Multirate iterative learning disturbance observer with measurement delay compensation for flexible spacecraft attitude stabilization subject to complex disturbances

Abstract

Complex disturbances with multifrequency components inevitably exist in the attitude control system of flexible spacecrafts. Multirate iterative learning disturbance observer (MILDO) is a promising solution to estimate and attenuate these disturbances much more accurately. However, measurement delay in the attitude measurement system may severely degrade the estimation performance of MILDO and even lead to instability of the control system. To suppress complex disturbances of flexible spacecrafts subject to a known measurement delay, a delay compensation-MILDO (DC-MILDO) is proposed in this study. First, an augmented model is constructed for the delayed disturbance by introducing slowly varying disturbance, multiple periodic disturbances, and the integral of the lumped disturbance as states. Then, a virtual measurement of the system is built by realigning the control torques with the delayed measurements in spacecraft attitude dynamics. According to the augmented model and the virtual measurement, a modified MILDO is designed by using multiple iterative learning structures. Based on the estimates of the modified MILDO, the lumped disturbance at current instant can be predicted by compensating time delay in the multifrequency components. Subsequently, a composite controller for flexible spacecraft is built by combining DC-MILDO with a robust controller based on Lyapunov–Krasovskii approach. Simulation results demonstrate the effectiveness of the proposed controller.

Keywords

Flexible spacecraft attitude control disturbance observer measurement delay iterative learning

1. Introduction

To accomplish complex space missions, various precision instruments are installed on spacecrafts, such as cameras, telescopes, laser transceivers, etc. Because these instruments are typically very susceptible to platform vibrations, high accuracy and high stability attitude control of spacecraft platforms are the basic requirements for them (Dennehy and Alvarez-Salazar, 2018). However, complex disturbances in the system may cause degradation of attitude control performance and even lead to instability of the system (Hu and Xiao, 2011; Qiao et al., 2019; Yan and Wu, 2019). Therefore, attenuation of complex disturbances is a primary task for spacecraft attitude controller design.

Generally, spacecrafts in orbit are affected by complex disturbances containing wide range of components in frequency. Environmental disturbances, vibrations from flexible appendages, uncertainties of inertial parameters, and output-torque errors of actuators mainly consist of slowly varying components and have been sufficiently studied (Song and Lu, 2019; Yan and Wu, 2017). Besides, there are internal periodic disturbances from equipment onboard, such as cryocoolers, drive motors, reaction wheels, etc (Liu et al., 2015). These disturbances also have significant adverse effects on the spacecraft attitude control.

To deal with complex disturbances of spacecrafts, different control schemes have been applied (Liu et al., 2012; Majumder and Patre, 2019; Zhang et al., 2014). Among them, disturbance observer-based control scheme (DOBC) is a potential solution (Chen et al., 2016; Guo and Chen, 2005; Liu et al., 2018). In DOBC, by integrating a disturbance observer with traditional controllers, the disturbance attenuation performance can be significantly improved whereas maintaining the nominal performance of baseline controllers (Back and Shim, 2008; Yang et al., 2011). However, traditional disturbance observers, such as nonlinear disturbance observer (Chen and Wen, 2018), extended disturbance observer (Li et al., 2019), and sliding mode disturbance observer (Yan and Wu, 2018) are mainly applied to estimate slowly varying disturbance. Recently, a multirate iterative learning disturbance observer (MILDO) is proposed to address multiple periodic disturbances as well as slowly varying disturbance (He and Wu, 2020). However, measurement delay is not fully considered in the observer design. It may lead to performance reduction and instability of the control system (Richard, 2003).

In practice, the acquisition of spacecraft attitude information is unavoidably affected by time delay. It typically arises from output delay of sensors, signal processing and transmission, and also in attitude determination (Bahrami and Namvar, 2015; Zhang et al., 2013; Zhu et al., 2018). The delay-robust controller design based on Lyapunov–Krasovskii approach has been adequately addressed and theoretical foundation has been provided (Ailon et al., 2003; Chunodkar and Akella, 2011). However, conservativeness of these controllers may be increased when significant disturbances exist. Alternatively, Smith predictor (García and Albertos, 2013) or communication disturbance observer Wang et al. (2018) can be applied by treating time delay as part of the disturbance. However, only slowly varying disturbance together with the small delay can be dealt with by these approaches.

Observers robust to time delay can be designed based on Lyapunov–Krasovskii theories (Cacace et al., 2010). The stability of observers can be guaranteed by choosing observer gains inside a conservative area (Qiao et al., 2019). Thus, the estimation performance is actually restricted. Besides, chain observers can be applied to reconstruct system states subject to large time delay (Kahelras et al., 2018). Nevertheless, the complexity of observer structure may not fully meet the demand of the spacecraft attitude control system with limited computing power. Due to these facts, disturbance observer for complex disturbance estimation of spacecraft subject to measurement delay still needs investigation.

For high performance attitude stabilization of a flexible spacecraft in the presence of a known measurement delay, a delay compensation-MILDO (DC-MILDO)-based robust composite controller is designed. First, an augmented model for delayed disturbance including slowly varying disturbance together with multiple periodic disturbances is constructed. The control torques and the delayed angular velocity measurements are realigned to fabricate the virtual disturbance measurement. Based on it, a modified MILDO is built to estimate the delayed disturbances. In the observer, disturbances with different frequency components are estimated separately. Then, the disturbance estimates of current instant are obtained by compensating time delay according to the disturbance frequencies. Second, DC-MILDO is combined with a robust controller to stabilize the spacecraft attitude dynamics. Furthermore, simulations are carried out to verify the effectiveness of proposed control scheme.

The main contributions of this work are described as follows

A MILDO with delay compensation, which consists of a modified MILDO and a prediction algorithm, is proposed to achieve accurate estimation of complex disturbances for flexible spacecrafts with measurement delay.

