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Multi-stage Anti-windup Design for Linear Systems with Saturation
Nonlinearity: Enlargement of the Domain of Attraction
Maopeng Ran ·Qing Wang ·Chaoyang Dong ·Maolin Ni
Received: 25 July 2014/ Revised: 25 January 2015
Abstract This paper considers the multi-stage anti-windup
(AW) design for linear systems with saturation nonlinearity,
with the objective of enlarging the domain of attraction of
the resulting closed-loop system. We present results for both
static and dynamic AW compensation gains, in terms of lin-
ear matrix inequalities (LMIs). Iterative algorithms are re-
spectively established to obtain the AW compensation gains
that maximize the estimate of the domain of attraction. In
addition, a particle swarm optimization (PSO) based sys-
tematic method is proposed to determine the design point of
the multi-stage AW scheme, and to set the initial conditions
of the established iterative algorithms. Using the proposed
method, benefits of the multi-stage AW on the domain of
attraction are illustrated through a benchmark example.
Keywords Multi-stage anti-windup ·Saturation nonlinear-
ity ·Domain of attraction ·Particle swarm optimization
1 Introduction
It has been well-recognized that saturation nonlinearity af-
fects virtually all practical control systems. Due to satura-
tion, the actual plant input will be different from the con-
troller output, which causes performance degradation and
may even induce instability. Designing a high-performance
M. Ran (4)·Q. Wang
School of Automation Science and Electrical Engineering, Beihang
University, Beijing 100191, China
E-mail: rmppinbo@asee.buaa.edu.cn, wangqing@buaa.edu.cn
C. Dong
School of Aeronautic Science and Engineering, Beihang University,
Beijing 100191, China
E-mail: dongchaoyang@buaa.edu.cn
M. Ni
Chinese Society of Astronautics, Beijing 100048, China
E-mail: niml@bice.org.cn
controller for dynamic systems subject to saturation nonlin-
earity has received an increasing attention in academia and
industry over the past several decades [1–6]. Various de-
sign methods for dealing with saturation nonlinearity have
been developed, which can be generally classified into two
broad classes: the one-step approach and the two-step ap-
proach. The one-step approach, i.e., an approach that takes
the saturation nonlinearity explicitly into account when de-
signing controllers. Although this methodology is satisfac-
tory in principle, it has often been criticized for its conser-
vatism [3]. In the two-step approach, a nominal controller
which not accounts for the saturation nonlinearity is first de-
signed to achieve some nominal performance requirements.
Then, a compensation term is added to the nominal con-
troller to minimize the adverse effects of saturation. Such
an approach is called anti-windup (AW) and the compensa-
tion term is referred to as AW compensator. In this paper, we
concentrate on the design of the AW compensator.
In traditional AW design, a single AW compensator (or
set of gains) is designed and set to be activated as soon as
saturation occurs. Almost all AW designs were based on this
paradigm (e.g., [7–10]). However, in recent years, several
AW schemes that use different activation mechanisms have
been proposed in the literature. In [11,12], a modified AW
scheme which called as delayed AW was proposed. The de-
layed AW is not to activate the AW compensator immedi-
ately when saturation is encountered, but instead to allow
saturated actuators act unassisted up to a pre-designed point.
In [13,14], Wu and Lin proposed a modified AW scheme op-
posites to the delayed AW which is referred to as anticipa-
tory AW. Considering the system dynamical nature, the basic
idea of the anticipatory AW is to activate the AW compen-
sator in anticipation of actuator saturation. Further modifi-
cation of the AW scheme can be found in [15,16], in which
the proposed AW scheme consists of two AW compensators:
one activated at the occurrence of actuator saturation (imme-
2 Maopeng Ran et al.
diate AW compensator) and the other activated in delayed of
actuator saturation (delayed AW compensator).
The modified AW schemes mentioned above add not
too much complexity, but indeed have the potential of fur-
ther improving the closed-loop performance in tracking ref-
erence signals. Except the transient performance, enlarging
the domain of attraction of the system with saturation non-
linearity is also an important index to measure the improve-
ment in AW design [17–21]. In [22], a linear matrix in-
equalities (LMIs) based analysis approach has been devel-
oped to enlarge the domain of attraction of the closed-loop
system under a pre-designed nominal controller. Based on
the work of [22], several improved analysis approaches have
been proposed to reduce the conservativeness of the result-
ing domains [23,24]. Typically, it has been shown that the
anticipatory AW could obtain a larger domain of attraction
than the delayed AW and the traditional AW, both in static
case [13] and dynamic case [25]. We note that the effects of
using multiple sets of AW gains on domain of attraction still
remain an open problem.
