Linear Parameter Varying Analysis of Switched Controllers for Attitude Control Systems

Abstract

T HE on-orbit life of a satellite is decomposed into phases, allowing the convergence from the separation state to the mission state, the orbit control, and the safe state in case of fault. For each phase ormode, a specific set of attitude pointing requirements is to be met by using a specific set of actuators and sensors. Inside a mode, several attitude control and estimation algorithms can be designed, for example, to meet increasing performance levels requirements. Over the past 30 years, the number of attitude and orbit control system (AOCS) modes has been reduced so that generic architectures now typically consist of three modes (one safe-hold mode, one mission mode, and one orbit control mode). In the same time, the software complexity has increased to meet ever more demanding requirements. Thus, the switching between different AOCS modes and software requires permanently enhanced robustness levels to cope with more stringent kinematics conditions (satellite angular rate and position)while also improving the pointing accuracy. In this Note, the problem of the reaction wheel control software switching inside the mission mode of the Centre National d’Etudes Spatiales (CNES) microsatellite DEMETER (detection of electromagnetic emissions transmitted from earthquake regions) [1–4] is considered. Because of the limited speed and torque capacities of the reaction wheels, the linear proportional-derivative controller specifically designed for small pointing errors can no longer be used when the latter get too large, which is typically the case when the mission mode is activated. Then, to avoid torque and speed saturations during this phase,whichwill be referred to as the rough pointing phase of the mission mode, a dedicated controller is used to maintain the angular rate around a constant target until the pointing error becomes small enough. More interesting, as soon as the speed target is reached, the commanded torque is null. The parameters of the control system are tuned in such a way that the delivered control input signal remains continuous when switching occurs for prespecified pointing and rate errors. However, because of external perturbations and AOCS sampling constraints, the nominal conditions are rarely met when the control system switches from rough to fine pointing phases, which locally induces some undesirable (and what it is hoped to be transient) behaviors. As a result, it becomes rather difficult to obtain stability and robustness proofs, which have to be addressed through extensive and time-consuming nonlinear simulations campaigns. In this context, the central idea of this Note consists of rewriting the switching-based control law in a quasi-linear-parameter-varying form thanks to which the following properties hold: 1) The continuity of the control signals is guaranteed at anytime, since the new system is based on a smooth transition from rough to fine pointing phases. 2) The stability and robustness proofs for the nonlinear closedloop plant can be obtained via enhanced linear-parameter-varying (LPV) analysis techniques, which are developed in the Note. While many methods are now available to evaluate the stability of LPVplants,most of thewell-established results are numerically quite demanding as soon as the size of the plant increases, togetherwith the complexity of the selected Lyapunov function. In this Note, a new algorithm based on parameter-dependent Lyapunov (PDL) functions is then developed, thanks to which the stability analysis can be performed at a reasonable cost. The key idea of the proposed iterative technique consists of a first step of optimizing the Lyapunov functions on a rough grid of the parameter space and, in a second step, to check the validity on the continuous space. The grid is then updated if necessary until the whole space is cleared. This approach may be viewed as an original adaptation of a similar idea [5], which was exploited 10 years ago in a -based robustness analysis. The Note is organized as follows. After a brief description of the AOCS control loops and switching-based controllers in Sec. II, the quasi-LPV formulation is detailed in Sec. III. Stability analysis of LPV systems via PDL functions is then described in Sec. IV, where the aforementioned algorithm is detailed. Section V is devoted to the presentation of stability analysis and time-domain simulation results. Finally, some concluding remarks end the Note.