- Email: [email protected]
Disturbance Decoupling of Affine Nonlinear Singular Control Systems
Copyright @ 1996 IFAC 13th Triennial World Congress. San Francisco, CSA
2b-075
DISTURBANCE DECOUPLING OF AFFINE NONLINEAR SINGULAR CONTROL SYSTEMS Xiaoping Liu I
Dep/
.f Aut.matic C. ntrol, N.rtheastern
Uni.... ity, Shen,ang, 110006, P.R .China liuxpe.ail.neu .• du.cn .
Abstract . This paper discusses disturbance decoupling problem for affine non linear singular systems. First , an algorithm is proposed, by which an affine nonlinear singular system can be changed into one that is equivalent by feedback to a regular nonlinear system. Then, another algorithm is presented, by which the relative degree of the singular system can be easily calculated. Finally, sufficient and necessary conditions for the solvability of tbe disturbance decoupling problem are derived.
Keywords. Singular systems, nonlinear systems, disturbance rejection, df'coupling problem} di fferential geometric method .
I. INTRODUCTION
Consider affine nonlinear singular systems of the form:
x = fl(x) + p,(x)z + 91(X)U + wl(x )d o = hex) + P2(x)z + 9,(X)U + w,(x)d y
= hex)
(1)
where x e Rn is the vector of differential variables, z E RIJ is the vector of algeb raic variables, u E Rm is the vector of inputs, d E H is the vector of disturbances, and y E R m is the vector of outputs, p;(x), g;(x ), and Wi(X), i = 1, 2, are matrix· valued an alytic functions with suitable sizes, h(x),h(x) and hex) are vector-valued analytic functions with dimensions n, s, and r.
In recent years, there has been a growing interest in singular systems due to their extensive ap plications in a variety of areas, such as robotics (You an d Chen, ]993), chemical processes (Kumar and Daolltidis, 1994), conThis work is supported by grants from NSF ot China, NSF ot Liaoning, China, State EduCAtion Commissiu n or China, Ministry or Metall\U'~ cal Industry ot Ch ina, Laboratnry or Robot ics, Chi· nese Acadenl,Y at Sciences, a nd State Key Lahoratary ot Industrial Control Technology, Hangzhou. China. I
strained mechanical systems (You and Chen, 1993), and so on . There has been substantial progress in dea1ing with many aspects of sjngular systems. Some of these successes have been listed recently in books (Campbell , 1982, Dai, 1989), and in survey papers (Campbell, 1980). For the case of linear singular systems, many papers and works dealing wi th the various control problems, which include solvabilit~y, controllability, observability, stability, pole-placement, observer design, deeoupling, optimal control and so on, have appeared. For a fairly comprehensive introduction to linear singular systems, see books (Campbell, 1982, Dai, 1989). However, for singular nonlinear control systems, a limited attention has been devoted to studying the solvability (Liu, 1995), controllability (Lin and Ahmed , 1991) ,control problems which include linearization (Kawaji and Taha, 1994, Liu , 1993), input-output decoupling (Liu and Celikovsky, 1995) and feedback stabilization (McClamroch, 1990, Chen and Shayman , 1992). In this paper, we will in detail discuss the disturbance decoupling problem for the system (1). We begin in Section 2 with the d efinition of the disturban ce decoupling problem . An algorithm is proposed in this section . Based
2120
on the algorithm, some assumptions are made, under which the system (I) admits differentiable solution8 to any differentiable disturbance d. Section 3 provides another algoritbm by wbicb one can calculate tbe relative degree of the system (I). Sufficient and necessary conditions are derived, under which the disturbance decou· pling problem is sclvable. NOTATIONS _ For analytic veclor-valued functions f(z) [f,(z), ... ,/.(z)JT and o(z) [g,(z), ... ,g.(z)jT, an analytic function h(z), and an analytic matrix-valued function p(:r) [P'(z), ... ,p'(:r)] with p;(z) [P\(z), ... , p~(x)IT, the following notations will be used:
=
=
=
dh(z)
=
= [8:;:), "', 8:::)1
L1h(z)
= h ex),
L;+'h(x)
L,h(z) = dh(x)f(x),
We now come to the algorithm that is essential in investigating sufficient conditions under which there exists tbe feedba
Algorithm 2.1 . Step 1. A.sume Ihal Ihe matnx [1'2("') g,(x) w,(",)] ha, constant rank/ say 81, in a neighborhood of %0. Then there exists an 8 x s nlJnsingular matrix R 1(z) such that
R,("')[P2(z)
g2(X)
ID,( .. ) 1=
[p~~z) iil~.. ) wlJr)]
wilh [p~(,,) iiHz) wl(x)] being an 8, x (s + m+ I) matrix of rank SI near ;'to. Multiplying the second equalion of (1) on the left by Ihe maIn" R.( .. ) leads 10 Ihe following relalion
+ [pHx)] z. [9H X)] u + [wl(r)] d [ li(",)] 1Hz) 0 0 0
= L/(L'h(z», L,L/h(x) = L,(L/h(,,»
LpL'h(,,) = [Lp. L'h(z), .. _, /' p.[.,h(x)]
(4)
wh ere, k = 0, 1, .... dh is called as the differential of h, and L,h, the Lie derivative of h along f . See Isidori (1995) for other differential geometric notations.
