# A Greedy Pursuit Hierarchical Iteration Algorithm
## 1 Introduction  
System identification is a crucial process in modern control theory that involves determining the mathematical model of a system based on its input-output measurement data. The system identification is summarized and extended in references [1; 2]. This modeling and estimation technique help engineers to estimate parameters that characterize system behavior, which can be used to design controllers for various applications. An efficient identification method is very important. The performance analysis of some algorithms for system identification is presented in [3], and some new identification methods based on multi-innovation theory are presented in [4]. Moreover, this technique has numerous practical applications across different fields such as aerospace engineering, robotics, and chemical processing plants. With the development of modern science and technology, the objects faced by system identification become increasingly complex. Those system models are defined as large-scale systems such as complex networks, neural networks, and artificial intelligence [5; 6]. The parameter estimation of large-scale systems is an important research direction in system identification, where the reduced-order and decomposition are effective methods for identification of large-scale systems [7; 8]. For instance, the neural network or artificial intelligence, which can be regarded as a class of typical large-scale systems, typically employ millions of interconnected neurons to simulate the brain. Similarly, the complex dynamic network control systems, characterized by a significant number of nodes and links, can also be considered as large-scale systems [9; 10]. In conclusion, system identification plays an essential role in modern control theory by providing accurate mathematical models for complex systems. It allows us to predict outputs accurately whileimproving overall performance and efficiency across various systems. However, the difficulty of identification will increase with the increasingly dimensionality of the large-scale systems, which is a challenge that has to be faced in the future [11]. Large-scale systems usually contain a large number of variables. When these systems have unknown input time delays, the model used to describe them often includes redundant information. The identification problem for large-scale systems is typically divided into two stages: first, test measures are taken to estimate the orders and time delays; second, some identification algorithms are used to estimate the parameters. Therefore, traditional methods require prior knowledge such as system orders and time delays before estimating parameters. For example, the expectation maximization approach is used to estimate the parameters when the system orders and time-delay are known [12], the Bayesian method is used to estimate the system when the time-delay is known [13], and so does reference [14]. The least squares (LS) algorithms are widely used in large-scale system identification due to their fast parameter convergence rate and high estimation accuracy. The hierarchical LS iterative method is used to estimate the input nonlinear system [15; 16], and the recursive LS estimation method is applied to identify the output nonlinear systems [17]. However, it should be noted that applying these methods directly can be difficult because there may be coupling items within large-scale systems. It is necessary to decouple the system before parameter estimation. A common approach is to decompose large-scale systems into some multi-variable subsystems; among these subsystems, multiple-input multiple-output (MIMO) systems are a typical class of multi-variable system. The MIMO systems can be decomposed into several single-input single-output (SISO) subsystems using decomposition techniques. Then, each of those subsystems can be identified one by one with enough sampled data [18], which comes with a huge computational cost.