esa-sceva/satcom-chunk-collection huggingface dataset:
## 3 Results and Discussion
### Galileo Scintillation Measurements  
Other than TEC and electron density, adding the Galileo system will produce more types of GNSS scintillation observations (e.g., different line-of-sights, frequency bands, signal power levels, and modulation schemes), which has advantages for multi-scale ionospheric remote sensing investigations. Several Galileo scintillation observations from a multi-constellation HIL simulation of the baseline ESF scenario based on the 13 July 2018 almanac are demonstrated here. Figure 11a-c present the vertical distribution of Galileo E1, E5a, and E5b effective 1-Hz S4 observed by the chief satellite receiver from all eight visible PRNs. Figure 12a-c show the horizontal view of Galileo E1, E5a, and E5b effective S4 level observed by the chief receiver following the satellite ground track, as the \"snapshots\" of 1-Hz S4 taken every minute from eight different visible PRNs. Similar observations from the deputy receiver of the fleet (not shown) can be obtained with similar macroscale behavior as the chief. In total, 10 Galileo PRNs were tracked by the deputy LEO receiver during the 1 h ESF scenario. Since the amplitude scintillation modeled in the GNSS simulator is confined in a cuboid region, the high S4 observations from Galileo mostly occur in the corresponding region (290-350 km, 20-21 LT, 0-10\({}^{\circ}\) S) near the JRO as anticipated. Therefore, the observation of S4 by SFF can potentially be designed as a tracer or new indicator of ionospheric irregularities.  
Because the GNSS simulator currently is only capable of modeling a simplified ionosphere, the amplitude scintillation affects all frequency bands of Galileo and GPS similarly. A multi-constellation, multi-frequency software GNSS receiver was developed and applied to ground-based ionospheric scintillation studies by [28]. Reference [29] utilized a ground-based, multi-constellation, multi-band GNSS data collection system to observe equatorial scintillation. Reference [30] analyzed the multi-frequency responses of GNSS receivers to ionospheric scintillations and characterized the difference in receiver behavior under scintillations across multiple frequency bands. This is currently an important area of investigation and, therefore, space-based, multi-constellation, multi-frequency-band GNSS scintillation measurements will facilitate studying multi-scale ionospheric irregularity impacts on GNSS signals from the perspective of different constellations, frequency bands, modulation schemes, chipping rates, powers, etc. [31].  
Figure 11: _Cont._  
Figure 11: Vertical distribution of 1-Hz S4 observed from different Galileo frequency bands on the chief’s receiver: **(a)** Galileo E1; **(b)** Galileo E5a; **(c)** Galileo E5b.  
Figure 12: _Cont._
## 6 Tracking  
During the tracking phase, the new position of the detected target is estimated sequentially, as shown in Figure 5A, followed by the temporal association of the new estimated track and target cluster to create a continuous target track, as shown in Figure 5B. The workflow of the proposed multiple-target-tracking-algorithm is shown in Figure 6. The components of the proposed target tracker are explored in detail below.
# Influence of Crack Geometry on Dynamic Damage of Cracked Rock: Crack Number and Filling Material
## 2 SHPB Test
### Specim Preparation  
Red sandstone was chosen as the testing material for the dynamic test. Three main steps were used to prepare the cracked rock. First, cylindrical specimens with dimensions of 100 mm \(\times\) 50 mm (diameter \(\times\) length) were carefully prepared. For consistency, all cylindrical specimens were made of the same sandstone block. Second, the size and location of cracks were digitalized and reconfigured using computer-aided design (CAD) techniques. In this study, the rectangular crack parallel to the longitudinal axis of the cylindrical specimen was designed with the dimensions of 50 \(\times\) 40 \(\times\) 4 mm. Subsequently, the cylindrical specimens were cut and engraved following the schematics of the specimen and using a computer numerical control (CNC) water-jet machine to obtain the cracked rock specimen. In this study, the man-made crack in the specimen was filled. The process of filling the crack was as follows: (1) Five types of filling materials were prepared: soil, sand, and materials with the weight ratio of soil/sand = 2, 1, and 0.5. (2) The filling material was poured into the crack three times. After each pour, the specimen was placed on the vibration table for 3 min of vibration. The schematics of the specimen with one crack is shown in Figure 3.  
Figure 3: Schematics of the specimen with one crack.  
