Let $T(s)$ be a transfer function that describes a mechanical system, where the input is force and the output is position. And let $[A,B,C,D]$ be the equivalent state-space representation of $T(s)$, where: $$ \dot{x}(t) = Ax(t) + Bu(t) \\ y(t) = Cx(t) + Du(t) $$ and let $[A_d,B_d,C_d,D_d]$ be the discretized model of $T(s)$, using zero-order hold on the inputs: $$ x_d(k+1) = A_dx_d(k) + B_du(k) \\ y(k) = C_dx_d(k) + D_du(k) $$
I know there are infinite possible realizations of $[A,B,C,D]$ that represent the same transfer function $T(s)$.
- How can I force the state vector, $x(t)$, to have a simple physical meaning? in this example, since it is a mechanical system, the elements of the state vector $x(t)$ would probably be position, velocity and acceleration. I can guess the matrix $A$ must have this form: $$ A = \left[\begin{array}{ccc} * & * & * \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right] $$ to express the derivative relationship between the state vector elements.
- Same question, but for the discrete time case: How can I force the state vector, $x_d(k)$, to have a simple physical meaning? In this case, I'm not sure how "velocity" and "acceleration" can be expressed.