Sage Journals: Discover world-class research

Abstract

This paper presents an adaptive neural network controller based on a disturbance observer to compensate the disturbance caused by neural network approximation for a class of unknown nonlinear systems. The proposed adaptive neural network control with an updated parameters mechanism is not subject to the restriction of compact set assumption for satisfying the universal approximation property. The neural network approximation error can be compensated online through the proposed disturbance observer. The proposed method eliminates the need to obtain an exact system model before applying the ideal controller. The effectiveness of the proposed method is validated by numerical simulations.

Keywords

Disturbance observer adaptive neural networks unknown nonlinear system

Introduction

Due to its universal approximation property, neural network control has been playing a very important role in the control community, especially for unknown nonlinear systems. With traditional control methods, the control algorithms are designed based on a system model to achieve stable tracking control. As the controlled objects become increasingly complex, it is usually impossible to obtain an accurate system model.^1–3

In recent years, many different neural network control methods have been proposed for a variety of unknown nonlinear systems.^4–9 These methods are implemented by an adaptive NN control strategy to compensate the uncertainties of the system and the external environment. A neural network controller was designed for an uncertain n-link robot with full-state constraints, and a barrier Lyapunov function was presented to guarantee uniform ultimate boundedness.⁸ A radial basis function neural network (RBFNN) was utilized as the prediction function for nonlinear model predictive control, with system stability guaranteed and network-induced delays and packet dropouts compensated.⁹

The compact set assumption is considered an important assumption in neural network control. In order for the assumption to be effective, the state variables of the system must be in a compact set range.^10,11 However, in the online control of nonlinear systems, it is often impossible to prove that the state of the system satisfies the compact set assumption. Therefore, in adaptive neural network control, the compact set assumption is not quite a feasible way to establish a stable control algorithm. In fact, the approximation error of the system can be seen as an unknown disturbance signal, which can change with time and the system state. This disturbance can be eliminated using a disturbance observer. To estimate the unknown disturbance and keep the system stable, it is important to add a control variable based on disturbance rejection, such as active disturbance rejection control and disturbance observer-based control, to the control loop. Several discrete-time variants of active disturbance rejection control have been studied and have found an increasingly wide range of applications, especially in power electronics and drives.¹² In Sira-Ramirez et al.,¹³ an active disturbance rejection control scheme for the angular velocity trajectory tracking task was proposed for a substantially perturbed, uncertain, and permanent magnet synchronous motor. The use of a disturbance observer for disturbance compensation offers improved robustness and has therefore been extensively investigated by many researchers.^14–16 Disturbance observer-based control has been well studied in both linear systems and nonlinear systems. For example, in Komada et al.,¹⁷ a disturbance observer was used for a manipulator control with no inverse dynamics. In Chen and Ge,¹⁵ a disturbance observer was designed to compensate unknown external disturbance, unknown nonsymmetric input saturation, and the approximation error of the neural network for a class of uncertain nonaffine nonlinear systems. Disturbance observer-based control methods have also been applied in power electronics.^18,19

In this paper, we release the compact set assumption to design an online neural network controller, and use a disturbance observer to deal with the approximation error of the system. First, we give a state-dependent upper bound assumption that can reflect the error between the neural network and the nonlinear system. Second, we apply the proposed approximation and a disturbance observer in the control method to compensate the approximation error without compact set assumption. Finally, we establish a backstepping-based adaptive neural network control algorithm with updated parameters for the high-order nonlinear system. The control scheme keeps the tracking error in a bounded function which depends on the initial state of the system.

The remainder of the paper is organized as follows. Section 2 formulates the control problem. Section 3 presents an adaptive neural network controller based on a disturbance observer. Some simulation examples are given in Section 4 to illustrate our theories. Concluding remarks are included in Section 5.

System description and problem formulation

To develop the proposed disturbance observer-based adaptive neural network controller, the control problem and system function approximation with a neural network are formulated as follows.

