Observer-based fault detection and diagnosis for the nonlinear stochastic distribution systems

Abstract

Our work describe a novel fault detection and diagnosis (FDD) problem for nonlinear stochastic distribution systems (SDS) with the help of the system output probability density functions (PDFs), and it can be obtained by rational square-root B-spline expansion. A new nonlinear FDD method based on observer is given by drawing into the adaptive tuning rule, in order that the residual signal can be sensitive to the system fault. And then, for the fault system, convergence and stability have been implemented by the fault detection and diagnosis. A simulation examples is shown to validates the efficiency of the proposed method and expecting results have been gained.

Keywords

Probability density functions stochastic distribution systems fault detection and diagnosis the adaptive tuning rule

1. Introduction

The FDD is a most important areas of research and application for the stochastic system. Over the past 20 years, all kind of methods have been investigated for the fault of the stochastic system [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. At present, two type of methods include the system identification and statistic method, Hypothesis test method and Bayesian theorem can be handle the FDD [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15].

Until recently, most of the published literature methods have only been dealed with the stochastic system submitted to Gaussian distribution [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. However, nonlinearity may give rise to non-Gaussian output, where the variance and mean of the system output are unable to sufficient to obtain the system statistical behavior [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34]. Therefore, we need investigate FDD methods that can be used to the SDS. In addition, with the development of image processing, the measurements are the system output PDFs instead of the actual values. For these systems, we define the stochastic distribution systems in the academic research [16, 17, 18, 19]. To be different from the previous methods, the system variables are not limited to be Gaussian and the system output PDFs should be taken into account, the target of FDD in the SDS is to utilize the measured PDFs to get message of the system faults. Many algorithms have been investigated in the SDS. For instance, FDD for the SDS with time delays have been established in [20], FDD for the singular time-delayed SDS have been presented in [21], Fault tolerant control for the SDS have been represented in [22, 23], Fault diagnosis and tolerant control for the SDS have been given in [24, 25], adaptive control approach have been proposed in [26], iterative learning algorithms for the SDS have been considered in [27]. SDC for Power system with High-Proportion Renewable Electtricity was investigated in [28], FDD control for the non-Gaussian singular SDS was present in [29], Robust tracking controller design for singular uncertainty SDS was investigated in [30], Fault detection for fuzzy SDS have been presented in [31], Collaborative operational fault tolerant control for SDS have been considered in [32], Adaptive fault-tolerant shape control for Lipschitz SDS have been proposed in [33], FDT for the non-Gaussian time-delayed SDS have been investigated in [34]. But, so far, few results are published for Observer-based FDD by the adaptive tuning rule in the SDS, this forms our work in this paper. To appraise the system fault in the SDS, an adaptive tuning observer is designed, and a fault tolerant algorithm is constructed, in order that the tracking error is less than a certain upper-bound.

The other parts of this paper is composed as follows. The system static models are introduced in Section 2. The Fault Detection was given in Section 3. In Section 4, the fault diagnosis was discussed. Section 5 presented a computer simulation. In Section 6, concluding remarks will be obtained.

2. Problem formulation

For a dynamic SDS, $x(t)\in R^{m}$ stands for the system input, $y\in[a,b]$ is the system output, ${X}(t)$ is the system fault. The FDD objective is to used the PDFs of the system input-output to devise an observer in order that the system fault ${X}(t)$ can be detected and diagnosed in the FDD context.

The system output PDFs $\gamma(y,x(t))$ can be obtained by the following expansions

$\displaystyle\sqrt{\gamma(y,x(t))}=\frac{\sum\limits_{i=1}^{n}{\pi_{i}(t)\pi_{% i}(t)}}{\sqrt{\sum\limits_{i,j=1}^{n}{\pi_{i}(t)\pi_{j}(t)\int_{a}^{b}{\xi_{i}% (y)\xi_{j}(y)dy}}}}=\xi(y)\omega(t)\pi(t)$ (1)

where $\xi_{i}(y)(i=1,\cdots,m)$ is a set of the known basis functions, $\pi(t)=\pi_{i}(x(t))(i=1,\cdots,m)$ represents the weights vector determined by $x(t)$ , $\omega(t){=}1\mathord{\left/{\vphantom{1{\sqrt{\pi^{T}(t)I\pi(t)}}}}\right.% \kern-1.2pt}{\sqrt{\pi^{T}(t)I\pi(t)}}$ . denote