Figure 2: Strain gauges used in SHPB testing system.  
To further study the properties of the filling materials (i.e., ISO sand and Liaoning soil), triaxial tests were carried out. The cohesive strength and strength of the ISO sand and Liaoning soil used as the filling materials were measured. The test results are shown in Table 1.  
The intact rock mass and cracked rock specimens containing one, two and three cracks were studied to investigate the effects of crack number on the dynamic damage of cracked rock, as shown in Figure 4a. Also, the cracked rock specimens with two cracks filled with sand, soil and the mixtures with the specified sand/soil ratios were studied to investigate the effects of filling material on the dynamic damage of cracked rock, as shown in Figure 4b. Each test group included three specimens to guarantee the repeatability of the experimental study.
## 4 Experimental Results and Discussion
### Performance of Data Augmentation and Rebalancing  
Three datasets are used and tested in this subsection, namely, original, augmented, and balanced datasets. The original dataset refers to the Chla in situ data acquired from Laguna Lake, the Philippines. The augmented and balanced datasets are the augmented in situ datasets without and with, respectively, the consideration of in situ Chla concentration unbalancing. The comparisons of the proposed transfer learning using these three datasets are shown in Figure 8. The results indicated that the RMSEs of the training using the original dataset ranged from 0.5 \(\upmu\)g/L to 1.0 \(\upmu\)g/L. By using the augmented dataset, the RMSEs of estimated Chla concentrations ranged from 7.5 \(\upmu\)g/L to 9.5 \(\upmu\)g/L. This is caused by the fact that more Chla samples in the augmented dataset are in the Chla concentration ranges 7.5-8 \(\upmu\)g/L and 9.5-10 \(\upmu\)g/L. Consequently, the sample imbalance on the Chla concentrations makes the performance of the trained model worse than that trained using the original dataset. When the data rebalancing that considers the distribution of samples' Chla concentrations in the augmented dataset is performed, the RMSEs of  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Fold**} & \multicolumn{3}{c}{**Transfer Learning Performance (RMSE in \(\upmu\)g/L)**} \\ \cline{2-4}  & **Pretraining Stage** & \multicolumn{2}{c}{**Transfer-Learning**} \\ \cline{2-4}  & **Main-Training Stage** & **Fine-Tuning Stage** \\ \hline
1 & 2.070 & 0.689 & 0.478 \\ \hline
2 & 2.144 & 0.562 & 0.229 \\ \hline
3 & 2.113 & 0.487 & 0.219 \\ \hline
4 & 2.089 & 0.606 & 0.430 \\ \hline
5 & 2.190 & 0.576 & 0.414 \\ \hline
6 & 2.120 & 0.517 & 0.284 \\ \hline
7 & 2.194 & 0.609 & 0.441 \\ \hline
8 & 2.228 & 0.560 & 0.387 \\ \hline
9 & 2.216 & 0.633 & 0.508 \\ \hline
10 & 2.079 & 0.581 & 0.336 \\ \hline
**Avg.** & 2.144 & 0.582 & 0.372 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Performance of training stages in the proposed transfer learning.  
the estimated Chla concentrations are improved to 0.5-0.7 \(\mathrm{\SIUnitSymbolMicro\SIUnitSymbolMicro\SIUnitSymbolMicro}\)/L. Similar statistical results are shown in Figure 9, where the correlation coefficient between the estimated and in situ Chla concentrations was improved when the data rebalancing was performed with data augmentation.  
Figure 6: Maps of estimated Chla concentrations using the trained ANN model. The Sentinel-3 images are shown in false color combination (R: Band 17; G: Band 5; B: Band 3).  
Figure 7: Feature maps in the trained ANN model. Chla concentration estimation map at 6 April 2019 (**left**) and the corresponding spatial features as the output of the feature extraction phase in WaterNet (**right**).
