Event-Triggering State and Fault Estimation for a Class of Nonlinear Systems Subject to Sensor Saturations

State estimation/filtering problems have always been one of the fundamental issues in the areas of target tracking, navigation and positioning, electric power systems, econometrics, biosystems, etc. Therefore, enormous research attention has been paid to the state estimation problems and some elegant work has been reported, see e.g., [1,2,3,4,5]. According to different performance indices, the current state estimation approaches include Kalman filtering (KF), extend Kalman filtering (EKF), $H_{\infty }$ filtering and so on. To be specific, the famous KF approach has been proposed in [2] by providing optimal state estimates in the sense of minimal mean-squared error under the assumption that system parameters and noise statistics are precisely known. The $H_{\infty }$ filtering method proposed [6,7] is capable to attenuate the influence from the exogenous disturbance to the filtering error. When it comes to the case that the system model is nonlinear or uncertain, the celebrated EKF approach has been shown to be a useful tool for the state estimation issues. For instance, in [8], the EKF approach has been developed to cope with the nonlinear systems subject to missing measurement. Moreover, in [9], the filtering approach has also been applied in the complex networks with incomplete measurements.

It is often the case that the faults are inevitable in practical applications because of a variety of reasons including component failure, ageing equipment, complex external environment, bandwidth limitation, etc. During the past few decades, fault detection and fault-tolerant control issues have gained considerable research enthusiasm due to the demand in reliability and security of the practical systems. It should be pointed out that the accurate information of the fault signals are hard to acquire, whilst the recently emerging state and fault estimation (SFE) approach provides a good solution to obtain the sufficient information of the state and fault signals simultaneously. By such a merit, increasing research attention has been paid on this aspect recently, and some inspiring work has been available in the literature [10,11,12]. For instance, an SFE algorithm relying on a recursive approach has been designed in [13] for uncertain systems with missing measurements and stochastic nonlinearities. $H_{\infty }$ SFE problems have also been studied for various dynamic systems, such as fuzzy systems [12,14], nonlinear systems [15,16] and 2-D systems [17,18]. Nevertheless, the SFE problems have not been thoroughly investigated yet and still have been a research hotspot in control/filtering community.

In reality, sensors may not always provide signals with unlimited amplitudes owing to the physical constraints. If the sensor saturation is not properly handled, it will severely decrease the system performance. The main challenge of research on the saturations in control community is how to design a filtering/control algorithm that can effectively dealing with the nonlinearities brought by the sensor saturations. As a consequence, the filtering/control issues subject to sensor saturations have gained initial research focus, see, e.g., [19,20]. For example, a recursive filtering issue has been solved for uncertain systems with faults and sensor saturations in [21]. In [22,23], the $H_{\infty }$ filtering issues have been settled for nonlinear systems with incomplete measurements and sensor saturations. In [20], mean-squared consensus control problem has been studied for stochastic multi-agent systems subject to sensor saturations where the desired controllers have been designed depending on the solutions to recursive matrix inequalities.

On another research front, the event-triggering mechanism (ETM) has become a research hotspot recently due mainly to its superiority of effectively reducing communication resources compared with the traditional time-triggering protocol. Under the ETM the current measurement will be transmitted only when the predefined triggering condition is met, and thereby the transmission numbers can be reduced largely. Based on this idea, various control and filtering issues under the ETM have been studied, see e.g., [6,24,25,26]. Very recently, considerable research attention has been paid on the event-triggering fault estimation issue owing to its vital role in the practical engineering. Accordingly, the event-triggering fault estimation problems have been investigated for various systems, such as nonlinear systems with missing measurements [27], stochastic systems subject to nonlinearities and packet dropouts [28], and stochastic systems with deception attacks [29]. However, to the best of the authors’ knowledge, the event-triggering state and fault estimation (ETSFE) problem for nonlinear systems with sensor saturations has not been fully studied, which constitutes the main motivation of this paper.

In terms of the methodologies, due to the effects brought by nonlinearities (including the saturation functions) and event-triggering protocol, it is almost impossible in the Kalman filtering framework to minimize the estimation error covariance through adjustment of the gain matrices. In [30], an alternative way has been proposed to handle the effects of norm bounded parameter uncertainties and a robust filter has been designed such that an upper bounded matrix of the estimation error covariance is minimized. Enlightened by this idea, such a filtering approach has been applied in various complex systems such as complex networks [9] and sensor networks [31]. However, it should be pointed out that, for the state and fault simultaneous estimation problem in the existing literature, the estimation error covariance minimization method has still been the main method which is incapable of dealing with more real complex phenomena. Therefore, it is the second motivation of this paper to develop the filtering approach proposed in [30] to handle the state and fault estimation problems with sensor saturations under the event-triggering strategy.

