Linear Processes Invariant in Time

Abstract

It is known that the usual analytical modeling of the linear processes invariant in time could be expressed by:

$$ {\dot{\mathbf{x}}} = {\mathbf{Ax}} + {\mathbf{Bu}} $$

(1.1)

$$ {\mathbf{y}} = {\mathbf{Cx}} + {\mathbf{Du}} $$

(1.2)

where: u = u(t), x = x(t) and y = y(t) represent the input vector, the state vector, the output vector respectively, and (A), (B), (C), and (D) correspond to the state matrix, the input-state matrix, the state-output matrix, to the input-output matrix respectively. These matrices are constant, and the initial conditions (IC), for t = t ₀, respectively x _IC = x(t ₀) are considered to be known. In the hypothesis that the known input vector u = u(t) presents a continuous evolution with respect to time (t), the solution of the ordinary differential equation (ode), in its vectorial form (1.1) respects the continuity conditions in the Cauchy sense.