Mathematical Problems in Engineering

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Models, Methods, and Applications of Dynamics and Control in Engineering Sciences: State of the Art

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Volume 2010 |Article ID 586169 | https://doi.org/10.1155/2010/586169

Mouhacine Benosman, "A Survey of Some Recent Results on Nonlinear Fault Tolerant Control", Mathematical Problems in Engineering, vol. 2010, Article ID 586169, 25 pages, 2010. https://doi.org/10.1155/2010/586169

A Survey of Some Recent Results on Nonlinear Fault Tolerant Control

Academic Editor: Jose Balthazar
Received19 Jun 2009
Accepted07 Sep 2009
Published01 Nov 2009

Abstract

Fault tolerant control (FTC) is the branch of control theory, dealing with the control of systems that become faulty during their operating life. Following the systems classification, as linear and nonlinear models, FTC can be classified in two different groups, linear FTC (LFTC) dealing with linear models, and the one of interest to us in this paper, nonlinear FTC (NFTC), which deals with nonlinear models. We present in this paper a survey of some of the results obtained in these last years on NFTC.

1. Introduction

Due to the complexity of modern engineering systems, it is increasingly important to ensure their reliability. This has motivated researchers to concentrate on FTC, which is primarily meant to ensure safety, that is, the stability of a system after the occurrence of a fault in the system. There are two approaches to synthesize controllers that are tolerant to system faults. One approach, known as passive FTC, aims at designing a controller which is a priori robust to some given expected faults. Another approach, known as active FTC, relies on the availability of a fault detection and diagnosis (FDD) block that gives, in real-time, information about the nature and the intensity of the fault. This information is then used by a control reconfiguration block to adjust online the control effort in such a way to maintain stability and to optimize the performance of the faulty system.

Passive FTC has the drawback to be reliable only for the class of faults expected and taken into account in the design of the passive FTC. Furthermore, the performances of the closed-loop are not optimized for each fault scenario. However, it has the advantage to avoid the time delay due to online diagnosis of the faults and reconfiguration of the controller, required in active FTC [1, 2], which is very important in practical situations where the time windows during which the system stays stabilizable is very short, for example, the unstable double inverted pendulum example [3]. In practical applications passive FTCs complement active FTC schemes. Indeed, passive FTCs are necessary during the fault detection and estimation phases [4], where passive FTCs are used to ensure the stability of the faulty system, before switching to active FTCs, that recover some performance after the fault is detected and estimated. Another scenario where passive FTC is used as a complement of active FTC is in the switching-based active FTC, where the active FTC switches between different passive FTC, each controller being designed off-line to cope with a finite number of expected faults and stored in a controller bank; see for example, [5]. Several passive FTC methods have been proposed, mainly based on robust theory, for example, multiobjective linear optimization and LMIs techniques [6], QFT method [7, 8], [3, 9], absolute stability theory [10], nonlinear regulation theory [11, 12], Lyapunov reconstruction [13, 14], and passivity-based FTC [15]. As for active FTC, many methods have been proposed for active LFTC, for example, [1621], as well as for NFTC, for example, [14, 2235]. As said before, this paper aim is to present some of the recent results on NFTC thus we will not further present LFTC here, and we refer the reader to other survey papers for linear systems; see for example, [3638].

In [11, 12] the nonlinear regulation theory was used to solve the NFTC problem for particular practical examples, that is, robot manipulators in [12] and induction motors in [11]. The faults treated were modelled as additive actuator faults. In [13, 39] Lyapunov reconstruction techniques were used to solve the problem of loss of actuator effectiveness for nonlinear models affine in the control. The main drawback of this scheme is that it is based on the apriori knowledge of a stabilizing feedback for the nominal safe model and the knowledge of the associated Lyapunov function in closed form. Furthermore, the problem of inputs saturation has not been solved in this work. In [15] the authors studied the case of nonlinear systems with multiplicative actuator faults, and considered the case of systems with inputs saturation. In active NFTC field we also quote [24], where the authors studied the NFTC problem for a particular class of continuous nonlinear models, that is, linear in the control, and proposed a new adaptive fault estimation module, complemented with a control reconfiguration block. In [14], the authors study a specific problem of active FTC, namely, the problem of graceful performance degradation. This problem aims to define online new performances for the faulty system, these performances having to be feasible by the faulty system within its states/actuators limits. Indeed, after the occurrence of a fault the faulty system is expected to be unable to perform the tasks required and planned initially for the safe system. Therefore, new tasks, less demanding, have to be generated online for the faulty system. The idea used there is based on two main stages. The first stage concerns online trajectory planning or reshaping, using online optimization scheme that generates online the closest trajectory to the nominal one, but without violating the new constraints of the faulty system. The second stage concerns the control reallocation problem, using nonlinear model predictive control (NMPC). This scheme deals with nonminimum phase nonlinear models affine in the control. We also refer to [28, 29] where uncertain nonlinear models with constrained inputs, were considered.

