Nonobservable or Noncontrollable Analog Schemes for the Solution of a Special Class of Differential Equations

The analog computer solution of linear time invariant differential equations of the type pⁿy + an-1p^n-1y +...+ a0y = bmp^mu + bm-1p^m-1u... + b0u; (p=d/dt) is treated for the case in which the polynomials Sⁿ + an-1S^n-1 +...+ a0 and bmS^m + bm-1S^m-1 + ... + b0 have common roots. It is shown that the ``dummy variable´´ [2] method of analog solution is not suitable if we want not only the ``forced´´ solution, but also the ``free´´ or ``homogeneous´´ solution (corresponding to arbitrary initial conditions y(0), py(0) ... p^n-1y(0)). It is in fact impossible to find the initial conditions x1(0) ... xn(0) at the output of the integrators in the analog scheme, corresponding to the ``dummy variable´´ method. The reason is found in the framework of linear system theory, which shows that this scheme represents a non observable system. An intuitive explanation of this fact is shown at the end of the paper. Other analog computer schemes are shown which do not present this difficulty, and which give the ``free´´ as well as the ``forced´´ solution of our equation.