Barrier Function Based Integral Sliding Mode Control for Attitude Control of Rigid Spacecraft

Spacecrafts often experience difficult conditions in space, represented by several forces that threaten their attitude and even their stability, such as gravitational force, aerodynamic drag, solar pressure, and others. In addition to this, there are the complexities of the nonlinear dynamics and Moment of Inertia (MOI) uncertainty. Given the importance of spacecraft, which have come to play a dominant role in several fields, such as communications, weather forecasting, monitoring, and guidance, the need to develop control over their positioning and stability has become indispensable.

Over the past decades, several methods have been developed to deal with these problems. These methods include the adaptive control [1-3], Sliding mode control [4-6], back-stepping control [7, 8], and fuzzy control [9, 10]. Through these approaches, the sliding mode control is one of the most effective strategies for attitude control of rigid spacecraft.

The use of Sliding Mode Control (SMC) for attitude controlling of rigid spacecraft was originally introduced in the study [11], after which several types of SMC control were introduced by researchers such as Conventional CSMC where convergence is asymptotic [12-15]. Unlike CSMC, Terminal SMC (TSMC) was used to ensure convergence at a specific time [16-18]. Then a modified TSMC was used to avoid singularity while ensuring convergence in finite time called Fast (FTSMC) [19-22]. However, in both cases, despite ensuring the speed of convergence and avoiding the singularity, global robustness cannot be guaranteed due to the presence of the reaching phase. To eliminate this situation, ISMC was resorted to, which is characterized by no reaching phase. Therefore, TISMC was introduced to ensure global robustness and convergence in a finite time [23, 24]. A fixed-time ISMC was introduced in the study [25] to eliminate the reaching phase operations and ensure convergence in a fixed time with global robustness.

However, all of the above studies rely on prior knowledge of the upper bound of the disturbance and uncertainties.

In the case where the upper bound of disturbance is unknown there are two main different ways for adaptive Sliding Mode Control (ASMC) [26]. The first method involves increasing the gain until the sliding mode (SM) is reached, after which the gain value stays stationary to ensure the ideal SM for a while. However, SM can be lost when disturbance increases and the gain value will increase to reattain the SM [27, 28]. Second method presented by the study [29] is based on the reference model and equivalent control to estimate the disturbances. A low pass filter was used for implementation, however, the choosing the filter gain which must be considerably lower than the inverse of the highest value of the first derivative of the disturbance is so difficult. So, many strategies are suggested by researchers, the core of these strategies is the adaptation of the controller gain as in studies [30-34] where the ASMC is proposed for attitude tracking, while a FTSMC is proposed in the study [35] to ensure finite time convergence. ASM Fault tolerant control [36] is developed as a controller for nonlinear systems, in this approach even with adaptation algorithm the dynamics of perturbation may require. While in the study [37] dynamic sliding variable and adaptation law are used to design adaptive dynamic SMC. In the study [38] Finite time attitude tracking by designing passivity _ ASMC is proposed, in this method the full state feedback is assumed, where it is practically unavailable. In addition, disturbance observer can be used to reject the disturbance and combined with ASMC for UAV trajectory tracking as in the study [39], this approach depends on parameters tuning which makes the computational process large. Moreover, fuzzy logic and neural network (NN) also played a role in attitude control, where Tkagi-Sogeno Fuzzy Algorithm combined with ASMC for spacecraft attitude controlling, the limitations of this strategy are the high computational process, chattering issue, and the precise selection of the membership function of fuzzy model [40, 41], and an adaptive fuzzy – Neural PID-Like Controller for Nonlinear MIMO Systems is presented in the study [42].

Most of Previously mentioned studies involve complex computational, in addition to reducing, rather than eliminating, the chattering. This has encouraged researchers to develop effective and simple strategies to cope with these problems, while ensuring robustness without knowing the details of the perturbation dynamics.

However, CSMC and adaptive SMC are the most effective method for controlling nonlinear systems, especially those suffering from disturbances and uncertainty, but through the reaching phase, the robust stability and performance may not be ensured. The ISMC is a special type of SMC, which is unique in that there is no reaching phase, which means the SM will be achieved from the beginning, meaning that the control system will be in nominal and full order from the first moment. However, like CSMC, it also suffers from chattering [43-46].