Problem formulation

We consider a class of uncertain SISO nonlinear systems, which can be described as

{\begin{matrix} {\overset{\cdot}{x}}_{1} = F_{1} ({\bar{x}}_{1}) + G_{1} x_{2} \\ {\overset{\cdot}{x}}_{2} = F_{2} ({\bar{x}}_{2}) + G_{2} x_{3} \\ \dots \\ {\overset{\cdot}{x}}_{n} = F_{r} ({\bar{x}}_{n}) + G_{n} u (t) \\ y (t) = x_{1}, \end{matrix}

(1)

where $x_{i} \in R, i \in {1, 2, \dots, n}$ are the states of the plant and ${\bar{x}}_{i} = {x_{1}, x_{2}, \dots, x_{i}}$ , $y (t)$ and $u (t)$ are the control input and output, respectively. $F_{i}, i \in {1, 2, \dots, n}$ are unknown nonlinear functions and $G_{i}, i \in {1, 2, \dots, r}$ are known constants. The desired trajectory is denoted as $y_{d}$ . In light of recent progress in control of uncertainty nonlinear system, we integrate a adaptive neural network control with disturbance observer, thus solving this tracking control problem through estimating the unknown nonlinear model and neural network approximation error. Here we present an assumption about the tracking signal.

Assumption 1. For all $t > 0$ , the desired trajectory $y_{d}$ and its $n - th$ orders are bounded, continuous and known.

Therefore, the problem of this paper can be presented to design a control input $u (t)$ such that the desired trajectory $y_{d}$ can be tracked. Because $F_{i}$ is unknown, we need to use a neural network to approximate it, and design a disturbance observer to deal with the neural network approximation error.

Neural network function approximation

Universal approximation theorem²⁰ expounds that any continuous function can be approximated by an ideal RBFNN over a compact set. Therefore, the continuous function $f_{i} (x)$ in equation (1) can be approximated by a neural network with arbitrary accuracy as

f_{i} ({\bar{x}}_{i}) = W_{i}^{T} Φ_{i} + ϵ_{i} ({\bar{x}}_{i})

(2)

where $W_{i} \in R^{n}$ represents the neural network weights and $ϵ_{i} ({\bar{x}}_{i}) \in R$ the reconstruction errors, respectively. $n$ is the number of neural nodes. $Φ_{i}$ is a Gaussian function in the form as

Φ_{i, j} = \exp (- \frac{∥ x_{i} - b_{i, j} ∥^{2}}{2 c_{i, j}^{2}})

(3)

where $b_{i, j}$ and $c_{i, j}$ are the center of the Gaussian function peak and the standard deviation. The optimal weight value of the RBFNN can be defined as

W_{i}^{*} = \underset{w_{i} \in R^{n}}{\arg min} [sup ∥ W_{i}^{T} Φ_{i} - f ({\bar{x}}_{i}) ∥]

(4)

where $R$ is the valid field of the parameter. To realize this approximation, online adaptive control is adopted in this paper. Specifically, the parameters in the radial basis function (RBF) are assumed to be constant and the weights will be updated through the proposed adaptive law.

When $W_{i} = W_{i}^{*}$ , that is, optimal approximation is achieved with an optimal weight value, we have $∥ ϵ_{i} ({\bar{x}}_{i}) ∥ \leq ϵ_{i}^{*}$ , where $ϵ_{i}^{*}$ is a small constant.²⁰ However, optimal parameters cannot be realized until the system becomes stable in the online neural network control. Therefore, we give an system state-based assumption about the reconstruction errors as follows.

Assumption 2. The neural network reconstruction error $ϵ_{i} ({\bar{x}}_{i}), (i = 1, 2, \dots, r)$ is bounded as:

∥ ϵ_{i} ({\bar{x}}_{i}) ∥ \leq ψ_{i, 1} + ψ_{i, 2} θ_{i} ({\bar{x}}_{i})

(5)

where $θ_{i} ({\bar{x}}_{i})$ is a known nonnegative function related to the system state, and $ψ_{i, j}, j = 1, 2$ are unknown and non-negative. Define $ψ_{i} = (ψ_{i, 1}, ψ_{i, 2})$ and $∥ {\overset{\cdot}{ψ}}_{i} ∥ \leq \nabla_{i}$ , $\nabla_{i}$ as positive constant. ²¹

