Adjustable dimension descriptor observer based fault estimation of nonlinear system with unknown input

doi:10.1016/j.amc.2020.125899

Applied Mathematics and Computation

Volume 396, 1 May 2021, 125899

https://doi.org/10.1016/j.amc.2020.125899 Get rights and content

Highlights

•
A novel adjustable dimension augmented descriptor observer is designed, and the system state, process and sensor faults can be estimated simultaneously.
•
Both disturbance decoupling and disturbance attenuation have been considered. one of disturbance, which is name unknown input in this paper, can be decoupled from the estimation results, and the other part can be attenuated.
•
The observer order can be selected in a certain range. Thus, we can select different observer orders for different conditions, which can obtain the tradeoff between estimation cost and precision.

Abstract

In this paper, the fault estimation problem is considered for nonlinear system with process fault, sensor fault and unknown input. A novel adjustable dimension augmented descriptor observer is designed. Based on the proposed observer, the system state, process and sensor faults can be estimated simultaneously, and the unknown input can be decoupled from the error dynamic. The observer parameters are calculated by solving LMI and matrix equations. The observer order can be selected in a certain range, which is helpful to achieve the compromise between the estimation cost and accuracy. At last, two simulation examples are listed to show the effectiveness of the proposed approach.

Introduction

In recent decades, the requirements for control accuracy and control reliability are more higher than ever before. It should be pointed out that in practice, various types of faults may inevitably occur with the increase of equipment operating time and the impact of different types of disturbance. Generally speaking, the fault can influence the plant performance, and in serious cases, the fault may undermine system stability. Fault diagnosis (FD) and fault-tolerant control (FTC) are effective ways to deal with the system fault, and they have attracted wide concern in recent years [1], [2], [3], [4], [5], [6].

As an important part of fault diagnosis, fault estimation can obtain much fault information, and it can be used to solve the FTC problem. Recently, fault estimation has become a hot issue [7], [8], [9], [10], [11]. Generally speaking, fault estimation is to reconstruct the fault by designing observer or filter. In [12], the unknown input observers have been considered to estimate the faults for switched systems with unknown input. In [13], sliding mode observer has been used, where the matching condition and the minimum phase condition might not be needed. In [14], [15], a novel sliding mode observer design method has been proposed to estimate the process fault in non-infinitely observable descriptor systems. In [16], [17], proportional-integral (PI) observer has been considered, and based on it, the process fault (actuator fault) in fuzzy systems and multi-agent systems could be reconstructed, respectively. In [18], a k-step PI observer design approach has been proposed.

It should be pointed out that the sensor fault was not considered in [13], [14], [15], [16], [17], [18]. In general, traditional observer design method cannot be used directly for sensor fault estimation, since the effect of sensor fault may be amplified due to the observer gain matrix [19]. In [20], based on the designed output filter, the sensor fault has been rewritten as the process fault in the augmented state system, then the proportional-proportional-integral(PPI) observer design approach has been reported to redraw the fault. In [19], a descriptor observer design approach has been reported, which could solve the problem that the sensor fault was amplified due to the observer gain. Based on the descriptor observer, the sensor fault could be estimated, even if both the sensor fault and its time-derivative were unbounded and completely unknown. In [21], [22], [23], the descriptor observer method has been applied in Lipschitz nonlinear system, fuzzy systems and Markov jump systems. In [24], a novel dynamic observer, which could unify the proportion observer (PO), proportional-integral observer (PIO) and the traditional dynamic observer (DO) under a unified framework, has been reported to estimate the fault in nonlinear systems.

In practice, sometimes process fault(or actuator fault) and sensor fault occur simultaneously [18], [25], and different types of observers always be applied simultaneously. In [26], the descriptor sliding mode observer has been designed to reconstruct the sensor and actuator faults. In [27], proportional-integral observer and the unknown input observer have been considered simultaneously to estimate system states and faults. Combining the descriptor observer and a novel PI observer, the actuator and sensor faults estimation approach has been reported in [28]. In [29], the disturbance estimator and the fault estimation observer have been considered simultaneously, in which the disturbance estimator could reduce the influence of the disturbance.

