# Distributed State Estimation for Dynamic Positioning Systems with Uncertain Disturbances and Transmission Time Delays

Authors: Li Liu; Tao Yao; Xin Hu; Chunjie Zhou; Dianli Hou; Shulin Feng; Hongyong Yang
Journal: Complexity
 (2020)

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## Abstract

The dynamic positioning system of unmanned underwater vehicles (UUVs) is a complex and large-scale system mainly due to the nonlinear dynamics, uncertainty in model parameters, and external disturbances. With the aid of the bio-inspired computing (BIC) method, the designed three-dimensional (3D) spatial positioning system is used for enlarging communication constraints and increasing signal coordination processing. With the growing of measurement scales, the issue of the networked high-precision positioning has been developed rapidly. Then, an information fusion estimation approach is presented for the distributed networked systems with data random transmission time delays and lost and disordered packets. To reduce the communication burden, an adaptive signal selection scheme is employed to reorganize the measurement sequence, and the parameter uncertainties as well as cross-correlated noise are used to describe the uncertain disturbances. Moreover, a reoptimal weighted fusion state estimation is designed to alleviate the information redundancy and maintain higher measurement accuracy. An illustrative example obtained from the 3D spatial positioning system is given to validate the effectiveness of the proposed method.


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## Body

## 1. Introduction

For the complex and large-scale systems in the field of energy, transportation, logistics, and so on, due to the emerging problems such as strongly nonlinear and highly coupling, they significantly challenge the current computational tools. To solve science and engineering problems based on complex biological mechanisms and living phenomena, bio-inspired computing (BIC) method is used to design and develop the computer algorithms and models, such as neural networks [1], genetic algorithms [2], and artificial immune systems [3].Taking the neural network as an example in the field of BIC, Tang et al. proposed a feedforward migration neural network to predict the battery aging trajectories, which arose from the highly nonlinear dynamics of battery aging [4]. In engineering application, the dynamic positioning system of unmanned underwater vehicles (UUVs) is a complex problem mainly due to the nonlinear dynamics, uncertainty in model parameters, and external disturbances. Using the adaptive method based on BIC computing is a promising solution to address these issues. Nevertheless, its full potential is yet to be realized, especially in underwater vehicle applications, mainly due to the highly nonlinear and time-varying nature of vehicle dynamics, unpredictable external disturbances, and the practical difficulty in accurately modeling the hydrodynamic effects [5]. In order to make the processing signal less sensitive to measurement noise, it suffers from scenarios such as the measurement information is inaccurate [6]. However, it is difficult to know the precise noise model in practice. Because of the absence of the full battery degradation model and the inevitable local aging fluctuations in the uncontrolled environments, a base model-oriented gradient-correction particle filter (GC-PF) [7] is proposed to predict aging trajectories of lithium-ion batteries. Therefore, as an estimation method that does not need to know the precise model of noise, such as H∞ filter, which considers the noise as random finite signals, has been developed in engineering applications like fault identification and estimation [6]. It should be noticed that both Kalman filter and H∞ filter are not suitable for nonlinear and non-Gaussian systems. In nonlinear systems, there are uncertainties caused by modeling errors and external disturbances.Parameter uncertainties are inevitable in practical systems, and they may lead to severe performance degradation or even instability of the closed-loop system. Therefore, they should be fully considered in designing an estimator [8]. Liu et al. employed the Gaussian process regression (GPR) technique to capture the underlying mapping among capacity, storage temperature, and state-of-charge [9]. Moreover, two related data-driven models are developed based on GPR technique [10]. The uncertainties stem from uncertain hydrodynamic parameters, modeling errors, and unknown forces due to the ocean currents in an underwater environment. Specifically, the disturbance observer is developed to estimate the lumped disturbance composed of parametric model uncertainties, modeling errors, and unknown environmental forces. The command governor is formulated as a quadratically constrained quadratic programming problem [11]. Moreover, as opposed to the one-step prediction, multistep prediction is used herein, and a better performance can be achieved [12, 13].As shown in Figure1, the ship operating in the ocean inevitably suffers the disturbances due to waves, wind, and currents. In practice, the ship dynamics is related to the ship’s own characteristics and the operating conditions; thus, there evidently exist parameter uncertainties in the ship motion mathematical model [14]. From Figure 1, it is believed that the concept of hybrid control can provide a scalable and stringent methodology for the design of real industrial control applications. A lot of research studies use the relevant BIC method to deal with several control objectives, changing environmental as well as operational conditions [15]. A ship’s speed measurement before GPS is important to the navigation system. In real-time, closed-loop control applications, there are problems such as noisy measurements and derivative processes. A variety of filtering methods are available to reduce the noise in the literature [16]. The Kalman filter is applied not only in signal processing but also in the estimation of velocity. The derivation of position signal data may be required not having another sensor for measuring velocity. A general derivative process is the Euler method, but this method may not give accurate results [16, 17].Figure 1
Frames and states of the ship.Dynamic networks are interconnected dynamic systems with measured node signals and dynamic modules reflecting the links between the nodes. Various approaches have been developed for identification of dynamic networks, roughly divided into three categories. The first approach considers the identification of a single module in the dynamic network in the situation that the interconnection structure, or topology, of the network is known. The second approach focuses on identification of the full network dynamics for a given topology, and the last category deals with the identification of the topology (and dynamics) of the network [18]. Due to the influence of underwater observation conditions, positioning gross error will often appear; a method of adding the conditional constraints and confidence assessment to EKF was put forward to filter the positioning value, which can make the filtering result more robust and smooth [19]. Sato and Toda [20] proposed adaptive algorithms of gain tuning for Kalman filters and switching Kalman and H∞ filters for discrete systems. Both gain tuning and switching rely on square means of innovations. Accurate estimation of the horizontal component of the crane force is essential for the feedforward solution, but it is extremely difficult to obtain it from a measurement due to its time-varying nature and the nonlinear system dynamics stemming from the ship-environment interaction [21].Considering the nonlinear and uncertain dynamic characteristics, a fuzzy neural network backstepping controller is designed to deal with the uncertainties, including estimation and compensation [22]. Brouwer et al. [23] presented an estimator to quantify the random uncertainty of the mean (RUM) from a stationary single time series, without performing actual repeat tests. To develop a robust ship dynamic modeling approach by making efforts on determining a common ship dynamic model and designing a robust identification method [24], the ship dynamic model should be suitable for cases which mean the sufficient balance between the complexity and the accuracy of the model.Filtering and state estimation are important features of a dynamic positioning system, and estimates of the velocities must be computed from the noisy position and heading measurements through a state observer [25]. To solve the problem of high-precision measurement and positioning, the space measurement and positioning systems with relevant measurement methods have been developed rapidly. With the increasing of the network scale, the scope of the measurement and positioning systems is more and more large; meanwhile, the coverage mode is more and more flexible. Considering the complexity of the communication links, the energy and computing power of each node are limited. The configuration structure uses the strategy of all nodes, shares, coordinates, and communicates with the signal processing workloads. Accordingly, the network performance does not increase with the size of the network attenuation. Therefore, the use of the distributed network structure can have the ability to increase the reliability of information and improve the precision of collaborative calculation. Moreover, the distributed network structure reduces the burden of communication [26].The distributed estimation strategy is applied to target tracking and positioning, fault diagnosis, and so on. The objective is that state estimation accuracy after fusion is higher than that of each local estimation. Research on filtering is the basis of the information perception and fusion; according to different performance indicators, relevant distributed filtering has aroused more and more attention, including distributed particle filtering [27], distributed H∞ filtering, and distributed Kalman filtering. Furthermore, distributed Kalman filtering is used for realizing the coordination and fusion of error estimation, which is an important scheme to solve the state estimation for large-scale systems [28]. A data-driven approach combining the RBF neural network (NN) and the extended Kalman filter (EKF) was proposed to estimate the internal temperature for lithium-ion battery thermal management [29, 30].Due to the restriction of communication facilities and scale for networked systems, the measurement information is inevitably affected by the unpredictable interference. Note that the fundamental Kalman filtering method cannot satisfy the performance requirement such that the robust Kalman filtering method is designed considering various noise sequences. Costa and Benites [31] investigated robust pattern independent filtering for discrete-time Markov jump systems with state-correlated noises. Considering the correlation for noise, Feng et al. and Yan et al. [32, 33] presented the distributed Kalman filtering fusion method with cross-correlation between measurement noise and process noise. In addition, for a class of noisy sequences with uncertain variance, Keshavarz-Mohammadiyan and Khaloozadeh [34] used the method of solving the least upper bound of all admissible uncertainties for its true filtering error covariance matrix. The noise variance is set to be suitable in the current system state during the iterative process, and the estimation precision is improved. To deal with multiple time-delay systems and reduce the computational complexity, the received information may be asynchronous in the presence of transmission time delays during network communication. The method of measurement transformation developed the Kalman filter using the recombination of the innovation sequence. Due to restructuring the measurement sequence, the system with time delay is transformed into the correlative one with free time-delay [35].In the engineering of the ship navigation applications, due to the complex issue of transmission over networks and external disturbances, the dynamic spatial positioning is difficult with the requirements of high-precision positioning and growing measurement errors. Motivated by the above analysis, based on the principle of spatial positioning of linear charge-coupled devices (CCD), the information measured by the three-dimensional (3D) photoelectric sensor is considered to be influenced by the uncertain factors, such as transmission time delays and cross-correlation noise, and the accuracy of state estimation is improved. In order to improve the robustness for the state estimation, the robust finite-horizon filtering is transformed into Kalman prediction to reduce the computational complexity using the method of reorganizing and transforming the measurement sequence as well as the innovation sequence. The proposed weighted fusion approach of filtering error covariance is used for coordinating and exchanging the measurement information between two subsystems. Therefore, based on the optimal information fusion criterion, the distributed weighted fusion estimation using an adaptive selection scheme is the reoptimization of the local estimation, which is able to obtain higher estimation precision than the local one.The remainder of the paper is organized as follows. The dynamic spatial positioning system is modelled in Section2. Section 3 presents the distributed weighted fusion method using an adaptive signal selection scheme. Simulation results and analysis for the dynamic spatial positioning system are presented in Section 4, and the concluding remarks are given in Section 5.


