Variable structure control (VSC) / Sliding mode control (SMC)

VSC systems are characterized by a suite of feedback control laws and a decision rule [87]. The decision rule, named switching function, decides which feedback controller should be used at a given instant, based on the system state. A VSC system can be regarded as a combination of subsystems, where each subsystem has a fixed control structure and is valid for a specific subset of system states. SMC is a particular type of VSC, where the state of the system is driven and constrained to lie in a neighbourhood of the switching function, with the advantage of making the system insensitive to a particular class of uncertainty. VSC/SMC has been used successfully in many applica- tions of FTC [9,10]. A reconfigurable SMC that achieves robust tracking after damage in an aircraft has been designed in [295] and [296]. A MF scheme, based on VSC, that possesses a fault tolerance property has been proposed by [160]. The combination of integral SMC methodology and observers with hypothesis testing has been used for FTC of a spark ignition engine in [162].

6.5

Recent developments of fault tolerant control

This section resumes some recent developments of FTC theory [372], highlighting some open issues that motivate further investigation in this topic.

• Redundancy: since the introduction of the concept of analytical redundancy, i.e. the use of a mathematical model of the system for FDI/FTC, important research efforts on how to efficiently utilize this concept have been made [119,220]. The following challenging issues regarding redundancy can be detected [372]: (i) the design of the overall fault tolerant and redundant system architecture; (ii) the optimal configuration of redundancy, achieving a tradeoff between specifications

and cost; (iii) the design and implementation of a fault tolerant controller that utilizes at best both the hardware and the analytical redundancy to achieve the control objectives; and (iv) the introduction of quantitative measures of the degree of redundancy [144,357,375].

• Integrated design of fault diagnosis/fault tolerant control: as remarked by [372], in order to obtain a functional active FTCS, it is important to ensure an adequate cooperation between the fault diagnosis and the FTC algorithms. In fact, an incor- rect information provided by the fault diagnoser can lead potentially to undesired behaviors and overall degradation of the FTCS performance. For this reason, the way how to integrate both the subsystems is an important topic of research, and recent papers have addressed this issue [129,152,310,363]. Also, another impor- tant issue worth investigating is how to mitigate the adverse interactions between each subsystem [84]. For further discussion about the integration of fault diagno- sis and fault tolerant control, the reader is referred to the survey paper in [371]. • Design for graceful performance degradation: recently, some techniques have

tried to achieve fault tolerance without aiming at recovering the original system performance, but accepting some performance degradation instead. These tech- niques are of particular interest in the case of actuator faults. In fact, once an actuator is affected by a fault, maintaining the original performance will typically increase the effort distributed on the remaining actuators, which is highly unde- sirable in practice, due to physical constraints on the actuators. Therefore, recent works have considered the design of FTCS with graceful performance degrada- tion, e.g. [143,370].

• Stability and stability robustness: in the case of active FTCS, stability require- ments are specified under different situations: (i) fault-free operation; (ii) tran- sient during reconfiguration; (iii) steady-state after reconfiguration. In all these situations, it is important to investigate the stability robustness [226]. As re- marked by [372], despite much work has been done, e.g. [191], stability analysis and stability robustness for real-time reconfigurable control systems in practical environments still need further investigation.

• FTC design for nonlinear systems: several strategies have been proposed to deal with nonlinear systems, such as feedback linearization [107], nonlinear dy- namic inversion [78], backstepping [360] and neural networks [365] among oth- ers. However, as stated by [372], the development of effective design methods for dealing with nonlinear FTCS issues is still an open research problem.

• FTC of constrained systems: the design of FTCS subject to actuator amplitude and rate saturation constraints has been investigated by a few works, such as

[41] and [221]. However, there are still many open problems in the framework of MIMO systems [372].

• Dealing with fault diagnosis uncertainties/delays: the presence of errors in the fault diagnosis process are inevitable [192]. Also, time delays and false alarms are associated with fault diagnosis decisions [196]. Hence, it is important to take into account these undesired effects to reduce their impact on the FTCS, and the development of new and practical approaches to accomplish this goal is an inves- tigation hot topic [372].

