Variance‐constrained resilient H∞ filtering for mobile robot localization under dynamic event‐triggered communication mechanism

This paper is concerned with the mobile robot localization problem subject to filter gain uncertainty under dynamic event‐triggered communication mechanism, and meanwhile, the H∞ filtering performance and the error variance constraint are guaranteed. For saving the sensor energy, a dynamic event‐triggered communication mechanism is introduced to manage the transmission of the measurement data from the sensor to the filter. To characterize the possible fluctuations of the desired filter gain, a resilient filter is constructed for the mobile robot localization. The aim of this paper is to find a solution to the mobile robot localization problem by designing a nonlinear resilient filter such that the filtering error dynamics satisfies both the H∞ performance requirement and the error variance constraint over a finite time horizon simultaneously. By resorting to the Lyapunov theory and the stochastic analysis technique, the sufficient conditions are established to guarantee that the error dynamic system satisfies both the H∞ performance requirement and the error variance constraint. Then, a recursive linear matrix inequality (RLMI) approach is employed to design the desired filter. Based on the proposed filter design scheme, the corresponding localization algorithm is presented. Finally, an experiment is conducted in the simulation environment to verify the effectiveness of the proposed localization algorithm.


INTRODUCTION
In the past few decades, the mobile robot has a wide range of applications in engineering, such as intelligent transportation, aerospace, and so on. As a fundamental issue in mobile robot research field, the localization problem has received much research attention, and a variety of results have been available in other studies [1][2][3][4][5][6][7]. It should be noted that, in the aforementioned results, the designed filter/estimator is assumed to be implemented accurately when mobile robot localization system operates. However, in practical mobile robot localization system, the acquisi-tion of the filter gain may be subject to possible parameter fluctuations resulting primarily from the finite resolution instrumentation, digital truncation errors, or component aging [8]. If the effect from the filter parameter variations is not taken into account adequately, the localization performance will be degraded as the working hours of the localization system increases. The resilient filtering scheme, as an effective solution, can tolerate the possible gain variations, and a variety of remarkable research results have been reported in the literature for power systems [9], time-varying systems [10], Markovian jump systems [11], nonlinear complex networks [12], neural networks [13,14], and other systems [15,16]. However, when it comes to the mobile robot localization problem, the corresponding results are very few. It is therefore the first motivation of the present study.
In mobile robot localization system, the sensor installed on the mobile robot platform can output the measurements and send them to the filter. The sensor is usually powered by battery with limited energy. In many practical situations, the working environment of the mobile robot is dangerous for human, which makes difficult for changing the battery. Therefore, it is needed to employ a strategy for the purpose of saving the sensor energy. In recent years, due to the advantage of saving energy, the event-triggered communication mechanism has been widely adopted in control and signal processing areas, see, for example, other studies [17][18][19][20][21][22][23]. Under such a mechanism, the measurements have a privilege to access to the transmission channel only when the event-triggering condition is met. Compared with the traditional periodic transmission, the unnecessary transmissions can be avoided, and the purpose of energy-saving can be achieved. For the aim of further saving, a dynamic event-triggered communication mechanism has been proposed in Girard [24], where it has demonstrated that the dynamic event-triggering strategy is capable of further reducing the energy consumption. The results in Girard [24] have been extended to the cases of complex networks [25], sensor networks [26], and neural networks [27,28]. It seems to be a natural idea to introduce the dynamic event-triggered communication mechanism to the mobile robot localization system. Nevertheless, to the best of our knowledge, the mobile robot localization problem under dynamic event-triggered communication mechanism has not yet received adequate attention despite its clear practical significance, and it is therefore the another motivation of this paper.
It is common that the upper bound of the filtering error variance is usually utilized to represent the filtering performance requirements [29]. In practical engineering, it is tolerable that the filtering error fluctuates in an acceptable range, which means that the filtering error variance is no longer required to be the minimum as long as the engineering requirements are met [30]. As such, during the past few decades, the variance-constrained filtering problem has been investigated widely in networked control systems, and a large number of results can be found in other studies [29][30][31][32][33][34][35]. Unfortunately, there have been very few results published on the mobile robot localization problem by employing variance-constrained filtering scheme, not to mention the constraint in the presence of H ∞ , filter gain uncertainty, and dynamic event-triggered communication mechanism, and this constructs the third motivation of our study. Note that, in Dong et al. [29], the robust H ∞ filtering problem has been studied for a class of uncertain nonlinear discrete time-varying stochastic systems subject to multiple missing measurements and error variance constraint, but the results cannot be used to solve the mobile robot localization problem.
Based on the above discussion, in this paper, we aim to investigate the mobile robot localization problem subject to filter gain uncertainty and dynamic event-triggered communication mechanism and meanwhile the H ∞ filtering performance and error variance constraint are satisfied. The main challenges we will encounter are identified as follows: (1) How to characterize the dynamic characteristics of the dynamic event-triggered communication mechanism and the phenomenon of filter gain fluctuation? (2) How to design a filter for mobile robot localization in the presence of the dynamic event-triggered communication mechanism and the possible filter gain fluctuations such that the H ∞ performance requirement and the error variance constraint are guaranteed simultaneously? In this paper, we endeavor to deal with the two identified challenges, and the main contributions of this paper are highlighted as follows: (1) The two objectives, the H ∞ performance requirement and error variance constraint, are considered simultaneously for the first time in mobil robot localization problem subject to filter gain uncertainty and dynamic event-triggered communication mechanism; (2) the dynamic behavior is characterized by an additional internal variable in the introduced dynamic event-triggered communication mechanism; (3) a nonlinear resilient filter is constructed for the addressed mobile robot localization problem under the dynamic event-triggered communication mechanism; (4) by resorting to the Lyapunov theory and the stochastic analysis technique, the sufficient conditions are established to guarantee that the error dynamic system satisfies both the H ∞ performance requirement and the error variance constraint simultaneously, then a recursive linear matrix inequality (RLMI) approach is employed to design the desired nonlinear resilient filter; and (5) based on the proposed filter design scheme, the corresponding mobile robot localization algorithm is given.
The remainder of this paper is organized as follows. Section 2 formulates the addressed mobile robot localization problem. In Section 3, the desired nonlinear resilient filter is designed, and the corresponding localization algorithm is given. In Section 4, the effectiveness of the provided localization algorithm is illustrated. Some conclusions are given in Section 5.

