# Multi-Innovation Stochastic Gradient Parameter and State Estimation Algorithm for Dual-Rate State-Space Systems withd-Step Time Delay

Authors: Ya Gu; Quanmin Zhu; Jicheng Liu; Peiyi Zhu; Yongxin Chou
Journal: Complexity
 (2020)

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## Abstract

This paper presents a multi-innovation stochastic gradient parameter estimation algorithm for dual-rate sampled state-space systems withd-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.


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## Body

## 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 withd-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: Section2 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 relationA≔X or X≔A means that A is defined as X; I (In) stands for an identity matrix of appropriate size (n×n); z denotes a unit forward shift operator: zxk=xk+1; T represents the matrix/vector transpose; θ^k is the estimate of θ at time k; 1n means that an n×1 vector whose elements are all unity; E denotes the expectation operator; adjX stands for the adjoint matrix of the square matrix X: adjX=detXX−1; detX=X represents the determinant of the square matrix X.Consider the following state-space system withd-step state-delay:(1)xk+1=Axk+Bxk−d+fuk,(2)yk=cxk+wk,(3)wk=∑r=1nggrvk−r+vk,(4)A≔010⋯0001⋱⋮⋮⋮⋱⋱000⋯01anan−1an−2⋯a1∈ℝn×n,B≔b1b2⋮bn∈ℝn×n,bi∈ℝ1×n,f≔f1,f2,…,fnT∈ℝn,c≔1,0,0,…,0∈ℝ1×n,where xk∈ℝn is the system state vector, uk∈ℝ is the system input, yk∈ℝ is the system output, vk∈ℝ is a random noise with zero mean, and A∈ℝn×n, B∈ℝn×n, f∈ℝn, and c∈ℝ1×n are the system parameter matrices/vectors. Assume that c,A is observable and uk=0 and yk=0 for k≤0. The system matrices/vectors A, B, and f are the unknown parameters to be estimated from the input-output data uk,yk. If we remove Bxk−d 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 xk is known, the system matrix/vector (A, b) is easy to identify. This paper considers the case that the state xk is completely unavailable. The objective is to propose new methods for jointly estimating the unknown states and parameters from the measurement data uk,yk:k=1,2,… 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(5)xik+1=xi+1k+bixk−d+fiuk,i=1,2,…,n−1,(6)xnk+1=anx1k+an−1x2k+⋯+a1xnk+bnxk−d+fnuk.Leta≔an,an−1,…,a1T∈ℝn, using the properties of the shift operator z, multiplying (5) by z−i and (6) by z−n, and adding all expressions give(7)x1k=axk−n+b1xk−d−1+b2xk−d−2+⋯+bn−1xk−d−n+1+bnxk−d−n+f1uk−1+f2uk−2+⋯+fnuk−n=axk−n+∑i=1nbixk−d−i+∑i=1nfiuk−i.Whend≤n−1, define the information vector φk and the parameter vector θ:(8)φk≔φ1Tk,φnTkT,φ1k≔xTk−d−1,xTk−d−2,…,xTk−n,…,xTk−n−d,uk−1,uk−2,…,uk−nT∈ℝn2+n,φnk≔vk−1,vk−2,…,vk−ngT∈ℝng,θ≔θ1T,θnTT,θ1≔b1,b2,…,a+bn−d,…,bn,fTT∈ℝn2+n,θn≔g1,g2,…,gngT∈ℝng.Whend≥n, define the information vector φk and the parameter vector θ:(9)φk≔φ1Tk,φnTkT,φ1k≔xTk−n,xTk−d−1,xTk−d−2,…,xTk−d−n+1,xTk−d−n,uk−1,uk−2,…,uk−nTT∈ℝn2+2n,φnk≔vk−1,vk−2,…,vk−ngT∈ℝng,θ≔θ1T,θnTT,θ1≔a,b1,b2,…,bn−1,bn,fTT∈ℝn2+2n,θn≔g1,g2,…,gngT∈ℝng.From (2) and (7), we have(10)yk=x1k+wk=φTkθ+wk.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 τ (τ≥2 is an integer), and thus, the measured input-output data are uk:k=0,1,2,… at the fast rate and ykτ:k=0,1,2,… at the slow rate. Replacing k in (10) with kτ gives(11)ykτ=x1kτ+wkτ=φTkτθ+wkτ.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 withd-step state-delay. The information vector φkτ consists of the state vector xkτ−i, the input ukτ−i, and the correlated noise wkτ−i, and the parameter vector θ consists of the parameters ai, bi, and fi 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 vectorθ in (11) and uses the observer to estimate the state vector xkτ of the system.