## 2. Dynamic Positioning System Model

The dynamic positioning system is designed by the 3D photoelectric sensor, and the principle of spatial positioning is shown in this section.


### 2.1. Principle of the 3D Photoelectric Sensor

Firstly, the Metris Krypton measurement approach based on the three-coordinate position measurement is developed. The 3D spatial position of each measuring point is obtained using the optical tracker and the special light emission device (LED). The measured target is made of a special LED, and its 3D coordinates are read from three sets of charge-coupled devices (CCD) on the optical tracker. Special LED placement can be moved arbitrarily, and it can be placed on any moving parts. And then, the processor is used for calculating the position change and direction from the LED at a certain time interval. As shown in Figure2(a), the 3D photoelectric sensor is composed of three linear CCDs, which are embedded three fixed-focus cylindrical optical mirrors. The three linear CCDs have a “Y” shape distribution on the same plane.Figure 2
3D photoelectric sensor. (a) Dynamic measurement device. (b) Spatial positioning principle.
(a)(b)The positioning principle of the 3D photoelectric sensor is shown in Figure2(b). The measured target is composed of the special LED. And then, each cylindrical optical mirror through light only projects to its corresponding plane under the CCD. A spatial point light source is projected into each CCD plane by a cylindrical optical mirror, which is straight line light intersecting with the CCD and is able to perceive its position information on the CCD. If the targets are in the measurement range of the 3D photoelectric sensor, it will be decomposed into three planes through the three CCD lenses. Therefore, the processor is used to compute the intersection point for the three planes, that is, the coordinates of the light source for the space points (X, Y, Z).


### 2.2. System Model Based on the Adaptive Signal Selection Scheme

Furthermore, considering the uncertainty of the distributed systems, the system model is established from the system state and measurement information, which is obtained by the sensor during the transmission process with uncertain disturbances and random transmission time delays.More points in the space are measured by optical spatial positioning systems. It is necessary to use the fusion data from the coordination for multisensors. For the tracking system of the space motion target, the distributed configuration scheme is established, and the optimal estimation is designed to obtain more accurate measurement information. The considered system model for each sensor is described by the following linear discrete-time system [36, 37]:(1)xk+1=Ak+ℱkFkEkxk+Bkwk,k=1,2,…,(2)zik=Cki+ℋkiFkEkixk+vki,i=1,…,L.Note thatxk∈ℝr represents the system state, and zik∈ℝmi expresses the measurement for each sensor at time k. Ak∈ℝr×r, Bk∈ℝr, Cki∈ℝmi×r, ℱk, Ek, ℋki, and Eki are known time-varying matrices, and Fk are the time-varying parameter uncertainties. wk∈ℝ and vki∈ℝmi denote the process noise and measurement noise of the i-th sensor, which are zero-mean white noise with covariance matrices Qk and Rki, respectively. The initial state x0 with mean μ0 and covariance P0 is assumed to be uncorrelated with other noise signals. Note that the uncertainties Fk satisfy FkFkT≤I.Due to the limited bandwidth, the transmission from sensors to processors inevitably generates the phenomenon of network congestion. To actively drop the disordered packets from the random transmission time delays, an adaptive signal selection scheme is designed (ASS).Assumed that the measurement is sampled with a constant sampling periodT, the sampling time instant is represented as t∈kT,k∈ℕ for any subsystem. Set the maximum of the transmission delay as N, and it does not exceed the current time instant, i.e., τk≤k. To further describe the networked system with random transmission time delays and lost and disordered packets, Figure 3 illustrates a typical scenario.Figure 3
Process of packet sequence resorting using the ASS.The current time instantk receives a signal with timestamp k1, and timestamp k2 is expressed as the most recent signal before transmitting. From the sensor to the processor, the corresponding transmission time delays are recorded as τk1∈ℕ and τk2∈ℕ, which denote the transmission time delays from the received packages using the ASS scheme or not at the sampling time k, which satisfies the following relationships:(3)k=τk1+k1=τk2+k2.Since the latest data packet during transmission is closer to the current data, a variable of delayβk≥0 is developed to express the relationship between k1 and k2, that is,(4)k1=k2+βk.From equations (3) and (4), τk1=τk2−βk and τk1≤τk2 are satisfied.When receiving a valid data packet with timestamp using the ASS, in order to reduce the computational burden, the stored signal is reorganized as(5)yASSk=zk−τk1.Note that the data packetyASSk is composed of the timestamp and transmission delay at each sampling time [38].


### 2.3. Uncertain Disturbances

To simplify the modeling of the wave and current for the UUV, considering unknown model parameters and uncertain time-varying disturbances in engineering applications, it is assumed that the bias model and the wave model are driven by zero-mean Gaussian noise. The offset of the vessel in surge, sway, and yaw is small as compared to the size of the ship. Suppose that the distributed systems involve the cross-correlation noise during networked transmission [36, 37, 39]. Meanwhile, the process noise wk and measurement noise vki for each subsystem are interrelated at the same instant. Similarly, the measurement noise vki for the i-th subsystem and the measurement noise vkj for the j-th subsystem are cross-related. The statistical attributes satisfy the following relationships:(6)Ewk=0,Evki=0,EwkvkiwlTvkjT=Qkδk,lSkjδk,lSkiTδk,lRki,jδk,l,where the covariance satisfies Qk=QkT, Rki,j=Rkj,iT, and Ski,j=Skj,iT. Note that δk,l is the Kronecker function.Since the stored measurement sequence contains random transmission time delays and lost and disordered packets, the distributed estimation problem is transformed into solving the optimal state estimationx^ASSkk using the ASS scheme, which fuses and compensates each local estimation x^ASSitt with t=k−τk1.


## 2.1. Principle of the 3D Photoelectric Sensor

Firstly, the Metris Krypton measurement approach based on the three-coordinate position measurement is developed. The 3D spatial position of each measuring point is obtained using the optical tracker and the special light emission device (LED). The measured target is made of a special LED, and its 3D coordinates are read from three sets of charge-coupled devices (CCD) on the optical tracker. Special LED placement can be moved arbitrarily, and it can be placed on any moving parts. And then, the processor is used for calculating the position change and direction from the LED at a certain time interval. As shown in Figure2(a), the 3D photoelectric sensor is composed of three linear CCDs, which are embedded three fixed-focus cylindrical optical mirrors. The three linear CCDs have a “Y” shape distribution on the same plane.Figure 2
3D photoelectric sensor. (a) Dynamic measurement device. (b) Spatial positioning principle.
(a)(b)The positioning principle of the 3D photoelectric sensor is shown in Figure2(b). The measured target is composed of the special LED. And then, each cylindrical optical mirror through light only projects to its corresponding plane under the CCD. A spatial point light source is projected into each CCD plane by a cylindrical optical mirror, which is straight line light intersecting with the CCD and is able to perceive its position information on the CCD. If the targets are in the measurement range of the 3D photoelectric sensor, it will be decomposed into three planes through the three CCD lenses. Therefore, the processor is used to compute the intersection point for the three planes, that is, the coordinates of the light source for the space points (X, Y, Z).


