IEEE TRANSACTIONS ON AUTOMATIC CONTROL,VOL.50,NO.9,SEPTEMBER20051455
A CKNOWLEDGMENT
The authors thank the referees for their very careful reading of their
manuscript,and for their many insightful and constructive suggestions
which lead to a signi?cant improvement of the manuscript.
R EFERENCES
[1]G.B.Di Masi,Y.M.Kabanov,and W.J.Runggaldier,“Mean-variance
hedging of options on stocks with Markov volatility,”Theory Probab.
Appl.,vol.39,pp.173–181,1994.
[2]X.Guo,“Inside information and option pricings,”Quant.Finance,vol.
1,pp.38–44,2001.
[3],“An explicit solution to an optimal stopping problem with regime
switching,”J.Appl.Probab.,vol.38,pp.464–481,2001.
[4]I.Karatzas,“On the pricing of American options,”Appl.Math.Optim.,
vol.17,pp.37–60,1988.
[5]S.D.Jacka,“Optimal stopping and the American put,”Math.Finance,
vol.1,pp.1–14,1991.
[6]H.P.McKean,“A free boundary problem for the heat equation arising
from a problem in mathematical economics,”Indust.Manage.Rev.,vol.
60,pp.32–39,1965.
[7] B.?ksendal,Stochastic Differential Equations,4th ed.New York:
Springer-Verlag,1995.
[8]G.Yin,R.H.Liu,and Q.Zhang,“Recursive algorithms for stock liq-
uidation:A stochastic optimization approach,”SIAM J.Optim.,vol.13,
pp.240–263,2002.
[9]Q.Zhang,“Stock trading:An optimal selling rule,”SIAM J.Control
Optim.,vol.40,pp.67–84,2001.
Iterative Learning Control for Systems
With Input Deadzone
Jian-Xin Xu,Jing Xu,and Tong Heng Lee
Abstract—Most iterative learning control(ILC)schemes proposed hith-
erto were designed and analyzed without taking the input deadzone into
account.Input deadzone is a kind of nonsmooth and nonaf?ne-in-input
factor widely existing in actuators or mechatronics devices.It gives rise to
extra dif?culty due to the presence of singularity in the input channels.In
this note,we disclose that ILC methodology remains effective for systems
with input deadzone that could be nonlinear,unknown and state-depen-
dent.Through rigorous proof,it is shown that despite the presence of the
input deadzone,the simplest ILC scheme retains its ability of achieving the
satisfactory performance.
Index Terms—Convergence analysis,input deadzone,iterative learning
control,nonlinear dynamics.
N OMENCLATURE
R n denotes the Euclidean space of dimension n;R denotes
the set of real numbers;Z+denotes the set of nonnegative in-
tegers;j g j denotes the absolute value of g;k g k denotes the
Euclidean norm of g;
K f0;1;...;N g where N is a?-
nite integer;k2K denotes the time instance;i2Z+de-notes the iteration number;l g denotes the Lipschitz constant of Manuscriptreceived July7,2004;revised December20,2004and May9, 2005.Recommended by Associate Editor Z.-P.Jiang.
The authors are with the Department of Electrical and Computer Engi-neering,National University of Singapore,Singapore117576,Singapore (e-mail:elexujx@https://www.doczj.com/doc/f08952275.html,.sg).
Digital Object Identi?er10.1109/TAC.2005.854658g(x;k)or g(x;k);
c k c k;
b
max x;k k b(x;k)k;
u max k j u d(k)j; v max k j v d(k)j;
max x;k j r(x;k)j;
m max x;k f m l(x;k);m r(x;k)g;In all notations,a quantity a associ-ated with the subscripts i;D;r;i,and d belongs respectively to the left region of the deadzone,the deadzone,the right region of the deadzone,the real system at the i-th iteration,and the desired system; Let32f i;d g; l;3(k
) l(x3;k),and r;3(k
) r(x3;k)are the leftand rightboundary point s of a deadzone;m l;3(k )m l(x3;k) and m r;3(k
)m r(x3;k)are the left and right slopes of a deadzone;
I D;3[ l;3(k); r;3(k)]is the deadzone;I l;3(01; l;3(k)]is the left region of the deadzone;I r;3[ r;3(k);+1)is the right region of the deadzone; l;
3
10 cb(x3;k)m l;3(k); r;3 10 cb(x3;k)m r;3(k);f i(k
)f(x i;k),and b i(k )b(x i;k).
