Fault Detection and Isolation of Sensors in Airplane Systems by Sliding Mode Observer with Stability Transformation

1. Introduction

Fault is appeared possibly in the airplane system due to the aging phenomenon, environmental erosion and assembly damage. Generally, the malfunction in an airplane system is categorized into actuator fault and sensor fault by the different location on the control loop[1,2,3]. Compared with other faults, the sensor is more vulnerable to fault as automation system generally relies on the precise readings of sorts of transducers to acquire state information, thus the detection of sensor faults plays an assisting role in the control loop[4,5,6]. To refine the healthy level of aerial system, the research on the faults detection and isolation(FDI) tends to be crucial that the accessibility of performance index of system is satisfied.

By using some transformation technique on system matrix or maintaining the health conditions by index functions, the FDI research of sensors in the modeled system has been broadly studied, including the sliding mode observer(SMO) which helps cancel faults in model system. By exploiting the structure of uncertainty, the estimation and reconstruction of fault is accomplished by SMO in [7]. [8] addressed the FDI design by regrouping the inputs of system into a suitable structure, with the fault isolation index proposed for FDI design. [9] emphasized the numerical tractability of SMO in the structure and analysis to diagnose faults in system. Moreover, [10,11] researched the fault signal reconstruction by equivalent output error injection based on the detected faults.

Similarly, by joining filter structures in auxiliary system design, the FDI of system is derived and solved by obtaining the state estimate. In [12], a fault detection filter is designed to surmount the residual error of observable subsystem by reconstructing the disturbance inputs. [1] studied the reconstruction of the sensor fault by a simple filter, and transformed the sensor fault into a pseudo-actuator fault to wipe out the fault effect. Besides, [13] designed the parameters of observer for fault detection issue using LMI and line filter techniques such that generated residuals of SMO are robust to uncertainties.

In the application of SMO to physical systems, the FDI with regards to the system characteristics is addressed, especially in improving system performance. For example, [14] proposed a model based FDI scheme and a second-order SMO for rigid manipulators. [15] designed SMO for a modular multi-level converter with half-bridge structure to assume and verify the location of the fault. [5,16] researched the FDI for power electronics and VTOL aircraft, respectively, by the designed $H-\infty$ performance index, the stability of system is judged. [4,17,18] improved the fault detection, reconstruction and control scheme for a motor system, and SMO & SMC are combined to find the sensorless control strategy. Moreover, [19] applied the fault detection and tolerant control scheme on a flight simulator, with the system trajectories attained along the sliding surface in finite time. However, the influence of pattern in the distribution of system faults has not been thoroughly researched in terms of the stability of SMO.

In this paper, FDI scheme with sensor malfunction and environmental disturbance is discussed by addressing the distribution of the faults in system equations, in the configured system structures. On one hand, under the assumptions on coefficients of fault states, the faulty terms for unobserved subsystem are canceled by state transform, and the SMO is designed to detect and isolate the fault effect at observed port with the stability of observer verified using Lyapunov function. On the other hand, in the general situation of fault distribution, the faults are confined in observed subsystem by similarity transform on system matrix, and SMO is designed to detect the faults in model system by employing inequality transformation. However, the existence of disturbance causes stable oscillation of unobserved state error around zero. Finally, the designed SMO is applied on a fixed-wing airplane system to validate its effect, and show the fitness of proposed SMO to be applied to system in-the-loop with suitable structure. The substantial innovation in this paper can be summarized into the following terms:

1.

As the coefficients of system faults and disturbances meet the counteraction assumption, SMO with compensating terms allocated in output states is designed, such that observed error of system states is convergent, and model system is stable via feedback control;

2.

In the general situation when perturbations are randomly distributed, the similarity transform on system matrices is designed to retain the faults at observed port in the transformed system, and the observed error by SMO is stabilized to the neighborhood of origin, which is verified by Lyapunov function and Inequality transform.

3.

The designed FDI method is applied on an airplane model with different disturbance conditions, and the effect of fault based control is validated in simulation results, including the convergence of observed error, the boundedness of state trajectory under fault and disturbance condition, and the fault detection responses.

Notation. 

In this paper, the capitalized bold font $\mathit{X}$ denotes a matrix variable, the lowercase $\mathit{x}$ denotes a vector, the vector $\widehat{\mathit{x}}$ denotes the estimating variable by observer, $\parallel ·\parallel$ denotes any matrix norm, ${\mathbb{R}}^{n\times m}$ denotes real numbered matrices domain with n rows and m columns, ${sup}_{t_1}f(t_1,t_2,\dots ,t_n)$ denotes the superior bound of function f with respect to variable $t_1$, $\overrightarrow{\mathbf{1}}\in {\mathbb{R}}^n$ denotes a vector where each entry is 1.

2. Preliminary Results

2.1. Airplane Model

Consider the fixed-wing airplane model shown in Figure 1 in lateral motion, the relevant coordinates include ground reference frame $O-X_g{Y}_g{Z}_g$, wind reference frame $O-X_w{Y}_w{Z}_w$ and body reference frame $O-X_b{Y}_b{Z}_b$, where $X_b$ directs forward along longitudinal axis, $Y_b$ directs to right normal to longitudinal plane, and $Z_b$ is obtained via right-hand triad. The lateral motion in consideration is comprised by side-slip angle $\beta$, rolling angle $\varphi$, rolling velocity p yaw velocity r, as well as washed out filter state $\zeta$ of r[20] which is associated to slewing rate. The control input of flight dynamics is constituted by the deflection angles ${\delta }_a,{\delta }_r$ of aileron and rudder, respectively. Then the lateral dynamic equation at zero thrust is formulated in $O-X_b{Y}_b{Z}_b$ by

$$\begin{array}{c}mV\dot{\beta }+mV{r}_w-mgcos\mu sin\varphi -Y=0,\\ \dot{\varphi }-p_w(cos\alpha +tan\theta sin\alpha )-r_w(tan\theta cos\alpha -sin\alpha )=0,\end{array}$$

$$\begin{array}{cc}\hfill {\dot{p}}_w-\dot{\beta }{q}_w-& \dot{\alpha }{r}_w+C_1{p}_w{q}_w+C_2{q}_w{r}_w+\hfill \\ \hfill & C_3{L}_w+C_4{N}_w=0,\hfill \end{array}$$

$$\begin{array}{c}{\dot{r}}_w+\dot{\alpha }{p}_w+D_1{p}_w{q}_w+D_2{q}_w{r}_w+D_3{N}_w=0,\\ \dot{\zeta }-R_1{r}_w-R_2{\zeta }_w=0,\end{array}$$

where $p_w,q_w,r_w$ denote the angular velocity transformed from $O-X_w{Y}_w{Z}_w$, Y denotes lateral aerodynamic force, $L_w,N_w$ denote aerodynamic moment at roll and yaw channel, $\alpha ,\mu$ denote attack angle and track bank angle, $R_1,R_2$ are filtered coefficients, and $C_1,\dots ,C_4$, $D_1,\dots ,D_3$ are coefficients of moment of inertial. Moreover, to analyze the motion at the instant, the small-perturbed linearization is conducted on lateral dynamics on the reference state, say $\left[\begin{array}{ccccc}{\beta }_e& {\varphi }_e& p_e& r_e& {\zeta }_e\end{array}\right]$, such that the linearized equation appending disturbance and fault effect is obtained as[20]

$$\begin{array}{cc}\hfill {\mathit{E}}_d\dot{\mathit{X}}& ={\mathit{A}}_e\mathit{X}+{\mathit{B}}_e\mathit{U}+{\mathit{E}}_e\mathit{\xi }+{\mathit{D}}_e\mathit{d},\hfill \\ \hfill \mathit{y}& ={\mathit{C}}_e\mathit{X},\hfill \end{array}$$

