anti-windup solutions for input saturated control...

Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References

Anti-windup solutions forinput saturated control systems

Luca Zaccarian

LAAS-CNRS and University of Trento

thanks to A.R. TeelF. Dabbene, S. Galeani, A. Garulli, T. Hu, S. Tarbouriech, R. Tempo, M. Turner,

S. Gayadeen, J. Marcinkovski, S. Podda, V. Vitale, E. Weyer,D. Dai, S. Formentin, F. Forni, G. Grimm, F. Morabito,

S. Onori, F. Todeschini, G. Valmorbida

Genova, October 1, 20151 / 58


Outline

1 Saturation and its effects on the performance metric

2 Direct Linear approaches and static anti-windup

3 Incorporating robustness in Direct Linear Anti-Windup

4 Model recovery anti-windup solution

5 Some applications of Model Recovery Anti-Windup

6 Fully nonlinear Anti-windup for nonlinear plants

2 / 58


Active control provides extreme vibration isolation

Newport Corporation’s Elite 3TM

vibration isolation table

• Useful, for example, in• high-precision microscopy• semiconductor manufacturing

• Actuators: piezoelectric stack

• Sensors: geophones

A/D

DSP

CONTROLLERPLANT

D/APassive

Isolation

SensorsGeophones

ActuatorsPiezoelectric

Active

IsolationControl

Algorithm

3 / 58


Input saturation confuses the base control algorithm

• Extreme vibration suppression (40 dB) up to t = 23 s

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140

−1

0

1

Time [s]

Co

ntr

olle

r O

utp

ut

+SAT

−SAT

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140

−1

0

1

Time [s]

Pla

nt

Ou

tpu

t

• At t = 23 s someone walks close to the table

4 / 58


Performance with saturation depends on size of disturbance

• Saturation: an abrupt nonlinearity:• Small signals: sat(u) = u ⇒ no effect

• Large signals: sat(u) bounded ⇒ severeeffect

P

K

w z

u

sat(u) y

• Signal size (L2 norm): ‖z‖2 :=

(∫ ∞

0

|z(t)|2dt) 1

2

• z ∈ L2 (square integrable) if ‖z‖2 <∞

• Closed-loop performance measures:

• Finite L2 gain (linear H∞ norm): γwz ∈ R≥0:

‖z‖2 ≤ γwz‖w‖2 for all w ∈ L2

• Nonlinear L2 gain: a function s 7→ γwz (s): Megretski [1996]

‖z‖2 ≤ γwz (s)‖w‖2 for all w satisfying ‖w‖2 ≤ s

5 / 58


Example demonstrates relevance of nonlinear gains

Controller K cancels the plant dynamicsand stabilizes (before saturation)

P : z = az + sat(u) + w

K : u = −az − 10z

P

K

w z

u

sat(u) y

Three representative cases Sontag [1984], Lasserre [1992]

10-1

100

101

10-1

100

101

Nonlin

ear

Gain

s

L2 norm (size) of w

γwz

(i.e., H∞

norm of Pwz

)

a=-1

a=0

a=1

6 / 58


Nonlinear L2 gains are estimated using Lyapunov functionsHu et al. [2006], Dai et al. [2009b], Garulli et al. [2013]

• Quadratic functions (LMIs Boyd et al.

[1994])V1(x) = xTPx

• Max of quadratics (BMIs)V2(x) = max

j∈1,...,JxTPjx

• Convex Hull of quadratics (BMIs)

V3(x) = minγj≥0:

∑j γj =1

xT(∑

j γjQj

)−1

x

• Piecewise quadratic (LMI-BMI)

V4(x) =

[x

dz(u(x))

]T

P

[x

dz(u(x))

]

• Piecewise Polynomial (LMI-BMI)

V5(x) =

[x

dz(u(x))

]mTP

[x

dz(u(x))

]m

V +1

γdz (s)|z |2 − γdz (s)|w |2 < 0

A possible level set

7 / 58




[1994])V1(x) = xTPx


j∈1,...,JxTPjx


V3(x) = minγj≥0:

∑j γj =1

xT(∑

j γjQj

)−1

x


V4(x) =

[x

dz(u(x))

]T

P

[x

dz(u(x))

]


V5(x) =

[x

dz(u(x))

]mTP

[x

dz(u(x))

]m

V +1

γdz (s)|z |2 − γdz (s)|w |2 < 0


8 / 58




[1994])V1(x) = xTPx


j∈1,...,JxTPjx


V3(x) = minγj≥0:

∑j γj =1

xT(∑

j γjQj

)−1

x


V4(x) =

[x

dz(u(x))

]T

P

[x

dz(u(x))

]


V5(x) =

[x

dz(u(x))

]mTP

[x

dz(u(x))

]m

V +1

γdz (s)|z |2 − γdz (s)|w |2 < 0


9 / 58




[1994])V1(x) = xTPx


j∈1,...,JxTPjx


V3(x) = minγj≥0:

∑j γj =1

xT(∑

j γjQj

)−1

x


V4(x) =

[x

dz(u(x))

]T

P

[x

dz(u(x))

]


V5(x) =

[x

dz(u(x))

]mTP

[x

dz(u(x))

]m

V +1

γdz (s)|z |2 − γdz (s)|w |2 < 0


10 / 58




[1994])V1(x) = xTPx


j∈1,...,JxTPjx


V3(x) = minγj≥0:

∑j γj =1

xT(∑

j γjQj

)−1

x


V4(x) =

[x

dz(u(x))

]T

P

[x

dz(u(x))

]


V5(x) =

[x

dz(u(x))

]mTP

[x

dz(u(x))

]m

V +1

γdz (s)|z |2 − γdz (s)|w |2 < 0


11 / 58


H∞ design generalizes to linear input-saturated plantsDai et al. [2009a]

Pd z

u

sat(u) y

+ AW

C

static

K

• Given P linear, design K, namely• C linear plant-order• AW static: linear gain

• Performance objective:given s∗, minimize γdz (s∗)

• Linear controller K equations

xc = Axc + By + E1(sat(yc )− yc )

yc = Cxc + Dy + E2(sat(yc )− yc )

• Synthesis is a convex problem (generalizes LMI-H∞ Gahinet and Apkarian

[1994], Iwasaki and Skelton [1994])

• Synthesis without AW is nonconvex

• Generalized hybrid approaches for sampled-data design Dai et al. [2010]

12 / 58


Pure linear anti-windup design is also convexGrimm et al. [2003a], Gomes da Silva Jr and Tarbouriech [2005], Hu et al. [2008]

Pd z

u

sat(u) y

+ AW

CK

static ordynamic

• Given P linear, C linear, design only• AW linear static or plant-order


• Necessary conditions:• linear feedback (P, C) exp stable

(∃V ([ xpxc

]) = [ xpxc

]T[

Qp Qpc

QTpc Qc

]−1

[ xpxc

])

• ∃K s.t. Ap + Bp,uK exp stable

(VK (xp) = xTp Qp

−1xp, |K | s∗→∞−→ 0)

• Static anti-windup construction (convex, LMIs)• feasible if Qp = Qp: quasi-common quadratic Lyapnuov function

• Plant-order anti-windup construction (convex, LMIs)• always feasible as long as VK and V above exist

13 / 58


Direct Linear static linear anti-windup design (LMI)Mulder et al. [2001], Grimm et al. [2003a], Gomes da Silva Jr and Tarbouriech [2005], Hu et al.[2008]

Pw z

u

sat(u) y

+ Daw

CK

u

v

dz(u)

• Given P linear, C linear, design only

• linear anti-windup gain Daw =[

Daw,1

Daw,2

]


• Linear controller K equations

xc = Axc + By + Daw ,1(u − sat(u))

yc = Cxc + Dy + Daw ,2(u − sat(u))

• LMI-based design Mulder et al. [2001], Grimm

et al. [2003a], Gomes da Silva Jr and Tarbouriech

[2005], Hu et al. [2008]

• Preserve of small signal response (Daw multiplies dz(u) = u − sat(u))

Asymptotically recover large signal response (global not always possible)

• Results generalize nontrivially to the plant-order case

14 / 58


Compact representation of the closed-loop system

Pw z

u

sat(u) y

+ Daw

CK

u

xcl

z

u

w

v HDaw

dz(u)

sat(u)u−v

dz(u)

H :

˙xcl = Aclxcl + Bcl,d (u − sat(u)) + Bcl,vv + Bcl,wwu = Ccl,uxp + Dcl,ud (u − sat(u)) + Dcl,uvv + Dcl,uwwz = Ccl,zxp + Dcl,zd (u − sat(u))︸︷︷︸

dz(u)

+Dcl,zvv + Dcl,zww ,

15 / 58


Quadratic analysis conditions are convexMulder et al. [2001], Gomes da Silva Jr and Tarbouriech [2005], Hu et al. [2008]

Proposition: Given the NOMINAL system and s > 0, if the LMI problem

γ2(s) = minγ2,Q,Y ,U

γ2 subject to Q = QT > 0, U > 0 diagonal,

He

AclQ Bcl,dU + Bcl,vDawU + Y T Bcl,w 0Ccl,uQ Dcl,udU + Dcl,uvDawU − U Dcl,uw 0

0 0 −I/2 0

Ccl,zQ Dcl,zdU + Dcl,zvDawU Dcl,zw −γ2

2 I

≺0,

[Q Y[k]

T

Y[k] u2k/s

2

] 0,

k = 1, . . . , nu

is feasible, then the following holds for the saturated closed-loop:

1 [Stab] the origin is locally exponentially stable with region ofattraction containing the set E(Q, s) := x : xTQ−1x ≤ s2;

2 [Reach] the reachable set from x(0) = 0 with ‖w‖2 ≤ s iscontained in E(Q, s);

