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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 .
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.
In the state-transition table, all possible inputs to the finite-state machine are enumerated across the columns of the table, while all possible states are enumerated across the rows. If the machine is in the state S 1 (the first row) and receives an input of 1 (second column), the machine will stay in the state S 1.
A Kripke structure is a variation of the transition system, originally proposed by Saul Kripke, [1] used in model checking [2] to represent the behavior of a system. It consists of a graph whose nodes represent the reachable states of the system and whose edges represent state transitions, together with a labelling function which maps each node ...
In control theory, a state observer, state estimator, or Luenberger observer is a system that provides an estimate of the internal state of a given real system, from measurements of the input and output of the real system. It is typically computer-implemented, and provides the basis of many practical applications.
In control theory, we may need to find out whether or not a system such as ˙ = + () = + is controllable, where , , and are, respectively, , , and matrices for a system with inputs, state variables and outputs.
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.
Linear Time Invariant (LTI) Systems are those systems in which the parameters , , and are invariant with respect to time. One can determine if the LTI system is or is not observable simply by looking at the pair ( A , C ) {\displaystyle ({\boldsymbol {A}},{\boldsymbol {C}})} .