(1)

where

$${\mathit{E}}_d=\left[\begin{array}{ccccc}{V}_e& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right],{\mathit{B}}_e=\left[\begin{array}{cc}{Y}_{{\delta }_a}& Y_{{\delta }_r}\\ 0& 0\\ L_{{\delta }_a}& L_{{\delta }_r}\\ N_{{\delta }_a}& N_{{\delta }_r}\\ 0& 0\end{array}\right],$$

$${\mathit{A}}_e=\left[\begin{array}{ccccc}{Y}_{\beta }& gcos{\mu }_e& Y_p& Y_r-V_e& 0\\ 0& 0& {\displaystyle \frac{cos{\mu }_e}{cos{\theta }_e}}& {\displaystyle \frac{sin{\mu }_e}{cos{\theta }_e}}& 0\\ L_{\beta }^{*}& 0& L_p^{*}& L_r^{*}& 0\\ N_{\beta }^{*}& 0& N_p^{*}& N_r^{*}& 0\\ 0& 0& 0& R_1& R_2\end{array}\right],$$

$${\mathit{C}}_e=\left[\begin{array}{ccccc}0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right],$$

denote Jacobian matrix at linearized point comprised by aerodynamic derivatives, ${\mathit{E}}_e,{\mathit{D}}_e$ denotes coefficient matrix of malfunction and disturbance, respectively, $\mathit{y}$ denotes the output, and perturbed values

$$\mathit{X}={\left[\begin{array}{ccccc}\Delta \beta & \Delta \varphi & \Delta p& \Delta r& \Delta \xi \end{array}\right]}^T,\mathit{U}={\left[\begin{array}{cc}\Delta {\delta }_a& \Delta {\delta }_r\end{array}\right]}^T$$

comprise state and control vectors.

2.2. Structural Decomposition of System

Consider the possible environmental disturbance and the malfunction in the sensors, which may cause instability of state response. In order to constrain the effect of malfunctions, appropriate structural transform and analysis are implemented as basic preparation in FDI design. Since coefficient ${\mathit{E}}_d$ is nonsingular, we can rewrite the linearized equation (1) in the general form as follows

$$\dot{\mathit{x}}=\mathit{A}\mathit{x}+\mathit{B}\mathit{u}+\mathit{E}\mathit{\xi }+\mathit{D}\mathit{d},$$

(2)

where $\mathit{x}\in {\mathbb{R}}^n$ denotes state vector, $\mathit{u}\in {\mathbb{R}}^m$ denotes control input, $\mathit{\xi }\in {\mathbb{R}}^{{\ell }_e}$ and $\mathit{d}\in {\mathbb{R}}^{{\ell }_d}$ denote sensor malfunction and environmental disturbance on system, respectively, and $\mathit{A}\in {\mathbb{R}}^{n\times n}$,$\mathit{B}\in {\mathbb{R}}^{n\times m}$,$\mathit{E}\in {\mathbb{R}}^{n\times {\ell }_e}$,$\mathit{D}\in {\mathbb{R}}^{n\times {\ell }_d}$ denote the coefficient matrix. To facilitate the FDI research of system (2), the following assumption on the magnitude of fault and disturbance is proposed.

Assumption 1. 

The perturbation terms caused by malfunction of sensors and environmental disturbance are restricted with $\parallel \mathit{\xi }\parallel <{\delta }_{\xi }$, $\parallel \mathit{d}\parallel <{\delta }_d$ for some real numbers ${\delta }_{\xi },{\delta }_d$.

To estimate the states of system under the malfunction of sensor, the following observer system is formulated

$$\dot{\widehat{\mathit{x}}}=\mathit{A}\widehat{\mathit{x}}+\mathit{B}\mathit{u}+\mathit{E}\mathit{v}+\mathit{H}\mathit{C}(\widehat{\mathit{x}}-\mathit{x}),$$

(3)

where $\mathit{v}\in {\mathbb{R}}^{{\ell }_e}$ is designed to cancel the divergence caused by $\mathit{\xi }$ in (2), $\mathit{C}\in {\mathbb{R}}^{p\times n}$ denotes output coefficient matrix, and $\mathit{H}\in {\mathbb{R}}^{n\times p}$ is the feedback coefficient of output error in observer. By subtracting (2) from (3), the equation of observed error $\mathit{e}=\widehat{\mathit{x}}-\mathit{x}$ is obtained as follows

$$\dot{\mathit{e}}=(\mathit{A}+\mathit{H}\mathit{C})\mathit{e}+\mathit{E}(\mathit{v}-\mathit{\xi })-\mathit{D}\mathit{d}.$$

(4)

By designing $\mathit{H}$, the error $\mathit{e}$ tends to be stabilized as the perturbations $\mathit{v},\mathit{d}$ are constricted by compensating term $\mathit{v}$, hence the following assumption is required[21].

Assumption 2. 

The system (2) is output observable, namely, $(\mathit{A},\mathit{C})$ forms an observable pair.

If the state vector is split into ${\left[\begin{array}{cc}{\mathit{x}}_1^T& {\mathit{x}}_2^T\end{array}\right]}^T$, where ${\mathit{x}}_2\in {\mathbb{R}}^p$ corresponds to observable part in (2), the system (2) is rewritten as

$$\left[\begin{array}{c}{\dot{\mathit{x}}}_1\\ {\dot{\mathit{x}}}_2\end{array}\right]=\left[\begin{array}{cc}{\mathit{A}}_{11}& {\mathit{A}}_{12}\\ {\mathit{A}}_{21}& {\mathit{A}}_{22}\end{array}\right]\left[\begin{array}{c}{\mathit{x}}_1\\ {\mathit{x}}_2\end{array}\right]+\left[\begin{array}{c}{\mathit{B}}_1\\ {\mathit{B}}_2\end{array}\right]\mathit{u}+\left[\begin{array}{c}{\mathit{E}}_1\\ {\mathit{E}}_2\end{array}\right]\mathit{\xi }+\left[\begin{array}{c}{\mathit{D}}_1\\ {\mathit{D}}_2\end{array}\right]\mathit{d},$$

(5)

corresponding to two states. Then state transform is enforced to unobserved part ${\mathit{x}}_1$ as

$${\overline{\mathit{x}}}_1={\mathit{x}}_1+{\Xi }_1^{-1}{\Xi }_2{\mathit{x}}_2,$$

(6)

where ${\Xi }_1\in {\mathbb{R}}^{(n-p)\times (n-p)}$ is positive definite, and ${\Xi }_2\in {\mathbb{R}}^{(n-p)\times p}$ is suitably configured such that

$${\mathit{A}}_{11}+{\Xi }_1^{-1}{\Xi }_2{\mathit{A}}_{21}$$

is a Hurwitz matrix[21], which is important in the convergence of observer system.

Moreover, the controllability of pair $(\mathit{A},\mathit{B})$ is needed to design control law to assure the stability of system. By control law of linear system, the linear feedback control $\mathit{u}=-\mathit{K}\mathit{x}$ can be derived such that $\mathit{A}-\mathit{B}\mathit{K}$ is Hurwitz, and the state $\mathit{x}$ is stabilized without the effect of perturbation.

3. Main Results

In this section, the observer to deal with sensor faults is designed via state transformation. Specifically, for perturbations satisfying the proposed assumptions, the state transform (6) is used in the derivation of observer. As for the general case, further coordinate transform is proposed and applied to constitute the observer.

