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.
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 time ) and unavailability (unavailability time ), and to generate an estimate of the plant's state during this 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 state , at the end of each interval, is a monotonic descent sequence. Moreover, if the plant state is perturbed by a class of vanishing perturbations , 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, Section 2 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 at the end of each 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 Figure 1 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.
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 state is being correctly measured by the sensors or not and commands the switch signal . 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 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 Figure 2, divided in alternate intervals where all sensors operate correctly (, with ), and where, at least, one of them fails (, with ) being replaced by the model estimate. Note that the index does not represent discrete instants of time, but is rather used to enumerate both the availability, , and the unavailability, , 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 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 . Therefore, in a complete time sequence there are intervals, where is an even number.
The model initial sate is made equal to the last available observation of the state when an interrupted observation occurs (), since the sate is no longer available.
3. Stability Results
Three distinct situations regarding system's stability are considered. In the first case, the nonlinear function does not exist (; see Figure 3). Moreover, the perturbation function is also considered not to exist (, see Figure 3). The second case is referred to the feedback system with nonlinear actuator function but, also, without the perturbation function ; see Figure 1. The third case considers the existence of the perturbation function in both feedback systems, with linear actuator, and with nonlinear actuator.
In all the situations the reference signal, , is considered to be zero, for all (regulation problem).
Throughout the text, matrices norms are the ones induced by the Euclidean norm of vectors, being given by their largest singular value ().
3.1. System with Linear Input
Consider Figure 3 with , and , for all . The plant and the model depicted are described in the state-space form by (1) and (2), respectively with and , accessible for direct measurement, , , , , and are of appropriate dimensions, and is controllable. Moreover, and represent modeling uncertainties. It is assumed that the plant is time invariant, and open-loop unstable. The state feedback of signal , yielded by the sensor is implemented by , a matrix of feedback gains assumed to stabilize the model. Furthermore, during availability intervals, when all sensors are working properly, and during unavailability intervals, when measuring interruptions take place.
During availability intervals the plant state equation is and during unavailability intervals the plant state equation is
Define the plant closed-loop dynamics matrix as the model closed-loop dynamics matrix as the plant open and closed-loop transition matrices and the model open and closed-loop transition matrices
Theorem 1. Consider the closed-loop system of Figure 3 with the unstable model in open-loop (bounded by with and , finite constants) rendered stable in closed-loop ( with a finite constant and ), through proper design of . Consider, also, that , , and that model uncertainties are bounded , and . The system with initial condition is globally uniformly stable provided that the total unavailability time , up to discrete time inside the unavailability interval , satisfies the bound with , a finite constant, and is such that verifies is the total availability time.
A result derived from the previous theorem is stated on the following corollary.
Corollary 1. Under the assumptions of Theorem 1, , for , (the state norm at the beginning of each availability interval) is a monotonic descent sequence provided that the unavailability interval satisfies and is such that verifies is the availability time previous to .
Concerning global uniform exponential stability, consider the next corollary.
Corollary 2. Under the assumptions of
Theorem 1, the system with initial condition is globally
uniformly exponentially stable provided that the total unavailability time , up to discrete time inside the
unavailability interval , satisfies with , a finite constant, and
is such that
verifies and is a constant
constrained to
is the total
availability time.
A proof of the theorem and of the corollaries is presented in Appendix A.1.
Remark 1. The constraint is imposed to assure that the plant closed-loop transition matrix is such that with (see the proof in Appendix A.1).
Remark 2. Notice that since and , then the bound on has a monotonous crescent linear relation with in the result from Theorem 1, and also has a monotonous crescent linear relation with in the result from Corollary 1.
Remark 3. The constant (in Corollary 2) represents an upper bound on the rate of exponential decay of the overall system. If , then the result of Corollary 2 would indicate a negative solution for , which, clearly, is not possible, since . Being , then the bound on has also a monotonous crescent linear relation with , as mentioned in the previous remark.
Remark 4. Concerning Theorem 1 and Corollary 2, constants and represent an offset term for the upper bound function on the evolution of . The bigger these constants are, the more conservative is the referred upper bound on uniform stability and uniform exponential stability, respectively.
Remark 5. Theorem 1 and Corollaries 1 and 2 present only conservative sufficient stability conditions for the system of Figure 3.
3.2. System with Nonlinear Input
Consider Figure 1 with , and , for all . The plant and the model depicted are described in the state-space form by (17) and (18), respectively, with and , accessible for direct measurement, and , , , , and are of appropriate dimensions, and is controllable. Moreover, and represent modeling uncertainties. It is assumed that the plant is time invariant, and open-loop unstable.
The vector represents the nonlinear input to both the plant and the model, . A memoryless nonlinearity, , is said to satisfy a sector condition globally [14] if for all , for all , for some real matrices and , where is a positive definite symmetric matrix. The nonlinearity is said to belong to a sector .
Proposition 1. Consider and with a finite positive constant. The nonlinearity can be decomposed in a linear component and a nonlinear component, [15] where represents the nonlinear component and verifies the sector condition
Proof. This result is straightforward using (20) in (19), and considering matrix definition.
