A SMALL-GAIN THEOREM FOR A WIDE CLASS OF FEEDBACK
SYSTEMS WITH CONTROL APPLICATIONS
Iasson Karafyllis* and Zhong-Ping Jiang1**
*
**
Department of Environmental Engineering, Technical University of Crete,
73100, Chania, Greece
email: [email protected]
Department of Electrical and Computer Engineering, Polytechnic University,
Six Metrotech Center, Brooklyn, NY 11201, U.S.A.
email: [email protected]
Abstract
A Small-Gain Theorem, which can be applied to a wide class of systems that
includes systems satisfying the weak semigroup property, is presented in the
present work. The result generalizes all existing results in the literature and
exploits notions of weighted, uniform and non-uniform Input-to-Output Stability
(IOS) property. Applications to partial state feedback stabilization problems
with sampled-data feedback applied with zero order hold and positive sampling
rate, are also presented.
Keywords: Input-to-Output Stability, Control Systems, Small-Gain Theorem.
1. Introduction
A common feature of stability analysis for complex interconnected systems is the application of small-gain results.
Small-gain theorems for continuous-time finite-dimensional systems expressed in terms of “nonlinear gain functions”
have a long history (see [10,27,44,45] and the references therein). A nonlinear small-gain result was presented in [10],
which allowed numerous applications to feedback stabilization problems. The methodology presented in [10] was
followed by many researchers (see [11,12,18,20,40,43]). A common characteristic of current research on nonlinear
small-gain results in Mathematical Systems Theory is the use of the notion of uniform Input-to-State Stability (ISS),
introduced by E.D. Sontag in [37] for systems described by Ordinary Differential Equations, or the notion of uniform
Input-to-Output Stability (IOS), introduced by E.D. Sontag and Y. Wang in [39] (also see [10]) and extended in [8].
Small-Gain theorems for discrete-time systems can be found in [13,14,15].
Extensions of small-gain results were presented recently in the literature. A non-uniform in time small-gain
theorem for continuous-time finite-dimensional systems was presented in [18]. Moreover, in [20,40] small-gain
results for wide classes of systems were provided. The classes of systems considered in [20,40] satisfy the classical
“semigroup property” (see [20,21,22,38]). Small-gain results for hybrid systems satisfying the classical “semigroup
property” were recently presented in [26].
An important feature of certain hybrid systems is that they do not satisfy the classical “semigroup property”: for
example, the solution x(t ) of a system Σ with initial condition x(t 0 ) = x 0 does not coincide (in general) for
t ≥ t1 > t 0 with the solution ~
x (t ) of Σ with initial condition ~
x (t1 ) = x(t1 ) . Such systems arise when sampled-data
feedback laws are applied to finite-dimensional control systems or when numerical discretization schemes are applied
for the numerical solution of a system of ordinary differential equations. However, from a mathematical point of
view, these structures cannot be considered as “systems” in the sense given in [16,20,38]. This feature has important
consequences, since the researcher cannot use the tools developed for systems theory and mathematical control
theory. In [21,22] the notion of a system was relaxed so that the “semigroup property” does not hold in a strict sense.
1
Corresponding author
1
Moreover, the modification introduced allows the results obtained in [20] to hold. Thus we are in a position to
develop a complete stability theory, which covers the systems that satisfy the so-called “weak semigroup property”.
The purpose of the present work is to present a small-gain result (Theorem 3.1 and Corollary 3.4), which
∗ can be applied to a very general class of systems (including systems that do not satisfy the classical “semigroup”
property).
∗ unifies all existing results, which make use of uniform or non-uniform and weighted notions of Input-to-State
Stability (ISS) or Input-to-Output Stability (IOS).
∗ can be used directly for the solution of sampled-data feedback stabilization problems or problems of numerical
stability of discretization schemes.
∗ can be applied to uncertain time-varying systems with vanishing or non-vanishing perturbations.
We believe that the main result of the present work is a valuable tool for establishing stability and will be used
frequently in future research. However, we would like to emphasize the theoretical significance of our main result: it
is a method for establishing qualitative properties expressed in a very general framework that unifies works from
various fields as well as different stability notions. The results presented in the paper can be extended without much
difficulty to the case of local stability notions.
The contents of this paper are presented as follows. In Section 2 we provide the definitions of the notions used and
several examples of systems that have the “Boundedness-Implies-Continuation” (BIC) property. In Section 3 the
main result is stated and proved. In Section 4, it is shown how the main result of the present work can be applied to an
ISS stabilization problem of a certain class of systems with partial-state sampled-data feedback. It should be
emphasized that sampled-data control systems cannot be handled with Small-Gain results that have appeared so far in
the literature, since sampled-data control systems do not satisfy the classical semigroup property. Finally, Section 5
contains the conclusions of the paper. The proofs of some basic results are given in the Appendix.
Notations Throughout this paper we adopt the following notations:
∗ We denote by K + the class of positive, continuous functions defined on ℜ + . We say that a function
ρ : ℜ + → ℜ + is of class N , if ρ is continuous, non-decreasing with ρ (0) = 0 . By K we denote the set of
positive definite, increasing and continuous functions. We say that a positive definite, increasing and continuous
function ρ : ℜ + → ℜ + is of class K ∞ if lim ρ ( s ) = +∞ . By KL we denote the set of all continuous functions
s → +∞
σ = σ ( s, t ) : ℜ + × ℜ + → ℜ + with the properties: (i) for each t ≥ 0 the mapping σ ( ⋅ , t ) is of class K ; (ii) for each
s ≥ 0 , the mapping σ ( s, ⋅ ) is non-increasing with lim σ ( s, t ) = 0 .
t → +∞
∗ By
X
, we denote the norm of the normed linear space X . By
{
U ⊆ X with 0 ∈ U . By BU [0, r ] := u ∈ U ; u
X
≤r
we denote the Euclidean norm of ℜ n . Let
} we denote the intersection of U ⊆ X
with the closed
sphere of radius r ≥ 0 , centered at 0 ∈ U .
∗ Let a set U be a subset of a normed linear space U , with 0 ∈ U . By M (U ) we denote the set of all locally
bounded functions u : ℜ + → U . By u 0 we denote the identically zero input, i.e., the input that satisfies
∞
(ℜ+ ;U ) denotes the space of measurable, locally bounded
u 0 (t ) = 0 ∈ U for all t ≥ 0 . If U ⊆ ℜ n then Lloc
functions u : ℜ + → U .
The following convention will be adopted throughout the paper: the Cartesian product of two normed linear spaces
C := X × Y will be considered to be endowed with the norm ( x, y )
C
:=
x
2
X
+ y
2
Y
, unless stated otherwise.
Furthermore, our results can be extended to the case of measurable and locally essentially bounded inputs (where the
“sup” operator is to be understood as “essential supermum”).
2
2. Input-to-Output Stability in a System-Theoretic Framework
In this section we first give the notion of a control system with outputs. We emphasize that we consider control
systems which do not necessarily satisfy the classical “semigroup property” (see [16,20,38]).
Definition 2.1: A control system Σ := (X, Y , M U , M D , φ , π , H ) with outputs consists of
(i) a set U (control set) which is a subset of a normed linear space U with 0 ∈ U and a set M U ⊆ M ( U )
(allowable control inputs) which contains at least the identically zero input u 0 ,
(ii) a set D (disturbance set) and a set M D ⊆ M ( D) , which is called the “set of allowable disturbances”,
(iii) a pair of normed linear spaces X, Y called the “state space” and the “output space”, respectively,
(iv) a continuous map H : ℜ + × X × U → Y that maps bounded sets of ℜ + × X × U into bounded sets of Y , called
the “output map”,
(v) a set-valued map ℜ + × X × M U × M D ∋ (t 0 , x 0 , u , d ) → π (t 0 , x 0 , u , d ) ⊆ [t 0 ,+∞) , with t 0 ∈ π (t 0 , x 0 , u , d ) for all
(t 0 , x 0 , u, d ) ∈ ℜ + × X × M U × M D , called the set of “sampling times”
(vi) and the map φ : Aφ → X where Aφ ⊆ ℜ + × ℜ + × X × M U × M D , called the “transition map”, which has the
following properties:
1) Existence: For each (t 0 , x 0 , u, d ) ∈ ℜ + × X × M U × M D , there exists t > t 0 such that [t 0 , t ] × (t 0 , x 0 , u , d ) ⊆ Aφ .
2) Identity Property: For each (t 0 , x 0 , u , d ) ∈ ℜ + × X × M U × M D , it holds that φ (t 0 , t 0 , x 0 , u, d ) = x 0 .
3) Causality: For each (t , t 0 , x 0 , u, d ) ∈ Aφ with
~
(u~ (τ ), d (τ )) = (u (τ ), d (τ ))
for
all
τ ∈ [t 0 , t ] ,
~
~
φ (t , t 0 , x 0 , u , d ) = φ (t , t 0 , x 0 , u , d ) .
t > t0
it
and
holds
for
each
that
~
(u~, d ) ∈ M U × M D
~
(t , t 0 , x 0 , u~, d ) ∈ Aφ
with
with
4) Weak Semigroup Property: There exists a constant r > 0 , such that for each t ≥ t 0 with (t , t 0 , x 0 , u , d ) ∈ Aφ :
(a) (τ , t 0 , x 0 , u , d ) ∈ Aφ for all τ ∈ [t 0 , t ] ,
(b) φ (t , τ , φ (τ , t 0 , x0 , u, d ), u, d ) = φ (t , t 0 , x0 , u, d ) for all τ ∈ [t 0 , t ] ∩ π (t 0 , x 0 , u , d ) ,
(c) if (t + r , t 0 , x 0 , u, d ) ∈ Aφ , then it holds that π (t 0 , x 0 , u , d ) ∩ [t , t + r ] ≠ ∅ .
(d) for all τ ∈ π (t 0 , x 0 , u , d ) with (τ , t 0 , x0 , u, d ) ∈ Aφ we have π (τ , φ (τ , t 0 , x 0 , u , d ), u , d ) = π (t 0 , x 0 , u , d ) ∩ [τ ,+∞ ) .
In order to develop stability notions for a control system with outputs we need to clarify the notions of an
equilibrium point as well as certain other important notions and classes of systems that characterize the dynamic
behavior of the system (see [20,21]).
Definition 2.2: Let T > 0 . A control system Σ := (X, Y , M U , M D , φ , π , H ) with outputs is called T-periodic, if:
a) H (t + T , x, u ) = H (t , x, u ) for all (t , x, u ) ∈ ℜ + × X × U ,
b) for every (u, d ) ∈ M U × M D and integer k there exist inputs PkT u ∈ M U , PkT d ∈ M D with (PkT u )(t ) = u (t + kT )
and (PkT d )(t ) = d (t + kT ) for all t + kT ≥ 0 ,
c) for each (t , t 0 , x 0 , u , d ) ∈ Aφ with t ≥ t 0 and for each integer k with t 0 − kT ≥ 0 it follows that
(t − kT , t 0 − kT , x 0 , PkT u, PkT d ) ∈ Aφ
and
π (t 0 − kT , x 0 , PkT u , PkT d ) =
φ (t , t 0 , x 0 , u , d ) = φ (t − kT , t 0 − kT , x 0 , PkT u, PkT d ) .
3
∪
{τ − kT }
τ ∈π ( t0 , x0 ,u , d )
with
Definition 2.3: A control system Σ := (X, Y , M U , M D , φ , π , H ) with outputs is called time-invariant or
autonomous, if:
a) the output map is independent of t , i.e., H (t , x, u ) ≡ H ( x, u ) ,
b) for every (θ , u, d ) ∈ ℜ × M U × M D there exist inputs Pθ u ∈ M U , Pθ d ∈ M D with (Pθ u )(t ) = u (t + θ ) and
(Pθ d )(t ) = d (t + θ ) for all t + θ ≥ 0 ,
c) for each (t , t 0 , x 0 , u , d ) ∈ Aφ with t ≥ t 0 and for each θ ∈ (−∞, t 0 ] it follows that (t − θ , t 0 − θ , x 0 , Pθ u , Pθ d ) ∈ Aφ
and π (t 0 − θ , x 0 , Pθ u, Pθ d ) =
{τ − θ } with φ (t , t 0 , x 0 , u , d ) = φ (t − θ , t 0 − θ , x 0 , Pθ u , Pθ d ) .
∪
τ ∈π (t 0 , x0 ,u , d )
Definition 2.4: Consider a control system Σ := (X, Y , M U , M D , φ , π , H ) with outputs. We say that system Σ
(i)
has
the
“Boundedness-Implies-Continuation”
(BIC)
property
if
for
each
(t 0 , x 0 , u , d ) ∈ ℜ + × X × M U × M D , there exists a maximal existence time, i.e., there exists
t max := t max (t 0 , x 0 , u , d ) ∈ (t 0 ,+∞] , such that Aφ =
∪+
[t 0 , t max ) × {(t 0 , x 0 , u, d )} . In
(t0 , x0 ,u , d )∈ℜ ×X × M U × M D
addition, if t max < +∞ then for every M > 0 there exists t ∈ [t 0 , t max ) with φ (t , t 0 , x 0 , u , d )
X
>M.
is robustly forward complete (RFC) from the input u ∈ M U if it has the BIC property and for every
r ≥ 0 , T ≥ 0 , it holds that
(ii)
{
sup φ (t 0 + s, t 0 , x 0 , u , d )
X
; u ∈ M ( BU [0, r ]) ∩ M U , s ∈ [0, T ] , x 0
X
}
≤ r , t 0 ∈ [0, T ] , d ∈ M D < +∞
Definition 2.5: Consider a control system Σ := (X, Y , M U , M D , φ , π , H ) and suppose that H (t ,0,0) = 0 for all t ≥ 0 .
We say that 0 ∈ X is a robust equilibrium point from the input u ∈ M U for Σ if
(i)
for every (t , t 0 , d ) ∈ ℜ + × ℜ + × M D with t ≥ t 0 it holds that φ (t , t 0 ,0, u 0 , d ) = 0 .
(ii)
for every ε > 0 , T , h ∈ ℜ + there exists δ := δ (ε , T , h) > 0 such that for all (t 0 , x, u ) ∈ [0, T ] × X × M U ,
τ ∈ [t 0 , t 0 + h] with x
X
+ sup u (t )
t ≥0
{
sup φ (τ , t 0 , x, u , d )
X
U
< δ it holds that (τ , t 0 , x, u , d ) ∈ Aφ for all d ∈ M D and
}
; d ∈ M D , τ ∈ [t 0 , t 0 + h] , t 0 ∈ [0, T ] < ε
Remark 2.6: Consider a control system Σ := (X, Y , M U , M D , φ , π , H ) with the BIC property. It follows that Σ
satisfies the (classical) semigroup property (see [20,38]) if the weak semigroup property holds with
π (t 0 , x 0 , u , d ) = [t 0 , t max ) , where t max ∈ (t 0 ,+∞] is the maximal existence time of the transition map for Σ that
corresponds to (t 0 , x 0 , u, d ) ∈ ℜ + × X × M U × M D , i.e.,
“for each t ∈ [t 0 , t max ) it holds that φ (t , τ , φ (τ , t 0 , x 0 , u, d ), u, d ) = φ (t , t 0 , x 0 , u, d ) for all τ ∈ [t 0 , t ] ”
(Classical Semigroup Property)
The following example shows the difference between the classical semigroup property and the weak semigroup
property for simple systems.
Example 2.7: Consider the following system:
x& (t ) = − x (τ i ) , t ∈ [τ i , τ i +1 )
τ i +1 = τ i + 1
x(t ) ∈ ℜ
4
(2.1)
with initial condition x(t 0 ) = x 0 ∈ ℜ and τ 0 = t 0 ≥ 0 . Such systems will be characterized as hybrid systems with
sampling partition generated by the system (see Example 2.11) and they satisfy the BIC property. In this case we can
determine analytically the transition map for all t ≥ t 0 ( u , d in this example are irrelevant):
⎧(1 − t + t 0 ) x 0 , for t ∈ [t 0 , t 0 + 1)
for t ≥ t 0 + 1
0 ,
⎩
φ (t , t 0 , x 0 ) = ⎨
It is clear that the state space is ℜ and that the classical semigroup property does not hold for this system. On the
other hand the weak semigroup property holds for this system with π (t 0 , x 0 ) = {t 0 , t 0 + 1, t 0 + 2,...} . Notice that the
set of sampling times (the sampling partition) π (t 0 , x 0 ) = {t 0 , t 0 + 1, t 0 + 2,...} is generated by the system itself and
depends on the initial condition. Furthermore, according to Definition 2.3, system (2.1) is autonomous.
Next, consider the following system:
x& (t ) = − x(τ i ) , t ∈ [τ i , τ i +1 )
x(t ) ∈ ℜ , π = {τ i }i∞= 0 = {0,1,2,...}
(2.2)
Such systems will be characterized as hybrid systems with impulses at fixed times (see Example 2.12) and they
satisfy the BIC property. Notice that if the initial time t 0 is not a member of the partition π = {τ i }i∞=0 = {0,1,2,...} ,
then it is not possible to determine the solution of (2.2) based only on the initial condition x(t 0 ) = x 0 ∈ ℜ and the
transition map is not well-defined. In order to be able to determine the solution of (2.2), we need to know
( x(t 0 ), x ([t 0 ]) = x 0 = ( x1,0 , x 2,0 ) ∈ ℜ 2 (where [t 0 ] denotes the integer part of t 0 ). Indeed, we have:
x1,0 − (t − t 0 ) x 2,0 ,
t 0 ≤ t < [t 0 ] + 1
⎧
⎪
x(t ) = ⎨(2 − t + [t 0 ])(x1,0 − ([t 0 ] + 1 − t 0 ) x 2,0 ) , [t 0 ] + 1 ≤ t < [t 0 ] + 2 , if t 0 ∉ π
⎪
t ≥ [t 0 ] + 2
0
,
⎩
⎧(1 − t + t 0 ) x1,0 , for t ∈ [t 0 , t 0 + 1)
x(t ) = ⎨
, if t 0 ∈ π
0 ,
for t ≥ t 0 + 1
⎩
In this case the state space is ℜ 2 and the state of (2.2) at time t ≥ t 0 is φ (t , t 0 , x 0 ) = ( x(t ), x([t ])) ∈ ℜ 2 . Furthermore,
notice that the classical semigroup property holds and that the partition π = {τ i }i∞=0 = {0,1,2,...} is fixed and does not
depend on the initial condition. Finally, according to Definitions 2.2 and 2.3, system (2.2) is T -periodic with T = 1
but it is not autonomous.