In this paper, we further investigate the possible benefits
of the multi-stage AW scheme proposed in [15,16], in terms
of domain of attraction. Our work is based on extending
the AW synthesis approach established in [22,26] to include
a static immediate AW compensator and a static/dynamic
delayed AW compensator. The actual saturation element is
modeled as a time varying gain, and the artificial satura-
tion element in the multi-stage AW scheme is modeled by
polytopic representation method. Then, iterative LMI-based
algorithms are established to design the AW compensators
that lead to the largest estimate of the domain of attrac-
tion of the resulting closed-loop system. Both static and dy-
namic AW compensation gains are considered. Finally, a
population-based optimization technique, particle swarm op-
timization (PSO), is utilized to determine the design point of
the multi-stage AW scheme, and search for the best initial-
ization of the established iterative algorithms. Unlike the tra-
ditional methods in which these free design parameters can
only be selected by trial and error according to the computa-
tional results, the PSO-based approach provides a systematic
way to determine these parameters.
The reminder of this paper is organized as follows. In
Section 2 we give a general description of the multi-stage
AW scheme. In Section 3 we establish the LMI-based iter-
ative algorithms to design the AW compensation gains that
maximize the estimate of the domain of attraction. Section 4
presents the PSO-based parameter selection method. Section
5 presents the simulation results of a well-known numerical
example. A brief conclusion in Section 6 ends the paper.
The notation in this paper is standard. Ris the set of real
numbers. ATis the transpose of a real matrix A. The ma-
trix inequality A>B(A≥B)means that Aand Bare square
Hermitian matrices and A−Bis positive (semi-) definite. I
and 0 denote the identity matrix and zero matrix with ap-
propriate dimensions, respectively. A block diagonal matrix
with sub-matrices X1,X2,.. . ,Xpin its diagonal will be de-
noted by diag{X1,X2,...,Xp}. For a non-Hermitian real ma-
trix, He(A) = A+AT.
2 Problem statement
Consider the following linear plant with saturation nonlin-
earity
P:˙xp=Apxp+Bpsath(u)
y=Cpxp
(1)
where xp∈Rnpis the plant state, u∈Rnuis the control input,
y∈Rnyis the measured output, and Ap,Bp, and Cpare real
constant matrices of appropriate dimensions. The function
sath(u):Rnu→Rnuis the standard decentralized saturation
function defined as
sath(u) = [sath1(u1),...,sathnu(unu)]T,
sathi(ui) = sign(ui)min{hi,|ui|} (2)
with h=diag{h1,...,hnu},hiis the saturation limit for i-th
input .
Assume that a linear controller of the form
C:˙xc=Acxc+Bcy
u=Ccxc+Dcy(3)
has been designed. Here xc∈Rncis the controller state, and
Ac,Bc,Cc, and Dcare real constant matrices of appropri-
ate dimensions. The linear controller guarantees the stability
of the closed-loop system and meets some performance re-
quirements in the absence of actuator saturation.
In the traditional AW design, a correction term propor-
tional to the difference between the controller output and the
actual plant input q=u−sath(u)is added to the linear con-
troller, that is
˙xc=Acxc+Bcy−Eq
u=Ccxc+Dcy(4)
where Eis the static AW compensation gain. It is straight-
forward to see that the compensator is activated as soon as
saturation occurs (i.e.,q6=0 ).
In [16], the multi-stage AW scheme that consists of an
immediate AW compensator and a delayed AW compen-
sator was proposed (see Fig.1). An artificial saturation el-
ement with a higher saturation bound h/gdwas added, here
0<gd<1 is a design variable specified by designer. When
the two AW compensators both have static gains, the result-
ing compensated controller can be written as
˙xc=Acxc+Bcy−Eq −Edqd
u=Ccxc+Dcy(5)
Multi-stage AW Synthesis: Enlargement of the Domain of Attraction 3
Fig. 1 Multi-stage AW scheme: P,C,AW and AWdare the plant, the
linear controller, the immediate AW compensator and the delayed AW
compensator, respectively.
where q=ud−ˆu,qd=u−ud,Eand Edare the immediate
AW compensation gain and the delayed AW compensation
gain, respectively.
Letting the delayed AW compensator has dynamic gains,
that is
˙xaw =Aawxaw +Baw qd
η=Cawxaw +Dawqd
(6)
where xaw ∈Rnaw is the delayed AW compensator state, Aaw,
Baw,Caw, and Daw are real constant matrices of appropri-
ate dimensions, ηis the output of the compensator, then the
compensated controller can be correspondingly written as
˙xc=Acxc+Bcy−Eq +η
u=Ccxc+Dcy(7)
In this paper, we will evaluate how the multi-stage AW
scheme affects the size of the achievable domain of attrac-
tion of the closed-loop system. Both the controller (5) and
(7) will be considered. For simplicity, we first concentrate
on single actuator plants, and the results can be readily ex-
tended to multi actuator plants, to be presented latter.
