On the stability of linear systems with uncertain delay - IEEE Xplore

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[9] P. Lancaster and D. Veiner, “Lectures on Linear algebra, control, and stability,” Dept. Math. Statist., Univ. Calgary, Calgary, AB, Canada, 1999. [10] A. M. Lyapunov, “The general problem of the stability of motion,” Int. J. Control, vol. 55, no. 3, pp. 531–773, 1892. [11] V. A. Utkin, “Variable structure systems with sliding modes,” IEEE Trans. Automat. Contr., vol. AC-22, pp. 212–222, Apr. 1977. [12] W. M. Wonham, Linear Multivariable Control: A Geometric Approach, 2nd ed. New York: Springer-Verlag, 1979. [13] K.-K. D. Young, P. V. Kokotovic´, and V. I. Utkin, “A singular pertubation analysis of high-gain feedback systems,” IEEE Trans. Automat. Contr., vol. AC-22, pp. 931–938, June 1977.

On the Stability of Linear Systems With Uncertain Delay Vladimir L. Kharitonov and Silviu-Iulian Niculescu

(

) = f00:5 6 2ig, 9 = [ 00:75 + 1:25i 1 + i ]. Matched

Fig. 4.  A disturbances present.

made that allowed for the sliding manifold S to be selected as a linear subspace of n in the same manner that S typically selected in n . By doing this, and then projecting the resulting system into 2n , the closed-loop system recovers all the dynamics lost in the SLMC design process and more. That is, (21) verified that the closed-loop system will have, at most, 2( 0 ) nonzero eigenvalues. This method of closed-loop system design appears to be novel. An algorithm was then given for the selection of an additional switching term on the control law that guaranteed both state convergence to the sliding mode, as well as robustness to bounded, matched disturbances in sliding. It should be noted, however, that the proposed controller does have some practical drawbacks, in that a perfect relay-type switch is needed to implement the controller in a laboratory environment. It is a well known fact that such a switching controller can sometimes excite unmodeled system dynamics, and lead to poor system performance. However, since this consideration is not important to the topic of this note (i.e., we have only considered a possible means of designing S as a submanifold of n ), this consideration has been intentionally omitted. The robustness of the closed-loop system was demonstrated.

n m

REFERENCES [1] J. W. Brewer, “Kronecker products and matrix calculus in system theory,” IEEE Trans. Circuits Syst., vol. CAS-25, pp. 772–781, Sept. 1978. [2] R. A. Decarlo, S. H. Zak, and G. P. Matthews, “Variable structure control of nonlinear multivariable systems: A tutorial,” Proc. IEEE, vol. 76, pp. 212–232, Mar. 1988. [3] R. A. Decarlo, “Sliding mode control design via Lyapunov approach,” in Proc. 33rd IEEE Conf. Decision Control, Lake Buena Vista, FL, 1994, pp. 1925–1930. [4] B. Draženovic´, “The invariance conditions in variable structure systems,” Automatica, vol. 5, no. 3, pp. 287–295, 1969. [5] M. J. L. Hautus, “Controllability and observability conditions of linear autonomous systems,” Indagationes Math., vol. 12, pp. 443–448, 1969. [6] H. K. Khalil, Nonlinear Systems, 2nd ed. Upper Saddle River, NJ: Prentice-Hall, 1996. [7] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. New York: Wiley, 1963. [8] P. Lancaster and M. Tismenetsky, The Theory of Matrices: Second Edition with Applications. New York: Academic, 1985.

Abstract—This note focuses on the stability of some class of delay systems, including uncertainty in the delays. More precisely, we are interested in guaranteeing the stability of perturbed delay systems by assuming the stability of the nominal system. If the delay perturbation is constant, necessary and sufficient conditions are derived in terms of generalized eigenvalue distribution of some (finite-dimensional) constant matrix pencil. If the delay perturbation is time-varying, some sufficient stability conditions are derived using “exact” Lyapunov–Krasovskii functionals. Illustrative examples are also included. Index Terms—Delay, Lyapunov functionals, matrix pencils, uncertainty.

