Tilamuuttujiksi

Tilamuuttujiksi is a Finnish term used in mathematics, engineering and computer science to describe the representation of a dynamical system in terms of state variables. The phrase roughly means turning the system into state variables or expressing it with state variables. The goal is to obtain a state-space description that captures the system's dynamics with first-order differential or difference equations.

In continuous time, the system is described by a state vector x(t) and an input vector u(t). The evolution is given by dx/dt = f(x, u, t) and the output by y = h(x, u, t). If the system is linear and time-invariant, this becomes dx/dt = Ax + Bu, y = Cx + Du, with x containing the state variables. The choice of state variables is not unique; a suitable set models the system completely, and often with minimal order.

Example: a mass–spring–damper with m x'' + b x' + k x = u. Let x1 = x and x2 = x'. Then the state equations are dx1/dt = x2, dx2/dt = (1/m)(-b x2 - k x1 + u), and the output y = x1. In matrix form: dx/dt = [ [0, 1], [-k/m, -b/m] ] x + [0, 1/m]^T u; y = [1, 0] x.

Applications: tilamuuttujiksi is used in control design, simulation, and state estimation, and in analysis of controllability and observability. It enables standardized methods such as Kalman filtering and observer design, and facilitates discretization for digital implementation.

See also: state-space representation, state variable, differential equations.