$\displaystyle\xi(y)=[\xi_{1}(y),\cdots,\xi_{n}(y)]$ $\displaystyle I=\int_{a}^{b}{\xi^{T}(y)\xi(y)dy}$ $\displaystyle\pi(t)=[\pi_{1}(x(t)),\cdots,\pi_{n}(x(t))]^{T}\quad\forall(\pi% \neq 0)$

To meet the practical requirement, nonlinearity should be follow with interest, the weighting model can be given by

$\displaystyle\left\{\begin{array}[]{l}\dot{\pi}(t)=Hx(t)+Gg(\pi(t))+A\pi(t)+DX% (t)\\ \sqrt{\gamma(y,x(t))}={\xi(y)\omega(t)\pi(t)}\\ \end{array}\right.$ (2)

where $H$ , $G$ , $A$ and $D$ are the known matrix of coefficients. The fault $X(t)$ is defined as

$\displaystyle X(t)=\left\{\begin{array}[]{ll}X,&\forall t\geqslant T>0\\ 0,&\forall 0<t<T\\ \end{array}\right.$ (3)

For the known matrix $U$ and vector function $\zeta(\chi)$ , the Lipschitz condition is satisfied

$\displaystyle\left\|{\zeta(\hat{\chi}(t))-\zeta(\chi(t))}\right\|\leqslant% \left\|{U(\hat{\chi}(t)-\chi(t))}\right\|$ (4)

It is noted that less FDD approaches are available presently due to the nonlinear function existing. In the following, a new FDD observer design will be discussed.

3. Fault detection

Since the system output information is the PDFs, the system fault can be detected by based on the change of the system output, then the system nonlinear observer is as following

$\displaystyle\left\{\begin{array}[]{l}\dot{\hat{\pi}}(t)=Hx(t)+Gg(\hat{\pi}(t)% )+A\hat{\pi}(t)+K\varepsilon(t)\\ \varepsilon(t)=\int_{a}^{b}{(\sqrt{\hat{\gamma}(y,x(t)}-\sqrt{\gamma(y,x(t)}})% dy\\ \sqrt{\hat{\gamma}(y,u(t))}={\chi(y)\hat{\omega}(t)\hat{\pi}(t)}\\ \end{array}\right.$ (5)

where $\hat{\pi}(t)$ is the estimated value of the vector $\pi(t)$ , $K$ is a adaptive gain, $\hat{\omega}{=}1\mathord{\left/{\vphantom{1{\sqrt{\hat{\pi}^{T}(t)I\hat{\pi}(t% )}}}}\right.\kern-1.2pt}{\sqrt{\hat{\pi}^{T}(t)I\hat{\pi}(t)}}$ .

Let $e(t)=\hat{\pi}(t)-\pi(t)$ , $\hat{g}(t)=g(\hat{\pi}(t))-g(\pi(t))$ . Then, the estimated error model can be written as

$\displaystyle\left\{\begin{array}[]{l}\dot{e}(t)=Ae(t)+G{\hat{g}}(t)+K% \varepsilon(t)-DX\\ \varepsilon(t)=\int_{a}^{b}\xi(y)[\hat{\omega}\hat{\pi}(t)-\omega\pi(t)]dy\\ \end{array}\right.$ (6)

Denote $\sum=\int_{a}^{b}{\xi(y)dy}$ and assume that the pair ( $A,\Sigma$ ) can be observed. Based on the second Equation of the Eq. (6), it can be further written as

$\displaystyle\varepsilon(t)=\sum\hat{\omega}e+\sum\pi(t)(\hat{\omega}-\omega)$ (7)

In order to simplify Eq. (3), the following lemma is given.