3 [L2Perf] for each w such that ‖w‖2 ≤ s, the zero state solutionsatisfies the L2 gain bound:

‖z‖2 ≤ γ(s)‖w‖2

16 / 58


Quadratic analysis conditions easily lead to synthesisMulder et al. [2001], Gomes da Silva Jr and Tarbouriech [2005], Hu et al. [2008]

Proposition: Given the NOMINAL system and s > 0. If the LMI problem

γ2(s) = minγ2,Q,Y ,U


He

AclQ Bcl,dU + Bcl,vDawU + Y T Bcl,w 0Ccl,uQ Dcl,udU + Dcl,uvDawU − U Dcl,uw 0

0 0 −I/2 0

Ccl,zQ Dcl,zdU + Dcl,zvDawU Dcl,zw −γ2

2 I

≺0,

[Q Y[k]

T

Y[k] u2k/s

2

] 0,

k = 1, . . . , nu

is feasible, then the following holds for the saturated closed-loop:




‖z‖2 ≤ γ(s)‖w‖2

17 / 58


Quadratic synthesis conditions are convexMulder et al. [2001], Gomes da Silva Jr and Tarbouriech [2005], Hu et al. [2008]

Proposition: Given the NOMINAL system and s > 0. If the LMI problem

γ2(s) = minγ2,Q,Y ,U,X


He

AclQ Bcl,dU + Bcl,vX + Y T Bcl,w 0Ccl,uQ Dcl,udU + Dcl,uvX − U Dcl,uw 0

0 0 −I/2 0

Ccl,zQ Dcl,zdU + Dcl,zvX Dcl,zw −γ2

2 I

≺0,

[Q Y[k]

T

Y[k] u2k/s

2

] 0,

k = 1, . . . , nu

is feasible, then, selecting the static AW gain asDaw = XU−1




‖z‖2 ≤ γ(s)‖w‖2

18 / 58


Cross-directional dynamics application: SynchrotronQueinnec et al. [2015]

• Model is based on a suitable Singular Value Decomposition (SVD)

p(z). . .

p(z)

ΣΨT Φ

dy

c(z). . .

c(z)

ΨΣ−1

dz

-+

++P (z)

C(z)

ζup

ν uc

sat u

ycΦT

• Equivalent dynamics requires generalized nonlinearity

p(z). . .

p(z)

d

y

c(z). . .

c(z)

dz

-+

++

ζ

ν

up

uc

ΣΨT

ΨΣ−1 ΦT

Φ

dΦ

++ yΦ

• Collaboration with Diamond Light Source synchrotron (Oxforshire, UK)19 / 58


Closed loop now depends on uncertain parameter q ∈ QTurner et al. [2007], Grimm et al. [2004], Formentin et al. [2013, 2014]

P(q)w z

u

sat(u) y

+ Daw

CK

u

xcl

z

u

w

v H(q)Daw

dz(u)

sat(u)u−v

dz(u)

H(q) :

˙xcl = Acl (q)xcl + Bcl,d (q)(u − sat(u)) + Bcl,v (q)v + Bcl,w (q)wu = Ccl,u(q)xp + Dcl,ud (q)(u − sat(u)) + Dcl,uv (q)v + Dcl,uw (q)wz = Ccl,z (q)xp + Dcl,zd (q) (u − sat(u))︸︷︷︸

dz(u)

+Dcl,zv (q)v + Dcl,zw (q)w ,

• Robust designs follow a deterministic worst case paradigm, imposingstrong convexity conditions Turner et al. [2007], Grimm et al. [2004]

20 / 58


Static AW synthesis based on scenario with certificatesFormentin et al. [2013, 2014]

Theorem (Robust static AW using scenario with certificates)

Fix a positive value s ≥ ‖w‖2, ε ∈ (0, 1), β ∈ (0, 1), and select N satisfyingB(N, ε, nθ) ≤ β, with nθ = 1 + nu + nu(nu + nc )

Extract N samples of the uncertain matrices according to the probability distribution

Solve

γ2sc (s) = min

γ2,U,X,Qi ,Yiγ2, subject to Qi = Qi

T > 0, U > 0 diagonal,

He

A

(i)cl Qi B

(i)cl,d U + B

(i)cl,v X + Y T

i B(i)cl,w 0

C(i)cl,uQi D

(i)cl,ud U +D

(i)cl,uv X−U D

(i)cl,uw 0

0 0 −I/2 0

C(i)cl,z Qi D

(i)cl,zd U + D

(i)cl,zv X D

(i)cl,zw − γ

2

2I

<0,

[Qi Y T

i [k]

Yi [k] u2k/s2

]≥ 0,

∀k = 1, . . . , nu

∀i = 1, . . . ,N

If the above LMIs are feasible, select the static anti-windup gain Daw = XU−1

Then, for each ‖w‖2 < s, the zero initial state solution of the closed loop satisfiesPr(‖z‖2 > γsc (s) ‖w‖2) ≤ ε, with probability no smaller than 1− β.

