0 )
4
φg (t, z0 )
3
2
u
1
Rg
0
−1
−2
Rn
−3
−4
−5
−2.5
Justin Teo (ACL, MIT)
−2
−1.5
−1
−0.5
0
x
0.5
1
Gradient Projection Anti-windup Scheme
1.5
2
2.5
Dec. 20, 2010
21 / 35
Input Constrained Planar LTI Systems
Region of Attraction Containment
Numerical Results II, Rn ⊂ Rg
Unstable plant, stable controller, 1.5 = umax > −umin = 1
5
φn (t, z0 )
4
φg (t, z0 )
3
2
u
1
Rg
0
−1
−2
Rn
−3
−4
−5
−2.5
Justin Teo (ACL, MIT)
−2
−1.5
−1
−0.5
0
x
0.5
1
Gradient Projection Anti-windup Scheme
1.5
2
2.5
Dec. 20, 2010
21 / 35
Input Constrained Planar LTI Systems
Region of Attraction Containment
Numerical Results III, Rn ⊂ Rg
Stable plant, unstable controller, umax = −umin
3
2
u
1
0
−1
−2
−3
−5
Justin Teo (ACL, MIT)
−4
−3
−2
−1
0
x
1
2
Gradient Projection Anti-windup Scheme
3
4
5
Dec. 20, 2010
22 / 35
Input Constrained Planar LTI Systems
Region of Attraction Containment
Numerical Results III, Rn ⊂ Rg
Stable plant, unstable controller, umax = −umin
3
Rn
2
u
1
0
−1
−2
−3
−5
Justin Teo (ACL, MIT)
−4
−3
−2
−1
0
x
1
2
Gradient Projection Anti-windup Scheme
3
4
5
Dec. 20, 2010
22 / 35
Input Constrained Planar LTI Systems
Region of Attraction Containment
Numerical Results III, Rn ⊂ Rg
Stable plant, unstable controller, umax = −umin
3
2
Rg
u
1
0
−1
−2
−3
−5
Justin Teo (ACL, MIT)
−4
−3
−2
−1
0
x
1
2
Gradient Projection Anti-windup Scheme
3
4
5
Dec. 20, 2010
22 / 35
Input Constrained Planar LTI Systems
Region of Attraction Containment
Numerical Results III, Rn ⊂ Rg
Stable plant, unstable controller, umax = −umin
3
Rn
2
Rg
u
1
0
−1
−2
−3
−5
Justin Teo (ACL, MIT)
−4
−3
−2
−1
0
x
1
2
Gradient Projection Anti-windup Scheme
3
4
5
Dec. 20, 2010
22 / 35
Input Constrained Planar LTI Systems
Region of Attraction Containment
Numerical Results III, Rn ⊂ Rg
Stable plant, unstable controller, umax = −umin
3
Rn
2
Rg
u
1
0
−1
φn (t, z0 )
−2
φg (t, z0 )
−3
−5
Justin Teo (ACL, MIT)
−4
−3
−2
−1
0
x
1
2
Gradient Projection Anti-windup Scheme
3
4
5
Dec. 20, 2010
22 / 35
Input Constrained Planar LTI Systems
Region of Attraction Containment
Numerical Results III, Rn ⊂ Rg
Stable plant, unstable controller, umax = −umin
3
Rn
2
Rg
u
1
0
−1
φn (t, z0 )
−2
φg (t, z0 )
−3
−5
Justin Teo (ACL, MIT)
−4
−3
−2
−1
0
x
1
2
Gradient Projection Anti-windup Scheme
3
4
5
Dec. 20, 2010
22 / 35
Input Constrained Planar LTI Systems
Region of Attraction Containment
Numerical Results III, Rn ⊂ Rg
Stable plant, unstable controller, umax = −umin
3
Rn
2
Rg
u
1
0
−1
φn (t, z0 )
−2
φg (t, z0 )
−3
−5
Justin Teo (ACL, MIT)
−4
−3
−2
−1
0
x
1
2
Gradient Projection Anti-windup Scheme
3
4
5
Dec. 20, 2010
22 / 35
Input Constrained Planar LTI Systems
Region of Attraction Containment
Numerical Results III, Rn ⊂ Rg
Stable plant, unstable controller, umax = −umin
3
Rn
2
Rg
u
1
0
−1
φn (t, z0 )
−2
φg (t, z0 )
−3
−5
Justin Teo (ACL, MIT)
−4
−3
−2
−1
0
x
1
2
Gradient Projection Anti-windup Scheme
3
4
5
Dec. 20, 2010
22 / 35
Input Constrained Planar LTI Systems
Region of Attraction Containment
Numerical Results IV, Rn ⊂ Rg
Unstable plant, stable controller
Symmetric constraints
Asymmetric constraints
5
5
φn (t, z0 )
4
