State Estimation for a Class of Singular Systems Using Shifted Legendre Polynomials

For linear time-invariant continuous-time singular systems, two new simple approaches, the Kronecker method and the recursive method, are developed in order to estimate the states from the system input-output information via shifted Legendre polynomials (SLP) and a simple observer in the descriptor form. Sufficient conditions for the existence of the present observer are given. The two methods simplify the system of state equations into the solution of a set of linear algebraic equations. Furthermore, the proposed algorithms can be easily implemented in a digital computer and the solutions can be obtained for any length of time. Further, these approaches include the filtering and the smoothing effect which can reduce the influence of zero-mean measurement noise on estimation. Simulation results of a given numerical example demonstrate the effect of the proposed approaches.