Probabilistic stability margins and their application to AOCS validation
Authors:   Onera, The French Aerospace Lab, Toulouse, France, Franca Somers Miss , Roos Clément , Sanfedino Francesco , Bennani Samir , Preda Valentin 
Published: January 2024

Current validation and verification (V&V) activities in aerospace industry mostly rely on time-consuming simulation-based tools. These tools can give a measure of probability for sufficiently frequent phenomena, but they may fail in detecting rare but critical combinations of parameters. As the complexity of modern space systems increases, this limitation plays an ever-increasing role. In recent years, model-based worst-case analysis methods have reached a good level of maturity. Without the need of simulations, these tools can fully explore the space of all possible combinations of uncertain parameters and provide guaranteed mathematical bounds on robust stability and worst-case performance levels. However, they give no measure of probability and can therefore be overly conservative. Introduced more recently probabilistic μ-analysis combines worst-case information with probability measure. As such, it tempts to bridge the analysis gap between Monte Carlo simulations and deterministic μ-analysis [4].
The STOchastic Worst-case Analysis Toolbox (STOWAT), is a toolbox dedicated to probabilistic μ-analysis, developed by ONERA, The French Aerospace Lab. The original version of the toolbox, released by [9] and [1], only allowed for probabilistic robust stability and H∞ performance analysis. However, for the STOWAT to be fully convincing for industry, it should be as efficient and versatile as possible. For this purpose, focus has been on efficiency improvement ever since [3]. Furthermore, the toolbox was recently equipped with four probabilistic stability margin algorithms, devoted to probabilistic gain, phase, disk and delay margin analysis [8], [2], [7]. All four can be classified as μ-analysis based Branch-and-Bound (B&B) algorithms. At each iteration sufficient μ-analysis based conditions are evaluated to ascertain if the considered margin is guaranteed to be below (violation test) or above (satisfaction test) a desired threshold on a given set of uncertainties. If no conclusions can be drawn, the uncertainty set is split into two subsets and the analysis is repeated on each of them. These tools are limited to Single-Input Single-Output (SISO) system analysis. But since most industrial problems involve Multiple-Input Multiple-Output (MIMO) systems, this contribution first focuses on extending the algorithms to MIMO system analysis. Only adjustments need to be made to the conditions used to determine whether the satisfaction test or violation test should be applied. For SISO systems these conditions mostly rely on grid-based methods. However, in [8] it was already shown that gridding is usually very efficient for SISO and loop-at-a-time margin analysis, but gets quickly slower as the number of input/output channels increases. An alternative approach for MIMO systems, using μ-based tools was already proposed for disk margin analysis in [8]. This approach is used again here for MIMO phase, gain and delay margin analysis. However, it should be noted that the μ-based algorithm used should be adapted to the type of uncertainties (real/complex) in the studied stability margin problem. The developed MIMO analysis algorithms are all implemented in the STOWAT.
Besides MIMO analysis, there is also an increased interest in multivariable margin analysis. This is because most realistic systems are subject to multiple perturbations at the same time. Multivariable analysis can for instance be an alternative to disk margin analysis [6], [8] in the case of simultaneous analysis of gain and phase perturbations. A probabilistic multivariable margin analysis algorithm is proposed in this contribution and implemented in the STOWAT. It was developed to overcome the conservatism provided by the deterministic worst-case equivalent at the end of the distribution tail. The STOWAT implementation allows users to specify multiple desired stability margins and determine the probability of multivariable margin violation. Analysis can be performed for both SISO and MIMO systems, where for MIMO systems different margin requirements can be set for each input and/or output. The heart of the existing algorithms remains the same, but two main modifications are needed. First a few additional matrix operations to construct the perturbed system used by the B&B algorithm should be included. Then new μ-analysis based conditions involving multiple real and complex uncertainties should be defined to determine whether the satisfaction or violation test should be performed.
To demonstrate the added value of the developed tools, they are applied to analyse two satellite models: an academic model and a realistic benchmark. The academic model represents the spinning satellite adapted from [10] and the realistic one concerns the satellite with two flexible solar panels, previously introduced in [5].
References
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