Extended Lie Brackets for Nonlinear Time-Delay Systems

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Systems

Claudia Califano, Luis-Alejandro Marquez Martinez, Claude Moog

To cite this version:

Claudia Califano, Luis-Alejandro Marquez Martinez, Claude Moog. Extended Lie Brackets for

Non-linear Time-Delay Systems. IEEE Transactions on Automatic Control, Institute of Electrical and

Electronics Engineers, 2011, 56 (9), pp.2213-2218. �10.1109/TAC.2011.2157405�. �hal-00625401�

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Extended Lie Brackets for Nonlinear Time-Delay Systems Claudia Califano, Member, IEEE,

Luis Alejandro Marquez-Martinez, and Claude H. Moog, Fellow, IEEE

Abstract—In this note the Extended Lie bracket operator is introduced

for the analysis and control of nonlinear time-delay systems (NLTDS). This tool is used to characterize the integrability conditions of a given submodule. The obtained results have two fundamental outcomes. First, they define the necessary and sufficient conditions under which a given set of nonlinear one-forms in the -dimensional delayed variables ( ) ( ), with constant but unknown, are integrable, thus generalizing the well known fundamental Frobenius Theorem to delay systems. Secondly, they set the basis for the extension to this context of the geometric approach used for delay-free systems. The effectiveness of the results is shown by solving the problem of the equivalence of a NLTDS to an accessible Linear Time-Delay System (LTDS) by bicausal change of coordinates.

Index Terms—Delay systems, geometric approach, linear equivalence,

nonlinear continuous-time systems.

I. INTRODUCTION

Time-delay systems are recently gaining more and more attention due to their importance in several applications such as those concerning the delay in the signal transmission over communication networks. The literature on the topic is extensive but concerns mainly linear time-delay systems [1], [2], [8], [14], [19], [22], [25], [26] and does not make use of geometric tools which are instead largely used in the delay free-case both in the linear and nonlinear context, [10], [21], [27]. A first attempt to extend these tools to this class of infinite dimensional systems has been pursued in [24] and [23] with reference to the input-output linearization problem and the feedback linearization problem for a particular class of delay systems.

In this note we will consider nonlinear time-delay systems with com-mensurate constant delays on the state and input variables—see for ex-ample [6], [7], [12]—and we will introduce an Extended Lie bracket operation to deal with this class of systems. As shown in Section III, a major technical achievement is that this operation fully character-izes the integrability of a distribution defined on the extended state space. This tool is then used to derive necessary and sufficient inte-grability conditions of the given submodule. Let us note that a first at-tempt to deal with the integrability of one-forms was pursued in [17] where a necessary condition was derived and only some special cases were treated. The effectiveness of the proposed approach is shown by characterizing the necessary and sufficient conditions under which a NLTDS is equivalent to an accessible LTDS by bicausal change of co-ordinates, a largely studied problem in the delay free case, as proven by the extensive literature on the topic (e.g. [3], [11], [13], [15], [16],

Manuscript received February 17, 2010; revised August 18, 2010; accepted April 19, 2011. Date of publication May 23, 2011; date of current version September 08, 2011. Recommended by Associate Editor H. Ito.

C. Califano is with the Dip. di Inform. e Sistemistica “Antonio Ruberti,” Università di Roma “La Sapienza,” Rome 00185, Italy (e-mail: claudia.cali-fano@uniroma1.it).

L. A. Marquez-Martinez is with the CICESE Research Center, Ensenada 22860, Mexico (e-mail: marquez@cicese.mx).

C. H. Moog is with the L’UNAM, IRCCyN, UMR C.N.R.S. Nantes 44321, France (e-mail: moog@ieee.org).

Color versions of one or more of the figures in this technical note are available online at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TAC.2011.2157405

[20]). With respect to [5], [24] we will consider a more general class of systems where there is no assumption on the delay of the input and we will study the effect of bicausal change of coordinates on the given system. For notational simplicity, and without loss of generality, we will consider as maximal delay the largest between the maximal de-lays on the state and input variables. Preliminary results can be found in [4].

The technical note is organized as follows. Some fundamental no-tions on time-delay systems are recalled and some notano-tions are in-troduced in Section II. In Section III geometric tools for dealing with time-delay systems are introduced and discussed. In Section IV the pro-posed approach is used to solve the linear equivalence problem under bicausal change of coordinates.

