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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.
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
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 .
In computer science, state space enumeration are methods that consider each reachable program state to determine whether a program satisfies a given property. [1] As programs increase in size and complexity, the state space grows exponentially. The state space used by these methods can be reduced by maintaining only the parts of the state space ...
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 ...
Consider a physical system modeled in state-space representation. A system is said to be observable if, for every possible evolution of state and control vectors, the current state can be estimated using only the information from outputs (physically, this generally corresponds to information obtained by sensors). In other words, one can ...
For the simplest example of a continuous, LTI system, the row dimension of the state space expression ˙ = + determines the interval; each row contributes a vector in the state space of the system. If there are not enough such vectors to span the state space of x {\displaystyle \mathbf {x} } , then the system cannot achieve controllability.
In quantum mechanics and computing, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system , named after the physicist Felix Bloch. [ 1 ] Mathematically each quantum mechanical system is associated with a separable complex Hilbert space H {\displaystyle H} .