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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 applied mathematics, the Rosenbrock system matrix or Rosenbrock's system matrix of a linear time-invariant system is a useful representation bridging state-space representation and transfer function matrix form. It was proposed in 1967 by Howard H. Rosenbrock. [1]
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 .
By Gelfand representation, every commutative C*-algebra A is of the form C 0 (X) for some locally compact Hausdorff X. In this case, S(A) consists of positive Radon measures on X, and the pure states are the evaluation functionals on X. More generally, the GNS construction shows that every state is, after choosing a suitable representation, a ...
If the size of the state space is finite, calculating the size of the state space is a combinatorial problem. [4] For example, in the Eight queens puzzle, the state space can be calculated by counting all possible ways to place 8 pieces on an 8x8 chessboard. This is the same as choosing 8 positions without replacement from a set of 64, or
State-space representation is especially powerful as it allows complex multi-order differential system to be solved as a system of first-order equations instead. The general form of the state equation is ˙ = + where () is a column matrix of the state variables, or the unknowns of the system.
In control engineering, a state-space representation is a mathematical model of a physical system as a set of input, output, and state variables, related by first-order differential equations. The dynamic evolution of a nonlinear, non-autonomous system is represented by
State variables can be redundant (dependent); a state-space model with the number of state variables is a minimal realization, which is a special (yet important) case. One should define "realization" first, noting that it isn't unique (see some of my previous contributions).