II. PRELIMINARIES ANDNOTATIONS

The following notations and definitions, borrowed from [9], [18], [28], will be used:K denotes the field of meromorphic functions of a finite number of elements infx(t0 i); u(t0i); 1 1 1 ; u(k)(t0i); i; k 2 INg; d is the standard differential operator that maps elements from K to E = spanKfdx(t 0 i); du(t 0 i); 1 1 1 ; du(k)(t 0 i); i;k 2 INg;

 is the backward time-shift operator: for a(1); f(1) 2 K, (a(t)df(t)) = a(t 0 1)df(t) = a(t 0 1)df(t 0 1). We will denote bydeg1 the polynomial degree in  of its argument; IR[] is the ring of polynomials in with coefficients in IR; K(] is the (left) ring of polynomials in with coefficients in K. Every element of K(] may be written as (] = ri=0 i(t)i, with i 2 K, and r = deg( (]). Addition and multiplication on this ring, which is non-commutative but Euclidean [28], are defined by (] + (] = maxfr ;r gi=0 ( i(t) +

i(t))i and (] (] = ri=0 rj=0 i(t) j(t 0 i)i+j. Given x(t) 2 IRn, xT

[s] = (xT(t); 1 1 1 xT(t 0 s)) and (x[s](0p))T =

(xT(t 0 p); 1 1 1 xT(t 0 s 0 p)), accordingly f(x

[l])jx(0j) :=

f(x(t 0 j); 1 1 1 ; x(t 0 j 0 l)); u[s]; u[s](0p); z[s], andz[s](0p) are

defined in a similar vein. When no confusion is possible the subindex will be omitted so thatx will stand for x[s] andx(0p) will stand forx[s](0p). Finally we will denote by u[j] := (u; _u; 1 1 1 u(j)) for j  0 while u[01]:= ;. Given T(x) = [

1(x); 1 1 1 ; n(x)], we will

denote byT(x)(@=@x(t 0 j)) = ni=1i(x)(@=@xi(t 0 j)) the

submodule element which has componenti(x) in position nj + i.

1 = spanK(]fr1(x; ); 1 1 1 ; rj(x; )g, is the right module spanned

overK(] by the column elements r1(x; ); 1 1 1 ; rj(x; ) 2 Kn(].

Finally, let us recall that a polynomial matrixA(x; ) 2 Kn2n(] is called unimodular if it has a polynomial inverse. Ifs = deg(A) then deg(A01)  (n 0 1)s, [4].

In the following we will consider a nonlinear dynamics with integer delays6, represented as:1

6 : _x(t) = F x[s] + s j=0

Gj x[s] u(t 0 j) (1)

withx 2 IRn,u 2 IR. We have that 6L, the variational or differential form representation of6, is given by

6L: d _x = f x[s]; u[s];  dx + g x[s];  du (2)

with g(x[s]; ) = sj=0Gj(x[s])j and f(x[s]; u[s]; ) =

s

i=0((@F (x[s])=@x(t 0 i))+ sj=0u(t 0 j)(@Gj(x[s])=@x(t 0

i)))i.

1Note that considering integer delays is not restrictive since any

commensu-rable delay system, where all the delays are multiples of a base constant delay D, can be put into this form by time scaling.

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In this context the following definition of bicausal change of coordi-nates [18] must be recalled.

Definition 1: Consider the dynamics6 with state coordinates x. z = (x[ ]);  2 Kn is a bicausal change of coordinates for6 if

there exist an integer` 2 IN and a function 01(z[`]) 2 Knsuch that x(t) = 01(z

[`]).

Letdz = T (x[ ]; )dx be the differential form representation of the bicausal change of coordinatesz(t) = (x[ ]):

• T (x[ ]; ) = i=0(@(x[ ])=@x(t 0 i))i= i=0Ti(x[ ])i is unimodular and  .

• The inversex = 01(z[`]) is characterized by `  (n 0 1) , and its differential representation is given bydx = T01(z; )dz, with T01(z; ) = `i=0(@01(z[`])=@z(t 0 i))i= T01(x; )j

x= (z ).

• The differential form (2) is transformed into d _z(t) = ~ f(z; u; )dz + ~g(z; )du, with ~ f(z; u; ) = T (x; )f(x; u; ) + _T (x; ) 2 T01(x; ) jx= (z) ~g(z; ) = (T (x; )g(x; ))jx= (z):

III. GEOMETRY OFTIME-DELAYSYSTEMS

The Extended Lie bracket is now introduced. It is shown how, through its application, it is possible to answer to some basic questions in control theory which were open problems up to now, such as: when a given set of one-forms depending on the state variablex(t) and its delaysx(t 0 i); i 2 [1; s], is integrable; when a given square matrixT (x; ) is the differential representation of a bicausal change of coordinates.