I.I NTRODUCTION
In many industrial processes,often a task is repeated over a?nite operation cycle and the perfect tracking is required from the very be-ginning.Iterative learning control(ILC)has been proposed to deal with this class of control problems.However,most ILC schemes proposed hitherto were designed and analyzed without taking the input dead-zone into account.Input deadzone is a kind of nonsmooth and non-af?ne-in-input factor widely existing in actuators or mechatronics de-vices.It gives rise to extra dif?culty in control due to the presence of singularity in the input channels.The input deadzone problem has been studied by many researchers.Some useful techniques for over-coming deadzone are variable structure control and dithering.Moti-vated by pursuing better control performance,several adaptive inverse approaches were proposed[1].Recently,soft computing such as fuzzy logic and neural network-based control algorithms has also been ap-plied to handle problems relevant to deadzones[2],[3].In these works, the input deadzone was assumed to be independent of the system oper-ating conditions.Such an assumption does not hold when we deal with a control process with high precision requirement.
In this work,we disclose a?nding that ILC schemes[4],[5],orig-inally designed for systems without input deadzone,can effectively compensate the nonlinear deadzone through control repetitions.In a rigorous mathematical manner,we prove that,despite the presence of the input deadzone,the simplest ILC scheme retains its ability of achieving the satisfactory performance in tracking control.The non-linear state-dependent deadzone presents a new challenging problem to control theory in general,and to ILC convergence analysis in particular.To address this issue,the learning convergence is derived in two phases.First,the learning convergence to the desired regions is derived.In this phase,we prove that the actual control input sequence will enter the correct region after a?nite number of iterations,then stay forever.In the second phase,we will use the mathematical induction principle to prove the uniform convergence of the learning control inputsequence.
This note is organized as follows.The nomenclature of this note are ?rst listed.In Section II,the control problem is formulated.The re-gional learning convergence is proven in Section III and the asymptotic tracking convergence is analyzed in Section IV.
II.P ROBLEM F ORMULATION
Consider the following discrete-time dynamic system:
x i(k+1)=f(x i(k);k)+b(x i(k);k)u i(k)
u i(k)=DZ[v i(k)]
y i(k+1)=cx i(k+1)(1)
0018-9286/$20.00?2005IEEE
1456IEEE TRANSACTIONS ON AUTOMATIC CONTROL,VOL.50,NO.9,SEPTEMBER 2005
where i 2Z +;x i 2R n is the system state;y i 2R is the physically accessible system output;u i 2R is the plant input,but not available for control;v i 2R is the actual system input;k 2K ;f :R n 2K !R n ;b :R n 2K !R n ;c 2R 12n ,and DZ [1]represents the input deadzone.The characteristic of DZ [1]is de?ned as
u i (k )=DZ [v i (k )]
=m r;i (k )[v i (k )0 r;i (k )];v i (k )2I r;i (k )
0;
v i (k )2I D;i (k )m l;i (k )[v i (k )0 l;i (k )];v i (k )2I l;i (k )
(2)
where m l;i (k )>0;m r;i (k )>0; l;i (k ) 0,and r;i (k ) 0.The dynamical system (1)and the deadzone (2)are assumed to satisfy the following assumptions.
Assumption 1:f i (k );b i (k ); l;i (k ); r;i (k );m l;i (k ),and m r;i (k )are global Lipschitz continuous (GLC)with respect to x i (k )with un-known Lipschitz constants l f ;l b ;l ;l ;l ;r ;l m;l ,and l m;r respectively.
Denote l m =max f l m;l ;l m;r g . m
max x ;k
f m l;i (k );m r;i (k )
g is known.
Assumption 2:System (1)satis?es the identical initial condition
(i.i.c.):8i 2Z +; x i
(0)
x d (0)0x i (0)=0.Hence e i
(0)y d (0)0y i (0)=0.
Assumption 3:The prior information about c and b i (k )are the sign of cb i (k )and the bounds of c and b i (k ),i.e.,k c k c and k b i (k )k b .Assume cb i (k )>0in this note.