## 2.2. System Model Based on the Adaptive Signal Selection Scheme

Furthermore, considering the uncertainty of the distributed systems, the system model is established from the system state and measurement information, which is obtained by the sensor during the transmission process with uncertain disturbances and random transmission time delays.More points in the space are measured by optical spatial positioning systems. It is necessary to use the fusion data from the coordination for multisensors. For the tracking system of the space motion target, the distributed configuration scheme is established, and the optimal estimation is designed to obtain more accurate measurement information. The considered system model for each sensor is described by the following linear discrete-time system [36, 37]:(1)xk+1=Ak+ℱkFkEkxk+Bkwk,k=1,2,…,(2)zik=Cki+ℋkiFkEkixk+vki,i=1,…,L.Note thatxk∈ℝr represents the system state, and zik∈ℝmi expresses the measurement for each sensor at time k. Ak∈ℝr×r, Bk∈ℝr, Cki∈ℝmi×r, ℱk, Ek, ℋki, and Eki are known time-varying matrices, and Fk are the time-varying parameter uncertainties. wk∈ℝ and vki∈ℝmi denote the process noise and measurement noise of the i-th sensor, which are zero-mean white noise with covariance matrices Qk and Rki, respectively. The initial state x0 with mean μ0 and covariance P0 is assumed to be uncorrelated with other noise signals. Note that the uncertainties Fk satisfy FkFkT≤I.Due to the limited bandwidth, the transmission from sensors to processors inevitably generates the phenomenon of network congestion. To actively drop the disordered packets from the random transmission time delays, an adaptive signal selection scheme is designed (ASS).Assumed that the measurement is sampled with a constant sampling periodT, the sampling time instant is represented as t∈kT,k∈ℕ for any subsystem. Set the maximum of the transmission delay as N, and it does not exceed the current time instant, i.e., τk≤k. To further describe the networked system with random transmission time delays and lost and disordered packets, Figure 3 illustrates a typical scenario.Figure 3
Process of packet sequence resorting using the ASS.The current time instantk receives a signal with timestamp k1, and timestamp k2 is expressed as the most recent signal before transmitting. From the sensor to the processor, the corresponding transmission time delays are recorded as τk1∈ℕ and τk2∈ℕ, which denote the transmission time delays from the received packages using the ASS scheme or not at the sampling time k, which satisfies the following relationships:(3)k=τk1+k1=τk2+k2.Since the latest data packet during transmission is closer to the current data, a variable of delayβk≥0 is developed to express the relationship between k1 and k2, that is,(4)k1=k2+βk.From equations (3) and (4), τk1=τk2−βk and τk1≤τk2 are satisfied.When receiving a valid data packet with timestamp using the ASS, in order to reduce the computational burden, the stored signal is reorganized as(5)yASSk=zk−τk1.Note that the data packetyASSk is composed of the timestamp and transmission delay at each sampling time [38].


## 2.3. Uncertain Disturbances

To simplify the modeling of the wave and current for the UUV, considering unknown model parameters and uncertain time-varying disturbances in engineering applications, it is assumed that the bias model and the wave model are driven by zero-mean Gaussian noise. The offset of the vessel in surge, sway, and yaw is small as compared to the size of the ship. Suppose that the distributed systems involve the cross-correlation noise during networked transmission [36, 37, 39]. Meanwhile, the process noise wk and measurement noise vki for each subsystem are interrelated at the same instant. Similarly, the measurement noise vki for the i-th subsystem and the measurement noise vkj for the j-th subsystem are cross-related. The statistical attributes satisfy the following relationships:(6)Ewk=0,Evki=0,EwkvkiwlTvkjT=Qkδk,lSkjδk,lSkiTδk,lRki,jδk,l,where the covariance satisfies Qk=QkT, Rki,j=Rkj,iT, and Ski,j=Skj,iT. Note that δk,l is the Kronecker function.Since the stored measurement sequence contains random transmission time delays and lost and disordered packets, the distributed estimation problem is transformed into solving the optimal state estimationx^ASSkk using the ASS scheme, which fuses and compensates each local estimation x^ASSitt with t=k−τk1.


## 3. ASS-Based Estimation

For the considered distributed uncertain system in equations (1) and (2), a distributed state estimation approach is studied in this section based on robust finite-horizon filtering.


### 3.1. Reorganized Filter and Prediction Vector of the State

First of all, assume that the stored packet isyASSik for the i-th subsystem at the current sampling time k. Depending on the role of the ASS, the received valid packet is zik−τk1 with delay τk1 before transmission. From equations (2) and (5), the reorganized measurement is denoted as(7)yASSik=zit=Cti+ℋtiFtEtixt+vti.Based on the stored packet sequenceyASSi0,…,yASSik−1,yASSik, the optimal state estimation x^ASSkk with random transmission time delays can be obtained by the following local recursive method:(8)x^ASSitt=x^ASSitt−1+KASS,tizit−C^ASS,tix^ASSitt−1,(9)x^ASSit+1t=A^ASS,tix^ASSitt−1+LASS,tizit−C^ASS,tix^ASSitt−1,in which x^ASSitt and x^ASSit+1t represent the filter and the predictor, respectively, at the time instant t. Meanwhile, C^ASS,ti, KASS,ti, A^ASS,ti, and LASS,ti are filtering parameters.In order to solve the upper boundary for the filter and prediction covariance matrix, the augmented vectors of the state are defined as follows:(10)ΨASSit=eASSitx^ASSitt,Ψ˜ASSit+1=e˜ASSit+1x^ASSit+1t,where eASSit≜xt−x^ASSitt and e˜ASSit+1≜xt+1−x^ASSit+1t. In addition, combining with the upper boundary, the augmented system modelled from equations (1) and (8)–(10) is rewritten as(11)ΨASSit=AASS,t1i+HASS,t1iFtEASS,t1iΨ˜ASSit+DASS,t1ivti,and(12)Ψ˜ASSit+1=AASS,t2i+HASS,t2iFtEASS,t2iΨ˜ASSit+BASS,t2wt+DASS,t2ivti.The symbols are defined as(13)AASS,t1i=I−KASS,tiCtiKASS,tiC^ASS,ti−CtiKASS,tiCtiI+KASS,tiCti−C^ASS,ti⋅,HASS,t1i=−KASS,tiℋtiKASS,tiℋti,EASS,t1i=EASS,t2i=EtiEti,DASS,t1i=−KASS,tiKASS,ti,HASS,t2i=ℱti−LASS,tiℋtiLASS,tiℋti,AASS,t2i=At−LASS,tiCtiAt−A^ASS,ti+LASS,tiC^ASS,ti−CtiLASS,tiCtiA^ASS,ti+LASS,tiCti−C^ASS,ti,BASS,t2=Bt0,DASS,t2i=−LASS,tiLASS,ti.Next, the covariance matrices are defined asΘ˜ASSit=EΨASSitΨASSitT and Σ˜ASSit+1≜EΨ˜ASSit+1Ψ˜ASSit+1T in equations (11) and (12). At the same time, the estimated error covariance matrices are derived from equations (11)–(15), and they are as follows:(14)Θ˜ASSit=AASS,t1i+HASS,t1iFtEASS,t1iΣ˜ASSitAASS,t1i+HASS,t1iFtEASS,t1iT+DASS,t1iRtiDASS,t1iT,(15)Σ˜ASSit+1=AASS,t2i+HASS,t2iFtEASS,t2iΣ˜ASSit×AASS,t2i+HASS,t2iFtEASS,t2iT+BASS,t2QtBASS,t2T+DASS,t2iRtiDASS,t2iT+BASS,t2StiDASS,t2iT+DASS,t2iStiTBASS,t2T.Finally, to find the upper boundary for the error covariance matrices, the filtering covariance is defined from the given equation (10):(16)Σ˜ASSit=Σ¯ASSit00Pt−Σ¯ASSit.Note that the corresponding definitions of covariance matrices areΣ¯ASSit=Ee˜ASSite˜ASSitT and Pt=ExtxTt.