2. PRELIMINARY RESULTS We start this section with the definition of the disturbance decoupling problem.
Assume Ihal fl(,,) ha. con.Ianl rank I, in a neighborhood of Zo. Then locally around zo, M. fHx) Dj is an (n - II)- dimensional •• bmanifold. Differen/i. aling Ihe lasl s - s, equalions of N) yields
={" :
0=
[9HZ)] u + [wHz)] wl(") d
fJ(X)] [pl(z)] [IHx) + pl(z) ,+ 9j(Z)
L/Ji(._.,J(X)]T and
Disturbance Decoupling Problem. Given an affine nonlinear singular system (l) and an initial point zo, find a regular s~atic state feedback control Jaw
u
= o(z) + P(x)v + ,),(x)z
L"JJ,(x)
L,._fMr) L,._ fi(._., )(,,)
z = fI(z) + g,( .. )<>(x) + [P,( .. ) + O,(,,)')'(x)],
L.. "f4,(x)
+ w,(x )d
0= J,(x) + g,( .. )o(x) + (P,(x) +g,(x)p(x)v + w,(z)d
+ g,("h( .. )]z
]
]
3
( )
Lw" fi(._.,)( x) with n;(x) being Ihe i-Ih 'ntry of fHz), PH(Z),O';(z), andwH(x} are th,i-th column ofp,(x),g,(r), andw,(",), respectively_
has Ih e following proper/;es (a) For any differentiable v(/) and d(/), Ihere erisls a unique differe ntiable solution {,,( "0, I, v, d), z("o, I, v , d)} to th, syslem (3) such Ihat z(O)
Step k+1. Suppose thal from ./ep' 1 Ihrough k w. have
= "'"
(b) y(l) i. decoupled from d(t) , Ihal ~tale :to and (.ach input v , lhe output
]
Lp.Ji(._.)r)
(2)
defined in a neighborhood U of"o wilh P(,,) nonsing.lar on U I such that the corresponding cJlJsed-Joop system
+9, (r)p(x)v
=
IS,
for Ihe inilial
0=
[p~(x)] [ii~("')] [w~(z)] [!l(,,)] f:(x) + pl(x) z + ii~(x) u + w~(r) d (5)
y,(I):= It(r(ro,l,v , dd) = y,(t) := h("("o , I,v , d,» for every pair 0/ disturbance vector fun ctions d •. d z , and f(Jr every tim e t > 0 for which the s(J/utilJns z(zo, t, V, dd, i 1,2, are defined.
with the matrix
pHz) [ pH,,)
=
2121
has constant rank 81:+1 in a ueighbor'hood oJ Xc in Ml;. Then it is possible to find an s x s nonsingular matrix R'+I(x) such that
RH I ( Z)
-. () [ ,,~(X) P2 Z
ii~(x)
-. ( )
zero dynamic algorithm (Isidori, 1995). In addition, similar method is used to derive the canonical/orm for a general class of nonlinear singular systems (R08chon at al., 1992). Notice that a necessary condition for the existence of
92 X
ditFerentiable solutions to the system (1) is that the initial condition "'0 satisfies I~(xo) = 0 for every k of the algorithm. If #(zo) = 0 and the two constant rank as-
= [,,~+1(Z)
g;+I(X) w~+I(X)] 000
where [,,~+I(x) ii~+'(z) ,jj~+I(x) I is an 8HI X (8 + m + I) matrix and has rank SI:+1 in a neighborhood 0/ Zo in M •. Multiplying (5) on the left by the matrix R.+I(z) gives
(6)
sumptions are satisfied at every step of the algorithm, then it easily follows from construction that the algorithm terminates after k· < n iterations, where k* is the least integer such that Mft- :;::: M1c-+l. Now we make the following assumptions whi ch guarantee that the two
assumptions of tbe algorithm do not depend OD tbe particular choice of matrices R!:(x) at the each iteration of the algorithm. Assulllption 2.3.
Ass.me that [f~(x)T , ... ,f~+'(x)TV has constant ronk t. in a neighborhood of %0 in M •. Now set M.+ I {z E M. : f;+'(x) 0). Then it is easily seen that M.+ I is an (11 -lie)· dimensional submanifold. Differentiating the last s - "HI equations of (6), one can get th e following algebraic equatiou.'i:
. [M(X) () 1 t he matrices V,(x)
W~(x)l h
g~(,,)
w~(x)
(7)
<
e+1.
(2) [fH,,)T •... ,f~+'(>:)TV has conslanl rankforallx E M. around Xo and for all k
< k' + I.
. ["f+'(z) iif+'(X) (3) the malrl" P2 -"+I() -"H() x 92 X mnk s.
0=
as co ..tant
rank for all x E M. around Xo and for all k
=
=
ii~(x)
wf+'(x)] - "+I() has w2 z
For convenience, we call Zo a regular point of Algorithm 2.1 if the assumptions of Assumption 2.3 are satisfied.
Remark 2.4. fn assumption 2.3, (I) and (2) do not depen d on th e particular cJwice of matrices introduced where and
at each iteration of the algorithm if (3) i. sali.fied, I.idori (1995).
i,Hl (z) = [ L!.f~t' (xl
g;+1(x)
see
Remark 2,5. The algorithm 2.1 results in a set Of'[;~~, (s - s,:) algebraic constrajnts which is independent of %, namely
=
W;+I(z) =
L, .. f~,+'(z) [L
L,,~f;t'(z)
...
g"
fHI
2(. - ••• ,)
(x)
L.,,, f:t ' (x)
.. .
[ L fk+ ' (xl 'lUll 2(A -.I1o+d
1 f,(x)
]
L
(x) "m JI::+I 2(.- ••• ,) L.,,, f;t' (x) ]
. ..
L 'IU,1 f'+' (x) 2('-'1o+a)
with I;'+'(x) being the i-th entry of f;+'(>:) .
Remark 2.2. The algorithm 2. 1 is a modified version of algorithmic procedure which is introduced by Kumar and Dao.lidis (199D and Liu (19950) to construct the algebraic variable z. 011 onc hand, Kumar and Daoutidis (199.{) devetoped such an algorithm based on Hirschorn's inversion algorithm ( Hirschoru , 1979)J on the other handJ Liu (1995a) introduced a similar algorithm based on Ihe
= O.
2 f 2(>:)
= O•... ,f2e
(x)
=0
Under the. conditions list ed in assumption 2.3, these. conslminis define a controlled-inMriant manifold:
M
={x E Rn : fj (x) = 0,
li(x)
= 0, ... ,fnz) =O}
of dimension n - '[;~~I (s - ,;). It i. worth noting that th e state x of the system (1) evolves on this manifold if initial slate Zo belongs to M . Now let us give the following conclusion which provides sufficient and necessary conditions which guarantee the
system (I) is solvable by feed back for any differentiable d, that is, there exi.ts a feedback of the form (2) such that the closed-loop system (3) admits a unique differ-
entiable solution for any differentiable v and d.