<
The following examples provide classes of control systems with the BIC property and a robust equilibrium point.
The examples help the reader to understand that the notions defined by Definitions 2.4-2.5 are typical for a wide class
of systems under minimal assumptions.
Example 2.8 (Finite-Dimensional Control Systems Described by Ordinary Differential Equations-ODEs):
Consider the class of systems described by ODEs of the form
x& (t ) = f (t , x(t ), u (t ), d (t ))
Y (t ) = H (t , x(t ), u (t ))
(2.3)
x(t ) ∈ ℜ , u (t ) ∈ U , d (t ) ∈ D , t ≥ t 0
n
where U ⊆ ℜ m , D ⊆ ℜ l , with 0 ∈ U and f : ℜ + × ℜ n × U × D → ℜ n , H : ℜ + × ℜ n × U → ℜ k are two locally
bounded mappings with H (t ,0,0) = 0 , f (t ,0,0, d ) = 0 for all (t , d ) ∈ ℜ + × D that satisfy the following hypotheses:
(A1) The mapping ( x, u, d ) → f (t , x, u, d ) is continuous for each fixed t ≥ 0 , measurable with respect to t ≥ 0 for
each fixed ( x, u, d ) ∈ ℜ n × U × D and such that for every bounded sets I ⊆ ℜ + , S ⊂ ℜ n × U , there exists a constant
L ≥ 0 such that:
5
(x − y )′ ( f (t , x, u, d ) − f (t , y, u, d )) ≤ L x − y 2
∀t ∈ I , ∀( x, u , y, u ) ∈ S × S , ∀d ∈ D
(A2) The mapping H : ℜ + × ℜ n × U → ℜ k is continuous.
(A3)
There
exist
+
functions
γ ∈K+,
a ∈ K∞
such
that
f (t , x, u , d ) ≤ γ (t )a ( x + u )
for
all
(t , x, u , d ) ∈ ℜ × ℜ × U × D .
n
The theory of ordinary differential equations guarantees that under hypotheses (A1-3), for each (t 0 , x 0 ) ∈ ℜ + × ℜ n
and for each pair of measurable and locally bounded inputs (u, d ) ∈ M ( U ) × M ( D) there exists a unique absolutely
continuous mapping x(t ) that satisfies a.e. the differential equation (2.3) with initial condition x(t 0 ) = x 0 ∈ ℜ n .
Moreover, certain results from the theory of ordinary differential equations guarantee that (2.3) is a control system
Σ := (ℜ n , ℜ k , M U , M D , φ , π , H ) with outputs that satisfies the BIC property with M U , M D the sets of all
measurable and locally bounded mappings u : ℜ + → U , d : ℜ + → D , respectively. Furthermore, the classical
semigroup property is satisfied for this system, i.e., we have π (t 0 , x 0 , u , d ) = [t 0 , t max ) , where t max > t 0 is the
maximal existence time of the solution. Finally, hypotheses (A1-3) guarantee that 0 ∈ ℜ n is a robust equilibrium
<
point from the input u ∈ M U for Σ .
The following example presents a class of neutral functional equations described by continuous time difference
equations. Such systems were recently studied in [22,35]. The importance of functional difference equations in
applications is explained in [35].
Example 2.9 (Control Systems described by Functional Difference Equations-FDEs): Consider the class of
systems described by FDEs of the form
x(t ) = f (t , Tr −τ (t ) (t − τ (t )) x, u (t ), d (t ))
Y (t ) = H (t , Tr (t ) x, u (t ))
(2.4)
x(t ) ∈ ℜ n , Y (t ) ∈ Y , u (t ) ∈ U , d (t ) ∈ D , t ≥ t 0
where r > 0 is a constant, τ : ℜ + → (0,+∞ ) is a positive continuous function with sup τ (t ) ≤ r , D ⊂ ℜ l , U ⊆ ℜ m
t ≥0
with 0 ∈ U are non-empty sets, Tr −τ (t ) (t − τ (t )) x := x(t − τ (t ) + θ ) ; θ ∈ [− r + τ (t ),0] , Tr (t ) x := x(t + θ ) ; θ ∈ [−r ,0] and
H , f : Ω × U × D → ℜ n , where Ω = ∪ {t} ×Ft and Ft denotes the set of bounded functions x : [−r + τ (t ),0] → ℜ n ,
t ≥0
are locally bounded mappings which satisfy the following hypotheses:
⎛
⎞
(R1) There exist functions γ ∈ K + , a ∈ K ∞ such that f t , Tr −τ (t ) (−τ (t )) x, u, d ≤ γ (t )a⎜⎜ sup x(θ ) + u ⎟⎟ for all
⎝ θ ∈[ − r , −τ (t )]
⎠
(
)
(t , x, u , d ) ∈ ℜ + × X × U × D , where X is the normed linear space of the bounded functions x : [−r ,0] → ℜ n with
x
X
:= sup x(θ ) .
θ ∈[ − r , 0 ]
(R2) The output map H : ℜ + × X × U → Y , where Y is a normed linear space, is a continuous mapping that maps
bounded sets of ℜ + × X × U into bounded sets of Y with H (t ,0,0) = 0 for all t ≥ 0 .
It should be clear that under the hypotheses stated above, for each (t 0 , x 0 ) ∈ ℜ + × X and for each pair of locally
bounded functions u : ℜ + → U , d : ℜ + → D there exists a unique locally bounded mapping x(t ) that satisfies the
difference equations (2.4) with initial condition x(t 0 + θ ) = x 0 (θ ) ; θ ∈ [−r ,0] . Consequently, (2.4) describes a control
Σ := (X, Y , M U , M D , φ , π , H )
with
outputs
and
evolution
map
φ
defined
by
system
6
φ (t , t 0 , x 0 , u , d ) = x(t + θ ) ; θ ∈ [−r ,0] , where U := ℜ m , M U the set of all locally bounded functions u : ℜ + → U
and M D the set of all functions d : ℜ + → D .
Systems described by functional difference evolution equations of the form (2.4) are considered in [4,22,35].
Working exactly in the same way as in [22], it can be shown that system (2.4) is RFC from the input u ∈ M U and
that 0 ∈ X is a robust equilibrium point from the input u ∈ M U for system (2.4).
Notice that a major advantage of allowing the output to take values in abstract normed linear spaces is that we are in
a position to consider:
• outputs with no delays, e.g. Y (t ) = h(t , x(t ), u (t )) with Y = ℜ k ,
• outputs with discrete or distributed delay, e.g. Y (t ) = h(t , x(t ), x(t − r ), u (t )) or Y (t ) = sup h(t , θ , x(θ ), u (t )) with
θ ∈[t − r ,t ]
Y =ℜ ,
• functional
k
Y (t ) = h(t , θ , x(t + θ )) ; θ ∈ [−r ,0]
outputs with memory, e.g.
Y (t ) = Tr (t ) x = x(t + θ ) ; θ ∈ [− r ,0] with Y = X .
or
the
identity
output
Finally, notice that the classical semigroup property is satisfied for this system, i.e., we have
π (t 0 , x 0 , u, d ) = [t 0 ,+∞) .
<
The following example is an immediate consequence of Theorems 2.2 and 3.2 in [4], concerning continuous
dependence on initial conditions and continuation of solutions of retarded functional differential equations,
respectively.
Example 2.10 (Control Systems described by Retarded Functional Differential Equations-RFDEs): Consider
the class of systems described by RFDEs of the form
x& (t ) = f (t , Tr (t ) x, u (t ), d (t ))
Y (t ) = H (t , Tr (t ) x, u (t ))
(2.5)
x(t ) ∈ ℜ , u (t ) ∈ U , d (t ) ∈ D , t ≥ t 0
n
where Tr (t ) x := x(t + θ ) ; θ ∈ [−r ,0] , D ⊆ ℜ l is a non-empty set, U ⊆ ℜ m is a non-empty set with 0 ∈ U ,
f : ℜ + × C 0 ([− r ,0] ; ℜ n ) × U × D → ℜ n , H : ℜ + × C 0 ([−r ,0] ; ℜ n ) × U → Y ( Y is a normed linear space) are locally
bounded mappings with f (t ,0,0, d ) = 0 , H (t ,0,0) = 0 for all (t , d ) ∈ ℜ + × D , that satisfy the following hypotheses:
(S1) The mapping ( x, u, d ) → f (t , x, u, d ) is continuous for each fixed t ≥ 0 and such that for every bounded
I ⊆ ℜ + and for every bounded S ⊂ C 0 ([−r ,0] ; ℜ n ) × U , there exists a constant L ≥ 0 such that:
(x(0) − y (0) )′ ( f (t , x, u, d ) − f (t , y, u, d ) ) ≤ L
max x(τ ) − y (τ )
2
τ ∈[ − r , 0]
∀t ∈ I , ∀( x, u , y, u ) ∈ S × S , ∀d ∈ D
(S2)
There
exist
+
functions
γ ∈K+,
a ∈ K∞
(t , x, u , d ) ∈ ℜ × C ([− r ,0]; ℜ ) × U × D , where
x
r
0
n
x
such
that
(
f (t , x, u , d ) ≤ γ (t )a x
r
r
+u
)
for
all
denotes the sup-norm of the space C ([−r ,0] ; ℜ ) , i.e.,
0
n
:= max x(θ ) .
θ ∈[ − r , 0]
(S3) There exists a countable set A ⊂ ℜ + , which is either finite or A = {t k ; k = 1,..., ∞} with t k +1 > t k > 0 for all
k = 1,2,... and lim t k = +∞ , such that mapping (t , x, u , d ) ∈ (ℜ + \ A) × C 0 ([− r ,0]; ℜ n ) × U × D → f (t , x, u , d ) is
7
continuous.
Moreover,
for
lim+ f (t , x, u, d ) = f (t 0 , x, u, d ) .
each
fixed
(t 0 , x, u , d ) ∈ ℜ + × C 0 ([− r ,0]; ℜ n ) × U × D ,
we
have
t →t 0
(S4) The mapping
H : ℜ + × C 0 ([− r ,0] ; ℜ n ) × U → Y is a continuous mapping that maps bounded sets of
ℜ + × C 0 ([−r ,0] ; ℜ n ) × U into bounded sets of Y .
The theory of retarded functional differential equations guarantees that under hypotheses (S1-4), for each
(t 0 , x 0 ) ∈ ℜ + × C 0 ([− r ,0] ; ℜ n ) and for each pair of measurable and locally bounded inputs (u, d ) ∈ M ( U ) × M ( D)
there exists a unique absolutely continuous mapping x(t ) that satisfies a.e. the differential equation (2.5) with initial
condition x(t 0 ) = x 0 ∈ C 0 ([−r ,0] ; ℜ n ) . Moreover, certain results from the theory of retarded functional differential
equations (Theorems 2.2 and 3.2 in [4]) guarantee that (2.5) is a control system
Σ := (C 0 ([−r ,0] ; ℜ n ), Y , M U , M D , φ , π , H ) with outputs that satisfies the BIC property with M U , M D the sets of all
measurable and locally bounded mappings u : ℜ + → U , d : ℜ + → D , respectively. Furthermore, the classical
semigroup property is satisfied for this system, i.e., we have π (t 0 , x 0 , u , d ) = [t 0 , t max ) , where t max > t 0 is the
maximal existence time of the solution. Finally, hypotheses (S1-4) guarantee that 0 ∈ C 0 ([−r ,0] ; ℜ n ) is a robust
equilibrium point from the input u ∈ M U for Σ . Again notice that a major advantage of allowing the output to take
values in abstract normed linear spaces is that we are in a position to consider various output cases (see previous
example).
<
The following example presents an important class of systems that does not satisfy the classical “semigroup
property”.
Example 2.11 (Hybrid Systems with sampling partition generated by the system): Consider the class of systems
described by impulsive differential equations of the form
x& (t ) = f (t , τ i , x(t ), x(τ i ), u (t ), u (τ i ), d (t ), d (τ i )) , t ∈ [τ i , τ i +1 )
τ 0 = t 0 , τ i +1 = τ i + h(τ i , x(τ i ), u (τ i ), d (τ i )) , i = 0,1,...
⎛
⎞
x(τ i +1 ) = R⎜⎜τ i , lim− x(t ), x(τ i ), u (τ i +1 ), u (τ i ), d (τ i +1 ), d (τ i ) ⎟⎟
⎝ t →τ i +1
⎠
Y (t ) = H (t , x (t ), u (t ))
(2.6)
where D ⊆ ℜ l , U ⊆ ℜ m is a closed set with 0 ∈ U , h : ℜ + × ℜ n × U × D → (0, r ] is a positive function which is
bounded by certain constant r > 0 , f : ℜ + × ℜ + × ℜ n × ℜ n × U × U × D × D → ℜ n , H : ℜ + × ℜ n × U → ℜ p and
R : ℜ + × ℜ n × ℜ n × U × U × D × D → ℜ n is a triplet of vector fields that satisfy the following hypotheses:
(P1) f (t , τ , x, x 0 , u , u 0 , d , d 0 ) is measurable with respect to t ≥ 0 , continuous with respect to ( x, d , u ) ∈ ℜ n × D × U
and such that for every bounded S ⊂ ℜ + × ℜ + × ℜ n × ℜ n × U × U there exists constant L ≥ 0 such that
(x − y )′ ( f (t , τ , x, x 0 , u, u 0 , d , d 0 ) − f (t , τ , y, x 0 , u, u 0 , d , d 0 )) ≤ L x − y 2
∀(t , τ , x, x 0 , u , u 0 , d , d 0 ) ∈ S × D × D , ∀(t , τ , y, x 0 , u , u 0 , d , d 0 ) ∈ S × D × D
(P2) There exist functions γ ∈ K + , a ∈ K ∞ such that
f (t , τ , x, x0 , u, u 0 , d , d 0 ) ≤ γ (t ) a( x + x0 + u + u 0 ) , ∀(τ , u, u0 , d, d0 , x, x0 ) ∈ℜ+ ×U ×U × D× D×ℜn ×ℜn , ∀t ≥ τ
R (t , x, x 0 , u , u 0 , d , d 0 ) ≤ γ (t ) a ( x + x 0 + u + u 0 ) , ∀(t , u , u 0 , d , d 0 , x, x 0 ) ∈ ℜ + × U × U × D × D × ℜ n × ℜ n
(P3) H : ℜ + × ℜ n × U → ℜ p is a continuous map with H (t ,0,0) = 0 for all t ≥ 0 ,
8
(P4) There exist a positive, continuous and bounded function hl : ℜ + × ℜ n × U → (0, r ] and a partition π = {Ti }i∞= 0 of
ℜ + , i.e., an increasing sequence of times with T0 = 0 and Ti → +∞ such that:
h(t , x, u , d ) ≥ min{pπ (t ) − t , hl (t , x, u ) } , ∀(t , x, u, d ) ∈ ℜ + × ℜ n × U × D
where pπ (t ) := min{T ∈ π ; t < T } .
Hybrid systems of the form (2.6) under hypotheses (P1-4) are considered in [21,22], where it is shown that for each
(t 0 , x 0 ) ∈ ℜ + × ℜ n and for each pair of measurable and locally bounded inputs u : ℜ + → U and d : ℜ + → D there
exists a unique piecewise absolutely continuous function t → x(t ) ∈ ℜ n with initial condition x(t 0 ) = x 0 , which is
produced by the following algorithm:
Step i :
1) Given τ i and x(τ i ) , calculate τ i +1 using the equation τ i +1 = τ i + h(τ i , x (τ i ), u (τ i ), d (τ i )) ,
2) Compute the state trajectory x(t ) , t ∈ [τ i , τ i +1 ) as the solution of the differential equation
x& (t ) = f (t , τ i , x (t ), x(τ i ), u (t ), u (τ i ), d (t ), d (τ i )) ,
⎛
⎞
3) Calculate x(τ i +1 ) using the equation x(τ i +1 ) = R⎜⎜τ i , lim− x(t ), x(τ i ), u (τ i +1 ), u (τ i ), d (τ i +1 ), d (τ i ) ⎟⎟ ,
τ
t
→
i
1
+
⎝
⎠
4) Compute the output trajectory Y (t ) , t ∈ [τ i , τ i +1 ] using the equation Y (t ) = H (t , x(t ), u (t ))
For i = 0 we take τ 0 = t 0 and x(τ 0 ) = x 0 (initial condition).
In [21] it is shown that system (2.6) under hypotheses (P1-4) is a control system Σ := (X, Y , M U , M D , φ , π , H ) with
outputs with the BIC property for which 0 ∈ ℜ n is a robust equilibrium point from the input u ∈ M U . Particularly,
we have X = ℜ n , Y = ℜ p , U = ℜ m and M U , M D the sets of measurable and locally bounded inputs u : ℜ + → U
and d : ℜ + → D , respectively. The set π (t 0 , x 0 , u, d ) ⊆ [t 0 ,+∞) involved in the weak semigroup property consists of
the sequence π = {τ i }i∞=0 generated by the recursive relation τ i +1 = τ i + h(τ i , x(τ i ), u (τ i ), d (τ i )) , i = 0,1,... with
τ 0 = t 0 . Notice that the control system (2.6) fails to satisfy the classical semigroup property.
Notice that if
h(τ + T , x, u , d ) = h(τ , x, u , d ) ,
f (t + T , τ + T , x, x 0 , u, u 0 , d , d 0 ) = f (t , τ , x, x 0 , u , u 0 , d , d 0 ) ,
R(τ + T , x, x 0 , u , u 0 , d , d 0 ) = R (τ , x, x 0 , u, u 0 , d , d 0 ) and H (t + T , x, u ) = H (t , x, u ) for certain T > 0 and for
(t , τ , u , u 0 , d , d 0 , x, x 0 ) ∈ ℜ + × ℜ + × U × U × D × D × ℜ n × ℜ n with t ≥ τ , then system (2.6) is T-periodic. Moreover,
if h(τ , x, u, d ) = h( x, u, d ) , f (t, τ , x, x0 , u, u0 , d , d 0 ) = f (t −τ , x, x0 , u, u0 , d , d 0 ) , R(τ , x, x0 , u, u0 , d , d 0 ) = R(x, x0 , u, u0 , d , d 0 )
and H (t , x, u) = H ( x, u) for (t , τ , u, u 0 , d , d 0 , x, x 0 ) ∈ ℜ + × ℜ + × U × U × D × D × ℜ n × ℜ n with t ≥ τ then system
(2.6) is autonomous.