Lemma 1. For a given $\mu$ ( $\delta_{1}\leqslant|\mu|\leqslant\delta_{2}$ ), the following equation holds

$\displaystyle(1/\omega)-(1/\hat{\omega})=\mu(||\pi(t)||-||\hat{\pi}(t)||)$

where $\delta_{1}=\lambda_{\min}(I)/\lambda_{\max}(I)$ , $\delta_{2}=\lambda_{\max}(I)/\lambda_{\min}(I)$ .

For the observable ( $A,\Sigma$ ), we suppose that there exist a const matrix $L$ such that $N=A+L\Sigma$ is Hurwiz matrix. It is two matrices $Q=Q^{T}>0$ and $P=P^{T}>0$ satisfying

$\displaystyle N^{T}P+PN=-Q$

$K$ is the observer gain, and can be given by

$\displaystyle K=L/{\omega(t)}$

Based on lemma 1, we can know that there exists $\mu_{1}$ satisfying

$\displaystyle(1/\omega)-(1/\hat{\omega})=\mu_{1}(||\pi(t)||-||\hat{\pi}(t)||)$

So, observer-based fault detection objective is to find the observer gain $K$ and we have

the system (6) is asymptotic stability;

the $\varepsilon(t)$ is sensitive to fault $F$ .

Under the absence of $F$ , the system (6) can be re-written as

$\displaystyle\left\{\begin{array}[]{l}\dot{e}(t)=Ae(t)+G{\hat{g}}(t)+L\sum e(t% )+L\sum\omega(t)\pi(t)\mu_{1}(\left\|{\pi(t)}\right\|-\left\|{\hat{\pi}(t)}% \right\|)\\ \varepsilon(t)=\sum\hat{\omega}(t)e(t)+\sum\hat{\omega}(t)\pi(t)-\sum\omega(t)% \pi(t)\\ \end{array}\right.$ (8)

when no fault, the system (8) is stability.

Let $\left\|{\tilde{g}(t)}\right\|\leqslant\mu_{2}^{2}\left\|{e(t)}\right\|$ , the Lyapunov function is defined by

$\displaystyle\phi(t)=e^{T}(t)Pe(t)+\mu_{3}^{3}\int_{0}^{t}[\left\|{Ue(\tau)}% \right\|^{2}-\left\|{\tilde{g}(\tau)}\right\|^{2}]d\tau$ (9)

From Eq. (9), we have that $\phi(t)>0$ . Under the absence of $F$ , the first-order derivative of $\phi(t)$ is obtained by

$\displaystyle\dot{\phi}(t)=e^{T}(t)\left(\left(L\sum+A\right)^{T}P+P\left(A+L% \sum\right)\right)e(t)+2PG\tilde{g}(t)e(t){}+2PL\sum\pi\frac{\mu_{1}(\left\|{% \hat{\pi}-e(t)}\right\|-\left\|\hat{\pi}\right\|)}{\sqrt{\pi^{T}I\pi}}e(t)+\mu% _{3}^{2}\left\|{Ue(t)}\right\|^{2}-\mu_{3}^{2}\left\|{\hat{g}(t)}\right\|% \leqslant-e^{T}(t)Qe(t)+2PG\hat{g}(t)e(t)+2PL\sum\pi\frac{\mu_{1}(\left\|{\hat% {\pi}-e(t)}\right\|-\left\|\hat{\pi}\right\|)}{\sqrt{\pi^{T}I\pi}}e(t)+\mu_{3}% ^{2}\left\|{Ue(t)}\right\|^{2}\leqslant-\lambda(Q)\left\|{e(t)}\right\|^{2}+2u% _{2}^{2}\left\|{PG}\right\|\left\|{e(t)}\right\|^{2}+\frac{2\delta_{2}\left\|{% PL\sum}\right\|\left\|\pi\right\|\left\|{e(t)}\right\|^{2}}{\left\|\pi\right\|% \sqrt{\left\|I\right\|}}+\mu_{3}^{2}\left\|U\right\|\left\|{e(t)}\right\|^{2}=% -(\lambda(Q)-\varpi)\left\|{e(t)}\right\|^{2}$

where $\varpi=2u_{2}^{2}\left\|{PG}\right\|+\frac{2\mu_{1}\delta_{2}\left\|{PL\sum}% \right\|}{\sqrt{\left\|I\right\|}}+\mu_{3}^{2}\left\|U\right\|$ .

