Shock and Vibration

Volume 2019, Article ID 3818539, 9 pages

https://doi.org/10.1155/2019/3818539

## A New Scheduling Quantitative Feedback Theory-Based Controller Integrated with Fault Detection for Effective Vibration Control

^{1}Smart Structures and System Laboratory, Department of Mechanical Engineering, Inha University, Incheon 22212, Republic of Korea^{2}Faculty of Mechanical and Automotive Engineering, Keimyung University, Daegu, Republic of Korea

Correspondence should be addressed to Seung-Bok Choi; rk.ca.ahni@kobgnues

Received 4 March 2019; Accepted 31 March 2019; Published 7 May 2019

Academic Editor: Mario Terzo

Copyright © 2019 R. Jeyasenthil et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this work, a new integrated fault detection and control (IFDC) method is presented for single-input/single-output systems (SISOs). The idea is centered on comparing the closed-loop output between the faulty system and fault-free one to schedule/switch the feedback control once the fault occurs. The problem addressed in this work is the output disturbance rejection. The set of feedback controllers are designed using quantitative feedback theory (QFT) for fault-free and faulty systems. In the context of QFT-based IFDC, the proposed active approach is novel, simple, and easy to implement from an engineering point of view. The efficiency of the proposed method is assessed on a flexible smart structure system featuring a piezoelectric actuator. The actuator and sensor faults considered are the multiplicative type with both fixed and time-varying magnitudes. In the fixed magnitude fault case, the actuator/sensor output delivering capability is reduced by 50% (multiplying a factor of 0.5 to its actual output), while in the time-varying magnitude case, it becomes 60% to 50% for a particular time interval. In both cases, the proposed control method identifies the fault and activates the required controller to satisfy the specification with less control effort as opposed to the passive QFT design featured by faulty system design alone.

#### 1. Introduction

It is necessary to design control systems that are reliable, fault tolerant, and safe operation of modern and advanced technological systems including flexible structures. The goal of fault-tolerant control systems (FTCSs) is to maintain the performance and stability in the presence of system and sensor/actuator faults. There are two approaches known in the literature such as the passive and active methods, respectively [1]. In passive FTCSs, the design is performed offline using the robust control methods with respect to uncertainty and faults. Active FTCSs involve automatic controller reconfiguration (or) switching mechanism using a fault detection and identification (FDI) module. The passive FTCSs can result in very conservative design during the fault-free phase, especially when the hardware is fully checked a priori by the engineer before start-up [2]. And also when the number of fault scenario increases, the passive FTCSs performance become less and less effective for each fault [3]. This is the main motivation for the active method-based FTCS designs. Active FTCSs use a model-based FDI strategy which has received increasing attention in recent years [1, 3]. A model-based FDI uses a mathematical model of the system and online measurement to draw conclusions about the faults on the system. The core concept of a model-based FDI is to generate a so-called “residual” which are signals that are zero in the fault-free case and nonzero otherwise [4]. In practice, the system is uncertain and the characteristics of the disturbances/noise are unknown, so the residuals are never zero [5]. One of the widely adopted approaches is to choose a so-called threshold value for the fault detection (FD) purpose [6, 7]. This enables us to make a reliable decision in the sense that the false alarm rate due to the model uncertainty/disturbances becomes small. These steps are known as residual generation and evaluation in an FDI module [1, 5]. In [8], the robust dissipative FTC of a discrete time system is addressed. Recently in [9], a randomly occurring actuator fault in complex dynamical networks is considered. It is noted here that the idea to identify the state between the fault-free and faulty model is to monitor the closed-loop output. If the closed-loop output is within the user-defined tolerance, the considered model is a fault-free model.