φn (t, z0 )
4
φg (t, z0 )
φg (t, z0 )
3
3
2
2
1
Rn = Rg
0
u
u
1
Rg
0
−1
−1
−2
−2
−3
−3
−4
−4
−5
−2.5
−5
−2.5
−2
−1.5
−1
−0.5
0
x
0.5
1
1.5
2
2.5
Rn
−2
−1.5
−1
−0.5
0
x
0.5
1
1.5
2
2.5
Stable plant, unstable controller
3
Rn
2
Rg
u
1
0
−1
φn (t, z0 )
−2
φg (t, z0 )
−3
−5
Justin Teo (ACL, MIT)
−4
−3
−2
−1
0
x
1
2
3
4
5
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
23 / 35
Input Constrained Planar LTI Systems
New Paradigm for Anti-windup Problem
New Paradigm for Anti-windup Problem
Claim (Global Asymptotic Stability of Nominal System)
If both open-loop plant and nominal controller are marginally or strictly
stable, then the origin of Σn is globally asymptotically stable (GAS) and
locally exponentially stable (LES), i.e. Rn = R2
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
24 / 35
Input Constrained Planar LTI Systems
New Paradigm for Anti-windup Problem
New Paradigm for Anti-windup Problem
Claim (Global Asymptotic Stability of Nominal System)
If both open-loop plant and nominal controller are marginally or strictly
stable, then the origin of Σn is globally asymptotically stable (GAS) and
locally exponentially stable (LES), i.e. Rn = R2
Corollary (Global Asymptotic Stability of GPAW System)
If both open-loop plant and nominal controller are marginally or strictly
stable, then the origin of Σg is GAS and LES (Rg ⊃ Rn = R2 )
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
24 / 35
Input Constrained Planar LTI Systems
New Paradigm for Anti-windup Problem
New Paradigm for Anti-windup Problem
Claim (Global Asymptotic Stability of Nominal System)
If both open-loop plant and nominal controller are marginally or strictly
stable, then the origin of Σn is globally asymptotically stable (GAS) and
locally exponentially stable (LES), i.e. Rn = R2
Corollary (Global Asymptotic Stability of GPAW System)
If both open-loop plant and nominal controller are marginally or strictly
stable, then the origin of Σg is GAS and LES (Rg ⊃ Rn = R2 )
Some anti-windup results are of the form of preceding Corollary
Such results tells nothing about advantages of anti-windup method
Same result obtained as Corollary of ROA containment
ROA containment result shows true advantage
Propose new paradigm to search for relative results
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
24 / 35
Input Constrained Planar LTI Systems
Need to Consider Asymmetric Saturation Constraints
Need to Consider Asymmetric Constraints
Conjecture (Relaxing Constraints Imply ROA Enlargement)
Let Rn1 be ROA for some saturation limits umin1 , umax1 , and Rn2 be ROA
for umin2 , umax2 . If [umin1 , umax1 ] ⊂ [umin2 , umax2 ], then Rn1 ⊂ Rn2
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
25 / 35
Input Constrained Planar LTI Systems
Need to Consider Asymmetric Saturation Constraints
Need to Consider Asymmetric Constraints
Conjecture (Relaxing Constraints Imply ROA Enlargement)
Let Rn1 be ROA for some saturation limits umin1 , umax1 , and Rn2 be ROA
for umin2 , umax2 . If [umin1 , umax1 ] ⊂ [umin2 , umax2 ], then Rn1 ⊂ Rn2
Conjecture intuitively appealing, but WRONG!
