Additional dynamics for general class of time-delay ... - Semantic Scholar
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• For minimum potential delay controllers, the local and global stability conditions are quite different for both REM and finitebuffer virtual queue. However, local nonoscillatory stability implies global stability for reasonable choice of parameters for the marking functions we considered. REFERENCES [1] S. Athuraliya, D. E. Lapsley, and S. H. Low, “Random early marking for Internet congestion control,” in Proc. IEEE GLOBECOM, 1999, pp. 1264–1268. [2] J. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, 2nd ed. New York, NY: Springer-Verlag, 1991. [3] C. V. Hollot and Y. Chait, “Nonlinear stability analysis for a class of TCP/AQM schemes,” in Proc. IEEE Conf. Decision Control, Dec. 2001, pp. 2309–2314. [4] C. V. Hollot, V. Misra, D. Towsley, and W. Gong, “On designing improved controllers for AQM routers supporting TCP flows,” in Proc. INFOCOM 2001, Anchorage, AK, Apr. 2001, pp. 1726–1734. [5] R. Johari and D. Tan, “End-to-end congestion control for the internet: Delays and stability,” IEEE/ACM Trans. Networking, vol. 9, pp. 818–832, Dec. 2001. [6] F. P. Kelly, A. Maulloo, and D. Tan, “Rate control in communication networks: Shadow prices, proportional fairness and stability,” J. Oper. Res. Soc., vol. 49, pp. 237–252, 1998. [7] F. P. Kelly, “Models for a self-managed Internet,” Phil. Trans. Royal Soc., vol. A358, pp. 2335–2348, 2000. , “Mathematical modeling of the Internet,” in Mathematics Unlim[8] ited—2001 and Beyond, B. Engquist and W. Schmid, Eds. Berlin, Germany: Springer-Verlag, 2001, pp. 685–702. [9] S. Kunniyur and R. Srikant, “End-to-end congestion control: Utility functions, random losses and ECN marks,” presented at the INFOCOM 2000, Tel Aviv, Israel, Mar. 2000. , “Analysis and design of an adaptive virtual queue algorithm for [10] active queue management,” presented at the ACM Sigcomm, San Diego, CA, Aug. 2001. [11] L. Massoulie, “Stability of distributed congestion control with heterogeneous feedback delays,” IEEE Trans. Automat. Contr., vol. 47, pp. 895–902, June 2002. [12] S. Kunniyur and R. Srikant, “A time-scale decomposition approach to adaptive ECN marking,” presented at the IEEE INFOCOM 2001, Anchorage, AK, Apr. 2001. [13] L. Massoulie and J. Roberts, “Bandwidth sharing: Objectives and algorithms,” presented at the IEEE INFOCOM 1999, New York, NY, Mar. 1999. [14] F. Paganini, J. Doyle, and S. Low, “Scalable laws for stable network congestion control,” in Proc. IEEE Conf. Decision Control, Dec. 2001, pp. 185–190. [15] S. Shakkottai and R. Srikant, “Mean FDE models for Internet congestion control,” Coordinated Science Lab., Univ. Illinois, Tech Rep., 2002. [16] S. Shakkottai, R. Srikant, and S. Meyn, “Bounds on the throughput of congestion controllers in the presence of feedback delay,” in Proc. IEEE Conf. Decision Control, Dec. 2001, pp. 616–621. [17] G. Vinnicombe, “On the stability of end-to-end congestion control for the Internet,” Univ. Cambridge, Cambridge, U.K., Tech. Rep., 2001.
Additional Dynamics for General Class of Time-Delay Systems Vladimir Kharitonov and Daniel Melchor-Aguilar Abstract—In this note, some recent results on additional dynamics introduced by transformations of time-delay systems are extended to the case of general time-varying systems with delay. Sufficient stability conditions for the additional dynamics are also given. Index Terms—Robust stability, stability, time-delay systems.
I. INTRODUCTION In recent works [1], [2], [6], it has been shown for the case of time-invariant systems with discrete delays that basic transformations, commonly used to obtain delay-dependent stability conditions, introduce some additional dynamics responsible for the lack of equivalence in stability property of the original and transformed systems. It is important to mention that these transformations are based on the Newton–Leibnitz formula which allows to replace time-delay terms in the right hand side of the delay system. In [7], it has been stated that the additional dynamics are introduced by transformations only if one replaces the derivative under the integrals by the right-hand side of the original system. In [7], some results about the additional dynamics have been extended to the case of time-varying delay systems. In this note, we are going to study the general class of time-varying systems with distributed delay. In particular we propose a new form for equations which describe the transformed systems. Two examples are given to illustrate the fact that the additional dynamics do not always impose new restrictions on stability of the transformed systems. II. TIME-INVARIANT SYSTEM Consider the following time-invariant delay system:
x1 (t) =
0
0h
dG()x(t + )
(1)
where all elements of G() have bounded variation on [0h, 0]. We denote by x(t; t0 ; ') the solution of (1) with the initial condition
x(t0 + ; t0 ; ') = '();
for
2 [0h; 0]
where ' 2 C n [0h; 0]. We will use the euclidian vector norm kxk, and the uniform norm
k'kh =
sup
2[0h;0]
k'()k
for functions. Definition 1: We say that (1) is stable if it is exponentially stable, that is, there exist constants 1 and > 0 such that every solution, x(t; t0 ; ') of the system satisfies the inequality
kx(t; t ; ')k k'kh e0 t0t 0
(
)
8t t : 0
(2)