Remark 1:Regarding Assumption 2,the robustness of ILC with shift or uncertain initial conditions has been discussed in [4]and [6].In practice,a homing mechanism using a PI controller is able to achieve the identical initialization condition with suf?cient accuracy.
The control target is to ?nd a control input sequence v i (k ),such that y i (k )converges to y d (k )as i !1.Here we assume a desired control input v d (k )exists such that
x d (k +1)=f d (k )+b d (k )u d (k )
u d (k )=DZ [v d (k )]y d (k )=cx d (k )
(3)
where f d (k
)
f (x d ;t );b d (k
)b (x d ;k ),and u d (k )is the ideal control input without deadzone.To guarantee the realizability,assume 8k 2K ;j v d (k )j is bounded by an unknown constant v .Accordingly there is an unknown upper bound for u d (k ),denoted by u .
Remark 2:According to (2),if u d (k )=0;v d (k )could be any value belonging to I D;d (k ).
Now,we derive several relations which link x i (k )and e i (k )with u i (j
)u d (j )0u i (j )for j =0;111;k 01.According to (1),(3)and Assumption 1,8k 2K ,we have
k x i (k )k k f d (k 01)0f i (k 01)k
+k b d (k 01)0b i (k 01)kj u d (k 01)j +k b i (k 01)kj u i (k 01)j L k x i (k 01)k + b j u i (k 01)j
(4)
where L =l f +l b u .Using (4)repeatedly and considering x i (0)=0,we obtain
k x i (k )k b
k 0
1
j =0
L k 010j j u i (j )j :
(5)
Moreover,from the previous equation,the following can be derived straightforward:
j e i (k )j j c x i (k )j c k x i (k )k
c b
k 0
1
j =0
L k 010j j u i (j )j
k f d (k )0f i (k )k l f k x i (k )k l f b
k 0
1
j =0L k 010j j u i (j )j (6)
k b d (k )0b i (k )k l b k x i (k )k l b b
k 0
1
j =0
L k 010j j u i (j )j (7)k d (k )0 i (k )k l k x i (k )k l b
k 0
1
j =0
L k 010j j u i (j )j (8)
where
i (k )denotes either l;i (k )or r;i (k ).III.R EGIONAL L EARNING C ONVERGENCE A NALYSIS
The simplest ILC control law is adopted
v i (k )=v i 01(k )+ e i 01(k +1)
(9)
where >0is a learning gain.v 01(k )and e 01(k )can be arbi-trary bounded functions.Since m l;i (k )and m r;i (k )are positive and upper bounded by m ,there exists satisfying 0<10 c b m <
3;i (k
)
10 cb i (k )m 3;i (k )<1;8k 2K ,and 3=l;r .In this section,we derive that if u i (j )converges att he ?rst k instants to u d (j ),then v i (k +1)will converge to the correct region among I l;i (k +1);I D;i (k +1),and I r;i (k +1).
Lemma 1:If lim i !1j u i (j )j =0(j =0;...;k and 0 k N 01)and u d (k +1)>0,then v i (k +1)will enter and stay in I r;i (k +1)after a ?nite number of iterations.