### 4.1. The SG Algorithm

Defining and minimizing the cost function,(12)Jθ≔ykτ−φTkτθ2,and using the gradient search principle, we may obtain a stochastic gradient algorithm:(13)θ^kτ=θ^kτ−τ+φkτrkτykτ−φTkτθ^kτ−τ,(14)rkτ=rkτ−τ+φkτ2,r0=1,where 1/rkτ is the step-size or convergence factor. The choice of rkτ guarantees that the parameter estimation error converges to zero. However, difficulties arise in that the information vector φkτ contains the unknown state vector xkτ−ii=1+d,2+d,…,n+d and the SG algorithm in (13) and (14) cannot compute the estimate of θ in (11). The approach here is to replace the unknown xkτ−i in φkτ with its x^kτ−i. Based on the identification model in (11), we can obtain the following stochastic gradient parameter estimation algorithm for estimating θ:(15)θ^kτ=θ^kτ−τ+φ^kτrkτykτ−φ^Tkτθ^kτ−τ,(16)rkτ=rkτ−τ+φ^kτ2,r0=1.Whend≤n−1, we have(17)φ^kτ=φ^1Tkτ,φ^nTkτT,φ^1kτ=x^Tkτ−d−1,x^Tkτ−d−2,…,x^Tkτ−n,…,x^Tkτ−n−d,ukτ−1,ukτ−2,…,ukτ−nT,φ^nkτ=v^kτ−1,v^kτ−2,…,v^kτ−ngT∈ℝng.Whend≥n, we have(18)φ^kτ=φ^1Tkτ,φ^nTkτT,φ^1kτ=x^Tkτ−n,x^Tkτ−d−1,…,x^Tkτ−d−n+1,x^Tkτ−d−n,ukτ−1,ukτ−2,…,ukτ−nT,φ^nkτ=v^kτ−1,v^kτ−2,…,v^kτ−ngT∈ℝng.