Analysis conditions can also be easily formulated

21 / 58


Illustrative example: A passive networkGrimm et al. [2003b]

Uncertain parameters with (known) Gaussian distribution

parameter mean std dev parameter mean std devR1 313 Ω ± 10 R5 10 F ± 10R2 20 Ω ± 10 C1 0.01 F ± 10R3 315 Ω ± 10 C2 0.01 F ± 10R4 17 Ω ± 10 c3 0.01 F ± 10

Input generator voltage constrained:u(t) = Vi (t) ∈ [−u, u] = [−1Volt, 1Volt]

Design parameters are ε = 0.01, β = 10−6, s = 0.003, nθ = 35⇒ N = 2270 (not 7565) for design based on sequential algorithm

22 / 58


Deterministic and Randomized nonlinear L2 gains

Robust compensator shows better robust performance (red curves)

The nominal behavior slightly deteriorated (thin curves)

10-2.6

10-2.5

10-2.4

10-2.3

0

5

10

15

20

25

30

35

s

γ

no AW (r)

no AW (n)

nAW (r)

nAW (n)

rAW (r)

rAW (n)

Without anti-windup (black dashed), with nominal anti-windup (bluedashed-dotted) and with robust anti-windup (red solid)

23 / 58


Time responses confirm nonlinear L2 gain trends

0 5 10 15 20 25

−2

−1

0

1

2

Time [−]

uncon

no AW

nAW

rAW

0 10 20−4

−2

0

2

4unconstrained

0 10 20−4

−2

0

2

4no AW

0 10 20−4

−2

0

2

4nominal AW

Time [−]0 10 20

−4

−2

0

2

4randomized AW

Time [−]

10-2.6

10-2.5

10-2.4

10-2.3

0

5

10

15

20

25

30

35

s

γ

no AW (r)

no AW (n)

nAW (r)

nAW (n)

rAW (r)

rAW (n)

24 / 58


Anti-windup extends to fully nonlinear compensationTeel and Kapoor [1997], Zaccarian and Teel [2002, 2011]

Pd z

sat(u) y

+

AW

CKyc`

xaw

y`

v

P+yaw

+

Model Recovery Anti-Windup (MRAW)• Framework for nonlinear AW:• AW is a model P of P• v = k(xaw ) is a (nonlinear) stabilizer

whose construction depends on P• AW is controller-independent:• any (nonlinear) C allowed

• Useful feature of MRAW:• C “receives” linear plant output y`• ⇒ C “delivers” linear plant input yc`

• Reduced order P possible (tested on adaptive noise suppression)

• MRAW allows for bumpless transfer among controllers

• MRAW generalizes to rate and curvature saturation

• MRAW generalizes to dead time plants (Smith predictor)

25 / 58


Anti-windup extends to fully nonlinear compensationTeel and Kapoor [1997], Zaccarian and Teel [2002, 2011]

Pd z

sat(u) y

+

AW

CKyc`

xaw

y`

v

P+yaw

+








26 / 58


Anti-windup extends to fully nonlinear compensationPagnotta et al. [2007]

Pd z

sat(u) y

+

AW

CKyc`

xaw

y`

v

P+yaw

+








27 / 58


Anti-windup extends to fully nonlinear compensationZaccarian and Teel [2005]

Pd z

sat(u) y

+

AW

CKyc`

xaw

y`

v

P+yaw

+








28 / 58


Anti-windup extends to fully nonlinear compensationForni et al. [2012, 2010]

Pd z

sat(u) y

+

AW

CKyc`

xaw

y`

v

P+yaw

+








29 / 58


Anti-windup extends to fully nonlinear compensationZaccarian et al. [2005]

Pd z

sat(u) y

+

AW

CKyc`

xaw

y`

v

P+yaw

+








30 / 58


Ad hoc gain adaptation induces very slow isolation recovery

• Effect of a footstep at the side of the table (recovery > 1 minute)

0 10 20 30 40 50 60 70 80 90

−1

0

1

Time [s]

Contr

olle

r O

utp

ut

+SAT

−SAT

0 10 20 30 40 50 60 70 80 90−0.4

−0.2

0

0.2

0.4

Time [s]

Pla

nt O

utp

ut

31 / 58


MRAW dramatically reduces isolation recovery timeTeel et al. [2006], Zaccarian et al. [2000]

• Effect of a footstep at the side of the table (recovery ≈ 4 s)

−5 −2.5 0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 25

−1

0

1

Time [s]

Co

ntr

olle

r O

utp

ut

+SAT

−SAT

−5 −2.5 0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 25−0.4

−0.2

0

0.2

0.4

Time [s]

Plant Output

Controller input

32 / 58


Even a bat strike does not confuse the MRAW controllerTeel et al. [2006], Zaccarian et al. [2000]

Hitting with a baseball bat the table leg (recovery ≈ 5 s)

−5 −2.5 0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 25

−1

0

1

Time [sec]

