Gradient Projection Anti-windup Schemeacl.mit.edu/papers/TeoMITScD11_slides.pdf · Anti-windup...
Transcript of Gradient Projection Anti-windup Schemeacl.mit.edu/papers/TeoMITScD11_slides.pdf · Anti-windup...
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Gradient Projection Anti-windup SchemeThesis Defense
Justin Teo (MIT Aero/Astro)
Thesis Committee: Jonathan P. How (Chair) (MIT Aero/Astro)Emilio Frazzoli (MIT Aero/Astro)Steven R. Hall (MIT Aero/Astro)Eugene Lavretsky (Boeing)
Thesis Readers: Luca F. Bertuccelli (MIT Aero/Astro)Louis Breger (Draper)
Department Representative: Wesley L. Harris (MIT Aero/Astro)
December 20, 2010
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme Dec. 20, 2010 1 / 35
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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
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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)
Mild effects [Visioli 2006]:
sluggish response
large overshoots
long settling times
Severe effects: instability
0 2 4 6 8 10 12 14 16 18 20−2
0
2
4
x
Stable Plant, Unstable Controller
0 2 4 6 8 10 12 14 16 18 20−3
−2
−1
0
1
2
umax
umin
u
time (s)
unconstrained
saturated
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme Dec. 20, 2010 3 / 35
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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)
Mild effects [Visioli 2006]:
sluggish response
large overshoots
long settling times
Severe effects: instability
0 2 4 6 8 10 12 14 16 18 20−2
0
2
4
x
Stable Plant, Unstable Controller
0 2 4 6 8 10 12 14 16 18 20−3
−2
−1
0
1
2
umax
umin
u
time (s)
unconstrained
saturated
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme Dec. 20, 2010 3 / 35
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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)
Mild effects [Visioli 2006]:
sluggish response
large overshoots
long settling times
Severe effects: instability
0 2 4 6 8 10 12 14 16 18 20−2
0
2
4
x
Stable Plant, Unstable Controller
0 2 4 6 8 10 12 14 16 18 20−3
−2
−1
0
1
2
umax
umin
u
time (s)
unconstrained
saturated
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme Dec. 20, 2010 3 / 35
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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 fighteraircraft [Butterworth-Hayes 1994, Stein 2003]
1992 crash of YF-22 1989 and 1993 crashes of Saab Gripen
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme Dec. 20, 2010 4 / 35
YF22_1992_crash.mpgMedia File (video/mpeg)
Gripen_1989_1993_crashes.mpgMedia File (video/mpeg)
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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 approachaccounts for saturation in design of nominal controller - complexoften conservative and hard to tune [Tarbouriech and Turner 2009,Sofrony et al. 2006, Mulder et al. 2009]
two-step approach or anti-windup compensationignores 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 [Tarbouriechand 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
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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 approachaccounts for saturation in design of nominal controller - complexoften conservative and hard to tune [Tarbouriech and Turner 2009,Sofrony et al. 2006, Mulder et al. 2009]
two-step approach or anti-windup compensationignores 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 [Tarbouriechand 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
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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 approachaccounts for saturation in design of nominal controller - complexoften conservative and hard to tune [Tarbouriech and Turner 2009,Sofrony et al. 2006, Mulder et al. 2009]
two-step approach or anti-windup compensationignores 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 [Tarbouriechand 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
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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, Tarbouriechand Turner 2009]
sat(u)u ẋ = Ax + Bv
y = Cx + Dv
Unconstrained plant
Σ̃cũ
Σ̃aw
r v y
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
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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, Tarbouriechand Turner 2009]
sat(u)u ẋ = Ax + Bv
y = Cx + Dv
Unconstrained plant
Σ̃cũ
Σ̃aw
r v
−
w
yaw1
yaw2
y
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
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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, Tarbouriechand Turner 2009]
sat(u)u ẋ = Ax + Bv
y = Cx + Dv
Unconstrained plant
Σ̃cũ
Σ̃aw
r v
−
w
yaw1
yaw2
y
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 approximationsJustin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme Dec. 20, 2010 6 / 35
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Introduction Problem Statement
Problem Statement
Saturated Plant:
Σp :
{ẋ = f(x, sat(u))
y = g(x, sat(u))
Nominal Controller:
Σc :
{ẋc = fc(xc, y, r)
u = gc(xc, y, r)
AW Compensated Controller:
Σaw :
{ẋaw = faw(xaw, y, r)
u = gaw(xaw, y, r)
Nominal system Σn: feedbackinterconnection (FI) of Σp, Σc
Anti-windup (AW) compensatedsystem Σaws: FI of Σp, Σaw
General Anti-windup Problem
Design Σaw and determined initializationxaw(0) such that Σaws satisfies:
when no controls saturate for Σn,then nominal performancerecovered, i.e. Σaws ≡ Σnwhen some controls saturate,stability and performance of Σaws isno worse than that of Σn
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme Dec. 20, 2010 7 / 35
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Introduction Literature Review
Literature Review
Anti-windup methods (partial citations) applicable to nonlinear systems:
