Prescribed Performance Spacecraft Attitude Control with Multiple Convergence Rates

Abstract

This paper proposes a prescribed performance control policy for spacecraft attitude tracking control. The designed controller employs a kind of performance envelope function that can configure the convergence rate arbitrarily. Moreover, the controller design is based on the barrier Lyapunov function and indirect adaptive methods, which can suppress the fluctuation of external disturbances to obtain lower steady-state errors. Compared to prior studies on spacecraft attitude tracking, the proposed policy can arbitrarily configure the system convergence performance to achieve performance trade-offs in different scenarios. Moreover, the barrier Lyapunov function design and adaptive methods can achieve low steady-state errors with low computational resource consumption. Simulations that verify its effectiveness and superiority are provided herein.

1. Introduction

1.1. Motivations

For spacecraft control, a large number of control methods have been used for spacecraft attitude and orbit control. For example, for spacecraft attitude control, there are adaptive control [1], event-triggered control [2,3], and sliding mode control [4] methods; for spacecraft formation control, there are nominal steering control [5] and event-triggered control [6] methods. However, these methods are not specifically designed for spacecraft state convergence performance. In many studies on spacecraft control [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18], the prescribed performance control (PPC) policy in [19] has been introduced to constrain the spacecraft attitude state convergence performance [10,11,12,13,14,15,16,17,18], including transient behavior and steady-state performance indicators. This method can constrain the convergence performance of the spacecraft’s attitude/orbit control systems, and has potential application value in practical engineering scenarios such as space station rendezvous and docking, satellite attitude tracking, and rocket attitude stabilization. However, most existing studies on PPC-based spacecraft control employ the exponential or hyperbolic cotangent performance envelope functions, which have fixed convergence rates and cannot be freely adjusted. Thus, it has been impossible to fully meet the different requirements for control performance across various scenarios. Moreover, the initial PPC method is usually sensitive to external disturbances, and its robustness to state evolution constraints has potential to be improved. This paper focuses on the prescribed performance convergence rate and robustness of spacecraft attitude control to design PPC policies. Its main concerns are as follows:

• Since most existing PPC studies on spacecraft attitude control are designed with fixed convergence rates, more general performance envelope functions are needed to achieve multiple convergence rates to meet more flexible control objectives and performance requirements.

• For the traditional static PPC scheme, the controller’s robustness against uncertainties is insufficient, which results in systems’ susceptibility to divergence due to external disturbances [20]. Thus, it is important to design PPC policies that can improve systems’ robustness and produce better steady-state performance.

In order to solve the aforementioned issues, this article intends to provide a more general PPC solution that enables the configuration of multiple convergence rates and steady-state performance improvements for state convergence.

1.2. Related Works

The PPC scheme was demonstrated in [19], which focuses on MIMO nonlinear systems, and was then extended to spacecraft control [10,11,12,13,14,15,16,17,18]. In [10], the authors provided learning-based PPC solutions for multiple spacecraft coordinated control. Meanwhile, ref. [11] provided PPC policies with observers for spacecraft attitude control systems. Considering systems’ states convergence performance, refs. [12,13] investigated an appointed-time PPC method. In [14], the authors explored a neural network adaptive finite-time PPC strategy with fault-tolerant ability. To achieve superior steady-state performance, Hu et al. [15] provided a PPC policy with high convergence accuracy based on barrier Lyapunov function (BLF) design for spacecraft attitude tracking fault-tolerant control. In [16,17], Shao et al. extended this PPC policy to input saturation scenarios and 6-DOF scenarios considering relative position tracking. Liu et al. [18] investigated a PPC method with fitted high-order polynomial performance envelope functions to achieve appointed-time fault-tolerant attitude tracking control. These results have partially taken into account the abovementioned issues. However, their performance envelope functions are characterized by exponential or fast finite-time convergence rates but are not suitable for scenarios where the convergence rate of state evolution needs to be adjusted. Therefore, their applicability is insufficient for different convergence performance requirements across distinct scenarios. Furthermore, external disturbances affect PPC controllers’ state constraint capability. To further suppress the fluctuation of external disturbances to obtain better steady-state performance, the adaptive PPC controllers with BLFs have been applied in spacecraft attitude control [15,16,17]. However, these adaptive methods require regression fitting to fit external disturbances, which relies on computational power consumption and is incompatible with online computing on low-power embedded computers. Thus, designing a PPC policy for spacecraft attitude control that can achieve multiple convergence rates freely, high steady-state performance, robustness to external disturbances, and less computational resource occupancy is an important challenge.

1.3. Contributions

The paper’s main contributions are as follows:

• This paper provides a general PPC method with adjustable convergence rate properties for spacecraft attitude tracking systems, in which the designed controller can arbitrarily appoint multiple state convergence rates to constrain tracking errors. Compared with the existing results on spacecraft attitude PPC policies [10,11,12,13,14,15,16,17,18], the proposed policy has greater applicability under different convergence performance requirements.

• The proposed PPC design method is based on BLF and indirect adaptive methods. It enhances robustness against external disturbances to obtain better steady-state performance. Compared with traditional PPC methods [10,11,12,13,19], the investigated PPC policy can achieve higher convergence accuracy. Moreover, compared with the existing adaptive PPC methods [10,15,16,21], the employed indirect adaptive method is superior in terms of computing resource occupation and design complexity, making it suitable for online implementation in embedded systems.

