Latest revision as of 11:46, 19 March 2024

Abstract

In this paper, a class of nonlinear systems with mismatched disturbance and actuator failure is investigated. A disturbance observer is proposed to estimate the disturbance first and the error of the estimation converges to zero exponentially. By introducing an integral sliding mode surface, the disturbance observer-based integral sliding mode fault tolerant control scheme is proposed to attenuate the disturbance and guarantee the stability of the system. In particular, the control law is designed for decoupling the partial disturbance and attenuating the disturbance that cannot be decoupled. Finally, two examples are given to illustrate the effectiveness of the proposed method.

Keywords: Actuator fault, fault tolerant control (FTC), disturbance observer, adaptive integral sliding mode control, nonlinear system

1. Introduction

Faults frequently occur in the engineering system because of the increasing complexity and scalability of industrial applications. Unexpected deviations of performance or system parameters can induce serious damage and even break down the system in the presence of a fault. With the growing demand for higher reliability, safety, and maintainability, it is desired that the fault can be detected at the early stage, determine the location and magnitude of the fault, identify the severity of the fault, and then accommodate the effects on the system and provide an acceptable performance. Abundant of results have been reported on the theme and many achievements have been applied to industrial systems such as aircraft systems [1], electric systems [2], and motor systems [3-4], etc. Many excellent methods were exploited such as robust control [5], sliding mode control [6-7], observer-based control [8], intelligent learning control [9], and adaptive control [10-11].

As aforementioned, the fault is vulnerable to occur in the practical system, which certainly reduces the nominal performance, results in vibration, destroys the stability of the system, and even causes catastrophic accidents. Actuator fault as the most common fault has caused wide attention. In Ye and Yang [12], the flight tracking control system with actuator fault was investigated; an adaptive controller was designed. In Zuo et al. [13], a class of singular systems with actuator saturation was developed; an adaptive controller was designed to compensate for the fault effects through the linear matrix inequality (LMI) technique. A class of nonlinear large-scale systems with actuator fault was considered and an observer-based fuzzy adaptive control was developed in Tong et al. [14]. The problem of part loss of effectiveness of the actuator was addressed in Li and Yang [15]. In Van et al. [16], a class of nonlinear systems with partial loss of actuator was considered; a third-order sliding model control was developed to compensate for the actuator faults. These results have considered linear or nonlinear systems, however, the nonlinear function either satisfies the matched condition [16], i.e., the nonlinear function is in the control channel.

It is worth pointing out that disturbance exists widely in many industrial processes, which affects the stability of the system seriously. The basic characteristics of disturbance are its uncertainty, nonlinearity, and complexity. In some cases, the fault is considered as an additional uncertainty, disturbance, or nonlinear function in the system [16,17]. The common method is to make a compensator, i.e. construct an anti-disturbance mechanism to compensate for the disturbance. As the characteristics of the disturbance, observer design is necessary and popular in the existing literature, i.e., disturbance observer (DO). In Kempf and Kobayashi [18], a precision positioning table system was studied and a discrete time-tracking controller was designed based on the DO. In Chen et al. [19], the ball mill grinding circuits system was investigated and a DO was designed to estimate the strong disturbance, the controller was designed based on the observer to compensate for the disturbance. In Chen [20], a class of nonlinear systems was investigated, by introducing a disturbance generator; the DO-based control law was established. Yao and Guo [21] addressed a class of Markovian jump systems with multiple disturbances; the control law was designed by integrating the DO output information and state feedback control. Wu et al. [22] take an insight into a generic hypersonic vehicle system with modeled and unmodeled disturbances. In these results, the disturbance matrix ${\displaystyle B_{d}}$ and the control matrix ${\displaystyle B}$ are assumed to satisfy the match condition, i.e., ${\displaystyle rank\left(B\right)=rank\left(\left[{\begin{array}{cc}B&B_{d}\end{array}}\right]\right)}$, this limitation may not be applicable in some control processes.

It should be pointed out that the mismatched disturbance, that is, the disturbance enters the system in different channels, is more practical in industrial dynamic systems. In Zhang et al. [23], a class of nonlinear systems with actuator fault, sensor fault, and mismatched disturbance was considered. In Wei and Guo [24], the disturbance was considered as two parts, the matched disturbance was compensated by the DO while the mismatched disturbance was attenuated by variable structure control. In Yang et al. [25], by constructing a nonlinear disturbance observer (NDO), the sliding mode control scheme was developed to counteract the mismatched disturbance.

Based on the aforementioned analysis, this paper attempts to solve the FTC problem for a class of nonlinear systems with actuator fault and mismatched disturbance. The main works can be summarized as follows. First of all, an improved nonlinear disturbance observer is designed to estimate the mismatched disturbance. Then, the integral sliding mode controller is presented based on the observer, where the reachability of sliding motion is proved. Furthermore, the mismatched disturbance is divided into two parts, in which the matched part is compensated by the disturbance information while the remaining part is attenuated by the adaptive controller. Finally, the adaptive control law is proposed, which can adaptively adjust controller parameters to compensate for the fault and disturbance. The effectiveness of the method is verified through two examples.

The remaining part of this paper is organized as follows. In section 2, the system description and some assumptions are presented. In section 3, the construction of the observer and the design of the controller are presented. In section 4, two simulation examples are used to illustrate the effectiveness of the method, and some conclusions are obtained in section 5.