Co

ntr

olle

r O

utp

ut

+SAT

−SAT

−5 −2.5 0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 25−0.4

−0.2

0

0.2

0.4

Time [s]

Plant Output

Controller input

33 / 58


Bumpless transfer enables smooth controller activationTeel et al. [2006], Zaccarian et al. [2000]

• Controller is gradually activated in bumpless transfer scheme

0 5 10 15 20 25 30 35 40

−1

0

1

Time [sec]

Co

ntr

olle

r O

utp

ut

System power−on

0 5 10 15 20 25 30 35 40−0.4

−0.2

0

0.2

0.4

Time [sec]

Pla

nt

Ou

tpu

t

0 5 10 15 20 25 30 35 40−0.4

−0.2

0

0.2

0.4

Time [sec]

Co

ntr

olle

r In

pu

t

34 / 58


Rate Saturated McDonnell Douglas TAFA dynamicsBarbu et al. [2005]

• Linearized longitudinal dynamics (α=angle of attack; q=pitch rate)

z :=

[αq

]=

[Zα Zq

Mα Mq

]z +

[0Mδ

]δ

=: A z + Bu δ

• Saturation: M = 20 deg , R = 40 deg/s.

δ = R sgn[M sat

( u

M

)− δ],

• Study a flight trim condition with one exp unstable mode

x :=

[xs

xu

]=

[−4 00 1

]x +

[bs

bu

]δ

35 / 58


Problems due to magnitude saturation

• Unconstrained trajectory may exit the null-controllability region

−80 −60 −40 −20 0 20 40 60 80−80

−60

−40

−20

0

20

40

60

80

Angle of atack [deg]

Pitch

ra

te [

de

g/s

ec]

• Unconstrained (−−), possible desired trajectories (− and − · −)36 / 58


Problems due to magnitude+rate saturation

• Unconstrained trajectory may exit the null-controllability region

−40 −30 −20 −10 0 10 20 30 40

−300

−200

−100

0

100

200

300

δ [deg]

5α

+ q

• Unconstrained (−−), possible desired trajectories (− and − · −)37 / 58


Close the position loop using a pilot model

• Use a simple crossover model

Closed-loop

Antiwindup

System

Pilot

d

Model

-

+

q

d

q

R

• Study the maneuverability of the aircraft with anti-windup

• Study the possible occurrence of PIOs (Pilot Induced Oscillations)

• Compare the response to the optimal response using static commandlimiting

• Use a step reference θd = 40 deg

38 / 58


Piloted flight simulation

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

10

20

30

40

50

Time [s]

Pitch

An

gle

[d

eg

]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−30

−20

−10

0

10

20

30

Time[s]

Co

ntr

ol [d

eg

]

(unconstrained −−, anti-windup −, optimal trajectory with staticlimiting − · −)

39 / 58


Unconstrained response information (linear case)

• Plant Px = Ax + Bd d + Bu uz = Cz x + Ddz d + Duz uy = Cy x + Ddy d + Duy u

• Anti-windup filter Pxaw = Axaw + Bu (u − yc )yaw = Cy xaw + Duy (u − yc )

• Unconstrained controller Cxc = Ac xc + Bcu uc + Bcr ryc = Cc xc + Dcu uc + Dcr r

• Interconnectionsu = sat(yc + v),uc = y − yaw

v1: to be selected!

• Coordinate transformation: (x`, xc , xaw ) = (x − xaw , xc , xaw )

• Unconstrained dynamics P − P:

x` = Ax` + Bd d + Bu yc

y − yaw = Cy x` + Ddy d + Duy yc

• ⇒ Information about the unconstrained response embedded within thescheme!

40 / 58


The unconstrained response information (nonlinear case)

• Plant Px = f (x , u)z = h(x , sat(u))

• Anti-windup filter Pxaw = f (x , u)− f (x − xaw , yc )yaw = xaw

• Unconstrained controller Cxc = g(xc , uc , r)yc = k(xc , uc , r)

• Interconnectionsu = sat(yc + v),uc = x − xaw

v1: to be selected!

• Coordinate transformation: (x`, xc , xaw ) = (x − xaw , xc , xaw )

• Unconstrained dynamics P − P:

x` = f (x`, yc )uc = x − xaw = x`

• ⇒ Information about the unconstrained response embedded within thescheme!