Systems of the form (2.6) under hypotheses (P1-4) arise frequently in certain applications in mathematical control
theory and numerical analysis. Specifically, they arise when
i)
ii)
a (not necessarily continuous) sampled-data feedback law (with possibly variable sampling rate) is
applied to a finite-dimensional control system. For example, state-dependent sampling rates were related
in [2] with the classical work on discontinuous stabilizability in [1] while feedback stabilization
problems with zero order hold and constant positive sampling rate were considered in [28-34] and timevarying sampling rates were considered in [6,7].
a numerical discretization method (with possibly variable integration step sizes) is applied in order to
obtain the numerical solution of a given system of ordinary differential equations; see [3,41] for the case
of constant integration step sizes and [19,21] for the case of variable integration step sizes.
For a unified description of the above problems, see [21,22].
9
<
In contrast with the previous example, it should be noted that hybrid systems with impulses at fixed times satisfy the
classical “semigroup property”. The following example illustrates this case.
Example 2.12 (Hybrid Systems with Impulses at fixed times): Consider the class of systems described by
impulsive differential equations of the form
x& (t ) = f (t , d (t ), d (τ i ), x(t ), x(τ i ), u (t ), u (τ i )) , τ i ≤ t < τ i +1
⎛
⎞
x(τ i +1 ) = R⎜⎜τ i , lim− x(t ), x(τ i ), u (τ i +1 ), u (τ i ), d (τ i +1 ), d (τ i ) ⎟⎟
→
τ
t
i +1
⎝
⎠
Y (t ) = H (t , x(t ), u (t ))
(2.7)
x(t ) ∈ ℜ n , Y (t ) ∈ ℜ k , u (t ) ∈ V ⊆ ℜ m , t ≥ 0 , d (t ) ∈ D
where D ⊆ ℜ l , V ⊆ ℜ m is a closed set with 0 ∈ V , π = {τ i }i∞= 0 is a partition of ℜ + with diameter r > 0 , i.e., an
increasing sequence of times with τ 0 = 0 , sup{τ i +1 − τ i ; i = 0,1,2,... } = r and τ i → +∞ ,
d (t ) represents the
disturbance vector or the vector of time-varying uncertainties taking values in the set D ⊂ ℜ , Y (t ) represents the
l
output of the system and u (t ) ∈ V represents the input vector. A wide class of systems described by impulsive
differential equations with impulses at fixed times, as well as hybrid systems of the form:
x& (t ) = f (t , x(t ), u (t ), w(i)) , τ i ≤ t < τ i +1
(2.8)
w(i ) = g (i, x(τ i ), u (τ i ))
where π = {τ i }i∞=0 is a partition of ℜ + of diameter r > 0 , can be represented by the time-varying case (2.7).
Fundamental properties of the solutions of systems of the form (2.8) are studied in [24,25].
Consider system (2.7) under the following assumptions:
(Q1) π = {τ i }i∞= 0 is a partition of ℜ + with finite diameter r > 0 , i.e., an increasing sequence of times with τ 0 = 0 ,
sup{τ i +1 − τ i ; i = 0,1,2,... } = r and τ i → +∞ .
(Q2) H : ℜ + × ℜ n × V → ℜ k is continuous with H (t ,0,0) = 0 , for all t ≥ 0 .
(Q3) f (t , d , d 0 , x, x 0 , u , u 0 ) is measurable with respect to t ≥ 0 , continuous with respect to ( x, d , u ) ∈ ℜ n × D × V
and such that for every compact S ⊂ ℜ n × ℜ n × V × V and for every compact I ⊂ ℜ + there exists constant L ≥ 0
such that
(x − y )′ ( f (t , d , d 0 , x, x 0 , u, u 0 ) − f (t , d , d 0 , y, x 0 , u, u 0 ) ) ≤ L x − y 2
∀t ∈ I , ∀(d , d 0 ) ∈ D × D , ∀( x, x 0 , u , u 0 ) ∈ S , ∀( y, x 0 , u, u 0 ) ∈ S
(Q4) There exist functions γ ∈ K + ,
a ∈ K∞
such that
f (t , d , d 0 , x, x 0 , u, u 0 ) ≤ γ (t ) a ( x 0 + x + u + u 0 ) ,
R (t , x, x 0 , u , u 0 , d , d 0 ) ≤ γ (t ) a ( x + x 0 + u + u 0 ) , for all (t , d , d 0 , x, x 0 , u , u 0 ) ∈ ℜ + × D × D × ℜ n × ℜ n × V × V .
Systems of the form (2.7) with R (t , x, x 0 , u, u 0 , d , d 0 ) ≡ x (impulse free case) were considered in [23]. Special
classes of impulsive systems of the form (2.7) were studied in [5]. Using the method of steps on consecutive intervals,
it is clear that system (2.7) under hypotheses (Q1-4) defines a control system Σ := (X, Y , M U , M D , φ , π , H ) with
outputs and the BIC property, with state space X = ℜ n × ℜ n , output space Y = ℜ k , set of structured uncertainties
~
~
M D being the set of mappings t ∈ ℜ + → d (t ) = d (t + θ ) ; θ ∈ [− r ,0] where d : ℜ → D is any measurable and
{
}
locally bounded function, input space U the normed linear space of measurable and bounded functions on [−r ,0]
taking values in ℜ m endowed with the sup norm, U ⊆ U the set of measurable and bounded functions on [−r ,0]
taking
values
in
V ⊆ ℜm
and
set
of
external
10
inputs
MU
being
the
set
of
mappings
t ∈ ℜ + → u (t ) = { u~ (t + θ ) ; θ ∈ [−r ,0] }∈ U , where u~ : ℜ → ℜ m is a measurable and locally bounded function. The
reader may be surprised by the complicated definition of M D and M U , but it should be emphasized that this
definition guarantees that the causality property of the control system (2.7) holds. Notice that the classical semigroup
property is satisfied for this system, i.e., we have π (t 0 , x 0 , u , d ) = [t 0 , t max ) , where t max > t 0 is the maximal
existence time of the solution. However notice that if the vector fields f and R are independent of
d (τ i ), x(τ i ), u (τ i ) (this is the case studied in [5]) then system (2.7) under hypotheses (Q1-4) defines a control system
Σ := (X, Y , M U , M D , φ , π , H ) with outputs and the BIC property, with state space X = ℜ n , output space Y = ℜ k ,
set of structured uncertainties M D being the set of measurable and locally bounded functions d : ℜ → D , input
space U = ℜ m and M U being the set of measurable and locally bounded functions u : ℜ → U .
Let qπ (t ) = max{τ i ; τ i ∈ π , τ i ≤ t } . For all (t 0 , x 0 , x1 , d , u ) ∈ ℜ + × ℜ n × ℜ n × M D × M U , we denote by
x(t ) = φ (t , t 0 , x 0 , x1 ; d , u ) ∈ ℜ n the solution of (2.7) at time t ≥ t 0 with initial condition x(t 0 ) = x 0 and the additional
condition x(qπ (t 0 )) = x1 , which holds only for the case t 0 ∉ π , corresponding to inputs (d , u ) ∈ M D × M U (this
solution is unique by virtue of property (Q3)). Notice that the actual state of system (2.7) at time t ≥ t 0 is given by
~
φ (t , t 0 , x 0 , x1 ; d , u ) = (φ (t , t 0 , x 0 , x1 ; d , u ), φ (qπ (t ), t 0 , x 0 , x1 ; d , u )) ∈ ℜ n × ℜ n .
Hypotheses (Q3-4) can be used in order to show that 0 ∈ ℜ n × ℜ n is a robust equilibrium point from the input
u ∈ M U , exactly in the same way with the proof of the analogous result in [23]. Notice that if
f (t + T , x, x 0 , u , u 0 , d , d 0 ) = f (t , x, x 0 , u, u 0 , d , d 0 ) ,
R(t + T , x, x 0 , u, u 0 , d , d 0 ) = R(t , x, x 0 , u, u 0 , d , d 0 ) ,
H (t + T , x, u ) = H (t , x, u )
π = {iT }i∞=0
and
+
for
T >0
certain
and
for
all
(t , u , u 0 , d , d 0 , x, x 0 ) ∈ ℜ × V × V × D × D × ℜ × ℜ , then system (2.7) is T-periodic. Moreover, it should be noted
that there system (2.7) fails to be autonomous for every possible selection of the sets D , V , vector fields f , R, H
and partition π .
<
n
n
For control systems with the BIC property the following lemma provides a useful characterization of the RFC
property. Its proof is provided in the Appendix.
Lemma 2.13: System Σ := (X, Y , M U , M D , φ , π , H ) is RFC from the input u ∈ M U if and only if for every β ∈ K +
there exist functions μ , c ∈ K + , a, p ∈ K ∞ (depending only on β ∈ K + ) such that the following estimate holds for
all (t 0 , x 0 , d , u ) ∈ ℜ + × X × M D × M U :
β (t ) φ (t , t 0 , x 0 , u , d )
X
(
⎧⎪
≤ max ⎨ μ (t − t 0 ) , c(t 0 ) , a x 0
⎪⎩
X
) , sup p( u(τ ) ) ⎫⎪⎬⎪ , ∀t ≥ t
U
t 0 ≤τ ≤t
⎭
0
(2.9)
Next we present the Input-to-Output Stability property for the class of systems described by Definition 2.1.
Definition 2.14: Consider a control system Σ := (X, Y , M U , M D , φ , π , H ) with outputs and the BIC property and for
which 0 ∈ X is a robust equilibrium point from the input u ∈ M U . Suppose that Σ is RFC from the input u ∈ M U .
∗ If there exist functions σ ∈ KL , β , δ ∈ K + , γ ∈ N such that the following estimate holds for all u ∈ M U ,
(t 0 , x 0 , d ) ∈ ℜ + × X × M D and t ≥ t 0 :
H (t , φ (t , t 0 , x 0 , u , d ), u (t ))
Y
(
≤ σ β (t 0 ) x 0
X
)
(
, t − t 0 + sup γ δ (τ ) u (τ )
t 0 ≤τ ≤ t
U
)
(2.10)
then we say that Σ satisfies the Weighted Input-to-Output Stability (WIOS) property from the input u ∈ M U with
gain γ ∈ N and weight δ ∈ K + . Moreover, if β (t ) ≡ 1 then we say that Σ satisfies the Uniform Weighted Input-toOutput Stability (UWIOS) property from the input u ∈ M U with gain γ ∈ N and weight δ ∈ K + .
11
∗ If there exist functions σ ∈ KL , β ∈ K + , γ ∈ N such that the following estimate holds for all u ∈ M U ,
(t 0 , x 0 , d ) ∈ ℜ + × X × M D and t ≥ t 0 :
H (t , φ (t , t 0 , x 0 , u , d ), u (t ))
(
≤ σ β (t 0 ) x 0
Y
X
)
(
, t − t 0 + sup γ u (τ )
t 0 ≤τ ≤ t
U
)
(2.11)
then we say that Σ satisfies the Input-to-Output Stability (IOS) property from the input u ∈ M U with gain γ ∈ N .
Moreover, if β (t ) ≡ 1 then we say that Σ satisfies the Uniform Input-to-Output Stability (UIOS) property from the
input u ∈ M U with gain γ ∈ N .
Finally, for the special case of the identity output mapping, i.e., H (t , x, u ) := x , the (Uniform) (Weighted) Input-toOutput Stability property from the input u ∈ M U is called (Uniform) (Weighted) Input-to-State Stability property
from the input u ∈ M U .
{
}
Remark 2.15: Using the inequalities max{a, b} ≤ a + b ≤ max a + ρ (a), b + ρ −1 (b) (which hold for all ρ ∈ K ∞ and
a, b ≥ 0 ), it should be clear that the WIOS property for Σ := (X, Y , M U , M D , φ , π , H ) can be defined by using an
estimate of the form:
⎫⎪
⎪⎧
H (t , φ (t , t 0 , x 0 , u , d ), u (t )) Y ≤ max ⎨σ β (t 0 ) x 0 X , t − t 0 , sup γ δ (τ ) u (τ ) U ⎬
(2.10’)
⎪⎩
⎪⎭
t 0 ≤τ ≤ t
instead of (2.10). Similarly, the IOS property for Σ := (X, Y , M U , M D , φ , π , H ) can be defined by using an estimate
of the form:
⎪⎧
⎪⎫
H (t , φ (t , t 0 , x 0 , u , d ), u (t )) Y ≤ max ⎨σ β (t 0 ) x 0 X , t − t 0 , sup γ u (τ ) U ⎬
(2.11’)
⎪⎩
⎪⎭
t 0 ≤τ ≤ t
instead of (2.11).
(
)
(
(
)
)
(
)
The following lemmas provide ε − δ characterizations of the WIOS and UWIOS properties, which are going to be
used in the following section of the paper. Their proofs are provided in the Appendix.
Lemma 2.16: Consider a control system Σ := (X, Y , M U , M D , φ , π , H ) with outputs and the BIC property and for
which 0 ∈ X is a robust equilibrium point from the input u ∈ M U . Suppose that Σ is RFC from the input u ∈ M U .
Furthermore, suppose that there exist functions V : ℜ + × X × U → ℜ + with V (t ,0,0) = 0 for all t ≥ 0 , γ ∈ N and
δ ∈ K + such that the following properties hold:
P1 For every s ≥ 0 , T ≥ 0 , it holds that
⎧⎪
sup ⎨ V (t , φ (t , t 0 , x 0 , u, d ), u (t )) − sup γ δ (τ ) u (τ )
⎪⎩
t 0 ≤τ ≤t
(
U
); t ≥ t
0
, x0
X
⎫⎪
≤ s , t 0 ∈ [0, T ] , d ∈ M D , u ∈ M U ⎬ < +∞
⎪⎭
P2 For every ε > 0 and T ≥ 0 there exists a δ := δ (ε , T ) > 0 , such that:
(
⎧⎪
sup ⎨ V (t , φ (t , t 0 , x 0 , u, d ), u (t )) − sup γ δ (τ ) u (τ )
⎪⎩
t 0 ≤τ ≤t
U
); t ≥ t
0
, x0
X
⎫⎪
≤ δ , t 0 ∈ [0, T ] , d ∈ M D , u ∈ M U ⎬ ≤ ε
⎪⎭
P3 For every ε > 0 , T ≥ 0 and R ≥ 0 , there exists a τ := τ (ε , T , R ) ≥ 0 , such that:
(
⎧⎪
sup ⎨ V (t , φ (t , t 0 , x 0 , u, d ), u (t )) − sup γ δ (τ ) u (τ )
⎪⎩
t 0 ≤τ ≤t
U
); t ≥ t
0
+ τ , x0
X
⎫⎪
≤ R , t 0 ∈ [0, T ] , d ∈ M D , u ∈ M U ⎬ ≤ ε
⎪⎭
Then there exist functions σ ∈ KL and β ∈ K + such that the following estimate holds for all u ∈ M U ,
(t 0 , x 0 , d ) ∈ ℜ + × X × M D and t ≥ t 0 :
12
(
V (t , φ (t , t 0 , x 0 , u , d ), u (t )) ≤ σ β (t 0 ) x 0
X
)
(
, t − t 0 + sup γ δ (τ ) u (τ )
t 0 ≤τ ≤ t
U
)
(2.12)
≤ a (V (t , x, u )) for all (t , x, u ) ∈ ℜ + × X × U , then for every
ρ ∈ K ∞ , Σ satisfies the WIOS property from the input u ∈ M U with gain γ~ ∈ N and weight δ ∈ K + , where
γ~ ( s ) := a(γ ( s ) + ρ (γ ( s ) )) .
Moreover, if there exists a ∈ N such that H (t , x, u )
Y
Lemma 2.17: Consider a control system Σ := (X, Y , M U , M D , φ , π , H ) with outputs and the BIC property and for
which 0 ∈ X is a robust equilibrium point from the input u ∈ M U . Suppose that Σ is RFC from the input u ∈ M U .
Furthermore, suppose that there exist functions V : ℜ + × X × U → ℜ + with V (t ,0,0) = 0 for all t ≥ 0 , γ ∈ N and
δ ∈ K + such that the following properties hold:
P1 For every s ≥ 0 , it holds that
⎧⎪
sup ⎨ V (t , φ (t , t 0 , x 0 , u, d ), u (t )) − sup γ δ (τ ) u (τ )
⎪⎩
t 0 ≤τ ≤t
(
U
); t ≥ t
0
, x0
X
⎫⎪
≤ s , t 0 ≥ 0 , d ∈ M D , u ∈ M U ⎬ < +∞
⎪⎭
P2 For every ε > 0 there exists a δ := δ (ε ) > 0 , such that:
(
⎧⎪
sup ⎨ V (t , φ (t , t 0 , x 0 , u, d ), u (t )) − sup γ δ (τ ) u (τ )
⎪⎩
t 0 ≤τ ≤t
U
); t ≥ t
0
, x0
X
⎫⎪
≤ δ , t0 ≥ 0 , d ∈ M D , u ∈ M U ⎬ ≤ ε
⎪⎭
P3 For every ε > 0 and R ≥ 0 , there exists a τ := τ (ε , R ) ≥ 0 , such that:
(
⎧⎪
sup ⎨ V (t , φ (t , t 0 , x 0 , u, d ), u (t )) − sup γ δ (τ ) u (τ )
⎪⎩
t 0 ≤τ ≤t
U
); t ≥ t
0
+ τ , x0
X
⎫⎪
≤ R , t0 ≥ 0 , d ∈ M D , u ∈ M U ⎬ ≤ ε
⎪⎭
Then there exists a function σ ∈ KL such that estimate (2.12) holds for all u ∈ M U , (t 0 , x 0 , d ) ∈ ℜ + × X × M D and
t ≥ t0
with
β (t ) ≡ 1 . Moreover, if there exists
a ∈N
such that
H (t , x, u )
Y
≤ a(V (t , x, u ))
for all
+
(t , x, u ) ∈ ℜ × X × U , then for every ρ ∈ K ∞ , Σ satisfies the UWIOS property from the input u ∈ M U with gain
γ~ ∈ N and weight δ ∈ K + , where γ~ ( s ) := a(γ ( s ) + ρ (γ ( s ) )) .