Manuscript received May 16, 2002; revised September 11, 2002 and October 17, 2002. Recommended by Associate Editor M. E. Valcher. This work was supported in part by CONACyT-México. The authors are with the Department of Automatic Control, CINVESTAVIPN, 07300 Mexico City, México (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2003.813200 0018-9286/03$17.00 © 2003 IEEE
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Remark 2: For the case of time-invariant systems, we assume that the initial time-instant t0 = 0, and in this case we will use notation x(t; ') for the solution with the following initial condition:
x(; ') = '();
for
2 [0h; 0]:
0
x1 (t + )d;
0:
y1 (t) =
0h
dG() y(t)0
0
0
0h
0
dG()
0
0h
dG( )y(t + + )d :
(4)
It is important to mention that the time-delay of the transformed system is twice that of the original one. Remark 3: Stability of (4) implies that of (1). Proof: Direct calculations show that every solution of (1) is also a solution of (4), and, therefore, the stability of (4) implies that of (1). The following statements will be proven in the next section for the case of time-varying systems. Theorem 4: Equation (4) and the system
y1 (t) = 00 h dG()y(t + ) + z (t) z (t) = 00 h dG() 0 z (t + )d
(5)
are equivalent in the sense that every solution of (4) generates a solution of system (5), and vice versa. Theorem 5: Equation (4) is stable if and only if (1) and 0
z (t) =
dG()
0h
0
z (t + )d
0
0h
0
d G(t; ) y(t)0 0h 0 0 0 d G(t; ) d G(t + ; )y(t + + )d : 0h 0h
kdG()k < 1:
y1 (t) = 00 h d G(t; )y(t + ) + z (t) z (t) = 00 h d G(t; ) 0 z (t + )d
z (t) =
0
0h
d G(t; )x(t + )
(7)
where h > 0 and matrix G(t; ) is continuous with respect to t for t and has a bounded variation with respect to on [0h, 0] uniformly with respect to t
0
0
0h
kd G(t; )k m:
(8)
In the following, we say that system is stable if it is exponentially stable, that is, the solutions of (7) satisfy inequality (2).
0
0h
d G(t; )
0
z (t + )d
where
z (t) = y1 (t) 0
0
0h
d G(t; )y(t + ):
These two equations form (10). Sufficiency Part: From the first equation of (10), we have that
z (t) = y1 (t) 0
0
d G(t; )y(t + ): 0h Substituting this expression for z (t) into the second equation of (10), we find that
y1 (t) =
0
0h
d G(t; ) y(t) 0
0
d G(t; )2 0h 0 0 d G(t + ; )y(t + + )d : 0h
2
Remark 8: The additional dynamics are generated by the second equation of (10) and enter into the first equation of the system in the additive way. Theorem 9: System (10) is stable if and only if (7) and
III. TIME-VARYING SYSTEM
x1 (t) =
(10)
are equivalent in the sense that every solution af (9) generates a solution of system (10), and vice versa. Proof: 0 Necessity Part: Substracting the term 0h d G(t; )y (t + ) from both sides of (9), after simple manipulations we arrive at
z (t) =
Let us consider the time-varying system
(9)
In the following theorem, a new equivalent form for (9) is proposed. The form shows it explicitly how the additional dynamics affect the original system. Theorem 7: Equation (9) and
(6)
are stable. Theorem 6: Equation (6) is stable if the following inequality holds:
h
y1 (t) =
(3)
Substituting (3) in (1) and replacing the derivative under the integral by the right-hand side of (1), we obtain the transformed system 0
Applying the transformation to system (7) we arrive at
0
In order to obtain simple delay-dependent stability conditions of (1), a model transformation is usually applied; see [4]. The transformation is based on the equality
x(t + ) = x(t) 0
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0
0h
d G(t; )
0
z (t + )d
(11)
are stable. Proof: Necessity Part: First, observe that (11) coincides with the second equation of (10). Therefore, the stability of system (10) implies that of (11). If we assume that the initial function for z (t) is identically zero, then the first equation of system (10) coincides with (7) and its stability is also implied by the stability of system (10). Sufficiency Part: Now, we suppose that both (7) and (11) are stable, that is, there exist constants 1 1, 2 1, 1 > 0 and 2 > 0 such that for all t t0 we have
kx(t; t ; ')k k'kh e0 t0t ; for t t kz(t; t ; )k k kh e0 t0t ; for t t 0
0
1 2
(
)
(
)
0 0
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where ' is the initial function for x(t), x(t0 + ) = '(), 2 [0h; 0], and is the initial function for z (t), z (t0 + ) = (), 2 [0h; 0]. Without any loss of generality we may assume that 1 6= 2 . Applying the Cauchy formula to the first equation of (10), we can write every solution of this equation (see [3]) as
t
y(t; t0 ; ) =X (t; t0 )(t0 ) + X (t; s)z (s)ds+ (12) t 0 t 0 + X (t; s)d G(s; )(s + ) ds 0h t (13) where X (t; t0 ) is the fundamental matrix of (7) and function for y (t). Stability of (7) implies that
kX (t; t0 )k 1 e0 (t0t ) ;
is the initial
t3 (; ) = inf t t0 j kz (t; t0 ;
kz(t3 (; ); t ; )k = k kh e0 t ; 0t : (17) For every > 0, there exists ( ) 1 such that for all ( ) kz(t; t ; )k k kh e0 t0t ; t 2 [t ; T ]: Therefore, t3 (; ) T . For such values of , we observe from (11)
(
1
)
where
m kk 1 kkh + 1 h e h 0 1 1
(
(
)
0
0h
Comparing (17) with the last inequality and calculating the integrals, we obtain
h
h kd G(t3 (; ); )k e h0 1 : 0h 0
lim
0 e0 t0t : (
)
e h 0 1 h
)
(14)
= 1 and
lim
e h 0 1 3 h
=1+
+
k kh j 0 j
h
21 2 1
h
0h
0
0
h t!1
=1
0h Then, there exists T > t0 , such that
h
0
0h
kd G(t; )k
0
:
2
1 0 4 :
This contradiction proves our statement that there exist > 0 such that (16) holds.