When a neural network is used in the control field, the compact set assumption must be met. This assumption requires that all input signals of the neural network be in a compact set, which is a difficult thing to achieve in the online control process. The above assumption depends on the system state without requiring boundedness of the neural network input, which has better practicability.²¹

Adaptive neural network controller with disturbance observer

The objective of this section is to design a adaptive neural controller which can promise the stability of the system. First, we assume that complete information about all the states are available, a model-based control with a low-frequency learning neural network is introduced to maintain the system’s performance. Then, a nonlinear disturbance observer is introduced based on the proposed adaptive neural network control method.

Adaptive neural network controller

We define new error variables $z_{i}$ as

\begin{matrix} z_{1} = x_{1} - y_{d} \\ z_{2} = x_{2} - α_{1} \\ \dots \\ z_{n} = x_{r} - α_{n - 1} \end{matrix}

(6)

where $α_{i}$ is a virtual control to $z_{1}$ in the backstepping process. The system dynamics (1), therefore, can be re-expressed as

{\begin{matrix} {\overset{\cdot}{z}}_{1} = F_{1} ({\bar{x}}_{1}) + G_{1} (z_{2} + α_{1}) - {\overset{\cdot}{y}}_{d} \\ {\overset{\cdot}{z}}_{2} = F_{2} ({\bar{x}}_{2}) + G_{2} (z_{3} + α_{2}) - {\overset{\cdot}{α}}_{1} \\ \dots \\ {\overset{\cdot}{z}}_{n} = F_{r} ({\bar{x}}_{n}) + G_{n} u (t) - {\overset{\cdot}{α}}_{n - 1} \end{matrix}

(7)

Now we summarize how a basic adaptive backstepping controller with a low-frequency learning neural network is designed through a standard recursive procedure.

Step 1.

Based on RBFNN approximation (4), let $F_{1} = w_{1}^{T} ϕ_{1} + ϵ_{1} ({\bar{x}}_{1})$ . Specifically, $w_{1} \in R^{n}$ , $ϕ_{1} \in R^{n}$ , and $ϵ_{1} \in R$ , which are the ideal weight vector, the RBF, and the network reconstruction error, respectively. Then let $∥ ϵ_{1} ∥ \leq ψ_{1, 1} + ψ_{1, 2} θ_{1}$ , where $ψ_{1, 1}$ and $ψ_{1, 2}$ are unknown positive constants and $θ_{1}$ is a known function as stated in Assumption 2. Hence, from (6) we have

{\overset{\cdot}{z}}_{1} = {\overset{\cdot}{x}}_{1} - {\overset{\cdot}{y}}_{d} = w_{1}^{T} ϕ_{1} + ϵ_{1} ({\bar{x}}_{1}) + G_{1} x_{2} - {\overset{\cdot}{y}}_{d}

(8)

The virtual control $α_{1} = x_{2} - z_{2}$ is given by

α_{1} = - {G_{1}}^{- 1} (z_{1} + {\hat{w}}_{1}^{T} ϕ_{1} - {\overset{\cdot}{y}}_{d} - {\hat{ψ}}_{1} Θ_{1})

(9)

where ${\hat{ψ}}_{1}$ is the disturbance estimation, which will be determined in the next subsection, $Θ_{1} = [1, θ_{i}]$ , and ${\hat{w}}_{1}$ is the estimate of $w_{1}$ .

Remark 1 : Although the neural network can handle the unknown system state well, there will still be a certain approximation error as shown in Narendra and Mukhopadhyay⁴ and Assumption (2). Therefore, the approximation error in this paper is considered as an disturbance signal which will be eliminated through disturbance observer to obtain better control performance.

