State space model identification of multirate processes with time-delay using the expectation maximization

doi:10.1016/j.jfranklin.2018.08.030

Journal of the Franklin Institute

Volume 356, Issue 3, February 2019, Pages 1623-1639

https://doi.org/10.1016/j.jfranklin.2018.08.030 Get rights and content

Abstract

This paper presents the problems of state space model identification of multirate processes with unknown time delay. The aim is to identify a multirate state space model to approximate the parameter-varying time-delay system. The identification problems are formulated under the framework of the expectation maximization algorithm. Through introducing two hidden variables, a new expectation maximization algorithm is derived to estimate the unknown model parameters and the time-delays simultaneously. The effectiveness of the proposed algorithm is validated by a simulation example.

Introduction

Parameter estimation approaches have wide applications in many areas such as signal processing [1], [2], [3], [4] and system identification [5], [6]. Mathematical models are the foundation of system control. The state space model can be utilized to describe dynamic systems [7], [8], [9]. In industrial systems, many controlled objects need to be abstracted into state space models of the system [10], [11]. In general, a physical phenomenon with time delay (fixed or varying) accompanies with the controlled object. In practice, the irregularly sampled outputs are often available with random delays due to manual analysis in laboratory [12], [13]. Moreover, the associated sampling delays are uncertain since only the arrival time of the measurements is recorded, while the time instant that the samples are actually taken is unknown or not accurately recorded.

Traditional methods assume the delay is a fixed value and it can be treated as a parameter in the identification problem [14], [15], [16]. So, the delay estimation may be solved by the recursive algorithms [17], [18], [19] and the iterative algorithms [20], [21] along with the model parameters. However, the delay is usually associated with some process variable transmission (e.g. liquid flow rate). The fast flow rate results in small time delay while a slow flow rate results in a long time delay, thus the varying delay is more reasonable at most situations [22], [23], [24]. Xie et al. proposed the FIR model identification of multirate processes with random delays using the EM algorithm [25]; Chen et al. presented the robust identification of continuous-time models with arbitrary time-delay from irregularly sampled data [26].

The elements of a variety of measurement sampling functions and control signal operate at different sampling rates which lead to multirate (MR) systems [27], [28], [29]. The theoretical research for such multirate sampled systems started in the 1950’s. The application in the field is concerned with computer signal processing, network control system and process control. Various methods have been proposed to model the MR systems and infer the unmeasurable or missing outputs [30], [31]. The first important work for the MR system was performed by Kranc on the switch decomposition technique, later termed as the lifting technique, which has become a standard tool to transform a periodically time-varying system into a time-invariant one. For example, Zamani et al. studied the tall discrete-time linear systems with multirate outputs [32], Zhang et al. presents the finite-time filtering problem for a class of wireless networked multirate systems with fading channels [33]. Some estimation methods have been proposed for linear systems [34], [35], pseduo-linear systems [36], bilinear systems [37], [38] and bilinear-parameter systems [39].

This paper develops a multirate model based identification procedure of state space systems, taking into account uncertain random delays associated with the irregularly sampled outputs. The expectation maximization (EM) algorithm computes the maximum likelihood estimates of unknown parameters in probabilistic models involving latent variables. The EM algorithm is an iterative method that alternates between computing a conditional expectation and solving a maximization problem [40], [41], [42]. We will in this work derive the EM algorithm and show that it provides a maximum likelihood estimate. There are two main motivations: to simultaneously estimate the discrete time delay and continuous states; to estimate the parameters of the state space model. To solve the above problems, we will use the EM algorithm. Under the EM framework, it will be shown that, instead of the point estimation of the scheduling variable, the complete probability distribution of the estimation of the scheduling variable is what one really needs for the estimation of system parameters. The states estimation are given by their expectation. The parameter identification methods can be applied to many engineering areas [43], [44], [45], [46], [47], [48].

This paper presents new expectation maximization methods for jointly estimating system parameters and time delay. This work differs from the method in [10] based on the iterative identification idea for state-delay systems; the recent work in [14] which is based on the robust identification for Wiener time-delay system.

The main contributions of this paper are as follows.

•
This paper proposes identification of state space model with unknown time delay, which includes conventional non-uniformly sampled-data systems and multirate systems as special cases.
•
By introducing two hidden variables, this paper presents the derived EM algorithm to estimate the unknown model parameters and the time-delays simultaneously.
•
By using a numerical example, this paper demonstrates the performances of the proposed algorithm, including the estimation errors of the EM algorithm for finite measurement data.

This paper is organized as follows. Section 2 describes the state space systems with time-varying time delay. Section 3 proposes the expectation maximization algorithm. Section 4 gives the model identification using the EM algorithm. A simulation example is used to illustrate the results in Section 5. Finally, we offer several concluding remarks in Section 6.