1
and
IV. EXAMPLES We present here two examples which illustrate some results of this note. Example 11: Let G() be
G() =
0 ; for some > 0:
0; A;
if if
2 [0h; 0 ) 2 [0 ; 0]
then (1) is of the form
10 ; 2
t T:
(15)
We need to prove that there exist constants 1 and > 0 such that
kz(t; t ; )k k
1
2
kd G(t; )k < 1:
kd G(t; )k
=+
kd G(t3 (; 3 ); )k 1 0 2 1
Proof: Let us suppose that lim
>0
0h because, as we know, t3 (; 3 ) T . Therefore, from (18), we obtain
2
Inequality (14) proves stability of (10). The following theorem gives a sufficient condition for stability of (11). Theorem 10: Equation (11) is stable if lim
(18)
On the other hand, from (15), we see that
= min f 1 ; 2 g :
t!1
)
0
e h 0 1 !+0 !+1 h we have to conclude that there exists 3 > 0 such that
+
and
0
0
0
Therefore, the following inequality holds:
ky(t; t0 ; )k e0 (t0t
)
kz(t3 (; ); t ; )k kd G(t3 (; ); )k kz(t3 (; ) + ; t ; )k d 0h k kh e0 t ; 0t 2 2 kd G(t3 (; ); )k e0 d:
1
e h 0 1
1
)
)
Taking into account the fact that, for any h
)
1 2 k kh 0 (t0t + j 1 0 2 j e
(
0
(
0
)
(
(
0
0
t
1
:
)
It is clear that
Considering (8) and calculating the integrals, we obtain 0
(
0
t t0 :
ky(t; t0 ; )k 1 kkh e0 (t0t ) + 1 kkh t +h 0 2 e0 (t0s) kd G(s; )k ds+ t 0h t e0 (t0s)0 (s0t ) ds: + 1 2 k k ky(t; t ; )k kkh e0 t0t + m kkh e0 t0t +
k > k kh e0 t0t
)
that
Then, we have from (13) that
h
1 and > 0, there
If such and do not exist, then for every exists
kh e0 (t0t ) ;
for t t0 :
(16)
x1 (t) = Ax(t 0 ):
(19)
The characteristic function of the system is of the form
f1 (s) = det(sI 0 e0 s A) =
n
(s
k=1
0 e0 sk )
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Fig. 1. Domains 0 and 9 .
where k , k = 1; 2; . . . ; n, are the eigenvalues of matrix A. Applying the D-decomposition method (see [5]) to g1 (s) = s
0 e0 s
one can easily verify that all roots of the function lie in the open left-half complex plane if and only if the complex number lies in the open domain 9 , whose boundary admits the following parametrization: @ 9
=
0! sin(! ) + i! cos(! ) ! 2 0
; 2 2
:
See the domain on Fig. 1. Therefore, the system (19) is stable if and only if all eigenvalues of matrix A lie in domain 9 . The additional dynamics associated with the system are described by z (t) = A
0
0
Domains 8 and .
Fig. 2.
one can easily verify that all roots of the function lie in the open left half complex plane if and only if the complex number lies in the open domain 8h , see Fig. 2, whose boundary is defined by the following expression: @ 8h
=
=
2
z (t) = A
z (t + )d:
(20)
!
2 0 2 ; 2
f3 (s) =
0 :
0
0h
x(t + )d:
(21)
0hs
0 1 0 se
A
=
n
s k=1
0hs
0 1 0 se
k
where k , k = 1; 2; . . . ; n, are the eigenvalues of matrix A. Applying again the D-decomposition method to g2 (s) = s
( + h) z (t + )d:
(22)
0hs 1 0 e 021 + hs k
n
s
k=1
0hs
0 1 0 se
0hs 0 1 + hs
0e
s2
one can easily verify that all roots of the function lie in the open left-half complex plane if and only if the complex number lies in the open domain h ; see Fig. 2, whose boundary is defined by the following parametrization: @ h
=
0
! 2 [1 cos(h! )] + [1 cos(h!)]2 + [sin(h!) h!]2 ! 2 [sin(h! ) h! ] ! +i [1 cos(h!)]2 + [sin(h!) h!]2
0
0
The characteristic function of the equation is f2 (s) = det sI
0h
g3 (s) = 1
So, in this case, the additional dynamics do not impose new restrictions on stability of the transformed system. Example 12: If we select G() = A , then (1) takes the form
1
0
where k , k = 1; 2; . . . ; n, are the eigenvalues of matrix A. Applying again the D-decomposition method to
:
It follows from Fig. 1 that
x(t) = A
:
The characteristic function of the equation is of the form
! sin(! ) + i! 2 [1 0 cos(! )] 2
9
2
Therefore, (19) is stable if and only if all eigenvalues of matrix A lie in domain 8h ; see Fig. 2. Now, the additional dynamics introduced by the transformation, are represented by the following equation:
It has been shown in [6] that (20) is exponentially stable if and only if all eigenvalues of matrix A lie in the open domain 0 (see Fig. 1) whose boundary admits the following parametrization: @ 0
h! ) 0 !2 + i 2 [1! 0sin( !2 0 ; cos(h!)] h h
0
0
0
2 (01; 1)
:
Therefore, (19) is stable if and only if all eigenvalues of matrix A lie in domain h . It follows from Fig. 2 that
8h h : So, again the additional dynamics do not introduce additional restrictions on stability of the transformed system.
