1. Introduction
As the demand for heightened performance in contemporary industrial systems grows, particularly in terms of system safety, dependability, ease of maintenance, and the ability to withstand adverse conditions, the likelihood of system failures also rises. Disruptions and defects in control elements, such as actuators and sensors, can lead to subpar performance or even instability, constraining the applicability of conventional control techniques that were once well-established. To overcome these limitations, novel control strategies have surfaced, aiming to ensure the stability of the feedback system even in the face of potential component malfunctions. A novel approach known as fault-tolerant control (FTC) has gained traction, involving the dynamic adjustment of the control law guided by fault estimations impacting the system. The problems of fault estimation (FE) and FE-based FTC have been widely investigated and fruitful results have been reported in several books [
1,
2,
3], survey papers [
4,
5,
6], and the references therein.
At another research forefront, in the last two decades, linear parameter-varying (LPV) systems have attracted much attention owing to their wide practical applications. The main reason arises from the fact the LPV approach provides efficient ways to deal with some nonlinearities. In the realm of LPV systems, the transformation of a nonlinear system into an array of interpolated, localized models governed by convex weighting functions is a fundamental approach. In a broader context, LPV systems are considered as nonlinear systems that experience a momentary linearization along a path defined by a parameter vector. This parameter vector is commonly assumed to be continuously and dynamically measurable in real-time, evolving within a polytopic region. Hence, scholarly focus in the academic sphere has significantly gravitated towards the creation of FE and FTC techniques based on LPV models, as indicated by references [
7,
8,
9].
Within this landscape of diverse approaches, the sliding mode observer (SMO) emerges as a notably effective technique for addressing FE challenges, particularly in the presence of disturbances. This method has been demonstrated to have the capability of concurrently estimating faults and states by capitalizing on the injection of an equivalent output error signal, vital for sustaining sliding motion within the state space of estimation errors, as evidenced by references [
10,
11,
12,
13,
14].
For instance, in [
10], a novel FE scheme predicated on an innovative SMO design method was introduced for LPV systems afflicted by sensor faults. A filtering mechanism for the discontinuous term was introduced to counteract chattering phenomena. The authors in [
11] tackled the intertwined issues of FE and FTC for LPV descriptor systems grappling with actuator faults and time delays, engineering a synthesized SMO to simultaneously estimate system states and actuator faults.
In ref. [
13], a pair of polytopic SMOs was ingeniously designed for an LPV system, offering the ability to estimate actuator and sensor faults simultaneously through the injection of two equivalent output error signals. However, it is important to note that this approach is primarily tailored for incipient fault signals.
Furthermore, in [
14], an integrated sensor fault-tolerant control scheme was proposed for LPV systems, where sensor faults were reconstructed using the SMO framework. This approach involved the simultaneous synthesis of gains in the LPV observer and the LPV static feedback controller to optimize the closed-loop system’s performance.
Nevertheless, there remains a conspicuous research gap pertaining to the challenge of FE and FTTC rooted in dynamic output feedback controllers for LPV systems subjected to simultaneous external disturbances, actuator faults, and sensor faults. It is precisely this void that serves as the impetus for the present study reported in this paper.
In this paper, we tackle the challenge of simultaneously dealing with disturbances, actuator faults, and sensor faults in LPV systems. We begin by employing a clever coordinate transformation to distinguish between actuator and sensor faults. We then proceed to devise an LPV adaptive sliding mode observer (ASMO) that accomplishes the concurrent estimation of system state, actuator faults, and sensor faults. Notably, this design allows us to reconstruct actuator faults through equivalent output error injection, while sensor FE is carried out through an adaptive algorithm driven by a proportional and integral component of the output estimation error. This innovative FE algorithm significantly enhances both the speed and accuracy of fault detection.