To cope these problems, BF based ISMC (BFISMC) has been proposed. In addition to exploiting characteristics of ISMC, the features of the presented method are:

1- The knowledge of the ultimate limit of perturbation and model uncertainties or their derivative are not required in advance.

2- There is only one control parameter to be tuned to improve the steady state error.

3- Chattering elimination because BF is continuous.

The structure of this study is organized as follows: Section 2 explains the mathematical model of a rigid spacecraft, Section 3 provides the details of ISMC, BF with ASMC (BFASMC) and the proposed BFISMC. The results and comparison will be presented in Section 4, while the conclusions will be in the Section 5.

In this section ISMC and BFASMC will be introduced which are both robust controllers for the purpose of comparison and proving the superiority of the proposed controller which will be presented in the last part of this section.

3.1 Integral Sliding Mode Controller (ISMC)

CSMC consist of two phases, the reaching phase, which is the trajectory of the system from its initial state to the moment it reaches the sliding surface, at which point it enters the sliding phase, as illustrated in Figure 1. The property of robustness against disturbances or parameters variation is achieved only through the SM. Therefore, robustness cannot be guaranteed over the reaching phase [50]. The removal of the reaching phase in the CSMC is done by using the ISMC, in which the states are in the sliding phase from the first moment [51]. In ISMC the order of system dynamic in sliding mode does not change, therefore it’s also called full order SMC (FOSMC), while in CSMC the order reduction of the system is due to numbers of control signals.

Figure 1. Sliding surface [52]

To design the ISMC, the error equations are:

$e_1=x_1-x_{1 d}$                 (18)

$e_2=x_3-x_{3 d}$                 (19)

$e_3=x_5-x_{5 d}$                 (20)

where, $\mathrm{x}_{1 \mathrm{d}}, \mathrm{x}_{3 \mathrm{d}}$ and $\mathrm{x}_{5 \mathrm{d}}$ are the desired Roll, Pitch and Yaw rotation angles and will be chosen later as constants. The time derivative of the error equations are:

$\dot{e}_1=\dot{x}_1-\dot{x}_{1 d}=x_2$                 (21)

$\dot{e}_2=\dot{x}_3-\dot{x}_{3 d}=x_4$                 (22)

$\dot{e}_2=\dot{x}_5-\dot{x}_{5 d}=x_6$                 (23)

The sliding surfaces corresponding to three control inputs are designed as:

$s_1=\dot{e}_1+\rho e_1$                 (24)

$s_2=\dot{e}_2+\rho e_2$                 (25)

$s_3=\dot{e}_3+\rho e_3$                 (26)

where, $\rho>0$ is convergence rate. The time derivative of sliding surface equations are:

$\dot{s}_1=\dot{x}_2+\rho x_2$                 (27)

$\dot{s}_2=\dot{x}_4+\rho x_4$                 (28)

$\dot{s}_3=\dot{x}_6+\rho x_6$                 (29)

by substitution Eqs. (13), (15), and (17) in Eqs. (27-29) the following equations are obtained:

$\dot{s}_1=\frac{1}{J_1}\left[\left(J_2-J_3\right) x_4 x_6+u_1+d_1\right]+\rho x_2$                (30)

$\dot{s}_2=\frac{1}{J_2}\left[\left(J_3-J_1\right) x_6 x_2+u_2+d_2\right]+\rho x_4$                (31)

$\dot{s}_3=\frac{1}{J_3}\left[\left(J_1-J_2\right) x_4 x_2+u_3+d_3\right]+\rho x_6$                (32)

Let the control law taken as:

$u_i=J_i\left(u_{n i}+u_{s i}\right) i=1,2$ and 3               (33)

$u_{n i}$ is the nominal control to make a stabilization to system nominal dynamics, while the disturbance term will addressed by switching control law $u_{si}$ [50].