Definition 2: Letr (x; u; ) = sj=0rj (x; u)j, with = 1; 2 and setr j(1; u) = 0, for j > s. Then, 8k; l  0, the Extended Lie bracket[r1k(1; u); rl2(1; u)]Ei, onIR(i+1)n,i  0, is defined as

rk1(1; u); r2l(1; u) Ei = min(k;l;i) j=0 rk0j 1 (1; u); rl0j2 (1; u) E0 T j(x(0j);u(0j)) 2@x(t 0 j)@ (3) with rk 1(1; u); r2l(1; u) E0= k i=0 @rl 2(x; u)

@x(t 0 i)rk0i1 (x(0i); u(0i)) +

0

l i=0

@rk 1(x; u)

@x(t 0 i)r2l0i(x(0i); u(0i)) : (4)

By definition the Extended Lie Bracket is antisymmetric.

Remark: The Extended Lie bracket definition is related to the infi-nite dimensional system

_x(t) = F x[s] + s j=0 Gj x[s] u(t 0 j) _x(t 0 1) = F x[s](01) + s j=0 Gj x[s](01) u(t 0 j 0 1) .. . (5)

naturally associated to the delay system (1). As a consequence if ri(x; ), i 2 [1; j] is associated to the dynamics _x(t) =

F (x[s])+ sj=0Gj(x[s])u(t0j), then to _x(t01) = F (x[s](01))+ s

j=0Gj(x[s](01))u(t 0 j 0 1) it will be associated ri(x; ) and

so on, that is r0 r1 1 1 1 rs 0 1 1 1 0 r0(01) 1 1 1 rs01(01) rs(01) 0 . .. .. . . .. . .. ... ... . .. 1 1 1 0 1 1 1 0 r0(0s) r1(0s) 1 1 1 . .. .. . ... ... ... ... ... 1 1 1 (6) with ri(0p) = (r1i(x(0p)); 1 1 1 ; rij(x(0p))). De-spite the infinite dimensionality of the state space xe = (xT(t); xT(t 0 1); xT(t 0 2); 1 1 1)T these elements are

finite-dimen-sional so that we can thus consider the Lie bracket[Rj1 ; Rj2 ] with Rj

l = ji=0(rj 0il (x(0i)))T(@=@x(t 0 i)), l = 1; 2. Denoting

byj = min(j1; j2), [Rj1 ; R2j ] = [rj1 (1); r2j (1)]Ej. The Extended

Lie bracket (3) is thus a projection on the(i + 1)n-dimensional state

space. /

Remark: The delayed state bracket introduced in [24] does not yield a criterion for the integrability of one-forms intrinsically depending on the delayed state variables as no decisive difference is made between the non integrable one-form!1(x; )dx = dx10 x2(t 0 1)dx2and the exact one-form!2(x; )dx = dx10 x2(t 0 1)dx2. The use of the Extended Lie bracket allows to make the difference as shown in the

Examples 1 and 2. /

To deal with the integrability of one-forms, we need now to introduce the following definition of integrable submodule. Let rl(x; ) = si=0rli(x)i, l 2 [1; j] and set 1 = spanK(]fr1(x; ); 1 1 1 ; rj(x; )g, and

x0 = (x0(t)T; 1 1 1 ; x0(t 0 )T)T.

Definition 3: The submodule1 is nonsingular locally around x0if rank(1(x)) = j, 8x 2 U0an open and dense subset ofx0.

Definition 4: The submodule1 nonsingular locally around x0, is integrable if there exist n 0 j independent functions l(x[ ]), l 2 [1; n 0 j] such that rank(@(x)=@x(t)) = n 0 j and

p=0 @l(x) @x(t 0 p)p s k=0 rki(x)k=0; 8l 2 [1; n 0 j]; 8i 2 [1; j]:

Definition 5: Consider the bicausal change of coordinates z = (x[ ]), with dz = T (x; )dx. In the new coordinates the

submodule elementr(x; u; ) is transformed as

~r(z; u; ) = [T (x; )r(x; u; )]jx= (z): (7) From (7), settingTj = 0 for j > = deg(T (x; )) and rj = 0 for j > deg(r(x; )) one has

~rl(z; u) = l p=0

Tp(x)rl0p(x(0p); u(0p))

jx= (z): (8)

Accordingly, from (8), the following result holds true.