A composite controller combining the DC-MILDO with a robust controller is proposed to realize high-precision attitude stabilization for flexible spacecrafts in the presence of measurement delay and complex disturbances.

This study is organized as follows: The spacecraft dynamics is introduced in Section 2. In Section 3, the DC-MILDO is constructed and analyzed. Then, a DC-MILDO-based controller is built in Section 4. Afterward, simulation results are presented in Section 5. At last, a conclusion is given in Section 6.

2. Mathematical model

2.1. Notations

In this study, $| \cdot |$ denotes the absolute value of a scalar; $‖ \cdot ‖$ denotes the Euclidean norm of a vector or a matrix. The function $int {x}$ denotes the nearest integer of $x$ toward zero; $I_{n}$ denotes an identity matrix with the dimension of $n \times n$ ; $x^{\times}$ denotes the cross product matrix $[\begin{matrix} 0 & - x_{3} & x_{2} \\ x_{3} & 0 & - x_{1} \\ - x_{2} & x_{1} & 0 \end{matrix}]$ for vector $x = {[x_{1}, x_{2}, x_{3}]}^{T}$ .

2.2. Spacecraft attitude dynamics

Because modified Rodrigues parameter (MRP) is a singularity-free unique global attitude description, we are using MRP to describe the attitude orientation of a spacecraft with respect to the inertial frame. Then, the spacecraft kinematics is described as (Hughes, 2004)

\dot{p} = 0.25 [(1 - p^{T} p) I_{3} + 2 p^{\times} + 2 p p^{T}] ω

(1)

where the MRP attitude parameterization

p \in ℝ^{3}

is defined as

p = tan (ϕ / 4) \hat{e}

;

ϕ

is the principal rotation angle about the principal rotation axis

\hat{e}

; for simplicity, it is denoted that

F (p) = 0.25 [(1 - p^{T} p) I_{3} + 2 p^{\times} + 2 p p^{T}]

By using $ω \in ℝ^{3}$ to describe the body-frame angular velocity of a flexible spacecraft with respect to the inertial frame, the attitude dynamics of a flexible spacecraft is given as (Hughes, 2004)

J \dot{ω} = - ω^{\times} (Jω + δ^{T} \dot{η}) - δ^{T} \ddot{η} + u + d_{e}

(2)

\ddot{η} + D \dot{η} + Sη + δ \dot{ω} = 0

(3)

where

J \in ℝ^{3 \times 3}

u \in ℝ^{3}

, and

d_{e} \in ℝ^{3}

denote the inertia matrix, the control torque, and the environmental disturbance, respectively;

δ \in ℝ^{n \times 3}

and

η \in ℝ^{n \times 1}

denote the flexible coupling matrix and modal coordinate vector, respectively;

D = diag (2 ξ_{1} ζ_{1}, \dots, 2 ξ_{1} ζ_{n})

and

S = diag (ζ_{1}^{2}, \dots, ζ_{n}^{2})

denote the damping matrix and stiffness matrix, respectively;

ζ

and

ξ

denote natural frequency and associated damping ratio, respectively.

2.3. Disturbances and measurement delay

Spacecraft vibration test results show that the attitude vibrations cover a wide range of frequency components. However, the low-frequency vibrations are dominant in energy. Besides, there are several significant periodic vibrations. Because high-frequency vibrations are generally damped or suppressed by isolation equipment onboard, their energy is relatively low. As a result, the major sources of the spacecraft attitude vibrations are the slowly varying disturbances and period disturbances (Jedrich et al., 2002; Sudey and Schulman, 1984; Toyoshima et al., 2010). These disturbances can be further described according to their sources as follows.

The environmental disturbance $d_{e}$ mainly consists of gravity gradient torque, solar radiation torque, geomagnetic torque, aerodynamic torque, etc. Their frequencies are typically related to the orbiting of the spacecrafts (Ruiter et al., 2012). Thus, the slowly varying components of these disturbances are dominant. In this study, equivalent disturbance results from flexible vibrations and inertia matrix uncertainties are denoted as $d_{f} = - ω^{\times} δ^{T} \dot{η} - δ^{T} \ddot{η}$ and $d_{j} = - J_{e} \dot{ω} - ω^{\times} J_{e} ω$ , respectively, where $J_{e}$ is the unknown inertial matrix. The deviation between the control torque command $u_{c}$ and the actual actuator torque $u$ is denoted as $u_{f} = u - u_{c}$ . It is also treated as a kind of disturbance in the system (Xiao et al., 2013). Generally, the major components of these equivalent disturbances are also slowly varying considering the sampling frequency of the system. According to Ginoya et al. (2014), these slowly varying disturbances can be estimated by designing an observer with bandwidth larger than their frequency components.

Actually, there are also significant internal periodic disturbances $d_{p}$ which come from moving components of equipment onboard. According to orbit test, multiple periodic disturbances of high energy may induce attitude vibrations of spacecraft with relevant frequencies (Jedrich et al., 2002). However, they cannot be easily estimated by simply increasing the bandwidth of a typical observer. The reason is that too large observer gains would cause heavier noise problem and even affect the stability of the system. Thus, these periodic disturbances need to be molded and estimated separately.

Following Sun et al. (2017), abovementioned disturbances are lumped into a total disturbance $d$ as $d= d_{e} + d_{f} + d_{j} + u_{f} + d_{p}$ . Then, equations (2) and (3) are rewritten as

J_{0} \dot{ω} = - ω^{\times} J_{0} ω + u_{c} + d

(4)

where

J_{0} = J - J_{e}

is the known inertial matrix.

Based on the analysis above, the following assumption is given for the lumped disturbance $d$ .

Assumption 1

The lumped disturbance $d$ is a combination of slowly varying component $d_{s}$ , periodic disturbances $d_{i}$ with known fundamental frequencies, and unmodeled disturbance $δ_{d}$ . The sampling frequency of the system is much larger than the frequency component of $d_{s}$ . $‖ δ_{d} ‖ \leq δ_{d}$ , $‖ {\dot{d}}_{s} ‖ \leq v_{d}$ ; $δ_{d}$ and $v_{d}$ are small positive constants.

