A neural network adaptive interval observer design for nonlinear systems

IET Control Theory & ApplicationsVolume 16, Issue 6 p. 615-624 ORIGINAL RESEARCH PAPEROpen Access A neural network adaptive interval observer design for nonlinear systems Zeren Yi, Zeren Yi orcid.org/0000-0002-1809-1962 School of Automation Science and Engineering, South China University of Technology, Guangzhou, People's Republic of China School of Electrical Engineering, Guangxi University, Nanning, People's Republic of ChinaSearch for more papers by this authorWei Xie, Corresponding Author Wei Xie weixie@scut.edu.cn School of Automation Science and Engineering, South China University of Technology, Guangzhou, People's Republic of China Guangdong Provincial Key Laboratory of Technique and Equipment for Macromolecular Advanced Manufacturing, South China University of Technology, China Correspondence Wei Xie, School of Automation Science and Engineering, South China University of Technology, Guangdong Provincial Key Laboratory of Technique and Equipment for Macromolecular Advanced Manufacturing, South China University of Technology, Guangzhou, 510641, People's Republic of China. Email: weixie@scut.edu.cnSearch for more papers by this authorLongwen Liu, Longwen Liu School of Automation Science and Engineering, South China University of Technology, Guangzhou, People's Republic of ChinaSearch for more papers by this authorBugong Xu, Bugong Xu School of Automation Science and Engineering, South China University of Technology, Guangzhou, People's Republic of ChinaSearch for more papers by this author Zeren Yi, Zeren Yi orcid.org/0000-0002-1809-1962 School of Automation Science and Engineering, South China University of Technology, Guangzhou, People's Republic of China School of Electrical Engineering, Guangxi University, Nanning, People's Republic of ChinaSearch for more papers by this authorWei Xie, Corresponding Author Wei Xie weixie@scut.edu.cn School of Automation Science and Engineering, South China University of Technology, Guangzhou, People's Republic of China Guangdong Provincial Key Laboratory of Technique and Equipment for Macromolecular Advanced Manufacturing, South China University of Technology, China Correspondence Wei Xie, School of Automation Science and Engineering, South China University of Technology, Guangdong Provincial Key Laboratory of Technique and Equipment for Macromolecular Advanced Manufacturing, South China University of Technology, Guangzhou, 510641, People's Republic of China. Email: weixie@scut.edu.cnSearch for more papers by this authorLongwen Liu, Longwen Liu School of Automation Science and Engineering, South China University of Technology, Guangzhou, People's Republic of ChinaSearch for more papers by this authorBugong Xu, Bugong Xu School of Automation Science and Engineering, South China University of Technology, Guangzhou, People's Republic of ChinaSearch for more papers by this author First published: 24 February 2022 https://doi.org/10.1049/cth2.12258AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Abstract A design method of neural network adaptive interval observer for the continuous-time unknown nonlinear system is proposed. The bounds of unknown nonlinear functions are described by the state and the input, so that the bounds of the nonlinear function vector are unmeasurable. Therefore, this method can be used to observe systems with higher