3.1. Fault Isolation by Perturbation Counteracting

Consider the model system (2) which is split into the form of (5), and the coefficient matrix $\mathit{D},\mathit{E}$ satisfying Assumption 3, the system equation after state transform

$${\overline{\mathit{x}}}_1={\mathit{x}}_1+{\Xi }_1^{-1}{\Xi }_2{\mathit{x}}_2,{\overline{\mathit{x}}}_2={\mathit{x}}_2$$

is rewritten as

$$\begin{array}{cc}\hfill \left[\begin{array}{c}{\dot{\overline{\mathit{x}}}}_1\\ {\dot{\overline{\mathit{x}}}}_2\end{array}\right]& =\left[\begin{array}{cc}{\overline{\mathit{A}}}_{11}& {\overline{\mathit{A}}}_{12}\\ {\overline{\mathit{A}}}_{21}& {\overline{\mathit{A}}}_{22}\end{array}\right]\left[\begin{array}{c}{\overline{\mathit{x}}}_1\\ {\overline{\mathit{x}}}_2\end{array}\right]+\left[\begin{array}{c}{\mathit{B}}_1+{\Xi }_1^{-1}{\Xi }_2{\mathit{B}}_2\\ {\mathit{B}}_2\end{array}\right]\mathit{u}+\left[\begin{array}{c}{\mathit{E}}_1+{\Xi }_1^{-1}{\Xi }_2{\mathit{E}}_2\\ {\mathit{E}}_2\end{array}\right]\mathit{\xi }\hfill \\ \hfill & +\left[\begin{array}{c}{\mathit{D}}_1+{\Xi }_1^{-1}{\Xi }_2{\mathit{D}}_2\\ {\mathit{D}}_2\end{array}\right]\mathit{d},\hfill \end{array}$$

(7)

where

$$\begin{array}{c}{\overline{\mathit{A}}}_{11}={\mathit{A}}_{11}+{\Xi }_1^{-1}{\Xi }_2{\mathit{A}}_{21},\\ {\overline{\mathit{A}}}_{12}={\mathit{A}}_{12}+{\Xi }_1^{-1}{\Xi }_2{\mathit{A}}_{22}-({\mathit{A}}_{11}+{\Xi }_1^{-1}{\Xi }_2{\mathit{A}}_{21}){\Xi }_1^{-1}{\Xi }_2,\\ {\overline{\mathit{A}}}_{21}={\mathit{A}}_{21},{\overline{\mathit{A}}}_{22}={\mathit{A}}_{22}-{\mathit{A}}_{21}{\Xi }_1^{-1}{\Xi }_2.\end{array}$$

To be continued, the following assumption is needed to extend the system transform.

Assumption 3. 

The coefficients of malfunction $\mathit{\xi }$ and disturbance $\mathit{d}$ can be transformed such that ${\Xi }_1{\mathit{D}}_1+{\Xi }_2{\mathit{D}}_2=\mathbf{0}$ and ${\Xi }_1{\mathit{E}}_1+{\Xi }_2{\mathit{E}}_2=\mathbf{0}$.

The condition in the assumption corresponds to the situation when perturbations are contained in sensor channels, which can be similarly transformed on the coefficient matrix as fault signals.

By Assumption 3 the perturbations are disappeared in the equation of ${\overline{\mathit{x}}}_1$, and the observer for (7) is designed as

$$\left[\begin{array}{c}{\dot{\widehat{\overline{\mathit{x}}}}}_1\\ {\dot{\widehat{\overline{\mathit{x}}}}}_2\end{array}\right]=\left[\begin{array}{cc}{\overline{\mathit{A}}}_{11}& {\overline{\mathit{A}}}_{12}\\ {\overline{\mathit{A}}}_{21}& {\overline{\mathit{A}}}_{22}\end{array}\right]\left[\begin{array}{c}{\widehat{\overline{\mathit{x}}}}_1\\ {\widehat{\overline{\mathit{x}}}}_2\end{array}\right]+\left[\begin{array}{c}{\mathit{B}}_1+{\Xi }_1^{-1}{\Xi }_2{\mathit{B}}_2\\ {\mathit{B}}_2\end{array}\right]\mathit{u}+\left[\begin{array}{c}-{\overline{\mathit{A}}}_{12}\\ \mathit{H}\end{array}\right]{\mathit{e}}_y+\left[\begin{array}{c}\mathbf{0}\\ \mathit{v}\end{array}\right],$$

(8)

in which ${\mathit{e}}_y={\widehat{\overline{\mathit{x}}}}_2-{\overline{\mathit{x}}}_2$ denotes output error, and an amending term $\mathit{v}$ is added with $\mathit{v}=-\rho sgn\left({\mathit{e}}_2\right)$, where

$$sgn\left({\mathit{e}}_2\right)={\left[\begin{array}{cccc}sgn\left(e_2^1\right)& sgn\left(e_2^2\right)& \dots & sgn\left(e_2^p\right)\end{array}\right]}^T,$$

and

$$sgn\left(e_2^i\right)=\left\{\begin{array}{cc}1\hfill & e_2^i>0\hfill \\ 0\hfill & e_2^i=0\hfill \\ -1\hfill & e_2^i<0\hfill \end{array}\right..$$

On the other hand, the linear feedback control

$$\mathit{u}=-\mathit{K}\widehat{\overline{\mathit{x}}}$$

(9)

is designed such that $\overline{\mathit{A}}-\overline{\mathit{B}}\mathit{K}$ is Hurwitz. The fault detection and isolation of (7) is discussed as follows.

Theorem 1. 

Consider the linear block-partitioned system (7) with perturbations $\mathit{\xi },\mathit{d}$, with SMO proposed in (8) for the FDI scheme of sensors, the observed error is convergent to zero if the following conditions are met

1.

${\mathit{A}}_{11}+{\Xi }_1^{-1}{\Xi }_2{\mathit{A}}_{21}$ is Hurwitz;

2.

${\overline{\mathit{A}}}_{22}+\mathit{H}$ is Hurwitz;

3.

$\rho >\parallel {\mathit{E}}_2\parallel {\delta }_{\xi }+\parallel {\mathit{D}}_2\parallel {\delta }_d$.

In consequence, the magnitude of perturbations is identically indicated in the element of $\mathit{v}$. Moreover, the states of (7) is stable via control (9).

Proof. 

By subtracting (7) from (8), the equation of observed error is obtained as

$$\left[\begin{array}{c}{\dot{\mathit{e}}}_1\\ {\dot{\mathit{e}}}_2\end{array}\right]=\left[\begin{array}{cc}{\overline{\mathit{A}}}_{11}& \mathbf{0}\\ {\overline{\mathit{A}}}_{21}& {\overline{\mathit{A}}}_{22}+\mathit{H}\end{array}\right]\left[\begin{array}{c}{\mathit{e}}_1\\ {\mathit{e}}_2\end{array}\right]+\left[\begin{array}{c}\mathbf{0}\\ \mathit{v}-{\mathit{E}}_2\mathit{\xi }-{\mathit{D}}_2\mathit{d}\end{array}\right],$$