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Proposition 2.6 . Ass"me that Xo is a reg"lar point of Al90rilhm 2.1. Then. Ihere exist •• al lea.1 locally. a feedback of Ihe form (2) ..ch Ih.1 Ihe clo ..d-Ioop .ys/em (9) admib Cl lInigue differentiable solution for any di//erefl. liable v .nd d if .nd onlJ if Ihe malrix
if+1(z) [ p;"+l(X)
change the problem for the system (1) into one for a regular system which is composed of only differeotial equatioo by introducin~ feedback law . For conveoieoce in notation, denote the system resulting from Algorithm
2.1 by
=
gr+l(x)] iir+l(x)
i: !I(z) + PI(Z)Z + 91(Z)" + wl(x)d 0= h(z) + l'2(z)z + g,(z)" + w,(z)d y = her)
has rank s. Proof: Necessity: Assume that the rank of the matrix
i{+l(X) [ p~ +I(x)
g~:+l(X)] g~ +l(x)
pf+1(x)
gf+l(X) -'"+l() 92 X
with [p,(",) g,(",)) having full row rank in a neighborhood of Zo in M. Because the matrix [1'2(z)
is 1... than s. Then. it easily follows from (3) of the assumption 2,3 that there exists an s X s nonsingular matrix H (z) such that 1f(x) [ P2 -."+l() X
(9)
wf+1(x)] . '"+l() %
in a neighborhood of
Zo
g,(z)) has full row rank in M I there exists an s X m
matrix 'Y(x) such that p,(z) + 9'(Z)oy(Z) is nonsingular in a neighborhood of Zo in M. Therefore the feedback of the form u = 'Y( x)z + v can render the closed-loop system
W2
= !I(z) + (P1(r) + gl(:rh(:r))z+ gl(z)v + w,(x)d o = h(z) + (P,(x) + g,(zh(x))z+ g,(z )v + w,(,,)d
i;
= [p;"+I(X) iif+'(X) uf+'(x)]
o
0
W(x)
As a result~ the corresponding algebraic equation becomes as follows
0= (8)
y
(10)
= hex)
have a unique differentiable solution for any differentiable v and d. In fact,z can be fixed uniquely from the
second equation of (10). namely z
= -Iv,(z) + g,(xh(zW1[j,(z) + g,(z)v + W2(X)d]
Substituting this into the fir. t equation of (10) leads to In order to make the second equations of (8) hold for any d, W(r) must be identically zero which is contrary to (3) of the assumption 2.3.
= !I(x) -
6(x)h(x) + (g,(z) - 6(x)g,(:r)}v +(WI(Z ) - 6(z)w,( :r)}d (11) y = hex)
i:
Sufficiency: Assume that the matrix
pf+l(X) [ pf+'(x)
with 5(x)
gf+'(X)] gr+l(x)
Up to now, the disturbance decoupling problem for the
has rank s. Then. there exists a mat.rix 'Y(x) such that the matrix
pf +'( x)] [ pf+l(z)
+
t·
+ 'Y(z)z can make
the corresponding closed-loop system have a. unique differentiab le solution for any differentiable v and d. 0
Assumption 2.7. The matrix [ pf.+'(z) +I(x) has rank s .
p;
3.
system (\ ) has been changed into one for the system (11). The later problem was well posed and solved by using the concept of a controlled invariant distribution
[!If+l(X)] iif+l(x) 'Y(x)
is nonsingular. So the feedback" =
= (PI(Z) + gl(:r)oy(z)]IP2(z) + g,(xh(x)]-l .
g~".+I(Z)]
g;
+I (Z)
or a rdati ve degree (Isidori. 199[)) . Based on the results given by Isidori (1995). one can easily draw the following conclusion.
Theorem 3.1. Suppose Ihal x o is a regular poinl of Algorithm 2.1, Assumptions 2.7 holds, and there exists a malrix r(z). which render the malrix P2(,,)+g,(z)r(z) nonsinguJar at .ro, such that the system (11) has a vector relative degree {PI , "',Pm} at Zo - Th en th e disturbance deco up/ing problem is solvab/~ if and only if
MAIN RESULTS
In this sect.ion, we wi.ll solve the disturbance decoupling
problem . The natural idea of solving this problem is to
LwL,h,(z)
=0
k
= O..... P, -1.i =1. .... m(12)
with f = ft(z) - 6(x)h«) .Hd W = Wl(X) - 6(x)w,(x) . In addilion, the feedback
2123
(13) can solve the disturbance decoupling problem where
A(",)
=
LgL?~lhl("') [
]
:
_ [ Lj'h l (",) ] b(:r) :
LgLjm-Ih~(x)
Ljmh~(r)
It is easily seen that the solvability of the disturbance decoupling problem apparently depends on the choice of the matrix ,Cz). However, we will see in a moment that this is not the case under some conditions. To this end, the following algorithm is helpful.
It follows from Liu and Celikovsky (1995) that the following lemma holds. Lemma 3.4. Suppose that Xo is a regular point oJ Algorithm 2.1, and A.sumptions 2.7 and 3.3 hold. Then, for
any ')'(x) which rende .. p,(",) + g,(x),),(,,) non8ingular, the closed-loop 8ystem (11) has a vector relative degree {PI, ... ,p~} at "'0. In addition, the matrices A(x) and b(",) can be calculated by A(",) = Du("') - [D,(x) + Du(x)')'(",)JtJ(X)9'(Z) (14) b(z) = D(,,) - [D,(x) + Du(",)')'(x)JtJ(x)h(z) with q(",) = [P,(",) + g,(",)')'(",)]-I.
Algorithm 3.2.
Now the main results can be stated as follows. Step 1. Set ",?(r) hi("') and calculllte Lp,tI>?(x), Lg,tI>?(x) Theoreln 3.5. Suppose that Xo is a regular point of and Lf,,,,7(:r). If the matrix Algorithm fU, and Assumptions 2.7 and 9.3 hold. Then, the disturbance decoupling problem is solvable if and only P2(X) g,(:r) if [ Lp,"'?("') Lg,,,,?(x)
=
1
has constant rank s in a neighborhood of Xo in M, then there exists a unique vector-valued analytic function E7(x)
o~ k
~
Pi - 1. 1 ~ i
~ m
(15)
of dimension s such that [Lp,.p7(x) Denote tl>i(x) Pi
Lg, tl>7(x)] = E?(x)[p,(x)
= L"tI>?(x) -
g2(:r)]
for all r in a neighborhood of Xo in M. In addition, the
feedback
EiO("')h(:r). Otherwise. set
u
= 1 and quit the algorithm.
