ISS CONTROL LAWS FOR STABILIZABLE RETARDED SYSTEMS

BY MEANS OF THE LIAPUNOV-KRASOVSKII METHODOLOGY

Pierdomenico Pepe ¤

ABSTRACT A disturbance adding to the control lawis a typical situation in practice because of actuator er-rors. In this paper, state feedback control laws whichprovide input-to-state stability of the closed loop systemwith respect to a disturbance adding to the control laware investigated for state feedback stabilizable (in thecase of disturbance equal to zero) retarded nonlinearsystems. The formulas for the input-to-state stabiliz-ing state feedback control law are provided by employ-ing the Liapunov-Krasovskii methodology. An exampletaken from the past literature is investigated in details,showing the e®ectiveness and the applicability of theproposed control design.

Keywords: Input-to-State Stabilizability, RetardedNonlinear Systems, Liapunov-Krasovskii Methodology.

INTRODUCTION

In 1989 Sontag showed in the paper [23] that nonlin-ear systems which are (smooth) feedback stabilizable,are also (smooth) input-to-state stabilizable with re-spect to disturbances adding to the control input. Aswell known, those disturbances are very frequent inpractice, because of actuator errors. Many contribu-tions concerning the state feedback stabilization and theinput-output state feedback linearization of nonlinearretarded systems can be found in the literature (see,for instance, [3,6,8,12,14,15,17,18,26,30]). Liapunov-Krasovskii methodologies for the input-to-state stabil-ity of retarded nonlinear systems have been studied in[25] and in [10,19,28], respectively. As far as the input-to-state stabilizability of stabilizable retarded nonlinearsystems is concerned, a contribution is given in [27],where, besides the main results dealing with the re-lationship between the input-to-state stability and theexponential stability in the unforced case, the input-to-state stabilizability of retarded nonlinear systems whichare transformable by a state feedback control law into alinear, delay-free, exponentially stable system is consid-ered, and the formula for the input-to-state stabilizingstate feedback control law is provided.

This work is supported by the Italian MIUR projectPRIN 2005.

¤ Dipartimento di Ingegneria Elettrica, Univer-sitµa degli Studi dell'Aquila, Monteluco di Roio, 67040L'Aquila, Italy, e-mail: pepe@ing.univaq.it

In this paper a general theory for the input-to-statestabilizability of state feedback stabilizable retardednonlinear systems is provided. It is proved that a statefeedback stabilizable (in the case of disturbance equalto zero) retarded nonlinear system admits an input-to-state stabilizable state feedback control law (i.e. ad-mits a state feedback control law such that the closedloop system is input-to-state stable with respect to adisturbance adding to the control law), provided thatthe control system obtained by closing the loop withthe stabilizing (in the case of disturbance equal to zero)state feedback control law admits a Liapunov-Krasovskiifunctional, with a suitable property, by which its globalasymptotic stability can be proved. The formula forsuch input-to-state stabilizing state feedback control lawis provided. It is assumed that the disturbance is mea-surable and locally essentially bounded. The essentialboundedness of the disturbance in [0;+1) is not as-sumed.

The e®ectiveness of the methodology here proposedis shown in details on an example of stabilizable (in thecase of disturbance equal to zero) retarded nonlinearsystems taken from the past literature.

Notations

R denotes the set of real numbers, R? denotes the ex-tended real line [¡1;+1], R+ denotes the set of nonnegative reals [0;+1). The symbol j ¢ j stands for theEuclidean norm of a real vector, or the induced Eu-clidean norm of a matrix. The essential supremumnorm of an essentially bounded function is indicatedwith the symbol k ¢ k1. A function v : R+ ! Rm,m positive integer, is said to be essentially bounded ifess supt¸0 jv(t)j < 1. For given times 0 · T1 < T2, weindicate with v[T1;T2) : R+ ! Rm the function given byv[T1;T2)(t) = v(t) for all t 2 [T1; T2) and = 0 elsewhere.An input v is said to be locally essentially bounded if,for any T > 0, v[0;T ) is essentially bounded. For a pos-itive real ¢, C([¡¢; 0];Rn) denotes the space of thecontinuous functions mapping [¡¢; 0] into Rn, n posi-tive integer. For positive integers m, n; Im denotes theidentity matrix in Rm£m, 0m;n denotes a matrix of ze-ros in Rm£n. A functional F : C([¡¢; 0];Rn) ! Rm£p,m;n; p positive integers, is said to be completely contin-uous if it is continuous and maps closed bounded sets ofC([¡¢; 0];Rn) into bounded sets of Rm£p. Let us hererecall that a function ° : R+ ! R+ is: positive de¯nite

