Observability of linear differential-algebraic equations in the class of Chebyshev functions

We consider linear time-varying systems of first order ordinary differential equations unresolved with respect to the derivative of the unknown function and identically degenerate in the domain. Such systems are called differential-algebraic equations (DAE). The unsolvability measure with respect to the derivatives for some DAE is an integer that is called the index of the DAE. We admit an arbitrarily high unsolvability index not more than the order of the system. The analysis is carried out under the assumption of existence of a structural form with separated differential and algebraic subsystems. This structural form is equivalent to the input system in the sense of solution, and the operator transforming the DAE into the structural form possesses the left inverse operator. The finding of the structural form is constructive and do not use a change of variables. In addition the problem of consistency of the initial data is solved automatically. The approach uses the concept of $r$-derivative array equations, where r is the unsolvability index. The existence of a nonsingular minor of order $n(r + 1)$ in the matrix describing derivative array equations is necessary and sufficient for existence of this structural form (n is the dimension of DAE under consideration). We investigate observability of DAE by a given scalar output. The problem of observability consists in finding the state vector of the system on the basis of incomplete data on its components determined by the output function. As a class of functions of resolving operations, that is, solving the observability problem, in addition to piecewise continuous ones, we consider the class of generalized Chebyshev functions. It is obtained a sufficient condition of $R$-observability (observability in the reachable set) of linear non-stationary systems of DAE in the class of Chebyshev polynomials. We consider the example illustrating the obtained results.