Parameter Identification of Fractional-order Systems with Unknown Both State and Input Delays Based on Block Pulse Functions

In this paper, we propose a method for identification of continuous-time fractional-order systems with unknown states and input delays. In practice, many systems are modeled accurately with fractional differential equations. In particular, many systems are modeled as fractional differential equations with input delay and state delay. Since the geometric and physical meaning of fractional calculus is not clear, it is difficult to model the real system directly to fractional order systems based on mechanical analysis. Thus, the identification of fractional order systems is the main method for constructing fractional order models and is the subject of the main research by many scientists. To solve the identification problem of systems with input delay and state delay, we use the fact that the fractional integral operator matrix by the block pulse functions is an upper triangular Toeplitz matrix. We have presented an efficient method to identify the linear and nonlinear parameters separably by using the commutativity and nilpotent property for multiplication between upper triangular Toeplitz matrices. We also have presented an efficient algorithm to newly approximate the Jacobian of the variable projection functional to solve the least squares problem with nonlinear parameters. Several simulation examples have been used to verify the effectiveness of the proposed method. It is shown that the input delay and the state delay have a significant effect on the output characteristics of the system, especially the state delay has a larger effect than the input delay.