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Online-learning control with weakened saturation response to attitude tracking: A variable learning intensity approach

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Abstract

This brief investigates the problem of attitude tracking control using a variable learning intensity (VLI) online-learning control (OLC) scheme. The unique specialty of the proposed VLI-OLC scheme is that it achieves control performance enhancement via learning the previous control information online. The implementation is performed by a simple algebraic equation, which achieves decent control robustness while avoiding a complex control design and saving computational resources. The OLC's saturation response caused by the extensive system error during execution is noticeably weakened by introducing a VLI approach. Compatibility with previous algorithms can be guaranteed. The application example shows that control performance and saturation reduction are guaranteed concurrently.

Introduction

Traditional control schemes typically use complex and exquisite structures to improve control performance, robustness, and accuracy. For instance, to achieve control performance enhancement, the commonly used adaptive tools [1], [2], [3], [4], [5] and observer tools [6], [7], [8], [9] make the designed algorithms more complicated and necessitate the use of numerous parameters, which increases the controller's conservativeness. However, practical scenarios may not provide all of the ideal conditions required for control systems. In practical spacecraft systems to be in service, there are certain constraints including energy or computational resources. The pressure of launch costs requires designing systems that are simple and efficient to use. This paper aims to design a control scheme with a simple structure that consumes little computational resources for application in spacecraft attitude control systems.

Consider a typical non-linear systemx˙(t)=f(t,x)+u(t)+d(t) in which x(t) is the state vector whose domain DRn contains the origin; u(t) denotes the control inputs; d(t) denotes the disturbances; f:[0,+)×DRn is a piecewise continuous function that has Lipschitz continuity with respect to x. The main task of designing a controller is to investigate an algorithm u(t) that guarantees the stability of the system. To this end, advanced control tools (such as an event-based approach [10], [11], prescribed performance approach [3], [12], [13]; quantization approach [14], [15]; and data-driven approach [16]) are utilized while designing the control schemes. Nevertheless, the use of these complex structures in a mathematical operation leads to an increase in the complexity of the algorithm, making it difficult to implement and increasing computational demands.

An effective way to deal with the aforementioned issues is the OLC approach [17], and the control scheme is given byu(t)=k1u(tτ)+k2v where k1u(tτ) and k2v are the learning portion and updating portion, respectively; τ>0 is the learning interval; k1 and k2 are positive constants; especially, k1 is termed as the learning intensity, which represents the proportion of previous information utilized in generating control instructions; and v is a control law that can stabilize the system dynamics (1). As shown in (2), OLC directly learns the previous control input information via a low-complexity algebraic equation, unlike other traditional algorithms that require analysis of system dynamics or a complex control algorithm design. In this way, though v can be selected using simple designs, control robustness, low computational costs and long-term energy saving can be guaranteed [17].

Nevertheless, the OLC scheme designed in [17] has the following deficiencies. Since the essence of OLC is equivalent to shifting the baseline of the control commands computation, when v is updated to the controller with a large amount (that is, when the system error is large, it will be swiftly learned into the new control commands), the control input u will quickly reach saturation situation. The above analysis is verified in [17, Figure 6]. It can be seen from the simulation result that instantaneous energy consumption is excessive at the beginning of the control process due to the short-term saturation. If the aforementioned undesirable property is improved, then the applicability of OLC to practical scenarios will be significantly enhanced. The main obstacles to solve this problem are two twofold.

  • 1)

    Maintenance of the simple structure and low computational complexity to OLC.

  • 2)

    Ensuring long-term robustness and decent control performance.

Based on the above analysis and the need to improve OLC, the contributions are threefold.

  • 1)

    The proposed VLI-OLC design mitigates the noticeable saturation response when encountering a large system error in comparison with [17], without using complex algorithms and introducing many parameters.

  • 2)

    A satisfactory attitude control performance and high robustness can be guaranteed, including easy parameter tuning, low computational costs, short- and long-term energy-saving, and a weakened saturation response.

  • 3)

    The proposed VLI-OLC has decent compatibility with conventional control algorithms and can be regarded as a useful extension to improve control performance, as shown from the proof process.

Section snippets

Attitude dynamics

To begin with, we give the dynamics (3) and kinematics (4), (5) of spacecraft [10]Jω˙=(Jω)×ω+u+dq˙=12(q×+q0I3)ωq˙0=12qω where ωR3 denotes the angular velocity of the body-fixed frame (B) with respect to the inertia frame (I) and is expressed in B; JR3×3 is a symmetric positive-definite matrix which is the inertia dyadic of the body relative to the center of mass resolved in B; Jmax and Jmin are the maximum and minimum eigenvalues of J; the nominal values of the inertia matrix of the

Variable learning intensity online-learning control scheme

We first present the VLI-OLC design and introduce its ideas. Subsequently, we provide the proof of attitude tracking stability under the proposed VLI-OLC scheme.

Parameter configuration

To test the robustness of the VLI-OLC algorithm, the uncertainty of inertia is also considered in the simulation.

  • Denote J=J¯+ΔJ (kg⋅m2) in (3) where

    Nominal value [24] J¯=[2020.92170.50.90.515],

    Inertia uncertainty ΔJ = diag((3+sin(0.5t))e0.1t+1(4+cos(0.5t))e0.1t+2(5+sin(0.5t))e0.1t1);

  • Disturbance [25] d(t)=[3cos(ϕt)6sin(0.3ϕt)+31.5sin(ϕt)3cos(0.5ϕt)23sin(ϕt)+8sin(0.4ϕt)1]103 (N⋅m) where ϕ=0.5+ω, in (3);

  • Controller u(t) parameters, γ1=4, γ2=2, ε=0.1, k2=1, k3=2, σ=1, in (9), (10), (11);

Conclusions

In this paper, a novel variable learning intensity online learning control scheme is proposed to achieve attitude tracking control. The most significant nature of VLI-OLC is that it inherits OLC's excellent control performance properties, with a high control accuracy and a low-complexity structure, avoiding the usage of complex tools such as adaptive and observers. Specifically, by introducing a novel VLI approach, the proposed VLI-OLC efficiently reduces the saturation induced by large

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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    This research is partially supported by the National Natural Science Foundation of China (No. 62003112), the National Research Foundation of Korea (NRF) grant funded by the Korea government (Ministry of Science and ICT) (No. NRF-2020R1A2C1005449), and the Shenzhen Governmental Basic Research Grant (JCY20170412151226061, JCY20170808110410773, JCYJ20180507182241622).

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