Global stabilization of oscillators with bounded delayed input

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Systems & Control Letters 53 (2004) 415 – 422

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Global stabilization of oscillators with bounded delayed input Fr'ed'eric Mazenca;∗ , Sabine Mondi'eb , Silviu-Iulian Niculescuc a Project

MERE, INRIA-INRA, UMR Analyse des Systemes et Biometrie, 34, 2 pl. Viala, 34060 Montpellier, France b Depto. de Control Autom atico, CINVESTAV-IPN. A.P. 14-740, 07360 Mexico, D.F., Mexico c Heudiasyc, UTC, UMR CNRS 6599, BP 20529, 60205 Compi egne, France Received 14 March 2002; received in revised form 14 April 2004; accepted 26 May 2004

Abstract The problem of globally asymptotically stabilizing by bounded feedback of an oscillator with an arbitrary large delay in the input is solved. A 4rst solution follows from a general result on the global stabilization of null controllable linear systems with delay in the input by bounded control laws with a distributed term. Next, it is shown through a Lyapunov analysis that the stabilization can be achieved as well when neglecting the distributed terms. It turns out that this main result is intimately related to the output feedback stabilization problem. c 2004 Elsevier B.V. All rights reserved.  Keywords: Global stability; Oscillator; Delay; Razumikhin theorem

1. Introduction The family of the linear systems described by x(t) ˙ = Ax(t) + Bu(t − );

(1)

where A ∈ Rn×n , B ∈ Rn×m ,  is a delay, u is the input and the initial condition is x( ) = ( );

∈ [ − ; 0]

is one of the simplest class of models with delay. Nevertheless, they describe properly a number of phenomena commonly present in controlled

∗ Corresponding author. Tel.: +33-499612498; fax: +33467521427. E-mail addresses: [email protected] (F. Mazenc), [email protected] (S. Mondi'e), [email protected] (S.-I. Niculescu).

processes such as transport of information or products, lengthy computations, information processing, delays inherent to sensors, etc. Well known approaches for the control of these systems include in an explicit or implicit manner a predictor of the state at time t + . Some of the more widely used are the Smith predictor [14,13], process-model control schemes [17], and 4nite spectrum assignment techniques [6,4]. A common drawback, linked to the internal instability of the prediction, is that they fail to stabilize unstable systems [10]. As shown in [11], it is possible to overcome this problem by introducing a periodic resetting of the predictor. A common concern in practical problems is the use of bounded control laws. It is well known that for linear systems with poles in the open right-half plane, only locally stabilizing control laws can be obtained if there is a bound on the input, and that global asymptotic stability can be achieved only for

c 2004 Elsevier B.V. All rights reserved. 0167-6911/$ - see front matter
doi:10.1016/j.sysconle.2004.05.018

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F. Mazenc et al. / Systems & Control Letters 53 (2004) 415 – 422

systems with poles in the closed left-half plane, named null-controllable systems. As shown in [15], the problem can be decomposed into two fundamental subproblems, namely, the control of chains of integrators of arbitrary 4nite length, and that of oscillators of arbitrary 4nite multiplicity. Solutions to this problem in the framework of systems with no delay are given in [16,15,8]. From the above, it follows that the study of linear systems whose inputs are both delayed and bounded is useful in many applications. In this paper, we focus our attention on the stabilization of a simple oscillator with arbitrarily large delay in the input described by x˙1 (t) = x2 (t); x˙2 (t) = −x1 (t) + u(t − ):

(2)

We restrict our analysis to the case where the frequency is one for the sake of simplicity, but all the results that are presented can be adapted easily to the system x˙1 (t) = x2 (t); x˙2 (t) = −x1 (t) + u(t − );

