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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.
The set of possible combinations of state variable values is called the state space of the system. The equations relating the current state of a system to its most recent input and past states are called the state equations, and the equations expressing the values of the output variables in terms of the state variables and inputs are called the ...
The continuous-time case is similar to the discrete-time case but now one considers differential equations instead of difference equations: ˙ = + (), = + ().An added complication now however is that to include interesting physical examples such as partial differential equations and delay differential equations into this abstract framework, one is forced to consider unbounded operators.
A state-space model is a representation of a system in which the effect of all "prior" input values is contained by a state vector. In the case of an m-d system, each dimension has a state vector that contains the effect of prior inputs relative to that dimension. The collection of all such dimensional state vectors at a point constitutes the ...
The state space is then factorized according to =, where is the space of 'external' states that are 'hidden' from the agent (in the sense of not being directly perceived or accessible), is the space of sensory states that are directly perceived by the agent, is the space of the agent's possible actions, and is a space of 'internal' states that ...
For example, if the dimension of the state space is greater than the dimension of the output, then there will be a set of possible state configurations for each individual output. That is, the system can have significant zero dynamics , which are trajectories of the system that are not observable from the output.
Example of a simple MDP with three states (green circles) and two actions (orange circles), with two rewards (orange arrows) A Markov decision process is a 4-tuple (,,,), where: is a set of states called the state space. The state space may be discrete or continuous, like the set of real numbers.
System in open-loop. If the closed-loop dynamics can be represented by the state space equation (see State space (controls)) _ ˙ = _ + _, with output equation _ = _ + _, then the poles of the system transfer function are the roots of the characteristic equation given by