### 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:(19)Ep,kτ=ekτekτ−1⋮ekτ−p+1∈ℝp,where the positive integer p represents the innovation length, and(20)ekτ−i=ykτ−i−φ^Tkτ−iθ^kτ−i−τ.In general, one may think that the estimateθ^kτ−τ is closer to θ than θ^kτ−τ at time kτ−ii=2,3,4,…,p−1. Thus, the innovation vector is taken more reasonably to be(21)Ep,kτ=ykτ−φ^Tkτθ^kτ−τykτ−1−φ^Tkτ−1θ^kτ−τ⋮ykτ−p+1−φ^Tkτ−p+1θ^kτ−τ∈ℝp.Defining the information matrixΦ^p,kτ and stacking output vector Yp,kτ as(22)Φ^p,kτ≔φ^kτ,φ^kτ−1,…,φ^kτ−p+1∈ℝn2+n+ng×p,Yp,kτ≔ykτ,ykτ−1,…,ykτ−p+1T∈ℝp,the innovation vector Ep,kτ can be equivalently expressed as(23)Ep,kτ=Yp,kτ−Φ^Tp,kτθ^kτ−τ.Furthermore, we can obtain the following multi-innovation stochastic gradient algorithm with the innovation lengthp:(24)θ^kτ=θ^kτ−τ+Φ^p,kτrkτEp,kτ,(25)Ep,kτ=Yp,kτ−Φ^Tp,kτθ^kτ−τ,(26)rkτ=rkτ−τ+φ^p,kτ2,r0=1,(27)Yp,kτ=ykτ,ykτ−1,…,ykτ−p+1T,(28)Φ^p,kτ=φ^kτ,φ^kτ−1,…,φ^kτ−p+1.Whend≤n−1, we have(29)φ^kτ=φ^1Tkτ,φ^nTkτT,(30)φ^nkτ=x^Tkτ−d−1,x^Tkτ−d−2,…,x^Tkτ−n,…,x^Tkτ−n−dukτ−1,ukτ−2,…,ukτ−nT,(31)φ^nkτ=v^kτ−1,v^kτ−2,…,v^kτ−ngT∈ℝng.Whend≥n, we have(32)φ^kτ=φ^1Tkτ,φ^nTkτT,(33)φ^1kτ=x^Tkτ−n,x^Tkτ−d−1,…,x^Tkτ−d−n+1,x^Tkτ−d−n,ukτ−1,ukτ−2,…,ukτ−nT,(34)φ^nkτ=v^kτ−1,v^kτ−2,…,v^kτ−ngT∈ℝng.When the innovation lengthp=1, 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 vkτ is random noise with zero mean and variance σ2 that is uncorrelated with the input ukτ, and the mean square is bounded, that is, (A1) Evkτ=0, Evkτ2=σ2kτ≤σ2; the existence of constant 0<α≤β<∞ and integer p≥n make the following condition: (A2) αI≤1/p∑i=1pφkτ−i+1φTkτ−i+1≤βI, a.s., kτ≥p. Then, the mean square error of the parameter estimation error given by MISG algorithm is bounded, that is, limsupkτ⟶∞Eθ^kτ−θ2≤4β3σ2/α4.


### 4.3. The State Estimation Algorithm

Using the parameter estimation vectorθ^kτ to form the system matrices/vectors A^kτ, B^kτ, and f^kτ and based on the canonical state-space model in (1)-(2), we can use the following observer to estimate the state vector xkτ:(35)x^kτ+τ=A^kτx^kτ+B^kτx^kτ−dτ+f^kτukτ,(36)A^kτ=01⋯000⋱⋮⋮⋮⋱00001a^nkτa^n−1kτ⋯a^1kτ,(37)B^kτ=b^1kτb^2kτ⋮b^nkτ,f^kτ=f^1kτf^2kτ⋮f^nkτ,(38)θ^kτ=b^1kτ,b^2kτ,…,a^kτ+b^n−dkτ,…,b^nkτ,f^TkτT,d≤n−1,(39)θ^kτ=a^kτ,b^1kτ,b^2kτ,…,b^n−1kτ,b^nkτ,f^TkτT,d≥n.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 estimateθ^kτ in (23)–(28) and the state estimate x^kτ+τ in (34)–(39) are listed in the following:(1)
Letk=1 and set the initial values θ^0=1n2+2n/p0, r0=1, and p0=106.(2)
Collect the input-output dataukτ and ykτ and form φ^kτ by (29) or (32), Φ^p,kτ by (28), and Yp,kτ by (27), respectively.(3)
ComputeEp,kτ by (25) and rkτ by (26).(4)
Update the parameter estimation vectorθ^kτ by (24).(5)
Reada^ikτ, b^ikτ, and f^kτ from θ^kτ according to the definition of θ^kτ.(6)
FormA^kτ and B^kτ by (36) and (37).(7)
Compute the state estimation vectorx^kτ+τ by (35).(8)
Increasek 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 Tables1–3. Total floating-point operation (flop) numbers of the MISG and the SG algorithms are N1=n2+n+ng4p+3 and N2=7n2+n+ng, respectively. The difference between the MISG algorithm and the SG algorithm is(40)N1−N2=n2+n+ng4p+3−7n2+n+ng=4p−4n2+n+ng>0.
Thus, the SG algorithm has smaller computational efforts than the MISG algorithm.Table 1
The computational burden of the MISG algorithm.
Computational sequencesNumber of multiplicationsNumber of additionsθ^kτ=θ^kτ−τ+Φ^p,kτ/rkτEp,kτn2+n+ngp+1n2+n+ngpEp,kτ=Yp,kτ−Φ^Tp,kτθ^kτ−τn2+n+ngpn2+n+ngprkτ=rkτ−τ+φ^p,kτ2n2+n+ngn2+n+ngSum2n2+n+np+1n2+n+ng2p+1Total flopsn2+n+n4p+3Table 2
The computational efficiency of the SG algorithm.
Computational sequencesNumber of multiplicationsNumber of additionsθ^kτ=θ^kτ−τ+φ^kτ/rkτykτ−φ^Tkτθ^kτ−τ3n2+n+ng2n2+n+ngrkτ=rkτ−τ+φ^kτ2n2+n+nn2+n+nSum4n2+n+ng3n2+n+ngTotal flops7n2+n+nTable 3
Comparison of the computational efficiency of the MISG and the SG algorithms.
AlgorithmsNumber of multiplicationsNumber of additionsTotal flopsMISG2n2+n+ngp+1n2+n+ng2p+1n2+n+ng4p+3SG4n2+n+ng3n2+n+ng7n2+n+ng


