Continuous-Time Subspace Identification with Prior Information Using Generalized Orthonormal Basis Functions

Abstract

This paper presents a continuous-time subspace identification method utilizing prior information and generalized orthonormal basis functions. A generalized orthonormal basis is constructed by a rational inner function, and the transformed noises have ergodic properties. The lifting approach and the Hambo system transform are used to establish the equivalent nature of continuous and transformed discrete-time stochastic systems. The constrained least squares method is adopted to investigate the incorporation of prior knowledge in order to further increase the subspace identification algorithm’s accuracy. The input–output algebraic equation derives an optimal multistep forward predictor, and prior knowledge is expressed as equality constraints. In order to solve an optimization problem with equality constraints characterizing the prior knowledge, the proposed method reduces the computational burden. The effectiveness of the proposed method is provided by numerical simulations.

1. Introduction

Subspace identification methods are extensively studied as effective tools for modeling linear dynamical systems [1,2,3]. Most of the identification methods focus on discrete-time systems. To obtain the continuous-time system models, the discrete-time system identification methods are first carried out, and then the system models are transformed into the continuous-time models.

Actually, the research on continuous-time subspace identification methods is significant. Firstly, it is more visible on system characteristics from model parameters, which illustrates that continuous-time models have wide application. Secondly, an equivalent continuous-time model can be obtained by the conversion of discrete-time models, because the identification method of discrete-time models is mature. However, this conversion may lead to some disadvantages, such as the complicated calculation of the matrix logarithm and the difficult selection of sampling time [4]. Many identification approaches have been provided to handle the problems above for obtaining the continuous-time models accurately. Ref. [5] estimated continuous-time models in an alternative domain by using a bilinear transformation. The continuous-time models were identified by the bias-eliminated least squares method, which incorporated the Poisson moment functionals (PMF) approach [6]. Using the generalized singular value decomposition, ref. [7] presented a subspace method based on PMF. The continuous-time stochastic systems were derived by a novel subspace identification approach, in which the time-derivatives were described via random distribution theory [8]. Ref. [9] studied a toroidal continuous variable transmission, which adopted the PMF method and linear integral filter method to deduce the two submodels of the original continuous-time models, respectively. Using generalized orthogonal basis functions, ref. [10] provided the continuous-time state-space model identification methods. Ref. [11] used the Laguerre filters and Laguerre projection to deal with the two kinds of identification problems and proposed the subspace identification methods. Ref. [12] presented the subspace identification method for continuous-time models based on Laguerre filters. A subspace identification approach via fractional Laguerre generating functions was investigated for fractional commensurate order systems with nonuniform data [13]. Ref. [14] provided a continuous-time subspace identification method based on generalized orthonormal bases. Using nuclear norm minimization and PMF, ref. [15] estimated the continuous-time models and expanded them to models with missing output situations. Ref. [16] proposed a continuous-time subspace identification method via generalized PMF, which fixed the size of the data matrices in the identification process. Based on the invariant subspace, ref. [17] presented a new system identification method that obtained a novel form of regression. It can be clearly seen that the linear filtering method is usually used in subspace identification. Compared with the Laguerre basis method, the transformation equations’ singularity is avoided by generalized orthonormal basis functions. It is also important to investigate the identification of nonlinear systems besides the linear systems discussed above. Ref. [18] proposed further results for nonlinear state-space system identification. Ref. [19] provided a consistent system identification method, which solved the nonlinearity problems. According to the range of information, a recurrent neural network (RNN) was designed for a multiagent containment control system in [20]. The nonlinear autoregressive moving average with eXogenous input (NARMAX) representation was used to describe a nonaffine discrete-time system with unknown nonlinear system dynamics in [21]. Ref. [22] provided a new long short-term memory (LSTM) architecture based on the classification and regression for nonuniformly sampled sequential data. From the aforementioned literature, it is still difficult to incorporate prior knowledge into the identification methods.

In general, the quality of the input–output data is key to the accuracy of the identified models. The subspace identification methods also call for the confirmation of a particular excitation, which can guarantee the consistency and precision of the identified models. The input–output data, however, may not provide enough information in many circumstances due to insufficient excitation or noise. Obviously, these factors may influence the accuracy of the estimated models. To improve the accuracy of the estimated model, the prior information can be integrated into the models in the identification process. Ref. [23] incorporated prior information into the state-space model realization algorithm, which used Kung’s singular value decomposition (SVD) realization to estimate the model parameters. Incorporating prior information, ref. [24] investigated subspace identification methods with constrained least squares. As a multi-step-ahead predictor, the subspace identification method was improved by solving an optimization problem under equality constraints. Based on the constrained recursive least squares, ref. [25] developed a new recursive subspace identification algorithm that took prior information into account. Ref. [26] considered prior information expressed as constraints on the impulse response, which was solved by active-set optimization methods. Compared with the classical subspace algorithms, the subspace identification with prior information provided more precision for identifying the models. Ref. [27] proposed the closed-loop subspace identification approach exploiting prior information. Ref. [28] presented a new closed-loop subspace identification method by using prior information. Ref. [29] provided unbiased parameter estimation and had better robustness to white measurement noise and presented a subspace identification approach with disturbances. Although integrating prior information into subspace identification algorithms can improve the accuracy of the identified model, very few studies have been focused on continuous-time subspace system identification.

In many circumstances, such as the identification of airplanes and rotorcraft (see [9,30]), system identification of continuous-time models are actually useful. In terms of generalized orthonormal basis functions, we propose continuous-time subspace identification with prior information. The main contributions can be summarized as follows: (1) Generalized orthogonal basis functions are adopted to transform the continuous-time stochastic system into the discrete-time stochastic system; (2) The proposed method translates prior information into an equality constraint, and the constrained least squares problems are solved by using a weighting method. It can generate unbiased process models with improved computational efficiency.

The remainder of this paper is organized as follows. Section 2 presents the preliminaries. The system transformation is given in Section 3. The main subspace system identification algorithm is presented in Section 4. Section 5 provides the numerical simulations to evaluate the proposed method. Some conclusions are provided in Section 6.

2. Preliminaries

2.1. Signal Spaces and Multiplication Operator

Let $L^2(-\infty ,\infty )$ denote the space of square integrable functions over $(-\infty ,\infty )$, and the inner product is obtained.

$$\langle u,v\rangle ={\int }_{-\infty }^{\infty }v{\left(t\right)}^{*}u\left(t\right)dt,$$

(1)

where superscript ${}^{*}$ denotes the conjugate transpose.

Let $L^2\left(j\mathbb{R}\right)$ denote the space of square integrable functions of frequency $j\omega \in j\mathbb{R}$, and the inner product is given.

$$\langle f,g\rangle =\frac{1}{2\pi }{\int }_{-\infty }^{\infty }g{\left(\omega \right)}^{*}f\left(\omega \right)d\omega .$$

(2)

The space of analytical functions on the right half-plane is known as the Hardy space $H^2$, and

$$\parallel f\parallel =\underset{\nu >0}{\sup }(\frac{1}{2\pi }{\int }_{-\infty }^{\infty }{\left|f(\nu +j\omega )\right|}^2{d\omega )}^{\frac{1}{2}}<\infty ,$$

(3)

where $|·|$ denotes the Euclidean norm.

