# Stability of Discrete Systems Controlled in the Presence of Intermittent Sensor Faults

Authors: Rui Vilela Dionísio; João M. Lemos
Journal: Journal of Control Science and Engineering
 (2008)

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## Abstract

This paper presents sufficient conditions for stability of unstable discrete time
invariant models, stabilized by state feedback, when interrupted observations due to
intermittent sensor faults occur. It is shown that the closed-loop system with
feedback through a reconstructed signal, when, at least, one of the sensors is unavailable,
remains stable, provided that the intervals of unavailability satisfy a certain time
bound, even in the presence of state vanishing perturbations. The result is first
proved for linear systems and then extended to a class of Hammerstein systems.


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## Body

## 1. Introduction

In recent years,
the mass advent of digital communication networks and systems has boosted the
integration of teleoperation in feedback control systems. Applications like
unmanned vehicles [1]  or internet-based real time control [2] provide significant examples raising, in turn, new
problems.This paper deals with one of such problems, if the
communication channel through which feedback information passes is not
completely reliable, sensors' measurements may not be available to the
controller during some intervals of time. In such a situation, one has to
couple the controller with a block, hereafter called supervisor, which is able
to discriminate between intervals of signal availability (availability timeTai) and
unavailability (unavailability time Tui+1), and to
generate an estimate of the plant's state during this Tui+1 intervals.
Methods for detection and estimation for abruptly changing systems [3] can be applied in the problem considered here. For
that purpose an algorithm based on Bayesian decision could be implemented, for
example.Somehow related with the problem of temporary sensor
unavailability presented in this paper are the problem of data packet dropout,
and the problem of network-induced delay, in networked control systems [4, 5].Moreover, the approach suggested in this paper can be
compared with different techniques based, for example, on the idea of the unknown
input observer, as suggested in [6]. On the other hand, it is obvious to exploit Kalman
filters and fuzzy logic for sensor fusion, applied to autonomous underwater
vehicle systems, as described in [7]. It was, also, shown in [8, 9] that the design of fault-tolerant observers can be
successfully applied to the control of rail traction drives. Finally, the
stability analysis for a real application example in the presence of
intermittent faults is described in [10].Biomedical applications provide, as well, examples in
that the sensor used for feedback is intermittently unavailable. In [11] the artifacts in the neuromuscular blockade level
measurement in patients subject to general anaesthesia are modeled as sensor
faults. The occurrence of these faults is detected with a Bayesian algorithm
and, during the periods of unavailability of the signal, the feedback
controller is fed with an estimate generated by a model.It is shown, throughout the paper, that with the above
described scheme, the controlled open-loop unstable plant will be stable (in
some sense, to be defined later) if the time interval, during which at least
one of the sensors measure is unavailable, is somehow “small”, and that the
Euclidean norm of the statex(k), at the end of each Tui+1 interval, is a
monotonic descent sequence. Moreover, if the plant state is perturbed by a
class of vanishing perturbations δx(k,x), similar stability results are derived.The contributions of the paper consist in providing
sufficient conditions for stability of feedback controlled open-loop unstable
systems with intermittent sensors faults. Linear as well as nonlinear systems
are considered.This paper is organized in four sections and two
appendices. After this introduction, Section2 makes a detailed system
description referring the functionality of the supervisor in terms of detection
and estimation of the state, and the way the feedback system with linear as
well as with nonlinear actuators behaves when intermittent sensors faults occur.
Section 3 presents two theorems with sufficient conditions, one for uniform
stability of the system with linear and nonlinear actuators, and respective
corollaries, also with sufficient conditions, and the other for uniform
exponential stability and descent monotonicity of the Euclidean norm of the
state x(k) at the end of each Tui+1 interval.
Moreover, Section 3 presents two other theorems, again with sufficient
conditions, that prove that the system with linear and nonlinear actuators, subject
to a vanishing perturbation, is asymptotically stable. In
Section 4 conclusions
are drawn. Appendix A.1 gives a full proof of Theorem 1 and Corollaries 1 and
2, and Appendix A.2 gives a full proof of Theorem 2 and Corollaries 3 and
4.