Lemma 1: Let r (x; u; ) = sj=0r j(x; u)j, = 1; 2. Under the bicausal change of coordinates z = (x[ ]), with dz = T (x; )dx =

j=0Tj(x)jdx one has, for k  l, 0  i  l

~rk 1(z; u); ~rl2(z; u) Ei= 0 l;i(x) rk 1(x; u); r2l(x; u) El jx= (z)

where, settingTj = 0 for j >

0l;i(x)=

T0(x) 1 1 1 1 1 1 1 1 1 Tl(x)

0 . .. ...

0 0 T0(x(0i)) 1 1 1 Tl0i(x(0i))

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The proof, omitted for space reasons, can be carried out by substi-tuting in[~r1k(z; u); ~rl2(z; u)]E0 the relations given by (8) and by re-calling that@(Tp(x))T=@x(t 0 i) = (@(Ti(x))T=@x(t 0 p))T. A. Integrability Conditions of One-Forms

We will now address the problem of defining the necessary and suf-ficient conditions under which some given one-forms are integrable. The obtained results are based on the consideration that though when dealing with delay systems one ends up on the infinite dimensional system (5), the elements that one considers are characterized by a fi-nite number of components. This is enlightened in next Lemma which displays the properties of the Extended Lie bracket operator.

Lemma 2: Let 1 = spanK(]fr1(x; ); 1 1 1 ; rj(x; )g with

ri(x; ) = sl=0rli(x[ ])l,s = maxf ; deg(ri(x; )); i = 1; 2g.

Then for any couple of integersl  k, and for any integer ;  ; p  0 such that  l and   l + ps, the following properties hold true:

rl i; rjk E  r l i; rkj El = l =0 rl0 i (1); rk0 t (1) E0 T j(x(0 )) 2@x(t 0 )@ (9) rl+ps i ; rk+psj E  rl+psi ; rk+psj E;l+ps = l =0 rl0 i (1); rk0 j (1) E0 T j(x(0 0ps)) 2@x(t 0 0 ps)@ ; l  s (10) rl i; rjk E  r l i; rkj El= 0; k 0 l > 2s: (11)

Sketch of Proof: The proof of (9)4 (11) is immediate if one considers the elementsRkj(x) = ki=0rjk0i(x(0i))(@=@x(t 0 i)) defined on (6). In fact on the extended state space(@Rli=@xe)Rkj 0 (@Rk

j=@xe)Rli= 0 whenever jk 0 lj > 2s, while the others yield the

same equations, only time and coordinates -shifted. / Next theorem, whose proof is reported in the Appendix, enlightens the conditions under whichn one-forms are exact and define a bicausal change of coordinates. The conditions are given on the corresponding submodule elements. It is shown that the nilpotency condition of a given Lie Algebra which is the key point in the case of nonlinear sys-tems without delays is transformed into a nilpotency condition on the given submodule which takes into account not only the state variable x(t) but also the delayed variables. The bound on the delay is defined by the state dimension and maximal delay.

Theorem 1: LetPn(x; ) = [r1(x; ); 1 1 1 ; rn(x; )] 2 Kn2n(] be a full rank matrix, with ri(x; ) = sj=0rji(x[ ])j,

s = maxf ; deg(ri(x; )); i 2 [1; n]g, which can be

factor-ized as Pn(x; ) = T (x; )Q(), with T (x; ) unimodular. Then

T01(x; ) defines a bicausal change of coordinates iff

ri j(x); rlk(x) E;2s= 0 8l; j 2 [1; n] and 8i  k 2 [0; 2s] or equivalently ri j(x); rkl(x) E0= 0 8l; j 2 [1; n] and 8i  k 2 [0; 2s]: (12) Consider now Pj(x; ) = [r1(x; ); 1 1 1 ; rj(x; )] = s

l=0Pjl(x)lwithPj0(x) of rank j and rk(x; ) = sl=0rsk(x)s,

k 2 [1; j]. Consider the distributions 1i and 10i, i  0

defined on IR(i+1)n and with vector fields parameterized by

x(t 0 i 0 1); 1 1 1 ; x(t 0 i 0 s) 1i= spanK l=0 r 0l k (x(0l)) T @ @x(t 0 l); k 2 [1; j] 2 [0; i] ; 10i= spanK min( ;i) l=0 r 0lk (x(0l)) T @ @x(t 0 l); k 2 [1; j] 2 [0; i + s] : (13)