## 4.1. The SG Algorithm

Defining and minimizing the cost function,(12)Jθ≔ykτ−φTkτθ2,and using the gradient search principle, we may obtain a stochastic gradient algorithm:(13)θ^kτ=θ^kτ−τ+φkτrkτykτ−φTkτθ^kτ−τ,(14)rkτ=rkτ−τ+φkτ2,r0=1,where 1/rkτ is the step-size or convergence factor. The choice of rkτ guarantees that the parameter estimation error converges to zero. However, difficulties arise in that the information vector φkτ contains the unknown state vector xkτ−ii=1+d,2+d,…,n+d and the SG algorithm in (13) and (14) cannot compute the estimate of θ in (11). The approach here is to replace the unknown xkτ−i in φkτ with its x^kτ−i. Based on the identification model in (11), we can obtain the following stochastic gradient parameter estimation algorithm for estimating θ:(15)θ^kτ=θ^kτ−τ+φ^kτrkτykτ−φ^Tkτθ^kτ−τ,(16)rkτ=rkτ−τ+φ^kτ2,r0=1.Whend≤n−1, we have(17)φ^kτ=φ^1Tkτ,φ^nTkτT,φ^1kτ=x^Tkτ−d−1,x^Tkτ−d−2,…,x^Tkτ−n,…,x^Tkτ−n−d,ukτ−1,ukτ−2,…,ukτ−nT,φ^nkτ=v^kτ−1,v^kτ−2,…,v^kτ−ngT∈ℝng.Whend≥n, we have(18)φ^kτ=φ^1Tkτ,φ^nTkτT,φ^1kτ=x^Tkτ−n,x^Tkτ−d−1,…,x^Tkτ−d−n+1,x^Tkτ−d−n,ukτ−1,ukτ−2,…,ukτ−nT,φ^nkτ=v^kτ−1,v^kτ−2,…,v^kτ−ngT∈ℝng.


