Academic literature on the topic 'Probabilistic μ-Analysis'

A Markov chain framework is developed for analyzing a wide variety of selection techniques used in genetic algorithms (GAs) and evolution strategies (ESs). Specifically, we consider linear ranking selection, probabilistic binary tournament selection, deterministic s-ary (s = 3,4, …) tournament selection, fitness-proportionate selection, selection in Whitley's GENITOR, selection in (μ, λ)-ES, selection in (μ + λ)-ES, (μ, λ)-linear ranking selection in GAs, (μ + λ)-linear ranking selection in GAs, and selection in Eshelman's CHC algorithm. The analysis enables us to compare and contrast the various selection algorithms with respect to several performance measures based on the probability of takeover. Our analysis is exact—we do not make any assumptions or approximations. Finite population sizes are considered. Our approach is perfectly general, and following the methods of this paper, it is possible to analyze any selection strategy in evolutionary algorithms.

Model uncertainty, if ignored, can seriously degrade the performance of an otherwise well-designed control system. If the level of this uncertainty is extreme, the system may even be driven to instability. In the context of structural control, performance degradation and instability imply excessive vibration or even structural failure. Robust control has typically been applied to the issue of model uncertainty through worst- case analyses. These traditional methods include the use of the structured singular value (μ-analysis), as applied to the small gain condition, to provide estimates of controller robustness. However, this emphasis on the worst-case scenario has not allowed a probabilistic understanding of robust control. Because of this, an attempt to view controller robustness as a probability measure is presented. As a result, a much more intuitive insight into controller robustness can be obtained. In this context, the joint probability distribution is of dimension equal to the number of uncertain parameters, and the failure hypersurface is defined by the onset of instability of the closed-loop system in the eigenspace. A first-order reliability measure (FORM) of the system is computed and used to estimate controller robustness. It is demonstrated via an example that this FORM method can provide accurate results on the probability of failure despite the potential complexity of the closed-loop. In addition to the FORM method, a probabilistic measure of robustness is developed based on the fundamentals of μ-analysis. It is shown that the μ-analysis based method is inferior to the FORM method and can only have qualitative value when assessing control system robustness in a probabilistic framework.

An experimental-probabilistic analysis of variations in the load-bearing capacity of bending reinforced concrete elements with matrices reinforced by polypropylene fibres was carried out. A numerical ex-periment was conducted using the normative methodology of multi-link and layerwise modelling of el-ement cross-sections and experimental "σ-ε" diagrams of fibrocomposites in initial and post-cyclic (50 cycles with η = 0.8 amplitude and zero asymmetry) states. Probabilistic changes in the load-bearing capacity of bending elements subjected to cyclic loads were estimated by the numerical strength modelling of rectangular beams (b × h = 100 × 200 mm) with the one-sided reinforcement (A400 class) of varying intensity. The observed high value of fatigue life of reinforced concrete ele-ments with fibre-reinforced matrices was found to be associated with the presence of mechanisms compensatory for structural changes, i.e., a decrease in the strength is accompanied by an increase in the ability to redistribute internal forces. A post-cyclic reduction in the strength of concrete causes practically no effect on the load-bearing capacity of bending elements with a large and economically preferable range of their structural reinforcement. The reliability kinetics of elements, estimated by the level of the realised concrete strength potential, was analysed. Moderate (μ ≤ μR) reinforcement was found to result in objective conditions for increasing the completeness of the stress diagram in the compressed cross-sectional part due to the redistribution of forces along the height. In this case, de-spite a significant decrease in the strength of concrete, the load-bearing capacity of elements at μ ≤ 2.5% reinforcement remains practically the same after cyclic effects.