## 2. System Description

The system
depicted in Figure1 is composed of two sub-systems: (i) the supervisor responsible
for detection of sensors' measures interruptions, and for switching state
feedback from plant to model and from model to plant; (ii) the plant and the
model rendered stable through state feedback.Figure 1
Block diagram of a discrete feedback system with nonlinear actuator and
interrupted observations supervisor.An example of supervisor based on Bayesian inference
is provided in [12, 13]. It is beyond the scope of this paper to detail this
block.The supervisor decides whether the statex(k) is being
correctly measured by the sensors or not and commands the switch signal σ(k). During the time intervals in which the sensors do
not provide a reliable measure of the actual plant state (it is admitted that
the state coincides with the output) one possibility is to replace it by an
estimate x^(k) obtained from a
plant model. This yields a loss of performance with respect to the ideal
situation in which the sensors are always available, and may pose stability
problems if the plant is open-loop unstable.In order to understand the system functioning,
consider the time line of operation, depicted in Figure2, divided in alternate intervals where all sensors
operate correctly (Taj, with j=1,3,5,7,...,i), and where,
at least, one of them fails (Tuj, with j=2,4,6,...,i−1,i+1) being
replaced by the model estimate. Note that the index j does not
represent discrete instants of time, but is rather used to enumerate both the
availability, Taj, and the unavailability, Tuj, intervals. These intervals are identified in script
font in the upper part of the time line of Figure 2. The time instants corresponding to the beginning of
each interval, wether it is an availability or an unavailability interval, are
represented in the lower part of the time line of Figure 2. Let k0 denote the
beginning of one such intervals. It is assumed that the first interval always
corresponds to an availability interval, and that the intervals are open at
their end. Furthermore, the time analysis always finishes in an unavailability
interval at time k. Therefore, in a complete time sequence there are (i+1) intervals,
where (i+1) is an even
number.Figure 2
Operation time line with availability intervals alternating with unavailability
intervals.The model initial satex^ is made equal
to the last available observation of the state x when an
interrupted observation occurs (x^(k0)=x(k0)=x(k0−1)), since the sate x is no longer
available.


## 3. Stability Results

Three distinct
situations regarding system's stability are considered. In the first case, the
nonlinear functionψ(u(k)) does not exist
(ψ(u(k))=I; see Figure 3). Moreover, the perturbation function δx(k,x) is also
considered not to exist (δx(k,x)=0, see Figure 3). The second case is referred to the feedback system
with nonlinear actuator function ψ(u(k)) but, also,
without the perturbation function δx(k,x); see Figure 1. The third case considers the existence of the
perturbation function δx(k,x) in both
feedback systems, with linear actuator, and with nonlinear actuator.Figure 3
Block diagram of a discrete feedback system with linear actuator, and
interrupted observations supervisor.In all the situations the reference signal,r(k), is considered to be zero, for all k≥0 (regulation
problem).Throughout the text, matrices norms are the ones
induced by the Euclidean norm of vectors, being given by their largest singular
value (∥A∥=σmax[A]=σA≥0).