$u_{n 1}=-\frac{1}{J_1}\left(J_2-J_3\right) x_4 x_6-\rho x_2-\beta_1 s_1$               (34)

$u_{n 2}=-\frac{1}{J_2}\left(J_3-J_1\right) x_6 x_2-\rho x_4-\beta_2 s_2$               (35)

$u_{n 3}=-\frac{1}{J_3}\left(J_1-J_2\right) x_4 x_2-\rho x_6-\beta_3 s_3$               (36)

$\beta_{1,2 \text { and } 3}$ are positive gains to be designed according to required performance. For ISMC the auxiliary sliding variable is designed as:

$\sigma_i=s_i+z_i$               (37)

$s_i, z_i$ are conventional sliding manifold and integral term (auxiliary sliding variable) respectively such that $\sigma_i, s_i, z_i \in \mathrm{R}^1$ with $\mathrm{z}(0)=-\mathrm{s}(0)$ which leads to $\sigma_i(0)=0$ from the first instant.

$\dot{\sigma}_i=u_{n i}+u_{s i}+d_i+\dot{z}_i$               (38)

let $\dot{z}_i=-u_{n i}$ and this leads to

$\dot{\sigma}_i=u_{s i}+d_i$               (39)

by applying equivalent control to Eq. (39) yields

$\begin{gathered}\left.\dot{\sigma}_i\right|_{e q}=0=\left.u_{s i}\right|_{e q}+d_i \\ \left.u_{s i}\right|_{e q}=-d_i\end{gathered}$

This means, the disturbance will be eliminated by discontinuous control in equivalent mode from the first moment $(t \geq 0)$.

Consequently, the equivalent close loop control systems in Eqs. (13), (15) and (17) can be given by:

$\dot{x}_2=\frac{1}{J_1}\left[\left(J_2-J_3\right) x_4 x_6+u_{n 1}\right]=-\beta_1 s_1$               (40)

$\dot{x}_4=\frac{1}{J_2}\left[\left(J_3-J_1\right) x_6 x_2+u_{n 2}\right]=-\beta_2 s_2$               (41)

$\dot{x}_6=\frac{1}{J_3}\left[\left(J_1-J_2\right) x_4 x_2+u_{n 3}\right]=-\beta_3 s_3$               (42)

It can be noticed that all control systems are nominal model systems with nominal control $u_{n i}$ which can be designed to satisfy the desired performance for all $(t \geq 0)$.

For the discontinuous control, it’s the same of CSMC and it’s given by:

$\mathrm{u}_{\mathrm{si}}=-\mathrm{k} \operatorname{sign}(\sigma)$               (43)

Substituting in Eq. (39)

$\dot{\sigma}_i=-k \operatorname{sign}(\sigma)+d_i$               (44)

To obtain k for globally symptomatically stability, the Lyapunov candidate function:

$\mathrm{V}=\frac{1}{2} \sigma^2>0$              (45)

$\dot{\mathrm{V}}=\sigma \dot{\sigma}=\sigma(-\mathrm{ksign}(\sigma)+\mathrm{d})$              (46)

$\begin{gathered}\dot{\mathrm{V}}_{\mathrm{i}} \leq-\left|\sigma_{\mathrm{i}}\right|\left(\mathrm{k}_{\mathrm{i}}-\left|\mathrm{d}_{\mathrm{i}}\right|\right) \\ \dot{\mathrm{V}}_{\mathrm{i}} \leq 0, \text { when }\left(\mathrm{k}_{\mathrm{i}}-\left|\mathrm{d}_{\mathrm{i}}\right|\right)>0 \\ \left|\mathrm{d}_{\mathrm{i}}\right| \leq \mathrm{d}_{\max } \\ \left(\mathrm{k}_{\mathrm{i}}-\mathrm{d}_{\max }\right)>0\end{gathered}$

For negative definite, $\mathrm{k}_{\mathrm{i}}>\mathrm{d}_{\max }$.