degree of nonlinearity and any priori knowledge about systems. In this work, the hyper basis function neural network (HBFNN) is presented to estimate the boundary (including lower and upper bounds) of the unknown nonlinear function vector. Furthermore, the non-negativity and stability of the error dynamic system are given, and the convergence and non-negativity are also guaranteed. Compared with the traditional interval observer, the efficiency of the proposed approach is verified through a numerical simulation example. 1 INTRODUCTION System state estimation plays a crucial role in fault diagnosis and enhancing control performance. However, most physical systems are inherently non-linear. Such systems are usually processed by linearization or quasi-linearization methods under appropriate conditions, and the purpose is to more easily obtain the system state and limit the state observation error to a smaller dynamic range. In the past decades, several nonlinear observer methods have been proposed, such as extended Kalman filters, adaptive observers and interval observers etc., [1-3]. However, most of them have depended completely on the understanding of the priori knowledge of the system's nonlinearity. Thus, some model-based observers are difficult or impossible to be applied to the nonlinear systems. So the interval observer was proposed, which was different from the previous point estimation observer technique. In general, the interval observer generates lower and upper bounds on state estimations, while the point estimation asymptotically converges to the real trajectory. Since the applicability of interval observers is generally weaker than that of traditional point estimators, this relaxation of the estimation target is allowed in some applications. Therefore, interval observers are more inclusive for nonlinear systems. Gouze et al. originally proposed the concept of interval observer [4], and some progress has also been made in the interval observers' application for nonlinear systems [5-11]. When the system is not cooperative, the matrix similarity transformation method has been adopted, and the non-Metzler matrix is converted to the Metzler, such as [5]. Combined with the precise linearization method, this result is further extended to nonlinear systems. Considering both the stability and non-negativity, it was difficult to give an observer gain calculation method. In response to this problem, a loop verification algorithm was elaborated in [6] to deal with this kind of problem. Another option can solve the LMI using an iterative method or a grid method. Through the interval observer, the feedback stabilization design of nonlinear systems is studied [7, 8]. For linear parameter-varying (LPV) systems and uncertain systems, corresponding interval observers are designed [9, 10]. In [11], the feedback gain of the nonlinear system based on the Luenberger-like interval observer was designed using the LMI method. Fault diagnosis concept for nonlinear systems was addressed in [12]. Menasria et al. designed an interval observer for nonlinear uncertain Takagi-Sugeno (T-S) systems [13]. This observer was issued by pole-placement and LMI tools. In [14], the concept of integral input-state stability was integrated into the interval observer design, and the boundedness and convergence of interval estimates in the case of disturbances were given. The construction of