Section snippets

Problem statement

The mathematical formulation of state space model with time-varying time delay is as follows: $x_{t + 1} = A x_{t} + b u_{t} + w_{t},$ $y_{t} = c x_{t - d_{t}} + v_{t},$ where {x_t} is the unmeasurable state; ${u_{t}, t = 1, 2, \dots, T}$ is the input and available at every sampling period Δt; T represents the number of data points that have been collected. ${y_{t}, t = T_{1}, T_{2}, \dots, T_{N}}$ is the irregularly sampled output and only available at time instant $t = T_{i} \cdot Δ t$ with unknown time delay $d_{T_{i}} \cdot Δ t$ (i.e., the delay can vary in each data sample); $w_{t} \in R^{n \times 1}$ is process noise and

Model identification using the EM algorithm

The mathematical form of the Q function of the EM algorithm for the state space system with time-varying time delay can be written as $\begin{matrix} Q (Θ | Θ^{k}) & = & E_{C_{m i s} | C_{o b s}, Θ^{k}} {\log [p (C_{o b s}, C_{m i s} | Θ)]} \\ = & \int_{C_{m i s}} \log [p (C_{o b s}, C_{m i s} | Θ] p (C_{m i s} | C_{o b s}, Θ^{k}) d C_{m i s}, \end{matrix}$ where Θ denotes the system parameters: A, $b$ and $c,$ Θ^k represents the parameter estimation results from the previous iteration which are used to compute the expectation of the complete data likelihood. The observed data set C_obs are ${y_{T_{1}}, \dots, y_{T_{N}}}$ and ${u_{1}, \dots, u_{T}},$ while the hidden

Simulation study

Consider the following multirate state space system with time-varying time delay: $\begin{matrix} x_{t} & = & 1.8 x_{t} - u_{t} + w_{t}, \\ y_{T_{i}} & = & 0.4 x_{T_{i} - d_{T_{i}}} + v_{T_{i}}, \end{matrix}$ where the fast-rate input sequence {u_t} is generated from Gaussian distribution $N (0, σ_{u}^{2})$ with $σ_{u}^{2} = 1$ shown in Fig. 2; the slow-rate output ${y_{T_{i}}}$ is available at time instant T_i · Δt $(T_{i} = 5 i)$ with random time delay $d_{T_{i}} \cdot Δ t$ $(d_{T_{i}} \in [0, 2])$ ; the variance of the process noise {w_t} and measurement noise ${v_{T_{i}}}$ are $σ_{v}^{2} = 0.01$ and $σ_{w}^{2} = 0.01,$ respectively; thus the noise to signal ratio is $δ_{ns} = \sqrt{σ_{v}^{2} / σ_{u}^{2}}$

Conclusions

This paper considers identification of MR systems with unknown random delays and continuous states, which includes conventional non-uniformly sampled-data systems and multirate systems as special cases. Due to the impact of random delay, the irregularly sampled output cannot appropriately indicate current noise-free output. Thus traditional identification methods like least squares algorithm fail to identify such MR systems if the uncertain delay problem is overlooked. To address this

References (69)

DingF.
State filtering and parameter estimation for state space systems with scarce measurements
Signal Process.
(2014)
N.G. Bretas et al.
A two steps procedure in state estimation gross error detection, identification, and correction
Int. J. Electr. Power Energy Syst.
(2015)
ChenJ. et al.
Identification methods for time-delay systems based on the redundant rules
Signal Process.
(2017)
ZhengG. et al.
Identification of the delay parameter for nonlinear time-delay systems with unknown inputs
Automatica
(2013)
XuL. et al.
A multi-innovation state and parameter estimation algorithm for a state space system with d-step state-delay
Signal Process.
(2017)
DingF. et al.
A recursive least squares parameter estimation algorithm for output nonlinear autoregressive systems using the input-output data filtering
J. Frankl. Inst.
(2017)
DingF. et al.
A hierarchical least squares identification algorithm for hammerstein nonlinear systems using the key term separation
J. Frankl. Inst.
(2018)
YangX.Q. et al.
Multiple model approach to linear parameter varying time-delay system identification with EM algorithm
J. Frankl. Inst.
(2014)
YangX.Q. et al.
Robust identification of wiener time-delay system with expectation-maximization algorithm
J. Frankl. Inst.
(2017)
ChenF.W. et al.
Robust identification of continuous-time models with arbitrary time-delay from irregularly sampled data
J. Process Control
(2015)

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^☆: This work was supported by Natural Science Foundation of Jiangsu Province (BK20181033, BK20170436), Natural Science Fundamental Research Project of Colleges and Universities in Jiangsu province (18KJB120001, 17KJB120001) and National Natural Science Foundation of China (61803049).

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State space model identification of multirate processes with time-delay using the expectation maximization☆

Abstract

Introduction

Section snippets

Problem statement

Model identification using the EM algorithm

Simulation study

Conclusions

Signal Process.

Int. J. Electr. Power Energy Syst.

Signal Process.

Automatica

Signal Process.

J. Frankl. Inst.

J. Frankl. Inst.

J. Frankl. Inst.

J. Frankl. Inst.

J. Process Control

J. Frankl. Inst.

Automatica

J. Process Control

Digit. Signal Process.

J. Frankl. Inst.

J. Frankl. Inst.

Mech. Syst. Signal Process.

Stat. Probab. Lett.

Digital Signal Process.

Futur. Gener. Comput. Syst.

Appl. Math. Lett.

Appl. Math. Lett.

J. Comput. Appl. Math.

Stat. Probab. Lett.

J. Comput. Appl. Math.

Insur. Math. Econ.

Comput. Electr. Eng.

Signal Process.

Recursive least squares and multi-innovation stochastic gradient parameter estimation methods for signal modeling

Circ. Syst. Signal Process.

Iterative parameter estimation for signal models based on measured data

Circu. Syst. Signal Process.

The parameter estimation algorithms based on the dynamical response measurement data

Adv. Mech. Eng.

Hierarchical parameter estimation for the frequency response based on the dynamical window data

Int. J. Control Autom. Syst.

Parameter estimation algorithms for dynamical response signals based on the multi-innovation theory and the hierarchical principle

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Parameter estimation for control systems based on impulse responses

Int. J. Control Autom. Syst.