Proof:The proof consists of three parts.Part A gives some boundedness conditions.In Part B ,the learning convergence from outside to I r;i (k +1)is derived.Part C shows that the actual control input will not escape the region after entering the region I r;i (k +1).Part A :There exist two arbitrarily small constants and such that u d (k +1) ((l f c +(2l ;r = ))0k )=(cb d (k +1))+ >0,where
0k
b k j =0L
k 0j
.As lim i !1j u i (j )j =0(j =0;...;k ),for the given ,there exists a ?nite number p such that 8i p;j u i (j )j .From (5),we have
k x i (k +1)k b
k
j =0
L k 0j =0k :
(10)
Therefore,considering Assumption 1,for any i p
0l f 0k
f d (k +1)0f i (k +1) l f 0k (11)0l ;r 0k
r;d (k +1)0 r;i (k +1) l ;r 0k (12)
j r;i +1(k +1)0 r;i (k +1)j =j r;i +1(k +1)0 r;d (k +1)j +j r;d (k +1)0 r;i (k +1)j 2l ;r 0k :
(13)
IEEE TRANSACTIONS ON AUTOMATIC CONTROL,VOL.50,NO.9,SEPTEMBER20051457
Part B:Substituting(9)into(1)yields
x i(k+1)=f i(k)+b i(k)DZ[v i01(k)
+ c(x d(k+1)0x i01(k+1))]:
Considering(9)and the GLC property of(1),the boundedness of v i(k)
can be ensured for any?nite number of iterations.Since N is?nite,if
i is?nite,both x i(k)and y i(k)are bounded.As u d(k+1) (l f c+
(2l ;r= ))(0k=(cb d(k+1)))+ ,from(3),itcan be derived t hat
y d(k+2)0cf d(k+1)
l f c+2l ;r
0k+cb d(k+1) :(14)
Assume v i(k+1)2I r;i(k+1)where i p.From the updating law (9),itcan be derived t hat
v i+1(k+1)
=v i(k+1)+ f y d(k+2)0cf d(k+1)+cf d(k+1)
0cf i(k+1)0cb i(k+1)DZ[v i(k+1)]g: Considering(11)and(14),we obtain
v i+1(k+1) v i(k+1)+2l ;r0k+ cb d(k+1)
0 cb i(k+1)DZ[v i(k+1)]
v i(k+1)+ cb d(k+1) :(15)
Since p is?nite,v p(k)is bounded.From(12),8i p; r;i(k+1) r;d(k+1)+l ;r0k which is?nite.According to(15),there exists a ?nite number n>p such that v n(k+1)2I r;n(k+1).
Part C:Now we prove that8i n,if v i(k+1)2I r;i(k+1),then v i+1(k+1)2I r;i+1(k+1).Considering(2)and u d(k+1) (l f c+(2l ;r= ))(0k=(cb d(k+1)))+ ,in I r;d(k+1),we have m r;d(k+1)[v d(k+1)0 r;d(k+1)]
c l f0k
cb d(k+1)+2l ;r0k
cb d(k+1)
+ :(16)
From the updating law(9),we have
v i+1(k+1)=v i(k+1)+ f cf d(k+1)+cb d(k+1)
2m r;d(k+1)[v d(k+1)0 r;d(k+1)]
0cf i(k+1)0cb i(k+1)
2m r;i(k+1)[v i(k+1)0 r;i(k+1)]g(17)
r;i(k+1) r;i(k+1)0 c l f0k
+ cb d(k+1)m r;d(k+1)
2[v d(k+1)0 r;d(k+1)]
+ cb i(k+1)m r;i(k+1) r;i(k+1)
= r;i(k+1)0 c l f0k
+ cb d(k+1)m r;d(k+1)[v d(k+1)
0 r;d(k+1)]:(18) Substituting(16)into(18)yields
v i+1(k+1)> r;i(k+1)+2l ;r0k:(19) Using(13),(19)shows that v i+1(k+1)> r;i+1(k+1),i.e.,8i n;v i+1(k+1)2I r;i+1(k+1).
Remark3:If u d(k+1)<0,by following the same derivation procedure in Lemma1,we can prove that the system input v i(k+1)2 I l;i(k+1)after a?nite number of iterations.
Lemma2:If lim i!1 u i(j)=0where j=0;...;k,and u d(k+ 1)=0,then at the time instant k+1;lim i!1 u i(k+1)=0and in the sequel v i(k+1)2I D;d(k+1).
Proof:Let us check the system input v i+1(k+1)according to the following three cases.