## 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:(19)Ep,kτ=ekτekτ−1⋮ekτ−p+1∈ℝp,where the positive integer p represents the innovation length, and(20)ekτ−i=ykτ−i−φ^Tkτ−iθ^kτ−i−τ.In general, one may think that the estimateθ^kτ−τ is closer to θ than θ^kτ−τ at time kτ−ii=2,3,4,…,p−1. Thus, the innovation vector is taken more reasonably to be(21)Ep,kτ=ykτ−φ^Tkτθ^kτ−τykτ−1−φ^Tkτ−1θ^kτ−τ⋮ykτ−p+1−φ^Tkτ−p+1θ^kτ−τ∈ℝp.Defining the information matrixΦ^p,kτ and stacking output vector Yp,kτ as(22)Φ^p,kτ≔φ^kτ,φ^kτ−1,…,φ^kτ−p+1∈ℝn2+n+ng×p,Yp,kτ≔ykτ,ykτ−1,…,ykτ−p+1T∈ℝp,the innovation vector Ep,kτ can be equivalently expressed as(23)Ep,kτ=Yp,kτ−Φ^Tp,kτθ^kτ−τ.Furthermore, we can obtain the following multi-innovation stochastic gradient algorithm with the innovation lengthp:(24)θ^kτ=θ^kτ−τ+Φ^p,kτrkτEp,kτ,(25)Ep,kτ=Yp,kτ−Φ^Tp,kτθ^kτ−τ,(26)rkτ=rkτ−τ+φ^p,kτ2,r0=1,(27)Yp,kτ=ykτ,ykτ−1,…,ykτ−p+1T,(28)Φ^p,kτ=φ^kτ,φ^kτ−1,…,φ^kτ−p+1.Whend≤n−1, we have(29)φ^kτ=φ^1Tkτ,φ^nTkτT,(30)φ^nkτ=x^Tkτ−d−1,x^Tkτ−d−2,…,x^Tkτ−n,…,x^Tkτ−n−dukτ−1,ukτ−2,…,ukτ−nT,(31)φ^nkτ=v^kτ−1,v^kτ−2,…,v^kτ−ngT∈ℝng.Whend≥n, we have(32)φ^kτ=φ^1Tkτ,φ^nTkτT,(33)φ^1kτ=x^Tkτ−n,x^Tkτ−d−1,…,x^Tkτ−d−n+1,x^Tkτ−d−n,ukτ−1,ukτ−2,…,ukτ−nT,(34)φ^nkτ=v^kτ−1,v^kτ−2,…,v^kτ−ngT∈ℝng.When the innovation lengthp=1, 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 vkτ is random noise with zero mean and variance σ2 that is uncorrelated with the input ukτ, and the mean square is bounded, that is, (A1) Evkτ=0, Evkτ2=σ2kτ≤σ2; the existence of constant 0<α≤β<∞ and integer p≥n make the following condition: (A2) αI≤1/p∑i=1pφkτ−i+1φTkτ−i+1≤βI, a.s., kτ≥p. Then, the mean square error of the parameter estimation error given by MISG algorithm is bounded, that is, limsupkτ⟶∞Eθ^kτ−θ2≤4β3σ2/α4.