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REFERENCES [1] K. Gu and S.-I. Niculescu, “Additional dynamics introduced in model system transformation,” IEEE Trans. Automat. Contr., vol. 45, pp. 572–575, Mar. 2000. , “Further remarks on additional dynamics in various model trans[2] formations of linear delay systems,” IEEE Trans. Automat. Contr., vol. 46, pp. 497–500, Mar. 2001. [3] J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations. New York: Springer-Verlag, 1993. [4] V. B. Kolmanosvkii and J.-P. Richard, “Stability of some linear systems with delays,” IEEE Trans. Automat. Contr., vol. 44, pp. 984–989, May 1999. [5] V. Kolmanosvkii and A. Myshkis, Applied Theory of Functional Differential Equations. Norwell, MA: Kluwer, 1992. [6] V. L. Kharitonov and D. Melchor-Aguilar, “On delay dependent stability conditions,” Syst. Control Lett., vol. 40, pp. 71–76, Mar. 2000. , “On delay dependent stability conditions for time varying sys[7] tems,” Syst. Control Lett., vol. 46, pp. 173–180, July 2002.
On the Synthesis of Safe Control Policies in Decentralized Control of Discrete-Event Systems Kurt Rohloff and Stéphane Lafortune
Abstract—State estimation and safe controller synthesis for a general form of decentralized control architecture for discrete-event systems is investigated. For this architecture, controllable events are assigned to be either “conjunctive” or “disjunctive.” A new state estimator that accounts for past local control actions when calculating the set of estimated system states is presented. The new state estimator is applied to a previous general decentralized control law. The new control method generates a controlled language at least as large as that generated by the original method if a safety condition is satisfied. An algorithm for generating locally maximal control policies for a given state estimate is also discussed. The algorithm allows an amount of “steering” of the controlled system through an event priority mechanism. Index Terms—Decentralized control, discrete-event systems, maximal behavior.
I. INTRODUCTION Decentralized control is commonly the most efficient or only method to control a system too large or complex to be controlled by a single centralized controller. Decentralized control systems are implemented as a number of local controllers that have authority over a subset of a system’s controllable events. As system operation evolves, the local controllers make observations of the system’s behavior and maintain local estimates of the system’s state. The state information is used to during the calculation of local control actions that are combined globally to determine allowed system behavior. Several decentralized control laws for discrete-event systems have been introduced in the literature. See, for example, [1], [2], [5]–[8], Manuscript received November 15, 2001; revised November 10, 2002 and December 18, 2002. Recommended by Associate Editor A. Giua. This work was supported in part by the National Science Foundation under Grant CCR0082784. The work of K. Rohloff was supported by a GAANN Fellowship for the 2000–2001 academic year. The authors are with the Department of Electrical Engineering and Computer Science, The University of Michigan, Ann Arbor, MI 48109-2122 USA (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TAC.2003.812812
[11], [13], and [14]. The solvability problem for safe and nonblocking decentralized controllers is undecidable [5], [12]. We restrict our attention to the safety problem and attempt to synthesize safe decentralized controllers that allow as much behavior as possible. This note builds upon Yoo and Lafortune’s [14] work on the control structures for the so-called “general decentralized architecture” where the set of controllable events is partitioned into a set of disjunctive events that follow a “fusion by union” rule and a set of conjunctive events that follow a “fusion by intersection” rule. Given the necessary brevity of this technical note we assume the reader has an understanding of the material presented in [14]. A more thorough introduction is also presented in this note’s companion technical report [9], which is available for download. Previously, many control schemes have used an inverse projection operation on the locally observed strings to estimate the current system state. However, this state estimation method does not make full use of a controller’s knowledge; controllers may have memory of past local control actions. We investigate the properties of a memory-based state estimator that has knowledge of past control actions. We use the estimator to construct control laws that locally allow more behavior than the gdec control law in [14] for a given partition of controllable events when the legal behavior may not be controllable and coobservable. The local state estimator we discuss makes no assumptions on the specific control actions of other controllers except to assume that other controllers always enable all events. Our first control law is called gmdec. For sufficient safety conditions, gmdec always allows at least as much behavior as gdec if the same partition of the controllable events is used for both control laws. The authors in [5] also propose the use of information on applied control patterns in processing partial observations but their approach is different from ours. We also develop a local greedy control policy that enables as many events as possible locally for a given state estimate. Ben Hadj-Alouane et al.[3] have done work with maximal control policies for centralized control systems and we apply their methods to general decentralized controllers. Our new control policy, the second variable lookahead policy for general decentralized memory-based control (VLP-GM2), enforces locally maximal control actions for a given state estimate. In Section II, the general decentralized control work in [14] is briefly reviewed. In Section III, the new state estimator is introduced. In the fourth section, we develop the new gmdec control policy. In Section V, the VLP-GM2 algorithm is presented and discussed. Some concluding remarks are made in Section VI. We use the notation employed in [14] that we briefly review in the next section. The proofs of all results in this note can be read in the companion technical report [9] along with more in-depth discussions of the topics presented here. II. PROBLEM FORMULATION AND PREVIOUS WORK We identify some of the terminology used in this note. The “state” of the system is the string of events that has been generated so far by the system. We consider a state estimate to be a set of possible states the system may be in at a given instance. A “state estimator” is a function that calculates a state estimate. Traditionally, many observers and controllers have used the inverse projection operation (P 01 (P (1))) as a state estimator. We restrict the possible and legal languages of the systems in this note to be prefix-closed because we are only concerned with safety issues. We also specify that the possible and legal languages be regular so that when the discussed state estimators are implemented, the local state estimations can be determined by the states of some finite observer automata.