By construction1i 10i. Leti= rank(10i) locally around x0, then 10

i= spanfl(x); l 2 [1; i]g  IR(i+1)nwhile its elements depend on the variablesx[i+s]. Let us thus consider the series development of lwith respect to the parametersx(0i 0 1) locally around x0(0i 0 1) which without loss of generality can be assumed to be the origin, that is l(x) = l0 x[i] + s j=1 n =1 l1 x[i] x (0i 0 j) + 12 n ; =1 s j;k=1

l2 x[i] x (0i 0 j)x (0i 0 k) + 1 1 1

and consider the possibly infinite set of distributions 10 i0= span fl0; l 2 [1; k]g 10 i1= span l1; l 2 [1; k]; j 2 [1; s]; 2 [1; n] .. . (14)

Seti0 = rank( k010ik). We can now state the main result con-cerning integrability.

Theorem 2: Let 1 = spanK(]fr1(x; ); 1 1 1 ; rj(x; )g with ri(x; ) = sl=0rli(x[ ])l,s = maxfdeg(ri(x; ); i 2 [1; j]; g

and such that the matricesPj(x; ) = (r1(x; ); 1 1 1 ; rj(x; )) =

s

l=0Pjl(x)l andPj0(x) are of rank j. Let 10iand ( k010ik)

be the associated set of distributions defined, respectively, by (13) and (14) which are assumed to be locally non singular on x0 = (x0(t)T; 1 1 1 x0(t 0 i)T)T with i = rank10i and

i0 = rank( k010ik) (with 01 = 01;0 = 0). Then 1 is

integrable iff there exists an index such that:

a) 8l; k 2 [1; j] and t  p  i + s, [rtl(1); rkp(1)]Ei 2 10i; i 2 [0; ];

b)  0  01 = j;

c)  00  01;0= j:

Proof: Necessity: Assume that there existn0j independent func-tionsi(x), i 2 [1; n 0 j] with rank(@=@x(t)) = n 0 j which

are in the kernel ofPj(x; ). In the set of all possible independent solutions, consider the one characterized by the minimum delay. De-note by the maximum delay in the functions i(x), then di(x) =

k=0!ki(x)kdx satisfies k=0 !k i(x)k s l=0 Pjl(x)l = 0:

It is easily seen that for any index  0, [rtl(1); rkp(1)]E 2 1 , 8t  p  . As a consequence [rt

l(1); rkp(1)]Ei2 10i,8t  p  i + s

which proves a).

Let be the maximum delay in the functions i(x). If

 01 = n 0 j 01, then there existj 01independent one-forms !(x; )dx = 01

k=0!k(x)dx(t 0 k) (not necessarily exact) such that

[!0(x); 1 1 1 ; ! 01(x)]10

01= 0. As a consequence

!0(x) !1(x) 1 1 1 ! (x)

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withrank[!0(x); !1(x); 1 1 1 ; ! (x)] = n 0 j, which implies that  = n( + 1) 0 (n 0 j) 0 j 01, and thus b) must hold.

Let us finally consider the distributions ( k010 01;k) and ( k010

;k), and let  01;0= n 0 k1, and ;0= n( + 1) 0 k2,

k1; k2 0. Then there exist k1 0 exact and independent one-forms

d~i(x[ 01]) = 01k=0~!ik(x)dx(t 0 k) such that

~!0(x); 1 1 1 ; ~! 01(x) k0

10

01;k = 0:

On the other hand then 0 j functions i(x(t); 1 1 1 ; x(t 0 )) being independent onx(t 0 k) for k > , implies that di(x(t); 1 1 1 ; x(t 0

)) = k=0~!ik(x)dx(t 0 k) satisfies !0(x); 1 1 1 ; ! (x) k0 10 ;k = 0:

Since also[0; ~!0(x(01)); 1 1 1 ; ~! 01(x(01))]( k010 ;k) = 0 then k2 = k1+ n 0 j and c) follows.

Sufficiency: Assume that conditions a)4 c) are satisfied. Then by assumption due to a) and b) there exists an index such that  0

 01 = ( + 1)n 0 k10 k2 0 (n 0 k1) = j, that is, there exist

k2= n 0 j independent one-forms k=0!ik(x)dx(t 0 k) such that

!0(x) 1 1 1 ! (x) 10 = 0

withrank(!0(x)) = n 0 j locally around x0. Finally according to c) the one-forms!i(x; )dx can be chosen to be exact, that is, there exist

n 0 j functions i(x) such that

@i(x)

@x(t) ; 1 1 1 ; @i (x)

@x(t 0 ) 10 = 0 withrank(@(x)=@x(t)) = n 0 j, which ends the proof.