Assumption 2

There is no control over the initial condition time interval, that is, $u_{c} (t) = O_{3 \times 1}$ and t ∈ (−T, 0).

In addition, because the attitude measurement is affected by the quality of sensors, signal transmission, and attitude determination algorithm utilized in the system, the obtained attitude information is generally time delayed. When the angular velocity and attitude information are aligned to be synchronous in the attitude determination system, the attitude measurement output is described as (Chunodkar and Akella, 2011)

{\begin{cases} ω_{m} = ω (t - T) \\ p_{m} = p (t - T) \end{cases}

(5)

where

T

is the delay time.

Because the delay time $T$ can be obtained by ground test or on-orbit delay identification, an assumption is given as follows.

Assumption 3

Time delay $T$ is assumed as a known constant.

3. MILDO with delay compensation

To obtain accurate estimates of the abovementioned disturbances in the presence of measurement delay, a DC-MILDO is built in this section.

3.1. Delayed disturbance model

According to Assumption 1, the lumped disturbance $d (t)$ can be expressed as follows

d (t) = d_{s} (t) + \sum_{i = 1}^{r} d_{i} (t) + δ_{d} (t), i = 1,2, \dots, r

(6)

where the fundamental frequency of $d_{i} (t)$ is $f_{i}$ , $i = 1,2, \dots, r$ .

For periodic disturbances $d_{i} (t)$ , it is easy to have that $d_{i} (t) = d_{i} (t - (1 / f_{i}))$ . For slowly varying disturbance $d_{s} (t)$ , it can be obtained that $d_{s} (t) \approx d_{s} (t - 1 / f_{0})$ , where $f_{0}$ is the sampling frequency of the control system. Therefore, equation (6) can be rewritten as

{\begin{cases} d (t) = d_{s} (t) + d_{1} (t) + \dots + d_{r} (t) + d_{d} (t) \\ d_{s} (t) = d_{s} (t - τ_{0}) \\ d_{i} (t) = d_{i} (t - τ_{i}), i = 1,2, \dots, r \end{cases}

(7)

where

τ_{i} = (1 / f_{i})

i = 0,1,2, \dots, r

Then, an integral state $d_{0} (t) = \int_{0}^{t} d (τ) d τ$ is introduced. According to equation (4), $d_{0} (t)$ can be obtained by utilizing the measured angular velocity $ω_{m}$ and the control torque command $u_{c}$ . However, $ω_{m}$ and $u_{c}$ are actually mismatched due to the measurement delay. To avoid this problem, we realign $ω_{m}$ and $u_{c}$ by using delayed control input $u_{c} (t - T)$ in the observer. By defining $d_{0 l} (t) = d_{0} (t - T)$ , we have

d_{0 l} (t) = J_{0} ω_{m} + \int_{0}^{t} [ω_{m}^{\times} J_{0} ω_{m} - u_{c} (τ - T)] d τ

(8)

By defining $d_{s l} (t) = d_{s} (t - T)$ , $d_{i l} (t) = d_{i} (t - T)$ , and $δ_{d l} (t) = δ_{d} (t - T)$ , the delayed disturbance model can be constructed based on equations (7) and (8) as

{\begin{cases} {\dot{d}}_{0 l} (t) = d_{s l} (t) + d_{1 l} (t) + \dots + d_{r l} (t) + δ_{d l} (t) \\ d_{s l} (t) = d_{s l} (t - τ_{0}) \\ d_{i l} (t) = d_{i l} (t - τ_{i}), i = 1,2, \dots, r \end{cases}

(9)

Remark 1

Considering that $d_{0 l} (t)$ can be obtained through $ω_{m}$ and $u_{c} (t - T)$ , it is utilized as the virtual disturbance measurement of the system (9). Because the asynchrony due to the measurement delay is avoided by realigning of the virtual measurement, the destructive effect of measurement delay on the observer can be alleviated.

3.2. Modified MILDO

A modified MILDO for delayed disturbance is then designed in Theorem 1 following method in He and Wu (2020) as follows.

Theorem 1

For the system (9), the modified MILDO is designed as

{\begin{cases} {\dot{\hat{d}}}_{0 l} (t) = {\hat{d}}_{s l} (t) + {\hat{d}}_{1 l} (t) + \dots + {\hat{d}}_{r l} (t) \\ {\hat{d}}_{s l} (t) = {\hat{d}}_{s l} (t - τ_{0}) + l_{0} {\tilde{d}}_{0 l} (t) + m_{0} {\hat{d}}_{l}^{*} (t) \\ {\hat{d}}_{i l} (t) = {\hat{d}}_{i l} (t - τ_{i}) + l_{i} {\tilde{d}}_{0 l} (t) + m_{i} {\tilde{d}}_{l}^{*} (t), i = 1,2, \dots, r \end{cases}

(10)

where

{\tilde{d}}_{l}^{*} (t) = [{\tilde{d}}_{0 l} (t) - {\tilde{d}}_{0 l} (t - τ_{0})] / τ_{0}

. Then, the estimation error of the observer converges to a bounded area by choosing observer gains as

{\begin{cases} l_{i} >0, m_{i} > 0 \\ α = m_{i} / l_{i}, 0 < α < 1, i = 0,1,2, \dots, r \\ l_{0} : l_{1} : l_{2} : \dots : l_{r} = τ_{0} : τ_{1} : τ_{2} : \dots : τ_{r} \\ \sqrt{(r + 1) (1 - α) l_{i} / τ_{i}} = ω_{t}, i = 0,1,2, \dots, r \end{cases}

(11)

where

{\hat{d}}_{s l} (t)

and

{\hat{d}}_{i l} (t)

are the estimates of

d_{s l} (t)

and

d_{i l} (t)

;

{\tilde{d}}_{0 l} (t) = d_{0 l} (t) - {\hat{d}}_{0 l} (t)

;

l_{i}

and

m_{i}

are observer gains;

α

is a damping ratio for ensuring the robustness of observer; and

ω_{t}

is the observer bandwidth.