41 / 58


Anti-windup for nonlinear systems: resulting scheme

r

uc

UnconstrainedController

+

NonlinearitySaturation x

CompensatorAnti-windup

upycα Nonlinear

plant

x

xaw

yaw-

• Need extra plant state measurements

• Recall that xaw = x − x`: very useful information

worry about stability looking at x

worry about performance looking at xaw

• A few application examples:

Anti-windup for robot manipulators Morabito et al. [2004]

Anti-windup for Brake-by-Wire systems Todeschini et al. [2015]

42 / 58


A SCARA robot manipulator exampleMorabito et al. [2004]

• SCARA robot with limited torque/force inputsLink 1 2 3 4mi 55 Nm 45 Nm 70 N 25 Nm

Robot

Dynami

inversion

+

+

Controller

PID

r

q; _qu

P I D

121 7.5 17.830 10 8.2

150 1 24.7150 0.5 20.1

• Feedback linearizing controller+PID action (computed torque) inducesdecoupled linear performance (for small signals)

43 / 58


A slight saturation can be disastrous

• The reference is r = [6 deg , −4 deg , 4 cm, 8 deg ]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0

2

4

6

8

Positio

n 1

Time [s]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

−4

−2

0

Positio

n 2

Time [s]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.05

0.1

Positio

n 3

Time [s]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

5

10

Positio

n 4

Time [s]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

−50

0

50

Torq

ue 1

Time [s]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

−50

0

50

Torq

ue 2

Time [s]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

−50

0

50

Forc

e 3

Time [s]

Without anti−windupWith anti−windupWithout saturation

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

−20

0

20

Torq

ue 4

Time [s]

• Stability is recovered, performance is almost fully preserved

44 / 58


Anti-windup injects signals and then fades out

• The reference is a sequence of little step followed by a large step

0 5 10 15 20

0

5

10

Po

sitio

n 1

Time [s]

0 5 10 15 20−5

0

5

Po

sitio

n 2

Time [s]

0 5 10 15 20−0.1

0

0.1

Po

sitio

n 3

Time [s]

0 5 10 15 20−0.2

0

0.2

Po

sitio

n 4

Time [s]


0 5 10 15 20

−50

0

50

Torq

ue 1

Time [s]

0 5 10 15 20−50

0

50

Torq

ue 2

Time [s]

0 5 10 15 20−40

−20

0

20

40

AW

action (

v1)

1Time [s]

0 5 10 15 20

−10

0

10

AW

action (

v1)

2

Time [s]

• The anti-windup action dies away to recover the unconstrainedclosed-loop

45 / 58


SCARA: large signals (nonlinear v1)

• The reference is r = [150 deg , −100 deg , 1 m, 200 deg ]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0

100

200

Positio

n 1

Time [s]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−200

−100

0

100

Positio

n 2

Time [s]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

Positio

n 3

Time [s]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

100

200

Positio

n 4

Time [s]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

−50

0

50

Torq

ue 1

Time [s]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

−50

0

50

Torq

ue 2

Time [s]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

−50

0

50

Forc

e 3

Time [s]


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

−20

0

20

Torq

ue 4

Time [s]

• Performance is dramatically improved (input authority is almost fullyexplotied)

46 / 58


MRAW applies to nonlinear fully actuated robots

Example: a SCARA robot (planar robot) following a circular motion• Saturated “computed torque” controller goes postal (unstable)• Nonlinear MRAW provides slight performance degradation

0 1 2 3 4 5 6 7 8 9 10

−100

−50

0

50

100

Torq

ue 1

[N

m]

Time [s]

0 1 2 3 4 5 6 7 8 9 1050

100

150

200

Positio

n 1

[deg]

Time [s]

0 1 2 3 4 5 6 7 8 9 10−50

0

50

Torq

ue 2

[N

m]

Time [s]

0 1 2 3 4 5 6 7 8 9 10−200

−150

−100

−50

Positio

n 2

[deg]

Time [s]

Without anti−windupWithout saturationWith anti−windup

0 0.05 0.1 0.15 0.2 0.25 0.30

0.05

0.1

0.15

0.2

0.25

Y a

xis

[m

]

X axis [m]

With anti−windupWithout saturationWithout anti−windup

47 / 58


Nonlinear anti-windup for a Brake By Wire SystemTodeschini et al. [2015]

• Brake-by-wire system in motorcycles corresponds to a nonlinear plant

• The main nonlinear effect can be easily isolated in the model:

1s2+k2s+k1ku

u yp(x)x

kp0 1 2 3 4 5 6

−20

−10

0

10

20

x [mm]

y [bar]

Experimental Data

p(x)

pl(x)

48 / 58


BBW solution uses nonlinear MRAW

• “Deadzone compensation” scheme provides nonlinear baseline controller• Fully Nonlinear anti-windup addresses saturation with nonlinear plant

and nonlinear controller

LCy

BBWul

DC

+

--

u

udc

ydc

yl

rknc

yBBW

unc

AW

u

r

Knc

++

-

--

uawxawyaw

yncxnc

σ(u)

x

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

10

20

30

Pre

ssure

[bar]

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4−20

−10

0

10

20

Time [s]

Curr

ent [A

]

r

PID + SAT

PID + DC + SAT

PID + DC + SAT + AW

PID + DC + SAT + IMC

• Step response reveals successfulanti-windup action• Driver would get confused by largeovershoots• Alternative existing solutions(nonlinear IMC-based anti-windup)are unacceptably slow (black)

49 / 58


Anti-windup designs apply to additional applicationsZaccarian et al. [2007], Vitelli et al. [2010]

Control of irrigation channels

• Gate saturation problems:• bumpless transfer from manual control• with small flows in the pools• with large disturbances (rain, etc)

• Challenge: plant is ANCBC (poles in 0)

Gate 2

Gate 1

y1

h1

p1 yd y2

h2

p2

Small signal nonlinearity compensation inhigh-power circulating current amps

• Thyristors have a min current threshold:• below the treshold: circulating current• this generates a undesired nonlinearity• possibly destabilizing outer feedback

• Challenge: reverse anti-windup problem

50 / 58


Summary of the presented works with references

. Summary of the proposed solutions in Galeani et al.