I. INTRODUCTION It is relatively well known that the existence of a delay in a dynamical system may induce instability or bad performances in open or closed-loop schemes [13], [17]. Furthermore, there exist control schemes which are very sensitive with respect to delay uncertainty, as, for example, the Smith predictor (see, for instance, [20] and the references therein). Note also that the delay-independent stability problem (stability guaranteed for all delays value) is ill-posed [3], [16] in terms of delay uncertainty if the delays are commensurate (rationally dependent). All these aspects argue for a deeper analysis of delay uncertainty effects on stability. More precisely, we shall consider an asymptotically stable linear delay system and we are interested in characterizing the stability of the same system if the delay is perturbed. The aim of the note is to derive various “practical” stability conditions easy to check. The problem will be treated in both frequency and time-domain frameworks. In the frequency-domain, we shall give some necessary and sufficient conditions based on the computation of the generalized eigenvalues distribution [1], [19] of some appropriate matrix pencils. In the time-domain, we are interested on the computation of ‘exact’ Lyapunov–Krasovskii functional [14] (see also [5] and [6] for numerical aspects). Note that the delay perturbation is considered constant in

Manuscript received October 1, 2001; revised April 8, 2002. Recommended by Associate Editor Y. Ohta. This note was presented in part at the 2002 American Control Conference, Anchorage, AK, May 8–10. V. L. Kharitonov is with the Departamento de Control Autómatico, CINVESTAV-IPN, Mexico City 07360, Mexico (e-mail: [email protected]). S.-I. Niculescu is with the HEUDIASYC (UMR CNRS 6599), Centre de Recherche de Royallieu, Université de Technologie de Compiègne, 60205 Compiègne, France (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2002.806665

0018-9286/03$17.00 © 2003 IEEE

128

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the frequency-domain case and continuous time-varying and bounded in the time-domain case. The note is organized as follows: the problem statement is presented in Section II. The frequency- and time-domain methods are proposed in Sections III and IV, respectively. Illustrative examples are also included. Some concluding remarks end the note. The notations used are standard. II. PROBLEM STATEMENT Consider the delay system

x_ (t) = Ax(t) + Bx(t 0  )

(1)

where x 2 Rn , A; B are constant n 2 n matrices, with initial condition

 2 [0; 0]

x() = ();

where  2 C ([0; 0]; Rn ). Consider now the following perturbed system:

y_ (t) = Ay(t) + By(t 0  + (t))

(2)

where the perturbed delay  is a continuous and bounded function, such that j (t)j   , for all t  0. Without loss of generality, we shall rewrite (2) as follows:

y_ (t) = Ay(t) + By(t 0 "(t))

(3)

where "(t) =  0  (t)  0 is a continuous and bounded function. We shall assume that " verifies the condition: 0  "(t)   . We are interested on the following problem. Problem 1: Find conditions on the perturbed delay "(t) guaranteeing the stability of (3) if (1) is asymptotically stable. III. A FREQUENCY-DOMAIN APPROACH In the sequel, we shall consider  0  (t) = "(t) = " constant and we are interested in characterizing the delay-intervals including  and ensuring asymptotic stability. Since the nominal system (1) is asymptotically stable, the characteristic equation associated with (1)

det sI 0 A 0 Be0  = 0 has no solutions with Re(s)  0. s

n

(4)

Consider now the characteristic equation of (2) with  constant, that

is

det sI 0 A 0 Be0 ( 0) = 0 s

n

(5)

or similarly the characteristic equation of (3)

det sIn 0 A 0 Be0s" = 0:

(6)

Then, using the continuity principle [2], [4], it follows that the stability problem considered above can be reduced to finding conditions on  (or ") such that the characteristic equation (5) (or (6), respectively) has no roots on the imaginary axis. In other words, we are interested in finding all " such that

det j!I 0 A 0 Be0 "! 6= 0 n

j

8! 2 R:

(7)