Example 1: Let1 = spanKfx2(t01)(@=@x1(t))+(@=@x2(t))g. From Theorem 2 it follows that1 is not integrable. In fact

10= 100= spanK x2(t 0 1) @@x 1(t)+

@ @x2(t)

which satisfies[r10; r10]E0= 0 2 10, while 11=101=10+spanK x2(t 0 2)@x @

1(t 0 1)+

@ @x2(t 0 1)

for which [r01; r11]E1 = 0(@=@x1(t)) 62 101, thus proving that the

one-form!1(x; )dx = dx10 x2(t 0 1)dx2which lies in the left kernel of1 is not integrable.

Example 2: Consider the submodule 1 = spanK(]fx2(t 0 1)x1(t 0 1)(@=@x1(t))+x1(t)(@=@x2(t))g. According to Theorem

2, to check if there exists a one-form which lies in the left kernel of1, we must consider 100=spanK x1(t) @@x 2(t); x2(t 0 1)x1(t 0 1) @@x1(t) ; 10 1=spanK x1(t) @@x 2(t); x2(t 0 2)x1(t 0 2) @ @x1(t 0 1) + spanK x1(t 0 1)x2(t 0 1) @@x 1(t) +x1(t 0 1)@x @ 2(t 0 1) ; .. .

since0 = 00= rank(100) = 2 locally around x0 6= 0, then condi-tion a) of Theorem 2 holds true for100. As for101, it is readily seen that

it is independent ofx(t 0 i) i  2 and is involutive, so that condition a) is satisfied for101; moreover1 = 1;0 = rank(101) = 3 so that

also condition b) and c) are satisfied being10 0= 100 00= 1. Thus there exists one integrable one-form in the left kernel of1 given by!2(x; )dx = dx10 x2(t 0 1)dx2= x1(t) 0 (1=2)[x2(t 0 1)]2.

IV. APPLICATION TOLINEAREQUIVALENCE OFTDS It is now shown how the results derived in Section III can be ef-ficiently used to solve the linear equivalence problem under bicausal change of coordinates.

Problem Statement (The Linear Equivalence Problem): Given system (1), find, if possible, a bicausal change of coordinates z = (x[ ]) such that _z = l j=1 Ajz(t 0 j) + l j=1 Bju(t 0 j) (15)

withA() = lj=1Ajj,B() = lj=1Bjj a weakly accessible pair.

Some preliminary definitions related to the accessibility properties of a NLTDS are in order.

A. Accessibility Submodules

Let g1(x[s]; ) := g(x[s]; ). The module generators gk(x; u[k02]; ), k  2 are recursively defined as

gk x; u[k02];  =f(x; u; )gk01 x; u[k03];  0_gk01 x; u[k03];  :

Definition 6: The accessibility submodulesRiof6, are defined as Ri= spanK(]fg1(x; ) 1 1 1 gi(x; u[i02]; )g; i  1.

Remark: Note that for nonlinear time-delay systems, the left-kernel ofRi is the left-submoduleHi+1, as shown in [28]. For nonlinear systems without delays

gk x(t); u(t); 1 1 1 ; u(k02)(t)

= (01)k01 adk01

f g + g; adk02f g u + 1 1 1

while for the linear time-varying and time-invariant cases,Rnreduces to the corresponding accessibility matrices[B(t) A(t)B(t)0 _B(t) 1 1 1],

[B AB 1 1 1 An01B] resp. /

The following results hold true [4].

Proposition 1: If gi+1(x; u[i01]; ) 2 Ri then 8j  1, gi+j(x; u[i+j02]; ) 2 Ri.

Proposition 2: Under the bicausal change of coordinates z = (x[ ]), with dz = T (x; )dx the accessibility

submod-ules elementsgj(1) are transformed, for j  1, as

~gj z; u[j02];  = T (x; )gj x; u[j02]; 

x= (z): (16)

Corollary 1: Under a bicausal change of coordinatesz = (x[ ]) Ri= spanK(] g1(x; ) 1 1 1 gi x; u[i02];   ~Ri

= spanK(] ~g1(z; ) 1 1 1 ~gi z; u[i02];  :

B. Linear Equivalence of Time-Delay Systems

Theorem 3: System (1) is equivalent, under a bicausal change of coordinates, to a linear weakly accessible delay system if and only if there exist a unimodular matrixT01(x; ) and a full rank matrix Q(), such that

