Stability of Perturbed Delay Differential Equations and Stabilization of Nonlinear Cascade Systems.
In this paper  the effect of bounded input perturbations on the stability of nonlinear globally asymptotically stable delay  differential equations is analyzed. We investigate under which conditions global stability is preserved and if not, whether semiglobal stabilization is possible by controlling the size or shape of the perturbation. These results are used to study the stabilization of partially linear cascade systems with partial state feedback.
Introduction
The stability analysis of the series (cascade) interconnection of two stable
nonlinear systems described by ordinary differential equations is a classical
subject in system theory ([13], [14], [17]).
stable stable
Contrary to the linear case, the zero input global asymptotic stability of
each subsystem does not imply the zero input global asymptotic stability of
the interconnection. The output of the first subsystem acts as a transient
input disturbance which can be sufficient to destabilize the second subsys-
tem. In the ODE case, such destabilizing mechanisms are well understood.
they can be subtle but are almost invariably associated to a finite escape
time in the second subsystem (Some states blow up to infinity in a finite
time). The present paper explores similar instability mechanisms generated
by the series interconnection of nonlinear DDEs. In particular we consider
the situation where the destabilizing effect of the interconnection is delayed
and examine the difference with the ODE situation.
In the first part of the paper we study the effect of external (affine)
perturbations w on the stability of nonlinear time delay systems
whereby we assume that
globally asymptotically stable.
We consider perturbations which belong to both L 1 and L1 and
investigate the region in the space of initial conditions which give rise to
bounded solutions under various assumptions on the system and the perturbation

First, we consider global results: in the ODE-case, an obstruction is
formed by the fact that arbitrary small input perturbations can cause the
state to escape to infinity in a finite time, for instance when the interconnection
term \Psi(z) is nonlinear in z. This is studied extensively in the literature
in the context of stability of cascades, i.e. when the perturbation in (1)
is generated by another ODE, see e.g. [15] [13] and the references therein.
Even though delayed perturbations do not cause a finite escape time, we
explain a similar mechanism giving rise to unbounded solutions, caused by
nonlinear delayed interconnection terms.
In a second part, we allow situations whereby unbounded solutions are
inevitable and we investigate under which conditions trajectories can be
bounded semi-globally in the space of initial conditions, in case the perturbation
is parametrized, i.e. a). Hereby we let the parameter a
regulate the L 1 or L1 norm of the perturbation. We also consider the effect
of concentrating the perturbation in an arbitrary small time-interval.
As an application, we consider the special case whereby the perturbation
is generated by a globally asymptotically stable ODE. This allows us
to strengthen the previous results by the application of a generalization of
LaSalle's theorem [7] to the DDE-case: the convergence to zero of a solution
is implied by its boundedness. We also show that the origin of the
cascade is stable. We will concentrate on the following synthesis problem:
the stabilization of a partial linear cascade,!
with the SISO-system (A; B; C) controllable, using only partial state feed-back
laws of the form which allows to influence the shape and size
of the input 'perturbation' y to the nonlinear delay equation.
In the ODE-case this stabilization problem is extensively studied in the
literature, for instance in [16][1][15][8]. Without any structural assumption
on the interconnection term, achieving global stabilization is generally
not possible because the output of the linear subsystem, which acts as a
destabilizing disturbance to the nonlinear subsystem, can cause trajectories
to escape to infinity in a finite time. Therefore one concentrates on semi-global
stabilization, i.e. the problem of finding feedback laws making the
origin asymptotically stable with a domain of attraction containing any pre-set
compact region in the state space. An instructive way to do so is to drive
the 'perturbation' y fast to zero. However, a high-gain control, placing all
observable eigenvalues far into the LHP, will not necessarily result in large
stability regions, because of the fast peaking obstacle [15] [13]. Peaking is
a structural property of the -subsystem whereby achieving faster convergence
implies larger overshoots which can in turn destabilize the cascade.
Semi-global stability results are obtained when imposing structural assumptions
on the -subsystem (a nonpeaking system) or by imposing conditions
on the z-subsystem and the interconnection term \Psi: for example in [15] one
imposes a linear growth restriction on the interconnection term and requires
global exponential stability of the z-subsystem.
In this paper the classical cascade results are obtained and analysed in
the more general framework of bounded input perturbations and generalized
to the time-delay case.
Preliminaries The state of the delay equation (1) at time t can be described
as a vector z(t) 2 R n or as a function segment z t defined by
z
Therefore delay equations form a special class of functional differential equations
[3][5][6].