Notation
The notation used here is fairly standard except where otherwise stated. ||A|| refers to the norm of a matrix A. The notation X ≥ Y (respectively, X > Y), where X and Y are real symmetric matrices, means that X − Y is positive semidefinite (respectively, positive definite). M T represents the transpose of the matrix M. I denotes the identity matrix of compatible dimension. E{x} stands for the expectation of the stochastic variable x. diag{ … } stands for a block-diagonal matrix. diag n {·} means the block-diagonal matrix with n blocks. " * " stands for a term induced by symmetry in symmetric block matrices.

Mobile robot kinematic model
In this paper, consider a wheeled mobile robot which is shown in Figure 1. The mobile robot kinematic model can be described as follows [36]: where (x(t), y(t)) is the position of mobile robot, the orientation angle (t) is the angle between x G axis and mobile robot forward axis y ′ R , v(t) is the mobile robot displacement velocity, (t) is the mobile robot angular velocity. The displacement and angular velocities can be obtained from the odometric measures. We assume that the displacement and angular velocities of the robot received by the odometric measures are constant over the sampling period. Then the continuous-time system (1) can be discretized to the following system: where ΔT is the sampling period.
Before carrying out the addressed mobile robot localization problem, we need to handle the nonlinear system (3). By denoting the state estimate of X(k) asX(k) and expanding the nonlinear function f (X(k), u(k)) about the estimatê X(k), Equation (3) can be further expressed by where A(k) = (X(k),u(k)) provide an extra degree of freedom to tune the filter, and Γ(k) ∈ R n Γ is an unknown time-varying matrix satisfying Γ(k)Γ T (k) ≤ I to take into account the linearization errors of the nonlinear system (3). The initial value X(0) is a Gaussian distributed random variable with E{X(0)} =X(0) and E{(X(0) − X(0))(X(0) −X(0)) T } = R(0) withX(0) and R(0), respectively, being a given vector and a symmetric matrix.

Doppler radar measurement model
M(x M , y M ) denotes the position of the landmark. At time instant k, the Doppler frequency shift d(k) and the azimuth (k) are treated as measurements. According to the definition of the Doppler frequency shift in Battistelli et al. [37], the measurement produced by the Doppler radar form the landmark can be expressed as follows [38]: where with f c and c being, respectively, the radar carrier frequency and the speed of light. (k) is the measurement noise which is assumed to be a zero mean Gaussian white noise sequence with covariance W(k) > 0. The initial state X(0), (k) and (k) are uncorrelated mutually. By using the Taylor series expansions again, the nonlinear measurement model (5) can be expanded as where the expression of C(k) can be found in the appendix, G(k) is a problem-dependent scaling matrix, F(k) is introduced to provide an extra degree of freedom to tune the filter, and unknown time-varying matrix Λ(k) ∈ R n Λ satisfies Λ(k)Λ T (k) ≤ I.