### 3.2. Upper Boundary for Error Covariance

Based on the augmented vector of the state, the corresponding filter and prediction error covariance matrices are designed by Theorem1 to detect the appropriate filtering parameters.Theorem 1.
For the unified form ofA+HFEXA+HFET, if there is a positive scale αt>0 as well as a symmetrical positive matrix ΣASSit, they satisfy αt−1I−EASS,t2iΣASSitEASS,t2iT>0, and the inequality is established, i.e., Θ˜ASSit≤ΘASSit and Σ˜ASSit+1≤ΣASSit+1. Therefore, the upper boundaries for the error covariance matrices can be calculated by the following recursive formulas:(17)ΘASSit=AASS,t1iΣASSitAASS,t1iT+αt−1HASS,t1iHASS,t1iT+DASS,t1iRtiDASS,t1iT+AASS,t1iΣASSitEASS,t1iT×αt−1I−EASS,t1iΣASSitEASS,t1iT−1EASS,t1iΣASSitAASS,t1iTΣASSit+1=AASS,t2iΣASSitAASS,t2iT+αt−1HASS,t2iHASS,t2iT+AASS,t2iΣASSitEASS,t2iTαt−1I−EASS,t2iΣASSitEASS,t2iT−1×EASS,t2iΣASSitAASS,t2iT+BASS,t2iQtiBASS,t2iT+DASS,t2iRtiDASS,t2iT+BASS,t2StiDASS,t2iT+DASS,t2iStiTBASS,t2T.Proof.
The proof is similar to the derivation process in [13, 38].
And then, to probe the appropriate filtering parameters, the upper boundaries of the error covariance matrices are derived separately:(18)EeASSiteASSitT=I0Θ˜ASSitI0≤I0ΘASSitI0=Θ¯ASSit,Ee˜ASSit+1e˜ASSit+1T=I0Σ˜ASSit+1I0≤I0ΣASSit+1I0=Σ¯ASSit+1.
Then, the optimal estimation based on the robust finite-horizon filtering in equations (8) and (9) is derived by the following Theorem 2.Theorem 2.
Suppose that the received valid measurement isyASSik with τk1-step transmission time delays at the current time k, and variable αt is a positive scalar. Matrices Pt and Σ¯ASSit are positive solutions, and they are derived from the following iteration formulas:(19)Θ¯ASSit=Σ¯ASSit+Σ¯ASSitEtiTM˜ASS,ti−1EtiΣ¯ASSit−ΛASSitΞASSit−1ΛASSitT,(20)Σ¯ASSit+1=AtΣ¯ASSitI+EtiTMASS,ti−1EtiΣ¯ASSitAtT−ΔASSitΞASSit−1ΔASSitT+BtQtBtT+αt−1ℱtℱtT,(21)Pt+1=AtP−1t−αtEtiTEti−1AtT+αt−1ℱtℱtT+BtQtBtT,in which ΛASSit=Σ¯ASSitCtiT+Σ¯ASSitEtiTMASS,ti−1EtiΣ¯ASSitCtiT, and ΔASSit=AtΣ¯ASSitI+EtiTMASS,ti−1EtiΣ¯ASSitCtiT+αt−1ℱtℋtiT+BtSti. Note that the inequalities P−1t−αtEtiTEti>0 and MASS,ti=αt−1I−EtiΣ¯ASSitEtiT>0 are satisfied.
Therefore, the results of the filtering parameters are solved as follows:(22)C^ASS,ti=CtiI+Σ¯ASSitEtiTMASS,ti−1Eti,KASS,ti=ΛASSitΞASSit−1,A^ASS,ti=AtI+Σ¯ASSitEtiTMASS,ti−1Eti,LASS,ti=ΔASSitΞASSit−1.
From equation (22), we define the following symbols as ΞASSit=CtiΣ¯ASSitI+EtiTMASS,ti−1EtiΣ¯ASSitCtiT+αt−1ℋtiℋtiT+Rti and M˜ASS,ti=αt−1I−EtiPtEtiT.Proof.
The upper boundary evolution is obtained by minimizing the estimation error covariance matrix, which is similar to solving the filtering parameters in [12, 13].


### 3.3. State Estimation Based on Weighted Fusion Criteria

For each subsystem, taking into account the network-induced random transmission time delays and lost and disordered packets, the current estimated state is calculated by Theorem2. In order to further improve the accuracy of state estimation, distributed weighted fusion estimation is studied in this section. Based on the weighted fusion criteria, the minimized estimation error cross-covariance matrix is used to perform the information exchange between any two subsystems.Theorem 3.
For a class of uncertain discrete time-varying systems, at the current sampling timek, the upper boundary for the filter error cross-covariance matrix Θ¯ASSi,jt between the i-th subsystem and the j-th subsystem as well as the prediction error cross-covariance matrix Σ¯ASSi,jt+1 are expressed as follows:(23)Θ¯ASSi,jt=I0ΘASSi,jtI0=I−KASS,tiCtiΣ¯ASSi,jtI−KASS,tjCtjT+KASS,tiC^ASS,ti−CtiPt−Σ¯ASSi,jtKASS,tjC^ASS,tj−CtjT+Σ¯ASSi,jt+KASS,tiC^ASS,tiPt−Σ¯ASSi,jt−CtiPtEtiTM˜ASS,ti,j−1Etj×Σ¯ASSi,jt+KASS,tjC^ASS,tjPt−Σ¯ASSi,jt−CtjPtT+αt−1KASS,tiℋtiℋtjTKASS,tjT+KASS,tiRti,jKASS,tjT,(24)Σ¯ASSi,jt+1=I0ΣASSi,jt+1I0=At−LASS,tiCtiΣ¯ASSi,jtAt−LASS,tjCtjT+BtQtBtT+LASS,tiRti,jLASS,tjT+At−A^ASS,ti+LASS,tiC^ASS,ti−CtiPt−Σ¯ASSi,jt×At−A^ASS,tj+LASS,tjC^ASS,tj−CtjT+At−LASS,tiCtiΣ¯ASSi,jt+At−A^ASS,ti+LASS,tiC^ASS,ti−Cti×Pt−Σ¯ASSi,jtEtiTM˜ASS,ti,j−1EtjΣ¯ASSi,jtAt−LASS,tjCtjT+Pt−Σ¯LZi,jtAt−A^LZ,tj+LLZ,tjC^LZ,tj−CtjT+αt−1ℱt−LLZ,tiℋtiℱt−LLZ,tjℋtjT−BtStjLLZ,tjT−LLZ,tiStiTBtT,in which M˜ASS,ti,j=αt−1I−EtiPtEtjT and MASS,ti,j=αt−1I−EtiΣ¯ASSi,jtEtjT.Proof.
Its derivation process is similar to the solution of Theorem2.
The optimal weighted fusion estimation can be used to solve the trace of the estimation error by minimizing the cross-covariance matrix [33, 40]. Set x^ASSikt as a local estimation with the r-dimensional random vector xk, which contains the network-induced factor. The distributed fusion estimator x^ASSkk is designed as a r-dimensional vector:(25)x^ASSkk=Ωk1x^ASS1kk+⋯+ΩkLx^ASSLkk.
It is worth noting that solvingx^ASSikk from x^ASSikt develops the one-step prediction method to guarantee the precision of the estimation [35, 41].
Thus, the distributed weighted fusion estimation performance is optimal only if the weighted matrix given by equation (25) meets the following criteria [37, 41], i.e.,(26)Ωk1,…,ΩkL=I0TΠASS−1kI0−1I0TΠASS−1k,in which I0=Ir,…,Ir︸LT is a rL×r-dimensional matrix, and ∑i=1LΩki=Ir. In equation (26), ΠASSk is the symmetrical positive matrix, which is composed of ΠASSi,jk=EeASSiteASSjtT,i=1,…,L;j=1,…,L. Meanwhile, the corresponding optimal information fusion estimation of the cross-covariance is calculated by(27)Π˜kk=Exk−x^kkxk−x^kkT=I0TΠASS−1kI0−1.
Note that wheni=j, Π˜kk≤ΠASSik≤Π¯ASSik is satisfied for the i-th subsystem.
So far, the solution process of the distributed weighted fusion estimation method based on the robust finite-horizon filtering using the ASS scheme is shown in Algorithm1.Algorithm 1: Distributed weighted fusion estimation using the ASS scheme.
Input:The initial statex0, measurement z10,…,zL0, estimated state x^00, and error covariance P00.Output:The local state estimationx^ASS1kk,…,x^ASSLkk and the fused state estimation x^kk.(1) Perceive the measurementsz1k,…,zLk from sensor data packets with timestamp at the current time instant k, which are transmitted over the networked systems.(2) Judge the received valid signals using the ASS scheme. The reorganized measurement is denoted asyit for the i-th subsystem.(3) Probe the filtering parametersC^ASS,ti, KASS,ti, A^ASS,ti, and LASS,ti according to Theorem 2.(4) Solve the filterx^ASSitt and predictor x^ASSit+1t in equations (8) and (9).(5) Obtain the weighted fusion estimationx^tt from equation (25).(6) Updatek=k+1 at the next time instant. Return to Step 2.(7)returnx^ASSitt and x^ASSit+1t in equations (8) and (9), respectively. Θ¯ASSit, Σ¯ASSit+1, and Pt+1 from equations (19)–(21).Remark 1.
The distributed uncertain systems propose the ASS scheme to deal with the network-induced random transmission time delays and lost and disordered packet phenomena. As mentioned before, for the random transmission delay, the one-step prediction method is used to compensate the missing data packets. On the contrary, in order to realize the information exchange, the distributed state estimation based on the weighted fusion criteria is studied using the linear minimum covariance matrix to obtain the consistent expression of the system state. As a theoretical basis, the fusion scheme has higher estimation accuracy than each local subsystem.


