*2.2. Sample Prepare*

The irradiation test was carried out at No.2 cobalt source facility of Heilongjiang Institute of Atomic Energy (activity, 200 kCi; temperature, 20 ± 1 ◦C; average dose rate, 10 Gy/h; inhomogeneity < 5%). The dried pea fiber samples were crushed by ST-G200 highspeed mill (Beijing Xuxinshengke Co., LTD., Beijing, China), and passed through a 60-mesh sieve. The irradiation experiments were carried out after the samples were packed in Ziplock bags (polyethylene). The irradiation doses were 0.5, 1, 2, 3 and 5 kGy, respectively. The irradiation dose of the samples was tracked by a silver dichromate chemical dosimeter (RadEye GF-10, Thermo Fisher Scientific, Waltham, MA, USA), and the absorbed dose was measured by a UV–visible spectrophotometer (SPECORD®210 PLUS, Analytik Jena AG, Jena, GER). After irradiation, samples were stored in sealed polyethylene bags at room temperature.

#### *2.3. Determination of Structural Properties of Pea Fiber*

#### 2.3.1. Determination of Main Components

Referring to the research method of Guo et al. [11], cellulose and hemicellulose were prepared by alkali method (24% KOH for 2 h and 10% KOH for 16 h), and lignin was prepared using the acid method (72% H2SO4 for 1 h). The separated cellulose and lignin were dried at 105 ◦C for 2 h and hemicellulose was dried at 60 ◦C for 24 h for further analysis.

#### 2.3.2. Determination of Particle Size and Specific Surface Area

The 3 g sample was dissolved in absolute ethanol, and ultrasonically dispersed for 3 min using an MS2000 Masterizer particle-size analyzer (Malvern company, Malvern, UK) to determine the volume-average particle size and specific surface area of pea fibers.

#### 2.3.3. Determination of Microstructure

The method of determination of microstructure is modified according to Li et al. [12]. The microstructure of pea fiber was determined by SU8010 field emission scanning electron microscope (Hitachi, Japan). The sample particles of about 1 mg were placed on the tape of the circular aluminum sample sub and coated with palladium for 90 s at a current of 15 mA. The specimen stubs were then placed in the observation room. These samples were observed at an accelerator potential of 5 kV with a 6000x increase.

#### 2.3.4. Determination of Fourier Transform Infrared (FTIR) Spectroscopy

According to the method of Guo et al. [13], the pea fiber and KBr (1:250 *w*/*w*) were fully mixed and pressed into a disk, and then Scimitar 2000 FTIR spectrometer (Agilent, Santa Clara, CA, USA) was used to obtain the infrared spectrum of the sample with a wave number of 400–4000 cm<sup>−</sup>1. Each sample was scanned 32 times.

#### 2.3.5. Determination of X-ray Diffraction

The crystal structure determination of pea fiber was slightly modified with reference to the research method of Yang et al. [14]. Scanning from 10 to 40◦ (2θ◦) at 1 π/min was performed with an XPert Powder Multifunctional Powder X-ray diffractometer (Dandong Tonda Technology Co., Ltd., Dandong, China). The generator voltage was 40 kV, and the incident current was 40 mA. The formula of relative crystallinity (RC) of pea fiber is as follows:

$$\text{RC} \left( \% \right) = \frac{\text{A}\_{\text{c}}}{\text{A}\_{\text{c}} + \text{A}\_{\text{a}}} \times 100 \tag{1}$$

where Ac is the area of the crystalline region; Aa is the area of the amorphous region.

#### *2.4. Determination of Functional Properties of Pea Fiber*

#### 2.4.1. Determination of Oil-Holding Capacity

Referring to the oil-holding capacity determination method of Liu et al. [15], 0.5 g (M0) pea fiber was mixed with 5 mL soybean oil. After standing in a centrifuge tube for 24 h, the samples were centrifuged for 20 min at 4000 rpm in a refrigerated high-speed centrifuge (X1R, Thermo Fisher Scientific, Waltham, MA, USA). The supernatant was removed, and the weight of the residue was recorded as M1. The oil-holding capacity (OHC) of pea fiber is calculated as follows:

$$\text{OHC} \left( \text{g/g} \right) = \frac{M\_1 - M\_0}{M\_0} \tag{2}$$

where, *M0* and *M1* are the weights of pea dietary fiber before and after oil absorption (g).

2.4.2. Determination of Swelling and Water-Holding Capacity

According to the determination method of Wang et al. [16], the pea fiber suspension was hydrated for 24 h after diluting 0.3 g (*m0*) of pea fiber to 15 mL (*V0*) with deionized water; the volume was recorded as V1. Then, the pea fiber suspension was centrifuged at 6000 rpm for 20 min. The supernatant was removed and wet fiber weight, m1, was recorded. Swelling (SC) and water-holding capacity (WHC) were calculated using the following formula:

$$\text{SC}(\text{mL}/\text{g}) = \frac{V\_1 - V\_0}{m\_0} \tag{3}$$

where *V0* is the dilution volume of pea fiber 15 mL; V1 is the volume of the suspension after pea fiber hydration for 24 h (mL).

$$\text{WHC}(\text{g/g}) = \frac{m\_1 - m\_0}{m\_0} \tag{4}$$

where *m0* and *m1* are the weight of pea fiber before and after water absorption (g).


**2.3 Future climate change data**

The statistically downscaled Regional Climate Model (RCM) Bias-corrected Coordinated Regional Climate Downscaling Experiment (CORDEX) precipitation, min/mean/max temperature for Ethiopia, under RCP 4.5 and RCP 8.5, downloaded from(https://dataservices.gfzpotsdam.de/pik/showshort.php?id=escidoc:3124935) is provided as input data for hydrological modeling of this study. This dataset contributes bias-corrected daily precipitation, min/mean/max air temperature of ten CORDEX RCM runs covering the country of Ethiopia for historical (1970–1999) and over the 21st century for RCP 4.5 and RCP 8.5 [24]. For this study, daily rainfall and maximum & minimum temperature data of historical (1990–2013) of eight climatic stations were obtained from the Meteorological Agency of Ethiopia, and other eight climatic stations were from the global database of Climate Forecast System Reanalysis (CFSR) after filling missing data, consistency, and outlier checked [1].

Accordingly, all the bias correction has improved the simulation of precipitation and temperature before using CORDEX-RCM outputs for any climate impact modeling. The study used climate model data for hydrological modeling CMhyd to extract CORDEX-NetCDF and bias correction of precipitation, minimum and maximum temperature to predict climate change-induced temperature changes in the Genale catchment.

**Figure 2.** *Steps of the implementation of the SWAT model for the study area. Analysis of SWAT input data.*

#### *Evaluation of Climate Change-Induced Impact on Streamflow and Sediment Yield of Genale… DOI: http://dx.doi.org/10.5772/intechopen.98515*

SWAT model was used to simulate water yield using the RCM under the future emission scenarios of two representative concentration pathways (RCPs) (medium emission scenario (RCP4.5) and high emission scenario (RCP-8.5)). Climate data for different periods are input into the SWAT model with the other components unchanged. The period of 1990–2013 is set as the baseline period.

Digital elevation model was downloaded from USGS Earth Explorer (http:// earthexplorer.usgs. gov/) SRTM (Shuttle Radar Topography Mission) 90 m\*90 m and used for watershed delineation, sub-basin, slope calculation/ classification, and extract stream networks. The spatial land use/cover collected from the Ethiopian Ministry of Water, Irrigation, and Electricity (MoWIE) GIS department of the year 2013 used for SWAT input, and the dominant land use/cover is range brushland (RNGB) accounts for about 71% of the area. This study's soil map/type is from the Food and Agricultural Organization (FAO) Digital Soil Map of the World (http:// www.fao.org/geonetwork/srv/en/metadata) the scale of 1/5000000 for 2007. The soil data that integrated into the SWAT model are: the available water content, the texture, the hydraulic conductivity, the apparent density of the different soil layers, and the dominant soil type in the study watershed was Rc19-bc-204 (Calcaric Regosols), and it accounts about 40% of the catchment.

The climate data required for this paper has been taken from the National Meteorological Agency of Ethiopia, http://www.ethiomet.gov.et/etms. These data subsist of precipitation, max and min temperatures, wind energy, solar radiation, and relative humidity daily and covered the period from 1990 to 2013 for sixteen stations. The discharge data from the Ethiopian Ministry of Water, Irrigation, and Electricity (MoWIE) Hydrology department Genale @ Halwen gauging station a bit upstream of the outlet, and then transferred to the outlet, and arranged for SWAT language for the period from 1990 to 2013.

**Figure 3**, shows different 16 (sixteen) meteorological stations were distributed in the watershed, hydrological gauging station, watershed outlet, stream reach, and basin mark of the study area. The stations which were designated as; GMS1- Gridded Meteorological station-1, GMS2- Gridded Meteorological station-2, GMS3- Gridded Meteorological station-3, 4,5,6,7, & 8 respectively (**Figure 3**).

**Figure 4** shows the distribution of rainfall in the study area for the selected different gauge stations.

From **Figure 5b**, the details of maximum, average, and minimum yearly temperature of the study area were pinpointed as 24.6, 19, & 12.93 °C, respectively, for 1990–2013 (**Figure 6**).

#### **2.4 SWAT-CUP(SUFI-2) description**

The automatic calibration and validation adjustment in the SWAT model achieved using the SWAT-CUP (SUFI-II) public user software developed by [25]. The SWAT-CUP has interfaced with five algorithms: (1) sequential uncertainty fitting (SUFI-2, (2) generalized likelihood uncertainty estimation (GLUE), (3) parameter solution (ParaSol), (4) Markov chain Monte Carlo (MCMC), and (5) particle swarm optimization (PSO) [26].

In this study, the analysis of uncertainty, calibration, validation was conducted using the SUFI-2 optimization algorithm; this algorithm needs less simulation number, faster, and one of the most used in the automatic calibration of model for several basins the semi-arid region like Genale Basin.

Assessment of performance criteria for the model is; Nash-Sutcliffe Efficiency (NSE), PBIAS, and Coefficient of Determination (R2 ) has been used as the efficiency criteria to evaluate the performance of models in the Genale watershed.

#### **Figure 3.**

*Meteorological and hydrological gauging station distribution sub-basin wise for Genale River.*

**Figure 4.**

*Mean monthly rainfall for selected stations in the study area over 1990–2013.*

The first three objective functions are mainly used for daily and monthly streamflow /sediment calibration–validation uncertainty analysis.

Coefficient of Determination (R2 )

$$R^2 = \frac{\left[\sum\_{i=1}^{n} (Q\_{si} - Q\_{sm})(Q\_{oi} - Q\_{om})\right]^2}{\sum\_{i=1}^{n} (Q\_{si} - Q\_{sm})^2 \sum\_{i=1}^{n} (Q\_{oi} - Q\_{om})^2} \tag{6}$$

*Evaluation of Climate Change-Induced Impact on Streamflow and Sediment Yield of Genale… DOI: http://dx.doi.org/10.5772/intechopen.98515*

#### **Figure 5.**

*Daily and yearly average maximum, minimum, and average daily temperatures in the study area, respectively.*

**Figure 6.** *DEM, LULC, soil type, and slope classes of the Genale watershed, respectively.*

Where, *Qsi* is the simulated value, *Qoi* is the measured value, *Qom*, is the average observed value and Qsm is an average simulated value? Nash Sutcliffe Efficiency (NSE).

$$\text{NSE} = \mathbf{1} - \frac{\sum\_{i=1}^{n} \left(Q\_{oi} - Q\_{si}\right)^2}{\sum\_{i=1}^{n} \left(Q\_{oi} - Q\_{om}\right)^2} \tag{7}$$

where, *Qoi* is the observed, *Qsi* is the simulated, and Qom is the observed discharge.

$$\text{Percent Bias (PBIAS)}; \text{PBIAS} = 100\text{\%} \times \frac{\sum\_{i=1}^{n} (Q\_{oi} - Q\_{si})}{\sum\_{i=1}^{n} Q\_{oi}} \tag{8}$$


**2.3 Contributions traditional Indigenous knowledge offers for climate-change risk communication**

In this section, I will discuss traditional indigenous knowledge contributions that it can offer to the scientific research community while focusing on five "areas of complementarity" given by Moller et al. [17] to demonstrate traditional indigenous systems' practicability. Traditional indigenous knowledge systems can contribute to climate-change risk communication in many ways: one is adding value to the risk communicators' knowledge. Indigenous Peoples' understanding of changes and variability occurring in their environments guide their responses to environmental changes. This knowledge can help climate-change adaptation policymakers and practitioners in articulating possible coping and adaptive options, priorities, and opportunities to manage climatic risks. In this respect, the following five "areas of complementarity" demonstrate the application of traditional indigenous knowledge that it can offer to science by integrating scientific knowledge and traditional indigenous knowledge systems.