In this work, the problem of integrated fault detection and control is addressed using quantitative feedback theory (QFT). Basically, QFT is a frequency-domain-based robust control method and uses two-degree-of-freedom (2-DOF) structure to satisfy the performance specifications such as robust tracking/disturbance rejection and also robust stability requirements [10]. The main objective of QFT is to reduce the effects of plant parametric uncertainties with the help of feedback. And the amount of feedback is directly related to the extent of plant uncertainty and unknown external disturbances. The plant uncertainties may be the result of modeling error or linearization of a plant around different operating conditions and/or occurrence of faults. Both QFT and *H*_{∞} performance problems have a common design philosophy except one significant difference being in the representation of uncertainty [11]. The inherent conservatism in *H*_{∞}-based design can lead to high-order controller design unlike the QFT-based low-order controller (use of both loop gain and phase in the loopshaping). So far, QFT has successfully been applied to many engineering applications [11–17]. For the past few decades, QFT-based FTC has drawn considerable attention in the QFT community. A brief review of the existing literature relating the QFT-based passive and active FTCSs is now presented. The QFT-based passive FTCS has been applied to a flight control system with the control surface damage [18, 19]. QFT-based position controller design for an electrohydraulic actuator (EHA) with a faculty position sensor and servo valve is presented in [20]. They account the sensor calibration gain fault by using redundant sensors. A passive QFT-based fault-tolerant position controller is designed for the servo-hydraulic positioning system in the presence of fluid leakage across the actuator piston seal. Recently, another application of QFT to passive FTC is presented for the position control of EHA with a faulty cylinder position seal in [21]. They further designed a QFT-actuating pressure controller for an EHA system with a leaky piston seal [22]. For a comprehensive list of QFT-based FTC work, the reader can refer the papers given in the reference of [21–23]. As far as the application of QFT to active FTC design is concerned, the work in [23] proposed a fault tolerant QFT position controller for electrohydraulic actuators against actuator piston fault. The idea is to design a bank of QFT controllers for the multiple linear models obtained from different leakage levels (piston orifice area) and then apply the aggregated control action by switching based on the smooth Gaussian function. It is remarked that multiplicative type of faults such as actuator and sensor faults are handled by the proposed fault model. In practice, the actuator delivering capability is reduced either by internal or external fault. Similarly, the sensor readings can be erroneous due to internal or external fault.

In fact, the idea of FDI-based QFT was initiated in [24]. The approach is centered on designing the feedback controller to satisfy the robust performance/stability at low-frequency range and maximize the faults (actuator/sensor) for FD purpose at the high-frequency range where the faults are likely to be concentrated. The FD filter design is posed as a model matching problem wherein the filter has to track the prespecified residual reference model. The main difficulty is the selection of the reference model of significant physical meaning for the FD point of view [6]. On the contrary, the study on the integrated fault detection (IFD) and QFT-based control for the multivariable system was undertaken in [2]. In this work, FD objective has been formulated as a constraint on sensitivity maximization over the high-frequency range wherein the fault energy is concentrated. In particular, the sensitivity function with a multiplicative fault (actuator/sensor) was chosen for the fault direction. It has been shown that control performances are fair, but the limitation of the formulations occurred in terms of the feedback cost in the high-frequency gain which can cause excessive control effort. Therefore, in order to resolve the above limitation, a novel IFDC method is proposed in this work. Consequently, the main technical contributions are summarized as follows:(i)A novel IFDC with scheduling QFT is proposed for SISO uncertain system to solve the output disturbance rejection problem.(ii)The faults are detected based on the difference between the closed-loop response of the faulty system and fault-free system (or) tolerance whichever is maximum. To the best of the author’s knowledge, the proposed approach is first of its kind wherein the systematic integration of IFDC with the scheduling QFT controller is presented.(iii)A flexible smart structure system is used to demonstrate the effectiveness of the proposed approach. Both actuator and sensor multiplicative faults with fixed and varying magnitude are considered for vibration control.(iv)The stability of the scheduling QFT controller is analyzed using the existing frequency-domain condition. The proposed method gives the less control effort with the better vibration-control performance as compared to the QFT-based passive FTC.

The paper is organized as follows. Section 2 states the problem statement and key components of the proposed IFDC followed by the frequency-domain stability condition for the scheduling/switching controller. The proposed design procedure for IFDC is given in Section 3 with the validation example of the smart flexible structure featuring the piezoelectric actuator and sensor. Section 4 presents control results with comparative works between the proposed method and existing approach in the time domain. The conclusion of the work is drawn in Section 5.

#### 2. Problem Formulation

The problem to be solved is closed-loop stability and limit cycle avoidance in systems with actuator input amplitude saturation taking into account uncertainty in the plant. The problem to be solved is output disturbance rejection/stability and fault detection in systems taking into account uncertainty in the plant. Our method can be viewed as a complement to [2, 24].