Case1.v i(k+1)2I r;i(k+1):Using(9)and the fact u d(k+1)=0, we have
v i+1(k+1)0 r;i+1(k+1)
=v i(k+1)+ c[f d(k+1)0f i(k+1)]
0 cb i(k+1)m r;i(k+1)[v i(k+1)0 r;i(k+1)]
0 r;i(k+1)+ r;i(k+1)0 r;i+1(k+1)
= r;i(k+1)[v i(k+1)0 r;i(k+1)]
+ c[f d(k+1)0f i(k+1)]
+ r;i(k+1)0 r;i+1(k+1):
From(5),j c[f d(k+1)0f i(k+1)]+ r;i(k+1)0 r;i+1(k+ 1)j 1i(k+1)( l f c+2l )
b
k
j=0
L k0j j u i(j)j,where l =max f l ;l;l ;r g.Hence,the following can be derived: r;i(k+1)[v i(k+1)0 r;i(k+1)]01i(k+1)
v i+1(k+1)0 r;i+1(k+1)
r;i(k+1)[v i(k+1)0 r;i(k+1)]+1i(k+1):(20)
Case2.v i(k+1)2I l;i(k+1):Analogous to Case1,we can derive l;i(k+1)[v i(k+1)0 l;i(k+1)]01i(k+1)
v i+1(k+1)0 l;i+1(k+1)
l;i(k+1)[v i(k+1)0 l;i(k+1)]+1i(k+1):(21)
Case3.v i(k+1)2I D;i(k+1):As v i(k+1)2I D;i;u i(k+1)= u d(k+1)=0.Hence,v i+1(k+1)=v i(k+1)+ c[f d(k+1)0 f i(k+1)].Analogous to preceding cases,j c[f d(k+1)0f i(k+1)]j 1i(k+1),and the mapping between v i+1(k+1)and v i(k+1)is v i(k+1)01i(k+1) v i+1(k+1)
v i(k+1)+1i(k+1):(22)
Notice the?niteness of( l f c+2l ) b;L,and k N,we have lim i!11i(k+1)=0if lim i!1j u i(j)j=08j2f0;...;k g. According to(20)–(22)and Lemma4,we obtain that as i!1;v i(k+ 1)2I D;i(k+1).On the other hand,since j d(k+1)0 i(k+ 1)j l
b
k
j=0
L k0j j u i(j)j;lim i!1 i(k+1)= d(k+1), and lim i!1I D;i(k+1)=I D;d(k+1)can be derived.Hence,as i!1;v i(k+1)2I D;d(k+1),and lim i!1u i(k+1)=u d(k+1).
IV.A SYMPTOTIC L EARNING C ONVERGENCE A NALYSIS Theorem1:Under Assumptions1–3,the learning law(9)guaran-tees that y i(k)and u i(k)converge to y d(k)and u d(k)respectively for every k2K.In particular,the system input signal v i(k)converges to v d(k)if u d(k)=0,otherwise v i(k)converges to I D;d(k).
Proof:The proof contains two steps.Step1)shows the learning convergence of u i(0)and y i(1).In Step2),assume that the conver-gence has been ensured for u i(j)and y i(j+1)(j=1;...;k and k N02).Then prove the convergence of u i(k+1)and y i(k+2). Hence,by the mathematical induction principle,the convergence for every k2K can be derived.
Step1)The convergence of u i(0)and y i(1)will be examined in two cases.
1458IEEE TRANSACTIONS ON AUTOMATIC CONTROL,VOL.50,NO.9,SEPTEMBER 2005
Case 1.u d (0)=0:From the updating law (9),for any i 2Z +we have
v i +1(0)=v i (0)+ f cf d (0)+cb d (0)u d (0)
0cf i (0)0cb i (0)DZ [v i (0)]g :
(23)
x i (0)=x d (0)leads to f i (0)=f d (0);b i (0)=b d (0),and I 3;i (0)=I 3;d (0)(3=l;D;r ).Hence,(23)can be rewritten as v i +1(0)=v i (0)0 cb d (0)DZ [v i (0)].If v i (0)2I D;d (0);v i +1(0)=v i (0)2I D;d (0)can be derived.If v i (0)2I r;d (0),we have
v i +1(0)=v i (0)0 cb d (0)m r;d (0)[v i (0)0 r;d (0)]
)v i +1(0)0 r;d (0)= r;d (0)[v i (0)0 r;d (0)]:(24)
As 0< r;d (0)<1;v i +1(0)2I r;d (0)can be derived.Hence,(24)implies that lim i !1v i (0)= r;d (0)2I D;d (0).Similarly,if v i (0)2I l;d (0);lim i !1v i (0)= l;d (0)2I D;d (0).There-fore,lim i !1v i (0)2I D;d (0);lim i !1u i (0)=u d (0),and lim i !1y i (1)=lim i !1[cf i (0)+cb i (0)u i (0)]=y d (1).