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a) the module generatorsgi’s do not depend onu[i02], that is, for 1  i  n, gi(1) := gi(x; );

b) Rn(x) = (g1(x; ); 1 1 1 gn(x; )) = T01(x; )Q(); c) gn+1(1) := gn+1(x; )2 spanIR(]fg1(x; ); 1 1 1 ; gn(x; )g;

d) denoting bys = max(deg(gl(x[ ]; )); l 2 [1; n]; ), for i; j 2 [1; n] and r  k 2 [0; 2s] gr i(x); gjk(x) E;2s= 0 or equivalently gir(x); gkj(x) E0= 0: (17)

The proof, omitted for space reasons, can be carried out as in [4], by first showing that the conditions are satisfied by a weakly accessible LTDS and are invariant under bicausal change of coordinates, and then by showing that, if the conditions are satisfied, according to Theorem 1 the matrixT01(x; ) defines a bicausal change of coordinates and that in these new coordinates the system is linear.

Corollary 2: System (1) is equivalent, under a bicausal change of coordinates, to a delay-free linear accessible system iff conditions a) and d) of Theorem 3 are satisfied, and

b’)Rn(x) = (g1(x; ); 1 1 1 gn(x; )) is unimodular

c’)gn+1(1) := gn+1(x; ) 2 spanIRfg1(x; ); 1 1 1 gn(x; )g.

Example 3: Consider the dynamics

_x1(t) = x2(t) 0 x2(t 0 1) + 2x2(t 0 1) 2 p=1 u(t 0 p) _x2(t) = u(t) + u(t 0 1) for which g1(1) = [2x2(t 0 1)( + 1)](@=@x1(t))+( + 1)(@=@x2(t)), g2(1) = (1 0 2)(@=@x1(t)) and g3(1) = 0.

Condition a) of Theorem 3 is satisfied and the accessibility matrix Rn(x) is given by Rn(x) = 2x2(t 0 1)( + 1) 1 0  2  + 1 0 = 2x2(t 0 1) 11 0  + 10 1 0 0 2 = T01(x; )Q():

Rn(x) is not unimodular but it is factorized in the product of an

uni-modular matrixT01(x; ) and a full rank matrix Q(), which shows that condition b) is verified. Condition c) is also satisfied so we must only check condition d) withs = 2. We can equivalently consider the Extended Lie brackets[gil; gjk]E0withl; k 2 [1; 2] and i  j 2 [0; 4] withg01 = (@=@x2(t)), g11 = 2x2(t 0 1)(@=@x1(t)) + (@=@x2(t)),

g2

1= 2x2(t01)(@=@x1(t)),g13= g14= 0, g02= (@=@x1(t)), g21= 0,

g2

2 = 0(@=@x1(t)), g23= g42= 0. Since these Extended Lie brackets

are zero, the matrixT01(x; ) defines the bicausal change of coordi-natesdz = (d(x2(t)); d(x1(t) 0 x22(t 0 1)))T and yields

_z1(t) = u(t) + u(t 0 1); _z2(t) = z1(t) + z1(t 0 1):

V. CONCLUSION

In this note a geometric approach for the analysis of the properties of time-delay systems has been presented. The proposed approach re-lies on the introduction of the Extended Lie bracket operation which reduces to the standard Lie bracket for systems without delays. This operation has allowed to define integrability conditions for modules as-sociated to systems with delays. As an example the linear equivalence problem for time-delay systems was addressed and solved.

APPENDIX

Proof of Theorem 1: Let!i(x; ) = sk=0!ik(x)k be thei-th

row of the matrixT01(x; ). Setting = s + s, by construction 8 ; l 2 [1; n], ! (x; )rl(x; ) = i=0Qi li, that is, fori =

0 1 1 1 , ij=0!j

(x)ri0jl (x(0j)) = Qi l. We thus have8p  0,

i 2 [0; ] @ @x(t 0 p) i j=0 ! j(x)ri0jl (x(0j)) = 0: (18)

Equation (18) immediately implies

q p=0 @ x(t 0 p) k =0 ! (x)rk0l (x(0)) 0 k p=0 @ @x(t 0 p) q =0 ! (x)rq0j (x(0)) = 0: (19)

Assuming without loss of generality q  k and setting ip = (@(!i (x))T=@x(t 0 p)) 0 (@(!p (x))T=@x(t 0 i))T, (19) leads to q p=0 k i=0 rk0i l (x(0i)) T iprq0pj (x(0p)) + !0 (x) 1 1 1 !q (x) rqj; rkl Eq= 0: (20)

Necessity: Assume now that! (x; )dx = si=0!i (x)idx is

exact, then for anyi; p 2 [0; ], ip = 0. Assuming without loss of generalityq  k, from (20) we get that necessarily

!0(x) 1 1 1 !q(x)

0 . .. ...