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if it is continuous, zero at zero and °(s) > 0 for all s > 0;of class G if it is continuous, zero at zero, and nonde-creasing; of class K if it is of class G and strictly increas-ing; of class K1 if it is of class K and it is unbounded; ofclass L if it monotonically decreases to zero as its argu-ment tends to +1. A function ¯ : R+£R+ ! R+ is ofclass KL if ¯(¢; t) is of class K for each t ¸ 0 and ¯(s; ¢)is of class L for each s ¸ 0. With Ma is indicated anyfunctional mapping C([¡¢; 0];Rn) into R+ such that,for some K1 functions °a; ¹°a, the following inequalitieshold

°a(jÁ(0)j) ·Ma(Á) · ¹°a(kÁk1); 8Á 2 C([¡¢; 0];Rn)(1)

For example, the k ¢ kM2norm, given by (see [2])

kÁkM2=³jÁ(0)j2 +

R 0¡¢jÁ(¿ )j2d¿

´ 12

, is a Ma func-

tional. As usual, ISS stands for both input-to-state sta-ble and input-to-state stability.

PRELIMINARIES

In this section, for the reader's convenience, some pre-viously published results which are fundamental for theunderstanding of the novel results which will be pro-vided in next sections are brie°y reported, with someslight modi¯cations for the purposes of this paper. Letus consider the following retarded nonlinear system

_x(t) = f(xt) + g(xt)v(t); t ¸ 0; a:e:;

x(¿ ) = »0(¿ ); ¿ 2 [¡¢; 0];(2)

where x(t) 2 Rn, v(t) 2 Rm is the input function,measurable and locally essentially bounded, for t ¸ 0xt : [¡¢; 0] ! Rn is the standard function (see Section2.1, pp. 38 in [5]) given by xt(¿ ) = x(t + ¿), ¢ is themaximum involved delay, f is a locally Lipschitz, com-pletely continuous functional mapping C([¡¢; 0];Rn)into Rn, g is a locally Lipschitz, completely contin-uous functional mapping C([¡¢; 0];Rn) into Rn£m,»0 2 C([¡¢; 0];Rn). It is here supposed that f(0) = 0,thus ensuring that x(t) = 0 is the trivial solution forthe unforced system _x(t) = f(xt) with zero initial con-ditions. Multiple discrete non-commensurate as well asdistributed delays can appear in (2). In the following,the continuity of a functional V : C([¡¢; 0];Rn) !R+ is intended with respect to the supremum norm.Given a locally Lipschitz continuous functional V :C([¡¢; 0];Rn) ! R+, the upper right-hand derivativeD+V of the functional V is given by (see [1], De¯nition4.2.4, pp. 258)

D+V (Á; v) = lim suph!0+

1

h(V (Áh)¡ V (Á)) ; (3)

where Áh 2 C([¡¢; 0];Rn) is given by

Áh(µ) =

½Á(µ + h); s 2 [¡¢;¡h);

Á(0) + (f(Á) + g(Á)v)(µ + h); µ 2 [¡h; 0](4)

De¯nition 1: ([23,19]) The system (2) is said to beinput-to-state stable (ISS) if there exist a KL function¯ and a K function ° such that, for any initial state»0 and any measurable, locally essentially bounded inputv, the solution exists for all t ¸ 0 and furthermore itsatis¯es

jx(t)j · ¯ (k»0k1; t) + °¡kv[0;t)k1

¢(5)

Theorem 2: ([19]) If there exist a locally Lipschitzfunctional V : C([¡¢; 0];Rn) ! R+, functions ®1, ®2of class K1, and functions ®3, ½ of class K such that:

H1) ®1(jÁ(0)j) · V (Á) · ®2(Ma(Á)); 8 Á 2C([¡¢; 0];Rn);

H2) D+V (Á; v) · ¡®3(Ma(Á));

8 Á 2 C([¡¢; 0];Rn); v 2 Rm : Ma(Á) ¸ ½(jvj);

then, the system (2) is input-to-state stable with ° =®¡11 ± ®2 ± ½.