#### *2.3.1 Synchronic-diachronic information*

In climate change research, data collection typically relies on a small sample size over a large area, with short temporal scope, which may likely miss vital information due to the small sample size, sampling technique used, and focus on a large area. Moller et al. [17] call this process "synchronic". In contrast, indigenous observations are based on smaller areas over a longer time period, which provides more opportunities to collect "diachronic" information, thereby minimizing the chances of any change being overlooked. Using synchronic and diachronic approaches together will provide complete information on both temporal and spatial scales [17]. However, developing a mechanism that provides a practical tool to combine both techniques can be challenging. Nevertheless, diachronic information offered by traditional indigenous knowledge will be a valuable resource, particularly when scientists are facing decreasing resources for research projects.

#### *2.3.2 Average-extreme events*

Science primarily depends on numerical and quantitative data, and statistical analyses [17]; therefore, scientific knowledge is based on averages, which makes the system vulnerable to overlooking critical information. Conversely, traditional indigenous knowledge, however, captures extreme events in a better way by closely observing both changes and variations. The two techniques can be integrated, which can also help narrow the trust deficit between Indigenous Peoples and non-Indigenous researchers.

#### *2.3.3 Quantitative-qualitative data*

According to Moller et al. [17], science examines quantitative aspects of parts of a system, while relying on numbers or numerical data. Conversely, traditional indigenous knowledge deals with a qualitative understanding of the whole system. As the two perspectives are complementary, their combination will enable scientists to use qualitative measures that are less costly and more rapid; however, this may come at the "cost of precision". Combining both quantitative and qualitative approaches will require a framework that allows the two data sources to be integrated without compromising the value or integrity of each. Qualitative data for climate and environmental risk management research offers valuable insights, particularly in mixed research methods as qualitative data can be collated with quantitative one for a better understanding of climate-change risk communication issues.


**3. Proposed BES System and Control**

Figure 5 shows the structure of the proposed PV-BES system. The BES connects at the terminals of the PV and executes ramping of the combined power of the PV-BES system. The BES either generates locally or receives the ramp-up-rate (RUR) and ramp-down-rate (RDR) signals from the grid operator and adjusts its operation accordingly. This can (1) prevent the abrupt PV disturbances, and (2) re-shape the daily load profile, as will be demonstrated in this paper.

**Figure 5.** Structure of proposed PV-battery energy storage (BES).

#### *3.1. Defining Operational Scenario of BES*

Figure 6a shows the scenario where PV power (*P*pvo) experiences an abrupt fall followed by an abrupt rise. The BES power, shown in Figure 6b, compensates this sudden disturbance and the combined power of PV-BES system, *P*o, has controlled ramp down/up. The charging and discharging of the BES causes rise/fall of its state-of-charge (SOC), as shown in Figure 6c. The BES control system should be equipped with an SOC control that charges/discharges the battery after the response to the disturbance is completed to regulate its SOC to the desired level.

**Figure 6.** Operational scenario of the BES.


**1. Introduction**

## **1.1 Liquid crystal-based photonic crystal devices**

Photonic crystals (PCs) are structural materials with periodically varying dielectric permittivity. The term PC was coined three decades ago, and fascinating properties of PCs have attracted numerous scientists to put in a great deal of effort. PCs consist of dielectric materials. In PCs, the index of refraction varies periodically in space. PCs were invented since 1987, when both Yablonovitch and John published their results independently [1, 2]. The most important characteristic of PCs is the photonic band gap (PBG). Furthermore, the PBG of PCs is an optical analog to the electronic bandgap in semiconductor materials, which means photons will be localized or forbidden in the PCs. Based on this characteristic, PCs can be used as various photonic applications [3–7]. However, if a defect layer is introduced to a PC structure, disrupting its periodicity, the transmission of photons at specific wavelengths will be induced within the PBG; these narrow transmission bands within PBG are called defect modes. Based on this special design of PCs with defect layers, many photonic device applications were proposed: PC lasers [8], PC optical fibers [9], and other optical devices [10, 11]. In addition, the spectral properties of the PCs can be controlled if the defect layer is tunable, for instance, liquid crystal (LC).

LCs are anisotropic materials. The optic axis (or molecular orientation) can be controlled by applying electric field, magnetic field, or temperature. Based on optical anisotropy, LC can be used as a phase retarder or an optical polarization rotator, of which the refractive indices can be tuned with external fields. LCs are also used in many other applications such as displays, smart windows, and optical fibers. Furthermore, by inserting a LC layer as a defect layer in PCs, the tunable defect modes can be achieved. The first tunable LC-based PC hybrid structure was developed by Ozaki et al. They employed a planar-aligned nematic LC as a central defect layer sandwiched between two one-dimensional (1D) PCs [12]. **Figure 1** shows the setup of the designed hybrid PC/LC cell. This idea was then extended to PC/cholesteric LC (CLC) structures for tunable laser applications [13–17]. CLCs are self-organized PCs, in which the molecular chirality forms helical structures with the optic axis continuously twisted along the helical axis. The effects of angle of incidence [18], temperature [19], and magnetic field [20] on the optical properties of PC/LC devices are investigated.

Zyryanov et al. investigated the hybrid PC/LC device between the crossed polarizers. Their experimental results show attractive features of the tunable defect modes within the PBG. The wavelengths of defect modes are shifted by the change in effective refractive index (neff) [21]. Larger number of defect modes can be induced by using high refractive index LC or by increasing the thickness of the LC defect layer. Moreover, they experimentally and theoretically demonstrated that the interference of defect modes can be achieved by placing the cell between crossed polarizers. The orthogonal polarization components through vector sum in the projection direction along the axis of the analyzer lead to shift of defect modes. As a result, the transmittance is increased when the defect mode wavelength of an extraordinary component overlaps that of an ordinary one in PCs [20].

In addition, the next milestone is electrically tunable photonic device based on PC/CLC and PC/polymer-stabilized CLC (PSCLC) hybrid structures [22–24]. With the PC/PSCLC structure, not only the wavelength of defect mode is switchable among multistable states by voltage pulses, but also the optical intensity of defect modes can be electrically tuned through switching among different metastable states. **Figure 2** shows the sandwiched structure of the PC/PSCLCs in the three stable states. It is wavelength switchable and intensity tunable of defect modes among three stable states. The unique optical tristability in the defect modes reduces power consumption and enhances flexibility. From then on, the PC and CLC combined

**53**

*Hybrid Liquid-Crystal/Photonic-Crystal Devices: Current Research and Applications*

devices became a hot topic. Many types of PC/chiral LC hybrid devices are invented such as PC/BHN, PC/THN, and PC/PSCT. Recently, many scientists employ hybrid

An overview of the development of PC/LC is presented. The optical properties, operation principles, and applications of liquid crystal-based photonic crystal devices are discussed. In Section 1, I introduce the basic knowledge of LC and PC physics to make the reader understand the next section. Section 2 details the operation principles of nonchiral and chiral LC modes to help the understanding of switching mechanisms described in two following sections. In addition, the optical properties of defect modes in different LC states are reported in Section 2. Section 2.1 is "Photonic crystals with a nonchiral nematic liquid crystal," Section 2.2 is "Chiral-tilted homeotropic nematic liquid crystal-based photonic crystal devices," Section 2.3 is "Chiral nematic and cholesteric liquid crystal with photonic crystal devices," and Section 2.4 is "Tristable photonic crystal devices with polymer-stabilized cholesteric textures." The configuration of many types of PC/ LC cell applications, including the design of PC multilayers for applications is schematically depicted in Session 3. To realize the PC applications; Session 3.1 introduces "Electrically switchable liquid crystal-based photonic crystals for a white light laser," Session 3.2 shows "Liquid crystal-based photonic crystals for pulse compression and signal enhancement in fluorescence applications," and Session 3.3 demonstrates "Photo-manipulated photonic devices based on tristable chiraltilted homeotropic nematic liquid crystal." Finally, I will summarize the results and

**2. Operation method of liquid crystal-based photonic crystal devices**

Nowadays, available LC materials can be classified into two categories: nonchiral and chiral system, according to their different operation functions on their molecule arrangement. The nonchiral LC owns only one stable arrangement state, which is determined by the use of the alignment layer. In addition, nonchiral LC molecule is continuously oriented, when the voltage is applied. On the contrary, two or multistable states are exhibited in the chiral LC system. Moreover, the multistable states can be switched and controlled from one to another based on the condition of applied voltage pulse. Based on the properties of the nonchiral LC system, it can serve as a phase retarder or optical rotator tuned by electrically

**2.1 Photonic crystals with a nonchiral nematic liquid crystal**

*DOI: http://dx.doi.org/10.5772/intechopen.82833*

PC/CLC to achieve the lasing applications [25, 26].

*Schematic of the PC/CLC hybrid device in three stable states.*

**1.2 Aim of this chapter**

**Figure 2.**

conclude about this chapter.

**Figure 1.** *Schematic of the PC/LC hybrid device in an electro-optical setup.*

*Hybrid Liquid-Crystal/Photonic-Crystal Devices: Current Research and Applications DOI: http://dx.doi.org/10.5772/intechopen.82833*

**Figure 2.**

*Photonic Crystals - A Glimpse of the Current Research Trends*

of PC/LC devices are investigated.

LCs are anisotropic materials. The optic axis (or molecular orientation) can be controlled by applying electric field, magnetic field, or temperature. Based on optical anisotropy, LC can be used as a phase retarder or an optical polarization rotator, of which the refractive indices can be tuned with external fields. LCs are also used in many other applications such as displays, smart windows, and optical fibers. Furthermore, by inserting a LC layer as a defect layer in PCs, the tunable defect modes can be achieved. The first tunable LC-based PC hybrid structure was developed by Ozaki et al. They employed a planar-aligned nematic LC as a central defect layer sandwiched between two one-dimensional (1D) PCs [12]. **Figure 1** shows the setup of the designed hybrid PC/LC cell. This idea was then extended to PC/cholesteric LC (CLC) structures for tunable laser applications [13–17]. CLCs are self-organized PCs, in which the molecular chirality forms helical structures with the optic axis continuously twisted along the helical axis. The effects of angle of incidence [18], temperature [19], and magnetic field [20] on the optical properties

Zyryanov et al. investigated the hybrid PC/LC device between the crossed polarizers. Their experimental results show attractive features of the tunable defect modes within the PBG. The wavelengths of defect modes are shifted by the change in effective refractive index (neff) [21]. Larger number of defect modes can be induced by using high refractive index LC or by increasing the thickness of the LC defect layer. Moreover, they experimentally and theoretically demonstrated that the interference of defect modes can be achieved by placing the cell between crossed polarizers. The orthogonal polarization components through vector sum in the projection direction along the axis of the analyzer lead to shift of defect modes. As a result, the transmittance is increased when the defect mode wavelength of an

extraordinary component overlaps that of an ordinary one in PCs [20].

In addition, the next milestone is electrically tunable photonic device based on PC/CLC and PC/polymer-stabilized CLC (PSCLC) hybrid structures [22–24]. With the PC/PSCLC structure, not only the wavelength of defect mode is switchable among multistable states by voltage pulses, but also the optical intensity of defect modes can be electrically tuned through switching among different metastable states. **Figure 2** shows the sandwiched structure of the PC/PSCLCs in the three stable states. It is wavelength switchable and intensity tunable of defect modes among three stable states. The unique optical tristability in the defect modes reduces power consumption and enhances flexibility. From then on, the PC and CLC combined

**52**

**Figure 1.**

*Schematic of the PC/LC hybrid device in an electro-optical setup.*

*Schematic of the PC/CLC hybrid device in three stable states.*

devices became a hot topic. Many types of PC/chiral LC hybrid devices are invented such as PC/BHN, PC/THN, and PC/PSCT. Recently, many scientists employ hybrid PC/CLC to achieve the lasing applications [25, 26].