[2009], Zaccarian and Teel [2011]

. Lyapunov certificates of performance in Dai et al. [2009b], Garulli et al. [2013],

Hu et al. [2006]

. LMI-based (Direct Linear) anti-windup proposed in Mulder et al. [2001],

Tarbouriech et al. [2011], Grimm et al. [2003a,b], Hu et al. [2008]

• Generalization to saturated H∞ in Dai et al. [2009a, 2010]

• Robust extensions of the approaches Formentin et al. [2013, 2014], Grimm

et al. [2004], Turner et al. [2007]

• Application to synchrotron control Queinnec et al. [2015]

. Model-Recovery anti-windup proposed in Teel and Kapoor [1997], Zaccarian

and Teel [2002]

• Bumpless transfer extensions Zaccarian and Teel [2005]

• Generalizations to rate and curvature saturations Forni et al. [2010, 2012]

• Applications: flight Barbu et al. [2005], vibration isolation Teel et al. [2006]

• Nonlinear MRAW for NL plants in Morabito et al. [2004], Todeschini et al. [2015]51 / 58


Bibliography I

C. Barbu, S. Galeani, A.R. Teel, and L. Zaccarian. Nonlinear anti-windup formanual flight control. Int. J. of Control, 78(14):1111–1129, September 2005.

S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan. Linear MatrixInequalities in System and Control Theory. Society for Industrial an AppliedMathematics, 1994.

D. Dai, T. Hu, A.R. Teel, and L. Zaccarian. Output feedback design forsaturated linear plants using deadzone loops. Automatica, 45(12):2917–2924, 2009a.

D. Dai, T. Hu, A.R. Teel, and L. Zaccarian. Piecewise-quadratic Lyapunovfunctions for systems with deadzones or saturations. Systems and ControlLetters, 58(5):365–371, 2009b.

D. Dai, T. Hu, A.R. Teel, and L. Zaccarian. Output feedback synthesis forsampled-data system with input saturation. In American Control Conference,pages 1797–1802, Baltimore (MD), USA, June 2010.

S. Formentin, S.M. Savaresi, L. Zaccarian, and F. Dabbene. Randomizedanalysis and synthesis of robust linear static anti-windup. In Conference onDecision and Control, pages 4498–4503, Florence (Italy), December 2013.

52 / 58


Bibliography II

S. Formentin, F. Dabbene, R. Tempo, L. Zaccarian, and S.M. Savaresi.Scenario optimization with certificates and applications to anti-windupdesign. In IEEE Conference on Decision and Control, pages 2810–2815, LosAngeles (CA), USA, December 2014.

F. Forni, S. Galeani, and L. Zaccarian. An almost anti-windup scheme forplants with magnitude, rate and curvature saturation. In American ControlConference, pages 6769–6774, Baltimore (MD), USA, June 2010.

F. Forni, S. Galeani, and L. Zaccarian. Model recovery anti-windup forcontinuous-time rate and magnitude saturated linear plants. Automatica, 48(8):1502–1513, 2012.

P. Gahinet and P. Apkarian. A linear matrix inequality approach to H∞control. Int. J. Robust and Nonlinear Control, 4:421–448, 1994.

S. Galeani, S. Tarbouriech, M.C. Turner, and L. Zaccarian. A tutorial onmodern anti-windup design. European Journal of Control, 15(3-4):418–440,2009.

A. Garulli, A. Masi, G. Valmorbida, and L. Zaccarian. Global stability and finiteL2m-gain of saturated uncertain systems via piecewise polynomial Lyapunovfunctions. IEEE Transactions on Automatic Control, 58(1):242–246, 2013.

53 / 58


Bibliography III

J.M. Gomes da Silva Jr and S. Tarbouriech. Anti-windup design withguaranteed regions of stability: an LMI-based approach. IEEE Trans. Aut.Cont., 50(1):106–111, 2005.

G. Grimm, J. Hatfield, I. Postlethwaite, A.R. Teel, M.C. Turner, andL. Zaccarian. Antiwindup for stable linear systems with input saturation: anLMI-based synthesis. IEEE Transactions on Automatic Control, 48(9):1509–1525, September 2003a.

G. Grimm, I. Postlethwaite, A.R. Teel, M.C. Turner, and L. Zaccarian. Casestudies using linear matrix inequalities for optimal anti-windup synthesis.European Journal of Control, 9(5):459–469, 2003b.