We assume that the right-hand side of (1) is continuous in all of its
arguments and Lipschitz in z and z(t \Gamma  ). Then a solution is uniquely
defined by specifying as initial condition a function segment z 0 whereby z 0 2
C([\Gamma; 0]; R n ), the Banach space of continuous bounded functions mapping
the delay-interval [\Gamma; 0] into R n and equipped with the supremum-norm
k:k s .
Sufficient conditions for stability of a functional differential equation are
provided by the theory of Lyapunov functionals [3] [6], a generalization of
the classical Lyapunov theory for ODEs: for functional differential equations
of the form
a mapping R is called a Lyapunov functional on a set G if V is
continuous on G and
0 on G. Here
V is the upper-right-hand derivative
of V along the solutions of (2), i.e.
The following theorem, taken from [3], provides sufficient conditions for
stability:
Theorem 1.1 Suppose z = 0 is a solution of (2) and
and there exist nonnegative functions a(r) and b(r) such that a(r) !1
as r !1 and
Then the zero solution is stable and every solution is bounded. If in addition,
b(r) is positive definite, then every solution approaches zero as t !1.
Instead of working with functionals, it is also possible to use classical Lyapunov
functions when relaxing the condition
This approach, leading
to the so-called Razumikhin-type theorems [6], is not considered in this paper

In most of the theorems of the paper, the condition of global asymptotic
stability for the unperturbed system (equation (1) with
cient. When the dimension of the system is higher than one, we sometimes
need precise information about the interaction of different components of
the state z(t). This information is captured in the Lyapunov functional, associated
with the unperturbed system. Therefore, when necessary, we will
restrict ourself to a specific class of functionals, with the following assumption

Assumption 1.2 The unperturbed system
delay-independent
globally asymptotically stable (i.e. GAS for all values of the delay) with a
Lyapunov-functional of the form
radially unbounded and such that the
conditions of theorem 1.1 (with b(r) positive definite) are satisfied.
This particular choice is motivated by the fact that such functionals are used
for a class of linear systems [3][6] and, since we assume delay-independent
stability, the time-derivative of the functional should not depend explicitly
on  . Choosing a delay-independent stable unperturbed system also allows
us to investigate whether the results obtained in the presence of perturbations
are still global in the delay. Note that in the ODE-case (3) reduces to
hardly forms any restriction because under mild conditions
its existence is guaranteed by converse theorems.
The perturbation j(t) 2 L p ([0; 1)) when 9M such that kjk
We assume j in (1) to be continuous and to belong to both L 1 and L1 .
When the perturbation is generated by an autonomous ODE,
b() with a and b continuous and locally Lipschitz, with
is globally asymptotically and locally exponentially stable (GAS and LES),
these assumptions are satisfied.
In the paper we show that when the unperturbed system is delay-independent
stable with a functional of the form (3) and the initial condition is bounded
(i.e. kz 0 k s  R ! 1), arbitrary small perturbations can cause unbounded
trajectories provided the delay is large enough. Therefore it is instructive to
treat the delay as the (n + 1)-th state variable when considering semi-global
results: with a parametrized perturbation j(t; a), we say for instance that
the trajectories of (1) can be bounded semi-globally in z and semi-globally
in the delay if for each compact
region\Omega ae R n , and 8
a positive number  a such that all initial conditions z
with z 0 (')
, give rise to bounded trajectories when
a   a.
belongs to class , if it is strictly increasing
and The symbol k:k is used for the Euclidean norm in R n and by
kx; yk we mean
.
2 The mechanism of destabilizing perturbations
In contrast to linear systems, small perturbations (in the L 1 or L1 sense) are
sufficient to destabilize nonlinear differential equations. In the ODE-case,
the nonlinear mechanism for instability is well known: small perturbations
suffice to make solutions escape to infinity in a finite time, for instance when
the interconnection term \Psi is nonlinear in z. This is illustrated with the
following example:
which can be solved analytically for z to give
ds
escapes to infinity in a finite time t e which is given by
log
This last expression shows that the escape time becomes smaller as the
initial conditions are chosen larger, and, as a consequence, however fast j(t)
would be driven to zero in the first equation of (4), z(0) could always be
chosen large enough for the solution to escape to infinity in finite time.
In the simple example (4), the perturbation is the output of a stable
linear system. Its initial condition j(0) dictates the L1-norm of the per-
turbation, while the parameter a controls its L 1 -norm. Making these norms
arbitrary small does not result in global stability. This is due to the nonlinear
growth of the interconnection term.