• Self-designing FTC: recent research activities have focused on FTC laws which rely on online estimation of plant parameters [207,299,342]. There are still some challenges in this field, e.g. dealing with poor input excitation and the adverse interactions between the identification and the control in a closed-loop setting [372].

• Control allocation: in presence of actuator redundancy, control allocation tech- niques aim at choosing how to use the available actuators in order to achieve a specified objective. In case of actuator faults, these techniques try to make the best use of the remaining healthy actuators [66,374]. As remarked by [146], there are still computational issues in the application of control allocation to nonlinear systems.

• Transient management: undesired transients during the reconfiguration process may be harmful to the safe operation of an FTCS, and lead to undesired conse- quences, such as saturations in the actuators, and damage to the components. For this reason, these transients should be minimized as much as possible which, in spite of a few results available in the literature, e.g. [117], is still an open problem. • Real-time issues: all the subsystems in an AFTCS should operate in real-time, and for this reason there should be hard deadlines for controller reconfiguration, in order to avoid risky situations. This issue, despite its criticality, has not been dealt with satisfactorily, and there are only a few works addressing it, e.g. [119]. • Fault-tolerant networks: FTC in networked control systems is a challenging prob-

lem due to timing issues, network-induced delays and packet losses, whose ef- fects should be carefully taken into consideration [135,283].

6.6

Conclusions

In this chapter, a review of the available FTC approaches has been performed. The ability to maintain stability and acceptable performance in spite of faults has moti- vated a lot of effort in developing FTC strategies. FTC techniques can be divided into three categories. Hardware redundancy approaches achieve fault tolerance by exploiting extra components, providing a simple but expensive solution. Different passive FTC approaches have been recalled (reliable linear quadratic control, H∞control, and LMI- based design). They all increase the robustness of the nominal controller against some predefined set of faults, and share the advantage of needing neither fault diagnosis nor controller reconfiguration, but at the expense of limited fault tolerance capabilities. Fi- nally, the active FTC approaches (reconfigured linear quadratic control, pseudo-inverse method, intelligent control, gain-scheduling, model following, adaptive control, mul- tiple model, integrated diagnostics and control, eigenstructure assignment, feedback linearization/dynamic inversion, model predictive control, quantitative feedback the- ory, variable structure control/sliding mode control) use the information given by the fault diagnosis to perform some automatic adjustments after the fault appearance, in order to achieve fault tolerance.

Despite the strong development of FTC theory in the last decades, a lot of open issues, resumed in the last part of the chapter, motivate further investigation in this topic.

Fault tolerant control of LPV

systems using robust state-feedback

control

The content of this chapter is based on the following works:

• [249] D. Rotondo, F. Nejjari, V. Puig. Passive and active FTC comparison for poly- topic LPV systems. In Proceedings of the 12th European Control Conference (ECC), pages 2951-2956, 2013.

• [250] D. Rotondo, F. Nejjari, V. Puig. Fault tolerant control design for polytopic uncertain LPV systems. In Proceedings of the 21st Mediterranean Control Conference (MED), pages 66-72, 2013.

• [251] D. Rotondo, F. Nejjari, A. Torren, V. Puig. Fault tolerant control design for polytopic uncertain LPV systems: application to a quadrotor. In Proceedings of the 2nd International Conference on Control and Fault-Tolerant Systems (SysTol), pages 643-648, 2013.

• [265] D. Rotondo, F. Nejjari, V. Puig. Robust quasi-LPV model reference FTC of a quadrotor UAV subject to actuator faults. International Journal of Applied Mathe- matics and Computer Science, 25(1):7-22, 2015.

7.1

Introduction

In Chapter 4, the idea of the robust LPV polytopic technique, obtained merging known results from the robust polytopic control area and the traditional LPV polytopic control

area, has been introduced. In the proposed technique, the vector of varying parameters is used to schedule between uncertain LTI systems. The resulting approach consists in using a double-layer polytopic description to take into account both the variability due to the varying parameter vector and the uncertainty. The first polytopic layer man- ages the varying parameters and is used to obtain the vertex uncertain systems, where the vertex controllers are designed. The second polytopic layer is built at each vertex system to take into account the model uncertainties and add robustness in the design step.

