International Journal of Aerospace Engineering

Volume 2018, Article ID 2615203, 9 pages

https://doi.org/10.1155/2018/2615203

## Hankel Matrix Correlation Function-Based Subspace Identification Method for UAV Servo System

^{1}School of Automation, Chongqing University, Chongqing, China^{2}Computer College, Chongqing College of Electronic Engineering, Chongqing, China

Correspondence should be addressed to Minghong She; moc.621@ehs_hm

Received 2 October 2017; Revised 24 December 2017; Accepted 14 January 2018; Published 5 April 2018

Academic Editor: Hikmat Asadov

Copyright © 2018 Minghong She and Pengju Zhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

For the identification problem of closed-loop subspace model, we propose a zero space projection method based on the estimation of correlation function to fill the block Hankel matrix of identification model by combining the linear algebra with geometry. By using the same projection of related data in time offset set and LQ decomposition, the multiplication operation of projection is achieved and dynamics estimation of the unknown equipment system model is obtained. Consequently, we have solved the problem of biased estimation caused when the open-loop subspace identification algorithm is applied to the closed-loop identification. A simulation example is given to show the effectiveness of the proposed approach. In final, the practicability of the identification algorithm is verified by hardware test of UAV servo system in real environment.

#### 1. Introduction

In the past twenty years, the subspace model identification (SMI) has received great attention, not only because of its excellent convergence and simple numerical calculation, but also the suitability for the application in the estimation, prediction, and control algorithm. In the early references, most subspace identification methods have open-loop recognition characteristics. Considering stability, safety, and control-oriented identification problems, researchers have been trying to apply these subspace methods to closed-loop identification [1–4].

The main difficulty in closed-loop system identification is that the correlation between device input and interference leads to a bias in the parameter estimation of the system model [5]. So far, it has developed many subspace identification methods, such as the literature [6–11], which can obtain consistent estimates and closed-loop data. It is noticed that most subspace system identification methods are based on the input and output data in time domain, and the frequency response methods of some linear time invariant systems are often based on the signal correlation function [12, 13].

Subspace method is extended to frequency response function estimation, and continuous and discrete time models are determined by auxiliary variables [14]. Two frequency statistical properties and subspace convergence analysis methods are proposed in the literature [15]. For a linear closed-loop system, where the external input is independent of the observed noise, the cross correlation function of the output and the external input signals is equal to the cross correlation function of the input and external signals of the dynamic system [16]. By using correlation function sequence as the interface function, the important information has been carried by the correlation function in data compression which has been hidden in the sequence, and by extracting the parameter information of interface function, it provides the basis for parameter identification [17].

The above algorithms can solve the problem of correlation between input and interference of the equipment to some extent. The unbiased parameter estimation can be obtained under arbitrary noise characteristics [18]. But the subspace identification algorithm can solve the relationship between the input and interference at the same time, because the relation between the two related function sequences cannot be completely determined, which leads to the higher dimension of the input matrix of the subspace computation matrix [19].

In order to simplify the calculation and improve the calculation accuracy, we design the input deletion strategy based on the zero space projection on the basis of block Hankel matrix and related data estimation separately [20]. This paper also uses LQ framework to solve the identification process and simplifies the calculation of the algorithm [21]. In this paper, a new subspace identification algorithm based on the estimation of correlation function is proposed, and the unbiased parameter estimation of closed-loop dynamics under linear closed-loop conditions is obtained.

For the EIV (errors-in-variables) model structure, namely, the input and output, both are affected by noise pollution. Chou and Verhaegen put forward a new method of subspace identification [12]. The method eliminates noise effects by regarding the past input/output data as auxiliary variables. Gustafsson [22] changed the steps of the traditional subspace identification method and proposed a new subspace auxiliary variable method (subspace-based identification using instrumental variables (SIVs)). The algorithm presented in the literature [12] was included, and the identification precision was improved after the algorithm was modified. In terms of the algorithm itself, the input being independent of noise assumption is not involved in the two methods; thus, it seems that they can be applied for identification in a closed-loop system. However, this is not the case based on simulation examples; these two types of algorithms used in closed-loop identification do not obtain consensus estimates in some cases.

On the other hand, for closed-loop systems subject to the input and output measurement noises, [23] developed a new closed-loop subspace identification algorithm (SOPIM) by adopting the EIV model structure of SIMPCA and proposed an orthogonal projection approach to avoid identifying the parity space of the feedback controller. In the literature [24], it proposed the use of parity space and principal component analysis (SIMPCA) for EIV identification with colored input excitation can also be applied to closed-loop identification. Instead of preestimating the Markov parameters or eliminating them via noncausal projections, SIMPCA reformulates the SIM problem in parity space. In the process of algorithm implementation, SVD should be applied twice to solve the orthogonal complement space in the two methods. It is difficult to determine the dimensions of the orthogonal complement space when the system order is unknown. The contribution of this paper is to use a zero space projection method based on the estimation of correlation function to fill the block Hankel matrix of identification model by combining the linear algebra with geometry. By using the same projection of related data in time offset set and LQ decomposition, the multiplication operation of projection is achieved and dynamics estimation of the unknown equipment system model is obtained. Simulation shows that this algorithm has higher accuracy, which is another contribution of this paper.

The outline of this paper is as follows: we start in Section 2 with the statement of the problem and notation, and then a state space model based on estimation of correlation function is obtained. In Section 3, we present a block Hankel matrix, related data estimation equations, and deletion of input items based on null space projection for subspace model identification process. In Section 4, both a simulation example and a real hardware verification experiment are presented. We end this paper with our conclusions in Section 5.

#### 2. Closed-Loop Subspace Identification

##### 2.1. Statement of the Problem and Notation

In the UAV servo closed-loop control system, there is an unknown device, as shown in Figure 1. The device model contains deterministic part , and random parts are obtained by filtering white noise sequence with the noise filter . Therefore, the device model can be represented as