Symmetric
Asymmetric
5
5
φn (t, z0 )
4
φn (t, z0 )
4
φg (t, z0 )
φg (t, z0 )
3
3
2
2
1
Rn = Rg
0
u
u
1
−1
−2
−2
−3
−3
−4
−4
−5
−2.5
−2
−1.5
−1
−0.5
0
x
0.5
1
1.5
umax = −umin = 1
2
2.5
−5
−2.5
Not pathological
Rg
0
−1
Rn
−2
−1.5
−1
−0.5
0
x
0.5
1
1.5
2
2.5
umax = 1.5, umin = −1
Need to consider asymmetric saturation constraints
Most literature (less [Hu et al. 2002]) considers only symmetric constraints
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
25 / 35
An ROA Comparison Result
Outline
1
Introduction
2
GPAW Compensated Controller
3
Input Constrained Planar LTI Systems
4
An ROA Comparison Result
5
A Numerical Comparison
6
Conclusions
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
26 / 35
An ROA Comparison Result
An ROA Comparison Result
Plant, Σp
ẋ = f (x, sat(u))
Nominal Controller, Σc
ẋc = fc (xc , y)
y = g(x, sat(u))
u = gc (xc )
Justin Teo (ACL, MIT)
GPAW Controller, Σgpaw
ẋg = RI ∗ fc (xg , y)
Gradient Projection Anti-windup Scheme
u = gc (xg )
Dec. 20, 2010
27 / 35
An ROA Comparison Result
An ROA Comparison Result
Plant, Σp
ẋ = f (x, sat(u))
Nominal Controller, Σc
ẋc = fc (xc , y)
GPAW Controller, Σgpaw
ẋg = RI ∗ fc (xg , y)
y = g(x, sat(u))
u = gc (xc )
u = gc (xg )
Assume zeq ∈ K \ ∂K is asymptotically stable equilibrium for Σn and Σg
Nominal System:
Σn : Σ p + Σ c
ROA: Rn (zeq )
GPAW System: Σg : Σp + Σgpaw ROA: Rg (zeq )
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
27 / 35
An ROA Comparison Result
An ROA Comparison Result
Plant, Σp
ẋ = f (x, sat(u))
Nominal Controller, Σc
ẋc = fc (xc , y)
GPAW Controller, Σgpaw
ẋg = RI ∗ fc (xg , y)
y = g(x, sat(u))
u = gc (xc )
u = gc (xg )
Assume zeq ∈ K \ ∂K is asymptotically stable equilibrium for Σn and Σg
Nominal System:
Σn : Σ p + Σ c
ROA: Rn (zeq )
GPAW System: Σg : Σp + Σgpaw ROA: Rg (zeq )
ROA estimate: ΩV = {z̄ | V (z̄) ≤ c} ⊂ Rn (zeq ) for some Lyapunov
function V (z) = V (x, xc ) of Σn
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
27 / 35
An ROA Comparison Result
An ROA Comparison Result
Plant, Σp
ẋ = f (x, sat(u))
Nominal Controller, Σc
ẋc = fc (xc , y)
GPAW Controller, Σgpaw
ẋg = RI ∗ fc (xg , y)
y = g(x, sat(u))
u = gc (xc )
u = gc (xg )
Assume zeq ∈ K \ ∂K is asymptotically stable equilibrium for Σn and Σg
Nominal System:
Σn : Σ p + Σ c
ROA: Rn (zeq )
GPAW System: Σg : Σp + Σgpaw ROA: Rg (zeq )
ROA estimate: ΩV = {z̄ | V (z̄) ≤ c} ⊂ Rn (zeq ) for some Lyapunov
function V (z) = V (x, xc ) of Σn
Theorem (ROA Bounds for GPAW Compensated System)
If there exists a Γ = ΓT > 0 such that
∂V (x̄, x̄c )
∂V (x̄, x̄c )
RI ∗ fc ≤
fc ,
∂xc
∂xc
∀(x̄, x̄c ) ∈ ΩV ∩ (Rn × K)
then Σg with Γ has ROA satisfying (ΩV ∩ (Rn × K)) ⊂ Rg (zeq )
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
27 / 35
An ROA Comparison Result
ROA Comparison Indicates True Advantage
Existing anti-windup results are in “absolute” sense
may not indicate any advantages of anti-windup scheme
ROA comparison result is in “relative” sense
directly shows advantage of GPAW scheme
first in new anti-windup paradigm
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
28 / 35
An ROA Comparison Result
ROA Comparison Indicates True Advantage
Existing anti-windup results are in “absolute” sense
may not indicate any advantages of anti-windup scheme
ROA comparison result is in “relative” sense
directly shows advantage of GPAW scheme
first in new anti-windup paradigm
States loosely that ROA of Σg is not less than ROA estimate ΩV
Applies for asymmetric saturation constraints
Specialized with additional assumptions (e.g. LTI)
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
28 / 35
An ROA Comparison Result
ROA Comparison Indicates True Advantage
Existing anti-windup results are in “absolute” sense
may not indicate any advantages of anti-windup scheme
ROA comparison result is in “relative” sense
directly shows advantage of GPAW scheme
first in new anti-windup paradigm
States loosely that ROA of Σg is not less than ROA estimate ΩV
Applies for asymmetric saturation constraints
Specialized with additional assumptions (e.g. LTI)
Main condition:
∂V (x̄,x̄c )
∗