3 Multi-stage AW compensation gain design
3.1 Static AW gains
To rewrite the closed-loop system depicted in Fig.1, we first
replace the actual saturation element with the following time
varying gain k(t),
k(t) = ˆu(t)
ud(t)(8)
When |u|<h, no saturation occurs, k(t) = 1. When h≤ |u|<
h/gd, only the actual saturation is in effect, gd<k(t)≤1.
When |u| ≥ h/gd, both saturation elements are activated,
k(t) = gd. Thus, we have k(t)∈[gd,1]. Considering sath/gd(u)
=h
gdsat1(gd
hu), the closed-loop system can be written as
˙x= (A−BF)x+B1sat1(F1x)
u=Fx (9)
where
x=xp
xc,
A=Ap0
BcCpAc,B=0
Ed,
B1="Bpkh
gd
(Ed−E+EK )h
gd#,
F=DcCpCc,F1=gd
hF.
Remark 1 Modeling the saturation element as a time vary-
ing gain has been attempted before [11,13]. In this paper,
the synthesis results will be formulated and solved as some
LMI-based optimization problems. Due to linearity, we only
need to check the resulting LMIs on the extreme values of
k(t): 1 and gd.
Define the symmetric polyhedron L(F1) = {x∈Rnp+nc|
|f1ix| ≤ 1,i=1,2,...,nu}, here f1iis the i-th row of the
matrix F1. Note that L(F1)stands for the unsaturated zone
of the closed-loop system (9).
Similar to [22,26], we use a contractively invariant el-
lipsoid to estimate the domain of attraction of the closed-
loop system. Define V(x) = xTPx,P∈R(np+nc)×(np+nc)is a
positive-definite matrix. The ellipsoid Ω(P) = {x∈Rnp+nc|
xTPx ≤1}is said to be contractively invariant if ∀x∈Ω(P)\0,
˙
V(x) = 2xTP((A−BF)x+B1sat1(F1x)) <0 (10)
Clearly, if Ω(P)is contractively invariant, then it is inside
the domain of attraction of the closed-loop system.
For any two matrices F1,H∈Rnu×(np+nc)and a vector
v∈V,V={v∈Rnu|vi=1 or 0}, denote
M(v,F1,H) = diag{v1,v2,...,vnu}F1
+(I−diag{v1,v2,...,vnu})H(11)
Let hibe the i-th row of the matrix H. We arrive at the
following lemma.
Lemma 1 Given an ellipsoid Ω(P), if there exists an H∈
Rnu×(np+nc)such that
(A−BF +B1M(v,F1,H))TP+P(A−BF
+B1M(v,F1,H)) <0,∀v∈V,k∈ {1,gd}(12)
and Ω(P)⊂L(H),i.e.,|hix| ≤ 1, i=1,2,...,nu, for all
x∈Ω(P), then Ω(P)is a contractively invariant set of the
closed-loop system (9).
Let χR=co{x1,x2,...,xl}be a reference shape set. Here
x1,x2,... ,xlare some priori given points in Rnp+nc, co{·}
denotes the convex hull of a set. With Lemma 1, we can
4 Maopeng Ran et al.
choose the largest ellipsoid through the following optimiza-
tion problem,
max
P>0,Hα,
s.t. a)αχR⊂Ω(P),
b) (A−BF +B1M(v,F1,H))TP+P(A−BF
+B1M(v,F1,H)) <0,∀v∈V,k∈ {1,gd},
c)|hix| ≤ 1,i=1,2,...,nu,∀x∈Ω(P).
(13)
Define Q=P−1,γ=α−2, and G=HQ. Let the i-th
row of Gbe gi. Then M(v,F1,H)Q=M(v,F1Q,G). The op-
timization problem (13) can be rewritten as
min
Q>0,Gγ,
s.t. a)γxT
i
xiQ≥0,i=1,2,...,l,
b)Q(A−BF)T+ (A−BF)Q+MT(v,F1Q,G)BT
1
+B1M(v,F1Q,G),∀v∈V,k∈ {1,gd},
c)1gi
gT
iQ≥0,i=1,2,...,nu.
(14)
Note that the AW compensation gains Eand Edare em-
bedded in Band B1, thus the optimization (14) cannot be
transformed into LMIs in terms of variables E,Ed,Q, and
G. Denote
P=P
1P
12
PT
12 P
2,M(v,F1,H) = [M1M2](15)
where P
1∈Rnp×np,P
12 ∈Rnp×nc,P
2∈Rnc×nc,M1∈Rnu×np,
and M2∈Rnu×nc. Then the condition b) in (13) can be changed
to
Σ=HeΣ11 Σ12
Σ21 Σ22 <0,∀v∈V,k∈ {1,gd}(16)
where
Σ11 =P
1Ap+P
12BcCp−P
12EdDcCp+P
1Bpkh
gd
M1
+P
12(Ed−E+E k)h
gd
M1,
Σ12 =P
12Ac−P
12EdCc+P
1Bpkh
gd
M2
+P
12(Ed−E+E k)h
gd
M2,
Σ21 =PT
12Ap+P
2BcCp−P
2EdDcCp+PT
12Bpkh
gd
M1
+P
2(Ed−E+Ek)h
gd
M1,
Σ22 =P
2Ac−P
2EdCc+PT
12Bpkh
gd
M2
+P
2(Ed−E+Ek)h
gd
M2.