It is clear that we shall have two subproblems. — First, (7) is guaranteed for all "  0. Then, we shall have a delay-independent type problem. In terms of nominal and perturbed delay systems, it follows that any constant perturbation    will guarantee the asymptotic stability of the perturbed system (2) if the nominal system (1) is asymptotically stable. Note that the stability independent of delay was largely treated in the literature. Further remarks and comments can be found in [11]. — Second, (7) does not hold for all delays ", then we will have a delay-interval guaranteeing stability, that is a delay-dependent type problem (see also [11] and [19]). We do not assume the asymptotic stability of the system free of delay. Some results in that direction have been proposed in [9] (see also [10]), where -analysis was used for giving some bounds on the delay uncertainty. Their conditions are only sufficient. The approach here is quite different and is inspired by [1] and [19] on matrix pencils. Furthermore, our results are necessary and sufficient stability conditions. The idea of using matrix pencils in the analysis of delay-independent or first delay-interval of the form [0; max ) stability was originally proposed in [1] (see also [19] and the comments therein). Since we do not assume the asymptotic stability of the system free of delay, but the stability of the system for a given delay  , one needs to appropriately adapt the idea of [1] in this case. Let us consider the characteristic equation (4) where  is replaced by  . Introduce now the “quantities” as shown in (8) at the bottom of the page, with u = 01 and l = +1 if the corresponding sets are empty (“u” for upper and “l” for lower). It is clear that these quantities give the real parts of the corresponding eigenvalues (if there exists any) “closest” to the imaginary axis j R. The numbers u and l are continuous in and  (see also [2] and [8]). The stability of the nominal system (1) implies that u1 < 0 (strictly) and l1 = +1. Using the continuity argument cited above, it follows that the stability property holds for some -interval including 1. In fact, the stability will be lost when the corresponding characteristic equation in  will have roots on the imaginary axis. This fact is equivalent to some roots distribution of some matrix polynomials with respect to the unit circle, whose linearizations give exactly the matrix pencil proposed in [1] using Kronecker products and sums or in [19] using more general matrix tensor products and sums. Let us introduce the following matrix pencil:

0 3(z) =z I0p  (B;

In ) +  I 0; (B )T  0A;IpAT

n 8

(9)

where  and 8 define the corresponding matrix tensor product and sum [19] (see also the Appendix). Now introduce the sets

3;+ = (k ; k ) : k = !k >  : e0j 2 (3) ki j!ki 2  A + e0j B 0 f0g 1  k  2p; 1  i  n

u = max Re(s)  0 : det sIn 0 A 0 Be0s  = 0 l = min Re(s)  0 : det sIn 0 A 0 Be0s  = 0

(10)

(8)

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 1, JANUARY 2003

3;0 = (k ; k ) : k = !k <  : e0j 2 (3) ki 0 j j!k i 2  A + e B 0 f0g 1  k  2p; 1  i  n

129

Example 1: Consider the following second-order exponentially stable system:

_ ( ) = 002 01 x(t) + 000:4 00 x(t 0 4): (13) Here,  = 4. Simple computations give the following matrix pencil (p = n(n 0 1)=2 = 1 and 3 2 C2p22p ): 3(z) = z 10 00:4 + 000:4 001 (14) which has z = 61 as generalized eigenvalues on the unit circle. x t

(11)

where  (1) denotes the set of (generalized) eigenvalues of the corresponding matrix (pencil). With all these remarks and definitions, some tedious but simple computations will lead to the following result. Theorem 1: The linear system (2) including a constant delay perturbation    is robustly stable if and only if the nominal system (1) is asymptotically stable and the following inequalities hold:

Then, the roots on the imaginary axis of the characteristic equation associated to (13) are !1

= 1:9596

!2

= 2:0396

which give the delay values (k a nonnegative integer)

 0 inf fh : (h; ) 2 3;+ g <  1 and < 1, respectively. With these arguments, it follows the sufficiency, i.e., the asymptotic stability is guaranteed for all delay uncertainty satisfying (12). To prove the necessity, we can simply use a standard argument by contradiction.