A continuous-time multiplication operator ${\Lambda }_h$: $L^2\left(j\mathbb{R}\right)\to L^2\left(j\mathbb{R}\right)$ is defined as

$$\left({\Lambda }_{h}f\right)\left(j\omega \right)=h\left(j\omega \right)f\left(j\omega \right),$$

(4)

where $h\in L^2\left(j\mathbb{R}\right)$, $L^{\infty }\left(j\mathbb{R}\right)$ is the space of matrix-valued functions of frequency $j\omega \in j\mathbb{R}$ with the norm

$${\parallel h\parallel }_{\infty }=\mathrm{ess}\sup {\sigma }_{max}\left(h\left(j\omega \right)\right).$$

(5)

2.2. Generalized Orthonormal Basis Functions

The inner function $\varphi \left(s\right)$ denotes a continuous-time, single-input, single-output rational transfer function. In the closed right half-plane, not even the infinity has a pole, and $\left|\varphi \right(j\omega \left)\right|=1$ for any $j\omega \in j\mathbb{R}$.

For a continuous-time inner function, $\varphi \left(s\right)$ is in $H^{\infty }$, and the multiplication operator ${\Lambda }_{\varphi }:L^2\left(j\mathbb{R}\right)\to L^2\left(j\mathbb{R}\right)$ is unitary. Let $\varphi \left(s\right)\in H^{\infty }$ be a nonconstant inner function. Consider the orthogonal complement of the invariant subspace ${\Lambda }_{\varphi }{H}^2$ in $H^2$, that is $S=H^2\ominus {\Lambda }_{\varphi }{H}^2$.

$$H^2=\underset{m=0}{\overset{\infty }{⨁}}{\Lambda }_{\varphi }^{m}S,L^2\left(j\mathbb{R}\right)=\underset{m=-\infty }{\overset{\infty }{⨁}}{\Lambda }_{\varphi }^{m}S,$$

(6)

where ⨁ denotes the direct sum.

A function $f, \tilde{f}\in H^2$ can be represented as follows.

$$f=\sum _{m=0}^{\infty }{\Lambda }_{\varphi }^m{f}_m,\tilde{f}=\sum _{m=-\infty }^{\infty }{\Lambda }_{\varphi }^m{f}_m,f_m\in S.$$

(7)

Let $\{v_1,\cdots ,v_{n_{\varphi }}\}$ be an orthonormal basis for S, and

$$\left\{v_1\cdots v_{n_{\varphi }},{\Lambda }_{\varphi }{v}_1,\cdots ,{\Lambda }_{\varphi }{v}_{n_{\varphi }},\cdots ,{\Lambda }_{\varphi }^m{v}_1,\cdots ,{\Lambda }_{\varphi }^m{v}_{n_{\varphi }},\cdots \right\}$$

(8)

is an orthonormal basis for $H^2$. Based on the the inverse Fourier transform, the set is identified with an orthonormal basis for $L^2(0,\infty )$. This set is called a generalized orthonormal basis. Then, we take

$$\varphi \left(s\right)=\frac{s-p}{s+p},$$

(9)

the $S=\mathrm{span}\left\{v\right\}$, where

$$v\left(s\right)=\frac{\sqrt{2p}}{s+p}.$$

(10)

An orthonormal basis for $H^2$ can be obtained as

$$\left\{\frac{\sqrt{2p}}{s+p},\frac{\sqrt{2p}(s-p)}{{(s+p)}^2},\cdots ,\frac{\sqrt{2p}{(s-p)}^{m-1}}{{(s+p)}^m},\cdots \right\}.$$

(11)

Based on the inverse Fourier transform, the Laguerre polynomials in $L^2(0,\infty )$ are

$$\left\{\sqrt{2p}{e}^{-pt},\sqrt{2p}{e}^{pt}\frac{d}{dt}\left(t{e}^{-2pt}\right),\cdots ,\sqrt{2p}{e}^{pt}\frac{d^{m-1}}{d{t}^{m-1}}\left(\frac{t^{m-1}}{(m-1)!}{e}^{-2pt}\right),\cdots \right\}.$$

(12)

A generalized orthonormal basis can be constructed by any finite-dimensional Blashke product.

3. System Transformation

In this section, we first introduce the deterministic system transformation, which is the Hambo system transform. inverse Fourier transform

3.1. Deterministic System Transformation

Consider a deterministic continuous-time system

$$\begin{array}{c}\frac{d}{dt}x=\mathit{A}x+\mathit{B}u,\hfill \\ y=\mathit{C}x+\mathit{D}u,\hfill \end{array}$$

(13)

where $\mathit{A}\in {\mathbb{R}}^{n\times n}$, $\mathit{B}\in {\mathbb{R}}^{n\times n_u}$, $\mathit{C}\in {\mathbb{R}}^{n_y\times n}$ and $\mathit{D}\in {\mathbb{R}}^{n_y\times n_u}$.

Signals in $L^2(0,\infty )$ have a representation (7) based on the Fourier transform given an inner function $\varphi \left(s\right)$. The system’s input u and output y can be regarded as $S{n}_u$- and $S{n}_y$-valued sequences, respectively. inverse Fourier transform

$$(u_0,u_1,\cdots ,u_m,\cdots ),(y_0,y_1,\cdots ,y_m,\cdots ).$$

(14)

Based on the sequences, the following linear discrete-time system is

$$\begin{array}{c}\xi (t+1)=\mathit{A}\xi \left(t\right)+\mathit{B}u\left(t\right),\hfill \\ y\left(t\right)=\mathit{C}x\left(t\right)+\mathit{D}u\left(t\right),\hfill \end{array}$$

(15)

where $\mathit{A}:{\mathbb{R}}^n\to {\mathbb{R}}^n$, $\mathit{B}:S^{n_u}\to {\mathbb{R}}^n$, $\mathit{C}:{\mathbb{R}}^n\to S^{n_y}$ and $\mathit{D}:S^{n_u}\to S^{n_y}$ are defined as

$$\begin{array}{c}\mathit{A}\xi ={\varphi }^{\sim }\left(\mathit{A}\right)\xi ,\hfill \\ {\displaystyle \mathit{B}u={\int }_{-\infty }^0{e}^{-A\tau }\mathit{B}\left({\mathcal{F}}^{-1}{\Lambda }_{{\varphi }^{\sim }}u\right)\left(\tau \right)d\tau ,}\hfill \\ \left(\mathit{C}\xi \right)\left(s\right)=\mathit{C}{(s\mathit{I}-\mathit{A})}^{-1}\xi -\varphi \left(s\right)\mathit{C}{(s\mathit{I}-\mathit{A})}^{-1}{\varphi }^{\sim }\left(\mathit{A}\right)\xi ,\hfill \\ \left(\mathit{D}u\right)\left(s\right)=h\left(s\right)u\left(s\right)-\varphi \left(s\right)\mathit{C}{(s\mathit{I}-\mathit{A})}^{-1}\mathit{B}u,\hfill \end{array}$$

(16)

and ${\mathcal{F}}^{-1}$ is the inverse Fourier transform, ${\varphi }^{\sim }\left(\mathit{A}\right)$ is functional calculus of the matrix $\mathit{A}$, ${\varphi }^{\sim }\left(s\right)=\overline{\varphi (-\tilde{s})}{}^T$ is the paraconjugate of the function $h\left(s\right)$.

