Parameter identification of fractional-order systems with time delays based on a hybrid of orthonormal Bernoulli polynomials and block pulse functions

In this paper, we present effective and efficient identification methods with a higher accuracy for fractional order system (FOS) with time delays and nonzero initial condition. We construct a new hybrid of orthonormal Bernoulli polynomials and block pulse functions (HOBPBPFs), and then derive the explicit representation for fractional integral operational matrix (FIOM) based on the HOBPBPFs via Laplace transformation. The orthogonality of polynomials gives us a higher accuracy of the proposed method. Based on it, new identification methods for FOS with input or state delay as well as nonzero initial condition are proposed. In particular, the variable least squares identification method for FOS with state delay using a hybrid seems to be the first within our knowledge. The effectiveness and efficiency of the proposed method is illustrated in several simulations via the comparison with methods using block pulse functions (BPFs) or a hybrid of Bernoulli polynomials and BPFs.