#### Abstract

This paper presents a multi-innovation stochastic gradient parameter estimation algorithm for dual-rate sampled state-space systems with -step time delay by the multi-innovation identification theory. Considering the stochastic disturbance in industrial process and using the gradient search, a multi-innovation stochastic gradient algorithm is proposed through expanding the scalar innovation into an innovation vector in order to obtain more accurate parameter estimates. The difficulty of identification is that the information vector in the identification model contains the unknown states. The proposed algorithm uses the state estimates of the observer instead of the state variables to realize the parameter estimation. The simulation results indicate that the proposed algorithm works well.

#### 1. Introduction

The mathematical model can represent the basic features of the system, and system identification applies the statistical methods to set up the mathematical models of dynamic systems from available data [1–4]. There exist some identification methods for state-space models with and without state-delay [5, 6], such as the recursive least squares (RLS) algorithm and the stochastic gradient (SG) algorithm. The SG algorithm is used in adaptive control because of its small computation. Recently, Chen et al. presented an Aitken based the modified Kalman filtering stochastic gradient algorithm for dual-rate nonlinear models [7]. Many identification methods have been developed for linear stochastic systems [8–11], bilinear stochastic systems [12–14], and nonlinear systems with colored noises [15–17].

In system identification, some algorithms concentrate on reducing the calculation amount and improving the recognition accuracy [18–20]. Since the gradient optimization method only requires calculating the first-order derivative, its computation is small [21]. However, the calculation accuracy of the gradient algorithm is low [22]. In order to improve the calculation accuracy, many improved gradient algorithms have been proposed. Although these improved algorithms can improve the parameter estimation accuracy, the computational complexity is large. The innovation is effective information to improve the parameter estimation accuracy [23, 24]. It can promote the convergence of the algorithm in the recursive process. In order to improve the estimation accuracy through using more innovation, the multi-innovation theory has been used in system recognition.

The stabilities and identification of time-delay systems have drawn a great deal of attention of many researchers in system control and system analysis [25]. In industrial processes and control systems, time delays are difficult to avoid due to material transmission and signal interruptions. The time delay makes it difficult for the control system to respond to the input changes in time [26]. In addition, the time delay can cause instability and unsatisfactory performance of the controlled process. The recognition of time-delay systems has been a hot topic [27]. For example, Sanz et al. studied an observation and stabilization of LTV systems with time-varying measurement delay [28]. Li et al. discussed the local discontinuous Galerkin method for reaction-diffusion dynamical systems with time delays [29].

This paper studies identification problems of a dual-rate state-space model with -step time delay. The main contributions of this paper are as follows. The input-output representation is derived from a canonical state-space model of the state-delay system for the identification through eliminating the state variables in the systems, to derive a joint parameter and state estimation algorithm by means of the multi-innovation identification theory and the state observer for reducing the computational burden and improving the parameter estimation accuracy and the convergence speed.

This paper is organized as follows: Section 2 gives the canonical state-space model for state-delay systems; Section 3 introduces the identification model; Section 4 presents a combined multi-innovation stochastic gradient (MISG) parameter and state estimation algorithm; Section 5 provides an illustrative example; and finally, we offer some concluding remarks in Section 6.

#### 2. The Canonical State-Space Model for State-Delay Systems

Let us introduce some symbols. The relation or means that is defined as ; () stands for an identity matrix of appropriate size (); denotes a unit forward shift operator: ; represents the matrix/vector transpose; is the estimate of at time ; means that an vector whose elements are all unity; denotes the expectation operator; stands for the adjoint matrix of the square matrix : ; represents the determinant of the square matrix .

Consider the following state-space system with -step state-delay:where is the system state vector, is the system input, is the system output, is a random noise with zero mean, and , , , and are the system parameter matrices/vectors. Assume that is observable and and for . The system matrices/vectors , , and are the unknown parameters to be estimated from the input-output data . If we remove in equation (1), then it becomes the conventional standard state-space model.

*Remark 1. *For the system in (1) and (2), if the state vector is known, the system matrix/vector (, ) is easy to identify. This paper considers the case that the state is completely unavailable. The objective is to propose new methods for jointly estimating the unknown states and parameters from the measurement data and to study the performance of the proposed methods.

#### 3. The Identification Model

This section derives the identification model of the canonical state-space model in (1) and (2). From (1), we have

Let , using the properties of the shift operator , multiplying (5) by and (6) by , and adding all expressions give

When , define the information vector and the parameter vector :

When , define the information vector and the parameter vector :

Multirate systems include dual-rate systems [30, 31] and nonuniformly sampled systems [32, 33] as a special cases. In the dual-rate scheme, the observed output is sampled by the sampler and the sampling period is multiple of the input retention period. Assume that the sampling interval is ( is an integer), and thus, the measured input-output data are at the fast rate and at the slow rate. Replacing in (10) with gives

The identification model is important for parameter estimation, and many estimation algorithms have been proposed based on the identification models from observation data [34, 35] such as the gradient algorithms, the least squares algorithms, and the Newton algorithms [36].

*Remark 2. *This is the identification model of the dual-rate state-space system with -step state-delay. The information vector consists of the state vector , the input , and the correlated noise , and the parameter vector consists of the parameters , , and of the state-space model in (1) and (2).

*Remark 3. *In what follows, a SG algorithm is derived for the state-space system with colored noise. Furthermore, a MISG algorithm is presented to reduce the computational burden and enhance the parameter estimation accuracy. A simulation example is provided to evaluate the estimation accuracy and the computational efficiency of the proposed algorithms.