*2.3. Hydrology*

The Kunduz River is a tributary of the Amu Darya River in North Afghanistan. The upper part of the KRB is characterized by high mountains and steep valleys. In the upper part, the KRB is fed by the rainfall, snow, and small glaciers of the Koh-e-Baba range and the Hindu-Kush mountains [24] (Figure 2). The KRB has a number of tributaries, including the Khinjan, Andarab, and Bamyan rivers [28]. Upstream of the Kunduz province, the Kunduz river is called the Pul-I Khumri River. Another small tributary, the Nahrin River, has its sources in Nahrin district and it joins the Kunduz river near the town Baghlan-i Kohna. Finally, the Kunduz River reaches the Amu Darya main stream at Qala-i Zal (Figure 3). The KRB covers all of Baghlan province, the western part of Bamiyan province, and parts of Kunduz and Takhar Provinces [23]. Two hydropower dams have been built on the Pul-i Khumri in 1943 [29].

The hydrology of the KRB is mainly controlled by the high mountains of the Hindukush. Upstream, channels are generally narrow and deep and flowing throughout the whole year [12]. The runoff regimes are largely controlled by snow-melt, with high discharge from April to June and only close to glaciers in the upstream parts of the catchment, the small glaciered area has significant influence on the flow regime (e.g., Doab station). Precipitation in the KRB mainly occurs in the form of rain, drizzle, snowfall, and hail, and it is high during the winter months [24]. The water carried by the river supports an intensive irrigated agriculture, which is the main economic basis of the region. There are a number of river gauging stations within the watershed, as shown in Figure 2.

Figure 4 presents the mean monthly discharge of the recent five years from 2014 to 2018 recorded in the four main gauging stations, Doab, Puli-Khumri, Char-Dara, and Kulukh-Tepa (for locations, see Figure 2). Historically, the monthly peak flows generally occurred during April through July, which resulted in very high discharge at the downstream drainage outlet (Figure 4). The Doab gauge is located in the most upper part at 1468 m a.s.l.  The peak monthly discharge at that gauge from 2014 to 2018 was 36 m3/s during June. The gauge at Puli-Khamri is downstream at 634 m a.s.l. and its peak monthly discharge during this period was 199 m3/s. Char-Dara gauge, further downstream at 401 m a.s.l., the peak discharge is 138 m3/s and 177 m3/s at Kulukh-Tepa gauge (320 m a.s.l.). The Kulukh-Tepa gauging station is located at the confluence of the Kunduz River and the Amu Darya mean stream (Figure 3). The peak average monthly discharge at Puli-Khamri gauge is higher than the Kulukh-Tepa

at the outlet of the KRB. This can be explained by the high temperature and related high evaporation during June, July, and August in the lowland downstream area and diverging small portion of the stream to irrigation as well.

**Figure 4.** Comparison of flow discharge at Doab, Char-Dara, Puli-Khamri, and Kulokh-Tepa gauges.


**Sustainable Management of Soil Potassium – A Crop Rotation Oriented Concept**

Witold Grzebisz, Witold Szczepaniak, Jarosław Potarzycki and Remigiusz Łukowiak

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/53185


**Abstract**

This research aims to analyze the role played by firm visibility in moderating the relationship between Corporate Social Responsibility (CSR) and Firm Financial Performance (FFP). Based on the legitimacy theory, a firm's responses to stakeholder's expectations would be affected by its public visibility; we hypothesize a positive link between CSR and firm visibility. Moreover, visibility is expected to moderate the CSR-FFP relationship. We applied a Moderated Regression Analysis using the aggregate ESG scores as a CSR proxy on a panel data of listed French Companies (SBF120) over the period 2008–2017. Our findings are in line with legitimacy theory, suggesting that social initiatives would be mean to strengthen the legitimacy and to secure "license to operate". Furthermore, firm visibility would be a contingency variable that moderates positively CSR-FFP relationship.

**Keywords:** corporate social responsibility (CSR), financial performance, firm visibility, legitimacy theory, SBF 120


**3. System optimization with parameter estimation**

Now, an expanded optimal control problem, which combines the real plant and the cost function in Problem (P) into Problem (M) and is referred to as Problem (E), is introduced by

$$\begin{aligned} \min\_{\boldsymbol{u}(k)} I\_2(\boldsymbol{u}) &= \frac{1}{2} \mathbf{x}(\boldsymbol{N})^\mathrm{T} \mathbf{S}(\boldsymbol{N}) \mathbf{x}(\boldsymbol{N}) + \boldsymbol{\gamma}(\boldsymbol{N}) + \sum\_{k=0}^{N-1} \frac{1}{2} \left( \mathbf{x}(k)^\mathrm{T} \mathbf{Q} \mathbf{x}(k) + \boldsymbol{u}(k)^\mathrm{T} \mathbf{R} \boldsymbol{u}(k) \right) + \boldsymbol{\gamma}(\boldsymbol{k}) \\ &+ \frac{1}{2} r\_1 ||\boldsymbol{u}(k) - \boldsymbol{v}(k)||^2 + \frac{1}{2} r\_2 ||\boldsymbol{x}(k) - \boldsymbol{z}(k)||^2 \end{aligned}$$

subject to

these algorithms were introduced by Robert [9–11], and Robert and Becerra [12–14], respectively. The basic idea of DISOPE is applying the model-based optimal control, which has different structures and parameters compared to the original optimal control problem, to obtain the correct optimal solution of the original optimal control problem, in spite of model-reality differences. Recently, this algorithm has been extended to cover both deterministic and stochastic versions, and it is known as an integrated optimal control and parameter estimation (IOCPE) algorithm [15, 16]. On the other hand, the application of the optimization techniques, particularly, using the conjugate gradient method for solving the optimal control problem [17–19] has also been studied, where the open-loop control strategy

In this chapter, the conjugate gradient approach [17, 19] is employed to solve the linear model-based optimal control problem for obtaining the optimal solution of the original optimal control problem. In our approach, the simplified model, which is adding the adjusted parameters, is formulated initially. Then, an expanded optimal control problem, which combines the system dynamic and the cost function from the original optimal control problem into the simplified model, is introduced. By defining the Hamiltonian function and the augmented cost function, the

corresponding necessary conditions for optimality are derived. Among these necessary conditions, a set of necessary conditions is for the modified model-based optimal control problem, a set of necessary conditions defines the parameter estimation problem, and a set of necessary conditions calculates the multipliers [15]. By virtue of the modified model-based optimal control problem, an equivalence optimization problem is defined, and the related gradient function is determined. With an initial control sequence, the initial gradient and the initial search direction are computed. Then, the control sequences are updated through the line search technique, where the gradient and search direction would satisfy the conjugacy condition [17, 18]. During the iteration, the state and the costate are updated by the control sequence obtained from the conjugate gradient approach. When the convergence is achieved within a tolerance given, the iterative solution approximates to the correct optimal solution of the original optimal control problem, in spite of model-reality differences. For illustration, examples of linear and nonlinear cases, which are damped harmonic oscillator [7] and continuous stirred-tank chemical

The chapter is organized as follows. In Section 2, the problem statement is described in detail, where the original optimal control problem and the simplified model are discussed. In Section 3, the methodology used is further explained. The necessary conditions for optimality are derived, and the use of the conjugate gradient method is delivered in solving the equivalence optimization problem. In Section 4, examples of a damped harmonic oscillator and a continuous stirred-tank chemical reactor are studied. The results show the efficiency of the algorithm

Consider a general class of the discrete-time nonlinear optimal control problem,

*x k*ð Þ¼ þ 1 *f xk* ð Þ ð Þ, *u k*ð Þ, *k* , *x*ð Þ¼ 0 *x*0

*N*�1

*Lxk* ð Þ ð Þ, *u k*ð Þ, *k*

(1)

*k*¼0

*u k*ð Þ *J*0ð Þ¼ *u φ*ð Þþ *x N*ð Þ, *N* X

is concerned [3, 8].

*Control Theory in Engineering*

tank [8], are studied.

**2. Problem statement**

given by

**82**

subject to

proposed. Finally, concluding remarks are made.

min

$$\begin{aligned} x(k+1) &= Ax(k) + Bu(k) + a(k), \; x(0) = x\_0 \\ &\quad \frac{1}{2} z(N)^T S(N) x(N) + \gamma(N) = \rho(z(N), N) \\ &\quad \frac{1}{2} \left( z(k)^T Q z(k) + v(k)^T R v(k) \right) + \gamma(k) = L(z(k), v(k), k) \\ &\quad Ax(k) + Bv(k) + a(k) = f(z(k), v(k), k) \\ v(k) &= u(k) \\ z(k) &= x(k) \end{aligned}$$

(3)

where *v k*ð Þ∈ *ℜm*, *k* ¼ 0, 1, ⋯, *N* � 1, and *z k*ð Þ∈ *ℜn*, *k* ¼ 0, 1, ⋯, *N*, are introduced to separate the sequences of control and state in the optimization problem from the respective signals in the parameter estimation problem, and k�k denotes the usual Euclidean norm. The terms 1 2 *r*1k k *u k*ð Þ� *v k*ð Þ 2 and 1 2 *r*2k k *x k*ð Þ� *z k*ð Þ 2 with *r*1,*r*2 ∈ *ℜ* are introduced to improve the convexity and to facilitate the convergence of the resulting iterative algorithm. Here, we clarify that the algorithm is designed such that the constraints *v k*ð Þ¼ *u k*ð Þ and *z k*ð Þ¼ *x k*ð Þ are satisfied upon termination of the iterations, assuming that convergence is achieved. Moreover, the state constraint *z k*ð Þ and the control constraint *v k*ð Þ are used for the computation of the parameter estimation and matching scheme, while the corresponding state constraint *x k*ð Þ and control constraint *u k*ð Þ are reserved for optimizing the model-based optimal control problem. Therefore, system optimization and parameter estimation are declared and mutually integrated.

(c) State equation:

(d) Boundary conditions:

(f) Modifier equations:

(g) Separable variables:

(e) Adjusted parameter equations:

*DOI: http://dx.doi.org/10.5772/intechopen.89711*

with *ξ*ð Þ¼ *k* 1 and *μ*ð Þ¼ *k p k* ^ð Þ þ 1 .

**3.2 Modified model-based optimal control problem**

*x N*ð ÞT*S N*ð Þ*x N*ð Þþ ΓT*x N*ð Þþ *γ*ð Þþ *N* X

*r*1k k *u k*ð Þ� *v k*ð Þ 2 þ

referred to as Problem (MM), is defined by

2

*x k*ð Þ¼ þ 1 *Ax k*ð Þþ *Bu k*ð Þþ *α*ð Þ*k* , *x*ð Þ¼ 0 *x*0

1 2

þ*γ*ð Þþ *k* 1

min *u k*ð Þ *J*3ð Þ¼ *u*

**85**

subject to

*Lzk* ð Þ¼ ð Þ, *v k*ð Þ, *k* 1

*φ*ð Þ¼ *z N*ð Þ, *N* 1

2

*Conjugate Gradient Approach for Discrete Time Optimal Control Problems with Model-Reality…*

*λ*ð Þ¼� *k* ∇*v k*ð Þ*L* � *Rv k*ð Þ � � � *∂ f*

*β*ð Þ¼� *k* ∇*z k*ð Þ*L* � *Qz k*ð Þ � � � *∂ f*

*x k*ð Þ¼ þ 1 *Ax k*ð Þþ *Bu k*ð Þþ *α*ð Þ*k* (8)

*p N*ð Þ¼ *S N*ð Þ*x N*ð Þþ Γ and *x*ð Þ¼ 0 *x*0 (9)

*f zk* ð Þ¼ ð Þ, *v k*ð Þ, *k Az k*ð Þþ *Bv k*ð Þþ *α*ð Þ*k* (12)

*∂v k*ð Þ � *B* � �T

*∂z k*ð Þ � *A* � �T

*v k*ð Þ¼ *u k*ð Þ, *z k*ð Þ¼ *x k*ð Þ, *p k* ^ð Þ¼ *p k*ð Þ*:* (16)

*N*�1

1 2

*k*¼0

Notice that for the optimality necessary conditions obtained above, they are divided into three sets of necessary conditions. The first set of necessary conditions in (6)–(9) is the necessary conditions for the system optimization problem. The second set of necessary conditions in (10)–(12) defines the parameter estimation problem. The third set of necessary conditions in (13)–(15) provides the computation of multipliers. In fact, the necessary conditions, which are defined in (6)–(9), are the optimality for the modified model-based optimal control problem, and the adjusted parameters, which are calculated from the necessary conditions in (10)–(12), measure the differences between the real plant and the model used.