Notations: Let $\parallel ·\parallel$ represent the Euclidean vector norm or the induced matrix norm. Let $B^{\top }$ represent the matrix or vector B’s transposition. Let ${\mathit{r}}^{\times }\in {\mathbb{R}}^{3\times 3}$ represent a skew-symmetric matrix of vector $\mathit{r}$ = $[r_1$ $r_2$ $r_3{]}^{\top }$. Let $\eta \left(t\right):{\mathbb{R}}^N\to {\mathbb{R}}^N$ be a measurable function. Let ${\parallel \eta \left(t\right)\parallel }_{\infty }$ = $\mathrm{ess}sup\parallel \eta \left(t\right)\parallel$ be its function $\infty -$norm. Here, $\mathrm{ess}sup\parallel \eta \left(t\right)\parallel$ is provided to represent $\eta \left(t\right)$’s essential supremum. ${\eta \left(t\right)|}_{t\ge 0}\in {\mathcal{L}}^{\infty }[0,+\infty )$ if and only if ${\parallel \eta \left(t\right)\parallel }_{\infty }$ < ∞. For a measurable function $\beta (·):{\mathbb{R}}_{t\ge 0}$→${\mathbb{R}}_{t\ge 0}$, if it satisfies the properties of continuous, strictly increasing, having an initial value of 0 and $\underset{\varphi \to +\infty }{lim}\beta \left(\varphi \right)\to +\infty$, it is considered as a class ${\mathcal{K}}_{\infty }$ function [22].

2. Problem Formulation

2.1. Spacecraft Attitude Dynamics

Let $\mathcal{I}$ and $\mathcal{B}$ denote the inertial frame and the body-fixed frame, respectively. Along three coordinate axes, the assembled actuators can provide control torques. The dynamics of the attitude tracking system are given by

(1)$\begin{array}{cc}\hfill & {\dot{\mathit{q}}}_e=0.5\left({\mathit{q}}_e^{\times }+q_{e0}{\mathit{I}}_3\right){\mathit{\omega }}_e,\hfill \end{array}$

(2)$\begin{array}{cc}\hfill & {\dot{q}}_{e0}=-0.5{\mathit{q}}_e^{\top }{\mathit{\omega }}_e,\hfill \end{array}$

(3)$\begin{array}{cc}\hfill & \mathit{J}{\dot{\mathit{\omega }}}_e=-{\mathit{\omega }}^{\times }\mathit{J}\mathit{\omega }+\mathit{J}\left({\mathit{\omega }}_e^{\times }\mathit{C}{\mathit{\omega }}_d-\mathit{C}{\dot{\mathit{\omega }}}_d\right)+\mathit{u}+\mathit{d},\hfill \end{array}$

where $({\mathit{q}}_e,q_{e0})\in {\mathbb{R}}^3\times \mathbb{R}$, $\mathit{\omega }\in {\mathbb{R}}^3$, ${\mathit{\omega }}_d\in {\mathbb{R}}^3$, $\mathit{J}\in {\mathbb{R}}^{3\times 3}$, $\mathit{C}\in SO\left(3\right)$, $\mathit{u}\in {\mathbb{R}}^3$, and $\mathit{d}\in {\mathbb{R}}^3$ denote the unit quaternion representing the attitude tracking error quaternion in $\mathcal{B}$ with respect to $\mathcal{I}$, the angular velocity of the spacecraft expressed in $\mathcal{B}$, the desired angular velocity, the inertia matrix, the direction cosine matrix, the control input, and the external disturbances, respectively. All time variables ${\mathit{q}}_e\left(t\right),q_{e0}\left(t\right),{\mathit{\omega }}_e,\mathit{C}\left(t\right),{\mathit{\omega }}_d\left(t\right),\mathit{u}\left(t\right),\mathit{d}\left(t\right)$ are abbreviated to the forms without t. Suppose that $\mathit{J}$ is positive definite and symmetric, which implies that ${\mathit{J}}^{\top }=\mathit{J}$. Here, $\mathit{C}=\left(q_{e0}^2-{\mathit{q}}_e^T{\mathit{q}}_e\right){\mathit{I}}_3+2{\mathit{q}}_e{\mathit{q}}_e^T-2q_{e0}{\mathit{q}}_e^{\times }$, ${\mathit{q}}_e^T{\mathit{q}}_e+q_{e0}^2=1$, ${\mathit{q}}_e$ = $q_{d0}\mathit{q}-q_0{\mathit{q}}_d+{\mathit{q}}^{\times }{\mathit{q}}_d$, $q_{e0}$ = $q_{d0}{q}_0+{\mathit{q}}_d^{\times }\mathit{q}$, and ${\mathit{\omega }}_e=\mathit{\omega }-\mathit{C}{\omega }_d$, where $({\mathit{q}}_d,q_{d0})\in {\mathbb{R}}^3\times \mathbb{R}$ represents the desired attitude angle quaternion. Suppose that there exist positive scalars $J_{max}$, $J_{min}$, and $d_{max}$ such that $0<J_{min}{\mathit{y}}^{\top }\mathit{y}\le {\mathit{y}}^{\top }\mathit{J}\mathit{y}\le J_{max}{\mathit{y}}^{\top }\mathit{y}$ for any $\mathit{y}\in {\mathbb{R}}^3$. The external disturbance vector $\mathit{d}$ satisfies $\parallel \mathit{d}\parallel \le d_{max}$. Suppose $\parallel {\mathit{\omega }}_d\parallel \le {\varpi }_1$ and $\parallel d{\omega }_d/dt\parallel \le {\varpi }_2$, with unknown bounded constants ${\varpi }_1>0$ and ${\varpi }_2>0$.