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• For minimum potential delay controllers, the local and global stability conditions are quite different for both REM and finitebuffer virtual queue. However, local nonoscillatory stability implies global stability for reasonable choice of parameters for the marking functions we considered. REFERENCES [1] S. Athuraliya, D. E. Lapsley, and S. H. Low, “Random early marking for Internet congestion control,” in Proc. IEEE GLOBECOM, 1999, pp. 1264–1268. [2] J. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, 2nd ed. New York, NY: Springer-Verlag, 1991. [3] C. V. Hollot and Y. Chait, “Nonlinear stability analysis for a class of TCP/AQM schemes,” in Proc. IEEE Conf. Decision Control, Dec. 2001, pp. 2309–2314. [4] C. V. Hollot, V. Misra, D. Towsley, and W. Gong, “On designing improved controllers for AQM routers supporting TCP flows,” in Proc. INFOCOM 2001, Anchorage, AK, Apr. 2001, pp. 1726–1734. [5] R. Johari and D. Tan, “End-to-end congestion control for the internet: Delays and stability,” IEEE/ACM Trans. Networking, vol. 9, pp. 818–832, Dec. 2001. [6] F. P. Kelly, A. Maulloo, and D. Tan, “Rate control in communication networks: Shadow prices, proportional fairness and stability,” J. Oper. Res. Soc., vol. 49, pp. 237–252, 1998. [7] F. P. Kelly, “Models for a self-managed Internet,” Phil. Trans. Royal Soc., vol. A358, pp. 2335–2348, 2000. , “Mathematical modeling of the Internet,” in Mathematics Unlim[8] ited—2001 and Beyond, B. Engquist and W. Schmid, Eds. Berlin, Germany: Springer-Verlag, 2001, pp. 685–702. [9] S. Kunniyur and R. Srikant, “End-to-end congestion control: Utility functions, random losses and ECN marks,” presented at the INFOCOM 2000, Tel Aviv, Israel, Mar. 2000. , “Analysis and design of an adaptive virtual queue algorithm for [10] active queue management,” presented at the ACM Sigcomm, San Diego, CA, Aug. 2001. [11] L. Massoulie, “Stability of distributed congestion control with heterogeneous feedback delays,” IEEE Trans. Automat. Contr., vol. 47, pp. 895–902, June 2002. [12] S. Kunniyur and R. Srikant, “A time-scale decomposition approach to adaptive ECN marking,” presented at the IEEE INFOCOM 2001, Anchorage, AK, Apr. 2001. [13] L. Massoulie and J. Roberts, “Bandwidth sharing: Objectives and algorithms,” presented at the IEEE INFOCOM 1999, New York, NY, Mar. 1999. [14] F. Paganini, J. Doyle, and S. Low, “Scalable laws for stable network congestion control,” in Proc. IEEE Conf. Decision Control, Dec. 2001, pp. 185–190. [15] S. Shakkottai and R. Srikant, “Mean FDE models for Internet congestion control,” Coordinated Science Lab., Univ. Illinois, Tech Rep., 2002. [16] S. Shakkottai, R. Srikant, and S. Meyn, “Bounds on the throughput of congestion controllers in the presence of feedback delay,” in Proc. IEEE Conf. Decision Control, Dec. 2001, pp. 616–621. [17] G. Vinnicombe, “On the stability of end-to-end congestion control for the Internet,” Univ. Cambridge, Cambridge, U.K., Tech. Rep., 2001.
Additional Dynamics for General Class of Time-Delay Systems Vladimir Kharitonov and Daniel Melchor-Aguilar Abstract—In this note, some recent results on additional dynamics introduced by transformations of time-delay systems are extended to the case of general time-varying systems with delay. Sufficient stability conditions for the additional dynamics are also given. Index Terms—Robust stability, stability, time-delay systems.
I. INTRODUCTION In recent works [1], [2], [6], it has been shown for the case of time-invariant systems with discrete delays that basic transformations, commonly used to obtain delay-dependent stability conditions, introduce some additional dynamics responsible for the lack of equivalence in stability property of the original and transformed systems. It is important to mention that these transformations are based on the Newton–Leibnitz formula which allows to replace time-delay terms in the right hand side of the delay system. In [7], it has been stated that the additional dynamics are introduced by transformations only if one replaces the derivative under the integrals by the right-hand side of the original system. In [7], some results about the additional dynamics have been extended to the case of time-varying delay systems. In this note, we are going to study the general class of time-varying systems with distributed delay. In particular we propose a new form for equations which describe the transformed systems. Two examples are given to illustrate the fact that the additional dynamics do not always impose new restrictions on stability of the transformed systems. II. TIME-INVARIANT SYSTEM Consider the following time-invariant delay system:
x1 (t) =
0
0h
dG()x(t + )
(1)
where all elements of G() have bounded variation on [0h, 0]. We denote by x(t; t0 ; ') the solution of (1) with the initial condition
x(t0 + ; t0 ; ') = '();
for
2 [0h; 0]
where ' 2 C n [0h; 0]. We will use the euclidian vector norm kxk, and the uniform norm
k'kh =
sup
2[0h;0]
k'()k
for functions. Definition 1: We say that (1) is stable if it is exponentially stable, that is, there exist constants 1 and > 0 such that every solution, x(t; t0 ; ') of the system satisfies the inequality
kx(t; t ; ')k k'kh e0 t0t 0
(
)
8t t : 0
(2)
