State, Energy and Power

Abstract

If at any instant of time the system dynamics is fully determined by a finite number of variables x₁,…,x_N, we call them state variables of the system. The time instantaneous values of these variables give information on the present state of the system as well as on its past. Thus the state is represented by a state vector $\bar{X}\left( t \right) = {{\left( {{{X}_{1}}\left( t \right),...,{{X}_{N}}\left( t \right)} \right)}^{T}}$ corresponding to a variable point in the state space ℝ ^N that describes the motion of the system. There t is the independent time variable t ≥ t₀, $\rlap{--}{V}{{t}_{0}} \in \mathbb{R}$, where as before t₀ is the initial instant. We let Δ be a given bounded set (or its closure) in ℝ^N, representing the formal (mathematical) and physical constraints imposed upon the state $\bar X\left( t \right)$. It will often be called a set of admissible states. The motion may proceed indefinitely in Δ, i.e. for t ∈ ℝ⁺,, ℝ⁺ = [t_°,∞), or it may terminate at the finite instant The latter is either arbitrary or stipulated. Correspondingly ${{\bar{x}}^{0}}\underline{\underline \vartriangle } \bar{x}\left( {{{t}_{0}}} \right),{{\bar{x}}^{f}}\underline{\underline \vartriangle } \bar{x}\left( {{{t}^{f}}} \right)$ denote the initial and terminal states.