det zB A

0

Remark 1: Assume now that the system free of delay is asymptotically stable. Then, 30;0 is useless since   0 and we need to focus on 30;+ . If this set is empty, we recover the delay-independent stability result presented in [1]. If 30;+ contains at least one element, then the first delay-interval has the form [0; ), where  = minf : ( ; ) 2 0;+ g, which slightly corrects the bound in [1] (see also [19]). In conclusion, we recover the results of [1] for the first-delay interval (finite or not) guaranteeing stability, but with the difference that the corresponding matrix pencil has lower size than the one proposed in [1]. The other novelty of our approach is the introduction of the set 3;0 if  > 0 and the characterization of the corresponding delay lower bound guaranteeing stability. Note that the search of this bound is done in a similar way to the upper bound case. Remark 2: If the intersection 3;0 \ [0; +1) is empty, then 3;+ gives the first-delay interval guaranteeing stability (see also the previous remark). Furthermore, if 3;+ is also empty, then we have a delay-independent type result and (12) becomes  2 (01;  ], etc.

h h

3

regular matrix  ((zB + A); (zB + A) ) for all j z j= 1 and z not a solution of det(zB + A) = 0 [19] corresponds to a delay-independent type property. The stability argument follows exactly the same steps as in [1] and it is omitted. 1A

k;1

(12)

Furthermore, note that the system free of delay is not stable (a pair of roots on the imaginary axis) and there exists one more delay-interval guaranteeing asymptotic stability. Note also that the system is unstable for large delays. IV. A TIME-DOMAIN APPROACH In this section, we shall assume that  (t) is a time varying function satisfying the following assumptions. Assumption 1: There exists 0 2 [0;  ] such that j (t)j  0 for all t  0. Assumption 2: There exists _ 0 2 [0; 1) such that j_ (t)j  _ 0 . We assume in this section that the nominal system (1) is exponentially stable and we will try to find conditions on 0 and _ 0 under which the perturbed system remains stable for all such delay perturbations. Let us rewrite the perturbed system (1) as follows:

y_ (t) = Ay (t) + By (t 0  ) + B [y (t 0 

+ (t)) 0 y(t 0  )]: (16) Given positive–definite n 2 n matrices Wi , i = 1; 2, let us define on C ([03; 0]; Rn ) the functional w(') = '(0)T W1 '(0) +

0

03

'()T W2 '()d:

(17)

For the functional w defined above exists the Lyapunov–Krasovskii functional v (') such that, along the solutions of the nominal system (1), we have the equality d v (x(t + 1)) = 0w(x(t + 1)): dt The Lyapunov–Krasovskii functional is of the form

V (x(t + 1)) =x(t)T U (0)x(t)

0

+ 2x(t)T + +

0

0

0 0 0

0

0

U T (

+ )Bx(t + )d

x (t + 1 )T B T U (1 0 2 ) Bx (t + 2 ) d1 d2

x(t + )T (3

+ )W2 x(t + )d

where the matrix valued function U () is defined for 

U () =

1

0

K (t)

T

(18)

2 R as follows:

(W1 + 3 W2 ) K (t + )dt:

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Here, K (t) is the fundamental matrix associated to the nominal system (1) (see also [12]). Since (1) is exponentially stable, the matrix valued function U () is well defined. Using the same steps as in [12], it follows that the time derivative of v along the solutions of the perturbed system (16) has the form

1

dV (y (t + )) = dt

0

U (0)y (t) +

0

T U ( + )By (t + )d :

 (t)

T

2 y (t

= 0

+ y (t

0

 (t)

1

B

T

2

2y (t

0

2

T y (t

0

0 2 +  )

B

B 2

2

T

T

U (0)y (t)d 2

M2 B y

T

1 (t 0 2 +  )d +  y(t)T U (0)M 0 U (0)y(t): 2

1

Let us consider now the term B

2

T

U (0)y (t)d

T 2 y (t

0  + (t)) 0 yT (t 0  ) BT 0

1

0

T U ( + )By (t + )d  (t)

0

0 2 +  + (t 0  + ))

=

0

0

0

0

0

U (0)y (t)d:

=

Now, let us select a symmetric and definite–positive matrix M1 . Then  (t)

2

0

T T T  +  ) A B U (0)y (t)d

T 2y (t

+ 0

0

2y (t

=

1M B

2

0 2 +  + (t 0  + ))

0

T T  +  )A B

0 2 +  + (t 0  + ))

T

 (t)

0

T 2y (t

 1 01 _

( )

 (t)