Now we design

\begin{matrix} {\overset{\cdot}{\hat{w}}}_{1} = Γ_{w_{1}} [ϕ_{1} z_{1} - μ ({\hat{w}}_{1} - {\hat{w}}_{1, f}) - k_{1} {\hat{w}}_{1}] \\ {\overset{\cdot}{\hat{w}}}_{1, f} = Γ_{w_{1, f}} [{\hat{w}}_{1} - {\hat{w}}_{1, f} - k_{1, f} {\hat{w}}_{1, f}] \end{matrix}

(10)

where $μ$ , $k_{1}$ , and $k_{1, f}$ are positive constants. $Γ_{w_{1}} \in R^{n \times n}$ and $Γ_{w_{1, f}} \in R^{n \times n}$ , which are symmetric positive matrices. ${\hat{w}}_{1, f}$ is a filtered value of ${\hat{w}}_{1}$ obtained by a low-pass filter. The filtered value ${\hat{w}}_{1, f}$ can be used to reduce the learning rate of the update law of ${\hat{w}}_{1}$ . A high-gain learning rate in adaptive control will lead to high-frequency oscillations of control output. The low-pass filter parameter is introduced to reduce the oscillations of the system response as in Yucelen and Haddad.²² Define a Lyapunov function as

V_{1} = \frac{1}{2} z_{1}^{2} + \frac{1}{2} {\tilde{w}}_{1}^{T} Γ_{w_{1}}^{- 1} {\tilde{w}}_{1} + \frac{1}{2} {\tilde{w}}_{1, f}^{T} Γ_{w_{1, f}}^{- 1} {\tilde{w}}_{1, f}

(11)

where ${\tilde{w}}_{1} = w_{1} - {\hat{w}}_{1}$ and ${\tilde{w}}_{1, f} = w_{1} - {\hat{w}}_{1, f}$ .

The derivative of $V_{1}$ along trajectory (11) is given by

\begin{matrix} {\overset{\cdot}{V}}_{1} = z_{1} {\overset{\cdot}{z}}_{1} + {\tilde{w}}_{1}^{T} Γ_{w_{1}}^{- 1} {\overset{\cdot}{\tilde{w}}}_{1} + {\tilde{w}}_{1, f}^{T} Γ_{w_{1, f}}^{- 1} {\overset{\cdot}{\tilde{w}}}_{1, f} \\ = z_{1} (w_{1}^{T} ϕ_{1} + ϵ_{1} ({\bar{x}}_{1}) + G_{1} (z_{2} + α_{1}) - {\overset{\cdot}{y}}_{d}) \\ - {\tilde{w}}_{1}^{T} [ϕ_{1} z_{1} - μ ({\hat{w}}_{1} - {\hat{w}}_{1, f}) - k_{1} {\hat{w}}_{1}] \\ - {\tilde{w}}_{1, f}^{T} [{\hat{w}}_{1} - {\hat{w}}_{1, f} - k_{1, f} {\hat{w}}_{1, f}] \\ \leq - z_{1}^{2} + k_{1, f} {\tilde{w}}_{1, f}^{T} {\hat{w}}_{1, f} - μ ({\hat{w}}_{1} - {\hat{w}}_{1, f})^{T} ({\hat{w}}_{1} - {\hat{w}}_{1, f}) \\ + z_{1} G_{1} z_{2} + k_{1} {\tilde{w}}_{1}^{T} {\hat{w}}_{1} + {\tilde{ψ}}_{1} Θ_{1} \\ = - z_{1}^{2} - k_{1} {\tilde{w}}_{1}^{T} {\tilde{w}}_{1} - k_{1, f} {\tilde{w}}_{1, f}^{T} {\tilde{w}}_{1, f} \\ + k_{1} w_{1}^{T} w_{1} + k_{1, f} w_{1, f}^{T} w_{1, f} + z_{1} G_{1} z_{2} + {\tilde{ψ}}_{1} Θ_{1} \\ = - β_{1} V_{1} + z_{1} G_{1} z_{2} + {\tilde{ψ}}_{1} Θ_{1} \end{matrix}

(12)

where ${\tilde{ψ}}_{1} = {\hat{ψ}}_{1}$ , and $β_{1}$ is a designed positive constant.