As a consequence, the modified model-based optimal control problem, which is

1 2

*z N*ð ÞT*S N*ð Þ*z N*ð Þþ *γ*ð Þ *N* (10)

2 *z k*ð ÞT*Qz k*ð Þþ *v k*ð ÞT*Rv k*ð Þ � � þ *γ*ð Þ*k* (11)

Γ ¼ ∇*z N*ð Þ*φ* � *S N*ð Þ*z N*ð Þ (13)

*p k* ^ð Þ þ 1 (14)

*p k* ^ð Þ þ 1 (15)

*x k*ð ÞT*Qx k*ð Þþ *u k*ð ÞT*Ru k*ð Þ � �

(17)

*r*2k k *x k*ð Þ� *z k*ð Þ 2 � *λ*ð Þ*k* T*u k*ð Þ� *β*ð Þ*k* T*x k*ð Þ

#### **3.1 Necessary conditions for optimality**

Define the Hamiltonian function for Problem (E), given by:

$$\begin{aligned} H\_2(k) &= \frac{1}{2} \left( \mathbf{x}(k)^\mathrm{T} \mathbf{Q} \mathbf{x}(k) + \boldsymbol{u}(k)^\mathrm{T} \mathbf{R} \boldsymbol{u}(k) \right) + \boldsymbol{\chi}(k) + \frac{1}{2} r\_1 ||\boldsymbol{u}(k) - \boldsymbol{v}(k)||^2 \\ &+ \frac{1}{2} r\_2 ||\boldsymbol{x}(k) - \boldsymbol{z}(k)||^2 + p(k+1)^\mathrm{T} (\boldsymbol{A} \mathbf{x}(k) + \boldsymbol{B} \mathbf{u}(k) + \boldsymbol{a}(k)) - \boldsymbol{\lambda}(k)^\mathrm{T} \boldsymbol{u}(k) \\ &- \boldsymbol{\beta}(k)^\mathrm{T} \mathbf{x}(k) \end{aligned} \tag{4}$$

where *λ*ð Þ*k* ∈ *ℜm*, *k* ¼ 0, 1, ⋯, *N* � 1, *β*ð Þ*k* ∈ *ℜn*, *k* ¼ 0, 1, ⋯, *N*, and *p k*ð Þ∈ *ℜn*, *k* ¼ 0, 1, ⋯, *N*, are modifiers. Using this Hamiltonian function in (4), write the cost function in (3) to be the augmented cost function, that is,

$$\begin{aligned} f\_2'(\boldsymbol{u}) &= \frac{1}{2} \mathbf{x}(N)^T \mathbf{S}(N) \mathbf{x}(N) + \boldsymbol{\gamma}(\boldsymbol{N}) + p(\mathbf{0})^T \mathbf{x}(\mathbf{0}) - p(\mathbf{N})^T \mathbf{x}(N) \\ &+ \boldsymbol{\xi}(\boldsymbol{N}) \left( \boldsymbol{\rho}(\boldsymbol{z}(N), \boldsymbol{N}) - \frac{1}{2} \mathbf{z}(N)^T \mathbf{S}(N) \mathbf{z}(\boldsymbol{N}) - \boldsymbol{\gamma}(\boldsymbol{N}) \right) + \boldsymbol{\Gamma}^T(\boldsymbol{x}(N) - \boldsymbol{z}(N)) \\ &+ \sum\_{k=0}^{N-1} H\_2(k) - p(k)^T \mathbf{x}(k) + \boldsymbol{\lambda}(\boldsymbol{k})^T \boldsymbol{v}(k) + \boldsymbol{\beta}(\boldsymbol{k})^T \boldsymbol{z}(k) \\ &+ \boldsymbol{\xi}(\boldsymbol{k}) \left( L(\boldsymbol{z}(k), \boldsymbol{v}(k), k) - \frac{1}{2} \left( \boldsymbol{z}(k)^T \mathbf{Q} \boldsymbol{z}(k) + \boldsymbol{v}(k)^T \mathbf{R} \boldsymbol{v}(k) \right) - \boldsymbol{\gamma}(\boldsymbol{k}) \right) \\ &+ \boldsymbol{\mu}(\boldsymbol{k})^T (f(\boldsymbol{z}(k), \boldsymbol{v}(k), k) - A \boldsymbol{z}(k) - B\boldsymbol{v}(k) - \boldsymbol{a}(k)) \end{aligned} \tag{5}$$

where *p k*ð Þ, *ξ*ð Þ*k* , *λ*ð Þ*k* , *β*ð Þ*k* , *μ*ð Þ*k* and Γ are the appropriate multipliers to be determined later.

Applying the calculus of variation [7, 9, 11, 13, 15] to the augmented cost function in (5), the following necessary conditions for optimality are obtained:

(a) Stationary condition:

$$R u(k) + B^T p(k+1) - \lambda(k) + r\_1(u(k) - v(k)) = 0 \tag{6}$$

(b) Co-state equation:

$$p(k) = \mathbb{Q}\mathbf{x}(k) + A^T p(k+1) - \beta(k) + r\_2(\mathbf{x}(k) - \mathbf{z}(k)) \tag{7}$$

*Conjugate Gradient Approach for Discrete Time Optimal Control Problems with Model-Reality… DOI: http://dx.doi.org/10.5772/intechopen.89711*

(c) State equation:

where *v k*ð Þ∈ *ℜm*, *k* ¼ 0, 1, ⋯, *N* � 1, and *z k*ð Þ∈ *ℜn*, *k* ¼ 0, 1, ⋯, *N*, are introduced to separate the sequences of control and state in the optimization problem from the respective signals in the parameter estimation problem, and k�k denotes

*r*1,*r*2 ∈ *ℜ* are introduced to improve the convexity and to facilitate the convergence of the resulting iterative algorithm. Here, we clarify that the algorithm is designed such that the constraints *v k*ð Þ¼ *u k*ð Þ and *z k*ð Þ¼ *x k*ð Þ are satisfied upon termination of the iterations, assuming that convergence is achieved. Moreover, the state constraint *z k*ð Þ and the control constraint *v k*ð Þ are used for the computation of the parameter estimation and matching scheme, while the corresponding state constraint *x k*ð Þ and control constraint *u k*ð Þ are reserved for optimizing the model-based optimal control problem.  The terms 1

*Control Theory in Engineering*

are declared and mutually integrated.

*H*2ð Þ¼ *k* 1

*J* 0 2ð Þ¼ *u* 2

þ 1 2

1 2

þX *N*�1

determined later.

**84**

*k*¼0

(a) Stationary condition:

(b) Co-state equation:

� *β*ð Þ*k* T*x k*ð Þ

**3.1 Necessary conditions for optimality**

Define the Hamiltonian function for Problem (E), given by:

where *λ*ð Þ*k* ∈ *ℜm*, *k* ¼ 0, 1, ⋯, *N* � 1, *β*ð Þ*k* ∈ *ℜn*, *k* ¼ 0, 1, ⋯, *N*, and *p k*ð Þ∈ *ℜn*, *k* ¼ 0, 1, ⋯, *N*, are modifiers. Using this Hamiltonian function in (4),

write the cost function in (3) to be the augmented cost function, that is,

� �

*H*2ð Þ� *k p k*ð ÞT*x k*ð Þþ *λ*ð Þ*k* T*v k*ð Þþ *β*ð Þ*k* T*z k*ð Þ

where *p k*ð Þ, *ξ*ð Þ*k* , *λ*ð Þ*k* , *β*ð Þ*k* , *μ*ð Þ*k* and Γ are the appropriate multipliers to be

Applying the calculus of variation [7, 9, 11, 13, 15] to the augmented cost function in (5), the following necessary conditions for optimality are obtained:

2

þ *μ*ð Þ*k* Tð Þ *f zk* ð Þ� ð Þ, *v k*ð Þ, *k Az k*ð Þ� *Bv k*ð Þ� *α*ð Þ*k*

*x k*ð ÞT*Qx k*ð Þþ *u k*ð ÞT*Ru k*ð Þ � �

*x N*ð ÞT*S N*ð Þ*x N*ð Þþ *γ*ð Þþ *N p*ð Þ 0 T

þ *ξ*ð Þ *N φ*ð Þ� *z N*ð Þ, *N* 1

þ *ξ*ð Þ*k Lzk* ð Þ� ð Þ, *v k*ð Þ, *k* 1

$$\mathbf{x}(k+1) = A\mathbf{x}(k) + Bu(k) + a(k) \tag{8}$$

(d) Boundary conditions:

$$p(\mathbf{N}) = \mathbf{S}(\mathbf{N})\mathbf{x}(\mathbf{N}) + \Gamma \text{ and } \mathbf{x}(\mathbf{0}) = \mathbf{x}\_0 \tag{9}$$

(e) Adjusted parameter equations:

$$\varphi(\mathbf{z}(\mathbf{N}), \mathbf{N}) = \frac{1}{2} \mathbf{z}(\mathbf{N})^{\mathrm{T}} \mathbf{S}(\mathbf{N}) \mathbf{z}(\mathbf{N}) + \boldsymbol{\chi}(\mathbf{N}) \tag{10}$$

$$L(z(k), v(k), k) = \frac{1}{2} \left( z(k)^T Q z(k) + v(k)^T R v(k) \right) + \gamma(k) \tag{11}$$

$$f(z(k), v(k), k) = Az(k) + Bv(k) + a(k)\tag{12}$$

(f) Modifier equations:

$$
\Gamma = \nabla\_{\mathbf{z}(N)} \varphi - \mathbf{S}(N)\mathbf{z}(N) \tag{13}
$$

$$\lambda(k) = -\left(\nabla\_{v(k)}L - Rv(k)\right) - \left(\frac{\partial f}{\partial v(k)} - B\right)^{\mathrm{T}} \hat{p}(k+1) \tag{14}$$

$$\boldsymbol{\beta}(k) = -\left(\nabla\_{\boldsymbol{x}(k)}\boldsymbol{L} - \mathbb{Q}\boldsymbol{z}(k)\right) - \left(\frac{\partial \boldsymbol{f}}{\partial \boldsymbol{z}(k)} - \boldsymbol{A}\right)^{\mathrm{T}} \boldsymbol{\hat{p}}(k+1) \tag{15}$$

with *ξ*ð Þ¼ *k* 1 and *μ*ð Þ¼ *k p k* ^ð Þ þ 1 .

(g) Separable variables:

$$
\nu(k) = \mu(k), \boldsymbol{z}(k) = \boldsymbol{x}(k), \hat{p}(k) = p(k). \tag{16}
$$

Notice that for the optimality necessary conditions obtained above, they are divided into three sets of necessary conditions. The first set of necessary conditions in (6)–(9) is the necessary conditions for the system optimization problem. The second set of necessary conditions in (10)–(12) defines the parameter estimation problem. The third set of necessary conditions in (13)–(15) provides the computation of multipliers. In fact, the necessary conditions, which are defined in (6)–(9), are the optimality for the modified model-based optimal control problem, and the adjusted parameters, which are calculated from the necessary conditions in (10)–(12), measure the differences between the real plant and the model used.