(10)

by constructing a Lyapunov function $V={\mathit{e}}^T\mathit{P}\mathit{e}$, where

$$\mathit{P}=\left[\begin{array}{cc}{\mathit{P}}_1& \mathbf{0}\\ \mathbf{0}& {\mathit{P}}_2\end{array}\right]\in {\mathbb{R}}^{n\times n}$$

is positive definite, we have

$$\begin{array}{cc}\hfill \dot{V}& ={\mathit{e}}_1^T({\overline{\mathit{A}}}_{11}^T{\mathit{P}}_1+{\mathit{P}}_1{\overline{\mathit{A}}}_{11}){\mathit{e}}_1+2{\mathit{e}}_2^T{\mathit{P}}_2{\overline{\mathit{A}}}_{21}{\mathit{e}}_1+{\mathit{e}}_2^T({({\overline{\mathit{A}}}_{22}+\mathit{H})}^T{\mathit{P}}_2+\hfill \\ & {\mathit{P}}_2({\overline{\mathit{A}}}_{22}+\mathit{H})){\mathit{e}}_2+2{\mathit{e}}_2^T{\mathit{P}}_2(\mathit{v}-{\mathit{E}}_2\mathit{\xi }-{\mathit{D}}_2\mathit{d})\hfill \\ \hfill & \le {\mathit{e}}_1^T({\overline{\mathit{A}}}_{11}^T{\mathit{P}}_1+{\mathit{P}}_1{\overline{\mathit{A}}}_{11}+ϵ{\overline{\mathit{A}}}_{21}{\mathit{P}}_2){\mathit{e}}_1+{\mathit{e}}_2^T({({\overline{\mathit{A}}}_{22}+\mathit{H})}^T{\mathit{P}}_2+{\mathit{P}}_2({\overline{\mathit{A}}}_{22}+\mathit{H})\hfill \\ \hfill & +{\displaystyle \frac{1}{ϵ}}{\overline{\mathit{A}}}_{21}{\mathit{P}}_2){\mathit{e}}_2+2\parallel {\mathit{e}}_2\parallel \parallel {\mathit{P}}_2\parallel (-\rho +\parallel {\mathit{E}}_2\parallel {\delta }_{\xi }+\parallel {\mathit{D}}_2\parallel {\delta }_d)\hfill \\ \hfill & \le {\mathit{e}}_1^T(-{\mathit{Q}}_1+ϵ{\overline{\mathit{A}}}_{21}{\mathit{P}}_2){\mathit{e}}_1+{\mathit{e}}_2^T(-{\mathit{Q}}_2+{\displaystyle \frac{1}{ϵ}}{\overline{\mathit{A}}}_{21}{\mathit{P}}_2){\mathit{e}}_2,\hfill \end{array}$$

(11)

where Young’s Inequality[22] is applied in the first inequality with some positive real number $ϵ$, and

$${\overline{\mathit{A}}}_{11}^T{\mathit{P}}_1+{\mathit{P}}_1{\overline{\mathit{A}}}_{11}=-{\mathit{Q}}_1,$$

$${({\overline{\mathit{A}}}_{22}+\mathit{H})}^T{\mathit{P}}_2+{\mathit{P}}_2({\overline{\mathit{A}}}_{22}+\mathit{H})=-{\mathit{Q}}_2,$$

(12)

as well as Condition 3) is used in the second inequality. By Conditions 1),2), the matrix $-{\mathit{Q}}_1+ϵ{\overline{\mathit{A}}}_{21}{\mathit{P}}_2$, $-{\mathit{Q}}_2+{\displaystyle \frac{1}{ϵ}}{\overline{\mathit{A}}}_{21}{\mathit{P}}_2$ is kept negative definite for the appropriately configured eigenvalues on the left hand plane of complex number coordinates, therefore, we have $\dot{V}<0$ as the Conditions 1)-3) are satisfied. Namely, the observed errors ${\mathit{e}}_1,{\mathit{e}}_2$ are gradually stabilized in the presence of perturbations.

Besides, for the faulty output channel, the corresponding entry in $\mathit{v}$ is switched on its sign, while the entries keep at zero for normal outputs. This means that we can distinguish the fault channel by the response of $\mathit{v}$. Furthermore, by taking control (9) into (8), we have

$$\dot{\widehat{\overline{\mathit{x}}}}=(\overline{\mathit{A}}-\overline{\mathit{B}}\mathit{K})\widehat{\overline{\mathit{x}}}+{\Delta }_0,$$

(13)

where

$${\Delta }_0=\left[\begin{array}{c}-{\overline{\mathit{A}}}_{12}\\ \mathit{H}\end{array}\right]{\mathit{e}}_y+\left[\begin{array}{c}\mathbf{0}\\ \mathit{v}\end{array}\right],$$

because ${\mathit{e}}_y$ is convergent, the term ${\Delta }_0$ is bounded, such that the state $\widehat{\overline{\mathit{x}}}$ by control is solved as

$$\begin{array}{cc}\hfill \widehat{\overline{\mathit{x}}}\left(t\right)& =exp\left((\overline{\mathit{A}}-\overline{\mathit{B}}\mathit{K})t\right)\widehat{\overline{\mathit{x}}}\left(0\right)+{\int }_0^{t}exp(\overline{\mathit{A}}-\overline{\mathit{B}}\mathit{K})(t-\tau ){\Delta }_{0}d\tau \hfill \\ \hfill & \le exp\left((\overline{\mathit{A}}-\overline{\mathit{B}}\mathit{K})t\right)\widehat{\overline{\mathit{x}}}\left(0\right)+{(\overline{\mathit{A}}-\overline{\mathit{B}}\mathit{K})}^{-1}(exp\left((\overline{\mathit{A}}-\overline{\mathit{B}}\mathit{K})t\right)-\mathit{I}){\delta }_{*},\hfill \end{array}$$

where $\parallel {\Delta }_0\parallel \le {\delta }_{*}$ is used. By the Hurwitz matrix property of $\overline{\mathit{A}}-\overline{\mathit{B}}\mathit{K}$, the response $\widehat{\overline{\mathit{x}}}$ is bounded as follows

$$\parallel \widehat{\overline{\mathit{x}}}\parallel \le \parallel {\mathcal{A}}_r\parallel {\delta }_{*},$$

where ${\mathcal{A}}_r={(\overline{\mathit{A}}-\overline{\mathit{B}}\mathit{K})}^{-1}$, identically, the state $\overline{\mathit{x}},\mathit{x}$ of model system are also bounded and stable. □

Remark 1. 

In observer system (8) the output feedback is designed as

$$\left[\begin{array}{cc}{\overline{\mathit{A}}}_{11}& {\overline{\mathit{A}}}_{12}\\ {\overline{\mathit{A}}}_{21}& {\overline{\mathit{A}}}_{22}\end{array}\right]+\left[\begin{array}{c}-{\overline{\mathit{A}}}_{12}\\ \mathit{H}\end{array}\right]\left[\begin{array}{cc}\mathbf{0}& \mathit{I}\end{array}\right],$$

(14)

where $\mathit{I}\in {\mathbb{R}}^{p\times p}$, and the Hurwitz stability of ${\overline{\mathit{A}}}_{11}$ and ${\overline{\mathit{A}}}_{22}+\mathit{H}$ guarantees the stability of overall system matrix since the resulting matrix is in block-partitioned lower triangular form. Furthermore, the state feedback control is taken on the following transformed input matrix

$$\overline{\mathit{B}}=\left[\begin{array}{c}{\mathit{B}}_1+{\Xi }_1^{-1}{\Xi }_2{\mathit{B}}_2\\ {\mathit{B}}_2\end{array}\right],$$

and general form is constructed by firstly extracting the coefficients of eigen-polynomial of $\overline{\mathit{A}}$ as

$$|s\mathit{I}-\overline{\mathit{A}}|=a_1{s}^4+a_2{s}^3+\dots +a_5,$$

$$\mathit{a}=\left[\begin{array}{ccc}{a}_5& \dots & a_2\end{array}\right],$$

then by constructing the controllability matrix

$${\mathit{T}}_1=\left[\begin{array}{cccc}{\overline{\mathit{A}}}^3{\mathit{b}}_i& {\overline{\mathit{A}}}^2{\mathit{b}}_i& \overline{\mathit{A}}{\mathit{b}}_i& {\mathit{b}}_i\end{array}\right],$$