## 3.1. Reorganized Filter and Prediction Vector of the State

First of all, assume that the stored packet isyASSik for the i-th subsystem at the current sampling time k. Depending on the role of the ASS, the received valid packet is zik−τk1 with delay τk1 before transmission. From equations (2) and (5), the reorganized measurement is denoted as(7)yASSik=zit=Cti+ℋtiFtEtixt+vti.Based on the stored packet sequenceyASSi0,…,yASSik−1,yASSik, the optimal state estimation x^ASSkk with random transmission time delays can be obtained by the following local recursive method:(8)x^ASSitt=x^ASSitt−1+KASS,tizit−C^ASS,tix^ASSitt−1,(9)x^ASSit+1t=A^ASS,tix^ASSitt−1+LASS,tizit−C^ASS,tix^ASSitt−1,in which x^ASSitt and x^ASSit+1t represent the filter and the predictor, respectively, at the time instant t. Meanwhile, C^ASS,ti, KASS,ti, A^ASS,ti, and LASS,ti are filtering parameters.In order to solve the upper boundary for the filter and prediction covariance matrix, the augmented vectors of the state are defined as follows:(10)ΨASSit=eASSitx^ASSitt,Ψ˜ASSit+1=e˜ASSit+1x^ASSit+1t,where eASSit≜xt−x^ASSitt and e˜ASSit+1≜xt+1−x^ASSit+1t. In addition, combining with the upper boundary, the augmented system modelled from equations (1) and (8)–(10) is rewritten as(11)ΨASSit=AASS,t1i+HASS,t1iFtEASS,t1iΨ˜ASSit+DASS,t1ivti,and(12)Ψ˜ASSit+1=AASS,t2i+HASS,t2iFtEASS,t2iΨ˜ASSit+BASS,t2wt+DASS,t2ivti.The symbols are defined as(13)AASS,t1i=I−KASS,tiCtiKASS,tiC^ASS,ti−CtiKASS,tiCtiI+KASS,tiCti−C^ASS,ti⋅,HASS,t1i=−KASS,tiℋtiKASS,tiℋti,EASS,t1i=EASS,t2i=EtiEti,DASS,t1i=−KASS,tiKASS,ti,HASS,t2i=ℱti−LASS,tiℋtiLASS,tiℋti,AASS,t2i=At−LASS,tiCtiAt−A^ASS,ti+LASS,tiC^ASS,ti−CtiLASS,tiCtiA^ASS,ti+LASS,tiCti−C^ASS,ti,BASS,t2=Bt0,DASS,t2i=−LASS,tiLASS,ti.Next, the covariance matrices are defined asΘ˜ASSit=EΨASSitΨASSitT and Σ˜ASSit+1≜EΨ˜ASSit+1Ψ˜ASSit+1T in equations (11) and (12). At the same time, the estimated error covariance matrices are derived from equations (11)–(15), and they are as follows:(14)Θ˜ASSit=AASS,t1i+HASS,t1iFtEASS,t1iΣ˜ASSitAASS,t1i+HASS,t1iFtEASS,t1iT+DASS,t1iRtiDASS,t1iT,(15)Σ˜ASSit+1=AASS,t2i+HASS,t2iFtEASS,t2iΣ˜ASSit×AASS,t2i+HASS,t2iFtEASS,t2iT+BASS,t2QtBASS,t2T+DASS,t2iRtiDASS,t2iT+BASS,t2StiDASS,t2iT+DASS,t2iStiTBASS,t2T.Finally, to find the upper boundary for the error covariance matrices, the filtering covariance is defined from the given equation (10):(16)Σ˜ASSit=Σ¯ASSit00Pt−Σ¯ASSit.Note that the corresponding definitions of covariance matrices areΣ¯ASSit=Ee˜ASSite˜ASSitT and Pt=ExtxTt.


## 3.2. Upper Boundary for Error Covariance

Based on the augmented vector of the state, the corresponding filter and prediction error covariance matrices are designed by Theorem1 to detect the appropriate filtering parameters.Theorem 1.
For the unified form ofA+HFEXA+HFET, if there is a positive scale αt>0 as well as a symmetrical positive matrix ΣASSit, they satisfy αt−1I−EASS,t2iΣASSitEASS,t2iT>0, and the inequality is established, i.e., Θ˜ASSit≤ΘASSit and Σ˜ASSit+1≤ΣASSit+1. Therefore, the upper boundaries for the error covariance matrices can be calculated by the following recursive formulas:(17)ΘASSit=AASS,t1iΣASSitAASS,t1iT+αt−1HASS,t1iHASS,t1iT+DASS,t1iRtiDASS,t1iT+AASS,t1iΣASSitEASS,t1iT×αt−1I−EASS,t1iΣASSitEASS,t1iT−1EASS,t1iΣASSitAASS,t1iTΣASSit+1=AASS,t2iΣASSitAASS,t2iT+αt−1HASS,t2iHASS,t2iT+AASS,t2iΣASSitEASS,t2iTαt−1I−EASS,t2iΣASSitEASS,t2iT−1×EASS,t2iΣASSitAASS,t2iT+BASS,t2iQtiBASS,t2iT+DASS,t2iRtiDASS,t2iT+BASS,t2StiDASS,t2iT+DASS,t2iStiTBASS,t2T.Proof.
The proof is similar to the derivation process in [13, 38].
And then, to probe the appropriate filtering parameters, the upper boundaries of the error covariance matrices are derived separately:(18)EeASSiteASSitT=I0Θ˜ASSitI0≤I0ΘASSitI0=Θ¯ASSit,Ee˜ASSit+1e˜ASSit+1T=I0Σ˜ASSit+1I0≤I0ΣASSit+1I0=Σ¯ASSit+1.
Then, the optimal estimation based on the robust finite-horizon filtering in equations (8) and (9) is derived by the following Theorem 2.Theorem 2.
Suppose that the received valid measurement isyASSik with τk1-step transmission time delays at the current time k, and variable αt is a positive scalar. Matrices Pt and Σ¯ASSit are positive solutions, and they are derived from the following iteration formulas:(19)Θ¯ASSit=Σ¯ASSit+Σ¯ASSitEtiTM˜ASS,ti−1EtiΣ¯ASSit−ΛASSitΞASSit−1ΛASSitT,(20)Σ¯ASSit+1=AtΣ¯ASSitI+EtiTMASS,ti−1EtiΣ¯ASSitAtT−ΔASSitΞASSit−1ΔASSitT+BtQtBtT+αt−1ℱtℱtT,(21)Pt+1=AtP−1t−αtEtiTEti−1AtT+αt−1ℱtℱtT+BtQtBtT,in which ΛASSit=Σ¯ASSitCtiT+Σ¯ASSitEtiTMASS,ti−1EtiΣ¯ASSitCtiT, and ΔASSit=AtΣ¯ASSitI+EtiTMASS,ti−1EtiΣ¯ASSitCtiT+αt−1ℱtℋtiT+BtSti. Note that the inequalities P−1t−αtEtiTEti>0 and MASS,ti=αt−1I−EtiΣ¯ASSitEtiT>0 are satisfied.
Therefore, the results of the filtering parameters are solved as follows:(22)C^ASS,ti=CtiI+Σ¯ASSitEtiTMASS,ti−1Eti,KASS,ti=ΛASSitΞASSit−1,A^ASS,ti=AtI+Σ¯ASSitEtiTMASS,ti−1Eti,LASS,ti=ΔASSitΞASSit−1.
From equation (22), we define the following symbols as ΞASSit=CtiΣ¯ASSitI+EtiTMASS,ti−1EtiΣ¯ASSitCtiT+αt−1ℋtiℋtiT+Rti and M˜ASS,ti=αt−1I−EtiPtEtiT.Proof.
The upper boundary evolution is obtained by minimizing the estimation error covariance matrix, which is similar to solving the filtering parameters in [12, 13].