The stability analysis of observer (12) is given in Appendix 1.

Remark 2

According to (31) in Appendix 1, the unmodeled disturbance $δ_{d} (t)$ will cause estimation error. However, considering that the majority of the complex disturbances in the spacecraft system has been modeled by (9) and $δ_{d} (t)$ is bounded, the estimation error caused by $δ_{d} (t)$ can be reduced by choosing the observer gains appropriately.

Remark 3

Because not only the slowly varying disturbance but also the periodic disturbances are modeled in (9), the estimation performance of observer (10) is not just determined by the observer bandwidth. Accurate estimation of periodic disturbance is achieved in limited bandwidth by adopting iterative learning technique.

Remark 4

In observer (10), $[{\tilde{d}}_{0 l} (t) - {\tilde{d}}_{0 l} (t - τ_{0})] / τ_{0}$ is utilized to approximate ${\tilde{d}}_{l} (t)$ . The reason is that although ${\tilde{d}}_{l} (t)$ might be derived according to equations (4) and (5) as ${\tilde{d}}_{l} (t) = [J_{0} {\dot{ω}}_{m} + ω_{m}^{\times} J_{0} ω_{m} - u (t - T)] - {\hat{d}}_{l} (t)$ , the differential of $ω_{m}$ will cause complex noise problem in engineering. The approximation error of utilizing ${\tilde{d}}_{l}^{*} (t)$ instead of ${\tilde{d}}_{l} (t)$ is $δ_{d e} = {\tilde{d}}_{l} (t) - {\tilde{d}}_{l}^{*} (t)$ . Given the fact that the sample interval $τ_{0}$ is a relatively small value, $δ_{d e}$ is assumed to be bounded by a positive constant value $δ_{d e}$ .

By defining $g_{i} = l_{i} + m_{i} / τ_{0}$ , $k_{i} = m_{i} / τ_{0}$ , $i = 0,1,2, \dots, r$ , observer (10) is simplified as

{\begin{cases} {\dot{\hat{d}}}_{0 l} (t) = {\hat{d}}_{s l} (t) + {\hat{d}}_{1 l} (t) + \dots + {\hat{d}}_{r l} (t) \\ {\hat{d}}_{s l} (t) = {\hat{d}}_{s l} (t - τ_{0}) + g_{0} {\tilde{d}}_{0 l} (t) - k_{0} {\tilde{d}}_{0 l} (t - τ_{0}) \\ {\hat{d}}_{i l} (t) = {\hat{d}}_{i l} (t - τ_{i}) + g_{i} {\tilde{d}}_{0 l} (t) - k_{i} {\tilde{d}}_{0 l} (t - τ_{0}), i = 1,2, \dots, r \end{cases}

(12)

The estimate of $d_{l} (t)$ is

{\hat{d}}_{l} (t) = {\hat{d}}_{s l} (t) + \sum_{i = 1}^{r} {\hat{d}}_{i l} (t)

(13)

Because ${\hat{d}}_{l} (t)$ is the estimate of delayed disturbance, it cannot be utilized for disturbance compensation directly. To obtain disturbance estimates of current instant, prediction based on ${\hat{d}}_{l} (t)$ is applied. It is given in the following subsection.

3.3. Estimation performance and delay compensation

According to equations (12) and (13), the disturbance estimation error can be described in s-domain as follows

{\begin{cases} {\hat{d}}_{l} (s) = {\hat{d}}_{s l} (s) + \sum_{i = 1}^{r} {\hat{d}}_{i l} (s) \\ {\hat{d}}_{s l} (s) = \frac{g_{0} - k_{0} e^{- τ_{0} s}}{1 - e^{- τ_{0} s}} \frac{{\tilde{d}}_{l} (s)}{s} \\ {\hat{d}}_{i l} (s) = \frac{g_{i} - k_{i} e^{- τ_{0} s}}{1 - e^{- τ_{i} s}} \frac{{\tilde{d}}_{l} (s)}{s}, i = 1,2, \dots, r \end{cases}

(14)

We define

G_{i} (s) = \frac{g_{i} - k_{i} e^{- τ_{i} s}}{s (1 - e^{- τ_{i} s})}, i = 0,1,2, \dots, r

(15)

Substituting equation (15) into (14) and rearranging, we have

{\hat{d}}_{l} (s) = \sum_{i = 0}^{r} G_{i} (s) {\tilde{d}}_{l} (s)

(16)

Because ${\tilde{d}}_{l} (s) = d (s) - {\hat{d}}_{l} (s)$ , equation (16) is rewritten as

{\tilde{d}}_{l} (s) = \frac{1}{1 + \sum_{i = 0}^{r} G_{i} (s)} d_{l} (s)

(17)

Then, the transfer function from $d_{l}$ to ${\tilde{d}}_{l}$ can be obtained from equation (17) as $G (s) = 1 / [1 + \sum_{i = 0}^{r} G_{i} (s)]$ . According to the frequency characteristics of $G (s)$ , the estimation performance of observer (12) can be evaluated.

For periodic disturbances with frequencies $ω = 2 n π f_{i}$ , $n = 1,2, \dots$ , we have $e^{- j ω τ_{i}} = 1$ and $G_{i} (j ω) = \infty$ . Considering that $G_{j} (j ω)$ is a complex number with bounded amplitude when $j \neq i$ , we have $G (j ω) = 0$ . It indicates that accurate estimates for $d_{i l}$ can be obtained by the observer (12).