Dynamic event-triggered communication mechanism
In order to reduce sensor energy consumption, a dynamic event-triggered communication mechanism is employed. Denote the transmission time instants as 0 = r 0 < r 1 < r 2 < … < r p < … , which can be determined by the following event-triggering condition where and are known positive scalars, (k) is defined by (k) = z(k) − z(r p ), and (k) is an internal dynamic variable satisfying with ∈ (0, 1) being a known constant and (0) = 0 ≥ 0 being the initial condition.
Remark 1. By introducing an internal variable (k), the threshold value of the adopted dynamic event-triggering condition (7) dynamically changes in light of the evolution of the internal variable (k) in Equation (8). It is noted that, in Equation (7), the parameter determines the triggering frequency which increases as increases monotonically. Let approach to +∞, it can be seen that the adopted dynamic event-triggering condition (7) reduce to a traditional static counterpart proposed in Liu et al. [39]. Moreover, the value of internal variable (k) is determined by (0) which is crucial to let z T (k)z(k) − T (k) (k) not always nonnegative. The parameters and (0) can be adjusted in terms of the localization performance requirement.
Remark 2. Because of the introduction of the internal variable (k) in dynamic event triggering condition (7), the unnecessary data transmission can be reduced effectively, and then, the communication burden can be alleviated, and the sensor energy can be saved. These merits cater for the requirement of sensor energy saving in mobile robot localization system.
In this paper, to perform the mobile robot localization, we construct the filter of the following form: whereX(k) is the estimate of X(k) and K(k) is the filter gain to be designed. The gain uncertainty matrix ΔK(k) is assumed to be of the following form: where T and Q are the known real constant matrices and Π(k) ∈ R n Π denotes the unknown time-varying matrix satisfying Π T (k)Π(k) ≤ I. Denoting the filtering error by e(k) = X(k) −X(k), the error dynamic system can be obtained as follows: where The filtering error covariance matrix is defined as In this paper, our purpose is to find a solution to the addressed mobile robot localization problem by designing a filter of structure (9) such that the filtering error dynamic system (11) satisfies the following two constraints simultaneously over a finite horizon [0, N − 1].

R1:
Given the disturbance attenuation level , matrix  ≥ 0 and initial error e(0), the following H ∞ performance criterion is satisfied: R2: At each time instant k, the error covariance satisfies where Ξ(k) is a sequence of positive definite matrices that are prespecified according to the localization accuracy requirement.

MAIN RESULTS
In this section, a solution to the addressed mobile robot localization problem is found. Before proceeding further, we introduce the following lemmas which will be needed for the derivation of our main results.

H ∞ performance analysis
Let us start with the H ∞ performance analysis for the filtering error dynamic system (11). In the following theorem, a sufficient condition is given under which the H ∞ performance requirement is guaranteed. where (k), ℬ(k) and (k) are given in Equation (12), and the initial condition Proof. Choose the following Lyapunov functional candidate: We construct the following equation: From the triggering condition (7), we have Then, one has Taking Equations (6), (8), and (11) into consideration and noticing the uncorrelatedness between (k) and (k), we have

× (h(X(k)) + C(k)e(k) + G(k)Λ(k)F(k)e(k))
where and ℋ (k) is given in Equation (16). Summing up both sides of Equation (18) from 0 to N − 1 with respect to k and according to the H ∞ performance requirement (13), we obtain Noticing that V(N) ≥ 0 and the initial condition (17), we can have that the H ∞ performance requirement (13) is satisfied when Equation (15) holds. The proof is now completed.

Variance analysis
Having discussed the H ∞ performance analysis for the error dynamic system (11), then, we will focus our attention on the variance analysis.
with initial condition (0) = (0). Here, Proof. Noticing the filtering error dynamic system (11), the corresponding evolution of (k) is governed by Then, applying the elementary inequality ab T + ba T ≤ aa T + bb T , one has Now, we are ready to complete the rest of the proof by induction. First, it is obvious that (0) ≤ (0). Then, assuming (k) ≤ (k), we now acquire that (k + 1) ≥ Ψ( (k)) ≥ Ψ((k)) ≥ (k + 1) which indicates that the inequality holds at the step k + 1. This completes the proof.
Remark 5. Until now, the solution to the addressed mobile robot localization problem has been found by designing the nonlinear resilient filter which has been designed according to the results in Theorem 6. Then, the corresponding mobile robot localization algorithm has been summarized and shown in Algorithm 1. In the next section, an experiment is conducted in the simulation environment to verify the effectiveness of the proposed localization algorithm.