Remark: Notice that Lemmata 2.16 and 2.17 can be very useful for the demonstration of the (U)WIOS property,
because in practice we show properties (P1-3) for some Lyapunov function V and not necessarily for the norm of the
output map. Moreover, notice that it is not required that V is continuous. If V : ℜ + × X × U → ℜ + is a continuous
functional that maps bounded sets of ℜ + × X × U into bounded sets of ℜ + , then Lemmata 2.16 and 2.17 guarantee
that Σ satisfies the WIOS and the UWIOS properties with V as output, respectively, from the input u ∈ M U with
gain γ ∈ N and weight δ ∈ K + .
Finally, we end this section with some useful observations for T-periodic control systems. It turns out that
periodicity guarantees uniformity with respect to the initial times. The following lemmas should be compared with
Lemma 1.1, page 131 in [4]. Their proofs are provided in the Appendix.
Lemma 2.18: Suppose that Σ := (X, Y , M U , M D , φ , π , H ) is T-periodic. If Σ satisfies the WIOS property from the
input u ∈ M U , then Σ satisfies the UWIOS property from the input u ∈ M U .
Lemma 2.19: Suppose that Σ := (X, Y , M U , M D , φ , π , H ) is T-periodic. If Σ satisfies the IOS property from the
input u ∈ M U , then Σ satisfies the UIOS property from the input u ∈ M U .
13
3. A Small-Gain Theorem for a Wide Class of Systems
The main result of the present work is stated next.
Theorem 3.1: Consider the system Σ := (X, Y , M U , M D , φ , π , H ) with the BIC property and suppose that there exist
maps V1 : ℜ+ × X × U → ℜ+ , V2 : ℜ+ × X × U → ℜ+ , with Vi (t ,0,0) = 0 for all t ≥ 0 ( i = 1,2 ) such that the following
hypotheses hold:
L1 : ℜ+ × X → ℜ+ with
(H1) There exist functions σ 1 ∈ KL , β 1 , μ1 , c1 , δ 1 , δ 1u , q1u ∈ K + , γ 1 , γ 1u , a1 , p1 , p1u ∈ N ,
L1 (t ,0) = 0
for
all
t ≥0,
such
that
for
(t 0 , x 0 , u, d ) ∈ ℜ + × X × M U × M D
every
the
mapping
t → V1 (t , φ (t , t 0 , x 0 , u, d ), u (t )) is locally bounded on [t 0 , t max ) and the following estimates hold for all
t ∈ [t 0 , t max ) :
(
U
(
U
V1 (t , φ (t , t 0 , x 0 , u, d ), u (t )) ≤ σ 1 (β1 (t 0 ) L1 (t 0 , x 0 ) , t − t 0 ) + sup γ 1 (δ 1 (τ ) V2 (τ ) ) + sup γ 1u δ 1u (τ ) u (τ )
t0 ≤τ ≤t
(
⎧⎪
⎪⎩
β 1 (t ) L1 (t , φ (t , t 0 , x 0 , u , d )) ≤ max ⎨ μ1 (t − t 0 ) , c1 (t 0 ) , a1 x 0
where V 2 (t ) = V 2 (t , φ (t , t 0 , x 0 , u , d ), u (t ) ) and t max
X
t0 ≤τ ≤t
) , sup
p1 (V 2 (τ ) ) , sup p1u q1u (τ ) u (τ )
)
(3.1)
) ⎫⎪⎬⎪
⎭
is the maximal existence time of the transition map of Σ .
t 0 ≤τ ≤t
t 0 ≤τ ≤t
(3.2)
(H2) There exist functions σ 2 ∈ KL , β 2 , μ 2 , c 2 , δ 2 , δ 2u , q 2u ∈ K + , γ 2 , γ 2u , a 2 , p 2 , p 2u ∈ N , L2 : ℜ+ × X → ℜ+ with
L2 (t ,0) = 0
for
all
t ≥0,
such
that
for
(t 0 , x 0 , u , d ) ∈ ℜ + × X × M U × M D
every
the
mapping
t → V2 (t , φ (t , t 0 , x 0 , u, d ), u(t )) is locally bounded on [t 0 , t max ) and the following estimates hold for all
t ∈ [t 0 , t max ) :
(
)
(3.3)
(
)
(3.4)
V2 (t ,φ (t , t0 , x0 , u, d ), u(t )) ≤σ 2 (β2 (t0 ) L2 (t0 , x0 ) , t − t0 ) + sup γ 2 (δ 2 (τ ) V1(τ )) + sup γ 2u δ 2u (τ ) u(τ ) U
t 0 ≤τ ≤t
(
⎧⎪
⎪⎩
) , sup
⎫⎪
p2 ( V1(τ )) , sup p2u q2u (τ ) u (τ ) U ⎬
⎪⎭
t 0 ≤τ ≤t
t 0 ≤τ ≤t
is the maximal existence time of the transition map of Σ .
β 2 (t ) L2 (t , φ (t , t0 , x0 , u, d )) ≤ max⎨ μ2 (t − t0 ) , c2 (t0 ) , a2 x0
where V1 (t ) = V1 (t , φ (t , t 0 , x 0 , u , d ), u (t ) ) and t max
t 0 ≤τ ≤t
X
(H3) There exist a function ρ ∈ K ∞ and a constant M > 0 such that:
δ 1 (t ) ≤ M , ∀t ≥ 0
where g i ( s ) := γ i ( s ) + ρ (γ i ( s ) ) , i = 1,2 .
(3.5)
g 1 (δ 1 (t ) g 2 (δ 2 (τ ) s )) ≤ s , ∀t , s ≥ 0 and τ ∈ [0, t ]
(3.6)
(H4) There exists a function a ∈ N such that the following inequality holds for all (t , x, u ) ∈ ℜ + × X × U :
H (t, x, u) Y ≤ a( V1 (t, x, u) + γ 1 (δ1 (t )V2 (t, x, u)) )
(3.7)
(H5) There exists a function b ∈ N such that the following inequality holds for all (t , x) ∈ ℜ + × X :
( )
x X ≤ b(L1(t , x) + L2 (t , x) ) ; max(L1(t , x), L2 (t , x)) ≤ b x X
(3.8)
Then there exists a function γ ∈ N such that system Σ satisfies the WIOS property from the input u ∈ M U with gain
γ ∈ N and weight δ ∈ K + , where
δ (t ) := max{δ 1u (t ), δ 2u (t ), q1u (t ) , q 2u (t )}
(3.9)
Moreover, if β 1 , β 2 , c1 , c 2 , δ 2 ∈ K + are bounded then system Σ satisfies the UWIOS property from the input
u ∈ M U with gain γ ∈ N and weight δ ∈ K + .
14
Remark 3.2:
(a) It should be clear that Theorem 3.1 takes into account all possible cases (weights, non-uniformity with
respect to initial times) and thus is applicable to a very wide class of systems.
(b) When γ 1 ∈ N (or γ 2 ∈ N ) is identically zero, it follows that (3.6) is automatically satisfied. This is the case
of systems in cascade (see [10]). On the other hand, if γ i ( s ) = K i s for certain constants K i ≥ 0 ( i = 1,2 )
⎛
⎞
then inequality (3.6) is satisfied if K 1 K 2 sup⎜ δ 1 (t ) max δ 2 (τ ) ⎟ < 1 . Moreover, if γ i ( s ) = K i s for certain
τ ∈[ 0,t ]
⎠
t ≥0 ⎝
constants K i ≥ 0 ( i = 1,2 ) and δ 1 (t ) ≡ δ 2 (t ) ≡ 1 then hypothesis (H3) is satisfied if K 1 K 2 < 1 . This is the
case of the classical Small-Gain Theorem.
(c) If,
instead
of
hypothesis
(H4),
there
exists
a
function
a ∈N
such
that
H (t, x, u) Y ≤ a( V2 (t, x, u) + γ 2 (δ 2 (t )V1 (t, x, u)) ) holds for all (t , x, u ) ∈ ℜ × X × U , then indices 1 and
+
2 must be changed in hypothesis (H3). Furthermore, in this case system Σ satisfies the UWIOS property
from the input u ∈ M U with gain γ ∈ N and weight δ ∈ K + , if functions β 1 , β 2 , c1 , c 2 , δ 1 ∈ K + are
bounded.
(d) In classical Small-Gain Theorems (e.g. [10]), the functions L1 : ℜ+ × X → ℜ+ and L2 : ℜ+ × X → ℜ+ take
the form L1 (t , x) = x1 and L 2 (t , x) = x 2 , where x = ( x1 , x 2 ) . It follows that hypothesis (H5) automatically
holds with b( s ) := s . This is the case in Corollary 3.4 below.
Since Small-Gain results are frequently applied to feedback interconnections of control systems, we need to
clarify the notion of the feedback interconnection of two control systems. However, the fact that we do not require the
classical semigroup property for each of the interconnected subsystems, creates technical difficulties: for example, the
determination of the set of sampling times for the composite system is not trivial. In order to guarantee the existence
of a set of sampling times for the composite system, we assume that the sampling times of the composite system are
the common sampling times of the interconnected subsystems. The details are given in the following definition.
~
Consider
a
pair
of
control
systems
Σ 1 = (X1 , Y1 , M S 2 ×U , M D , φ1 , π 1 , H 1 ) ,
~
Σ 2 = (X 2 , Y2 , M S1 ×U , M D , φ 2 , π 2 , H 2 ) with outputs H 1 : ℜ + × X1 × U → S1 ⊆ Y1 , H 2 : ℜ + × X 2 × Y1 × U → S 2 ⊆ Y2
Definition
3.3:
and the BIC property and for which 0 ∈ Xi i = 1,2 are robust equilibrium points from the inputs (v 2 , u ) ∈ M S 2 ×U ,
(v1 , u ) ∈ M S1 ×U , respectively. Suppose that there exists a unique pair of a map φ = (φ1 , φ 2 ) : Aφ → X and a set-
valued map ℜ + × X × M U × M D ∋ (t 0 , x 0 , u , d ) → π (t 0 , x 0 , u , d ) ⊆ [t 0 ,+∞) , where Aφ ⊆ ℜ + × ℜ + × X × M U × M D ,
X = X1 × X 2 , such that for every (t , t 0 , x 0 , u , d ) ∈ Aφ with t ≥ t 0 , x 0 = ( x1,0 , x 2,0 ) ∈ X1 × X 2 it holds that:
“there exists a pair of external inputs v i ⊆ M ( S i ) i = 1,2 with v1(τ ) = H1(τ , φ1(τ , t0 , x0 , u, d ), u(τ )) ,
v 2 (τ ) = H 2 (τ , φ 2 (τ , t 0 , x0 , u, d ), v1 (τ ), u(τ ))
for all
τ ∈ [t 0 , t ] ,
(v i , u ) ∈ M Si ×U
i = 1,2 ,
~
π (t0 , x0 , u, d ) = π1(t0 , x1,0 , (v2 , u ), d ) ∩ π 2 (t0 , x2,0 , (v1, u ), d ) and φ1(τ , t0 , x0 , u, d ) = φ1(τ , t0 , x1,0 , (v2 , u), d ) ,
~
φ 2 (τ , t 0 , x 0 , u , d ) = φ 2 (τ , t 0 , x 2,0 , (v1 , u ), d ) for all τ ∈ [t 0 , t ] .”
Moreover, let Y be a normed linear space and H : ℜ + × X × U → Y a continuous map that maps bounded sets of
ℜ + × X × U into bounded sets of Y , with H (t ,0,0) = 0 for all t ≥ 0 and suppose that Σ := (X, Y , M U , M D , φ , π , H )
is a control system with outputs and the BIC property, for which 0 ∈ X is a robust equilibrium point from the input
u ∈ M U . Then system Σ is said to be the feedback connection or the interconnection of systems Σ 1 and Σ 2 .
It should be emphasized that the feedback interconnection of two systems may create a system, which has different
qualitative properties from each of the interconnected subsystems. For example, if we interconnect a subsystem
described by RFDEs (see Example 2.10) with a hybrid subsystem with impulses at fixed times (see Example 2.12),
then the overall system will be a system with both “memory” and impulses (discontinuous systems described by
RFDEs-see [42]).
15
We are now in a position to state our main result for feedback interconnections of control systems. It is a direct
consequence of Theorem 3.1 and its proof is omitted.
Suppose that Σ := (X, Y , M U , M D , φ , π , H ) is the feedback connection of
~
~
and
with
Σ 1 = (X1 , Y1 , M S 2 ×U , M D , φ1 , π 1 , H 1 )
Σ 2 = (X 2 , Y2 , M S1 ×U , M D , φ 2 , π 2 , H 2 )
Corollary
systems
3.4:
outputs
H 1 : ℜ + × X1 × U → S1 ⊆ Y1 , H 2 : ℜ + × X2 × Y1 × U → S 2 ⊆ Y2 . We assume that:
(H1’) Subsystem Σ 1 satisfies the WIOS property from the inputs v 2 ∈ M ( S 2 ) and u ∈ M U . Particularly, there exist
functions σ 1 ∈ KL , β 1 , μ1 , c1 , δ 1 , δ 1u , q1u ∈ K + , γ 1 , γ 1u , a1 , p1 , p1u ∈ N , such that the following estimate holds for all
(t 0 , x1 , (v 2 , u 0 ), d ) ∈ ℜ + × X1 × M S 2 ×U × M D and t ≥ t 0 :
~
H 1 (t , φ1 (t , t 0 , x1 , (v 2 , u), d ), u(t ))
Y1
(
≤ σ 1 β1 (t 0 ) x1
X1
)
(
, t − t 0 + sup γ 1 δ 1 (τ ) v 2 (τ )
t0 ≤τ ≤t
)+ sup γ (δ
Y2
u
1
t0 ≤τ ≤t
u
1 (τ )
u(τ ) U
)
(3.10)
~
β 1 (t ) φ1 (t , t 0 , x1 , (v 2 , u ), d )
X1
(
⎧⎪
≤ max ⎨ μ1 (t − t 0 ) , c1 (t 0 ) , a1 x1
⎪⎩
X1
), sup p ( v (τ ) ), sup p (q
t 0 ≤τ ≤t
1
Y2
2
u
1
t 0 ≤τ ≤t
u
1 (τ )
u (τ )
U
) ⎫⎪⎬⎪
⎭
(3.11)
(H2’) Subsystem Σ 2 satisfies the WIOS property from the inputs v1 ∈ M ( S1 ) and u ∈ M U . Particularly, there exist
functions σ 2 ∈ KL , β 2 , μ 2 , c 2 , δ 2 , δ 2u , q 2u ∈ K + , γ 2 , γ 2u , a 2 , p 2 , p 2u ∈ N , such that the following estimate holds for
all (t 0 , x 2 , (v1 , u 0 ), d ) ∈ ℜ + × X 2 × M S1 ×U × M D and t ≥ t 0 :
~
H 2 (t ,φ2 (t, t0 , x2 , (v1, u), d ), v1(t ),u(t ))
~
β 2 (t ) φ2 (t , t0 , x2 , (v1, u), d )
X2
Y2
(
≤σ 2 β2 (t0 ) x2
X2
)
(
)
(
, t − t0 + sup γ 2 δ 2 (τ ) v1(τ ) Y + sup γ 2u δ 2u (τ ) u(τ ) U
(
⎧⎪
≤ max⎨ μ2 (t − t0 ) , c2 (t0 ) , a2 x2
⎪⎩
X2
t0 ≤τ ≤t
1
t0 ≤τ ≤t
)
) , sup p ( v (τ ) ) , sup p (q (τ ) u(τ ) ) ⎫⎪⎬⎪
t 0 ≤τ ≤t
2
1
Y1
t 0 ≤τ ≤t
u
2
u
2
U
⎭
(3.12)
(3.13)
Moreover, assume that hypothesis (H3) of Theorem 3.1 holds and there exists a function a ∈ N such that the
following inequality holds for all (t , x, u ) ∈ ℜ + × X × U with x = ( x1 , x 2 ) ∈ X1 × X 2 :
(
(
H (t, x, u) Y ≤ a H1 (t, x1 , u) Y + γ 1 δ1 (t ) H 2 (t, x2 , H1 (t, x1 , u), u) Y
1
2
))
(3.14)
Then there exists a function γ ∈ N such that system Σ satisfies the WIOS property from the input u ∈ M U with gain
γ ∈ N and weight δ ∈ K + , where δ ∈ K + is defined by (3.9). Moreover, if β 1 , β 2 , c1 , c 2 , δ 2 ∈ K + are bounded then
system Σ satisfies the UWIOS property from the input u ∈ M U with gain γ ∈ N and weight δ ∈ K + .
Remark 3.5:
(a) When δ 1 (t ) ≡ δ 2 (t ) ≡ 1 and γ 1 ∈ K ∞ then the result of Corollary 3.4 guarantees the WIOS property from
the input u ∈ M U for the output H (t , x, u ) := ( H 1 (t , x1 , u ), H 2 (t , x 2 , H 1 (t , x1 , u ), u )) , i.e., for the output that
combines the outputs of each individual subsystem. Moreover, if in addition the functions q iu (t ) ( i = 1,2 )
are bounded, then the result of Corollary 3.4 guarantees the IOS from the input u ∈ M U for the output
H (t , x, u ) := ( H 1 (t , x1 , u ), H 2 (t , x 2 , H 1 (t , x1 , u ), u )) .
Finally,
if
in
addition
the
functions β 1 , β 2 , c1 , c 2 , δ 2 ∈ K + are bounded then the result of Corollary 3.4 guarantees the UIOS from the
input u ∈ M U for the output H (t , x, u ) := ( H 1 (t , x1 , u ), H 2 (t , x 2 , H 1 (t , x1 , u ), u )) . This particular case
coincides with the prior result of the nonlinear ISS Small-Gain Theorem presented in [10] for control
systems described by ODEs.
16
(b) Conditions (3.11) and (3.13) hold automatically, when each one of the subsystems Σ 1 and Σ 2 satisfy the
WISS property.