## 3.3. State Estimation Based on Weighted Fusion Criteria

For each subsystem, taking into account the network-induced random transmission time delays and lost and disordered packets, the current estimated state is calculated by Theorem2. In order to further improve the accuracy of state estimation, distributed weighted fusion estimation is studied in this section. Based on the weighted fusion criteria, the minimized estimation error cross-covariance matrix is used to perform the information exchange between any two subsystems.Theorem 3.
For a class of uncertain discrete time-varying systems, at the current sampling timek, the upper boundary for the filter error cross-covariance matrix Θ¯ASSi,jt between the i-th subsystem and the j-th subsystem as well as the prediction error cross-covariance matrix Σ¯ASSi,jt+1 are expressed as follows:(23)Θ¯ASSi,jt=I0ΘASSi,jtI0=I−KASS,tiCtiΣ¯ASSi,jtI−KASS,tjCtjT+KASS,tiC^ASS,ti−CtiPt−Σ¯ASSi,jtKASS,tjC^ASS,tj−CtjT+Σ¯ASSi,jt+KASS,tiC^ASS,tiPt−Σ¯ASSi,jt−CtiPtEtiTM˜ASS,ti,j−1Etj×Σ¯ASSi,jt+KASS,tjC^ASS,tjPt−Σ¯ASSi,jt−CtjPtT+αt−1KASS,tiℋtiℋtjTKASS,tjT+KASS,tiRti,jKASS,tjT,(24)Σ¯ASSi,jt+1=I0ΣASSi,jt+1I0=At−LASS,tiCtiΣ¯ASSi,jtAt−LASS,tjCtjT+BtQtBtT+LASS,tiRti,jLASS,tjT+At−A^ASS,ti+LASS,tiC^ASS,ti−CtiPt−Σ¯ASSi,jt×At−A^ASS,tj+LASS,tjC^ASS,tj−CtjT+At−LASS,tiCtiΣ¯ASSi,jt+At−A^ASS,ti+LASS,tiC^ASS,ti−Cti×Pt−Σ¯ASSi,jtEtiTM˜ASS,ti,j−1EtjΣ¯ASSi,jtAt−LASS,tjCtjT+Pt−Σ¯LZi,jtAt−A^LZ,tj+LLZ,tjC^LZ,tj−CtjT+αt−1ℱt−LLZ,tiℋtiℱt−LLZ,tjℋtjT−BtStjLLZ,tjT−LLZ,tiStiTBtT,in which M˜ASS,ti,j=αt−1I−EtiPtEtjT and MASS,ti,j=αt−1I−EtiΣ¯ASSi,jtEtjT.Proof.
Its derivation process is similar to the solution of Theorem2.
The optimal weighted fusion estimation can be used to solve the trace of the estimation error by minimizing the cross-covariance matrix [33, 40]. Set x^ASSikt as a local estimation with the r-dimensional random vector xk, which contains the network-induced factor. The distributed fusion estimator x^ASSkk is designed as a r-dimensional vector:(25)x^ASSkk=Ωk1x^ASS1kk+⋯+ΩkLx^ASSLkk.
It is worth noting that solvingx^ASSikk from x^ASSikt develops the one-step prediction method to guarantee the precision of the estimation [35, 41].
Thus, the distributed weighted fusion estimation performance is optimal only if the weighted matrix given by equation (25) meets the following criteria [37, 41], i.e.,(26)Ωk1,…,ΩkL=I0TΠASS−1kI0−1I0TΠASS−1k,in which I0=Ir,…,Ir︸LT is a rL×r-dimensional matrix, and ∑i=1LΩki=Ir. In equation (26), ΠASSk is the symmetrical positive matrix, which is composed of ΠASSi,jk=EeASSiteASSjtT,i=1,…,L;j=1,…,L. Meanwhile, the corresponding optimal information fusion estimation of the cross-covariance is calculated by(27)Π˜kk=Exk−x^kkxk−x^kkT=I0TΠASS−1kI0−1.
Note that wheni=j, Π˜kk≤ΠASSik≤Π¯ASSik is satisfied for the i-th subsystem.
So far, the solution process of the distributed weighted fusion estimation method based on the robust finite-horizon filtering using the ASS scheme is shown in Algorithm1.Algorithm 1: Distributed weighted fusion estimation using the ASS scheme.
Input:The initial statex0, measurement z10,…,zL0, estimated state x^00, and error covariance P00.Output:The local state estimationx^ASS1kk,…,x^ASSLkk and the fused state estimation x^kk.(1) Perceive the measurementsz1k,…,zLk from sensor data packets with timestamp at the current time instant k, which are transmitted over the networked systems.(2) Judge the received valid signals using the ASS scheme. The reorganized measurement is denoted asyit for the i-th subsystem.(3) Probe the filtering parametersC^ASS,ti, KASS,ti, A^ASS,ti, and LASS,ti according to Theorem 2.(4) Solve the filterx^ASSitt and predictor x^ASSit+1t in equations (8) and (9).(5) Obtain the weighted fusion estimationx^tt from equation (25).(6) Updatek=k+1 at the next time instant. Return to Step 2.(7)returnx^ASSitt and x^ASSit+1t in equations (8) and (9), respectively. Θ¯ASSit, Σ¯ASSit+1, and Pt+1 from equations (19)–(21).Remark 1.
The distributed uncertain systems propose the ASS scheme to deal with the network-induced random transmission time delays and lost and disordered packet phenomena. As mentioned before, for the random transmission delay, the one-step prediction method is used to compensate the missing data packets. On the contrary, in order to realize the information exchange, the distributed state estimation based on the weighted fusion criteria is studied using the linear minimum covariance matrix to obtain the consistent expression of the system state. As a theoretical basis, the fusion scheme has higher estimation accuracy than each local subsystem.


## 4. Numerical Simulations

### 4.1. Simulation of Spatial Positioning

The spatial position structure of the 3D photoelectric sensor is shown in Figure2(a). Set the length of the three equivalent linear CCDs as 8.8mm, which are embedded in the 3D photoelectric sensor. The focal length of the column mirror is f=20mm, and the distance from the inner end of each CCD to the center of the coordinates is set as 10mm. The simulation results are demonstrated as follows.


#### 4.1.1. Each Intersection of the Projection Line on the CCD

It is assumed that a spatial luminous pointP0x0,y0,z0 is used as the spatial positioning point, and the measurable range is 4000∗4000∗6000mm3 by the 3D photoelectric sensor. P0x0,y0,z0 passes through each focal point. i.e., f1,f2,f3. And then, the light is divergent from the spatial luminous point. Note that the light is dispersed into a straight line through a cylindrical prism based on the refraction of light principle. The three CCDs project onto the X-Y plane, and it is vertical between each straight line and the corresponding CCD. Since the straight line over the CCD point can be perceived, there are four points which are coplanar for each CCD. That is, the spatial point light, the CCD through the lens of the light point, the lens focus, and the projection point on the X-Y plane. As a result, the three CCDs are extended into three planes, where only the spatial point light is communal. Note that the spatial point light positioning information is the intersection of the three planes. The positioning principle is shown in Figure 2(b).


#### 4.1.2. Measurable Range on the Plane

The dynamic positioning system is verified using the following three cases:Case 1: according to the principle of the spatial positioning system, every point on thez=100mm plane, and the range of 81×81mm2, the values are obtained on the spatial plane, and the points perceived by the sensor are the spatial points in the three CCD measurable ranges. As shown in Figure 4, the red area represents the measurable point, and the cyan area represents the undetectable point.Case 2: in the determinable spaceX=−2000,2000,Y=−2000,2000,Z=6000, the measurable spatial points are represented by red dots on the plane of Z=6000mm as shown in Figure 5.Case 3: measurable range of multilayer planes: in the case of the determined spaceX=−2000,2000,Y=−2000,2000,Z=6000, the space plane is set from Z=100mm to Z=6000mm, defining the measurable range of each spatial point be 500mm as the additional steps. Figure 6(a) shows the measurable range for each spatial layer. Similarly, Figure 6(b) describes the results due to reducing the addition steps as 10mm, i.e., the measurable range from Z=90mm to Z=300mm on the X-Y plane. Figure 6 shows that the measurable range of spatial points is presented from point, line to triangle, until the polygon, which is dynamically changing.Similarly, it is obvious that the more precise the pace on the CCD is, the more perceptible the number of points could be sensed. Therefore, the spatial positioning system possesses a wider measurement range. From the current simulation results, the range of measurable points in each spatial layer is stillY-axis symmetry, and the distribution profile of each layer point is from one point to the triangle, until the polygon measurement interval. Note that the schematic diagram of the dynamic measurement device is symmetry on the Y-axis as shown in Figure 2(a). According to the principle of the 3D photoelectric sensor, the projection from the spatial point source is also symmetry on each CCD plane.Figure 4
Measurable range on thez=100mm plane.Figure 5
Measurable range on thez=6000mm plane.Figure 6
Measurable range of the sensor multilayer plane. (a)Z = {100, 600, …, 6000}. (b) Z = {90, 100, …, 300}.
(a)(b)