The following specifications are considered for the QFT-based feedback controller design.

##### 2.1. Robust Stability

The robust stability of the closed-loop system with the fault is ensured by designing the feedback controller such that the nominal loop transmission function (*L*_{0} = *k*_{a}*k*_{s}*P*_{0}*G*_{f}) does not penetrate the universal high frequency bound at *ω* = [0, *ω*_{h}]:where *k*_{a} and *k*_{s} represent the actuator and sensor fault magnitude, respectively.

##### 2.2. Robust Output Disturbance Rejection

The closed-loop output disturbance transfer functions should fall below a priori defined disturbance specification within [0, *ω*_{l}] in order to minimize the effects on the output. The closed-loop system should satisfy the following inequality:

For the fault-free case, the actuator and sensor fault magnitudes become *k*_{a} = *k*_{s} = 1 in the specification inequalities (1) and (2).

The key component of the proposed approach is fault detection. The concept involved for fault detection here is simple and direct one. From the specification (2), the closed-loop output response always lies below a user-defined tolerance (*B*_{d}) for any plant element from the uncertainty . So, the idea is to compare the maximum of the tolerance (*B*_{d}) and the worst case fault-free closed-loop output (*y*_{max}) with that of the actual closed-loop system (*y*_{f}). If the actual output is more than the maximum of (*B*_{d}, *y*_{max}), then the fault occurs otherwise there is no fault. The reason for considering the maximum of (*B*_{d}, *y*_{max}) instead of *B*_{d} alone is that the worst case output (fault-free system), sometimes, may exceed the tolerance *B*_{d} by a very small value especially at the steady state. This is due to the fact that there is no direct translation from the frequency domain to the time domain and vice versa [10]. Because of this, the chances of detecting this as a fault become satisfactorily small due to the model uncertainty. The goal of robust residual evaluation in FDI also emphasizes this point of reliable decision-making [5].

The following condition captures the proposed idea for the IFDC:where *y*_{f} and *y*_{max} are denoted as faulty and fault-free (worst case) closed-loop outputs, respectively. And the corresponding feedback controllers are denoted as *G* and *G*_{f}, respectively. This is similar to the model-based FDI wherein residual generation and evaluation is used to identify and isolate the faults. But, here we use it for scheduling/switching between the feedback controllers in order to reduce the fault effect on the output as opposed to the FD purpose (switch ON the alarm) alone.

In this work, we consider the multiplicative fault (actuator/sensor) as it affects the stability of the closed-loop system depending on its size. So, the stability of the closed loop can be analyzed using the following result from the paper [25] for the switching controller. The stability of the two closed-loop systems with tracking transfer functions and under switching is given by the following inequality:where the loop transmission function of systems 1 and 2 are denoted by and , where and denotes the numerator and denominator polynomials, respectively.

Remark. In QFT, the tolerance (*B*_{d}) can be specified as a discrete magnitude (dB) at each discrete performance frequency. Condition (3) checking to identify the fault becomes difficult one as it is required to compare the output continuously. To resolve this issue, it is suggested to compare the magnitude of *y*_{max} with *y*_{f}.

#### 3. Design Procedures for IFDC

##### 3.1. Design of Feedback Controller Using QFT Principle

Design a set of linear feedback controllers (*G* and *G*_{f}) for fault-free and faulty systems, respectively, to satisfy the desired specifications (step 1) using the loopshaping technique. The nominal loop transmission function (fault-free case, *L*_{0} = *P*_{0}*G* and faulty case L_{0f} = *k*_{a}*k*_{s}*P*_{0}*G*_{f}) is shaped such that it should lie on or above the open bounds (performance bounds) and lies outside the closed (stability) bounds at each design frequency. For this purpose, MATLAB QFT toolbox [26] or QFTCT software [27] provides an interactive environment.

##### 3.2. Integrated fault Detection and Control

Next, the actual closed-loop system (with controller *G*) output is continuously compared with the maximum of (*B*_{d}, *y*_{max}), as explained in Section 3 to identify the faults. Once the fault occurs, then activate the controller *G*_{f} into the feedback loop. The proposed design strategy is shown in Figure 1.