3.2 Adaptive SMC based on Barrier Function (BFASMC)

This approach is introduced by Obeid et al. [26], in this method, and based on the magnitude of sliding manifold which is affected by disturbances and uncertainties, the increasing and decreasing of the gain is allowed. Unlike the CSMC and ISMC, the BFASMC, the knowledge of the maximum value of uncertain parameters and external disturbances is not requested in advance.

3.2.1 BF

Definition [26]: Assume that there is $\epsilon>0$ can be selected and reformed.

BF is defined as even continuous function such as:

$B: X \in[-\epsilon, \epsilon] \rightarrow B(X) \in[N, \infty]$ and strictly increasing on $[0, \epsilon]$

$\lim _{|x| \rightarrow \varepsilon} B(x)=+\infty$

$\mathrm{B}(\mathrm{x})$ has a unique minimum at zero and $\mathrm{B}(0)=\mathrm{N} \geq 0$

There are two types of BF:

1- Positive definite BF (PBF), $B_p=\frac{\epsilon N}{\epsilon-|x|}$, such as $B_p(0)=N>0$

2-Positive semi definite BF (PSBF), $\mathrm{B}_{\mathrm{ps}}(\mathrm{x})=\frac{|\mathrm{x}|}{\mathrm{\epsilon}-|\mathrm{x}|}$, such as $\mathrm{B}_{\mathrm{PS}}(0)=0$

3.2.2 Adaptation algorithm

The adaptive algorithm of this method is partitioned into two stages or phases [26]. The first phase, the gain will increase to a pre-determined time $t^*$, at which time the gain value becomes equal to the maximum value of the disturbance to ensure a finite time convergence of sliding variable to neighbourhood of zero. Then it turns into the second phase, which is the BF.

Phase 1 (Constant gain Adaptation $\mathrm{t}<\mathrm{t}^*$)

Let’s consider the control law as:

$\mathrm{u}_{\mathrm{si}}=-\mathrm{k}_{\mathrm{ci}} \operatorname{sign}(\mathrm{s})$

and the adaptive gain law as:

$\mathrm{k}_{\mathrm{ci}}=\mathrm{k}_{\mathrm{ci}}+\alpha\left|\mathrm{s}_{\mathrm{i}}\right|$                        (47)

$\mathrm{k}_{\mathrm{ci}}$ is directly increasing to $\left|s_i\right|$ to ensure rapid initial adaptation, and $\alpha$ is the adaptation rate.

Phase $2\left(\right.$BF gain $\left.t \geq t^*\right)$

A positive semi-definite BF will be used in this paper

$k_{c i}=\frac{\left|s_i\right|}{\epsilon-\left|s_i\right|} \operatorname{PSDBF}$                        (48)

$k_{c i}$ adapt to counteract disturbance and ensures $s_i$ to stay in $\left(\left|s_i\right|<\epsilon\right)$.

When $s$ approaches zero and $\epsilon>0$ them exist $t^*$ which is the smallest root of inequality $|\mathrm{s}(\mathrm{t})| \leq \frac{\epsilon}{2}$ such that for all $t \geq t^*$, the inequality the $|s|<\epsilon$ is satisfied [52].

3.3 The proposed BF based Integral SMC (BFISMC)

In an integral sliding controller, unlike CSMC, there is no reaching phase , but there is still a need to know the maximum value of the disturbance in advance. On the contrary, when using BFASM the advanced knowledge of the ultimate limit of the disturbance is not required, but at the same time there is a time to reach the region close to zero (reaching phase). In contrast, the proposed method, illustrated in Figure 2, provides a solution by utilizing the ISMC with the use of the BF as explained below:

From Eq. (43) the discontinuous control law for ISMC:

$\mathrm{u}_{\mathrm{si}}=-\operatorname{sign}(\sigma)$

Using positive semi definite continuous BF as a controller gain such as:

$\mathrm{u}_{\mathrm{si}}=-\frac{\left|\sigma_{\mathrm{i}}\right|}{\epsilon-\left|\sigma_{\mathrm{i}}\right|} \operatorname{sign}\left(\sigma_{\mathrm{i}}\right)=-\mathrm{k}_{\mathrm{b}} \frac{\sigma_{\mathrm{i}}}{\epsilon-\left|\sigma_{\mathrm{i}}\right|}$                        (49)

where, $1 \gg \epsilon>0$ represents the boundary layer thickness and $\mathrm{k}_{\mathrm{b}}>0$ is the gain of the controller that could be chosen for steady state error regulation and robustness improvement. The advantages of the proposed approach could be outlined as follows:

- No knowledge of the maximum value of the disturbance is to be known in advance.

