**5. Experimental Manifestations of Spin-Torsion Coupling**

Experimental search for the nontrivial torsion effects is naturally embedded into the broader framework of the studies of the spin-dependent long-range forces [21–40]. By making use of the corresponding experimental techniques, it is possible to find strong limits on the values of the gauge gravity spin-torsion coupling constants and on the torsion field itself. A good example of an efficient approach in this respect gives an observation of the nuclear spin precession in gaseous spin polarized 3He or 129Xe samples with the help of a highly sensitive low-field magnetometer [61–63] detecting a sidereal variation of the relative spin precession frequency in a new type of 3He/129Xe clock comparison test. In a similar experiment [64], the ratio of nuclear spin-precession frequencies of 199Hg and 201Hg atoms was measured in the magnetic field and the Earth's gravitational field. Based on the corresponding experimental data from [64], the analysis of dynamics of the minimally coupled Dirac fermion [19] in external electromagnetic and gravitational fields revealed the strong bounds on the possible background space-time torsion:

$$\frac{c}{2} \left| \tilde{T} \right| \cdot \left| \cos \theta \right| < 6.45 \times 10^{-6} \text{s}^{-1} \text{ } \tag{55}$$

where *θ* is the angle between the magnetic *B* and torsion *T***ˇ** fields. On the other hand, by making use of the experimental data from [61], one finds the restriction:

$$\frac{c}{2} \left| \check{T} \right| \cdot \left| \cos \theta \right| < 3.59 \times 10^{-7} s^{-1}. \tag{56}$$

These results are consistent with the alternative empirical estimates for the torsion limits [30–40]. As another powerful tool one can mention the use of the quantum interferometry to probe the spacetime structure, including the search for possible post-Riemannian deviations, focusing on the detection of the phase shift and polarization rotation effects for the neutron and atom beams [65–69].

As an application of the quantum hydrodynamics formalism, let us investigate a simple model of a continuous medium of particles with spin and consider the dynamics of the spin waves in such a particle system. Neglecting the spin-thermal coupling in the spin dynamical Equation (46) and assuming the small perturbations of the spin *s* = *s*0 + *δs* around an undisturbed value |*s*0| = *h*¯ /2, we find, in the first order,

$$
\delta\partial\_t\delta\mathbf{s} = \mathbf{\hat{D}}^{(0)} \times \delta\mathbf{s} + \mathbf{\hat{D}}^{(1)} \times \mathbf{s}\_0. \tag{57}
$$

Here, the equilibrium values of the external background fields are encoded in **Ωˆ** (0) = − *ω*0 − *q m B*0 − *c* 2 *T***ˇ**, where *B*0, *ω*0 and *T***ˇ** are the external uniform magnetic field, the Earth's angular velocity and the background torsion, respectively, and the small disturbance reads:

$$
\hat{\mathbf{M}}^{(1)} = -\frac{\Delta\delta\mathbf{s}}{m},
\tag{58}
$$

Assuming that the perturbations of the spin vary as *δs* ∼ exp(−*iωst* + *ik* · *r*), we then derive the dispersion law relating the wave frequency *ωs* and the wave vector *k* for the spin waves excited in the external magnetic and torsion fields:

$$
\omega\_s^2 = \Omega\_c^2 + \frac{c^2}{4}\vec{T}^2 + \omega\_0^2 + 2\Omega\_c\omega\_0\cos\theta\_1 + c\,\Omega\_c\vec{T}\cos\theta\_2 + c\,\omega\_0\vec{T}\cos(\theta\_2 - \theta\_1). \tag{59}
$$

Here, Ω*c* = *qB*0 *m* + *hk*¯ 2 2*m* , whereas *θ*1 and *θ*2 are, respectively, the angle between the external magnetic field and Earth's angular velocity, and the angle between *B*0 and the background torsion *T***ˇ**. Equation (59) is a generalization for the dispersion relation of spin waves found in Reference [70] and it takes into account the contribution of the spin part of the quantum Bohm potential as an additional spin torque due to the self-action inside the system of particles, which leads to the propagation of spin waves. The square of the frequency *ω*2 *s* encompasses a contribution proportional to the square of the modulus of the wave vector ∼ *k* 2 . As we can see from the dispersion relation (59) the torsion effect is maximal when the pseudovector field *T***ˇ** is aligned along the external magnetic field.

### **6. Discussion and Conclusions**

In this paper, we for the first time developed the quantum hydrodynamics for the many-particle system of massive Dirac fermion spin-1/2 particles interacting with external electromagnetic, metric gravitational/inertial and torsion fields. This essentially extends the single-particle quantum hydrodynamical approach which was developed for the flat spacetime, see [71–76] and the references therein. Taking as the basis of the earlier general formalism [19], the consistent hydrodynamical formulation was constructed for the manyparticle quantum system of fermions, and the explicit relations between the microscopic and macroscopic fluid variables were derived with help of the Madelung decomposition approach. In the present study, we have focused on the physically important situation with *T*ˇ 0ˆ = 0, a more exotic case with *T*ˇ 0ˆ 6= 0 (which may be realized in cosmology, for example, see the earlier work [15], or in the more general models) will be considered elsewhere.

The resulting system of hydrodynamical equations consists of the continuity Equation (44), the momentum balance Equation (45) and the spin dynamics Equation (46). The momentum balance equation includes the contributions in the form of the quantum Bohm potential and a new spin part of the Bohm quantum potential which are proportional to the square of Planck's constant. In addition, the dynamical equations take into account the thermal effects resulting from the fluctuations of the spin and velocity near their average values. As an application of the formalism, we evaluated the possible effects of the spacetime torsion and the spin part of quantum Bohm potential on the dispersion characteristics of the spin waves (59) excited in the many-particle fermion system. The

developed hydrodynamical model can be used in the future studies of various types of transport phenomena in spinning matter with an account of external electromagnetic and gravitational fields.

**Author Contributions:** All authors discussed the results and contributed to the final manuscript. All authors have read and agreed to the published version of the manuscript.

**Funding:** The work of MIT was supported by the Russian Science Foundation under the grant 19-72-00017, and the research of YNO was supported in part by the Russian Foundation for Basic Research (Grant No. 18-02-40056-mega).

**Conflicts of Interest:** The authors declare no conflict of interest.

### **References**


*3.2.1. Inverted OLEDs incorporated with the combination of Li2CO3:PTCDA and Li2CO3:BCP*

Here first we will describe the optical and structural properties of n-doped materials and then the prperties of OLEDs fabricated from these materials.

#### **<IP_ADDRESS>. Optical and structural properties of Li2CO3:PTCDA and Li2CO3:BCP**

152 Organic Light Emitting Devices

performance.

of acceptor.

*Li2CO3:BCP*

IBOLEDs as described below.

**layers**

mechanism of the ITO/PTCDA:CuPc anode is described as follows: when the ITO is biased positive, holes confined at the ITO/PTCDA:CuPc interface may effectively polarize PTCDA:CuPc pairs, efficiently generating holes in CuPc and electrons in PTCDA, and then holes are transferred into the NPB layer by CuPc aggregates and electrons into the ITO by PTCDA aggregates, thus generating a very efficient hole current. The static-inducement for very efficient hole-electron pair generation in PTCDA:CuPc is the reason why the ITO/PTCDA:CuPc anode structure is superior to the ITO/CuPc anode structure. For the ITO/ PTCDA:CuPc anode structure, if the thickness of the PTCDA: CuPc composite (such as 5 nm) is smaller than the optimal thickness of 10–20 nm, the number of PTCDA:CuPc pairs involved in hole-electron pair generation is smaller, hence generating less efficient hole current and resulting in reduced device performance. If the thickness of the PTCDA:CuPc composite is larger (e.g. 30 nm) than the optimal thickness of 10–20 nm, there is no more enhancement of hole-electron pairs generation, but the absorption of PTCDA:CuPc composite over Alq3 emission becomes remarkable, causing the degradation in device

*3.1.3. The significance and future development of the donor/ acceptor interface in OLEDs*

Very recently, the planar and mixed donor/acceptor interfaces have also been used as very efficient interconnecting structures for tandem OLEDs [27], showing the versatility of the donor/acceptor pairs. However, it should be noted that all the donors and acceptors currently used to form the hole-generating interfaces in OLEDs exhibit significant absorption in the visible-light range, thus reducing the light out-coupling from the devices. Therefore, it is of great importance to develop wide-bandgap donors and acceptors. In addition, in order to increase the amplitude of hole current at the donor/acceptor interfaces, it is of much necessity to narrow down the offset between the HOMO of donor and LUMO

**3.2. Increasing electron current in IBOLEDs via the combination of two n-doped**

The n-doped organic electron acceptors, e.g., n-NTCDA, n-PTCDA, n-C60, possess markedly higher conductivities but markedly lower capabilities of injecting electrons into electron transport layer (such as BCP, Alq3, etc.), as compared to the frequently used n-doped materials (such as n-BCP, n-Alq3, etc.) in OLEDs. However, recent work [13-16] shows that the combination of the above two classes of n-doped materals may increase current in

Here first we will describe the optical and structural properties of n-doped materials and

*3.2.1. Inverted OLEDs incorporated with the combination of Li2CO3:PTCDA and*

then the prperties of OLEDs fabricated from these materials.

The optical properties of 1:2 Li2CO3:PTCDA and 1:4 Li2CO3:BCP are studied here and presented in Fig. 8. As seen in Fig. 8(a), compared to the intrinsic PTCDA, Li2CO3:PTCDA exhibits much lower intermolecular optical absorption at 558 nm [26], indicating severe distortion of the ordered - stack of PTCDA molecules caused by the doping with Li2CO3. In addition, a new sub-gap absorption centered at a wavelength of 686 nm appears, indicating the formation of the charge transfer state between the matrix and dopant in Li2CO3:PTCDA composite [28]. Fig. 8(b) shows that the edge of the optical absorption for 1:4 Li2CO3:BCP was slightly blue-shifted relative to that of intrinsic BCP, which implies the occurrence of electron transfer from the O2- bonded to Li to BCP 1:4 Li2CO3:BCP composite [29].

**Figure 8.** UV-vis absorption spectra for the 20 nm PTCDA and 1:2 Li2CO3:PTCDA thin films (a), and the 20 nm BCP and 1:2 Li2CO3:PTCDA thin films (b). The inset in (b) is the molecular structure of PTCDA.

#### **<IP_ADDRESS>. Electrical properties of Li2CO3:PTCDA and Li2CO3:BCP**

For studying the electrical properties we have fabricated the following two devices: Device 6: ITO (anode)/ 1:4 Li2CO3:BCP 10 nm/ 1:2 Li2CO3:PTCDA 5 nm/ Alq3 65 nm/ Al (cathode); Device 7: ITO (anode)/ 1:4 Li2CO3:BCP 5 nm/ 1:2 Li2CO3:PTCDA 10 nm/ Alq3 65 nm/

Al (cathode).

In device 6 with the structure of ITO (anode)/1:4 Li2CO3:BCP *x* nm/1:2 Li2CO3:PTCDA 15-*x* nm/Alq3 65 nm/Al (cathode), the electron current passes through Alq3 to 1:2 Li2CO3:PTCDA and to 1:4 Li2CO3:BCP connected in series. Therefore, the relative conductivity of 1:2 Li2CO3:PTCDA to 1:4 Li2CO3:BCP can be detected by observing the influence of 1:4 Li2CO3:BCP thickness on the device current. Fig. 9 exhibits that the current of device 6 is much lower than that of device 7 at a given driving voltage, demonstrating that 1:2 Li2CO3:PTCDA is much more conductive than 1:4 Li2CO3:BCP. This can be due to the following two factors: firstly, the charge carriers are more efficiently generated in 1:2 Li2CO3:PTCDA than in 1:4 Li2CO3:BCP due to the lower LUMO level of PTCDA than that of BCP. Secondly, the electron mobility of intrinsic PTCDA (310-6 cm2·V-1·s-1) is much higher than that of BCP (610-7 cm2·V-1·s-1). Therefore, it might be feasible to improve the performance of IBOLED via the introduction of the Li2CO3:PTCDA composite onto the ITO cathode for enhancing current conduction.

The Advanced Charge Injection Techniques Towards the Fabrication of High-Power Organic Light Emitting Diodes 155

**Figure 10.** (a) I-V characteristics, (b) L-V characteristics, (c) current and power efficiencies versus current density of devices 8 (filled squares) and 9 (filled triangles). Both the devices give nearly the

Figs. 10 (a) and (b) show that device 8 performs better that device 9, which may be attributed to the enhanced electron current due to the following : (i) In device 8, the electron injection is realized not only via tunneling, but also via thermionic emission due to relatively smaller energy barrier from ITO into the LUMO of PTCDA. (ii) The conductivity of 1:2 Li2CO3:PTCDA is higher than that of 1:4 Li2CO3:BCP, leading to reduced ohmic loss of

same Alq3 emission.

**Figure 9.** I-V characteristics for devices 6 and 7. As there is no Alq3 emission observed from the two devices in the measurement range, the current in both the two devices is considered to be comprised of electrons only.

#### **<IP_ADDRESS>. Inverted OLED using the combination of Li2CO3:PTCDA and Li2CO3:BCP**

We have fabricated the following three inverted devices and studied their characteristics.

Device 8: ITO (cathode)/ 1:2 Li2CO3:PTCDA 5 nm/ 1:4 Li2CO3:BCP 5 nm/ Alq3 40 nm/ NPB 40 nm/ MoO3 10 nm/Al (anode);

Device 9: ITO (cathode)/ 1:4 Li2CO3:BCP 10 nm/ Alq3 40 nm/

NPB 40 nm/ MoO3 10 nm/Al (anode);

Device 10: ITO (cathode)/ 1:2 Li2CO3:PTCDA 5 nm/ BCP 5 nm/ Alq3 40 nm/ NPB 40 nm/ MoO3 10 nm/Al (anode).