Abstract The validity of the Riemann hypothesis (RH) on the location of the non-trivial zeros of the Riemann ζ-function is directly related to the growth of the Mertens function M ( x ) = ∑ k = 1 x μ ( k ) , where μ(k) is the Möbius coefficient of the integer k; the RH is indeed true if the Mertens function goes asymptotically as M(x) ∼ x 1/2+ϵ , where ϵ is an arbitrary strictly positive quantity. We argue that this behavior can be established on the basis of a new probabilistic approach based on the global properties of the Mertens function, namely, based on reorganizing globally in distinct blocks the terms of its series. With this aim, we focus attention on the square-free numbers and we derive a series of probabilistic results concerning the prime number distribution along the series of square-free numbers, the average number of prime divisors, the Erdős–Kac theorem for square-free numbers, etc. These results point to the conclusion that the Mertens function is subject to a normal distribution as much as any other random walk. We also present an argument in favor of the thesis that the validity of the RH also implies the validity of the generalized RH for the Dirichlet L-functions. Next we study the local properties of the Mertens function, i.e. its variation induced by each Möbius coefficient restricted to the square-free numbers. Motivated by the natural curiosity to see how closely to a purely random walk any sub-sequence is extracted by the sequence of the Möbius coefficients for the square-free numbers, we perform a massive statistical analysis on these coefficients, applying to them a series of randomness tests of increasing precision and complexity; together with several frequency tests within a block, the list of our tests includes those for the longest run of ones in a block, the binary matrix rank test, the discrete Fourier transform test, the non-overlapping template matching test, the entropy test, the cumulative sum test, the random excursion tests, etc, for a total of 18 different tests. The successful outputs of all these tests (each of them with a level of confidence of 99% that all the sub-sequences analyzed are indeed random) can be seen as impressive ‘experimental’ confirmations of the Brownian nature of the restricted Möbius coefficients and the probabilistic normal law distribution of the Mertens function analytically established earlier. In view of the theoretical probabilistic argument and the large battery of statistical tests, we can conclude that while a violation of the RH is strictly speaking not impossible, it is however extremely improbable.

The paper considers the problem of analysis of the information transmission process by multi-element communication systems in presence of a multipath signal propagation channel. To generalize the propagation effects, the model of the κ–μ fading channel with correlated shadowing was assumed, and the technology used for organizing a multi-element system was the SIMO system, equipped with the maximum-ration combiner of the signal on the receiving side. To describe the characteristics of the information transfer process, an approach based on the higher-order statistics of the ergodic capacity was used. Closed-form analytical expressions for arbitrary-order capacity higher-order statistics were obtained for the channel model under consideration. The behavior of the first four statistics (ergodic capacity, its reliability, skewness and kurtosis coefficients) is analyzed depending on the channel parameters (the number of multipath propagation clusters, the ratio of power of the dominant components to the total power of multipath waves, the degree of shadowing of the dominant components, and the shadowing correlation coefficient). Within the framework of the study, 4 distinct situations of the assumed channel model behavior were considered, which significantly differ in their properties. It is noted that, in contrast to the capacity, its higher-order statistics are significantly more sensitive to the channel parameters and, as a result, are more significant indicators of fluctuations in the information transfer rate within the communication channel. The existence of a pronounced extremum (minimum) of the reliability ergodic capacity dependence from the signal-to-noise ratio was established. It should be accounted for in practical applications, when the requirements of the signal-to-noise ratio that guarantees the desired communication link quality are set.

Purpose. Structuring of experimental data of mine observations of the performance of arched flexible supports to establish a probabilistic assessment of the state of haulage drifts of steep coal seams in the excavation areas of a coal mine. Methods. The methodological basis of the research is an integrated approach, including the analysis and generalization of scientific achievements on the problem under study; mine observations of the state of development workings; analytical calculations using the basic provisions of probability theory, mathematical statistics using differential equations. Results. A probabilistic assessment of the state of the haulage drift under different protection methods, used to predict the stability of section development workings, was obtained from the experimental data of mine observations (observation time t = 4280 hours) of the performance of the arched pliable lining installed in the working along the length of the excavation section. After the sample was formed, the calculation of a statistical assessment of the reliability of the support operation was performed: the indicators of the failure rate λ and the recovery rate μ were established. Dependencies are obtained that allow assessing the state of development workings along the length of the excavation section and in the zone of influence of the stopping operations. It is proved that in the steady mode of operation of the haulage drift, with an increase in the failure rate of the arch support by 5 times, the availability factor, as a function of the reliability and maintainability of the support, changes from 0.9 to 0.5, which leads to a deterioration in time, approximately by 60 %, operational characteristics of the support in the supported development. Novelty. The probabilistic assessment of the loss of stability of the haulage drift along the length of the excavation section is based on the structuring of the data on the performance of arched supports and depends on the method of protection of the working, the intensity of failures of the arched supple support and the frequency of its damage, when in the desired interval of observations when forming the sample, the position of the stopping front is taken into account. Practical relevance. For the reuse of development workings with a store method of preparing steep seams and a descending order of mining floors, a conceptual model is proposed that allows predicting the state of haulage drifts along the length of the excavation area, taking into account the method of protection.

