transitionmodell

A transitionmodell describes how a system moves from one state to another over time. In discrete time it is often written as x_{t+1} = f(x_t, u_t), where x_t denotes the system state and u_t represents inputs or actions. In stochastic or probabilistic settings the model specifies a distribution over possible next states, P(x' | x, u). In continuous time the evolution is described by differential equations, such as dx/dt = f(x, u). A transitionmodell typically involves a state space X, an input or control space U, and possibly an output or observation function.

Deterministic and stochastic variants are common. Deterministic models yield a unique next state given the current state and input, while stochastic models assign probabilities to different next states. Discrete-time and continuous-time formulations address different temporal resolutions. Many models assume the Markov property, where the next state depends only on the current state and input, not on the past.

Applications and usage vary. In control engineering and robotics, the transitionmodell is used for simulation, planning, and feedback control. In reinforcement learning, the transition model P(s' | s, a) describes the environment’s dynamics; model-based approaches learn or use this transition to plan actions. In statistics and epidemiology, transitionmodels describe progression between states such as susceptible, infected, and recovered. They also serve as building blocks for state estimation, filtering, and prediction when observations are noisy or incomplete.

Design and challenges include selecting an appropriate level of detail, handling non-stationarity, dealing with high-dimensional state spaces, and balancing data requirements with model fidelity. Accurate transitionmodels enable realistic simulation, robust control, and informed decision-making in dynamic systems.