Fig.10 shows the performance of devices 8 and 9. It can be seen in Fig. 10(a) that the driving voltage of device 8 for achieving a given current was reduced than that of device 9. To generate a current density of 100 mA/cm2, device 8 needs a driving voltage of 7.1 V, smaller than that needed by device 9 (8.3 V). Device 8 also gives higher luminance than device 9 as shown in Fig. 10(b). At a driving voltage of 8 V, the luminance of device 8 is 6983 cd/m2, in contrast to that of 1354 cd/m2 for device 9. Fig. 10(c) shows that the current efficiency of device 8 is slightly lower than that of device 9, mostly because 1:2 Li2CO3:PTCDA exhibits some absorption over Alq3 emission, while 1:4 Li2CO3:BCP is nearly transparent in the visible-light range. Nevertheless, due to the marked reduction of driving voltage, device 8 provides higher power efficiency than device 9. It can be concluded that the two-layer combination of 1:2 Li2CO3:PTCDA and 1:4 Li2CO3:BCP outperforms the single 1:4 Li2CO3:BCP with regard to promoting the current conduction for the IBOLED.

electrons only.

characteristics.

for the IBOLED.

NPB 40 nm/ MoO3 10 nm/Al (anode);

NPB 40 nm/ MoO3 10 nm/Al (anode);

NPB 40 nm/ MoO3 10 nm/Al (anode).

Device 9: ITO (cathode)/ 1:4 Li2CO3:BCP 10 nm/ Alq3 40 nm/

cathode for enhancing current conduction.

than that of BCP (610-7 cm2·V-1·s-1). Therefore, it might be feasible to improve the performance of IBOLED via the introduction of the Li2CO3:PTCDA composite onto the ITO

**Figure 9.** I-V characteristics for devices 6 and 7. As there is no Alq3 emission observed from the two devices in the measurement range, the current in both the two devices is considered to be comprised of

We have fabricated the following three inverted devices and studied their

Fig.10 shows the performance of devices 8 and 9. It can be seen in Fig. 10(a) that the driving voltage of device 8 for achieving a given current was reduced than that of device 9. To generate a current density of 100 mA/cm2, device 8 needs a driving voltage of 7.1 V, smaller than that needed by device 9 (8.3 V). Device 8 also gives higher luminance than device 9 as shown in Fig. 10(b). At a driving voltage of 8 V, the luminance of device 8 is 6983 cd/m2, in contrast to that of 1354 cd/m2 for device 9. Fig. 10(c) shows that the current efficiency of device 8 is slightly lower than that of device 9, mostly because 1:2 Li2CO3:PTCDA exhibits some absorption over Alq3 emission, while 1:4 Li2CO3:BCP is nearly transparent in the visible-light range. Nevertheless, due to the marked reduction of driving voltage, device 8 provides higher power efficiency than device 9. It can be concluded that the two-layer combination of 1:2 Li2CO3:PTCDA and 1:4 Li2CO3:BCP outperforms the single 1:4 Li2CO3:BCP with regard to promoting the current conduction

**<IP_ADDRESS>. Inverted OLED using the combination of Li2CO3:PTCDA and Li2CO3:BCP**

Device 8: ITO (cathode)/ 1:2 Li2CO3:PTCDA 5 nm/ 1:4 Li2CO3:BCP 5 nm/ Alq3 40 nm/

Device 10: ITO (cathode)/ 1:2 Li2CO3:PTCDA 5 nm/ BCP 5 nm/ Alq3 40 nm/

**Figure 10.** (a) I-V characteristics, (b) L-V characteristics, (c) current and power efficiencies versus current density of devices 8 (filled squares) and 9 (filled triangles). Both the devices give nearly the same Alq3 emission.

Figs. 10 (a) and (b) show that device 8 performs better that device 9, which may be attributed to the enhanced electron current due to the following : (i) In device 8, the electron injection is realized not only via tunneling, but also via thermionic emission due to relatively smaller energy barrier from ITO into the LUMO of PTCDA. (ii) The conductivity of 1:2 Li2CO3:PTCDA is higher than that of 1:4 Li2CO3:BCP, leading to reduced ohmic loss of current conduction. Fig. 10(c) shows that the current efficiency of device 8 is slightly lower than that of device 9, mostly because 1:2 Li2CO3:PTCDA exhibits some absorption over Alq3 emission, while 1:4 Li2CO3:BCP is nearly transparent in the visible-light range. Nevertheless, due to the marked reduction of driving voltage, device 8 provides higher power efficiency than device 9. In terms of device 8, the issue of the efficient electron transfer from 1:2 Li2CO3:PTCDA into 1:4 Li2CO3:BCP needs to be addressed, since the LUMO level (4.6 eV) of PTCDA is 1.6 eV lower than that (3.0 eV) of BCP. In quest of the mechanism of electron transport over such a big Schottky barrier, device 10 is fabricated and characterized. As seen in Fig. 11, device 10 shows poor current-voltage and luminancevoltage characteristics. At a driving voltage of 10 V, it exhibits a current of 2.3 mA/cm2 and a luminance of 4.1 cd/m2, in clear contrary to 531 mA/cm2 and 10492 cd/m2 achieved at a driving voltage of 8.4 V in device 8. This greatly improved performance of device 8 relative to device 10 implies that the energy barrier for electron injection from PTCDA into BCP can be significantly reduced by fulfilling the double-sided n-doping to the Schottky interface.

The Advanced Charge Injection Techniques Towards the Fabrication of High-Power Organic Light Emitting Diodes 157

dipole 1=-0.65 eV, resulting a 0.65 eV downwards shift of the vacuum level (VL) on the BCP side LUMO as shown in Fig. 12(i). Thus, the electron injection barrier from the LUMO of 1:2 Li2CO3:PTCDA to hat of BCP reduces roughly to 0.95 eV. This shows that the electron transport through the 1:2 Li2CO3:PTCDA/BCP interface can be small in device 10. For the 1:2 Li2CO3:PTCDA/1:4 Li2CO3:BCP heterojunction, the term (1/1+1/2) is assumed to be 0.2, because the low-frequency dielectric constants of the doped materials are larger compared to the undoped materials. Provided the CNL for n-doped PTCDA-4.8 eV and the CNL for ndoped BCP3.2 eV [31], Eq. (2) yields an interface dipole 2= -1.44 eV, leading to a 1.44 eV downwards shift of the vacuum level (VL) in the doped BCP side, as shown in Fig. 12(ii). Accordingly, the electron injection barrier at the 1:2 Li2CO3:PTCDA/1:4 Li2CO3:BCP heterojunction is estimated to be 0.16 eV, suggesting that the double n-doping can induce significant realignment of molecular levels of organic-organic heterojunction. Thus, the contact becomes ohmic, resulting in efficient electron current and increasing device

**Figure 12.** Schematic diagram for depicting the electronic structures of the interfaces of 1:2

*3.2.2. Inverted OLEDs incorporating the combination of MoO3 and Li2CO3:BCP*

Li2CO3:PTCDA/BCP (i) and 1:2 Li2CO3:PTCDA/1:4 Li2CO3:BCP (ii), both deposited onto the ITO substrates. For simplicity, the gap states are not shown. The EF, HOMO, and horizontal dotted arrow represent the Fermi level, highest occupied molecular orbital, and the electron-injecting process, respectively.

There has been some controversy about the intrinsic property of MoO3, which has been intensively investigated in OLEDs for improving the hole injection. It is considered by some groups that MoO3 should act as a hole conductor with highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) levels of 5.4 eV and 2.3 eV, respectively [32]. Recently, however, it is suggested [33] that due to being doped by oxygen vacancy defects MoO3 intrinsically offers n-typed conduction with deep-lying HOMO and LUMO levels of 9.7 eV and 6.7 eV, respectively. The enhanced hole injection by the MoO3 intervention may be attributed to the process that one electron in the HOMO level of organic hole-transporting material can be easily transferred into the LUMO level of MoO3 because the Fermi level of MoO3 is necessarily pinned close to the HOMO level of p-typed organic conductor [33]. Here, we investigate MoO3 as electron injection layer in IBOLEDs and show

performance (the case of device 8).

**Figure 11.** The I-V and L-V characteristics for device 10.

#### **<IP_ADDRESS>. The working mechanism of the combination of Li2CO3:PTCDA and Li2CO3:BCP**

The electronic structures of the interfaces of 1:2 Li2CO3:PTCDA/BCP and 1:2 Li2CO3:PTCDA/1:4 Li2CO3:BCP are schematically shown in Fig. 5. The LUMO-LUMO offset at organic heterojunction can be varied by the interfacial dipole given by [30]:

$$
\Delta = \left(1 - \frac{1}{2}(\frac{1}{\varepsilon\_1} + \frac{1}{\varepsilon\_2})\right) \text{(CNL}\_1 - \text{CNL}\_2\text{)}\_{initial} \text{ \textdegree \tag{2}
$$

where 1 and 2 are the low frequency dielectric constants for the two organic materials, CNL1 and CNL2 represent their charge neutrality levels. In the case of 1:2 Li2CO3:PTCDA/BCP heterojunction, considering that the dielectric constant of n-doped PTCDA is larger compared to that of the undoped PTCDA (1.9) and taking BCP1.4, (1/1+1/2) is estimated to be 0.7. Provided that the CNL for n-doped PTCDA is roughly equal to that of undoped PTCDA(-4.8 eV) due to the Fermi level pinning [22] and CNL of BCP-3.8 eV, Eq. (2) yields an interface dipole 1=-0.65 eV, resulting a 0.65 eV downwards shift of the vacuum level (VL) on the BCP side LUMO as shown in Fig. 12(i). Thus, the electron injection barrier from the LUMO of 1:2 Li2CO3:PTCDA to hat of BCP reduces roughly to 0.95 eV. This shows that the electron transport through the 1:2 Li2CO3:PTCDA/BCP interface can be small in device 10. For the 1:2 Li2CO3:PTCDA/1:4 Li2CO3:BCP heterojunction, the term (1/1+1/2) is assumed to be 0.2, because the low-frequency dielectric constants of the doped materials are larger compared to the undoped materials. Provided the CNL for n-doped PTCDA-4.8 eV and the CNL for ndoped BCP3.2 eV [31], Eq. (2) yields an interface dipole 2= -1.44 eV, leading to a 1.44 eV downwards shift of the vacuum level (VL) in the doped BCP side, as shown in Fig. 12(ii). Accordingly, the electron injection barrier at the 1:2 Li2CO3:PTCDA/1:4 Li2CO3:BCP heterojunction is estimated to be 0.16 eV, suggesting that the double n-doping can induce significant realignment of molecular levels of organic-organic heterojunction. Thus, the contact becomes ohmic, resulting in efficient electron current and increasing device performance (the case of device 8).

156 Organic Light Emitting Devices

**Figure 11.** The I-V and L-V characteristics for device 10.

current conduction. Fig. 10(c) shows that the current efficiency of device 8 is slightly lower than that of device 9, mostly because 1:2 Li2CO3:PTCDA exhibits some absorption over Alq3 emission, while 1:4 Li2CO3:BCP is nearly transparent in the visible-light range. Nevertheless, due to the marked reduction of driving voltage, device 8 provides higher power efficiency than device 9. In terms of device 8, the issue of the efficient electron transfer from 1:2 Li2CO3:PTCDA into 1:4 Li2CO3:BCP needs to be addressed, since the LUMO level (4.6 eV) of PTCDA is 1.6 eV lower than that (3.0 eV) of BCP. In quest of the mechanism of electron transport over such a big Schottky barrier, device 10 is fabricated and characterized. As seen in Fig. 11, device 10 shows poor current-voltage and luminancevoltage characteristics. At a driving voltage of 10 V, it exhibits a current of 2.3 mA/cm2 and a luminance of 4.1 cd/m2, in clear contrary to 531 mA/cm2 and 10492 cd/m2 achieved at a driving voltage of 8.4 V in device 8. This greatly improved performance of device 8 relative to device 10 implies that the energy barrier for electron injection from PTCDA into BCP can be significantly reduced by fulfilling the double-sided n-doping to the Schottky interface.

**<IP_ADDRESS>. The working mechanism of the combination of Li2CO3:PTCDA and Li2CO3:BCP**

at organic heterojunction can be varied by the interfacial dipole given by [30]:

1 2

The electronic structures of the interfaces of 1:2 Li2CO3:PTCDA/BCP and 1:2 Li2CO3:PTCDA/1:4 Li2CO3:BCP are schematically shown in Fig. 5. The LUMO-LUMO offset

11 1 1 ( )( ) 2

where 1 and 2 are the low frequency dielectric constants for the two organic materials, CNL1 and CNL2 represent their charge neutrality levels. In the case of 1:2 Li2CO3:PTCDA/BCP heterojunction, considering that the dielectric constant of n-doped PTCDA is larger compared to that of the undoped PTCDA (1.9) and taking BCP1.4, (1/1+1/2) is estimated to be 0.7. Provided that the CNL for n-doped PTCDA is roughly equal to that of undoped PTCDA(-4.8 eV) due to the Fermi level pinning [22] and CNL of BCP-3.8 eV, Eq. (2) yields an interface

1 2

*CNL CNL initial*

, (2)

**Figure 12.** Schematic diagram for depicting the electronic structures of the interfaces of 1:2 Li2CO3:PTCDA/BCP (i) and 1:2 Li2CO3:PTCDA/1:4 Li2CO3:BCP (ii), both deposited onto the ITO substrates. For simplicity, the gap states are not shown. The EF, HOMO, and horizontal dotted arrow represent the Fermi level, highest occupied molecular orbital, and the electron-injecting process, respectively.

#### *3.2.2. Inverted OLEDs incorporating the combination of MoO3 and Li2CO3:BCP*

There has been some controversy about the intrinsic property of MoO3, which has been intensively investigated in OLEDs for improving the hole injection. It is considered by some groups that MoO3 should act as a hole conductor with highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) levels of 5.4 eV and 2.3 eV, respectively [32]. Recently, however, it is suggested [33] that due to being doped by oxygen vacancy defects MoO3 intrinsically offers n-typed conduction with deep-lying HOMO and LUMO levels of 9.7 eV and 6.7 eV, respectively. The enhanced hole injection by the MoO3 intervention may be attributed to the process that one electron in the HOMO level of organic hole-transporting material can be easily transferred into the LUMO level of MoO3 because the Fermi level of MoO3 is necessarily pinned close to the HOMO level of p-typed organic conductor [33]. Here, we investigate MoO3 as electron injection layer in IBOLEDs and show that MoO3 in association with an n-doped layer of BCP can enable the efficient electron injection.