Remark 3: With respect to the published literature:Theorem 2 makes use of theMa functional instead of thek ¢ ka norm used in [19], thus weakening the hypotheses.²

MAIN RESULTS

Let us consider now the following retarded nonlinearsystem, corresponding to (2) when the input v is givenas the sum of the control input and of the disturbance:

_x(t) = f(xt) + g(xt)(u(t) + d(t)); t ¸ 0; a:e:;

x(¿) = »0(¿ ); ¿ 2 [¡¢; 0];(6)

where u(t) 2 Rm is the control input, d(t) 2 Rm

is the disturbance, measurable and locally essentiallybounded.

Theorem 4: Let there exists a locally Lipschitz, com-pletely continuous functional k : C([¡¢; 0];Rn) ! Rm,such that the closed loop system (6) with u(t) = k(xt),and no disturbance, described by the equations

_x(t) = f(xt) + g(xt)k(xt); (7)

is globally asymptotically stable. Let, for Á 2C([¡¢; 0];Rn), h 2 (0;¢);

Áah(µ) =½

Á(µ + h); µ 2 [¡¢;¡h);Á(0) + (µ + h) (f(Á) + g(Á)k(Á)) ; µ 2 [¡h; 0];

±gh(µ) =

½0n£m; µ 2 [¡¢;¡h);

(µ + h)g(Á); µ 2 [¡h; 0](8)

5265

Let there exist a locally Lipschitz continuous functionalV0 : C([¡¢; 0];Rn) ! R+, functions ®1, ®2 and ®3of class K1, a locally Lipschitz, completely continuousfunctional r : C([¡¢; 0];Rn) ! Rm such that, for anyÁ 2 C([¡¢; 0];Rn) and any M 2 Rm:

i) ®1(jÁ(0)j) · V0(Á) · ®2(Ma(Á));

ii) the functional D+a V0 : C([¡¢; 0];Rn) ! R?, de¯ned

as

D+a V0(Á) = lim sup

h!0+

V0(Áah)¡ V0(Á)

h(9)

is locally Lipschitz, completely continuous and,moreover, D+

a V0(Á) · ¡®3(Ma(Á));

iii)

lim suph!0+

V0(Áah + ±

ghM)¡ V0(Á

ah)

h= rT (Á)M: (10)

Then, the feedback control law

u(t) = k(xt) + p(xt); (11)

with p : C([¡¢; 0];Rn)! Rm de¯ned as

p(Á) =1

2D+a V0(Á)r(Á); (12)

is such that the closed loop system (6), (11), describedby the following equations

_x(t) = f(xt)+g(xt)k(xt)+g(xt)p(xt)+g(xt)d(t); (13)

is input-to-state stable with respect to the measurableand locally essentially bounded disturbance d(t).

Proof.

In order to prove the input-to-state stability of theclosed loop system (13) with respect to the disturbanced(t), let us apply Theorem 2 by using the locally Lips-chitz continuous functional V = V0. The functional Vsatis¯es the hypothesis H1 in Theorem 2. As far as thehypothesis H2 in Theorem 2 is concerned, the deriva-tive D+V (Á; d) of the functional V with respect to theclosed loop system (13), according to (3), is as follows

D+V (Á; d) = lim suph!0+

V (Áah + ±gh(p(Á) + d))¡ V (Á)

h

(14)The following equalities/inequalities hold

D+V (Á; d) = lim suph!0+

V (Áah + ±gh(p(Á) + d))¡ V (Áah)

h+

V (Áah)¡ V (Á)

h· ¡®3(Ma(Á)) + rT (Á)(p(Á) + d) ·

¡ ®3(Ma(Á))¡1

2®3(Ma(Á))r

T (Á)r(Á) + rT (Á)d

(15)

If ®3(Ma(Á)) ¸ jdj (this is not an hypothesis aboutsuitable essential boundedness of the disturbance, butjust concerns the hypothesis H2 in Theorem 2), thenthe following inequalities hold

D+V (Á; d) · ¡®3(Ma(Á))¡1

2®3(Ma(Á))r

T (Á)r(Á)+

1

2®3(Ma(Á))r

T (Á)r(Á) +1

2®3(Ma(Á));

(16)and, therefore, the inequality follows

D+V (Á; d) · ¡1

2®3(Ma(Á)) (17)