One may wonder whether the instability mechanism encountered in the
ODE situation (4) will persist in the DDE situation
ae
In contrast to (4), system (7) exhibits no finite escape time. This can be
proven by application of the method of steps, i.e. from the boundedness of
of
and thus of z('). Nevertheless the exponentially decaying input j still causes
unbounded solutions in (7): this particular system is easily seen to have an
exponential solution z e
at . The instability mechanism can be
explained by the superlinear divergence of the solutions of
Theorem 2.1
has solutions which diverge faster than any exponential function.
Proof 2.1 Take as initial condition a strictly positive solution segment z 0
over [\Gamma; 0] with z(0) ? 1. For t  0, the trajectory is monotonically
increasing. This means that in the interval [k;
z
The solution at point k; k  1 is bounded below by the sequence satisfying
z
which has limit +1. The ratio R
z
satisfies
R
and consequently (R tends to infinity. However for an exponential
function e at , R = e a and (R \Gamma 1)R is constant.
As a consequence, for the system (7), arbitrarily fast exponential decay of
cannot counter the blow-up caused by the nonlinearity in z(t \Gamma  ), and
hence the system is not globally asymptotically stable.
The instability mechanism illustrated by (4) and (7) can be avoided by
imposing suitable growth restrictions on the interconnection term \Psi. When
the unperturbed system is scalar, it is sufficient to restrict the interconnection
term to have linear growth in both of its arguments, i.e.
This linearity condition is by itself not sufficient however, if the unperturbed
system has dimension greater than one. In that case, the interaction of the
different components of the state z(t) can still cause "nonlinear" effects
leading to unbounded solutions. An illustration of this phenomenon is given
by the following system!
which was shown in [13] to have unbounded solutions, despite the linearity
of the interconnection. The instability is caused by the mutual interaction
between z 1 and z 2 when j 6= 0.
The following theorem, inspired by theorem 4.7 in [13], provides sufficient
conditions for bounded solutions. To prevent the instability mechanism due
to interacting states, conditions are put on the Lyapunov functional of the
unperturbed system.
Theorem 2.2 Assume that the system
satisfies Assumption 1.2 and that the interconnection term \Psi(z;
grows linearly in its arguments, i.e. satisfies (8). Furthermore if the perturbation
(ii) jj dk
dz jj jjzjj  ck(z),
then all trajectories of the perturbed system are bounded, for all values of
the time delay.
Condition (ii) is sometimes called a polynomial growth condition because it
is satisfied if k(z) is polynomial in z.
Proof 2.2 Along a trajectory z(t) we have:
dz
kzk
cff 1=fl
r
ff 2=fl'
cff 1=fli
cff 1=fli
For cannot escape to infinity because k(z(t\Gamma )) is bounded
(calculated from the initial condition) and the above estimate can be integrated
over the interval since the right hand side is linear in V and j 2 L 1 .
For t   we can use the estimate k(z(t \Gamma
Because this estimate for
V is increasing in both of its argument, an upper
bound for V (t) along the trajectory is described by
with as initial condition W (z  the method of steps, it is clear
that W cannot escape to infinity in a finite time. From
monotonically increasing. As a consequence, for t  2 , W (t)  W
and
and this estimate can be integrated leading to boundedness of lim t!1 sup V (t)
because Hence the trajectory z(t) is bounded.
Remark: When the interconnection term \Psi is undelayed, i.e. \Psi(z), condition
(i) in theorem 2.2 can be dropped [13].
3 Semi-global results for parametrized perturbation

Although no global results can be guaranteed in the absence of growth con-
ditions, the examples in the previous section suggest that one should be
able to bound the solutions semi-globally in the space of initial conditions
by decreasing the size of the perturbation. Therefore we assume that the
perturbation is parametrized,
We will consider two cases: a) parameter a controls the L 1 - or the L1-norm
of j and b) a regulates the shape of a perturbation with fixed L 1 -norm.
3.1 Modifying the L 1 and the L1 norm of the perturbation
We first assume that the L 1 -norm of j is parametrized. We have the following
result:
Theorem 3.1 Consider the system
and suppose that the unperturbed system is GAS with the Lyapunov functional
Assumption 1.2. If furthermore kj(t;
a !1, then the trajectories can be bounded semi-globally both in z and the
delay  , by increasing a.
Proof 3.1 Let   0 be fixed and denote
by\Omega the desired stability domain in
R n , i.e. such that all trajectories starting in z 0 with z 0 (')
2\Omega for ' 2 [\Gamma; 0]
are bounded. Let V c , sup z0
As long as V (t)  2V c , z(t) and belong to a compact set. Hence
When a ! 1, the increase of V tends to zero. As a consequence the assumption
is valid for 8t  0. Hence the trajectories with initial
condition
in\Omega are bounded.