In brake systems, some dynamic phenomena can worsen the performance (e.g., fading, hot banding), but a major part of the research concerns phenomena which reduce driving comfort (e.g., squeal, judder, or creep groan). These dynamic phenomena are caused by specific instabilities that lead to self-excited oscillations. In practice, these instabilities can be investigated using the Complex Eigenvalues Analysis (CEA), in which positive real parts of the eigenvalues are identified to characterize instable regions. Measurements on real brake test benches or tribometers show that the coefficient of friction (COF), μ , is not a constant, but dynamic, system variable. In order to consider this aspect, the Method of Augmented Dimensioning (MAD) has been introduced and implemented, which couples the mechanical degrees of freedom of the brake system with the degrees of freedom of the friction dynamics. In addition to this, instability prediction techniques can often determine whether a system is stable or instable, but cannot eliminate the instability phenomena on a real brake system. To address this, the current work deals with the quantification of the relevant polymorphic uncertainty of the friction dynamics, wherein the aleatory and epistemic uncertainties are described simultaneously. Aleatory uncertainty is concerned with the stochastic variability of the friction dynamics and incorporated with probabilistic methods (e.g., a Monte Carlo simulation), while the epistemic uncertainty resulting from model uncertainties is modeled via fuzzy methods. The existing measurement data are collected and processed through Data Driven Methods (DDM) for the identification of the dynamic friction models and corresponding parameters. Total Variation Regularization is used for the evaluation of derivatives within noisy data. Using an established minimal model for brake squealing, this paper addresses the question of probabilities for instabilities and the degree of certainty with which this conclusion can be made. The focus is on a comparison between the conventional Coulomb friction model and a dynamic friction model in combination with the MAD. This shows that the quality of the predictive accuracy improves dramatically with the more precise friction model.

In this article, we give the sharp bounds of probabilistic Kolmogorov N,δ-widths and probabilistic linear N,δ-widths of the multivariate Sobolev space W2A with common smoothness on a Sq norm equipped with the Gaussian measure μ, where A⊂Rd is a finite set. And we obtain the sharp bounds of average width from the results of the probabilistic widths. These results develop the theory of approximation of functions and play important roles in the research of related approximation algorithms for Sobolev spaces.

In this paper, we study the probabilistic Gel’fand N,δ-width of multivariate Sobolev spaces MW2rTd with mixed derivative that are equipped with Gaussian measure μ in LqTd. The sharp asymptotic estimates are determined by employing the discretization method.