∂xc RI fc
≤
∂V (x̄,x̄c )
∂xc fc
independent of sat(·)
Can be used in two ways: comparison against ROA estimate of
unconstrained system or nominal system
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
28 / 35
An ROA Comparison Result
Application of ROA Comparison Result
Example nonlinear planar system [Khalil 2002]

(
ẋ = − sat(u)


(
ẋ = − sat(u)
Σn :
Σ
:
0,
if A1
gs

u̇ = x + (x2 − 1)u
 u̇ =
2
x + (x − 1)u, otherwise
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
29 / 35
An ROA Comparison Result
Application of ROA Comparison Result
Example nonlinear planar system [Khalil 2002]

(
ẋ = −u


(
ẋ = −u
Σu :
Σ
:
0,
if A1
g

u̇ = x + (x2 − 1)u
 u̇ =
2
x + (x − 1)u, otherwise
Compare with ROA estimate ΩV of unconstrained system:
4
3
uncompensated
GPAW
2
x
Ru (zeq )
2
0
1
u
ΩV
−2
0
2
0
1
2
3
1
2
3
4
5
6
7
8
9
10
4
5
6
7
8
9
10
1
u
−1
−2
−3
−3
Rg (zeq )
−2
uncompensated
GPAW
−1
Justin Teo (ACL, MIT)
0
x
1
2
0
−1
−2
0
3
Gradient Projection Anti-windup Scheme
time (s)
Dec. 20, 2010
29 / 35
An ROA Comparison Result
Application of ROA Comparison Result
Example nonlinear planar system [Khalil 2002]

(
ẋ = −u


(
ẋ = −u
Σu :
Σ
:
0,
if A1
g

u̇ = x + (x2 − 1)u
 u̇ =
2
x + (x − 1)u, otherwise
Compare with ROA estimate ΩV of unconstrained system:
4
3
uncompensated
GPAW
2
x
Ru (zeq )
2
0
1
u
ΩV
−2
0
2
0
1
2
3
1
2
3
4
5
6
7
8
9
10
4
5
6
7
8
9
10
1
u
−1
−2
−3
−3
Rg (zeq )
−2
uncompensated
GPAW
−1
0
x
1
2
0
−1
−2
0
3
time (s)
Toy example defeats methods for LTI systems, feedback linearizable
systems, and nonlinear anti-windup [Morabito et al. 2004]
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
29 / 35
A Numerical Comparison
Outline
1
Introduction
2
GPAW Compensated Controller
3
Input Constrained Planar LTI Systems
4
An ROA Comparison Result
5
A Numerical Comparison
6
Conclusions
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
30 / 35
A Numerical Comparison
Numerical Comparison with Robot Example
ROA comparison and stability results still too conservative
Compare GPAW vs [Yoon et al. 2008] (feedback linearizable systems)
vs [Morabito et al. 2004] (nonlinear anti-windup) without stability
guarantees
Feedback linearizable nonlinear plant with disturbance input w [Yoon
et al. 2008]:
(
#
"
x2
ẋ1
Σp : ẋ =
= −10x1 −0.1x31 −48.54x2 −w+sat(u) ,
y=x
ẋ2
6.67(1+0.1 sin x )
1
Feedback linearizing PID controller: Σc
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
31 / 35
A Numerical Comparison
Numerical Comparison with Robot Example
ROA comparison and stability results still too conservative
Compare GPAW vs [Yoon et al. 2008] (feedback linearizable systems)
vs [Morabito et al. 2004] (nonlinear anti-windup) without stability
guarantees
Feedback linearizable nonlinear plant with disturbance input w [Yoon
et al. 2008]:
(
#
"
x2
ẋ1
Σp : ẋ =
= −10x1 −0.1x31 −48.54x2 −w+sat(u) ,
y=x
ẋ2
6.67(1+0.1 sin x )
1
Feedback linearizing PID controller: Σc
Nominal system:
Feedback linearized AW System [Yoon et al. 2008]:
Nonlinear AW system [Morabito et al. 2004]:
GPAW compensated system:
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
NS: Σp + Σc
FL
NAW
GPAW
Dec. 20, 2010
31 / 35
A Numerical Comparison
disturbance
GPAW Achieves Comparable Performance
200
100
0
−100
−200
0
2
4
6
2
4
6
8
10
12
14
8
10
12
14
output
1.5
NS
FL
16 NAW
18
GPAW
20
1
0.5
0
0
16
18
20
time (s)
GPAW achieves comparable performance with state-of-the-art methods
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
32 / 35
Conclusions
Outline
1
Introduction
2
GPAW Compensated Controller
3
Input Constrained Planar LTI Systems
4
An ROA Comparison Result
5
A Numerical Comparison
6
Conclusions
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
33 / 35
Conclusions
GPAW in Context
In context (some dates from [Tarbouriech and Turner 2009]):
problem as old as control theory itself (James Watt’s governor - 1788)
windup problem recognized (1930s)
ad-hoc schemes devised and adopted (LTI) (1930s)
academic studies (1950s)
provably stable “modern” anti-windup schemes (LTI) (late 1990s)
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
34 / 35