An important part of FTC is the one specializing in actuator faults. Indeed, FTCs dealing with actuator faults are relevant for practical application and have already been the subject of many publications [13, 15, 18, 19, 25, 33, 3949]. The nonlinear case has been studied in [25], where active FTC with respect to additive actuator faults was studied for nonlinear systems affine in the control. Constrained actuators were considered, and state-feedback as well as output-feedback FDDs/FTCs were proposed. In [4, 50], an active NFTC has been proposed for the class of SISO nonlinear systems, with incipient faults. The structure of the FTC was based on three controllers: a nominal controller for the safe system, that guarantees the system trajectories boundedness until the fault is detected. Then, the NFTC was reconfigured to the second controller that recovers some control performances before the fault is isolated. After the isolation of the fault, a third controller was used based on the faulty model, to improve the control performances. The reconfiguration of the controllers was based on adaptive backstepping approaches. In [14, 33, 48] the authors used model predictive controllers (linear case in [48], and nonlinear case [14, 33]) to reconfigure the controller online after the isolation and estimation of the fault. Finally, we quote [51], where a class of delayed nonlinear systems, modelled with linear terms added to Lipschitz-like nonlinearities with delay terms have been studied. The authors proposed an adaptive LMI-based active NFTC to ensure the stability of the faulty model as well as some optimal performances.

We do not pretend here to present in details all the work quoted above, instead, we will concentrate on some of these results and point out pros and cons of each scheme. We also underline, that we will not report the proofs of the results here, the reader will be refereed to the corresponding paper for the detailed proofs.

This paper is structured as follows. In Section 2, we introduce some notations, and recall some definitions that will be used throughout the paper. Section 3, concerns passive NFTC, followed by active NFTC in Section 4. Finally, we conclude the paper in Section 5, by pointing out some open problems in NFTC.

2. Preliminaries

Throughout the paper we will use the norm denoted by , that is, for we define . The notation denotes the standard Lie derivative of a scalar function along a vector function . We also denote by the hyperbolic tangent function and by the th-order-derivative of the scalar function . Let us introduce now some definitions from [52], that will be frequently used in the sequel.

Definition 2.1 (see [52, page 45]). The solution of the system , locally Lipschitz, is stable conditionally to , if , and for each there exists such that If, furthermore, there exist , s.t. , the solution is asymptotically stable conditionally to . If , the stability is global.

Definition 2.2 (see [52, page 48]). Consider the system , , , , with zero inputs, that is, and let be its largest positively invariant set contained in . We say that is globally zero-state detectable (GZSD) if is globally asymptotically stable conditionally to . If , the system is zero-state observable (ZSO).

Definition 2.3 (see [52, page 27]). We say that is dissipative in containing , if there exists a function such that for all for all and all such that , where the function , called the supply rate, is locally integrable for every , that is, . is called the storage function. If the storage function is differentiable, the previous conditions write as The system is said to be passive if it is dissipative with the supply rate .

Remark 2.4. The definitions of (ZSD) and (ZSO) are simply an extension to the nonlinear case of the classical notions of detectability and observability for linear systems; see for example, [53].

We will also need the following definition to study the case of time-varying faults in Section 3.

Definition 2.5 (see [54]). A function is called a limiting solution of the system , and a smooth vector function, with respect to an unbounded sequence in , if there exist a compact and a sequence of solutions of the system such that the associated sequence converges uniformly to on every compact subset of .

Definition 2.6 (see [55, page 144]). A continuous function is said to belong to class if it is strictly increasing and . A continuous function is said to belong to class if for each fixed the mapping belongs to class with respect to and for each fixed the mapping is decreasing with respect to and as .

Definition 2.7 (53). A system is said of nonminimum phase, if it has internal dynamics, and their associated zero dynamics are unstable in the Lyapunov sense.