interval observers for Lipschitz nonlinear systems was implemented in [15, 16]. In [17], the boundary of the nonlinear system with disturbances was estimated by the disturbance interval observer method. Although the aforementioned interval observer can obtain good performance, its works on the bounds of nonlinear systems are usually based on the implicit assumption of the model linearization or satisfied Lipschitz condition. However, these are difficult to achieve in real life. Apparently, when the nonlinear systems are unknown, especially, with unmeasurable state, the following two essential issues should be dealt with: (1) how to design a suitable interval observer depending only on unknown nonlinear functions; (2) how to estimate the bounds of the unknown nonlinear dynamics in the presence of unmeasurable states? The neural network (NN) can be widely used to estimate the unknown nonlinear functions due to its universal approximation. In the past few decades, scholars have been studying nonlinear system observers based on neural networks and fuzzy logic systems (FLSs). Taking advantage of neural network's powerful learning capabilities, various neural network observers are designed[18-25]. For example, by using the HBFNN to approximate the nonlinear state, the complexity of traditional nonlinear system observer design method is reduced, and the complex nonlinear modeling problem is solved in [26]. In the NN observer design of nonlinear systems, the methods in [27, 28] have improved the learning speed of neural network in the training process. Thus, neural network is expected to estimate nonlinear functions for interval observer design in a similar way. In [29], a new two-hidden-layer recurrent neural network (THLRNN) structure containing two hidden layers was designed to approximate the unknown nonlinearities. The fuzzy double hidden layer recurrent neural network (FDHLRNN) is presented and applied to improve the accuracy of a nonlinear approximation in [30, 31]. Furthermore, [32] investigated the design of a fuzzy state observer through FLSs to estimate the unmeasurable states. In [33], during the control design of a class of multi-input and multi-output (MIMO) nonlinear non-strict feedback systems, the fuzzy state observer is also designed to estimate the unmeasured states by approximating the unknown nonlinear functions with FLSs. Therefore, the NN is extensively regarded as an effective tool for dealing with unknown nonlinear systems with immeasurable states. Motivated by the above considerations, the NN is used to estimate unknown nonlinear functions in the constructed interval observer. The purpose of the proposed neural network interval observer is to solve the bounds of unknown nonlinear systems with immeasurable states. How to design the NN interval observer for unknown nonlinear systems is an open and challenging problem. So far, there are no available results to be reported. This paper investigates the HBFNN interval observer design. Compared with traditional interval observer design methods, our methods can weaken restrictions that the known structure condition of nonlinear functions should be satisfied. Moreover, the main contributions of this work are: (a) The NN is combined with the interval observer design, which leads to a NN interval estimation method for unknown nonlinear systems. So it has the advantages of both neural networks and interval