Case 2.u d (0)=0:Here,only consider u d (0)>0.Similar results can be derived for u d (0)<0.Assume v i (0)2I r;d (0).Analogous to (23),we have v i +1(0) v i (0)+ cb d (0)u d (0).As cb d (0)u d (0)>0,there is a ?nite number p >0such that v p 01(0)2I r;d (0)while v p (0)2I r;d (0).Assume v i (0)> r;d (0)and i >p .According to (9),we have
v i +1(0)=v i (0)+ c f [f d (0)0f i (0)]
+[b d (0)m r;d (0)v d (0)0b i (0)m r;i (0)v i (0)]0[b d (0)m r;d (0) r;d (0)0b i (0)m r;i (0) r;i (0)]g =[10 cb d (0)m r;d (0)]v i (0)+ cb d (0)m r;d (0)v d (0)(25)
>[10 cb d (0)m r;d (0)] r;d (0)+ cb d (0)m r;d (0) r;d (0)= r;d (0):
Hence,8i p;v i (0)> r;d (0).From (25),for any i p; v i +1(0)=
r;d (0) v i (0),where v i (0)=v d (0)0v i (0).Since 0< r;d (0)<1;lim i !1 v i (0)=0,consequently lim i !1 u i (0)=0.From (8),lim i !1 u i (0)=0leads to lim i !1e i (1)=0.
Step 2)Assume that lim i !1 u i (j )=0and lim i !1e i (j +1)=0,
where j =0;...;k and 1 k N 01.L etus check t
he convergence property for the time instant k +1.
Case 1.u d (k +1)=0:From Lemma 2,the learning convergence can be derived directly.
Case 2.u d (k +1)=0:If u d (k +1)<0or u d (k +1)>0,according to Lemma 1and Remark 3,a ?nite number n can be found such that 8i n;v i (k +1)2I l;i (k +1)or v i (k +1)2I r;i (k +1).Here,only u d (n +1)>0is considered.Referring to (17)
v i +1(k +1)
= v i (k +1)0 cb i (k +1)m r;i (k +1) v i (k +1)0 c [f d (k +1)0f i (k +1)]0 c [b d (k +1)m r;d (k +1)
0b i (k +1)m r;i (k +1)]v d (k +1)0 c 2[b d (k +1)m r;d (k +1) r;d (k +1)0b i (k +1)m r;i (k +1) r;i (k +1)]:
(26)
Let us evaluate this equation item by item.First,j v i (k +1)0cb i (k +1)m r;i (k +1) v i (k +1)j = r;i (k +1)j v i (k +1)j .Second,using
(5),j c [f d (k +1)0f i (k +1)]j c l f
b k j =0L
k 0j
j u i (j )j .To evaluate the last two items in (26),we use j a 1b 10a 2b 2j j a 10a 2jj b 1j +j a 2jj b 10b 2j and j a 1b 1c 10a 2b 2c 2j j a 10a 2jj b 1c 1j +
j a 2jj b 10b 2jj c 1j +j a 2b 2jj c 10c 2j .Hence,considering (5),we have j c [b d (k +1)m r;d (k +1)0b i (k +1)m r;i (k +1)]v d (k +1)j c k b d (k +1)0b i (k +1)kj m r;d (k +1)v d (k +1)j + c k b i (k +1)kj m r;d (k +1)0m r;i (k +1)jj v d (k +1)j
c (l b m + b l m ) v
b k j =0L
k 0j
j u i (j )j .Similarly,j c [b d (k +1)m r;d (k +1) r;d (k +1)0b i (k +1)m r;i (k +1) r;i (k +1)]j
c (l b m ;r + b l m ;r + b m l ;r )
b k j =0L
k 0j
j u i (j )j .Therefore,j v i +1(k +1)j r;i (k +1)j v i (k +1)j +
M k j =0L
k 0j
j u i (j )j ,where M = c [l f +(l b m + b l m ) m +(l b m ;r + b l m ;r + b m l ;r )] b is a ?nite unknown con-stant.Based on Lemma 3(in the Appendix),lim i !1j u i (j )j =0(j =0;...;k )leads to lim i !1j v i (k +1)j =0,hence lim i !1j u i (k +1)j =0.According to (8),lim i !1j e i (k +2)j =0can be obtained.