The Advanced Charge Injection Techniques Towards the Fabrication of High-Power Organic Light Emitting Diodes 159

Device 16: ITO/ MoO3 5 nm/ 1:4 Li2CO3:BCP 5 nm/ Alq3 50 nm/ NPB 60 nm/ MoO3 10 nm/Al; Device 17: ITO/ MoO3 5 nm/ 1:4 Li2CO3:BCP 5 nm/ Alq3 60 nm/ NPB 60 nm/ MoO3 10 nm/Al; Device 18: ITO/ MoO3 5 nm/ 1:4 Li2CO3:BCP 5 nm/ Alq3 70 nm/ NPB 60 nm/ MoO3 10 nm/Al; Fig. 14 shows the performances of devices 13 and 14. Device 13 shows nearly the same I-V characteristics as device 14, but the luminance and current efficiency of device 13 are higher than those of device 14. At a luminance of 100 cd/m2, device 13 achieves a driving voltage of 7.51 V and a current efficiency of 1.39 cd/A, in comparison with 7.72 V and 1.08 cd/A, respectively, reached in device 14. At a luminance of 1000 cd/m2, device 13 exhibits a driving voltage of 8.86 V and a current efficiency of 2.0 cd/A, in comparison with 9.04 V and 1.62 cd/A obtained in device 14. The improved performance of device 13 over device 14 may be attributed to the reduced dopant diffusion from Li2CO3:BCP into Alq3 as a result of the

**Figure 14.** (a) I-V and L-V characteristics and (b) current efficiency versus current density

**<IP_ADDRESS>. Mechanism of operation in the combination of MoO3 and Li2CO3:BCP**

Despite that MoO3 exhibits better electron conduction than 1:4 Li2CO3:BCP composite, device 13 produces nearly the identical electron current as device 14, demonstrating that there must be some voltage drop across the MoO3/1:4 Li2CO3:BCP interface in device 13. In an effort to investigate the mechanism of the electron transfer from MoO3 to 1:4 Li2CO3:BCP, the characteristics of device 15 is studied as presented in Fig. 15, which shows that device 15 exhibits poor I-V characteristics with no Alq3 electroluminescence observed, suggesting there is a big Schottky barrier for electron injection from MoO3 to BCP. The marked performance difference between devices 13 and 15 indicates that the LUMO-LUMO offset at the MoO3/BCP interface can be remarkably lowered due to doping Li2CO3 into BCP, which can be explained by the interfacial dipole calculated by Eq. (2). In the case of the MoO3/BCP heterojunction, taking MoO320 and BCP1.4 [30], the item of (1/1+1/2) is estimated to be 0.76. Considering that the CNL (work function) of MoO3 is -6.9 eV [33] and that of BCP-3.8 eV, Eq. (2) yields an interface dipole 1=-1.92 eV. This results in a 1.92 eV downward shift of the vacuum level on the BCP side. Thus, the Schottky barrier for the electron injection from the LUMO of MoO3 to

characteristics for devices 13 (circles) and 14 (squares).

thinner Li2CO3:BCP in device 13.


**3. The work principle of the pesticide biosensor with nanoporous Si layer**

#### **3.1. Work principle**

The proposed pesticide biosensor works with acetylcholinesterase noted by AChE, with the code—EC <IP_ADDRESS>. This receptor element is used to degrade agonists of the acetylcholine (AcH) neurohormone. In the living matter, AcH is present at neuromuscular junction and in the cholinergic nervous system, modulating the electrical pulse transmission at synaptic spaces, as other neurotransmitters [25]. The AChE has a very high catalytic affinity for acetylcholine and for its agonists as parasympathomimetic pesticides. This property opens the door of pesticide-selective detection by AChE-based enzyme biosensors [26].

The pesticides are intensively used in agriculture usually as organochlorines, carbamates, and organophosphate. Paraoxon belongs to the organophosphate class, being an oxon and the active metabolite of the parathion pesticide. Their working principle on the pests is based on the inhibition of AChE, allowing acetylcholine to transfer nerve impulses indefinitely and causing paralysis. Paraoxon is a novel generation of pesticide, which reacts as an inhibitor of AChE. Pesticides from this group act directly by stimulating the nicotinic receptors or indirectly by the inhibition of cholinesterase, as an acetylcholinesterase inhibitor, abbreviated as AChEI. Paraoxon is one of the most potent acetylcholinesterase inhibitor available in insecticide [27]. In water solvent, it stands for a high risk of poisoning for humans or animals, due to its simply absorption through teguments in contact with the contaminated water from environment. As pesticide, the parathion is dissolved in water and usually is applied by treatment. It is frequently sprayed to rice and fruits. The usual concentrations are 0.05 and 0.1% [28]. After the rain, the pesticide is accumulated in water and soil. This parathion degradation in time produces multiple water-soluble products.


Discrete-Time Nonlinear Attitude Tracking Control of Spacecraft

*Yuichi Ikeda*

## **Abstract**

Recent space programs require agile and large-angle attitude maneuvers for applications in various fields such as observational astronomy. To achieve agility and large-angle attitude maneuvers, it will be required to design an attitude control system that takes into account nonlinear motion because agile and large-angle rotational motion of a spacecraft in such missions represents a nonlinear system. Considerable research has been done about the nonlinear attitude tracking control of spacecraft, and these methods involve a continuous-time control framework. However, since a computer, which is a digital device, is employed as a spacecraft controller, the control method should have discrete-time control or sampled-data control framework. This chapter considers discrete-time nonlinear attitude tracking control problem of spacecraft. To this end, a Euler approximation system with respect to tracking error is first derived. Then, we design a discrete-time nonlinear attitude tracking controller so that the closed-loop system consisting of the Euler approximation system becomes input-to-state stable (ISS). Furthermore, the exact discrete-time system with a derived controller is indicated semiglobal practical asymptotic (SPA) stable. Finally, the effectiveness of proposed control method is verified by numerical simulations.

**Keywords:** spacecraft, attitude tracking control, discrete-time nonlinear control

#### **1. Introduction**

Recent space programs require agile and large-angle attitude maneuvers for applications in various fields such as observational astronomy [1–3]. To achieve agility and large-angle attitude maneuvers, it will be required to design an attitude control system that takes into account nonlinear motion because agile and largeangle rotational motion of a spacecraft in such missions represents a nonlinear system.

Considerable research has been done about the nonlinear attitude tracking control of spacecraft [4–12], and these methods involve a continuous-time control framework. However, since a computer, which is a digital device, is employed as a spacecraft controller, the control method should have discrete-time control or sampled-data control framework.

Although a sampled-data control method for nonlinear system did not advance because it is difficult to discretize a nonlinear system, a control method based on the Euler approximate model has been proposed in recent years [13, 14] and is applied to ship control [15]. Although our research group has proposed a sampled-data

control method using backstepping [16] and a discrete-time control method based on sliding mode control [17] for spacecraft control problem, these methods are disadvantageous because control input amplitude depends on the sampling period *T* as the control law is of the form *u* ¼ *a x*ð Þþ ð Þ *b x*ð Þ*=T* .

*ωe*ðÞ¼ *t ω*ðÞ�*t C t*ð Þ*ωd*ð Þ*t ,* (4)

1 þ k k *σe*ð Þ*t* 2 � �2 *:* (5)

*G*ð Þ *σe*ð Þ*s ωe*ð Þ*s ds,* (8)

*,* ∀*wk* ∈ R3

*,*

(9)

*σ*\_*e*ðÞ¼ *t G*ð Þ *σe*ð Þ*t ωe*ð Þ*t ,* (6)

where *C t*ð Þ∈R3�3 is the direction cosine matrix from f g*b* to f g*d* that expresses

Substituting Eqs. (3) and (4) into Eqs. (1) and (2) using the identity

*C t* \_ðÞ¼�*ωe*ð Þ*t* �*C t*ð Þ yields the following relative motion equations:

*ω*\_ *d*ð Þ*t* , and the disturbance *w t*ð Þ, the following assumption is made.

*σe,k*þ1 ¼ *σe,k* þ

�*J Ckω*\_ *d,k* � *ωe,k*

**3. Discrete-time nonlinear attitude tracking control**

satisfies the following equation:

k k *xk*þ1 ≤*ρ*ð Þþ k k *x*0 *; k γ*ð Þ k k *wk ,* ∀*xk* ∈ R3

ðð Þ *k*þ1 *T kT*

8 *σe*ð Þ*t* � ð Þ2 � 4 1 � k k *σe*ð Þ*t* 2 � �*σe*ð Þ*t* �

½�f g *ωe*ðÞþ*t C t*ð Þ*ωd*ð Þ*t* �*J*f g *ωe*ðÞþ*t C t*ð Þ*ωd*ð Þ*t*

Hereafter, we assume that the variables of spacecraft *σ*ð Þ*t* and *ω*ð Þ*t* are directly measurable and *J* is known. In addition, regarding the desired states *σd*ð Þ*t* , *ωd*ð Þ*t* ,

**Assumption 1**: the desired states *σd*ð Þ*t* , *ωd*ð Þ*t* , and *ω*\_ *d*ð Þ*t* are uniformly continuous and bounded ∀*t*∈ ½ Þ 0*;* ∞ . The disturbance *w t*ð Þ is uniformly bounded ∀*t* ∈½ Þ 0*;* ∞ . From Eqs. (A4) and (A5) in Appendix, the exact discrete-time model of relative

> ðð Þ *k*þ1 *T kT*

�*J Cs*ð Þ*ω*\_ *d*ðÞ�*s ωe*ð Þ*s* � f g *C s*ð Þ*ωd*ð Þ*s* þ *uk* þ *wk*�*ds*

and the Euler approximate model of relative motion equations are obtained as

*ωe,k*þ1 ¼ *ωe,k* � *TJ*�1 �f g *ωe,k* þ *Ckωd,k* � ½ *J*f g *ωe,k* þ *Ckωd,k*

We derive a controller based on the backstepping approach that makes the closed-loop system consisting of the Euler approximate modes (10) and (11) become input-to-state stable (ISS), i.e., the state variable of closed-loop system

where *ρ*ð Þ� is the class KL function and *γ*ð Þ� is the class K function. To this end, assume that *ωe,k* is the virtual input to subsystem (10), and derive the stabilizing function *αk* that *σe,k* is asymptotic convergence to zero. Then, derive the control

�*J Ct*ð Þ*ω*\_ *d*ðÞ�*t ωe*ð Þ*t* � f g *C t*ð Þ*ωd*ð Þ*t* þ *u t*ð Þþ *w t*ð Þ� (7)

�f g *ωe*ðÞþ*s C s*ð Þ*ωd*ð Þ*s* � ½ *J*f g *ωe*ðÞþ*s C s*ð Þ*ωd*ð Þ*s*

*σe,k*þ1 ¼ *σe,k* þ *TG* ð Þ *σe,k ωe,k,* (10)

� f g *Ckωd,k* þ *uk* þ *wk*�*:* (11)

the following Eq. [7]:

*C t*ðÞ¼ *I*3 þ

*Discrete-Time Nonlinear Attitude Tracking Control of Spacecraft*

*DOI: http://dx.doi.org/10.5772/intechopen.87191*

�1

*ω*\_ *e*ðÞ¼ *t J*

motion equations is obtained as

*ωe,k*þ1 ¼ *ωe,k* þ

*xk* ¼ *σT*

**63**

*e,kωT e,k* h i*T*

For these facts, about the spacecraft attitude control problem that requires agile and large-angle attitude maneuvers, this chapter proposed a discrete-time nonlinear attitude tracking control in which the control input amplitude is independent of the sampling period *T*. The effectiveness of proposed control method is verified by numerical simulations.

The following notations are used throughout the chapter. Let R and N denote the real and the integer numbers. Rn and R*n*�*m* are the sets of real vectors and matrices. For real vector *a*∈R*n*, *aT* is the vector transpose, k k*a* denotes the Euclidean norm, and *a*� ∈R3�3 is the skew symmetric matrix

$$a^\times = \begin{bmatrix} \mathbf{0} & -a\_3 & a\_2 \\ a\_3 & \mathbf{0} & -a\_1 \\ -a\_2 & a\_1 & \mathbf{0} \end{bmatrix}$$

derived from vector *a*∈R3 . For real symmetric matrix *A*, *A* . 0 means the positive definite matrix. The identity matrix of size 3 � 3 is denoted by *I*3. *λ*max *A* ∈R and *λ*min *A* ∈R are the maximal and the minimal eigenvalues of a matrix *A*, respectively.

#### **2. Relative equation of motion and discrete-time model for spacecraft**

In this chapter, as the kinematics represents the attitude of the spacecraft with respect to the inertia frame f g*i* , the modified Rodrigues parameters (MRPs) [5] are used. The rotational motion equations of the spacecraft's body-fixed frame f g*b* are given by the following equations:

$$\dot{\sigma}(t) = G(\sigma(t))o(t), \tag{1}$$

$$G(\sigma(t)) = \frac{1}{2} \left\{ \frac{1 - \left\| \sigma(t) \right\|^2}{2} I\_3 + \sigma(t)\sigma(t)^T + \sigma(t)^\times \right\},$$

$$\dot{o}(t) = f^{-1} \{-o(t)^\times J o(t) + u(t) + w(t)\}, \tag{2}$$

where Eq. (1) is the kinematics that represents the attitude of f g*b* with respect to the f g*i* , Eq. (2) is the rotation dynamics, *σ*ð Þ*t* ∈R3 [�] is the MRPs, *ω*ð Þ*t* ∈ R3 [rad/s] is the angular velocity, *u t*ð Þ∈ R3 [Nm] is the control torque (input), *w t*ð Þ∈R3 [Nm] is the disturbance input, and *J* ∈ R3�3 [kg m2 ] is the moment of inertia.

