Sliding Mode Control with Application to Fault-Tolerant Control: Assessment and Open Problems
Abstract
:1. Context and Motivation
- (i)
- the commonly invoked assumption of about the continuity of the fault profile that breaks the sliding motion;
- (ii)
- the proof of stability when an anti-windup strategy is embedded in the SMC architecture (when such a unit is considered, which is rarely the case); and
- (iii)
- the non-existence of a separation principle when using a fault estimator to schedule the control law, which is the most encountered solution to ensure fault tolerance with SMC techniques, as shown below.
2. The Genesis of Sliding Mode Control Theory
3. The Fundamental Principle of SMC Techniques
- The linear part of the controller is in charge of maintaining the sliding motion. Typically, the nominal equivalent control or a state feedback is employed for . It is designed based on the nominal system, that is with .
- The nonlinear (discontinuous) part is in charge of compensating and inducing the sliding motion.
4. The Five Generations of SMC Techniques
4.1. First Order Sliding Mode Control (FOSMC)
4.2. Second-Order Sliding Mode Approaches
4.2.1. Twisting Algorithm TA
4.2.2. Terminal Sliding Mode TSM
4.2.3. Super-Twisting Algorithm STA
4.2.4. Variable Gain Super-Twisting Algorithm VGSTA
4.2.5. Generalized Super-Twisting Algorithm GSTA
4.2.6. Differentiator
4.3. Arbitrary-Order Sliding Mode Approaches
4.3.1. Arbitrary-Order Differentiator
4.3.2. Continuous Nested Sliding Mode Algorithm (CNSMA)
4.4. A Few Remarks
- Adaptive Sliding Mode Controllers: In the previous sections, the tuning of the described controllers is shown to be dependent on the bound of the disturbance (see, e.g., (34)). One may assume that, when the bound of the disturbance is unknown or variable, the gains of the controllers can be selected with an overestimated bound. The consequence would be an increment in the chattering effect. Adaptive Sliding Mode Controllers (ASMC)s were developed with the aim of having a robust controller, even when the bound of the disturbance is unknown or in the case the disturbance is time-varying. Their main design principle is to adjust the gains of the controller to maintain the sliding motion, depending only on the information that is available. In the literature, different approaches towards applying adaptation to SMC can be found (see, e.g., [42,43] where the coefficients of the switching plane are varied without information of the plant with the aim of improving the systems response). Recent research in this field is dedicated to the proposal of a solution that considers reducing the chattering effect (see, e.g., [44,45,46,47,48,49,50] where the adaptation principles are applied to STA, TA, arbitrary order SMC, TSM, and observers).
- Output Tracking: For the output tracking problem, the control design procedure is the same as explained in the previous sections. The main difference is the definition of the switching function, given that it is based on the tracking error. Following the example shown by Shtessel et al. [7], consider the following system:
5. Fault Tolerant Control and SMC Techniques
- Passive techniques consider that possible system failures are known. The controller is thus developed to cover the a priori known characteristics of a set of pre-specified faults. Given that the controller stays fixed during the systems operation, this makes passive approaches less complex, considering that the robustness properties of the so-designed controller are exploited (see Figure 4). As a consequence, the type of faults that the robust controller can compensate is limited. However, their lack of complexity is an advantage during implementation, given that they have fewer software/hardware requirements.
- Active techniques reconfigure the control parameters in the presence of a fault. They rely on a Fault Detection and Isolation (FDI) unit (see Figure 5). FDI is in charge of the constant monitoring of the status of the system and its components. In this way, when the FDI unit identifies a fault, a reconfiguration is carried out in the controller. As a result, a wider range of faults can be compensated and many control techniques have the potential to be used for fault tolerant control, e.g., both within the LTI and linear parameter varying setting, control allocation, dynamic inversion, adaptive methods, neural networks, and model predictive control, to mention a few. One limitation of this scheme is that it has limited time to perform FDI followed by control reconfiguration. In addition, the accuracy of FDI affects the reconfiguration process. In other words, despite the existence of some stability and performance proofs for active FTC techniques, the main problem lies in guaranteeing stability and performances of the overall fault-tolerant scheme taking into account FDI performances (detection delay, possible false alarms, etc.), control specifications, and reconfiguration mechanism. Solutions to this problem are considered in [56,57]. The authors proposed to use the theory of switched systems based on the dwell-time concept to guarantee stability and fault accommodation, even in the case of false fault identification.