Conditioning Technique [Hanus et al. 1987] - computationallyprohibitive and severely limited for nonlinear systems
Feedback Linearizable Nonlinear Systems [Yoon et al. 2008] - requiresfeedback linearizable plant and feedback linearizing controller
For some Particular Controllers [Hu and Rangaiah 2000, Johnson andCalise 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] - someconservatism introduced
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme Dec. 20, 2010 8 / 35
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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 forgeneral saturated systems
developed region of attraction (ROA) comparison and stability resultsfor GPAW compensated (nonlinear) systems
demonstrated viability of GPAW scheme as a candidate anti-windupscheme for general systems
related GPAW compensated systems to projected dynamical systemsand linear systems with partial state constraints
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme Dec. 20, 2010 9 / 35
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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 forgeneral saturated systems
developed region of attraction (ROA) comparison and stability resultsfor GPAW compensated (nonlinear) systems
demonstrated viability of GPAW scheme as a candidate anti-windupscheme for general systems
related GPAW compensated systems to projected dynamical systemsand linear systems with partial state constraints
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme Dec. 20, 2010 9 / 35
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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
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GPAW Compensated Controller Conditional Integration
Conditional Integration
Conditional integration (CI) for PID controllers [Fertik and Ross 1967]
ėi = e
u = Kpe+Kiei +Kdė
CI−→ėi =
0, if u ≥ umax ∧ e > 00, if u ≤ umin ∧ e < 0e, otherwise
u = Kpe+Kiei +Kdė
Stop integration when nominal update will aggravate saturationconstraints, or stop integration when departing unsaturated region
K(e, ė) = {ēi ∈ R | sat(Kpe+Kiēi +Kdė) = Kpe+Kiēi +Kdė}Attempts to achieve controller state-output consistency sat(u) = u
Extends easily to decoupled nonlinear controllersẋci = fci(xci, y, r)
uci = gci(xci, y, r)
For coupled nonlinear controllers, need projection op-erator - project onto K(e, ė) analogue
ẋc = fc(xc, y, r)
uc = gc(xc, y, r)
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme Dec. 20, 2010 11 / 35
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GPAW Compensated Controller Conditional Integration
Conditional Integration
Conditional integration (CI) for PID controllers [Fertik and Ross 1967]
ėi = e
u = Kpe+Kiei +Kdė
CI−→ėi =
0, if u ≥ umax ∧ e > 00, if u ≤ umin ∧ e < 0e, otherwise
u = Kpe+Kiei +Kdė
Stop integration when nominal update will aggravate saturationconstraints, or stop integration when departing unsaturated region
K(e, ė) = {ēi ∈ R | sat(Kpe+Kiēi +Kdė) = Kpe+Kiēi +Kdė}Attempts to achieve controller state-output consistency sat(u) = u
Extends easily to decoupled nonlinear controllersẋci = fci(xci, y, r)
uci = gci(xci, y, r)
For coupled nonlinear controllers, need projection op-erator - project onto K(e, ė) analogue
ẋc = fc(xc, y, r)
uc = gc(xc, y, r)
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme Dec. 20, 2010 11 / 35
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GPAW Compensated Controller Gradient Projection Method for Nonlinear Programming
Gradient Projection Method for NonlinearProgramming [Rosen 1960, 1961]
Nonlinear program:minx∈Rq
J(x)
subject to h̃(x) ≤ 0Feasible region:K̃ = {x̄ | h̃(x̄) ≤ 0}
Boundaries:H1, H2, G3
Projections: z1, z2, z3
K̃
H1
H2
H3 (x
3 )
∇h̃1 ∇h̃2
∇h̃3 (x
3 )G
3
x0
−∇J(x0)
z1x1
−∇J(x1)
z2
zdx2
−∇J(x2)z3
x3
−∇J(x3)
Extended to continuous-time to yield projection operator - requiressolution to combinatorial optimization subproblem at each point
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme Dec. 20, 2010 12 / 35
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GPAW Compensated Controller Gradient Projection Method for Nonlinear Programming
Gradient Projection Method for NonlinearProgramming [Rosen 1960, 1961]
Nonlinear program:minx∈Rq
J(x)
subject to h̃(x) ≤ 0Feasible region:K̃ = {x̄ | h̃(x̄) ≤ 0}
Boundaries:H1, H2, G3
Projections: z1, z2, z3
K̃
H1
H2
H3 (x
3 )
∇h̃1 ∇h̃2
∇h̃3 (x
3 )G
3
x0
−∇J(x0)
z1x1
−∇J(x1)
z2
zdx2
−∇J(x2)z3
x3
−∇J(x3)
Extended to continuous-time to yield projection operator - requiressolution to combinatorial optimization subproblem at each point
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme Dec. 20, 2010 12 / 35
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GPAW Compensated Controller GPAW Compensated Controller
GPAW Compensated Controller
Gradient projection anti-windup (GPAW) compensated controller:
obtained by applying projection operator from continuous-timegradient 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
RI(xg) =
{I − ΓNI(NTI ΓNI)−1NTI (xg), if I 6= ∅I, otherwise
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme Dec. 20, 2010 13 / 35
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GPAW Compensated Controller GPAW Compensated Controller
GPAW Compensated Controller
Gradient projection anti-windup (GPAW) compensated controller:
obtained by applying projection operator from continuous-timegradient 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
RI(xg) =
{I − ΓNI(NTI ΓNI)−1NTI (xg), if I 6= ∅I, otherwise
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme Dec. 20, 2010 13 / 35
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GPAW Compensated Controller GPAW Compensated Controller
GPAW Compensated Controller
Gradient projection anti-windup (GPAW) compensated controller:
obtained by applying projection operator from continuous-timegradient 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∗!