For slowly varying disturbance, it can be derived that $\lim_{ω \to 0} G_{i} (j ω) = \infty$ , $i = 0,1,2, \dots, r$ . Thus, we have $\lim_{ω \to 0} G (j ω) = 0$ . Accurate estimation for slowly varying disturbance can also be achieved. However, it is noted that subsystems related to ${\hat{d}}_{s l} (t)$ and ${\hat{d}}_{i l} (t)$ , $i = 1,2, \dots, r$ all contribute to the estimation of slowly varying disturbance. According to equation (16), the contributions of them can be evaluated by $G_{i} (s)$ , $i = 0,1,2, \dots, r$ . If $G_{0} (s)$ and $G_{i} (s)$ are taken as example, we have

\begin{array}{l} \lim_{ω \to 0} \frac{G_{i} (j ω)}{G_{0} (j ω)} = \lim_{ω \to 0} \frac{g_{i} - k_{i} e^{- j ω τ_{0}}}{j ω (1 - e^{- j ω τ_{i}})} / \frac{g_{0} - k_{0} e^{- j ω τ_{0}}}{j ω (1 - e^{- j ω τ_{0}})} \\ = \lim_{ω \to 0} \frac{(g_{i} - k_{i}) (1 - e^{- j ω τ_{0}})}{(g_{0} - k_{0}) (1 - e^{- j ω τ_{i}})} \\ = \lim_{ω \to 0} \frac{j τ_{0} (g_{i} - k_{i}) e^{- j ω τ_{0}}}{j τ_{i} (g_{0} - k_{0}) e^{- j ω τ_{i}}} \\ = \frac{τ_{0} l_{i}}{τ_{i} l_{0}} \end{array}

(18)

According to equation (11), $l_{i} / τ_{i} = l_{0} / τ_{0}$ , $i = 0,1,2, \dots, r$ . Thus, it can be derived that $\lim_{ω \to 0} (G_{i} (j ω) / G_{0} (j ω)) = 1$ . Therefore, the contributions of ${\hat{d}}_{s l} (t)$ and ${\hat{d}}_{i l} (t)$ , $i = 1,2, \dots, r$ for slowly varying disturbance are approximately the same to each other.

Based on the analysis above, the estimates of slowly varying disturbance and periodic disturbances are given by $(r + 1) {\hat{d}}_{s l} (t)$ and ${\hat{d}}_{i l} (t) - {\hat{d}}_{s l} (t)$ , $i = 1,2, \dots, r$ , respectively. Thus, ${\hat{d}}_{s l} (t)$ and ${\hat{d}}_{i l} (t)$ , $i = 1,2, \dots, r$ are separately obtained. Because $d_{s} (t)$ is slowly varying disturbance, ${\hat{d}}_{s l} (t)$ can be utilized as the estimation of $d_{s} (t)$ without introducing significant compensation error. Then, the disturbance estimate is reconstructed according to delay time $T$ and disturbance frequencies $f_{i}$ as follows

\begin{array}{l} \hat{d} (t) = {\hat{d}}_{s} (t) + \sum_{i = 1}^{r} {\hat{d}}_{i} (t) \\ = (r + 1) {\hat{d}}_{s l} (t) + \sum_{i = 1}^{r} [{\hat{d}}_{i l} (t - T_{c}) - {\hat{d}}_{s l} (t - T_{c})] \end{array}

(19)

where

T_{c} = (n + 1) τ_{i} - T

n = int {T / τ_{i}}

Noted that the delays of periodic disturbances are compensated, the total disturbance compensation error of current instant is derived as

\begin{array}{l} \tilde{d} (t) = d (t) - \tilde{d} (t) \\ = d_{s} (t) + \sum_{i = 1}^{r} d_{i} (t) - (r + 1) {\tilde{d}}_{s l} (t) - \sum_{i = 1}^{r} [{\tilde{d}}_{i l} (t - T_{c}) - {\tilde{d}}_{s l} (t - T_{c})] \\ = (r + 1) [d_{s} (t) - d_{s l} (t) + {\tilde{d}}_{s l} (t)] + \sum_{i = 1}^{r} [{\tilde{d}}_{i} (t) - {\tilde{d}}_{s} (t)] \\ \leq (r + 1) (‖ d_{s} (t) - d_{s l} (t) ‖ + ‖ {\tilde{d}}_{s l} (t) ‖) + r ‖ {\tilde{d}}_{s} (t) ‖ + ‖ \sum_{i = 1}^{r} {\tilde{d}}_{i} (t) ‖ \end{array}

(20)

In view of Assumption 1 and equation (31), we have $‖ \tilde{d} (t) ‖ \leq (r + 1) [v_{d} T + 2 (e_{d l} + δ_{d e})] = e_{d}$ . Thus, the disturbance compensation error $\tilde{d}$ is ultimately bounded.

Remark 5

It should be noted that small deviation between the obtained $T$ and the actual value may increase the estimation error, and large deviation will affect the stability of the designed observer. Thus, accurate acquisition of measurement delay from ground test or on-orbit delay identification is necessary.

Remark 6

The conventional MILDO is redesigned into the form of (10) and (11) for realizing the separated estimates of the components for the slowly varying disturbance and periodic disturbances which is the basis of the prediction algorithm (19).

4. DC-MILDO-based robust composite controller

To stabilize the spacecraft attitude system with robustness to delayed state feedback, a DC-MILDO-based robust composite controller is designed following Lyapunov–Krasovskii approach (Chunodkar and Akella, 2011; Samiei and Butcher, 2013).

For the simplicity of controller design, the spacecraft system (1) and (4) can be rewritten as

{\begin{cases} \dot{p} = \frac{1}{4} ω+ [F (p) - \frac{1}{4} I_{3}] ω \\ \dot{ω} = J_{0}^{- 1} u + J_{0}^{- 1} (- ω^{\times} J_{0} ω + d) \end{cases}

(21)

Consider a controller as follows

u = - 4 J_{0} [k_{p} p_{m} + k_{d} ω_{m}] - \tilde{d}

(22)

where

\tilde{d}

is the disturbance estimates given by equation (19).