We consider a control problem in which a spacecraft tracks a desired attitude (MRPs) *σd*ð Þ*t* ∈ R3 and angular velocity *ωd*ð Þ*t* ∈R3 in fixed frame f g*d* . The MRPs of the relative attitude *σe*ð Þ*t* ∈ R3 and the relative angular velocity *ωe*ð Þ*t* ∈ R3 in the frame f g*b* are given by

$$\sigma\_{\varepsilon}(t) = \frac{N\_{\varepsilon}(t)}{1 + \left\|\sigma(t)\right\|^2 \left\|\sigma\_d(t)\right\|^2 + 2\sigma\_d(t)^T \sigma(t)},\tag{3}$$

$$N\_{\varepsilon}(t) = \left(1 - \left\|\sigma\_d(t)\right\|^2\right)\sigma(t) - \left(1 - \left\|\sigma(t)\right\|^2\right)\sigma\_d(t) + 2\sigma(t)^\times \sigma\_d(t),$$

*Discrete-Time Nonlinear Attitude Tracking Control of Spacecraft DOI: http://dx.doi.org/10.5772/intechopen.87191*

control method using backstepping [16] and a discrete-time control method based on sliding mode control [17] for spacecraft control problem, these methods are disadvantageous because control input amplitude depends on the sampling period *T*

For these facts, about the spacecraft attitude control problem that requires agile and large-angle attitude maneuvers, this chapter proposed a discrete-time nonlinear attitude tracking control in which the control input amplitude is independent of the sampling period *T*. The effectiveness of proposed control method is verified by

The following notations are used throughout the chapter. Let R and N denote the real and the integer numbers. Rn and R*n*�*m* are the sets of real vectors and matrices. For real vector *a*∈R*n*, *aT* is the vector transpose, k k*a* denotes the Euclidean norm,

> 0 �*a*3 *a*2 *a*3 0 �*a*1 �*a*2 *a*1 0

3 7 5

. For real symmetric matrix *A*, *A* . 0 means the

*σ*\_ðÞ¼ *t G*ð Þ *σ*ð Þ*t ω*ð Þ*t ,* (1)

�1 �*ω*ð Þ*t* � f g *Jω*ðÞþ*t u t*ðÞþ *w t*ð Þ *,* (2)

] is the moment of inertia.

k k *σd*ð Þ*t* 2 þ 2*σd*ð Þ*t Tσ*ð Þ*t ,* (3)

*σd*ðÞþ*t* 2*σ*ð Þ*t* �*σd*ð Þ*t ,*

*,*

*I*3 þ *σ*ð Þ*t σ*ð Þ*t T* þ *σ*ð Þ*t* �

( )

where Eq. (1) is the kinematics that represents the attitude of f g*b* with respect to the f g*i* , Eq. (2) is the rotation dynamics, *σ*ð Þ*t* ∈R3 [�] is the MRPs, *ω*ð Þ*t* ∈ R3 [rad/s] is the angular velocity, *u t*ð Þ∈ R3 [Nm] is the control torque (input), *w t*ð Þ∈R3 [Nm]

We consider a control problem in which a spacecraft tracks a desired attitude (MRPs) *σd*ð Þ*t* ∈ R3 and angular velocity *ωd*ð Þ*t* ∈R3 in fixed frame f g*d* . The MRPs of the relative attitude *σe*ð Þ*t* ∈ R3 and the relative angular velocity *ωe*ð Þ*t* ∈ R3 in the

*Ne*ð Þ*t*

*σ*ðÞ�*t* 1 � k k *σ*ð Þ*t* 2 � �

*A* ∈R

as the control law is of the form *u* ¼ *a x*ð Þþ ð Þ *b x*ð Þ*=T* .

and *a*� ∈R3�3 is the skew symmetric matrix

derived from vector *a*∈R3

given by the following equations:

*G*ð Þ¼ *σ*ð Þ*t*

is the disturbance input, and *J* ∈ R3�3 [kg m2

*σe*ðÞ¼ *t*

*Ne*ðÞ¼ *t* 1 � k k *σd*ð Þ*t* 2 � �

1 þ k k *σ*ð Þ*t* 2

frame f g*b* are given by

**62**

1 2

*ω*\_ðÞ¼ *t J*

1 � k k *σ*ð Þ*t* 2 2

and *λ*min

respectively.

*a*� ¼

2 6 4

positive definite matrix. The identity matrix of size 3 � 3 is denoted by *I*3. *λ*max

*A* ∈R are the maximal and the minimal eigenvalues of a matrix *A*,

**2. Relative equation of motion and discrete-time model for spacecraft**

In this chapter, as the kinematics represents the attitude of the spacecraft with respect to the inertia frame f g*i* , the modified Rodrigues parameters (MRPs) [5] are used. The rotational motion equations of the spacecraft's body-fixed frame f g*b* are

numerical simulations.

*Gyroscopes - Principles and Applications*

$$
\rho\_{\mathbf{t}}(\mathbf{t}) = \boldsymbol{\alpha}(\mathbf{t}) - \mathbf{C}(\mathbf{t})\boldsymbol{\alpha}\_d(\mathbf{t}), \tag{4}
$$

where *C t*ð Þ∈R3�3 is the direction cosine matrix from f g*b* to f g*d* that expresses the following Eq. [7]:

$$C(t) = I\_3 + \frac{8\left(\sigma\_\epsilon(t)^\times\right)^2 - 4\left(1 - \left\|\sigma\_\epsilon(t)\right\|^2\right)\sigma\_\epsilon(t)^\times}{\left(1 + \left\|\sigma\_\epsilon(t)\right\|^2\right)^2}. \tag{5}$$

Substituting Eqs. (3) and (4) into Eqs. (1) and (2) using the identity *C t* \_ðÞ¼�*ωe*ð Þ*t* �*C t*ð Þ yields the following relative motion equations:

$$
\dot{\sigma}\_{\varepsilon}(t) = G(\sigma\_{\varepsilon}(t)) a\_{\varepsilon}(t), \tag{6}
$$

$$\dot{\boldsymbol{\alpha}}\_{\varepsilon}(t) = \boldsymbol{f}^{-1}[-\{\boldsymbol{\alpha}\_{\varepsilon}(t) + \mathbf{C}(t)\boldsymbol{\alpha}\_{d}(t)\}^{\boldsymbol{\upbeta}}] \boldsymbol{f}\{\boldsymbol{\alpha}\_{\varepsilon}(t) + \mathbf{C}(t)\boldsymbol{\alpha}\_{d}(t)\} \tag{7}$$

$$-J\{\mathbf{C}(t)\dot{o}\_d(t) - a\_\epsilon(t)^\times \mathbf{C}(t)a\_d(t)\} + \omega(t) + w(t)\tag{7}$$

Hereafter, we assume that the variables of spacecraft *σ*ð Þ*t* and *ω*ð Þ*t* are directly measurable and *J* is known. In addition, regarding the desired states *σd*ð Þ*t* , *ωd*ð Þ*t* , *ω*\_ *d*ð Þ*t* , and the disturbance *w t*ð Þ, the following assumption is made.

**Assumption 1**: the desired states *σd*ð Þ*t* , *ωd*ð Þ*t* , and *ω*\_ *d*ð Þ*t* are uniformly continuous and bounded ∀*t*∈ ½ Þ 0*;* ∞ . The disturbance *w t*ð Þ is uniformly bounded ∀*t* ∈½ Þ 0*;* ∞ .

From Eqs. (A4) and (A5) in Appendix, the exact discrete-time model of relative motion equations is obtained as

$$
\sigma\_{\epsilon,k+1} = \sigma\_{\epsilon,k} + \int\_{kT}^{(k+1)T} G(\sigma\_{\epsilon}(s)) \alpha\_{\epsilon}(s) \, ds,\tag{8}
$$

$$\begin{aligned} \boldsymbol{\dot{\alpha}\_{\epsilon,k+1}} &= \boldsymbol{\alpha\_{\epsilon,k}} + \int\_{kT}^{(k+1)T} [-\{\boldsymbol{\alpha}\_{\epsilon}(\boldsymbol{\varsigma}) + \mathbf{C}(\boldsymbol{\varsigma})\boldsymbol{\alpha\_{d}}(\boldsymbol{\varsigma})\}^{\times}] \boldsymbol{f} \{\boldsymbol{\alpha}\_{\epsilon}(\boldsymbol{\varsigma}) + \mathbf{C}(\boldsymbol{\varsigma})\boldsymbol{\alpha\_{d}}(\boldsymbol{\varsigma})\} \\ &- \boldsymbol{f} \{\mathbf{C}(\boldsymbol{\varsigma})\boldsymbol{\dot{\alpha}\_{d}}(\boldsymbol{\varsigma}) - \boldsymbol{\alpha\_{\epsilon}}(\boldsymbol{\varsigma})^{\times}\mathbf{C}(\boldsymbol{\varsigma})\boldsymbol{\alpha\_{d}}(\boldsymbol{\varsigma})\} + \boldsymbol{u\_{k}} + \boldsymbol{w\_{k}}] d\boldsymbol{s} \end{aligned} \tag{9}$$

and the Euler approximate model of relative motion equations are obtained as

$$
\sigma\_{\varepsilon,k+1} = \sigma\_{\varepsilon,k} + T\mathcal{G}\left(\sigma\_{\varepsilon,k}\right)a\_{\varepsilon,k} \tag{10}
$$

$$\begin{split} \boldsymbol{\alpha}\_{\boldsymbol{\epsilon},k+1} &= \boldsymbol{\alpha}\_{\boldsymbol{\epsilon},k} - \boldsymbol{T} \boldsymbol{\beta}^{-1} [-\{\boldsymbol{\alpha}\_{\boldsymbol{\epsilon},k} + \mathbf{C}\_{k} \boldsymbol{\alpha}\_{d,k}\}^{\times}]^{\times} \boldsymbol{f} \{\boldsymbol{\alpha}\_{\boldsymbol{\epsilon},k} + \mathbf{C}\_{k} \boldsymbol{\alpha}\_{d,k}\} \\ &- \boldsymbol{J} \{\mathbf{C}\_{k} \boldsymbol{\dot{\alpha}}\_{\boldsymbol{d},k} - \boldsymbol{\alpha}\_{\boldsymbol{\epsilon},k} \times \mathbf{C}\_{k} \boldsymbol{\alpha}\_{d,k}\} + \boldsymbol{u}\_{k} + \boldsymbol{w}\_{k} \big]. \end{split} \tag{11}$$

#### **3. Discrete-time nonlinear attitude tracking control**

We derive a controller based on the backstepping approach that makes the closed-loop system consisting of the Euler approximate modes (10) and (11) become input-to-state stable (ISS), i.e., the state variable of closed-loop system *xk* ¼ *σT e,kωT e,k* h i*T* satisfies the following equation:

$$||\varkappa\_{k+1}|| \le \rho(||\varkappa\_0||,k) + \gamma(||\varkappa\_k||), \quad \forall \varkappa\_k \in \mathbb{R}^3, \quad \forall w\_k \in \mathbb{R}^3,$$

where *ρ*ð Þ� is the class KL function and *γ*ð Þ� is the class K function. To this end, assume that *ωe,k* is the virtual input to subsystem (10), and derive the stabilizing function *αk* that *σe,k* is asymptotic convergence to zero. Then, derive the control

input *uk* that closed-loop system becomes ISS. Here, regarding the variable *σe,k*, the following assumption is made.

*Tf* 1*bk*

*Discrete-Time Nonlinear Attitude Tracking Control of Spacecraft*

*DOI: http://dx.doi.org/10.5772/intechopen.87191*

0 , *f* 1 ,

0 , *f* 1 ,

Therefore, if *f* 1 satisfies Eq. (17) and *ωe,k* ! *αk*ð Þ *k* ! ∞ , then *σe,k* ! 0.

The error variable between the state *ωe,k* and *αk* is defined as

2 *Tbk*

4

In addition, since 2≤ð Þ 1*=bk* ≤4 under Assumption 2, the range of *f* 1 that holds

The control input *uk* that makes the closed-loop system becomes ISS is derived.

*αk* � *αk*þ1 ¼ *Tf* 1 *G*ð Þ *σe,k zk* � *f* 1*bkσe,k*

� *J C*f g *kω*\_ *d,k* � ð Þ� *zk* þ *α,k Ckωd,k* þ *uk* þ *wk*�*:*

*uk* ¼ f g *zk* þ *α,k* þ *Ckωd,k* � *J z*f g *k* þ *α,k* þ *Ckωd,k*

� *f* 2*Jzk,*

*zk* þ *TJ*�1

2 ¼ k k *Xk*

2

*, Xk* ¼ *σT*

*e,kzT k T*

where *f* 2 ∈R is the feedback gain. The candidate Lyapunov function for

þ *J C*f g *kω*\_ *d,k* � ð Þ� *zk* þ *α,k Ckωd,k*

� *f* 1*J G*ð Þ *σe,k zk* � *f* 1*bkσe,k*

*zk*þ1 ¼ 1 � *Tf* 2

*V*2ð Þ¼ *k V*1ð Þþ *k* k k *zk*

is obtained as

Eq. (15) is obtained as

**3.2 Derivation of control input** *uk*

From Eq. (18), subsystem (10) becomes

From Eqs. (18) and (19) and the following equation

the discrete-time equation with respect to *zk* is

*zk*þ1 ¼ *zk* þ *Tf* 1 *G*ð Þ *σe,k zk* � *f* 1*bkσe,k*

þ *TJ*�1

Now, by setting *uk* to

Eq. (20) becomes

**65**

Eqs.  Assumption 2 is equivalent to *ηe,k* ∈½ � 0*;* 1 .

In addition, Lemmas when using the derivation of the control law are shown below.

**Lemma 1:** for all *σ* ∈R3 , the following equations hold [5]:

$$b\sigma^T G(\sigma) = b\sigma^T, \quad G(\sigma)^T G(\sigma) = b^2 I\_3, \quad \left(b = \frac{\mathbf{1} + \left\|\sigma\right\|^2}{4} > \mathbf{0}\right).$$

**Lemma 2:** when the quadratic equation

$$a\mathfrak{x}^2 + b\mathfrak{x} + c = \mathbf{0} \\ (a, b, c \in \mathbb{R}).$$

has two distinct real roots *x* ¼ *α, β α*ð Þ , *β* , if *a* . 0, then the solution of the quadratic inequality

$$a\mathbf{x}^2 + b\mathbf{x} + c \le \mathbf{0}$$

is *α* , *x* , *β*.

