Journal ArticleDOI

# A Partial Solution of the Aizerman Problem for Second-Order Systems With Delays

07 Oct 2008--Vol. 53, Iss: 9, pp 2158-2160
TL;DR: It is proved that for retarded systems with a single delay the Aizerman conjecture is true, and for systems with multiple delays, a delay-dependent class of systems is found, for which the AIZerman conjectureIs true.
Abstract: This paper considers the Aizerman problem for second-order systems with delays. It is proved that for retarded systems with a single delay the Aizerman conjecture is true. For systems with multiple delays, a delay-dependent class of systems is found, for which the Aizerman conjecture is true. The proof is based on the Popov's frequency-domain criterion for absolute stability.

### I. INTRODUCTION

• The Aizerman problem has a very long history.
• 1) The real parts of all the roots of the polynomial P (s) are negative.
• The next task is to compare these stability conditions with those for nonlinear systems.

### IV. DELAY INVOLVING THE FIRST DERIVATIVE

• For systems with a single delay involving first derivative, the situation is somewhat more complicated.
• If the first of the inequalities (11) holds, then the inequalities (12) are necessary and sufficient conditions for stability of the zero solution of (5) .
• Once again, the authors can define the function f(x) by ( 14 Therefore, the Aizerman conjecture is true for all second-order systems with a single delay involving the first derivative.

### VI. CONCLUSION

• The results obtained can be summarized as follows.
• For retarded systems with a single delay, the Aizerman problem is solved completely-the conjecture is proved to be true.
• For systems with multiple delays, the frequency-domain inequality yields a delay-dependent stability criterion.
• The problem is still open for neutral systems.
• Another open question is the possibility of improving the result in Section V since the estimate used in the derivation is rather coarse.