The inequality above is LMI in P
1,Eand Ed. This in-
dicates that with fixed P
12,P
2and H, one can determine
the AW compensation gains Eand Edto make the region
{xp∈Rnp|xT
pP
1xp≤1}as large as possible. Based on the
above analysis and the iterative algorithm in [22], we estab-
lish the following iterative algorithm.
Algorithm 1 (Iterative algorithm for determining AW com-
pensation gains E and Ed)
Step 1) For a given reference set χRand E=0, Ed=0,
solve the optimization problem (14). Denote the solution
as γ0,Q0and G0. Set χR=γ−1/2
0χR.
Step 2) Set i=1 and γopt =1. Set Eand Edwith initial
values E0and E0
d, respectively.
Step 3) Solve the optimization problem (14) for γ,Qand G.
Denote the solution as γi,Qand G.
Step 4) Let γopt =γiγo pt,χR=γ−1/2
iχR,P=Q−1and H=
GQ−1.
Step 5) IF |γi−1|<δ, a pre-determined tolerance, GOTO
Step 7), ELSE GOTO Step 6).
Step 6) Solve the following LMI optimization problem
min
P
1>0,E,Ed
γ,
s.t. a)γxT
i
xiP−1≥0,i=1,2,...,l,
b)Σ<0,∀v∈V,k∈ {1,gd},
c)1hi
hT
iP≥0,i=1,2,...,nu.
(17)
Set the solution as Eand Ed, and i=i+1, then GOTO
Step 3).
Step 7) IF γopt ≤1, then α=γ−1/2
opt and Eand Edare feasi-
ble solutions and STOP, ELSE set Eand Edwith another
initial values and GOTO Step 2).
3.2 Combination of Dynamic and Static AW
With a static immediate AW compensator and a dynamic
delayed AW compensator, the closed-loop system depicted
in Fig.1 can be written as
˙x= (A−BF)x+B1sat1(F1x)
u=Fx (18)
where
x=
xp
xc
xaw
,
Multi-stage AW Synthesis: Enlargement of the Domain of Attraction 5
A=
Ap0 0
BcCpAcCaw
0 0 Aaw
,B=
0
−Daw
−Baw
,
B1=
Bpkh
gd
(−E+Ek −Daw )h
gd
−Baw h
gd
,
F=DcCpCc0,F1=gd
hF.
Define the ellipsoid Ω(P) = {x∈Rnp+nc+naw |xTPx ≤1}
and let χR=co{x1,x2,...,xl}be a reference shape set. Here
x1,x2,... ,xlare some priori given points in Rnp+nc+naw .
Based on Lemma 1, we arrive at the following optimization
problem to enlarge the estimate of the domain of attraction
of the closed-loop system (18),
max
P>0,Hα,
s.t. a)αχR⊂Ω(P),
b) (A−BF +B1M(v,F1,H))TP+P(A−BF
+B1M(v,F1,H)) <0,∀v∈V,k∈ {1,gd},
c)|hix| ≤ 1,i=1,2,...,nu,∀x∈Ω(P).
(19)
Here, hiis the i-th row of the matrix H,H∈Rnu×(np+nc+naw).
Define Q=P−1,γ=α−2, and G=HQ. Let the i-th row
of Gbe gi. The optimization problem (19) can be rewritten
as
min
Q>0,Gγ,
s.t. a)γxT
i
xiQ≥0,i=1,2,...,l,
b)Q(A−BF)T+ (A−BF)Q+MT(v,F1Q,G)BT
1
+B1M(v,F1Q,G),∀v∈V,k∈ {1,gd},
c)1gi
gT
iQ≥0,i=1,2,...,nu.