Step k+l. Assume that we have defined a sequence of .p7(x), ... ,.p7(x). Now calculate L p,.p7(",), Lg,tI>7("') and L"tI>hx). Tt the matrix
p,(x) [ Lp,.pf(x)
g,(",) Lg,.p7(",)
1
Lg,,,,f(x)]
= Ef(x)[p,(x)
f(x)g,(",)]-I[D(z) - f(x)h(x)] -[Du(") - f(",)g,(xn-Iv + -y(x)z
achieves the goal of disturbance decoupling where -y(x) is such that p,(x) + g,(x),),(x) is nonsingular, where
f(x)
= [D,(x) + Du(x),),(z)]"(,,,)
Proof: It follows from Algorithm 3.2 that the relation
has constant rank s in a neighborhood of Zo in M, then there exists a unique vector-valued analytic function Et(x) of dimension s such that
[Lp'
= -[Du("') -
[Lp,.p:(x)
Lg,tI>7(x)] =: Ef(",)[p,(",)
It is well-known that the equation above is equivalent to the following equation
g,(",)] [L p ,,,,7(x)
Denote q;~+I(X) = Lf,.pf(x) - E[(",)h(x). Otherwise, set Pi = k + 1 and quit the algorithm.
=
Performing Algorithm 3.2 for i 1, ... , rn, produces integers Pt, ... ,Pm. Now, introduce the following matrices;
g,(",)](16)
= E[(x)[p,(x)
Lg,,;f(,,,)]
~l
[.J",)
9'(")] [')'("')
~1
that is
[Lp,.pf(x)
+ Lg,tI>rc"'h("') Lg,tI>:("')]
= Ef(",)[p,(x) + g,(x)-y(x)
g,(x)]
Now for any ')'(x) such that [P,(",) + g,(",)-y(x)] is nonsingular, Ef(x) can be uniquely determined by
Ef(",) . 3 T"ne matnx . [p,(X) AssumptIOn 3.. D~(x)
g,(",) Du(x)
1is non-
= [Lp, ,;7(",) + L g , q'>f(x)-y(x)]q(",)
(17)
Substituting Ef(,,) into (16) gives the following relations:
singular at Xa.
2124
L., "'f(x)
= [Lp, "'f(x) + Lg, "'f(x)-r(x)Mx)g2(x)
Campbell, S.L. (1982). Singular System. of Diffe,..ntial Equations, Vo/.Il, London: Pitman.
By the construction of .p:+l(x), it is easily seen that
Chen, X.L. and M.A. Shayman (1992). Dynamics and
",:+1(x) = [Lp, "'f(x) + L" "'f(xh(x)J~(x)h(x)
control of constrained nonlinear systems with application to robotics, Proc. of A CC, 2. pp.2962-2966.
Now] according to (17), calculate
Lwh,(x) = Lw,h,(x) - [Lp,h,(x) + Lg,h,("h(x)J~(X)W2(X) = Lw, q,?(x) - [Ll" ",?(x) + L., "'?(xh(x)J~(X)W2(X) = Lw, q,?(x) - E;'(x)w,(x) and
L,h,(x) = L"h,(x) - [Lp,h,(x) + Lg,h,(xh(x)J~(x)h(x) = Lh",?(x) - [LO',,,,?(x) + L.,,,,?(xh(x)J'1(x)h(x) Lh",?(x) - E, (x)h(x) = (x)
"'1
=
Then, it follows that L,h,(,,) = "'1(,,) and Lwh,(,,) is equivalent to Lw,"'?(x) = E?(x)w,(zj.
=
°
Now, assume that LwLih,(z) = O,t = O, ... ,k -1 are equivalent to Lw,"'l(z) = El(x)w,(x),t = O, ... ,k - I, and Lih,(,,) = "';(x). t = I, ... k. Then, one has
and
= L''''f(x)
= L,,"':(") - [Lp, "'f(x) + L g ,"'f(xh(x)J'1(x)h(x)
= ""
linear differential-algebraic process systems, Proc.
0/
ACC, 1, pp.33Q..335, Baltimore. Lin, Y.L. and N. U.Ahmed (L991) Approach to controllability problems for singular system, I. J. Control, 22, pp.675-690. Liu X.P. (1993). On linearization of nonlinear singular control systems, Proc. of A CC, 3, pp.2284-2287, San Francisco.
LwL1h,(z) = Lw"'f(x) = Lw,"':(x) - [Lp, "'f(x) + Lg,,,,f(x);-(x)Mx)w,(x) = Lw,q,f(x) - Et(z)w,(x) L~+lh,(x)
Dai, L.Y.(1989). Singular Control Systems, Springer. Hirschorn, R.N. (1979). Invertibility of multivariable nonlinear control systems, IEEE Tran8. Auto, Control, AC-24, pp.855-865. Isidori, A. (1995). Nonlinear Contml Systems, 3rd Edition, Springer-Verlag. Kawaji, S. and E.Z. Taha (1994). Feedback linearization of a class of nonlinear descriptor systems, Proc. of the 33rd CDC, 4, ppA035-4037, Lake Buena Vista, FL. Kumar, A. and P. Daoutidis (1994). Control of non-
+I(X)
Thus, (12) is equivalent to (15), which means that the first part of the theorem holds. In addition, it follows froln (13) and (14) that the state0 ment about the feedback is true.
4. CONCLUSION
Liu X.P. (1995). Solvability of nonlinear singuLar systems: part 11, the case with inputs, Proc. 0/ ACe, to appear.
Liu X.P. and S. Celikovsky (1995). Noninteracting control of affine nonlinear singular control systems. Research Report, No.1852, UTIA AV CR, 1995. (ftp:ftp.utia.cas.cz/pub/reports/ utiaI852.ps.z). McClamroch, N.R. (1990). Feedback stabilization of control systems described by a class of nonlinear differential-algebraic equations, Systems fj Control Let·
ters, 15. Rouchon, P., M.Fliess and J.Levine (1992). Kronecker's canonical forms for nonlinear implicit differential sys-
tems, 2nd [FAC Whork.hop on System Str.cture and Control, pp.248-251, Prague, Czechoslovkia. You, L.S. and B.S.Chen (L993). Tracking control designs for both holonomic and non-holonomic constrained mechanical systems: a unified viewpoint, /. J. Control,
This paper has investigated the disturbance decoupling problems for affine nonlinear singular systems. An algorithm is proposed, by which the original system can
58, pp.587-6L2.
be changed into a system which is solvable by feedback. Another algorithm is provided. Based on this algorithm, sufficient and necessary conditions ror the solvability of the disturbance decoupling problems has been derived.