(3)

where and  are strictly positive or strictly negative real numbers because this system can be transformed into system (2) by a time rescaling and a change of coordinates. To the best of authors’ knowledge, the issue of the stabilization of the oscillator was 4rst discussed in [9] in the 1940s for a second-order (delayed) friction equation. Further comments and remarks on delayed oscillatory systems can also be found in [2,3]. The nature of oscillators is signi4cantly diJerent from that of chains of integrators: for example, when the input is zero, the solutions of oscillators are trigonometric functions of the time while those of integrators are polynomial functions of the time. Not surprisingly, the technique of proof we use is signi4cantly diJerent from the approach of [7] for chains of integrators with bounded delayed inputs based on the use of saturated control laws introduced in [16] that only require the knowledge of an upper bound on the delay. In the case of oscillators, we take advantage of the properties of the explicit solutions to zero inputs to determine expressions of the control laws which

depend explicitly on the exact value of the delay. The proof of the result relies on the celebrated Lyapunov–Razumikhin theorem. The key feature of our control design are that for arbitrarily large delays it allows us to determine a family of globally asymptotically stabilizing state feedbacks which contains elements arbitrarily small in norm. The reinterpretation of this result in the context of output feedback stabilization leads to the following result: for any linear output one can 4nd an arbitrarily large delay for which the problem of globally asymptotically stabilizing the oscillator by bounded feedback is solvable. The paper is organized as follows. In Section 2 a general result on the stabilization of null-controllable systems is established via distributed control laws. The procedure is applied to the oscillator. Next, a stabilizing bounded state feedback is proposed. In Section 3 this last result is embedded in the output feedback context. Some concluding remarks are presented in Section 4. Razumikhin’s Theorem is recalled in the appendix, and an illustrative example is given. Preliminaries: • For a real-valued C 1 function k(·), we denote by k  (·) its 4rst derivative. • A function : [0; +∞) → [0; +∞) is said to be of class K∞ if it is zero at zero, strictly increasing and unbounded. • We denote by (·) an odd nondecreasing smooth saturation such that ◦ (s) = s when s ∈ [0; 1], ◦ (s) = 32 when s ¿ 2, ◦ −s 6 (s) 6 s, 0 6  (s) 6 1.

2. Stabilization by bounded delayed control laws We establish 4rst a general result on the stabilization of null-controllable linear systems with delayed bounded inputs, using a control law that contains distributed elements. Next, we particularize this result to the cases of the oscillator. Finally, we exploit this last result in order to determine state feedback laws which do not contain distributed elements.

F. Mazenc et al. / Systems & Control Letters 53 (2004) 415 – 422

2.1. Distributed control laws for null-controllable systems Lemma 2.1. Consider a controllable linear multivariable system with delay in the input described by x(t) ˙ = Ax(t) + Bu(t − );

(4)

where A ∈ Rn×n ; B ∈ Rn×m is such that (A; B) is null-controllable, and the delays in the input is  ¿ 0, with initial conditions x( ) = ( );

∈ [ − ; 0];

where (·) is a continuous function. Then the following problems are equivalent: (i) The control law u(t) = (x)

(5)

globally asymptotically stabilizes system (4) when  = 0. (ii) The distributed control law
u(t − ) =  eA x(t − )  +

t

t−

e(t−s)A Bu(s − ) ds

 (6)

globally asymptotically stabilizes system (4).

417

where  is a strictly positive parameter and (·) is the bounded function de@ned in the preliminaries. Using the Cauchy formula for the oscillator x1 (t) = cos()x1 (t − ) + sin()x2 (t − )  t − sin(s − t)u(s − ) ds; t−

x2 (t) = −sin()x1 (t − ) + cos()x2 (t − )  t − cos(s − t)u(s − ) ds t−

(9)

(10)

and Lemma 2.1, it follows straightforwardly that the distributed delayed control law
u(t − ) = − −sin()x1 (t − ) + cos()x2 (t − )  −

t

t−

 cos(s − t)u(s − ) ds

(11)

globally asymptotically stabilizes system (2). 2.3. Delayed state feedbacks for the oscillator