(20)
Note that the optimization problem above is linear in
terms of variables Qand G. Next, define
P=
P
1P
12 P
13
PT
12 P
2P
23
PT
13 PT
23 P
3
,M(v,F1,H) = [M1M2M3](21)
where P
1∈Rnp×np,P
12 ∈Rnp×nc,P
13 ∈Rnp×naw ,P
2∈Rnc×nc,
P
23 ∈Rnc×naw ,P
3∈Rnaw×naw ,M1∈Rnu×np,M2∈Rnu×nc,
M3∈Rnu×naw . Then, condition b) in (19) is equivalent to
Θ=He
Θ11 Θ12 Θ13
Θ21 Θ22 Θ23
Θ31 Θ32 Θ33
<0,∀v∈V,k∈ {1,gd}(22)
where
Θ11 =P
1Ap+P
12BcCp−P
12Daw DcCp
−P
13Baw DcCpP
1Bpkh
gd
M1
+P
12(−E+E k −Daw)h
gd
M1−P
13Baw
h
gd
M1,
Θ12 =P
12Ac−P
12DawCc−P
13BawCcP
1Bpkh
gd
M2
+P
12(−E+E k −Daw)h
gd
M2−P
13Baw
h
gd
M2,
Θ13 =P
12Caw +P
13Aaw P
1Bpkh
gd
M3
P
12(−E+E k −Daw)−P
13Baw
h
gd
M3,
Θ21 =PT
12Ap+P
22BcCp−P
22Daw DcCp
−P
23Baw DcCpPT
12Bpkh
gd
M1
+P
22(−E+E k −Daw)h
gd
M1−P
23Baw
h
gd
M1,
Θ22 =P
22Ac−P
22DawCc−P
23BawCc+PT
12Bpkh
gd
M2
+P
22(−E+E k −Daw)h
gd
M2−P
23Baw
h
gd
M2,
Θ23 =P
22Caw +P
23Aaw PT
12Bpkh
gd
M3
+P
22(−E+E k −Daw)h
gd
M3−P
23Baw
h
gd
M3,
Θ31 =PT
13 +PT
23BcCp−PT
23Daw DcCp
−P
33Baw DcCpPT
13Bpkh
gd
M1
+PT
23(−E+E k −Daw)h
gd
M1−P
33Baw
h
gd
M1,
Θ32 =PT
23Ac−PT
23DawCc−P
33BawCcPT
13Bpkh
gd
M2
+PT
23(−E+E k −Daw)h
gd
M2P
33Baw
h
gd
M2,
Θ33 =PT
23Caw +P
33Aaw PT
13Bpkh
gd
M3
+PT
23(−E+E k −Daw)h
gd
M3−P
33Baw
h
gd
M3.
The inequality above is linear in terms of variables P
1,
E,Aaw,Baw ,Caw, and Daw . Thus, we arrive at the following
iterative algorithm to design the static immediate AW com-
pensation gain and the dynamic delayed AW compensation
gains to make the estimate of the domain of attraction of the
closed-loop system (18) as large as possible.
Algorithm 2 (Iterative algorithm for determining AW com-
pensation gains E, Aaw , Baw, Caw , and Daw)
6 Maopeng Ran et al.
Step 1) For a given reference set χRand E=0, Aaw =0,
Baw =0, Caw =0, and Daw =0, solve the optimization
problem (20). Denote the solution as γ0,Q0and G0. Set
χR=γ−1/2
0χR.
Step 2) Set i=1 and γopt =1. Set E,Aaw ,Baw,Caw, and
Daw with initial values E0,A0
aw,B0
aw,C0
aw, and D0
aw, re-
spectively.
Step 3) Solve the optimization problem (20) for γ,Qand G.
Denote the solution as γi,Qand G.
Step 4) Let γopt =γiγo pt,χR=γ−1/2
iχR,P=Q−1and H=
GQ−1.
Step 5) IF |γi−1|<δ, a pre-determined tolerance, GOTO
Step 7), ELSE GOTO Step 6).
Step 6) Solve the following LMI optimization problem
min
P
1>0,E,Aaw,Baw ,Caw,Daw
γ,
s.t. a)γxT
i
xiP−1≥0,i=1,2,...,l,
b)Θ<0,∀v∈V,k∈ {1,gd},
c)1hi
hT
iP≥0,i=1,2,...,nu.
(23)
Set the solution as E,Aaw,Baw ,Caw, and Daw, and i=
i+1, then GOTO Step 3).
Step 7) IF γopt ≤1, then α=γ−1/2
opt and E,Aaw ,Baw,Caw,
and Daw are feasible solutions and STOP, ELSE set E,
Aaw,Baw ,Caw, and Daw with another initial values and
GOTO Step 2).
Remark 2 For multiple input systems (i.e.,nu>1), the time
varying gain k(t)and the design variable gddefined before
respectively become the following nu×nudiagonal matri-
ces:
K(t) = diag{k1(t),k2(t),...,knu(t)},
Gd=diaggd1,gd2,...,gdnu
where ki(t)∈[gdi,1]. Here 0 <gdi<1, i=1,2,...,nu, is the
design point chosen for the i-th actuator. Define
K=K(t) = diag{k1(t),k2(t),...,knu(t)}|ki(t) = gdior 1
(24)
Due to linearity, any K(t)can be represented as a linear com-
bination of extreme values evaluated at the corners of the
parameter hypercube K,
K(t) =
2nu
∑
i=1
αi(t)Ki(25)
where Ki∈K,i=1,2,...,2nu, and αi(t)≥0 with ∑2nu
i=1αi(t)
=1. Then, in the optimization problems established in this
paper, we only need to check the LMIs at the vertices ob-
tained from K(t)(i.e.,Ki∈K,i=1,2,...,2nu).