In this chapter, it is shown that the proposed framework can be used for FTC, with the advantage that, depending on how much information is available, it gives rise to differ- ent strategies. If the faults are considered as though as they were perturbations, a pas- sive FTC would arise. On the other hand, if the faults are used as additional scheduling parameters, an active FTC would be obtained. Finally, if the fault estimation uncertainty is taken into account explicitly during the design step, the robust LPV polytopic tech- nique would lead to a hybrid FTC. The different controllers are obtained using LMIs, in order to achieve regional pole placement and H∞performance constraints. Results obtained using a quadrotor UAV simulator are used to show the effectiveness of the proposed approach.

7.2

Problem formulation

Consider the following slight modification of the LPV system (2.1):

σ.x(τ ) = A (θ(τ )) x(τ ) + B (θ(τ )) u(τ ) + c(τ ) (7.1) where c(τ ) is a known exogenous input, and let us consider the problem of designing a control scheme in order to achieve the goal of tracking a desired trajectory. The con- ditions provided in Chapter2for the analysis and state-feedback controller design for LPV systems cannot be directly applied, since they refer to the stability of the origin. In order to assure the convergence of the system trajectory to the desired one, the idea of LPV model reference control is considered. Originally developed for the LTI case [81], this technique has been successfully extended to the LPV case [3,51] and relies on the use of a reference model, as follows:

σ.xref(τ ) = A (θ(τ )) xref(τ ) + B (θ(τ )) uref(τ ) + c(τ ) (7.2) where xref ∈ Rnx is the reference state vector and uref ∈ Rnu is the reference input vector. The reference model gives the trajectories to be followed by the real system.

Thus, considering the tracking error, defined as e(τ ) , xref(τ ) − x(τ ), the following error system is obtained:

σ.e(τ ) = A (θ(τ )) e(τ ) + B (θ(τ )) ∆u(τ ) (7.3) with ∆u(τ ) = uref(τ ) − u(τ ).

Then, the results presented in Chapter 2 can be applied for the design of an error- feedback control law of the form:

∆u(τ ) = K (θ(τ )) e(τ ) (7.4)

that constitutes a slight modification of (2.134):

u(τ ) = K (θ(τ )) x(τ ) (7.5)

Hence, the control action to be applied to the system will be made up by the sum of two components, the feedforward one uref(τ ), and the feedback one ∆u(τ ).

However, in presence of faults, the equation (7.1), in the following denoted as nominal system, does not describe correctly the system dynamics anymore, and the obtained results in terms of stability and performance could not hold anymore. In particular, in this chapter, two types of faults are considered: i) parametric faults, affecting the matrix A (θ(τ )) and changing it into Af(θ(τ ), f (τ )); and ii) actuator faults, affecting the matrix B (θ(τ )) and changing it into Bf(θ(τ ), f (τ )). Hence, under fault occurrence, the equation (7.1) becomes:

σ.x(τ ) = Af(θ(τ ), f (τ )) x(τ ) + Bf(θ(τ ), f (τ )) u(τ ) + c(τ ) (7.6) that will be referred to as the faulty system, and the error system changes into:

σ.e(τ ) = A (θ(τ )) xref(τ ) + B (θ(τ )) uref(τ ) − Af(θ(τ ), f (τ )) x(τ ) − Bf(θ(τ ), f (τ )) u(τ ) (7.7) from which is evident that a controller in the form (7.4), designed to behave in some desired way when applied to (7.1), could lead to a very different behavior when applied to the error faulty system (7.7).

In the following section, the problem to be solved is the one of adding fault tolerance to the control scheme. This will be done by redefining the reference model and by designing the error-feedback controller using some results about the robust feedback control of uncertain LPV systems presented in Chapter4.

7.3

Fault tolerant control