## 3.1. System with Linear Input

Consider Figure3 with δx(k,x)=0, and r(k)=0,  for all k≥0. The plant and the model depicted are described in
the state-space form by (1) and (2), respectively (1)x(k+1)=(A+δA)x(k)+(B+δB)u(k),(2)x^(k+1)=Ax^(k)+Bu(k)  with x and x^∈ℝn, accessible for direct measurement, u∈ℝp, A, B, δA, and δB are of
appropriate dimensions, and (A,B) is
controllable. Moreover, δA and δB represent
modeling uncertainties. It is assumed that the plant is time invariant, and
open-loop unstable. The state feedback of signal z(k), yielded by the sensor (3)u(k)=−Lz(k),  is implemented
by L, a matrix of feedback gains assumed to stabilize the
model. Furthermore, z(k)=x(k) during
availability intervals, when all sensors are working properly, and z(k)=x^(k) during
unavailability intervals, when measuring interruptions take place.During availability intervals the plant state equation
is(4)x(k+1)=[(A+δA)−(B+δB)L]x(k)  and during
unavailability intervals the plant state equation is (5)x(k+1)=(A+δA)x(k)−(B+δB)Lx^(k).Define the plant closed-loop dynamics matrix as(6)AδCL:=(A+δA)−(B+δB)L=Aδ−BδL,  the model
closed-loop dynamics matrix as (7)ACL:=A−BL,  the plant open
and closed-loop transition matrices (8)Φδ(k,k0):=Aδk−k0,ΦδCL(k,k0):=AδCLk−k0,  and the  model
open and closed-loop transition matrices (9)Φ(k,k0):=Ak−k0,ΦCL(k,k0):=ACLk−k0.Theorem 1.
Consider the closed-loop system of Figure3 with the unstable model in open-loop (bounded by ∥Φ(k,k0)∥≤αβk−k0 with α≥1 and β>1, finite constants) rendered stable in closed-loop (∥ΦCL(k,k0)∥≤γλk−k0 with γ≥1 a finite
constant and 0≤λ<1), through
proper design of L. Consider, also, that σB:=∥B∥, σL:=∥L∥, and that model uncertainties are bounded ∥δA∥≤σδA, and ∥δB∥≤σδB. The system with initial condition x(0)=x0 is globally
uniformly stable provided that the total unavailability time Tu, up to discrete time k inside the
unavailability interval Tui+1, satisfies the bound (10)Tu<logM1log(β+α⋅σδA)−(i+1)2⋅log[(1+(σB+σδB)σL⋅γ/(β+α⋅σδA−λ))αγ]log(β+α⋅σδA)−Talog(λ+γΣ)log(β+α⋅σδA) with M1≥1, a finite constant, and Σ:=σδA+σδB⋅σL is such that
verifies (11)0≤Σ<1−λγ,Ta
is the total
availability time.A result derived from the previous theorem is stated
on the following corollary.Corollary 1.
Under the assumptions of
Theorem1,  ∥x(Tj)∥, for  j=0,2,4,...,i−1, (the state norm at the beginning of each
availability interval) is a monotonic descent sequence provided that the unavailability
interval Tui−1  satisfies   (12)Tui−1<log[(1+(σB+σδB)σL⋅γ/(β+α⋅σδA−λ))αγ]log(β+α⋅σδA)−Tai−2log(λ+γΣ)log(β+α⋅σδA)and Σ:=σδA+σδB⋅σL  is such that
verifies   (13)0≤Σ<1−λγ,Tai−2
is the
availability time previous to  Tui−1.Concerning global uniform exponential stability,
consider the next corollary.Corollary 2.
Under the assumptions of
Theorem1, the system with initial condition  x(0)=x0 is globally
uniformly exponentially stable provided that the total unavailability time  Tu, up to discrete time k  inside the
unavailability interval Tui+1, satisfies   (14)Tu<logM2log((β+α⋅σδA)/N2)−(i+1)2⋅log[(1+(σB+σδB)σL⋅γ/(β+α⋅σδA−λ))αγ]log((β+α⋅σδA)/N2)−Talog((λ+γΣ)/N2)log((β+α⋅σδA)/N2) with M2≥1, a finite constant, and Σ:=σδA+σδB⋅σL
is such that
verifies   (15)0≤Σ<1−λγ and 0≤N2<1   is a constant
constrained to
(16)N2>λ+γΣ,Ta
is the total
availability time.A proof of the theorem and of the corollaries is
presented in AppendixA.1.Remark 1.
The constraintΣ<(1−λ)/γ is imposed to
assure that the plant closed-loop transition matrix is such that ∥ΦδCL(k,k0)∥≤γ(λ+γΣ)k−k0 with 0≥(λ+γΣ)<1 (see the proof in
Appendix A.1).Remark 2.
Notice that since(β+α⋅σδA)>1 and 0≤(λ+γΣ)<1, then the bound on Tu has a
monotonous crescent linear relation with Ta in the result
from Theorem 1, and Tui−1 also has a
monotonous crescent linear relation with Tai−2 in the result
from Corollary 1.Remark 3.
The constantN2 (in Corollary
2) represents an upper bound on the rate of exponential decay of the overall
system. If N2<λ+γΣ, then the result
of Corollary 2 would indicate a negative solution for Tu, which, clearly, is not possible, since Tu∈[0,∞[. Being N2>λ+γΣ, then the bound on Tu has also a
monotonous crescent linear relation with Ta, as mentioned in the previous remark.Remark 4.
Concerning Theorem1 and Corollary 2,
constants M1 and M2 represent an
offset term for the upper bound function on the evolution of ∥x(k)∥. The bigger these constants are, the more
conservative is the referred upper bound on uniform stability and uniform
exponential stability, respectively.Remark 5.
Theorem1 and Corollaries 1 and 2
present only conservative sufficient stability conditions for the system of
Figure 3.