Then, it is shown that $\dot{\phi}(t)<0$ holds if and only if

$\displaystyle\lambda Q>\varpi=2u_{2}^{2}\left\|{PG}\right\|+\frac{2\delta_{2}% \left\|{PL\sum}\right\|}{\sqrt{\left\|I\right\|}}+\mu_{3}^{2}\left\|U\right\|$ (10)

Furthermore,we can conclude that $\mathop{\lim}\limits_{x\to\infty}e(t)=0$ . Under absence of $F$ , the system (6) is asymptotic stability.

After designing observer-based fault detection, the following objective is how to obtain the evaluation of the $\varepsilon(t)$ . For fault detection, we choose a so-called threshold $\xi_{th}>0$ , and using the following logical relationship

$\displaystyle||\varepsilon(t)||>\xi_{th}\to\text{Fault}\to\text{Alarm};$ $\displaystyle||\varepsilon(t)||<\xi_{th}\to\text{No fault}.$

For fault diagnosis, a assumption is given as follows

Assumption 1. Assume that the fault is bound and given by

$\displaystyle\left\|X\right\|\leqslant x/2$

where $x>$ 0 is constant.

4. Fault diagnosis

After the fault is detected, we should diagnose the system fault and take the stock of the size. And then, the fault diagnosis observer is given by

$\displaystyle\left\{\begin{array}[]{l}\dot{\hat{\pi}}(t)=Hx(t)+A\hat{\pi}(t)+% Gg(\hat{\pi}(t))+K\varepsilon(t)+D\hat{X}(t)\\ \dot{\hat{X}}(t)=-\Upsilon\sqrt{\hat{\pi}^{T}I\hat{\pi}}\varepsilon(t)\quad% \forall\left\|\hat{X}\right\|\geqslant\frac{x}{2}\\ \dot{\hat{X}}(t)=0\quad\forall\left\|\hat{X}\right\|<\frac{x}{2}\\ \end{array}\right.$ (11)

where $\hat{X}$ is the estimated value, $\Upsilon>0$ is a pre-specified operator, $x/2$ is supremum. Then, the second equation of the system (11) presents a adaptive tuning rule.

Denote $e(t)=\hat{\pi}(t)-\pi(t),\tilde{X}(t)=\hat{X}(t)-X(t)$ , the system (8) can be given by

$\displaystyle\dot{e}(t)=L\sum e(t)+Ae(t)+G\tilde{g}(t)+L\sum\omega(t)\pi(t)\mu% _{1}(\left\|{\pi(t)}\right\|-\left\|{\hat{\pi}(t)}\right\|){}+K\varepsilon(t)+% D\tilde{X}(t)$ (12)

The fault diagnosis is to find the estimated value of $F$ , and satisfying $\mathop{\lim}\limits_{x\to\infty}e(t)=0$ . For the purpose, we introduce the following theorem.

Theorem 1. Suppose that $\left\|X\right\|\leqslant x/2$ and the rule $\hat{X}=-\Upsilon\sqrt{\hat{\pi}^{T}(t)I\hat{\pi}(t)}\varepsilon(t)$ in the system (11) take $\left\|\hat{X}\right\|\leqslant x/2$ to hold, then the system (11) is asymptotic stability and satisfies

$\displaystyle\mathop{\lim}\limits_{x\to\infty}\left\|{e(t)}\right\|\leqslant% \frac{fk_{2}(\kappa_{1}+\sqrt{\kappa_{1}})}{2\kappa_{1}}$ (13)

Proof. Lyapunov function is then defined as

$\displaystyle\phi(t)=e^{T}(t)Pe(t)+\mu_{3}^{3}\int_{0}^{t}[\left\|{Ue(\tau)}% \right\|^{2}-\left\|{\tilde{g}(\tau)}\right\|^{2}]d\tau+\tilde{X}^{T}\tilde{X}$ (14)