2.2. Prescribed Performance Control

For the above dynamics of spacecraft attitude tracking, this paper designs the controller $\mathit{u}$ to achieve the performance constraints on the spacecraft attitude states. In order to show the superiority of the studied policy more intuitively, this paper will constrain the combined state of spacecraft attitude angular velocity and attitude angular position. The performance envelopes are designed for the combined states. Let $\mathit{s}\left(t\right)={\mathit{\omega }}_e+\kappa {\mathit{q}}_e$, abbreviated as $\mathit{s}$, be a filtered error variable vector with $\kappa >0$. For $i=1,2,3$, let each

(4)$\begin{array}{c}\hfill P_{li}\left(t\right)<s_i\left(t\right)<P_{ui}\left(t\right)\end{array}$

hold with $P_{ui}\left(t\right)$ = $(l_{si}+(1-l_{si})l_{oi}){\eta }_i\left(t\right)$ and $P_{li}\left(t\right)$ = $(l_{si}-1-l_{si}{l}_{oi}){\eta }_i\left(t\right)$, where $\mathit{s}$ $=[s_1$ $s_2$ $s_3{]}^{\top }$. The provided parameters $0<l_{oi}\le 1$, $l_{si}=1$ if $s_i\left(0\right)\ge 0$, and $l_{si}=0$ if $s_i\left(0\right)<0$. The performance functions ${\eta }_i\left(t\right)$, $i=1,2,3$, satisfy

(5)$\begin{array}{c}\hfill {\dot{\eta }}_i\left(t\right)=-{\alpha }_i\left({\eta }_i\left(t\right)-{\eta }_{\infty }^i\right),{\eta }_i\left(0\right)={\eta }_0^i>0.\end{array}$

where ${\eta }_i\left(t\right)$ is the argument of ${\alpha }_i(·)$. Let ${\alpha }_i(·)$ be a locally Lipschitz continuous ${\mathcal{K}}_{\infty }$ function, where ${\eta }_i\left(0\right)={\eta }_0^i>0$ represents the initial value and ${\eta }_{\infty }^i>0$ represents the final value. Moreover, the properties of each measurable function ${\eta }_i\left(t\right)$ are provided by Lemma 1.

Lemma 1

([23]). The function ${\eta }_i\left(t\right):\mathbb{R}\to \mathbb{R}$ described by (5) has the properties: (1) ${\eta }_i\left(t\right)\in {\mathcal{L}}^{\infty }[0,+\infty )$; and (2) $\underset{t\to \infty }{lim}{\eta }_i\left(t\right)={\eta }_{\infty }^i$ holds.

3. Main Results

3.1. Adaptive PPC Controller Design

For $i=1,2,3$, let each element of the normalized attitude tracking error vector $\widehat{\mathit{s}}$ = $[{\widehat{s}}_1\left(t\right)$ ${\widehat{s}}_2\left(t\right)$ ${\widehat{s}}_3{\left(t\right)]}^{\top }$ be

(6)$\begin{array}{c}\hfill {\widehat{s}}_i\left(t\right)=\frac{2s_i\left(t\right)-\left(P_{ui}\left(t\right)+P_{li}\left(t\right)\right)}{P_{ui}\left(t\right)-P_{li}\left(t\right)}.\end{array}$

Here, if one has all constraints $|{\widehat{s}}_i\left(t\right)|<1$, $i=1,2,3$, this can lead to the attitude tracking error vector $\mathit{s}$ being constrained within the regions described by (4). Let ${\mathcal{D}}_{\widehat{s}}$ = $\{\widehat{\mathit{s}}\in {\mathbb{R}}^3:-1<{\widehat{s}}_i<1,i=1,2,3\}$ be the prescribed performance set.

Calculating ${\widehat{s}}_i\left(t\right)$’s derivative and abbreviating all time variables to the forms without t, one has

(7)$\begin{array}{c}\hfill {\dot{\widehat{s}}}_i=\frac{2{\dot{s}}_i-\left({\dot{P}}_{ui}+{\dot{P}}_{li}\right)-{\widehat{s}}_i\left({\dot{P}}_{ui}-{\dot{P}}_{li}\right)}{P_{ui}-P_{li}}\end{array}$

and $\dot{\widehat{\mathit{s}}}$ = $\mathbf{\Lambda }(2\dot{\mathit{s}}-\mathit{p}-\mathcal{P}\widehat{\mathit{s}})$, where $\mathbf{\Lambda }={\mathrm{diag}}_{i\in \{1,2,3\}}(1/(P_{ui}-P_{li}))$, $\mathcal{P}$ = ${\mathrm{diag}}_{i\in \{1,2,3\}}$ $({\dot{P}}_{ui}-{\dot{P}}_{li})$, and $\mathit{p}$ = $[p_1$ $p_2$ $p_3{]}^{\top }$ with $p_i$ = ${\dot{P}}_{ui}+{\dot{P}}_{li}$. Construct a symmetric barrier Lyapunov function as follows:

(8)$\begin{array}{c}\hfill V_1\left(t\right)=\frac{c_0}{2}log\left(\prod _{i=1}^3\frac{1}{1-{\widehat{s}}_i^2}\right),\end{array}$

where $0<c_0\le 1/2$. The time derivative of $V_1\left(t\right)$ is given by

(9)$\begin{array}{c}\hfill {\dot{V}}_1\left(t\right)=-c_0{\widehat{\mathit{s}}}^{\top }\mathbf{\Gamma }\mathbf{\Lambda }\mathcal{P}\widehat{\mathit{s}}+c_0{\widehat{\mathit{s}}}^{\top }\mathbf{\Gamma }\mathbf{\Lambda }\left(2\dot{\mathit{s}}-\mathit{p}\right)\end{array}$

with $\mathbf{\Gamma }$ = ${\mathrm{diag}}_{i\in \{1,2,3\}}\left(1/(1-{\widehat{s}}_i^2)\right)$.

Next, an adaptive law is provided. By (1)–(3), one has

(10)$\begin{array}{c}\hfill \mathit{J}\dot{\mathit{s}}=\mathit{M}+\mathit{u}\end{array}$

with $\mathit{M}$ = $-{\mathit{\omega }}^{\times }\mathit{J}\mathit{\omega }$+$\mathit{J}\left({\mathit{\omega }}_e^{\times }\mathit{C}{\mathit{\omega }}_d-\mathit{C}{\dot{\mathit{\omega }}}_d\right)$+$0.5\kappa \mathit{J}({\mathit{q}}_e^{\times }$+$q_{e0}{\mathit{I}}_3){\mathit{\omega }}_e$+$\mathit{d}$. Note that the assumptions related to the variables in (1)–(3) can lead to $\parallel \left({\mathit{q}}_e^{\times }+q_{e0}{\mathit{I}}_3\right){\mathit{\omega }}_e\parallel$ = $\parallel \left({\mathit{q}}_e^{\times }+q_{e0}{\mathit{I}}_3\right)(\mathit{\omega }-\mathit{C}{\mathit{\omega }}_d)\parallel$≤$\parallel \mathit{\omega }\parallel +b_{\omega }$ for an unknown nonnegative scalar $b_{\omega }$, which implies that $\parallel \mathit{M}\parallel$≤$b_0$+$b_1\parallel \mathit{\omega }\parallel$+$b_2{\parallel \mathit{\omega }\parallel }^2$≤$b\Omega$, where the unknown parameters $b,b_0,b_1,b_2\ge 0$ and $\Omega$ = $1+\parallel \mathit{\omega }\parallel +{\parallel \mathit{\omega }\parallel }^2$. Let $\widehat{b}\left(t\right)$ be the estimated value of b. The adaptive law is given by

(11)$\left\{\begin{array}{c}\dot{\widehat{b}}\left(t\right)=-{\delta }_1\widehat{b}\left(t\right)+\frac{{\delta }_2\Omega {\parallel {\mathit{J}}^{-1}\mathbf{\Lambda }\mathbf{\Gamma }\widehat{\mathit{s}}\parallel }^2}{\parallel {\mathit{J}}^{-1}\mathbf{\Lambda }\mathbf{\Gamma }\widehat{\mathit{s}}\parallel +\varsigma },\hfill \\ \varsigma =\frac{\nu }{1+\Omega },\hfill \end{array}\right.$

where the scalars ${\delta }_1,{\delta }_2,\nu >0$ must be designed. Construct the adaptive estimation Lyapunov function as follows:

(12)$\begin{array}{c}\hfill V_2\left(t\right)=\frac{{\left(b-\widehat{b}\left(t\right)\right)}^2}{2{\delta }_2}.\end{array}$

Then, the overall Lyapunov function $V\left(t\right)$ = $V_1\left(t\right)+V_2\left(t\right)$ and its time derivative satisfies

(13)$\begin{array}{cc}\hfill \dot{V}\left(t\right)=& -c_0{\widehat{\mathit{s}}}^{\top }\mathbf{\Gamma }\mathbf{\Lambda }\mathcal{P}\widehat{\mathit{s}}+2c_0{\widehat{\mathit{s}}}^{\top }\mathbf{\Gamma }\mathbf{\Lambda }{\mathit{J}}^{-1}(\mathit{M}+\mathit{u})-c_0{\widehat{\mathit{s}}}^{\top }\mathbf{\Gamma }\mathbf{\Lambda }\mathit{p}-\frac{\left(b-\widehat{b}\left(t\right)\right)\dot{\widehat{b}}\left(t\right)}{{\delta }_2}\hfill \\ \hfill \le & -c_0{\widehat{\mathit{s}}}^{\top }\mathbf{\Gamma }\mathbf{\Lambda }\mathcal{P}\widehat{\mathit{s}}+2c_{0}b\Omega \parallel {\mathit{J}}^{-1}\mathbf{\Lambda }\mathbf{\Gamma }\widehat{\mathit{s}}\parallel +2c_0{\widehat{\mathit{s}}}^{\top }\mathbf{\Gamma }\mathbf{\Lambda }{\mathit{J}}^{-1}\mathit{u}-c_0{\widehat{\mathit{s}}}^{\top }\mathbf{\Gamma }\mathbf{\Lambda }\mathit{p}-\frac{{\delta }_1}{2{\delta }_2}{\left(b-\widehat{b}\left(t\right)\right)}^2+\frac{{\delta }_1{b}^2}{2{\delta }_2}\hfill \\ & -\frac{\left(b-\widehat{b}\left(t\right)\right)\Omega {\parallel {\mathit{J}}^{-1}\mathbf{\Lambda }\mathbf{\Gamma }\widehat{\mathit{s}}\parallel }^2}{\parallel {\mathit{J}}^{-1}\mathbf{\Lambda }\mathbf{\Gamma }\widehat{\mathit{s}}\parallel +\varsigma }.\hfill \end{array}$