### **3.1 Derivation of virtual input** *αk*

Assume that *ωe,k* is the virtual input to subsystem (10), and define the stabilizing function such that

$$a\_{\varepsilon,k} = a\_k = -f\_{\,1} \sigma\_{\varepsilon,k},\tag{12}$$

where *f* 1 ∈ R is the feedback gain. The candidate Lyapunov function for (10) is defined as

$$\left|V\_1(k) = \left|\sigma\_{\epsilon,k}\right|\right|^2. \tag{13}$$

From Lemma 1, the difference of Eq. (13) along the trajectories of the closedloop system is given by

$$
\Delta V\_1(k) = V\_1(k+1) - V\_1(k) = \left\{ \left( T f\_1 b\_k \right)^2 - 2T f\_1 b\_k \right\} ||\sigma\_{\varepsilon,k}||^2. \tag{14}
$$

From Lemma 2, *ΔV*1ð Þ*k* becomes negative, i.e., the range of *f* 1 that holds the following equation

*Discrete-Time Nonlinear Attitude Tracking Control of Spacecraft DOI: http://dx.doi.org/10.5772/intechopen.87191*

$$\left(\left(Tf\_1b\_k\right)^2 - 2Tf\_1b\_k < 0\right) \tag{15}$$

is obtained as

input *uk* that closed-loop system becomes ISS. Here, regarding the variable *σe,k*, the

0≤k k *σe,k* ≤1*,* ∀*k:*

**Assumption 2:** *σe,k* lies in the region that satisfies the following equation:

*σe,k* ¼ *εe,k*

�≤1*,* ∀*k*Þ. Assumption 2 is equivalent to *ηe,k* ∈½ � 0*;* 1 . In addition, Lemmas when using the derivation of the control law are shown

, the following equations hold [5]:

*ax*2 þ *bx* þ *c* ¼ 0ð Þ *a; b;c*∈R

has two distinct real roots *x* ¼ *α, β α*ð Þ , *β* , if *a* . 0, then the solution of the

*ax*2 þ *bx* þ *c* , 0

Assume that *ωe,k* is the virtual input to subsystem (10), and define the

where *f* 1 ∈ R is the feedback gain. The candidate Lyapunov function for (10) is

2

� �2 � 2*Tf* 1*bk* n o

*V*1ð Þ¼ *k* k k *σe,k*

From Lemma 1, the difference of Eq. (13) along the trajectories of the closed-

From Lemma 2, *ΔV*1ð Þ*k* becomes negative, i.e., the range of *f* 1 that holds the

*ΔV*1ð Þ¼ *k V*1ð Þ� *k* þ 1 *V*1ð Þ¼ *k Tf* 1*bk*

1 þ *ηe,k*

*,*

� � �

*I*3*, b* ¼ 1 þ k k*σ* 2

*ωe,k* ¼ *αk* ¼ �*f* 1*σe,k,* (12)

*:* (13)

k k *σe,k* 2

*:* (14)

4

!

. 0

*:*

�

*e,kηe,k* h i*T* �

� � � � ¼

following assumption is made.

*Gyroscopes - Principles and Applications*

1*,* k k *εe,k* ≤1*, ηe,k* � � �

quadratic inequality

is *α* , *x* , *β*.

defined as

**3.1 Derivation of virtual input** *αk*

stabilizing function such that

loop system is given by

following equation

**64**

**Lemma 1:** for all *σ* ∈R3

below.

**Remark 1:** from the relational expression

where *εe,k* ∈R3 and *ηe,k* ∈R are the quaternion *εT*

*σTG*ð Þ¼ *σ bσT, G*ð Þ *σ TG*ð Þ¼ *σ b*2

**Lemma 2:** when the quadratic equation

$$0 < f\_1 < \frac{2}{Tb\_k}.\tag{16}$$

In addition, since 2≤ð Þ 1*=bk* ≤4 under Assumption 2, the range of *f* 1 that holds Eq. (15) is obtained as

$$0 < f\_1 < \frac{4}{T}.\tag{17}$$

Therefore, if *f* 1 satisfies Eq. (17) and *ωe,k* ! *αk*ð Þ *k* ! ∞ , then *σe,k* ! 0.

#### **3.2 Derivation of control input** *uk*

The error variable between the state *ωe,k* and *αk* is defined as

$$z\_k \coloneqq \alpha\_{\varepsilon,k} - \alpha\_k. \tag{18}$$

The control input *uk* that makes the closed-loop system becomes ISS is derived. From Eq. (18), subsystem (10) becomes

$$
\sigma\_{\varepsilon,k+1} = \sigma\_{\varepsilon,k} + T\mathcal{G}\left(\sigma\_{\varepsilon,k}\right)(z\_k + a\_k). \tag{19}
$$

From Eqs. (18) and (19) and the following equation

$$a\_k - a\_{k+1} = T f\_1 \{ G(\sigma\_{e,k}) z\_k - f\_1 b\_k \sigma\_{e,k} \},$$

the discrete-time equation with respect to *zk* is

$$\begin{split} z\_{k+1} &= z\_k + T f\_1 \left\{ G(\sigma\_{\epsilon,k}) z\_k - f\_1 b\_k \sigma\_{\epsilon,k} \right\} \\ &+ T f^{-1} [-\{ z\_k + a\_{,k} + C\_k a\_{d,k} \} \times f \{ z\_k + a\_{,k} + C\_k a\_{d,k} \} \\ &- f \{ C\_k \dot{\alpha}\_{d,k} - (z\_k + a\_{,k}) \times C\_k a\_{d,k} \} + u\_k + w\_k ]. \end{split} \tag{20}$$

Now, by setting *uk* to

$$\begin{aligned} u\_k &= \{z\_k + a\_{,k} + \mathcal{C}\_k a\_{d,k}\} \times f\{z\_k + a\_{,k} + \mathcal{C}\_k a\_{d,k}\} \\ &+ f\{\mathcal{C}\_k \dot{\alpha}\_{d,k} - (z\_k + a\_{,k}) \times \mathcal{C}\_k a\_{d,k}\} \\ &- f\, f\{\mathcal{G}(\sigma\_{e,k}) z\_k - f\, \_1 b\_k \sigma\_{e,k}\} - f\, \_2 Iz\_k, \end{aligned}$$

Eq. (20) becomes

$$z\_{k+1} = (1 - Tf\_2)z\_k + Tf^{-1}w\_k.\tag{21}$$

where *f* 2 ∈R is the feedback gain. The candidate Lyapunov function for Eqs. (19) and (21) is defined as

$$V\_2(k) = V\_1(k) + \|\mathbf{z}\_k\|^2 = \|X\_k\|^2,\\ X\_k = \left[\sigma\_{e,k}^T \mathbf{z}\_k^T\right]^T. \tag{22}$$

As Eq. (14) is given by

$$\Delta V\_1(k) = \left(Tb\_k\right)^2 \left\|z\_k\right\|^2 + \left\{ \left(Tf\_1b\_k\right)^2 - 2Tf\_1b\_k \right\} \left\|\sigma\_{\epsilon,k}\right\|^2 + 2Tb\_k\left(1 - Tf\_1b\_k\right)z\_k^T\sigma\_{\epsilon,k}^T$$

must hold true in order to obtain a real number. As the denominator of Eq. (27)

in order to hold Eq. (27). From Lemma 2, the range of *f* 1 that holds for Eq. (28) is

1 þ

2 . 0 ) 0 , *T* ,

must hold in order to have the real number. As 2≤ ð Þ 1*=bk* ≤4 under Assumption

2

2

2 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 � *T*2 p

Therefore, if *f* 1 satisfies Eq. (32) under Assumption 2, Eqs. (27) and (28) hold.

*T* �

*T* þ

*Tf* 2

2*T*2

Therefore, if *f* 1 and *f* 2 satisfy Eqs. (32) and (33) under Assumption 2, then

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � ð Þ *Tbk*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � ð Þ *Tbk*

q

*Tbk*

¼ 2 � ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 � *T*2 p *T ,*

¼ 2 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 � *T*2 p *T*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*f* 1 *Tf* 1 � 4 � � s

1 � 4*f* 1 þ *T*

*f* 1 *Tf* 1 � 4 � � s

1 � 4*f* 1 þ *T*

*Tf* 2

2*T*2

*Tf* 2

2*T*2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*f* 1 *Tf* 1 � 4 � � s

1 � 4*f* 1 þ *T*

, *f* 1 ,

1 � 2*f* 1 þ *Tbk* , 0 (28)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � ð Þ *Tbk*

> 1 *bk*

0 , *T* , 2*:* (31)

*T* ð Þ 0 , *T* , 2 *:* (32)

*,*

*,*

ð Þ 0 , *T* , 2 *:* (33)

2

*,* (29)

(30)

is the same as Eq. (24), the following equation must hold

*Discrete-Time Nonlinear Attitude Tracking Control of Spacecraft*

1 �

*DOI: http://dx.doi.org/10.5772/intechopen.87191*

and the following Eq.

2, *T* must satisfy the condition

In addition, since

Furthermore, since

1 *T* �

*Qk* , 0.

**67**

q

max*bk*

min*bk*

2 � ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 � *T*2 p *T*

max *bk*

min *bk*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*f* 1 *Tf* 1 � 4 � � s

1 � 4*f* 1 þ *T*

*Tf* 2

2*T*2

1 �

1 þ

under Assumption 2, the condition (29) is given by

2 � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2ð Þ � *ck*

2 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2ð Þ � *ck*

p

p

under Assumption 2, the condition (26) is given by.

, *f* 2 ,

q

q

*Tbk*

*Tbk*

, *f* 1 ,

2*T* ¼ 1

2*T* ¼ 1

1 *T* þ

Summarizing the above, the following theorem can be obtained.

obtained as

*Tbkf* 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � ð Þ *Tbk*

*Tbk*

1 � ð Þ *Tbk*

2

from Eq. (18), by using completing square, the difference of Eq. (22) along the trajectories of the closed-loop system is given by

$$\begin{split} \Delta V\_{2}(k) &= \left(T^{2}f\_{2}^{2} - 2Tf\_{2} + T^{2}b\_{k}^{2}\right) \|x\_{k}\|^{2} + \left\{\left(Tf\_{1}b\_{k}\right)^{2} - 2Tf\_{1}^{\prime}b\_{k}\right\} \|\sigma\_{\epsilon,k}\|^{2} \\ &\quad + 2Tb\_{k}\left(1 - Tf\_{1}b\_{k}\right)z\_{k}^{T}\sigma\_{\epsilon,k}^{T} + T^{2}w\_{k}^{T}f^{-2}w\_{k} + 2T(1 - Tf\_{2})w\_{k}^{T})^{-1}x\_{k} \\ &\leq \left(2T^{2}f\_{2}^{2} - 4Tf\_{2} + T^{2}b\_{k}^{2} + 1\right) \|x\_{k}\|^{2} + \left\{\left(Tf\_{1}b\_{k}\right)^{2} - 2Tf\_{1}b\_{k}\right\} \|\sigma\_{\epsilon,k}\|^{2} \\ &\quad + 2Tb\_{k}\left(1 - Tf\_{1}b\_{k}\right)z\_{k}^{T}\sigma\_{\epsilon,k}^{T} + 2\left(\frac{T}{\lambda\_{l}}\right)^{2} \|w\_{k}\|^{2} \\ &= X\_{k}^{T}Q\_{k}X\_{k} + 2\left(\frac{T}{\lambda\_{l}}\right)^{2} \|w\_{k}\|^{2}, \end{split} \tag{23}$$

where

$$\lambda\_l = ||f||,\\\ Q\_k = \begin{bmatrix} Q\_{11,k} & Q\_{12,k} \\ Q\_{12,k}^T & Q\_{22,k} \end{bmatrix},\\\ Q\_{11,k} = \left\{ \left( T f\_1 b\_k \right)^2 - 2 T f\_1 b\_k \right\} I\_{34},$$

$$\mathcal{Q}\_{12,k} = T b\_k \left( 1 - T f\_1 b\_k \right) I\_{34},\\\ Q\_{22,k} = \left( 2 T^2 f\_2^2 - 4 T f\_2 + T^2 b\_k^2 + 1 \right) I\_{34}.$$

In Eq. (23), if *Qk* , 0, then

$$
\Delta V\_2(k) \le -\left|\lambda\_{Q\_k}^{\min}\right| \|X\_k\|^2 + 2\left(\frac{T}{\lambda\_I}\right)^2 \|w\_k\|^2,
$$

where *λ*min *Qk* , 0∈R is the minimum eigenvalue of *Qk* and the condition of ISS holds [18]. Hereafter, conditions of *f* 1 and *f* 2 which the matrix *Qk* holds *Qk* , 0 are derived under Assumption 2.

From Schur complement, condition *Qk* , 0 is equivalent to the following equations:

$$\left(\left(Tf\_1 b\_k\right)^2 - 2Tf\_1 b\_k \le 0,\tag{24}$$

$$-2T^2f\_2^2 - 4Tf\_2 + c\_k < 0 \quad \left(c\_k = \frac{Tb\_kf\_1^2 - 2f\_1 - Tb\_k}{Tb\_kf\_1^2 - 2f\_1}\right). \tag{25}$$

Condition (24) is the same as Eq. (15), and assume that Eq. (24) holds. From Lemma 2, the range of *f* 2 that holds for Eq. (25) is obtained as

$$\frac{2 - \sqrt{2(2 - c\_k)}}{2T} < f\_2 < \frac{2 + \sqrt{2(2 - c\_k)}}{2T},\tag{26}$$

and the following Eq.

$$2 - c\_k > 0 \quad \Rightarrow \quad \frac{Tb\_k f\_1^2 - 2f\_1 + Tb\_k}{Tb\_k f\_1^2 - 2f\_1} > 0 \tag{27}$$

must hold true in order to obtain a real number. As the denominator of Eq. (27) is the same as Eq. (24), the following equation must hold

$$\left\| Tb\_{b}f\_{1}^{2} - \mathfrak{H}\_{1} + Tb\_{k} \le 0 \right\| \tag{28}$$

in order to hold Eq. (27). From Lemma 2, the range of *f* 1 that holds for Eq. (28) is obtained as

$$\frac{\mathbf{1} - \sqrt{\mathbf{1} - \left(Tb\_k\right)^2}}{Tb\_k} < f\_1 < \frac{\mathbf{1} + \sqrt{\mathbf{1} - \left(Tb\_k\right)^2}}{Tb\_k},\tag{29}$$

and the following Eq.