Let $\bar{P}=GP-\Upsilon\sum$ , Along the trajectories of Eq. (9), we obtain that

$\displaystyle\dot{\phi}(t)=e^{T}(t)\left(P^{T}\left(A+L\sum\right){+}\left(A+L% \sum\right)^{T}P\right)e(t){}+2PL\sum\omega\pi e^{T}(t)\mu_{1}(\left\|{\hat{% \pi}-e(t)}\right\|-\left\|\hat{\pi}\right\|)+e^{T}(t)PD\tilde{X}+2\tilde{X}^{T% }\tilde{X}+\mu_{3}^{2}\left\|{Ue(t)}\right\|^{2}{}-\mu_{3}^{2}\left\|{\tilde{g% }(t)}\right\|+2PG\tilde{g}(t)e(t)\leqslant-e^{T}(t)Qe(t)+2e^{T}(t)PG\tilde{g}(% t)+2PL\sum\omega\pi e^{T}(t)\mu_{1}(\left\|{\hat{\pi}-e(t)}\right\|-\left\|% \hat{\pi}\right\|){}+\mu_{3}^{2}\left\|{Ue(t)}\right\|^{2}+2e^{T}(t)PD\tilde{X% }-2\tilde{X}^{T}\Upsilon\left(\sum e+\sum\pi-\sum\pi\frac{\omega}{\hat{\omega}% }\right)\leqslant-\lambda_{\min}(Q)\left\|{e(t)}\right\|^{2}+2u_{2}^{2}\left\|% {PG}\right\|\left\|{e(t)}\right\|^{2}+\frac{2\delta_{2}\left\|{PL\sum}\right\|% \cdot\left\|\pi\right\|\left\|{e(t)}\right\|^{2}}{\left\|\pi\right\|\sqrt{% \left\|I\right\|}}{}+\mu_{3}^{2}\left\|U\right\|\left\|{e(t)}\right\|^{2}+2e^{% T}(t)\tilde{X}\bar{P}-2\tilde{X}^{T}\Upsilon\sum V\left(1-\frac{\omega}{\hat{% \omega}}\right)\leqslant-\left(\lambda_{\min}(Q)-\frac{2\delta_{2}\left\|P% \right\|\cdot\left\|L\right\|\cdot\left\|\sum\right\|}{\sqrt{\left\|I\right\|}% }-2u_{2}^{2}\left\|{PG}\right\|-\mu_{3}^{2}\left\|U\right\|\right)\left\|{e(t)% }\right\|^{2}{}+\left\|\tilde{X}\right\|\left(2\left\|\bar{P}\right\|+\frac{2% \delta_{2}\left\|\Upsilon\right\|\cdot\left\|\sum\right\|}{\sqrt{\left\|I% \right\|}}\right)\left\|{e(t)}\right\|=-\kappa_{1}\left\|{e(t)}\right\|^{2}+% \kappa_{2}\left\|\tilde{X}\right\|\cdot\left\|{e(t)}\right\|$ (15)

where

$\displaystyle\kappa_{1}=\lambda_{\min}(Q)-\frac{2\delta_{2}\left\|P\right\|% \cdot\left\|L\right\|\cdot\left\|\sum\right\|}{\sqrt{\left\|I\right\|}}-2u_{2}% ^{2}\left\|{PG}\right\|-\mu_{3}^{2}\left\|U\right\|$ $\displaystyle\kappa_{2}=\left(2\left\|\bar{P}\right\|+\frac{2\delta_{2}\left\|% \Upsilon\right\|\cdot\left\|I\right\|}{\sqrt{\left\|I\right\|}}\right)$

The supremum of the fault is of $x/2$ and adaptive tuning rule (the second and third Eq. (11)) has been used such that $\left\|\hat{X}\right\|<\frac{x}{2}$ , then Eq. (13) can be rewritten as

$\displaystyle\dot{\phi}(t)\leqslant-\kappa_{1}\left(\left\|{e(t)}\right\|-% \frac{\kappa_{2}x}{2\kappa_{1}}\right)^{2}+\frac{\kappa_{2}^{2}x^{2}}{4\kappa_% {1}}<0$ (16)

Thus, $\dot{\phi}(t)<0$ holds if and only if $\left\|{e(t)}\right\|\geqslant\frac{xk_{2}(\kappa_{1}+\sqrt{\kappa_{1}})}{2% \kappa_{1}}$ is satisfied. This means that the system (12) is asymptotic stability.