Proposition 2. For the defined matrices and , the memoryless sector nonlinearity is bounded by , for all , for all .
Proof. Replacing and by their respective values in (19), and since is a scalar, yields By definition , , and , which implies
In order to find a bound on starting from (20), using (23), and definition, it follows that
The state feedback of signal , yielded by the sensor is implemented by , a matrix of feedback gains assumed to stabilize the model. Furthermore, during availability intervals, when all sensors are working properly, and during unavailability intervals, when measuring interruptions take place.
During availability intervals, the plant state equation is and during unavailability intervals, the plant state equation is
Define the plant closed-loop dynamics matrix as the model closed-loop dynamics matrix as the plant open and closed-loop transition matrices as the model open and closed-loop transition matrices as and matrix , considered to stabilize the model in closed loop.
Theorem 2. Consider the closed-loop system of Figure 1 where the model is unstable in open-loop (bounded by , with , and , finite constants), rendered stable in closed-loop (, with a finite constant, and ), through proper design of . The nonlinearity satisfies for all , for all . Consider, also, that , , and that model uncertainties are bounded and . The system with initial condition is globally uniformly stable provided that the total unavailability time , up to discrete time inside the unavailability interval , satisfies the bound with , a finite constant, and is such that verifies and is the less of the following two inequalities: is the total availability time.
As in the previous subsection the following two corollaries are derived.
Corollary 3. Under the assumptions of Theorem 1, , for , (the state norm at the beginning of each availability interval) is a monotonic descent sequence provided that the unavailability interval satisfies and is such that verifies and is the less of the following two inequalities: is the availability time previous to .
Corollary 4. Under the assumptions of Theorem 2, the system with initial condition is globally uniformly exponentially stable provided that the total unavailability time , up to discrete time inside the unavailability interval , satisfies with , a finite constant, and is such that verifies and is the less of the following two inequalities: and is a constant constrained to is the total availability time.
A proof of the theorem and of the corollaries is presented in Appendix A.2.
Remark 6. Notice that since then it must be , which leads to (34), (37), and (40), so that the bound on has a monotonous crescent linear relation with in the result from Theorem 2, and also has a monotonous crescent linear relation with in the result from Corollary 3.
Remark 7. The constant (in Corollary 4) represents an upper bound on the rate of exponential decay of the overall system. If , then the result of Corollary 4 would indicate a negative solution for , which, clearly, is not possible, since . Being , then the bound on has also a monotonous crescent linear relation with , as mentioned in the previous remark.
Remark 8. Concerning Theorem 2 and Corollary 4, constants and represent, once again, an offset term for the upper bound function on the evolution of . The bigger these constants are, the more conservative is the referred upper bound on uniform stability and uniform exponential stability, respectively.
Remark 9. Theorem 2 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 Figures 1 and 3, suffer the influence of perturbation , where is piecewise continuous in and locally Lipschitz in on , and is a domain that contains the origin . Also, for all for all , and is a nonnegative constant, meaning that the perturbation satisfies a linear growth bound, therefore, considering a vanishing perturbation, [14].
During availability intervals , for , both systems can be represented by the autonomous equation where , for the system depicted in Figure 3, is and for the system depicted in Figure 1, is Clearly, in both situations (from (19) and matrices’ and definition in Proposition 1, the sector memoryless nonlinearity verifies ). Recalling the state equations (5) and (27) during unavailability intervals , for , and the fact that the initial model state is made equal to the last available observation of the state when an interrupted observation occurs, (), 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 branch related with the unavailability interval is not of obvious writing in terms only of . It has an easier writing in terms of and of . 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, 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 that for all , (including availability and unavailability intervals).
Also, linear and nonlinear systems were proved to be globally uniformly exponentially stable, under the conditions of Corollaries 2 and 4, respectively, therefore, both are Lipschitz not only near the origin, but in , and verify .
Combining the results from Corollaries 2 and 4 with the above comments, and with the result presented in [16], is reproduced in the next theorem.
Theorem 3. Let satisfy a Lipschitz condition in a neighborhood of the origin, with . If the origin is an exponentially stable fixed point of , it is an asymptotically stable fixed point of the perturbed system .
This leads to the next two theorems.
Theorem 4. The nonperturbed system from Figure 3, , verifying Corollary 2 and Theorem 3 sufficient conditions, has a globally asymptotically stable fixed point of the perturbed system in the origin, and is piecewise continuous in and locally Lipschitz in on , and is a domain that contains the origin . Also, for all for all with a nonnegative constant satisfies a linear growth bound.
Theorem 5. It is the same redaction of Theorem 4, but considering the system from Figure 1.
Remark 10. These results are global since both are Lipschitz continuous in , 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 Figure 1, 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.
Appendix
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 2. The intervals where the sensors yield correct measures are designated as , with , and the intervals where the observations are interrupted are designated as , with . Let the discrete time instant denote the beginning of a generic interval.
Since it will be often used in the following proofs, a Gronwall-Bellman type of inequality for sequences is presented [17].