#### **3.2 Modified model-based optimal control problem**

As a consequence, the modified model-based optimal control problem, which is referred to as Problem (MM), is defined by

$$\min\_{\boldsymbol{\mu}(k)} J\_3(\boldsymbol{\mu}) = \frac{1}{2} \mathbf{x}(\boldsymbol{N})^\mathrm{T} \mathbf{S}(\boldsymbol{N}) \mathbf{x}(\boldsymbol{N}) + \boldsymbol{\Gamma}^\mathrm{T} \mathbf{x}(\boldsymbol{N}) + \boldsymbol{\gamma}(\boldsymbol{N}) + \sum\_{k=0}^{N-1} \frac{1}{2} \left( \mathbf{x}(k)^\mathrm{T} \mathbf{Q} \mathbf{x}(k) + \boldsymbol{u}(k)^\mathrm{T} \mathbf{R} \boldsymbol{u}(k) \right)$$

$$+ \boldsymbol{\gamma}(\boldsymbol{k}) + \frac{1}{2} r\_1 \left\| \boldsymbol{u}(\boldsymbol{k}) - \boldsymbol{v}(\boldsymbol{k}) \right\|^2 + \frac{1}{2} r\_2 \left\| \boldsymbol{x}(\boldsymbol{k}) - \boldsymbol{x}(\boldsymbol{k}) \right\|^2 - \boldsymbol{\lambda}(\boldsymbol{k})^\mathrm{T} \boldsymbol{u}(\boldsymbol{k}) - \boldsymbol{\beta}(\boldsymbol{k})^\mathrm{T} \mathbf{x}(\boldsymbol{k})$$

subject to

$$\varkappa(k+1) = A\varkappa(k) + Bu(k) + a(k), \ \varkappa(0) = \varkappa\_0$$

(17)

with the specified *α*ð Þ*k* , *γ*ð Þ*k* , *λ*ð Þ*k* , *β*ð Þ*k* , Γ, *v k*ð Þ and *z k*ð Þ, where the boundary conditions are given by *x*0 and *p N*ð Þ with the specified multiplier Γ.

It is obvious that Problem (MM), which is derived from Problem (E), is a modification of optimal control problem and is also known as a modified linear quadratic regular problem. Importantly, the set of the necessary conditions in (6)–(9) for Problem (E) is the necessary conditions that are satisfied by Problem (MM). In addition, due to the quadratic criterion feature of the objective function, the conjugate gradient method [17, 18], which is one of the numerical optimization techniques, could be applied to solve Problem (MM).

#### **3.3 Conjugate gradient algorithm**

For simplicity [19], establish Problem (MM) as a nonlinear optimization problem with the initial control given by *u*ð Þ 0 ¼ *u k*ð Þ0 as follows:

$$\min\_{u(k)} J\_3(u) \text{ subject to } u = u(k) \in \mathfrak{R}^m \text{ for } k = 0, 1, \dots, N - 1 \tag{18}$$

Let this problem as Problem (Q). Moreover, the Hamiltonian function defined in (4) is taken into consideration as an equivalent objective function. Hence, this Hamiltonian function allows the evaluation of the gradient function, which is the stationary condition in (6), and by using the iterative solution *u*ð Þ*i* ¼ *u k*ð Þ*i* to satisfy the state Eq. (8), which is solved forward in time, and the co-state Eq. (7), which is solved backward in time.

Define the gradient function *g* : *ℜm* ! *ℜm* as

$$\mathbf{g}\left(\boldsymbol{u}^{i}\right) = \nabla\_{\boldsymbol{u}} J\_{\mathfrak{I}}\left(\boldsymbol{u}^{i}\right) \tag{19}$$

*bi* ¼ *g*ð Þ *i*þ1 T � *g*ð Þ *i*þ1

*Conjugate Gradient Approach for Discrete Time Optimal Control Problems with Model-Reality…*

From the discussion above, we present the result as a proposition given as follows: Proposition 1. *Consider Problem* (*Q*)*. The control sequence u*ð Þ*i , which is defined in* (*22*)

*u*ð Þ*i* ¼ ð Þ *u*ð Þ 0 T,ð Þ *u*ð Þ1 T, ⋯,ð Þ *u N*ð Þ � 1 T h i,

*is generated through a set of the search direction vector d*ð Þ*i whose components are*

Step 0: Compute the initial gradient *g*ð Þ 0 from (20) and the initial search direc-

Step 1: Solve the state Eq. (8) forward in time from *k* ¼ 0 to *k* ¼ *N* with the

, *k* ¼ 0, 1, ⋯, *N:* Step 2: Solve the costate Eq. (7) backward in time from *k* ¼ *N* to *k* ¼ 0 with the

Step 6: Evaluate the gradient *g*ð Þ *i*þ1 and the search direction *d*ð Þ *i*þ1 , respectively, from (24) and (25) with computing *bi* from (26). If the gradient *g*ð Þ *i*þ1 ¼ *g*ð Þ*i* , within

a. Step 0 is the preliminary step for setting the initial search direction based on

b. Steps 1, 2, and 3 are performed to solve the system optimization by using the

c. Steps 4, 5, and 6 are the computation steps in implementing the conjugate

Accordingly, Problem (Q ) is solved by using the conjugate gradient algorithm. Indeed, the solution procedure for system optimization with parameter estimation could be described by joining the conjugate gradient algorithm with the parameters estimated. A summary of the calculation procedure including the principle of

Data: *A*, *B*, *Q*, *R*, *S N*ð Þ, *N*, *x*0, *r*1, *r*2, *kv*, *kz*, *kp*, *f*, *L:* Note that *A* and *B* could be determined based on the linearization of *f* at *x*0 or from the linear terms of *f*. Step 0: Compute a nominal solution. Assume that *α*ð Þ¼ *k* 0, *k* ¼ 0, 1, ⋯, *N* � 1,

,

the gradient direction in using the conjugate gradient algorithm.

for *i* = 0, 1, 2, … represents the iteration numbers.

*linearly independent. Also, the direction d*ð Þ*i is conjugacy.*

**Conjugate gradient algorithm**

*DOI: http://dx.doi.org/10.5772/intechopen.89711*

initial condition (9) to obtain *x k*ð Þ*i*

tion *d*ð Þ 0 from (21), respectively. Set *i* ¼ 0*:*

The conjugate gradient algorithm is summarized below:

boundary condition (9), where *p k*ð Þ*i* is the solution obtained.

Step 4: Solve (23) to obtain the step size *ai*. Step 5: Calculate the control *u*ð Þ *i*þ1 from (22).

a given tolerance, stop, else set *i* ¼ *i* þ 1, go to Step 1.

corresponding control sequence *u*ð Þ*i* .

Data: Choose the arbitrary initial control *u*ð Þ 0 and the tolerance *ε*.

Step 3: Calculate the value of the cost functional *J*3 *u*ð Þ*i* � � from (17).

*and is represented by*

**Remark 1:**

direction.

**87**

**3.4 Iterative calculation procedure**

model-reality differences is listed as follows:

**Iterative algorithm based on model-reality differences**

and *r*1 ¼ *r*2 ¼ 0*:* Solve Problem (M) defined by (2) to obtain *u k*ð Þ0

*g*ð Þ*i* T � *g*ð Þ*i* (26)

which is represented by the stationary condition in (6). For arbitrary initial control *u*ð Þ 0 ∈ *ℜm*, the initial gradient and the initial search direction are calculated from

$$\mathbf{g}^{(0)} = \mathbf{g}\left(\boldsymbol{u}^{(0)}\right) \tag{20}$$

$$d^{(0)} = -\mathbf{g}^{(0)}.\tag{21}$$

The following line search equation is applied to update the control sequence:

$$u^{(i+1)} = u^{(i)} + a\_i \cdot d^{(i)}\tag{22}$$

where *ai* ∈ *ℜ* is the step size, and its value can be determined from

$$\mathfrak{a}\_{i} = \underset{a \geq 0}{\arg\min} \mathcal{J}\_{\mathfrak{z}} \Big( \mathfrak{u}^{(i)} + \mathfrak{a} \cdot \boldsymbol{d}^{(i)} \Big). \tag{23}$$

After that, the gradient and the search direction are updated by

$$\mathbf{g}^{(i+1)} = \mathbf{g}\left(\boldsymbol{\mu}^{(i+1)}\right) \tag{24}$$

$$d^{(i+1)} = -\mathbf{g}^{(i+1)} + b\_i \cdot d^{(i)}\tag{25}$$

with

*Conjugate Gradient Approach for Discrete Time Optimal Control Problems with Model-Reality… DOI: http://dx.doi.org/10.5772/intechopen.89711*

$$b\_i = \frac{\mathbf{g}^{(i+1)\mathbf{T}} \cdot \mathbf{g}^{(i+1)}}{\mathbf{g}^{(i)\mathbf{T}} \cdot \mathbf{g}^{(i)}}\tag{26}$$

for *i* = 0, 1, 2, … represents the iteration numbers.

From the discussion above, we present the result as a proposition given as follows: Proposition 1. *Consider Problem* (*Q*)*. The control sequence u*ð Þ*i , which is defined in* (*22*) *and is represented by*

$$\boldsymbol{u}^{(i)} = \left[ \left( \boldsymbol{u}(\mathbf{0}) \right)^{\mathrm{T}}, \left( \boldsymbol{u}(\mathbf{1}) \right)^{\mathrm{T}}, \cdots, \left( \boldsymbol{u}(N-1) \right)^{\mathrm{T}} \right],$$

*is generated through a set of the search direction vector d*ð Þ*i whose components are linearly independent. Also, the direction d*ð Þ*i is conjugacy.*

The conjugate gradient algorithm is summarized below:

#### **Conjugate gradient algorithm**

Data: Choose the arbitrary initial control *u*ð Þ 0 and the tolerance *ε*.

Step 0: Compute the initial gradient *g*ð Þ 0 from (20) and the initial search direction *d*ð Þ 0 from (21), respectively. Set *i* ¼ 0*:*

Step 1: Solve the state Eq. (8) forward in time from *k* ¼ 0 to *k* ¼ *N* with the initial condition (9) to obtain *x k*ð Þ*i* , *k* ¼ 0, 1, ⋯, *N:*

Step 2: Solve the costate Eq. (7) backward in time from *k* ¼ *N* to *k* ¼ 0 with the boundary condition (9), where *p k*ð Þ*i* is the solution obtained.

Step 3: Calculate the value of the cost functional *J*3 *u*ð Þ*i* � � from (17).

Step 4: Solve (23) to obtain the step size *ai*.

Step 5: Calculate the control *u*ð Þ *i*þ1 from (22).

Step 6: Evaluate the gradient *g*ð Þ *i*þ1 and the search direction *d*ð Þ *i*þ1 , respectively, from (24) and (25) with computing *bi* from (26). If the gradient *g*ð Þ *i*þ1 ¼ *g*ð Þ*i* , within a given tolerance, stop, else set *i* ¼ *i* þ 1, go to Step 1.

**Remark 1:**

with the specified *α*ð Þ*k* , *γ*ð Þ*k* , *λ*ð Þ*k* , *β*ð Þ*k* , Γ, *v k*ð Þ and *z k*ð Þ, where the boundary

For simplicity [19], establish Problem (MM) as a nonlinear optimization prob-

Let this problem as Problem (Q). Moreover, the Hamiltonian function defined in

(4) is taken into consideration as an equivalent objective function. Hence, this Hamiltonian function allows the evaluation of the gradient function, which is the stationary condition in (6), and by using the iterative solution *u*ð Þ*i* ¼ *u k*ð Þ*i* to satisfy the state Eq. (8), which is solved forward in time, and the co-state Eq. (7), which is

which is represented by the stationary condition in (6). For arbitrary initial control *u*ð Þ 0 ∈ *ℜm*, the initial gradient and the initial search direction are calculated

*g*ð Þ 0 ¼ *g u*ð Þ 0

The following line search equation is applied to update the control sequence:

*g*ð Þ *i*þ1 ¼ *g u*ð Þ *i*þ1

*J*3 *u*ð Þ*i* þ *a* � *d*ð Þ*i*

where *ai* ∈ *ℜ* is the step size, and its value can be determined from

*ai* ¼ arg min *a* ≥0

After that, the gradient and the search direction are updated by

*u k*ð Þ *J*3ð Þ *u* subject to *u* ¼ *u k*ð Þ∈ *ℜm* for *k* ¼ 0, 1, ⋯, *N* � 1 (18)

*g ui* ¼ ∇*uJ*3 *ui* (19)

*d*ð Þ 0 ¼ �*g*ð Þ 0 *:* (21)

*u*ð Þ *i*þ1 ¼ *u*ð Þ*i* þ *ai* � *d*ð Þ*i* (22)

*d*ð Þ *i*þ1 ¼ �*g*ð Þ *i*þ1 þ *bi* � *d*ð Þ*i* (25)

*:* (23)

(20)

(24)

It is obvious that Problem (MM), which is derived from Problem (E), is a modification of optimal control problem and is also known as a modified linear quadratic regular problem. Importantly, the set of the necessary conditions in (6)–(9) for Problem (E) is the necessary conditions that are satisfied by Problem (MM). In addition, due to the quadratic criterion feature of the objective function, the conjugate gradient method [17, 18], which is one of the numerical optimization

conditions are given by *x*0 and *p N*ð Þ with the specified multiplier Γ.

techniques, could be applied to solve Problem (MM).