Les activités actuelles de vérification et validation (V&V) dans l'industrie aérospatiale reposent principalement sur des outils de simulation qui prennent beaucoup de temps. Ces approches classiques de type Monte-Carlo sont largement utilisées depuis des décennies pour évaluer les performances des systèmes de guidage, de navigation et de contrôle (GNC) et des systèmes de contrôle d'attitude et d'orbite (SCAO) contenant de multiples paramètres incertains. Elles permettent de quantifier la probabilité d'occurrence de phénomènes suffisamment fréquents, mais peuvent échouer dans la détection de combinaisons rares, mais critiques, de paramètres. Au fur et à mesure que la complexité des systèmes spatiaux modernes augmente, cette limitation joue un rôle de plus en plus important. Ces dernières années, les méthodes d'analyse des pires cas basées sur des modèles ont atteint un bon niveau de maturité. Sans avoir recours à des simulations, ces outils peuvent explorer l'espace de toutes les combinaisons possibles de paramètres incertains et fournir des limites mathématiques garanties sur les marges de stabilité robustes et les niveaux de performance pire-cas. Les configurations problématiques, identifiées à l'aide de ces méthodes, peuvent être utilisées pour guider les campagnes Monte-Carlo finales, ce qui raccourcit considérablement le processus V&V standard. L'une des limites des méthodes classiques d'analyse pire-cas basées sur des modèles est qu'elles supposent que les paramètres incertains peuvent prendre n'importe quelle valeur dans un intervalle donné avec une probabilité égale. La probabilité d'occurrence d'une combinaison de paramètres pire-cas n'est donc pas mesurée et la conception d'un système peut ainsi être rejetée sur la base d'un scénario très rare et extrêmement improbable. Cette recherche se concentre sur μ-analyse probabiliste pour développer de nouveaux outils efficaces et fiables afin d'améliorer la caractérisation d'événements rares mais néanmoins possibles. Ceci permet de resserrer l'écart d'analyse V&V ci-dessus entre les méthodes basées sur la simulation et les approches pire-cas déterministes basées sur des modèles
Current verification and validation (V&V) activities in aerospace industry mostly rely on time-consuming simulation-based tools. These classical Monte Carlo approaches have been widely used for decades to assess performance of Guidance, Navigation and Control (GNC) algorithms and Attitude and Orbit Control Systems (AOCS) containing multiple uncertain parameters. They are able to quantify the probability of 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 more important 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 margins and worst-case performance levels. Problematic parameter configurations, identified using these methods, can be used to guide the final Monte Carlo campaigns, thereby drastically shortening the standard V&V process. A limitation of classical model-based worst-case analysis methods is that they assume the uncertain parameters can take any value within a given range with equal probability. The probability of occurrence of a worst-case parameter combination is thus not measured and a control architecture can be rejected based on a very rare and extremely unlikely scenario. This PhD research makes advances in probabilistic μ-analysis to develop new efficient and reliable tools to improve the characterization of rare but nonetheless possible events. This to tighten the aforementioned V&V analysis gap between simulation-based methods and deterministic model-based worst-case approaches

AbstractIt is well-known that the winning region of a parity game with n nodes and k priorities can be computed as a k-nested fixpoint of a suitable function; straightforward computation of this nested fixpoint requires $$\mathcal {O}(n^{\frac{k}{2}})$$ O ( n k 2 ) iterations of the function. Calude et al.’s recent quasipolynomial-time parity game solving algorithm essentially shows how to compute the same fixpoint in only quasipolynomially many iterations by reducing parity games to quasipolynomially sized safety games. Universal graphs have been used to modularize this transformation of parity games to equivalent safety games that are obtained by combining the original game with a universal graph. We show that this approach naturally generalizes to the computation of solutions of systems of any fixpoint equations over finite lattices; hence, the solution of fixpoint equation systems can be computed by quasipolynomially many iterations of the equations. We present applications to modal fixpoint logics and games beyond relational semantics. For instance, the model checking problems for the energy $$\mu $$ μ -calculus, finite latticed $$\mu $$ μ -calculi, and the graded and the (two-valued) probabilistic $$\mu $$ μ -calculus – with numbers coded in binary – can be solved via nested fixpoints of functions that differ substantially from the function for parity games but still can be computed in quasipolynomial time; our result hence implies that model checking for these $$\mu $$ μ -calculi is in $$\textsc {QP}$$ QP . Moreover, we improve the exponent in known exponential bounds on satisfiability checking.

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 [1] J.-M. Biannic, C. Roos, S. Bennani, F. Boquet, V. Preda, and B. Girouart, “Advanced probabilistic μ-analysis techniques for AOCS validation,” European Journal of Control, vol. 62, pp. 120–129, 2021. [2] F. Somers, C. Roos, F. Sanfedino, S. Bennani, and V. Preda, “Probabilistic delay margin analysis,” Submitted to the American Control Conference, 2023. [3] C. Roos, J.-M. Biannic, and H. Evain, “A new step towards the integration of probabilistic μ in the aerospace V&V process,” in Proceedings of the 6th CEAS Conference on Guidance, Navigation and Control, 2022. [4] C. Roos, F. Sanfedino, V. Preda, and S. Bennani, “Phd position in analysis of aerospace control systems: Enhanced probabilistic tools to improve verification and validation of space control systems,” 2021. [5] F. Sanfedino, D. Alazard, E. Kassarian, and F. Somers, “Satellite dynamics toolbox library: a tool to model multi body space systems for robust control synthesis and analysis,” Submitted to the IFAC World Congress, 2023. [6] P. Seiler, A. Packard, and P. Gahinet, “An introduction to disk margins [lecture notes],” IEEE Control Systems Magazine, vol. 40, no. 5, pp. 78–95, 2020. [7] F. Somers, C. Roos, F. Sanfedino, S. Bennani, and V. Preda, “Comparative study of new probabilistic delay margin analysis techniques,” Submitted to International Journal of Robust and Nonlinear Control, 2023. [8] F. Somers, S. Thai, C. Roos, J.-M. Biannic, S. Bennani, V. Preda, and F. Sanfedino, “Probabilistic gain, phase and disk margins with application to AOCS validation,” in Proceedings of the 10th IFAC Symposium on Robust Control Design, 2022. [9] S. Thai, C. Roos, and J.-M. Biannic, “Probabilistic μ-analysis for stability and H∞ performance verification,” in Proceedings of the American Control Conference, 2019. [10] K. Zhou, J. Doyle, and K. Glover, Robust and optimal control. Prentice-Hall, New Jersey, 1996.