Lemma 1. Suppose the scalar sequences and are such that for , and where and are constants with . Then
Consider, also, the sum of the () terms of a geometric progression with ratio , If , then, as , (A.3) becomes
A.1. Stability Proofs for System with Linear Input
Proof of Theorem 1. Consider the system depicted in Figure 3. During availability time intervals , with , it is , and the plant state evolves
according to
On the other
hand, during unavailability time intervals , with , it is , the model state evolves
according to and the plant
state evolves
according to Replacing (A.6) in (A.7), and knowing that the model initial sate is made equal
to the last available observation of the state when an
interrupted observation occurs (), the plant state evolution is It is assumed
that the model in closed-loop is stable and bounded by , , with and , and that the model is unstable in open-loop, but
bounded by , , with and .
For bounded model uncertainties , and considering the bound on , with (this
corresponds to assume an unfavorable situation), it can be proved through the
use of Lemma 1, if is seen as a
perturbation in the system , [17], that , with . This means, as expected, that if the model dynamics
are open-loop unstable, then there will be a such that the
plant dynamics will be open-loop unstable (the use of a continuity
argumentation could also explain such assertion). A similar proof can be given
for the stability of the plant in closed-loop since the model is stable in
closed-loop (, with ). Again,
recurring to Lemma 1, and considering that is seen as a
perturbation in the system , it can be proved that , , with , and .
Upper bounds for (A.5) during availability time intervals, and for (A.8) during unavailability time intervals, are obtained, respectively Starting from (A.9), yields and for (A.10), recalling that , , and , Since , and considering (A.4), after some calculations The complete
state evolution from time instant , up to the final time instant at , is given by the
alternate product of (A.5) by (A.8), where is considered.
Applying results (A.11) and (A.13) to this product originates where and represent the
entire duration of all unavailability and availability time intervals,
respectively, and In order for
the system to be uniformly stable, it must verify , , with . Therefore, from (A.14) Replacing (A.15) in (A.16) gives the desired result from Theorem 1 subject to
the constraint , and the result holds globally since it is valid for
any .
Proof of Corollary 1. Consider the Euclidean norm of at discrete times , and , at the end of the unavailability intervals , and , respectively. In order for , for , to be a monotonic descent sequence, it should verify equivalently, from the first two lines of (A.14), and considering (A.15) or, since , and Replacing (A.15) in (A.19) gives the desired result from Corollary 1 subject to the constraint , and the result holds globally since it is valid for any .
Proof of Corollary 2. In order for the system to be uniformly exponentially stable, it must verify , , with , and . Therefore, from (A.14) and considering , and , Replacing (A.15) in (A.20) gives the desired result from Corollary 2 subject to the constraint , and the result holds globally since it is valid for any .
A.2. Stability Proofs for System with Nonlinear Input
Proof of Theorem 2. Consider the system depicted in Figure 1. During availability time intervals , with , it is , and the plant state evolves
according to On the other
hand, during unavailability time intervals , with , it is , the model initial sate is made equal
to the last available observation of the state when an interrupted
observation occurs (), the model state evolves
according to and the plant
state evolves according
to
It is assumed that the model in closed-loop is stable
and bounded by , , with and , and that the model is unstable in open-loop, but
bounded by , , with and .
For bounded model uncertainties , and considering the bound on with (this
corresponds to assume an unfavorable situation), it was proved in Appendix A.1
that with . A similar proof can be given for the stability of
the plant in closed-loop since the model is stable in closed-loop ( with ). Recurring to
Lemma 1, and considering that is seen as a
perturbation in the system , it can be proved that , with , and , [17].
Upper bounds for (A.21) during availability time intervals, and for (A.23) during unavailability time intervals, are obtained,
respectively From (A.24), recalling that , , and considering (24) yield, respectively Applying Lemma
1 to (A.25) gives An upper bound
for (A.22) is obtained from Applying Lemma
1 and recalling that yield Using (A.29) in (A.26), Making use of (A.3) in (A.30) gives Providing , and considering (A.4), after some calculations, (A.31) yields From this point
on, the demonstration follows closely the one of Theorem 1 (see Appendix A.1),
and applying results (A.27) and (A.32) originates where and represent the
entire duration of all unavailability and availability time intervals, respectively,
and In order for
the system to be uniformly stable, it must verify , with . Therefore, from (A.33) Replacing (A.34) in (A.35) gives the desired result from Theorem 2 subject to
the constraints , and . The result holds globally since it is valid for any .
Proof of Corollary 3. Consider the Euclidean norm of at discrete times , and , at the end of the unavailability intervals , and , respectively. In order for , for , to be a monotonic descent sequence, it should verify This proof is outlined in the very same way as Corollary 1 proof, therefore, the following equation yields naturally after Theorem 2 proof calculations: Replacing (A.34) in (A.37) gives the desired result from Corollary 3 subject to the constraints , and . The result holds globally since it is valid for any .
Proof of Corollary 4. In order for the system to be uniformly exponentially stable, it must verify , , with , and . Therefore, from (A.33), and considering , and , Replacing (A.34) in (A.38) gives the desired result from Corollary 4 subject to the constraints and . The result holds globally since it is valid for any .
Acknowledgment
This work was produced in the framework of the project IDEA—Integrated Design for Automated Anaesthesia, PTDC/EEA-ACR/69288/2006.