Conclusions
GPAW in Context
In context (some dates from [Tarbouriech and Turner 2009]):
problem as old as control theory itself (James Watt’s governor - 1788)
windup problem recognized (1930s)
ad-hoc schemes devised and adopted (LTI) (1930s)
academic studies (1950s)
provably stable “modern” anti-windup schemes (LTI) (late 1990s)
provably stable classes of nonlinear systems (mid 2000s)
provably stable general nonlinear systems (GPAW - 2010)
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
34 / 35
Conclusions
GPAW in Context
In context (some dates from [Tarbouriech and Turner 2009]):
problem as old as control theory itself (James Watt’s governor - 1788)
windup problem recognized (1930s)
ad-hoc schemes devised and adopted (LTI) (1930s)
academic studies (1950s)
provably stable “modern” anti-windup schemes (LTI) (late 1990s)
provably stable classes of nonlinear systems (mid 2000s)
provably stable general nonlinear systems (GPAW - 2010)
less conservative stability results (???)
Future work (partial list):
search for less conservative stability results
consider robustness issues due to presence of noise, disturbances, time
delays, and unmodeled dynamics
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
34 / 35
Conclusions
Conclusions
Contributions of this research include:
developed general purpose anti-windup scheme
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
35 / 35
Conclusions
Conclusions
Contributions of this research include:
developed general purpose anti-windup scheme
motivated new paradigm for anti-windup problem
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
35 / 35
Conclusions
Conclusions
Contributions of this research include:
developed general purpose anti-windup scheme
motivated new paradigm for anti-windup problem
demonstrated need to consider asymmetric saturation constraints for
general saturated systems
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
35 / 35
Conclusions
Conclusions
Contributions of this research include:
developed general purpose anti-windup scheme
motivated new paradigm for anti-windup problem
demonstrated need to consider asymmetric saturation constraints for
general saturated systems
developed region of attraction (ROA) comparison and stability results
for GPAW compensated (nonlinear) systems
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
35 / 35
Conclusions
Conclusions
Contributions of this research include:
developed general purpose anti-windup scheme
motivated new paradigm for anti-windup problem
demonstrated need to consider asymmetric saturation constraints for
general saturated systems
developed region of attraction (ROA) comparison and stability results
for GPAW compensated (nonlinear) systems
demonstrated viability of GPAW scheme as a candidate anti-windup
scheme for general systems
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
35 / 35
Conclusions
Conclusions
Contributions of this research include:
developed general purpose anti-windup scheme
motivated new paradigm for anti-windup problem
demonstrated need to consider asymmetric saturation constraints for
general saturated systems
developed region of attraction (ROA) comparison and stability results
for GPAW compensated (nonlinear) systems
demonstrated viability of GPAW scheme as a candidate anti-windup
scheme for general systems
related GPAW compensated systems to projected dynamical systems
and linear systems with partial state constraints
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
35 / 35
Conclusions
Conclusions
Contributions of this research include:
developed general purpose anti-windup scheme
motivated new paradigm for anti-windup problem
demonstrated need to consider asymmetric saturation constraints for
general saturated systems
developed region of attraction (ROA) comparison and stability results
for GPAW compensated (nonlinear) systems
demonstrated viability of GPAW scheme as a candidate anti-windup
scheme for general systems
related GPAW compensated systems to projected dynamical systems
and linear systems with partial state constraints
Questions?