Manuscript received May 16, 2002; revised September 11, 2002 and October 17, 2002. Recommended by Associate Editor M. E. Valcher. This work was supported in part by CONACyT-México. The authors are with the Department of Automatic Control, CINVESTAVIPN, 07300 Mexico City, México (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2003.813200 0018-9286/03$17.00 © 2003 IEEE
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 6, JUNE 2003
Remark 2: For the case of time-invariant systems, we assume that the initial time-instant t0 = 0, and in this case we will use notation x(t; ') for the solution with the following initial condition:
x(; ') = '();
for
2 [0h; 0]:
0
x1 (t + )d;
0:
y1 (t) =
0h
dG() y(t)0
0
0
0h
0
dG()
0
0h
dG( )y(t + + )d :
(4)
It is important to mention that the time-delay of the transformed system is twice that of the original one. Remark 3: Stability of (4) implies that of (1). Proof: Direct calculations show that every solution of (1) is also a solution of (4), and, therefore, the stability of (4) implies that of (1). The following statements will be proven in the next section for the case of time-varying systems. Theorem 4: Equation (4) and the system
y1 (t) = 00 h dG()y(t + ) + z (t) z (t) = 00 h dG() 0 z (t + )d
(5)
are equivalent in the sense that every solution of (4) generates a solution of system (5), and vice versa. Theorem 5: Equation (4) is stable if and only if (1) and 0
z (t) =
dG()
0h
0
z (t + )d
0
0h
0
d G(t; ) y(t)0 0h 0 0 0 d G(t; ) d G(t + ; )y(t + + )d : 0h 0h
kdG()k < 1:
y1 (t) = 00 h d G(t; )y(t + ) + z (t) z (t) = 00 h d G(t; ) 0 z (t + )d
z (t) =
0
0h
d G(t; )x(t + )
(7)
where h > 0 and matrix G(t; ) is continuous with respect to t for t and has a bounded variation with respect to on [0h, 0] uniformly with respect to t
0
0
0h
kd G(t; )k m:
(8)
In the following, we say that system is stable if it is exponentially stable, that is, the solutions of (7) satisfy inequality (2).
0
0h
d G(t; )
0
z (t + )d
where
z (t) = y1 (t) 0
0
0h
d G(t; )y(t + ):
These two equations form (10). Sufficiency Part: From the first equation of (10), we have that
z (t) = y1 (t) 0
0
d G(t; )y(t + ): 0h Substituting this expression for z (t) into the second equation of (10), we find that
y1 (t) =
0
0h
d G(t; ) y(t) 0
0
d G(t; )2 0h 0 0 d G(t + ; )y(t + + )d : 0h
2
Remark 8: The additional dynamics are generated by the second equation of (10) and enter into the first equation of the system in the additive way. Theorem 9: System (10) is stable if and only if (7) and
III. TIME-VARYING SYSTEM
x1 (t) =
(10)
are equivalent in the sense that every solution af (9) generates a solution of system (10), and vice versa. Proof: 0 Necessity Part: Substracting the term 0h d G(t; )y (t + ) from both sides of (9), after simple manipulations we arrive at
z (t) =
Let us consider the time-varying system
(9)
In the following theorem, a new equivalent form for (9) is proposed. The form shows it explicitly how the additional dynamics affect the original system. Theorem 7: Equation (9) and
(6)
are stable. Theorem 6: Equation (6) is stable if the following inequality holds:
h
y1 (t) =
(3)
Substituting (3) in (1) and replacing the derivative under the integral by the right-hand side of (1), we obtain the transformed system 0
Applying the transformation to system (7) we arrive at
0
In order to obtain simple delay-dependent stability conditions of (1), a model transformation is usually applied; see [4]. The transformation is based on the equality
x(t + ) = x(t) 0
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0
0h
d G(t; )
0
z (t + )d
(11)
are stable. Proof: Necessity Part: First, observe that (11) coincides with the second equation of (10). Therefore, the stability of system (10) implies that of (11). If we assume that the initial function for z (t) is identically zero, then the first equation of system (10) coincides with (7) and its stability is also implied by the stability of system (10). Sufficiency Part: Now, we suppose that both (7) and (11) are stable, that is, there exist constants 1 1, 2 1, 1 > 0 and 2 > 0 such that for all t t0 we have
kx(t; t ; ')k k'kh e0 t0t ; for t t kz(t; t ; )k k kh e0 t0t ; for t t 0
0
1 2
(
)
(
)
0 0
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where ' is the initial function for x(t), x(t0 + ) = '(), 2 [0h; 0], and is the initial function for z (t), z (t0 + ) = (), 2 [0h; 0]. Without any loss of generality we may assume that 1 6= 2 . Applying the Cauchy formula to the first equation of (10), we can write every solution of this equation (see [3]) as
t
y(t; t0 ; ) =X (t; t0 )(t0 ) + X (t; s)z (s)ds+ (12) t 0 t 0 + X (t; s)d G(s; )(s + ) ds 0h t (13) where X (t; t0 ) is the fundamental matrix of (7) and function for y (t). Stability of (7) implies that
kX (t; t0 )k 1 e0 (t0t ) ;
is the initial
t3 (; ) = inf t t0 j kz (t; t0 ;
kz(t3 (; ); t ; )k = k kh e0 t ; 0t : (17) For every > 0, there exists ( ) 1 such that for all ( ) kz(t; t ; )k k kh e0 t0t ; t 2 [t ; T ]: Therefore, t3 (; ) T . For such values of , we observe from (11)
(
1
)
where
m kk 1 kkh + 1 h e h 0 1 1
(
(
)
0
0h
Comparing (17) with the last inequality and calculating the integrals, we obtain
h
h kd G(t3 (; ); )k e h0 1 : 0h 0
lim
0 e0 t0t : (
)
e h 0 1 h
)
(14)
= 1 and
lim
e h 0 1 3 h
=1+
+
k kh j 0 j
h
21 2 1
h
0h
0
0
h t!1
=1
0h Then, there exists T > t0 , such that
h
0
0h
kd G(t; )k
0
:
2
1 0 4 :
This contradiction proves our statement that there exist > 0 such that (16) holds.