Also, throughout this paper it is said that a statement holds a.e. if the Lebesgue measure of the set is zero [54]. We also mean by semiglobal stability of the equilibrium point for the autonomous system with a smooth function, that for each compact set containing , there exist a locally Lipschitz state feedback, such that is asymptotically stable, with a basin of attraction containing (see [56, Definition  3, page 1445]).

3. Passive NFTC

Let us start first with some passive NFTC algorithms. As we said before, these types of FTCs are not expected to “do all the job alone”, since in practice they have to be associated with some active FTCs to obtain an efficient controller tolerant to faults.

3.1. Lyapunov-Reconstruction-Based Passive NFTC

We first consider nonlinear systems of the form

where and represent, respectively, the state and the input vectors. The vector fields , columns of are supposed to satisfy the classical smoothness assumptions, with . We also assume the system (3.1), locally reachable (in the sense of [57, Definition  5, page 400]). Adding to the previous classical assumptions, we need also the following to hold.

Assumption 3.1. We assume the existence of a nominal closed-loop control , such that the solutions of the closed-loop system satisfy , where and is a class function.

Assumption 3.2. We assume here two types of actuator faults.(i)Firstly, one considers faults that enter the system in an additive way; that is, the faulty model writes as where represents the actuator fault and s.t. , where is a nonnegative continuous function.(ii)Secondly, one considers loss of actuator effectiveness, represented by a multiplicative matrix as where is a diagonal continuous time variant matrix, with the diagonal elements s.t. .

The authors in [39] proved the following propositions.

Proposition 3.3. The control law where is s.t. Assumption 3.1 is satisfied, is the associated Lyapunov function, is defined in Assumption 3.2, and denotes the vector function, s.t. ; ensures that the equilibrium point is locally UAS in for the closed-loop system (3.3) and (3.5).

Proposition 3.4. The control law where s.t. Assumption 3.1 is satisfied, is the associated Lyapunov function, and denotes the function; ensures that the equilibrium point is locally UAS in for the closed-loop system (3.4) and (3.6).

These two controllers ensure robust stabilization with respect to additive as well as multiplicative actuators' faults; however, they are discontinuous; that is, due to the function, therefore the authors in [39] proposed the following two “continuous” versions of the previous propositions.

Proposition 3.5. The control law ensures that the solutions of the closed-loop system (3.3) and (3.7) satisfy where, for a vector , and are class functions in and is class .

Proposition 3.6. The control law ensures that the solutions of the closed-loop system (3.4) and (3.10) satisfy where, for a vector , and are class functions in and is class .

The two continuous controllers (3.7) and (3.10) and do not guarantee the local UAS anymore. However, they guarantee that the closed-loop trajectories are bounded by a class function, and that this bound can be made as small as desired by choosing a small in the definition of the function . The passive NFTC recalled above is in closed form and thus easy to implement. However, they have two main drawbacks. Firstly, they are based on the availability of the closed-from expression of the Lyapunov function associated with the nominal stabilizing law, and secondly, they do not consider input saturations in the control design. Therefore, trying to overcome these limitations, other controllers have been proposed and are recalled hereinafter.

3.2. Passivity-Based NFTC

In [15], the passivity theory has been used to develop some new NFTC dealing with actuator multiplicative faults. These results are reported hereinafter.

Theorem 3.7. Consider the closed-loop system that consists of the faulty system (3.4), with constant unknown matrix , and the dynamic state feedback: where is a radially unbounded, positive semidefinite function, s.t. , and . Consider the fictitious system If the system (3.14) is (G)ZSD with the input and the output , then the closed-loop system (3.4) with (3.13) admits the origin as (globally) asymptotically stable ((G)AS) equilibrium point.

In Theorem 3.7, one of the necessary conditions is the existence of , s.t. the uncontrolled part of (3.3) satisfies . To avoid this condition that may not be satisfied for some practical systems, the authors proposed the following Theorem.

Theorem 3.8. Consider the closed-loop system that consists of the faulty system (3.4), with constant unknown matrix , and the dynamic state feedback: where and the function s.t. a radially unbounded, positive semidefinite function satisfying Consider the fictitious system If (3.17) is (G)ZSD with the input and the output , for for all s.t. , . Then, the closed-loop system (3.4) with (3.15) admits the origin as (G)AS equilibrium point.

The previous theorems may guaranty global AS. However, the conditions required may be difficult to satisfy for some systems. Thus, the authors in [15] introduced the following control law that ensures, under less demanding conditions, semiglobal stability instead of global stability.

