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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 Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
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
Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
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
Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
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
Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
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
Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
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
Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
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
Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
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
8 / 58
Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
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
9 / 58
Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
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
10 / 58
Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
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
11 / 58
Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
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
Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
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
• Performance objective:given s∗, minimize γdz (s∗)
• 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
Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
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
]
• Performance objective:given s∗, minimize γdz (s∗)
• 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
Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
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
Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
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
Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
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
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
17 / 58
Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
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
γ2 subject to Q = QT > 0, U > 0 diagonal,
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
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
18 / 58
Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
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
Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
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
Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
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
Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
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
Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
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
Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
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
Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
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
Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
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)
26 / 58
Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
Anti-windup extends to fully nonlinear compensationPagnotta et al. [2007]
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)
27 / 58
Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
Anti-windup extends to fully nonlinear compensationZaccarian and Teel [2005]
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)
28 / 58
Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
Anti-windup extends to fully nonlinear compensationForni et al. [2012, 2010]
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)
29 / 58
Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
Anti-windup extends to fully nonlinear compensationZaccarian et al. [2005]
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)
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Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
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
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Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
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
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Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
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
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Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
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
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Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
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
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Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
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
Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
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
Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
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
Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
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
Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
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!
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Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
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!
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Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
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
Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
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)
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Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
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
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Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
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]
Without anti−windupWith anti−windupWithout saturation
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
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Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
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]
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]
• Performance is dramatically improved (input authority is almost fullyexplotied)
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Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
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
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Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
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
Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
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)
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Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
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
Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
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
Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References
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