Projection operator RI∗ defined by Γ = ΓT > 0, online solution to
combinatorial optimization subproblem I∗, and projection matrix
RI(xg) =
{I − ΓNI(NTI ΓNI)−1NTI (xg), if I 6= ∅I, otherwise
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme Dec. 20, 2010 13 / 35
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GPAW Compensated Controller GPAW Compensated Controller
GPAW Compensated Controller
Gradient projection anti-windup (GPAW) compensated controller:
obtained by applying projection operator from continuous-timegradient 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
RI(xg) =
{I − ΓNI(NTI ΓNI)−1NTI (xg), if I 6= ∅I, otherwise
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme Dec. 20, 2010 13 / 35
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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(xg) =[
gc(xg)−umax−gc(xg)+umin
]≤ 0 ⇔ xg ∈ K := {x̄ | h(x̄) ≤ 0}
Non-“strictly proper” nonlinear controllers can be approximated arbitrarilywell to be “strictly proper” (singular perturbation theory [Khalil 2002])
ẋc = fc(xc, y, r)
uc = gc(xc, y, r)
a∈(0,∞)−−−−−→≈
˙̃xc = f̃c(x̃c, y, r)
uc = g̃c(x̃c)
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme Dec. 20, 2010 14 / 35
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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(xg) =[
gc(xg)−umax−gc(xg)+umin
]≤ 0 ⇔ xg ∈ K := {x̄ | h(x̄) ≤ 0}
Non-“strictly proper” nonlinear controllers can be approximated arbitrarilywell to be “strictly proper” (singular perturbation theory [Khalil 2002])
ẋc = fc(xc, y, r)
uc = gc(xc, y, r)
a∈(0,∞)−−−−−→≈
˙̃xc = f̃c(x̃c, y, r)
uc = g̃c(x̃c)
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme Dec. 20, 2010 14 / 35
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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(xg) =[
gc(xg)−umax−gc(xg)+umin
]≤ 0 ⇔ xg ∈ K := {x̄ | h(x̄) ≤ 0}
Non-“strictly proper” nonlinear controllers can be approximated arbitrarilywell to be “strictly proper” (singular perturbation theory [Khalil 2002])
ẋc = fc(xc, y, r)
uc = gc(xc, y, r)
a∈(0,∞)−−−−−→≈
˙̃xc = f̃c(x̃c, y, r)
uc = g̃c(x̃c)
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme Dec. 20, 2010 14 / 35
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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(xg) =[
gc(xg)−umax−gc(xg)+umin
]≤ 0 ⇔ xg ∈ K := {x̄ | h(x̄) ≤ 0}
Non-“strictly proper” nonlinear controllers can be approximated arbitrarilywell to be “strictly proper” (singular perturbation theory [Khalil 2002])
ẋc = fc(xc, y, r)
uc = gc(xc, y, r)
a∈(0,∞)−−−−−→≈
˙̃xc = f̃c(x̃c, y, r)
uc = g̃c(x̃c)
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme Dec. 20, 2010 14 / 35
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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(sat(u)− u) Figure
Theorem (GPAW Controller State-output Consistency)
For GPAW compensated controller, if there exists a T ∈ R such thatsat(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)
saturation function sat(·) eliminated: significant simplificationall complications arising from saturation accounted for by RI∗
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme Dec. 20, 2010 15 / 35
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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(sat(u)− u) Figure
Theorem (GPAW Controller State-output Consistency)
For GPAW compensated controller, if there exists a T ∈ R such thatsat(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)
saturation function sat(·) eliminated: significant simplificationall complications arising from saturation accounted for by RI∗
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme Dec. 20, 2010 15 / 35
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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(sat(u)− u) Figure
Theorem (GPAW Controller State-output Consistency)
For GPAW compensated controller, if there exists a T ∈ R such thatsat(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)
saturation function sat(·) eliminated: significant simplificationall complications arising from saturation accounted for by RI∗
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme Dec. 20, 2010 15 / 35
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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(sat(u)− u) Figure
Theorem (GPAW Controller State-output Consistency)
For GPAW compensated controller, if there exists a T ∈ R such thatsat(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)
xg(0)∈K−−−−−→ẋ = f(x, gc(xg))
ẋg = RI∗fc(x, xg, gc(xg), r)
saturation function sat(·) eliminated: significant simplificationall complications arising from saturation accounted for by RI∗
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme Dec. 20, 2010 15 / 35
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GPAW Compensated Controller Controller State-output Consistency
GPAW Scheme Visualization
Nominal controller:ẋc = fc(xc, y, r)
uc = gc(xc)
GPAW controller:ẋg = RI∗fc(xg, y, r)
ug = gc(xg)
Boundaries:H1, H2, G3
Gradients:∇hi(xg) = ±∇gci(xg)
K
H1
H2
H3 (x
g3 )
∇h1 ∇h2
∇h3 (x
g3 )
G3
xg0
fc0
fg1xg1
fc1
fg2
xg2
fc2fg3
xg3
fc3
Unsaturated region: K := {x̄ | sat(gc(x̄)) = gc(x̄)}Nominal update: fci := fc(xgi, y(ti), r(ti)) for xgi := xg(ti)
Projections: fgi := RI∗fci
Justin Teo (ACL, MIT) 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), u̇ = cx+ du
GPAW compensated system:
u̇ =
0, if u ≥ umax, cx+ du > 00, if u ≤ umin, cx+ du < 0cx+ du, otherwise