## 3.2. System with Nonlinear Input

Consider Figure1 with δx(k,x)=0, and r(k)=0, for all k≥0. The plant and the model depicted are described in
the state-space form by (17) and (18), respectively, (17)x(k+1)=(A+δA)x(k)+(B+δB)unl(k),(18)x^(k+1)=Ax^(k)+Bunl(k), with x and x^∈ℝn, accessible for direct measurement, u and unl∈ℝp, A, B, δA, and δB are of
appropriate dimensions, and (A,B) is
controllable. Moreover, δA and δB represent
modeling uncertainties. It is assumed that the plant is time invariant, and
open-loop unstable.The vectorunl represents the
nonlinear input to both the plant and the model, unl(k)=ψ(k,u). A memoryless nonlinearity, ψ:[0,∞[×ℝp→ℝp, is said to satisfy a sector condition globally [14] if (19)[ψ(k,u)−Kminu(k)]T[ψ(k,u)−Kmaxu(k)]≤0   for all t≥0, for all u∈ℝp, for some real matrices Kmin and Kmax, where K=Kmax−Kmin is a positive
definite symmetric matrix. The nonlinearity ψ(k,u) is said to
belong to a sector [Kmin,Kmax].Proposition 1.
ConsiderKmin=−(γ2/2)I and Kmax=(γ2/2)I with γ2 a finite
positive constant. The nonlinearity ψ(k,u) can be
decomposed in a linear component and a nonlinear component, [15] (20)ψs(k,u)=ψ(k,u)−Kminu(k),  where ψs(k,u) represents the
nonlinear component and verifies the sector condition (21)ψsT(k,u)[ψs(k,u)−Ku(k)]≤0.Proof.
This result is straightforward using (20) in (19), and considering matrix K definition.Proposition 2.
For the defined matricesKmin and Kmax, the memoryless sector nonlinearity ψ(k,u) is bounded by ∥ψ(k,u)∥≤(γ2/2)∥u(k)∥, for all t≥0, for all u∈ℝp.Proof.
ReplacingKmin and Kmax by their
respective values in (19), and since ψT(k,u)u(k) is a scalar,
yields (22)∥ψ(k,u)∥2−(γ22)2∥u(k)∥2≤0. By definition ∥ψ(k,u)∥≥0, ∥u(k)∥≥0, and γ2/2>0, which implies (23)∥ψ(k,u)∥≤γ22∥u(k)∥.In order to find a bound onψs(k,u) starting from (20), using (23), and Kmin definition, it
follows that (24)∥ψs(k,u)∥≤γ2∥u(k)∥.The state feedback of signalz(k), yielded by the sensor (25)u(k)=−Lz(k),  is implemented
by L, a matrix of feedback gains assumed to stabilize the
model. Furthermore, z(k)=x(k) during
availability intervals, when all sensors are working properly, and z(k)=x^(k) during
unavailability intervals, when measuring interruptions take place.During availability intervals, the plant state equation
is(26)x(k+1)=[(A+δA)−(B+δB)KminL]x(k)+(B+δB)ψs(−Lx(k))  and during
unavailability intervals, the plant state equation is
(27)x(k+1)=(A+δA)x(k)+(B+δB).(ψs(−Lx^(k))−KminLx^(k)).Define the plant closed-loop dynamics matrix as(28)A¯δCL:=(A+δA)−(B+δB)KminL=Aδ−BδKminL,  the model
closed-loop dynamics matrix as (29)A¯CL:=A−BKminL,  the plant open
and closed-loop transition matrices as(30)Φδ(k,k0):=(A+δA)k−k0=Aδk−k0,Φ¯δCL(k,k0):=A¯δCLk−k0,  the model open
and closed-loop transition matrices as(31)Φ(k,k0):=Ak−k0,Φ¯CL(k,k0):=A¯CLk−k0,  and matrix P=KminL, considered to stabilize the model in closed loop.Theorem 2.
Consider the closed-loop system of Figure1 where the model is unstable in open-loop (bounded by ∥Φ(k,k0)∥≤αβk−k0, with α≥1, and  β>1, finite constants), rendered stable in closed-loop (∥Φ¯CL(k,k0)∥≤γλk−k0, with γ≥1 a finite
constant, and  0≤λ<1), through
proper design of P. The nonlinearity  ψs(k,u)  satisfies ∥ψs(k,u)∥≤γ2∥u(k)∥ for all t≥0, for all u∈ℝp. Consider, also, that σB:=∥B∥, σL:=∥L∥, σP:=∥P∥  and that model