SIMULATION RESULTS
In this section, the effectiveness of the proposed mobile robot localization algorithm is demonstrated in the simulation environment.
The length of finite horizon N is set as N = 610. Set the sampling period of mobile robot's odometer be 150 ms. When 0 ≤ k ≤ 130 and 430 ≤ k ≤ N, set the displacement velocity be 400 mm/s and angular velocity be 2 45 rad∕s. When 130 < k < 430, set the displacement velocity be 400 mm/s and angular velocity be − 2 45 rad∕s. The covariance R(k) and covariance W(k) are set as R(k) = 0.01I and W(k) = 0.01I, respectively. Set L(k), H(k) and G(k), F(k) in Equations (4) and (6)  In the mobile robot localization process, the mobile robot starts to move from initial statē X(0) = [0.2m, 0.2m, 0.2rad] T . Once the mobile robot movement, the Doppler radar mounted on the mobile robot plant can output the Doppler frequency shift and the azimuth. Then, the sensor sends the measurements to the localization center, which is governed by the dynamic event-triggered mechanism, and then, the localization center can process these data according to the developed algorithm and obtain the position and the attitude of the mobile robot. We conduct the experiment in a simulation environment. By using Matlab (with the YALMIP 3.0) and implementing Algorithm 1, the RLMIs in Theorem 6 can be solved recursively. The corresponding simulation results are shown in Figures 2-7, where Figure 2 shows the trajectory of mobile robot and its estimate, Figure 3 shows the angle of mobile robot and its estimate, Figures 4-6 depict the filtering error variances and their upper bounds for the x(k), y(k) and (k), respectively, and Figure 7 shows the filtering errors for the x(k), y(k), and (k), respectively.
From Figure 2, the blue dotted line denotes the actual trajectory of mobile robot (i.e., (x(k) and y(k))), and the red dotted line denotes the estimated trajectory of mobile robot (i.e., (x(k) and̂(k))). In Figure 3, the actual orientation angle of mobile robot (t) is shown by using blue dotted line, and the red dotted line denotes the estimated orientation anglê(t). From Figures 4-8, it can be seen that the errors satisfy simultaneously the prespecified performance requirements.   In addition, Figure 8 shows the dynamic event-based release instants and the corresponding release intervals. It can be seen from Figure 8 that updates under the dynamic event-triggered communication mechanism are only executed 134 times over the finite time horizon [0, 610] which means the energy consumption of sensor is effectively reduced as the communication times is reduced. Moreover, the H ∞ performance constraint is calculated under three cases at each time instant k which is shown in Table 1. All the calculated H ∞ performance constraints are less than the given disturbance attenuation level = 8.
Moreover, in order to clarify the effects from different noise intensities on the localization accuracy under the given H ∞ disturbance attenuation level, by implementing the proposed localization algorithm twice, the obtained localization errors have been depicted in Figures 9 and 10 under the different noise intensities, respectively.
From the above simulation results and discussions, we can clearly see that the designed filter has a satisfactory performance for the addressed mobile robot localization problem and the effectiveness of the proposed mobile robot localization algorithm is well demonstrated.

CONCLUSION
In this paper, the mobile robot localization problem subject to filter gain uncertainty under dynamic event-triggered The filtering errors under W(k) = 0.05I

FIGURE 10
The filtering errors under W(k) = 0.1I communication mechanism has been studied and meanwhile the H ∞ filtering performance and the error variance constraint are guaranteed. For the sake of saving the sensor energy, a dynamic event-triggered communication mechanism has been introduced to manage the transmission of the measurement data. To characterize the possible fluctuations of the desired filter gain, the nonlinear resilient filter has been constructed. The solution to the mobile robot localization problem has been found by designing a nonlinear resilient filter such that the filtering error dynamics satisfies both the H ∞ performance requirement and the error variance constraint over a finite time horizon simultaneously. By resorting to the Lyapunov theory and the stochastic analysis technique, the sufficient conditions have been established to guarantee that the error dynamic system satisfies both the H ∞ performance requirement and the error variance constraint. Then, a RLMI approach has been employed to design the desired nonlinear resilient filter. Based on the proposed filter design scheme, the corresponding mobile robot localization algorithm has been presented. Finally, an experiment has been conducted in the simulation environment to verify the effectiveness of the proposed localization algorithm. Future research topics would include mobile robot localization under denial-of-service attacks [40], mobile robot localization under coding-decoding communication protocol [41,42], and the extension of the developed algorithm to other more complex systems, such as artificial neural networks [43], sensor networks [44], and power systems [9].