Then, by defining $q = ω / 4$ , $x = {[p, q]}^{T}$ , and substituting equation (22) into (21), the closed-loop system dynamics can be given by

{\begin{cases} \dot{x} = Ax + A_{d} x (t - T) + g (x) + \bar{d} \\ x (s) = x_{0} (s), s \in [- T, 0] \end{cases}

(23)

where $A = [\begin{matrix} O_{3} & I_{3} \\ O_{3} & O_{3} \end{matrix}]$ , $A_{d} = [\begin{matrix} O_{3} & O_{3} \\ - k_{p} I_{3} & - 4 k_{d} I_{3} \end{matrix}]$ , $g (x) = [\begin{array}{l} [4 F (p) - I_{3}] q \\ - 4 J_{0}^{- 1} q^{\times} J_{0} q \end{array}]$ , and $\bar{d} = 1 / 4 B J_{0}^{- 1} \tilde{d}$ , $B = [\begin{array}{l} O_{3} \\ I_{3} \end{array}]$ ; $x_{0} (s)$ is the initial function; it is assumed that the system states do not escape from the domain of attraction during the time evolution $[- T, 0]$ .

Assumption 4

The nonlinear term $g (x)$ is assumed to satisfy locally Lipschitz condition in a certain vicinity of the origin as follows

‖ g [x (t)] ‖ \leq γ (‖ x (t) ‖) ‖ x (t) ‖

(24)

where

γ (‖ x (t) ‖) > 0

is a Lipschitz parameter which is a function of

‖ x (t) ‖

, and

γ (‖ x (t) ‖)

is bounded by

γ_{\max}

Theorem 2

For the spacecraft system (23) with complex disturbances and measurement delay $T$ , the controller (22) guarantees that the system asymptotically converges to a bounded area including the origin under Assumption 4, if there are positive symmetric matrices $M$ , $N$ , $K_{1}$ , $K_{2}$ and matrices $R$ , $R_{1}$ , $Y$ such that the following linear matrix inequality (LMI) holds

Λ_{1} = [\begin{matrix} Φ_{1} & Γ & T A^{T} \\ * & - N & T A_{d}^{T} \\ * & * & - T K_{1}^{- 1} \end{matrix}] < 0

(25)

[\begin{matrix} R & Y \\ Y^{T} & K_{1} \end{matrix}] \geq 0, [\begin{matrix} R_{1} & Y \\ Y^{T} & K_{2} \end{matrix}] \geq 0

(26)

where

Φ_{1} = A^{T} M + MA + 2 γ_{\max} M + T (γ_{\max}^{2} K_{2} + R + R_{1}) + N + Y^{T} + Y

;

Γ = M A_{d} - Y

; * represents an ellipsis for the terms induced by symmetry.

Proof

See Appendix 2.

Remark 7

According to equation (46), the bounded area of system states is enlarged by the disturbance estimation error. When significant disturbances exist, the spacecraft attitude control performance will be reduced. With the help of DC-MILDO, most of the disturbances can be accurately compensated. Consequently, the conservativeness of the feedback controller can be reduced. It is noted that the computation consumption of the designed approach may be slightly increased by the extra observer. Nevertheless, it is acceptable considering the simplicity of the observer and the improvement of the attitude control performance.

5. Simulation results

To verify the effectiveness of a proposed controller, simulations on a flexible spacecraft are carried out on the MATLAB platform. Following Di Gennaro (2003), the parameters of an orbiting spacecraft with rigid body and a flexible array are given in Table 1. Considering that the low-frequency flexible modes are generally dominant, only the first three modes are considered in the simulation. According to Chunodkar and Akella (2011), measurement delay of

p

and

ω

is assumed as

T = 0.12 s

. The initial condition of the spacecraft is in the form of

p (0) = {[0.3, - 0.4, - 0.5]}^{T}

with

ω (0) = {[0,0,0]}^{T}

deg/s. Complex disturbances in the system are a combination of slowly varying disturbance

d_{s} = {[d_{s φ}, d_{s θ}, d_{s ψ}]}^{T}

and periodic disturbances shown in Table 2. The fundamental frequencies of periodic disturbances

d_{p 1} = {[d_{1 φ} + d_{2 ϕ}, d_{1 θ} + d_{2 θ}, d_{1 ψ} + d_{2 ψ}]}^{T}

and

d_{p 2} = {[d_{3 φ} + d_{4 ϕ}, d_{3 θ} + d_{4 θ}, d_{3 ψ} + d_{4 ψ}]}^{T}

that come from a cryocooler and a driven motor are assumed as 8 Hz and 10 Hz, respectively, with reference to Jedrich et al. (2002) and Toyoshima et al. (2010).

Table 1.

Parameters of spacecraft.

Parameter	Values
$J_{0}$	$[\begin{matrix} 350 & 3 & 4 \\ 3 & 280 & 10 \\ 4 & 10 & 190 \end{matrix}] kg \cdot m^{2}$
$J_{e}$	$diag [60,40,20] [1 + e^{- 0.01 t} + 2 g (t - 40) - 4 g (t - 60)] kg \cdot m^{2}$ , $g (t < 0) = 0$ , $g (t \geq 0) = 1$
$δ$	$[\begin{matrix} 6.45637 & 1.27815 & 2.15629 \\ - 1.25819 & 0.91756 & - 1.67264 \\ 1.11687 & 2.48901 & - 0.83674 \end{matrix}] k g^{1 / 2} \cdot m^{2} / s^{2}$
$[ξ_{1}, ξ_{2}, ξ_{3}]$	$[0.056, 0.086, 0.13]$
$[ζ_{1}, ζ_{2}, ζ_{3}]$	$[0.7681, 1.1038, 1.8733] rad / s$

Table 2.

Disturbances.