uncertainties are bounded ∥δA∥≤σδA and ∥δB∥≤σδB. The system with initial condition x(0)=x0  is globally
uniformly stable provided that the total unavailability time Tu, up to discrete time  k  inside the
unavailability interval Tui+1, satisfies the bound
(32)Tu<logM1log(β+α⋅σδA)−(i+1)2.log[(1+(σB+σδB)(σP+γ2⋅σL)γ(β+α⋅σδA)−(λ+γσδB⋅γ2⋅σL))αγ]log(β+α⋅σδA)−Talog[(λ+γΣ)+γ(σB+σδB)γ2⋅σL]log(β+α⋅σδA) with M1≥1, a finite constant, and Σ:=σδA+σδB⋅σP
is such that
verifies   (33)0≤Σ<1−λγand γ2
is the less of
the following two inequalities:   (34)γ2<1−(λ+γΣ)γ(σB+σδB)σL,γ2<β+α⋅σδA−λγ⋅σδB⋅σL,Ta
is the total
availability time.As in the previous subsection the following two
corollaries are derived.Corollary 3.
Under the assumptions of
Theorem1,  ∥x(Tj)∥, for  j=0,2,4,...,i−1, (the state norm at the beginning of each
availability interval) is a monotonic descent sequence provided that the unavailability
interval  Tui−1  satisfies   (35)Tui−1<−log[(1+(σB+σδB)(σP+γ2⋅σL)γ(β+α⋅σδA)−(λ+γσδB⋅γ2⋅σL))αγ]log(β+α⋅σδA)−Tai−2log[(λ+γΣ)+γ(σB+σδB)γ2⋅σL]log(β+α⋅σδA) and Σ:=σδA+σδB⋅σL
is such that
verifies   (36)0≤Σ<1−λγ and γ2
is the less of
the following two inequalities:(37)γ2<1−(λ+γΣ)γ(σB+σδB)σL,γ2<β+α⋅σδA−λγ⋅σδB⋅σL,Tai−2
is the
availability time previous to  Tui−1.Corollary 4.
Under the assumptions of
Theorem2, the system with initial condition  x(0)=x0 is globally
uniformly exponentially stable provided that the total unavailability time  Tu, up to discrete time k  inside the
unavailability interval Tui+1, satisfies   (38)Tu<logM2log((β+α⋅σδA)/N2)−(i+1)2.log[(1+(σB+σδB)(σP+γ2⋅σL)γ(β+α⋅σδA)−(λ+γσδB⋅γ2⋅σL))αγ]log((β+α⋅σδA)/N2)−Talog(((λ+γΣ)+γ(σB+σδB)γ2⋅σL)/N2)log((β+α⋅σδA)/N2) with M1≥1, a finite constant, and Σ:=σδA+σδB⋅σP
is such that
verifies   (39)0≤Σ<1−λγand γ2
is the less of
the following two inequalities:   (40)γ2<1−(λ+γΣ)γ(σB+σδB)σL,γ2<β+α⋅σδA−λγ⋅σδB⋅σL,  and 0≤N2<1
is a constant
constrained to   (41)N2>(λ+γΣ)+γ(σB+σδB)γ2⋅σL,Ta
is the total
availability time.A proof of the theorem and of the corollaries is
presented in AppendixA.2.Remark 6.
Notice that since(β+α⋅σδA)>1 then it must be (λ+γΣ)+γ(σB+σδB)γ2⋅σL<1, which leads to (34), (37), and (40), so that the bound on Tu has a
monotonous crescent linear relation with Ta in the result
from Theorem 2, and Tui−1 also has a
monotonous crescent linear relation with Tai−2 in the result
from Corollary 3.Remark 7.
The constantN2 (in Corollary
4) represents an upper bound on the rate of exponential decay of the overall
system. If N2<(λ+γΣ)+γ(σB+σδB)γ2⋅σL, then the result
of Corollary 4 would indicate a negative solution for Tu, which, clearly, is not possible, since Tu∈[0,∞[. Being N2>(λ+γΣ)+γ(σB+σδB)γ2⋅σL, then the bound on Tu has also a
monotonous crescent linear relation with Ta, as mentioned in the previous remark.Remark 8.
Concerning Theorem2 and Corollary 4,
constants M1 and M2 represent, once
again, an offset term for the upper bound function on the evolution of ∥x(k)∥. The bigger these constants are, the more
conservative is the referred upper bound on uniform stability and uniform
exponential stability, respectively.Remark 9.
Theorem2 and Corollaries 3 and 4
present only conservative sufficient stability conditions for the system of
Figure 1.