REFERENCES Campbell, S.L. (1990). Descriptor systems in the 90's, Proc. of the 29th IEEE CDC, 1, pp.442-447.
2125
2b-075
DISTURBANCE DECOUPLING OF AFFINE NONLINEAR SINGULAR CONTROL SYSTEMS Xiaoping Liu I
Dep/
.f Aut.matic C. ntrol, N.rtheastern
Uni.... ity, Shen,ang, 110006, P.R .China liuxpe.ail.neu .• du.cn .
Abstract . This paper discusses disturbance decoupling problem for affine non linear singular systems. First , an algorithm is proposed, by which an affine nonlinear singular system can be changed into one that is equivalent by feedback to a regular nonlinear system. Then, another algorithm is presented, by which the relative degree of the singular system can be easily calculated. Finally, sufficient and necessary conditions for the solvability of tbe disturbance decoupling problem are derived.
Keywords. Singular systems, nonlinear systems, disturbance rejection, df'coupling problem} di fferential geometric method .
I. INTRODUCTION
Consider affine nonlinear singular systems of the form:
x = fl(x) + p,(x)z + 91(X)U + wl(x )d o = hex) + P2(x)z + 9,(X)U + w,(x)d y
= hex)
(1)
where x e Rn is the vector of differential variables, z E RIJ is the vector of algeb raic variables, u E Rm is the vector of inputs, d E H is the vector of disturbances, and y E R m is the vector of outputs, p;(x), g;(x ), and Wi(X), i = 1, 2, are matrix· valued an alytic functions with suitable sizes, h(x),h(x) and hex) are vector-valued analytic functions with dimensions n, s, and r.
In recent years, there has been a growing interest in singular systems due to their extensive ap plications in a variety of areas, such as robotics (You an d Chen, ]993), chemical processes (Kumar and Daolltidis, 1994), conThis work is supported by grants from NSF ot China, NSF ot Liaoning, China, State EduCAtion Commissiu n or China, Ministry or Metall\U'~ cal Industry ot Ch ina, Laboratnry or Robot ics, Chi· nese Acadenl,Y at Sciences, a nd State Key Lahoratary ot Industrial Control Technology, Hangzhou. China. I
strained mechanical systems (You and Chen, 1993), and so on . There has been substantial progress in dea1ing with many aspects of sjngular systems. Some of these successes have been listed recently in books (Campbell , 1982, Dai, 1989), and in survey papers (Campbell, 1980). For the case of linear singular systems, many papers and works dealing wi th the various control problems, which include solvabilit~y, controllability, observability, stability, pole-placement, observer design, deeoupling, optimal control and so on, have appeared. For a fairly comprehensive introduction to linear singular systems, see books (Campbell, 1982, Dai, 1989). However, for singular nonlinear control systems, a limited attention has been devoted to studying the solvability (Liu, 1995), controllability (Lin and Ahmed , 1991) ,control problems which include linearization (Kawaji and Taha, 1994, Liu , 1993), input-output decoupling (Liu and Celikovsky, 1995) and feedback stabilization (McClamroch, 1990, Chen and Shayman , 1992). In this paper, we will in detail discuss the disturbance decoupling problem for the system (1). We begin in Section 2 with the d efinition of the disturban ce decoupling problem . An algorithm is proposed in this section . Based
2120
on the algorithm, some assumptions are made, under which the system (I) admits differentiable solution8 to any differentiable disturbance d. Section 3 provides another algoritbm by wbicb one can calculate tbe relative degree of the system (I). Sufficient and necessary conditions are derived, under which the disturbance decou· pling problem is sclvable. NOTATIONS _ For analytic veclor-valued functions f(z) [f,(z), ... ,/.(z)JT and o(z) [g,(z), ... ,g.(z)jT, an analytic function h(z), and an analytic matrix-valued function p(:r) [P'(z), ... ,p'(:r)] with p;(z) [P\(z), ... , p~(x)IT, the following notations will be used:
=
=
=
dh(z)
=
= [8:;:), "', 8:::)1
L1h(z)
= h ex),
L;+'h(x)
L,h(z) = dh(x)f(x),
We now come to the algorithm that is essential in investigating sufficient conditions under which there exists tbe feedba
Algorithm 2.1 . Step 1. A.sume Ihal Ihe matnx [1'2("') g,(x) w,(",)] ha, constant rank/ say 81, in a neighborhood of %0. Then there exists an 8 x s nlJnsingular matrix R 1(z) such that
R,("')[P2(z)
g2(X)
ID,( .. ) 1=
[p~~z) iil~.. ) wlJr)]
wilh [p~(,,) iiHz) wl(x)] being an 8, x (s + m+ I) matrix of rank SI near ;'to. Multiplying the second equalion of (1) on the left by Ihe maIn" R.( .. ) leads 10 Ihe following relalion
+ [pHx)] z. [9H X)] u + [wl(r)] d [ li(",)] 1Hz) 0 0 0
= L/(L'h(z», L,L/h(x) = L,(L/h(,,»
LpL'h(,,) = [Lp. L'h(z), .. _, /' p.[.,h(x)]
(4)
wh ere, k = 0, 1, .... dh is called as the differential of h, and L,h, the Lie derivative of h along f . See Isidori (1995) for other differential geometric notations.