G. Grimm, A.R. Teel, and L. Zaccarian. Robust linear anti-windup synthesis forrecovery of unconstrained performance. Int. J. Robust and NonlinearControl, 14(13-15):1133–1168, 2004.

T. Hu, A.R. Teel, and L. Zaccarian. Stability and performance for saturatedsystems via quadratic and non-quadratic Lyapunov functions. IEEETransactions on Automatic Control, 51(11):1770–1786, November 2006.

54 / 58


Bibliography IV

T. Hu, A.R. Teel, and L. Zaccarian. Anti-windup synthesis for linear controlsystems with input saturation: achieving regional, nonlinear performance.Automatica, 44(2):512–519, 2008.

T. Iwasaki and R.E. Skelton. All controllers for the general H∞ controlproblem: LMI existence conditions and state space formulas. Automatica, 30(8):1307–17, August 1994.

J.B. Lasserre. Reachable, controllable sets and stabilizing control ofconstrained linear systems. Automatica, 29(2):531–536, 1992.

A. Megretski. L2 BIBO output feedback stabilization with saturated control. In13th Triennial IFAC World Congress, pages 435–440, San Francisco, USA,1996.

F. Morabito, A.R. Teel, and L. Zaccarian. Nonlinear anti-windup applied toEuler-Lagrange systems. IEEE Trans. Rob. Aut., 20(3):526–537, 2004.

E.F. Mulder, M.V. Kothare, and M. Morari. Multivariable anti-windupcontroller synthesis using linear matrix inequalities. Automatica, 37(9):1407–1416, September 2001.

55 / 58


Bibliography V

L. Pagnotta, L. Zaccarian, A. Constantinescu, and S. Galeani. Anti-windupapplied to adaptive rejection of unknown narrow band disturbances. InEuropean Control Conference, pages 150–157, Kos (Greece), July 2007.

I. Queinnec, S. Tarbouriech, S. Gayadeen, and L. Zaccarian. Static anti-windupdesign for discrete-time large-scale cross-directional saturated linear controlsystems. In IEEE Conference on Decision and Control, Osaka, Japan,December 2015.

E.D. Sontag. An algebraic approach to bounded controllability of linearsystems. Int. J. of Control, 39(1):181–188, 1984.

S. Tarbouriech, G. Garcia, J.M. Gomes da Silva Jr., and I. Queinnec. Stabilityand stabilization of linear systems with saturating actuators. Springer-VerlagLondon Ltd., 2011.

A.R. Teel and N. Kapoor. The L2 anti-windup problem: Its definition andsolution. In European Control Conference, Brussels, Belgium, July 1997.

A.R. Teel, L. Zaccarian, and J. Marcinkowski. An anti-windup strategy foractive vibration isolation systems. Control Engineering Practice, 14(1):17–27, 2006.

56 / 58


Bibliography VI

F. Todeschini, S. Formentin, G. Panzani, M. Corno, S. Savaresi, andL. Zaccarian. Nonlinear pressure control for bbw systems via dead zone andanti-windup compensation. IEEE Transactions on Control SystemsTechnology, to appear, 2015.

M.C. Turner, G. Herrmann, and I. Postlethwaite. Incorporating robustnessrequirements into antiwindup design. IEEE Transactions on AutomaticControl, 52(10):1842–1855, 2007.

R. Vitelli, L. Boncagni, F. Mecocci, S. Podda, V. Vitale, and L. Zaccarian. Ananti-windup-based solution for the low current nonlinearity compensation onthe FTU horizontal position controller. In Conference on Decision andControl, pages 2735–2740, Atlanta (GA), USA, December 2010.

L. Zaccarian and A.R. Teel. A common framework for anti-windup, bumplesstransfer and reliable designs. Automatica, 38(10):1735–1744, 2002.

L. Zaccarian and A.R. Teel. The L2 (l2) bumpless transfer problem: itsdefinition and solution. Automatica, 41(7):1273–1280, 2005.

L. Zaccarian and A.R. Teel. Modern anti-windup synthesis: controlaugmentation for actuator saturation. Princeton University Press, Princeton(NJ), 2011.

57 / 58


Bibliography VII

L. Zaccarian, A.R. Teel, and J. Marcinkowski. Anti-windup for an activevibration isolation device: theory and experiments. In Proceedings of theAmerican Control Conference, pages 3585–3589, Chicago (IL), USA, June2000.

L. Zaccarian, D. Nesic, and A.R. Teel. L2 anti-windup for linear dead-timesystems. Systems and Control Letters, 54(12):1205–1217, 2005.

L. Zaccarian, E. Weyer, A.R. Teel, Y. Li, and M. Cantoni. Anti-windup formarginally stable plants and its application to open water channel controlsystems. Control Engineering Practice, 15(2):261–272, 2007.

58 / 58

anti-windup solutions for input saturated control...

Documents