## 3 Results and Discussion
### Imaging on a Hazy Day  
As is well known, the presence of disturbances on the transmission path, such as clouds, rain, and haze, poses significant challenges to the effectiveness of optical imaging. Starting from 4 p.m. in the winter, we conducted intensity imaging measurements of a haze scene with a PM 2.5 particle concentration of 167 \(\upmu\)g/m\({}^{3}\) based on data from the China Air Quality Index. The utilization of gating technology and polarization filtering technology enables this system to achieve imaging detection beyond the line-of-sight, as shown in Figure 3. The roof platform of a structure was captured at an approximate distance of 870 m, utilizing a resolution of 128 \(\times\) 128 pixels. Subsequently, the logo of a building was imaged at a distance of around 4665 m, again at the same resolution. Additional  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Sampling Rate** & **50\%** & **20\%** & **10\%** & **5\%** \\ \hline Sampling & & & & \\ Pattern & & & & \\ \hline Compressed & & & & \\ Sensing & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 2: Capability of the compression-aware algorithm to reconstruct the image as the sampling rate diminishes.  
objects, i.e., a building and a TV tower, were imaged at distances of roughly 6000 m and 6570 m, respectively, both at a resolution of 128 \(\times\) 128 pixels. In hazy conditions, both the incident and echo photons may be scattered by haze particles, leading to a weaker signal and stronger noise photons. This effect becomes more pronounced as the distance increases.
# Driverless Bus Path Tracking Based on Fuzzy Pure Pursuit Control with a Front Axle Reference
## 3 Simulations
### Simulation of Lateral Control  
#### 3.1.1 Stability Analysis  
In order to verify the stability of the improved method, PPC-FAR and PPC-FAR are compared in lane-keeping. The initial lateral error of bus to the desired path is 1 m, and the initial heading error is 0. Referring to Equation (11), \(k\) is identified as 1. The preview point is calculated according to the look-ahead distance. As Figures 14 and 15 show, the simulations are carried out in different look-ahead distances and velocities. Additionally, look-ahead distance is described as the distance between the reference point and the preview point.  
The performances of PPC-RAR are indicated in Figure 14. The desired path and the path of the driverless bus are illustrated in Figure 14a. Meanwhile, the steering angle is shown in Figure 14b. To highlight the stability of PPC-FAR, the heading error is shown in Figure 14c and the lateral error is indicated in Figure 14d. It is obvious that the controller oscillates when the lateral error is small. The velocity is 10 km/h and the look ahead distance is 20 m. The oscillation is more obvious in Figure 14b,c. The reason is that the kinematic model ignores the lateral dynamics of the bus, and the influences are exaggerated in the driverless bus.  
The same as Figure 14, the performances of PPC-FAR are illustrated in Figure 15. Figure 15a shows the desired path and the path of the driverless bus. Figure 15b illustrates the steering angle. Figure 15c indicates the heading error, and the lateral error is shown in Figure 15d. Notably, the driverless bus tracks the desired path without oscillation when the velocity is below 30 km/h. However, the system oscillates when the velocity increase to 60 km/h. The simulation results verify that the stability of PP is improved in PPC-FAR when the velocity is lower than 30 km/h. That also means that PPC-FAR is suitable for low-velocity conditions in the driverless bus.  
Meanwhile, PPC-FAR uses less distance to merge the lateral error, but the maximum heading error of PPC-FAR is larger than PPC-FAR. That is because aggressive steering strategies eliminate the lateral error more quickly.  
Moreover, Table 4 lists the final state of system and final state lateral error with different look-ahead distances. Both PPC-RAR and PPC-FAR control the bus to track the desired path. This illustrates that the assumptions about the kinematic model are suitable to driverless buses. However, the oscillations verify that the ignorance should be considered in the path-tracking controller. Meanwhile, the high accuracy of PPC-FAR illustrates that the modification to PP is meaningful and suitable to driverless buses.  
Figure 15: Performance of PPC-FAR. (**a**) Desired path and bus’s path under different conditions. (**b**) Steering angle. (**c**) Heading error. (**d**) Lateral error.
## 1 Introduction  
Following the 2007 study linking hyperactivity in 300 children to consumption of foodstuff and beverages using synthetic azo dyes along with preservative sodium benzoate [1], the six azo dyes synthetically derived from tar (sunset yellow, quinoline yellow, carmoisine, allura red, tartrazine, ponceau 4R) widely employed as pigment food additives are increasingly replaced by naturally derived colorants [2].  
Water-soluble sunset yellow (E110 in the European labeling for food additives, Yellow 6 in the USA), quinoline yellow (E104, Yellow 10 in the USA), and tartrazine (E102, Yellow 5 in the USA) water-soluble \"aza\" dyes can be replaced by combinations of natural colorants such as curcumin, paprika, anntato, lutein, caroten, and crocin.  
