State ﬁltering and parameter estimation for two-input two-output systems with time delay

This paper focuses on presenting a new identiﬁcation algorithm to estimate the parameters and state variables for two-input two-output dynamic systems with time delay based on canonical state space models. First, the related input-output equation is determined and transformed into an identiﬁcation oriented model, which does not involve in the unmeasurable states, and then a residual based least squares identiﬁcation algorithm is presented for the estimations. After the parameters being estimated, the system states are subsequently estimated by using the estimated parameters. Through theoretical analysis, the convergence of the algorithm is derived to provide assurance for applicability. Finally, a selected simulation example is given for a meaningful case study to show the effectiveness of the proposed algorithm.


INTRODUCTION
Parameter estimation and state identification have been the core issues of data driven modelling, signal filtering and controller design [1][2][3][4]. The mathematical model is the foundation to quantitatively represent the system, and the system identification makes use of statistical algorithms to develop the mathematical model of the dynamic systems from the known information [5][6][7][8]. The iterative/recursive algorithms are the typical parameter identification algorithms [9], which have a wide range of applications in seeking the roots of the equation and developing parameter estimation methods [10][11][12][13][14]. Ansari and Bernstein proposed the deadbeat unknown-input state estimation and input reconstruction for linear discrete-time systems [15]; Xu et al. presented a hierarchical Newton and least squares iterative estimation algorithm for dynamic systems based on the impulse responses [16]. Recently [17]. Telmoudi et al discussed the modelling and state of health estimation of nickel-metal hydride battery using an EPSO-based fuzzy c-regression model [18] and the parameter estimation of nonlinear systems using a robust possibilistic c-regression model algorithm [19]. In this work, the differential equations are expanded to the two-input twooutput model with time delay, which is difficult from the point of view of identification and the time delay system is generally ubiquitous in industry.
Regarding the model structures, state space models have been predominantly adopted in system identification and control system design [20][21][22][23][24], and thus have received much research attention in parameter and state estimation over decades [25,26] and witnessed the applications [27]. In engineering applications, it should be noted that the states of some systems are not completely known due to various reasons (e.g. no available sensors, heave cost in measurements, etc.) Therefore, the system state estimation has played an important role in control design and system identification. There are many state and parameter estimation algorithms: Meurer et al. proposed the nonlinear state estimation for the Czochralski process based on the weighing signal using an extended Kalman filter [28]; Alessandri and Gaggero discussed the fast moving horizon state estimation for discrete-time systems using single and multi iteration descent methods [29].
The complexity and uncertainty of the analysis or sampling often accompanied with the outputs subject to uncertain delays [30,31]. The existence of time delay makes it difficult for the control system to respond to timely changes in input [32,33]. Besides, the time delay can lead to instability and poor performance of controlled processes. The parameter and delay estimation of such systems is a challenging problem and meaningful in academic research and applications which has attracted a lot of attention, especially in the case of measurement noise. Some useful techniques have been introduced in this aspect. Stojanovic discussed the robust finite-time stability of discrete time systems with interval time-varying delay and nonlinear perturbations [34]. For multivariate delayed state space models, more research has paid attention to parameter estimation and status estimation, ignored the computation demanding of the algorithms [35]. The expectation maximization (EM) algorithm has been a widely used for computing maximum likelihood estimates of unknown parameters in probabilistic models involving latent variables. The EM algorithm takes up an iterative process that alternates between computing a conditional expectation and solving a maximization problem. However EM cannot be directly used to estimate state variables.
This article investigates a residual-based least squares identification algorithm to simultaneously estimate states and parameters of a class of two-input two-output systems. Based on the thought of decomposition in the identification model, the twoinput two-output system is decomposed into two less dimension and variables two-input single-output subsystems, again to identify each subsystem. To overcome the difficulty of the information matrix including unmeasurable noise terms, the unknown noise terms are replaced with their estimated residuals, which are computed through the preceding parameter estimates. The simulation results are provided to show the computational experimental validity tests.
The contributions of the study, in terms of reducing computational complexity/demanding in parameter and state estimation, lie in three aspects: • The two-input two-output model with time delay is decomposed into two two-input single-output models with few dimensions and few variables based on the idea of identification model decomposition. • The presented algorithm can make full use of all data to generate highly accurate parameter estimates. • The deducing process of the identification model is simplified to reduce the computational load of multivariable system identification.
Regarding the related research, it should be noted that the gradient iterative identification algorithm has a small amount of calculation, but low estimation accuracy, and slow convergence speed [36]. The least squares algorithm has high accuracy, but it has heavy computational demand [37]. This study presents a hierarchical identification algorithm to decompose the identification system into two subsystems, reducing the dimensionality of the covariance matrix, reducing the computational load, and improving the estimation accuracy by filtering the input and output data from noise.
The communique is organized as follows. Section 2 gives the identification model for two-input two-output state space system with time delay. Section 3 derives a parameter identification algorithm for canonical state space systems with time delay. Section 4 presents the state estimation identification algorithm. Section 5 provides an illustrative example for the results in this study. Finally, we offer some concluding remarks in Section 6.