Remark 3 Different from [15,16], in which the authors vali-
dated the superiority of the multi-stage AW scheme by com-
paring the tracking performance, the results in this section
allow one to further investigate the possible benefits of the
multi-stage AW scheme in enlarging the domain of attrac-
tion of the resulting closed-loop system. On the other hand,
since the multi-stage AW scheme consists of a traditional
AW loop and a delayed AW loop, the obtained results can
be also readily extended to the traditional AW scheme and
the delayed AW scheme. Let Ed=0, then Algorithm 1 will
be reduced to the algorithm obtained in [22] which is for the
traditional AW scheme. Let E=0, then Algorithm 1 will be
reduced to the algorithm obtained in [13] which is for the
static delayed AW scheme, and Algorithm 2 will be reduced
to the algorithm obtained in [25] which is for the dynamic
delayed AW scheme.
4 PSO-based parameter selection
It should be pointed out that the multi-stage AW scheme can
provide better results than the traditional AW scheme, but
strongly depends on the choice of the design point Gd. How-
ever, no systematic method that determines Gdis available
in the literature until now. On the other hand, it is well rec-
ognized that for the nonlinear optimization problems solved
by Algorithm 1 and Algorithm 2, the optimization results
depend on the given initial conditions. In the original paper
[22], the initialization was left to be given by trial and error
based on the obtained optimization results. Such a problem
also exists in [23]. In a recent research [24], Li and Lin use
an optimal solution from the work of [19] as the initializa-
tion, but such a selection does not guarantee the solution is
global either.
On the other hand, PSO is a population-based global op-
timization technique developed by Kennedy and Eberhart in
1995 [27]. In PSO, the system is initialized with a popula-
tion of random solutions and searches for optima by updat-
ing generations. In recent several years, PSO has become
quite popular in control engineering [28–32]. As far as our
knowledge goes, no work of applying PSO to anti-windup
problems has been reported in the literature before. In this
paper, we use the PSO algorithm to decide the design point
Gdand to give the initialization values of the established it-
erative algorithms. With the application of PSO algorithm,
one can obtain an optimal selection of the design point Gd
and the initialization values of the iterative algorithms, in a
relatively more systematic way. In addition, the algorithm
is easy to be implemented, and it’s benefits will grow when
the dimension of the problem increases. For example, for
a system with nc=10 and nu=10, there will be 210 ele-
ments in Gd,E0, and E0
d(510 elements in Gd,E0,A0
aw,B0
aw,
C0
aw, and D0
aw), and thus it is almost impossible to determine
these parameters by trial and error. However, the PSO algo-
Multi-stage AW Synthesis: Enlargement of the Domain of Attraction 7
rithm is a computational intelligence-based technique that is
not largely affected by the size of the optimization problem
[35].
In PSO each particle is treated as a potential solution of
the optimization problem, and initialized with a random po-
sition and velocity. Then, all the particles “fly” in the search
space to track the feasible solutions. During the fly progress,
a particle will track two best positions: one is the best posi-
tion so far found by the particle itself, and the other is the
best position so far found by the swarm. The search path is
different for each particle, thus guarantees a wide area of the
search space is explored and increases the chance of finding
the optimal solution. The ith particle in the swarm updates
its velocity and position according to the following iterative
equations:
vk+1
i j =ωvk
i j +c1r1(pk
i j −xk
i j) + c2r2(pk
g j −xk
i j)(26)
xk+1
i j =xk
i j +vk+1
i j (27)
where i=1,2,...,N,Nis the population size, j=1,2,...,D,
Dis the dimension of the search space, k=1,2,...,kmax,
kmax is the maximum iteration, vk
i j and xk
i j are the velocity
and position of jth dimension of particle iat iteration k, re-
spectively, pk
i j is the best position of jth dimension found by
article iuntil iteration k,pk
g j is the best position of jth di-
mension found by the swarm until iteration k,r1and r2are
independent random numbers between 0 and 1, c1and c2
are learning factors, ωis the inertia weight specifying how
much the current velocity will affect the new velocity vector.
The inertia weight ωat iteration kcan be given as [33]
ωk=ωstart −ωstart −ωend
kmax
×k(28)
where ωstart and ωend are the initial value and terminal value
of ω, respectively.