Proposition 1 

([31]). Consider an inner function ϕ and the subspace $S=H^2\ominus {\Lambda }_{\varphi }{H}^2$. If $\varphi \left(s\right)={\mathit{D}}_{\varphi }+{\mathit{C}}_{\varphi }{(s\mathit{I}-{\mathit{A}}_{\varphi })}^{-1}{\mathit{B}}_{\varphi }$ is the balanced realization of an inner function for a continuous-time system such that ${\mathit{D}}_{\varphi }=\mathit{I}$, ${\mathit{A}}_{\varphi }+{\mathit{A}}_{\varphi }^T+{\mathit{B}}_{\varphi }^T{\mathit{B}}_{\varphi }=\mathbf{0}$, then

$$\begin{array}{ccc}v\left(s\right)\hfill & =\hfill & [v_1\left(s\right),\cdots ,v_{n_u{n}_{\varphi }}\left(s\right)],\hfill \\ \hfill & =\hfill & ({\mathit{I}}_n\otimes {\mathit{C}}_{\varphi }){(s\mathit{I}-({\mathit{I}}_n\otimes {\mathit{A}}_{\varphi }))}^{-1}\hfill \end{array}$$

is an orthonormal basis for $S^{n_u}$, where ⊗ means the Kronecker product.

Applying Proposition 1 to the input and the output in Equation (15), Equation (14) can be regarded as the sequences

$$({\tilde{u}}_0,{\tilde{u}}_1,\cdots ,{\tilde{u}}_k,\cdots ),({\tilde{y}}_0,{\tilde{y}}_1,\cdots ,{\tilde{y}}_k,\cdots ),$$

(17)

where

$$\begin{array}{c}{\tilde{u}}_k=\left[\begin{array}{c}\langle {\Lambda }_{\varphi }^k{v}_1,u\rangle \\ ⋮\\ \langle {\Lambda }_{\varphi }^k{v}_{n_u{n}_{\varphi }},u\rangle \end{array}\right],{\tilde{y}}_m=\left[\begin{array}{c}\langle {\Lambda }_{\varphi }^k{v}_1,u\rangle \\ ⋮\\ \langle {\Lambda }_{\varphi }^k{v}_{n_u{n}_{\varphi }},u\rangle \end{array}\right].\hfill \end{array}$$

(18)

Using the the orthonormal basis in Proposition 1, Equation (15) has the following realization:

$$\begin{array}{c}\xi (t+1)=\tilde{\mathit{A}}\xi \left(t\right)+\tilde{\mathit{B}}\tilde{u}\left(t\right),\hfill \\ \tilde{y}\left(t\right)=\tilde{\mathit{C}}\xi \left(t\right)+\tilde{\mathit{D}}\tilde{u}\left(t\right),\hfill \end{array}$$

(19)

where

$$\begin{array}{c}\tilde{\mathit{A}}={\varphi }^{\sim }\left(\mathit{A}\right),\tilde{\mathit{B}}=\mathit{X},\tilde{\mathit{C}}=\mathit{Y},\hfill \\ \tilde{\mathit{D}}=\left[\begin{array}{ccc}{h}_{11}^{\sim }\left({\mathit{A}}_{\varphi }^T\right)& \cdots & h_{1n_u}^{\sim }\left({\mathit{A}}_{\varphi }^T\right)\\ ⋮& ⋮& ⋮\\ h_{n_{y}1}^{\sim }\left({\mathit{A}}_{\varphi }^T\right)& \cdots & h_{n_y{n}_u}^{\sim }\left({\mathit{A}}_{\varphi }^T\right)\end{array}\right],\hfill \end{array}$$

(20)

$h_{ij}\left(s\right)$ is the $(i,j)$-th element of the transfer function $h\left(s\right)=\mathit{D}+\mathit{C}{(s\mathit{I}-\mathit{A})}^{-1}\mathit{B}$, and the Sylvester equations are obtained as

$$\begin{array}{c}\mathit{AX}+\mathit{X}{({\mathit{I}}_{n_u}\otimes {\mathit{A}}_{\varphi })}^T+\mathit{B}{({\mathit{I}}_{n_u}\otimes {\mathit{B}}_{\varphi })}^T=0,\hfill \\ {({\mathit{I}}_{n_u}\otimes {\mathit{A}}_{\varphi })}^T\mathit{Y}+\mathit{YA}+{({\mathit{I}}_{n_u}\otimes {\mathit{C}}_{\varphi })}^T\mathit{C}=0.\hfill \end{array}$$

(21)

where $\mathit{X}$ and $\mathit{Y}$ are the unique solution of Equation (21). The transformation of the system (13) to (19) denotes the Hambo transform.

3.2. Stochastic System Transformation

Consider a continuous-time system with process and observation noises,

$$\begin{array}{c}dx=\mathit{A}xdt+{\mathit{B}}_{1}dw+{\mathit{B}}_{2}udt,x\left(0\right)=x_0\hfill \\ d\eta =\mathit{C}xdt+{\mathit{D}}_{1}dw+{\mathit{D}}_{2}udt,\hfill \end{array}$$

(22)

where $\mathit{A}\in {\mathbb{R}}^{n\times n}$, ${\mathit{B}}_1\in {\mathbb{R}}^{n\times n_w}$, ${\mathit{B}}_2\in {\mathbb{R}}^{n\times n_u}$, $\mathit{C}\in {\mathbb{R}}^{n_y\times n}$, ${\mathit{D}}_1\in {\mathbb{R}}^{n_y\times n_w}$, ${\mathit{D}}_2\in {\mathbb{R}}^{n_y\times n_u}$, u is a deterministic signal, and w is a Wiener process.

Define

$$\begin{array}{c}{\displaystyle {\tilde{u}}_k={\int }_0^{\infty }{\Lambda }_{\varphi }^{k}v{\left(t\right)}^{T}u\left(t\right)dt,}\hfill \\ {\displaystyle {\tilde{w}}_k={\int }_0^{\infty }{\Lambda }_{\varphi }^{k}v{\left(t\right)}^{T}dw\left(t\right),}\hfill \\ {\displaystyle {\tilde{y}}_k={\int }_0^{\infty }{\Lambda }_{\varphi }^{k}v{\left(t\right)}^{T}d\eta \left(t\right).}\hfill \end{array}$$

(23)

Notice that ${\tilde{w}}_k$ and ${\tilde{y}}_k$ are stochastic processes.

Theorem 1. 