THE PROBLEM FORMULATION
Consider the following model describing two-input two-output state space system with time delay, where x(t ) ∈ ℝ n is the unmeasurable state vector, T ∈ ℝ 2 is the uncorrelated stochastic noise with zero mean. Assume that n is known, and u(t ) = 0 for t ⩽ 0. Since it is a multivariable system with coupling, the matrices A ∈ ℝ n×n , B ∈ ℝ n×n , F ∈ ℝ n×2 and c ∈ ℝ 2×n are the system parameter matrices to be identified.

A ∶=
[ Here, n i ⩾ 1 are the observability indices, satisfying n 1 + T ∈ ℝ n , x i (t ) ∈ ℝ n i . Because the model (1)-(2) contains the unknown parameter vectors/matrices of the system and unmeasurable state vectors, which is the difficulty of identification, the idea of this paper is to replace the state vector with a measurable input and output. First analyze the first subsystem, we have Since the model (1)-(2) is in the observable canonical form, the decomposed subsystem is still the observable canonical model. According to the special structure of the matrix A 1 and e T 1 , it is known that the observability matrix T is an identity matrix: Post-multiplying both sides of the above equation by the matrix A 12 yields: Observing the structure of matrix A 12 , we get Observing Equation (6), From Equations (3) and (4), we have Combining the observable matrix in (5) and the above equation gives For the second subsystem, we have For n 12 ⩽ n 1 and n 12 ⩽ min{n 1 , n 2 }, the number of non-zero elements of e T 1 A 12 , now construct a vector associated with it: similarly, In order to get the parameter estimates, define the information set at time t by 1 (t ) and the parameter vector 1 as T ∈ ℝ n+2n 1 +nn 1 , Substituting the state variable in (7)-(8) into (4) gives Replacing t in (12) with t − n 1 can be simplified as the following regression model, The proposed parameter estimation algorithms are based on this identification model. Many identification methods are derived based on the identification models of the systems [38][39][40][41][42][43] and can be used to estimate the parameters of other linear systems and nonlinear systems [44][45][46][47][48][49][50] and can be applied to other fields [51][52][53][54][55][56][57] such as chemical process control systems.
Remark 1. The above equation is the identification model of the two-input two-output state space system with time delay. For research convenience, assume t is the current moment, {u(t ), y(t ) ∶ t = 0, 1, 2, …} is the measurable input-output information, y(t ) and (t ) are the current information,
Remark 2. For the information vector 1 (t ) in (14)-(16) contains the unknown noise item v 1 (t − i ) and the state vector , the above algorithm cannot be realized, which is the difficulty in identification. This section adopts the basic idea of replacing the unknown noise item v 1 (t − i ) and the state vector Use the estimatesv i (t ) andx(t ) of v i (t ) and x(t ) to construct the estimatesV i (t ) andX (t ) of V i (t ) and X (t ): T is the estimate of T at time t . According to Equation (13), Thus, replacing the unknown variable 1 (t ) on the right-hand sides of algorithm (14)-(16) with its corresponding estimatê 1 (t ), replacing the unknown 1 with its estimatê1(t − 1) at the previous time t − 1, we obtain the following parameter estimation based recursive least squares algorithm to calculate 1 : Similar to the derivation process of the parameter vector 1 , the second subsystem is obtained as follows.
Define l 2 and h 2 as When computing one parameter vector, others are replaced with their estimates, then we get the parameter estimation based recursive least squares algorithm as follows: wherê2(t ) ∶= [̂T 21 (t ),̂T 22 (t ),̂T 23
Remark 3. Since we consider a multivariable system, the coupling of the system needs to be analyzed during the decomposition, which is to realize the decoupling of the system: the multivariable system that makes the input and output are correlated to each other realizes that each output is only controlled by the corresponding input. (1) and (2), the identification model in (13) and the least squares algorithm in (17)- (26), suppose that {v(t )} is a white noise sequence with zero mean and variance 2 , that is,