and the lower triangular matrix

$${\mathit{T}}_2=\left[\begin{array}{cccc}{a}_1& 0& \dots & 0\\ a_2& a_1& \ddots & ⋮\\ ⋮& \ddots & \ddots & 0\\ a_n& \dots & a_2& a_1\end{array}\right],$$

where ${\mathit{b}}_i$ is the arbitrary column of $\overline{\mathit{B}}$ such that ${\mathit{T}}_1$ is full-column rank, the feedback coefficient is configured as follows

$$\mathit{K}=\left[\begin{array}{c}{\mathit{O}}_1\\ (\mathit{b}-\mathit{a}){\mathit{T}}^{-1}\\ {\mathit{O}}_2\end{array}\right],$$

where $\mathit{b}\in {\mathbb{R}}^n$ denotes the coefficient row vector of expected eigen-polynomial, $\mathit{T}={\mathit{T}}_1{\mathit{T}}_2$, and ${\mathit{O}}_1\in {\mathbb{R}}^{(i-1)\times n}$, ${\mathit{O}}_2\in {\mathbb{R}}^{(m-i)\times n}$ denote zero matrix. Besides, because of the convergence of observed error, feedback control by $-\mathit{K}\widehat{\overline{\mathit{x}}}$ guarantees the stability of both systems (7) and (8).

Remark 2. 

The compensating term in SMO (8) can be directly imposed on fault channel as follows

$$\left[\begin{array}{c}\mathbf{0}\\ {\mathit{E}}_2\end{array}\right]\mathit{v},$$

then

$$\mathit{v}=\left\{\begin{array}{cc}-\overline{\rho }{\displaystyle \frac{\Lambda {\mathit{e}}_2}{\parallel \Lambda {\mathit{e}}_2\parallel }}\hfill & {\mathit{e}}_2\ne \mathbf{0}\hfill \\ \mathbf{0}\hfill & {\mathit{e}}_2=\mathbf{0},\hfill \end{array}\right.$$

(15)

where $\Lambda ={\mathit{P}}_2{\mathit{E}}_2$ is nonsingular for some positive definite matrix ${\mathit{P}}_2$, and $\overline{\rho }$ satisfies the relation

$$\overline{\rho }>{\delta }_{\xi }+\parallel {\mathit{E}}_2{\parallel }^{-1}\parallel {\mathit{D}}_2\parallel {\delta }_d,$$

such that similar stability criterion can be derived as in (8).

According to the separation principle of control $\mathit{u}$ and observer (8), the convergence of state $\mathit{x}$ and observed error $\mathit{e}$ is obtained, respectively, in a concurrent manner. On the other hand, in a more general situation when the Assumption 3 is not satisfied on malfunction $\mathit{\xi }$ and disturbance $\mathit{d}$, the fault isolation can be accomplished with limited fault distribution via further state transform.

3.2. Fault Isolation with State Transform

For system equation (5), the fault effect appeared in unobserved subsystem is canceled by state transform, such that the fault of sensor is detected and isolated in the observed port. Moreover, the system matrix over unobserved state is adjusted by another state under the detectability condition.

Firstly, by splitting the state ${\mathit{x}}_2$ into two segments

$${\left[\begin{array}{cc}{\mathit{x}}_{21}^T& {\mathit{x}}_{22}^T\end{array}\right]}^T,$$

we assume that the fault phenomena mainly distribute in ${\mathit{x}}_{22}\in {\mathbb{R}}^{{\ell }_e}$ and the fault-output port satisfies $p>{\ell }_e$, then ${\mathit{x}}_{21}\in {\mathbb{R}}^{p-{\ell }_e}$ is irrelevant to malfunction. Furthermore, the coefficient matrix ${\mathit{E}}_2$ is full column rank.

The following transformation matrix is proposed

$${\mathit{P}}_a=\left[\begin{array}{cc}\mathit{I}& \left[\begin{array}{cc}\mathbf{0}& -{\mathit{E}}_1{\Delta }^{-1}\end{array}\right]\Psi \\ \mathbf{0}& \Psi \end{array}\right],$$

(16)

where $\Psi \in {\mathbb{R}}^{p\times p}$ is designed such that

$$\Psi {\mathit{E}}_2=\left[\begin{array}{c}\mathbf{0}\\ \Delta \end{array}\right],$$

where $\Delta \in {\mathbb{R}}^{{\ell }_e\times {\ell }_e}$ is non-singular. By multiplying ${\mathit{P}}_a$ with $\mathit{E}$, we have

$${\mathit{P}}_a\mathit{E}=\left[\begin{array}{c}{\mathit{E}}_1+\left[\begin{array}{cc}\mathbf{0}& -{\mathit{E}}_1{\Delta }^{-1}\end{array}\right]\Psi {\mathit{E}}_2\\ \Psi {\mathit{E}}_2\end{array}\right]=\left[\begin{array}{c}\mathbf{0}\\ \Delta \end{array}\right].$$

The system matrix multiplied by ${\mathit{P}}_a$ is partitioned and denoted as follows

(17)

where ${\mathit{A}}_{211}\in {\mathbb{R}}^{(p-{\ell }_e)\times q_1}$, ${\mathit{A}}_{212}\in {\mathbb{R}}^{(p-{\ell }_e)\times q_2}$, ${\mathit{A}}_{23}\in {\mathbb{R}}^{{\ell }_e\times (n-p)}$ and $q_1+q_2=n-p$. In order to stabilize the observed error in unobservable subsystem of ${\mathit{x}}_1$, let ${\mathit{C}}_a=\left[\begin{array}{cc}{\mathit{A}}_{211}& {\mathit{A}}_{212}\end{array}\right]$, the following assumption is applied on system[21].

Assumption 4. 

For partitioned system matrix ${\mathit{A}}_a$, $({\mathit{A}}_{11},{\mathit{C}}_a)$ forms a detectable pair.

Remark 3. 

By using Assumption 4, the disturbance and fault are not required in coincident distribution on their coefficients. The amended assumption hints the detectability of ${\mathit{x}}_1$ in the following equation

$${\mathit{y}}_a=\left[\begin{array}{cc}{\mathit{A}}_{211}& {\mathit{A}}_{212}\end{array}\right]{\mathit{x}}_1,$$

which can be met in most situations as

(18)

is full column rank.

By Assumption 4, ${\mathit{A}}_{11}$ can be stabilized by output feedback. Then another row-transform is imposed on both sides of (5), where the transform matrix is

(19)

where ${\mathit{H}}_b\in {\mathbb{R}}^{(n-p)\times (p-{\ell }_e)}$, and ${\mathit{I}}_1\in {\mathbb{R}}^{(n-p)\times (n-p)}$, ${\mathit{I}}_2\in {\mathbb{R}}^{p\times p}$ are identity matrix, and the multiplication of ${\mathit{P}}_b$ and ${\mathit{A}}_a$ results in

(20)

where ${\tilde{\mathit{A}}}_{11}={\mathit{A}}_{11}+{\mathit{H}}_b{\mathit{C}}_a$ and