Design the controller $\mathit{u}$ as follows:

(14)$\left\{\begin{array}{c}\mathit{u}=-\left({\varrho }_0+{\varrho }_1\left(t\right)\right){\mathit{J}}^{-1}\mathbf{\Lambda }\mathbf{\Gamma }\widehat{\mathit{s}}+0.5\mathit{J}\mathit{p}\hfill \\ {\varrho }_1\left(t\right)=\frac{\widehat{b}\left(t\right)\Omega }{\parallel {\mathit{J}}^{-1}\mathbf{\Lambda }\mathbf{\Gamma }\widehat{\mathit{s}}\parallel +\varsigma }\hfill \end{array}\right.$

where ${\varrho }_0>0$ and ${\varrho }_1\left(t\right)$ is a variable determined by the above equation. Substituting $\mathit{u}$ into the inequality (13), one can derive

(15)$\begin{array}{cc}\hfill \dot{V}\left(t\right)\le & -c_0{\widehat{\mathit{s}}}^{\top }\mathbf{\Gamma }\mathbf{\Lambda }\mathcal{P}\widehat{\mathit{s}}+2c_{0}b\Omega \parallel {\mathit{J}}^{-1}\mathbf{\Lambda }\mathbf{\Gamma }\widehat{\mathit{s}}\parallel -2c_0\left({\varrho }_0+\frac{\widehat{b}\left(t\right)\Omega }{\parallel {\mathit{J}}^{-1}\mathbf{\Lambda }\mathbf{\Gamma }\widehat{\mathit{s}}\parallel +\varsigma }\right){\widehat{\mathit{s}}}^{\top }\mathbf{\Gamma }\mathbf{\Lambda }{\mathit{J}}^{-2}\mathbf{\Lambda }\mathbf{\Gamma }\widehat{\mathit{s}}+\frac{{\delta }_1{b}^2}{2{\delta }_2}\hfill \\ & -\frac{{\delta }_1}{2{\delta }_2}{\left(b-\widehat{b}\left(t\right)\right)}^2-\frac{\left(b-\widehat{b}\left(t\right)\right)\Omega {\parallel {\mathit{J}}^{-1}\mathbf{\Lambda }\mathbf{\Gamma }\widehat{\mathit{s}}\parallel }^2}{\parallel {\mathit{J}}^{-1}\mathbf{\Lambda }\mathbf{\Gamma }\widehat{\mathit{s}}\parallel +\varsigma }\hfill \\ \hfill \le & -c_0{\widehat{\mathit{s}}}^{\top }\mathbf{\Gamma }\mathbf{\Lambda }\mathcal{P}\widehat{\mathit{s}}-2c_0{\varrho }_0{\widehat{\mathit{s}}}^{\top }\mathbf{\Gamma }\mathbf{\Lambda }{\mathit{J}}^{-2}\mathbf{\Lambda }\mathbf{\Gamma }\widehat{\mathit{s}}+\frac{{\delta }_1{b}^2}{2{\delta }_2}+\frac{2c_{0}b\Omega {\parallel {\mathit{J}}^{-1}\mathbf{\Lambda }\mathbf{\Gamma }\widehat{\mathit{s}}\parallel }^2}{\parallel {\mathit{J}}^{-1}\mathbf{\Lambda }\mathbf{\Gamma }\widehat{\mathit{s}}\parallel +\varsigma }-\frac{2c_0\widehat{b}\left(t\right)\Omega {\widehat{\mathit{s}}}^{\top }\mathbf{\Gamma }\mathbf{\Lambda }{\mathit{J}}^{-2}\mathbf{\Lambda }\mathbf{\Gamma }\widehat{\mathit{s}}}{\parallel {\mathit{J}}^{-1}\mathbf{\Lambda }\mathbf{\Gamma }\widehat{\mathit{s}}\parallel +\varsigma }\hfill \\ & +b\varsigma \Omega -\frac{\left(b-\widehat{b}\left(t\right)\right)\Omega {\parallel {\mathit{J}}^{-1}\mathbf{\Lambda }\mathbf{\Gamma }\widehat{\mathit{s}}\parallel }^2}{\parallel {\mathit{J}}^{-1}\mathbf{\Lambda }\mathbf{\Gamma }\widehat{\mathit{s}}\parallel +\varsigma }-\frac{{\delta }_1}{2{\delta }_2}{\left(b-\widehat{b}\left(t\right)\right)}^2\hfill \\ \hfill \le & -c_0{\widehat{\mathit{s}}}^{\top }\mathbf{\Gamma }\mathbf{\Lambda }\mathcal{P}\widehat{\mathit{s}}-2c_0{\varrho }_0{\widehat{\mathit{s}}}^{\top }\mathbf{\Gamma }\mathbf{\Lambda }{\mathit{J}}^{-2}\mathbf{\Lambda }\mathbf{\Gamma }\widehat{\mathit{s}}+\frac{{\delta }_1{b}^2}{2{\delta }_2}+b\nu -\frac{{\delta }_1}{2{\delta }_2}{\left(b-\widehat{b}\left(t\right)\right)}^2.\hfill \end{array}$

where $\parallel {\mathit{J}}^{-1}\mathbf{\Lambda }\mathbf{\Gamma }\widehat{\mathit{s}}\parallel /(\parallel {\mathit{J}}^{-1}\mathbf{\Lambda }\mathbf{\Gamma }\widehat{\mathit{s}}\parallel +\varsigma )$≤ 1 for $\varsigma \ge 0$ and $\Omega /(1+\Omega )$≤ 1.

