As the 2019 novel coronavirus disease (COVID-19) pandemic rages globally, its impact has been felt in the stock markets around the world. Amidst the gloomy economic outlook, certain sectors seem to have survived better than others. This paper aims to investigate the sectors that have performed better even as market sentiment is affected by the pandemic. The daily closing stock prices of a total usable sample of 1,567 firms from 37 sectors are first analyzed using a combination of hierarchical clustering and shape-based distance (SBD) measures. Market sentiment is modeled from Google Trends on the COVID-19 pandemic. This is then analyzed against the time series of daily closing stock prices using augmented vector autoregression (VAR). The empirical results indicate that market sentiment towards the pandemic has significant effects on the stock prices of the sectors. Particularly, the stock price performance across sectors is differentiated by the level of the digital transformation of sectors, with those that are most digitally transformed, showing resilience towards negative market sentiment on the pandemic. This study contributes to the existing literature by incorporating search trends to analyze market sentiment, and by showing that digital transformation moderated the stock market resilience of firms against concern over the COVID-19 outbreak. The onslaught of the 2019 novel coronavirus disease (COVID-19) pandemic has impacted stock markets worldwide, with the stock prices of many firms seeing unprecedented fall. Many governments around the world imposed a shutdown on their cities in attempts to implement social distancing practices and slow down the spread of the life-threatening virus. Such shutdowns have brought what is classified as non-essential corporeal economic activities close to complete standstill. This has resulted in well-known brands, such as Hertz, JC Penny, and J Crew, filing for bankruptcy, and sparking concerns of large-scale economic recession or even economic collapse  Industry Digitalization Framework  have influenced a large number of economies worldwide. All of these epidemics have resulted in severe damage to the economy and loss of lives, e.g., 5% gross domestic product (GDP) of the US was reduced owing to the Influenza Palese ; as resuHowever, the new digital world has made the COVID-19 pandemic’s impact on the global economy different from that of previous epidemics. The development of digital technologies in the past two decades has changed the way people and firms interact with each other. With the fast-growing Internet users worldwide, online shopping has become one of the most popular online activities, and the emergence of e-business platforms such as Amazon and Alibaba contribute to the rapid growth of online sales. In 2019, e-retail sales accounted for 14.1% of all retail sales worldwide, expected to reach 22% in 2023  to validate the groups of firms identified in the MGI Industry Digitalization Framework. It followed similar approaches employed by Sardá-Espinosa  and PapaPhase 2 modeled Google search trends and stock price changes jointly in an augmented VAR. In this phase, the analysis compared and contrasted the stock performance of firms across the three levels of digital transformation, i.e., low, medium, and high, based on the MGI Industry Digitalization Framework. The analysis in the two phases built on each other, supporting the notion that market sentiment towards the COVID-19 pandemic as reflected in Google search trends affects the stock prices.Using data from FactSet, we selected nearly 2000 firms listed on the NASDAQ based on their sectoral information in 2020. NASDAQ is one of the world’s largest stock exchange based on market capitalization. Among the scholarly community, NASDAQ represents one of the most widely investigated stock exchange databases . The average market capitalization for the sampled stocks is USD8,587 million. The sample period  is chosen because it allows a reasonable amount of time lag for the COVID-19 outbreak to generate public attention and stir market sentiment. Additionally, the current research focuses on how sectoral digital transformation mitigates the effect of negative events on the stock market. We are more interested in the period from the outbreak of the COVID-19 to the eventual stock market crash (the affected period). There might be other potential confounding factors coming into play during the recovery period. For example, the U.S. coronavirus relief funds, established under the Coronavirus Aid, Relief, and Economic Security (CARES) Act, were announced in early April 2020. The stock market is likely to respond to such information on economic support. The chosen sample period ends when the CARES Act was announced, minimizing the potential confounding effect of government support and intervention on the stock market. For every entry, three data fields are obtained, namely, the daily price change, trading volume, and market cap. These data are used in the models in both Phase 1 and Phase 2. Table In Phase 2, the search variable is operationalized as an “interest over time,” measured on a relative scale of 0–100, with peak popularity coinciding with a value of 100. While Google’s indicator of interest, over time, has been used as a source of data in numerous studies   on March 11, 2020. Thus, this cluster is named as the Resilient Cluster.By analyzing the extracted centroid, the times series patterns of the two clusters are then examined. For firms in Cluster 1, it is observed that a steeper declining trend of market capitalization occurs amid the COVID-19 outbreak. This is thus termed as the Sensitive Cluster and the Resilient Cluster, with a high, medium, and low level of digital transformation following the MGI Industry Digitalization Framework, to answer the research question and to validate the sectoring. To further simplify the research model and to enable in-depth investigation in Phase 2, Categories 2 to 5 in the original MGI Industry Digitalization Framework, which represent sectors with the potential to transform digitally further, are merged into one group of firms with a medium level of digital transformation. Firms under the most digital sectors (Category 1) and the most lagging sectors (Category 6) are left unchanged. It is then analyzed to assess if the level of digital transformation is associated with the two identified clusters. A descriptive summary of the sensitive and resilient clusters can be found in Table The firms are cross-tabulated with percentages of their market capitalization decline under the Resilient Cluste. The majority of the firms in the lowest rung (60%), including agriculture, hotels, and healthcare, fall under the Sensitive Cluster. Firms with mid-level of sectoral digital transformation fall in between, with about half of the firms (48%) being in the Resilient Cluster. Additionally, Pearson χ2 statistic is used to test the two-way associations with frequencies in the cells (χ2 (1) =16.667, p < 0.05). The results provide some directional support to the relative impact of digital transformation. It also validates the relevance of the level of digital transformation  as a grouping variable to be used in the analysis in Phase 2.Consistent with expectation, more than two-thirds of the firms (68%) within the most digitally transformed sectors, such as telecommunications equipment, personal services, financial conglomerates, and advertising/marketing services, fall under the Google trends index represents the search volume of keywords by Google users, and it serves to indicate market sentiment and the public’s attention to an event or incident , a negatively correlated movement of the two variables for the three groups of firms is observed. Stock prices move downwards when the search for coronavirus increased. We do not test correlation, given that it does not necessarily reflect causality, and Granger causality is used to test the leading relationship between these two variables. However, the typical Granger causality test cannot be relied on when one or both time series are non-stationary, which could lead to spurious causality ,,3), Yt r for Eq.  is H0: bX, Y, and Z removes the unit root. Thus, the maximum order of integration is set to be 1, denoted by I (1). As a next step, the VAR model is set up using the levels of the data without differencing and determining the appropriate lag length for the variables X and Y. Based on the information criteria of Akaike Information Criterion Hannan Quinn, Schwarz Criterion, and Final Prediction Error, it is decided that six lags are used in the analyses that followed. As noted by Toda and Yamamoto ,,3), giveX in Eq.  and the Table As the focus is on the impact that Google searches impose on stock price changes and also to differentiate the impact across the different groups, a detailed investigation into the results of Eq.  is then The results of Eq.  examine This study aims to investigate the sectors that have performed better even as market sentiment is affected by the COVID-19 pandemic. Market sentiment on the COVID-19 is modeled with Google search trends. The sectors are organized according to the MGI Industry Digitalization Framework. Performance is quantified by the stock prices of firms in the sectors. The analysis is carried out over two phases.In the first phase, a three-group model on how stock prices adjust to market sentiment towards the sudden emergence of the COVID-19 pandemic is established. The stock prices of a majority of firms across sectors with a higher level of digital transformation are found to have remained resilient to the impact of market sentiment, while sectors that lag across most digital transformation dimensions are among the most negatively affected.In the second phase , market sentiment on the COVID-19 pandemic, as reflected in Google search trends, is affirmed to be a predictor of stock performance. There is also further evidence to show that the digital transformation of firms mitigates the negative impact of market sentiment induced by large-scale unanticipated incidents. As the stock price of firms dips amid the coronavirus pandemic, firms with mid to high level of digital transformation have out-performed others. Analysis in the second phase also suggests that a rise in stock price can affect market sentiment.In answering “what are the sectors that have performed better even as market sentiment is affected by the pandemic,” the analysis reveals that sectors with a higher degree of digital transformation have performed better. Thus, during the COVID-19 pandemic, the market has reacted more favorably towards firms and sectors with a higher level of digital transformation and more negatively towards the laggards. An explanation for this is thus offered.The spread of the COVID-19 virus has led governments around the world to shut down their cities to slow down the rate and magnitude at which the pandemic is developing. Even as such shutdowns have brought many corporeal economic activities to a near-complete standstill, consumer purchases and even trade not only continue online, but they have increased considerably. Demand for Internet bandwidth as well as portable computing devices such as laptops and tablets also intensify as many people switch to working and learning online. Many are also predicting the emergence of a post-COVID-19 period, where digital transformation will be prominent. Given