By constructing state transform

$$\tilde{\mathit{x}}={\mathit{P}}_b{\mathit{P}}_a\mathit{x},$$

(21)

system (5) is rewritten as follows

$$\left[\begin{array}{c}{\dot{\tilde{\mathit{x}}}}_1\\ {\dot{\tilde{\mathit{x}}}}_2\end{array}\right]=\left[\begin{array}{cc}{\tilde{\mathit{A}}}_{11}& {\tilde{\mathit{A}}}_{12}\\ {\tilde{\mathit{A}}}_{21}& {\tilde{\mathit{A}}}_{22}\end{array}\right]\left[\begin{array}{c}{\tilde{\mathit{x}}}_1\\ {\tilde{\mathit{x}}}_2\end{array}\right]+\left[\begin{array}{c}{\tilde{\mathit{B}}}_1\\ {\tilde{\mathit{B}}}_2\end{array}\right]\mathit{u}+\left[\begin{array}{c}\mathbf{0}\\ \Delta \end{array}\right]\mathit{\xi }+\left[\begin{array}{c}{\tilde{\mathit{D}}}_1\\ {\tilde{\mathit{D}}}_2\end{array}\right]\mathit{d},$$

(22)

accordingly, the observer is formulated as

$$\left[\begin{array}{c}{\dot{\widehat{\tilde{\mathit{x}}}}}_1\\ {\dot{\widehat{\tilde{\mathit{x}}}}}_2\end{array}\right]=\left[\begin{array}{cc}{\tilde{\mathit{A}}}_{11}& {\tilde{\mathit{A}}}_{12}\\ {\tilde{\mathit{A}}}_{21}& {\tilde{\mathit{A}}}_{22}\end{array}\right]\left[\begin{array}{c}{\widehat{\tilde{\mathit{x}}}}_1\\ {\widehat{\tilde{\mathit{x}}}}_2\end{array}\right]+\left[\begin{array}{c}{\tilde{\mathit{B}}}_1\\ {\tilde{\mathit{B}}}_2\end{array}\right]\mathit{u}+\left[\begin{array}{c}-{\tilde{\mathit{A}}}_{12}\\ \tilde{\mathit{H}}\end{array}\right]{\tilde{\mathit{e}}}_y+\left[\begin{array}{c}\mathbf{0}\\ \Delta \end{array}\right]\tilde{\mathit{v}},$$

(23)

in which

$$\tilde{\mathit{v}}=\left\{\begin{array}{cc}-\tilde{\rho }{\displaystyle \frac{\Pi {\tilde{\mathit{e}}}_y}{\parallel \Pi {\tilde{\mathit{e}}}_y\parallel }}\hfill & {\tilde{\mathit{e}}}_y\ne \mathbf{0}\hfill \\ \mathbf{0}\hfill & {\tilde{\mathit{e}}}_y=\mathbf{0}\hfill \end{array}\right.,$$

(24)

where ${\tilde{\mathit{e}}}_y={\widehat{\tilde{\mathit{x}}}}_2-{\tilde{\mathit{x}}}_2$, and $\tilde{\mathit{H}}\in {\mathbb{R}}^{p\times p}$ denotes coefficient of output feedback. Besides, the feedback control

$$\mathit{u}=-\tilde{\mathit{K}}\widehat{\tilde{\mathit{x}}}$$

(25)

is designed such that $\tilde{\mathit{A}}-\tilde{\mathit{B}}\tilde{\mathit{K}}$ is Hurwitz. The convergence of observed error ${\tilde{\mathit{e}}}_y$ as well as state $\mathit{x}$ is discussed as follows.

Theorem 2. 

Consider the model system (5) with malfunction $\mathit{\xi }$ and disturbance $\mathit{d}$, which is transformed to (22) by state transform (21). The sensor fault is detected and isolated by SMO (23) if the following conditions are met:

1.

${\mathit{A}}_{11}+{\mathit{H}}_b{\mathit{C}}_a$ is Hurwitz matrix;

2.

${\tilde{\mathit{A}}}_{22}+\tilde{\mathit{H}}\mathit{C}$ is Hurwitz matrix;

3.

there exists positive definite matrix ${\mathit{P}}_{\delta }\in {\mathbb{R}}^{p\times p}$ such that

$${\mathit{P}}_{\delta }\left[\begin{array}{c}\mathbf{0}\\ \Delta \end{array}\right]={\Pi }^T.$$

4.

the positive real number coefficient $\tilde{\rho }$ in (24) satisfies

$$\tilde{\rho }>{\delta }_{\xi }+{\parallel \Pi \parallel }^{-1}\parallel {\mathit{P}}_{\delta }{\mathit{D}}_2\parallel {\delta }_d.$$

Moreover, the state $\mathit{x}$ of (5) is stable via feedback control (25).

Proof. 

By subtracting (22) from (23), the equation of observed error is obtained as

$$\left[\begin{array}{c}{\dot{\tilde{\mathit{e}}}}_1\\ {\dot{\tilde{\mathit{e}}}}_2\end{array}\right]=\left[\begin{array}{cc}{\tilde{\mathit{A}}}_{11}& \mathbf{0}\\ {\tilde{\mathit{A}}}_{21}& {\tilde{\mathit{A}}}_{22}+\tilde{\mathit{H}}\end{array}\right]\left[\begin{array}{c}{\tilde{\mathit{e}}}_1\\ {\tilde{\mathit{e}}}_2\end{array}\right]+\left[\begin{array}{c}\mathbf{0}\\ \Delta \end{array}\right](\tilde{\mathit{v}}-\mathit{\xi })-\left[\begin{array}{c}{\tilde{\mathit{D}}}_1\\ {\tilde{\mathit{D}}}_2\end{array}\right]\mathit{d},$$

(26)

namely, ${\tilde{\mathit{e}}}_1\in {\mathbb{R}}^{n-p}$ is decoupled with $\mathit{\xi }$, with the solution as follows

$$\begin{array}{cc}\hfill {\tilde{\mathit{e}}}_1\left(t\right)& =exp\left({\tilde{\mathit{A}}}_{11}\left(t\right)\right){\tilde{\mathit{e}}}_1\left(0\right)+{\int }_0^{t}exp\left({\tilde{\mathit{A}}}_{11}(t-\tau )\right){\tilde{\mathit{D}}}_1\mathit{d}d\tau \hfill \\ \hfill & \le exp\left({\tilde{\mathit{A}}}_{11}\left(t\right)\right){\tilde{\mathit{e}}}_1\left(0\right)+(exp\left({\tilde{\mathit{A}}}_{11}\left(t\right)\right)-\mathit{I}){\tilde{\mathit{A}}}_{11}^{-1}{\tilde{\mathit{D}}}_1{\delta }_d\overrightarrow{\mathbf{1}}\hfill \\ \hfill & =exp\left({\tilde{\mathit{A}}}_{11}\left(t\right)\right)({\tilde{\mathit{e}}}_1\left(0\right)+{\tilde{\mathit{A}}}_{11}^{-1}{\tilde{\mathit{D}}}_1{\delta }_d)-{\tilde{\mathit{A}}}_{11}^{-1}{\tilde{\mathit{D}}}_1{\delta }_d\overrightarrow{\mathbf{1}},\hfill \end{array}$$

(27)

since ${\tilde{\mathit{A}}}_{11}$ is Hurwitz by condition 1), the first term in (27) is gradually convergent to zero as t approaches to infinity. Therefore, the steady state satisfies the relation

$$\parallel {\tilde{\mathit{e}}}_1(\infty )\parallel \le \parallel {\tilde{\mathit{A}}}_{11}^{-1}\parallel \parallel {\tilde{\mathit{D}}}_1\parallel {\delta }_d.$$

(28)