Note that for a fixed
region\Omega increases with  and this influences
both the value M in the estimation of jk 0 (z)\Psi(z; z(t \Gamma )j and the
critical value
a of a in order to bound the trajectories. However when
belongs to a compact interval [0;   ], we can take a  sup 2[0;
hence bound the trajectories semi-globally in both the state and the delay.
The result given above is natural because for a given initial condition, a
certain amount of energy is needed for destabilization, expressed mathematically
by kjk 1 . However global stability in the state is not possible because
the required energy can become arbitrary small provided the initial condition
is large enough, see for instance example (4). Later we will discuss why
the trajectories cannot be bounded globally in the delay.
Now we consider the case whereby the L1-norm of the perturbation is
parametrized.
Theorem 3.2 Consider the system
Suppose that the unperturbed system is GAS with the Lyapunov functional
Assumption 1.2. If kj(t; a)k trajectories
of the perturbed system can be bounded semi-globally in both z and the
delay  .
Proof 3.2 As in the proof of Theorem 3.1, it is sufficient to prove semi-global
stability in the state for a fixed   0.
Let\Omega and V c be defined as in
Theorem 3.1.
\Omega , with ffl ? 0 small.
The time derivative of V satisfies
When z(t)
n\Omega ffl we have, since b is positive definite,
Mkjk1  \GammaN for some number N ? 0 provided kjk 1 is small enough.
Only when z(t)
the value of V can increase with the estimate
Mkjk1 .
Now we prove by contradiction that all trajectories with initial condition
in\Omega are bounded for small suppose that a solution starting
in\Omega
(with it has to cross the level set 2V c . Assume
that this happens for the first time at t   . Note that for small kjk 1 , t   is
large. During the interval [t  increase and decrease, but
increases, z(t)
2\Omega ffl and the increase \DeltaV is
limited: \DeltaV  Mkjk1  . When z(t) would be
outside\Omega ffl for a time-interval
Hence by reducing kj(t; a)k 1 we can make the time-interval \Deltat arbitrary
small. On the other hand (for large a),
dz
when z t is
inside\Omega 2 , because f and \Psi map bounded sets into bounded sets.
Hence with jt \Deltat we have L\Deltat. Because of (12)
we can increase a (reduce kj(t; a)k 1 ) such that L\Deltat  ffl and consequently
we have:
If ffl was chosen such
lies
inside\Omega , we have a contradiction because
this implies W (t   )  W c . Hence a trajectory can never cross the level set
2W c and is bounded.
The results of Theorems 3.1 and 3.2 are not global in the delay, although
the unperturbed system is delay-independent stable. Global results in the
delay are generally not possible: we give an example whereby it is impossible
to bound the trajectories semi-globally in the state and globally in the delay,
even if we make the size of the perturbation arbitrary small w.r.t. the L 1
and L1 -norm.
Example 3.1 Consider the following system:
z
The unperturbed system, i.e. (13) with delay-independent stable.
This is proven with the Lyapunov functional
Its time derivative
\Gammaz 2(z 1
z 2+1
z 2+1
is negative definite: when z 1 62 [1; 3], both terms are negative and in the
other case the second term is dominated, because it saturates in z 2 . From
this it follows that the conditions of Assumption 1.2 are satisfied.
With the perturbation
whereby increasing a leads to a reduction of both kjk 1 and kjk 1 , we can not
bound the trajectories semi-globally in the state and globally in  : for each
value of a we can find a bounded initial condition (upper bound independent
of a), leading to a diverging solution, provided  is large enough: the first
equation of (13) has a solution z \Gammaff is the real
solution of equation
boundedness in  of this solution over the interval [\Gamma; 0] (initial condition)
is guaranteed. Choose z 2
The above solution for z 1 satisfies:
when
ff log 5
3 ] and thus
A rather lengthy calculation shows that with z 2 and the perturbation
(14), the solution of (15) always escapes to infinity in a finite time t f (a).
Hence this also holds for the solution of the original system when the delay
is large enough such
log 5
This result is not in contradiction with the intuition that a perturbation
with small L 1 -norm can only cause escape in a finite time when the initial
condition is far away from the origin, as illustrated with example (4): in
the system (13) with driven away from the origin as long as
By increasing the delay in the first equation, we can keep
z 1 in this interval as long as desired. Thus the diverging transient of the
unperturbed system is used to drive the state away from the origin, far
enough to make the perturbation cause escape.