Lemma 2.2. Consider the system

The control law (11) involves distributed elements. The implementation of such terms using numerical approximation is time consuming and, in some cases, may lead to instability [10]. One can observe that in (10) the distributed term is of the order of  whereas the other term does not depend on , which implies that, roughly speaking, the control law (11) is equal to a term of order  plus a distributed term of order 2 . This remark leads us to conjecture that as  decreases, the inLuence of the distributed term decreases as well. Then, the question that arises naturally, is whether or not stability is preserved if, when , the integral terms are neglected, that is to say, if instead of (6), the control law

x˙1 (t) = x2 (t);

u(t − ) = −(−sin()x1 (t − )

Proof. The result follows in a straightforward manner from the fact that  t x(t) = eA x(t − ) + eA(t−s) Bu(s − ) ds: t−

2.2. Distributed control laws for the oscillator As a preliminary step to the main part of the work, we specialize this general result to the case of an oscillator. Recall, 4rst, a very simple result whose proof is omitted.

x˙2 (t) = −x1 (t) + u;

(7)

where u is the input. This system is globally asymptotically stabilized by the bounded control law u(x1 (t); x2 (t)) = −(x2 (t));

(8)

+ cos()x2 (t − ))

(12)

is used. The main result of the work, stated below, gives an answer to this question. In Appendix B, we perform simulations to validate this result.

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F. Mazenc et al. / Systems & Control Letters 53 (2004) 415 – 422

Theorem 2.3. Let  be a positive number. The origin of system (2) is globally asymptotically stabilized by the control law u(x1 ; x2 ) = −(−sin()x1 + cos()x2 ); where    1 1 1 ;  ∈ 0; min ; 2 3242 40

(13)

(14)

and (·) is the bounded function de@ned in the preliminaries. Proof. System (2) in closed loop with feedback (13) rewrites x˙1 (t) = x2 (t); x˙2 (t) = −x1 (t) − (x2 (t) + B2 (t)) with



B2 (t) =

t

t−



=

(15)

cos(s − t)us (x1 (s − ); x2 (s − )) ds

t

t−

cos(s − t)(−sin()x1 (s − )

+ cos()x2 (s − )) ds:

(16)

x˙1 (t) = x2 (t);

with



(t) = B2 (t)

0

1

 (x2 (t) + lB2 (t)) dl:

2 (x12 + x22 )x2 (t)2 (x2 )

+ x22 − 12 x12 − 2 (x2 )2 + (t)2 :

(20)

According to the properties of (·) and the fact that  6 12 , it follows that U˙ (x1 ; x2 ) 6 −  (x12 + x22 )x2 (x2 )    2 2 (x1 + x22 )x2 + 1 (t)2 + 54 x22 − 12 x12 : (21) + (x2 ) Let now choose  s l (s) = 2k dl; (l) 0

(22)

where k ¿ 12 . The properties satis4ed by (·) ensure that this function is well de4ned, of class ∞ , and such that  (s) ¿ 0. Then, U˙ (x1 ; x2 )



x12 + x22 k x2 (x2 ) − 2 x22 |(x2 )| 2 2 (x1 + x2 ) (x2 )    x12 +x22 x2  4k  (x12 +x22 )   2 5 2 1 2  + +1  (t) + 4 x2 − 2 x1 (x ) 2  

(17)

(18)

The result is established through a LyapunovRazumikhin approach. Consider the function U (x1 ; x2 ) = (x12 + x22 ) + x1 x2 ;

6 −  (x12 + x22 )x2 (x2 ) +

6 − k

One can check readily that

x˙2 (t) = −x1 (t) − (x2 (t)) + (t)

+ x22 − x12 − x1 (x2 ) + x1 (t)

(19)

where (·) is a function of class ∞ such that (s) ¿ s; s ¿ 0 to be speci4ed later. With such a choice for (·); U (x1 ; x2 ) is a positive de4nite and radially unbounded function. Its time derivative along the trajectories of system (17) satis4es U˙ (x1 ; x2 ) = − 2 (x12 + x22 )x2 (x2 ) + 2 (x12 + x22 )x2 (t)