observers. In this work, this method is used to estimate the bounds of the unknown nonlinear function, and further realize the unmeasurable state interval estimation. (b) The HBFNN can approximate the boundary of nonlinear function with arbitrary precision by regulating variable weight connection. The designed HBFNN interval observer adjusts interval width by network weight correction and network error, which can generate a narrower interval width. The efficiency and the high performance of the proposed observer are due to the parallel architecture of the HBFNN and its high capacity of training, which can observe the bounds of highly nonlinear systems with any priori knowledge. The general structure of this paper is as follows. The system description and some basic knowledge of interval observers and the NN are given in Section 2. In Section 3, the main results are illustrated: (i) the NN adaptive interval observer is designed for a class of unknown nonlinear systems, (ii) the interval state and error stability analysis are given. A simulation example is given in Section 4, which shows the results for the interval state estimator and illustrates the effectiveness of the method. Finally, concluding remarks of this work are made in Section 5. 2 SYSTEM DESCRIPTION AND PRELIMINARIES 2.1 Model formulation for nonlinear systems Consider the following nonlinear system x ̇ = A x + g x , u + E d y = C x , \begin{equation} {\begin{cases} \dot{x} = Ax + g{\left({x, u } \right)} + Ed \\ y = Cx , \end{cases}} \end{equation} (1)where A, E and C are constant matrices. x ∈ R n $x\in {\mathbb {R}^n}$ , u ∈ R p $u\in {\mathbb {R}^p}$ , y ∈ R m $y\in {\mathbb {R}^m}$ denote the state, the input and the output vectors, respectively. g ( x , u ) $g(x,u)$ is the unknown nonlinear function. d is the bounded time-varying disturbances. 2.2 Notations, definitions and lemmas Some mathematical notations to be used throughout this paper are given below. About a variable x, x ̲ $\underline{x}$ is the lower bound, and x ¯ $\overline{x}$ is the upper bound, such as x ∈ [ x ̲ , x ¯ ] $x\in {[\underline{x},\overline{x}]}$ . If all the off-diagonal elements are nonnegative regarding a matrix A ∈ R n × n $A\in {\mathbb {R}^{n\times n}}$ , it is called Metzler. The Euclidean norm of a vector x ∈ R n $x\in {\mathbb {R}^n}$ is ∥ x ∥ $\Vert x\Vert$ . Regarding a matrix A ∈ R n × n $A\in {\mathbb {R}^{n\times n}}$ , let the symbol ∥ A ∥ F ${\Vert A\Vert }_F$ denotes the Frobenius norm of the matrix. For a symmetric matrix A, λ m i n ( A ) ${\lambda }_{min}(A)$ denotes the minimum eigenvalue. Definition 1.The HBFNN is the improved radial basis function neural network (RBFNN). The HBFNN is the 3-layer forward NN with only a single hidden layer. Hidden Layer adopts a radial basis function. There are many radial basis transfer functions, such as the Gaussian function. It turns out that any continuous function can be approximated by the HBFNN with arbitrary precision. We use it to approximate the nonlinear function with high precision, such as estimating the function vector g ( x , u ) $g(x,u)$ . The network's input is z = [ x T , u T ] T $z={[x^{\rm T},u^{\rm T}]^{\rm T}}$ , z i ∈ R n x ${z_i}\in {\mathbb {R}^{n_x}}$ is the i th $i^{\text{th}}$ input of Input Layer, w j ${w_j}$ is the j th $j^{\text{th}}$ connection weights from Hidden to Output Layer, h j ( ∗ ) ${{h_j}(*)}$ is the basic function of the