THE STATE ESTIMATION ALGORITHM
The relationship between the parameter vector 1 and the matrices/vector A 1 , A 12 , B 1 , e T 1 has been established; postmultiplying the observable matrix in (5) on both sides by B 1 gives From Equation (27) and the definition of B 1 , we have e T 1 A k−1 1 B 1 = b 1k , k = 1, 2, … , n 1 .
Equations (37) and (41) can be expressed as W F = K, W B = J. Applying the estimatesâ i ,F (t ) andB(t ) to set up: Replacing t − n in (7) and (8) to t yields UsingM 1 ,M 2 ,Q 1 ,Q 2 ,V 1 andV 2 to replace M 1 , M 2 , Q 1 , Q 2 , V 1 and V 2 in the above equations, we get the estimates of state vectors: Under the known̂1(t ) and̂2(t ), according to the least squares principle, the state estimation algorithm of the two-input twooutput systems with time delay is summarized as follows: Remark 4. According to the state equation at different time t , the state vector is represented by measurable input and output variables, and the identification model of the system is derived. Then, the single-input single-output model algorithm is generalized, and its corresponding residualbased augmented least squares algorithm is derived. The estimated parameters are used to identify system status. The proposed algorithm is computationally intensive and highly accurate.
Remark 5. The convergence properties of the proposed algorithm can be analyzed by means of the martingale convergence theorem [58][59][60].

EXAMPLE
Consider the two-input two-output state space system with 2step state-delay, the parameters are In simulation, the input {u(t )} is generated from uniform distribution and is taken as an uncorrelated persistent excitation signal sequence with zero mean and unit variance, and {v(t )} as a white noise sequence is generated from Gaus-sian distribution with zero mean and variances 2 = 0.  From Tables 1-2 and Figures 1-6, we can draw the following conclusions.
• The parameter estimation errors become smaller (in general) with the increasing of t . • In the case of the same zero mean variance, the parameter estimation accuracy improves as the data length t increases. • The data converges faster when the noise variance is lower.
• The state estimates are close to their true values with t increasing.

CONCLUSIONS
The basic algorithm derivation principle of this paper is similar to the corresponding multi-input single-output model, but the number of parameters of this model is large, the dimension The parameter estimation errors 2 versus t is complex, and there is coupling. When calculating the system state according to the hierarchical identification principle, it is necessary to combine the identification of two subsystems. The parameters make the identification more difficult, and because the recursive algorithm calculates the inverse of the matrix in the calculation process, the calculation amount is relatively large,   The state and state estimatex 3 (t ) versus t which affects the identification accuracy. This paper starts with a bivariate model with few dimensions to study the recursive least squares algorithm based on residuals. The proposed algorithms here can combine other estimation methods [61][62][63][64][65][66] to study parameter identification problems of different systems [67][68][69][70][71][72] and can be applied to other fields.