these, the markets may have given eminence and greater confidence to firms and sectors with a higher level of digital transformation as they are positioned better to sustain operations not only amid the pandemic, but also to recover faster in the post-COVID-19 period. Thus, sectors with a higher level of digital transformation have been more resilient in their performance and also performed better relative to sectors with lower levels of digital transformation.Further, this study contributes to understanding the role of digital transformation in the firms’ stock market performance, particularly due to large-scale unanticipated incidents such as a pandemic. While previous research has investigated whether the use of technology can improve organizational performance, this research is among the first to demonstrate how digital transformation may give rise to stock market performance. This suggests that digital transformation is an important factor for investors.The findings from this study also suggest that investors consider recent stock prices when conducting further online search about COVID-19, and such search behaviors will drive changes in stock prices. However, the direction, magnitude, and timing of such effects depend on the level of digital transformation.Regarding the research method, this study demonstrates the feasibility of using a search trend index to model market sentiment, especially when investigating how large-scale anticipated incidents can affect the stock prices for a large number of firms or across sectors. This presents researchers with another approach to assess market sentiment and expands the research arsenal to include search indexes such as Google trends, which are readily available for use.This paper has limitations that offer opportunities for future research. First, our study examines how sectoral digital transformation mitigates the effect of market sentiment on negative events  on the stock market, focusing on the period from the outbreak of the COVID-19 to the eventual stock market crash. Since early April 2020, much of the stock markets’ main indices have regained much of their lost territory for the year. The aggressive stimulus packages roll out by central banks and governments to boost growth, such as the U.S. coronavirus relief funds announced around early April 2020, have fueled recovery. Such confounding effects of government support and intervention on the stock market are difficult to account for in the current research model. Future research can look into how the intervention of governments and support of central banks restore investors’ confidence and accelerate the recovery phase of the market cycle.The reopening of businesses and economies is also underpinning market optimism. It is plausible that sectoral digital transformation can also influence how firms perform during the recovery period. Therefore, a possible extension of our research is to investigate how digital transformation moderates the speed and extent of the stock market rebound, as it is of interest to both firms and investors. Firms in the most digitally advanced sectors are likely to be better positioned for the recovery and the post-pandemic period.Second, the literature on the state of digital transformation in sectors suggests a large and growing gap between companies within the sectors (Manyika et al. Lastly, this paper does not investigate the causes behind the growing gap between sectors in digital transformation. Future research may try to uncover how the different conditions lead to a widening gap between the “haves” and the “have-mores”: companies and sectors that are using their digital capabilities far more than others to innovate and transform how they operate.


This paper developed an adaptive backstepping fuzzy sliding control (ABFSC) approach for a micro gyroscope. Based on backstepping design, an adaptive fuzzy sliding mode control was proposed to adjust the fuzzy parameters with self-learning ability and reject the system nonlinearities. With the Lyapunov function analysis of error function and sliding surface function, a comprehensive controller is derived to ensure the stability of the proposed control system. The proposed fuzzy control scheme does not need to know the system model in advance and could approximate the system nonlinearities well. The adaptive fuzzy control method has self-learning ability to adjust the fuzzy parameters. Simulation studies were implemented to prove the validity of the proposed ABFSMC strategy, showing that it can adapt to the changes of external disturbance and model parameters and has a satisfactory performance in tracking and approximation. Micro gyroscopes are widely used in inertial navigation and guidance systems. However the error and disturbances existing in micro gyroscopes may decrease the accuracy and sensitivity of the system. With the advancement of microelectromechanical system (MEMS) technology, it is possible to produce chip-based sensors such as accelerometers, gyroscopes, and magnetometers, in combination with the miniaturization of electronic devices. Micro gyroscopes are small, lightweight, have low energy consumption, long service life, and are extremely low cost.Many control strategies have been investigated to compensate the error and disturbance from micro gyroscopes ,4,5,6,7.The universal approximation theorem indicates that the fuzzy system is a new universal approximator in addition to polynomial function approximators and neural network approximators. As the universal approximation theory of fuzzy system can approximate any nonlinear model and realize arbitrary nonlinear control law, it is widely used in control systems. Adaptive sliding mode control with a neural estimator and adaptive control with fuzzy compensator for a micro gyroscope was investigated in ,32. In , a fuzzyHowever, the fuzzy control strategy is not combined with an adaptive backstepping controller in the application of micro gyroscope and a backstepping controller has not been incorporated into a fuzzy sliding mode control system for a micro gyroscope. As such, the adaptive backstepping fuzzy sliding mode control approach has not been proposed in the control of a micro gyroscope. Motivated by the above research, this paper presents an adaptive backstepping fuzzy sliding controller to adjust the fuzzy parameters with self-learning ability and reject the system nonlinearities. Comparing other existing methods, the main contributions of this article can be summarized as follows:(1) Backstepping is a nonlinear control approach by means of the recursion process. A major advantage of a backstepping controller is its flexibility to avoid cancellations of useful nonlinearities and achieve regulation and tracking properties. The gyroscope equations are transformed into an analogically cascade system where the backstepping approach can be implanted.(2) Backstepping design and adaptive fuzzy sliding mode control were applied to a micro gyroscope. The proposed adaptive fuzzy controller not only does not rely on the system model, but also makes the algorithm adjust the fuzzy parameters with self-learning ability.(3) The proposed sliding mode control adds additional compensators to improve the stability, hence obtaining desired system characteristics. Thus, the entire closed-loop system meets the dynamic and static performance indicators and achieves accurate tracking performance.In this section, the dynamic model of a micro gyroscope is presented. The characteristics of micro gyroscopes are similar to traditional gyroscopes, mainly through the Coriolis force to achieve the miniaturization of equipment. The dynamics model of the micro gyroscope can be regarded as a damping-spring-mass system, as shown in The driving electrode generates electrostatic force to drive the base mass block back and forth to maintain stable oscillatory momentum. The induction device is used to sense the movement of the base mass block in the vertical driving direction, and to extract the external angular velocity from the vibration information. Referring to Park , with soThe asymmetric spring and damping terms, Dividing both sides of Equation (1) by the mass Define Taking into account the system nonlinearities, Equation (2) can be expressed as:Motivated by the research results in ,20,21,28The reference model is defined as In Equation (5), We selected the first Lyapunov function as:The time derivative of the Define a function of sliding surface asDefine the second Lyapunov function asFrom Equation (9), we can getThen, substituting Equation (10) into the time derivative of In the controller design, we used exponential reaching law asBy Equations (11) and (12), we designed a comprehensive controller asSubstituting Equation (13) into Equation (11) yieldsHowever, since the system model The singleton fuzzifier mapping was used, where The output of the fuzzy system is written by a center-average defuzzifier, product inference, and singleton fuzzifier as:We defined the optimal approximation constant Since the system model In this section, the stability analysis of the proposed control system is discussed. First, we chose a Lyapunov function candidate asThen, the time derivative of From the derivation in (11),Substituting (18) into (21) yieldsSince To make Substituting Equation (24) into Equation (23) yieldsAccording to the Inequality According to the inequality Substituting Equation (27) into Equation (26) yieldsDefine Solving Equation (29) yieldsRemark 1.In order to overcome this problem, a proper adaptation law can be proposed to estimator the upper boundto realize the adaptive upper bound control, weakening the chattering and ensuring the stability of the control system.Remark 2.Since there is a switching functions sign in the proposed controller (13), we can use saturation function or hyperbolic function to replace sign function to decrease chattering.In this section, the proposed ABFSMC scheme was evaluated on the lumped model of a micro gyroscope sensor. The simulation experiment of the proposed scheme was carried out on a MATLAB/SIMULINK software platform. Meanwhile, the superiority of the proposed can be further confirmed by comparing it with the adaptive backstepping controller .The parameters of the micro gyroscope were selected as in The angular velocity of micro gyroscope was assumed to be Initial conditions were An adaptive backstepping control (ABC) technique for a microscope was presented in . In ordeFor the quantity discussion, the root mean square errors (RMSE) of the tracking error of the two axes of the micro gyroscope using these three difference controllers are compared in In summary, the introduction of the ABFSMC approach can adapt to the changes in the external disturbance and model parameters and maintain a satisfactory performance in tracking and approximation.The ABFSMC strategy was investigated in a micro gyroscope through theoretical discussion and numerical simulation. The mathematical model of the micro gyroscope was transformed for the handiness of the backstepping control design. A backstepping approach was adopted to deal with the model uncertainties, disturbances, and unknown parameters of the micro gyroscope. The fuzzy parameters were updated online to approximate the nonlinear dynamics in the micro gyroscope. Simulation studies were investigated to demonstrate the advantages of the proposed ABFSMC strategy in tracking and approximation performance.


