Ohjausenergia

Ohjausenergia, or control energy, is a concept in control theory that describes the amount of energy contained in the external control input required to drive a dynamical system from a given initial state to a specified final state within a set time horizon. It is commonly defined as the integral of the squared control signal, typically ∫_0^T ||u(t)||^2 dt, over the control interval.

In mathematical terms, for a system described by x'(t) = f(x(t), u(t), t), the minimum energy control problem seeks to minimize the control energy subject to the system dynamics and boundary conditions x(0) = x0 and x(T) = xF. In linear time-invariant systems, x' = Ax + Bu, the problem can be analyzed using the reachability (or controllability) Gramian W_r(T) = ∫_0^T e^{A t} B B^T e^{A^T t} dt. The minimum energy required to reach xF from the origin in time T is E_min = xF^T W_r(T)^{-1} xF, assuming suitable controllability. In general, optimal control formulations may combine energy terms with state penalties, as in the linear quadratic regulator (LQR) problem, where the cost includes both x^T Q x and u^T R u.

Ohjausenergia is applicable in robotics, aerospace, electrical engineering, and any field involving precise state transfers under energy constraints. It highlights the trade-off between time, energy, and accuracy, and its computation can be straightforward for linear systems but becomes more complex for nonlinear dynamics, often requiring numerical methods.