2. PRELIMINARY RESULTS We start this section with the definition of the disturbance decoupling problem.
Assume Ihal fl(,,) ha. con.Ianl rank I, in a neighborhood of Zo. Then locally around zo, M. fHx) Dj is an (n - II)- dimensional •• bmanifold. Differen/i. aling Ihe lasl s - s, equalions of N) yields
={" :
0=
[9HZ)] u + [wHz)] wl(") d
fJ(X)] [pl(z)] [IHx) + pl(z) ,+ 9j(Z)
L/Ji(._.,J(X)]T and
Disturbance Decoupling Problem. Given an affine nonlinear singular system (l) and an initial point zo, find a regular s~atic state feedback control Jaw
u
= o(z) + P(x)v + ,),(x)z
L"JJ,(x)
L,._fMr) L,._ fi(._., )(,,)
z = fI(z) + g,( .. )<>(x) + [P,( .. ) + O,(,,)')'(x)],
L.. "f4,(x)
+ w,(x )d
0= J,(x) + g,( .. )o(x) + (P,(x) +g,(x)p(x)v + w,(z)d
+ g,("h( .. )]z
]
]
3
( )
Lw" fi(._.,)( x) with n;(x) being Ihe i-Ih 'ntry of fHz), PH(Z),O';(z), andwH(x} are th,i-th column ofp,(x),g,(r), andw,(",), respectively_
has Ih e following proper/;es (a) For any differentiable v(/) and d(/), Ihere erisls a unique differe ntiable solution {,,( "0, I, v, d), z("o, I, v , d)} to th, syslem (3) such Ihat z(O)
Step k+1. Suppose thal from ./ep' 1 Ihrough k w. have
= "'"
(b) y(l) i. decoupled from d(t) , Ihal ~tale :to and (.ach input v , lhe output
]
Lp.Ji(._.)r)
(2)
defined in a neighborhood U of"o wilh P(,,) nonsing.lar on U I such that the corresponding cJlJsed-Joop system
+9, (r)p(x)v
=
IS,
for Ihe inilial
0=
[p~(x)] [ii~("')] [w~(z)] [!l(,,)] f:(x) + pl(x) z + ii~(x) u + w~(r) d (5)
y,(I):= It(r(ro,l,v , dd) = y,(t) := h("("o , I,v , d,» for every pair 0/ disturbance vector fun ctions d •. d z , and f(Jr every tim e t > 0 for which the s(J/utilJns z(zo, t, V, dd, i 1,2, are defined.
with the matrix
pHz) [ pH,,)
=
2121
has constant rank 81:+1 in a ueighbor'hood oJ Xc in Ml;. Then it is possible to find an s x s nonsingular matrix R'+I(x) such that
RH I ( Z)
-. () [ ,,~(X) P2 Z
ii~(x)
-. ( )
zero dynamic algorithm (Isidori, 1995). In addition, similar method is used to derive the canonical/orm for a general class of nonlinear singular systems (R08chon at al., 1992). Notice that a necessary condition for the existence of
92 X
ditFerentiable solutions to the system (1) is that the initial condition "'0 satisfies I~(xo) = 0 for every k of the algorithm. If #(zo) = 0 and the two constant rank as-
= [,,~+1(Z)
g;+I(X) w~+I(X)] 000
where [,,~+I(x) ii~+'(z) ,jj~+I(x) I is an 8HI X (8 + m + I) matrix and has rank SI:+1 in a neighborhood 0/ Zo in M •. Multiplying (5) on the left by the matrix R.+I(z) gives
(6)
sumptions are satisfied at every step of the algorithm, then it easily follows from construction that the algorithm terminates after k· < n iterations, where k* is the least integer such that Mft- :;::: M1c-+l. Now we make the following assumptions whi ch guarantee that the two
assumptions of tbe algorithm do not depend OD tbe particular choice of matrices R!:(x) at the each iteration of the algorithm. Assulllption 2.3.
Ass.me that [f~(x)T , ... ,f~+'(x)TV has constant ronk t. in a neighborhood of %0 in M •. Now set M.+ I {z E M. : f;+'(x) 0). Then it is easily seen that M.+ I is an (11 -lie)· dimensional submanifold. Differentiating the last s - "HI equations of (6), one can get th e following algebraic equatiou.'i:
. [M(X) () 1 t he matrices V,(x)
W~(x)l h
g~(,,)
w~(x)
(7)
<
e+1.
(2) [fH,,)T •... ,f~+'(>:)TV has conslanl rankforallx E M. around Xo and for all k
< k' + I.
. ["f+'(z) iif+'(X) (3) the malrl" P2 -"+I() -"H() x 92 X mnk s.
0=
as co ..tant
rank for all x E M. around Xo and for all k
=
=
ii~(x)
wf+'(x)] - "+I() has w2 z
For convenience, we call Zo a regular point of Algorithm 2.1 if the assumptions of Assumption 2.3 are satisfied.
Remark 2.4. fn assumption 2.3, (I) and (2) do not depen d on th e particular cJwice of matrices introduced where and
at each iteration of the algorithm if (3) i. sali.fied, I.idori (1995).
i,Hl (z) = [ L!.f~t' (xl
g;+1(x)
see
Remark 2,5. The algorithm 2.1 results in a set Of'[;~~, (s - s,:) algebraic constrajnts which is independent of %, namely
=
W;+I(z) =
L, .. f~,+'(z) [L
L,,~f;t'(z)
...
g"
fHI
2(. - ••• ,)
(x)
L.,,, f:t ' (x)
.. .
[ L fk+ ' (xl 'lUll 2(A -.I1o+d
1 f,(x)
]
L
(x) "m JI::+I 2(.- ••• ,) L.,,, f;t' (x) ]
. ..
L 'IU,1 f'+' (x) 2('-'1o+a)
with I;'+'(x) being the i-th entry of f;+'(>:) .
Remark 2.2. The algorithm 2. 1 is a modified version of algorithmic procedure which is introduced by Kumar and Dao.lidis (199D and Liu (19950) to construct the algebraic variable z. 011 onc hand, Kumar and Daoutidis (199.{) devetoped such an algorithm based on Hirschorn's inversion algorithm ( Hirschoru , 1979)J on the other handJ Liu (1995a) introduced a similar algorithm based on Ihe
= O.
2 f 2(>:)
= O•... ,f2e
(x)
=0
Under the. conditions list ed in assumption 2.3, these. conslminis define a controlled-inMriant manifold:
M
={x E Rn : fj (x) = 0,
li(x)
= 0, ... ,fnz) =O}
of dimension n - '[;~~I (s - ,;). It i. worth noting that th e state x of the system (1) evolves on this manifold if initial slate Zo belongs to M . Now let us give the following conclusion which provides sufficient and necessary conditions which guarantee the
system (I) is solvable by feed back for any differentiable d, that is, there exi.ts a feedback of the form (2) such that the closed-loop system (3) admits a unique differ-
entiable solution for any differentiable v and d.