Here, we take the case where the two AW compensators
both have static gains as example to demonstrate the imple-
mentation of PSO. As we hope to obtain a largest domain
of attraction, the objective function in PSO algorithm can be
straightforward defined as
J=1
αopt
(29)
The decision variables are the design point Gd=diag{gd1,
... , gdnu}and the given initial values E0= [e0
i j]nc×nuand
E0
d= [e0
d,i j]nc×nu, and in PSO algorithm, they will be ex-
pressed as
X= [gd1,...,gdnu,e0
11,...,e0
ncnu,e0
d,11,...,e0
d,ncnu](30)
To prevent the values of the obtained AW compensation gains
Eand Edfrom being too large, we can constrain E= [ei j]nc×nu
and Ed= [ed,i j]nc×nuelement-by-element by setting
ϕi j ≤eij ≤ψi j ,ϕd,i j ≤ed,ij ≤ψd,i j (31)
Accordingly, in the PSO algorithm, the search range for the
design point Gdis set to be gdi∈(0 1),i=1,2,...,nu,
and for the AW compensation gains Eand Edare set to be
ei j ∈[ϕi j ψi j ]and ed,i j ∈[ϕd,i j ψd,i j ],i=1,2,...,nc,j=
1,2,...,nu, respectively.
Thus, the PSO-based iterative algorithm to decide Gd,
E0, and E0
dcan be stated as follows:
Algorithm 3 (Iterative algorithm for determining Gd, E0,
and E0
d)
Step 1) Set the swarm population size N, the maximal search
speed Vmax, the maximum search iteration number kmax,
the initial and terminal values of inertia weight ωsatrt and
ωend, and the search range of the decision variables. Ini-
tialize the position and velocity of each particle.
Step 2) Set p1
i j =x1
i j and p1
g j =p1
m j, where p1
m j satisfies
J(p1
m j) = min{J(p1
1j),...,J(p1
N j)}.
Step 3) Set k=k+1. Update the velocity and the position of
each particle, and the inertia weight ωaccording to (26),
(27) and (28), respectively. If vk
i j >Vmax, then vk
i j =Vmax.
If vk
i j <−Vmax, then vk
i j =−Vmax. Constrain the position
of each particle in the given search range.
Step 4) For each particle, calculate the objective function
(29) by using Algorithm 1.
Step 5) Update the particle itself best position and the swarm
best position respectively according to
pk+1
i j =(pk
i j,if J(pk
i j)≤J(xk+1
i j )
xk+1
i j ,else (32)
pk+1
g j =(pk
g j,if J(pk
g j)≤min{J(pk+1
1j),...,J(pk+1
N j )},
pk+1
m j ,else
(33)
where pk+1
m j satisfies J(pk+1
m j ) = min{J(pk+1
1j),...,J(pk+1
N j )}.
Step 6) IF the number of iteration kreaches the maximum
value kmax, then GOTO next step, else GOTO Step 3).
Step 7) The latest swarm best position pkmax
g j is an optimal
solution to Gd,E0, and E0
d, and J(pkmax
g j )is the optimal
objective function value.
Remark 4 When applying PSO, several settings must be
taken into account to facilitate the convergence and avoid
fall into local optimal. The search range of the decision vari-
ables is decided by the optimization problem itself. For ex-
ample, in this paper, the design point gdi,i=1,2,...,nu, is
constrained by 0 <gdi<1, and the AW compensation gains
can be limited by ±100. Vmax is the parameter that limits the
velocity of the particles. If the value of Vmax is too high, the
particles may fly past good solutions. If the value of Vmax
is too small, the particles’ movements are limited and may
not explore sufficiently beyond local solutions. In general,
8 Maopeng Ran et al.
Vmax is set at 10−20% of the search range of the variable
on each dimension [34]. The learning factors are often set to
be c1=c2=2. Population sizes of 20-50 are most common.
The inertia weight controls the exploration property of the
algorithm, a larger ωleads to a more global behavior and
a smaller ωresults in facilitating a more local behavior. In
general, ωstart and ωend are set to be 0.9 and 0.4, respectively.
5 Numerical example
We consider here a benchmark example also studied in [22].
The plant and linear controller matrices are given by
Ap=−0.1 0
0−0.1,Bp=1.5 4
1.2 3 ,Cp=1 0
0 1 ,
Ac=0 0
0 0 ,Bc=−1 0
0−1,Cc=0.3333 0
0−0.1,
Dc=−3.333 0
0 1 .
We first consider the case that all the AW compensation
gains are static. Let χR= [0.6 0.400]T,δ=10−4, and con-
strain each element of the compensation gains by ±100. Fig.
2 demonstrates the obtained ellipsoids by different settings
of the design point Gdand the initialization E0and E0
d. We
see that the obtained ellipsoids strongly depend on the given
values of Gd,E0and E0
d. As there are 10 elements in these
three parameters, it is not easy to find the best group of Gd,
E0and E0
dthat leads to the largest domain of attraction by a
trail and error method.