## 4.3. The State Estimation Algorithm

Using the parameter estimation vectorθ^kτ to form the system matrices/vectors A^kτ, B^kτ, and f^kτ and based on the canonical state-space model in (1)-(2), we can use the following observer to estimate the state vector xkτ:(35)x^kτ+τ=A^kτx^kτ+B^kτx^kτ−dτ+f^kτukτ,(36)A^kτ=01⋯000⋱⋮⋮⋮⋱00001a^nkτa^n−1kτ⋯a^1kτ,(37)B^kτ=b^1kτb^2kτ⋮b^nkτ,f^kτ=f^1kτf^2kτ⋮f^nkτ,(38)θ^kτ=b^1kτ,b^2kτ,…,a^kτ+b^n−dkτ,…,b^nkτ,f^TkτT,d≤n−1,(39)θ^kτ=a^kτ,b^1kτ,b^2kτ,…,b^n−1kτ,b^nkτ,f^TkτT,d≥n.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 estimateθ^kτ in (23)–(28) and the state estimate x^kτ+τ in (34)–(39) are listed in the following:(1)
Letk=1 and set the initial values θ^0=1n2+2n/p0, r0=1, and p0=106.(2)
Collect the input-output dataukτ and ykτ and form φ^kτ by (29) or (32), Φ^p,kτ by (28), and Yp,kτ by (27), respectively.(3)
ComputeEp,kτ by (25) and rkτ by (26).(4)
Update the parameter estimation vectorθ^kτ by (24).(5)
Reada^ikτ, b^ikτ, and f^kτ from θ^kτ according to the definition of θ^kτ.(6)
FormA^kτ and B^kτ by (36) and (37).(7)
Compute the state estimation vectorx^kτ+τ by (35).(8)
Increasek 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 Tables1–3. Total floating-point operation (flop) numbers of the MISG and the SG algorithms are N1=n2+n+ng4p+3 and N2=7n2+n+ng, respectively. The difference between the MISG algorithm and the SG algorithm is(40)N1−N2=n2+n+ng4p+3−7n2+n+ng=4p−4n2+n+ng>0.
Thus, the SG algorithm has smaller computational efforts than the MISG algorithm.Table 1
The computational burden of the MISG algorithm.
Computational sequencesNumber of multiplicationsNumber of additionsθ^kτ=θ^kτ−τ+Φ^p,kτ/rkτEp,kτn2+n+ngp+1n2+n+ngpEp,kτ=Yp,kτ−Φ^Tp,kτθ^kτ−τn2+n+ngpn2+n+ngprkτ=rkτ−τ+φ^p,kτ2n2+n+ngn2+n+ngSum2n2+n+np+1n2+n+ng2p+1Total flopsn2+n+n4p+3Table 2
The computational efficiency of the SG algorithm.
Computational sequencesNumber of multiplicationsNumber of additionsθ^kτ=θ^kτ−τ+φ^kτ/rkτykτ−φ^Tkτθ^kτ−τ3n2+n+ng2n2+n+ngrkτ=rkτ−τ+φ^kτ2n2+n+nn2+n+nSum4n2+n+ng3n2+n+ngTotal flops7n2+n+nTable 3
Comparison of the computational efficiency of the MISG and the SG algorithms.
AlgorithmsNumber of multiplicationsNumber of additionsTotal flopsMISG2n2+n+ngp+1n2+n+ng2p+1n2+n+ng4p+3SG4n2+n+ng3n2+n+ng7n2+n+ng