Define the gradient function *g* : *ℜm* ! *ℜm* as

lem with the initial control given by *u*ð Þ 0 ¼ *u k*ð Þ0 as follows:

**3.3 Conjugate gradient algorithm**

*Control Theory in Engineering*

min

solved backward in time.

from

with

**86**

#### **3.4 Iterative calculation procedure**

Accordingly, Problem (Q ) is solved by using the conjugate gradient algorithm. Indeed, the solution procedure for system optimization with parameter estimation could be described by joining the conjugate gradient algorithm with the parameters estimated. A summary of the calculation procedure including the principle of model-reality differences is listed as follows:

#### **Iterative algorithm based on model-reality differences**

Data: *A*, *B*, *Q*, *R*, *S N*ð Þ, *N*, *x*0, *r*1, *r*2, *kv*, *kz*, *kp*, *f*, *L:* Note that *A* and *B* could be determined based on the linearization of *f* at *x*0 or from the linear terms of *f*.

Step 0: Compute a nominal solution. Assume that *α*ð Þ¼ *k* 0, *k* ¼ 0, 1, ⋯, *N* � 1,

and *r*1 ¼ *r*2 ¼ 0*:* Solve Problem (M) defined by (2) to obtain *u k*ð Þ0 , *k* ¼ 0, 1, ⋯, *N* � 1, and *x k*ð Þ0 , *p k*ð Þ0 , *k* ¼ 0, 1, ⋯, *N:* Then, with *α*ð Þ¼ *k* 0, *k* ¼ 0, 1, ⋯, *N* � 1, and using *r*1, *r*2 from the data. Set *i* ¼ 0, *v k*ð Þ0 ¼ *u k*ð Þ0 , *z k*ð Þ0 ¼ *x k*ð Þ0 and *p k* ^ð Þ0 ¼ *p k*ð Þ0 *:*

Step 1: Compute the parameters *γ*ð Þ*k i* , *k* ¼ 0, 1, ⋯, *N*, and *α*ð Þ*k i* , *k* ¼ 0, 1, ⋯, *N* � 1, from (10)–(12). This is called the parameter estimation step.

Step 2: Compute the modifiers Γ*i* , *λ*ð Þ*k i* and *β*ð Þ*k i* , *k* ¼ 0, 1, ⋯, *N* � 1, from (13)– (15). Notice that this step requires taking the derivatives of *f* and *L* with respect to *v k*ð Þ*i* and *z k*ð Þ*i :*

Step 3: With *γ*ð Þ*k i* , *α*ð Þ*k i* , Γ*i* , *λ*ð Þ*k i* , *β*ð Þ*k i* , *v k*ð Þ*i* , and *z k*ð Þ*i* , solve Problem (Q ) using the conjugate gradient algorithm. This is called the system optimization step.

Step 4: Test the convergence and update the optimal solution of Problem (P). In order to provide a mechanism for regulating convergence, a simple relaxation method is employed:

$$\boldsymbol{v}(\boldsymbol{k})^{i+1} = \boldsymbol{v}(\boldsymbol{k})^{i} + \boldsymbol{k}\_{\boldsymbol{v}} \left(\boldsymbol{u}(\boldsymbol{k})^{i} - \boldsymbol{v}(\boldsymbol{k})^{i}\right) \tag{27}$$

*x*\_ ¼ 0 1

*x*0 ¼ ð Þ 10 10 T*:* Define the state *x* ¼ ð Þ *x*1 *x*2

*DOI: http://dx.doi.org/10.5772/intechopen.89711*

*J*ð Þ¼ 0 1 2 ð9*:*4 0*:*0

min*u J u*ð Þ¼ X

*x k*ð Þ¼ þ 1

min*u J u*ð Þ¼ X

*x k*ð Þ¼ þ 1

10

1 2

1 0 0 1 !*x k*ð Þþ

99% of the cost reduction to obtain the final cost of 128.50 units.

with the initial state *x*0 ¼ ð Þ 10 10 T, and the adjusted parameters *γ*ð Þ*k* , *k* ¼ 0, 1, ⋯, *N*, and *α*ð Þ*k* , *k* ¼ 0, 1, ⋯, *N* � 1, are supplied to the model used. By using the algorithm proposed, the simulation result is shown in **Table 1**. Notice that the minimum cost for Problem (M) is 546.05 units without adding the adjusted parameters. Once the adjusted parameters are taken into consideration, the iterative solution approximates to the true optimal solution of the original optimal control problem, in spite of model-reality differences. It is highlighted that there is a

**Figures 1** and **2** show the trajectories of control and state, respectively. With this

**Number of iteration Initial cost Final cost Elapsed time (s)** 20 17053.11 128.50 1.38021

control effort, the state reaches at the steady state after 4 units of time, which presents the oscillator stopped from moving. **Figure 3** shows the changes of the

*k*¼0

for the discretization transform.

10

1 2

1*:*00 0*:*94 �0*:*60 0*:*85 !*x k*ð Þþ

with the initial state *x*0 ¼ ð Þ 10 10 T*:* and the sampling time Δ*t* ¼ 0*:*94 s is taken

Consider the model-based optimal control problem, which is regarded as Prob-

*x*1ð Þ*k* 2 þ *x*2ð Þ*k* 2 þ *u k*ð Þ2 � � þ *γ*ð Þ*k* � �Δ*t*

*u k*ð Þþ *α*ð Þ*k*

1 0

!

*k*¼0

function

Problem (P), is given by:

subject to

lem (M), given by:

subject to

**Table 1.**

**89**

*Simulation result, Example 1.*

�*ω*2 �2*δω* � �*x* þ

*Conjugate Gradient Approach for Discrete Time Optimal Control Problems with Model-Reality…*

the velocity. For the purpose of controlling this oscillator, the following objective

with the natural frequency *ω* = 0.8, the damping ratio *δ* = 0.1, and the initial state

is minimized. This problem is a continuous-time linear optimal control problem, and the equivalence discrete time optimal control problem, which is regarded as

Δ*t x*1ð Þ*k* 2 þ *x*2ð Þ*k* 2 þ *u k*ð Þ2 � �

0 1

ð Þ *x*1ð Þ*t* 2 þ ð Þ *x*2ð Þ*t* 2 þ ð Þ *u t*ð Þ 2 � �*dt* (31)

0*:*00 0*:*94 !*u k*ð Þ (32)

(33)

� �*u* (30)

T, where *x*1 is the displacement and *x*2 is

$$z(k)^{i+1} = z(k)^i + k\_z \left(\varkappa(k)^i - z(k)^i\right) \tag{28}$$

$$\left(\hat{p}(k)^{i+1} = \hat{p}(k)^{i} + k\_{p}\left(p(k)^{i} - \hat{p}(k)^{i}\right)\right.\tag{29}$$

where *kv*, *kz*, *kp* ∈ð � 0, 1 are scalar gains. If *v k*ð Þ*i*þ1 ¼ *v k*ð Þ*i* , *k* ¼ 0, 1, ⋯, *N* � 1, and *z k*ð Þ*i*þ1 ¼ *z k*ð Þ*i* , *k* ¼ 0, 1, ⋯, *N*, within a given tolerance, stop; else set *i* ¼ *i* þ 1, and repeat the procedure starting from Step 1.

#### **Remark 2:**


**Author details**

Teresa Mouga MARE – Marine and Environmental Sciences Centre, Polytechnic Institute of Leiria, Leiria, Portugal

\*Address all correspondence to<EMAIL_ADDRESS>
© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


**Metal Oxide Nanoparticle Preparation by Pulsed Laser Ablation of Metallic Targets in Liquid Metal Oxide Nanoparticle Preparation by Pulsed Laser Ablation of Metallic Targets in Liquid**

Valery A. Svetlichnyi, Anastasiia V. Shabalina, Ivan N. Lapin and Daria A. Goncharova Valery A. Svetlichnyi, Anastasiia V. Shabalina, Ivan N. Lapin and Daria A. Goncharova

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/65430

### **Abstract**

[41] S. Y. Wei and K. Saitow. In situ multipurpose time-resolved spectrometer for monitoring nanoparticle generation in a high-pressure fluid. *Review of Scientific Instruments*, 83(7):

244 Applications of Laser Ablation - Thin Film Deposition, Nanomaterial Synthesis and Surface Modification

[42] L. V. Zhigilei, P. B. S. Kodali, and B. J. Garrison. A microscopic view of laser ablation. *Journal of Physical Chemistry B*, 102(16):2845–2853, 1998. doi: 10.1021/

073110, 2012. doi: 10.1063/1.4737886.

jp9733781.

The basic mechanisms of pulsed laser ablation in liquids (PLAL) as a method for the synthesis of nanoparticles (NPs) were considered. Physical and chemical processes occurring during the PLAL that determine the formation, composition and structure of the nanoparticles obtained are described. The influence of the composition and properties of the target material, the solvent and the characteristics of the laser irradiation on the efficiency of the synthesis of nanoparticles is discussed. Separately, an influence of the absorption and scattering (including nonlinear) of laser radiation in the dispersion of nanoparticles on the primary synthetic processes and secondary transformations inside the colloidal solution is examined. The specificity of the characterization of the colloidal solutions of oxide particles produced by PLAL is highlighted. The most promising practical applications of nanomaterials obtained are identified and the examples of their successful use in catalytic research and biomedicine are provided.

**Keywords:** pulsed laser ablation, nanocolloids, chemical reactions, composition and structure of nanoparticles, metal oxide nanoparticles


**7. Recommendations for safety and efficacy of exercise training in HD patients**

Exercising should be done within the first hour of dialysis; do not exercise if you are hypertensive, cramping, or have a volume overload. Avoiding excessive weights on the vascular access limb; physicians in charge of a workout program Intradialytic or interdialytic exercise is preferable. Strength training is also impractical during dialysis [42].

Aerobic activities dominate intraadialytic exercise regimes. AE is performed with a cycle ergometer or bicycle training. There are several workout plans that use varied exercise frequency, intensities, and durations. It has been demonstrated that AE lasting from 8 weeks to 6 months improved peak VO2 by around 17% in individuals with ESRD [43].

During dialysis, both endurance and RE training regimes have been routinely used. Most published research papers use intraadialytic cycling as the principal modality of endurance training. Patients bike on a cycle ergometer while receiving HD. Many research have employed intraadialytic resistance training as the predominant modality of exercise [44].

Whole body resistance training, in which muscles are progressively stressed by increasing weights or resistance over time, is the most effective way for enhancing muscular development, strength, and function. Unfortunately, patients confined to a dialysis chair or bed find it challenging to exercise, hence the majority of intradialytic resistance training regimens have relied on low- to moderate-intensity exercises employing ankle weights or elastic bands [45].

Because it affects intradialytic hemodynamic stress, exercise during dialysis session is best done within first 2 hours of dialysis. Exercise commonly raises blood pressure and results in post-exercise hypotension, as is widely recognized. The post-exercise hypotension is particularly alarming since it may raise the likelihood of harmful ischemia episodes, especially in the latter stages of HD when ultrafiltration is reducing the total blood volume [42].