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2158 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 53, NO. 9, OCTOBER 2008
A Partial Solution of the Aizerman Problem for
Second-Order Systems With Delays
Dmitry A. Altshuller
Abstract—This paper considers the Aizerman problem for second-order
systems with delays. It is proved that for retarded systems with a single
delay the Aizerman conjecture is true. For systems with multiple delays, a
delay-dependent class of systems is found, for which the Aizerman conjec-
ture is true. The proof is based on the Popov’s frequency-domain criterion
for absolute stability.
Index Terms—Absolute stability, Aizerman problem, delay systems, fre-
quency-domain methods.
I. I
NTRODUCTION
The Aizerman problem has a very long history. For systems without
delays, the matter is completely settled: the Aizerman conjecture is
true for second-order systems and, generally, false for systems of order
three and higher [1]. For systems with delays, the problem is unsolved
[2] except that Rasvan himself proved that the Aizerman conjecture is
true for ﬁrst-order systems with a single delay, independently of the
delay [3].
In this paper, we consider the second-order retarded system de-
scribed by the scalar delay-differential equation
x
(
t
)+
a
1
_
x
(
t
)+
'
(
x
)+
b
1
_
x
(
t
0
)+
bx
(
t
0
)=0
:
(1)
It is assumed that the function
'
(
x
)
, hereafter called the nonlinearity,
satisﬁes the sector condition
0
<
'
(
x
)
x
:
(2)
For the linear terms in (1), we can deﬁne a transfer function
W
(
s
)=
s
2
+
a
1
s
+(
b
1
s
+
b
)
e
0
s
0
1
:
(3)
In proving the results of this paper, we are going to rely extensively
on the Popov’s frequency-domain stability criterion: the zero solution
of (1) is globally asymptotically stable (GAS) if there exists a constant
, such that for all values of
!
, including inﬁnity, the following in-
equality holds:
0
1
+Re[(1+
i!
)
W
(
i!
)]
>
0
:
(4)
In addition to (1), we are also going to consider the linear equation
x
(
t
)+
a
1
_
x
(
t
)+
ax
+
b
1
_
x
(
t
0
)+
bx
(
t
0
)=0
:
(5)
The problem under investigation requires comparing the values of
, for which the zero solution of (1) is GAS, with the values of
a
, for
which such solution of (5) is GAS. The Aizerman conjecture states that
these values are the same. The question is if this conjecture is true.
Manuscript received January 30, 2008; revised April 8, 2008 and July 31,
2008. Current version published October 8, 2008. Recommended by Associate
Editor G. Feng.
D. A. Altshuller is with the Crane Aerospace and Electronics, Burbank, CA
91510 USA (e-mail: altshuller@ieee.org).
Color versions of one or more of the ﬁgures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identiﬁer 10.1109/TAC.2008.930193
The ﬁrst step in answering this question is to determine stability con-
ditions for (5). This will be carried out in Section II. Delay not involving
derivatives
(
b
1
=0)
is considered in Section III. Delay involving
the ﬁrst derivative is investigated in Section IV. Finally, in Section V,
we extend some of the results of Section III to systems with multiple
delays.
II. L
INEAR
SYSTEMS
For the system described by (5), we deﬁne the transfer function
W
L
(
s
)=
P
(
s
)+
Q
(
s
)
e
0
s
0
1
:
(6)
In this equation
P
(
s
)=
s
2
+
a
1
s
+
a
;
Q
(
s
)=
b
1
s
+
b:
(7)
It is well known that the zero solution of (5) is GAS if and only if
all the poles of
W
L
(
s
)
have negative real parts. An immediate conse-
quence of the results of Pontryagin [4] is that the following inequality
constitutes a necessary condition for this to be true:
j
b
j
<
j
a
j
:
(8)
If this inequality holds, then the necessary and sufﬁcient condition
for the zero solution of (5) to be GAS for all nonnegative values of the
delay
is that the following two conditions are met [5].
1) The real parts of all the roots of the polynomial
P
(
s
)
are negative.
This is true if and only if both coefﬁcients
a
1
and
a
are positive.
2) For any
!>
0
j
Q
(
i!
)
j
<
j
P
(
i!
)
j
:
(9)
These stability conditions can be reduced as follows. Both of the
following inequalities are the necessary conditions:
j
b
j
<a
;
j
b
1
j
<a
1
:
(10)
If (10) are satisﬁed, then the necessary and sufﬁcient condition for
stability is that that one of the following inequalities is satisﬁed:
j
b
j
<
a
2
1
0
b
2
1
2
;
a>
4
b
2
+
a
2
1
0
b
2
1
2
4(
a
2
1
0
b
2
1
)
:
(11)
These conditions can be represented graphically in the plane of the
parameters
a
and
j
b
j
shown in Fig. 1.
The stability region is the shaded area, bounded by the abscissa axis,
the diagonal
j
b
j
=
a
, and the curve given by the equation
a
=
4
b
2
+
a
2
1
0
b
2
1
2
4(
a
2
1
0
b
2
1
)
:
(12)
at the point of tangency
j
b
j
=(
a
2
1
0
b
2
1
)
=
2
.
This provides a complete answer to the delay-independent stability
problem for the second-order linear systems with a single delay (with
the exception of neutral systems). The next task is to compare these
stability conditions with those for nonlinear systems.
III. D