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 dynamicalsystem [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 Projected Dynamical System
Input Constrained Planar LTI Systems
Simplest possible feedback system, 1st order LTI plant and controller:
Σplant : ẋ = ax+ b sat(u), u̇ = cx+ dufeedback−−−−−→Σplant
Σn
GPAW compensated system:
u̇ =
0, if u ≥ umax, cx+ du > 00, if u ≤ umin, cx+ du < 0cx+ du, otherwise
feedback−−−−−→Σplant
Σg
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 dynamicalsystem [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 Projected Dynamical System
Input Constrained Planar LTI Systems
Simplest possible feedback system, 1st order LTI plant and controller:
Σplant : ẋ = ax+ b sat(u), u̇ = cx+ dufeedback−−−−−→Σplant
Σn
GPAW compensated system:
u̇ =
0, if u ≥ umax, cx+ du > 00, if u ≤ umin, cx+ du < 0cx+ du, otherwise
feedback−−−−−→Σplant
Σg
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 dynamicalsystem [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 | limt→∞
φn(t, z̄) = 0} Rg := {z̄ ∈ R2 | limt→∞
φg(t, z̄) = 0}
Anti-windup schemes aim to improve performance only when saturated
Require ROA to be maintained/enlarged to be valid anti-windupscheme, i.e. Rn ⊂ Raw
Proposition (ROA Containment)
The ROA of the origin of system Σn is contained within the ROA of theorigin of system Σg, i.e. Rn ⊂ Rg
ROA containment is a strong result:
valid for all system parameters and saturation limitsindependent of any Lyapunov functionimplies 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
Region of Attraction Containment
Region of Attraction (ROA) limits utility of systems, defined as:
Rn := {z̄ ∈ R2 | limt→∞
φn(t, z̄) = 0} Rg := {z̄ ∈ R2 | limt→∞
φg(t, z̄) = 0}
Anti-windup schemes aim to improve performance only when saturated
Require ROA to be maintained/enlarged to be valid anti-windupscheme, i.e. Rn ⊂ Raw
Proposition (ROA Containment)
The ROA of the origin of system Σn is contained within the ROA of theorigin of system Σg, i.e. Rn ⊂ Rg
ROA containment is a strong result:
valid for all system parameters and saturation limitsindependent of any Lyapunov functionimplies 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
Region of Attraction Containment
Region of Attraction (ROA) limits utility of systems, defined as:
Rn := {z̄ ∈ R2 | limt→∞
φn(t, z̄) = 0} Rg := {z̄ ∈ R2 | limt→∞
φg(t, z̄) = 0}
Anti-windup schemes aim to improve performance only when saturated
Require ROA to be maintained/enlarged to be valid anti-windupscheme, i.e. Rn ⊂ Raw
Proposition (ROA Containment)
The ROA of the origin of system Σn is contained within the ROA of theorigin of system Σg, i.e. Rn ⊂ Rg
ROA containment is a strong result:
valid for all system parameters and saturation limitsindependent of any Lyapunov functionimplies 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
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−5
−4
−3
−2
−1
0
1
2
3
4
5
x
u
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme 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
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−5
−4
−3
−2
−1
0
1
2
3
4
5
x
u
Rn
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme 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
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−5
−4
−3
−2
−1
0
1
2
3
4
5
x
u
Rg
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme 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
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−5
−4
−3
−2
−1
0
1
2
3
4
5
x
u Rn = Rg
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme 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
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−5
−4
−3
−2
−1
0
1
2
3
4
5
x
u Rn = Rg
φn(t, z0)
φg(t, z0)
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme 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
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−5
−4
−3
−2
−1
0
1
2
3
4
5
x
u Rn = Rg
φn(t, z0)
φg(t, z0)
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme 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
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−5
−4
−3
−2
−1
0
1
2
3
4
5
x
u Rn = Rg
φn(t, z0)
φg(t, z0)
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme 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
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−5
−4
−3
−2
−1
0
1
2
3
4
5
x
u Rn = Rg
φn(t, z0)
φg(t, z0)
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme 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
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−5
−4
−3
−2
−1
0
1
2
3
4
5
x
u
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme 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
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−5
−4
−3
−2
−1
0
1
2
3
4
5
x
u
Rn
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme 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
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−5
−4
−3
−2
−1
0
1
2
3
4
5
x
u
Rg
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme 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
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−5
−4
−3
−2
−1
0
1
2
3
4
5
x
u
Rn
Rg
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme 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
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−5
−4
−3
−2
−1
0
1
2
3
4
5
x
u
Rn
Rg
φn(t, z0)
φg(t, z0)
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme 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