Wireless virtual reality (VR) is a promising direction for future VR systems that offloads heavy computation to a remote processing entity and wirelessly receives high-quality streams. WiGig and WiFi are representative solutions to implement wireless VR; however, they differ in communication bandwidth and reliability. Our testbed experiments show that the performance of WiGig and VR traffic generation strongly correlates with and consequently can be predicted from a user’s motion. Based on this observation, we develop a wireless VR system that exploits the benefits of both links by switching between them and controlling the VR frame encoding for latency regulation and image quality enhancement. The proposed system predicts the performance of the links and selects the one with a higher capacity in an opportunistic manner. It adjusts the encoding rate of the host based on the motion-aware prediction of the frame size and estimated latency of the selected link. By evaluating the testbed data, we demonstrate that the proposed system outperforms a WiGig-only system with a fixed encoding rate in terms of latency regulation and image quality. Virtual reality (VR) services let a user immerse in a virtual world by enabling the user to explore the virtual space, which is rendered as stereoscopic images, in the way he/she does in the real world, i.e., through head movements . This prVR headset systems are generally classified into two types: tethered and untethered. Tethered headsets  use a powerful PC (or a gaming console) to process the VR content computations, which offer the highest-quality VR experiences of all types with high resolution and frame rate. However, today’s tethered headsets use cables to transmit display and motion data, and the length, weight, and tension of the required wire harness disturb the user’s mobility, which decreases the immersiveness, restricts the scale of supported services, and creates a tripping hazard. Untethered headsets  have a processing unit inside the headset (either an attached smartphone or an embedded processor) and provide VR services with no wires; thus, they are portable and convenient but have a lower content quality due to the limited processing power.wireless VR in this paper. The data traffic of wireless VR has a bidirectional nature; the VR image frames rendered by the host are transferred to the VR headset in the downlink, and the motion (IMU) data of the headset are fed back to the host in the uplink, so that the next VR image frame is rendered for the user’s latest viewport. Due to the ultrashort viewing distance through the magnifying lenses of VR headsets, a high pixel resolution of the VR frames is required . Moreover, to increase immersion with reduced juddering and motion sickness, a high frame rate  is desired. Therefore, delivering a VR video service to VR headsets over wireless connectivity requires a large bandwidth.A promising direction of evolution for both types of VR headset systems is to offload VR processing to a high-end host  and wirelessly stream the rendered VR image frames to a VR headset, which we call latency and motion sickness [Timewarp technique [WiGig  and WiFisickness . Recentlsickness  have shoechnique ,10, wherMotion-awareWiGig-WiFiInterplay system forVirtualReality  requires an increased data size. Therefore, the VR system must be designed to be able to transfer large frame sizes as frequently as possible.Regulated latency: Since the total latency becomes excessive, Timewarp produces noticeable black borders, which disturb the immersion. Overfilling [rfilling  is the sIn this paper, we propose a wireless VR solution to exploit the benefits of both WiGig and WiFi in a user-motion- and performance-aware manner for high-quality and reliable VR services, which is called the MWThrough testbed experiments, we observe that the WiGig performance greatly depends on the distance and angular direction, while the WiFi performance is stably maintained in a room-scale VR environment. The generated frame size is also proportional to the head angular speed due to the changing user viewport. Based on these observations, MWDesign of the prediction scheme for the WiGig throughput and the VR frame size to be generated based on the motion awareness of a VR user,Design of the joint control mechanism of interface switching and encoding rate adjustment for enhanced VR frame quality and latency regulation,Experimental evaluation of the integrated wireless VR system to show the performance gain of the proposed design over various conventional approaches.In summary, the main contributions of our work are listed as follows:For evaluation, we build a wireless VR system testbed and collect the data of the user motion, link signal levels, and generated frame size. The evaluation of these testbed data shows that MWThe remainder of the paper is organized as follows: In There have been recent attempts to design communication and networking schemes for wireless VR in unlicensed spectra. Abari et al.  proposedThere is increasing research on delivering VR services in cellular networks such as 5G systems. Elbamby et al.  discusseOther research directions for wireless VR include VR content compression methods ,24, wireThe wireless VR system under consideration is illustrated in x265 library [In our testbed, a local PC is used as a processing host. A Netgear Nighthawk X10 and an Acer Travelmate laptop (equipped with the Sparrow 11ad module) are used. An HTC VIVE headset is connected to the laptop via HDMI and USB cables. The motion data of the headset are captured and transferred to the host using VirtualHere software . The com library , and theTo observe the performance of WiGig and WiFi in wireless VR scenarios, we perform throughput measurement tests. Within a VR service, the VR user walks around and rotates his/her head to explore the virtual world. In other words, when the user moves, the position and directional angle of the headset with respect to the connected AP change. For performance measurement, we establish a line-of-sight WiGig/WiFi connection between the AP and the client, run iPerf tests, and observe throughput changes for varying distances and directional angles.The throughput performance of WiGig for varying distances is shown in iPerf [In the second experiment, we continuously change the orientation of the laptop during the iPerf  test at iPerf ). The coUnlike WiGig, the performance of WiFi is not sensitive to user motion at the room scale. In the same environment as the WiGig test (performed at midnight to minimize interference), the throughput performance of WiFi remains unchanged with distance and direction, as shown in Due to the limited bandwidth of wireless connectivity, the wireless VR system under consideration transfers a stream of encoded VR frames to a user headset. The principle of video encoding is to extract the differential information between frames as motion vectors; larger differences between frames result in more information, which increases the size of the resulting encoded frame. If we can predict the sizes of upcoming frames, this will be important information in MWIn wireless VR, the major source of differences between frames is the user’s motion. To investigate the relationship between the user motion and the amount of generated VR traffic to be delivered, we simultaneously collect both data during the playing of VR content in our wireless VR system. p-value is commonly used as a measure of correlation; it is defined as the probability of obtaining the observed correlation under the hypothesis of no correlation, i.e., the true correlation is zero. Then, a sufficiently small p-value  implies that such a correlation is unlikely observed under this hypothesis, and the new hypothesis of a significant correlation should be accepted as true. For the samples in the figure, the p-value is To quantitatively observe the correlation between the two parameters, we show the scatter plot in The VR content may have moving objects in its scenes. Therefore, although the user is in a steady state, the frame size may change with time. The user’s motion will cause additional changes in the frame size. The amount of VR traffic to transmit will not increase immediately after the user’s motion due to the latency components between the occurrence of the user’s motion and the transmission of the corresponding VR traffic, such as motion data transfer, content simulation, rendering, and encoding.Based on the observations in the previous sections, we know that WiGig and WiFi are complementary to each other, so cooperatively utilizing them can help to achieve high image quality and regulated latency. MWWe will describe each algorithm of MWMWWhen the WiFi link is in use for VR streaming, MWAccording to the experimental observations in When the WiGig link is not in use for VR streaming, it is turned off to reduce the power consumption of the client. The average power consumption of the WiGig interface was measured using the Wattman power consumption analyzer (HPM-100A)  as 8.4 wL be the mean size of the generated VR frames. Latency D in delivering a VR frame to the client using the selected link at rate MWv is the head speed and v, and As observed in To satisfy the target latency In this section, we evaluate the performance of MWGreat Power [Legacy VR with a fixed interface: The wireless interface in use is not changed during VR service. The encoding rate is fixed as well. The cases of WiGig-only and WiFi-only are considered.Interface switching: Switching between WiGig and WiFi interfaces is made for higher throughput during VR service. The switching algorithm of MWEncoding adjustment: The encoding rate of a VR service is adjusted for latency regulation but without interface switching. The adjustment algorithm of MWTo evaluate the performance of MWat Power  and the The cumulative distribution function (CDF) in The generated and delivered frame sizes over time are shown in The CDF of the frame size is shown in We developed a wireless VR system that incorporated WiGig and WiFi links for latency regulation and image quality enhancement, which is called MW


