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An efficient algorithm to state-delay identification based on a collocation integrator and adjoint sensitivity propagation

For nonlinear systems with multiple state-delays, this paper proposes a numerical algorithm for delay identification based on dynamic optimization. An implicit continuous Runge-Kutta integrator is applied to evaluate the output prediction error. Then, adjoint sensitivity propagation is proposed to compute the gradient following the idea of internal numerical differentiation. Compared with state sensitivity propagation, this adjoint-based algorithm is more efficient when there are more than one delays to be estimated. Furthermore, the integral step-length is not restricted to be shorter than the delays. Thus, very dense time grids can be avoided, and computational cost can be reduced as well. With the proposed function evaluation and gradient computation method, a sequential solution algorithm can be designed conveniently utilizing optimization routines off-the-shelf. An identification problem on a chemical reaction in a continuously-stirred tank reactor demonstrates that the designed algorithm can estimate the state delay efficiently with satisfactory accuracy.