(c) If,
instead
(
of
hypothesis
(3.14),
there
(
exists
a
))
H (t, x, u) Y ≤ a H 2 (t, x2 , H1 (t, x1 , u), u) Y + γ 2 δ 2 (t ) H1 (t, x1 , u) Y
2
a ∈N
function
1
holds
such
that
for
all
(t , x, u ) ∈ ℜ + × X × U , then indices 1 and 2 must be changed in hypothesis (H3). Furthermore, in this case
system Σ satisfies the UWIOS property from the input u ∈ M U with gain γ ∈ N and weight δ ∈ K + , if
functions β 1 , β 2 , c1 , c 2 , δ 1 ∈ K + are bounded.
Proof of Theorem 3.1: The proof consists of three steps:
Step 1: We show that Σ is RFC from the input u ∈ M U .
1
Step 2: Let ϕ~ ( s ) := s + ρ ( s ) , where ρ ∈ K ∞ is the function involved in hypothesis (H3). We show that properties
2
~
P1 and P2 of Lemma 2.16 hold for system Σ with V = V1 or V = ϕ (γ 1 (δ1 (t )V2 )) , for appropriate γ~ ∈ N and
δ ∈ K + as defined by (3.9). Moreover, if β 1 , β 2 ∈ K + are bounded we show that properties P1 and P2 of Lemma
~
2.17 hold for system Σ with V = V1 or V = ϕ (γ 1 (δ1 (t )V2 )) , for appropriate γ~ ∈ N and δ ∈ K + as defined by
(3.9).
Step 3: We show that property P3 of Lemma 2.16 holds for system Σ with V = V1 or V = ϕ (γ 1 (δ1 (t )V2 )) , for
appropriate γ~ ∈ N and δ ∈ K + as defined by (3.9). Moreover, if β 1 , β 2 , c1 , c 2 , δ 2 ∈ K + are bounded, we show that
~
property P3 of Lemma 2.17 holds for system Σ with V = V or V = ϕ (γ (δ (t )V )) , for appropriate γ~ ∈ N and
~
1
δ ∈K
+
1
1
2
as defined by (3.9).
It then follows from Lemma 2.16 that exist functions σ ∈ KL and β ∈ K + such that the following estimate holds for
all u ∈ M U , (t 0 , x 0 , d ) ∈ ℜ + × X × M D and t ≥ t 0 :
(
V1 (t , φ (t , t 0 , x 0 , u , d ), u (t )) ≤ σ β (t 0 ) x 0
(
X
)
(
, t − t 0 + sup γ δ (τ ) u (τ )
ϕ~ (γ 1 (δ 1 (t )V 2 (t , φ (t , t 0 , x 0 , u , d ), u (t )) )) ≤ σ β (t 0 ) x 0
X
t 0 ≤τ ≤ t
)
(
U
)
, t − t 0 + sup γ δ (τ ) u (τ )
t 0 ≤τ ≤ t
(3.15a)
U
)
(3.15b)
Moreover, if β 1 , β 2 , c1 , c 2 , δ 2 ∈ K + are bounded it follows from Lemma 2.17 that estimates (3.15a,b) hold with
β (t ) ≡ 1 .
Thus, using (3.7) and the fact that ϕ~ ( s ) ≥ s for all s ≥ 0 , we conclude that Σ satisfies the WIOS property from the
input u ∈ M U with gain γ ( s ) := a(4γ~ ( s ) ) ∈ N and weight δ ∈ K + . Moreover, if β 1 , β 2 , c1 , c 2 , δ 2 ∈ K + are bounded
we conclude that Σ satisfies the UWIOS property from the input u ∈ M with gain γ ( s ) := a(4γ~ ( s ) ) ∈ N and weight
U
+
δ ∈K .
Step 1:
Let arbitrary (t , t 0 , x 0 , u , d ) ∈ Aφ with t ≥ t 0 , x 0 ∈ X and let t max ∈ (t 0 ,+∞] the maximal existence time of the
transition map φ of Σ := (X, Y , M U , M D , φ , π , H ) that corresponds to (t 0 , x 0 , u , d ) ∈ ℜ + × X × M U × M D . Notice
that by virtue of the BIC property, if t max < +∞ then for every M > 0 there exists t ∈ [t 0 , t max ) with
17
φ (t , t 0 , x 0 , u , d )
X
>M .
We
V1(τ ) = V1(τ ,φ (τ , t0 , x0 , u, d ),u(τ )) ,
define
L1(τ ) = L1(τ ,φ (τ , t0 , x0 , u, d ))
and
V2 (τ ) = V2 (τ ,φ (τ , t0 , x0 , u, d ),u(τ )) , L2 (τ ) = L2 (τ ,φ (τ , t0 , x0 , u, d )) for all τ ∈ [t 0 , t ] .
The previous definitions in conjunction with (3.1), (3.2), (3.3), (3.4) imply the following inequalities for all
t ∈ [t 0 , t max ) :
(
V1 (t ) ≤ σ 1 (β 1 (t 0 ) L1 (t 0 ) , t − t 0 ) + sup γ 1 (δ 1 (τ ) V 2 (τ ) ) + sup γ 1u δ 1u (τ ) u (τ )
t 0 ≤τ ≤t
(
⎧⎪
⎪⎩
β 1 (t ) L1 (t ) ≤ max ⎨ μ1 (t − t 0 ) , c1 (t 0 ) , a1 x 0
X
t 0 ≤τ ≤t
) , sup
t 0 ≤τ ≤t
(
p1 ( V 2 (τ ) ) , sup p1u q1u (τ ) u (τ )
t 0 ≤τ ≤t
(
(
X
t 0 ≤τ ≤t
(3.17)
⎭
p 2 ( V1 (τ ) ) , sup p 2u q 2u (τ ) u (τ )
U
t 0 ≤τ ≤ t
) , sup
) ⎫⎪⎬⎪
)
t 0 ≤τ ≤ t
⎧⎪
⎪⎩
(3.16)
U
V 2 (t ) ≤ σ 2 (β 2 (t 0 ) L 2 (t 0 ) , t − t 0 ) + sup γ 2 (δ 2 (τ ) V1 (τ ) ) + sup γ 2u δ 2u (τ ) u (τ )
β 2 (t ) L 2 (t ) ≤ max ⎨ μ 2 (t − t 0 ) , c 2 (t 0 ) , a 2 x 0
U
)
U
(
t 0 ≤τ ≤t
(3.18)
) ⎫⎪⎬⎪
(3.19)
⎭
Let ρ ∈ K ∞ the function involved in hypothesis (H3) and define κ ( s ) := s + ρ −1 (s ) , ϕ ( s ) := s + ρ (s ) . Using the
inequality r + s ≤ max{κ (r ) ; ϕ ( s )} (which holds for all r , s ≥ 0 ) as well as the equality g 2 ( s ) = ϕ (γ 2 ( s ) ) , we obtain
from (3.18) for all t ∈ [t 0 , t max ) :
⎧⎪ ⎛
V 2 (t ) ≤ max ⎨κ ⎜⎜ σ 2 (β 2 (t 0 ) L 2 (t 0 ) , t − t 0 ) + sup γ 2u δ 2u (τ ) u (τ )
t 0 ≤τ ≤t
⎪⎩ ⎝
(
)⎞⎟⎟ ; sup g
U
⎠
t 0 ≤τ ≤ t
⎫
2
(δ 2 (τ ) V1 (τ ) ) ⎪⎬
(3.20)
⎪⎭
Notice that inequality (3.6) implies that γ 1 (δ 1 (t ) g 2 (δ 2 (τ ) s )) ≤ ϕ −1 ( s ) , ∀t , s ≥ 0 and τ ∈ [0, t ] . Thus (3.20) in
conjunction with (3.5) and the previous observation implies the following estimate which holds for all t ∈ [t 0 , t max ) :
⎧ ⎛
⎪⎩ ⎜⎝
⎛
)⎞ ⎞⎟
(
⎛
⎞⎫
⎝ t0 ≤τ ≤t
⎠ ⎪⎭
⎪
⎪
γ 1(δ1(t )V2 (t )) ≤ max⎨γ 1⎜ Mκ ⎜⎜ σ 2 (β 2 (t0 ) L2 (t0 ) , t − t0 ) + sup γ 2u δ 2u (τ ) u (τ ) U ⎟⎟ ⎟ ; ϕ −1⎜⎜ sup V1(τ ) ⎟⎟ ⎬
⎝
t 0 ≤τ ≤t
⎠⎠
(3.21)
Combining estimate (3.16) with (3.21), we obtain:
(
sup V1 (τ ) ≤ σ 1 (β 1 (t 0 ) L1 (t 0 ) , 0 ) + sup γ 1u δ 1u (τ ) u (τ )
t 0 ≤τ ≤t
t 0 ≤τ ≤ t
U
)
⎧⎪ ⎛
⎛
+ max ⎨γ 1 ⎜ Mκ ⎜⎜ σ 2 (β 2 (t 0 ) L 2 (t 0 ) , 0 ) + sup γ 2u δ 2u (τ ) u (τ )
⎜
t 0 ≤τ ≤ t
⎪⎩ ⎝
⎝
(
Distinguishing
the
cases
⎛
⎜
⎝
⎛
U
⎛
⎞⎞
⎞⎫
⎟ ⎟ ; ϕ −1 ⎜ sup V1 (τ ) ⎟ ⎪⎬
⎜ t ≤τ ≤t
⎟⎟
⎟
⎝ 0
⎠⎠
⎠ ⎪⎭
(3.22)
)
(
)⎞ ⎞⎟
⎛
⎞
γ 1 ⎜ Mκ ⎜⎜ σ 2 (β 2 (t 0 ) L 2 (t 0 ) , 0 ) + sup γ 2u δ 2u (τ ) u (τ ) U ⎟⎟ ⎟ ≥ ϕ −1 ⎜⎜ sup V1 (τ ) ⎟⎟ ,
⎝
t 0 ≤τ ≤t
⎠⎠
⎝
t 0 ≤τ ≤t
⎠
⎛
⎛
⎞⎞
⎛
⎞
⎟ ⎟ ≤ ϕ −1 ⎜ sup V1 (τ ) ⎟ ,
using
the
identity
U
⎟⎟
⎜ t ≤τ ≤t
⎟
⎜
t 0 ≤τ ≤t
⎝
⎠⎠
⎝ 0
⎠
⎝
s − ϕ −1 ( s ) = κ −1 ( s ) and the fact that κ ( s ) ≥ s in conjunction with (3.22) and (3.8) (which implies Li (t 0 ) ≤ b x 0 X ,
(
γ 1 ⎜ Mκ ⎜⎜ σ 2 (β 2 (t 0 ) L 2 (t 0 ) , 0 ) + sup γ 2u δ 2u (τ ) u (τ )
)
(
i = 1,2 ) gives the following estimate which holds for all t ∈ [t 0 , t max ) :
18
)
sup V1(τ ) ≤
t 0 ≤τ ≤t
⎧
⎪
⎪
max⎨
⎪σ β (t ) b x
0
⎪ 1 1 0
⎩
(
(
(
⎛
(
κ ⎜⎜σ1 β1(t0 ) b x0
⎝
X
) , 0 ) + sup γ (δ
t 0 ≤τ ≤t
u
1 (τ )
u
1
X
) , 0 ) + sup γ (δ
t 0 ≤τ ≤t
u
1 (τ )
u
1
)
⎞
u(τ ) U ⎟⎟
⎠
⎛
⎛
u(τ ) U + γ 1⎜ Mκ ⎜⎜σ 2 β2 (t0 ) b x0
⎜
⎝
⎝
)
(
(
X
), 0 )+
(
sup γ 2u δ 2u (τ ) u(τ ) U
t 0 ≤τ ≤t
⎫
⎪
⎪
⎬
⎞ ⎞⎪
⎟⎟
⎟ ⎟⎪
⎠ ⎠⎭
(3.23)
)
We show next that Σ is RFC from the input u ∈ M U by contradiction. Suppose that t max < +∞ . Then by virtue of
φ (t , t 0 , x 0 , u , d )
the BIC property for every M > 0 there exists t ∈ [t 0 , t max ) with
X
> M . On the other hand
estimate (3.23) in conjunction with the hypothesis t max < +∞ shows that there exists M 1 ≥ 0 such
that sup V1 (τ ) ≤ M 1 . The fact that V1 (t ) is bounded in conjunction with estimates (3.18) and (3.19), implies that
t 0 ≤τ <t max
sup V 2 (τ ) ≤ M 2 and
there exist constants M 2 , M 3 ≥ 0 such that
sup
t 0 ≤τ <t max
t 0 ≤τ < t max
L 2 (τ ) ≤ M 3 . Finally, the fact that
V 2 (t ) is bounded in conjunction with estimate (3.17), implies that there exists a constant M 4 ≥ 0 such that
sup L1 (τ ) ≤ M 4 . It follows from (3.8) and inequality φ (t , t0 , x0 , u, d ) X ≤ b(L1(t ) + L2 (t )) that the transition map of
t 0 ≤τ <t max
Σ , i.e., φ (t , t 0 , x 0 , u, d ) , is bounded on [t 0 , t max ) and this contradicts the requirement that for every M > 0 there
exists t ∈ [t 0 , t max ) with φ (t , t 0 , x 0 , u , d )
X
> M . Hence, we must have t max = +∞ .
Let arbitrary R ≥ 0 , T ≥ 0 . For every u ∈ M ( BU [0, R ]) ∩ M U , s ∈ [0, T ] , x 0
X
≤ R , t 0 ∈ [0, T ] , d ∈ M D estimate
(3.23) shows that there exists M 1 (T , R) ≥ 0 such that V1 (t 0 + s ) ≤ M 1 (T , R ) , for all s ∈ [0, T ] . The previous
(
observation in conjunction with estimates (3.18), (3.19) and (3.8) (which gives Li (t 0 ) ≤ b x 0
that
there
M 2 (T , R ), M 3 (T , R ) ≥ 0
exist
u ∈ M ( BU [0, R ]) ∩ M U , s ∈ [0, T ] , x 0
X
such
≤ R , t 0 ∈ [0, T ] , d ∈ M D
we
X
that
) , i = 1,2 ), implies
for
every
V 2 (t 0 + s ) ≤ M 2 (T , R )
have
and
L 2 (t 0 + s ) ≤ M 3 (T , R ) , for all s ∈ [0, T ] . Finally, inequality V 2 (t 0 + s ) ≤ M 2 (T , R ) in conjunction with estimate
(3.17),
implies
that
there
exists
u ∈ M ( BU [0, R ]) ∩ M U , s ∈ [0, T ] , x 0
s ∈ [0, T ] .
It
follows
from
(3.8)
u ∈ M ( BU [0, R ]) ∩ M U , s ∈ [0, T ] , x 0
satisfies
φ (t , t 0 , x 0 , u , d )
X
X
a
M 4 (T , R) ≥ 0
≤ R , t 0 ∈ [0, T ] , d ∈ M D
and
X
constant
inequality
such
that
for
every
L1 (t 0 + s ) ≤ M 4 (T , R) , for all
we have
φ (t , t0 , x0 , u, d ) X ≤ b(L1(t ) + L2 (t ))
that
for
every
≤ R , t 0 ∈ [0, T ] , d ∈ M D the transition map of Σ , i.e., φ (t , t 0 , x 0 , u, d ) ,
≤ b(M 3 (T , R) + M 4 (T , R ) ) < +∞ and this according to Definition 2.2 implies that Σ is
RFC from the input u ∈ M U .