For example, the 2011 guidelines for the replacement of artificial dyes commissioned to industry by the Food Standards Agency in Scotland highlighted that \"smoked haddock used to be colored with either tartrazine or quinoline yellow, but these are now a combination of anntato and curcumin, while another company uses curcumin and paprika in its haddock fillets\" [3].  
The need to access natural yellow dyes of higher stability, lower cost, and abundant supply is widespread, especially considering that tartrazine is not only used in foods, but also in personal care and cosmetic and medication products [4], and that carcinogenic contaminants in Yellow 5 such as benzidine, whose limit in the USA is 1 part per billion, in certain dye batches were found to actually amount to 83 ppb (free and bound benzidine) [5].  
The main problems with biocolorants derived from natural sources are the lower chemical and physical stability and higher cost (2-10-fold higher [2]) compared to synthetic counterparts.  
Chemical technologies such as microencapsulation or dye stabilization with naturally derived polyphenols can be used to develop biocolorant formulations whose stability is comparable to that of artificial dyes. In the field of red colorants, examples include lycopene micronized crystals formulated in glycerol [6] and betanin in water from _Opuntia focus-indica_ peel stabilized by the fruit peel biophenols [7]. In 2019, some of us identified and isolated limocitrol 3-O-6\"-[3-hydroxyl-3-methylglutary]]-\(\beta\)-p-glucopyranoside, namely a flavonol glycoside called yellow #15, in the ethanol extract of _Citrus limon_ fruit zest [8]. In this study, we investigate the technical and economic feasibility of yellow #15 production from waste lemon peel, namely a by-product of the citrus industry widely employed as raw material to manufacture highly valued pectin [9] and lemon essential oil, in the context of the rapidly unfolding lemon bioeconomy [10].
## 3 Results and Analysis
### Pozzolan Reactivity  
As shown in Tables 4 and 5, the pozzolan reactivity index differed significantly among the different types of mineral admixtures. Higher compressive strength led to a larger index of pozzolan activity. At an age of 7 d, except for FA10, which had an index of \(-0.10\), the indices of the admixtures with a 10% ratio were higher than the admixtures with a 30% mixing ratio, and the indices of DLS10 and GLP10 were 1.02 and 1.53, respectively. DLS showed no pozzolan reactivity, possibly because hydration derived CaCO\({}_{3}\) increased the concrete strength [69]. As hydrated glass powder contains a high content of SiO\({}_{2}\), the reactivity index of GLP10 was greater. After 28 d, all mineral admixtures with a 30% mixing ratio had higher pozzolan reactivity indices than those of the same admixture with a 10% mixing ratio. Moreover, the pozzolan reactivity index of FA30 increased the most from 7 d to 120 d among admixtures with a replacing ratio of 30%, and the pozzolan reactivity index increased from 0.45 to 1.11. This is the only sample with a greater reactivity index than that of the same admixture at a mixing ratio of 10%, indicating the occurrence of a pozzolan reaction.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multirow{2}{*}{**Sample**} & \multirow{2}{*}{**Comenti(\%)**} & \multirow{2}{*}{**Compressive Strength (MPa)**} & \multirow{2}{*}{**Strength/Cement Ratio (\%)**} & \multirow{2}{*}{**Pazzolan Strength/Cement Ratio (\%)**} & \multirow{2}{*}{**Offer of Polarization Polarization Activity (\%)**} \\  & & & & & & & & \\ \hline \multirow{8}{*}{7} & MA0 & 100 & 32.48 & 0.325 & 0 & 1 & 0 & — \\  & FA10 & 90 & 28.96 & 0.322 & \(-0.003\) & 0.990 & \(-1.0\) & \(-0.10\) \\  & FA30 & 70 & 26.26 & 0.375 & 0.050 & 1.154 & 13.4 & 0.45 \\  & BFS10 & 90 & 32.51 & 0.361 & 0.036 & 1.111 & 10.0 & 1.00 \\  & BFS30 & 70 & 31.11 & 0.444 & 0.119 & 1.367 & 26.9 & 