МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ ТА АНАЛІЗ КЕРОВАНОСТІ СИНГУЛЯРНИХ ГІБРИДНИХ СИСТЕМ В ПРИКЛАДНИХ ЗАДАЧАХ

The article investigates singular hybrid systems (SHS), which combine differential-algebraic structures with hybrid dynamics involving mode switching. A mathematical model is presented, approaches to stability and controllability analysis are substantiated, and numerical simulation is performed. Such systems naturally arise in complex engineering applications – from energy systems to biomedical devices – where the system behavior changes due to faults, mode transitions, or internal structural variations. A mathematical model of SHS is proposed based on matrices, which depend on the discrete mode), with potentially being singular, leading to a differential-algebraic formulation. The controllability and stability of such systems are analyzed with a focus on the specific challenges of differential-algebraic formulation – including algebraic constraints, impulsive responses, and non-uniqueness of solutions. A case study of a two-mode system is presented: one regular mode with full dynamics, and one singular mode where certain state variables become constrained. The transition between modes demonstrates the shift from dynamic to algebraic behavior. Approaches to stability analysis are proposed, including hybrid Lyapunov functions, Weierstrass decomposition, and algebraic controllability criteria that account for matrix singularity. Numerical simulation is performed in Python using a state-feedback control law. The results confirm the model's capability to accurately capture SHS behavior, explore the effects of singularity on state evolution, and highlight the need for careful control design under mode transitions. Plots illustrate the characteristic response of the system, particularly adaptation to algebraic constraints and loss of controllability. The proposed model and simulation outcomes offer a basis for developing robust and adaptive control strategies in systems where singular hybrid dynamics are intrinsic to the control process.