3.2 Modifying the shape of the perturbation
We assume that the shape of a perturbation with a fixed L 1 -norm can be
controlled and consider the influence of an energy concentration near the
origin. In the ODE case this does not allow to improve stability proper-
ties. This is illustrated with the first equation of example (4): instability
occurs when z(0)  1
R te \Gammas j(s)ds
and by concentrating the perburbation the
stability domain may even shrink, because the beneficial influence of damping
is reduced. In the DDE-case however, when the interconnection term is
linear in the undelayed argument, it behaves as linear during one delay interval
preventing escape. Moreover, starting from a compact region of initial
conditions, the reachable set after one delay interval can be bounded independently
of the shape of the perturbation (because of the fixed L 1 -norm).
After one delay interval we are in the situation treated in Theorem 3.1. This
is expressed in the following theorem. As in Theorem 2.2 the polynomial
growth condition prevents hidden nonlinearities due to interacting states.
Theorem 3.3 Consider
and suppose that the unperturbed system is GAS with the Lyapunov functional
Assumption 1.2. Let k(z)
satisfy the polynomial growth condition k dk
dz kkzk  ck(z). Assume that \Psi has
linear growth in z(t), kj(t; a)k independent of a and lim a!1
the trajectories of (16) can be bounded semi-globally in z
and for all
Proof 3.3 Consider a fixed
let\Omega be the desired stability
domain in R n and let R be such that z 0 (')
The interconnection term has linear growth in z, i.e. there exist two
class- functions fl 1 and fl 2 such that
The time-derivative of the Lyapunov function V satisfies
dz
jj(t; a)j
cV (z t )jj(t; a)j
During the interval [0;
to\Omega . Therefore, when kzk  R,
one can bound
by a factor c 2 independent of
a. Thus
and when a trajectory leaves the set fz : kzk  Rg at t   , because
t   jj(s;a)jds
for some constant M , independently of a. In the above expression, V
As a consequence, also k(z) and kz(t)k can be bounded, uniformly in
a. Hence at time  the state z  , i.e. z(t); t 2 [0;  ] belongs to
a compact
region\Omega 2 independently of a.
Now we can translate the original problem over one delay interval: at
time  the initial conditions belong to the bounded
region\Omega 2 and with t
jj(s; a)jds
Because of Theorem 3.1, we can increase a such that all solutions starting
in\Omega 2 are bounded.
Until now we assumed a fixed  . But because is compact, we can
take the largest threshold of a for bounded solutions over this interval.
Applications: stability of cascades and partial
state feedback
In this section we use the results given above to study the stability of cascade
systems:
whereby the subsystem
with a functional of
the form (3),
globally asymptotically and locally exponentially
stabilizable, h() is continuous and locally Lipschitz,
partial state feedback laws investigate in which situations the
equilibrum (z; can be made semiglobally asymptotically stable.
4.1 LaSalle's theorem
Because the 'perturbation' y in (17) is generated by a GAS ODE, we can
strengthen the boundedness results in the previous section to asymptotic
stability results, by applying a generalization to the time-delay case of the
classical LaSalle's theorem [7]:
Theorem 4.1 (LaSalle Invariance Principle)
Let\Omega be a positively invariant
set of an autonomous ODE, and suppose that every solution starting
in\Omega converges to a set E
2\Omega . Let M be the largest invariant set contained
in E, then every bounded solution starting
in\Omega converges to M as t !1.
This theorem is generalized to functional differential equations by Hale
[3]: with the following definition of an invariant set,
Definition 4.1 Let fT (t); t  0g be the solution semigroup associated to
the functional differential equation , then a set Q ae C is called invariant if
LaSalle's theorem can be generalized to:
Theorem 4.2 If V is a Lyapunov functional on G and x t is a bounded
solution that remains in G, then x t converges to the largest invariant set in
We will now outline its use in the context of stability of cascades. Suppose
that in the cascade (17)-(18), with a particular feedback law the
-subsystem is GAS. Hence there exists a Lyapunov function V () such that
we use this Lyapunov function as a
Lyapunov functional for the whole cascade, we can conclude from Theorem
4.2 that every solution that remains bounded converges to the largest invariant
set where and by the GAS of
this is
the equilibrium point (z; solutions are either unbounded or
converge to the origin.
By means of a local version of Theorem 3.1 one can show that the origin
of (17)-(18) is stable.
Hence the theorems of the previous section are strengthened to asymptotic
stability theorems. For example under the conditions of Theorem 3.1,
one can achieve semi-global asymptotic stability in both the state and the
delay.