1
and
IV. EXAMPLES We present here two examples which illustrate some results of this note. Example 11: Let G() be
G() =
0 ; for some > 0:
0; A;
if if
2 [0h; 0 ) 2 [0 ; 0]
then (1) is of the form
10 ; 2
t T:
(15)
We need to prove that there exist constants 1 and > 0 such that
kz(t; t ; )k k
1
2
kd G(t; )k < 1:
kd G(t; )k
=+
kd G(t3 (; 3 ); )k 1 0 2 1
Proof: Let us suppose that lim
>0
0h because, as we know, t3 (; 3 ) T . Therefore, from (18), we obtain
2
Inequality (14) proves stability of (10). The following theorem gives a sufficient condition for stability of (11). Theorem 10: Equation (11) is stable if lim
(18)
On the other hand, from (15), we see that
= min f 1 ; 2 g :
t!1
)
0
e h 0 1 !+0 !+1 h we have to conclude that there exists 3 > 0 such that
+
and
0
0
0
Therefore, the following inequality holds:
ky(t; t0 ; )k e0 (t0t
)
kz(t3 (; ); t ; )k kd G(t3 (; ); )k kz(t3 (; ) + ; t ; )k d 0h k kh e0 t ; 0t 2 2 kd G(t3 (; ); )k e0 d:
1
e h 0 1
1
)
)
Taking into account the fact that, for any h
)
1 2 k kh 0 (t0t + j 1 0 2 j e
(
0
(
0
)
(
(
0
0
t
1
:
)
It is clear that
Considering (8) and calculating the integrals, we obtain 0
(
0
t t0 :
ky(t; t0 ; )k 1 kkh e0 (t0t ) + 1 kkh t +h 0 2 e0 (t0s) kd G(s; )k ds+ t 0h t e0 (t0s)0 (s0t ) ds: + 1 2 k k ky(t; t ; )k kkh e0 t0t + m kkh e0 t0t +
k > k kh e0 t0t
)
that
Then, we have from (13) that
h
1 and > 0, there
If such and do not exist, then for every exists
kh e0 (t0t ) ;
for t t0 :
(16)
x1 (t) = Ax(t 0 ):
(19)
The characteristic function of the system is of the form
f1 (s) = det(sI 0 e0 s A) =
n
(s
k=1
0 e0 sk )
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Fig. 1. Domains 0 and 9 .
where k , k = 1; 2; . . . ; n, are the eigenvalues of matrix A. Applying the D-decomposition method (see [5]) to g1 (s) = s
0 e0 s
one can easily verify that all roots of the function lie in the open left-half complex plane if and only if the complex number lies in the open domain 9 , whose boundary admits the following parametrization: @ 9
=
0! sin(! ) + i! cos(! ) ! 2 0
; 2 2
:
See the domain on Fig. 1. Therefore, the system (19) is stable if and only if all eigenvalues of matrix A lie in domain 9 . The additional dynamics associated with the system are described by z (t) = A
0
0
Domains 8 and .
Fig. 2.
one can easily verify that all roots of the function lie in the open left half complex plane if and only if the complex number lies in the open domain 8h , see Fig. 2, whose boundary is defined by the following expression: @ 8h
=
=
2
z (t) = A
z (t + )d:
(20)
!
2 0 2 ; 2
f3 (s) =
0 :
0
0h
x(t + )d:
(21)
0hs
0 1 0 se
A
=
n
s k=1
0hs
0 1 0 se
k
where k , k = 1; 2; . . . ; n, are the eigenvalues of matrix A. Applying again the D-decomposition method to g2 (s) = s
( + h) z (t + )d:
(22)
0hs 1 0 e 021 + hs k
n
s
k=1
0hs
0 1 0 se
0hs 0 1 + hs
0e
s2
one can easily verify that all roots of the function lie in the open left-half complex plane if and only if the complex number lies in the open domain h ; see Fig. 2, whose boundary is defined by the following parametrization: @ h
=
0
! 2 [1 cos(h! )] + [1 cos(h!)]2 + [sin(h!) h!]2 ! 2 [sin(h! ) h! ] ! +i [1 cos(h!)]2 + [sin(h!) h!]2
0
0
The characteristic function of the equation is f2 (s) = det sI
0h
g3 (s) = 1
So, in this case, the additional dynamics do not impose new restrictions on stability of the transformed system. Example 12: If we select G() = A , then (1) takes the form
1
0
where k , k = 1; 2; . . . ; n, are the eigenvalues of matrix A. Applying again the D-decomposition method to
:
It follows from Fig. 1 that
x(t) = A
:
The characteristic function of the equation is of the form
! sin(! ) + i! 2 [1 0 cos(! )] 2
9
2
Therefore, (19) is stable if and only if all eigenvalues of matrix A lie in domain 8h ; see Fig. 2. Now, the additional dynamics introduced by the transformation, are represented by the following equation:
It has been shown in [6] that (20) is exponentially stable if and only if all eigenvalues of matrix A lie in the open domain 0 (see Fig. 1) whose boundary admits the following parametrization: @ 0
h! ) 0 !2 + i 2 [1! 0sin( !2 0 ; cos(h!)] h h
0
0
0
2 (01; 1)
:
Therefore, (19) is stable if and only if all eigenvalues of matrix A lie in domain h . It follows from Fig. 2 that
8h h : So, again the additional dynamics do not introduce additional restrictions on stability of the transformed system.