ELAY NOT INVOLVING DERIVATIVES
In case of a single delay not involving derivatives, the inequalities
(11) simplify to
j
b
j
<
a
2
1
2
;
a>
4
b
2
+
a
4
1
4
a
2
1
:
(13)
0018-9286/$25.00 © 2008 IEEE Authorized licensed use limited to: Dmitry Altshuller. Downloaded on October 7, 2008 at 17:50 from IEEE Xplore. Restrictions apply. “© 2008 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works for resale or 1 redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.” IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 53, NO. 9, OCTOBER 2008 2159 Fig. 1. Stability boundaries for the linear system (5). If the rst of these inequalities holds, then a> j b j is a necessary and sufcient condition for stability of the zero solution of (5). Therefore, the nonlinearity ' ( x ) must lie in the sector ( j b j ; + 1 ) and it makes sense to dene the function f ( x ) by f ( x )= ' ( x ) 0j b j x: (14) The new nonlinearity f ( x ) satises the sector condition (2) with 0 1 =0 and the transfer function of the linear terms becomes W ( s )= s 2 + a 1 s + j b j + be 0 s 0 1 : (15) Expansion of (4) shows that it holds for all > 0 if j b j +( a 1 0 1) ! 2 0 b! sin ! + b cos ! 0 : (16) Using a well-known trigonometric identity, this inequality can be rewritten in the form j b j +( a 1 0 1) ! 2 + ( b! ) 2 + b 2 sin( ! + ) 0 : (17) It is easy to show that as long as the rst of the inequalities (13) holds, (17) holds for all values of ! if the constant is chosen to satisfy a 1 j b j 0 a 2 1 0 2 j b j b 2 << a 1 j b j + a 2 1 0 2 j b j b 2 : (18) This shows that the Aizerman conjecture is true in this case. Of course, if b =0 , we have a system without delays, and the Aiz- erman conjecture is known to be true. Let us turn our attention to the case when a> 4 b 2 + a 4 1 4 a 2 1 : (19) Instead of (14), we now have f ( x )= ' ( x ) 0 4 b 2 + a 2 1 4 a 2 1 x: (20) Similarly, (15) is replaced with W ( s )= s 2 + a 1 s + 4 b 2 + a 4 1 4 a 2 1 + be 0 s 0 1 : (21) Expansion of (4) shows that it holds for all if a 4 1 +16 b 2 +4 a 2 1 ( a 1 0 1) ! 2 +4 a 2 1 b (cos ! 0 ! sin ! ) 0 : (22) Following the same procedure as in the previous case, it can be shown that this inequality holds for all values of ! if is chosen to satisfy both of the following inequalities: 16 a 1 + 128 a 2 1 + a 6 1 b 4 + 24 a 2 1 b 2 + a 3 1 b 2 > 4 (23) 128 a 2 1 + a 6 1 b 4 + 24 a 2 1 b 2 +4 > 16 a 1 + a 3 1 b 2 : (24) Therefore, in this case the Aizerman conjecture is true as well. This proves that it is true for all second-order systems with a single delay not involving derivatives. IV. D ELAY INVOLVING THE FIRST DERIVATIVE For systems with a single delay involving rst derivative, the sit- uation is somewhat more complicated. If the rst of the inequalities (11) holds, then the inequalities (12) are necessary and sufcient con- ditions for stability of the zero solution of (5). Once again, we can de- ne the function f ( x ) by (14). The transfer function of the linear terms becomes W ( s )= s 2 + a 1 s + j b j +( b + b 1 s ) e 0 s 0 1 : (25) Expansion of (4) shows that it holds for all > 0 if j b j +( a 1 0 1) ! 2 +( b 1 0 b ) ! sin ! + b + b 1 ! 2 cos ! 0 : (26) It can be shown by the same process as in the previous section that as long as the inequalities (10) and the rst of the inequalities (11) hold, (26) holds for all values of ! if the constant is chosen to satisfy On the other hand, if b> 0 , we choose to satisfy a 1 j b j 0 a 2 1 0 2 j b j0 b 2 1 b 2 << a 1 j b j + a 2 1 0 2 j b j0 b 2 1 b 2 : (27) If b =0 , we replace (26) with ( a 1 0 1) ! 2 + b 1 ! sin ! + b 1 ! 2 cos ! 0 : (28) Following the same procedure as before, we nd that (28) holds for all values of ! if we choose to satisfy 1+ b 2 1 0 a 2 1 > 0 : (29) This proves the Aizerman conjecture for the case of a single delay involving the rst derivative if the rst of the inequalities (11) holds. If the second of the inequalities (11) holds, then we dene B = 4 b 2 + a 2 1 0 b 2 1 2 4( a 2 1 0 b 2 1 ) ; (30) Instead of (26), we have B +( a 1 0 1) ! 2 +( b 1 0 b ) ! sin ! + b + b 1 ! 2 cos ! 0 : (31) It is easy to see that B b . Therefore, if (26) holds, then (31) holds as well. Authorized licensed use limited to: Dmitry Altshuller. Downloaded on October 7, 2008 at 17:50 from IEEE Xplore. Restrictions apply. 2160 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 53, NO. 9, OCTOBER 2008 Therefore, the Aizerman conjecture is true for all second-order sys- tems with a single delay involving the rst derivative. V. M ULTIPLE DELAYS Let us extend some of our results to systems with multiple delays. We are only going to consider the case of delays not involving derivatives x ( t )+ a 1 _ x ( t )+ ' ( x )+ m j =1 b j x ( t 0 j )=0 : (32) We are not going to investigate in depth the stability of the corre- sponding linear system except to note that the necessary condition (8) now becomes [5] m j =1 j b j j < j a j : (33) Therefore, instead of (14), we have f ( x )= ' ( x ) 0 x m j =1 j b j j : (34) Instead of (16), we now obtain m j =1 j b j j +( a 1 0 1) ! 