4.2 Stabilization of partially linear cascades
In the rest of this paper we assume a SISO linear driving system (the -
with controllable and consider linear partial state feedback control
laws . From the previous sections it is clear that the input y of the
z-subsystem can act as a destabilizing disturbance. However, the control
can drive the output of the linear system fast to zero. We will investigate
under which conditions this is sufficient to stabilize the whole cascade. An
important issue in this context is the so-called fast peaking phenomenon
[15]. This is a structural property of the -system whereby imposing faster
convergence of the output to zero implies larger overshoots which can in
turn destabilize the cascade and may form an obstacle to both global and
semi-global stabilizability. We start with a short description of the peaking
phenomenon, based on [15], and then apply the results of the previous
section to the stabilization of the cascade system.
4.2.1 The peaking phenomenon
When in the system,
the pair (A; B) is controllable, one can always find state feedback laws
F  resulting in an exponential decay rate with exponent \Gammaa. Then the
output of the closed loop system satisfies
where fl depends on the choice of the feedback gain. We are interested
in the lowest achievable value of fl among different feedback laws and its
dependence upon a.
Denote by F(a) the collection of all stabilizing feedback laws
with the additional property that all observable 1 eigenvalues  of (C; A F ),
with A \Gammaa. For a given a and F 2 F(a), define
the smallest value of fl in (22) as
ky(t)ke at
where the supremum is taken over all t  0 and all initial conditions satisfying
k(0)k  1. Now denote by (a) = inf F2F(a)  F . The output of system
(21) is said to have peaking exponent s when there exists constants ff
such that
In [15] one places all eigenvalues to the left of the line
for large a. When the output is said to be nonpeaking.
The peaking exponent s is a structural property related to the zero-
dynamics: when the system has relative degree r, it can be transformed (in-
cluding a preliminary feedback transformation) into the normal form [4][1]:
ae
which can be interpreted as an integrator chain linearly coupled with the
zero-dynamics subsystem
. Using state feedback the output of an
integrator chain can be forced to zero rapidly without peaking [13]. Because
of the linear interconnection term, stability of the zero-dynamics subsystem
implies stability of the whole cascade. On the contrary, when the zero
dynamics are unstable, some amount of energy, expressed by
is needed for its stabilization and therefore the output must peak. More
precisely we have the following theorem, proven in the appendix.
Theorem 4.3 The peaking exponent s equals the number of eigenvalues in
the closed RHP of the zero-dynamics subsystem.
The definition of the peaking exponent (23) is based on an upper bound of
the exponentially weighted output, while its L 1 -norm is important in most
of the theorems of section 3. But because the overshoots related to peaking
occur in a fast time-scale ( at), there is a connection. For instance we have
the following theorem, based on a result of Braslavsky and Middleton [10]:
Theorem 4.4 When the output y of system (21) is peaking (s  1), ky(t)k 1
can not be reduced arbitrarily.
Proof 4.1 Denote by z 0 an unstable eigenvalue of the zero-dynamics of
(21). When a feedback is stabilizing the relation between y and
in the Laplace-domain is given by
with
. The first term vanishes at z 0 because the eigenvalues of the
zero dynamics appear as zeros in the corresponding transfer function H(s)
and since the feedback F is stabilizing, no unstable pole-zero cancellation
occurs at z 0 . Hence
4.2.2 Nonpeaking cascades
When the -subsystem is minimum-phase and thus nonpeaking, one can find
state feedback laws resulting in
and the L 1 -norm of the output can be made arbitrary small. So by Theorem
3.1, the cascade (20) can be stabilized semi-globally in the state and in the
delay.
4.2.3 Peaking cascades
When the -subsystem is nonminimum phase, the peaking phenomenon
forms an obstacle to semi-global stabilizability, because the L 1 -norm of the
output cannot be reduced (Theorem 4.4).