Step 2:
Using (3.23) in conjunction with the inequality q(r + s ) ≤ q( κ (r ) ) + q( ϕ ( s ) ) (which holds for all r , s ≥ 0 and q ∈ N )
gives the following estimate, which holds for all t ≥ t 0 :
(( (
+ sup κ (ϕ (γ (δ
(
V1 (t ) ≤ κ κ σ 1 β 1 (t 0 ) b x 0
t 0 ≤τ ≤t
u
1
u
1 (τ )
u (τ )
X
U
) , 0 )) )+ γ (Mκ (κ (σ (β
)))+ sup γ (Mκ ( ϕ (γ (δ
t 0 ≤τ ≤t
( x ) , 0 )) ))
u (τ ) ))))
1
2
2 (t 0 ) b
1
u
2
u
2 (τ )
0 X
(3.24)
U
Moreover, combining estimates (3.21) and (3.23) and using the equalities ϕ −1 (κ ( s ) ) = ρ ( s ) and ϕ ( s ) := s + ρ (s ) as
(
well as the inequalities ϕ −1 (s ) ≤ s and (3.8) (which gives Li (t 0 ) ≤ b x 0
which holds for all t ≥ t 0 :
19
X
) , i = 1,2 ), gives the following estimate,
⎛
⎜
⎝
(
⎛
(
γ 1 (δ 1 (t )V 2 (t ) ) ≤ γ 1 ⎜ M κ ⎜⎜ σ 2 β 2 (t 0 ) b x 0
⎛
+ ϕ ⎜⎜ σ 1 β 1 (t 0 ) b x 0
⎝
(
(
⎝
X
) , 0 ) + sup
t 0 ≤τ ≤ t
γ 1u
X
(
δ 1u
), 0 )+
(
sup γ 2u δ 2u (τ ) u (τ )
t 0 ≤τ ≤ t
(τ ) u (τ )
U
U
)⎞⎟⎟ ⎞⎟⎟
⎠⎠
(3.25)
)
⎞
⎟
⎟
⎠
Using (3.25) in conjunction with the inequality q(r + s ) ≤ q( κ (r ) ) + q( ϕ ( s ) ) (which holds for all r , s ≥ 0 and q ∈ N )
gives the following estimate, which holds for all t ≥ t 0 :
( (( (
( ) , 0 )) )) + sup γ (Mκ ( ϕ (γ (δ
τ
+ ϕ (κ (σ (β (t ) b( x ) , 0 )) ) + sup ϕ (ϕ (γ (δ (τ ) u(τ ) )))
γ 1(δ1(t )V2 (t )) ≤ γ 1 Mκ κ σ 2 β2 (t0 ) b x0
1
0 X
1 0
X
u
1
t 0 ≤τ ≤t
u
1
u
2
1
t 0 ≤ ≤t
u
2 (τ )
u(τ ) U
))))
(3.26)
U
1
Let ϕ~ ( s ) := s + ρ ( s ) . Using (3.26) in conjunction with the inequality q(r + s ) ≤ q( b(r ) ) + q( ϕ ( s ) ) (which holds for
2
all r , s ≥ 0 and q ∈ N ), we obtain the following estimate, which holds for all t ≥ t 0 :
ϕ~(γ (δ (t )V (t ))) ≤ ϕ~ κ γ Mκ κ σ β (t ) b x
,0
+ ϕ κ σ β (t ) b x
,0
1
1
(( ( (( (
( ) )) )) ( ( (
((( ( (
))) ( ( ( (
))))))
2
1
2
0 X
2 0
1
1 0
(
0 X
) )) )))
(3.27)
+ sup ϕ~ ϕ ϕ ϕ γ 1u δ1u (τ ) u(τ ) U + γ 1 Mκ ϕ γ 2u δ 2u (τ ) u(τ ) U
t0 ≤τ ≤t
Estimates (3.24), (3.27) show that properties P1 and P2 of Lemma 2.16 hold for system Σ with V = V1 or
V = ϕ~(γ 1 (δ1 (t )V2 )) , for appropriate γ~ ∈ N and δ ∈ K + as defined by (3.9). Moreover, if β 1 , β 2 ∈ K + are
bounded then estimates (3.24), (3.27) show that properties P1 and P2 of Lemma 2.17 hold for system Σ with V = V1
~
or V = ϕ (γ (δ (t )V )) , for appropriate γ~ ∈ N and δ ∈ K + as defined by (3.9). Particularly, γ~ ∈ N satisfies
1
1
2
((( (
)) ( ( (
))))) and γ~(s) ≥ κ (ϕ (γ
γ~ ( s ) ≥ ϕ~ ϕ ϕ ϕ γ 1u (s ) + γ 1 Mκ ϕ γ 2u (s )
u
1
(s )))+ γ 1 (Mκ ( ϕ (γ 2u (s )))) , for all
s≥0
(3.28)
Step 3:
1
Let ϕ~ ( s ) := s + ρ ( s ) , κ~ ( s ) := s + ρ −1 (2s ) . Exploiting estimates (3.16), (3.21) in conjunction with the inequality
2
r + s ≤ max{κ~(r ) ; ϕ~ ( s )} (which holds for all r , s ≥ 0 ), we obtain:
⎧⎪ ⎛
V1 (t ) ≤ max ⎨ κ~⎜⎜ σ 1 (β 1 (t 0 ) L1 (t 0 ) , t − t 0
⎪⎩ ⎝
⎧ ⎛ ⎛
⎜ ⎜
⎪⎩ ⎝ ⎝
)+
(
sup γ 1u δ 1u (τ ) u (τ )
t 0 ≤τ ≤t
⎛
U
)⎞⎟⎟ ; sup ϕ~(γ
⎠
t 0 ≤τ ≤t
⎞
)⎞ ⎞⎟ ⎟
(
⎫
1
(δ 1 (τ ) V2 (τ ) )) ⎪⎬
⎪⎭
⎛
⎜
⎝
⎛
(3.29)
⎞⎞ ⎫
⎟
⎠ ⎠ ⎪⎭
(3.30)
⎪
⎪
ϕ~(γ 1(δ1 (t )V2 (t ))) ≤ max⎨ϕ~⎜ γ 1⎜ Mκ ⎜⎜σ 2 (β 2 (t0 ) L2 (t0 ) , t − t0 ) + sup γ 2u δ 2u (τ ) u (τ ) U ⎟⎟ ⎟ ⎟ ; ϕ~⎜ϕ −1⎜⎜ sup V1(τ ) ⎟⎟ ⎟ ⎬
⎝
t 0 ≤τ ≤t
⎠⎠⎠
⎝ t0 ≤τ ≤t
Let arbitrary ξ ∈ π (t0 , x0 , u, d ) and t ≥ ξ . Estimates (3.29), (3.30) in conjunction with estimates (3.17), (3.19) and
the weak semigroup property imply:
⎧
⎪ κ~ κ σ a x
1 1
0 X + c1 (t 0 ) + μ1 (ξ − t 0 ) , t − ξ
⎪
⎪
V1 (t ) ≤ max ⎨ sup κ~ κ σ 1 p1u q1u (τ ) u (τ ) U , 0 , sup κ~ (κ (σ 1 ( p1 ( V 2 (τ ) ) , t − ξ
t 0 ≤τ ≤ξ
⎪t0 ≤τ ≤ξ
⎪
u
~
~
ϕ (γ 1 (δ 1 (τ ) V 2 (τ ) )) , sup κ ϕ γ 1 δ 1u (τ ) u (τ ) U
⎪ξsup
ξ ≤τ ≤t
⎩ ≤τ ≤t
(( ( (
)
(( ( (
) )))
(( (
20
)))
)))
⎫
⎪
⎪
))) ⎪⎬
⎪
⎪
⎪
⎭
(3.32)
⎫
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎭
⎧
⎪
⎪ϕ~ γ Mκ κ σ a x
2 2
0 X + c2 (t0 ) + μ 2 (ξ − t0 ) , t − ξ
⎪ 1
⎪
u u
~
~
ϕ (γ 1(δ1 (t )V2 (t ))) ≤ max⎨ sup ϕ γ 1 Mκ ϕ γ 2 δ 2 (τ ) u (τ ) U , sup ϕ~ γ 1 Mκ κ σ 2 p2u q2u (τ ) u (τ ) U , 0
t 0 ≤τ ≤ξ
⎪ξ ≤τ ≤t
⎪ ⎛
⎞
⎪ ϕ~⎜ ϕ −1⎛⎜ sup V (τ ) ⎞⎟ ⎟ , sup ϕ~(γ (Mκ (κ (σ ( p ( V (τ )) , t − ξ )))))
1
2 2
1
⎪ ⎜ ⎜⎝ ξ ≤τ ≤t 1 ⎟⎠ ⎟ t0 ≤τ ≤ξ
⎠
⎩ ⎝
( ( (( ( ( )
(( ( ( (
)))))
( ( (( ( (
)))))
) )))))
(3.33)
Estimate (3.33) combined with estimate (3.24) gives:
⎧
⎪
⎪ϕ~ γ Mκ κ σ a x
2 2
0 X + c2 (t0 ) + μ 2 (ξ − t0 ) + p2 ( κ ( A)) , t − ξ
⎪ 1
⎪
ϕ~(γ 1(δ1 (t )V2 (t ))) ≤ max⎨ sup ϕ~ γ 1 Mκ ϕ γ 2u δ 2u (τ ) u (τ ) U , sup ϕ~ γ 1 Mκ κ σ 2 p2u q2u (τ ) u (τ ) U , 0
t 0 ≤τ ≤ξ
⎪ξ ≤τ ≤t
⎪ ⎛
⎞
⎪ ϕ~⎜ ϕ −1⎛⎜ sup V (τ ) ⎞⎟ ⎟ , sup ϕ~(γ (Mκ (κ (σ ( p ( ϕ (B )) , 0 )))))
1
1
2 2
⎟ ⎟ t ≤τ ≤t
⎪ ⎜ ⎜⎝ ξ ≤τ ≤t
⎠⎠ 0
⎩ ⎝
( ( (( ( ( )
(( ( ( (
where
(( (
B = κ (ϕ (γ (δ
(
A = κ κ σ 1 β 1 (t 0 ) b x 0
u
1
u
1 (τ )
u (τ )
X
U
)))))
( ( (( ( (
)))))
⎫
⎪
⎪
⎪
⎪
⎬ (3.34)
⎪
⎪
⎪
⎪
⎭
) )))))
) , 0 )) )+ γ (Mκ (κ (σ (β (t ) b( x ) , 0 )) ))
)))+ γ (Mκ ( ϕ (γ (δ (τ ) u(τ ) ))))
1
2
u
2
1
2
0 X
0
u
2
U
(
Similarly, estimate (3.18) combined with estimate (3.24) and inequality (3.8) (which implies Li (t 0 ) ≤ b x 0
i = 1,2 ), gives:
) , 0 ) + sup γ (δ (τ ) u (τ ) )
τ
~
) , 0 )) )+ γ (Mκ (κ (σ (β
+ γ (δ (t ) κ (κ (κ (σ (β (t ) b( x
~
+ sup γ (δ (t ) ϕ (κ (ϕ (γ (δ (τ ) u (τ ) ))) + γ (Mκ ( ϕ (γ (δ
(
(
V 2 (t ) ≤ σ 2 β 2 (t 0 ) b x 0
2
2
t 0 ≤τ ≤ t
1
2
X
1
where
t0 ≤ ≤t
u
2
0 X
0
u
1
2
u
2
u
1
U
X
),
U
( x ) , 0 )) ))))
u (τ ) ))))))
1
2
2 (t 0 ) b
1
u
2
u
2 (τ )
0 X
(3.35)
U
~
δ 2 (t ) := max δ 2 (τ )
0 ≤τ ≤t
Consequently, by combining estimates (3.32) and (3.35) we obtain:
⎧
⎪ κ~ κ σ a x
, κ~ (κ (σ 1 ( p1 ( κ (C ) ) , t − ξ )))
1 1
0 X + c1 (t 0 ) + μ1 (ξ − t 0 ) , t − ξ
⎪
⎪
V1 (t ) ≤ max ⎨ sup κ~ κ σ 1 p1u q1u (τ ) u (τ ) U , 0 , sup κ~ (κ (σ 1 ( p1 ( ϕ ( D) ) , 0 )))
t 0 ≤τ ≤t
⎪t0 ≤τ ≤ξ
⎪
u
~
~
ϕ (γ 1 (δ 1 (τ ) V 2 (τ ) )) , sup κ ϕ γ 1 δ 1u (τ ) u (τ ) U , κ~ (κ (σ 1 ( p1 ( E ) , t − ξ
⎪ξsup
ξ ≤τ ≤t
⎩ ≤τ ≤t
(( ( (
)
(( ( (
)))
) )))
(( (
)))
⎫
⎪
⎪
⎪
⎬
⎪
)))⎪⎪
⎭
where
C :=
(
(
σ 2 β 2 (t 0 ) b x 0
(
D := γ 2u δ 2u (τ )
), 0 ) + γ (δ~ (ξ ) κ (κ (κ (σ (β (t ) b( x ), 0 )))+ γ (Mκ (κ (σ (β (t ) b( x ), 0 ))))))
⎛ ⎛1
⎞⎞
u (τ ) ) + γ ⎜⎜ ϕ ⎜ ϕ (κ (ϕ (γ (δ (τ ) u (τ ) ))) + γ (Mκ ( ϕ (γ (δ (τ ) u (τ ) )))))⎟ ⎟⎟
2
X
2
2
1
2
U
2
⎝ ⎝
u
1
1
0
u
1
0 X
U
⎛ ⎛1 ~
⎞⎞
E := γ 2 ⎜⎜ κ ⎜ δ 22 (ξ ) ⎟ ⎟⎟
⎠⎠
⎝ ⎝2
21
1
1
2
u
2
2
u
2
0
0 X
U
⎠⎠
(3.36)
S1 , S 2 ∈ KL , continuous functions
From (3.34) and (3.36) we conclude that there exist functions
( )
M1, M 2 : ℜ
+ 3
→ ℜ + and γ~ ∈ N such that the following estimates hold for all ξ ∈ π (t0 , x0 , u, d ) and t ≥ ξ :
( (
)
)
⎧S2 M 2 t0 ,ξ − t0 , x0 X , t − ξ
⎪⎪
ϕ~(γ 1(δ1(t )V2 (t ) )) ≤ max⎨ ~⎛ −1⎛
⎞⎞
~
⎪ ϕ ⎜⎜ϕ ⎜⎜ sup V1 (τ ) ⎟⎟ ⎟⎟ , sup γ δ (τ ) u (τ ) U
t
≤
τ
≤
t
⎪⎩ ⎝ ⎝ ξ ≤τ ≤t
⎠⎠ 0
(
( (
)
)
⎧ S1 M 1 t 0 , ξ − t 0 , x 0 X , t − ξ
⎪
V1 (t ) ≤ max ⎨
sup ϕ~ (γ 1 (δ 1 (τ ) V 2 (τ ) )) , sup γ~ δ (τ ) u (τ )
⎪ξ ≤τ ≤t
t 0 ≤τ ≤t
⎩
(
U
)
⎫
⎪⎪
⎬
⎪
⎪⎭
)
(3.37)
⎫
⎪
⎬
⎪
⎭
(3.38)
where δ ∈ K + is defined by (3.9). Notice that if β 1 , β 2 , c1 , c 2 , δ 2 ∈ K + are bounded, then the functions
( )
3
M 1 , M 2 : ℜ + → ℜ + are independent of t 0 ∈ ℜ + (but still depend on ξ − t 0 ). Moreover, the function γ~ ∈ N in
addition to (3.28) satisfies for all s ≥ 0 :
{ (( (
γ~ ( s ) ≥ max κ~ κ σ 1 p1u (s ) , 0
{( ( ( (
))) , κ~ (κ (σ
1
)))) , ϕ~ (γ (Mκ (κ (σ (p
γ~ ( s ) ≥ max ϕ~ γ 1 Mκ ϕ γ 2u (s )
1
2
(((
u
2
( p1 ( ϕ ( D ( s )) ) , 0 ))) ,
((
)) }
κ~ ϕ γ 1u (s )
(3.39a)
(s ) , 0 ))))), ϕ~ (γ 1 (Mκ (κ (σ 2 ( p 2 ( ϕ (B ( s ) )) , 0 ))))) }
)) ( ( (
))))
((
)) ( ( (
(3.39b)
))) are
⎛ ⎛1
⎞⎞
where D( s ) := γ 2u (s ) + γ 2 ⎜⎜ ϕ ⎜ ϕ 2 κ ϕ γ 1u (s ) + γ 1 Mκ ϕ γ 2u (s ) ⎟ ⎟⎟ and B ( s ) := κ ϕ γ 1u (s ) + γ 1 Mκ ϕ γ 2u (s )
2
⎠⎠
⎝ ⎝
functions of class N .
We define:
a 1 ( h , T , R ) :=
⎧
sup ⎨ V1 (t 0 + h ) −
⎩
sup
t 0 ≤τ ≤ t 0 + h
(
γ~ δ (τ ) u (τ )
U
);
x0
X
U
);
⎫
≤ R , t 0 ∈ [ 0, T ] , d ∈ M D , u ∈ M U ⎬
⎭
(3.40)
a 2 ( h , T , R ) :=
⎧⎪
sup ⎨ ϕ~ (γ 1 (δ 1 ( t 0 + h )V 2 (t 0 + h ) )) − sup γ~ δ (τ ) u (τ )
⎪⎩
t 0 ≤τ ≤ t 0 + h
(
x0
X
≤ R , t 0 ∈ [0, T ] , d ∈ M
D
l1 := lim sup a1 (h, T , R) , l 2 := lim sup a 2 (h, T , R)
h → +∞
⎫⎪
,u ∈ MU ⎬
⎪⎭
(3.41)
(3.42)
h → +∞
By virtue of (3.24) and (3.27), the limits defined in (3.42) exist and are finite. Definition (3.42) implies that for every
ε > 0 , T ≥ 0 and R ≥ 0 , there exists a τ := τ (ε , T , R ) ≥ 0 , such that:
a1 (h, T , R) ≤ l1 + ε ; a 2 (h, T , R ) ≤ l 2 + ε , ∀h ≥ τ
(3.43)
By virtue of the Weak Semigroup Property for system Σ , there exists a constant r > 0 , such that for each
(t 0 , x 0 , u, d ) ∈ ℜ + × X × M U × M D
we
have
π (t 0 , x 0 , u , d ) ∩ [t 0 + τ , t 0 + τ + r ] ≠ ∅ .
Let
ξ ∈ π (t 0 , x 0 , u, d ) ∩ [t 0 + τ , t 0 + τ + r ] . Estimates (3.37), (3.38) in conjunction with definitions (3.40), (3.41) and
inequalities (3.43) give:
(
V1 (t ) − sup γ~ δ (τ ) u (τ )
t 0 ≤τ ≤t
U
) ≤ max{ S (M (t
1
22
1 0 ,ξ
− t 0 , x0
X
), t − ξ ); l
2
+ε
}
(3.44)
( (
)
)
⎧S2 M 2 t0 , ξ − t0 , x0 X , t − ξ
⎪⎪
~
~
ϕ (γ 1(δ1(t )V2 (t ) )) − sup γ δ (τ ) u (τ ) U ≤ max⎨ ~⎛ −1⎛
~
t 0 ≤τ ≤t
⎪ ϕ ⎜⎜ϕ ⎜⎜ l1 + ε + sup γ δ (τ ) u (τ ) U
t 0 ≤τ ≤t
⎪⎩ ⎝ ⎝
(
(
)
)
(
(
)⎞⎟⎟ ⎞⎟⎟ −
⎠⎠
(
sup γ~ δ (τ ) u (τ ) U
t 0 ≤τ ≤t
)
)
⎫
⎪⎪
⎬
⎪
⎪⎭
(3.45)
1
Using the identity ϕ~ ϕ −1 ( s ) = s − ρ ϕ −1 ( s ) and inequality (3.45) we get:
2
(
)
( (
⎧
ϕ~(γ 1(δ1 (t )V2 (t ))) − sup γ~ δ (τ ) u (τ ) U ≤ max⎨S2 M 2 t0 ,ξ − t0 , x0
t 0 ≤τ ≤t
⎩
X
) , t − ξ ) ; l + ε − 12 ρ (ϕ
−1
1
( l1 ))⎫⎬
⎭
(3.46)
The properties of the KL functions in conjunction with estimates (3.44), (3.46), the fact that ξ ∈ [t 0 + τ , t 0 + τ + r ]
and definitions (3.40), (3.41), (3.42), give for all ε > 0 :
1
l1 ≤ l 2 + ε ; l2 ≤ l1 + ε − ρ ϕ −1 ( l1 )
2
From the first inequality we obtain l1 ≤ l 2 . The second inequality implies ρ ϕ −1( l1 ) ≤ 2ε for all ε > 0 , which
directly gives l1 = l 2 = 0 .
(
)
(
)
Definitions (3.40), (3.41) and (3.42) imply that P3 of Lemma 2.16 holds for system Σ with V = V1 or
V = ϕ~(γ (δ (t )V )) , for appropriate γ~ ∈ N (which satisfies (3.28) and (3.39)) and δ ∈ K + as defined by (3.9).
