Identification of non-uniformly sampled-data systems with asynchronous input and output data☆
Introduction
In order to recover the continuous-time signals without any information loss, the Nyquist–Shannon sampling theorem advocates that the uniform sampling must have a sampling rate at least twice the Nyquist rate. For the high frequency signals, using the uniform sampling leads to a large number of samples, which could not only pose challenges to storage spaces and communication resources, but also significantly increase the computational burden in the data analysis phase [1]. Therefore, the non-uniform sampling is emerged as an alternative sampling technique to acquire cost-effective data, which can alleviate or even eliminate the negative impacts of aliasing when the average sampling rate is below the Nyquist rate [2]. Over the past few decades, the non-uniform sampling has received considerable attention in the area of signal processing [3], [4].
Due to hardware limitations, economic considerations and environmental constraints, the non-uniform sampling is frequently encountered in many practical applications [5]. For instance, in process industries, the conventional process variables are measured at fast rates, but the quality variables are often non-uniformly sampled through off-line laboratory analyses [6], [7]. Furthermore, in networked control systems, transmission delays and packet losses can generate non-uniformly sampled input–output data [8], [9], [10]. Such systems with non-uniformly spaced inputs and/or outputs are termed as non-uniformly sampled-data (NUSD) systems. In recent years, extensive research efforts have been dedicated to NUSD systems, and a variety of identification and control approaches are reported in the literature [11], [12], [13], [14].
For the NUSD systems with fast uniform updating inputs and slow non-uniform sampling outputs, Zhu et al. utilized the prediction error method to identify the fast-rate output error (OE) model, where the minimization problem was solved by the Gauss–Newton method [15]. Ding et al. proposed a gradient-based algorithm to interactively estimate the inter-sampled outputs and the parameters of the fast-rate OE model [16]. Raghavan et al. studied identification of the fast-rate state space model by combining the expectation-maximization algorithm with the Kalman filter [17]. Furthermore, Gillberg and Ljung reconstructed the outputs using B-spline functions and proposed a method to directly identify the continuous-time OE model without performing the system discretization [18].
The lifting technique is a benchmark tool to deal with multirate systems, which can transform a time-varying system into a slow-rate time-invariant system [19], [20], [21]. By means of this technique, Liu et al. derived the input–output representation of the NUSD systems with fast non-uniform updating inputs and slow uniform sampling outputs, and proposed a hierarchical least squares algorithm to estimate the model parameters [22]. In the presence of colored noise, Jing et al. investigated a recursive Bayesian algorithm with covariance resetting to identify the Box–Jenkins systems with non-uniform input data [23]. For the Wiener nonlinear systems, Zhou et al. established the mapping relationship between the non-uniform inputs and the slow outputs, and presented a gradient-based iterative algorithm for its identification [24].
The causality constraint problem must be carefully tackled when applying the lifting technique [25]. Taking this constraint into consideration, Ding and Lin derived the lifted state space model of the NUSD systems with synchronous input and output data, and further decomposed the measurement equation of which into several sub-equations, presented a modified subspace identification method [26]. Since the state space representations of systems are not unique, the lifted state space models are usually transformed into the equivalent lifted transfer function (L-TF) models for identification. For example, a partially coupled stochastic gradient algorithm and an auxiliary model based multi-innovation generalized extended stochastic gradient algorithm were proposed in [27], [28], respectively.
NUSD systems with asynchronous input and output data are more general and complicated than synchronous NUSD systems, studies on which can enrich and expand the results in the field of multirate systems [29]. Such asynchronous NUSD systems have been considered in [30], [31], where a time-varying backward shift operator δ−1 based transfer function (δ−1–TF) model was derived to describe the mapping relationship between the sampled input–output data, and the corresponding identification and control methods were presented based on this model. Although the δ−1–TF model is more concise in structure and includes fewer parameters than the L-TF model, it has an inherent disadvantage. According to Theorem 1 in [30], [31], the δ−1–TF model requires that the observation matrix of each subsystem is invertible. This condition depends on the specific system matrices and the non-uniform sampling intervals, which limits its application range. On the contrary, the L-TF model can be applied without such limitations. Therefore, it is important to develop new identification algorithms with better performances for the L-TF model of asynchronous NUSD systems.
Motivated by this fact, this paper focuses on the identification problem of asynchronous NUSD systems, the main contributions include:
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By using the lifting technique, the L-TF model is derived to describe asynchronous NUSD systems, and an auxiliary model based recursive least squares (AM-RLS) algorithm is developed for its parameter estimation.
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To solve the causality constraint problem, the L-TF model is decomposed into several subsystems for identification, and a coupled AM-RLS algorithm is proposed accordingly to enhance the identification accuracy and improve the computational efficiency.
The rest of the paper is organized as follows. 2 Problem formulation, 3 Model derivation present the problem formulation and derive the L-TF model, respectively. Three identification algorithms are derived in Section 4 and validated in Section 5 via two simulation examples. Finally, concluding remarks are given in Section 6.
Section snippets
Problem formulation
Consider a class of asynchronous NUSD systems as depicted in Fig. 1, where P is a continuous process, described by, u(t), w(t), y(t) and v(t) are the state vector, the process input, the noise-free output, the process output and the measurement noise, respectively; are the constant system matrices. and represent the non-uniform zero-order hold and sampler with updating and sampling pattern as shown in
Model derivation
For the notation simplicity, define and to represent the sampled-data at time () and (), respectively.
As illustrated in Fig. 1, the continuous-time input u(t) to the process is generated from an input sequence by a non-uniform zero-order hold , that is
Given an initial state , the state at time () is calculated by
Identification algorithms
This section first considers the direct identification of the L-TF model in Eq. (6). Subsequently, to solve the causality constraint problem and improve the computational efficiency, the L-TF model in Eq. (6) is decomposed into r subsystems for separate identification, and two recursive least squares algorithms are derived.
Examples
To illustrate the effectiveness of the proposed identification algorithms for the asynchronous NUSD systems, two simulation examples are given in this section. Example 1 Assume that the continuous-time process P has the following state space realization [27],
Assume that the input–output data are non-uniformly sampled r=2 times over the frame period T=1 s, and , , q0=0.1 s, . Therefore, the non-uniform sampling intervals
Conclusions
This paper proposes a coupled auxiliary model based recursive least squares (C-AM-RLS) algorithm for the non-uniformly sampled-data (NUSD) systems with asynchronous input and output data. The C-AM-RLS algorithm provides a higher parameter estimation accuracy than the AM-RLS algorithm developed for the subsystems of the lifted transfer function (L-TF) model. Furthermore, it can avoid the causality constraint problem and save lots of computational cost in comparison with the AM-RLS algorithm
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This work was supported by the National Natural Science Foundation of China (No. 61403166) and the Natural Science Foundation of Jiangsu Province (China, BK20140164).