2 0 ! m j =1 b j sin ! j + m j =1 b j cos ! j 0 : (35) We can take advantage of the easily veried estimate sin   2 , valid for > 0 , and state that (35) holds for all values of ! if there exists > 0 such that the following inequality holds for all values of ! : m j =1 ( j b j j + b j cos ! j ) + a 1 0 m j =1 j b j j j 0 1 ! 2 0 : (36) This can be assured by choosing > 0 to satisfy a 1 0 m j =1 j b j j j > 1 : (37) Clearly, this can be done if and only if the following inequality is true a 1 > m j =1 j b j j j : (38) Thus, in this case we have identied a delay-dependent class of sys- tems, for which the Aizerman conjecture is true. VI. C ONCLUSION The results obtained can be summarized as follows. For re- tarded systems with a single delay, the Aizerman problem is solved completelythe conjecture is proved to be true. For systems with multiple delays, the frequency-domain inequality yields a delay-de- pendent stability criterion. The problem is still open for neutral systems. Another open question is the possibility of improving the result in Section V since the estimate used in the derivation is rather coarse. Indeed, if we set m =1 , the resulting stability criterion is much weaker than the one obtained in Section III. REFERENCES [1] V. A. Pliss, Certain Problems in Theory of Stability of Motion in the Whole. Washington, DC: National Aeronautics and Space Adminis- tration, 1958. [2] V. Rasvan, Problem 6.6: delay-independent and delay-dependent Aiz- erman problem, in Unsolved Problems in Mathematical Systems and Control Theory, V. D. Blondel and A. Megretski, Eds. Princeton, NJ: Princeton Univ. Press, 2004, pp. 212220. [3] V. Rasvan, Absolute Stability of Automatic Control Systems With Time Delay (in Russian). Moscow, Russia: Nauka Publishing House, 1975. [4] L. S. Pontryagin, On zeros of some elementary transcendental func- tions, in L. S. Pontryagin. Selected Scholarly Works (in Russian), R. V. Gamkrelidze, Ed. Moscow: Nauka Publishing House, 1988, vol. 2, pp. 3550. [5] L. E. Elsgolts and S. B. Norkin, Introduction to the Theory and Ap- plications of Differential Equations With Deviating Arguments.New York: Academic, 1973, p. 176. Reversibility and Poincaré Recurrence in Linear Dynamical Systems Sergey G. Nersesov and Wassim M. Haddad AbstractIn this paper, we study the Poincaré recurrence phenomenon for linear dynamical systems, that is, linear systems whose trajectories re- turn inﬁnitely often to neighborhoods of their initial condition. Speciﬁcally, we provide several equivalent notions of Poincaré recurrence and review sufﬁcient conditions for nonlinear dynamical systems that ensure that the system exhibits Poincaré recurrence. Furthermore, we establish necessary and sufﬁcient conditions for Poincaré recurrence in linear dynamical sys- tems. In addition, we show that in the case of linear systems the absence of volume-preservation is equivalent to the absence of Poincaré recurrence implying irreversibility of a dynamical system. Finally, we introduce the notion of output reversibility and show that in the case of linear systems, Poincaré recurrence is a sufﬁcient condition for output reversibility. Index TermsIrreversibility, Lagrangian and Hamiltonian systems, output reversibility, Poincaré recurrence, volume-preserving ﬂows. I. I NTRODUCTION The Poincaré recurrence theorem states that every nite-dimen- sional, isolated dynamical system with volume-preserving ow and bounded trajectories will return arbitrarily close to its initial state innitely many times. This theorem was proven by Poincaré [1] and further studied by Birkhoff [2] for Lagrangian systems and Halmos [3] for ergodic systems. Poincaré recurrence has been the main source for the long and erce debate between the microscopic and macroscopic points of view of thermodynamics [4]. In thermodynamic models predicated on statistical mechanics, an isolated dynamical system will return arbitrarily close to its initial state of molecular positions and velocities innitely often. If the system entropy is determined by the Manuscript received November 14, 2006; revised February 11, 2008. Current version published October 8, 2008. This work was supported in part by the Air Force Ofce of Scientic Research under Grant FA9550-06-1-0240. Recom- mended by Associate Editor M. A. Demetriou. S. G. Nersesov is with the Department of Mechanical Engineering, Villanova University, Villanova, PA 19085 USA (e-mail: sergey.nersesov@vil- lanova.edu). W. M. Haddad is with the School of Aerospace Engineering, Georgia Insti- tute of Technology, Atlanta, GA 30332 USA (e-mail: wm.haddad@aerospace. gatech.edu). Digital Object Identier 10.1109/TAC.2008.930194 0018-9286/$25.00 © 2008 IEEE
Authorized licensed use limited to: Dmitry Altshuller. Downloaded on October 7, 2008 at 17:50 from IEEE Xplore. Restrictions apply.
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[...]

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