3.2. Convergence Results

Theorem 1.

Consider the system (1)–(3). Moreover, consider the performance envelope function ${\eta }_i\left(t\right)$ generated by the differential Equation (5) and the corresponding prescribed performance bounds imposed by (4) with the error transformation (7). When employing the adaptive law (11), the prescribed performance controller (14), and the control input (14), the following properties hold: (1) All signals in the considered attitude tracking system are bounded; and (2) The filtered errors $s_i\left(t\right)$, $i=1,2,3$, evolve strictly within the performance envelopes presented by (4).

Proof. 

The time derivative of the Lyapunov function candidate $V\left(t\right)$ satisfies the following inequality:

(16)$\begin{array}{cc}\hfill \dot{V}\left(t\right)\le & -c_0{\widehat{\mathit{s}}}^{\top }\mathbf{\Gamma }\mathbf{\Lambda }\mathcal{P}\widehat{\mathit{s}}-2c_0{\varrho }_0{\widehat{\mathit{s}}}^{\top }\mathbf{\Gamma }\mathbf{\Lambda }{\mathit{J}}^{-2}\mathbf{\Lambda }\mathbf{\Gamma }\widehat{\mathit{s}}+\frac{{\delta }_1{b}^2}{2{\delta }_2}+b\nu -\frac{{\delta }_1}{2{\delta }_2}{\left(b-\widehat{b}\left(t\right)\right)}^2\hfill \\ \hfill \le & {\mathrm{diag}}_{i\in \{1,2,3\}}\left(\frac{c_0{\varrho }_0}{2J_{max}(s_i-P_{li})(s_i-P_{ui})}-\frac{c_0\left({\dot{P}}_{ui}-{\dot{P}}_{li}\right)}{P_{ui}-P_{li}}\right)\times {\left[\frac{{\widehat{s}}_1^2}{1-{\widehat{s}}_1^2}\frac{{\widehat{s}}_2^2}{1-{\widehat{s}}_2^2}\frac{{\widehat{s}}_3^2}{1-{\widehat{s}}_3^2}\right]}^{\top }\hfill \\ & -{\delta }_1{V}_2\left(t\right)+\frac{{\delta }_1{b}^2}{2{\delta }_2}+b\nu .\hfill \end{array}$

Since the inequality (4) holds, there exist finite unknown positive scalars ${\gamma }_i$, $i=1,2,3$, such that $-{\gamma }_i\le (s_i-P_{li})(s_i-P_{ui})<0$, thereby deducing $0.5c_0{\varrho }_0/{\gamma }_i$≤$0.5c_0{\varrho }_0/\left(J_{max}(s_i-P_{li})(s_i-P_{ui})\right)$< 0, $i=1,2,3$. According to the properties of the function ${\eta }_i\left(t\right)$ described in Lemma 1, the items $-c_0({\dot{P}}_{ui}-{\dot{P}}_{li})/(P_{ui}-P_{li})$ = $-c_0{\alpha }_i\left({\eta }_i\left(t\right)-{\eta }_{\infty }^i\right)/{\eta }_i\left(t\right)$ have monotonically decreasing properties with the minimum values $-c_0{\alpha }_i({\eta }_0^i-{\eta }_{\infty }^i)/{\eta }_0^i$ and $\underset{t\to \infty }{lim}-c_0{\alpha }_i\left({\eta }_i\left(t\right)-{\eta }_{\infty }^i\right)/{\eta }_i\left(t\right)$= 0. Therefore, a positive scalar ${\psi }_0$ always exists such that ${\psi }_0$≤${min}_{i\in \{1,2,3\}}\{c_0({\dot{P}}_{ui}-{\dot{P}}_{li})/(P_{ui}-P_{li})$−$0.5c_0{\varrho }_0/\left(J_{max}(s_i-P_{li})(s_i-P_{ui})\right)\}$, implying that

(17)$\begin{array}{cc}\hfill \dot{V}\left(t\right)& \le -{\psi }_0{V}_1\left(t\right)-{\delta }_1{V}_2\left(t\right)+\frac{{\delta }_1{b}^2}{2{\delta }_2}+b\nu \hfill \\ & \le -{\psi }_{1}V\left(t\right)+{\psi }_2\hfill \end{array}$

where ${\psi }_1$ $:=$ $min\{{\psi }_0,{\delta }_1\}>0$ and ${\psi }_2$ = ${\delta }_1{b}^2/\left(2{\delta }_2\right)+b\nu <\infty$. Therefore, we have $V\left(t\right)\in {\mathcal{L}}^{\infty }$, $V_1\left(t\right)\in {\mathcal{L}}^{\infty }$ and $V_2\left(t\right)\in {\mathcal{L}}^{\infty }$, thereby deriving $\widehat{\mathit{s}}\in {\mathcal{D}}_{\widehat{s}}$ and $\widehat{b}\left(t\right)\in {\mathcal{L}}^{\infty }$. Furthermore, we may conclude that the elements of $\mathbf{\Lambda }$, $\mathbf{\Gamma }$, and $\mathcal{P}$ are bounded, and each $s_i\left(t\right)$ evolves strictly within the state trajectory envelopes presented by (4). □