Consider a stochastic system (22), and define the deterministic sequence ${\tilde{u}}_k$ and the stochastic processes ${\tilde{w}}_k$, ${\tilde{y}}_k$ by Equation (23). Then, they satisfy the following discrete-time stochastic system:

$$\begin{array}{c}\xi (k+1)=\tilde{\mathit{A}}\xi \left(k\right)+{\tilde{\mathit{B}}}_1{\tilde{w}}_k+{\tilde{\mathit{B}}}_2{\tilde{u}}_k,\xi \left(0\right)=x_0,\hfill \\ y_k=\tilde{\mathit{C}}\xi \left(k\right)+{\tilde{\mathit{D}}}_1{\tilde{w}}_k+{\tilde{\mathit{D}}}_2{\tilde{u}}_k,\hfill \end{array}$$

(24)

where

$$\begin{array}{c}\tilde{\mathit{A}}={\varphi }^{\sim }\left(\mathit{A}\right),{\tilde{\mathit{B}}}_1={\mathit{X}}_1,{\tilde{\mathit{B}}}_2={\mathit{X}}_2,\tilde{\mathit{C}}=\mathit{Y},\hfill \\ {\tilde{\mathit{D}}}_1=\left[\begin{array}{ccc}{h}_{111}^{\sim }\left({\mathit{A}}_{\varphi }^T\right)& \cdots & h_{11n_u}^{\sim }\left({\mathit{A}}_{\varphi }^T\right)\\ ⋮& ⋮& ⋮\\ h_{1n_{y}1}^{\sim }\left({\mathit{A}}_{\varphi }^T\right)& \cdots & h_{1n_y{n}_u}^{\sim }\left({\mathit{A}}_{\varphi }^T\right)\end{array}\right],\hfill \\ {\tilde{\mathit{D}}}_2=\left[\begin{array}{ccc}{h}_{211}^{\sim }\left({\mathit{A}}_{\varphi }^T\right)& \cdots & h_{21n_u}^{\sim }\left({\mathit{A}}_{\varphi }^T\right)\\ ⋮& ⋮& ⋮\\ h_{2n_{y}1}^{\sim }\left({\mathit{A}}_{\varphi }^T\right)& \cdots & h_{2n_y{n}_u}^{\sim }\left({\mathit{A}}_{\varphi }^T\right)\end{array}\right],\hfill \end{array}$$

(25)

$h_{1ij}\left(s\right)$ is the $(i,j)$-th element of the transfer function $h_1\left(s\right)={\mathit{D}}_1+\mathit{C}{(s\mathit{I}-\mathit{A})}^{-1}{\mathit{B}}_1$, $h_{2ij}\left(s\right)$ is the $(i,j)$-th element of the transfer function $h_2\left(s\right)={\mathit{D}}_2+\mathit{C}{(s\mathit{I}-\mathit{A})}^{-1}{\mathit{B}}_2$, ${\mathit{X}}_1$, ${\mathit{X}}_2$, and $\mathit{Y}$ are the unique solution to the following Sylvester equations:

$$\begin{array}{c}{\mathit{AX}}_1+{\mathit{X}}_1{({\mathit{I}}_{n_u}\otimes {\mathit{A}}_{\varphi })}^T+{\mathit{B}}_1{({\mathit{I}}_{n_u}\otimes {\mathit{B}}_{\varphi })}^T=0,\hfill \\ {\mathit{AX}}_2+{\mathit{X}}_2({({\mathit{I}}_{n_u}\otimes {\mathit{A}}_{\varphi })}^T+{\mathit{B}}_2{({\mathit{I}}_{n_u}\otimes {\mathit{B}}_{\varphi })}^T=0,\hfill \\ {({\mathit{I}}_{n_u}\otimes {\mathit{A}}_{\varphi })}^T\mathit{Y}+\mathit{YA}+{({\mathit{I}}_{n_u}\otimes {\mathit{C}}_{\varphi })}^T\mathit{C}=0.\hfill \end{array}$$

(26)

Proof. 

Assume that $n_u=n_y=1$. If $u\in S$, then from Proposition 1, it follows that $u={\mathit{C}}_{\varphi }{(s\mathit{I}-\mathit{A})}^{-1}\eta$ for some $\eta$. In terms of ${\mathit{C}}_{\varphi }^T{\mathit{C}}_{\varphi }=(s\mathit{I}-{\mathit{A}}_{\varphi })-(s\mathit{I}+{\mathit{A}}_{\varphi }^T)$, we have

$$\begin{array}{ccc}\hfill {\Lambda }_{\varphi }^{\sim }u& =\hfill & [{\mathit{D}}_{\varphi }^T-{\mathit{B}}_{\varphi }^T{(s\mathit{I}+{\mathit{A}}_{\varphi }^T)}^{-1}{\mathit{C}}_{\varphi }^T]·{\mathit{C}}_{\varphi }{(s\mathit{I}-{\mathit{A}}_{\varphi })}^{-1}\eta \hfill \\ \hfill & =\hfill & {\mathit{D}}_{\varphi }^T{\mathit{C}}_{\varphi }{(s\mathit{I}-{\mathit{A}}_{\varphi })}^{-1}\eta -{\mathit{B}}_{\varphi }^T{(s\mathit{I}+{\mathit{A}}_{\varphi }^T)}^{-1}\eta +{\mathit{B}}_{\varphi }^T{(s\mathit{I}-{\mathit{A}}_{\varphi })}^{-1}\eta \hfill \\ \hfill & =\hfill & -{\mathit{B}}_{\varphi }^T{(s\mathit{I}+{\mathit{A}}_{\varphi }^T)}^{-1}\eta .\hfill \end{array}$$

In view of Equation (16), it can be found that

$$\begin{array}{ccc}\mathit{B}u\hfill & =\hfill & {\displaystyle {\int }_{-\infty }^0{e}^{-\mathit{A}\tau }\mathit{B}\left({\mathcal{F}}^{-1}{\Lambda }_{\varphi }^{\sim }u\right)\left(\tau \right)d\tau ,}\hfill \\ \hfill & =\hfill & {\displaystyle {\int }_{-\infty }^0{e}^{-\mathit{A}\tau }{\mathit{BB}}_{\varphi }^T{e}^{-{\mathit{A}}_{\varphi }^T\tau }d\tau \eta =\mathit{X}\eta .}\hfill \end{array}$$

From $\mathit{C}\xi$, the orthogonal projection of $\mathit{C}{(s\mathit{I}-\mathit{A})}^{-1}$ onto S is obtained. The orthonormal basis for S is used for the following equation:

$$\begin{array}{ccc}\mathit{C}\xi \hfill & =\hfill & {\displaystyle {\mathit{C}}_{\varphi }{(s\mathit{I}-{\mathit{A}}_{\varphi })}^{-1}{\int }_0^{\infty }{e}^{-{\mathit{A}}_{\varphi }^T\tau }{\mathit{C}}_{\varphi }^T\mathit{C}{e}^{-\mathit{A}\tau }d\tau \eta ,}\hfill \\ \hfill & =\hfill & {\mathit{C}}_{\varphi }{(s\mathit{I}-{\mathit{A}}_{\varphi })}^{-1}Y\xi .\hfill \end{array}$$

Notice that $\mathit{D}u$ is the orthogonal projection of ${\Lambda }_{h}u$ onto S. Therefore,

$$\begin{array}{ccc}\hfill & {\displaystyle \frac{1}{2\pi }}\hfill & {\displaystyle {\int }_{-\infty }^{\infty }{(-j\omega \mathit{I}-{\mathit{A}}_{\varphi }^T)}^{-1}{\mathit{C}}_{\varphi }^{T}h\left(j\omega \right)·{\mathit{C}}_{\varphi }{(j\omega \mathit{I}-{\mathit{A}}_{\varphi })}^{-1}d\omega \eta }\hfill \\ \hfill & =\hfill & {\displaystyle \frac{1}{2\pi j}\oint h\left(s\right){(s\mathit{I}+{\mathit{A}}_{\varphi }^T)}^{-1}{C}_{\varphi }^T{C}_{\varphi }{(s\mathit{I}-{\mathit{A}}_{\varphi })}^{-1}ds\eta ,}\hfill \\ \hfill & =\hfill & {\displaystyle \frac{1}{2\pi j}\oint h\left(s\right)[{(s\mathit{I}+{\mathit{A}}_{\varphi }^T)}^{-1}-{(s\mathit{I}-{\mathit{A}}_{\varphi })}^{-1}]ds\eta ,}\hfill \\ \hfill & =\hfill & h(-{\mathit{A}}_{\varphi }^T)\eta =h^{\sim }\left({\mathit{A}}_{\varphi }^T\right)\eta .\hfill \end{array}$$

□

Remark 1. 