Polarized hyperspectral images can reflect the rich physicochemical characteristics of targets. Meanwhile, the contained plentiful information also brings great challenges to signal processing. Although compressive sensing theory provides a good idea for image processing, the simplified compression imaging system has difficulty in reconstructing full polarization information. Focused on this problem, we propose a two-step reconstruction method to handle polarization characteristics of different scales progressively. This paper uses a quarter-wave plate and a liquid crystal tunable filter to achieve full polarization compression and hyperspectral imaging. According to their numerical features, the Stokes parameters and their modulation coefficients are simultaneously scaled. The first Stokes parameter is reconstructed in the first step based on compressive sensing. Then, the last three Stokes parameters with similar order of magnitude are reconstructed in the second step based on previous results. The simulation results show that the two-step reconstruction method improves the reconstruction accuracy by 7.6 dB for the parameters that failed to be reconstructed by the non-optimized method, and reduces the reconstruction time by 8.25 h without losing the high accuracy obtained by the current optimization method. This feature scaling method provides a reference for the fast and high-quality reconstruction of physical quantities with obvious numerical differences. Due to reflecting the rich spectral and spatial characteristics, hyperspectral imaging is widely used in agricultural detection ,2, food In order to overcome the defects caused by the FTSP, the compressive sensing (CS) theory  has beenTo avoid the drawbacks of CCISP, we proposed the full polarization-compressed hyperspectral imaging (FPCHI) system. As shown in In this paper, we therefore propose a two-step reconstruction method for the FPCHI system to achieve the fast and efficient reconstruction of the full-Stokes parameters simultaneously under compressive measurement. It is well known that the first Stokes parameter represents the total light intensity, and the last three Stokes parameters represent various light intensity differences. Therefore, the first Stokes parameter and the last three Stokes parameters are not in the same order of magnitude. According to the inherent numerical difference of the four Stokes parameters, the reconstruction process is divided into two steps. Both steps are reconstructed based on the CS theory. Specifically, the first Stokes parameter comes from the first reconstruction, and the last three Stokes parameters come from the second reconstruction. Due to relying on the inherent properties of the four Stokes parameters, the two-step reconstruction method can be used to reconstruct the full-Stokes parameters of any targets. In addition, we verify this proposed method based on the polarized hyperspectral images measured in our laboratory. In order to further illustrate the effectiveness of the two-step reconstruction method, it is also compared with the above optimization method and the non-optimized method.As shown in Let M makes up the modulation matrix of the proposed system. Moreover, the values of M are all non-zero values. Thus, after Full polarization compression can be achieved by the combination of QWP and LCTF. The Mueller matrices of QWP and LCTF are denoted as n is less than four. Then, the In order to realize the compression measurement of the four Stokes parameters, G is a known matrix obtained by CMOS, and H is determined by the current system parameters. In particular, F is an unknown matrix to be reconstructed.The compression measurements of each spatial pixel in each spectral band are combined to form the full polarization-compressed hyperspectral images Inspired by polarization compression, we consider the polarization reconstruction of one spatial pixel in one spectral band. Since the first Stokes parameter is much larger than the other three Stokes parameters, it is difficult to reconstruct the four Stokes parameters at the same time. In this paper, the four Stokes parameters are reconstructed in two steps using feature scaling.R times to obtain a new modulation matrix as follows:In the first step, suppose that the first Stokes parameter Therefore, Equation  can be rThen, the deformed polarization information is reconstructed based on the CS theory. The deformed polarization information is first sparsely represented byIn this equation, Equation  into EquEquation , we can lgorithm , the spa follows:θ^R=argmiEquation , we can In the second step, the last three Stokes parameters are reconstructed again. Substituting the deformed first Stokes parameter into Equation , we can In Equation , HL∈Rn×3ented byfL=ΨL×θL,Equation , the newUsing the TwIST algorithm  again, tThen according to Equation , the lasR times the deformed parameter. In other words, let Finally, the required four Stokes parameters can be obtained by combining the reconstruction results from the above two steps. The last three Stokes parameters are taken from the reconstruction results of the second step. The first Stokes parameter is obtained by deforming the reconstruction result of the first step. According to the scaling rule in the first step, the first Stokes parameter is In order to measure the polarized hyperspectral images as the simulation input of the FPCHI system, we first established the definition-based polarization hyperspectral imaging (DBPHI) system . As showIn the experiment, the SFW is equipped with 18 spectral filters  whose center wavelengths are from 520 nm to 690 nm with an interval of 10 nm. The LP  works effectively from 510 nm to 800 nm, and the QWP  operates from 325 nm to 1100 nm. The CMOS detector  has Then, full polarization compression and reconstruction are simulated on the measured images of each spatial pixel in each spectral band. Assume that polarization modulation is performed twice. The linear polarization angle of the incident surface of LCTF remains 0, and the fast axis angles of QWP are 105 and 150, respectively. That is, Therefore, full polarization-compressed hyperspectral images As described in The discrete W transform (DWT) matrix  severs aBased on Equations –8), we , we 8), In the second step, the modulation matrix corresponding to the last three Stokes parameters isThe DWT matrix  is also Then according to Equations –12), th, th12), By integrating the reconstruction results based on these two independent steps, we can obtain the ideal four Stokes parameters In order to compare with feature scaling, we also reconstruct the polarized hyperspectral images not only with the sparse basis optimized by particle swarm optimization (PSO) , but alsIt can be seen from Equation  that theFurthermore, The reconstruction effect can be seen not only from the image details, but also from the PSNR values shown in In particular, there are two reasons why the two-step reconstruction method can quickly and accurately reconstruct polarization information. On the one hand, the first Stokes parameter contributes much more to the compressed image than the last three Stokes parameters. As a result, when the four Stokes parameters are reconstructed at the same time, the weight flooding problem makes the reconstruction of the last three Stokes parameters seriously affected by the first Stokes parameter. The two-step reconstruction method solves this problem. In the first step where all four Stokes parameters are involved, only the reconstruction result of the first Stokes parameter is retained. Then, the last three Stokes parameters are reconstructed in the second step, avoiding the interference of the first Stokes parameter. On the other hand, the reconstruction error can be better controlled under the two-step reconstruction method. In the two steps, different reconstruction algorithms can be used or the regularization weight value can be adjusted to accurately improve the reconstruction accuracy of each Stokes parameter. Moreover, there is no need to learn to optimize the sparse basis, which not only greatly improves the reconstruction efficiency, but also is widely applicable to various targets with different polarization characteristics.The FPCHI system can achieve both compression measurement of full-Stokes parameters and hyperspectral imaging. The QWP with adjustable fast axis angle unites the LCTF with selectable linear polarization angle to modulate the four Stokes parameters. Meanwhile, the LCTF also enables flexible selection among a large number of fine spectral bands to meet hyperspectral imaging. Then for each spectral band, the two-step reconstruction method can recover four Stokes parameters from the compressed measurements based on feature scaling. In the two steps, the first Stokes parameter is extracted from the first reconstruction, and the last three Stokes parameters are from the second reconstruction.Based on the polarized hyperspectral images measured in our laboratory, both the polarization-compressed hyperspectral imaging of the FPCHI system and the two-step reconstruction method are simulated sequentially. The simulation results perform well in the evaluation of image display, grayscale distribution and PSNR values. Compared with the optimization method using machine learning, the reconstruction accuracy of the two-step reconstruction method does not decrease and still maintains a satisfactory level. Further in terms of reconstruction efficiency, the two-step reconstruction method shortens the 9.9 h required for the optimization method to 1.65 h, which means a time reduction of 83.33%.The grayscale value of the first Stokes parameter is quite different from the last three Stokes parameters, so the two-step reconstruction method separates their reconstruction processes. This method is conducive to more accurately reduce the reconstruction error of each step. Since it is based on the inherent properties of the Stokes parameters, this method is suitable for various targets with different polarization characteristics. Moreover, for the compression of the spatial domain or other domains, this provides a reference for the reconstruction of parameters that are originally at different orders of magnitude. However, this approach still faces some limitations. Next, we will focus on solving the intelligent and appropriate selection of feature scaling values. Through further optimization, such as optimizing the polarization encoding pattern, more efficient reconstruction will be achieved to apply the polarization hyperspectral imaging to actual target detection and other monitoring fields.