0 2 +  ) B T T y (t 0 2 +  )d

T y (t

and we arrive at the following result:

0

0  + (t)) 0 y(t 0  )]T BT U (0)y(t)  t T 2[B y_ (t 0  +  )] U (0)y (t)d

0

0

0

2

Using the Leibniz–Newton formula, one has

=

2

 1 01 _

0w(y(t + 1)) + 2 [B(y(t 0  + (t))

0y(t 0  ))]T 2[y (t

1 yT (t 0 2 +  ) 1 + _ (td0  + )

 (t)

2[y_ (t

0  + )]T BT U T ( + )By(t + )dd

0  + ) + By 1 (t 0 2 +  + (t 0  + ))]T 2[Ay (t

1 BT U T ( + )By(t + )dd:

0  + )T AT BT U (0)y(t)d

The first term on the right-hand side can be estimated as follows:  (t)

 

0

y (t

0

T T T  +  ) A B M1 BAy (t

T 01 +  (t) y (t) U (0)M1 U (0)y (t)  T T T y (t  +  ) A B M1 BAy (t 0 T 01 + 0 y (t) U (0)M1 U (0)y (t):

j

j

0

0  + )d

 (t)

0

0

T 2y (t

0

 (t)



0  + )d

0  + )AT BT U T ( + )By(t + )dd T y (t

0

j

0

j

+  (t)

0

In a similar way, for a given definite–positive matrix M2 we have  (t)

T 2y (t

0

0 2 +  + (t 0  + ))

 (t)



T y (t

0

B

T

2

0 2 + (t 0  + ))(B

2

T 2 ) M2 B

0

0  + )

then y (t

0

0 2 + (t 0  + ))

B

 (t)

0

Now, we change the integration variable in the right-hand side integral

T

y

T

0

0

2

T

M2 B

T 2y (t

0

 (t0 )

T y (t

1 By(t + )d T T (t 0  +  )A B M BAy (t 0  +  )d 3

T T 01 T y (t + )B U ( + )M3 U ( + )

1 By(t + )d

2

0 2 +  )

 (t)

T y (t

0

1M B j

j

+  (t)

B

2

T

M2 B

2

B

2

T

1 U T ( + )By(t + )dd 

1 yT (t 0 2 + (t 0  + ))d  (t)+ (t0 + (t))

T T 01 T y (t + )B U ( + )M3 U ( + )

0 2 +  + (t 0  + ))

4

=

3

where again matrix M3 is a positive–definite one. Now, the term

T 01 + 0 y (t) U (0)M2 U (0)y (t):

 (t)

0

+ 0

1 yT (t 0 2 + (t 0  + ))d

 =  +  (t



U (0)y (t)d



0  + )AT BT M BAy(t 0  + )d

2

0 2 +  + (t 0  + ))

y (t 0

0

B

2

T

0 2 +  0 (t 0  + ))d

T T 01 T y (t + )B U ( + )M4 U ( + )

1 By(t + )d where M4 is a positive–definite matrix.

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Once again, changing the integration variable in the first integral in the right-hand side as

 =  + (t 0  +  ) we arrive at the following estimation:

 (t) 0

y T (t 0 2 +  + (t 0  +  )) B 2 T M4 B 2

(t)+(t0 +(t)) (t0 )

then



 1 01 _

0

0

A=

+

+



3

0 1 BAy(t 0  + )d 1

T 0 _ 0 y (t 0 2 +  ) B 1 y(t 0 2 +  )d: 0

0

and B =

01

0

:

1

2

T

(21)

0

W1 = W2 =

[M2 + M4 ] B

1

0

0

1

and select matrices Mj as follows:

4

Mj

=



1

0

0

1

;

j = 1; 2; 3; 4

where  > 0 is a design parameter. For our case

yT (t 0  +  )AT B T [M1 + M3 ] 

1

(20)

We set

U T ( + )By(t + )d

0 M101 + M201 U (0)y(t) yT (t + )B T U ( + ) M 01 + M 01

2

0

01 02

x(t 0 1):

0 1

0

0 1 U T ( + )By(t + )d

1

 0:

j(t)j   < 0 and j_ (t)j  _ < 13 :

y(t)T U (0)

