The need for increasingly autonomous space missions with reduced development times requires efficient Attitude and Orbit Control Systems (AOCS), but also adequate Verification and Validation (V&V) techniques to assess performance in the presence of uncertainties and external disturbances affecting the system. An example is the Laser Interferometer Space Antenna (LISA), an ESA mission planned for launch in 2035. It will be the first space-based observatory to detect the gravitational waves emitted by the most powerful events in the Universe, such as pairs of black holes merging. The mission is composed of three identical spacecraft orbiting in a triangular configuration at a distance of 2.5 million km from each other. The attitude control aims to constrain the orientation of each spacecraft with respect to the incoming laser waves from the two other twins. The corresponding closed-loop chain has been modeled as a Multiple-Input Multiple-Output (MIMO) Linear Time Invariant (LTI) system affected by several uncertainties, which may significantly perturb the nominal performance. 

Monte Carlo simulations can be used to assess the robustness of the model in the presence of uncertainties, but they only provide soft bounds and may fail to detect rare events. On the contrary, other approaches like µ-analysis and Integral Quadratic Constraints (IQCs) allow computing guaranteed bounds on the system's stability and performance without any simulation. The latest algorithms developed in this domain at ONERA with the co-funding of CNES have recently been tested on the LISA attitude control model provided by ESA. In particular, the analysis has focused on two relevant metrics which are often addressed during the AOCS V&V process, namely the H_∞ and the H₂ norms. The paper is intended as a survey of these techniques and their achievable results.

The H_∞ norm is a well-known metric, typically used in the robust control field, which can physically be interpreted in two different ways. First, this norm is a measure of the maximum steady-state amplification that a sinusoidal input can experience through the system. Secondly, it also corresponds to the maximum energy of the system's output due to an input with bounded energy. A guaranteed upper bound on the H_∞ norm of an uncertain system can be computed with standard techniques such as µ-analysis. Furthermore, given a certain H_∞-performance requirement, it is also possible to compute hard bounds on the probability that such requirement is either met or not. Differently from Monte Carlo simulations, these probability bounds are guaranteed, which corresponds to a confidence level of 100%.

Another important metric often used to design and validate a control system is the H₂ norm, which has three main physical interpretations. First, this norm is a measure of how a white noise is amplified by the system in terms of its Power Spectral Density (PSD). This metric, therefore, represents one of the pivotal design criteria in applications like structural dynamics, micro-vibration, acoustics and colored noise disturbance rejection. Secondly, the H₂ norm also corresponds to the energy of the system's impulsive response, which is strictly related to its damping properties. Eventually, for SISO systems the H₂ norm is equivalent to the energy-to-peak gain, namely the worst-case amplitude of the output when the input has finite energy. 

This last interpretation of the H₂ norm for SISO systems is particularly interesting as it offers possibilities to evaluate transient effects by a powerful frequency-domain approach. Such an evaluation is of particular interest in the  context of the LISA mission for which extremely high pointing accuracy is required. Impacts due to micrometeorites, mode switching and transient mechanical loads may cause saturations of the optical instruments, resulting in a temporary loss of the link between the three spacecraft. These phenomena shall therefore be carefully examined to identify worst-case input signals along with critical combinations of uncertainties.

We thank Valentin Preda, Guidance, Navigation, and Control (GNC) Systems Engineer at ESA, for providing the LISA model used for the analysis.