Then, we use Algorithm 3 to decide Gd,E0and E0
d. The
settings of PSO algorithm are listed in Table 1. The search
range of gdi,i=1,2, is (0,1). The search range of e0
i j and
e0
d,i j,i=1,2, j=1,2, is [−100 100].Vmax is set to be 15%
of the search range. We run Algorithm 3 for 10 times, and
the evolution histories of the objective function are depicted
in Fig. 3. Based on the optimization results, we select
Gd=diag{0.9752,0.9747},
E0=70.9697 36.4084
55.8063 65.0102 ,E0
d=18.2149 45.6080
18.2149 84.7193 ,
which, leads to α=81.9708, and
E=70.3940 83.1007
1.2076 −84.3258 ,Ed=83.6765 −83.7532
96.1066 −99.8825 ,
P
1=10−3×2.8254 2.4486
2.4486 2.6776 .
Plotted in Fig. 4 are the obtained ellipsoid and a state
trajectory that starts from a point on its bound. Also plot-
ted in the figure in a dotted line is the ellipsoid obtained
in [22]. It can be observed that the multi-stage AW scheme
achieves a significantly larger domain of attraction than the
traditional AW scheme. System response corresponding to
the trajectory is depicted in Fig. 5 and Fig. 6. As these fig-
ures suggest, both the immediate AW compensator and the
delayed AW compensator are in effect.
We next consider the situation when the delayed AW
compensator has dynamic gains. Also let χR= [0.6 0.400]T,
δ=10−4, and constrain each element of the compensation
gains by ±100. Using Algorithm 3, we choose
Gd=diag{0.9979,0.9968},
E0=−39.9210 61.8891
40.9158 51.4248 ,
A0
aw =−81.3937 97.8870
−83.1400 77.6196 ,B0
aw =−63.8194 13.8084
83.0660 80.0112 ,
C0
aw =83.1783 −77.0453
−34.1644 72.50488 ,D0
aw =99.9305 −98.4908
69.2371 −2.4055 ,
which, leads to α=109.4719, and
E=38.1685 23.8691
34.7836 −45.4089 ,
Aaw =−49.6898 −66.3143
18.2308 −53.6246 ,Baw =8.0438 −24.0927
98.9166 −61.6883 ,
Caw =−9.4493 51.9200
−99.3915 −49.4815 ,Daw =−86.1278 86.9735
−98.9582 95.6214 ,
P
1=10−3×1.6143 −1.4575
−1.4575 1.6822 .
The obtained ellipsoid is plotted in Fig. 7. We note that
letting the delayed AW compensator to be dynamic leads to
a larger domain of attraction than that obtained by two static
AW compensators. The dashed curves in Fig. 7 are the state
trajectories with the state initial conditions on the bound-
ary of the ellipsoid and the controller initial state [xT
c(0)
xT
aw(0)]T= [0 0]T. Clearly, all trajectories converge to the
origin.
6 Conclusion
This paper considered the multi-stage AW design for linear
systems with saturation nonlinearity, with the objective of
enlarging the domain of attraction of the resulting closed-
loop system. Iterative algorithms were established to obtain
the AW compensation gains, and a PSO-based symmetric
method was proposed to decide the parameters that cannot
be easily determined before. Simulation results confirmed
that the multi-stage AW scheme has the potential of lead-
ing to larger domain of attraction than the traditional AW
scheme, and PSO algorithm can be a useful compensatory
tuner in multi-stage AW design.
Acknowledgements This work was supported by the National Natu-
ral Science Foundation of China (Grant Nos. 61273083 and 61374012).
Multi-stage AW Synthesis: Enlargement of the Domain of Attraction 9
−40 −30 −20 −10 0 10 20 30 40
−40
−30
−20
−10
0
10
20
30
40
x1
x2
−30 −15 0 15 30 45−45
−40
−30
−20
−10
0
10
20
30
40
x1
x2
(a) E0=50 50
50 50 ,E0
d=10 10
10 10 (b) E0=10 10
10 10 ,E0
d=50 50
50 50
−40 −30 −20 −10 0 10 20 30 40
−40
−30
−20
−10
0
10
20
30
40
x1
x2
−40 −30 −20 −10 0 10 20 30 40
−40
−30
−20
−10
0
10
20
30
40
x1
x2
(c) E0=20 20
20 20 ,E0
d=80 80
80 80 (d) E0=80 80
80 80 ,E0
d=20 20
20 20
Fig. 2 Ellipsoids with different settings of Gd,E0, and E0
d. Solid line is with Gd=diag{0.80 0.80}, dotted line is with Gd=diag{0.90 0.90},
dash-dotted line is with Gd=diag{0.95 0.95}, and dashed line is with Gd=diag{0.98 0.98}.
Table 1 Settings of PSO algorithm
Population size Maximal iteration Learning factors Initial inertia weight Terminal inertia weight
N kmax c1and c2ωstart ωend
30 100 2 0.9 0.4
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