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−5
−4
−3
−2
−1
0
1
2
3
4
5
x
u
Rn
Rg
φn(t, z0)
φg(t, z0)
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme 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
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−5
−4
−3
−2
−1
0
1
2
3
4
5
x
u
Rn
Rg
φn(t, z0)
φg(t, z0)
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme 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
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−5
−4
−3
−2
−1
0
1
2
3
4
5
x
u
Rn
Rg
φn(t, z0)
φg(t, z0)
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme 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
−5 −4 −3 −2 −1 0 1 2 3 4 5−3
−2
−1
0
1
2
3
x
u
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme 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
−5 −4 −3 −2 −1 0 1 2 3 4 5−3
−2
−1
0
1
2
3
x
u
Rn
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme 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
−5 −4 −3 −2 −1 0 1 2 3 4 5−3
−2
−1
0
1
2
3
x
u
Rg
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme 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
−5 −4 −3 −2 −1 0 1 2 3 4 5−3
−2
−1
0
1
2
3
x
u
Rn
Rg
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme 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
−5 −4 −3 −2 −1 0 1 2 3 4 5−3
−2
−1
0
1
2
3
x
u
Rn
Rg
φn(t, z0)
φg(t, z0)
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme 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
−5 −4 −3 −2 −1 0 1 2 3 4 5−3
−2
−1
0
1
2
3
x
u
Rn
Rg
φn(t, z0)
φg(t, z0)
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme 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
−5 −4 −3 −2 −1 0 1 2 3 4 5−3
−2
−1
0
1
2
3
x
u
Rn
Rg
φn(t, z0)
φg(t, z0)
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme 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
−5 −4 −3 −2 −1 0 1 2 3 4 5−3
−2
−1
0
1
2
3
x
u
Rn
Rg
φn(t, z0)
φg(t, z0)
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme Dec. 20, 2010 22 / 35
-
Input Constrained Planar LTI Systems Region of Attraction Containment
Numerical Results IV, Rn ⊂ Rg
Unstable plant, stable controllerSymmetric constraints Asymmetric constraints
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−5
−4
−3
−2
−1
0
1
2
3
4
5
x
u Rn = Rg
φn(t, z0)
φg(t, z0)
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−5
−4
−3
−2
−1
0
1
2
3
4
5
x
u
Rn
Rg
φn(t, z0)
φg(t, z0)
Stable plant, unstable controller
−5 −4 −3 −2 −1 0 1 2 3 4 5−3
−2
−1
0
1
2
3
x
u
Rn
Rg
φn(t, z0)
φg(t, z0)
Justin Teo (ACL, MIT) 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 strictlystable, then the origin of Σn is globally asymptotically stable (GAS) andlocally 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 strictlystable, 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 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 strictlystable, then the origin of Σn is globally asymptotically stable (GAS) andlocally 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 strictlystable, 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 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 strictlystable, then the origin of Σn is globally asymptotically stable (GAS) andlocally 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 strictlystable, 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 ROAfor umin2, umax2. If [umin1, umax1] ⊂ [umin2, umax2], then Rn1 ⊂ Rn2
Conjecture intuitively appealing, but WRONG!Symmetric Asymmetric
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−5
−4
−3
−2
−1
0
1
2
3
4
5
x
u Rn = Rg
φn(t, z0)
φg(t, z0)
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−5
−4
−3
−2
−1
0
1
2
3
4
5
x
u
Rn
Rg
φn(t, z0)
φg(t, z0)
Not pathological
umax = −umin = 1 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
-
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 ROAfor umin2, umax2. If [umin1, umax1] ⊂ [umin2, umax2], then Rn1 ⊂ Rn2
Conjecture intuitively appealing, but WRONG!Symmetric Asymmetric
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−5
−4
−3
−2
−1
0
1
2
3
4
5
x
u Rn = Rg
φn(t, z0)
φg(t, z0)
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−5
−4
−3
−2
−1
0
1
2
3
4
5
x
u
Rn
Rg
φn(t, z0)
φg(t, z0)
Not pathological
umax = −umin = 1 umax = 1.5, umin = −1
Need to consider asymmetric saturation constraints
Most literature (less [Hu et al. 2002]) considers only symmetric constraintsJustin 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 Nominal Controller, Σc GPAW Controller, Σgpawẋ = f(x, sat(u))
y = g(x, sat(u))
ẋc = fc(xc, y)
u = gc(xc)
ẋg = RI∗fc(xg, y)
u = gc(xg)
Assume zeq ∈ K \ ∂K is asymptotically stable equilibrium for Σn and ΣgNominal System: Σn : Σp + Σc ROA: Rn(zeq)
GPAW System: Σg : Σp + Σgpaw ROA: Rg(zeq)
ROA estimate: ΩV = {z̄ | V (z̄) ≤ c} ⊂ Rn(zeq) for some Lyapunovfunction 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)
∂xcRI∗fc ≤
∂V (x̄, x̄c)
∂xcfc, ∀(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
An ROA Comparison Result
Plant, Σp Nominal Controller, Σc GPAW Controller, Σgpawẋ = f(x, sat(u))
y = g(x, sat(u))
ẋc = fc(xc, y)
u = gc(xc)
ẋg = RI∗fc(xg, y)
u = gc(xg)
Assume zeq ∈ K \ ∂K is asymptotically stable equilibrium for Σn and ΣgNominal System: Σn : Σp + Σc ROA: Rn(zeq)
GPAW System: Σg : Σp + Σgpaw ROA: Rg(zeq)
ROA estimate: ΩV = {z̄ | V (z̄) ≤ c} ⊂ Rn(zeq) for some Lyapunovfunction 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)
∂xcRI∗fc ≤
∂V (x̄, x̄c)
∂xcfc, ∀(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