V .C ONCLUSION
This work provides a rigorous proof that the simplest ILC achieves uniform learning convergence for systems with input-deadzone.A new deadzone compensation approach based on ILC is established.More-over,at least for repeated control systems,we can now handle input-deadzone that is nonlinear,unknown,time varying and state-dependent.
A PPENDIX
Lemma 3:L ett hree sequences be f z i g R ;f a i g R ,and f i g 2R ,with i 2Z +.Assume that 8i 2Z +,the inequality j z i +10a i +1j i j z i 0a i j +j i j holds,where 0< i <1.Then lim i !1z i =a can be derived if lim i !1a i =a and lim i !1j i j =0.
Proof:De?ne a sequence ~
f ~ 0;...;~ i ;...
g ,where ~ n =sup fj n j ;...;j i j ;...g .Hence,~
n ~ n +1 0,and ~ n j n j .As lim i !1j i j =0;lim i !1~
i =https://www.doczj.com/doc/f08952275.html,ing j z i +10a i +1j j z i 0a i j +j i j repeatedly,we have j z i 0a i j i j z 00a 0j + i 01~
0+111+ ~
i 02+~ i 01.If i is even j z i 0a i j
j z 00a 0j +~
0+111+~ 0
1
+~
( 01
+111+ +1)
(j z 00a 0j +
i 2~ 0)+~
10 10
:Therefore,lim i !1j z i 0a i j =0.Similarly,when i is odd,
lim i !1j z i 0a i j =0.As lim i !1a i =a;lim i !1z i =a can be derived.
Lemma 4:For any i
2Z +,divide R into three intervals:I 1
;i (01;a i ];I 2;i [
a i ;
b i ],and I 3;i [b i ;1),where a i b i .Assume that 8i 2Z +,the following relations hold:
if z i 2I 1;i ; 1;i (z i 0a i )0j i j z i +10a i +1
(27)
1;i (z i 0a i )+j i j ;
if z i 2I 2;i ;z i 0j i j z i +1
z i +j i j
(28)if z i 2I 3;i ; 2;i (z i 0b i )0j i j
z i +10b i +1 2;i (z i 0b i )+j i j
(29)
where 0< 1;i <1;0< 2;i <1;lim i !1a i =a;lim i !1b i =b ,and lim i !1j i j =0.Then,for any z 02R ,as i approaches in?nity,z i 2I 2[a;b ]can be derived.
Proof:As lim i !1j i j =0;lim i !1a i =a ,and lim i !1b i =b ,for arbitrarily small >0,there exists a ?nite number M >0such that for any i M ;j i j ( =4);j a 0a i j ( =4),and j b 0b i j
( =4),where =min f ;
;10 ;10 g =min f ;10 g .Hence,
IEEE TRANSACTIONS ON AUTOMATIC CONTROL,VOL.50,NO.9,SEPTEMBER20051459
8i M ;j a i0a i+1j j a i0a j+j a0a i+1j ( =2),and similarly,
j b i0b i+1j ( =2).The proof consists of two parts.In Part A,it is
shown that a?nite number p M can be found such that,8i p;z i
will enter I02;i(
)[a i0(3 =4);b i+(3 =4)] [a0 ;b+ ].In Part
B,itis proven t hat z i2I02;i( )is guaranteed for any i p.Therefore, from the de?nition of limit,as i!1;z i2I2=[a;b]holds.
Part A:Suppose z M>b M+(3 =4).According to Lemma3and
(29),a?nite iteration number p>M can be found such that z p01>
b p01+(3 =4)and z p b p+(3 =4).On the other hand,from(29), we have z p0b p l(z p010b p01)0j p01j l(3 =4)0( =4)>
0.Hence,z p2(b p;b p+(3 =4)] I02;p( ).Similarly,for z M<
a M0(3 =4),a?nite constant p>M can be found such that z p2 [a p+(3 =4);a p) I02;p( ).Obviously,there exists a?nite p such that z p2I02;p( ).