### 4.2. Experimental Results

The effectiveness of the proposed distributed weighted fusion estimation strategy is validated by numerical examples. The dynamic positioning system for the intermittent measurement is considered as a distributed uncertain system [32, 37–39]:(28)xk+1=0.9TT2/200.9T000.9+ℱkFkEkxk+T2/2T1wk,k=1,2,…,zik=Cki+ℋkiFkEkixk+vki,i=1,2,3,wk=ηk,vki=ζiwk,where the sampling period is set as T=0.1s. The maximum value of the transmission time delay is set as 5 sampling periods. The time-varying parameter uncertainty is satisfied as Fk=sin0.6k. System state xk=sks˙ks¨kT is used to represent the values of position, velocity, and acceleration for the target at kT. Set parameters as ℱk=0.10.10.1T, Ek=0.020.020.02, Ck1=0.60.81, Ck2=10.80.5, Ck3=0.310.7, and ℋk1=ℋk2=ℋk3=0.8. Define ηk∈ℝ as zero-mean white noise with the variance ση2=0.09. Considering the cross-correlation between the process and measurement noise, ζi in equation (28) is set as ζ1=2, ζ2=0.8, and ζ3=1, respectively. The covariance of the process noise in equation (28) is wk with the covariance Qk. Meanwhile, the covariance Rki,j is represented as Rki,j=ζkiζkjQk, and the mutual covariance is denoted as Ski=ζkiQk in equation (28).The initial values are set asx^00=μ0=Ex0=111T and P00=0.01I3. For the numerical examples, 300 sampling points are selected, and the experimental results are verified based on 300 Monte Carlo simulation calculations. Comparing the traces of the estimated error covariance matrix, the results are shown in Figure 7. The estimated state x^kk is validated by the improved robust finite-horizon Kalman filtering (IRFHKF) in [38], as well as the proposed distributed weighted fusion state estimation using an adaptive signal selection scheme (DWFASS). First of all, taking into account the transmission with delay-free from Figure 7(a), the upper boundary of the estimation error covariance matrix is small compared with the IRFHKF method. It is worth noting that the upper boundary of the minimum value for the fused cross-covariance matrix is smaller than the upper boundary for the other covariance matrices, since the weighted fusion criterion is the optimized again in the sense of the minimal variance. Comparing the two methods shown in Figures 7(b)–7(d), the proposed method is more suitable for detecting the optimal filtering parameters. Therefore, the designed estimator is appropriate for the uncertainty in model parameters as well as external disturbances, such as random transmission time delays and lost and disordered packets.Figure 7
Comparing DWFASS and IRFHKF methods: (a) trace for the upper boundary, (b) comparison of the estimated position state, (c) comparison of the estimated velocity state, (d) comparison of the estimated acceleration state, and (e) comparison of MAEs for the position.
(a)(b)(c)(d)(e)In addition, to further analyze the performance of the proposed DWFASS method, the maximum absolute errors (MAE) [42] are obtained by(29)MAE=xk−x^ASSikk.On the one hand, the root mean square error (RMSE) [43] between the measured and estimated values is defined as(30)RMSE=1M∑k=1Mxk−x^ASSikk2,where M represents the number of the measured points and xk and x^ASSikk represent the measured and estimated states at the current sampling points k, respectively.Figure7(e) shows the MAEs comparing DWFASS and IRFHKF methods; meanwhile, the RMSEs are demonstrated in Table 1. Considering the cross-correlation for the noise, the dynamic tracking trajectory is closer to the actual state value if the noise is more correlated.Table 1
Comparing RMSEs between DWFASS and IRFHKF.
MethodsPositionVelocityAccelerationSensor 10.04690.09060.3182Sensor 20.04020.06970.3120Sensor 30.04260.07650.3140IRFHKF0.06810.09800.3831Next, for the distributed networked systems, the processing effect of each subsystem is different, which is determined by the cross-correlation for the system parameters and uncertain disturbances. Note that if the process noise is more cross-correlated with the measurement noise, the estimation performance is more accurate.Moreover, to further illustrate the effectiveness of the DWFASS method, the distributed weighted fusion results for state estimation are shown in Figure8. The results of state estimation are calculated from Theorems 2 and 3, and the method of the robust finite-horizon filtering is studied using the ASS scheme, as well as the distributed weighted fusion estimation criteria.Figure 8
Comparison of distributed weighted fusion estimation results: (a) estimated position state, (b) estimated velocity state, (c) estimated acceleration state, and (d) comparison of MAEs for the position.
(a)(b)(c)(d)Figure8 shows that the ASS scheme is able to actively discard data packets owing to the random transmission time delays and lost and disordered packets. At the same time, Table 2 enumerates the RMSEs of each sensor. Therefore, the distributed weighted fusion estimation uses the one-step prediction compensation for better performance and computational efficiency in target tracking.Table 2
Comparing RMSEs of each sensor.
MethodsPositionVelocityAccelerationSensor 10.16580.27490.5745Sensor 20.15080.25530.5464Sensor 30.12730.21280.4909


### 4.3. Numerical Simulation Analysis

According to the simulation results, the dynamic tracking results are compared by the method of recombination error covariance. The comparative results show that the proposed estimation instrument has better estimation accuracy, especially in the case of communication constraints with multistep random delays and lost and disordered network induction packets. Because the true estimation error covariance is lower than the upper boundary of the probe, the proposed distributed fusion estimation method has better performance in the phenomenon of network induction, including the ability to closely track the state of the system.


## 4.1. Simulation of Spatial Positioning

The spatial position structure of the 3D photoelectric sensor is shown in Figure2(a). Set the length of the three equivalent linear CCDs as 8.8mm, which are embedded in the 3D photoelectric sensor. The focal length of the column mirror is f=20mm, and the distance from the inner end of each CCD to the center of the coordinates is set as 10mm. The simulation results are demonstrated as follows.


### 4.1.1. Each Intersection of the Projection Line on the CCD

It is assumed that a spatial luminous pointP0x0,y0,z0 is used as the spatial positioning point, and the measurable range is 4000∗4000∗6000mm3 by the 3D photoelectric sensor. P0x0,y0,z0 passes through each focal point. i.e., f1,f2,f3. And then, the light is divergent from the spatial luminous point. Note that the light is dispersed into a straight line through a cylindrical prism based on the refraction of light principle. The three CCDs project onto the X-Y plane, and it is vertical between each straight line and the corresponding CCD. Since the straight line over the CCD point can be perceived, there are four points which are coplanar for each CCD. That is, the spatial point light, the CCD through the lens of the light point, the lens focus, and the projection point on the X-Y plane. As a result, the three CCDs are extended into three planes, where only the spatial point light is communal. Note that the spatial point light positioning information is the intersection of the three planes. The positioning principle is shown in Figure 2(b).


### 4.1.2. Measurable Range on the Plane

The dynamic positioning system is verified using the following three cases:Case 1: according to the principle of the spatial positioning system, every point on thez=100mm plane, and the range of 81×81mm2, the values are obtained on the spatial plane, and the points perceived by the sensor are the spatial points in the three CCD measurable ranges. As shown in Figure 4, the red area represents the measurable point, and the cyan area represents the undetectable point.Case 2: in the determinable spaceX=−2000,2000,Y=−2000,2000,Z=6000, the measurable spatial points are represented by red dots on the plane of Z=6000mm as shown in Figure 5.Case 3: measurable range of multilayer planes: in the case of the determined spaceX=−2000,2000,Y=−2000,2000,Z=6000, the space plane is set from Z=100mm to Z=6000mm, defining the measurable range of each spatial point be 500mm as the additional steps. Figure 6(a) shows the measurable range for each spatial layer. Similarly, Figure 6(b) describes the results due to reducing the addition steps as 10mm, i.e., the measurable range from Z=90mm to Z=300mm on the X-Y plane. Figure 6 shows that the measurable range of spatial points is presented from point, line to triangle, until the polygon, which is dynamically changing.Similarly, it is obvious that the more precise the pace on the CCD is, the more perceptible the number of points could be sensed. Therefore, the spatial positioning system possesses a wider measurement range. From the current simulation results, the range of measurable points in each spatial layer is stillY-axis symmetry, and the distribution profile of each layer point is from one point to the triangle, until the polygon measurement interval. Note that the schematic diagram of the dynamic measurement device is symmetry on the Y-axis as shown in Figure 2(a). According to the principle of the 3D photoelectric sensor, the projection from the spatial point source is also symmetry on each CCD plane.Figure 4
Measurable range on thez=100mm plane.Figure 5
Measurable range on thez=6000mm plane.Figure 6
Measurable range of the sensor multilayer plane. (a)Z = {100, 600, …, 6000}. (b) Z = {90, 100, …, 300}.
(a)(b)


## 4.1.1. Each Intersection of the Projection Line on the CCD

It is assumed that a spatial luminous pointP0x0,y0,z0 is used as the spatial positioning point, and the measurable range is 4000∗4000∗6000mm3 by the 3D photoelectric sensor. P0x0,y0,z0 passes through each focal point. i.e., f1,f2,f3. And then, the light is divergent from the spatial luminous point. Note that the light is dispersed into a straight line through a cylindrical prism based on the refraction of light principle. The three CCDs project onto the X-Y plane, and it is vertical between each straight line and the corresponding CCD. Since the straight line over the CCD point can be perceived, there are four points which are coplanar for each CCD. That is, the spatial point light, the CCD through the lens of the light point, the lens focus, and the projection point on the X-Y plane. As a result, the three CCDs are extended into three planes, where only the spatial point light is communal. Note that the spatial point light positioning information is the intersection of the three planes. The positioning principle is shown in Figure 2(b).