The novelties of this paper are emphasized as follows: (1) a novel ETSFE issue is, for the first time, addressed when the effects of sensor saturations, nonlinearities as well as ETM are simultaneously taken into consideration; (2) the state and fault estimator is designed such that the upper bounds on the error covariances of the state and fault estimation are respectively guaranteed at each time instant; and (3) the gain matrices are designed via two recursions which minimize the obtained upper bounds. Finally, two illustrative examples are utilized to verify the feasibility of the developed ETSFE algorithm.

The remaining part of this paper is organized as follows. In Section 2, the problem to be investigated is addressed. The main results are listed in Section 3 where the desired state and fault estimators is designed. In Section 4, two illustrative examples are given and the conclusion is drawn in Section 5.

Notations: In this paper, the notations mentioned are standard. ${\mathbb{R}}^n$ and ${\mathbb{R}}^{m\times n}$ respectively denote the n-dimensional Euclidean space and $m\times n$ real matrix. I is the identity matrix, while $\mathrm{diag}\{a_1,a_2,\dots ,a_N\}$ represents the block-diagonal matrix with matrices $a_1,a_2,\cdots ,a_N$. For symmetric matrices x and y, $x\ge y$$(x>y)$ means that $x-y$ is positive semi-define (positive definite) matrix. The superscript “T” and “$-1$” refer to matrix transposition and inverse, respectively. $\mathrm{R}\left(M\right)$ is the rank of the matrix M. $\mathbb{E}\left\{x\right\}$ denotes the mathematical expectation of the stochastic variable x. $\mathrm{tr}\left\{M\right\}$ denotes the trace of the matrix M. $\parallel ·\parallel$ stands for the Euclidean norm.

The estimation structure under consideration is shown in Figure 1 and the dynamics of the plant is given by where $x_k\in {\mathbb{R}}^{n_x}$ is the system state vector, $f_k\in {\mathbb{R}}^{n_f}$ represents a fault signal, $w_k\in {\mathbb{R}}^{n_x}$ is the process noise, and $B_k$ is a given compatible matrix.

The measurements with sensor saturation are described by where $y_k\in {\mathbb{R}}^{n_y}$ represents the measurement vector at time instant k, $v_k\in {\mathbb{R}}^{n_y}$ is the measurement noise, and $C_k$, $D_k$ are both appropriate-dimensional matrices.

The nonlinear function $h_k(·)$ satisfies the following condition where ${\nu }_k>0$ is a known matrix.

The noise signals $w_k$ and $v_k$ have the following statistical properties where $R_k>0$ and $Q_k>0$ are known appropriate-dimensional matrices, and ${\delta }_{kl}$ represents the Kronecker function with

The saturation function $\vartheta (·)$: ${\mathbb{R}}^{n_y}$↦${\mathbb{R}}^{n_y}$ is defined by with $\vartheta \left(s_i\right)$ = $\mathrm{sign}\left(s_i\right)\min \{{\varrho }_i,|s_i\left|\right\}$$(i=1,2,\dots ,n_y)$, where $s_i$ is the ith element of vector s, $\mathrm{sign}(·)$ represents the signum function, and ${\varrho }_i$ denotes the saturation level for ith element.

For the sake of reducing limited communication resource, the ETM is adopted to govern the transmission frequency between the sensor and the estimator. We denote the transmission instants by $0=k_0<k_1<k_2<\cdots <k_l<\cdots$, which is determined by where $\tau >0$ is a given scalar and $y_{k_l}$ is the measurement transmitted at the latest time.

For the purpose of estimating the state and fault simultaneously, we construct the estimator as follows where ${\widehat{f}}_k$, ${\widehat{x}}_k$ represent the estimates of fault and state respectively and $L_k$, $G_k$ are the estimator parameters respectively.

Let the state estimation error and fault estimation error be ${\tilde{x}}_k=x_k-{\widehat{x}}_k$ and ${\tilde{f}}_k=f_k-{\widehat{f}}_k$, respectively.

By noting (1) and (7), one has and where ${\epsilon }_k=y_k-y_{k_l}$.

Assuming that the constraint condition $L_k{D}_k=I$ is met, we eventually derive

Then, we define the estimation error covariances of the state and fault as follows

Our main objective of this paper is to develop an event-triggering state and fault estimator of the form (7) such that, for all nonlinearities as well as sensor saturations, the upper bounds $\left({\Sigma }_k^x\mathrm{and}{\Sigma }_k^f\right)$ for the estimation error covariances of state and fault are respectively guaranteed, that is

Moreover, the designed gain matrices $G_k$ and $L_k$ are expected to minimize the upper bound ${\Sigma }_k^x$ and ${\Sigma }_k^f$ simultaneously at each iteration.

In this section, the upper bounds on the estimation error covariances of the state and fault are expressed by means of recursions. Then, the proper gain matrices $G_k$ and $L_k$ are designed to minimize the upper bounds on the estimation error covariances and fault error covariances, respectively. The following lemmas will be used for obtaining the results.

In this section, two simulation examples are utilized to validate the usefulness of the developed ETSFE algorithm.