The photonic environment can significantly influence emission properties and interactions among atomic systems. In such scenarios, frequently the electric dipole approximation is assumed that is justified as long as the spatial extent of the atomic system is negligible compared to the spatial variations of the field. While this holds true for many canonical systems, it ceases to be applicable for more contemporary nanophotonic structures. To go beyond the electric dipole approximation, we propose and develop in this article an analytical framework to describe the impact of the photonic environment on emission and interaction properties of atomic systems beyond the electric dipole approximation. Particularly, we retain explicitly magnetic dipolar and electric quadrupolar contributions to the light-matter interactions. We exploit a field quantization scheme based on electromagnetic Green’s tensors, suited for dispersive materials. We obtain expressions for spontaneous emission rate, Lamb shift, multipole-multipole shift and superradiance rate, all being modified with dispersive environment. The considered influence could be substantial for suitably tailored nanostructured photonic environments, as demonstrated exemplarily. Hence, it has also been called the Purcell effect. The effect has been experimentally verified in various types of cavities or band-gap environments, including semiconductor microstructures2, photonic crystals3, and plasmonic nanoparticles, where the emission rate was enhanced up to three orders of magnitude5. Besides emission enhancement on its own, photonic environment equally affects the interactions between multiple atomic systems. In vacuum, examples such as dipole-dipole coupling6, or collective phenomena like Dicke superradiance have been explored8. Also, all these effects can be tailored by suitably engineering the photonic environment10.An excited atomic system, e.g. an atom, a molecule, a quantum dot, can decay radiatively from an excited to a ground state while releasing its energy into the photonic environment. The rate of this process depends on the properties of the environment, and in a pioneering work by Purcell the possibility to control the spontaneous emission lifetimes of atomic systems by tailoring their surroundings was first investigated11 or in Rydberg excitons12. Similarly, corrections beyond the electric dipole approximation may be necessary if the electromagnetic field is focused into spots comparable in size to the atomic system. The latter can be realized by nanoscopic environments and picocavities, capable to localize the electric field into nanometric spatial domains, providing high intensities and spatial modulations at the length scale of tens of nanometers. This tends to be comparable to the size scale of molecules or quantum dots15. Then the usual mismatch of size scales of photonic modes and atomic systems is reduced, which leads to enhanced interaction probability if the photonic modes and the atomic system overlap in space. The resulting need to include corrections beyond the electric dipole approximation to properly quantify light-matter coupling was demonstrated for quantum dots near plasmonic nanoparticles16.The influence of photonic environment on these phenomena is usually quantified in the electric dipole approximation. This is justified when the electric field shows a negligible spatial variation across the size of the atomic system. Steps beyond may be required if the atomic system is less than an order of magnitude smaller than the wavelength of light it is coupled to, as demonstrated in semiconductor quantum dots17. This prompts to consider not just electric-multipolar contributions but at the same time their magnetic counterparts. Until now, enhancement of the rate of a magnetic dipole emission by a nanostructure was considered18 and reported experimentally in lanthanide ions22. Large enhancement of quadrupole25 and even higher-order transitions was equally predicted. Transitions driven with several multipolar mechanisms have been considered26 and observed27 respectively in semiconductor quantum dots and transition-metal/lanthanide ions. Recently, it was pointed out that the simultaneous existence of several interaction channels enhanced in intensity with photonic nanostructures might open new avenues on the route to control spontaneous emission, related to their controlled interference28. To account for these effects, treatments beyond the point-dipole approximation were developed31, based on spatial integration of the transition elements expressed in terms of wavefunctions of the atomic system. Nonclassical corrections to these transition rates originating from effects such as nonlocality, electronic spill-out and Landau damping have been considered in ref. 32. Although such approaches take into account all multipolar orders of light-matter interaction, they require information on the atomic system’s wavefunctions, which are in general complex multidimentional structures. They are hardly accessible experimentally and their calculation may be a significant computational effort already in case of few-electron systems.Furthermore, and potentially more interestingly, nanostructured environments may also open light-matter interaction channels beyond that one corresponding to coupling of electric field with electric dipoles. For example, light concentration in nanoscopic regions causes modulations of electromagnetic field at spatial distances comparable to the size of atomic systems, which may couple to electric-quadrupolar or higher-order moments. In addition, due to their large refractive index, dielectric nanomaterials offer the possibility of strong concentration of magnetic fields34 or measured21,35. Contrary to the previous works mentioned above, we evaluate not just transition rates, but also energy shifts. In the case of an individual atomic system we evaluate Purcell enhancement of spontaneous emission and Lamb-shift modification in nanostructured dispersive environment. Additionally, we provide expressions to evaluate interaction strengths and collective emission rates of multiple atomic systems immersed in the same environment, while considering the magnetic-dipolar and electric-quadrupolar interaction channels. Our method exploits field quantization in dispersive media36, in which the optical properties of the environment are expressed in terms of the electromagnetic Green’s tensor, defined by the spatial and spectral dependence of the electric permittivity. Our result is an extension to those results previously obtained in electric-dipole approximation in refs. 37. Here, we include two next-order terms of the multipolar coupling Hamiltonian38, namely the magnetic dipole and the electric quadrupole. This unlocks qualitatively new routes to tailor light-matter coupling exploiting interference effects.In this work, we develop an analytic theory based on the multipolar coupling Hamiltonian, in which the spatial extent of the atomic system is taken into account in terms of multipolar transition moments rather than wavefunctions. These moments can be evaluated with smaller computational effort than full wavefunctions, found in literatureThe work is organized as follows: in the next section we introduce the general framework used to describe light-matter coupling in the presence of dispersive and absorptive structured materials beyond electric-dipole approximation. Examples of application of the theory to specific geometries are given at the end of the Results section. Detailed numerical results and lengthy calculations are provided in the Supplementary Information available online.In this section, we introduce the general framework used to describe light-matter coupling in the presence of dispersive and absorptive structured materials beyond electric-dipole approximation. After giving a statement of the problem, we offer expressions for spontaneous emission rate and Lamb shift of a single atomic system in such environment at first. Next, we generalize these expressions to many-atom systems. Finally, examples of application of the theory to specific geometries are given. Lengthy calculations are documented in Supplementary Information.We assume that the atomic system can be approximated by two active energy levels, separated by energy 42. We restrict ourselves to nonmagnetic matter with relative permeability In this work we follow the quantization scheme in dispersive and absorbing media, developed in refs. Since we investigate vacuum-induced effects, we are interested in the case of a vanishing mean electric field. Then, the only field is related to noise current fluctuations and can be expressed as The dyadic tensor 41The requirement for the canonical equal-time commutation relations of quantized fields to hold allows one to express the noise current in the form41Here, Finally, the electric field can be expressed in terms of bosonic operators as follows In the Schrödinger picture, the positive frequency part The free-field Hamiltonian reads 16, near picocavities or even in free space when geometrically large emitters like semiconductor quantum dots are considered31. For this reason, we include two higher-order terms of the multipolar coupling Hamiltonian, which include first-order spatial derivatives of the electric field38. The Hamiltonian is given in the rotating wave approximation, valid as long as the coupling strengths are small with respect to the transition frequency The degree of freedom related to the two-dimensional Hilbert space spanned by the dipole moment vectors and the quadrupole moment tensor have elements corresponding to the finally, the fields depend on position in space The electric dipole approximation is based on the assumption that the electric field can be approximated as constant across the spatial extent of the atomic system. In particular, this means that direct coupling of the emitter with the magnetic field is ignored and spatial modulations of the electric fields are neglected as well. Typically, the above assumption is very well met: the scale of spatial modulations of the electric field is set by its wavelength, usually in the optical or near-infrared range, while the modulations of the field’s envelope are even slower and thus negligible. This assumption holds true in free space or traditional cavities, if the atomic system is represented by an atom or a molecule. However, the assumption might no longer be applicable if the emitter is positioned within a subwavelength electromagnetic hotspot, e.g. in photonic crystal cavities, in close vicinity of plasmonic nanostructures9, where only the electric dipole interaction term was taken into account. Since the equations tend to be lengthy, we describe the consecutive steps in detail in Section 1 of the Supplementary Information. The effective evolution of the atomic system reads To derive the spontaneous emission rate of an atomic system, we proceed as follows: First, Heisenberg equations are found for the atomic and the field operators. The equations for the field are then formally integrated, so that the field operators are expressed through the atomic ones. The result is then inserted into atomic equations, so that field variables are completely eliminated from the description. The resulting