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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 .
Simultaneous transitions in multiple finite-state machines can be shown in what is effectively an n-dimensional state-transition table in which pairs of rows map (sets of) current states to next states. [1] This is an alternative to representing communication between separate, interdependent finite-state machines.
For such a system, the weighting pattern is (,) = (,) such that is the state transition matrix. The weighting pattern will determine a system, but if there exists a realization for this weighting pattern then there exist many that do so.
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.
Change-of-basis matrix, associated with a change of basis for a vector space. Stochastic matrix , a square matrix used to describe the transitions of a Markov chain . State-transition matrix , a matrix whose product with the state vector x {\displaystyle x} at an initial time t 0 {\displaystyle t_{0}} gives x {\displaystyle x} at a later time t ...
The state-transition equation is defined as the solution of the linear homogeneous state equation. The linear time-invariant state equation given by = + + (), with state vector x, control vector u, vector w of additive disturbances, and fixed matrices A, B, E can be solved by using either the classical method of solving linear differential equations or the Laplace transform method.
This state-space realization is called controllable canonical form (also known as phase variable canonical form) because the resulting model is guaranteed to be controllable (i.e., because the control enters a chain of integrators, it has the ability to move every state).
The matrix () = = () is nonsingular for any >. ... is the state transition matrix of ˙ = (), is nonsingular. Again, we have a similar method to determine if a system ...