The observability of linear singularly perturbed systems in state space (Q1339573)

A linear stationary singularly perturbed (LSSP) system of differential equations is considered. \[ \begin{aligned} x'(t) &= A_ 1 x(t) + A_ 2 y(t)\\y'(t) & = A_ 3 x (t) + A_ 4 y(t),\;t\geq t_ 0 \end{aligned} \tag{1} \] \(\mu\) a small parameter \((\mu > 0)\) and the output vector-valued function \(w(t) = D_ 1 x(t,\mu) + D_ 2y (t,\mu)\), \(t \in [t_ 0,t_ 1]\). The problem of complete observability is: Given measurements \(w(t)\), \(t \in [t_ 0,t_ 1]\) one can uniquely reconstruct the initial state \((x(t_ 0), y(t_ 0))\) of system (1). Criteria, necessary and sufficient conditions, phrased in terms of matrix ranks, are obtained for the observability problem; they involve the solutions recursive algebraic matrix equations. Duality principles are established between the LSSP observed systems and the control systems with coefficients having varying time-scales. An example is given.