Due to less limitations on the type, amount, and intensity of exercises that patients may conduct when they are not confined to a dialysis chair or bed, interdialytic exercise would appear to offer numerous advantages to intradialytic activities. IDE is supported primarily by the fact that it is highly time-effective for patients and that compliance can be thoroughly tracked [42].

Studies have shown that exercising while receiving dialysis treatments increases the effectiveness of the procedure. These studies' findings imply that intradialytic cycling can improve blood flow to the active leg muscles. This transfers the urea and other toxins that have been held in the muscle compartments to the blood stream for HD elimination. It has been proposed that an additional 20 minutes of dialysis might be equivalent to an hour of AE. Improvements in tiredness levels, sadness, quality of life, sleep, restless legs, inflammation, and hospitalization rates are also seen in the studies [12].

The following The Southern Alberta Renal Program (SARP) recommendations may aid in the delivery of IDE in HD units:

1.The physiotherapist should evaluate each patient's suitability for activity. ESRD comorbidities and etiology, all pertinent blood work, medications, cardiac history, bone health, symptoms (angina, shortness of breath, or pain), previous surgeries, injuries, hospitalizations, falls history, past/current exercise habits,

current living situation, ambulation aids, and ability to perform daily activities must all be covered in a thorough medical history.

	- Unstable cardiac state (arrhythmias, severe arterio-venous stenosis, decompensated congestive heart failure, and angina pectoris)
	- Physical conditions that would make using the bike difficult
	- Ineffective blood glucose regulation
	- A current disease or infection
	- An ineffective CVC or AVF/AVG
	- Aiming for an ultrafiltration rate (UFR) of less than 13 ml/h/kg
	- BP 180/100 or >100/50 mm Hg
	- Resting heart rate (BPM): 100
	- No illness or hospitalization in the previous week
	- AVF or AVG needing enough needling or a well working CVC
	- Absence of any unusual symptoms (headaches, nausea, dizziness, or the flu).
	- A minimum hemoglobin level of 9 g/dl is required; patients with more problematic cardiac histories may be put on hold until their hemoglobin levels rise.
	- Controlled blood sugar levels (between 126 and 252 mg/dl)
	- Without experiencing any symptoms, oxygen saturation levels should be over 90% at rest and above 88% during activity.


**Manuela Malatesta 1, Manuela Costanzo 1, Barbara Cisterna 1,\* and Carlo Zancanaro 1**

Anatomy and Histology Section, Department of Neurosciences, Biomedicine and Movement Sciences, University of Verona, Strada Le Grazie, 8 I-37134 Verona, Italy<EMAIL_ADDRESS>(M.M<EMAIL_ADDRESS>(M.C<EMAIL_ADDRESS>(C.Z.)

**\*** Correspondence<EMAIL_ADDRESS>
Received: 23 March 2020; Accepted: 21 April 2020; Published: 23 April 2020

**Abstract:** Satellite cells (SCs) participate in skeletal muscle plasticity/regeneration. Activation of SCs implies that nuclear changes underpin a new functional status. In hibernating mammals, periods of reduced metabolic activity alternate with arousals and resumption of bodily functions, thereby leading to repeated cell deactivation and reactivation. In hibernation, muscle fibers are preserved despite long periods of immobilization. The structural and functional characteristics of SC nuclei during hibernation have not been investigated yet. Using ultrastructural and immunocytochemical analysis, we found that the SCs of the hibernating edible dormouse, *Glis glis*, did not show apoptosis or necrosis. Moreover, their nuclei were typical of quiescent cells, showing similar amounts and distributions of heterochromatin, pre-mRNA transcription and processing factors, as well as paired box protein 7 (Pax7) and the myogenic differentiation transcription factor D (MyoD), as in euthermia. However, the finding of accumulated perichromatin granules (i.e., sites of storage/transport of spliced pre-mRNA) in SC nuclei of hibernating dormice suggested slowing down of the nucleus-to-cytoplasm transport. We conclude that during hibernation, SC nuclei maintain similar transcription and splicing activity as in euthermia, indicating an unmodified status during immobilization and hypometabolism. Skeletal muscle preservation during hibernation is presumably not due to SC activation, but rather to the maintenance of some functional activity in myofibers that is able to counteract muscle wasting.

**Keywords:** Hibernation; electron microscopy; immunocytochemistry


**5.3.2 In mission adaptation**

For these experiments, SAMP was given a static location in which to take control of the host vehicle. At this point, the mission planner agent generated a mission plan based on the knowledge available and the mission requirements. The instantiation of this mission plan is described in Fig. 13 using Planning Application Ontology concepts. The mission was then passed to the executive agent that took control of the vehicle for its execution.

While the mission was executed the status monitor agent maintained the knowledge base updated (in the OODA-loop sense) by reporting changes in the status of hardware components, such as batteries and sensors, and external parameters, such as water currents.

When observations indicated that some of these changes were affecting the mission under execution, the mission planner was activated in order to adapt the mission to the changes. This indication was detected by the planner agent by querying the knowledge base with the following question:

1 RSTA-I i.e., Reconnaissance/Surveillance/Target Acquisition & Identification capability concepts inherited from JAUS (SAE, 2006)

a)

The Path Towards Permanent Presence of Underwater Networks

b)

c)

d)

e)

action from the plan.

(m) during the mission, all plotted against mission time (s).

at the moment of initialising the survey of the areas.

memory usage, network activity and disk usage.

Fig. 15. Vehicle telemetry (top to bottom): a) vehicle velocity (m/s), b) compass heading (degrees), c) altitude (m), d) depth (m) and e) reconstructed profile of the seabed bathymetry

217 Embedded Knowledge and Autonomous Planning:

*SELECT ?Mission ?Action ?Param ?Status WHERE* { *plan:hasAction( ?Mission, ?Action)* ∧

*plan:hasExecParam( ?Action,?Param)* ∧ *plan:hasStatus( ?Param, ?Status)* }

An incipient *Status* of the action parameters indicates that the action can still be performed by adapting the way it is being executed, an execution repair. If both transducers were down, a critical status of the sidescan sensor is diagnosed and a plan repair adaptation of the mission plan would have been required instead. In that case, the adaptive mission planner would have looked for redundant components or similar capabilities to perform the action or to drop the

The same procedure was used after the transducer recovery was reported to adapt the survey action to the normal pattern during the second lawn mower survey. In a similar process, SAMP adapted the lawnmower pattern survey of the areas to the detected water current *Status*

The timeline of the mission executed using the SAMP approach is described in the following figures: Figure 14 represents the final trajectory of the vehicle in 2D and 3D using a North-East coordinate frame projection with the origin at the starting point of the mission. Figure 15 displays the vehicle's telemetry recorded during the mission. It includes vehicle's velocity, compass heading, altitude, depth measurements and processed bathymetry over time. Figure 16 shows subset of the variables being monitored by the status monitor agent that were relevant to this experiment. These variables include direction of water current, remaining battery power, the availability of the transducers in the sidescan sensor and the mission execution status. Figure 17 represents the system activity of the payload computer recorded during the mission. The system activity logs show percentage of processor usage,

(7)

Fig. 14. Left: Vehicle's track during mission in a North-East coordinate frame projection with the origin at the starting point of the mission. Right: Three-dimensional display of the vehicle's track during the mission (Note that depth coordinates are not to scale).

• *Are the observations coming from the environment affecting the mission currently under execution?*

In order to explain the reasoning process involved during the event detection, diagnosis and response phases of the mission adaptation process, a component fault as an internal event was temporarily simulated in the host vehicle. The fault simulated the gains of the starboard transducer of the sidescan sonar dropping to their minimum levels half way through the lawn mower survey of the first area.

For the detection phase, the low gain signals from the transducer triggered a symptom instance, which had an associated event level. This event level, represented in the Status Monitoring Application Ontology using a value partition pattern, plays a key role in the classification of the instances in the *Event* concept between being critical or incipient. This classification is represented axiomatically in the Eqs. 5 and 6.

$$\begin{array}{l}\text{status:} \text{CriticalEvent} \ \sqsubseteq \text{status:} \text{Event} \sqcap \text{status:} \text{causeedBySympatom} \dots \\\text{(status:} \text{Sympatom} \sqcap \text{status:} \text{hasEventLevel} \dots \\\text{(status:} \text{Level} \sqcap \text{status:} \text{High})\end{array} \tag{5}$$

status:IncipientEvent ... status:Event status:causedBySymptom . . . (status:Symptom status:hasEventLevel . . . (status:Level status:Med)) (6)

After the *Event* individuals were re-classified, the *Status* property of the related component in the Core Ontology was updated.

During the diagnosis event phase, a critical status of a component is only considered to be caused by a critical event. Therefore, due to the fact that the sidescan sonar component is composed of two transducers, port and starboard, one malfunctioned transducer was only diagnosed as an incipient *Status* of the overall sidescan sonar component.

During the response phase, the *Status* property of the Core Ontology components were used by the mission planner to perform the plan diagnosis of the mission under execution. The query to the knowledge base shown in Eq. 7 reported that the two survey actions in the mission plan were affected by the incipient status of the sidescan sonar.

18 Underwater Vehicles

East (m) East North

Fig. 14. Left: Vehicle's track during mission in a North-East coordinate frame projection with the origin at the starting point of the mission. Right: Three-dimensional display of the vehicle's track during the mission (Note that depth coordinates are not to scale).

• *Are the observations coming from the environment affecting the mission currently under execution?* In order to explain the reasoning process involved during the event detection, diagnosis and response phases of the mission adaptation process, a component fault as an internal event was temporarily simulated in the host vehicle. The fault simulated the gains of the starboard transducer of the sidescan sonar dropping to their minimum levels half way through the lawn

For the detection phase, the low gain signals from the transducer triggered a symptom instance, which had an associated event level. This event level, represented in the Status Monitoring Application Ontology using a value partition pattern, plays a key role in the classification of the instances in the *Event* concept between being critical or incipient.

After the *Event* individuals were re-classified, the *Status* property of the related component in

During the diagnosis event phase, a critical status of a component is only considered to be caused by a critical event. Therefore, due to the fact that the sidescan sonar component is composed of two transducers, port and starboard, one malfunctioned transducer was only

During the response phase, the *Status* property of the Core Ontology components were used by the mission planner to perform the plan diagnosis of the mission under execution. The query to the knowledge base shown in Eq. 7 reported that the two survey actions in the

North East

(5)

(6)

Depth

Depth

100 50 0 50 100 150 200

classification is represented axiomatically in the Eqs. 5 and 6.

(status:Level status:High))

(status:Symptom status:hasEventLevel . . .

status:IncipientEvent ...

(status:Level status:Med))

diagnosed as an incipient *Status* of the overall sidescan sonar component.

mission plan were affected by the incipient status of the sidescan sonar.

East (m)

mower survey of the first area.

the Core Ontology was updated.

North (m) North (m)

Fig. 15. Vehicle telemetry (top to bottom): a) vehicle velocity (m/s), b) compass heading (degrees), c) altitude (m), d) depth (m) and e) reconstructed profile of the seabed bathymetry (m) during the mission, all plotted against mission time (s).

*SELECT ?Mission ?Action ?Param ?Status WHERE* { *plan:hasAction( ?Mission, ?Action)* ∧ *plan:hasExecParam( ?Action,?Param)* ∧ *plan:hasStatus( ?Param, ?Status)* } (7)

An incipient *Status* of the action parameters indicates that the action can still be performed by adapting the way it is being executed, an execution repair. If both transducers were down, a critical status of the sidescan sensor is diagnosed and a plan repair adaptation of the mission plan would have been required instead. In that case, the adaptive mission planner would have looked for redundant components or similar capabilities to perform the action or to drop the action from the plan.

The same procedure was used after the transducer recovery was reported to adapt the survey action to the normal pattern during the second lawn mower survey. In a similar process, SAMP adapted the lawnmower pattern survey of the areas to the detected water current *Status* at the moment of initialising the survey of the areas.