complicated integro-differential equations are simplified in the Markovian approximation, where the memory effects are neglected. As a result, we obtain effective dynamics of the atomic system alone. The procedure is a generalization of the one introduced in ref. 43.where the fields are always evaluated at 44.The Lamb shift in Eq.  is expret in Eq.  is \docuBefore we move to the next section, we will make a few general comments.45.The emission and frequency shifts originate from coupling to quantized electromagnetic vacuum and can be derived in the quantum model, albeit their enhancement with respect to free space can be expressed with classical quantities. The enhancement depends on the geometry of the surroundings of the emitter, which determines the properties of the field, here accounted for in terms of the Green’s tensor. The tensor is governed by the classical electric permittivity. Such description does not take into account the electron tunneling effects which might appear between the atomic system and metallic nanoparticles for separations below 1 nm46. The homogeneous term corresponds to the response in free space or a homogeneous medium, while the scattered one describes influence of scatterers in the environment, e.g. extended interfaces among different media, nanostructured particles, or photonic crystals. The contribution of the homogeneous part of the Green’s tensor is already included in the homogeneous-medium spontaneous emission rate or the respective Lamb shift, while the contribution of the scattered part is of general interest. The scattered contribution is frequently finite at the origin, as The Green’s tensor 28.From an analysis of the generalized multipole moment, we find that the electric dipole component depends on the imaginary part of Green’s tensor, while the electric quadrupole and the magnetic dipole components are proportional to the sum and difference of the corresponding derivatives of the imaginary part of the Green’s function: Generalizing the expressions for the transition rate beyond the electric dipole approximation not only allows to consider corrections to the atomic systems’ dynamics in cavities of extreme geometries. More importantly, it is a tool to describe, e.g. optical activity of chiral molecules for which an interplay between the electric and magnetic dipolar coupling of matter and light, i.e. interference of the two transition mechanisms, plays a crucial role. In centrosymmetric systems, parity is a good quantum number, allowing one to identify transitions either described by the electric dipole mechanism or a combination of electric quadrupole and magnetic dipole. Such systems could be considered sources of photons with well-defined parity, corresponding to a given transition mechanism.The same formalism can be applied to the case where multiple two-level atomic systems, indexed with In the case of multiple atomic systems, the Hamiltonian from Eq.  should bSteps to derive evolution equations of atomic operators in the Markovian approximation are listed in Section 2 of the Supplementary Information. The resulting equations read For a better understanding, it is useful to note that the same equations can be derived for a collection of atomic systems described by an effective Hamiltonian of the form n in Eq. , the phowhere the principal value integral is no longer present and the integration is now performed along the imaginary axis. There, the Green’s tensor shows greater numerical stability due to its decaying rather than oscillating character.9. Due to the enhancement of off-diagonal elements of the Green’s tensor, long-range coupling of multipoles or, in general, arbitrary nonparallel orientations may be enabled. Due to the strong field localization, corrections beyond the electric dipole approximation in such systems may be significant, and coupling of different multipoles is possible. Interference of different interaction components may lead to further enhancement or suppression of interaction strength, resulting in a corresponding modification of interaction distance. Please note that due to the large width of the peak in the density of states, assumed in Section 2 of the Supplementary Information, atomic systems with slightly different transition frequencies may in general be coupled.The multipole-multipole interaction strength 47.The dissipators in Eq.  include 49. In the latter case, we demonstrate that in a suitably engineered environment contributions beyond electric dipole may have a significant impact on the atomic system’s dynamics.In this section we apply the formulas derived above first to the case of a homogeneous background dielectric and second to an exemplary selected nanostructured environment into which the atomic system is placed. In the first considered case, our goal is to retrieve familiar scaling of different multipolar components of spontaneous emission rates with different powers of refractive indexIn a homogeneous, isotropic, and infinitely extended medium the Green’s tensor takes the form via higher-order channels51, which scale with the third power of the refractive index49. Similarly, multipole-multipole interaction terms can be retrieved.In Section 4 of the Supplementary Information we show that away from the medium resonances the imaginary part of the Green’s tensor is diagonal in the limit of 50: a Maxwell equation solver based on the Boundary Element Method54. Full Maxwell equations were solved. Silver was modeled using data from ref. 55. The tensor is calculated at the frequency To offer also an example of a structured photonic environment, we consider here a pair of silver nanospheres of 40 nm diameter, separated by a 6 nm gap inside of which an atomic system is positioned. The chosen coordinate frame is shown in the system sketch in Fig. mentclass2pt{minim28. Among possible applications this suggests potential for enhancement of optical activity, in particular circular dichroism. Among the two considered higher-order channels, the magnetic dipole transition channel dominates by two orders of magnitude over the electric quadrupole one. However, the latter is manifested through interference, which we find always destructive in the investigated region, and whose contribution to the transition rate in channels beyond the ED is of the order of We now consider a two-level atomic system with transition frequency mentclass2pt{minimmentclass2pt{minimWe now evaluate the correction of the frequency shift, i.e. the Lamb shift, of the atomic system coupled to the quantum vacuum near the two nanospheres. Please note that the full Green’s tensor diverges at the origin. However, we are only interested to evaluate the correction of the shift with respect to free space, determined by the scattered part of the Green’s tensor. The result obtained for selected positions within the gap is shown in Fig. Finally, we evaluate the interaction strengths ∣ξ∣ Fig.  and collγ12 Fig.  for a paWe have studied dynamics of atomic systems coupled to a photonic environment in its vacuum state. The environment is described in terms of the electromagnetic Green’s tensor and the interaction contributions beyond the paradigmatic electric dipole approximation have been included. The derived formalism allows us to evaluate dynamical parameters characterizing optical properties of atomic systems: both the individual and the collective contributions to energy shifts and decay rates. Inclusion of terms beyond the electric dipole approximation allows us to study the role of higher multipolar channels, including enhancement or suppression through interference of different interaction mechanisms. Examples of phenomena to be investigated are optical activity, multipole-multipole interactions between atomic systems and spontaneous emission suppression due to interference of different mutipolar channels in tailored photonic environments.42. To derive equations presented in this work we apply the Markovian approximation, as described in detail in the Supplementary Information available online.Our approach is based on quantization scheme introduced in refs. 50: a Maxwell equation solver based on the Boundary Element Method54. Full Maxwell equations were solved. Silver was modeled using data from ref. 55.Green’s tensor corresponding to the pair of metallic nanospheres in the Examples section was calculated using the MNPBEM toolbox for MATLABSupplementary Information.


As one of the core issues of autonomous vehicles, vehicle motion control directly affects vehicle safety and user experience. Therefore, it is expected to design a simple, reliable, and robust path following the controller that can handle complex situations. To deal with the longitudinal motion control problem, a speed tracking controller based on sliding mode control with nonlinear conditional integrator is proposed, and its stability is proved by the Lyapunov theory. Then, a linear parameter varying model predictive control (LPV-MPC) based lateral controller is formulated that the optimization problem is solved by CVXGEN. The nonlinear active disturbance rejection control (ADRC) method is applied to the second lateral controller that is easy to be implemented and robust to parametric uncertainties and disturbances, and the pure pursuit algorithm serves as a benchmark. Simulation results in different scenarios demonstrate the effectiveness of the proposed control schemes, and a comparison is made to highlight the advantages and drawbacks. It can be concluded that the LPV-MPC has some trouble to handle uncertainties while the nonlinear ADRC performs slight worse tracking but has strong robustness. With the parallel development of the control theory and computing power, robust MPC may be the future direction. Autonomous driving technology has become a research and development hotspot recently, since it has great potential to improve active safety, alleviate traffic congestion, and reduce energy consumption ,2. It waThe key technologies of autonomous vehicles consist of environment perception, decision making, motion planning, and vehicle motion control, and the objective of vehicle motion control can be divided into path stabilization and trajectory stabilization . The difA longitudinal speed controller based on sliding mode control with nonlinear conditional integrator is proposed to realize the longitudinal motion control of autonomous vehicle during the path following process. It is well known that sliding mode controllers suffer from chattering. To solve this problem, we adopt the saturation function to replace the sign function, meanwhile, a nonlinear integral action is introduced to achieve zero steady-state error and improve the transient performance, which can also avoid integral divergence. Stability analysis is given to prove that the equilibrium point is globally asymptotically stable.The lateral controllers based on LPV-MPC (Linear Parameter Varying Model Predictive Control), ADRC (Active Disturbance Rejection Control), and PPM (Pure Pursuit Method)are designed, respectively. First, because the MPC can exploit available preview information and handles constraints, an MPC lateral control-based strategy that considers the soft constraints of the sideslip angle of the steer wheel is formulated based on the error dynamics model, which adopts