Disturbance	Values (Nm)
$d_{s}$	$d_{s ϕ} = D_{s} cos (ω_{s} t)$ , $d_{s θ} = - D_{s} sin (ω_{s} t)$ , $d_{s ψ} = D_{s} cos (ω_{s} t)$ , $D_{s} = 0.004$ , and $ω_{s} = 0.002 π rad / s$
$d_{p 1}$	$d_{1 ϕ} = D_{1} cos (ω_{01} t)$ , $d_{1 θ} = D_{1} sin (ω_{01} t)$ , $d_{1 ψ} = D_{1} cos (ω_{01} t)$ , $D_{1} = 0.004$ , and $ω_{01} = 16 π rad / s$
$d_{p 1}$	$d_{2 ϕ} = D_{2} cos (ω_{02} t)$ , $d_{2 θ} = D_{2} sin (ω_{02} t)$ , $d_{2 ψ} = D_{2} cos (ω_{02} t)$ , $D_{2} = 0.002$ , and $ω_{02} = 32 π rad / s$
$d_{p 2}$	$d_{3 ϕ} = D_{3} cos (ω_{03} t)$ , $d_{3 θ} = D_{3} sin (ω_{03} t)$ , $d_{3 ψ} = D_{3} cos (ω_{03} t)$ , $D_{3} = 0.0032$ , and $ω_{03} = 20 π rad / s$
$d_{p 2}$	$d_{4 ϕ} = D_{4} cos (ω_{04} t)$ , $d_{4 θ} = D_{4} sin (ω_{04} t)$ , $d_{4 ψ} = D_{4} cos (ω_{04} t)$ , $D_{4} = 0.0016$ , and $ω_{04} = 40 π rad / s$

5.1. Attitude stabilization performance

To evaluate the effectiveness of the designed approach, traditional extended disturbance observer (EDO) (Yan and Wu, 2017) and MILDO (He and Wu, 2020) are also included in the simulations. The gains of them are chosen based on the bandwidth tuning method in Gao (2003). Then, simulations of robust controller without disturbance compensation (Chunodkar and Akella, 2011), EDO-based robust controller (EDOBC), MILDO-based robust controller (MILDOBC), and DC-MILDO-based robust controller (DC-MILDOBC) are conducted. The parameters of robust controller are chosen as

k_{p}

= 1.02 and

k_{d}

= 2.02. The Euler angle

σ = {[ϕ, θ, ψ]}^{T}

and the angular velocity ω of spacecraft are given to demonstrate the attitude control performance. The standard deviation (STD) of

σ

, ω , and

\tilde{d}

in steady state are given in Table 3.

Table 3.

Control performance of the robust controller EDOBC and DC-MILDOBC.

Parameter		Robust controller	EDOBC	DC-MILDOBC
$σ (STD)$	$ϕ$ (deg)	1.69 × 10⁻⁴	8.15 × 10⁻⁶	2.01 × 10⁻⁶
	$θ$ (deg)	9.16 × 10⁻⁵	1.83 × 10⁻⁶	6.22 × 10⁻⁷
	$ψ$ (deg)	1.47 × 10⁻⁴	5.37 × 10⁻⁶	1.40 × 10⁻⁶
$ω (STD)$	$ω_{ϕ}$ (deg/s)	1.18 × 10⁻⁵	1.16 × 10⁻⁵	1.62 × 10⁻⁶
	$ω_{θ}$ (deg/s)	1.31 × 10⁻⁵	1.21 × 10⁻⁵	4.91 × 10⁻⁷
	$ω_{ψ}$ (deg/s)	2.02 × 10⁻⁵	1.86 × 10⁻⁵	1.18 × 10⁻⁶
$\tilde{d} (STD)$	${\tilde{d}}_{ϕ}$ (Nm)	—	4.51 × 10⁻³	2.58 × 10⁻⁴
	${\tilde{d}}_{θ}$ (Nm)	—	3.63 × 10⁻³	2.03 × 10⁻⁴
	${\tilde{d}}_{ψ}$ (Nm)	—	3.96 × 10⁻³	1.87 × 10⁻⁴

Note: EDOBC: EDO-based robust controller; DC-MILDOBC: DC-MILDO-based robust controller; STD: standard deviation.

As shown in Figure 1 and Table 3, the spacecraft attitude dynamics can be stabilized by the robust controller. $σ$ and ω in steady state are within 2.3 × 10⁻³ deg and 4.4 × 10⁻⁵ deg/s, respectively. However, the spacecraft attitude is still affected by complex disturbances. Especially, as shown in Figure 1(b), periodic vibrations of ω are induced by the periodic disturbances.

Figure 1.

Spacecraft attitude stabilization under robust controller without disturbance observer. (a) Euler angles; (b) angular velocities; (c) control torques.

Conventional EDO and MILDO can be applied to suppress complex disturbances. However, large measurement delay will cause instability of them. Robust modification to observer gain matrix in Qiao et al. (2019) may be applied to EDO. Nevertheless, the stability of MILDO is not easy to achieve by simply regulating the gains of conventional MILDO. Therefore, we modify the bandwidth of EDO as 1 rad/s for tolerating the measurement delay and then introduce DC-MILDO for comparison.

The simulation results of EDOBC are given in Figure 2. As shown in Figure 2, $σ$ and ω in steady state are within 5 × 10⁻⁶ deg and 4.5 × 10⁻⁵ deg/s, respectively. Evidently, the attitude stabilization performance is improved compared with robust controller in Figure 1. The reason is that most of the slowly varying disturbances are estimated by EDO and then compensated. However, as shown in Figure 2(a) and (b), there are significant vibrations related to periodic disturbances in $σ$ and ω . These periodic disturbances cannot be adequately estimated and suppressed.

Figure 2.

Spacecraft attitude stabilization of EDO-based robust controller. (a) Euler angles; (b) angular velocities; (c) disturbance estimation errors; (d) control torques.