 2  x12 +x22 5 x2 (x2 )+ − 2k 3 + 4 x2 2 2 (x1 +x2 )    x12 + x22 4k + (23) x2 +1 (t)2 : (x2 ) (x12 +x22 )

6 − 12 x12 −k

Choosing k = 4= (which, according to (14), is larger than 12 ), U˙ (x1 ; x2 )



x12 + x22 x2 (x2 ) (x12 + x22 )    2 2 x + x 16 1 2 − 43 x22 + 2 x2 + 1 (t)2 :  (x2 ) (x12 + x22 )

6 − 12 x12 − 4

(24)

F. Mazenc et al. / Systems & Control Letters 53 (2004) 415 – 422

On the other hand, the use of the inequalities 0 6  (s) 6 1 and (16) leads to  2 1 2 2 2  (t) =  B2 (t)  (x2 (t) + lB2 (t)) dl

and 2

2

(t) =  B2 (t)

0

6 4



t

t−

6 4

|(−sin()x1 (s − ) 2

+ cos()x2 (s − ))| ds

:

(25)

We distinguish now between two cases. First case: x12 + x22 ¿ 14 . Inequalities (24) and (25) imply that  x12 + x22 2 1 U˙ (x1 ; x2 ) 6 − 2 x1 − 4 x2 (x2 ) (x12 + x22   162 2  x12 +x22 3 2 2 − 4 x2 + 1+ 8 x2 : (x2 ) (x12 +x22 ) (26) One can check readily that s=(s) 6 1+s for all s ¿ 0. It follows that  x12 + x22 U˙ (x1 ; x2 ) 6 − 12 x12 − 4 x2 (x2 ) − 34 x22 (x12 + x22 )    2 + 1+ 8 162 2 (1+|x2 |) 1+x12 +x22  6 − 12 x12 − 4  + 1+

2 8



x12 + x22 (x12 + x22 )

x2 (x2 ) − 34 x22

162 2 10(x12 + x22 ):

(27)

When  6 1=40, the inequality   2 1 1 + 8 162 2 10 6 : (28) 4 holds, and  x12 + x22 2 1 x2 (x2 ) − 14 x22 : U˙ (x1 ; x2 ) 6 − 4 x1 − 4 (x12 + x22 ) (29) Second case: x12 + x22 6 14 . In this case |x1 | 6 12 ; |x2 | 6 12 ; (x2 ) = x2 ; (x12 + x22 ) = x12 + x22 . Then it follows from (24) that   2 U˙ (x1 ; x2 ) 6 − 12 x12 − 43 x22 + 16 (30) 2 + 1 (t)



2

t

t−



1

0

419

2 

 (x2 (t) + lB2 (t)) dl

  |x1 (s − )| + x2 (s − )| ds

2 : (31)

Combining (30) and (31) leads to U˙ (x1 ; x2 ) 6 − 12 x12 − 43 x22 + [162 + 4 ]  ×

t

t−

  |x1 (s − )| + x2 (s − )| ds

6 − 12 x12 − 43 x22 + 172 22  ×

2

(32)



sup [x1 (s − )2 + x2 (s − )2 ] :

s∈[t−; t]

(33) Next, we determine the values of  for which feedback (13) globally asymptotically stabilizes system (2) with the help of Razumikhin Theorem (see Appendix A). To do so we prove by exploiting (29) and (33) that when the inequality U (x1 (t + ); x2 (t + )) ¡ 2U (x1 (t); x2 (t))

(34)

holds for all ∈ [ − 2; 0], there exists a continuous, nondecreasing functions !(·) such that U˙ (x1 ; x2 ) 6 − !( x ) with x = (x1 ; x2 )T . Recall that U (·) is positive de4nite and radially unbounded. 1. When x1 (t)2 + x2 (t)2 ¿ 14 , it follows from (29) that U˙ (x1 (t); x2 (t)) 6 − 14 x(t) 2 . 2. When x1 (t)2 + x2 (t)2 6 14 , the saturations act in their linear regions hence, U (x1 (t); x2 (t)) 8 = (x1 (t)2 + x2 (t)2 ) + x1 (t)x2 (t): 