In the contemporary industrial landscape, where the demand for enhanced system performance and robustness continues to grow, the specter of system failures looms ever larger. Factors such as system safety, dependability, ease of maintenance, and the ability to withstand adverse conditions have become paramount. These factors underscore the critical need for control systems capable of functioning reliably in the presence of component malfunctions. Traditional control techniques, once the cornerstone of industrial control, are rendered less effective in the face of disruptions or defects in control elements, such as actuators and sensors. It is in response to this challenge that novel control strategies, with a particular emphasis on fault-tolerant control, have emerged.
Fault-tolerant control involves dynamically adjusting the control law in response to fault estimations, thus safeguarding the stability of the entire closed-loop system, even in the presence of faults. While substantial research has been conducted in the field of FE and FE-based FTC, the development of such techniques has primarily been centered on LPV systems. LPV systems have garnered significant attention in the past two decades due to their wide range of practical applications and their inherent ability to address some nonlinearities efficiently.
In the realm of LPV systems, a fundamental approach involves transforming nonlinear systems into a collection of interpolated, localized models defined by convex weighting functions. These models are governed by a continuously and dynamically measurable parameter vector, which evolves within a polytopic region. As a result, research in the academic sphere has seen a notable shift towards the development of FE and FTC techniques based on LPV models [
15,
16,
17].
Within this landscape of diverse approaches, the SMO has emerged as a particularly effective technique for addressing FE challenges, especially in the presence of disturbances. The SMO method is capable of concurrently estimating faults and system states through the injection of an equivalent output error signal, a vital component for sustaining sliding motion within the state space of estimation errors. Previous research has explored the application of SMO in various scenarios, from sensor fault estimation to the simultaneous estimation of actuator faults and system states [
18,
19,
20].
However, a notable research gap remains when it comes to dynamic output feedback controllers for LPV systems subjected to simultaneous external disturbances, actuator faults, and sensor faults. This specific research void serves as the driving force behind the study presented in this paper.
Our research endeavors to bridge this gap by addressing the simultaneous challenges posed by external disturbances, actuator faults, and sensor faults in LPV systems. We begin by employing a clever coordinate transformation to differentiate between actuator and sensor faults, a crucial step in the fault-tolerant process. Subsequently, we introduce a novel LPV ASMO, capable of simultaneously estimating system states, actuator faults, and sensor faults. This innovative design enables the reconstruction of actuator faults through the injection of equivalent output error, while sensor fault estimation is carried out through an adaptive algorithm driven by the proportional and integral components of the output estimation error. This novel FE algorithm enhances the speed and accuracy of fault detection. In summary, our study addresses the pressing need for a comprehensive solution to simultaneous disturbances and multiple fault scenarios in LPV systems, contributing to the growing body of research in the field of fault-tolerant control for complex industrial systems.
Following this, we formulate a tracking controller resilient to faults, with the goal of mitigating their effects and guaranteeing that the states of the closed-loop system accurately track their intended reference signals. The primary contributions of this paper can be succinctly summarized as follows:
In the case of LPV systems encountering concurrent external disturbances, actuator and sensor faults, we delve into the novel challenge of addressing FE and FTTC using a dynamic output feedback controller.
Compared with the results in [
7,
11,
20], the proposed ASMO can not only reconstruct the actuator faults by exploiting the equivalent output error injection, but can also estimate the sensor faults using an adaptive algorithm of the output estimation error.
Unlike the results reported in reference [
13], the proposed ASMO presents a more forgiving perspective on the structural prerequisites necessary for creating an observer-based estimation system that can effectively manage both actuator and sensor faults concurrently.
By employing performance standards, fresh adequate requirements for achieving the desired LPV ASMO and FTTC are extracted and formulated as a convex optimization challenge utilizing linear matrix inequalities (LMIs).
It is worth emphasizing that the design of the ASMO and FTTC is carried out separately—a design method that conveniently lessens computational intricacy.
2. System Description
Let us contemplate a continuous LPV system, influenced by actuator faults, sensor anomalies, and external disturbances.