1
1
2
Notice that if β 1 , β 2 , c1 , c 2 , δ 2 ∈ K + are bounded then definitions (3.40), (3.41) are modified as follows:
⎧
a 1 ( h , R ) := sup ⎨ V1 ( t 0 + h ) −
⎩
a 2 ( h , R ) :=
sup
t 0 ≤τ ≤ t 0 + h
⎧
sup ⎨ ϕ (γ 1 (δ 1 ( t 0 + h )V 2 ( t 0 + h ) )) −
⎩
(
γ~ δ (τ ) u (τ )
sup
t 0 ≤τ ≤ t 0 + h
(
U
);
γ~ δ (τ ) u (τ )
x0
U
X
);
≤ R , t0 ≥ 0 , d ∈ M
x0
X
D
⎫
,u ∈ MU ⎬
⎭
≤ R , t0 ≥ 0 , d ∈ M
D
⎫
,u ∈ MU ⎬
⎭
(3.47)
(3.48)
Similar arguments as above show that property P3 of Lemma 2.17 holds for system Σ with V = V1 or
V = ϕ~(γ (δ (t )V )) , for appropriate γ~ ∈ N (which satisfies (3.28) and (3.39)) and δ ∈ K + as defined by (3.9). The
1
1
2
proof is complete.
<
Remark 3.6: If the functions γ 1u , p1u , γ 2u , p 2u ∈ N are all identically zero then it follows that the gain function γ ∈ N
is identically zero. Indeed, the reader should notice that γ ∈ N may be selected as γ ( s ) := a(4γ~ ( s ) ) ∈ N , where
a ∈ N is the function involved in hypothesis (H4) and γ~ ∈ N is the function that satisfies (3.28), (3.39a,b).
Moreover, notice that for the input free case Theorem 3.1 and Corollary 3.4 imply (Uniform) Robust Global
Asymptotic Output Stability (RGAOS) for the corresponding system. The following example shows the applicability
of this particular remark to systems with impulses at fixed times.
Example 3.7: Consider the following system:
z& (t ) = Az (t ) + g ( x (t ))
(3.49a)
x& (t ) = f ( x(t )) , t ∉ π
⎛
⎞
x(τ i ) = h⎜⎜ lim− x(t ) ⎟⎟
⎝ t →τ i
⎠
z (t ) ∈ ℜ k , x(t ) ∈ ℜ n
23
(3.49b)
where A ∈ ℜ k ×k is a Hurwitz matrix, π = {τ i }i∞=0 is a partition of ℜ + with diameter r > 0 , f : ℜ n → ℜ n ,
g : ℜ n → ℜ k , h : ℜ n → ℜ n are continuous vector fields, f (x ) being locally Lipschitz with respect to x ∈ ℜ n , with
f (0) = 0 , g (0) = 0 , h(0) = 0 . Notice that subsystem (3.49a) is a system described by ODEs which satisfies
hypotheses (A1-3) of Example 2.8. Hence, subsystem (3.49a) satisfies the BIC property and 0 ∈ ℜ k is a robust
equilibrium point from the input x . Moreover, subsystem (3.49b) is a hybrid system with impulses at fixed times,
which satisfies hypotheses (Q1-4) of Example 2.12. Hence, subsystem (3.49b) satisfies the BIC property and 0 ∈ ℜ n
is a robust equilibrium point (from the zero input). We remark that since both subsystems (3.49a,b) satisfy the
classical semigroup property and consequently the composite system (3.49) can be regarded as the feedback
interconnection of subsystems (3.49a,b).
Since A ∈ ℜ k ×k is Hurwitz, it follows that subsystem (3.49a) satisfies the UISS property from the input x .
Moreover, if there exists a C 1 positive definite and radially unbounded function V : ℜ n → ℜ + and constants
c1 , c 2 ∈ ℜ with c 2 ≠ 0 , μ , λ > 0 , such that:
∇V ( x) f ( x) ≤ −c1V ( x) , ∀x ∈ ℜ n
V (h( x)) ≤ exp(−c 2 )V ( x ) , ∀x ∈ ℜ n
−c 2 card (π ∩ [ s, s + t ) ) ≤ μ + (c1 − λ )t ∀s, t ∈ ℜ +
(3.50a)
(3.50b)
(3.50c)
where card (S ) denotes the cardinal number of the set S , then Theorem 1 in [5] implies that 0 ∈ ℜ n is GAS for
subsystem (3.49b). Taking into account Remark 3.2(b) and Remark 3.6, we conclude that 0 ∈ ℜ k × ℜ n is uniformly
GAS for the composite system (3.49) under the hypotheses stated above.
<
4. Application to Partial State Sampled-Data Control
In this section we present applications of the Small-Gain results (Theorem 3.1 and Corollary 3.4) to partial state
sampled-data control problems. It should be emphasized that sampled-data control systems cannot be handled with
Small-Gain results that have appeared so far in the literature, since sampled-data control systems do not satisfy the
classical semigroup property (see Example 2.11).
Consider the following control system described by ODEs:
z& = f (t , d , z , x, u )
z ∈ ℜ k , d ∈ D , u ∈U , t ≥ 0
x& = Ax + Bv + Bg (t , d , z, u )
x ∈ ℜ n , v ∈ ℜ , d ∈ D , u ∈U , t ≥ 0
(4.1a)
(4.1b)
where ( A, B) is a controllable pair of matrices, D ⊂ ℜ l is a compact set, U ⊆ ℜ p is non-empty with 0 ∈ U and the
mappings f : ℜ + × D × ℜ k × ℜ n × U → ℜ k , g : ℜ + × D × ℜ k × U → ℜ are continuous, locally Lipschitz in ( z , x) ,
uniformly in d ∈ D with f (t , d ,0,0,0) = 0 , g (t , d ,0,0) = 0 for all (t , d ) ∈ ℜ + × D . The problem we consider is the
(W)ISS stabilization problem for (4.1) with sampled-data feedback applied with zero order hold and depending only
on x ∈ ℜ n , i.e., we want to find a function k : ℜ n → ℜ with k (0) = 0 and a constant r > 0 such that system (4.1a)
with
x& (t ) = Ax(t ) + Bk ( x(τ i )) + Bg (t , d (t ), z (t ), u (t )) , t ∈ [τ i , τ i +1 )
τ i +1 = τ i + exp(− w(τ i ) ) r , w(t ) ∈ ℜ +
24
(4.2)
satisfies the WISS property from the inputs (u, w) . Notice that the input w has been introduced in order to quantify
the uncertainty in sampling times, i.e., we have to guarantee stability properties for the closed-loop system (4.1a)(4.2) for all sampling schedules of diameter less than or equal to r > 0 . To this purpose we make the following
assumptions:
(A1) System (4.1a) satisfies the WISS property from the inputs x and u . Specifically, there exist functions σ ∈ KL ,
∞
∞
∞
β , δ 1u ∈ K + , γ 1 , γ 1u ∈ N such that for all (t 0 , z 0 , d , x, u ) ∈ ℜ + × ℜ k ×Lloc
(ℜ + ; D ) ×Lloc
(ℜ + ; ℜ n ) ×Lloc
(ℜ + ; U ) the
z (t 0 ) = z 0
corresponding
to
inputs
solution
of
(4.1)
with
initial
condition
∞
∞
∞
(d , x, u ) ∈Lloc
(ℜ + ; D) ×Lloc
(ℜ + ; ℜ n ) ×Lloc
(ℜ + ; U ) satisfies the following estimate for all t ≥ t 0 :
(
z (t ) ≤ σ (β (t 0 ) z 0 , t − t 0 ) + sup γ 1 ( x (τ ) ) + sup γ 1u δ 1u (τ ) u (τ )
t 0 ≤τ ≤ t
(A2) There exist functions
+
δ 2u ∈ K + , γ 2 , γ 2u ∈ N
t 0 ≤τ ≤ t
)
(4.3)
such that the following inequality holds for all
(t , z , d , u ) ∈ ℜ × ℜ × D × U :
k
(
g (t , d , z , u ) ≤ γ 2 ( z ) + γ 2u δ 2u (t ) u
)
(4.4)
(A3) There exist a function ρ ∈ K ∞ and a constant R ≥ 1 such that:
(
(
)) ( (
(
)))
γ 1 R −1γ 2 (s ) + ρ R −1γ 2 (s ) + ρ γ 1 R −1γ 2 (s ) + ρ R −1γ 2 (s ) ≤ s , ∀s ≥ 0
(4.5)
For example, hypothesis (A3) holds if γ i ( s ) = K i s , where K i ≥ 0 ( i = 1,2 ), i.e., if the gain functions are linear.
Next we show that the problem of WISS stabilization problem for (4.1) with sampled-data feedback applied with
zero order hold and depending only on x ∈ ℜ n is solvable under hypotheses (A1-3) by linear feedback. The proof of
this result will be made by making use of Corollary 3.4.
Notice that since ( A, B) is a controllable pair of matrices, it follows that for every μ > 0 , R ≥ 1 there exist a
symmetric positive definite matrix P ∈ ℜ n×n , a vector k ∈ ℜ n , constants Q1 , Q 2 > 0 such that the following
inequalities hold for all ( x, u ) ∈ ℜ n × ℜ :
Q1 x
2
≤ x ′Px ≤ Q 2 x
2
2 x ′P ( A + Bk ′) x + 2 x ′PBu ≤ −4 μ x ′Px +
Q1
4μ R 2
(4.6)
u2
Next we show the following claim.
Claim: For every μ > 0 , R ≥ 1 there exists a vector k ∈ ℜ n , constants M , r > 0 , such that for all
∞
∞
∞
∞
(t0 , x0 , d, z, u, w) ∈ℜ+ × ℜn ×Lloc
(ℜ+ ; D) ×Lloc
(ℜ+ ; ℜk ) ×Lloc
(ℜ+ ;U ) ×Lloc
(ℜ+ ; ℜ+ ) the solution of the hybrid system:
x& (t ) = Ax(t ) + Bk ′x (τ i ) + Bg (t , d (t ), z (t ), u (t )) , t ∈ [τ i , τ i +1 )
(4.7)
τ i +1 = τ i + exp(− w(τ i ) ) r , w(t ) ∈ ℜ +
initial
condition
x(t 0 ) = x 0
k
∞
+
∞
+
∞
+
∞
(d , z, u, w) ∈ Lloc (ℜ ; D) ×Lloc (ℜ ; ℜ ) ×Lloc (ℜ ;U ) ×Lloc
(ℜ+ ; ℜ+ )
with
corresponding
t0 ≤τ ≤t
where δ 2u ∈ K + , γ 2 , γ 2u ∈ N are the functions involved in (4.4).
25
inputs
satisfies the following estimate for all t ≥ t 0 :
(
x(t ) ≤ M exp(− μ (t − t 0 ) ) x 0 + sup R −1γ 2 ( z (τ ) ) + sup R −1γ 2u δ 2u (τ ) u (τ )
t 0 ≤τ ≤t
to
)
(4.8)
Proof of Claim: Let arbitrary μ > 0 , R > 1 . Since ( A, B) is a controllable pair of matrices, it follows that there
exists a symmetric positive definite matrix P ∈ ℜ n×n , a vector k ∈ ℜ n , constants Q1 , Q 2 > 0 such that inequalities
(4.6)
hold
for
+
Let
arbitrary
∞
∞
∞
∞
×Lloc
(ℜ + ; D) ×Lloc
(ℜ + ; ℜ k ) ×Lloc
(ℜ + ; U ) ×Lloc
(ℜ + ; ℜ + )
(t 0 , x 0 , d , z, u, w) ∈ ℜ × ℜ
x(t )
of
(4.7)
with
n
( x, u ) ∈ ℜ n × ℜ .
all
initial
x(t 0 ) = x 0
condition
and consider the solution
corresponding
to
inputs
∞
∞
∞
∞
(d , z, u, w) ∈ Lloc
(ℜ+ ; D) ×Lloc
(ℜ+ ; ℜk ) ×Lloc
(ℜ+ ;U ) ×Lloc
(ℜ+ ; ℜ+ )
(the solution exists for all t ≥ t 0 ). Finally, consider
the function V (t ) := x ′(t ) Px(t ) , which is absolutely continuous on [t 0 ,+∞) . By virtue of (4.6) the derivative of V (t )
satisfies a.e. on the interval [τ i , τ i +1 ) :
V& (t ) ≤ −4 μ V (t ) + 2 x ′PBk ′( x (τ i ) − x(t )) +
Q1
4μ R 2
g (t , d (t ), z (t ), u (t ))
2
(4.9)
Let r > 0 constant that satisfies:
r≤
2μ
1
; r≤
2
′
′
′
′
M Bk A + Bk (2 A + Bk + B ) + 2μ A + 2μ A + Bk
4 μR M B Bk ′ + A + A + Bk ′
where M :=
(4.10)
Q2
≥1.
Q1
It follows from (4.7) that
which directly implies
x(t ) − x(τ i ) ≤ r A sup x( s ) − x(τ i ) + r A + Bk ′ x(τ i ) + r B sup g ( s, d ( s ), z ( s ), u ( s )) ,
τ i ≤ s ≤t
x(t ) − x(τ i ) ≤
r A + Bk ′
1− r A
τ i ≤ s ≤t
x(τ i ) +
rB
1− r A
sup g ( s, d ( s ), z ( s ), u ( s )) , for all t ∈ [τ i , τ i +1 ) .
τ i ≤ s ≤t
Moreover, the previous inequality in conjunction with the triangle inequality x(τ i ) ≤ x(t ) − x(τ i ) + x(t ) implies the
estimate x(t ) − x(τ i ) ≤
r A + Bk ′
1 − r A − r A + Bk ′
x(t ) +
rB
1 − r A − r A + Bk ′
sup g ( s, d ( s ), z ( s ), u ( s )) , for all t ∈ [τ i , τ i +1 ) .
τ i ≤ s ≤t
Using the previous inequality in conjunction with (4.9) and completing the squares, we obtain for almost all
t ∈ [τ i , τ i +1 ) :
⎛
⎛
⎞
rM Bk ′ A + Bk ′ (2 A + Bk ′ + B ) ⎞
r B Q 2 Bk ′
⎟ V (t ) + ⎜ Q1 +
⎟ sup g ( s, d ( s ), z ( s ), u ( s )) 2
V& (t ) ≤ −⎜ 4μ −
2
⎜
⎟
⎜
1 − r ( A + A + Bk ′ )
1 − r ( A + A + Bk ′ ) ⎟⎠ τ i ≤ s ≤t
⎝
⎠
⎝ 4μ R
(4.11)
It follows from inequalities (4.10), (4.11) that the following estimate holds for the derivative of V (t ) a.e. on the
interval [τ i , τ i +1 ) :
V& (t ) ≤ −2 μ V (t ) +
Q1
2μ R 2
sup g ( s, d ( s), z ( s ), u ( s ))
t 0 ≤ s ≤t
2
(4.12)
Notice that since estimate (4.12) does not depend on the particular interval [τ i , τ i +1 ) , we may conclude that estimate
t ≥ t0 .
Estimate
(4.12)
implies
directly
that
(4.12)
holds
a.e.
for
V (t ) ≤ exp(− 2 μ (t − t 0 ) )V (t 0 ) +
Q1
R2
sup g ( s, d ( s ), z ( s ), u ( s ))
t0 ≤ s ≤t
2
for all t ≥ t 0 . Finally, estimate (4.8) is an immediate
consequence of the previous inequality, definitions V (t ) := x ′(t ) Px(t ) and M :=
and (4.6). The proof of the claim is complete.
<
26
Q2
≥ 1 as well as inequalities (4.4)
Q1
By making use of Corollary 3.4 and specifically Remark 3.5(b), we may conclude that the closed-loop system (4.1a)
with (4.7) satisfies the WISS property from the inputs u and w , when R > 1 is chosen to be greater than or equal to
the constant involved in hypothesis (A3). Moreover the gain function for the input w is identically zero.
Furthermore, if the functions δ 1u , δ 2u ∈ K + are bounded, then the closed-loop system (4.1a) with (4.7) satisfies the
ISS property from the inputs u and w . Finally, if in addition β ∈ K + is bounded, then the closed-loop system (4.1a)
with (4.7) satisfies the UISS property from the inputs u and w .
Example 4.1: The following planar system described by ODEs:
z& = − z 3 + z x
x& = d (t ) z 2 + u + v
(4.13)
( z , x) ′ ∈ ℜ 2 , u , v ∈ ℜ
1
then the feedback law v(t ) = − x(t ) , guarantees the
2
UISS property for the closed-loop system from the input u ∈ ℜ . The proof of this fact is made by using a slightly
modified version of the Small-Gain Theorem presented in [10]. Here we study the possibility of robustly globally
stabilizing the origin for system (4.13), using the following feedback law with zero order hold and positive sampling
rate:
is studied in [9] where it is shown that if d (t ) ≡ d with d <
v(t ) = − x(τ i ) , t ∈ [τ i , τ i +1 )
τ i +1 = τ i + exp(− w(τ i )) r , w(t ) ∈ ℜ +
(4.14)
for time-varying disturbances d (t ) ∈ D := [−δ , δ ] with δ ∈ (0,1) .
First notice that system (4.13) is a system of the form (4.1) with f (t , d , z , x, u ) = − z 3 + z x , g (t , d , z , u ) = d z 2 + u ,
B = [1] , A = [0] . Moreover, hypothesis (A2) holds with γ 2 ( s ) = δ s 2 , γ 2u ( s ) := s and δ 2u (t ) ≡ 1 .
Working exactly as in [9] it may be shown that for every ε ∈ (0,1) the subsystem Σ 1 :
z& = − z 3 + z x
satisfies the UISS property from the input x with gain function γ 1 ( s) :=
hypothesis (A1) with γ 1 ( s ) :=
(4.15)
s
. Thus the subsystem Σ 1 satisfies
1− ε
s
, β (t ) ≡ 1 , γ 1u ( s ) ≡ 0 and appropriate σ ∈ KL .
1− ε
For every δ ∈ (0,1) there exist ε ∈ (0,1) and L > 0 such that:
(1 + L)
(1 + L)δ
≤1
1− ε
(4.16)
Selecting ρ ( s ) := Ls with L > 0 , we conclude from (4.16) that (4.5) holds with R = 1 . Finally, since (4.6) holds with
Q1 = Q 2 = 1 , P = [1] , μ =
1
and k = −1 , we conclude that (4.13) with (4.14) satisfies the UISS property from the
4
inputs u and w . Moreover the gain function for the input w is identically zero. The maximum allowable sampling
<
period ( r ) may be determined by inequalities (4.10), which give r = 1 / 7 .