j th $j^{\text{th}}$ neuron. ( n + p ) $(n+p)$ is the number of input nodes, and the number of output nodes is n. J is the number of hidden nodes. The center of the j th $j^{\text{th}}$ basis function is c j $c_j$ . The similarity between z i $z_i$ and c j $c_j$ is σ ∈ R 1 $\sigma \in {\mathbb {R}^1}$ . The ith output of an HBFNN is y i = f i x , u = ∑ j = 0 J w i j h j z i , c j , σ j . \begin{align} {y_i} = {f_i}{\left({x,u} \right)} = \sum \limits _{j = 0}^J {{w_{ij}}{h_j}{\left({{z_i},{c_j},{\sigma _j}} \right)}}. \end{align} (2) When calculating the distance between the input neuron and the central neuron, we use the Mahalanobis-like distance for the HBF [34-36]. The form is as follows h j z i , c j , ∑ j = e − 0.5 x i − c j T ∑ j x i − c j , \begin{align} {h_j}{\left({{z_i},{c_j},{{\sum }_j}} \right)} = {e^{ - 0.5{{{\left({{x_i} - {c_j}} \right)}}^{\rm T}}{{\sum }_j}{\left({{x_i} - {c_j}} \right)}}}, \end{align} (3)where ∑ j = diag ( 1 / σ 1 2 , 1 / σ 2 2 , … , 1 / σ n x 2 ) , j = 1 , 2 , … , J ${{\sum }_j} = {\rm diag}\Big(1 \Big / {\sigma _1^{2}}, 1 \Big / {\sigma _2^{2}}, \ldots , 1 \Big / {\sigma _{n_x}^{2}} \Big),j = 1,2, \ldots ,J$ .For the HBFNN learning algorithm, this paper adopts the decision tree method to calculate the network center. Quinlan's C4.5 algorithm was used for calculation [27]. Lemma 1.[37] If x is a vector satisfying x ̲ ≤ x ≤ x ¯ $\underline{x}\le x \le \overline{x}$ , then A + x ̲ − A − x ¯ ≤ A x ≤ A + x ¯ − A − x ̲ , \begin{align} {A^ + }\underline{x} - {A^ - }\overline{x} \le Ax \le {A^ + }\overline{x} - {A^ - }\,\underline{x}, \end{align} (4)where A m × n $A^{m \times n}$ represents the constant matrix. A + = max { A , 0 } ${A^ + }= \max \lbrace {A, 0} \rbrace$ , A = A + − A − $A = {A^ + }-{A^ - }$ . Lemma 2.[38] The continuous-time system z ̇ = S z + r $\dot{z} = Sz + r$ is called cooperative, where S is a Metzler matrix and r ≥ 0 $r \ge 0$ . If z ( 0 ) ≥ 0 $z(0 ) \ge 0$ , then z ( t ) ≥ 0 $z(t ) \ge 0$ . Lemma 3.[5, 39] If there are a gain matrix L ∈ R ( n − q ) × ( p − q ) $L \in {\mathbb {R}_{({n - q} ) \times ({p - q} )}}$ and a corresponding transformation matrix H, then H ( A − L C ) H − 1 $H({A - LC} ){H^{ - 1}}$ is both Hurwitz and Metzler. Assumption 1.The weight matrices Θ ̲ $\underline{\Theta }$ , Θ and Θ ¯ $\overline{\Theta }$ are bounded, ∥ Θ ̲ ∥ F ≤ Θ ̲ M ${\Vert \underline{\Theta } \Vert _F} \le {\underline{\Theta }_M}$ , ∥ Θ ¯ ∥ F ≤ Θ ¯ M ${\Vert \overline{\Theta } \Vert _F} \le {\overline{\Theta }_M}$ , which satisfies Θ ̲ ≤ Θ ≤ Θ ¯ $\underline{\Theta }\le \Theta \le \overline{\Theta }$ . Assumption 2.For x ̲ ( t ) ≤ x ( t ) ≤ x ¯ ( t ) $\underline{x}(t)\le x(t) \le \overline{x}(t)$ , ∥ u ∥ ≤ U $\Vert u\Vert \le U$ , ∀ t ≥ 0 $\forall t \ge 0$ , there exist two functions h ̲ ( x ̲ , x ¯ , u ) $\underline{h}(\underline{x},\overline{x},u)$ , and h ¯ ( x ̲ , x ¯ , u ) $\overline{h}(\underline{x},\overline{x},u)$ satisfying − ∞ < h ̲ x ̲ , x ¯ , u ≤ h x , u ≤ h ¯ x ̲ , x ¯ , u < + ∞ , \begin{align} - \infty < \underline{h}{\left({\underline{x},\overline{x},u} \right)} \le h{\left({x,u} \right)} \le \overline{h}{\left({\underline{x},\overline{x},u} \right)} < + \infty , \end{align} (5)where h ̲ ( x ̲ , x ¯ , u ) $\underline{h}(\underline{x},\overline{x},u)$ and h ¯ ( x ̲ , x ¯ , u ) $\overline{h}(\underline{x},\overline{x},u)$ satisfy the following conditions: the vectors involved in inequalities should be understood in terms of components. 