For ODE-cascades, we illustrate the peaking obstruction with the following
example:
Example 4.1 In the cascade,
the peaking exponent of the -subsystem is 1 (zero dynamics
cascade cannot be stabilized semi-globally since the explicit solution of the
first equation is given by
whereby
Hence the solution reaches infinity in a
For DDE-cascades, we consider two cases:
Case 1: Peaking exponent=1 We can apply theorem 3.3 and obtain
semi-global stabilizability in the state and in the delay, when the interconnection
term is linear in the undelayed argument: besides (25) the L 1 -norm
of y can also be bounded from above since there exists feedback laws
such that
and because of the fast time-scale property, the energy can be concentrated
since 8t ? 0:
jy(s)jds
ds ! 0 as a !1:
Case 2: Peaking exponent ? 1 In this case, we expect the L 1 -norm of
y to grow unbounded with a, as suggested by the following example:
Example 4.2 When  k is considered as the output of the integrator chain,
the peaking exponent is
be reduced arbitrarily by achieving a faster exponential decay rate. In Proposition
4.32 of [13], it is shown that with the feedback-law
k=1 a n\Gammak+1 q solutions of
there exists a constant c independent of a such that
hence the particular feedback gain is able to achieve an upper
bound which corresponds to definition (23), for each choice of the output
. It is also shown in [13] that with the same feedback and with as
initial condition  1
d such that s such that
As a consequence,
while the peaking exponent of output
With the two following examples, we show that when the energy of an
exponentially decaying input perturbation ( e \Gammaat ) grows unbounded with
a, an interconnection term which is linear in the undelayed argument, is
not sufficient to bound the solutions semi-globally in the state. Because it
is hard to deal in general with outputs generated by a linear system with
peaking exponent s ? 1, we use an artificial perturbation a s e \Gammaat , which has
both the fast time-scale property and the suitable growth-rate of the energy
(a
Example 4.3 The solutions of equation
can not be bounded semi-globally in z by increasing a, for any  ? 0, if the
'peaking exponent' s is larger than one. \Sigma
Proof: Equation (26) has an exponential solution z e (t),
z e
( a
a s
ff
e a
Consider the solution z(t) with initial condition z 0 j L ? 0 on [\Gamma; 0]. For
and consequently coincides on [o;  ] with
For large a, expression (27) describes a decreasing lower bound on [; 2 ],
since y(t) reaches its maximum in t   (a) with t   ! 0 as a ! 1. Thus
imposing and from this
one can argue 2 that z(t)  z e (t); t   . Thus the trajectory starting with
initial condition L on [\Gamma; 0] is unbounded when
Le L ff a
a
a s
ff
e 3a
When s ? 2, for each value of L, the solution is unstable for large a, thus the
attraction domain of the stable zero solution shrinks to zero. When
a solution starting from L ?
\Theta 3
ffff is unstable for large a.

Even when the interconnection term contains no terms in z(t), but only
delayed terms of z, semi-global results are still not possible in general, as
shown with the following example.
2 Intersection at t   would imply
Example 4.4 The solutions of the system
with otherwise, can not be
bounded semi-globally in z by increasing a, for any  ? 0, when the 'peaking
exponent' s is greater than one. \Sigma
Proof: When z  1, equation (28) reduces to:
which has the following explicit solution,
z l (t) = at
When the initial condition of (28) is L on [\Gamma; 0], during one delay-interval,
one can find an lower bound of the solution by integrating
with solution
z
When a is chosen such that b  1, the expression for z l (t) is valid for
t  0. When z u (2) ? z l (2 ), one can argue that for large a, z u (t) ?
z l (t); t 2 [; 2 ] and z u (t) describes a lower bound for the solution starting
in L for reaches is maximum in t   (a) with t   ! 0 as
1). Consequently, the trajectory with initial condition L on [\Gamma; 0],
is unbounded when

4.2.4 Zero dynamics with eigenvalues on the imaginary axis
The situation where the zero dynamics possess eigenvalues on the imaginary
axis but no eigenvalues in the open RHP deserves special attention. According
to Theorem 4.3, the system is peaking, that is, the L 1 norm of the output
cannot be reduced arbitrarily. However this energy can be 'spread out' over
a long time interval: it is indeed well known that a system with all its ei-
genalues in the closed LHP can be stabilized with a low-gain feedback, as
expressed by the following theorem, taken from [13]:
Theorem 4.5 If a system
stabilizable and the eigenvalues
of A 0 are in the closed left half plane, then it can be stabilized with a
low-gain control law which for large a satisfies:
a
The infinity-norm of such a low-gain control signal can be arbitrary reduced,
which results, by theorem 3.2, in satisfactory stabilizability results when
it also acts as an input disturbance of a nonlinear system. This suggests
not to force the output of (21) exponentially fast ( e \Gammaat ) to zero, which
results in peaking, but to drive it rapidly without peaking to the manifold
on which the dynamics are controlled by the low-gain control
action. Mathematically, with and a feedback transformation
the normal form of the -subsystem is transformed into
Using a high-gain feedback driving e(t) to zero without peaking, as proven
in [13], proposition 4.37, one can always force the output to satisfy the
constraint
a
with fl independent of a. A systematic treatment of such high-low gain
control laws can be found in [8].