3.3. Design of Multiple Convergence Rates

The evolution trajectories of each ${\eta }_i\left(t\right)$ depends on the expression of each ${\alpha }_i(·)$. This subsection provides a design method for multiple convergence rates. Let ${\alpha }_i({\eta }_i\left(t\right)-{\eta }_{\infty }^i)$ be a power exponential function with the form ${\alpha }_i({\eta }_i\left(t\right)-{\eta }_{\infty }^i)$ = ${\gamma }_0{({\eta }_i\left(t\right)-{\eta }_{\infty }^i)}^{\delta }$, meaning that ${\dot{\eta }}_i\left(t\right)$ = $-{\gamma }_0{\left({\eta }_i\left(t\right)-{\eta }_{\infty }^i\right)}^{\delta }$ holds with the two positive scalars ${\gamma }_0$ and $\delta$. To illustrate, we present the following cases:

• With $\delta =1$, we can solve ${\eta }_i\left(t\right)$ = $\left({\eta }_i\left(t\right)-{\eta }_{\infty }^i\right)e^{-{\gamma }_{0}t}$+${\eta }_{\infty }^i$. Obviously, it has an exponential evolution rate. This setting is similar to the policies in [10,11,15,16,19].

• With $0<\delta <1$, compared with the case with $\delta =1$, the approximation rate ${\eta }_i\left(t\right)\to {\eta }_{\infty }^i$ is faster when t is smaller and slower when t is larger. This form is similar to the PPC policies in [12,13].

• With $\delta >1$, compared with the case with $\delta =1$, the evolution rate ${\eta }_i\left(t\right)\to {\eta }_{\infty }^i$ is slower when t is smaller and faster when t is larger.

Thus, the evolution rate adjustable properties of the performance constraint functions ${\eta }_i\left(t\right)$ can be achieved by arranging different locally Lipschitz continuous ${\mathcal{K}}_{\infty }$ functions ${\alpha }_i(·)$.

4. Simulations

Consider the scenario of a satellite orbiting the Earth. Assign its inertia $\mathit{J}$ = $\left[\begin{array}{ccc}20& 2& 0.9\\ 2& 17& 0.5\\ 0.9& 0.5& 15\end{array}\right]$ $\mathrm{kg}·{\mathrm{m}}^2$ and $\mathit{d}$ = $0.001\times \left[\begin{array}{c}3cos\left(\phi t\right)-6sin\left(0.3\phi t\right)+7\\ -3cos\left(\phi t\right)-6sin\left(0.3\phi t\right)+8\\ -3sin\left(\phi t\right)-6cos\left(0.3\phi t\right)+9\end{array}\right]$ $\mathrm{N}·\mathrm{m}$ with $\phi$= 2 + $\parallel \mathit{\omega }\parallel$. Such an inertia matrix is reasonable for satellites. Considering practical engineering scenarios in which satellites are affected by thin atmospheric drag, which is typically of the order of ${10}^{-4}$, such external disturbances are also reasonable. Let the initial values be $\mathit{q}\left(0\right)$ = $[-0.1$ $0.3$ ${-0.2]}^{\top }$, $q_0\left(0\right)=\sqrt{1-\mathit{q}{\left(0\right)}^{\top }\mathit{q}\left(0\right)}$ $=\sqrt{0.86}$, and $\mathit{\omega }\left(0\right)$ = $[0$ 0 ${0]}^{\top }$ rad/s. Let $q_{d0}=1$, ${\mathit{q}}_d$ = $[0$ 0 ${0]}^{\top }$ and ${\mathit{\omega }}_d$ = $0.01$×$[cos(0.1t)$ $-sin\left(0.1t\right)$ ${-cos\left(0.1t\right)]}^{\top }$ rad/s, respectively. Assign the parameters in $\mathit{s}$ be $\kappa =1$. These initial values ensure that the initial value of the combined state variable $\mathit{s}$ is within the envelope functions. Choose the control gain parameter ${\varrho }_0=12$ and adaptive parameters ${\sigma }_1=1$, ${\sigma }_2=2$, and $\mu =0.2$. Such control gain ensures that the satellite can track the target in about 10 s, which is in line with the actual engineering application scenario. In addition, the setting of the parameters in the adaptive law ensures that the adaptive gain has the same order of magnitude as the fixed control gain ${\varrho }_0$, which achieves the suppression of external disturbances. For $c,i=1,2,3$, let $l_{oi}=1$, the performance functions ${\eta }_{c,i}\left(t\right)$ satisfy each ${\dot{\eta }}_{1,i}\left(t\right)=-0.45{\left({\eta }_i^1\left(t\right)-0.025\right)}^{{\nu }_c}$. In all three simulations, let ${\nu }_1=0.2$, ${\nu }_2=1$, and ${\nu }_3=1.5$, respectively; let the initial values ${\eta }_i^c\left(0\right)=2$, for all $i,c=1,2,3$. Since the control torques in actual engineering scenarios have physical constraints, let the upper and lower bounds be $\pm 2.0$ N·m.