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An ROA Comparison Result
An ROA Comparison Result
Plant, Σp Nominal Controller, Σc GPAW Controller, Σgpawẋ = f(x, sat(u))
y = g(x, sat(u))
ẋc = fc(xc, y)
u = gc(xc)
ẋg = RI∗fc(xg, y)
u = gc(xg)
Assume zeq ∈ K \ ∂K is asymptotically stable equilibrium for Σn and ΣgNominal System: Σn : Σp + Σc ROA: Rn(zeq)
GPAW System: Σg : Σp + Σgpaw ROA: Rg(zeq)
ROA estimate: ΩV = {z̄ | V (z̄) ≤ c} ⊂ Rn(zeq) for some Lyapunovfunction 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)
∂xcRI∗fc ≤
∂V (x̄, x̄c)
∂xcfc, ∀(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
An ROA Comparison Result
Plant, Σp Nominal Controller, Σc GPAW Controller, Σgpawẋ = f(x, sat(u))
y = g(x, sat(u))
ẋc = fc(xc, y)
u = gc(xc)
ẋg = RI∗fc(xg, y)
u = gc(xg)
Assume zeq ∈ K \ ∂K is asymptotically stable equilibrium for Σn and ΣgNominal System: Σn : Σp + Σc ROA: Rn(zeq)
GPAW System: Σg : Σp + Σgpaw ROA: Rg(zeq)
ROA estimate: ΩV = {z̄ | V (z̄) ≤ c} ⊂ Rn(zeq) for some Lyapunovfunction 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)
∂xcRI∗fc ≤
∂V (x̄, x̄c)
∂xcfc, ∀(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 schemefirst 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)
∂xcfc independent of sat(·)
Can be used in two ways: comparison against ROA estimate ofunconstrained system or nominal system
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 schemefirst 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)
∂xcfc independent of sat(·)
Can be used in two ways: comparison against ROA estimate ofunconstrained system or nominal system
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 schemefirst 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)
∂xcfc independent of sat(·)
Can be used in two ways: comparison against ROA estimate ofunconstrained 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]
Σn :
{ẋ = − sat(u)u̇ = x+ (x2 − 1)u
Σgs :
ẋ = − sat(u)
u̇ =
{0, if A1
x+ (x2 − 1)u, otherwise
Compare with ROA estimate ΩV of unconstrained system:
−3 −2 −1 0 1 2 3−3
−2
−1
0
1
2
3
x
u
Ru(zeq)
Rg(zeq)
ΩV
uncompensated
GPAW
0 1 2 3 4 5 6 7 8 9 10−2
0
2
4
x
0 1 2 3 4 5 6 7 8 9 10−2
−1
0
1
2
u
time (s)
uncompensated
GPAW
Toy example defeats methods for LTI systems, feedback linearizablesystems, and nonlinear anti-windup [Morabito et al. 2004]
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme Dec. 20, 2010 29 / 35
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An ROA Comparison Result
Application of ROA Comparison Result
Example nonlinear planar system [Khalil 2002]
Σu :
{ẋ = −uu̇ = x+ (x2 − 1)u
Σg :
ẋ = −u
u̇ =
{0, if A1
x+ (x2 − 1)u, otherwiseCompare with ROA estimate ΩV of unconstrained system:
−3 −2 −1 0 1 2 3−3
−2
−1
0
1
2
3
x
u
Ru(zeq)
Rg(zeq)
ΩV
uncompensated
GPAW
0 1 2 3 4 5 6 7 8 9 10−2
0
2
4
x
0 1 2 3 4 5 6 7 8 9 10−2
−1
0
1
2
u
time (s)
uncompensated
GPAW
Toy example defeats methods for LTI systems, feedback linearizablesystems, and nonlinear anti-windup [Morabito et al. 2004]
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]
Σu :
{ẋ = −uu̇ = x+ (x2 − 1)u
Σg :
ẋ = −u
u̇ =
{0, if A1
x+ (x2 − 1)u, otherwiseCompare with ROA estimate ΩV of unconstrained system:
−3 −2 −1 0 1 2 3−3
−2
−1
0
1
2
3
x
u
Ru(zeq)
Rg(zeq)
ΩV
uncompensated
GPAW
0 1 2 3 4 5 6 7 8 9 10−2
0
2
4
x
0 1 2 3 4 5 6 7 8 9 10−2
−1
0
1
2
u
time (s)
uncompensated
GPAW
Toy example defeats methods for LTI systems, feedback linearizablesystems, 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
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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 stabilityguarantees
Feedback linearizable nonlinear plant with disturbance input w [Yoonet al. 2008]:
Σp :
{ẋ =
[ẋ1ẋ2
]=
[x2
−10x1−0.1x31−48.54x2−w+sat(u)6.67(1+0.1 sinx1)
], y = x
Feedback linearizing PID controller: Σc
Nominal system: NS: Σp + ΣcFeedback linearized AW System [Yoon et al. 2008]: FL
Nonlinear AW system [Morabito et al. 2004]: NAWGPAW compensated system: GPAW
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 stabilityguarantees
Feedback linearizable nonlinear plant with disturbance input w [Yoonet al. 2008]:
Σp :
{ẋ =
[ẋ1ẋ2
]=
[x2
−10x1−0.1x31−48.54x2−w+sat(u)6.67(1+0.1 sinx1)
], y = x
Feedback linearizing PID controller: Σc
Nominal system: NS: Σp + ΣcFeedback linearized AW System [Yoon et al. 2008]: FL
Nonlinear AW system [Morabito et al. 2004]: NAWGPAW compensated system: GPAW
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme Dec. 20, 2010 31 / 35
-
A Numerical Comparison
GPAW Achieves Comparable Performance
0 2 4 6 8 10 12 14 16 18 20−200
−100
0
100
200
disturbance
0 2 4 6 8 10 12 14 16 18 200
0.5
1
1.5
output
time (s)
NS
FL
NAW
GPAW
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)
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, timedelays, and unmodeled dynamics
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, timedelays, and unmodeled dynamics
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, timedelays, 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
motivated new paradigm for anti-windup problem
demonstrated need to consider asymmetric saturation constraints forgeneral saturated systems
developed region of attraction (ROA) comparison and stability resultsfor GPAW compensated (nonlinear) systems
demonstrated viability of GPAW scheme as a candidate anti-windupscheme for general systems
related GPAW compensated systems to projected dynamical systemsand linear systems with partial state constraints
Questions?