In this context, we have several variables at play: x representing the system state in , u denoting the control input in , and y indicating the measured output in . Additionally, we have signals like residing in to represent the actuator faults, in for the sensor faults, and in representing the disturbances.
The dynamics of the system are characterized by parameter-varying matrices denoted as , , , and , which depend affinely on the varying parameters vector . It is worth noting that is measured in real-time, facilitating dynamic adaptation.
Furthermore, we have fixed matrices in play: G has full column rank, C has full row rank, and .
In this paper, it is assumed that the varying parameters vector
is bounded and varies in a hypercube
ℑ such that
where
and
represent, respectively, the minimum and maximum values of
. Under the assumption of the affine dependence of the parameter vector
, the system (1) can be represented by a convex combination of
k vertices where each vertex
is defined as
The polytopic coordinates are denoted by
and vary within the convex set
.
To unveil the core findings, we lay the groundwork by introducing specific assumptions and lemmas.
Assumption 1. We presuppose that the varying matrix can be decomposed intowhere is fixed and is a varying matrix that is assumed to be invertible for all . Remark 1. The utilization of factorization (5) will contribute to the advancement of the intended observer. While this constraint is evident, it is noteworthy that this ideal factorization is frequently encountered in over-actuated systems or in fault-tolerant systems equipped with inherent redundancy, such as large civil aircraft. An illustration of a corresponding factorization for a large civil aircraft can be found in references [12,21,22]. Applying Equation (
5), we can express the system from Equation (
1) as
where
is the virtual actuator faults. In the following, an observer will be designed to firstly estimate
and then
will be recuperated by exploiting the fact that
. Assume that
where
is a known scalar function.
Assumption 2. We posit that the distribution matrix for virtual actuator faults, denoted as F, possesses a complete column rank, signifying: Assumption 3. For all , , F and C in (6) satisfy the rank conditionfor every complex number s with a non-negative real part. Assumption 4. The sensor fault , its derivative, and the disturbance ξ satisfywhere , , and are known positive constants. Moreover, the disturbance ξ belongs to . Remark 2. Remark 2 asserts that Assumptions 2 and 3 are both essential and comprehensive conditions for formulating a stable sliding-mode, particularly in scenarios involving disturbances, as documented in references [3,13]. Assumption 3 posits that all invariant zeros of the triple must reside in the left half-plane, or alternatively, the triple must exhibit minimum phase characteristics. Additionally, Assumption 4 mandates that the derivative of the sensor fault, as well as of the disturbances, remains bounded, which aligns with a general assumption in fault estimation (FE) and fault-tolerant control (FTC) methodologies, as elucidated in reference [13]. Lemma 1 ([
23])
. Suppose we have U and V, two matrices with suitable dimensions, , such that Lemma 2. According to reference [24], for a square matrix , it is possible to ascertain that the eigenvalues of lie within a circular region denoted as . This circular region is centered at the coordinates and exhibits a radius of β. This significant property remains valid, provided that there exists a positive-definite matrix such that the subsequent condition is satisfied: 5. Fault-Tolerant Tracking Controller Design
Employing the obtained FE data, we will proceed to formulate a dynamic output feedback controller tailored for fault-tolerant tracking. The objective is to guarantee that the specified output of the compromised LPV system faithfully follows the prescribed reference signal.
To counteract the repercussions stemming from sensor faults, we will employ the subsequent sensor compensation output:
where
is called the reliable output. The objective is to revamp the LPV system using an extended model, primarily encompassing the state vector
x and the state representing the tracking error, denoted as
. This error state is defined as the integral of the difference between the desired reference and the rectified output, with the objective of accomplishing the following:
where
is the tracking error,
is the desired reference, and
is used to define which output variable is considered to track the desired reference.
Considering
and
to be the outputs, the LPV augmented tracking system can be constructed in the following form:
For system (72), we propose the following structure of the FTTC law:
Here, we have several components at play:
is the state of the controller, residing in .
represents the closed-loop error signal, which is defined as . We introduce the matrix in to ensure that the dimensions of and align appropriately.