0.90 \\  & DI\(\downarrow\)S10 & 90 & 32.59 & 0.362 & 0.037 & 1.114 & 10.2 & 1.02 \\  & DL\(\downarrow\)S30 & 70 & 23.43 & 0.335 & 0.010 & 1.030 & 2.9 & 0.10 \\  & GLP10 & 90 & 34.53 & 0.384 & 0.059 & 1.181 & 15.3 & 1.53 \\  & GLP30 & 70 & 28.53 & 0.408 & 0.083 & 1.254 & 20.3 & 0.68 \\ \hline \multirow{8}{*}{28} & MA0 & 100 & 37.21 & 0.372 & 0 & 1 & 0 & — \\  & FA10 & 90 & 31.31 & 0.348 & 0.023 & 1.070 & 6.6 & 0.66 \\  & FA30 & 70 & 30.05 & 0.429 & 0.104 & 1.321 & 24.3 & 0.81 \\  & BFS10 & 90 & 36.2 & 0.402 & 0.077 & 1.238 & 19.2 & 1.92 \\  & BFS30 & 70 & 32.63 & 0.466 & 0.141 & 1.434 & 30.3 & 1.01 \\  & DI\(\downarrow\)S10 & 90 & 38.29 & 0.425 & 0.100 & 1.309 & 23.6 & 2.36 \\  & DLS0 & 70 & 26.87 & 0.384 & 0.059 & 1.181 & 15.3 & 0.51 \\  & GLP10 & 90 & 39.54 & 0.439 & 0.114 & 1.352 & 26.0 & 2.60 \\  & GLP30 & 70 & 31.56 & 0.451 & 0.126 & 1.387 & 27.9 & 0.93 \\ \hline \multirow{8}{*}{Note} & \multirow{8}{*}{**Sample**} & \multirow{8}{*}{**Comenti(\%)**} & \multirow{2}{*}{**Compressive Strength (MPa)**} & \multirow{2}{*}{**Strength/Cement Ratio (\%)**} & \multirow{2}{*}{**Pazzolan Strength/Cement Ratio (\%)**} & \multirow{2}{*}{**Offer of
# Optimal Meshing Degree Performance Analysis in a mmWave FWA 5G Network Deployment
## 2 Literature Review
### 5G Non-Public Networks and Fixed Wireless Access Deployments  
The use of 5G technologies to support deployments beyond traditional MNO networks opens a wide range of applications and scenarios for wireless technologies that can benefit from reduced costs and complexity while still providing good bandwidth and responsiveness. In the 3 GPP scope, these deployments are called private or non-public networks (NPNs) [9]. These deployments can, for example, make use of a combination of shared and private base stations and backend infrastructure through slicing to provide network capacity solely provisioned to a given use case.  
In order to support 5G NPNs, various scenarios have been envisioned wherein the 5G Core (5 GC) can be provisioned and controlled locally or remotely or through some combination of the above [10; 11]. Essentially, both the Control Plane (CP), which provides device and network control such as access control and management, session management, mobility management, and policy management, and the User Plane Functions (UPF), which deals with data routing and forwarding, need to be provisioned for the user in order to provide a 5G service [12]. Moreover, the heterogeneous nature of available RANtechnologies in 5G provides a powerful mechanism with which to support a range of devices and applications ranging from LoRa sensor devices to 4K streaming or VR-type applications. In addition to 5G NR, an operator might also make use of some version of LTE in addition to Wi-Fi, mmWave, and other WLAN technologies to provide appropriate coverage [13].  
An interesting deployment area for 5G is in public networks deployed by and for a community [13]. Such networks can, for example, be setup by a local government or organisation and provided to members in locations where MNO coverage is not available or suitable or can be tailored to a specific use case that meets a public need. In this case, there are, again, a wide range of potential deployment models that could be adopted depending on the specific circumstances. An organisation might apply for a specific portion of the available spectrum in their area to deploy their own base stations or adopt another unlicensed solution. For example, a town/city with an available fibre infrastructure (perhaps to support a CCTV platform) could simply look to extend it into the necessary areas using a Wi-Fi mesh or use more specialized and higher-capacity technologies as needed. It is in this context that we introduce the Liverpool 5G, which is an 802.11 ad based mesh backhaul network that was deployed as part of the DCMS 5G Testbeds and Trials Programme [14].