The Laguerre basis and the Kautz basis are the special subclasses in the class of generalized orthonormal basis [10,32]. Compared with the Laguerre basis method [11], the transformation equations’ singularity is avoided by the proposed approach when the system’s pole is also the pole of the Laguerre filter. Additionally, the Laguerre basis has less degrees of freedom than the generalized orthonormal basis.

4. Subspace System Identification

4.1. Derivation of Input–Output Algebraic Equation

In terms of the stochastic system (22), the following data can be constructed as

$$\begin{array}{c}{\displaystyle {\tilde{u}}_{k,i}={\int }_0^{\infty }{\Lambda }_{\varphi }^{k}v{\left(t\right)}^{T}u(t+t_i)dt,}\hfill \\ {\displaystyle {\tilde{w}}_{k,i}={\int }_0^{\infty }{\Lambda }_{\varphi }^{k}v{\left(t\right)}^{T}dw(t+t_i),}\hfill \\ {\displaystyle {\tilde{y}}_{k,i}={\int }_0^{\infty }{\Lambda }_{\varphi }^{k}v{\left(t\right)}^{T}d\eta (t+t_i),}\hfill \end{array}$$

(27)

where $0\le t_0<t_1<\cdots <t_i<\cdots$ is the sequence of time instances such that $t_{i+1}-t_i\ge t_{\min }$ for some $t_{\min }>0$.

Let $x_i=x\left(t_i\right)$. For fixed integers $p,q$, and N, define

$$\begin{array}{c}{\mathit{X}}_N=[x_0{x}_1\cdots x_{N-1}],{\mathit{X}}_{q,N}=[x_q{x}_{q+1}\cdots x_{q+N-1}],\hfill \\ {\mathit{U}}_{p,q,N}=\left[\begin{array}{cccc}{\tilde{u}}_{p,0}& {\tilde{u}}_{p,1}& \cdots & {\tilde{u}}_{p,N-1}\\ {\tilde{u}}_{p+1,0}& {\tilde{u}}_{p+1,1}& \cdots & {\tilde{u}}_{p+1,N-1}\\ ⋮& ⋮& ⋮& ⋮\\ {\tilde{u}}_{p+q-1,0}& {\tilde{u}}_{p+q-1,1}& \cdots & {\tilde{u}}_{p+q-1,N-1}\end{array}\right],\hfill \end{array}$$

(28)

$$\begin{array}{c}{\mathit{W}}_{p,q,N}=\left[\begin{array}{cccc}{\tilde{w}}_{p,0}& {\tilde{w}}_{p,1}& \cdots & {\tilde{w}}_{p,N-1}\\ {\tilde{w}}_{p+1,0}& {\tilde{w}}_{p+1,1}& \cdots & {\tilde{w}}_{p+1,N-1}\\ ⋮& ⋮& ⋮& ⋮\\ {\tilde{w}}_{p+q-1,0}& {\tilde{w}}_{p+q-1,1}& \cdots & {\tilde{w}}_{p+q-1,N-1}\end{array}\right],\hfill \end{array}$$

(29)

$$\begin{array}{c}{\mathit{Y}}_{p,q,N}=\left[\begin{array}{cccc}{\tilde{y}}_{p,0}& {\tilde{y}}_{p,1}& \cdots & {\tilde{y}}_{p,N-1}\\ {\tilde{y}}_{p+1,0}& {\tilde{y}}_{p+1,1}& \cdots & {\tilde{y}}_{p+1,N-1}\\ ⋮& ⋮& ⋮& ⋮\\ {\tilde{y}}_{p+q-1,0}& {\tilde{y}}_{p+q-1,1}& \cdots & {\tilde{y}}_{p+q-1,N-1}\end{array}\right].\hfill \end{array}$$

(30)

Let

$$\begin{array}{c}{\mathbf{\Gamma }}_q=\left[\begin{array}{c}\tilde{\mathit{C}}\\ \tilde{\mathit{C}}\tilde{\mathit{A}}\\ ⋮\\ \tilde{\mathit{C}}{\tilde{\mathit{A}}}^{q-1}\end{array}\right],\hfill \\ {\mathit{H}}_{1,q}=\left[\begin{array}{cccc}{\tilde{\mathit{D}}}_1& 0& \cdots & 0\\ \tilde{\mathit{C}}{\tilde{\mathit{B}}}_1& {\tilde{\mathit{D}}}_1& \ddots & 0\\ ⋮& \ddots & \ddots & ⋮\\ \tilde{\mathit{C}}{\tilde{\mathit{A}}}^{q-2}{\tilde{\mathit{B}}}_1& \tilde{\mathit{C}}{\tilde{\mathit{A}}}^{q-3}{\tilde{\mathit{B}}}_1& \cdots & {\tilde{\mathit{D}}}_1\end{array}\right],\hfill \\ {\mathit{H}}_{2,q}=\left[\begin{array}{cccc}{\tilde{\mathit{D}}}_2& 0& \cdots & 0\\ \tilde{\mathit{C}}{\tilde{\mathit{B}}}_2& {\tilde{\mathit{D}}}_2& \ddots & 0\\ ⋮& \ddots & \ddots & ⋮\\ \tilde{\mathit{C}}{\tilde{\mathit{A}}}^{q-2}{\tilde{\mathit{B}}}_2& \tilde{\mathit{C}}{\tilde{\mathit{A}}}^{q-3}{\tilde{\mathit{B}}}_2& \cdots & {\tilde{\mathit{D}}}_2\end{array}\right].\hfill \end{array}$$

(31)

From Theorem 1, the above matrices satisfy the input–output algebraic equation

$${\mathit{Y}}_{p,q,N}={\mathbf{\Gamma }}_q{\mathit{X}}_{q,N}+{\mathit{H}}_{1,q}{\mathit{W}}_{p,q,N}+{\mathit{H}}_{2,q}{\mathit{U}}_{p,q,N}.$$

(32)

4.2. Constrained Least Squares Approach

The state is obtained by the past input–output data as follows [33]:

$${\mathit{X}}_{q,N}={\mathit{L}}_N{\mathit{Z}}_N+{\tilde{\mathit{A}}}_K^N{\mathit{X}}_N,$$

(33)

where ${\mathit{L}}_N=[{\tilde{\mathit{B}}}_K{\tilde{\mathit{A}}}_K{\tilde{\mathit{B}}}_K\cdots {\tilde{\mathit{A}}}_K^{N-1}{\tilde{\mathit{B}}}_K],$

$$\begin{array}{c}{\mathit{Z}}_N=\left[\begin{array}{c}{\mathit{U}}_{0,p,N}\\ {\mathit{Y}}_{0,p,N}\end{array}\right].\hfill \end{array}$$

(34)

For a sufficiently large N, ${\tilde{\mathit{A}}}_K^N\simeq 0$. In terms of Equations (32) and (33), we have

$${\mathit{Y}}_{p,q,N}={\mathit{L}}_{q,N}{\mathit{Z}}_N+{\mathit{H}}_{1,q}{\mathit{W}}_{p,q,N}+{\mathit{H}}_{2,q}{\mathit{U}}_{p,q,N}.$$

(35)

where ${\mathit{L}}_{q,N}={\mathbf{\Gamma }}_q{\mathit{L}}_N$ denotes the subspace matrix corresponding to past input–output data.