For error ${\tilde{\mathit{e}}}_2$, we set up the Lyapunov function $V={\displaystyle \frac{1}{2}}{\tilde{\mathit{e}}}_2^T{\mathit{P}}_{\delta }{\tilde{\mathit{e}}}_2$ with ${\mathit{P}}_{\delta }\in {\mathbb{R}}^{p\times p}$ as shown in Condition 3), then its time derivative is

$$\begin{array}{cc}\hfill \dot{V}& ={\tilde{\mathit{e}}}_2^T{\mathit{P}}_{\delta }{\tilde{\mathit{A}}}_{21}{\tilde{\mathit{e}}}_1+{\displaystyle \frac{1}{2}}{\tilde{\mathit{e}}}_2^T({({\tilde{\mathit{A}}}_{22}+\tilde{\mathit{H}})}^T{\mathit{P}}_{\delta }+{\mathit{P}}_{\delta }({\tilde{\mathit{A}}}_{22}+\tilde{\mathit{H}})){\tilde{\mathit{e}}}_2\hfill \\ & +{\tilde{\mathit{e}}}_2^T{\mathit{P}}_{\delta }\Delta (\tilde{\mathit{v}}-\mathit{\xi })-{\tilde{\mathit{e}}}_2^T{\mathit{P}}_{\delta }{\tilde{\mathit{D}}}_2\mathit{d}\hfill \\ \hfill & \le {\tilde{\mathit{e}}}_2^T{\mathit{P}}_{\delta }{\tilde{\mathit{A}}}_{21}{\tilde{\mathit{e}}}_1-{\displaystyle \frac{1}{2}}{\tilde{\mathit{e}}}_2^T\tilde{\mathit{Q}}{\tilde{\mathit{e}}}_2-\tilde{\rho }\parallel \Pi \parallel \parallel {\mathit{e}}_2\parallel +\parallel {\tilde{\mathit{e}}}_2\parallel \parallel \Pi \parallel {\delta }_{\xi }+\parallel {\tilde{\mathit{e}}}_2\parallel \parallel {\mathit{P}}_{\delta }{\mathit{D}}_2\parallel {\delta }_d\hfill \\ \hfill & \le {\tilde{\mathit{e}}}_2^T{\mathit{P}}_{\delta }{\tilde{\mathit{A}}}_{21}{\tilde{\mathit{e}}}_1-{\displaystyle \frac{1}{2}}{\tilde{\mathit{e}}}_2^T\tilde{\mathit{Q}}{\tilde{\mathit{e}}}_2-({\delta }_{\xi }+\parallel {\mathit{P}}_{\delta }{\mathit{D}}_2{\parallel \parallel \Pi \parallel }^{-1}{\delta }_d)\parallel \Pi \parallel \parallel {\mathit{e}}_2\parallel \hfill \\ \hfill & +\parallel {\tilde{\mathit{e}}}_2\parallel \parallel \Pi \parallel {\delta }_{\xi }+\parallel {\tilde{\mathit{e}}}_2\parallel \parallel {\mathit{P}}_{\delta }{\mathit{D}}_2\parallel {\delta }_d\hfill \\ \hfill & \le {\displaystyle \frac{ϵ}{2}}{\tilde{\mathit{e}}}_1^T{\mathit{P}}_{\delta }{\tilde{\mathit{A}}}_{21}{\tilde{\mathit{e}}}_1+{\displaystyle \frac{1}{2}}(-{\lambda }_{min}\left(\tilde{\mathit{Q}}\right)+{\displaystyle \frac{1}{ϵ}}\parallel {\mathit{P}}_{\delta }{\tilde{\mathit{A}}}_{21}\parallel )\parallel {\tilde{\mathit{e}}}_2{\parallel }^2\hfill \end{array}$$

(29)

where conditions 2), 3) are used in the first inequality with

$${({\tilde{\mathit{A}}}_{22}+\tilde{\mathit{H}})}^T{\mathit{P}}_{\delta }+{\mathit{P}}_{\delta }({\tilde{\mathit{A}}}_{22}+\tilde{\mathit{H}})=-\tilde{\mathit{Q}}$$

satisfied, condition 4) and Young’s Inequality are employed in the third inequality, to counteract the terms of perturbation and the coupled term of ${\tilde{\mathit{e}}}_1$ and ${\tilde{\mathit{e}}}_2$. Consequently, as shown in (28), ${\tilde{\mathit{e}}}_1$ is bounded to the scale of ${\delta }_d$, then as the eigenvalue of $\tilde{\mathit{Q}}$ determined by ${\tilde{\mathit{A}}}_{22}+\tilde{\mathit{H}}$ is adjusted such that

$${\lambda }_{min}\left(\tilde{\mathit{Q}}\right)>{\displaystyle \frac{1}{ϵ}}\parallel {\mathit{P}}_{\delta }{\tilde{\mathit{A}}}_{21}\parallel ,$$

the observed error ${\tilde{\mathit{e}}}_2$ will be stabilized to bounded region

$$\parallel {\tilde{\mathit{e}}}_2\parallel \le \sqrt{{\displaystyle \frac{\parallel {\mathit{P}}_{\delta }{\tilde{\mathit{A}}}_{21}\parallel ϵ}{\gamma }}}\parallel {\tilde{\mathit{A}}}_{11}^{-1}\parallel \parallel {\tilde{\mathit{D}}}_1\parallel {\delta }_d,$$

(30)

where

$$\gamma ={\lambda }_{min}\left(\tilde{\mathit{Q}}\right)-{\displaystyle \frac{1}{ϵ}}\parallel {\mathit{P}}_{\delta }{\tilde{\mathit{A}}}_{21}\parallel ,$$

as $\dot{V}<0$ is effective. Namely, the influence of $\mathit{\xi }$ on the system is isolated by observer (23). Specially, by the switch phenomena of $\tilde{\mathit{v}}$, the possible fault element in $\mathit{\xi }$ can be detected.

On the other hand, with the designed control (25), the solution $\widehat{\tilde{\mathit{x}}}$ is solved as follows

$$\widehat{\tilde{\mathit{x}}}\left(t\right)\le exp\left((\tilde{\mathit{A}}-\tilde{\mathit{B}}\tilde{\mathit{K}})t\right)(\widehat{\tilde{\mathit{x}}}\left(0\right)+{(\tilde{\mathit{A}}-\tilde{\mathit{B}}\tilde{\mathit{K}})}^{-1}\Gamma )-{(\tilde{\mathit{A}}-\tilde{\mathit{B}}\tilde{\mathit{K}})}^{-1}\Gamma ,$$

where

$$\parallel \Gamma \parallel =\underset{t}{sup}\parallel \left[\begin{array}{c}-{\tilde{\mathit{A}}}_{12}\\ \tilde{\mathit{H}}\end{array}\right]{\tilde{\mathit{e}}}_y+\left[\begin{array}{c}\mathbf{0}\\ \Delta \end{array}\right]\tilde{\mathit{v}}\parallel$$

is bounded since ${\tilde{\mathit{e}}}_y$ and $\tilde{\mathit{v}}$ are stable, respectively, as shown in (24),(30). Therefore, as system matrix $\tilde{\mathit{A}}-\tilde{\mathit{B}}\tilde{\mathit{K}}$ is configured to be Hurwitz, the observed state $\parallel \widehat{\tilde{\mathit{x}}}\parallel$ is convergent to $\parallel {(\tilde{\mathit{A}}-\tilde{\mathit{B}}\tilde{\mathit{K}})}^{-1}\Gamma \parallel$ to be stable, so as to the model state $\tilde{\mathit{x}}$ and $\mathit{x}$. □

In the proof, the malfunction is transformed to output channel, and counteracted by designed switch function $\tilde{\mathit{v}}$ if specific stability conditions are met. Besides, the disturbance is considered in the stability discussion, which causes observed error evolved in the region around zero.

In summary, the FDI scheme is resolved by SMO according to the structure configuration of system, and both designs are capable of stabilizing the observed error and state trajectory by the Hurwitz stability of feedback coefficient matrix. The structure diagram of the SMO based FDI design is shown in Figure 2, where the transform on the system matrix is sorted in different conditions, whereas the SMO design is concatenated to both situations to detect and isolate fault in the control loop.

