State and Fault Estimation for T–S Fuzzy Nonlinear Systems Using an Ensemble UKF

Abstract

This paper introduces a new filter based on a Takagi–Sugeno (T–S) fuzzy augmented ensemble unscented Kalman filter (FAEnUKF) for a class of nonlinear stochastic systems with multiplicative fault and noise. Multiplying a nonlinear term on the fault signal generates a non-Gaussian noise which cannot be optimally estimated by the Kalman filter. One way to resolve this problem is to transform the nonlinear system to several T–S fuzzy systems with Gaussian noise. Using the sector nonlinearity model, the nonlinear term can be derived as constant matrices for each fuzzy rule. Thus, fuzzy augmented UKFs (AUKFs) are designed for state and fault estimation. Using Lyapunov’s stability theory, the convergence conditions of the developed filter algorithm are presented as a theorem. In addition, the boundedness of the error covariance matrix of the proposed algorithm is discussed theoretically. Finally, selected illustrative examples to evaluate the effectiveness of the FAEnUKF are presented. Comparisons between the FAEnUKF and the augmented extended Kalman filter (AEKF) and the AUKF are made in a numerical example. The simulation results showed the robustness of the fuzzy ensemble UKF for modeling the non-Gaussian noise. Despite the increase in the number of calculations in this method, the root-mean-square error (RMSE) is less than other filters.

Appendices

Appendix A

According to [2, 43], the computational complexity for the proposed filter algorithm can be expressed in Table 4, where ${n}_{a}$, $p$, and $m$ are the sizes of the augmented state, input, and output. $r$ is the number of fuzzy rules.

Appendix B

In this Sect., ${\lambda }_{k}$ and ${\mu }_{max}$ are employed to get Lemma 3. Now, by definition ${\lambda }_{k}$ is given as follows:

$$ \begin{gathered} \lambda _{{k + 1}} = \tilde{X}_{k} ^{T} \\ \left[ \begin{gathered} \mathop \sum \limits_{{i = 1}}^{r} h_{i} \left( \theta \right)\left( {\alpha _{{i,k}} F_{{i,k}} } \right)^{T} \left[ \begin{gathered} \left( {\Theta _{{k + 1}} - \beta _{{k + 1}} H_{{k + 1}} } \right)^{T} \hat{R}_{{k + 1}} ^{{ - 1}} ~\left( {\Theta _{{k + 1}} - \beta _{{k + 1}} H_{{k + 1}} } \right) \hfill \\ - \left( {\beta _{{k + 1}} H_{{k + 1}} } \right)^{T} \left( {\Theta _{{k + 1}} \hat{P}_{{X_{{k + 1,k}} }} \Theta _{{k + 1}} ^{T} + \hat{R}_{{k + 1}} } \right)^{{ - 1}} \beta _{{k + 1}} H_{{k + 1}} \hfill \\ \end{gathered} \right]\left( {\alpha _{{i,k}} F_{{i,k}} } \right) \hfill \\ + \left( {r - 1} \right)\hat{P}_{{X_{k} }} ^{{ - 1}} \hfill \\ \end{gathered} \right] \\ \tilde{X}_{k} /V_{k} \left( {\tilde{X}_{k} } \right) \end{gathered}$$

(52)

According to conditions (36) and (37), the following matrix inequality is obtained:

$$\sum_{i=1}^{r}{h}_{i}\left(\theta \right){\left({\alpha }_{i,k}{F}_{i,k}\right)}^{T}\left[{\left({\Theta }_{k+1}-{\beta }_{k+1}{H}_{k+1}\right)}^{T}{{\widehat{R}}_{k+1}}^{-1} \left({\Theta }_{k+1}-{\beta }_{k+1}{H}_{k+1}\right)\right]\left({\alpha }_{i,k}{F}_{i,k}\right)\ge 0$$

(53)