Part B:Assume z i2I02;i( )(i p).z i+1is evaluated according to three cases.Case1.z i2I2;i=[a i;b i]:From(28),we have a i0 ( =4) z i+1 b i+( =4),hence,a i+10(3 =4) R EFERENCES [1]G.Tao and P.V.Kokotovic′,Adaptive Control of Systems With Actuator and Sensor Nonlinearities.New York:Wiley,1996. [2]J.H.Kim,J.H.Park,S.W.Lee,and E.K.P.Cheng,“A two layered fuzzy logic controller for systems with deadzones,”IEEE Trans.Ind. Electron.,vol.41,no.2,pp.155–162,Apr.1994. [3]R.R.Selmic′and F.L.Lewis,“Deadzone compensation in motion con- trol systems using neural networks,”IEEE Trans.Autom.Control,vol. 45,no.4,pp.602–613,Apr.2000. [4]Y.Q.Chen and C.Y.Wen,Iterative Learning Control:Convergence, Robustness,and Applications.New York:Springer-Verlag,1999,vol. LNCIS-248. [5]J.X.Xu and Y.Tan,Linear and Nonlinear Iterative Learning Con- trol.Berlin,Germany:Springer-Verlag,2003.Lecture Notes in Con-trol and Information Sciences(291). [6]K.H.Lee and Z.Bien,“Initial condition problem of learning control,” in Proc.Inst.Elect.Eng.D,vol.138,1991,pp.525–528. Avoiding Constraints Redundancy in Predictive Control Optimization Routines Sorin Olaru and Didier Dumur Abstract—This note concentrates on removing redundancy in the set of constraints for the multiparametric quadratic problems(mpQP)related with the constrained predictive control.The feasible domain is treated as a parameterized polyhedron with a focus on its parameterized vertices. The goal is to?nd a splitting of the parameters(state)space corresponding to domains with regular shape(nonredundant constraints),resulting in a table of regions where the constraints have a minimal representation,so that the online optimization routines can act with better performances.The procedure can be seen as a preprocessor either for the classical QP methods or for the routines based on explicit solutions.For important degrees of redundancy,the proposed technique may bring computational gains for real-time application or on the complexity of the positioning mechanism for evaluating the explicit solution. Index Terms—Multiparametric optimization,parameterized polyhedra, predictive control. I.I NTRODUCTION Constrained predictive control implies the resolution of a multipara-metric quadratic problem(mpQP).For a long period,?nding this so-lution was based on online routines providing the optimal control se-quence using active-set or lately interior point methods.Recently,al-ternative methods based on the explicit solution[1],[5],[7]in terms of piecewise linear functions,have been given much attention.All ap-proaches are sensitive to the presence of redundancy on the constraints de?nition.The current note proposes as novelty a method to de?ne piecewise redundancy-free(RF)set of constraints for mpQPs within model predictive control(MPC). A geometrical approach is developed for the representation of the feasible space,based on the Motzkin double description principle[4] and the parameterized polyhedra[3].As the topology of the feasible domain changes as function of parameters,the limits of the available control actions in the optimization program are given by some parame-terized vertices.Further,based on these parameterized vertices,piece-wise RF feasible domains can be identi?ed. The resulting mpQP can be used either for developing the explicit solution,or in an online manner with classical QP solvers.Once the redundancy eliminated,the computational load required for the explicit solution is diminished with the cost of certain supplementary cuttings in the parameters space.These supplementary cuttings can be eliminated or used when constructing the positioning mechanism for the look-up table necessary in real-time implementation. The MPC computational amelioration depends on the complexity of the RF formulation and on the degree of redundancy for the original set. A large number of RF zones will add to the effective optimization ef-fort a supplementary part of positioning which might be overwhelming. However,if the set of constraints is separable in few subsets,eventu-ally disjunctive(large degree of redundancy),the gains are obvious as illustrated in an example. Manuscriptreceived July14,2004;revised February11,2005and May13, 2005.Recommended by Associate Editor L.Magni.This work was supported by the European Commission,D.G.for Research. The authors are with the Supélec—Automatic Control Department,Plateau de Moulon91192,Gif-sur-Yvette cedex,France(e-mail:sorin.olaru@https://www.doczj.com/doc/f08952275.html,; didier.dumur@supelec.fr). Digital Object Identi?er10.1109/TAC.2005.854659 0018-9286/$20.00?2005IEEE
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