## 5. Example

Consider the following dual-rate time-delay system withτ=2:(41)xt+1=01−0.01−0.22xt+0.19−0.080.16−0.12xt−2+0.8−0.8ut,yt=1,0xt+vt.The parameter vector to be identified is(42)θ=a2,a1,b11,b12,b21,b22,f1,f2T=−0.01,−0.22,0.19,−0.08,0.16,−0.12,0.8,−0.8T.In simulation, the inputut is taken as an uncorrelated persistent excitation signal sequence with zero mean and unit variance, and vt as a white noise sequence with zero mean and variances σ2=0.302 and σ2=1.002. The parameter estimation based MISG algorithm in (24)–(34) to estimate the parameter vector θ and the state estimation algorithm in (35)–(39) to estimate the state vector xt 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 δ versus t are shown in Figures 1 and 2 with p=1,2,5, respectively, and the state estimates x^1t and x^2t versus t are shown in Figures 3 and 4.Table 4
The parameter estimates and errors withσ2=0.302.
Algorithmsta2a1b11b12b21b22f1f2δ%SG (MISG,p=1)100−0.00912−0.150360.04307−0.066300.02505−0.010040.49833−0.3867447.52683200−0.00071−0.158970.03430−0.062260.03583−0.018740.51905−0.4167144.53744500−0.01158−0.162180.04174−0.074580.04926−0.025780.55028−0.4595839.909661000−0.01300−0.167420.04407−0.077790.05693−0.029350.57120−0.4880936.941002000−0.01608−0.172430.04797−0.081600.06551−0.032940.58943−0.5120434.325113000−0.01644−0.176160.04817−0.083120.07199−0.036210.59949−0.5255732.86166MISG,p=21000.00359−0.106780.07097−0.116810.07671−0.046560.66219−0.5408929.999622000.01599−0.121090.06185−0.111770.09003−0.055900.68106−0.5775926.689125000.00850−0.133090.07485−0.125150.10632−0.062200.70645−0.6235022.1653210000.01018−0.142600.07880−0.126290.11409−0.063840.72115−0.6518219.6247420000.00966−0.151290.08537−0.129160.12141−0.065840.73406−0.6741017.4759330000.01029−0.156880.08764−0.130420.12688−0.067870.74038−0.6856716.37969MISG,p=5100−0.01604−0.160630.16648−0.049420.12274−0.110180.77804−0.728339.28037200−0.00941−0.172610.16361−0.051830.13136−0.112730.78161−0.756816.94835500−0.01370−0.188670.17725−0.068840.14888−0.119530.79311−0.774463.863591000−0.00956−0.195520.17681−0.067200.15297−0.116850.79300−0.784993.001032000−0.00865−0.201730.18160−0.070540.15557−0.116470.79656−0.790772.104263000−0.00843−0.205390.18430−0.073110.15762−0.116750.79793−0.792821.61476True values−0.01000−0.220000.19000−0.080000.16000−0.120000.80000−0.80000Table 5
The parameter estimates and errors withσ2=1.002.
Algorithmsta2a1b11b12b21b22f1f2δ%SG (MISG,p=1)100−0.01550−0.154700.04028−0.062650.02094−0.016300.52050−0.3456549.20737200−0.00846−0.161140.03142−0.059000.02910−0.023020.53817−0.3816546.04580500−0.02283−0.164250.04190−0.075130.04531−0.032150.57022−0.4274441.044001000−0.02246−0.169470.04135−0.075210.05323−0.034290.58828−0.4595038.055252000−0.02481−0.175220.04532−0.078760.06177−0.037690.60570−0.4859435.314883000−0.02492−0.179670.04580−0.080540.06861−0.040990.61533−0.5007933.76525MISG,p=2100−0.00070−0.099010.08115−0.115490.07387−0.053780.68241−0.5152130.882442000.01133−0.110980.06966−0.109750.08403−0.060430.69602−0.5626527.171745000.00040−0.124740.08572−0.128220.10703−0.071770.72361−0.6097722.1994610000.00665−0.134820.08321−0.122540.11560−0.070750.73290−0.6420519.6704320000.00802−0.145270.08971−0.124750.12278−0.072320.74507−0.6665317.3468530000.00917−0.152330.09273−0.126660.12873−0.074270.75100−0.6789816.12797MISG,p=5100−0.02959−0.139140.19948−0.015310.08421−0.153410.76481−0.7327912.99066200−0.01861−0.153410.18733−0.014750.09415−0.149510.76310−0.7804010.55685500−0.02916−0.179500.20785−0.045990.13403−0.166180.79390−0.783716.835731000−0.01718−0.187740.19516−0.032220.14082−0.153780.78631−0.795426.023582000−0.01503−0.198040.20265−0.038260.14446−0.149830.79412−0.799445.034533000−0.01509−0.205010.20834−0.044620.14803−0.148640.79741−0.800054.46176True values−0.01000−0.220000.19000−0.080000.16000−0.120000.80000−0.80000Figure 1
The parameter estimation errorsδ versus t with σ2=0.302.Figure 2
The parameter estimation errorsδ versus t with σ2=1.002.Figure 3
The state and state estimatex^1t versus t; solid line: the true x1t; dots: the estimated x^1t.Figure 4
The state and state estimatex^2t versus t; solid line: the true x2t; dots: the estimated x^2t.From Tables4 and 5 and Figures 1–4, we can draw the following conclusions:(1)
The parameter estimates converge fast to their true values for largep, see Tables 4 and 5(2)
The MISG algorithm withp≥2 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 lengtht and the innovation length p increasing, see Tables 4 and 5 and Figures 1 and 2(4)
The state estimates are close to their true values witht 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.



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