The timeline of the mission executed using the SAMP approach is described in the following figures: Figure 14 represents the final trajectory of the vehicle in 2D and 3D using a North-East coordinate frame projection with the origin at the starting point of the mission. Figure 15 displays the vehicle's telemetry recorded during the mission. It includes vehicle's velocity, compass heading, altitude, depth measurements and processed bathymetry over time. Figure 16 shows subset of the variables being monitored by the status monitor agent that were relevant to this experiment. These variables include direction of water current, remaining battery power, the availability of the transducers in the sidescan sensor and the mission execution status. Figure 17 represents the system activity of the payload computer recorded during the mission. The system activity logs show percentage of processor usage, memory usage, network activity and disk usage.

Also, a peak on the CPU usage can be noted as this is the point where the mission partial plan

219 Embedded Knowledge and Autonomous Planning:

Symbol represents the point where the vehicle arrives to perform the survey of the area. At this point, the action survey gets instantiated based on the properties of the internal elements and external factors. Although the Loch waters where the trials were performed were very still (see Figure 16.b), note how the vehicle heading during the lawnmower pattern performed to survey the areas follows the water current direction sensed at the arrival (approx. 12*o*, Symbol - Figure 16.a) in order to minimize drag and maximise battery efficiency. The heading of the vehicle during the survey can be observed in Figure 14 and Figure 15.b. The link between the vehicle heading in relation to the water current direction and its effect on the battery consumption was expert orientation knowledge captured by a relationship property between

Symbol ♦ represents the point when the status monitor agent detects and reports a critical status in the starboard transducer of the sidescan sonar (Figure 16.d). It can be seen how the lawnmower pattern was adapted to cope with the change and to use the port transducer to cover the odd and even spacing of the survey. This pattern avoids gaps in the sidescan data under the degraded component configuration and maximises sensor coverage for the survey

Symbol � indicates the point where the starboard transducer recovery is diagnosed. It can be observed how the commands executing the action are modified in order to optimise the survey pattern and minimise distance travelled. Although also being monitored, the power status does not report any critical status during the mission that requires modification of the

Symbol � shows the location where all the mission goals are considered achieved and the control is given back to the mission control of the host vehicle (see Symbol � - Figure 16.e shows the mission is still active but the payload is not longer in control (0x01) ). From this point the host vehicle's control module takes the control back and drives the vehicle to the

The underwater domain is a challenging environment in which to maintain the operability of an UUV. Operability can be improved with the embedded adaptation of the mission plan. We implement a system capable of adapting mission plans autonomously in the face of events while during a mission. We do this by using a combination of ontological hierarchical representation of knowledge and adaptive mission plan repair techniques. The advantage of this approach is that it maximises robustness, system performance and response time. The system performance has been demonstrated in simulation. Additionally, the mission

• *Knowledge based framework:* We have presented a semantic-based framework that provides the core architecture for knowledge representation for service oriented agents in UUVs. The framework combines the initial expert orientation and the observations acquired during mission in order to improve the situation awareness in the vehicle. This is currently

• *Goal-oriented plan vs. waypoint-based plan:* The system uses a goal-oriented approach in which the mission is described in terms of 'what to do' instead of a 'how to do' it. The

adaptation capability is shown during an in-water field trial demonstration.

In our fully integrated experiments we achieved the following:

gets generated (Figure 17.a).

The Path Towards Permanent Presence of Underwater Networks

the two concepts in the Core Ontology.

while the transducer is down.

loiter at the recovery location.

unavailable in UUVs.

**6. Conclusion and future work**

actions (Figure 16.c).

Fig. 16. Status monitoring (top to bottom): a) direction of water current (degrees), b) speed of water current (m/s), c) battery power (Wh), d) sidescan sensor port and e) starboard transducers availability (on/off) and f) mission status binary flag, all plotted against mission time (s).

Fig. 17. System activity (top to bottom): a) % processor usage, b) % memory usage, c) network activity (packets/s) and d) disk usage (I/O sectors/s), all plotted against mission time (s).

Each of the symbols �, , ♦, � and � on the aforementioned figures represents a point during the mission where an event occurred. Symbol � represents the point where SAMP takes control of the vehicle. Note a change on the host platform mission status binary flag that becomes 0x05, i.e. the mission is active (0x01) and the payload is in control (0x04) (Figure 16.e). 20 Underwater Vehicles

0 500 1000 1500 2000

0 500 1000 1500 2000

Fig. 16. Status monitoring (top to bottom): a) direction of water current (degrees), b) speed of

transducers availability (on/off) and f) mission status binary flag, all plotted against mission

0 500 1000 1500 2000

Fig. 17. System activity (top to bottom): a) % processor usage, b) % memory usage, c) network activity (packets/s) and d) disk usage (I/O sectors/s), all plotted against mission

Each of the symbols �, , ♦, � and � on the aforementioned figures represents a point during the mission where an event occurred. Symbol � represents the point where SAMP takes control of the vehicle. Note a change on the host platform mission status binary flag that becomes 0x05, i.e. the mission is active (0x01) and the payload is in control (0x04) (Figure 16.e).

water current (m/s), c) battery power (Wh), d) sidescan sensor port and e) starboard

a)

b)

c)

d)

e)

f)

a)

b)

c)

d)

time (s).

time (s).

0.0 0.4 0.8

Also, a peak on the CPU usage can be noted as this is the point where the mission partial plan gets generated (Figure 17.a).

Symbol represents the point where the vehicle arrives to perform the survey of the area. At this point, the action survey gets instantiated based on the properties of the internal elements and external factors. Although the Loch waters where the trials were performed were very still (see Figure 16.b), note how the vehicle heading during the lawnmower pattern performed to survey the areas follows the water current direction sensed at the arrival (approx. 12*o*, Symbol - Figure 16.a) in order to minimize drag and maximise battery efficiency. The heading of the vehicle during the survey can be observed in Figure 14 and Figure 15.b. The link between the vehicle heading in relation to the water current direction and its effect on the battery consumption was expert orientation knowledge captured by a relationship property between the two concepts in the Core Ontology.

Symbol ♦ represents the point when the status monitor agent detects and reports a critical status in the starboard transducer of the sidescan sonar (Figure 16.d). It can be seen how the lawnmower pattern was adapted to cope with the change and to use the port transducer to cover the odd and even spacing of the survey. This pattern avoids gaps in the sidescan data under the degraded component configuration and maximises sensor coverage for the survey while the transducer is down.

Symbol � indicates the point where the starboard transducer recovery is diagnosed. It can be observed how the commands executing the action are modified in order to optimise the survey pattern and minimise distance travelled. Although also being monitored, the power status does not report any critical status during the mission that requires modification of the actions (Figure 16.c).

Symbol � shows the location where all the mission goals are considered achieved and the control is given back to the mission control of the host vehicle (see Symbol � - Figure 16.e shows the mission is still active but the payload is not longer in control (0x01) ). From this point the host vehicle's control module takes the control back and drives the vehicle to the loiter at the recovery location.


**6. Substitutional soft tissue grafts**

Although the results of PIS grafting with substitutional grafts, at the present are inferior to the results obtained after autogenous soft tissue grafting [5, 8, 65, 66, 93, 99], the absence of the donor site makes this treatment modality appealing to the patient and practitioners, as well [26, 42, 100].

The elimination of the harvesting procedure [25, 65] will lead to the reduction of surgical time [65], simplify the surgical procedure [42, 100], decrease the patient morbidity [26, 42, 65, 93], allow the unlimited supply of the soft tissue graft [26, 101, 102], and increase patient acceptance for the procedure [42, 65, 100, 103].

The augmentation procedure with substitutional soft tissue grafts will result in PIS with perfect color and texture blending to the adjacent soft tissue [65, 101].

Two types of substitutional grafts are available—xenogenic and allogeneic soft tissue grafts. Both of the grafts can be used for augmenting the volume and the width of keratinized mucosal band [25, 26, 65, 93, 99, 104].

The substitutional grafts are deprived of vital cells. During the manufacturing procedure, cells and antigenic components are removed, preserving only the extracellular matrix consisting mainly of collagen and elastin fibers. The three-dimensional structures of the aforementioned scaffold will promote fibroblast and keratinocyte migration and vascular ingrowth from the surrounding tissue [105–107]. This will result in an excellent color match since the keratinocytes are derived from the surrounding tissue. Nevertheless, compared to autogenous soft tissue grafts, they do not possess the ability to promote keratinization, limiting their application for increasing the width of KM [105, 106]. To overcome this drawback, a combination of an FGG graft with reduced apico-coronal dimensions to 2 mm and an XCM was proposed [6].

When used to augment PIS thickness, substitutional grafts are less resistant to compression of the overlaying flap compared to CTG. Loss of the initial volume of the substitutional graft can lead to the compromised outcome of the grafting procedure. To overcome this drawback, a volume stable collagen matrix was developed [105]. As a result of the cross-linking process of the collagen fibers, the collagen matrix becomes more volume stable and prone to withstand soft-tissue pressure [108, 109]. At the moment there is a lack of literature on the long-term stability of augmented PIS with the substitutional grafts (**Figures 39–66**) [105].

**Figure 39.** *Initial situation—Lateral view.*

**Figure 40.** *Initial situation.*

**Figure 41.** *Surgical stent.*

**Figure 42.** *Dehiscence bone defects around implant 16 and 15.*

#### **Figure 43.**

*GBR: A composite bone graft was used consisting of 50% autogenous and 50% xenogenic graft. The bone graft was applied in two layers—The internal layer which is covering the exposed implant surface, is made out of autogenous bone and the external layer is consisting of a xenogenic bone graft.*

**Figure 44.** *GBR: Native collagen membrane stabilized with resorbable sutures.*

#### **Figure 45.**

*Suturing in three layers;(1) palatal-apical position—Mattress sutures for membrane stabilization, (2) buccalapical—Mattress sutures for initial closing of the flap, and (3) bucco-coronal—Simple interrupted suture for the final closure of the flap.*

#### **Figure 46.**

*Four months after the GBR, the palatal displacement of the mucogingival line is evident. Occlusal view.*

*Peri-Implant Soft Tissue Augmentation DOI: http://dx.doi.org/10.5772/intechopen.101336*

**Figure 48.** *The surgical stent was used to determine the dimensions of the flap.*

#### **Figure 49.**

*The flap incision is made not at the mucogingival junction but 4 mm within the keratinized mucosa, therefore the buccal split-thickness flap will include a band of keratinized mucosa which is 4 mm wide.*

#### **Figure 50.**

*Finalized preparation of the buccal split-thickness flap. The most coronal part of the partial-thickness flap consists of keratinized mucosa.*

#### **Figure 51.**

*Apically positioned flap—Stabilization of the buccal partial-thickness flap in the new apical position. The exposed periosteal surface is completely surrounded by keratinized mucosa.*

**Figure 52.** *The exposed periosteal surface is covered with xenogenic collagen matrix (Mucoderm, Botiss gmbh, Berlin).*

#### **Figure 53.**

*Healing two months after the keratinized mucosa widening procedure. The gain of the keratinized mucosa is evident but the thickness of the gained tissue is unsatisfactory. Occlusal view.*

#### **Figure 54.**

*Healing two months after the keratinized mucosa widening procedure. The gain of the keratinized mucosa is evident but the thickness of the gained tissue is unsatisfactory. Lateral view.*

#### **Figure 55.**

*During the implant uncovering procedure, a primary flap was prepared on the palatal side. The minimal thickness of 1.5 mm of the primary flap was respected, and the connective tissue was exposed.*

#### **Figure 56.**

*Mesial, distal and apical incisions were made inside the connective tissue graft in order to completely dissect the CTG from the rest of the adjacent soft tissue.*

**Figure 57.** *The CTG is completely dissected from the rest of the adjacent soft tissue.*

#### **Figure 58.**

*On the left: CTG harvested from the palate (the harvesting procedure was displayed on the previous pictures), on the right additional CTG harvested from the tuberosity on the same side.*

#### **Figure 60.** *Appearance of the regenerated bone on the buccal side of the implants.*

**Figure 61.** *Both of the CTG grafts were stabilized with sutures to the buccal flap.*

**Figure 63.** *Final suturing of the flap.*

#### **Figure 64.**

*After 2 months of healing adequate quantity (soft tissue thickness) and quality (width of keratinized mucosa) of soft tissue surrounding the healing abutments.*

**Figure 65.** *Screw retained abutments.*

**Figure 66.** *Three units screw-retained bridge.*