(35)

One can check readily that 9 U (x1 (t); x2 (t)) 6 (x1 (t)2 + x2 (t)2 ): 

(36)

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F. Mazenc et al. / Systems & Control Letters 53 (2004) 415 – 422

On the other hand,  2 2 l 8 x1 +x2 dl + x1 x2 : U (x1 ; x2 ) =  0 (l)

with the output y = ax1 + bx2 ;

where a and b are arbitrary real numbers such that a2 + b2 ¿ 0. Then the system is globally asymptotically stabilized by a delayed feedback 
y(t − ) ; (43) u = − √ a2 + b 2

Since (l) 6 l; l ¿ 0, it follows that 8 U (x1 ; x2 ) ¿ (x12 + x22 ) + x1 x2  
8 1 (x12 + x22 ): ¿ −  2 In particular, this inequality implies that 14(x1 (t + )2 + x2 (t + )2 ) 6 U (x1 (t + ); x2 (t + )):

(37)

Combining (36), (34) and (37), 14(x1 (t + )2 + x2 (t + )2 ) 18 (x1 (t)2 + x2 (t)2 ):  It follows from (38) and (33) that 6

(38)

U˙ (x1 ; x2 ) 6 − 12 x12 − 43 x22 + 342 2 79 [x1 (t)2 + x2 (t)2 ] 6 − 12 x12 − 43 x22 + 1622 [x1 (t)2 + x2 (t)2 ]: (39)    1 1 , implies that Choosing  ∈ 0; min 12 ; 40 ; 648 2 U˙ (x1 ; x2 ) 6 − 14 x 2 :

(40)

This concludes the proof.

where (·) is the function de@ned in Theorem 2.3, and where  is a strictly positive number such that a = − a2 + b2 sin(); b = a2 + b2 cos()    1 1 . ; 648 and  ∈ 0; min 12 ; 40 2 An interesting and perhaps surprising particular case is when the output is x1 . In this case, although the system is not stabilizable when the delay is zero, it is for suitably chosen delays, as shown in Corollary 3.1. The explanation of this fact is that, for a delay  appropriately chosen, the past value of the output x1 (t − ) gives information on the value of x2 (t). An analysis of the problem of output feedback stabilization by linear feedback, based on the study of the corresponding characteristic equation [12], or on the Nyquist criterion [1] can be carried out. It leads to the following result that provides additional informations on our problem. Proposition 3.2 (Niculescu [12], Abdallah et al. [1]): The linear system

3. Output feedback stabilization

x˙1 (t) = x2 (t);

The above analysis shows that when the delay is , feedbacks depending only on

x˙2 (t) = −x1 (t) + u(t);

−sin()x1 (t − ) + cos()x2 (t − )

with the output

stabilize the oscillator. This leads straightforwardly to the following interpretation of Theorem 2.3 in an output feedback framework.

y(t) = x1 (t);

Proposition 3.1. Consider the system

u(t) = −ky(t − )

x˙1 (t) = x2 (t);

for all the pairs (k; ) satisfying simultaneously

x˙2 (t) = −x1 (t) + u(t);

(42)

(41)

(44)

(45)

can be stabilized by delayed output feedback

(i) the gain k ∈ (0; 1),

(46)

F. Mazenc et al. / Systems & Control Letters 53 (2004) 415 – 422

(ii) the delay  ∈ (i (k); Mi (k)) where (2i − 1)! ; i (k) = √ 1−k Mi (k) = √

2i! ; 1+k

(47)