+ 0

for 

We would like to find conditions on 0 and _ 0 which guarantee stability of the perturbed system for any continuous function  (t) satisfying the restrictions

0  + (t)) 0 y(t 0  ))]T

0

k e0 ;

Here,  = 1, matrices

yT (t 0 2 +  ) B 2 T

0

+ 3W2

0 1 0 01 02 x(t) + 01

x_ (t) =

0 1 M4 B2 y(t 0 2 +  )d:

1 U (0)y(t) + 

2

for t  0

Example 2: Let us consider the following exponentially stable system:

yT (t 0 2 +  ) B 2 T M4 B 2

Now, we collect the estimations 2[B (y (t

kU ()k  2 kW

1

1 y(t 0 2 +  ) 1 + _ (td0  + ) 2

to satisfy the first two inequalities in (19) one should choose 0 sufficiently small. To check the second inequality one has to know matrix U (). It can be easily calculated numerically, or one can also use an estimate of the matrix. For example, if we know that the solutions of the nominal system satisfy the inequality

kx(t; ')k   k'kh e0t ;

1 y(t 0 2 +  0 (t 0  + ))d =

131

W1 + 3W2 = 4

2

Remark 3: When 0 = 0 (no perturbation) all terms in the right hand side of the last inequality are equal to zero. It means that by continuity they should remain small enough for sufficiently small values of 0 . Remark 4: Matrices Mj , j = 1; 2; 3; 4 are free parameters and can be used for an optimization purpose. Simple computations lead to the following robustness result. Theorem 2: Let Assumptions 1 and 2 hold and assume additionally that 0 < (1=3) . Then, (16) is robustly stable for a continuous timevarying function  (t) if the following conditions hold simultaneously: i) the nominal system (1) is exponentially stable; ii) there exist positive–definite matrices Wi , i = 1; 2 and Mk , k = 1; 2; 3; 4, such that (19), as shown at the bottom of the page, holds. In conclusion, given matrices W1 and W2 , and value _ 0 2 (0; 1), then one can satisfy first to the last two inequalities in (19) at the expense of a correct choice of matrices Mj , j = 1; 2; 3; 4. Then, in order

1

0

0

1

and all components of U () are depicted in Fig. 1 We now start to check conditions of Theorem 2. Condition i) of Theorem 2 holds because (20) is exponentially stable. The first inequality from ii) takes the form 1

0

0

1

> 20 U 2 (0) = 320  

17

5

5

2

:

The inequality holds if

0
40 B T U (1 + )U T (1 + )  40 2 = u2 (1 + ) + u22 (1 +  )  21 for all  2 [01; 0]. 1

0

0

1

1

01

01 1

It follows from Fig. 1 that max

2[01;0]

2 2 u21 (1 +  ) + u22 (1 +  )

W1 > 0 U (0) M101 + M201 U (0) 1 W2 > 0 B T U ( + ) M301 + M401 U T ( + )B; 2 1 W2 > AT B T [M1 + M3 ] BA 2 1 W2 > 101_ B 2 T [M2 + M4 ] B 2 2

for 

2 [0; 0]

=

2 2 u21 (0) + u22 (0) = 32:

(19)

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where cij = 1=2(pi j qi j + pi2 j2 qi j 0 pi j qi j 0 pi j qi j ), with (i1 ; i2 ) the ith pair of the sequence (1; 2); (1; 3); . . . (1; m), (2; 3); . . . (2; m); . . . (m; m) and (j1 ; j2 ) is generated by duality. For P 8Q, we use the classical definition of the Kronecker sum P 8Q = P Im + Im Q. Algebraic properties of these new quantities can be found in [18] (tensor product) or in [21] (matrix computations). For the sake of the brevity, they are not included here.

ACKNOWLEDGMENT The authors would like to thank the referees and the Associate Editor for their useful comments.

Fig. 1. Matrix U ( ).

REFERENCES

Therefore, the second inequality holds if

0
4(BA)T BA = 4

1

3

3

9

and holds if


B2 B2 = 0 1 1 0 _ 0 1 0 _ 0 It holds when