${\lambda }_{min}$ is chosen by Assumptions 1 and 2, and using (11) and (25):

$$ \begin{gathered} \lambda _{{k + 1}} V_{k} \left( {\tilde{X}_{k} } \right) \ge \lambda _{{\min }} V_{k} \left( {\tilde{X}_{k} } \right) \hfill \\ \triangleq \tilde{X}_{k} ^{T} \left[ \begin{gathered} \left( {r - 1} \right)p_{{\max }} ^{{ - 1}} \hfill \\ - \mathop \sum \limits_{{i = 1}}^{r} h_{i} \left( \theta \right)\alpha _{{i,\max }} ^{2} .f_{{i,\max }} ^{2} \beta _{{\max }} ^{2} .h_{{\max }} ^{2} \left( \begin{gathered} \mathop \sum \limits_{{j = 1}}^{r} h_{j} \left( \theta \right)\theta _{{\min }} ^{2} .p_{{\min }} .\alpha _{{j,\min }} ^{2} .f_{{j,\min }} ^{2} \hfill \\ + \theta _{{\min }} ^{2} .\hat{q}_{{j,\min }} + \hat{r}_{{\min }} \hfill \\ \end{gathered} \right)^{{ - 1}} \hfill \\ \end{gathered} \right]\tilde{X}_{k} \ge 0 \hfill \\ \end{gathered} $$

(54)

With respect to (34), the following condition must also be confirmed:

$${\lambda }_{k+1}{V}_{k}\left({\tilde{X }}_{k}\right)-{V}_{k}\left({\tilde{X }}_{k}\right)<0\to {\lambda }_{k+1}<1$$

(55)

Therefore,

$$ \begin{gathered} \lambda _{{k + 1}} V_{k} \left( {\tilde{X}_{k} } \right) - V_{k} \left( {\tilde{X}_{k} } \right) \hfill \\ = \tilde{X}_{k} ^{T} \left[ {\mathop \sum \limits_{{i = 1}}^{r} h_{i} \left( \theta \right)\left( {\alpha _{{i,k}} F_{{i,k}} } \right)^{T} \left[ \begin{gathered} \left( {\Theta _{{k + 1}} - \beta _{{k + 1}} H_{{k + 1}} } \right)^{T} \hat{R}_{{k + 1}} ^{{ - 1}} ~\left( {\Theta _{{k + 1}} - \beta _{{k + 1}} H_{{k + 1}} } \right) \hfill \\ - \left( {\beta _{{k + 1}} H_{{k + 1}} } \right)^{T} \left( {\Theta _{{k + 1}} \hat{P}_{{X_{{_{{k + 1\left| k \right.}} }} }} \Theta _{{k + 1}} ^{T} + \hat{R}_{{k + 1}} } \right)^{{ - 1}} \beta _{{k + 1}} H_{{k + 1}} \hfill \\ \end{gathered} \right]}\right. \\ \left.{\vphantom{\left[ \begin{gathered} \left( {\Theta _{{k + 1}} - \beta _{{k + 1}} H_{{k + 1}} } \right)^{T} \hat{R}_{{k + 1}} ^{{ - 1}} ~\left( {\Theta _{{k + 1}} - \beta _{{k + 1}} H_{{k + 1}} } \right) \hfill \\ - \left( {\beta _{{k + 1}} H_{{k + 1}} } \right)^{T} \left( {\Theta _{{k + 1}} \hat{P}_{{X_{{_{{k + 1\left| k \right.}} }} }} \Theta _{{k + 1}} ^{T} + \hat{R}_{{k + 1}} } \right)^{{ - 1}} \beta _{{k + 1}} H_{{k + 1}} \hfill \\ \end{gathered} \right]}\left( {\alpha _{{i,k}} F_{{i,k}} } \right) + \left( {r - 2} \right)\hat{P}_{{X_{{_{k} }} }} ^{{ - 1}} } \right]\tilde{X}_{k} \hfill \\ \end{gathered} $$

(56)

For (56), derived from conditions (36) and (37), we obtain

$$ \begin{gathered} \lambda _{{k + 1}} V_{k} \left( {\tilde{X}_{k} } \right) - V_{k} \left( {\tilde{X}_{k} } \right) \hfill \\ \le \tilde{X}_{k} ^{T} \left[ {\mathop \sum \limits_{{i = 1}}^{r} h_{i} \left( \theta \right)\alpha _{{i,\min }} ^{2} .f_{{i,\min }} ^{2} \left[ \begin{gathered} \left( {\theta _{{\max }} - \beta _{{\min }} .h_{{\min }} } \right)^{2} \hat{r}_{{\min }} ^{{ - 1}} \hfill \\ - \beta _{{\min }} ^{2} .h_{{\min }} ^{2} \left( {\mathop \sum \limits_{{j = 1}}^{r} h_{j} \left( \theta \right)\left( \begin{gathered} \theta _{{\max }} ^{2} .p_{{\max }} .\alpha _{{j,\max }} ^{2} .f_{{j,\max }} ^{2} \hfill \\ + \theta _{{\max }} ^{2} .\hat{q}_{{j,\max }} + \hat{r}_{{\max }} \hfill \\ \end{gathered} \right)} \right)^{{ - 1}} \hfill \\ \end{gathered} \right] + \left( {r - 2} \right)p_{{\min }} ^{{ - 1}} } \right]\tilde{X}_{k} < 0 \hfill \\ \end{gathered} $$

