The paper presents a tutorial introduction to the complex structured singular value (μ), emphasizing its mathematical aspects and applications in analyzing the performance and robustness of linear feedback systems. It discusses μ-based methods for robust stability and performance tests, using Linear Fractional Transformations (LFTs) to unify frequency-domain and state-space approaches. Key concepts include the definition of μ, its relationship with Linear Matrix Inequalities (LMIs), and its application in robust control. The paper also covers the Main Loop Theorem, which is central to μ-based analysis, and explores how μ can be used to derive robustness and performance criteria for systems with structured uncertainty. It discusses the computation of μ, its bounds, and its connection to other control theory concepts like the spectral radius and maximum singular value. The paper also addresses the use of μ in various control problems, including robust stability tests and performance bounds, and highlights the importance of μ in ensuring system robustness against structured perturbations. Theoretical results are presented, along with practical implications for control system design and analysis.The paper presents a tutorial introduction to the complex structured singular value (μ), emphasizing its mathematical aspects and applications in analyzing the performance and robustness of linear feedback systems. It discusses μ-based methods for robust stability and performance tests, using Linear Fractional Transformations (LFTs) to unify frequency-domain and state-space approaches. Key concepts include the definition of μ, its relationship with Linear Matrix Inequalities (LMIs), and its application in robust control. The paper also covers the Main Loop Theorem, which is central to μ-based analysis, and explores how μ can be used to derive robustness and performance criteria for systems with structured uncertainty. It discusses the computation of μ, its bounds, and its connection to other control theory concepts like the spectral radius and maximum singular value. The paper also addresses the use of μ in various control problems, including robust stability tests and performance bounds, and highlights the importance of μ in ensuring system robustness against structured perturbations. Theoretical results are presented, along with practical implications for control system design and analysis.