Generally speaking, compared with disturbance attenuation, disturbance decouple can obtain more accurate result. While the unknown input cannot be decoupled by the descriptor observer design methods considered in [19], [21], [22], [23]. In fact, these observers cannot combine with the unknown input observer to reconstruct the fault and decouple the unknown input simultaneously, since the observer matching condition can not be satisfied. In addition, we all know that the estimation cost relates to the observer order. To reduce the cost, the reduced-order observers have been considered to reconstruct the possible fault [25], [26], [30]. At the same time, estimation precision may be affected by the observer order. Generally speaking, increasing the observer order can improve the estimation precision [1], [31]. Thus, for different practical problems, it is a challenging problem that how to design an adjustable dimension observer to achieve the tradeoff between estimated cost and estimated precision. These problems inspire our research.

In this paper, a fault estimation method is proposed for nonlinear system with unknown input. The contributions of the proposed method can be listed as follows: 1) According to the coordinate transformation, the augmented system is obtained, and then a novel robust descriptor observer is designed, which can reconstruct the system state, process and sensor faults, and decouple the unknown input from the estimation error, simultaneously. 2) The observer order can be selected in a certain range. Thus, we can select different observer orders for different conditions, which can obtain the tradeoff between estimation cost and precision.

The structure of the paper is as follows: Section 2 provides the problem description. Section 3 is the main results, where the coordinate transformation and the designed observer are introduced. At last of this section, the stability analysis of the error dynamic is provided. In Section 4, two examples are listed. Finally, Section 5 is the conclusions.

Section snippets

Problem formulation

Considering the following system: $\begin{matrix} \dot{x_{0}} (t) & = & A x_{0} (t) + G g (x_{0} (t)) + B_{u} u (t) + B_{p} f_{p} (t) \\ + E d (t) + F_{x} w_{x} (t) \end{matrix}$ $y (t) = C x_{0} (t) + D f_{s} (t) + F_{y} w_{y} (t)$ in which $x_{0} (t) \in R^{n}$ denotes the system state vector, $u (t) \in R^{n_{u}}$ and $y (t) \in R^{n_{y}}$ denote the input and output vectors, $w_{x} (t) \in R^{n_{w_{x}}},$ $w_{y} (t) \in R^{n_{w_{y}}}$ are disturbances in the dynamic equation and the measurement output equation, and it is assumed that both of them are with bounded energy. The possible process fault and sensor fault are denoted as $f_{p} (t) \in R^{n_{p}}$ and $f_{s} (t) \in R^{n_{s}},$ respectively. $d (t) \in R^{n_{d}}$ is the

Coordinate transformation

Introduce a new state $x (t),$ where $x (t) = [\begin{matrix} x_{0} (t) \\ f_{p} (t) \\ f_{s} (t) \end{matrix}],$ then we have $\begin{matrix} H_{0} \dot{x} (t) & = & A_{1} x (t) + G_{1} g (x_{0} (t)) + B_{u 1} u (t) + B_{p 1} f_{0 p} (t) \\ + D_{1} f_{s} (t) + E_{1} d (t) + F_{x 1} w_{x} (t) \end{matrix}$ $y (t) = C_{0} x (t) + D f_{s} (t) + F_{y} w_{y} (t) = C_{1} x (t) + F_{y} w_{y} (t)$ where $H_{0} = [\begin{matrix} I_{n} & (1 / a) B_{p} & O_{n \times n_{s}} \\ O_{n_{p} \times n} & I_{n_{p}} & O_{n_{p} \times n_{s}} \\ O_{n_{y} \times n} & O_{n_{y} \times n_{p}} & O_{n_{y} \times n_{s}} \end{matrix}], A_{1} = [\begin{matrix} A & O_{n \times n_{p}} & O_{n \times n_{s}} \\ O_{n_{p} \times n} & - a I_{n_{p}} & O_{n_{p} \times n_{s}} \\ O_{n_{y} \times n} & O_{n_{y} \times n_{p}} & - D \end{matrix}], G_{1} = [\begin{matrix} G \\ O_{n_{p} \times n_{g}} \\ O_{n_{y} \times n_{g}} \end{matrix}],$ $B_{u 1} = [\begin{matrix} B_{u} \\ O_{n_{p} \times n_{u}} \\ O_{n_{y} \times n_{u}} \end{matrix}], D_{1} = [\begin{matrix} O_{n \times n_{s}} \\ O_{n_{p} \times n_{s}} \\ D \end{matrix}], E_{1} = [\begin{matrix} E \\ O_{n_{p} \times n_{d}} \\ O_{n_{y} \times n_{d}} \end{matrix}], F_{x 1} = [\begin{matrix} F_{x} \\ O_{n_{p} \times n_{w_{x}}} \\ O_{n_{y} \times n_{w_{x}}} \end{matrix}], B_{p 1} = [\begin{matrix} B_{p} / a \\ I_{n_{p}} \\ O_{n_{y} \times n_{p}} \end{matrix}],$ $C_{0} = [\begin{matrix} C & O_{n_{y} \times n_{p}} & O_{n_{y} \times n_{s}} \end{matrix}], C_{1} = [\begin{matrix} C & O_{n_{y} \times n_{p}} & D \end{matrix}],$ and $f_{0 p} = a f_{p} (t) + {\dot{f}}_{p} (t),$ and $a$ is a small constant scalar. In the