As Eq. (14) is given by

*f* 2

≤ 2*T*2 *f* 2

¼ *XT*

where

where *λ*min

equations:

**66**

2 k k *zk*

*Gyroscopes - Principles and Applications*

2 þ *Tf* 1*bk*

*b*2 *k*

> *k σT e, k* þ *T*2

> > *b*2

*k σT e, k* þ 2

k k *wk* 2 *,*

12*,k Q*22*,k*

*ΔV*2ð Þ*k* ≤ � *λ*min

� �*I*3*, Q*22*,k* ¼ 2*T*2

*Qk* � � �

*Tf* 1*bk*

Lemma 2, the range of *f* 2 that holds for Eq. (25) is obtained as

2 � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2ð Þ � *ck*

2 � *ck* . 0 )

p 2*T*

2 � 4*Tf* 2 þ *ck* , 0 *ck* ¼ *Tbkf*

� � �k k *Xk*

" #

trajectories of the closed-loop system is given by

� �k k *zk*

� �*zT*

2 � 4*Tf* 2 þ *T*2

� �*zT*

*λJ* � �2

*λJ* ¼ k k*J , Qk* ¼ *Q*11*,k Q*12*,k QT*

*Q*12*,k* ¼ *Tbk* 1 � *Tf* 1*bk*

In Eq. (23), if *Qk* , 0, then

derived under Assumption 2.

and the following Eq.

2*T*2 *f* 2

*k* þ 1 � �k k *zk*

2 � 2*Tf* 2 þ *T*2

þ 2*Tbk* 1 � *Tf* 1*bk*

þ 2*Tbk* 1 � *Tf* 1*bk*

*k QkXk* þ 2 *T*

� �2 � 2*Tf* 1*bk* n o

from Eq. (18), by using completing square, the difference of Eq. (22) along the

2 þ *Tf* 1*bk*

*wT k J* �2

*T λJ* � �2 k k *σe,k*

� �2 � 2*Tf* 1*bk* n o

2 þ *Tf* 1*bk*

k k *wk* 2

*, Q*11*,k* ¼ *Tf* 1*bk*

*f* 2

2 þ 2 *T λJ* � �2

*Qk* , 0∈R is the minimum eigenvalue of *Qk* and the condition of ISS

holds [18]. Hereafter, conditions of *f* 1 and *f* 2 which the matrix *Qk* holds *Qk* , 0 are

Condition (24) is the same as Eq. (15), and assume that Eq. (24) holds. From

, *f* 2 ,

*Tbkf* 2

> *Tbkf* 2 1 � 2*f* 1

From Schur complement, condition *Qk* , 0 is equivalent to the following

*wk* þ 2*T* 1 � *Tf* 2

� �2 � 2*Tf* 1*bk* n o

2 þ 2*Tbk* 1 � *Tf* 1*bk*

k k *σe,k* 2

> *k J* �1 *zk*

> > k k *σe,k* 2

> > > *I*3*,*

*:* (25)

2*T ,* (26)

. 0 (27)

� �*wT*

� �2 � 2*Tf* 1*bk* n o

*b*2

2 � 4*Tf* 2 þ *T*2

k k *wk* 2 *,*

� �2 � 2*Tf* 1*bk* , 0*,* (24)

1 � 2*f* 1 � *Tbk*

2

!

*Tbkf* 2 1 � 2*f* 1

2 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2ð Þ � *ck*

p

1 � 2*f* 1 þ *Tbk*

*k* þ 1 � �*I*3*:*

� �*zT*

*k σT e,k*

(23)

*ΔV*1ð Þ¼ *k* ð Þ *Tbk*

*ΔV*2ð Þ¼ *k T*2

$$\mathbf{1} - (Tb\_k)^2 > \mathbf{0} \quad \Rightarrow \quad \mathbf{0} \le T \le \frac{1}{b\_k} \tag{30}$$

must hold in order to have the real number. As 2≤ ð Þ 1*=bk* ≤4 under Assumption 2, *T* must satisfy the condition

$$0 \le T \le 2.\tag{31}$$

In addition, since

$$\max\_{b\_k} \frac{1 - \sqrt{1 - \left(Tb\_k\right)^2}}{Tb\_k} = \frac{2 - \sqrt{4 - T^2}}{T},$$

$$\min\_{b\_k} \frac{1 + \sqrt{1 - \left(Tb\_k\right)^2}}{Tb\_k} = \frac{2 + \sqrt{4 - T^2}}{T}$$

under Assumption 2, the condition (29) is given by

$$\frac{2 - \sqrt{4 - T^2}}{T} < f\_1 < \frac{2 + \sqrt{4 - T^2}}{T} (0 < T < 2). \tag{32}$$

Therefore, if *f* 1 satisfies Eq. (32) under Assumption 2, Eqs. (27) and (28) hold. Furthermore, since

$$\max\_{b\_k} \frac{2 - \sqrt{2(2 - c\_k)}}{2T} = \frac{1}{T} - \sqrt{\frac{T f\_1^2 - 4f\_1 + T}{2T^2 f\_1 (T f\_1 - 4)}},$$

$$\min\_{b\_k} \frac{2 + \sqrt{2(2 - c\_k)}}{2T} = \frac{1}{T} + \sqrt{\frac{T f\_1^2 - 4f\_1 + T}{2T^2 f\_1 (T f\_1 - 4)}},$$

under Assumption 2, the condition (26) is given by.

$$\frac{1}{T} - \sqrt{\frac{T f\_1^2 - 4f\_1 + T}{2T^2 f\_1 (T f\_1 - 4)}} < f\_2 < \frac{1}{T} + \sqrt{\frac{T f\_1^2 - 4f\_1 + T}{2T^2 f\_1 (T f\_1 - 4)}} (0 < T < 2). \tag{33}$$

Therefore, if *f* 1 and *f* 2 satisfy Eqs. (32) and (33) under Assumption 2, then *Qk* , 0.

Summarizing the above, the following theorem can be obtained.

**Theorem 1:** if sampling period *T* and feedback gains *f* 1 and *f* 2 satisfy Eqs. (31), (32), and (33) under Assumption 2, then the closed-loop systems (10) and (11) with the following control law

$$u\_k = \{z\_k + a\_{,k} + \mathcal{C}\_k \boldsymbol{o}\_{d,k}\}^\times \{z\_k + a\_{,k} + \mathcal{C}\_k \boldsymbol{o}\_{d,k}\}$$

$$+ f\{\mathcal{C}\_k \dot{\boldsymbol{o}}\_{d,k} - (\mathbf{z}\_k + a\_{,k})^\times \mathcal{C}\_k \boldsymbol{o}\_{d,k}\}$$

$$- f\_1 f\{\mathcal{G}(\sigma\_{\epsilon,k}) \mathbf{z}\_k - f\_1 \mathbf{b}\_k \sigma\_{\epsilon,k}\} - f\_2 f\_2 \mathbf{z}\_k \tag{34}$$

$$= \boldsymbol{o}\_k \, ^\times f \boldsymbol{o}\_k + f\{\mathcal{C}\_k \dot{\boldsymbol{o}}\_{d,k} - (\mathbf{z}\_k + a\_{,k})^\times \mathcal{C}\_k \boldsymbol{o}\_{d,k}\}$$

$$- f\_1 f\_2 f \sigma\_{\epsilon,k} - f\{f\_1 \mathcal{G}(\sigma\_{\epsilon,k}) + f\_2 I\_3\} \boldsymbol{o}\_{\epsilon,k}$$

Euler approximate systems (10) and (11). Then, the following theorem can be

**Theorem 2:** control input (34) is SPA stabilizing for exact discrete-time systems

The properties of the proposed method are discussed in the numerical study. For

*, σ*ð Þ¼ 0

*, f* 1 ¼ 0*:*6*, f* 2 ¼ 0*:*8*:*

0 0 0 3 7 5*, ω*ð Þ¼ 0

0 0 0

2 6 4

2 6 4

3 7 5kgm2

The moment of inertia *J* is from [1]. The initial values *σ*ð Þ 0 correspond to Euler angles of 1–2-3 system of *θ*ð Þ¼ 0 ½ � *θ*1ð Þ 0 *θ*2ð Þ 0 *θ*3ð Þ 0 *T* ¼ ½ � 000 *T* ½ � deg . The feedback gains *f* 1 and *f* 2 satisfy Eqs. (25) and (28) for all cases of *T*. The desired states *σd*ð Þ*t* , *ωd*ð Þ*t* , and *ω*\_ *d*ð Þ*t* in this simulation are the switching maneuver as shown

1*:*0 : Case 1 0*:*5 : Case 2 0*:*1 : Case 3

obtained by Theorem A.1 in Appendix.

*DOI: http://dx.doi.org/10.5772/intechopen.87191*

this purpose, parameter setting of simulation is as follows:

*Discrete-Time Nonlinear Attitude Tracking Control of Spacecraft*

8 ><

>:

7050*:*0 �0*:*536 43*:*9 �0*:*536 2390 1640*:*0 43*:*9 1640*:*0 6130*:*0

*T* ¼

**4. Numerical simulation**

(8) and (9).

*J* ¼

in **Figure 1**.

**Figure 1.**

**69**

*Switching maneuver.*

2 6 4

becomes ISS.

Then, we show that the pair ð Þ *uk;V*2ð Þ*k* is semiglobal practical asymptotic (SPA) stabilizing pair for the Euler approximate systems (10) and (11). Hereafter, suppose that sampling period *T* and feedback gains *f* 1 and *f* 2 satisfy Eqs. (31), (32), and (33) under Assumption 2. By using the following coordinate transformation

$$X\_k = \begin{bmatrix} \mathbf{1} & \mathbf{0} \\ f\_1 & \mathbf{1} \end{bmatrix} \begin{bmatrix} \sigma\_{\epsilon,k} \\ a\_{\epsilon,k} \end{bmatrix} = Z \overline{X}\_k.$$

Lyapunov function *V*2ð Þ*k* and its difference *ΔV*2ð Þ*k* can be rewritten as

$$V\_2(k) = \overline{X}\_k^T Z^T Z \overline{X}\_k = \overline{X}\_k^T R \overline{X}\_k,$$

$$\Delta V\_2(k) = \overline{X}\_k^T Z^T Q\_k Z \overline{X}\_k + 2\left(\frac{T}{\lambda\_l}\right)^2 ||w\_k||^2 = \overline{X}\_k^T \overline{Q\_k} \overline{X}\_k + 2\left(\frac{T}{\lambda\_l}\right)^2 ||w\_k||^2.$$

Since *R* . 0 and *Qk* , 0, *V*2ð Þ*k* and *ΔV*2ð Þ*k* satisfy following equations:

$$\lambda\_R^{\min} \left|| \overline{X}\_k \right||^2 \le V\_2(k) \le \lambda\_R^{\max} \left|| \overline{X}\_k \right||^2,\tag{35}$$

$$
\Delta V\_2(k) \le -\left| \lambda\_{\overline{Q}\_k}^{\min} \right| \left|| \overline{X}\_k \right||^2 + 2 \left( \frac{T}{\lambda\_I} \right)^2 \left\| w\_k \right\|^2. \tag{36}
$$

In addition, *Xk* is bounded, and *V*2ð Þ*k* is radially unbounded from Eqs. (35) and (36). Hence, the control input (34) satisfies the following equation under Assumption 1:

$$\|\|u\_k\|\| \le M,\tag{37}$$

where *M* is a positive constant. Furthermore, *V*2ð Þ*k* also satisfies the following equation for all *x, z*∈ R6 with maxf g k k*x ;* k k*z* ≤ Δ:

$$\begin{aligned} \left| V\_2(\mathbf{x}) - V\_2(\mathbf{z}) \right| &= \left| \mathbf{x}^T \mathbf{R} \mathbf{x} - \mathbf{z}^T \mathbf{R} \mathbf{z} \right| = \left| (\mathbf{x} + \mathbf{z})^T \mathbf{R} (\mathbf{x} - \mathbf{z}) \right| \\ &= \lambda\_\mathcal{R}^{\max} ||\mathbf{x} + \mathbf{z}|| ||\mathbf{x} - \mathbf{z}|| \le 2 \Delta \lambda\_\mathcal{R}^{\max} ||\mathbf{x} - \mathbf{z}||, \end{aligned} \tag{38}$$

where Δ is a positive constant. Therefore, from Eqs. (35) to (38), Lyapunov function *V*2ð Þ*k* and control input *uk* satisfied Eqs. (A8)–(A11) in Definition 2 under Assumptions 1 and 2, and the pair ð Þ *uk;V*2ð Þ*k* becomes SPA stabilizing pair for the

Euler approximate systems (10) and (11). Then, the following theorem can be obtained by Theorem A.1 in Appendix.

**Theorem 2:** control input (34) is SPA stabilizing for exact discrete-time systems (8) and (9).

### **4. Numerical simulation**

**Theorem 1:** if sampling period *T* and feedback gains *f* 1 and *f* 2 satisfy Eqs. (31), (32), and (33) under Assumption 2, then the closed-loop systems (10) and (11) with

*uk* ¼ f g *zk* þ *α,k* þ *Ckωd,k* �*J z*f g *k* þ *α,k* þ *Ckωd,k*

� *f* 2*Jzk*

Then, we show that the pair ð Þ *uk;V*2ð Þ*k* is semiglobal practical asymptotic (SPA) stabilizing pair for the Euler approximate systems (10) and (11). Hereafter, suppose that sampling period *T* and feedback gains *f* 1 and *f* 2 satisfy Eqs. (31), (32), and (33)

> *ωe,k*

*k ZTZXk* ¼ *XT*

k k *wk*

≤*V*2ð Þ*k* ≤ *λ*max

In addition, *Xk* is bounded, and *V*2ð Þ*k* is radially unbounded from Eqs. (35) and (36). Hence, the control input (34) satisfies the following equation under Assump-

where *M* is a positive constant. Furthermore, *V*2ð Þ*k* also satisfies the following

where Δ is a positive constant. Therefore, from Eqs. (35) to (38), Lyapunov function *V*2ð Þ*k* and control input *uk* satisfied Eqs.  By using the following coordinate transformation

Lyapunov function *V*2ð Þ*k* and its difference *ΔV*2ð Þ*k* can be rewritten as

*λJ* 2

Since *R* . 0 and *Qk* , 0, *V*2ð Þ*k* and *ΔV*2ð Þ*k* satisfy following equations:

 *Xk* 2 þ 2 *T λJ* 2

*Qk*

*Xk* ¼ 1 0 *f* 1 1 *σe,k*

*V*2ð Þ¼ *k XT*

*k ZTQkZXk* þ 2 *T*

*λ*min *R Xk* 2

equation for all *x, z*∈ R6 with maxf g k k*x ;* k k*z* ≤ Δ:

*ΔV*2ð Þ*k* ≤ � *λ*min

j j *V*2ð Þ� *x V*2ð Þ*z* ¼ *xTRx* � *zTRz*

¼ *λ*max

¼ *ωk*

the following control law

*Gyroscopes - Principles and Applications*

becomes ISS.