5. Computer simulation

In Many production processes, for example the steel production process and paper production process molecular, the system output PDFs have two or there peaks. We assume that the system output PDFs can be expressed by

$\displaystyle\xi_{1}(y)=(y+1)^{2}I_{1}+(-y^{2}-3y-2)I_{2}+(y-1)^{2}I_{3}$ $\displaystyle\xi_{2}(y)=(y+2)^{2}I_{1}+(-y^{2}-2y-3)I_{2}+y^{2}I_{3}$ $\displaystyle\xi_{3}(y)=(y+1)^{2}I_{1}+(-y^{2}-y-1)I_{2}+(y-2)^{2}I_{3}$

Where $\xi_{i}(y)(i=1,2,3)$ , $I_{i}=\left\{\begin{array}[]{ll}1&\forall y\in[i-5,i-3](i=1,\ldots,5)\\ 0&\textit{other}\\ \end{array}\right.$

In the system (2), the coefficient matrices can be selected as following

$\displaystyle A=\left[{{\begin{array}[]{*{20}c}{-4}&{-1}&0\\ 1&{-2}&0\\ 0&0&{-3}\\ \end{array}}}\right]\quad H=\left[{{\begin{array}[]{*{20}c}{-1}&0&0\\ 0&{-0.8}&0\\ 0&0&{1.2}\\ \end{array}}}\right]\quad G=\left[{{\begin{array}[]{*{20}c}{0.5}&0&0\\ 0&0&1\\ 0&0&{-1.3}\\ \end{array}}}\right]\quad D=\left[{{\begin{array}[]{*{20}c}1&0&0\\ 0&2&0\\ 0&0&3\\ \end{array}}}\right]$

Figure 1.

The response of the control input with observer and no-observer.

Assume that the fault arise at 20 s, The value of $L$ is given by $L=[-0.1\,0.5\,-0.13]$ and

$\displaystyle Q=\left[{{\begin{array}[]{*{20}c}{0.8912}&{0.0956}&{0.1537}\\ {0.1057}&{0.4869}&{0.5318}\\ {0.1708}&{0.1567}&{0.4593}\\ \end{array}}}\right].$

The initial value of the system fault is then defined as [0.1 0.3 0.2] ${}^{T}$ and [0.2 0.1 0] ${}^{T}$ respectively.

Suppose that fault constructed is given by

$\displaystyle X(t)=\left\{{{\begin{array}[]{ll}1.0+0.02\sin(0.03t),&t\geqslant 2% 0\\ 0,&t<20\\ \end{array}}}\right.$

Through computer simulation, The PDFs with and without fault observer control are shown in Figs 1–3, respectively. By comparing Figs 1–3, after fault occurrence, we can conclude that the system fault can be well appraise by the fault observer, a better tracking performance has been obtained, and the desire results of the system fault diagnosis can be encouraged.

Figure 2.

The response of the system output with observer and no-observer.

Figure 3.

The response of the fault with observer and no-observer.

6. Conclusion

In our work, the FDD problem for nonlinear SDS via rational square-root B-spline expansion is presented. The system output PDF can be given by the rational square-root B-spline approximation, a nonlinear dynamical model can be given between the system input and the weights vector. A new adaptive observer is established to detect and diagnose the system fault by Lyapunov functional approach. In addition, a fault diagnosis observer be devoted to formulate by the adaptive tuning rule and require observer gain. Finally, computer simulations is shown to test and verify the point of the proposed method in this paper.

Footnotes

Acknowledgments

Shaanxi Provincial Department of Education Special Scientific Research Project (Item Number: 17JK1169).

XianYang Vocational Technical College Doctoral Research Startup Fund Project.

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