(57)

From (54) and (57), we get $0\le {\lambda }_{k+1}<1$.

Now, from (42), including noises ${{\omega }_{eq}}_{k}$ and ${\vartheta }_{k+1}$, and (29), ${\mu }_{k+1}$ will be:

$$ \begin{aligned} \mu _{{k + 1}} & = E\left\{ \begin{gathered} \omega _{{eq_{k} }} ^{T} \left[ \begin{gathered} \left( {\beta _{{k + 1}} H_{{k + 1}} } \right)^{T} \hat{R}_{{k + 1}} ^{{ - 1}} ~\Theta _{{k + 1}} \hat{P}_{{X_{{k + 1}} }} \Theta _{{k + 1}} ^{T} \hat{R}_{{k + 1}} ^{{ - 1}} \beta _{{k + 1}} H_{{k + 1}} \hfill \\ - \left( {\beta _{{k + 1}} H_{{k + 1}} } \right)^{T} \hat{R}_{{k + 1}} ^{{ - 1}} ~\Theta _{{k + 1}} \hat{P}_{{X_{{k + 1}} }} - \hat{P}_{{X_{{k + 1}} }} \Theta _{{k + 1}} ^{T} \hat{R}_{{k + 1}} ^{{ - 1}} \beta _{{k + 1}} H_{{k + 1}} + \hat{P}_{{X_{{k + 1}} }} ^{{ - 1}} \hfill \\ \end{gathered} \right]\omega _{{eq_{k} }} \hfill \\ + \vartheta _{{k + 1}} ^{T} \hat{R}_{{k + 1}} ^{{ - 1}} ~\Theta _{{k + 1}} \hat{P}_{{X_{{k + 1}} }} \Theta _{{k + 1}} ^{T} \hat{R}_{{k + 1}} ^{{ - 1}} \vartheta _{{k + 1}} \hfill \\ \end{gathered} \right\} \\ & = E\left\{ \begin{gathered} \omega _{{eq_{k} }} ^{T} \left( {I - \hat{P}_{{X_{{k + 1}} }} \Theta _{{k + 1}} ^{T} \hat{R}_{{k + 1}} ^{{ - 1}} \beta _{{k + 1}} H_{{k + 1}} } \right)^{T} \left( {\hat{P}_{{X_{{k + 1}} }} } \right)^{{ - 1}} ~\left( {I - \hat{P}_{{X_{{k + 1}} }} \Theta _{{k + 1}} ^{T} \hat{R}_{{k + 1}} ^{{ - 1}} \beta _{{k + 1}} H_{{k + 1}} } \right)\omega _{{eq_{k} }} \hfill \\ + \vartheta _{{k + 1}} ^{T} \hat{R}_{{k + 1}} ^{{ - 1}} \Theta _{{k + 1}} \hat{P}_{{X_{{k + 1}} }} \Theta _{{k + 1}} ^{T} \hat{R}_{{k + 1}} ^{{ - 1}} \vartheta _{{k + 1}} \hfill \\ \end{gathered} \right\} \\ \end{aligned} $$

(58)