*ΔV*2ð Þ¼ *k XT*

tion 1:

**68**

The properties of the proposed method are discussed in the numerical study. For this purpose, parameter setting of simulation is as follows:

$$J = \begin{bmatrix} 7050.0 & -0.536 & 43.9 \\ -0.536 & 2390 & 1640.0 \\ 43.9 & 1640.0 & 6130.0 \end{bmatrix} \text{kg} \text{m}^2, \sigma(\mathbf{0}) = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}, \rho(\mathbf{0}) = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \text{rad/s}$$

$$T = \begin{cases} 1.0 : \text{Case 1} \\ 0.5 : \text{Case 2}, f\_1 = 0.6, f\_2 = 0.8. \\ 0.1 : \text{Case 3} \end{cases}$$

The moment of inertia *J* is from [1]. The initial values *σ*ð Þ 0 correspond to Euler angles of 1–2-3 system of *θ*ð Þ¼ 0 ½ � *θ*1ð Þ 0 *θ*2ð Þ 0 *θ*3ð Þ 0 *T* ¼ ½ � 000 *T* ½ � deg . The feedback gains *f* 1 and *f* 2 satisfy Eqs. (25) and (28) for all cases of *T*. The desired states *σd*ð Þ*t* , *ωd*ð Þ*t* , and *ω*\_ *d*ð Þ*t* in this simulation are the switching maneuver as shown in **Figure 1**.

**Figure 1.** *Switching maneuver.*

#### **Figure 2.**

*Time histories of MRPs σ*ð Þ*t and σe*ð Þ*t (solid line, case 1; dashed-dotted line, case 2; dashed line, and case 3; dotted line, σd*ð Þ*t ).*

The results of the numerical simulation are shown in **Figures 2**–**5**. The relative attitude *σe*ð Þ*t* and relative angular velocity *ωe*ð Þ*t* converge to the neighborhood of ð Þ¼ *σe*ð Þ*t ;ωe*ð Þ*t* ð Þ 0*;* 0 , and the control input amplitude *u t*ð Þ does not depend on the sampling period *T* although there is a slight difference in the

*Time histories of control input u t*ð Þ *(solid line, case 1; dashed-dotted line, case 2; and dashed line, case 3).*

This chapter considers the spacecraft attitude tracking control problem that requires agile and large-angle attitude maneuvers and proposed a discrete-time nonlinear attitude tracking control that the amplitude of the control input does not depend on the sampling period *T*. The effectiveness of proposed control method is verified by numerical simulations. Extension to the guarantee of stability as

This section shows preliminary results for nonlinear sampled-data control

where *x t*ð Þ∈R*n* is the state variable and *x t*ð Þ∈R*m* is the control input. The function *f xt* ð Þ ð Þ*; u t*ð Þ in Eq. (A1) is assumed to be such that, for each initial condition and each constant control input, there exists a unique solution defined on some

and zero-order hold (D/A converter), and the control signal is assumed to be

The nonlinear system (A1) is assumed to be between a sampler (A/D converter)

*u t*ðÞ¼ *u kT* ð Þ ≕ *u k*ð Þ*,* ∀*t*∈½ � *kT;*ð Þ *k* þ 1 *T , k*∈f g0 ∪ N*,* (A2)

*x t* \_ðÞ¼ *f xt* ð Þ ð Þ*; u t*ð Þ *, x*ð Þ¼ 0 *x*0*, f*ð Þ¼ 0*;* 0 0*,* (A1)

sampled-data control system will be subject to future work.

*Discrete-Time Nonlinear Attitude Tracking Control of Spacecraft*

*DOI: http://dx.doi.org/10.5772/intechopen.87191*

**Appendix: sampled-data control of nonlinear system**

Let us consider the following nonlinear system:

maximal value of *u t*ð Þ.

**5. Conclusion**

**Figure 5.**

[13, 14, 19].

intervals of *x*½0*; τ*Þ.

**71**

piecewise constant, that is,

#### **Figure 3.**

*Time histories of attitude angles θ*ð Þ*t and θe*ð Þ*t (solid line, case 1; dashed-dotted line, case 2; dashed line, and case 3; dotted line, θd*ð Þ*t ).*

#### **Figure 4.**

*Time histories of angular velocities ω*ð Þ*t and ωe*ð Þ*t (solid line, case 1; dashed-dotted line, case 2; dashed line, and case 3; dotted line, ωd*ð Þ*t ).*

*Discrete-Time Nonlinear Attitude Tracking Control of Spacecraft DOI: http://dx.doi.org/10.5772/intechopen.87191*

#### **Figure 5.**

**Figure 2.**

**Figure 3.**

**Figure 4.**

**70**

*and case 3; dotted line, ωd*ð Þ*t ).*

*case 3; dotted line, θd*ð Þ*t ).*

*dotted line, σd*ð Þ*t ).*

*Gyroscopes - Principles and Applications*

*Time histories of MRPs σ*ð Þ*t and σe*ð Þ*t (solid line, case 1; dashed-dotted line, case 2; dashed line, and case 3;*

*Time histories of attitude angles θ*ð Þ*t and θe*ð Þ*t (solid line, case 1; dashed-dotted line, case 2; dashed line, and*

*Time histories of angular velocities ω*ð Þ*t and ωe*ð Þ*t (solid line, case 1; dashed-dotted line, case 2; dashed line,*

*Time histories of control input u t*ð Þ *(solid line, case 1; dashed-dotted line, case 2; and dashed line, case 3).*

The results of the numerical simulation are shown in **Figures 2**–**5**. The relative attitude *σe*ð Þ*t* and relative angular velocity *ωe*ð Þ*t* converge to the neighborhood of ð Þ¼ *σe*ð Þ*t ;ωe*ð Þ*t* ð Þ 0*;* 0 , and the control input amplitude *u t*ð Þ does not depend on the sampling period *T* although there is a slight difference in the maximal value of *u t*ð Þ.

### **5. Conclusion**

This chapter considers the spacecraft attitude tracking control problem that requires agile and large-angle attitude maneuvers and proposed a discrete-time nonlinear attitude tracking control that the amplitude of the control input does not depend on the sampling period *T*. The effectiveness of proposed control method is verified by numerical simulations. Extension to the guarantee of stability as sampled-data control system will be subject to future work.

#### **Appendix: sampled-data control of nonlinear system**

This section shows preliminary results for nonlinear sampled-data control [13, 14, 19].

Let us consider the following nonlinear system:

$$\dot{\boldsymbol{x}}(t) = f(\boldsymbol{x}(t), \boldsymbol{u}(t)), \boldsymbol{x}(0) = \boldsymbol{x}\_0, f(0, 0) = 0,\tag{A1}$$

where *x t*ð Þ∈R*n* is the state variable and *x t*ð Þ∈R*m* is the control input. The function *f xt* ð Þ ð Þ*; u t*ð Þ in Eq. (A1) is assumed to be such that, for each initial condition and each constant control input, there exists a unique solution defined on some intervals of *x*½0*; τ*Þ.

The nonlinear system (A1) is assumed to be between a sampler (A/D converter) and zero-order hold (D/A converter), and the control signal is assumed to be piecewise constant, that is,

$$u(t) = u(kT) =: u(k), \forall t \in [kT, (k+1)T], k \in \{0\} \cup \mathbb{N},\tag{A2}$$

where *T* . 0 is a sampling period. In addition, assume that the state variable

$$\mathfrak{x}(k) := \mathfrak{x}(kT) \tag{A3}$$

is measurable at each sampling instance. The exact discrete-time model and Euler approximate model of the nonlinear sampled-data systems (A1)–(A3) are expressed as follows, respectively:

$$\mathfrak{x}\_{k+1} = \mathfrak{x}\_k + \int\_{kT}^{(k+1)T} f(\mathfrak{x}(s), u\_k) \, ds =: F\_T^\epsilon(\mathfrak{x}\_k, u\_k), \tag{A4}$$

$$\varkappa\_{k+1} = \varkappa\_k + T f(\varkappa\_k, \mu\_k) =: F\_T^{Euler}(\varkappa\_k, \mu\_k), \tag{A5}$$

where we abbreviate *x k*ð Þ and *u k*ð Þ to *xk* and *uk*. For the stability of the exact discrete-time model (A4) (*Fe T*) and Euler approximate model (A5) (*FEuler T* ), the following definitions are used [13, 14, 19].

**Definition 1:** consider the following discrete-time nonlinear system:

$$\mathbf{x}\_{k+1} = F\_T(\mathbf{x}\_k, \boldsymbol{\mu}\_T(\mathbf{x}\_k)),\tag{A6}$$

where *xk* ∈ R*n* is the state variable and *uT*ð Þ *xk* ∈ R*m* is a control input. The family of controllers *uT*ð Þ *xk* SPA stabilizes the system (A6) if there exists a class KL function *β*ð Þ� such that for any strictly positive real numbers ð Þ *D; ν* , there exists *T*∗ . 0, and such that for all *T* ∈ 0*; T*∗ ð Þ and all initial state *x*0 with k k *x*0 ≤ *D*, the solution of the system satisfies

$$\|\mathbf{x}\_k\| \le \beta(\|\mathbf{x}\_0\|, kT) + \nu, \forall k \in \{\mathbf{0}\} \cup \mathbf{N}.\tag{A7}$$

**Definition 2:** let *T*^ . 0 be given, and for each *T* ∈ 0*; T*^ � �, let functions *VT* : R*n* ! R and *uT* : R*n* ! R*m* be defined. The pair of families ð Þ *uT;VT* is a SPA stabilizing pair for the system (A7) if there exist a class K∞ functions *α*1, *α*2, and *α*3 such that for any pair of strictly positive real numbers ð Þ Δ*; δ* , there exists a triple of strictly positive real numbers *T*∗ ð Þ *; L; M T*∗ ≤*T*^ � � such that for all *x, z*∈ R*n* with maxf g k k*x ;* k k*z* ≤ Δ, and *T* ∈ 0*; T* ∗ ð Þ:

$$a\_1(||\mathfrak{x}||) \le V\_T(\mathfrak{x}) \le a\_2(||\mathfrak{x}||),\tag{A8}$$

$$V\_T(F\_T(\mathfrak{x}, \mathfrak{u}\_T(\mathfrak{x}))) - V\_T(\mathfrak{x}) \le -a\_3(\|\mathfrak{x}\|) + T\delta,\tag{A9}$$

$$|V\_T(\mathbf{x}) - V\_T(\mathbf{z})| \le L||\mathbf{x} - \mathbf{z}||,\tag{A10}$$

$$\|\|\mu\_T(\mathbf{x})\|\| \le M. \tag{A11}$$

**Author details**

Shonan Institute of Technology, Fujisawa, Japan

provided the original work is properly cited.

\*Address all correspondence to<EMAIL_ADDRESS>
*Discrete-Time Nonlinear Attitude Tracking Control of Spacecraft*

*DOI: http://dx.doi.org/10.5772/intechopen.87191*

© 2019 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,

Yuichi Ikeda

**73**

In addition, if there exists *T*∗ ∗ . 0 such that Eqs. (A8)–(A11) with *δ* ¼ 0 hold for all *x*∈R*n* and *T* ∈ 0*; T* ∗ ∗ ð Þ, then the pair ð Þ *uT;VT* is globally asymptotic (GA) stabilizing pair for the system (A6).

Using the above definitions, the following theorem is obtained by literatures [13, 14, 19].

**Theorem A.1:** if the pair ð Þ *uT;VT* is SPA stabilizing for *FEuler T* , then *uT* is SPA stabilizing for *Fe T* .

Hence, if we can find a family of pairs of ð Þ *uT;VT* that is a GA or SPA stabilizing pair for *FEuler T* , then the controller *uT* will stabilize the exact model *Fe T*.

*Discrete-Time Nonlinear Attitude Tracking Control of Spacecraft DOI: http://dx.doi.org/10.5772/intechopen.87191*

where *T* . 0 is a sampling period. In addition, assume that the state variable

is measurable at each sampling instance. The exact discrete-time model and Euler approximate model of the nonlinear sampled-data systems (A1)–(A3) are

*f xs* ð Þ ð Þ*; uk ds* ≕ *Fe*

*T*) and Euler approximate model (A5) (*FEuler*

*xk*þ1 ¼ *FT*ð Þ *xk; uT*ð Þ *xk ,* (A6)

k k *xk* ≤*β*ð Þþ k k *x*0 *; kT ν,* ∀*k*∈f g0 ∪ N*:* (A7)

*α*1ð Þ k k*x* ≤*VT*ð Þ *x* ≤*α*2ð Þ k k*x ,* (A8)

j j *VT*ð Þ� *x VT*ð Þ*z* ≤ *L x*k k � *z ,* (A10)

k k *uT*ð Þ *x* ≤ *M:* (A11)

*T* , then *uT* is SPA

*T*.