the CVXGEN  solver tMultiple simulations in different scenarios are conducted to validate the effectiveness and capability of the proposed longitudinal speed controller and lateral path following controllers. A comparison of the three mentioned lateral controllers is made to highlight the advantages and drawbacks of each approach in path following. Finally, the possible development direction in the future is given.In this paper, we design, validate, and compare three path following schemes to track a predetermined path, using linear parameter varying model predictive control, nonlinear active disturbance rejection control , and theThe rest of this paper is organized as follows: Extensive work has been done to vehicle motion control of autonomous vehicles. In this section, we provide a brief review of the state of the art. Vehicle motion control can be divided into longitudinal control and lateral control. Further, lateral control methods can be divided into geometry based, kinematics based, and dynamics based.The longitudinal control manipulates drive and brake actuators to guide the autonomous vehicle to track a desired speed profile. The main control methods can be roughly classified into three categories: (1) Model-free control. The system dynamics are regarded as a black-box, and the control commands are generated from tracking errors only, e.g., the proportional-integral-derivative (PID) design . (2) ModThe lateral control manipulates the steering actuator to deal with the path following problem generally, and the direct yaw moment control can be used as an additional approach for distributed drive vehicles and vehicles equipped with differential braking. According to the difference of the system model, path tracking control methods can be roughly classified into three categories: (1) Geometry based. It used the geometric relationship between the vehicle and reference. Classical algorithms include the pure pursuit method and Stanley algorithm , which w problem . CompareIn order to describe the motion of an autonomous vehicle, the system dynamics of path following is defined, as illustrated in Considering a vehicle moving on an inclined road, a force balance along the vehicle longitudinal axis yields:The wheel dynamics can be expressed as the following equation:The objective of lateral control design is to eliminate the path error while guaranteeing vehicle stability. As shown in The preview lateral kinematics can be modeled as:The 2DOF vehicle dynamics model is used to design the lateral controller, which can be expressed as:The combined dynamics (3) and (4) can be written in the state space form, with the states The objective of longitudinal speed control is to track the desired speed profile accurately and smoothly, by manipulating the motor and electro-hydraulic brake system (EHB) in this paper. To improve the robustness to parametric uncertainties and provide disturbance rejection, a speed tracking controller via sliding mode control with nonlinear conditional integrator is developed. The control framework is shown in From the control block diagram in According to the desired longitudinal acceleration As shown in Equation (1), the road resistance concludes the resistance force from the vehicle’s rolling Then, the drag compensation is obtained from:The total feedforward longitudinal torque is expressed as:Robust asymptotic tracking can be achieved by the sliding mode control, therefore, it is applied here. To simplify the longitudinal dynamic model, assuming that the lateral velocity is zero, then we can get:Using the pure rolling assumption that Combining Equations (10) and (11):Define the tracking error It is typical to choose the sliding surface as Zero steady-state error can be achieved by including integral action in the controller. This was done by Khalil  by augmeIn this paper, the linear integrator is replaced by a nonlinear conditional integrator to avoid integral divergence and improve the transient performance:At the actuator level, the electric brake is used for slight braking case to improve response speed and control accuracy, and the hydraulic brake compensation is used for heavy braking case.To analyze the stability performance of the sliding mode controller with the conditional integrator, it is carried out from two aspects of inside the boundary layer and outside the boundary layer.Substituting Equation (15) into Equation (13), we can have:The closed-loop system has a unique equilibrium point Selecting a Lyapunov candidate function:Differentiating the left and right sides of Equation (20), it can be expressed as:Substituting Define a function Combining Equation (23) and If the inequality Substituting Equation (15) into Equation (13), the control law outside the boundary layer can be expressed as:It can be seen that the feedback control input converges to To obtain a simple, reliable, and robust path following controller that can handle complex situations, three lateral control schemes are proposed and compared. First, a LPV-MPC lateral control-based strategy is designed due to its ability of preview and handles constraints. Second, considering the simple control structure and good robustness, the nonlinear ADRC is applied to develop the second lateral controller. The third controller is based on the pure pursuit method, which is provided as a benchmark.Despite nonlinear MPC is attractive because of its ability to capture the nonlinearity of vehicle dynamics, the model complexity limited real-time implementation. Here, a LPV-MPC control law is designed.The continuous vehicle model Equation (5) is discretized at each time step with a zero-order hold approximation that assumes constant continuous inputs over the duration of each discrete time step. The resulting discretized vehicle model is:To eliminate the steady-state error, we change the control input Assuming that the current moment is To reduce the notational complexity, the predicted outputs can be expressed as:The objective of the lateral controller is to track the desired path accurately and smoothly. Therefore, define the cost function as:Substituting Equation (31) into Equation (32), the cost function can be rewritten as:Considering the constraint Considering the constraint Considering the outputs constraint Considering the states soft constraint of slip angle of front axle Combining Equations (35)–(38), the constraints of control increment sequence The final optimization problem takes the form:For the quadratic program representable problem as shown in Equation (40), CVXGEN provides a custom, high-speed solver that is library-free C code, which make an online solution as fast as possible. The optimization resolves each time step, as is standard with MPC, and the first element Since active disturbance rejection control has strong robustness to system uncertainties and external disturbances, and can be implemented easily, so the nonlinear ADRC is applied to design the lateral controller in this paper. The ADRC is evolved from proportional-integral-derivative (PID) and it inherits the core idea of PID that makes it such a success: the error driven, rather than model-based .As shown in Consider a single input and single output nonlinear n-order dynamic system:To provide the fastest tracking of Finally, the control law using nonlinear feedback combination is proposed as:It should be pointed out that the controller coefficients are not dependent on the mathematical model of the plant, thus making ADRC largely model independent.For the ADRC method, it is not suitable to process the single input multiple output coupling problem. Therefore, we only consider the lateral error stabilization in the ADRC scheme design. According to the lateral vehicle dynamics Equation (5), we can have:Rewrite the equation as the state space form:The ESO for lateral controller is designed as:The control law using nonlinear feedback combination can be expressed as:The pure pursuit method (PPM) is used as benchmark, shown in Based on the simple geometric bicycle model of an Ackerman steered vehicle, the pure pursuit control law is given as:In order to evaluate and compare the performance of the proposed path following controllers, a series of simulation experiments have been conducted. The main parameters of the autonomous vehicle are given as To validate the effectiveness of the proposed longitudinal speed tracking controller, stair step and sinusoidal tests have been carried out respectively, and simulation results are shown in To illustrate the steady state characteristics of the proposed lateral controllers for autonomous vehicles, the skid pad test was chosen. As shown as To investigate the transient response capability of the autonomous vehicle under the conditions of variable and large curvature, the double lane change path is designed, referring to the ISO 3888-2:2002 obstacle avoidance case, as shown in 2.In this case, the autonomous vehicle was set to pass through the test area at a constant longitudinal speed of 5 m/s, and the corresponding maximum lateral acceleration is about 1.2 m/sThe control results of the proposed lateral controllers are shown in 2 so that the tires are still in the linear zone.The desired longitudinal speed is set to 10 m/s in this case, and the corresponding maximum lateral acceleration is about 4 m/sThe responses are presented in 2 so that the tires show highly nonlinear characteristic.The desired longitudinal speed is set to 15 m/s in this case, and the corresponding maximum lateral acceleration is about 8 m/sThe simulation results are shown in As shown in An intuitive evaluation chart is formed by scoring the KPI (Key Performance Indication) of the simulation results, as shown in For model predictive control, further research is needed to build a good model that tradeoff between high fidelity and simple enough, and robust and stochastic MPC may be the future direction.In this paper, three control schemes have been designed, verified, and compared for the path following application of autonomous vehicles. A longitudinal speed tracking controller is proposed based on sliding mode control with nonlinear conditional integrator, and stability analysis is also given. Considering the ability of preview and handling multi-constraints, the first lateral controller is based on the linear parameter varying model predictive control, which shows excellent tracking accuracy in the simulation tests. Though it requires real-time computations to solve the optimization problem, using the CVXGEN can achieve high efficiency to solve the QP problem. Nonlinear active disturbance rejection control is applied in the second lateral controller since ADRC is robust against the parametric uncertainties of vehicles. However, the comprehensive tracking performance is slightly worse than the MPC that was shown in comparison. The lateral controller via the pure pursuit method is used for a benchmark, which performs well in low speed case. The effectiveness and ability of the proposed schemes are validated by simulations. It should be noted that the LPV-MPC controller is relatively sensitive to the parametric uncertainties and the vehicle dynamics is highly nonlinear in large acceleration cases, therefore, the robust nonlinear model predictive control is our future research direction.