2. Problem formulation

Consider the following nonlinear system with actuator fault and mismatched disturbance as

where ${\displaystyle x\left(t\right)\in R^{n}}$ is the state vector, ${\displaystyle y\left(t\right)\in R^{q}}$ is the output of the system, ${\displaystyle u\left(t\right)\in R^{p}}$ is the control input, ${\displaystyle \Delta f\left(x,t\right)}$ is a nonlinear function, which can be regarded as the un-modeled uncertainty, ${\displaystyle d\left(t\right)\in R^{v}}$ is the unmatched external disturbance, ${\displaystyle \xi \left(x,t\right)}$ is the mismatched nonlinearity, ${\displaystyle A}$, ${\displaystyle B}$, ${\displaystyle B_{d}}$, and ${\displaystyle C}$ are known constant coefficient matrices with appropriate dimensions.

In this paper, the actuator failure problem is concerned. The actuators can be divided into two parts ${\displaystyle u_{H}}$ and ${\displaystyle u_{F}}$, where ${\displaystyle u_{H}}$ stands for the health actuator and ${\displaystyle u_{F}}$ represent the actuator in a fault condition. Then system (1) becomes (2)

where ${\textstyle B_{H}}$ and ${\textstyle B_{F}}$ denote the healthy and fault matrices.

One can see that proposer designing of ${\displaystyle u_{H}}$ can guarantee the stability of the system despite the existence of external disturbance.

Assumption 1 [17]. The actuators in fault condition work abnormally, and the remaining actuators work normally.

With assumption 1, one can use ${\displaystyle {\overline {u}}_{F}}$ and ${\displaystyle u_{F}}$ represent the actual control and designed control respectively for the actuators in fault condition. Then system (2) becomes (3)

where ${\displaystyle \Delta u_{F}={\overline {u}}_{F}-u_{F}}$.

Assumption 2 [15]. There exists a known function ${\displaystyle {\overline {f}}\left(x\right)}$ and two unknown constants ${\displaystyle {\theta }_{0}}$ and ${\displaystyle {\upsilon }_{0}}$ , such that the following inequality holds,

Assumption 3. The disturbance and the derivative of the disturbance are bounded, i.e., ${\displaystyle \Vert d\left(t\right)\Vert \leq {\epsilon }_{1}}$, ${\displaystyle \Vert {\dot {d}}\left(t\right)\Vert \leq {\epsilon }_{d}}$, where ${\displaystyle {\epsilon }_{1}}$ and ${\displaystyle {\epsilon }_{d}}$ are two positive constants.

Assumption 4. The nonlinearity function ${\displaystyle \xi \left(x,t\right)}$ is bounded and satisfies the following condition

where ${\displaystyle R}$ and ${\displaystyle Q}$ are two positive symmetry matrices, ${\displaystyle {\tilde {\xi }}\left(x,t\right)=\xi \left(x_{1},t\right)-}$${\displaystyle \xi \left(x_{2},t\right)}$, ${\displaystyle {\tilde {x}}\left(x,t\right)=x_{1}\left(x,t\right)-x_{2}\left(x,t\right)}$.

Remark 1. The assumption 1 is general for the actuator failure condition [16,17], in which the actuator fault can be treated as an additional uncertainty or disturbance. Compared with the results in [26], the assumption 2 in this study has been much more relaxed. Assumption 3 is common in FTC control results [15]. Assumption 4 is more general compared with the traditional Lipschitz condition [16], it should be noted that if ${\displaystyle R{\mbox{=}}I}$, and ${\displaystyle Q=l_{f}{}^{2}I}$, then assumption 4 degenerates to the normal Lipschitz condition, where ${\displaystyle l_{f}}$ is the Lipschitz constant.

With assumptions 1 and 2, the object of the paper is to design a control law to compensate for the effects of the disturbance and fault so that the stability and convergence of the system can be guaranteed in normal and fault conditions.

3. Main results

In this part, an observer will be applied to estimate the external disturbance, and an observer-based integral sliding mode fault colorant control scheme will be designed.

3.1 Observer design

For the feasibility of the observer, the following assumption is necessary.

Assumption 5 [16]. The additional fault term ${\displaystyle \varphi \left(t\right)=B_{F}\Delta u_{F}\left(t\right)}$ satisfies the following condition, ${\displaystyle \Vert B_{F}\Delta u_{F}\left(t\right)\Vert \leq {\overline {\omega }}_{b}}$, where ${\displaystyle {\overline {\omega }}_{b}}$ is a positive scalar.

For system (3), the following observer is proposed in the form of

where ${\displaystyle p\left(t\right)}$ is the internal state of the observer, ${\displaystyle {\overset {\mbox{ˆ}}{d}}\left(t\right)}$ is the estimation of the disturbance ${\displaystyle d\left(t\right)}$, ${\displaystyle q\left(x\right)}$ is a nonlinear function to be designed, ${\displaystyle T_{n}}$ is a parameter matrix with proper dimensions, and ${\displaystyle g\left(x\right)}$ is the observer gain and satisfies the following condition

${\displaystyle {\delta }_{n}}$ is the error compensator and is defined by