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 forgeneral saturated systems
developed region of attraction (ROA) comparison and stability resultsfor GPAW compensated (nonlinear) systems
demonstrated viability of GPAW scheme as a candidate anti-windupscheme for general systems
related GPAW compensated systems to projected dynamical systemsand linear systems with partial state constraints
Questions?
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 forgeneral saturated systems
developed region of attraction (ROA) comparison and stability resultsfor GPAW compensated (nonlinear) systems
demonstrated viability of GPAW scheme as a candidate anti-windupscheme for general systems
related GPAW compensated systems to projected dynamical systemsand linear systems with partial state constraints
Questions?
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 forgeneral saturated systems
developed region of attraction (ROA) comparison and stability resultsfor GPAW compensated (nonlinear) systems
demonstrated viability of GPAW scheme as a candidate anti-windupscheme for general systems
related GPAW compensated systems to projected dynamical systemsand linear systems with partial state constraints
Questions?
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 forgeneral saturated systems
developed region of attraction (ROA) comparison and stability resultsfor GPAW compensated (nonlinear) systems
demonstrated viability of GPAW scheme as a candidate anti-windupscheme for general systems
related GPAW compensated systems to projected dynamical systemsand linear systems with partial state constraints
Questions?
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 forgeneral saturated systems
developed region of attraction (ROA) comparison and stability resultsfor GPAW compensated (nonlinear) systems
demonstrated viability of GPAW scheme as a candidate anti-windupscheme for general systems
related GPAW compensated systems to projected dynamical systemsand linear systems with partial state constraints
Questions?
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 forgeneral saturated systems
developed region of attraction (ROA) comparison and stability resultsfor GPAW compensated (nonlinear) systems
demonstrated viability of GPAW scheme as a candidate anti-windupscheme for general systems
related GPAW compensated systems to projected dynamical systemsand 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 projectionanti-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 ComparisonResults
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 GPAWCompensated Controllers
Appendix B Closed Form Expressions for GPAW Compensated Controllerswith 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 Dissertation Overview
Dissertation Overview
Covered Chapter 1, Introduction. Dissertation on gradient projectionanti-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 ComparisonResults
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 GPAWCompensated Controllers
Appendix B Closed Form Expressions for GPAW Compensated Controllerswith 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 Dissertation Overview
Dissertation Overview
Covered Chapter 1, Introduction. Dissertation on gradient projectionanti-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 ComparisonResults
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 GPAWCompensated Controllers
Appendix B Closed Form Expressions for GPAW Compensated Controllerswith 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 Dissertation Overview
Dissertation Overview
Covered Chapter 1, Introduction. Dissertation on gradient projectionanti-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 ComparisonResults
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 GPAWCompensated Controllers
Appendix B Closed Form Expressions for GPAW Compensated Controllerswith 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 Dissertation Overview
Dissertation Overview
Covered Chapter 1, Introduction. Dissertation on gradient projectionanti-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 ComparisonResults
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 GPAWCompensated Controllers
Appendix B Closed Form Expressions for GPAW Compensated Controllerswith 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 Dissertation Overview
Dissertation Overview
Covered Chapter 1, Introduction. Dissertation on gradient projectionanti-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 ComparisonResults
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 GPAWCompensated Controllers
Appendix B Closed Form Expressions for GPAW Compensated Controllerswith 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 Dissertation Overview
Dissertation Overview
Covered Chapter 1, Introduction. Dissertation on gradient projectionanti-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 ComparisonResults
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 GPAWCompensated Controllers
Appendix B Closed Form Expressions for GPAW Compensated Controllerswith 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 Dissertation Overview
Dissertation Overview
Covered Chapter 1, Introduction. Dissertation on gradient projectionanti-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 ComparisonResults
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 GPAWCompensated Controllers
Appendix B Closed Form Expressions for GPAW Compensated Controllerswith 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 Dissertation Overview
Dissertation Overview
Covered Chapter 1, Introduction. Dissertation on gradient projectionanti-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 ComparisonResults
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 GPAWCompensated Controllers
Appendix B Closed Form Expressions for GPAW Compensated Controllerswith 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
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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)
Adaptive sliding-mode (nominal) controller:
˙̂a = −ΘY Tsuc = Y â−KDs
feedback−−−−−→Σplant
Σnx1
x2
Approximate nominal controller:
˙̃xc = −ΘY Tsẋaug = a(z(y, r)− xaug)uc = Ŷ (xaug)x̃c −KDŝ(xaug)