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REFERENCES [1] K. Gu and S.-I. Niculescu, “Additional dynamics introduced in model system transformation,” IEEE Trans. Automat. Contr., vol. 45, pp. 572–575, Mar. 2000. , “Further remarks on additional dynamics in various model trans[2] formations of linear delay systems,” IEEE Trans. Automat. Contr., vol. 46, pp. 497–500, Mar. 2001. [3] J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations. New York: Springer-Verlag, 1993. [4] V. B. Kolmanosvkii and J.-P. Richard, “Stability of some linear systems with delays,” IEEE Trans. Automat. Contr., vol. 44, pp. 984–989, May 1999. [5] V. Kolmanosvkii and A. Myshkis, Applied Theory of Functional Differential Equations. Norwell, MA: Kluwer, 1992. [6] V. L. Kharitonov and D. Melchor-Aguilar, “On delay dependent stability conditions,” Syst. Control Lett., vol. 40, pp. 71–76, Mar. 2000. , “On delay dependent stability conditions for time varying sys[7] tems,” Syst. Control Lett., vol. 46, pp. 173–180, July 2002.
On the Synthesis of Safe Control Policies in Decentralized Control of Discrete-Event Systems Kurt Rohloff and Stéphane Lafortune
Abstract—State estimation and safe controller synthesis for a general form of decentralized control architecture for discrete-event systems is investigated. For this architecture, controllable events are assigned to be either “conjunctive” or “disjunctive.” A new state estimator that accounts for past local control actions when calculating the set of estimated system states is presented. The new state estimator is applied to a previous general decentralized control law. The new control method generates a controlled language at least as large as that generated by the original method if a safety condition is satisfied. An algorithm for generating locally maximal control policies for a given state estimate is also discussed. The algorithm allows an amount of “steering” of the controlled system through an event priority mechanism. Index Terms—Decentralized control, discrete-event systems, maximal behavior.
I. INTRODUCTION Decentralized control is commonly the most efficient or only method to control a system too large or complex to be controlled by a single centralized controller. Decentralized control systems are implemented as a number of local controllers that have authority over a subset of a system’s controllable events. As system operation evolves, the local controllers make observations of the system’s behavior and maintain local estimates of the system’s state. The state information is used to during the calculation of local control actions that are combined globally to determine allowed system behavior. Several decentralized control laws for discrete-event systems have been introduced in the literature. See, for example, [1], [2], [5]–[8], Manuscript received November 15, 2001; revised November 10, 2002 and December 18, 2002. Recommended by Associate Editor A. Giua. This work was supported in part by the National Science Foundation under Grant CCR0082784. The work of K. Rohloff was supported by a GAANN Fellowship for the 2000–2001 academic year. The authors are with the Department of Electrical Engineering and Computer Science, The University of Michigan, Ann Arbor, MI 48109-2122 USA (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TAC.2003.812812
[11], [13], and [14]. The solvability problem for safe and nonblocking decentralized controllers is undecidable [5], [12]. We restrict our attention to the safety problem and attempt to synthesize safe decentralized controllers that allow as much behavior as possible. This note builds upon Yoo and Lafortune’s [14] work on the control structures for the so-called “general decentralized architecture” where the set of controllable events is partitioned into a set of disjunctive events that follow a “fusion by union” rule and a set of conjunctive events that follow a “fusion by intersection” rule. Given the necessary brevity of this technical note we assume the reader has an understanding of the material presented in [14]. A more thorough introduction is also presented in this note’s companion technical report [9], which is available for download. Previously, many control schemes have used an inverse projection operation on the locally observed strings to estimate the current system state. However, this state estimation method does not make full use of a controller’s knowledge; controllers may have memory of past local control actions. We investigate the properties of a memory-based state estimator that has knowledge of past control actions. We use the estimator to construct control laws that locally allow more behavior than the gdec control law in [14] for a given partition of controllable events when the legal behavior may not be controllable and coobservable. The local state estimator we discuss makes no assumptions on the specific control actions of other controllers except to assume that other controllers always enable all events. Our first control law is called gmdec. For sufficient safety conditions, gmdec always allows at least as much behavior as gdec if the same partition of the controllable events is used for both control laws. The authors in [5] also propose the use of information on applied control patterns in processing partial observations but their approach is different from ours. We also develop a local greedy control policy that enables as many events as possible locally for a given state estimate. Ben Hadj-Alouane et al.[3] have done work with maximal control policies for centralized control systems and we apply their methods to general decentralized controllers. Our new control policy, the second variable lookahead policy for general decentralized memory-based control (VLP-GM2), enforces locally maximal control actions for a given state estimate. In Section II, the general decentralized control work in [14] is briefly reviewed. In Section III, the new state estimator is introduced. In the fourth section, we develop the new gmdec control policy. In Section V, the VLP-GM2 algorithm is presented and discussed. Some concluding remarks are made in Section VI. We use the notation employed in [14] that we briefly review in the next section. The proofs of all results in this note can be read in the companion technical report [9] along with more in-depth discussions of the topics presented here. II. PROBLEM FORMULATION AND PREVIOUS WORK We identify some of the terminology used in this note. The “state” of the system is the string of events that has been generated so far by the system. We consider a state estimate to be a set of possible states the system may be in at a given instance. A “state estimator” is a function that calculates a state estimate. Traditionally, many observers and controllers have used the inverse projection operation (P 01 (P (1))) as a state estimator. We restrict the possible and legal languages of the systems in this note to be prefix-closed because we are only concerned with safety issues. We also specify that the possible and legal languages be regular so that when the discussed state estimators are implemented, the local state estimations can be determined by the states of some finite observer automata.
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