gradient projection anti-windup schemeacl.mit.edu/papers/teomitscd11_slides.pdf · anti-windup...

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

Outline

1 Introduction

2 GPAW Compensated Controller

3 Input Constrained Planar LTI Systems

4 An ROA Comparison Result

5 A Numerical Comparison

6 Conclusions

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

sluggish response

large overshoots

long settling times

0 2 4 6 8 10 12 14 16 18 20−2

0

2

4

x

0 2 4 6 8 10 12 14 16 18 20−3

−2

−1

0

1

2

umax

umin

u

time (s)

unconstrained

saturated

sluggish response

large overshoots

long settling times

0 2 4 6 8 10 12 14 16 18 20−2

0

2

4

x

0 2 4 6 8 10 12 14 16 18 20−3

−2

−1

0

1

2

umax

umin

u

time (s)

unconstrained

saturated

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

YF22_1992_crash.mpgMedia File (video/mpeg)

Gripen_1989_1993_crashes.mpgMedia File (video/mpeg)

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

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

y = Cx + Dv

Unconstrained plant

Σ̃cũ

Σ̃aw

r v

−

w

yaw1

yaw2

y

Anti-windup compensated controller

Most practical control systems are nonlinear - LTI are approximations

y = Cx + Dv

Unconstrained plant

Σ̃cũ

Σ̃aw

r v

−

w

yaw1

yaw2

y

Most practical control systems are nonlinear - LTI are approximationsJustin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme Dec. 20, 2010 6 / 35

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

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

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

Introduction Contributions

Contributions

GPAW Compensated Controller

Outline

1 Introduction

6 Conclusions

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)

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)

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

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}

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

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

ẋc = fc(xc, y, r)

uc = gc(xc)

ug = gc(xg)

RI(xg) =

ẋc = fc(xc, y, r)

uc = gc(xc)

GPAW,Γ=ΓT>0−−−−−−−−−−→ẋg = fc(xg, y, r)

ug = gc(xg)

RI(xg) =

ẋc = fc(xc, y, r)

uc = gc(xc)

ug = gc(xg)

RI(xg) =

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)

Other Properties

h(xg) =[

]≤ 0 ⇔ xg ∈ K := {x̄ | h(x̄) ≤ 0}

ẋc = fc(xc, y, r)

uc = gc(xc, y, r)

a∈(0,∞)−−−−−→≈

˙̃xc = f̃c(x̃c, y, r)

uc = g̃c(x̃c)

Other Properties

h(xg) =[

]≤ 0 ⇔ xg ∈ K := {x̄ | h(x̄) ≤ 0}

ẋc = fc(xc, y, r)

uc = gc(xc, y, r)

a∈(0,∞)−−−−−→≈

˙̃xc = f̃c(x̃c, y, r)

uc = g̃c(x̃c)

Other Properties

h(xg) =[

]≤ 0 ⇔ xg ∈ K := {x̄ | h(x̄) ≤ 0}

ẋc = fc(xc, y, r)

uc = gc(xc, y, r)

a∈(0,∞)−−−−−→≈

˙̃xc = f̃c(x̃c, y, r)

uc = g̃c(x̃c)

GPAW Compensated Controller 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∗

xg(0)∈K−−−−−→ẋ = f(x, gc(xg))

ẋg = RI∗fc(x, xg, gc(xg), r)

GPAW Scheme Visualization

Nominal controller:ẋc = fc(xc, y, r)

uc = gc(xc)

GPAW controller:ẋg = RI∗fc(xg, y, r)

ug = gc(xg)

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

Input Constrained Planar LTI Systems

Outline

1 Introduction

6 Conclusions

Input Constrained Planar LTI Systems Projected Dynamical System

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]

Σplant : ẋ = ax+ b sat(u), u̇ = cx+ dufeedback−−−−−→Σplant

Σn

u̇ =

feedback−−−−−→Σplant

Σg

Σplant : ẋ = ax+ b sat(u), u̇ = cx+ dufeedback−−−−−→Σplant

Σn

u̇ =

Σg

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

Rn := {z̄ ∈ R2 | limt→∞

φn(t, z̄) = 0} Rg := {z̄ ∈ R2 | limt→∞

φg(t, z̄) = 0}

Rn := {z̄ ∈ R2 | limt→∞

φn(t, z̄) = 0} Rg := {z̄ ∈ R2 | limt→∞

φg(t, z̄) = 0}

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

−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

−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

−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

−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)

−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)

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

−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

−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

−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

−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)

−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)

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

−5 −4 −3 −2 −1 0 1 2 3 4 5−3

−2

−1

0

1

2

3

x

u

Rn

−5 −4 −3 −2 −1 0 1 2 3 4 5−3

−2

−1

0

1

2

3

x

u

Rg

−5 −4 −3 −2 −1 0 1 2 3 4 5−3

−2

−1

0

1

2

3

x

u

Rn

Rg

−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)

−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)

−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)

−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)

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)

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

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

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

6 Conclusions

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)

y = g(x, sat(u))

ẋc = fc(xc, y)

u = gc(xc)

u = gc(xg)

∂V (x̄, x̄c)

∂xcRI∗fc ≤

∂V (x̄, x̄c)

∂xcfc, ∀(x̄, x̄c) ∈ ΩV ∩ (Rn ×K)

y = g(x, sat(u))

ẋc = fc(xc, y)

u = gc(xc)

u = gc(xg)

∂V (x̄, x̄c)

∂xcRI∗fc ≤

∂V (x̄, x̄c)

∂xcfc, ∀(x̄, x̄c) ∈ ΩV ∩ (Rn ×K)

y = g(x, sat(u))

ẋc = fc(xc, y)

u = gc(xc)

u = gc(xg)

∂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

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

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]

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

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

A Numerical Comparison

Outline

1 Introduction

6 Conclusions

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

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

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

Conclusions

Outline

1 Introduction

6 Conclusions

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

Conclusions

GPAW in Context

Conclusions

GPAW in Context

Conclusions

Questions?

Conclusions

Questions?

Conclusions

Questions?

Conclusions

Questions?

Conclusions

Questions?

Conclusions

Questions?

Conclusions

Questions?Justin Teo (ACL, MIT) Gradient Projection Anti-windup Scheme Dec. 20, 2010 35 / 35

Backup Slides

Backup slides

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

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

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)

Σg Movies

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

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

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

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)

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

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

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

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

Conclusions

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

Conclusions

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. InProc. 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 generalizedreference 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:

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