3 MAIN RESULTS For the continuous-time nonlinear system (1), the Luenberger-based HBFNN adaptive interval observer structure is depicted in Figure 1. It is expressed as x ¯ ̇ = A x ¯ + g ¯ x ̲ , x ¯ , u + L y − C x ¯ + ϕ ¯ x ̲ ̇ = A x ̲ + g ̲ x ̲ , x ¯ , u + L y − C x ̲ + ϕ ̲ , \begin{equation} {\begin{cases} \dot{\overline{x}} = A\overline{x} + \overline{g}{\left({\underline{x},\overline{x},u} \right)} + L{\left({y - C\overline{x}} \right)} + \overline{\phi } \\[6pt] \dot{\underline{x}} = A\underline{x} + \underline{g}{\left({\underline{x},\overline{x},u} \right)} + L{\left({y - C\underline{x}} \right)}+ \underline{\phi }, \end{cases}} \end{equation} (6)where x ̲ $\underline{x}$ denotes x lower bound, and x ¯ $\overline{x}$ denotes x upper bound. L is the interval observer gain matrix. g ̲ ( x ̲ , x ¯ , u ) $\underline{g}({\underline{x},\overline{x},u} )$ represents g ( x , u ) $g(x,u)$ lower bound, and g ¯ ( x ̲ , x ¯ , u ) $\overline{g}({\underline{x},\overline{x},u} )$ represents g ( x , u ) $g(x,u)$ upper bound. The bounded disturbance ϕ = E d $\phi = Ed$ , ϕ ̲ $\underline{\phi }$ is ϕ lower bound, and ϕ ¯ $\overline{\phi }$ is ϕ upper bound. FIGURE 1Open in figure viewerPowerPoint Structure block diagram of the HBFNN adaptive interval observer Define the state observation error and residual, respectively, as follows e ¯ = x ¯ − x e ̲ = x − x ̲ , \begin{equation} {\begin{cases} \overline{e} = \overline{x} - x \\ \underline{e} = x - \underline{x}, \end{cases}} \end{equation} (7) r ¯ = y ¯ − y = C e ¯ r ̲ = y − y ̲ = C e ̲ . (8) Combining equations (6) and (7), e ¯ ̇ = A c e ¯ + g ¯ x ̲ , x ¯ , u − g x , u + ϕ ¯ − ϕ e ̲ ̇ = A c e ̲ + g x , u − g ̲ x ̲ , x ¯ , u + ϕ − ϕ ̲ , \begin{equation} {\begin{cases} \dot{\overline{e}} = {A_c}\overline{e} + \overline{g}{\left({\underline{x},\overline{x},u} \right)} - g{\left({x,u} \right)} + \overline{\phi } - \phi \\[6pt] \dot{\underline{e}} = {A_c}\underline{e} + g{\left({x,u} \right)} - \underline{g}{\left({\underline{x},\overline{x},u} \right)} + \phi - \underline{\phi }, \end{cases}} \end{equation} (9)where A c = A − L C ${A_c} = A - LC$ . The aim of this work is to satisfy: 1). The observation errors of (9) are stable. 2). g ̲ ( x ̲ , x ¯ , u ) $\underline{g}({\underline{x},\overline{x},u} )$ and g ¯ ( x ̲ , x ¯ , u ) $\overline{g}({\underline{x},\overline{x},u} )$ in (6) are estimated by the HBFNN. According to the basic idea that a nonlinear function can be approximated via using feed-forward neural networks with arbitrary precision, the unknown nonlinear function vector g ( x , u ) $g(x,u)$ is expressed as g x , u = Θ T h z + ε z , \begin{align} g{\left({x,u} \right)} = {{\Theta }^{\rm T}}h{\left({z} \right)} + \varepsilon {\left(z \right)}, \end{align} (10)where ΘT is the NN weight matrix. z = [ x T , u T ] T $z={[x^{\rm T},u^{\rm T}]^{\rm T}}$ . ε ( z ) $\varepsilon (z )$ is the given approximation error. h ( z ) $h(z)$ is the neural network transfer function. The HBFNN is used for adaptive learning of the nonlinear functions, and the network estimated value g ¯ $\overline{g}$ and g ̲ $\underline{g}$ are expressed as g ¯ = Θ ¯ T h ¯ z ̲ , z ¯ + ε ¯ z g ̲ = Θ ̲ T h ̲ z ̲ , z ¯ + ε ̲ z , \begin{equation} {\begin{cases} \overline{g} = {{\overline{\Theta }}^{\rm T}}\overline{h}{\left({\underline{z},\overline{z}} \right)} + \overline{\varepsilon }{\left(z \right)} \\[11pt] {\underline{g} = {{\underline{\Theta }}^{\rm T}}\underline{h}{\left({\underline{z},\overline{z}} \right)} + \underline{\varepsilon }{\left(z \right)}}, \end{cases}} \end{equation} (11)where Θ ¯ T ${\overline{\Theta }}^{\rm T}$ is the upper bound of ΘT, Θ ̲ T ${\underline{\Theta }}^{\rm T}$ is the lower bound of ΘT. h ¯ ( z ̲ , z ¯ ) $\overline{h}({\underline{z},\overline{z}} )$ , h ̲ ( z ̲ , z ¯ ) $\underline{h}({\underline{z},\overline{z}} )$ are the upper and the lower bounds of the neural network transfer function. ε ¯ ( z ) $\overline{\varepsilon }(z )$ is the upper bound of ε ( z ) $\varepsilon (z )$ , and ε ̲ ( z ) $\underline{\varepsilon }(z )$ is the lower bound of ε ( z ) $\varepsilon (z )$ , ε ̲ ( z ) ≤ ε ( z ) ≤ ε ¯ ( z ) $ \underline{\varepsilon }(z ) \le \varepsilon (z ) \le \overline{\varepsilon }(z )$ . We substitute Equations (10) and (11) into Equation (9) and get e ¯ ̇ = A c e ¯ + Θ ¯ T h ¯ z ̲ , z ¯ − Θ T h z + ε ¯ z − ε z + ϕ ¯ − ϕ e ̲ ̇ = A c e ̲ − Θ ̲ T h ̲ z ̲ , z ¯ + Θ T h z + ε z − ε ̲ z + ϕ − ϕ ̲ . \begin{align} {\begin{cases} \dot{\overline{e}} = {{{A_c}\overline{e} + \overline{\Theta }}^{\rm T}}\overline{h}{\left({\underline{z},\overline{z}} \right)} - {{\Theta }^{\rm T}}h{\left({z} \right)} \\[11pt] + \overline{\varepsilon }{\left(z \right)} - \varepsilon {\left(z \right)} + \overline{\phi }- \phi \\[11pt] \dot{\underline{e}} = {A_c}\underline{e} - \underline{\Theta } ^{\rm T}\underline{h}{\left({\underline{z},\overline{z}} \right)} + {{\Theta }^{\rm T}}h{\left({z} \right)} \\[11pt] + \varepsilon {\left(z \right)} - \underline{\varepsilon }{\left(z \right)} + \phi - \underline{\phi }. \end{cases}} \end{align} (12) The following theorem is proposed to design the interval observer (6). Theorem 1.Considering the nonlinear system (1), the neural network adaptive interval observer can be constructed as x ¯ ̇ = A c x ¯ + Θ ¯ T h ¯ z ̲ , z ¯ + ϕ ¯ + L y x ̲ ̇ = A c x ̲ + Θ ̲ T h ̲ z ̲ , z ¯ + ϕ ̲ + L y , \begin{equation} {\begin{cases} \dot{\overline{x}} = {{{A_c}\overline{x} + \overline{\Theta }}^{\rm T}}\overline{h}{\left({\underline{z},\overline{z}} \right)} + \overline{\phi }+ Ly\\[11pt] \dot{\underline{x}} = {A_c}\underline{x} + \underline{\Theta } ^{\rm T}\underline{h}{\left({\underline{z},\overline{z}} \right)} +\underline{\phi }+ Ly, \end{cases}} \end{equation} (13)where Θ ¯ $\overline{\Theta }$ and Θ ̲ $\underline{\Theta }$ satisfy Assumption 1. h ¯ ( z ̲ , z ¯ ) $\overline{h}({\underline{z},\overline{z}} )$ and h ̲ ( z ̲ , z ¯ ) $\underline{h}({\underline{z},\overline{z}} )$ satisfy Assumption 2. The initial conditions x ̲ ( 0 ) ≤ x ( 0 ) ≤ x ¯ ( 0 ) $\underline{x}(0)\le x(0) \le \overline{x}(0)$ are true.The parameter linearization weight correction rule of the neural network: Θ ¯ ̇ = − η ¯ ∂ J ¯ ∂ Θ ¯ − ρ ¯ r ¯ Θ ¯ Θ ̲ ̇ = − η ̲ ∂ J ̲ ∂ Θ ̲ − ρ ̲ r ̲ Θ ̲ , \begin{align} {\begin{cases} \dot{\overline{\Theta }} = - {\overline{\eta }}\frac{{\partial {\overline{J} }}}{{\partial \overline{\Theta }}} - {\overline{\rho }}{\left\Vert {\overline{r}} \right\Vert} \overline{\Theta }\\[11pt] \dot{\underline{\Theta }} = - {\underline{\eta }}\frac{{\partial {\underline{J} }}}{{\partial \underline{\Theta }}} - {\underline{\rho }}{\left\Vert {\underline{r}} \right\Vert} \underline{\Theta }, \end{cases}} \end{align} (14)where J ¯ = 1 2 r ¯ T r ¯ ${\overline{J} } = \frac{1}{2}{\overline{r}^{\rm T}}\overline{r}$ , J ̲ = 1 2 r ̲ T r ̲ ${\underline{J}} = \frac{1}{2}{\underline{r}^{\rm T}}\underline{r}$ are the upper and lower objective functions. η ¯ $\overline{\eta }$ , η ̲ $\underline{\eta }$ are the upper and lower learning rates. ρ ¯ $\overline{\rho }$ , ρ ̲ $\underline{\rho }$ are the upper and lower attenuation coefficients.If A c ${A_c}$ is the Hurwitz and Metzler matrix for some L ∈ R n × m $L \in {\mathbb {R}^{n \times m}}$ , then the neural network adaptive interval observer (13) is stable, and the following relationship is obtained: x ̲ ( t ) ≤ x ( t ) ≤ x ¯ ( t ) $\underline{x}(t)\le x(t) \le \overline{x}(t)$ , ∀ t ≥ 0 $\forall t \ge 0$ . Proof. e ¯ ̇ = A c e ¯ + Θ ¯ − Θ T h ¯ z ̲ , z ¯ + Θ T h ¯ z ̲ , z ¯ − h z + ε ¯ z − ε z + ϕ ¯ − ϕ e ̲ ̇ = A c e ̲ − Θ ¯ − Θ T h ̲ z ̲ , z ¯ + Θ T h z − h ̲ z ̲ , z ¯ + ε z − ε ̲ z + ϕ − ϕ ̲ . \begin{align} {\begin{cases} \dot{\overline{e}} = {A_c}\overline{e} + {\left({\overline{\Theta } - {\Theta }} \right)}^{\rm T}\overline{h}{\left({\underline{z},\overline{z}} \right)} \\[11pt] + {{\Theta }^{\rm T}}{\left[ \overline{h}{\left({\underline{z},\overline{z}} \right)} - h{\left({z} \right)} \right]} + \overline{\varepsilon }{\left(z \right)} - \varepsilon {\left(z \right)} + \overline{\phi }- \phi \\[11pt] \dot{\underline{e}} = {A_c}\underline{e} - {\left({\overline{\Theta } - {\Theta }} \right)}^{\rm T}\underline{h}{\left({\underline{z},\overline{z}} \right)}\\[11pt] + {{\Theta }^{\rm T}}{\left[h{\left({z} \right)} - \underline{h}{\left({\underline{z},\overline{z}} \right)} \right]} + \varepsilon {\left(z \right)} - \underline{\varepsilon }{\left(z \right)} + \phi - \underline{\phi }. \end{cases}} \end{align} (15) Define the lower bound of the NN weight estimation error as e ̲ Θ = Θ ̲ − Θ ${\underline{e}_{\Theta }} = \underline{\Theta } - {\Theta }$ , and the upper bound is e ¯ Θ = Θ ¯ − Θ ${\overline{e}_{\Theta }} = \overline{\Theta } - {\Theta }$ . Define the matrix A c ${A_c}$ as Hurwitz.Thus, e ¯ ̇ = A c e ¯ + e ¯ Θ T h ¯ z ̲ , z ¯ + ψ ¯ e ̲ ̇ = A c e ̲ + e ̲ Θ T h ̲ z ̲ , z ¯ + ψ ̲ , \begin{equation} {\begin{cases} \dot{\overline{e}} = {A_c}\overline{e} + \overline{e}_{\Theta }^{\rm T}\overline{h}{\left({\underline{z},\overline{z}} \right)} + \overline{\psi }\\[11pt] \dot{\underline{e}} = {A_c}\underline{e} + \underline{e}_{\Theta } ^{\rm T}\underline{h}{\left({\underline{z},\overline{z}} \right)} + \underline{\psi }, \end{cases}} \end{equation} (16)where ψ ¯ = Θ T h ¯ z ̲ , z ¯ − h z + ε ¯ z − ε z + ϕ ¯ − ϕ ψ ̲ = Θ T h z − h ̲ z ̲ , z ¯ + ε z − ε ̲ z + ϕ − ϕ ̲ . \begin{align} {\begin{cases} \overline{\psi }= {{\Theta }^{\rm T}}{\left[ \overline{h}{\left({\underline{z},\overline{z}} \right)} - h{\left({z} \right)} \right]} + \overline{\varepsilon }{\left(z \right)} - \varepsilon {\left(z \right)} + \overline{\phi }- \phi \\[11pt] \underline{\psi }= {{\Theta }^{\rm T}}{\left[h{\left({z} \right)} - \underline{h}{\left({\underline{z},\overline{z}} \right)} \right]} + \varepsilon {\left(z \right)} - \underline{\varepsilon }{\left(z \right)} + \phi - \underline{\phi }. \end{cases}} \end{align} (17)According to the error backpropagation algorithm [40] Θ ̲ ̇ = − η ̲ ∂ J ̲ ∂ Θ ̲ − ρ ̲ r ̲ Θ ̲ , \begin{align} \dot{\underline{\Theta }} = - {\underline{\eta }}\frac{{\partial {\underline{J} }}}{{\partial \underline{\Theta }}} - {\underline{\rho }}{\left\Vert {\underline{r}} \right\Vert} \underline{\Theta }, \end{align} (18)the lower bound of the modified network weight is Θ ̲