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
35 / 35
Backup Slides
Backup Slides
Backup slides
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
36 / 35
Backup Slides
Dissertation Overview
Dissertation Overview
Covered Chapter 1, Introduction. Dissertation on gradient projection
anti-windup (GPAW) scheme. Remaining chapters:
Chapter 2 Construction and Fundamental Properties
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
37 / 35
Backup Slides
Dissertation Overview
Dissertation Overview
Covered Chapter 1, Introduction. Dissertation on gradient projection
anti-windup (GPAW) scheme. Remaining chapters:
Chapter 2 Construction and Fundamental Properties
Chapter 3 Input Constrained Planar LTI Systems
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
37 / 35
Backup Slides
Dissertation Overview
Dissertation Overview
Covered Chapter 1, Introduction. Dissertation on gradient projection
anti-windup (GPAW) scheme. Remaining chapters:
Chapter 2 Construction and Fundamental Properties
Chapter 3 Input Constrained Planar LTI Systems
Chapter 4 Geometric Properties and Region of Attraction Comparison
Results
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
37 / 35
Backup Slides
Dissertation Overview
Dissertation Overview
Covered Chapter 1, Introduction. Dissertation on gradient projection
anti-windup (GPAW) scheme. Remaining chapters:
Chapter 2 Construction and Fundamental Properties
Chapter 3 Input Constrained Planar LTI Systems
Chapter 4 Geometric Properties and Region of Attraction Comparison
Results
Chapter 5 Input Constrained MIMO LTI Systems
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
37 / 35
Backup Slides
Dissertation Overview
Dissertation Overview
Covered Chapter 1, Introduction. Dissertation on gradient projection
anti-windup (GPAW) scheme. Remaining chapters:
Chapter 2 Construction and Fundamental Properties
Chapter 3 Input Constrained Planar LTI Systems
Chapter 4 Geometric Properties and Region of Attraction Comparison
Results
Chapter 5 Input Constrained MIMO LTI Systems
Chapter 6 Numerical Comparisons
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
37 / 35
Backup Slides
Dissertation Overview
Dissertation Overview
Covered Chapter 1, Introduction. Dissertation on gradient projection
anti-windup (GPAW) scheme. Remaining chapters:
Chapter 2 Construction and Fundamental Properties
Chapter 3 Input Constrained Planar LTI Systems
Chapter 4 Geometric Properties and Region of Attraction Comparison
Results
Chapter 5 Input Constrained MIMO LTI Systems
Chapter 6 Numerical Comparisons
Chapter 7 Conclusions and Future Work
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
37 / 35
Backup Slides
Dissertation Overview
Dissertation Overview
Covered Chapter 1, Introduction. Dissertation on gradient projection
anti-windup (GPAW) scheme. Remaining chapters:
Chapter 2 Construction and Fundamental Properties
Chapter 3 Input Constrained Planar LTI Systems
Chapter 4 Geometric Properties and Region of Attraction Comparison
Results
Chapter 5 Input Constrained MIMO LTI Systems
Chapter 6 Numerical Comparisons
Chapter 7 Conclusions and Future Work
Appendix A Closed Form Expressions for Single-output GPAW
Compensated Controllers
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
37 / 35
Backup Slides
Dissertation Overview
Dissertation Overview
Covered Chapter 1, Introduction. Dissertation on gradient projection
anti-windup (GPAW) scheme. Remaining chapters:
Chapter 2 Construction and Fundamental Properties
Chapter 3 Input Constrained Planar LTI Systems
Chapter 4 Geometric Properties and Region of Attraction Comparison
Results
Chapter 5 Input Constrained MIMO LTI Systems
Chapter 6 Numerical Comparisons
Chapter 7 Conclusions and Future Work
Appendix A Closed Form Expressions for Single-output GPAW
Compensated Controllers
Appendix B Closed Form Expressions for GPAW Compensated Controllers
with Output of Dimension Two
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
37 / 35
Backup Slides
Dissertation Overview
Dissertation Overview
Covered Chapter 1, Introduction. Dissertation on gradient projection
anti-windup (GPAW) scheme. Remaining chapters:
Chapter 2 Construction and Fundamental Properties
Chapter 3 Input Constrained Planar LTI Systems
Chapter 4 Geometric Properties and Region of Attraction Comparison
Results
Chapter 5 Input Constrained MIMO LTI Systems
Chapter 6 Numerical Comparisons
Chapter 7 Conclusions and Future Work