4. Simulation Results

In this section, the designed FDI method is tested on the airplane system (1). The aerodynamic parameters of (1) is shown in Table 1, the other parameters are chosen as $V_e=500$,${\theta }_e=0.5$,${\mu }_e=0.2$, $g=9.8$, and the initial states are chosen as

$${\mathit{x}}_0={\left[\begin{array}{ccccc}0.200& 0.100& 1.200& 1.500& 0.500\end{array}\right]}^T,$$

$${\widehat{\mathit{x}}}_0={\left[\begin{array}{ccccc}0.210& 0.101& 1.201& 1.505& 0.55\end{array}\right]}^T.$$

Moreover, the perturbations are chosen as

$$\mathit{\xi }={\left[\begin{array}{cc}asin\left(2t\right)& 0\end{array}\right]}^T,\mathit{d}={\left[\begin{array}{cc}0& -bcos\left(5t\right)\end{array}\right]}^T,$$

with positive real numbers $a,b$.

4.1. Perturbations Counteraction

In this case, the transform matrix is chosen as

$${\Xi }_1=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],{\Xi }_2=\left[\begin{array}{ccc}1.6669& 0& 0\\ 43.3818& 0& 0\end{array}\right],$$

(31)

coefficient matrix of perturbations and faults are

$$\mathit{F}=\left[\begin{array}{cc}-3.5005& -3.7006\\ -91.1017& -96.3075\\ 2.10& 2.22\\ 3.10& 3.02\\ 0.20& 0.50\end{array}\right],\mathit{D}=\left[\begin{array}{cc}-0.1667& -1.0002\\ -4.3882& -26.0291\\ 0.10& 0.60\\ 0.20& 0\\ 0& 0.40\end{array}\right],$$

that meet the Assumption 3. The observer is constructed as in (8) where observer gain and the coefficient of feedback control designed as per (12),(13) are given as follows

$$\begin{array}{c}\mathit{H}=\left[\begin{array}{ccc}-4.32& -2.33& 0\\ 3.41& -4.67& 0\\ 5.00& 3.50& -2.50\end{array}\right],\\ \mathit{K}=\left[\begin{array}{ccccc}-18.8739& -7.8928& -7.5372& 4.1912& -69.9220\\ 0& 0& 0& 0& 0\end{array}\right],\end{array}$$

moreover, fault estimator is constructed as

$$\mathit{v}=-\rho {\displaystyle \frac{\Lambda {\mathit{e}}_y}{\parallel \Lambda {\mathit{e}}_y\parallel }},$$

in which

$$\rho =a+\parallel {\mathit{F}}_2{\parallel }^{-1}\parallel {\mathit{D}}_2\parallel b+\upsilon ,$$

$$\Lambda ={\mathit{P}}_2{\mathit{F}}_2=\left[\begin{array}{ccc}14.2277& 4.2918& 1.1327\\ 14.8088& 4.4427& 1.2112\end{array}\right],$$

with $a=4,b=1,\upsilon =1$ as discussed in (15). In Figure 3 the plot of fault estimator show some obvious oscillations after $t=2.5$, and the two components nearly coincide with each other. Then in Figure 3, it is shown that observed errors are bounded in range $\left[\begin{array}{cc}-2.5& 2.5\end{array}\right]$. Moreover, in order to indicate the ability of fault detection of (8), the fault is posed at the fourth row of $\mathit{F}$ and $\mathit{D}$, and the responses of compensating term $\mathit{f}=\mathit{F}\mathit{v}\in {\mathbb{R}}^5$ are shown in Figure 4, where the elements $f_1,f_2$ are zero, and major oscillation lies at $f_4$. Namely, the fault is detected by the entries of $\mathit{f}$.

4.2. State Transform

In the general case when the cancellation of $\mathit{F}$ and $\mathit{D}$ are disabled, the coordinate transform as discussed in subSection 3.2 is used to place the faults in the sensor channels except the detected counter-part shown in (17). The fault coefficient is

$$\mathit{F}=\left[\begin{array}{cc}0.34& 0.12\\ 0& 0.11\\ 2.10& 2.22\\ 3.10& 3.02\\ 0.20& 0.50\end{array}\right],$$

the transform matrix in (21) are computed as

$$\begin{array}{c}{\mathit{P}}_a=\left[\begin{array}{ccccc}1& 0& 0& -0.0003& 0.0014\\ 0& 1& 0& 0.0233& -0.3605\\ 0& 0& 1& -0.6406& -0.5708\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right],\\ {\mathit{P}}_b=\left[\begin{array}{ccccc}1& 0& 1.043& 0& 0\\ 0& 1& 54.3125& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right],\end{array}$$

and the coefficient matrix $\Pi$ in (24) is calculated to be

$$\Pi =\left[\begin{array}{ccc}3.9972& 1.6283& 0.4467\\ 3.9937& 1.6270& 0.4860\end{array}\right].$$

The responses of states $\beta ,\varphi ,r,\zeta$ and the estimate states are shown in Figure 5Figure 6, where all the states are coincided with the estimate. The angle variables $\beta ,\varphi$ are confined within $\left[\begin{array}{cc}-5& 2\end{array}\right]$, and the filtered angular velocity $\zeta$ is considerably constrained on magnitude compared with r. In Figure 7, the observed errors are stabilized within $\left[\begin{array}{cc}-0.08& 0.08\end{array}\right]$, and the responses of compensating term $\mathit{f}=\mathit{F}\mathit{v}$ shows that main overshoots are distributed at terms $f_3,f_4,f_5$ in response to fault variation, as shown in Figure 7. This demonstrates that the proposed fault estimator is able to detect and isolate the faults via SMO and coordinate transform.

Furthermore, the distributions of ${\mathit{e}}_y$ in profiles (10) and (26), say ${\mathit{e}}_{ya}$, ${\mathit{e}}_{yb}$, are compared and shown by box plot in Figure 8. As shown in plot, the mean of ${\mathit{e}}_{yb}$ coincides with 0, and the whiskers spread to $\left[\begin{array}{cc}-0.05& 0.05\end{array}\right]$; meanwhile, the mean of ${\mathit{e}}_{ya}$ departs from 0, with approximating magnitude $0.6$, and the magnitudes whiskers reach to are ranged in $\{1,2,1.5\}$. Besides, the statistics of ${\mathit{e}}_{ya}$ and ${\mathit{e}}_{yb}$ are listed in Table 2, which is congruent to the trend in box plot. Namely, we can find that the observed precision by matrix transform in (23) is more advanced than (8) by state organization. By the FDI scheme with regards to specific system configuration, the application in reliable control of physical systems is possibly pushed forward.

5. Conclusion

In this paper, the FDI scheme with sensor malfunctions and environmental disturbances is researched to detect the location of malfunctions in sensor and guarantee the convergence of observer by isolating the faults. According to the distribution of faults, on one hand, the coefficients of faults and disturbances in unobserved subsystem are canceled by state transform, then the stability of the observed error is discussed with regards to the proposed sliding surface. Based on the state estimate provided by SMO, the stability of model system is obtained by feedback control. On the other hand, for the more general distribution of sensor faults and disturbances, the faults are retained at observed subsystem by similarity transform of system matrix, together with SMO to detect and isolate the faults. By using Lyapunov function and inequality transform, the boundedness of system states is proved, with the induced oscillation by disturbances. Finally, the researched methods are applied on a fixed-wing airplane system to validate the theoretical derivation. In the future, the performance of SMO including response time and observed precision will be further evaluated and compared.