Knowing that $tr\left(A+B\right)=tr\left(A\right)+tr(B)$ and Eq. (58) is scalar, traces are taken on both sides of (58)

$$ \begin{aligned} \mu _{{k + 1}} & = ~E\left\{ \begin{gathered} tr\left[ {\left( {I - \hat{P}_{{X_{{k + 1}} }} \Theta _{{k + 1}}^{T} \hat{R}_{{k + 1}}^{T} \beta _{{k + 1}} H_{{k + 1}} } \right)^{T} \left( {\hat{P}_{{X_{{k + 1}} }} } \right)^{{ - 1}} ~\left( \begin{gathered} I \hfill \\ - \hat{P}_{{X_{{k + 1}} }} \Theta _{{k + 1}}^{T} \hat{R}_{{k + 1}}^{{ - 1}} \beta _{{k + 1}} H_{{k + 1}} \hfill \\ \end{gathered} \right)} \right]Q_{{eq_{k} }} \hfill \\ + tr\left( {\hat{R}_{{k + 1}}^{{ - 1}} \Theta _{{k + 1}} \hat{P}_{{X_{{k + 1}} }} \Theta _{{k + 1}}^{T} \hat{R}_{{k + 1}}^{{ - 1}} } \right)R_{{k + 1}} \hfill \\ \end{gathered} \right\} \\ & \le \left\{ {\frac{{q_{{\max }} \left( {\hat{r}_{{\max }} - p_{{\min }} .\theta _{{\min }} .\beta _{{\min }} .h_{{\min }} } \right)^{2} }}{{p_{{\min }} .\hat{r}_{{\max }}^{2} }}L + \frac{{\theta _{{\max }} ^{2} .p_{{\max }} .r_{{\max }} }}{{\hat{r}_{{\min }}^{2} }}M} \right\} \\ \,\mathop = \limits^{\Delta } \mu _{{\max }} \\ \end{aligned} $$

(59)

Appendix C

This appendix presents the proof of Theorem 2.

Proof

For each fuzzy set, inserting (28) and (29) in (25) and rearranging the terms, we obtain

$$ \hat{P}_{{X_{{_{{\left. {i,k + 1} \right|k}} }} }} = \alpha_{i,k} F_{i,k} \left[ {\hat{P}_{{X_{{_{\left. k \right|k - 1} }} }} - \hat{P}_{{X_{{_{\left. k \right|k - 1} }} }} \Theta_{k}^{T} \left[ {\Theta_{k} \hat{P}_{{X_{{_{\left. k \right|k - 1} }} }} \Theta_{k}^{T} + \hat{R}_{k} } \right]^{ - 1} \Theta_{k} \hat{P}_{{X_{{_{\left. k \right|k - 1} }} }} } \right]F_{i,k}^{T} \alpha_{i,k} + \hat{Q}_{{eq_{{_{k} }} }} $$

(60)

By definition $\Psi = \Theta_{k} \hat{P}_{{X_{\left. k \right|k - 1} }} \Theta_{k}^{T}$ and$\Upsilon={\widehat{R}}_{k}$, and applying Lemma 4:

$$ \hat{P}_{{X_{{\left. {i,k + 1} \right|k}} }} \le \alpha_{i,k} F_{i,k} \left[ {\Theta_{k}^{ - 1} \hat{R}_{k} \Theta_{k}^{ - T} } \right]F_{i,k}^{T} \alpha_{i,k} + \hat{Q}_{{eq_{k} }} $$

(61)

Using Assumptions 1 and 2 for the upper bound (61), we obtain

$$ \hat{P}_{{X_{{_{{\left. {i,k + 1} \right|k}} }} }} \le \theta_{\min }^{ - 2} .\hat{r}_{\max } .\alpha_{i,\max }^{2} .f_{i,\max }^{2} + \hat{q}_{\max } I \triangleq \widehat{{p_{i} }}I $$

(62)

The upper bound of $\hat{P}_{{X_{{\left. {i,k + 1} \right|k}} }}$ is shown in (62). From (31) and Lemma 2, we obtain

$$ \hat{P}_{{X_{{_{k + 1} }} }} \le \hat{P}_{{X_{{_{{\left. {k + 1} \right|k}} }} }} \le \mathop \sum \limits_{i = 1}^{r} h_{i} \left( {\theta \left( {u,y} \right)} \right)\widehat{{p_{i} }}I $$

(63)

From ${h}_{i}\left(\theta \left(u,y\right)\right)\ge 0$, (18) and (63), the following is obtained:

$$ E\left\{ {\hat{P}_{{X_{{_{k + 1} }} }} } \right\} \le E\left\{ {\mathop \sum \limits_{i = 1}^{r} h_{i} \left( {\theta \left( {u,y} \right)} \right)\widehat{{p_{i} }}I} \right\} \triangleq {\mathcal{P}}_{\max } I $$

(64)

Thus, from Theorem 2, the upper bound ${\mathcal{P}}_{max}$ is obtained.