Appendix A Closed Form Expressions for Single-output GPAW
Compensated Controllers
Appendix B Closed Form Expressions for GPAW Compensated Controllers
with Output of Dimension Two
Appendix C Procedure to Apply GPAW Compensation
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
37 / 35
Backup Slides
Application on Nonlinear Two-link Robot
Application on Nonlinear Two-link Robot
Two-link robot (plant):
Σplant : H(xt )ẍt + C(xt , ẋt )ẋt = sat(u)
x2
Adaptive sliding-mode (nominal) controller:
â˙ = −ΘY T s
feedback
−−−−−→
Σn
Σplant
uc = Y â − KD s
Approximate nominal controller:
x̃˙ c = −ΘY T s
ẋaug = a(z(y, r) − xaug )
uc = Ŷ (xaug )x̃c − KD ŝ(xaug )



x1
(
≡


ẋc = fc (xc , y, r)
uc = gc (xc )
GPAW compensated controller:
ẋg = RI ∗ fc (xg , y, r)
ug = gc (xg )
Justin Teo (ACL, MIT)
feedback
−−−−−→
Σplant
Gradient Projection Anti-windup Scheme
Σg
Movies
Dec. 20, 2010
38 / 35
Backup Slides
Passivity Properties
Passivity Properties
Decompose Γ = ΦΦT , define:
PI (xg ) := Φ−1 RI (xg )Φ
SI (xg ) := I − PI (xg )
Passivity and L2 -gain of Projection Operators
PI ∗ (xg , y, r) and SI ∗ (xg , y, r) are passive and with L2 -gain less than 1
r
RI ∗
fc (xg , y, r)
ṽ
Φ−1
PI ∗
Φ
w̃
ẋg = w̃
u
u = gc (xg )
xg
y
ẋ = f (x, ũ)
y = g(x, ũ)
ũ
sat(u)
GPAW modifies uncompensated system with passive operator
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
39 / 35
Backup Slides
Passivity Properties
Passivity Properties
Decompose Γ = ΦΦT , define:
PI (xg ) := Φ−1 RI (xg )Φ
SI (xg ) := I − PI (xg )
Passivity and L2 -gain of Projection Operators
PI ∗ (xg , y, r) and SI ∗ (xg , y, r) are passive and with L2 -gain less than 1
r
fc (xg , y, r)
ṽ
Φ−1
v
PI ∗
w
Φ
w̃
ẋg = w̃
u
u = gc (xg )
xg
y
ẋ = f (x, ũ)
y = g(x, ũ)
ũ
sat(u)
GPAW modifies uncompensated system with passive operator
Can derive passivity and small-gain based stability results
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
39 / 35
Backup Slides
Geometric Properties
Geometric Bounding Condition
Let K be unsaturated region,
K = {x̄ | sat(gc (x̄)) = gc (x̄)}
Let fc (x, y, r), fg (x, y, r) = RI ∗ fc (x, y, r)
be the vector fields of nominal and GPAW
compensated controllers
fc1 fg1
x
ker(K)
fc2 f
g2
xker
K
Let Γ = ΓT > 0 be the GPAW parameter
Theorem (Geometric Bounding Condition)
If unsaturated region K is a star domain, then for any x ∈ K and any
xker ∈ ker(K),
hΓ−1 (x − xker ), fg (x, y, r)i ≤ hΓ−1 (x − xker ), fc (x, y, r)i
holds for all (y, r) and all Γ = ΓT > 0
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
40 / 35
Backup Slides
Geometric Properties
Star Domains
Examples and counterexamples of star domains in R2 :
Star, ker(Xi ) 6= ∅
NOT Star, ker(Xi ) = ∅
X6 = Y1 ∪ Y2
ker(X3 )
X4
ker(X1 )
X1
ker(X2 )
ker(X4 )
Y1
Y2
X4
X7
X2
X5
Any convex set X is also a star domain with ker(X) = X
For any non-convex star domain, ker(X) is a strict subset of X
If X is a star domain, then Rn × X is also a star domain with kernel
ker(Rn × X) = Rn × ker(X)
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
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Backup Slides
Geometric Properties
Geometric Interpretation
hΓ−1 (x − xker ), fg (x, y, r)i ≤ hΓ−1 (x − xker ), fc (x, y, r)i
fc1
pg
2
x
p c2
1
Justin Teo (ACL, MIT)
pg
GPAW controller:
fg = RI ∗ fc
p c1
Nominal controller:
fc
fg1
ker(K)
fc2
xker
fg2
Gradient Projection Anti-windup Scheme
K
Dec. 20, 2010
42 / 35
Backup Slides
Geometric Properties
GPAW in Context
Standard anti-windup structure:
r
Σ̃c
ũ
u
sat(u)
v
yaw1
Σ̃aw
y
y = Cx + Dv
−
yaw2
ẋ = Ax + Bv
Unconstrained plant
w
Anti-windup compensated controller
Virtually all anti-windup schemes are variants of above
GPAW scheme has additional “built-in” features
GPAW has single parameter, only for “fine tuning”
GPAW alone comparable to three state-of-the-art methods
GPAW has potential to be developed into truly general purpose
anti-windup scheme with better stability guarantees
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
43 / 35
Backup Slides
Geometric Properties
Conclusions
Anti-windup compensation for nonlinear systems is an open problem
Developed GPAW scheme, a general purpose anti-windup scheme:
achieves controller state-output consistency
several ways to realize
defined by passive operator
has clear geometric properties
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
44 / 35
Backup Slides
Geometric Properties
Conclusions
Anti-windup compensation for nonlinear systems is an open problem
Developed GPAW scheme, a general purpose anti-windup scheme:
achieves controller state-output consistency
several ways to realize
defined by passive operator
has clear geometric properties
Strong results for planar LTI systems:
ROA containment result independent of any Lyapunov function
shows qualitative weaknesses of existing results
motivated new anti-windup paradigm to search for “relative” results
shows need to consider asymmetric saturation constraints
establish link to projected dynamical systems
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
44 / 35
Backup Slides
Geometric Properties
Conclusions
Anti-windup compensation for nonlinear systems is an open problem
Developed GPAW scheme, a general purpose anti-windup scheme:
achieves controller state-output consistency
several ways to realize
defined by passive operator
has clear geometric properties
Strong results for planar LTI systems:
ROA containment result independent of any Lyapunov function
shows qualitative weaknesses of existing results
motivated new anti-windup paradigm to search for “relative” results
shows need to consider asymmetric saturation constraints
establish link to projected dynamical systems
Derived ROA comparison and stability results - first results to directly
indicate advantages of anti-windup
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
44 / 35
Backup Slides
Geometric Properties
Conclusions
Anti-windup compensation for nonlinear systems is an open problem
Developed GPAW scheme, a general purpose anti-windup scheme:
achieves controller state-output consistency
several ways to realize
defined by passive operator
has clear geometric properties
Strong results for planar LTI systems:
ROA containment result independent of any Lyapunov function
shows qualitative weaknesses of existing results
motivated new anti-windup paradigm to search for “relative” results
shows need to consider asymmetric saturation constraints
establish link to projected dynamical systems
Derived ROA comparison and stability results - first results to directly
indicate advantages of anti-windup
Even without stability proofs, ad-hoc methods can be used to design
GPAW controller yielding comparable performance with
state-of-the-art anti-windup methods
Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
44 / 35
Backup Slides
References
References I
D. S. Bernstein and A. N. Michel. A chronological bibliography on saturating actuators. Int. J. Robust Nonlinear Control, 5(5):
375 – 380, 1995. doi: 10.1002/rnc.4590050502.
P. Butterworth-Hayes. Gripen crash raises canard fears. Aerosp. Am., 32(2):10 – 11, Feb. 1994.
H. M. Do, T. Başar, and J. Y. Choi. An anti-windup design for single input adaptive control systems in strict feedback form. In
Proc. American Control Conf., volume 3, pages 2551 – 2556, Boston, MA, June/July 2004.
M. A. Dornheim. Report pinpoints factors leading to YF-22 crash. Aviat. Week Space Technol., 137(19):53 – 54, Nov. 1992.
P. Dupuis and A. Nagurney. Dynamical systems and variational inequalities. Ann. Oper. Res., 44(1):7 – 42, Feb. 1993. doi:
10.1007/BF02073589.
C. Edwards and I. Postlethwaite. Anti-windup and bumpless-transfer schemes. Automatica, 34(2):199 – 210, Feb. 1998. doi:
10.1016/S0005-1098(97)00165-9.
H. A. Fertik and C. W. Ross. Direct digital control algorithm with anti-windup feature. ISA Trans., 6(4):317 – 328, 1967.
E. Gilbert and I. Kolmanovsky. Nonlinear tracking control in the presence of state and control constraints: a generalized
reference governor. Automatica, 38(12):2063 – 2073, Dec. 2002. doi: 10.1016/S0005-1098(02)00135-8.
R. Hanus, M. Kinnaert, and J.-L. Henrotte. Conditioning technique, a general anti-windup and bumpless transfer method.
Automatica, 23(6):729 – 739, Nov. 1987. doi: 10.1016/0005-1098(87)90029-X.
Q. Hu and G. P. Rangaiah. Anti-windup schemes for uncertain nonlinear systems. IET Control Theory Appl., 147(3):321 – 329,
May 2000. doi: 10.1049/ip-cta:20000136.
T. Hu, A. N. Pitsillides, and Z. Lin. Null controllability and stabilization of linear systems subject to asymmetric actuator
saturation. In V. Kapila and K. M. Grigoriadis, editors, Actuator Saturation Control, Control Eng., chapter 3, pages 47 – 76.
Marcel Dekker, New York, NY, 2002.
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Backup Slides
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Justin Teo (ACL, MIT)
Gradient Projection Anti-windup Scheme
Dec. 20, 2010
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