≡{ẋc = fc(xc, y, r)
uc = gc(xc)
GPAW compensated controller:
ẋg = RI∗fc(xg, y, r)
ug = gc(xg)
feedback−−−−−→Σplant
Σg Movies
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme Dec. 20, 2010 38 / 35
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Backup Slides Passivity Properties
Passivity Properties
Decompose Γ = ΦΦT, define:
PI(xg) := Φ−1RI(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
fc(xg, y, r) Φ−1 PI∗ Φẋg = w̃
u = gc(xg)
sat(u)ẋ = f(x, ũ)
y = g(x, ũ)
ṽ w̃
ũ
r
u
y
xg
RI∗
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
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Backup Slides Passivity Properties
Passivity Properties
Decompose Γ = ΦΦT, define:
PI(xg) := Φ−1RI(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
fc(xg, y, r) Φ−1 PI∗ Φẋg = w̃
u = gc(xg)
sat(u)ẋ = f(x, ũ)
y = g(x, ũ)
ṽ v w w̃
ũ
r
u
y
xg
GPAW modifies uncompensated system with passive operatorCan derive passivity and small-gain based stability results
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme Dec. 20, 2010 39 / 35
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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 GPAWcompensated controllers
Let Γ = ΓT > 0 be the GPAW parameter
K
ker(K)
x
xker
fg1fc1
fg2fc2
Theorem (Geometric Bounding Condition)
If unsaturated region K is a star domain, then for any x ∈ K and anyxker ∈ ker(K),
〈Γ−1(x− xker), fg(x, y, r)〉 ≤ 〈Γ−1(x− xker), fc(x, y, r)〉
holds for all (y, r) and all Γ = ΓT > 0
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme Dec. 20, 2010 40 / 35
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Backup Slides Geometric Properties
Star Domains
Examples and counterexamples of star domains in R2:Star, ker(Xi) 6= ∅ NOT Star, ker(Xi) = ∅
X1
ker(X1)
ker(X2)X2
ker(X3)
ker(X4)X4
X5
Y2Y1
X6 = Y1 ∪ Y2
X7
X4
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 kernelker(Rn ×X) = Rn × ker(X)
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme Dec. 20, 2010 41 / 35
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Backup Slides Geometric Properties
Geometric Interpretation
〈Γ−1(x− xker), fg(x, y, r)〉 ≤ 〈Γ−1(x− xker), fc(x, y, r)〉
Nominal controller:fc
GPAW controller:fg = RI∗fc K
ker(K)
x
xker
fg1fc1
fg2fc2
pc1
pg1
pc2
pg2
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme Dec. 20, 2010 42 / 35
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Backup Slides Geometric Properties
GPAW in Context
Standard anti-windup structure:
sat(u)ẋ = Ax + Bv
y = Cx + Dv
Unconstrained plant
Σ̃c
Σ̃aw
r ũ u v
−
w
yaw1
yaw2
y
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 purposeanti-windup scheme with better stability guarantees
Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme Dec. 20, 2010 43 / 35
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Backup Slides Geometric Properties
Conclusions
Anti-windup compensation for nonlinear systems is an open problemDeveloped GPAW scheme, a general purpose anti-windup scheme:
achieves controller state-output consistencyseveral ways to realizedefined by passive operatorhas clear geometric properties
Strong results for planar LTI systems:ROA containment result independent of any Lyapunov functionshows qualitative weaknesses of existing resultsmotivated new anti-windup paradigm to search for “relative” resultsshows need to consider asymmetric saturation constraintsestablish link to projected dynamical systems
Derived ROA comparison and stability results - first results to directlyindicate advantages of anti-windupEven without stability proofs, ad-hoc methods can be used to designGPAW controller yielding comparable performance withstate-of-the-art anti-windup methods
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 problemDeveloped GPAW scheme, a general purpose anti-windup scheme:
achieves controller state-output consistencyseveral ways to realizedefined by passive operatorhas clear geometric properties
Strong results for planar LTI systems:ROA containment result independent of any Lyapunov functionshows qualitative weaknesses of existing resultsmotivated new anti-windup paradigm to search for “relative” resultsshows need to consider asymmetric saturation constraintsestablish link to projected dynamical systems
Derived ROA comparison and stability results - first results to directlyindicate advantages of anti-windupEven without stability proofs, ad-hoc methods can be used to designGPAW controller yielding comparable performance withstate-of-the-art anti-windup methods
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 problemDeveloped GPAW scheme, a general purpose anti-windup scheme:
achieves controller state-output consistencyseveral ways to realizedefined by passive operatorhas clear geometric properties
Strong results for planar LTI systems:ROA containment result independent of any Lyapunov functionshows qualitative weaknesses of existing resultsmotivated new anti-windup paradigm to search for “relative” resultsshows need to consider asymmetric saturation constraintsestablish link to projected dynamical systems
Derived ROA comparison and stability results - first results to directlyindicate advantages of anti-windup
Even without stability proofs, ad-hoc methods can be used to designGPAW controller yielding comparable performance withstate-of-the-art anti-windup methods
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 problemDeveloped GPAW scheme, a general purpose anti-windup scheme:
achieves controller state-output consistencyseveral ways to realizedefined by passive operatorhas clear geometric properties
Strong results for planar LTI systems:ROA containment result independent of any Lyapunov functionshows qualitative weaknesses of existing resultsmotivated new anti-windup paradigm to search for “relative” resultsshows need to consider asymmetric saturation constraintsestablish link to projected dynamical systems
Derived ROA comparison and stability results - first results to directlyindicate advantages of anti-windupEven without stability proofs, ad-hoc methods can be used to designGPAW controller yielding comparable performance withstate-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
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