for i = 1; 2; : : :. Furthermore, if  = i (k) or  = Mi (k), the corresponding characteristic equation in closed-loop has at least one eigenvalue on the imaginary axis. The regions of stabilizing k shrink as the delay  gets larger, and furthermore for each k there exists a value ∗ (k), such that for any  ¿ ∗ (k) the closed-loop system is unstable. Besides, the robustness margin of the closed-loop system decreases as  gets larger. Remark 3.3. Observe that in the case when y = x1 , the values of a and b in Eq. (42) in Corollary 3.1 are, respectively, 1 and 0. This implies that i =− 12 !+2i!, where i is an integer. Not surprisingly, these values are such that i = − 12 ! + 2i! ∈ ]i (k); Mi (k)[ when k is suNciently small: indeed, feedback (43) is linear in a neighborhood of the origin which implies that √ system (41) in closed-loop with −y(t − )= a2 + b2 is globally asymptotically stable when  = i . Remark 3.4. This observation gives a new way to establish that the delayed output feedback (46) satisfying (i) and (ii) are stabilizing. Indeed, boundaries of the regions of the plane k – described by (i) and (ii) correspond to crossing of the imaginary axis of the roots of the closed loop quasipolynomial. It follows from a continuity argument that in each region, the quasipolynomial has a 4xed number of roots in the right-half plane, hence it is either stable or unstable in the whole region. A consequence of Remark 3.3 is that we are able to exhibit a stabilizing element (ki ; i ) in each of the regions described by (i) and (ii), hence all the control laws and all the pairs (k; ) satisfying simultaneously (i) and (ii) are stabilizing as well. 4. Concluding remarks The problem of globally asymptotically stabilizing by bounded feedback an oscillator with an arbitrary

421

large delay in the input is solved. A 4rst solution follows from a general result on the global stabilization of null-controllable linear systems with delay in the input by bounded control laws with a distributed term. Next, it is shown through a Lyapunov analysis that the stabilization can be achieved as well when neglecting the distributed terms. It turns out that this main result is intimately related to the output feedback stabilization problem. The robustness problems due to the need for the exact knowledge of the delay can be investigated with the help of the Lyapunov function we have constructed. Appendix A. Razumikhin’s Theorem Theorem A.1 (Hale and Verduyu Lunel [5]): Consider the functional diBerential equation x(t) ˙ = f(t; xt );

t ¿ 0;

(A.1)

xt0 ( ) = ( );

∀ ∈ [ − ; 0]

(A.2)

(Note: xt ( ) = x(t + ); ∀ ∈ [ − ; 0].) The function f : R × Cn;  is such that the image by f of R×(a bounded subset of Cn;  ) is a bounded subset of Rn and the functions u; v; w : R+ → R+ are continuous, nondecreasing, u(s); v(s) positive for all s ¿ 0; u(0)= v(0) = 0 and v is strictly increasing. If there exists a function V : R × Rn → R such that (a) u( x ) 6 V (t; x) 6 v( x ); t ∈ R, x ∈ Rn , (b) V˙ (t; x) 6 − w( x ) if V (t + ; x(t + )) 6 V (t; x(t)); ∀ ∈ [ − ; 0] then the trivial solution of (A.1), (A.2) is uniformly stable. Moreover, if w(s) ¿ 0 when s ¿ 0; and there exists a function p : R+ → R+ , p(s) ¿ s when s ¿ 0 such that (a) u( x ) 6 V (t; x) 6 v( x ); t ∈ R, x ∈ Rn , (b) V˙ (t; x) 6 − w( x ) if V (t + ; x(t + )) ¡ p(V (t; x(t)));

∀ ∈ [ − ; 0]

(A.3)

then the trivial solution of (A.1), (A.2) is uniformly asymptotically stable.

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F. Mazenc et al. / Systems & Control Letters 53 (2004) 415 – 422

References

Fig. 1. Closed-loop states ( = !=4).

Fig. 2. Closed-loop states ( = 3!=4).

Fig. 3. Delayed input ( = 3!=4).

Appendix B. Illustrative example In this section, we carry out some computer simulations to validate the result presented in Section 2.3. In Fig. 1, one can observe the behavior of the oscillator when the delay is !=4 and when the control law is u = −0:1(−sin(!=4)x1 + cos(!=4)x2 ), which results from Theorem 2.3. Figs. 2 and 3 show that the same system in closed loop with the same feedback is unstable when the delay is not !=4 but the larger delay 3!=4.

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