During the development of a new attitude control system for ambitious satellite missions, the validation & verification phase represents a large part of the process. One difficulty is to detect worst case configurations. In such cases, when applicable, µ-analysis [1] offers a nice additional tool to be used before launching the Monte Carlo simulation campaign, but does not provide any quantification of the probability of occurrence of the identified worst-cases. A control system can then be invalidated on the basis of unlikely events. Probabilistic µ-analysis was introduced in this context 20 years ago to bridge the gap between the two techniques. It has been used for the first time in [2] in the challenging context of validation of launcher thrust vector control systems. But it appeared to be computationally very expensive. At that time indeed, no practical tool offering both good reliability and reasonable computational time was available, making this technique hardly usable in an industrial context. After the preliminary work of [3,4], strong improvements have been achieved by ONERA supported by ESA and CNES to develop the STOchastic Worst-case Analysis Toolbox (STOWAT). With the help of this new Matlab toolbox, probabilistic µ-analysis may now be considered as a very good candidate for integration in the aerospace V&V process in a near future, finding its place between Monte Carlo simulations – useful for quantifying the probability of sufficiently frequent phenomena – and worst-case μ-analysis – relevant for detecting extremely rare events. Recently tested on a series of AOCS benchmarks of increasing complexity [5,6,7], the most recent version of the toolbox is now evaluated for the first time on a more challenging and realistic attitude control problem. The analysis focuses both on the normal mode (MNO) and on the orbit control mode (MCO) of the CNES MicroCarb mission [8,9]. The paper compares and discusses the results which have been obtained with different V&V techniques, critically assessing the advantages of the innovative method with respect to more classical procedures. [1] C. Roos. Systems Modeling, Analysis and Control (SMAC) toolbox: an insight into the robustness analysis library. Proceedings of the IEEE CACSD Conference, Hyderabad, India, 2013. [2] A. Marcos, S. Bennani, C. Roux. Stochastic µ-analysis for launcher thrust vector control systems. Proceedings of the EuroGNC Conference, Toulouse, France, 2015. [3] A. Falcoz, D. Alazard, C. Pittet. Probabilistic µ-analysis for system performances assessment. Proceedings of the 20th IFAC World Congress, Toulouse, France, 2017. [4] S. Thai, C. Roos, J.M. Biannic. Probabilistic µ-analysis for stability and H∞ performance verification. Proceedings of the ACC, Philadelphia, PA, USA, 2019. [5] J.M. Biannic, C. Roos, S. Bennani, F. Boquet, V. Preda, B. Girouart. Advanced probabilistic µ-analysis techniques for AOCS validation. European Journal of Control, 62 (2021), pp. 120-129. [6] C. Roos, J-M. Biannic, and H. Evain. A new step towards the integration of probabilistic µ in the aerospace V&V process. Proceedings of the EuroGNC Conference, Berlin, Germany, 2022. [7] F. Somers, S. Thai, C. Roos,[ J-M. Biannic, S. Bennani, V. Preda, and F. Sanfedino. Probabilistic gain, phase and disk margins with application to AOCS validation. Proceedings of the IFAC ROCOND Symposium, Kyoto, Japan, 2022. [8] Arnaud Varinois and al., “MICROCARB: A micro-satellite for atmospheric CO2 monitoring”, 4S 2016 [9] Genin, F. and Viaud, F. “An innovative control law for Microcarb microsatellite”, 32nd annual AAS Guidance and Control Conference, 2018