#### 4. The Parameter and State Estimation Algorithm

This section derives a multi-innovation stochastic gradient algorithm to estimate the parameter vector in (11) and uses the observer to estimate the state vector of the system.

##### 4.1. The SG Algorithm

Defining and minimizing the cost function,and using the gradient search principle, we may obtain a stochastic gradient algorithm:where is the step-size or convergence factor. The choice of guarantees that the parameter estimation error converges to zero. However, difficulties arise in that the information vector contains the unknown state vector and the SG algorithm in (13) and (14) cannot compute the estimate of in (11). The approach here is to replace the unknown in with its . Based on the identification model in (11), we can obtain the following stochastic gradient parameter estimation algorithm for estimating :

When , we have

When , we have

##### 4.2. The MISG Algorithm

In order to improve the accuracy of the SG algorithm, we extend the SG algorithm and derive a multi-innovation stochastic gradient algorithm by expanding the innovation length.

Define an innovation vector:where the positive integer represents the innovation length, and

In general, one may think that the estimate is closer to than at time . Thus, the innovation vector is taken more reasonably to be

Defining the information matrix and stacking output vector asthe innovation vector can be equivalently expressed as

Furthermore, we can obtain the following multi-innovation stochastic gradient algorithm with the innovation length :

When , we have

When , we have

When the innovation length , the MISG algorithm degrades to the SG algorithm.

Theorem 1. *For the system in (11) and the MISG algorithm in (24)–(34), assume that the system noise is random noise with zero mean and variance that is uncorrelated with the input , and the mean square is bounded, that is, (A1) , ; the existence of constant and integer make the following condition: (A2) , a.s., . Then, the mean square error of the parameter estimation error given by MISG algorithm is bounded, that is, .*

##### 4.3. The State Estimation Algorithm

Using the parameter estimation vector to form the system matrices/vectors , , and and based on the canonical state-space model in (1)-(2), we can use the following observer to estimate the state vector :

The proposed algorithms in this paper can combine some mathematical tools [37] and other identification methods [38–40] to investigate the parameter identification methods of other linear and nonlinear systems [41–49] and can be applied to other literature studies [50–53] such as engineering application systems.

The steps of computing the parameter estimate in (23)–(28) and the state estimate in (34)–(39) are listed in the following:(1)Let and set the initial values , , and .(2)Collect the input-output data and and form by (29) or (32), by (28), and by (27), respectively.(3)Compute by (25) and by (26).(4)Update the parameter estimation vector by (24).(5)Read , , and from according to the definition of .(6)Form and by (36) and (37).(7)Compute the state estimation vector by (35).(8)Increase by 1 and go to step 2, and continue the recursive calculation.

*Remark 4. *The flop number is used to measure the calculation efficiency (calculation amount) of a complex algorithm. The total number of four floating-point operations required by an algorithm is defined as its calculation amount. Based on this, the calculation efficiency of the algorithm is evaluated as a benchmark, and an efficient and economical algorithm is sought. The calculation method is necessary to analyze the performance of the proposed algorithm.

The computational efficiency of the MISG and the SG algorithms is shown in Tables 1–3. Total floating-point operation (flop) numbers of the MISG and the SG algorithms are and , respectively. The difference between the MISG algorithm and the SG algorithm isThus, the SG algorithm has smaller computational efforts than the MISG algorithm.

#### 5. Example

Consider the following dual-rate time-delay system with :

The parameter vector to be identified is

In simulation, the input is taken as an uncorrelated persistent excitation signal sequence with zero mean and unit variance, and as a white noise sequence with zero mean and variances and . The parameter estimation based MISG algorithm in (24)–(34) to estimate the parameter vector and the state estimation algorithm in (35)–(39) to estimate the state vector of this example system are applied. The parameter estimates and their estimation errors are shown in Tables 4 and 5 and the parameter estimation errors versus are shown in Figures 1 and 2 with , respectively, and the state estimates and versus are shown in Figures 3 and 4.

From Tables 4 and 5 and Figures 1–4, we can draw the following conclusions:(1)The parameter estimates converge fast to their true values for large , see Tables 4 and 5(2)The MISG algorithm with has higher accuracy than the SG algorithm, see Figures 1 and 2(3)The parameter estimation errors given by the MISG algorithm become smaller with the data length and the innovation length increasing, see Tables 4 and 5 and Figures 1 and 2(4)The state estimates are close to their true values with increasing, see Figures 3 and 4

#### 6. Conclusions

This study has taken up a category of state-space models with state time delay as the research background and accordingly developed two folds of solutions for the model identification (parameter and state estimation in specific). The theoretical analysis has proved that the estimates converge to the real value under the condition of continuous excitation in modelling. The algorithms used in this paper can be applied to hybrid switched impulsive power networks and uncertain chaotic nonlinear systems with time delay [54–57] and can be applied to other literature studies [58–64]. The simulation case study has demonstrated that the proposed algorithms/procedures are effective and efficient in design and implementation.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

This work was supported by the National Natural Science Foundation of China (nos. 61903050 and 61673277), the Natural Science Foundation of Jiangsu Province (no. BK20181033), the Natural Science Fundamental Research Project of Colleges and Universities in Jiangsu Province (no. 18KJB120001), the Science and Technology Development Plan Project of Changshu under grant no. CR201711, and Jiangsu Planned Projects for Postdoctoral Research Funds.