For ${\mathit{W}}_{p,q,N}$ is the stochastic disturbance term in Equation (35), the optimal multi-step-ahead prediction is obtained by performing the least square method on Equation (35) by

$${\widehat{\mathit{L}}}_{q,N},{\widehat{\mathit{H}}}_{2,q}=\underset{{\mathit{L}}_{q,N},{\mathit{H}}_{2,q}}{\min }{∥{\mathit{Y}}_{p,q,N}-\left[{\mathit{L}}_{q,N}{\mathit{H}}_{2,q}\right]\left[\begin{array}{c}{\mathit{Z}}_N\\ {\mathit{U}}_{p,q,N}\end{array}\right]∥}_F^2,$$

(36)

The optimal prediction ${\widehat{\mathit{Y}}}_{p,q,N}$ can be given by

$${\widehat{\mathit{Y}}}_{p,q,N}={\widehat{\mathit{L}}}_{q,N}{\mathit{Z}}_N+{\widehat{\mathit{H}}}_{2,q}{\mathit{U}}_{p,q,N}=\left[{\widehat{\mathit{L}}}_{q,N}{\widehat{\mathit{H}}}_{2,q}\right]\left[\begin{array}{c}{\mathit{Z}}_N\\ {\mathit{U}}_{p,q,N}\end{array}\right].$$

(37)

According to reference [34] on page 110, this obtained multistep prediction is accurate by using the least square method, and the premise is that ${\mathit{W}}_{p,q,N}$ is uncorrelated with ${\mathit{Z}}_N$ and ${\mathit{U}}_{p,q,N}$.

To enforce the structure of ${\widehat{\mathit{H}}}_{2,q}$, the following relationship can be required.

$$\mathrm{vec}\left(\mathit{ABC}\right)=({\mathit{C}}^T\otimes \mathit{A})\mathrm{vec}\left(\mathit{B}\right),$$

(38)

where $\mathrm{vec}$ stands for the operation of forming a long vector from a matrix by stacking its columns one under another.

Based on Equations (37) and (38), we have

$$\mathrm{vec}\left({\widehat{\mathit{Y}}}_{p,q,N}\right)=\left(\left[{\mathit{Z}}_N^T{\mathit{U}}_{p,q,N}^T\right]\otimes {\mathit{I}}_{q{n}_y{n}_{\varphi }\times q{n}_y{n}_{\varphi }}\right)\mathrm{vec}\left(\left[{\widehat{\mathit{L}}}_{q,N}{\widehat{\mathit{H}}}_{2,q}\right]\right).$$

(39)

A matrix $\mathit{R}$ can be found as follows:

$$\mathrm{vec}\left(\left[{\widehat{\mathit{L}}}_{q,N}{\widehat{\mathit{H}}}_{2,q}\right]\right)=\mathit{R}\left[\begin{array}{c}{\theta }_l\\ {\theta }_g\end{array}\right],$$

(40)

where ${\theta }_l=\mathrm{vec}({\widehat{L}}_{q,N})\in {\mathbb{R}}^{q{n}_y{n}_{\varphi }(p{n}_y{n}_{\varphi }+p{n}_u{n}_{\varphi })}$ and ${\theta }_g=\left([g_0,\dots ,g_{q-1}]\right)\in {\mathbb{R}}^{q{n}_y{n}_{\varphi }{n}_u{n}_{\varphi }}$ are the distinct elements in ${\widehat{\mathit{L}}}_{q,N}$ and ${\widehat{\mathit{H}}}_{2,q}$. Note that ${\theta }_g$ corresponds to the impulse response coefficients.

Substituting (40) to (38) yields

$$\underset{y}{\underbrace{\mathrm{vec}({\widehat{\mathit{Y}}}_{p,q,N})}}=\underset{\mathit{Z}}{\underbrace{\left(\left[{\mathit{Z}}_N^T{\mathit{U}}_{p,q,N}^T\right]\otimes {\mathit{I}}_{q{n}_y{n}_{\varphi }\times q{n}_y{n}_{\varphi }}\right)\mathit{R}}}\underset{\theta }{\underbrace{\left[\begin{array}{c}{\theta }_l\\ {\theta }_g\end{array}\right]}}.$$

(41)

Equation (41) can be solved in a least squares sense, with added equality constraints representing the prior knowledge. It leads to the following problem,

$$\underset{\theta }{\min }{∥y-\mathit{Z}\theta ∥}_2^2,$$

(42)

which is subject to the equality constraints as follows:

$${\mathit{A}}_{\mathrm{eq}}\theta ={\mathit{b}}_{\mathrm{eq}},$$

(43)

where ${\mathit{A}}_{\mathrm{eq}}\in {\mathbb{R}}^{c\times (q{n}_y{n}_{\varphi }(p{n}_y{n}_{\varphi }+p{n}_u{n}_{\varphi })+q{n}_y{n}_{\varphi }{n}_u{n}_{\varphi })}$, ${\mathit{b}}_{\mathrm{eq}}\in {\mathbb{R}}^{c\times 1}$, and c is the number of constraints. There are many methods that can solve Equation (42), and the the Lagrange multipliers method [35] is used to help deal with this constrained least squares problem. The Lagrangian of Equations (42) and (43) is

$$J=\frac{1}{2}{(y-\mathit{Z}\theta )}^T(y-\mathit{Z}\theta )+\lambda ({\mathit{A}}_{\mathrm{eq}}\theta -{\mathit{b}}_{\mathrm{eq}}),$$

where $\lambda$ is the Lagrange multiplier.

Taking the partial derivatives of J with respect to $\theta$ and equating it to zero, we have

$$\widehat{\theta }={\left({\mathit{Z}}^T\mathit{Z}\right)}^{-1}{\mathit{Z}}^{T}y-{\left({\mathit{Z}}^T\mathit{Z}\right)}^{-1}{\mathit{A}}_{\mathrm{eq}}^T\lambda ,$$

where $\lambda ={({\mathit{A}}_{\mathrm{eq}}{({\mathit{Z}}^T\mathit{Z})}^{-1}{\mathit{A}}_{\mathrm{eq}}^T)}^{-1}({\mathit{A}}_{\mathrm{eq}}{({\mathit{Z}}^T\mathit{Z})}^{-1}{\mathit{Z}}^{T}y-{\mathit{b}}_{\mathrm{eq}})$. The estimated $\widehat{\theta }$, which integrates prior information through the equality constraints, represents an improved model parameter estimation.

According to a suitable choice of ${\mathit{A}}_{\mathrm{eq}}$ and ${\mathit{b}}_{\mathrm{eq}}$ in Equation (43), several pieces of prior information can be considered. In the following subsection, ${\mathit{A}}_{\mathrm{eq}}$ and ${\mathit{b}}_{\mathrm{eq}}$ are described as two types of prior information: steady-state gain and zero transfer function in the systems.

Remark 2. 