Theorem 3.9. Consider the closed-loop system that consists of the faulty system (3.4), with constant matrix , and the dynamic state feedback: where the nominal controller achieves semiglobal asymptotic and local exponential stability of for the safe system (3.1). Then, the closed-loop (3.4) with (3.18) admits the origin as semiglobal AS equilibrium point.

In [15], the practical problem of input saturation has been studied, and the following result on general nonlinear models, nonnecessarily affine on , has been proposed.

Theorem 3.10. Consider the closed-loop system that consists of the faulty system: for , and the static state feedback: where is a radially unbounded, positive semidefinite function, s.t. . Consider the fictitious system: If (3.21) is (G)ZSD, then the closed-loop system (3.19) with (3.20) admits the origin as (G)AS equilibrium point. Furthermore .

For the particular case of affine nonlinear systems, that is, , we have the following proposition, which is a direct consequence of Theorem 3.10.

Proposition 3.11. Consider the closed-loop system that consists of the faulty system (3.4), with constant unknown matrix , and the static state feedback: where is a radially unbounded, positive semidefinite function, s.t. . Consider the fictitious system: If (3.23) is (G)ZSD, then the closed-loop system (3.4) with (3.22) admits the origin as (G)AS equilibrium point. Furthermore .

The time-varying versions, that is, for time-varying faults, of the previous results have also been proven in [15] and are recalled hereinafter.

Theorem 3.12. Consider the closed-loop system that consists of the faulty system (3.4) with the dynamic state feedback: where is a function, s.t. , and is a , positive semidefinite function, such that(1);(2)the system is AS conditionally to the set ; (3)for all limiting solutions for the system with respect to unbounded sequence in , then if a.e., then either for some or is a -limit point of , that is, .
Then the closed-loop system (3.4) with (3.24) admits the origin as UAS equilibrium point.

Theorem 3.13. Consider the closed-loop system that consists of the faulty system: for , with the static state feedback: where is a , positive semidefinite function, such that (1);(2)the system is AS conditionally to the set ; (3)for all limiting solutions for the system with respect to unbounded sequence in , then if a.e., then either for some or is a -limit point of .
Then the closed-loop system (3.26) with (3.27) admits the origin as UAS equilibrium point. Furthermore

Proposition 3.14. Consider the closed-loop system that consists of the faulty system (3.4) with the static state feedback: where is a , positive semidefinite function, such that (1);(2)the system is AS conditionally to the set ; (3)for all limiting solutions for the system with respect to unbounded sequence in , then if a.e., then either for some or is a -limit point of .
Then the closed-loop system (3.4) with (3.29) admits the origin as UAS equilibrium point. Furthermore

These passive NFTC schemes are valid for a large class of nonlinear systems, not necessarily affine in the control, and take into account input saturations; however, the conditions to satisfy might be difficult to check when dealing with models having a large number of states.

4. Active NFTC

As we have explained in the introduction, passive FTCs cannot cope with the fault alone, they have to be associated with active FTCs. Indeed, passive FTCs first ensure, at least the stability of the faulty system, during the time period when the FDD is estimating the fault, then active FTC takes over the passive FTC and, using the estimated faulty model they try to optimize the performances of the faulty system. We present in this section some active NFTC schemes.

4.1. Optimization-Based Active NFTC

In [14], the authors studied the problem of graceful performance degradation for affine nonlinear systems. The method is an optimization-based scheme, that gives a constructive way to re-shape online the output reference for the postfault system, and explicitly take into account the actuators and states saturations. The online output reference reshaping is associated with an online, MPC-based, controller reconfiguration, that forces the postfault system to track the new output reference.

The model considered are affine in the control:

where , and represent respectively the state, the input and the controlled output vectors. The vector fields , columns of and function are supposed to satisfy the following classical assumptions.

Assumption 4.1. and the columns of are smooth vector fields on a compact set of and is a smooth function on with .

Assumption 4.2. System (3.1) has a well-defined (vector) relative degree at each point (see e.g., [58]).

Assumption 4.3. The system is fully or over-actuated, in the sense that the number of actuators is at least equal to the number of controlled outputs, that is, .

Assumption 4.4. We assume that assumptions 4.14.3 above, are preserved after the occurrence of a fault in the system.

Assumption 4.5. We assume additionally that the desired nominal trajectory is feasible by the nominal (safe) system, within its input/state limits.