The automatic detection of epilepsy is essentially the classification of EEG signals of seizures and nonseizures, and its purpose is to distinguish the different characteristics of seizure brain electrical signals and normal brain electrical signals. In order to improve the effect of automatic detection, this study proposes a new classification method based on unsupervised multiview clustering results. In addition, considering the high-dimensional characteristics of the original data samples, a deep convolutional neural network (DCNN) is introduced to extract the sample features to obtain deep features. The deep feature reduces the sample dimension and increases the sample separability. The main steps of our proposed novel EEG detection method contain the following three steps: first, a multiview FCM clustering algorithm is introduced, and the training samples are used to train the center and weight of each view. Then, the class center and weight of each view obtained by training are used to calculate the view-weighted membership value of the new prediction sample. Finally, the classification label of the new prediction sample is obtained. Experimental results show that the proposed method can effectively detect seizures. Epilepsy is a common mental disease in neurology. It is a chronic neurological disease caused by sudden and temporary disturbance of brain function due to the paroxysmal abnormal discharge of brain neurons . WorldwiAt present, the diagnosis of epilepsy is mainly through the patient and his relatives and friends, as much as possible to obtain a complete and detailed history of the patient's seizure and then be referred to the EEG for examination. The determination of seizures is mainly based on the patient's clinical manifestations and EEG records. EEG can record the abnormal electrical activity of neurons in patients' brains during epileptic seizures, so it has become an important tool for exploring the characteristics of epilepsy. It has been widely used in various epilepsy studies including epilepsy diagnosis, seizure location, qualitative research, seizure prediction, and seizure control. Until now, electroencephalogram examination is still mainly carried out by medical staff visual observation combined with their own experience. However, relying on manual recognition of epilepsy EEG has many drawbacks. First, with the advancement of medical conditions and the development of EEG acquisition equipment, the amount of EEG data is getting larger and larger, and the duration of EEG of a patient generally exceeds tens or even hundreds of hours. In the face of a large amount of EEG data, manual detection is time-consuming and efficient. Second, some features of epilepsy on EEG are very subtle and difficult to recognize by human eyes alone. Third, when medical staffs work for a long time, their judgment will be affected, which will easily lead to an increase in the false detection rate. With the in-depth research and development of computer technology, automatic detection and recognition of epilepsy EEG signals using computers as auxiliary tools have become an important auxiliary detection method. The study of automatic epilepsy detection technology becomes extremely important.In view of the fact that EEG signals are a type of time series, the time-frequency analysis method is widely used when researching automatic epilepsy detection algorithms. Such methods analyze the time domain or frequency domain of epilepsy EEG signals or combine the two to capture the characteristic information of epilepsy EEG signals. Some time-frequency analysis methods such as wavelet analysis , 5, S trConstruct the original multiview data set of epilepsy EEG. Different EEG analysis methods were used to obtain the data set under three perspectives of spectrum signal, time domain signal, and time domain signal of EEGExtract deep features. Use deep DCNN to extract deep features of EEG signals from 3 perspectives, respectively, so as to obtain deep features from each perspective. This deep feature has a better ability to recognize epilepsy. And construct a multiview data set with deep featuresTrain a multiview classifier. First, a multiview FCM clustering algorithm is introduced, and the training samples are used to train the center and weight of each view. Then, the class center and weight of each view obtained by training are used to calculate the view weighted membership value of the new prediction sample. Finally, the classification label of the sample is obtainedAt present, epilepsy detection technology has made great progress and many important research results have been achieved, but there are also some problems in the existing research. For example, (1) in the study of epilepsy detection based on EEG signals, different features such as the time domain and time-frequency domain of EEG are often directly combined with machine learning algorithms to identify different EEG patterns –26. DiffThe EEG signals of patients with epilepsy are generally divided into four periods, namely, interseizure period, preseizure period, seizure period, and late seizure period. The interval between seizures refers to the EEG signals collected by epilepsy patients when the seizures are not occurring. The preseizure period refers to the EEG signals that may be collected when epilepsy is about to occur. There is no clear regulation on the length of this period and the starting position. The seizure period refers to the EEG signals recorded during the patient's seizure. Late seizures generally refer to the collection of EEG signals when the brain is in a recovered state after the seizure has ended. The process of predicting seizures can generally be divided into the following steps. The first step is to preprocess the EEG signals. In the second step, feature extraction is performed on the preprocessed samples to reduce the sample dimensions. The third step is to use the training samples to train the classifier. The fourth step is to use the trained classifier to classify the test samples to identify whether they are epilepsy patients. The detection flow chart is shown in As an important tool for researchers to study brain function, EEG reflects the electrical activity of brain nerve cell groups on the surface of the cerebral cortex or scalp. EEG is obtained by recording continuous, spontaneous, rhythmic potential changes in the brain nerve cell population through electrodes placed in the cortex or scalp. Certain characteristics of EEG signals are usually used as an important basis for the diagnosis of brain diseases. Because it contains a large amount of physiological and pathological information, it has played a huge role in clinical medicine, especially for the location and characterization of epilepsy.The current mainstream EEG analysis methods are the time domain analysis method –29, freqIn the 1960s, when Hubel and Wiesel studied the cat's cerebral cortex, they discovered that the cat's neurons for direction selection and local sensitivity formed a unique network structure. This structure greatly reduces the complexity of the artificial neural network and the number of parameters . DCNN boIn this paper, the DCNN model is used to model the EEG data of patients with epilepsy to achieve automatic learning of epilepsy features. The DCNN structure used in the experiment is shown in Current data sets often have high dimensionality and complexity. Complex data sets usually contain many different and reasonable clustering patterns. For example, in document data classification, there are both topic-based models and writing styles. In movie classification, it can be classified according to the type of movie or according to the genre of the movie. In gene classification, clustering can be based on the structure and function of genes.When traditional clustering algorithms analyze data, they usually only focus on a single clustering pattern in the data. To understand data from a single perspective, we cannot have a comprehensive analysis and understanding of complex data. This problem has gradually attracted the attention of scholars and promoted the rapid development of multiview clustering. The multiview clustering algorithm hopes to mine multiple different and reasonable clustering structures from the data. There are two basic requirements. (1) The quality of the clustering result is high; that is, it can capture some reasonable internal structure contained in the data set itself. (2) There is no redundancy between different clustering results. Analyze the data from multiple perspectives, and discover multiple different clustering patterns in the data. The comparison between the traditional single-view clustering and multiview clustering schematics is shown in The algorithm framework is shown in DCNN is used to extract the features of the data from various perspectives to obtain the deep spectrum features, deep time domain features, and deep time frequency features. Construct a multiview data set of epilepsy EEG based on deep features. The structure of the DCNN model used when extracting deep features from various perspectives is shown in X = {x1, x2, ⋯, xK} is a multiview data set with N views. Xi represents the i-th view data, 1 ≤ i ≤ N. The u(1 ≤ u ≤ K) row vector represents the feature vector of the data object xu in the i-th view space, where ni is the characteristic dimension of the i-th perspective. The objective function of multiview FCM is as follows:m is the fuzzy index, which is used to adjust the fuzzy degree of membership. U = (ucμ)C×N is the membership matrix, where ucμ represents the membership degree of the object xμ belonging to the c-th cluster. Ok)( = {o1k)(} is the vector of the k-th class center. W = (wi)N1×, where wi represents the view i weights. d(xμpi) represents the Euclidean distance between xμ and oci), the, the8), Wi is obtained by Equation (Ui)((x) is obtained by Equation Equation .The algorithm execution steps are as follows.In order to verify the effectiveness of the proposed algorithm, the experiment mainly conducts comparative analysis from two aspects. (1) When the proposed classification method based on unsupervised multiview clustering results is used to analyze whether deep features are used, the trend of classification performance changes. The general feature extraction methods used here are PCA  and LDA The EEG database used in the experiment came from the Epilepsy Center of the University Hospital of Freiburg, Germany . For 16 In order to analyze the influence of depth features on classification performance, a classifier is fixed, and the multiview FCM is selected here. The classifier is used to process feature data obtained by different feature extraction methods. The experimental data in In order to analyze the classification performance of different classifiers on deep feature data, this paper selected three single-view classifiers and one multiview classifier. The different classifiers are applied to the deep feature data from different perspectives and the deep feature data from multiple perspectives for comparative analysis. This study proposes a new classification method based on unsupervised multiview clustering results. First, a multiview FCM clustering algorithm is introduced, and the training samples are used to train the center and weight of each view. Then the class center and weight of each view obtained by training are used to calculate the view weighted membership value of the new prediction sample. Finally, the classification label of the sample is obtained. The main contributions of the proposed method are as follows: (1) the deep feature can effectively reduce the data dimension while increasing the data separability; (2) the use of multiview data can improve the richness of data information, making decision results more stable and robust; and (3) the use of multiview classifiers improves the classification performance of traditional single-view classifiers. Experiments show that the proposed method is more automated, intelligent, and efficient than traditional EEG detection based on machine learning.