## 4.1.2. Measurable Range on the Plane

The dynamic positioning system is verified using the following three cases:Case 1: according to the principle of the spatial positioning system, every point on thez=100mm plane, and the range of 81×81mm2, the values are obtained on the spatial plane, and the points perceived by the sensor are the spatial points in the three CCD measurable ranges. As shown in Figure 4, the red area represents the measurable point, and the cyan area represents the undetectable point.Case 2: in the determinable spaceX=−2000,2000,Y=−2000,2000,Z=6000, the measurable spatial points are represented by red dots on the plane of Z=6000mm as shown in Figure 5.Case 3: measurable range of multilayer planes: in the case of the determined spaceX=−2000,2000,Y=−2000,2000,Z=6000, the space plane is set from Z=100mm to Z=6000mm, defining the measurable range of each spatial point be 500mm as the additional steps. Figure 6(a) shows the measurable range for each spatial layer. Similarly, Figure 6(b) describes the results due to reducing the addition steps as 10mm, i.e., the measurable range from Z=90mm to Z=300mm on the X-Y plane. Figure 6 shows that the measurable range of spatial points is presented from point, line to triangle, until the polygon, which is dynamically changing.Similarly, it is obvious that the more precise the pace on the CCD is, the more perceptible the number of points could be sensed. Therefore, the spatial positioning system possesses a wider measurement range. From the current simulation results, the range of measurable points in each spatial layer is stillY-axis symmetry, and the distribution profile of each layer point is from one point to the triangle, until the polygon measurement interval. Note that the schematic diagram of the dynamic measurement device is symmetry on the Y-axis as shown in Figure 2(a). According to the principle of the 3D photoelectric sensor, the projection from the spatial point source is also symmetry on each CCD plane.Figure 4
Measurable range on thez=100mm plane.Figure 5
Measurable range on thez=6000mm plane.Figure 6
Measurable range of the sensor multilayer plane. (a)Z = {100, 600, …, 6000}. (b) Z = {90, 100, …, 300}.
(a)(b)


## 4.2. Experimental Results

The effectiveness of the proposed distributed weighted fusion estimation strategy is validated by numerical examples. The dynamic positioning system for the intermittent measurement is considered as a distributed uncertain system [32, 37–39]:(28)xk+1=0.9TT2/200.9T000.9+ℱkFkEkxk+T2/2T1wk,k=1,2,…,zik=Cki+ℋkiFkEkixk+vki,i=1,2,3,wk=ηk,vki=ζiwk,where the sampling period is set as T=0.1s. The maximum value of the transmission time delay is set as 5 sampling periods. The time-varying parameter uncertainty is satisfied as Fk=sin0.6k. System state xk=sks˙ks¨kT is used to represent the values of position, velocity, and acceleration for the target at kT. Set parameters as ℱk=0.10.10.1T, Ek=0.020.020.02, Ck1=0.60.81, Ck2=10.80.5, Ck3=0.310.7, and ℋk1=ℋk2=ℋk3=0.8. Define ηk∈ℝ as zero-mean white noise with the variance ση2=0.09. Considering the cross-correlation between the process and measurement noise, ζi in equation (28) is set as ζ1=2, ζ2=0.8, and ζ3=1, respectively. The covariance of the process noise in equation (28) is wk with the covariance Qk. Meanwhile, the covariance Rki,j is represented as Rki,j=ζkiζkjQk, and the mutual covariance is denoted as Ski=ζkiQk in equation (28).The initial values are set asx^00=μ0=Ex0=111T and P00=0.01I3. For the numerical examples, 300 sampling points are selected, and the experimental results are verified based on 300 Monte Carlo simulation calculations. Comparing the traces of the estimated error covariance matrix, the results are shown in Figure 7. The estimated state x^kk is validated by the improved robust finite-horizon Kalman filtering (IRFHKF) in [38], as well as the proposed distributed weighted fusion state estimation using an adaptive signal selection scheme (DWFASS). First of all, taking into account the transmission with delay-free from Figure 7(a), the upper boundary of the estimation error covariance matrix is small compared with the IRFHKF method. It is worth noting that the upper boundary of the minimum value for the fused cross-covariance matrix is smaller than the upper boundary for the other covariance matrices, since the weighted fusion criterion is the optimized again in the sense of the minimal variance. Comparing the two methods shown in Figures 7(b)–7(d), the proposed method is more suitable for detecting the optimal filtering parameters. Therefore, the designed estimator is appropriate for the uncertainty in model parameters as well as external disturbances, such as random transmission time delays and lost and disordered packets.Figure 7
Comparing DWFASS and IRFHKF methods: (a) trace for the upper boundary, (b) comparison of the estimated position state, (c) comparison of the estimated velocity state, (d) comparison of the estimated acceleration state, and (e) comparison of MAEs for the position.
(a)(b)(c)(d)(e)In addition, to further analyze the performance of the proposed DWFASS method, the maximum absolute errors (MAE) [42] are obtained by(29)MAE=xk−x^ASSikk.On the one hand, the root mean square error (RMSE) [43] between the measured and estimated values is defined as(30)RMSE=1M∑k=1Mxk−x^ASSikk2,where M represents the number of the measured points and xk and x^ASSikk represent the measured and estimated states at the current sampling points k, respectively.Figure7(e) shows the MAEs comparing DWFASS and IRFHKF methods; meanwhile, the RMSEs are demonstrated in Table 1. Considering the cross-correlation for the noise, the dynamic tracking trajectory is closer to the actual state value if the noise is more correlated.Table 1
Comparing RMSEs between DWFASS and IRFHKF.
MethodsPositionVelocityAccelerationSensor 10.04690.09060.3182Sensor 20.04020.06970.3120Sensor 30.04260.07650.3140IRFHKF0.06810.09800.3831Next, for the distributed networked systems, the processing effect of each subsystem is different, which is determined by the cross-correlation for the system parameters and uncertain disturbances. Note that if the process noise is more cross-correlated with the measurement noise, the estimation performance is more accurate.Moreover, to further illustrate the effectiveness of the DWFASS method, the distributed weighted fusion results for state estimation are shown in Figure8. The results of state estimation are calculated from Theorems 2 and 3, and the method of the robust finite-horizon filtering is studied using the ASS scheme, as well as the distributed weighted fusion estimation criteria.Figure 8
Comparison of distributed weighted fusion estimation results: (a) estimated position state, (b) estimated velocity state, (c) estimated acceleration state, and (d) comparison of MAEs for the position.
(a)(b)(c)(d)Figure8 shows that the ASS scheme is able to actively discard data packets owing to the random transmission time delays and lost and disordered packets. At the same time, Table 2 enumerates the RMSEs of each sensor. Therefore, the distributed weighted fusion estimation uses the one-step prediction compensation for better performance and computational efficiency in target tracking.Table 2
Comparing RMSEs of each sensor.
MethodsPositionVelocityAccelerationSensor 10.16580.27490.5745Sensor 20.15080.25530.5464Sensor 30.12730.21280.4909


## 4.3. Numerical Simulation Analysis

According to the simulation results, the dynamic tracking results are compared by the method of recombination error covariance. The comparative results show that the proposed estimation instrument has better estimation accuracy, especially in the case of communication constraints with multistep random delays and lost and disordered network induction packets. Because the true estimation error covariance is lower than the upper boundary of the probe, the proposed distributed fusion estimation method has better performance in the phenomenon of network induction, including the ability to closely track the state of the system.


## 5. Conclusion

This paper, facing the spatial positioning principle of the linear CCD, introduces the workflow of the 3D photoelectric sensor dynamic positioning system and simulates the spatial positioning effect of the proposed DWFASS method. According to the measured spatial target, the trajectory of its motion is tracked and estimated, and a kind of distributed uncertain system with network-inducing factors is studied. In order to process the lost and disordered data packets by the random transmission time delays, the system is modelled based on the adaptive signal selection scheme. In order to overcome the influence of limited communication capacity and reduce the computational burden, this paper adopted the one-step prediction compensation strategy and proposed the optimal state estimation method to compensate for the missing packets. At the same time, the finite-horizontal filtering strategy is designed, and the real estimated error variance obtained by the DWFASS method is less than its upper boundary. Then, the distributed estimation method based on filtering is introduced, which introduces the uniform expression of the weighted fusion standard. Therefore, the distributed fusion estimation had higher estimation accuracy than that of each local subsystem. The numerical simulation is demonstrated using the spatial target positioning system containing three sensor devices, and the results show that the proposed distributed weighted fusion estimation has better tracking performance. To better process the data required for positioning and improve the accuracy of positioning, this paper has optimized the performance of positioning algorithms. The proposed method can be used in various motion control scenarios such as target tracking, trajectory tracking, path following, and formation control of AUVs subject to uncertainties and constraints. In addition, the future work will further focus on following some known distributions for the time delays or packet loss to further enhance the state estimation, such as the Gaussian or binary distribution based on the known bandwidth of the network.



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*Source: 7698504-2020-07-20.xml*