27
5. Conclusions
A Small-Gain Theorem, which can be applied to a wide class of systems that includes systems that satisfy the
weak semigroup property, is presented in the present work. The result generalizes all existing results in the literature
and exploits notions of weighted, uniform and non-uniform Input-to-Output Stability (IOS) property. Moreover, the
Small-Gain Theorem of the present work is a method for establishing qualitative properties expressed in a very
general framework unifying works from various fields as well as different stability notions. The results presented in
the paper can be extended without much difficulty to the case of local stability notions.
Applications to partial state feedback stabilization problems with sampled-data feedback applied with zero order
hold and positive sampling rate, are also presented. It should be emphasized that sampled-data control systems cannot
be handled with Small-Gain results that have appeared so far in the literature, since sampled-data control systems do
not satisfy the classical semigroup property. The results are illustrated by examples, which show the usefulness of the
main result for the stability analysis of interconnected systems.
Acknowledgments: This work is supported partly by the U.S. NSF under grants ECS-0093176, OISE-0408925
and DMS-0504462.
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29
Appendix
Proof of Lemma 2.13: Lemma 3.5 in [20] guarantees that the control system Σ := (X, Y , M U , M D , φ , π , H ) which
has the BIC property is RFC from the input u ∈ M (U ) if and only if there exist functions q ∈ K + , a ∈ K ∞ and a
constant R ≥ 0 such that the following estimate holds for all (t 0 , x 0 , d , u ) ∈ ℜ + × X × M D × M U and t ≥ t 0 :
φ (t , t 0 , x 0 , u, d )
X
⎛
≤ q (t ) a⎜⎜ R + x 0
⎝
X
+ sup u (τ )
U
t 0 ≤τ ≤t
⎞
⎟
⎟
⎠
(A1)
It should be emphasized that all results in [20] were proved under the assumption of the classical semigroup property
for the control system. However, the proof of Lemma 3.5 does not depend on the semigroup property and
consequently may be repeated as it stands for a system, which satisfies the weak semigroup property. Let β ∈ K +
arbitrary. Using (A1) we obtain:
β (t ) φ (t , t 0 , x 0 , u , d )
X
≤
⎧⎪
1 2
1
q (t ) β 2 (t ) + max ⎨ a 2 (3R ) ; a 2 3 x 0
2
2
⎪⎩
(
⎧⎪
≤ max ⎨ q 2 (t ) β 2 (t ) ; a 2 (3R) ; a 2 3 x 0
⎪⎩
(
⎧⎪
≤ max ⎨ γ (t ) ; a 2 3 x 0
⎪⎩
(
); sup a ( 3 u(τ ) )⎫⎪⎬⎪
2
X
U
t 0 ≤τ ≤t
⎭
); sup a (3 u(τ ) )⎫⎪⎬⎪
2
X
U
t 0 ≤τ ≤t
(A2)
⎭
); sup a ( 3 u(τ ) )⎫⎪⎬⎪
2
X
U
t 0 ≤τ ≤t
⎭
where γ (t ) = q 2 (t ) β 2 (t ) + a 2 (3R ) . Define:
a(T , s ) := max{ γ (t 0 + h) − γ (t 0 ) : h ∈ [0, s ] , t 0 ∈ [0, T ] }
(A3)
Clearly, definition (A3) implies that for each fixed s ≥ 0 a( ⋅ , s ) is non-decreasing and for each fixed T ≥ 0 a(T , ⋅ )
is non-decreasing. Furthermore, continuity of γ guarantees that for every T ≥ 0 lim+ a (T , s ) = a (T ,0) = 0 . It turns
s →0
out from Lemma 2.3 in [17], that there exist functions ζ ∈ K ∞ and κ ∈ K
+
such that
( )
2
a(T , s ) ≤ ζ (κ (T ) s ) , ∀(T , s ) ∈ ℜ +
(A4)
Combining definition (A3) with inequality (A4), we conclude that for all t 0 ≥ 0 and t ≥ t 0 , it holds that:
⎛1
⎝2
γ (t ) ≤ γ (t 0 ) + ζ (κ (t 0 ) (t − t 0 )) ≤ γ (t 0 ) + ζ ⎜ κ 2 (t 0 ) +
(
) (
)
{
1
(t − t 0 )2 ⎞⎟
2
⎠
(
) (
≤ γ (t 0 ) + ζ κ 2 (t 0 ) + ζ (t − t 0 )2 ≤ max 2γ (t 0 ) + 2ζ κ 2 (t 0 ) ; 2ζ (t − t 0 )2
)}
The above inequality in conjunction with (A2) implies that (2.9) holds for all (t 0 , x 0 , d , u ) ∈ ℜ + × X × M D × M U and
( )
(
)
t ≥ t 0 with μ (t ) := 2ζ t 2 , c(t ) := 2γ (t ) + 2ζ κ 2 (t ) , a( s ) := a 2 (3s ) and p( s ) := a 2 (3s ) . The proof is complete.
<
Proof of Lemma 2.16: As in the proof of Proposition 2.2 in [17], let T , h ≥ 0 , s ≥ 0 and define:
(
)
⎧⎪
a (T , s ) := sup ⎨ V (t ,φ (t , t0 , x0 , u , d ), u (t )) − sup γ δ (τ ) u (τ ) U ; x0
⎪⎩
t 0 ≤τ ≤ t
X
⎫⎪
≤ s , t ≥ t0 ∈ [0, T ] , d ∈ M D , u ∈ M U ⎬
⎪⎭
⎧⎪
M (h, T , s ) := sup ⎨ V (t0 + h,φ (t0 + h, t0 , x0 , u , d ), u (t )) − sup γ δ (τ ) u (τ ) U ; x0
⎪⎩
t 0 ≤τ ≤ t 0 + h
(
)
X
(A5)
⎫⎪
≤ s , t0 ∈ [0, T ] , d ∈ M D , u ∈ M U ⎬
⎪⎭
(A6)
30
First notice that by virtue of property P1 it holds that a(T , s ) < +∞ for all T ≥ 0 , s ≥ 0 . Moreover, notice that since
0 ∈ X is a robust equilibrium point from the input u ∈ M U and V (t ,0,0) = 0 for all t ≥ 0 , we have a (T , s ) ≥ 0 for
all T ≥ 0 , s ≥ 0 . Furthermore, notice that M is well-defined, since by definitions (A5), (A6) the following inequality
is satisfied for all T , h ≥ 0 and s ≥ 0 :
0 ≤ M (h, T , s ) ≤ a (T , s )
(A7)
Clearly, definition (A5) implies that for each fixed s ≥ 0 a( ⋅ , s ) is non-decreasing and for each fixed T ≥ 0 a(T , ⋅ )
is non-decreasing. Furthermore, property P2 asserts that for every T ≥ 0 lim+ a (T , s ) = 0 . Hence, the inequality
s →0
a(T ,0) ≥ 0 for all T ≥ 0 , in conjunction with lim+ a(T , s ) = 0 and the fact that a (T , ⋅ ) is non-decreasing implies
s →0
a( ⋅ ,0) = 0 . It turns out from Lemma 2.3 in [17], that there exist functions ζ ∈ K ∞ and q ∈ K + such that
( )
a(T , s ) ≤ ζ (q (T ) s ) , ∀(T , s ) ∈ ℜ +
2
(A8)
Without loss of generality we may assume that q ∈ K + is non-decreasing. Moreover, property P3 guarantees that for
every ε > 0 , T ≥ 0 and R ≥ 0 , there exists a τ = τ (ε , T , R ) ≥ 0 , such that
M (h, T , s ) ≤ ε for all h ≥ τ (ε , T , R ) and 0 ≤ s ≤ R
(A9)
Let
g ( s ) := s + s 2
(A10)
and let p be a non-decreasing function of class K + with p (0) = 1 and
lim p (t ) = +∞
(A11)
t →+∞
Define
⎧
⎫
M (h, T , s )
; T ≥ 0 , s > 0⎬
⎩ p (T ) g (ζ (q(T ) s ))
⎭
μ (h) := sup ⎨
(A12)
Obviously, by virtue of (A7), (A8) and (A10) the function μ : ℜ + → ℜ + is well defined and satisfies μ ( ⋅ ) ≤ 1 . We
show that lim μ (h) = 0 , equivalently, we establish that for any given ε > 0 , there exists a δ = δ (ε ) ≥ 0 such that
h → +∞
μ (h ) ≤ ε , for h ≥ δ (ε )
(A13)
Notice first, that for any given ε > 0 there exist constants a := a(ε ) and b := b(ε ) with 0 < a < b such that
x ∉ ( a, b) ⇒
x
x + x2
≤ε
(A14)
We next recall (A11), which asserts that, for the above ε for which (A14) holds, there exists a c := c(ε ) ≥ 0 such that
1
p(T ) ≥ for all T ≥ c . This by virtue of (A7), (A8), (A10) and (A14) yields:
ε
M (h, T , s )
≤ ε , ∀h ≥ 0 , when T ≥ c , or ζ (q (T ) s ) ∉ (a, b)
p (T ) g (ζ (q(T ) s ))
(A15)
Hence, in order to establish (A13), it remains to consider the case:
a ≤ ζ (q (T ) s ) ≤ b and 0 ≤ T ≤ c
31
(A16)
( )
Since, for each fixed (h, s ) ∈ ℜ +
have that
2
the mappings M (h, ⋅ , s ) , M (h, s, ⋅ ) , q( ⋅ ) and p ( ⋅ ) are non-decreasing, we
⎛
ζ −1 (b) ⎞⎟
M ⎜ h, c,
⎜
q (0) ⎟⎠
M (h, T , s )
⎝
≤
p (T ) g (ζ (q(T ) s ))
g (a )
(A17)
provided that (A16) holds. By using (A9) and (A17) with
ε := ε g (a ) , T := c , R :=
ζ −1 (b)
q ( 0)
it follows
⎛
⎛
ζ −1 (b) ⎞⎟
ζ −1 (b) ⎞⎟
M ⎜ h, c,
≤ ε g (a ) , for h ≥ δ (ε ) := τ ⎜ ε g (a ), c,
⎜
⎜
q(0) ⎟⎠
q(0) ⎟⎠
⎝
⎝
(A18)
By taking into account (A15), (A16), (A17), (A18) and definition (A12) of μ ( ⋅ ) it follows that (A13) holds with
δ = δ (ε ) as selected in (A18). Since ε > 0 was arbitrary we conclude that lim μ (h) = 0 . Consequently, there exists
h → +∞
+
a continuous strictly decreasing function μ : ℜ → (0,+∞) such that μ (h) ≥ μ (h) for all h ≥ 0 and lim μ (h) = 0 .
h → +∞
Thus, by recalling definition (A12) we obtain
( ) , ∀h ≥ 0
M (h, T , s ) ≤ μ (h)θ (T , s ) , ∀(T , s ) ∈ ℜ +
2
(A19)
where θ (T , s ) := p (T ) g (ζ (q (T ) s )) . Clearly, θ satisfies all hypotheses of Lemma 2.3 in [17] and therefore there exist
ζ 2 ∈ K ∞ and β ∈ K + such that
( )
θ (T , s ) ≤ ζ 2 ( β (T ) s) , ∀(T , s) ∈ ℜ +
2
(A20)
Thus definition (A6) implies that the following estimate holds for all u ∈ M U , (t 0 , x 0 , d ) ∈ ℜ + × X × M D and t ≥ t 0 :
V (t , φ (t , t 0 , x 0 , u , d ), u (t ))
(
≤ μ (t − t 0 ) ζ 2 β (t 0 ) x 0
Estimate (A21) implies (2.12) with σ ( s, t ) := μ (t )ζ 2 ( s ) .
X
)+ sup γ (δ (τ ) u (τ ) )
t 0 ≤τ ≤ t
(A21)
U
<
Proof of Lemma 2.17: As in the proof of Lemma 2.16, let h ≥ 0 , s ≥ 0 and define:
⎧⎪
a ( s ) := sup ⎨ V (t ,φ (t , t0 , x0 , u , d ), u (t )) − sup γ δ (τ ) u (τ ) U ; x0
⎪⎩
t 0 ≤τ ≤ t
(
)
X
⎫⎪
≤ s , t ≥ t0 ≥ 0 , d ∈ M D , u ∈ M U ⎬
⎪⎭
⎧⎪
M (h, s ) := sup ⎨ V (t0 + h,φ (t0 + h, t0 , x0 , u , d ), u (t )) − sup γ δ (τ ) u (τ ) U ; x0
⎪⎩
t 0 ≤τ ≤ t 0 + h
(
)
X
(A22)
⎫⎪
≤ s , t0 ≥ 0 , d ∈ M D , u ∈ M U ⎬
⎪⎭
(A23)
First notice that by virtue of property P1 it holds that a ( s ) < +∞ for all s ≥ 0 . Moreover, notice that since 0 ∈ X is a
robust equilibrium point from the input u ∈ M U and V (t ,0,0) = 0 for all t ≥ 0 , we have a( s ) ≥ 0 for all s ≥ 0 .
Furthermore, notice that M is well-defined, since by definitions (A22), (A23) the following inequality is satisfied for
all h ≥ 0 and s ≥ 0 :
0 ≤ M (h, s ) ≤ a ( s )
(A24)
Clearly, definition (A22) implies that a ( ⋅ ) is non-decreasing. Furthermore, property P2 asserts that lim+ a( s ) = 0 .
s →0
Hence, the inequality a(0) ≥ 0 , in conjunction with lim+ a( s ) = 0 and the fact that a ( ⋅ ) is non-decreasing implies
s →0
32
1
a (0) = 0 . It turns out that a can be bounded from above by the K ∞ function a~ defined by a~ ( s ) := s +
s
~
for s > 0 and a (0) = 0 . Define
2s
∫ a(w)dw
s
⎧ M (h, s )
⎫
; s > 0⎬
~
(
(
)
)
⎩g a s
⎭
μ (h) := sup ⎨
(A25)
where g is defined by (A10). Working exactly as in the proof of Lemma 2.16 we can show that the function
μ : ℜ + → ℜ + is well defined and satisfies μ ( ⋅ ) ≤ 1 ,
lim μ (h) = 0 . Consequently, there exists a continuous strictly
h → +∞
decreasing function μ : ℜ + → (0,+∞) such that μ (h) ≥ μ (h) for all h ≥ 0 and lim μ (h) = 0 . Thus, by recalling
h → +∞
definition (A25) we obtain
M (h, s ) ≤ μ (h) g (a~( s )) , ∀h, s ≥ 0
(A26)
Hence definition (A23) implies that the following estimate holds for all u ∈ M U , (t 0 , x 0 , d ) ∈ ℜ + × X × M D and
t ≥ t0 :
V (t , φ (t , t 0 , x 0 , u , d ), u (t )) ≤ μ (t − t 0 ) g a~ ( x 0 X ) + sup γ δ (τ ) u (τ ) U
(A27)
(
)
Estimate (A27) implies (2.12) with β (t ) ≡ 1 and σ ( s, t ) := μ (t ) g (a~ ( s )) .
(
t 0 ≤τ ≤ t
)
<
Proof of Lemmas 2.18-2.19: The proof is based on the following observation: if Σ := (X, Y , M U , M D , φ , π , H ) is
T − periodic then for all (t 0 , x 0 , u , d ) ∈ ℜ + × X × M U × M D it holds that φ(t, t0 , x0 , u, d) = φ(t − kT, t0 − kT, x0 , PkTu, PkT d )
and H (t, φ (t, t 0 , x0 , u, d ), u(t )) = H (t − kT, φ (t − kT, t 0 − kT, x0 , PkT u, PkT d ), (PkT u)(t − kT )) , where k := [t 0 / T ] denotes the
integer part of t 0 / T and the inputs PkT u ∈ M U , PkT d ∈ M D are defined in Definition 2.2.
Since Σ := (X, Y , M U , M D , φ , π , H ) satisfies the WIOS property from the input u ∈ M U , there exist functions
σ ∈ KL , β , δ ∈ K + , γ ∈ N such that (2.10) holds for all (t 0 , x 0 , u, d ) ∈ ℜ + × X × M U × M D and t ≥ t 0 .
Consequently, it follows that the following estimate holds for all (t 0 , x 0 , u, d ) ∈ ℜ + × X × M U × M D and t ≥ t 0 :
H (t , φ (t , t0 , x0 , u, d ), u(t )) Y
(
≤ σ β (t0 − kT ) x0
X
)
, t − t0 +
sup
τ ∈[t0 − kT ,t − kT ]
(
γ δ (τ ) (PkT u )(τ ) U
)
⎡t ⎤
Setting τ = s − kT and since 0 ≤ t 0 − ⎢ 0 ⎥T < T , for all t 0 ≥ 0 , we obtain
⎣T ⎦
H (t , φ (t , t0 , x0 , u, d ), u (t )) Y
(
≤ σ~ x0
X
)
(
, t − t0 + sup γ δ ( s − kT ) (PkT u )( s − kT ) U
s∈[t 0 ,t ]
)
(A28)
where σ~ ( s, t ) := σ (rs, t ) and r := max{ β (t ) ; 0 ≤ t ≤ T } . Estimate (A28) and the identity (PkT u )( s − kT ) = u ( s ) for all
s ≥ 0 , implies that the following estimate holds for all (t 0 , x 0 , u, d ) ∈ ℜ + × X × M U × M D and t ≥ t 0 :
H (t , φ (t , t0 , x0 , u, d ), u (t )) Y
(
≤ σ~ x0
X
)
(
~
, t − t0 + sup γ δ ( s) u ( s ) U
s∈[t 0 ,t ]
)
(A29)
~
where δ (t ) := max{δ ( s ) ; s ∈ [0, t ] } .
In case that Σ := (X, Y , M U , M D , φ , π , H ) satisfies the IOS property from the input u ∈ M U , then all arguments
above may be repeated with δ (t ) ≡ 1 . Thus we conclude that (A29) holds for all (t 0 , x 0 , u, d ) ∈ ℜ + × X × M U × M D
~
and t ≥ t 0 with δ (t ) ≡ 1 . The proof is complete. <
33