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Proposition 2.6 . Ass"me that Xo is a reg"lar point of Al90rilhm 2.1. Then. Ihere exist •• al lea.1 locally. a feedback of Ihe form (2) ..ch Ih.1 Ihe clo ..d-Ioop .ys/em (9) admib Cl lInigue differentiable solution for any di//erefl. liable v .nd d if .nd onlJ if Ihe malrix
if+1(z) [ p;"+l(X)
change the problem for the system (1) into one for a regular system which is composed of only differeotial equatioo by introducin~ feedback law . For conveoieoce in notation, denote the system resulting from Algorithm
2.1 by
=
gr+l(x)] iir+l(x)
i: !I(z) + PI(Z)Z + 91(Z)" + wl(x)d 0= h(z) + l'2(z)z + g,(z)" + w,(z)d y = her)
has rank s. Proof: Necessity: Assume that the rank of the matrix
i{+l(X) [ p~ +I(x)
g~:+l(X)] g~ +l(x)
pf+1(x)
gf+l(X) -'"+l() 92 X
with [p,(",) g,(",)) having full row rank in a neighborhood of Zo in M. Because the matrix [1'2(z)
is 1... than s. Then. it easily follows from (3) of the assumption 2,3 that there exists an s X s nonsingular matrix H (z) such that 1f(x) [ P2 -."+l() X
(9)
wf+1(x)] . '"+l() %
in a neighborhood of
Zo
g,(z)) has full row rank in M I there exists an s X m
matrix 'Y(x) such that p,(z) + 9'(Z)oy(Z) is nonsingular in a neighborhood of Zo in M. Therefore the feedback of the form u = 'Y( x)z + v can render the closed-loop system
W2
= !I(z) + (P1(r) + gl(:rh(:r))z+ gl(z)v + w,(x)d o = h(z) + (P,(x) + g,(zh(x))z+ g,(z )v + w,(,,)d
i;
= [p;"+I(X) iif+'(X) uf+'(x)]
o
0
W(x)
As a result~ the corresponding algebraic equation becomes as follows
0= (8)
y
(10)
= hex)
have a unique differentiable solution for any differentiable v and d. In fact,z can be fixed uniquely from the
second equation of (10). namely z
= -Iv,(z) + g,(xh(zW1[j,(z) + g,(z)v + W2(X)d]
Substituting this into the fir. t equation of (10) leads to In order to make the second equations of (8) hold for any d, W(r) must be identically zero which is contrary to (3) of the assumption 2.3.
= !I(x) -
6(x)h(x) + (g,(z) - 6(x)g,(:r)}v +(WI(Z ) - 6(z)w,( :r)}d (11) y = hex)
i:
Sufficiency: Assume that the matrix
pf+l(X) [ pf+'(x)
with 5(x)
gf+'(X)] gr+l(x)
Up to now, the disturbance decoupling problem for the
has rank s. Then. there exists a mat.rix 'Y(x) such that the matrix
pf +'( x)] [ pf+l(z)
+
t·
+ 'Y(z)z can make
the corresponding closed-loop system have a. unique differentiab le solution for any differentiable v and d. 0
Assumption 2.7. The matrix [ pf.+'(z) +I(x) has rank s .
p;
3.
system (\ ) has been changed into one for the system (11). The later problem was well posed and solved by using the concept of a controlled invariant distribution
[!If+l(X)] iif+l(x) 'Y(x)
is nonsingular. So the feedback" =
= (PI(Z) + gl(:r)oy(z)]IP2(z) + g,(xh(x)]-l .
g~".+I(Z)]
g;
+I (Z)
or a rdati ve degree (Isidori. 199[)) . Based on the results given by Isidori (1995). one can easily draw the following conclusion.
Theorem 3.1. Suppose Ihal x o is a regular poinl of Algorithm 2.1, Assumptions 2.7 holds, and there exists a malrix r(z). which render the malrix P2(,,)+g,(z)r(z) nonsinguJar at .ro, such that the system (11) has a vector relative degree {PI , "',Pm} at Zo - Th en th e disturbance deco up/ing problem is solvab/~ if and only if
MAIN RESULTS
In this sect.ion, we wi.ll solve the disturbance decoupling
problem . The natural idea of solving this problem is to
LwL,h,(z)
=0
k
= O..... P, -1.i =1. .... m(12)
with f = ft(z) - 6(x)h«) .Hd W = Wl(X) - 6(x)w,(x) . In addilion, the feedback
2123
(13) can solve the disturbance decoupling problem where
A(",)
=
LgL?~lhl("') [
]
:
_ [ Lj'h l (",) ] b(:r) :
LgLjm-Ih~(x)
Ljmh~(r)
It is easily seen that the solvability of the disturbance decoupling problem apparently depends on the choice of the matrix ,Cz). However, we will see in a moment that this is not the case under some conditions. To this end, the following algorithm is helpful.
It follows from Liu and Celikovsky (1995) that the following lemma holds. Lemma 3.4. Suppose that Xo is a regular point oJ Algorithm 2.1, and A.sumptions 2.7 and 3.3 hold. Then, for
any ')'(x) which rende .. p,(",) + g,(x),),(,,) non8ingular, the closed-loop 8ystem (11) has a vector relative degree {PI, ... ,p~} at "'0. In addition, the matrices A(x) and b(",) can be calculated by A(",) = Du("') - [D,(x) + Du(x)')'(",)JtJ(X)9'(Z) (14) b(z) = D(,,) - [D,(x) + Du(",)')'(x)JtJ(x)h(z) with q(",) = [P,(",) + g,(",)')'(",)]-I.
Algorithm 3.2.
Now the main results can be stated as follows. Step 1. Set ",?(r) hi("') and calculllte Lp,tI>?(x), Lg,tI>?(x) Theoreln 3.5. Suppose that Xo is a regular point of and Lf,,,,7(:r). If the matrix Algorithm fU, and Assumptions 2.7 and 9.3 hold. Then, the disturbance decoupling problem is solvable if and only P2(X) g,(:r) if [ Lp,"'?("') Lg,,,,?(x)
=
1
has constant rank s in a neighborhood of Xo in M, then there exists a unique vector-valued analytic function E7(x)
o~ k
~
Pi - 1. 1 ~ i
~ m
(15)
of dimension s such that [Lp,.p7(x) Denote tl>i(x) Pi
Lg, tl>7(x)] = E?(x)[p,(x)
= L"tI>?(x) -
g2(:r)]
for all r in a neighborhood of Xo in M. In addition, the
feedback
EiO("')h(:r). Otherwise. set
u
= 1 and quit the algorithm.