So, by theorem 2, the proof of the input-to-state stabil-ity of the closed loop system (13) is accomplished. ut

ILLUSTRATIVE EXAMPLE

Let us consider the retarded nonlinear system studied in[18] in the case of no disturbance adding to the controllaw, described by the following nonlinear equations

_x1(t) = ¡4x1(t)¡ x1(t¡¢)¡ x23(t¡¢)

_x2(t) = x1(t¡¢)¡ x2(t¡¢) + (1 + x21(t))(u(t) + d(t))

_x3(t) = x2(t) + x1(t)x3(t¡¢)(18)

In [18] the following change of coordinates is intro-duced

2

4z1(t)z2(t)»(t)

3

5 =

2

4x3(t)

x2(t) + x1(t)x23(t¡¢)

x1(t)

3

5 (19)

by which the system (18) can be rewritten as

·_z(t)_»(t)

¸=

2

4

·0 10 0

¸z(t) +

·01

¸T (zt; zt¡¢)

¡4»(t)¡ »(t¡¢)¡ z21(t¡¢)

3

5+

2

40

1 + »2(t)0

3

5 (u(t) + d(t));

(20)

where z(t) =

·z1(t)z2(t)

¸, T : C([¡¢; 0];R2) £

C([¡¢; 0];R2) ! R is a suitable continuous functional(see [18]). By the feedback control law

u(t) = k(zt; zt¡¢) =¡T (zt; zt¡¢) +GT z(t)

1 + »2(t); (21)

where G 2 R2 is such that the matrix H =

·0 10 0

¸+

·01

¸GT is Hurwitz (see (45) and the expression of v(t)

5266

in [18]), the closed loop system, in the new coordinates,is described by the following nonlinear equations

·_z(t)_»(t)

¸=

2

4 Hz(t) +

·0

(1 + »21(t))

¸d(t)

¡4»(t)¡ »(t¡¢)¡ z21(t¡¢)

3

5 (22)

Without loss of generality, let us suppose that the vec-tor G is chosen such that the matrix H has real negativedistinct eigenvalues ¡¸1, ¡¸2. In [18], as said before,the disturbance adding to the control law is not consid-ered, that is d(t) = 0, and it is proved that the unforcedclosed loop system (20),(21) is (locally) asymptoticallystable. Here we apply Theorem 4 in order to ¯nd out afeedback control law

u(t) = k(zt; zt¡¢) + p(zt) (23)

such that the forced closed loop system (20), (23) is(globally) input-to-state stable with respect to the dis-turbance d(t). By using a new partial change of coordi-

nates z(t) = Rz(t), by which RHR¡1 =

·¡¸1 00 ¡¸2

¸,

it results that the global asymptotic stability of theclosed loop system (22) (with disturbance equal tozero) is proved by means of the following Liapunov-Krasovskii functional V0 : C([¡¢; 0];R3) ! R+, given,

for Á =

2

4Á1Á2Á3

3

5 2 C([¡¢; 0];R3), by

V0(Á) = a¡k1(R1;1Á1(0) +R1;2Á2(0))

4+

k2(R2;1Á1(0) +R2;2Á2(0))4 + k3Á

23(0)+

Z 0

¡¢

®(¿)((R1;1Á1(¿ ) +R1;2Á2(¿))4+

(R2;1Á1(¿) +R2;2Á2(¿ ))4)d¿+

Z 0

¡¢

¯(¿)Á23(¿ )d¿

¶;

(24)

where a is any positive real (it may be a useful furthertuning parameter at the aim of keeping the new feedbackcontrol law suitably bounded), ®(¿) = ¡a1

¿¢ + a2

¿+¢¢ ,

¯(¿) = ¡b1¿¢

+ b2¿+¢¢

, a1; a2; b1; b2; k1; k2; k3 are posi-tive reals which are chosen (it is always possible) suchto satisfy

4k3maxf(R¡11;1)

4; (R¡11;2)4g < a1 < a2 <

4minfk1¸1; k2¸2g;

k3 < b1 < b2 < 5k3;

(25)

Ri;j , and R¡1i;j , i; j = 1; 2; are the elements in the ith

row and jth column of R and R¡1, respectively. For thefunctional V0, D

+a V0 is given by

D+a V0(Á) = a

¡(a2 ¡ 4k1¸1)(R1;1Á1(0) +R1;2Á2(0))