0 0 !0(x(0q))

rqj; rkl Eq= 0

which due to the full rank of!0(x), proves (12).

Sufficiency: By assumption Pn(x; ) = T (x; )Q() =

s

j=0Pnjj, that is Pnk = ki=0Ti(x)Qk0i(0i) = k

i=0Ti(x)Qk0i.

Letadj[Q()] = i=0Sii be the adjugate matrix associated to

Q() =

i=0Qii. Then, sinceQ()adj[Q()] = [Id]det(Q())=

[Id](q0 + q1 + 1 1 1 + qtt) , we have that li=0QiSl0i = 0

forl  0 1 while i=0QiS 0i = [Id]q0 6= 0. It follows that l

i=0Pni(0j)Sl0i = 0 for l  0 1 and i=0Pni(0j)S 0i =

T0(0j)q 0.

Consider now (20). Since condition (12) is satisfied, for any couple of indicesk; q q p=0 k i=0 rk0i l (x(0i)) T iprjq0p(x(0p)) = 0 (21)

that is,0qk(l) = qp=0 ki=0(rk0il (x(0i)))T ipPn;q0p(x(0p)) = 0. It follows that: q=0 0q k(l)S 0q= k i=0 rk0i l (x(0i)) T i0T0q0= 0

which due to the invertibility ofT0q0, implies that ~0k(l) = k i=0 rk0i l (x(0i)) T i0= 0:

For l 2 [1; n], and k 2 [0; ] we get ~0k =

k

j=0[Pn;k0j(x(0j))]T j0 = 0. Then i=0S 0iT ~0i =

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By induction, assume thatip= 0 8i  0 when p 2 [0; q0 1] and fori 2 [0; k 0 1] when p = q, then 8  0

0 = t=0 0q+t  k+ (l)S 0t =  k+ i=0 rk+ 0il (x(0i)) T iqT0(x(0q)) q0

that is, due to the invertibility of T0(x(0q))q0, i=0k+ [rlk+ 0i(x(0i))]T

iq = 0 or equivalently i=0

[r 0i

l (x(0k 0 i))]T k+i;q = 0. One thus gets 0 = =0 i=0

ST

0 [Pn; 0i(x(0k0i))]T k+i;qthat is0 = q0[T0(x(0k))]T k;q

which proves the thesis,q0[T0(x(0k))]T being invertible. REFERENCES

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The Bounded Real Lemma for Internally Positive Systems and H-Infinity Structured Static State Feedback

Takashi Tanaka and Cédric Langbort

Abstract—We consider the bounded real lemma for internally positive

linear time-invariant systems. We show that the norm of such systems can be evaluated by checking the existence of a certain diagonal quadratic storage function. Taking advantage of this fact, the problem of designing a structured static state feedback controller achieving internal stability, contractiveness, and internal positivity in closed loop becomes convex and tractable.

Index Terms—Decentralized control, linear matrix inequality (LMI),

positive systems.

I. INTRODUCTION

The design of structured and decentralized static state feedback con-trollers has attracted attention in the controls community for several decades. For arbitrary plants and controller structures, the problem is known to be NP-hard [2]. However, a number of situations have been identified recently in which the plant’s spatial invariance or quadrati-cally invariant structure make it possible to reformulate it as a convex program.

In this technical note, we identify another class of plants, the so-called internally positive systems, for which the design of struc-tured H1 static state feedback controllers can be reformulated as a convex problem. Roughly speaking (a rigorous definition is given in Section III), internally positive systems are such that their state and output lie in the first orthant of their respective spaces, when the disturbance input does, i.e., such that the positive cone of the state-and output-space is positively invariant under positive disturbances. Positive systems have been studied extensively in the past (see, e.g., [5] for a thorough study of their stability properties) and shown to be accurate models of dynamic processes taking place over networks

Manuscript received June 01, 2010; revised January 04, 2011, April 12, 2011, and April 18, 2011; accepted May 12, 2011. Date of publication May 23, 2011; date of current version September 08, 2011. This work was supported in part by National Science Foundation (NSF) Grant CMMI 08-26469. Recommended by Associate Editor D. Arzelier.

The authors are with the Department of Aerospace Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA (e-mail: ttanaka3@illi-nois.edu; langbort@illinois.edu).

Digital Object Identifier 10.1109/TAC.2011.2157394

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