In Figures 3 and 4 and Table 3, the simulation results of DC-MILDOBC are given. As shown in Figure 3, $σ$ and ω are within 2 × 10⁻⁶ deg and 1.5 × 10⁻⁶ deg/s, respectively. The vibrations of ω are reduced by about an order of magnitude compared with EDOBC. Also, $σ$ and ω contain much less vibrations induced by periodic disturbances. Thus, the attitude control stability and accuracy of spacecraft indicated by the STD of ω and $σ$ are improved.

Figure 3.

Spacecraft attitude stabilization of DC-MILDO-based robust controller. (a) Euler angles; (b) angular velocities; (c) disturbance estimation errors; (d) control torques.

Figure 4.

Disturbance estimates of DC-MILDO-based robust controller. (a) $d_{s ϕ}$ and ${\hat{d}}_{s ϕ}$ ; (b) $d_{p 1 ϕ}$ and ${\hat{d}}_{p 1 ϕ}$ ; (c) $d_{p 2 ϕ}$ and ${\hat{d}}_{p 2 ϕ}$ .

The improvement of the attitude stabilization via DC-MILDOBC relies on the accurate estimation and compensation of not only slowly varying disturbance but also of periodic disturbances. As shown in Figure 3(c), the disturbance estimation errors are reduced to 1 × 10⁻³ Nm. The STD of $\tilde{d}$ is reduced by an order of magnitude compared with EDO in Figure 3. Specifically, as shown in Figure 4, the slowly varying disturbance d _s, periodic disturbances d _p1 and d _p2 can be separately estimated by DC-MILDO. Thus, they can be accurately compensated separately. Therefore, the effectiveness of the designed DC-MILDOBC is verified.

5.2. Robustness to abrupt disturbances

The spacecrafts in orbit may also be affected by impacts of space debris or booting of an instrument onboard. To test the robustness of the designed approach to these disturbances, a strong abrupt disturbance $d_{a}$ is introduced during the attitude stabilization in this scenario. It is assumed that $d_{a}$ is in the form of $d_{a} = {[d_{a ϕ}, d_{a θ}, d_{a ψ}]}^{T} = {\begin{cases} {[1, - 2,2]}^{T} \times e^{- 0.05 (t - 50)}, t \geq 50 \\ 0, 0 \leq t < 50 \end{cases}$ . The attitude control performance of the robust controllers EDOBC and DC-MILDOBC is given in Figures 5 and 6.

Figure 5.

Attitude control via robust controller in the presence of $d_{a}$ .

Figure 6.

Attitude control in the presence of $d_{a}$ . (a) Euler angles of EDOBC; (b) disturbance estimation errors of EDO; (c) Euler angles of DC-MILDOBC; (d) disturbance estimation errors of DC-MILDOBC. Note: EDOBC: EDO-based robust controller; DC-MILDOBC: DC-MILDO-based robust controller.

As shown in Figure 5, the abrupt disturbance $d_{a}$ causes significant attitude vibrations. However, as shown in Figure 6(b) and (d), the effect of $d_{a}$ can be reduced by the disturbance observers. It shows that the estimation errors of both EDO and DC-MILDO converge to the neighborhoods of zero after a period of time. As a result, as shown in Figures 5, 6(a), and (c), the attitude vibrations caused by $d_{a}$ can be suppressed faster by EDOBC and DC-MILDOBC than the robust controller. Furthermore, the ultimate stabilization performance of DC-MILDOBC is also improved compared with EDOBC. It can be concluded that the designed approach shows adequate robustness to the abrupt disturbance.

5.3. Robustness to identification error of the delay time

In this scenario, the robustness of the designed approach to small identification error of the delay time is tested. The obtained delay time $T_{i}$ is assumed to be $T_{i} = T + Δ T$ , where $T$ is the actual delay time and $Δ T$ is the identification error. In the simulation, $T$ is assumed as 0.12 s. Simulation results for $Δ T = 5 % \times T$ , $Δ T = 10 % \times T$ , and $Δ T = 15 % \times T$ are given in Figure 7(a)–(c), respectively.

Figure 7.

Performance of delay compensation-MILDO in the presence of $Δ T$ . (a) $Δ T = 5 % \times T$ ; (b) $Δ T = 10 % \times T$ ; (c) $Δ T = 15 % \times T$

As shown in Figure 7(a) and (b), the disturbance estimation errors of DC-MILDO are slightly increased to $3 \times 10^{- 3}$ Nm and $5 \times 10^{- 3}$ Nm compared with the results in Figure 3. Thereafter, the stabilization performance of spacecraft is slightly reduced due to the identification error $Δ T$ . Nevertheless, because most disturbances including periodic disturbances can still be estimated and compensated when $Δ T$ is small, the attitude control performance via DC-MILDO is favorable. However, when $Δ T = 15 % \times T$ , the estimation errors of DC-MILDO severely increase as shown in Figure 7(c). It indicates that the stability of the observer cannot be maintained. Thus, it can be concluded that the designed approach is robust to small identification error of the delay time. To obtain stronger robustness of the control system, the online delay identification technique, such as in Bayrak and Tatlicioglu (2013), could be combined with the designed approach.

6. Conclusion

To suppress complex disturbances of flexible spacecrafts subject to measurement delay, a DC-MILDO-based composite controller is designed in this study. The designed observer is based on an augmented model of the delayed disturbances. A tuning rule for the observer gains is proposed to realize separated estimates of slowly varying disturbance and periodic disturbances. Then, a prediction algorithm is designed for obtaining the accurate disturbance estimation of current instant. To clarify the disturbance estimation performance, rigorous theoretical analysis is given in s-domain. By combining DC-MILDO with a robust controller, high performance attitude stabilization of a flexible spacecraft subject to complex disturbances and measurement delay is realized. Simulation results also indicate that the designed approach shows adequate robustness to the abrupt disturbances and small identification error of the delay time. In our future research, online delay identification technique may be combined with the designed approach for improving the robustness of the control system.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Tongfu He

Zhong Wu

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