In many cases, insufficient excitation or existent noise may lead to insufficient information in input–output data. The prior information can be integrated into the models, which can improve the identification accuracy. It can be challenging to adapt some methods, including [36,37], to other subspace identification techniques, since they only take certain limitations into account. Ref. [23] incorporated the prior information into subspace identification approaches using a Bayesian framework. Unfortunately, this leads to nonlinear least squares optimization and intensive computation [38]. The proposed method translates prior information into an equality constraint in the process of subspace identification to solve the problems above, which generates the unbiased process models with improved computational efficiency.

4.2.1. Known Steady-State Gain

Assume that it takes k sampling times for the system to settle and that q impulse response parameters, $g_0,g_1,\dots ,g_{q-1}$, are estimated. The constraints can be described as

$$\sum _{i=0}^{k-1}{g}_i={\mathit{K}}_{ss},g_k=g_{k+1}=\cdots =g_{q-1}={\mathbf{0}}_{n_y{n}_{\varphi }\times n_u{n}_{\varphi }},$$

(44)

where ${\mathit{K}}_{ss}\in {\mathbb{R}}^{n_y{n}_{\varphi }\times n_u{n}_{\varphi }}$ is the DC gain matrix. It can be described as

$${\mathit{K}}_{ss}=\left[\begin{array}{ccc}{\mathit{K}}_{11}& \cdots & {\mathit{K}}_{1n_u{n}_{\varphi }}\\ ⋮& ⋮& ⋮\\ {\mathit{K}}_{n_y{n}_{\varphi }1}& \cdots & {\mathit{K}}_{n_y{n}_{\varphi }{n}_u{n}_{\varphi }}\end{array}\right],$$

(45)

and ${\mathit{K}}_{ij}$ is the DC gain from the jth input to the ith output.

In view of Equation (38), applying this $\mathrm{vec}$ operation for both sides in each constraint in (44), we have

$$\mathit{Y}\times \underset{{\theta }_g}{\underbrace{\left[\begin{array}{c}\mathrm{vec}\left(g_0\right)\\ \mathrm{vec}\left(g_1\right)\\ ⋮\\ \mathrm{vec}\left(g_{k-1}\right)\\ \mathrm{vec}\left(g_k\right)\\ \mathrm{vec}\left(g_{k+1}\right)\\ ⋮\\ \mathrm{vec}\left(g_{q-1}\right)\end{array}\right]}}=\left[\begin{array}{c}\mathrm{vec}\left({\mathit{K}}_{ss}\right)\\ {\mathbf{0}}_{n_y{n}_{\varphi }{n}_u{n}_{\varphi }\times 1}\\ ⋮\\ {\mathbf{0}}_{n_y{n}_{\varphi }{n}_u{n}_{\varphi }\times 1}\end{array}\right]$$

(46)

where

$$\mathit{Y}=\left[\begin{array}{cccccc}{\mathit{I}}_{lm\times lm}& \cdots & {\mathit{I}}_{lm\times lm}& {\mathbf{0}}_{lm\times lm}& \cdots & {\mathbf{0}}_{lm\times lm}\\ {\mathbf{0}}_{lm\times lm}& \cdots & {\mathbf{0}}_{lm\times lm}& {\mathit{I}}_{lm\times lm}& \cdots & {\mathbf{0}}_{lm\times lm}\\ ⋮& ⋮& \ddots & \ddots & \ddots & \\ {\mathbf{0}}_{lm\times lm}& \cdots & {\mathbf{0}}_{lm\times lm}& {\mathbf{0}}_{lm\times lm}& \cdots & {\mathit{I}}_{lm\times lm}\end{array}\right],$$

(47)

The Equation (46) can be expressed as

$$\left[\begin{array}{cc}{\mathbf{1}}_{1\times k}\otimes {\mathit{I}}_{lm\times lm}& {\mathbf{0}}_{lm\times (q-k)lm}\\ {\mathbf{0}}_{(q-k)lm\times klm}& {\mathit{I}}_{(q-k)lm\times (q-k)lm}\end{array}\right]{\theta }_g=\left[\begin{array}{c}\mathrm{vec}\left({\mathit{K}}_{ss}\right)\\ {\mathbf{0}}_{(q-k)lm\times 1}\end{array}\right],$$

(48)

where symbol $l=n_y{n}_{\varphi },m=n_u{n}_{\varphi }$, ${\mathbf{1}}_{p\times q}$ is a matrix of ones of size $p\times q$, and ${\mathbf{0}}_{p\times q}$ is a matrix of zeros of size $p\times q$. In view of $\theta$, Equation (48) can be described as

$${\mathit{A}}_{\mathrm{eq}}\theta ={\mathit{b}}_{\mathrm{eq}},$$

(49)

where

$$\begin{array}{c}{\mathit{A}}_{\mathrm{eq}}=\left[\begin{array}{cc}{\mathbf{0}}_{(q-k+1)lm\times ql(pl+pm)}& \begin{array}{cc}{\mathbf{1}}_{1\times k}\otimes {\mathit{I}}_{lm\times lm}& {\mathbf{0}}_{lm\times (q-k)lm}\\ {\mathbf{0}}_{(q-k)lm\times klm}& {\mathit{I}}_{(q-k)lm\times (q-k)lm}\end{array}\end{array}\right],\hfill \\ {\mathit{b}}_{\mathrm{eq}}=\left[\begin{array}{c}\mathrm{vec}\left({\mathit{K}}_{ss}\right)\\ {\mathbf{0}}_{(q-k)lm\times 1}\end{array}\right].\hfill \end{array}$$

(50)

4.2.2. Zero Transfer Functions

The outputs of some MIMO systems are unaffected by the inputs. Accordingly, some transfer functions are zero. However, by identifying such systems based on noisy data, the zero transfer functions cannot be established. The transfer functions might be prespecified during the identification phase to address the aforementioned issue.

If the jth input and ith output have a zero transfer function, and all the impulse response coefficients at this channel are zeros,

$$g_0^{ij}=g_1^{ij}=\cdots =g_{q-1}^{ij}=0,$$

(51)

there are q constraints from Equation (51). The zero transfer function can be written as

$$({\mathbf{0}}_{1\times ql(pl+pm)}{\mathbf{0}}_{1\times klm+l(j-1)+i-1}10\dots 0)\theta =0.$$

(52)

Therefore, ${\mathit{A}}_{\mathrm{eq}}$ and ${\mathit{b}}_{\mathrm{eq}}$ can be described as

$$\begin{array}{c}{\mathit{A}}_{\mathrm{eq}}=\left[\begin{array}{cc}{\mathbf{0}}_{q\times ql(pl+pm)}& \begin{array}{ccccc}{\mathbf{0}}_{1\times l(j-1)+i-1}& 1& 0& \cdots & 0\\ {\mathbf{0}}_{1\times lm+l(j-1)+i-1}& 1& 0& \cdots & 0\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ {\mathbf{0}}_{1\times (q-1)lm+l(j-1)+i-1}& 1& 0& \cdots & 0\end{array}\end{array}\right]\hfill \\ \in {\mathbb{R}}^{q\times \left(ql\right(pl+pm)+qlm)},\hfill \\ {\mathit{b}}_{\mathrm{eq}}={\mathbf{0}}_{q\times 1}.\hfill \end{array}$$

(53)

Note that each zero transfer function needs q constraints.