## 3.3. Perturbed System with Linear and Nonlinear Inputs

Consider that
both systems depicted in Figures1 and  3, suffer the influence of perturbation δx(k,x), where δx:[0,∞[×D→ℝn is piecewise
continuous in k and locally
Lipschitz in x on [0,∞[×D, and D⊂ℝn is a domain
that contains the origin x=0. Also, ∥δx(k,x)∥≤ϵ∥x(k)∥ for all k≥0, for all x∈D, and ϵ is a
nonnegative constant, meaning that the perturbation satisfies a linear growth bound,
therefore, considering a vanishing perturbation, [14].During availability intervalsTaj, for j=1,3,5,...,i, both systems can be represented by the autonomous
equation (42)x(k+1)=F(k,x),  where F(k,x), for the system depicted in Figure 3, is (43)F(k,x)=AδCLx(k),k∈Taj,  and for the
system depicted in Figure 1, F(k,x) is
(44)F(k,x)=A¯δCLx(k)+(B+δB).ψs(−Lx(k)),k∈Taj.  Clearly, F(0)=0 in both
situations (from (19) and  matrices’ Kmin and Kmax definition in
Proposition 1, the sector memoryless nonlinearity verifies ψs(0)=0). Recalling
the state equations (5) and (27) during unavailability intervals Tuj, for j=2,4,6,...,i+1, and the fact that the initial model state x^ is made equal
to the last available observation of the state x when an
interrupted observation occurs, (x^(k0)=x(k0)=x(k0−1)), it is clearly understood that if the state becomes
zero during an availability interval, then it will remain zero for all time
instants belonging to any unavailability interval that may occur. The
function’s F(k,x) branch related
with the unavailability interval is not of obvious writing in terms only of x(k). It has an easier writing in terms of x(k) and of x^(k). Nevertheless, since these two states are related at
the switching time between availability and unavailability intervals (as
recalled above), it can be understood that during an unavailability interval, F(k,x) exists.It is important to stress out that an unavailability
interval cannot occur without having previously existed an availability
interval. Bearing this in mind, it is possible to state thatF(0)=0, for all k≥0, (including availability and unavailability
intervals).Also, linear and nonlinear systems were proved to be
globally uniformly exponentially stable, under the conditions of Corollaries2 and 4, respectively, therefore, both F(k,x) are Lipschitz
not only near the origin, but in ℝn, and verify ∥F(x1)−F(x2)∥≤Lv∥x1−x2∥.Combining the results from Corollaries2 and
4 with the above comments, and with the result presented in [16], is reproduced in the next theorem.Theorem 3.
LetF:ℝn→ℝn
satisfy a
Lipschitz condition in a neighborhood of the origin, with F(0)=0. If the origin is an exponentially stable fixed point
of x(k+1)=F(x(k)), it is an asymptotically stable fixed point of the
perturbed system x¯(k+1)=F(x¯(k))+δx(k,x¯).This leads to the
next two theorems.Theorem 4.
The nonperturbed system from Figure3,  x(k+1)=F(k,x), verifying Corollary 2 and Theorem 3 sufficient
conditions, has a globally asymptotically stable fixed point of the perturbed
system  x¯(k+1)=F(x¯(k))+δx(k,x¯)
in the origin,
and δx:[0,∞[×D→ℝn is piecewise
continuous in  k
and locally
Lipschitz in  x  on  [0,∞[×D, and  D⊂ℝn  is a domain
that contains the origin x=0. Also,  ∥δx(k,x)∥≤ϵ∥x(k)∥ for all k≥0, for all x∈D with ϵ
a nonnegative
constant satisfies a linear growth bound.Theorem 5.
It is the same redaction of Theorem4, but
considering the system from Figure 1.Remark 10.
These results are global since bothF(k,x) are Lipschitz
continuous in ℝn, and the original systems are uniformly exponentially
stable, [16].


## 4. Conclusions

The paper
presents and proves sufficient conditions that allow a discrete time analysis
of sensor unavailability (interrupted observations) intervals, bounding these
intervals in order to state that the unstable open-loop plant represented in
Figure1, when controlled in closed-loop, is globally
uniformly exponentially stable. These results are proved under the existence of
modeling uncertainties and if plant state vanishing perturbations occur, then
global asymptotical stability is achieved for the perturbed system. The results
were proved for either systems with linear actuators, or with memoryless sector
nonlinear actuators.It is interesting to note that in a related work [4], a similar conservative theoretical result regarding
uniform exponential stability is reported, showing that longer intervals of
unavailability can be reached in practice and that these theoretical results
might be too conservative for practical purposes.



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