Simulation study

Example 1

Consider the following system: $\begin{matrix} \dot{x_{0}} & = & [\begin{matrix} - 9.9477 & - 0.7476 & 0.2632 & 5.0337 \\ 52.1659 & 2.7452 & 5.5532 & - 24.4221 \\ 26.0922 & 2.6361 & - 4.1975 & - 19.2774 \\ 0 & 0 & 1.0000 & 0 \end{matrix}] x_{0} + [\begin{matrix} 0 \\ - 0.1 \\ 0 \\ 0 \end{matrix}] s i n (x_{04}) \\ + [\begin{matrix} 0.4422 & 0.1761 \\ 3.5446 & - 7.5922 \\ - 5.5200 & 4.4900 \\ 0 & 0 \end{matrix}] u + [\begin{matrix} 0.1761 \\ - 7.5922 \\ 4.4900 \\ 0 \end{matrix}] f_{p} + [\begin{matrix} 1 \\ 2 \\ - 1 \\ 1 \end{matrix}] d + [\begin{matrix} 1 \\ 0 \\ - 1 \\ 0.5 \end{matrix}] w_{x} \\ y & = & [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 1 & 1 \end{matrix}] x_{0} + [\begin{matrix} 1 \\ 2 \\ - 1 \\ 0 \end{matrix}] f_{s} + [\begin{matrix} 0 \\ 1 \\ - 2 \\ 1 \end{matrix}] w_{y} \end{matrix}$

It should be noted that if we ignore the nonlinear part, the nominal system of above system is the linearized longitudinal dynamic of the VTOL aircraft [29], [37], and the parameter matrices in above system are supposed

Conclusions

This paper has studied the fault estimation problem for nonlinear system with unknown input. Both the process and the sensor faults have been considered in this paper. Based on coordinate transformation, the augmented system has been obtained, where the original system state and faults are regarded as the augmented system state. An adjustable dimension observer has been proposed for the augmented system. Based on our observer, the unknown input can be decoupled from the estimation error, while

Acknowledgements

This work was supported by the National Natural Science Foundation of China (61803195, 61903173, 61973149, 61903174, 62003337), and Shandong Provincial Natural Science Foundation, China (ZR2019PF014, ZR2019BF016, ZR2019PF006, ZR2019PF009), and the Outstanding Youth Innovation Team Project of Shandong Higher Education Institution, China (2019KJN017).

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Adjustable dimension descriptor observer based fault estimation of nonlinear system with unknown input

Highlights

Abstract

Introduction

Section snippets

Problem formulation

Coordinate transformation

Simulation study

Conclusions

Acknowledgements

Appl. Math. Comput.

Nonlinear Anal. Hybrid Syst.

Fuzzy Sets Syst.

Appl. Math. Comput.

Automatica

Automatica

Appl. Math. Comput.

Automatica

Appl. Math. Comput.

J. Frankl. Inst.

J. Frankl. Inst.

Syst. Control Lett.

J. Frankl. Inst.

Automatica

ISA Trans.

Automatica

Appl Math Comput

Automatica

Model-Based Fault Diagnosis Techniques: Design Schemes, Algorithms, and Tools