Current and future space missions require increasingly more complex AOCS/GNC designs. Multiple drivers can be identified for this trend, such as larger deployable structures and the extended operational range due to more autonomous spacecraft. For each of these control problems, tailored solutions have to be developed with stringent performance requirements, while staying within time and cost constraints. The drivers highlight the need for efficient analysis tools that can support the Verification and Validation (V&V) process of AOCS/GNC systems. As pointed out in [1], the design phase of a project roughly takes about 20% of the total time, while the remaining 80% are used for planning and executing V&V activities. The increasing system complexity will only make this V&V gap worse. Thus, reducing the V&V effort through enhanced methods is a promising field for improved industrial efficiency. This paper presents the continued work on this topic building on the Airbus contribution to the multi-agency workshop & seminar series [2], [3]. A systematic and detailed framework to assess the robust stability and performance of uncertain systems is proposed, which aims to reduce the V&V engineering effort. Common V&V approaches often rely on large-scale simulation campaigns to cover the various system uncertainties in form of probabilistic Monte Carlo analyses. This process is very time consuming for complex simulations, does not give insights into the uncertainties that drive the stability and performance requirements, and might fail to identify critical scenarios. An alternative is to focus on worst-cases through analytical robustness analysis. Although techniques, such as the μ-analysis, have been available for decades, they have not been integrated into standard industrial processes. To facilitate the use of robustness analysis techniques, a step-by-step approach has been developed that results in an enhanced V&V framework. This framework combines various V&V techniques including worst-case analysis, sensitivity analysis, and optimization within both the time and frequency domain. It consists of 1) modelling the uncertainties through the Linear Fractional Transform (LFT) framework and forming the models from the LFT for stability and performance metrics, 2) defining the robustness metrics under analysis 3) identifying system driving uncertainties through a sensitivity analysis, 4) computing the worst-case degradation of the stability and performance metrics, and 5) performing analysis on the nonlinear simulator. The steps 1) - 4) are all performed within the frequency domain, whereas the last step is computed in the time domain. If the current control design fails the requirements, the obtained worst-cases can be used to synthesize a robust controller. To illustrate the steps in the enhanced V&V framework, an analysis on the robust attitude performance based on MetOp-SG (SAT-B) is presented. The uncertain multi-body dynamics are modelled according to the modular concept in [4] resulting in a compact LFT form. This model is used to analyse the degradation of stability metrics, such as gain, phase, and modulus margin, and performance metrics, such as pointing metrics defined in [5]. Additionally, the uncertainties that are driving these degradations are identified. This information is useful to further improve the control design and can be used as a model reduction technique for the LFT. Finally, the resulting worst-cases from the frequency domain are compared with nonlinear simulations in the time domain. Lastly, conclusions will be drawn how the enhanced V&V framework can potentially improve the current industrial standards. This comparison identifies future work that is necessary to pave the way for increased industrial efficiency in the future. [1] Dennehy, C., Bennani, S., Shankar, U., Vandersteen, J., and VanZwieten, T., “Verification and Validation (V&V) of Guidance & Control Systems: Restults From The First Inter-Agency Workshop [2] Winkler, S., Chapman, P., Juanpere, X. M., Ott, T., “AOCS/GNC Challenges, Solutions and Beyond: Engineering Passion versus for Industrial Efficiency”, Multi-Agency Workshop & Seminar Series, 2021. [3] Martin, M., Belien, F., Winkler, S., „Towards Increased AOCS/GNC Industrial Efficiency: Robust Performance Analysis Considering Real Mission Constraints”, Multi-Agency Workshop & Seminar Series, 2021. [4] Alazard, D., Sanfedino, F., “Satellite dynamics toolbox for preliminary design phase” In : 43rd Annual AAS Guidance and Control Conference, 2020. [5] ESSB-HB-E-003 Working Group, “ESA pointing error engineering handbook ESSB-HB-E-003,” Tech. rep., ESA, 2011.