Step k+l. Assume that we have defined a sequence of .p7(x), ... ,.p7(x). Now calculate L p,.p7(",), Lg,tI>7("') and L"tI>hx). Tt the matrix
p,(x) [ Lp,.pf(x)
g,(",) Lg,.p7(",)
1
Lg,,,,f(x)]
= Ef(x)[p,(x)
f(x)g,(",)]-I[D(z) - f(x)h(x)] -[Du(") - f(",)g,(xn-Iv + -y(x)z
achieves the goal of disturbance decoupling where -y(x) is such that p,(x) + g,(x),),(x) is nonsingular, where
f(x)
= [D,(x) + Du(x),),(z)]"(,,,)
Proof: It follows from Algorithm 3.2 that the relation
has constant rank s in a neighborhood of Zo in M, then there exists a unique vector-valued analytic function Et(x) of dimension s such that
[Lp'
= -[Du("') -
[Lp,.p:(x)
Lg,tI>7(x)] =: Ef(",)[p,(",)
It is well-known that the equation above is equivalent to the following equation
g,(",)] [L p ,,,,7(x)
Denote q;~+I(X) = Lf,.pf(x) - E[(",)h(x). Otherwise, set Pi = k + 1 and quit the algorithm.
=
Performing Algorithm 3.2 for i 1, ... , rn, produces integers Pt, ... ,Pm. Now, introduce the following matrices;
g,(",)](16)
= E[(x)[p,(x)
Lg,,;f(,,,)]
~l
[.J",)
9'(")] [')'("')
~1
that is
[Lp,.pf(x)
+ Lg,tI>rc"'h("') Lg,tI>:("')]
= Ef(",)[p,(x) + g,(x)-y(x)
g,(x)]
Now for any ')'(x) such that [P,(",) + g,(",)-y(x)] is nonsingular, Ef(x) can be uniquely determined by
Ef(",) . 3 T"ne matnx . [p,(X) AssumptIOn 3.. D~(x)
g,(",) Du(x)
1is non-
= [Lp, ,;7(",) + L g , q'>f(x)-y(x)]q(",)
(17)
Substituting Ef(,,) into (16) gives the following relations:
singular at Xa.
2124
L., "'f(x)
= [Lp, "'f(x) + Lg, "'f(x)-r(x)Mx)g2(x)
Campbell, S.L. (1982). Singular System. of Diffe,..ntial Equations, Vo/.Il, London: Pitman.
By the construction of .p:+l(x), it is easily seen that
Chen, X.L. and M.A. Shayman (1992). Dynamics and
",:+1(x) = [Lp, "'f(x) + L" "'f(xh(x)J~(x)h(x)
control of constrained nonlinear systems with application to robotics, Proc. of A CC, 2. pp.2962-2966.
Now] according to (17), calculate
Lwh,(x) = Lw,h,(x) - [Lp,h,(x) + Lg,h,("h(x)J~(X)W2(X) = Lw, q,?(x) - [Ll" ",?(x) + L., "'?(xh(x)J~(X)W2(X) = Lw, q,?(x) - E;'(x)w,(x) and
L,h,(x) = L"h,(x) - [Lp,h,(x) + Lg,h,(xh(x)J~(x)h(x) = Lh",?(x) - [LO',,,,?(x) + L.,,,,?(xh(x)J'1(x)h(x) Lh",?(x) - E, (x)h(x) = (x)
"'1
=
Then, it follows that L,h,(,,) = "'1(,,) and Lwh,(,,) is equivalent to Lw,"'?(x) = E?(x)w,(zj.
=
°
Now, assume that LwLih,(z) = O,t = O, ... ,k -1 are equivalent to Lw,"'l(z) = El(x)w,(x),t = O, ... ,k - I, and Lih,(,,) = "';(x). t = I, ... k. Then, one has
and
= L''''f(x)
= L,,"':(") - [Lp, "'f(x) + L g ,"'f(xh(x)J'1(x)h(x)
= ""
linear differential-algebraic process systems, Proc.
0/
ACC, 1, pp.33Q..335, Baltimore. Lin, Y.L. and N. U.Ahmed (L991) Approach to controllability problems for singular system, I. J. Control, 22, pp.675-690. Liu X.P. (1993). On linearization of nonlinear singular control systems, Proc. of A CC, 3, pp.2284-2287, San Francisco.
LwL1h,(z) = Lw"'f(x) = Lw,"':(x) - [Lp, "'f(x) + Lg,,,,f(x);-(x)Mx)w,(x) = Lw,q,f(x) - Et(z)w,(x) L~+lh,(x)
Dai, L.Y.(1989). Singular Control Systems, Springer. Hirschorn, R.N. (1979). Invertibility of multivariable nonlinear control systems, IEEE Tran8. Auto, Control, AC-24, pp.855-865. Isidori, A. (1995). Nonlinear Contml Systems, 3rd Edition, Springer-Verlag. Kawaji, S. and E.Z. Taha (1994). Feedback linearization of a class of nonlinear descriptor systems, Proc. of the 33rd CDC, 4, ppA035-4037, Lake Buena Vista, FL. Kumar, A. and P. Daoutidis (1994). Control of non-
+I(X)
Thus, (12) is equivalent to (15), which means that the first part of the theorem holds. In addition, it follows froln (13) and (14) that the state0 ment about the feedback is true.
4. CONCLUSION
Liu X.P. (1995). Solvability of nonlinear singuLar systems: part 11, the case with inputs, Proc. 0/ ACe, to appear.
Liu X.P. and S. Celikovsky (1995). Noninteracting control of affine nonlinear singular control systems. Research Report, No.1852, UTIA AV CR, 1995. (ftp:ftp.utia.cas.cz/pub/reports/ utiaI852.ps.z). McClamroch, N.R. (1990). Feedback stabilization of control systems described by a class of nonlinear differential-algebraic equations, Systems fj Control Let·
ters, 15. Rouchon, P., M.Fliess and J.Levine (1992). Kronecker's canonical forms for nonlinear implicit differential sys-
tems, 2nd [FAC Whork.hop on System Str.cture and Control, pp.248-251, Prague, Czechoslovkia. You, L.S. and B.S.Chen (L993). Tracking control designs for both holonomic and non-holonomic constrained mechanical systems: a unified viewpoint, /. J. Control,
This paper has investigated the disturbance decoupling problems for affine nonlinear singular systems. An algorithm is proposed, by which the original system can
58, pp.587-6L2.
be changed into a system which is solvable by feedback. Another algorithm is provided. Based on this algorithm, sufficient and necessary conditions ror the solvability of the disturbance decoupling problems has been derived.
REFERENCES Campbell, S.L. (1990). Descriptor systems in the 90's, Proc. of the 29th IEEE CDC, 1, pp.442-447.
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