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The state-transition matrix is used to find the solution to a general state-space representation of a linear system in the following form ˙ = () + (), =, where () are the states of the system, () is the input signal, () and () are matrix functions, and is the initial condition at .
Now if the machine is in the state S 1 and receives an input of 0 (first column), the machine will transition to the state S 2. In the state diagram, the former is denoted by the arrow looping from S 1 to S 1 labeled with a 1, and the latter is denoted by the arrow from S 1 to S 2 labeled with a 0.
The changes of state of the system are called transitions. The probabilities associated with various state changes are called transition probabilities. The process is characterized by a state space, a transition matrix describing the probabilities of particular transitions, and an initial state (or initial distribution) across the state space ...
The matrix on the left shows how probabilities corresponding to different states can be arranged in matrix form. A state diagram for a simple example is shown in the figure on the right, using a directed graph to picture the state transitions.
The state space or phase space is the geometric space in which the axes are the state variables. The system state can be represented as a vector, the state vector. If the dynamical system is linear, time-invariant, and finite-dimensional, then the differential and algebraic equations may be written in matrix form.
Figure 7: State roles in a state transition. In UML, a state transition can directly connect any two states. These two states, which may be composite, are designated as the main source and the main target of a transition. Figure 7 shows a simple transition example and explains the state roles in that transition.
Transition graph with transition probabilities, exemplary for the states 1, 5, 6 and 8. There is a bidirectional secret passage between states 2 and 8. The image to the right describes a discrete-time Markov chain modeling Pac-Man with state-space {1,2,3,4,5,6,7,8,9}. The player controls Pac-Man through a maze, eating pac-dots.
The fundamental matrix is used to express the state-transition matrix, an essential component in the solution of a system of linear ordinary differential equations. [3]