*VT*ð*FT*ð Þ *x; uT*ð Þ *x* Þ � *VT*ð Þ *x* ≤ � *α*3ð Þþ k k*x Tδ,* (A9)

ðð Þ *k*þ1 *T kT*

*xk*þ1 ¼ *xk* þ *Tf x*ð Þ *k; uk* ≕ *FEuler*

**Definition 1:** consider the following discrete-time nonlinear system:

of controllers *uT*ð Þ *xk* SPA stabilizes the system (A6) if there exists a class KL function *β*ð Þ� such that for any strictly positive real numbers ð Þ *D; ν* , there exists *T*∗ . 0, and such that for all *T* ∈ 0*; T*∗ ð Þ and all initial state *x*0 with k k *x*0 ≤ *D*, the

**Definition 2:** let *T*^ . 0 be given, and for each *T* ∈ 0*; T*^ � �, let functions *VT* : R*n* ! R and *uT* : R*n* ! R*m* be defined. The pair of families ð Þ *uT;VT* is a SPA stabilizing pair for the system (A7) if there exist a class K∞ functions *α*1, *α*2, and *α*3 such that for any pair of strictly positive real numbers ð Þ Δ*; δ* , there exists a triple of

strictly positive real numbers *T*∗ ð Þ *; L; M T*∗ ≤*T*^ � � such that for all *x, z*∈ R*n*

In addition, if there exists *T*∗ ∗ . 0 such that Eqs. (A8)–(A11) with *δ* ¼ 0 hold for all *x*∈R*n* and *T* ∈ 0*; T* ∗ ∗ ð Þ, then the pair ð Þ *uT;VT* is globally asymptotic (GA)

Using the above definitions, the following theorem is obtained by literatures

Hence, if we can find a family of pairs of ð Þ *uT;VT* that is a GA or SPA stabilizing

*T* , then the controller *uT* will stabilize the exact model *Fe*

**Theorem A.1:** if the pair ð Þ *uT;VT* is SPA stabilizing for *FEuler*

where we abbreviate *x k*ð Þ and *u k*ð Þ to *xk* and *uk*. For the stability of the exact

where *xk* ∈ R*n* is the state variable and *uT*ð Þ *xk* ∈ R*m* is a control input. The family

expressed as follows, respectively:

*Gyroscopes - Principles and Applications*

discrete-time model (A4) (*Fe*

solution of the system satisfies

with maxf g k k*x ;* k k*z* ≤ Δ, and *T* ∈ 0*; T* ∗ ð Þ:

stabilizing pair for the system (A6).

*T* .

[13, 14, 19].

stabilizing for *Fe*

pair for *FEuler*

**72**

following definitions are used [13, 14, 19].

*xk*þ1 ¼ *xk* þ

*x k*ð Þ ≔ *x kT* ð Þ (A3)

*T*ð Þ *xk; uk ,* (A4)

*T* ), the

*T* ð Þ *xk; uk ,* (A5)


Composite and Nanocomposite Materials From Knowledge to Industrial Applications

*Edited by Tri-Dung Ngo*

Composite and Nanocomposite Materials - From Knowledge to Industrial Applications *Edited by Tri-Dung Ngo*

Published in London, United Kingdom


**4. Experiments**

To provide a comprehensive overview of the strengths of the pipeline, it was separately evaluated on benchmark task-specific datasets. All the experiments were performed on a Linux Intel(R) Core(TM) i5-7440HQ CPU system with a RAM capacity of GB; the GPU was an NVidia GeForce 980 with 4 GB memory; and the operating system was Ubuntu version 16.04. The entire pipeline was implemented in Python 3.6 with the Pytorch deep learning library. The datasets and evaluation metrics used for evaluating our pipeline are explained in the following sections.


**Author details**

Rai Mukkram Ali Tahir\*1 , Noor-us-Sabah1 , Muhammad Afzal2 , Ghulam Sarwar1 and Ijaz Rasool Noorka1

1 Department of Soil and Environmental Sciences, College of Agriculture, University of Sargodha, Pakistan

2 Department of Agricultural Entomology, College of Agriculture, University of Sargodha, Pakistan

\*Address all correspondence to<EMAIL_ADDRESS>
© 2019 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.


**2. The concept of quality of life**

One of the topics that have not often gained widespread attention is quality of life, whether at the level of scientific application or in terms of broad applicability in our daily lives. Quality of life is prevalent in many scientific fields. It is sometimes used to describe improvements in the quality of the material and social services offered to society's citizens and to define people's awareness of the degree to which these services can meet their diverse requirements. However, despite the concept's modernity in handling, the users have yet to agree on a precise definition for this concept [11–16].

The definition of the concept of quality of life differed from one researcher to the next due to differences in research specialization, theoretical orientation, and research objectives to be attained in each researcher's study of the quality of life in humans with this term. On the one hand, it is challenging to reach an accord on quality of life since it is seen as an ambiguous notion and because there is not a broad consensus on a specific definition [16–21].

Numerous philosophers and academics have historically expressed an interest in the study of quality of life, according to Zuna et al. [22]. The Greek philosophers Plato and Aristotle first thought of the quality of life. As a result, many perspectives developed, each based on his field of specialization [23–25]. Because psychologists have neglected positive psychology for a long time, there is an increasing interest in investigating the quality of life and the related factors, such as life satisfaction, happiness, the meaning of life, self-efficacy, and meeting needs [1, 2].

Emerson claimed that a person's quality of life is based on how content they are with their values, goals, and wants resulting from developing their abilities and potential or adopting a particular way of living. According to Proshanky and Fabian [24], a person's quality of life is a function of the actual environmental conditions in which they live and how they feel and experience those surroundings. Tartar et al. [21] mention that quality of life is a complicated idea that includes a person's behavioral and cognitive talents, psychological health, capacity to interact with his surroundings, and social and professional responsibilities. Taylor and Bogdan reported that a person's quality of life is based on how happy they are on the inside and how satisfied they are with their life's circumstances and fortune. Reidulf et al. reported that a person's quality of life is determined by how much they use the real opportunities in their existence; feeling content and achieving some benefits are two components of that satisfaction.

Felce and Perry [26] defined the quality of life as the overall general happiness that results from the subjective and objective evaluation of physical, material, social, and emotional happiness side by side after a person's level of personal development and his purposeful activities, all of which are evaluated in terms of the person's value

#### *Quality of Life: The Concept and Interventions DOI: http://dx.doi.org/10.5772/intechopen.108610*

system. Similarly, Schalock defined quality of life as the demands that a person wishes or desires in their life that are connected to eight aspects of that person's life: bodily happiness, self-orientation, social security, and social rights. According to the World Health Organization, a person's understanding of where they fit into the society and system of values in which they live as well as how they relate to their own goals, standards, expectations, and interests make up their quality of life. Furthermore, "quality of life" is a multifaceted concept that depends on a person's values and includes both subjective and objective indicators, according to Hass.

Quality of life refers to a person's life in general rather than one specific component without the other, according to Hegerty et al. Rejeski and Mihalko [12] defined quality of life as the assessment or emotional, cognitive appraisal of a person's satisfaction with their life. Schalock et al. [14] mentioned that the notion of quality of life had become a focus of research and application in the last 20 years in the domains of education, special education, physical and behavioral health care, social services (for the elderly and those with special needs), and families. In light of this, he contends that even if the quality of life was divided into many dimensions, these measurements should display the composite or overall image of the quality of life. The person's physical, social, and psychological quality of life is affected by their personal experiences, beliefs, and perceptions of themselves, others, and the world in which they live [16, 27, 28].

There are several main trends in the theoretical perspective in defining the quality of life. According to the philosophical perspective, the idea of quality of life led to the placement of the concepts of happiness within the well-known pragmatic trilogy. This trilogy holds that an idea cannot become a belief unless it demonstrates its success on the practical level or the immediate value rather than the cash value (utilitarianism). The practical level is closer to the ideas of happiness and personal well-being than to any other idea. Another philosophical perspective views the quality of life as "a paradox of reality in the sense of imaginary and dreamy happiness in which a person lives in a state of freedom from the captivity of reality and soars into an ideal space that pushes man to transcend that stifling reality and let go for moments of a rich creative imagination." According to this perspective, the quality of life is "a state of freedom from the captivity of reality in which a person lives in a state of imaginary and dream" [1, 2, 26, 29–31].

Also, the term "quality of life" is defined from a social perspective, focusing on the family and society, interpersonal interactions, cultural norms, population, money, work, workplace demands, and other social elements. Numerous societal measures of people's quality of life include happiness, satisfaction with oneself and others, positive social relations, awareness of the feelings of others, control of emotions, the internal control of behavior, personal and social responsibility, participation in cooperative work, loyalty and belonging to family, city, and country, personal, social, health, family, and professional compatibility, and optimism [18, 19, 24].

A basic definition of quality of life in the medical field was provided by Walker and Rosser, who stated that it is a concept that encompasses a wide range of physical and psychological characteristics and limitations that describe a person's capacity for performance and satisfaction of his work-related desires. This means that quality of life refers to health-related issues, such as the impact of disease and available treatment on health beliefs and a better quality of life [6, 25, 32].

Also, the Encyclopedia of Psychology describes the quality of life as a psychological concept into seven axes: emotional balance consists of controlling positive and negative emotions, such as sadness, depression, anxiety, and psychological stress.

A dynamic notion, psychological well-being also involves many beneficial psychological elements and is related to an effort to track: How do people view or value the various facets of their psychological lives? How much do people feel they influence certain parts of their daily lives, for instance? How much do people believe their personal lives are meaningful and valuable? How much do people feel they have mutually beneficial social interactions with others? [1, 26, 27, 33, 34].

We can conclude that the concept of quality of life is multidimensional and differs from person to person depending on how they view elements of life, such as religious, psychological, medical, and social dimensions. According to this viewpoint, a person's quality of life is defined as living in good physical, mental, and emotional health to a degree of acceptance and satisfaction, being strong-willed and steadfast in the face of pressures, having high self and social efficiency, and being satisfied with his family, career, and community life [35, 36].

#### **3. Quality of life interventions**

The most valuable human asset is probably one's health. Despite the apparent importance of health, studies of people's preferences for various states of life reveal that almost everyone ranks health as the most essential. The first of two recurrent themes in definitions of health is early death, and the second is quality of life. Al-Ghamdi [37] mentioned that health psychologists have recently understood how crucial it is to assess and improve the quality of life. Numerous serious illnesses, such as arthritis, heart disease, diabetes, and digestive issues, are rated according to how they affect people's quality of life and life expectancy, affecting their mental health. For example, a person with heart disease could encounter restrictions on living activities due to their condition, and illness or incapacity might also make it less attractive. The degree to which a person's quality of life improves following therapy for specific disorders can also be used to evaluate the effectiveness of the treatment [16, 19, 32].

Societies today face many stresses and daily quarrels that affect citizens' quality of life. Quality of life is essential in interacting with individuals and their positivity with others and the society around them. Therefore, the availability of quality of life to a large extent has a positive impact on society, where individuals are more positive and more involved in achieving society's goals and contributing to their progress and prosperity. Quality of life is one of the critical topics related to mental health and positive psychology, which helps individuals adapt to the different situations they are exposed to in their daily lives.

Given the importance of the quality of life for individuals and societies to achieve sustainable personal and community development, the researchers sought to design programs based on behavioral therapy, cognitive-behavioral therapy, meaning therapy, reality therapy, and others to improve the quality of life for individuals.

By reviewing the literature about quality of life in the Arab world, a growing literature has examined the effectiveness of interventions in improving the quality of life for different individuals, such as mothers of cancer patients, as Abu-Ghazaleh's study [38]. Also, students such as El-Safty's study findings [39] demonstrated a program's ability to improve the social life of senior students. Al-Wolani's study [40] discovered the value of training in improving university students' quality of life. In addition, for the individuals with special needs and their siblings with down syndrome, the study conducted by Al-Ghamdi [37] sought to determine

#### *Quality of Life: The Concept and Interventions DOI: http://dx.doi.org/10.5772/intechopen.108610*

the efficacy of a counseling program in enhancing the quality of life for common siblings with down syndrome and its influence on the growth of adaptive behavior skills for mentally challenged siblings with down syndrome. The findings showed that the counseling program enhances common siblings with down syndrome quality of life.

Mustafa's study revealed that a counseling program could improve blind people's quality of life. Additionally, Ayesh [41] demonstrated that the counseling program increased the quality of life for newlywed wives between the ages 25–35. The Shaheen study [42] likewise sought to determine the efficacy of a training program in improving the quality of life for gifted kids, and the outcomes demonstrated the program's success in doing so. According to Al-Aour [43] and Al-Najaheh's study, counseling programs are successful at fostering optimism and improving students' quality of life. The goal of Al-Jawaziyyah's [44] study was to evaluate how well a counseling program may enhance the quality of life for secondary school teachers. The findings of Muhammad's study from 2019 demonstrated the value of a psychology curriculum based on an esthetic approach in helping secondary school pupils develop the aspects of their quality of life. The Al-Sibai study's findings from 2019 also show how a program's ability to foster emotional intelligence can enhance the quality of life for kids who struggle with reading. According to a recent study by Nasr [45], there is convincing evidence that moms of children with cerebral palsy experience psychological stress and a lower quality of life. The results demonstrated the program's success in enhancing the experimental group's quality of life and reducing psychological stress (**Table 1**).

It is clear from the results of the above studies that interventions to improve the quality of life among Arab citizens are effective for all groups of individuals of all ages and demographic characteristics. Also, these studies indicated that there is still a need for studies on quality-of-life interventions in the Arab world, especially for individuals who suffer from chronic diseases, patients with different types of cancer, diabetes patients, people with various mental and physical disabilities, and adults with psychological problems, Alzheimer's disease, and Dementia.

In developed countries, such as Europe and the United States of America, studies have found the effectiveness of quality-of-life interventions with different groups of individuals. Many studies revealed the effectiveness of interventions in improving the quality of life among cancer patients [17, 34, 46].

Many studies found that quality-of-life interventions, such as peer-led education, telephone and videophone, counseling and cognitive interventions, peer-counseling, and nurse-led cognitive-behavior therapy, are effective for breast cancer patients and breast cancer survivors [13, 18, 20, 47–55]. Also, [29] improved patients' quality of life with chronic conditions through integrated care interventions.

There is a growing literature on the effectiveness of quality-of-life interventions for patients with chronic obstructive pulmonary disease [25, 56–61].

The results of several studies indicated the effectiveness of interventions in improving the quality of life for dementia and Alzheimer's patients [8, 9, 62]. The study by Qian et al. [63] and Chen et al. [5] found a statistical effect of interventions to improve COVID-19 patients. Also, there is increasing theoretical literature indicating the effectiveness of interventions to improve overweight/obese women [4, 64] and also improve diabetes patients [65].

Interventions can also improve the quality of life among older people [7, 10, 32], as well as interventions are effectively to enhance young adults' quality of life [6].

### *Quality of Life: The Concept and Interventions DOI: http://dx.doi.org/10.5772/intechopen.108610*

#### **Table 1.**

*A review of studies that examined the effectiveness of quality-of-life interventions.*