5.1. Modeling the Faults
5.2. SMC Techniques as a Potential Solution for FTC
- If , then ; thus, it is immediate to see that from the FTC point of view that we are facing a large class of actuator faults since the fault profile f may be state-dependent. A particular case often encountered in the literature consists of faults being exogenous signals, i.e., . From the SMC point of view, f can be treated as (state-dependent) matched perturbations.
- If as in (77), then ; thus, we are faced with actuator faults where the ith column of K represents the fault mode of the ith actuator. From the SMC point of view, f is seen as unmatched perturbations.
- If , then the problem is concerned by component faults (see (74)). From a SMC point of view, we are facing unmatched perturbations.
- If f plays the role of matched perturbations, the control theories presented in Section 4.1, Section 4.2 and Section 4.3 can be applied with a few changes. That is why active and passive SMC-based FTC schemes have be proposed in the literature (see, e.g., [66,67,68,69,70,71] for passive and [72,73,74,75,76] for active FTC solutions). However, tuning the gains of the tSMC control law to ensure fault tolerance may lead to a high gain controller. This is in fact related to the outcomes discussed below about the assumption on and its successive time derivatives, and the problem of control saturation.
- If f plays the role of unmatched perturbations, it is required to use a fault reconstruction unit. This can be done by using either a (high-order sliding mode) observer or a differentiator on an adequate formulated problem (see Section 4.2.6 and Section 4.3.1). The models (77), (85), (86), (89), and (90) are generally used for that purpose (for studies that use this principle, see [18,48,64,65,77,78,79,80,81,82,83,84,85,86,87]).
- First, the following assumptions must be satisfied: the system (91)–(92) must be strongly observable, or, equivalently, the triplet has no invariant zeros; and depending on the SMC technique used for FTC and fault reconstruction, f and its derivatives up to a certain order (say r) may be bounded, i.e., , , or must be smooth at least.
- Second, because many SMC techniques are state-feedback control solutions, the state must be available. However, controllers such as the nested algorithm and quasi-continuous sliding-mode controller only require the relative degree and the measurement of one of the state variables.
- Third, fault accommodation is done at the expense of increasing the control actions, which may lead the control signal to go into saturation.
5.3. Applications of SMC for FTC in Real-World Systems
5.3.1. Robot Manipulators
5.3.2. Marine Vehicles
5.3.3. Aeronautical Applications
5.3.4. Space Applications
5.3.5. Notable Facts
- Almost all current SMC-based FTC solutions are based on the ASMC principle, as described in Section 5.2, i.e., a fault estimator is used to schedule a SMC law. It is interesting to note that the most popular control scheme is the TSM control solution and its variants, i.e., the fast and nonsingular fast versions.
- Many papers do not consider the problem of control signal saturation. The majority of contributions in this field are developed for space applications.
- No papers (or almost none) have proposed a solution to relax the assumption about the discontinuity of the fault profile. The considered faults are manly concerned by loss-of-efficiency or smooth fault profiles.
- Finally, no papers have established a formal proof of the existence of a separation principle when using a fault estimator, even if it is mainly used as the fault tolerance principle. This means that the coupling between the dynamics of the fault estimator and the SMC law is not sufficiently studied, in our opinion.
6. Conclusions
- (i)
- the case of discontinuous faults, since, by definition, such faults do not respect the assumption about the bound required by the sliding-mode controller to maintain the sliding-motion, and to not test again a transient process of finite-duration;
- (ii)
- the proof of stability when an anti-windup strategy is joint to the SMC architecture (when such a unit is considered, which is rarely the case); and
- (iii)
- the non-existence of a separation principle when using a fault estimator.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Zenteno-Torres, J.; Cieslak, J.; Dávila, J.; Henry, D. Sliding Mode Control with Application to Fault-Tolerant Control: Assessment and Open Problems. Automation 2021, 2, 1-30. https://doi.org/10.3390/automation2010001
Zenteno-Torres J, Cieslak J, Dávila J, Henry D. Sliding Mode Control with Application to Fault-Tolerant Control: Assessment and Open Problems. Automation. 2021; 2(1):1-30. https://doi.org/10.3390/automation2010001
Chicago/Turabian StyleZenteno-Torres, Jazmín, Jérôme Cieslak, Jorge Dávila, and David Henry. 2021. "Sliding Mode Control with Application to Fault-Tolerant Control: Assessment and Open Problems" Automation 2, no. 1: 1-30. https://doi.org/10.3390/automation2010001