The control objective is then, that to find a controller s.t. the nominal as well as the faulty systems' output vector tracks asymptotically a desired smooth feasible trajectory , while satisfying the actuators and states constraints:

where , and , are vectors of lower/upper actuators and states limits, respectively. To do so the authors formulate the problem as the following optimization problem:

under the constraints

where, , are positive definite weight matrices, is the vector of the polynomials coefficients, is the final motion time for the optimal trajectory vector , and is the final motion time for the nominal trajectory vector , and where hold for the modified vector field and matrix after the occurrence of the fault. The existence of solutions and the computation scheme was then studied for different cases, that is, without internal dynamics, with internal dynamics for minimum phase and with internal dynamics for nonminimum phase systems. The authors did not consider in this paper explicitly FDD synthesis for nonlinear systems. Instead they assumed the availability of an FDD module and they studied both cases: first where FDD provides a precise postfault model and, second the realistic case, where FDD gives a delayed imprecise postfault model. This optimization-based scheme, can deal with the general class of nonlinear models affine in the control, with state and input constraints, and include a stable inversion part to deal with nonminimum phase systems, however, the necessary online computation can be time consuming for large models.

4.2. Learning-Based Active NFTC

We report here the results presented in [59], where the author used a learning scheme to modify the feedback control so as to stabilize the system in the presence of a fault.

The author considers systems of the form

where, , and are the state and control vectors, respectively, and is an matrix function, are known smooth vector fields representing the nominal system dynamics, is a step function representing an abrupt fault occurring at an unknown time , represents the time-varying model uncertainties, and is the vector of state-dependent faults. The author assumes the existence of a nominal controller that guarantees uniform stabilization of the nominal system:

The scheme assumes also the availability of the closed form Lyapunov function associated with the nominal stable feedback system:

The author proposes then the NFTC:

where is a and represents the basis function for the neural network approximation of the fault by . Then, under the assumption of matching conditions, that is, are in the range space of , the author proves that the feedback controller (4.8) stabilize the faulty system (4.5). However, this control law, is based on the knowledge of the full state vector, and might lead to chattering effect if the parameter is chosen too small.

4.3. Adaptive Backstepping-Based Active NFTC

The scheme presented here is based on the results of [4, 50]. The systems studied are of the form

where is the state vector, , are the input and the output, respectively. The function is a nonzero smooth function, and are smooth functions. The control goal is to force the output to track a desired trajectory , where are known, piecewise continuous and bounded. As in Section 4.2, represent the model uncertainties and the expected faults, respectively, and , represent the time profile of the faults. Then, based on assumption of the availability of a FDD module that detects and estimates the fault, the authors propose the following three-stage controller:

where , and are the time of the fault detection and fault isolation, respectively. Based on the adaptive-backstepping approach, the authors propose the following expression for the three controllers.

(i) First for

with

(ii) Second for

with

and the parameter adaptive laws are

with

where are the basis functions of the linear approximation for the unknown fault function, that is, , and are given bounding control functions.

(iii) Third for

associated with the same update laws (4.16), except that the basis functions and the bounding control functions are different from the previous case, that is, for , since in this case they are specific to the isolated fault. Then, the authors proved that under the assumption of bounded uncertainties and bounded fault approximation-error, that is, bounded , that all the signals and parameter estimates are uniformly bounded, that is, , and are bounded . However, this approach is based on the special structure of the faulty model (4.9), and assumes the availability of the measurements of state vector for the feedback control. Eventually, the FDD and FTC presented here are based on the assumption of the fault being part of an apriori known set of expected fault's models.

4.4. Switched Control-Based Active NFTC

We report here the schemes introduced in [25, 26], where the authors consider both problems of FDD and FTC for a class of nonlinear systems, with input constraints. The model studied are of the form

where is the vector of state variables, is the vector of measurable variables, and denotes the control vector under the th configuration. The additive actuator faults are modelled by . The vector function and the matrices are assumed to be sufficiently smooth on their domains of definition. For each value of the system is controlled via a different set of manipulated inputs, which defines a given control configuration. The nonlinear model (4.18) is associated with the following assumption.

Assumption 4.6. Consider the system (4.18) in configuration under state-feedback. Then for every input , there exists a unique state , such that with as output, the relative degree of with respect to and only with respect to is equal to .