4.3. Summary of Subspace System Identification Algorithm

The proposed method CSIPI-RIOM (continuous-time subspace identification with prior information using realization of input–output maps) can be summarized as follows:

(1) Form the data matrices ${\mathit{Y}}_{p,q,N},{\mathit{U}}_{p,q,N},{\mathit{Y}}_{0,p,N},{\mathit{U}}_{0,p,N}$. Then, formulate $\mathit{R},\mathit{Z},y$ in Equation (41).

(2) Transform the prior information into ${\mathit{A}}_{\mathrm{eq}}$ and ${\mathit{b}}_{\mathrm{eq}}$ and solve the least squares optimization problem in Equation (42) with the constraint in Equation (43).

(3) Estimate the impulse response parameters ${\widehat{\theta }}_g$ and construct the following Hankel matrix $\mathit{T}$:

$$\mathit{T}=\left[\begin{array}{cccc}{\widehat{g}}_1& {\widehat{g}}_2& \cdots & {\widehat{g}}_{q/2}\\ {\widehat{g}}_2& {\widehat{g}}_3& \cdots & {\widehat{g}}_{q/2+1}\\ ⋮& ⋮& ⋮& ⋮\\ {\widehat{g}}_{q/2}& {\widehat{g}}_{q/2+1}& \cdots & {\widehat{g}}_{q-1}\end{array}\right].$$

(54)

(4) Factorize $\mathit{T}$ using Kung’s realization algorithm as

$$\mathit{T}={\mathit{USV}}^T.$$

(55)

(5) Determine the system order n by inspecting the singular value of $\mathit{S}$. Then, estimate the observability matrix $\mathbf{\Gamma }$ and controllability matrix $\mathbf{\Delta }$ as

$$\mathbf{\Gamma }=\mathit{U}(:,1:n){\mathit{S}}^{1/2},\mathbf{\Delta }={\mathit{S}}^{1/2}\mathit{V}{(:,1:n)}^T.$$

(56)

(6) Extract the system matrices using the $\mathbf{\Gamma }$ and $\mathbf{\Delta }$ as $\widehat{\mathit{A}}={\underline{\mathbf{\Gamma }}}^{†}\overline{\mathbf{\Gamma }},\underline{\mathbf{\Gamma }}=\mathbf{\Gamma }(1:l(q-1),:),\overline{\mathbf{\Gamma }}=\mathbf{\Gamma }(l+1;ql,:)$. $\widehat{\mathit{B}}=\mathbf{\Delta }(:,1:m)$, $\widehat{\mathit{C}}=\mathbf{\Gamma }(1:l,:)$, $\widehat{\mathit{D}}={\widehat{g}}_0$.

5. Numerical Simulation

5.1. Example 1: Known Steady-State Gain

Consider the following continuous-time system:

$${\displaystyle G\left(s\right)=\frac{42}{(s+6)(s+7)},}$$

(57)

and the following first-order inner function is used:

$${\displaystyle \varphi \left(s\right)=\frac{s-p}{s+p}.}$$

(58)

To validate the superiority of the new algorithm, CSIPI-RIOM is compared with CSI-RIOM (Ohta et al., 2005) [32] and PBSIL (Bergamasco et al., 2011) [11]. A total of 100 Monte Carlo simulations were run. They were excited using the unit variance, zero-mean white Gaussian process. The Hankel matrices have $N=81$ columns, and the horizons for the three methods were all set to $p=q=10$. The sample period was set to $0.01$ s.

In this case, the known steady-state gain is known during subspace identification. Figure 1 depicts the average step responses for each of the three white noise approaches. The blue line of tiny circles in Figure 1 shows that the step response from CSIPI-RIOM is closer to the step response of the actual system.

Figure 2 shows the averaged bode plot of the true and identified systems. From Figure 2, the red dotted lines of the CSIPI-RIOM method are closer to the black lines of the true system. This shows that the proposed approach produces better identification results.

For clarity, the mean and variance of the gain estimations are depicted in

Table 1. The mean and variance of the gain estimations for Example 1.

Approach	CSIPI-RIOM	CSI-RIOM	PBSIL
Mean	1.0085	1.0264	1.0847
Variance	0.0026	0.0087	0.0579

.

Table 1 shows that the estimate values of CSIPI-RIOM are better than those obtained from CSI-RIOM and PBSIL. It can be clearly seen that CSIPI-RIOM performs more consistently and accurately than the other two approaches.

5.2. Example 2: Zero Transfer Functions

Consider the following continuous-time system:

$$G\left(s\right)=\left[\begin{array}{cc}{G}_{11}\left(s\right)& G_{12}\left(s\right)\\ G_{21}\left(s\right)& G{\left(s\right)}_{22}\end{array}\right]=\left[\begin{array}{cc}{\displaystyle \frac{2}{(s+2)}}& 0\\ 0& {\displaystyle \frac{3}{(s+3)}}\end{array}\right],$$

(59)

and the following first-order inner function is used:

$${\displaystyle \varphi \left(s\right)=\frac{s-p}{s+p},}$$

(60)

which has two zero transfer functions $G_{12}$ and $G_{21}$, while $G_{11}$ and $G_{22}$ have unity gains.

The new algorithm, CSIPI-RIOM, is contrasted with CSI-RIOM (Ohta et al., 2005) [32] and PBSIL (Bergamasco et al., 2011) [11] to demonstrate its superiority and adaptability. A total of 100 Monte Carlo simulations were run. The zero-mean white Gaussian process with unit variance was used to excite the system. The Hankel matrices have $N=81$ columns, and the horizons for the three methods were all set to $p=q=10$. The sample period was set to $0.01$ s.

Figure 3 shows the average step responses from three methods with white noise, respectively. Obviously, the green lines reflecting the step response from CSIPI-RIOM are closer to the step response of the true system. Figure 4, Figure 5 and Figure 6 show the pole estimations for the identified system using the methods PBSIL, CSI-RIOM, and CSIPI-RIOM. In these figures, red points and blue points indicate the pole value of the true system and the identified system, respectively. It can be seen that the pole estimations of the CSIPI-RIOM method are closer to the pole of the true system.

For clarity, the mean and variance of the pole estimations are depicted in

Table 2. The numerical results of the pole estimations for Example 2.

Approach	Mean of ${\mathit{p}}_1$	Variance of ${\mathit{p}}_1$	Mean of ${\mathit{p}}_2$	Variance of ${\mathit{p}}_2$
CSIPI-RIOM	−1.9986	0.0048	−2.9917	0.0052
CSI-RIOM	−1.9860	0.0524	−2.9604	0.0624
PBSIL	−1.9674	0.2647	−3.0252	0.3147

.

Table 2 shows that the estimate values for CSIPI-RIOM are better than those obtained from CSI-RIOM and PBSIL. In comparison with the other two methods, the proposed method CSIPI-RIOM performs more accurately and consistently. As a result, according to the simulation results, CSIPI-RIOM performs better in the identification process.

6. Conclusions

A continuous-time subspace identification with prior information using generalized orthonormal basis functions is proposed. The method is based on converting a continuous-time stochastic system into a discrete-time stochastic system. The continuous-time system model is produced by the deterministic part’s inverse transform. Prior information is described as equality constraints based on the constrained least squares. The accuracy of the impulse response parameters is increased by incorporating prior knowledge, and system matrices can be obtained by applying Kung’s realization approach. The results of the simulation examples demonstrate that the proposed method indeed improves the accuracy of the identified models. It is important to extend the proposed method to more types of prior information, which will be future work.