Figure 1 shows the simulated external disturbance trajectories and the evolutionary trajectories of the adaptive gain in these three cases. Figure 2, Figure 3 and Figure 4 present the simulation results in the cases with ${\nu }_1=0.2$, ${\nu }_2=1$, and ${\nu }_3=1.5$, respectively. In Figure 1, the simulated external disturbance amplitudes are approximately $2\times {10}^{-2}$ and the amplitude’s magnitude is close to the simulation setting in [15]. The fluctuation ranges of the states in Figure 2, Figure 3 and Figure 4 are less than $\pm 1\times {10}^{-4}$. Compared with [15], the external disturbance rejection capabilities of our policy are similar. However, its indirect adaptive method only requires differential equations to estimate unknown parameters, which is easily implemented using the four-order Runge–Kutta method on embedded digital computers. It has low resource consumption and does not require neural network calculations (as in [10,14]) or regression fitting (as in [15]). Therefore, the designed controller can achieve excellent robustness and exhibits excellent steady-state performance with low computing power consumption. Moreover, the three different performance envelope functions have different convergence rates, resulting in different convergence rates of the considered system states. With ${\nu }_1=0.2$, the PPC functions’ convergence rate is faster than the exponential function with ${\nu }_2=1$, which can constrain the system states to converge to the given envelopes faster, and better suppress the state overshoot. In the case of ${\nu }_3=1.5$, the PPC functions’ convergence rate is slower than the exponential function with ${\nu }_2=1$, which can constrain the system states to converge more slowly to the given envelopes slower, but with smaller control torques. Thus, in actual engineering scenarios, the appropriate performance envelope functions can be set according to the distinct requirements of the state tracking rates and control torques. That is, compared with existing PPC results [10,11,12,13,14,15,16,17,18,19], the proposed PPC policy in this paper offers a trade-off between different types of resource consumption. In addition, the proposed PPC method has better steady-state performance compared to the methods described in the literature [10,15]. In this paper and in references [10,15], the magnitude of the external disturbances is on the order of ${10}^{-3}$. However, the convergence accuracy of ${\mathit{q}}_e$ and ${\mathit{\omega }}_e$ is ${10}^{-3}$ (within 35∼40 s) in [10] and ${10}^{-4}$ (within 90∼100 s) in [15]. The steady-state errors in the literature [10,15] are significantly worse than ${10}^{-5}$ (within 30∼40 s, $\nu =0.2$), ${10}^{-5}$ (within 30∼40 s, $\nu =1$), and ${10}^{-4}$ (within 30∼40 s, $\nu =1.5$) in this paper.

Remark 1.

The designed controller in this paper has the potential to be employed in practical engineering scenarios. The effectiveness and superiority of the designed controller can be verified on a semi-physical experimental platform. The 3-DOF air-floated micro-vibration isolation platform outputs attitude information to the control computer. The control computer outputs control commands to assign the three-axis inertial flywheel and outputs the control torque to the 3-DOF air-floated micro-vibration isolation platform. In this process, the acquisition of attitude information of the 3-DOF air-floated platform is crucial, which corresponds to the measurement of attitude information of satellites in real space scenarios. Inspired by references [24,25], the acquisition of satellite attitude information also needs to take into account the influence of white Gaussian noise. Thus, the EKF filtering algorithm needs to be used to eliminate the influence of noise and obtain accurate attitude information.

5. Conclusions

This paper examines more general PPC policies for spacecraft attitude control. Compared with previous studies, this paper designs a novel performance envelope function with multiple adjustable convergence rate characteristics, which can achieve different performance trade-offs in various scenarios, thereby having better potential for employment in different complex practical engineering scenarios. In addition, the BLF design method and indirect adaptive law are applied to the design of the controller to better suppress the influence of external disturbances and achieve excellent robustness. The paper’s numerical simulations support these conclusions.

The shortcoming of this study is the lack of practical experiments to verify the designed controller, which will be the focus of future work. In addition, in response to the theoretical research, future work will be based on two aspects: (1) Further improvement of PPC policies to achieve adaptive changes in performance envelopes to cope with more complex actual engineering scenarios. (2) Further design 6-DOF spacecraft attitude and orbit integrated control strategies based on advanced PPC methods, and extending them to multiple spacecraft coordinated control.

Author Contributions

Conceptualization, M.G. and C.Z.; Funding acquisition, M.-Z.D. and M.G.; Investigation, X.Z. and J.W.; Methodology, C.Z., J.W. and M.G.; Project administration, M.-Z.D. and M.G.; Supervision, M.-Z.D. and C.Z.; Writing—original draft, M.-Z.D. and M.G.; Writing—review and editing, M.-Z.D., X.Z. and C.Z. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

Data are contained with article.

Conflicts of Interest

Author Xing Zhang was employed by the company China Resources Digital Co., Ltd. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Footnotes

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Figures

Figure 1. External disturbances and adaptive gain.

Figure 2. With [Forumla omitted. See PDF.].

Figure 3. With [Forumla omitted. See PDF.].

Figure 4. With [Forumla omitted. See PDF.].

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