# A Greedy Pursuit Hierarchical Iteration Algorithm
## 1 Introduction
Another alternative approach for decomposing MIMO systems is to break them down into several multiple-input single-output (MISO) systems. The identification problem of MISO systems has been extensively studied, and a large number of effective research results are given. Along with the emergence of compressed sensing (CS) in recent years, reference [19] gives the relevant basic of CS theory, reference [20] mainly introduces the principle and application of CS, and reference [21] presents the convergence of an interesting orthogonal matching pursuit (OMP) algorithm. The CS theory provides a new idea and some useful methods for system identification. For MISO systems that contain unknown input time delays, the parameter vector to be identified of the system that is over-parameterized has high dimension and sparse characteristics since the information vector contains unknown input delays. The CS is an effective idea to solve the parameter estimation for sparse systems. Many scholars have carried out research in related fields and achieved a series of results. Inspired by the greedy search idea of the CS, some parameter identification algorithms for output error models were designed by combining the auxiliary model and the hierarchical identification principle. For MISO output error systems with unknown time-delays, reference [22] combines a high dimensional and sparse estimation model with an auxiliary identification model to present the auxiliary model iterative algorithm, which has less computational cost compared with the auxiliary model least squares iterative, and reference [23] gives an auxiliary gradient pursuit iterative algorithm using the gradient search principle to further reduce the amount of computation. An iterative algorithm has been applied to achieve the estimation of time delays and parameters for multivariate systems output error systems by combining basis pursuit and auxiliary models in [24]. For MISO finite impulse response systems with input time delays, reference [25] combines the gradient search and matching pursuit method to give a gradient pursuit iterative algorithm and realize the parameters and time-delay estimation, and the gradient pursuit iterative algorithm can reduce the computational complexity compared with the traditional method. For nonlinear systems, a block-oriented nonlinear system was parameterized into a linear identification model including a sparse parameter vector, and the parameter estimation algorithm was presented by combining the matching pursuit idea based on CS, the auxiliary model, and key variable separation technology in [26]. For nonlinear systems containing colored noise in general, an OMP iterative method was presented to obtain the true parameters and time delays of an input Hammerstein nonlinear system in [27]. In this article, the kernel matrix is constructed to normalize the information matrix and satisfy the application conditions of compressed sensing, and a hierarchical iterative algorithm based on a greedy pursuit search idea is investigated to realize the parameters and time-delay estimation effectively at the same time [28]. The main innovations in this article are listed as follows. * The multi-variable systems model is recast based on the framework of CS by using the hierarchical identification principle. * The unknown true internal noise items of the recast sparse model in the presented algorithm are replaced by their estimation values according to the hierarchical principle. * The presented algorithm constructs a kernel matrix to find the locations of key parameters and reduce the estimated dimension and computational cost by using greedy pursuit search, in which only limited sampled data are used. * The parameters and time delays are estimated simultaneously by using the presented algorithm. The remaining of this article is organized as follows. Section 2 introduces the multi-input system model with unknown input delays and colored noise.