- No reaching phase.

- Chattering is eliminated or (attenuated for smaller $\epsilon$), and this is due the continuity feature of the BF within boundary layer $|s|<\epsilon$.

Calling Eqs. (45) and (46) of Lyapunov candidate function:

$\begin{gathered}\mathrm{V}=\frac{1}{2} \sigma^2 \\ \dot{\mathrm{~V}}=\sigma \dot{\sigma}=\sigma(-\operatorname{sign}(\sigma)+\mathrm{d})\end{gathered}$  

$\dot{\mathrm{V}}=\sigma \dot{\sigma}=\sigma\left(-\mathrm{k}_{\mathrm{b}} \frac{\sigma_{\mathrm{i}}}{\epsilon-\left|\sigma_{\mathrm{i}}\right|}+\mathrm{d}\right)$           (50)

$\dot{\mathrm{V}} \leq-|\sigma|\left(\mathrm{k}_{\mathrm{b}} \frac{\left|\sigma_{\mathrm{i}}\right|}{\epsilon-\left|\sigma_{\mathrm{i}}\right|}-|\mathrm{d}|\right)$           (51)

For negative definiteness

$\left(k_b \frac{\left|\sigma_i\right|}{\epsilon-\left|\sigma_i\right|}-|d|\right)>0$           (52)

let $|\mathrm{d}| \leq \mathrm{d}_{\max }$

$\frac{\left|\sigma_i\right|}{\epsilon-\left|\sigma_i\right|}>d_{\max }$            (53)

$\mathrm{k}_{\mathrm{b}}\left|\sigma_{\mathrm{i}}\right|>\left(\epsilon-\left|\sigma_{\mathrm{i}}\right|\right) \mathrm{d}_{\max }$           (54)

$\mathrm{k}_{\mathrm{b}}\left|\sigma_{\mathrm{i}}\right|+\mathrm{d}_{\max }\left|\sigma_{\mathrm{i}}\right|>\mathrm{d}_{\max } \epsilon$           (55)

$\left|\sigma_{\mathrm{i}}\right|\left(\mathrm{k}_{\mathrm{b}}+\mathrm{d}_{\max }\right)>\mathrm{d}_{\max } \epsilon$           (56)

$\left|\sigma_{\mathrm{i}}\right|>\frac{\mathrm{d}_{\max } \epsilon}{\mathrm{k}_{\mathrm{b}}+\mathrm{d}_{\max }} \rightarrow \dot{\mathrm{V}}<0$           (57)

Which means the ultimate bound of $\left|\sigma_{\mathrm{i}}\right|$ is less than $\frac{d_{\text {max}}{ }^{\mathrm{\epsilon}}}{k_b+d_{\text {max}}}$ which is less than $\epsilon$, [51, 53].

Figure 2. Block diagram of BFISMC

This paper presents robust BF based Integral Sliding Mode Control (BFISMC) strategy for Attitude controlling of rigid spacecraft, the proposed method has no reaching phase because of the use of the ISMC, which is known for this property, and no need for prior knowledge of disturbance limits, making it more practical and effective in real-world applications. This method ensures robustness and eliminates chattering under varying conditions by using BF to regulate the gain smoothly. Moreover, this approach is effective in reducing the effect of disturbance, which makes the control system able to behave as nominal full order system from the beginning. In addition, the stability has been proven by Lyapunov criteria, which provides strong theoretical guarantees for the effectiveness of this method. A comparative simulation has been performed, explained that BFISMC outperforms ISMC and BFASMC in terms of adaptability, smoothness and control effort, whereas robust stability and performance were achieved with a steady state error of 0.0026 (rad) and control effort of 0.33 (nm).