This assumption means that each actuator is the only one influencing at least some state. This implies that the effect of a specific actuator on the system evolution is completely distinguishable, which allows fault isolation in that specific actuator. This sufficient fault detection/isolation condition, can be relaxed if the input enters the model in an “upper-triangular” or “lower-triangular” form (refer to [25, Remark  3]). The authors introduced a nonlinear FDD in the following theorem.

Theorem 4.7. Consider the model (4.18) in configuration which satisfies Assumption 4.6, under the control law: assuming that the set , contains the origin and a neighborhood of the origin.

Let the fault detection and isolation filter for the th manipulated input in the th configuration be described by

where denotes the th element of the vector , and the subscripts refer to the th state under the th control configuration. Let be the earliest time for which , then the fault detection and isolation filter of (4.20) ensures that . Also, only if .

Then, the NFTC has been introduced in the following theorem.

Theorem 4.8. Consider the closed-loop system (4.18), (4.19), and let for some , with being defined as: is a level set of . Let be the earliest time such that for some corresponding to a manipulated input in (4.20). Then, the following switching rule: guarantees asymptotic stability of the origin of the closed-loop system (4.18) and (4.19).

This active NFTC are applicable for the general class of nonlinear models affine in the control, and are based on a state-feedback (the authors proposed in the same papers an extension to the case of output feedback). However, they require Assumption 4.6 to hold to be able to detect and isolate the actuator fault. Another point is that this scheme does not consider multiplicative actuator faults.

4.5. Predictive Control-Based Active NFTC

The authors in [28, 29] study the problem of NFTC for nonlinear models affine in the control, with input constraints and uncertainties. The nature of faults treated is actuator faults, under the assumption of controllability of the faulty system. Let us recall below the main result of these work.

The models considered are of the form

where denotes the vector of state variables, , and denotes the vector of constrained inputs. The vector denotes the vector of time-varying uncertainties but bounded variables taking values in a nonempty compact convex subset of . The vector (s.t. ), the matrices , and , are assumed to be sufficiently smooth on their domain of definition. For each value of the index the process is controlled via a different manipulated input, which defines a given control configuration. Switching between the available control configuration is controlled by a higher-level supervisor, which ensures that only one control configuration is active at any given time, and allows only finite number of switches over any finite time interval of time. The main idea of this work is that the authors assume that after the occurrence of a fault, the system will be associated with one of the configuration and then they build off-line a bank of nonlinear model-predictive stabilizing controllers, and based on the value of the state vector at the time of fault occurrence, they switch among these controllers to ensure the stability of the faulty system. To make the presentation of the NFTC clear, we follow the same structure of the paper [28] and present first a Lyapunov-based switched controller, then we present the associated nonlinear model predictive controller (NMPC) and finally we present the NFTC based on this algorithm.

The Lyapunov-based controller associated with the system (4.22) is given by the bounded state feedback:

where, is a robust control Lyapunov function (RCLF) (as defined in [60, page 49]), , , , , , , and .

The following convergence result has been reported in [28] and proven in [61]: let and assume that , for some . Then, given any positive real number, , s.t.: and , s.t. if the solutions of the closed-loop system (4.22) and (4.23) satisfy and .

We also need to recall a convergence result from [28], that characterizes the behavior of the solutions of (4.22) and (4.23), when the continuous controller (4.23) is implemented in discrete time. The result is as follows: consider the system (4.22) for a fixed with , associated with the controller (4.23). Let . Then, , s.t. if and then and . Also, if then and if , then .

Next we report the Lyapunov-based predictive control associated with the Lyapunov-based controller (4.23). The following result has been reported in [28] and proven in [62]:

Consider the system (4.22), for a fixed value of , with , associated with the following NMPC controller:

where are as defined above, is the family of piecewise continuous functions with period mapping into , is the horizon of the optimization, and is RCLF that yields a stability region , under continuous implementation of the controller (4.23), with a fixed . Then, , , s.t., if and , then and .

Finally, we can report the predictive control-based NFTC as follows: Consider the system (4.22), for which the bounded controllers (4.23) and Lyapunov-based MPCs (4.24) have been designed and the stability regions , under the Lyapunov-based MPCs have been explicitly characterized. Let , as defined above, and let . Define , where is the earliest time at which the state of the closed-loop system under bounded controller enters the level set defined by . Then, let for some index and . Let be the earliest time at which a fault occurs. Furthermore, let , and let be such that . Then, the following switching rule

guarantees that and .

To avoid further overload this paper with long equations, we have reported here only the NFTC in the case without uncertainties, that is, <