A normal form for pure differential algebraic systems
The paper deals with differential algebraic equations (DAEs) of the form \[ E \dot x = Ax + bu, \quad y=cx, \eqno(1) \] where \(x \in {\mathbb R}^n\) and \(y \in {\mathbb R}^1\) are variables depending on the real time \(t\), \(\dot x = \frac{dx}{dt}\), \(E\) and \(A\) are \((n \times n)\)-matrices with constant coefficients, \(b\) is a constant column vector, \(c\) is a constant row vector, \(u\) is a scalar input (control). The equations \(E \dot x = Ax + bu\) are also called descriptor system, quasilinear systems or generalized vector field of a constant rang, the studying of such DAEs goes back to Weierstrass and Kronecker. In the case \(\det E \neq 0\) the DAE (1) turns into a standard ODE. In this paper the author mainly consider the antipodal case, so-called pure DAEs: \(\det E = 0\), \(\det A \neq 0\), and the matrix \(A^{-1}E\) is nilpotent. The rational function \(g(s) = c(Es-A)^{-1}b\) is called the transfer function of (1). In particular, the transfer functions of pure DAEs (1) are polynomials. The author proves that for a given rational function \(g(s)\) there exists DAE (1) with the transfer function \(g(s)\), which is a composition of some pure DAE and some ODE. The main result of the paper is the normal form (Theorem 20) for pure DAEs (1), which allows to simplify the matrices \(E, A\) and vectors \(b ,c\). Here, the equivalence of two DASs (1) with the same dimension \(n\) and corresponding \((c_1, E_1, A_1, b_1)\) and \((c_1, E_1, A_1, b_1)\) means that there exist invertible \((n \times n)\)-matrices \(S\) and \(T\) such that \(c_2=c_1 T\), \(E_2=S E_1 T\), \(A_2=S A_1 T\), \(b_2=S b_1\). Using this normal form, the author gives characterizations of impulsive controllability and impulsive observability for pure DAEs (1).
scientific article; zbMATH DE number 5505614
scientific article