For instance the system,
ae
is weakly minimum-phase (zero-dynamics
0). With the high-low gain
feedback the explicit solution of (30) for large a can be
approximated by:
a
a
a
Perturbations satisfying constraint (29) can be decomposed in signals
with vanishing L 1 and L1 -norm. This suggests the combination of theorems
3.1 and 3.2 to:
Theorem 4.6 Consider the interconnected system
Suppose that the z-subsystem is GAS with the Lyapunov functional V (z t )
satisfying Assumption 1.2 and the zeros of the -subsystem are in the closed
LHP. Then the interconnected system can be made semi-globally asymptotically
stable in both [z; ] and the delay, using only partial-state feedback.
Proof 4.2 As argumented in the beginning of section 4, the origin (z;
(0;
Let\Omega be the desired region of attraction in the (z; )-
space and choose R such that for all (z
R. Because of the
assumption on the -subsystem, there exist partial-state feedback laws such
that
with fl independent of a.
Consider the time-interval [0; 1]. Because
lim
a
one can show, as in the proof of theorem 3.1, that by taking a large, the
increase of V can be limited arbitrary. Hence for t  1, the trajectories can
be bounded inside a compact
region\Omega 2 . We can now translate the original
problem over one time-unit and since
sup
flR(e \Gammaat +a
we can, by theorem 3.2, increase a until the stability domain
Hence all trajectories starting
in\Omega are bounded and converge to the origin,
because of LaSalle's theorem.
Conclusions
We studied the effect of bounded input-perturbations on the stability of
nonlinear delay equations of the form (1).
Global stability results are generally not possible without structural assumptions
on the interconnection term because arbitrary small perturbations
can lead to unbounded trajectories, even when they are exponentially
decaying. In the ODE-case this is caused by the fact that superlinear
destabilizing terms can drive the state to infinity in a finite time. Superlinear
delayed terms cannot cause finite-escape but can still make trajectories
diverge faster than any exponential function.
In a second part we dropped most of the structural assumptions on
the unperturbed system and the interconnection term and considered semi-global
results when the size of the perturbation can be reduced arbitrary.
Here we assumed that the unperturbed system is delay-independent stable.
When the L 1 or the L1 norm of the perturbations is brought to zero,
trajectories can be bounded semi-globally in both the state and the delay.
By means of examples we explained mechanisms prohibiting global results in
the delay. We also considered the effect of concentrating a perturbation with
a fixed L 1 -norm near the origin. This leads to semi-global stabilizability in
the state and compact delay-intervals not containing the origin, when the
interconnection term is linear in its undelayed arguments.
As an application, we studied the stabilizability of partial linear cascades
using partial state feedback. When the interconnection term
is nonlinear, output peaking of the linear system can form an obstruction
to semi-global stabilizability because the L 1 -norm of the output cannot be
reduced by achieving a faster exponential decay rate. If we assume that
the interconnection term is linear in the undelayed argument and the peaking
exponent is one, we have semi-global stabilizability results, because the
of the output can be bounded from above while concentrating its
energy. Even with the above assumption on the interconnection term, higher
peaking exponents may form an obstruction. When the zeros of the linear
driving system are in the closed left half plane, we have satisfactory stability
results when using a high-low gain feedback, where the output of the
linear subsystem can be decomposed in two signals with vanishing L 1 and
L1 norm respectively.
The main contribution of this paper lies in placing the classical cascade
results in the more general framework of bounded input perturbations and
its generalization to a class of functional differential equations.

Acknowledgements

The authors thank W.Aernouts for fruitful discussions on the results presented
in the paper. This paper presents research results of the Belgian programme
on Interuniversity Poles of Attraction, initiated by the Belgian
State, Prime Minister's Office for Science, Technology and Culture (IUAP
P4/02). The scientific responsibility rests with its authors.



--R

Asymptotic stability of minimum phase non-linear systems
Sufficient conditions for stability and instability of autonomous functional-differential equations

Nonlinear control systems.
Introduction to the theory and application of functional differential equations
Stability of Functional Differential Equations
Stability theory of ordinary differential equations.

Robustness of nonlinear delay equations w.

How violent are fast controls?
Slow peaking and low-gain design for global stabilization of nonlinear systems
Constructive Nonlinear Control.
On the input-to-state stability properties
The peaking phenomenon and the global stabilization of nonlinear systems.
Tools for semiglobal stabilization by partial state and output feedback.
nonlinear small gain theorem for the analysis of control systems with saturation.

--TR
nonlinear control;delay equations;cascade systems