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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.

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 [

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 time

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 [

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 [

Biomedical applications provide, as well, examples in
that the sensor used for feedback is intermittently unavailable. In [

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 state

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, Section

The system
depicted in Figure

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 [

The supervisor decides whether the state

In order to understand the system functioning,
consider the time line of operation, depicted in Figure

Operation time line with availability intervals alternating with unavailability intervals.

The model initial sate

Three distinct
situations regarding system's stability are considered. In the first case, the
nonlinear function

Block diagram of a discrete feedback system with linear actuator, and interrupted observations supervisor.

In all the situations the reference signal,

Throughout the text, matrices norms are the ones
induced by the Euclidean norm of vectors, being given by their largest singular
value (

Consider Figure

During availability intervals the plant state equation
is

Define the plant closed-loop dynamics matrix as

Consider the closed-loop system of Figure

A result derived from the previous theorem is stated on the following corollary.

Under the assumptions of
Theorem

Concerning global uniform exponential stability, consider the next corollary.

Under the assumptions of
Theorem

A proof of the theorem and of the corollaries is
presented in Appendix

The constraint

Notice that since

The constant

Concerning Theorem

Theorem

Consider Figure

The vector

Consider

This result is straightforward using (

For the defined matrices

Replacing

In order to find a bound on

The state feedback of signal

During availability intervals, the plant state equation
is

Define the plant closed-loop dynamics matrix as

Consider the closed-loop system of Figure

As in the previous subsection the following two corollaries are derived.

Under the assumptions of
Theorem

Under the assumptions of
Theorem

A proof of the theorem and of the corollaries is
presented in Appendix

Notice that since

The constant

Concerning Theorem

Theorem

Consider that
both systems depicted in Figures

During availability intervals

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 that

Also, linear and nonlinear systems were proved to be
globally uniformly exponentially stable, under the conditions of Corollaries

Combining the results from Corollaries

Let

This leads to the next two theorems.

The nonperturbed system from Figure

It is the same redaction of Theorem

These results are global since both

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
Figure

It is interesting to note that in a related work [

Throughout the appendix, the matrices norms are the ones induced by the Euclidean norm of vectors, being given by their largest singular values.

Consider the discrete time line represented in Figure

Since it will be often used in the following proofs, a
Gronwall-Bellman type of inequality for sequences is presented [

Suppose the scalar sequences

Consider, also, the sum of the (

Consider the system depicted in Figure

For bounded model uncertainties

Upper bounds for (

Consider the
Euclidean norm of

In order for the system to be uniformly exponentially stable, it must verify

Consider the system depicted in Figure

It is assumed that the model in closed-loop is stable
and bounded by

For bounded model uncertainties

Upper bounds for (

Consider the
Euclidean norm of

In order for
the system to be uniformly exponentially stable, it must verify

This work was produced in the framework of the project IDEA—Integrated Design for Automated Anaesthesia, PTDC/EEA-ACR/69288/2006.