# RCS and Aero Surfaces Control Allocation Research on RLV's Re-Entry Phase
## 3 Control Allocation Scheme Design
### Stability Analysis  
After designing the control allocation scheme, the stability of the control system should be analyzed. Assuming the stability  
\[\begin{split}\dot{\mathbf{x}}&=\bigl{(}\mathbf{A}_{0}+ \Delta\mathbf{A}\bigl{(}\mathbf{x}\bigr{)}\bigr{)}\mathbf{x}+\mathbf{B}\bigl{(} \mathbf{x}\bigr{)}\bigl{(}\boldsymbol{\Phi}\bigl{(}\mathbf{u}\bigr{)}- \mathbf{u}\bigr{)}-\bigl{(}\mathbf{B}_{0}+\Delta\mathbf{B}\bigl{(}\mathbf{x} \bigr{)}\bigr{)}\bigl{(}\mathbf{K}_{0}+\Delta\mathbf{K}\bigl{(}\mathbf{x} \bigr{)}\bigr{)}\mathbf{x}\\ &=\bigl{(}\mathbf{A}_{0}-\mathbf{B}_{0}\mathbf{K}_{0}\bigr{)} \mathbf{x}+\mathbf{B}\bigl{(}\mathbf{x}\bigr{)}\bigl{(}\boldsymbol{\Phi} \bigl{(}\mathbf{u}\bigr{)}-\mathbf{u}\bigr{)}+\bigl{(}\Delta\mathbf{A}\bigl{(} \mathbf{x}\bigr{)}-\mathbf{B}_{0}\Delta\mathbf{K}\bigl{(}\mathbf{x}\bigr{)}- \Delta\mathbf{B}\bigl{(}\mathbf{x}\bigr{)}\mathbf{K}_{0}\bigr{)}\mathbf{x} \end{split} \tag{14}\]  
Denote that  
\[\mathbf{A}_{cl0}=\mathbf{A}_{0}-\mathbf{B}_{0}\mathbf{K}_{0},\mathbf{h}_{1}( \mathbf{x})=\mathbf{B}(\mathbf{x})(\boldsymbol{\Phi}(\mathbf{u})-\mathbf{u}), \mathbf{h}_{2}(\mathbf{x})=(\Delta\mathbf{A}(\mathbf{x})-\mathbf{B}_{0}\Delta \mathbf{K}(\mathbf{x})-\Delta\mathbf{B}(\mathbf{x})\mathbf{K}_{0})\mathbf{x} \tag{15}\]  
Then  
\[\dot{\mathbf{x}}=\mathbf{A}_{cl0}x+\mathbf{h}_{1}\bigl{(}\mathbf{x}\bigr{)}+ \mathbf{h}_{2}\bigl{(}\mathbf{x}\bigr{)} \tag{16}\]  
According to Equation (12),  
\[\begin{split}\|\mathbf{h}_{1}(\mathbf{x})\|\leq k\|\mathbf{B}( \mathbf{x})\|\|\mathbf{K}(\mathbf{x})\|\|\mathbf{x}\|^{2}\\ \|\mathbf{h}_{2}(\mathbf{x})\|\leq(\|\Delta\mathbf{A}(\mathbf{x}) \|+\|\mathbf{B}\|_{0}\|\|\Delta\mathbf{K}(\mathbf{x})\|+\|\Delta\mathbf{B}( \mathbf{x})\|\|\mathbf{K}_{0}\|)\|\mathbf{x}\|\end{split} \tag{17}\]  
Therefore,  
\[\lim_{\mathbf{x}\to\mathbf{x}_{0}\to 0}\frac{\|\mathbf{h}_{1}(\mathbf{x}- \mathbf{x}_{0})+\mathbf{h}_{2}(\mathbf{x}-\mathbf{x}_{0})\|}{\|\mathbf{x}- \mathbf{x}_{0}\|}=0 \tag{18}\]  
Thus, in the neighborhood \(\varphi\), when \(\mathbf{x}\to\mathbf{x}_{0}\), the stability of the closed-loop system will be determined by the constant matrix \(\mathbf{A}_{cl0}\). So assuming that the closed-loop system under the control law \(\mathbf{u}=-\mathbf{K}\mathbf{x}\) is locally asymptotically stable, the new closed-loop system with the control output \(\boldsymbol{\Phi}(\mathbf{u})\) can still maintain the local asymptotic stability. \(\Box\)