4+

(a2 ¡ 4k2¸2)(R2;1Á1(0) +R2;2Á2(0))4+

(b2 ¡ 8k3)Á23(0)¡ 2k3Á3(0)Á3(¡¢)

¡ 2k3Á3(0)Á21(¡¢)¡ a1(R1;1Á1(¡¢) +R1;2Á2(¡¢))4

¡ a1(R2;1Á1(¡¢) +R2;2Á2(¡¢))4 ¡ b1Á23(¡¢)

¡a2 ¡ a1

¢

Z 0

¡¢

((R1;1Á1(¿ ) +R1;2Á2(¿ ))4d¿¡

¡a2 ¡ a1

¢

Z 0

¡¢

(R2;1Á1(¿ ) +R2;2Á2(¿ ))4)d¿

¡b2 ¡ b1

¢

Z 0

¡¢

Á23(¿ )d¿

¶

(26)

As far as the lim suph!0+V0(Á

ah+±

g

hM)¡V0(Á

ah)

h; where

M 2 R, the following equality holds

lim suph!0+

V0(Áah + ±

ghM)¡ V0(Á

ah)

h=

a¡4k1(R1;1Á1(0) +R1;2Á2(0))

3(1 + Á3(0)2)M+

+4k2(R2;1(Á1(0) +R2;2Á2(0))3)(1 + Á3(0)

2)M¢;

(27)by which the functional r(Á) in iii) of Theorem 4 is de-¯ned as

r(Á) = 4ak1(R1;1Á1(0) +R1;2Á2(0))3(1 + Á3(0)

2)+

4ak2(R2;1Á1(0) +R2;2Á2(0))3(1 + Á3(0)

2)(28)

Therefore the new feedback control law is equal to

u(t) = k(zt; zt¡¢) + p(zt); (29)

where p : C([¡¢; 0];R3) ! R is de¯ned as

p(Á) = D+a V0(Á)(2ak1(R1;1Á1(0)+

R1;2Á2(0))3(1 + Á3(0)

2)+

2ak2(R1;2Á1(0) +R2;2Á2(0))3(1 + Á3(0)

2));

(30)

D+a V0(Á) given by (26). By Theorem 4, the closed loop

system (20), (29) is (globally) input-to-state stable withrespect to the disturbance d(t). As far as the originalsystem (18) is concerned, taking into account the changeof coordinates (19), the (global) input-to-state stabilityof the closed loop system (20), (29) implies the (global)input-to-state stability of the closed loop system (18),(19), (29) (with respect to the disturbance d(t)).

5267

¯g. 1: z(t) and »(t) (control law 21)

¯g. 2: z(t) and »(t) (control law 29)

The numerical simulations, performed by Matlab,validate the theoretical results. In the case reported

here, the matrix H =

·0 1¡2 ¡3

¸(chosen eigenvalues

¡1, ¡2), the disturbance is equal to d(t) = 3(1+sin(¼t))and the initial state variables are chosen constant in[¡¢; 0] and equal to [ 3 3 3 ]T . The delay ¢ is chosenequal to 0:1. The parameter a is chosen equal to 10¡4.The other parameters are chosen as k1 = 2, k2 = 2,k3 = 1, a1 = 5, a2 = 6, b1 = 2, b2 = 3. When the feed-back control law (21) is used, the variables z(t) and »(t)of the closed loop system (20), (21), with disturbance,diverge to1 (see ¯g. 1). When the feedback control law(29) is used, the variables z(t) and »(t) of the closed loopsystem (20), (29), with disturbance, are kept bounded(¯g. 2). In this case the feedback control law proposedin [18] for the unforced case (d(t) = 0) cannot work inthe forced case, while the feedback control law proposedin this paper yields very good performance also for thecase with disturbance.

CONCLUSIONS

In this paper, it is proved that retarded systems whichare stabilizable by means of a state feedback control law,

are also input-to-state stabilizable, by a state feedbackcontrol law, with respect to measurable and locally es-sentially bounded disturbances adding to the control in-put, provided that a suitable Liapunov-Krasovskii func-tional exists such that the asymptotic stability of theclosed loop unforced system can be proved. Such func-tional must have some smooth property with respect toa suitable variable, which is a very weak hypothesis. Anexample taken from the past literature is investigated indetails, showing the e®ectiveness and the applicabilityof the methodology here proposed.

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