, , , , and denote the controller matrices, which we will derive subsequently. These matrices have dimensions in , , , , and , respectively.
Substituting (72) into (73), we obtain
Then, substituting (74) into (72), we further obtain
The result (75) follows by selecting
, where
is the pseudo-inverse of
for all
. Hence,
is considered as a known gain in the derivation of the LMI-based FTTC design.
We derive the dynamic equation governing the closed-loop system as follows:
where
,
, and
We define the output performance
as
where
is part of the design.
The next step is to design the controller gains to make the closed loop system (75) stable, and to satisfy the following performance index:
where
is a positive scalar.
Theorem 3. Given the circular region , if there exist matrices , , , , , and such that the following inequalities hold:andwithFollowing this, the closed-loop system, as described by Equation (76), exhibits robust stability at the performance level denoted as , as indicated in (79). Additionally, the eigenvalues of the matrix are situated within the set . The determination of the controller gains is subsequently carried out using the following methodology:where U, satisfy . Proof. Supposing that
. We have
Let
Substituting (83) into (84), we have
Let us define
We have
where
. Substituting (87) into (85), we have
Using the Schur complement, it is easy to find that
if
We can find that the form (89) is not an LMI. As a result, we can change (89) to make it solvable.
Let us define
Since
, we can deduce that
. By performing matrix multiplication on both sides of Equation (
89) with
and its transpose, we can derive the following:
Taking into consideration the formulations of
,
, and
as provided in (77), and incorporating the subsequent change of variables:
Inequality (80) can, thus, be easily obtained.
Our subsequent objective is to formulate the conditions for regional pole constraints. When we substitute
and
into Lemma 2, it becomes evident that the eigenvalues of
lie within the sector shaped like a cone, denoted as
, if:
Next, by left- and right-multiplying (93) with
and its transpose, and subsequently employing the definitions of
,
,
, and
as presented in (92), we can readily derive the inequality (81). This concludes the proof. □
The structure of the proposed FE and FTTC strategy is shown in
Figure 1.
7. Conclusions
In this work, we explore the complex issues surrounding the domains of FE and FTTC within the framework of LPV systems. These systems confront the daunting task of coping with concurrent disturbances, actuator faults that vary over time, and sensor faults.
The LPV systems we are examining are depicted within a polytopic LPV framework, encompassing measurable gain scheduling functions. We introduce an adaptive LPV SMO designed to provide robust state and fault estimations, even in the presence of external disturbances. What sets our proposed approach apart from traditional SMO designs is its dual capability: it can directly reconstruct actuator faults by harnessing the equivalent output error injection required to maintain sliding motion, while also employing an adaptive algorithm to estimate time-varying sensor faults.
To address the FE and FTTC challenges in LPV systems plagued by simultaneous external disturbances, actuator faults, and sensor faults, we propose a novel approach based on dynamic output feedback controllers. Our contributions can be summarized as follows:
We venture into uncharted territory by investigating FE and FTTC problems specifically for LPV systems grappling with concurrent external disturbances, actuator faults, and sensor faults.
Our proposed SMO outperforms existing approaches by not only enabling direct actuator fault reconstruction through equivalent output error injection but also by effectively estimating time-varying sensor faults using an adaptive algorithm driven by the output estimation error.
Differing from prior research, our SMO takes a more flexible approach by alleviating specific structural constraints necessary for the design of an observer-based system capable of simultaneously estimating actuator and sensor faults.
We present novel adequate prerequisites for the establishment of the sought-after adaptive LPV SMO and FTTC, making use of performance standards. These prerequisites are reformulated as convex optimization challenges rooted in LMIs.
Notably, our SMO and FTTC designs are independent, reducing computational complexity, and enhancing their practical applicability.
In conclusion, our comprehensive investigation and innovative solutions pave the way for enhanced FE and FTTC in LPV systems, fostering greater reliability and performance in the face of complex real-world challenges.