Step j. $(2 \leq j < n)$

From (6) and (19), we have $α_{j - 1}$ is a function of $x_{1}, x_{2}, . . ., x_{j - 1}$ . ${\overset{\cdot}{α}}_{j - 1}$ is defined as

{\overset{\cdot}{α}}_{j - 1} = φ_{j - 1} ({\bar{z}}_{j - 1}, {\bar{\hat{w}}}_{j - 1}, {\bar{y}}_{d}^{(j)})

(13)

f_{r} ({\bar{x}}_{r}) - φ_{j - 1} ({\bar{z}}_{j - 1}, {\bar{\hat{w}}}_{j - 1}, {\bar{y}}_{d}^{(j)}) = w_{j}^{T} ϕ_{j} + ϵ_{j}

(14)

where ${\bar{z}}_{j} = [z_{1}^{T}, \dots, z_{j}^{T}]^{T}$ , ${\bar{\hat{w}}}_{j} = [{\hat{w}}_{1}^{T}, \dots, {\hat{w}}_{j}^{T}]^{T}$ , ${\bar{y}}_{d}^{(j)} = [{\bar{y}}_{d}^{(1)}, \dots, {\bar{y}}_{d}^{(j)}]^{T}$ . $w_{j} \in R^{n}$ , $ϕ_{j} \in R^{n}$ , and $ϵ_{j} \in R$ , which are the ideal weight vector, the RBF, and the network reconstruction error, respectively. Based on Assumption 2, let $∥ ϵ_{j} ∥ \leq ψ_{j, 1} + ψ_{j, 2} θ_{j}$ . Then

{\overset{\cdot}{z}}_{j} = {\overset{\cdot}{x}}_{j} - {\overset{\cdot}{α}}_{j - 1} = w_{j}^{T} ϕ_{j} + ϵ_{j} ({\bar{x}}_{j}) + G_{j} ({\bar{x}}_{j}) x_{j + 1}

(15)

In the $j$ th step, we define the following Lyapunov function

V_{j} = V_{j - 1} + \frac{1}{2} z_{j}^{T} z_{j} + \frac{1}{2} {\tilde{w}}_{j}^{T} Γ_{w_{j}}^{- 1} {\tilde{w}}_{j} + \frac{1}{2} {\tilde{w}}_{j, f}^{T} Γ_{w_{j, f}}^{- 1} {\tilde{w}}_{j, f}

(16)

where ${\tilde{w}}_{j} = w_{j} - {\hat{w}}_{j}$ , ${\tilde{w}}_{j, f} = w_{j} - {\hat{w}}_{j, f}$ , ${\tilde{ψ}}_{j, 1} = ψ_{j, 1} - {\hat{ψ}}_{j, 1}$ , and ${\tilde{ψ}}_{j, 2} = ψ_{j, 2} - {\hat{ψ}}_{j, 2}$ . $Γ_{w_{j}} \in R^{n \times n}$ , and $Γ_{w_{j, f}} \in R^{n \times n}$ , which are symmetric positive matrices. $γ_{j, 1}$ and $γ_{j, 2}$ are positive constants. The adaptive laws are given as

\begin{matrix} {\overset{\cdot}{\hat{w}}}_{j} = Γ_{w_{j}} [ϕ_{j} z_{j} - μ ({\hat{w}}_{j} - {\hat{w}}_{j, f}) - k_{j} {\hat{w}}_{j}] \\ {\overset{\cdot}{\hat{w}}}_{j, f} = Γ_{w_{j, f}} [{\hat{w}}_{j} - {\hat{w}}_{j, f} - k_{j, f} {\hat{w}}_{j, f}] \end{matrix}

(17)

where $k_{j}$ and $k_{j, f}$ are positive constants. Based on the similar analysis in Step 1, we can infer that

{\overset{\cdot}{V}}_{j} \leq - β_{j} V_{j} + \sum_{i = 1}^{j} {\tilde{ψ}}_{i} Θ_{i} + z_{j} G_{j} z_{j + 1}

(18)

Choose the following virtual control law

α_{j} = - G_{j}^{- 1} (G_{j - 1} z_{j - 1} + z_{j} + {\hat{w}}_{j}^{T} ϕ_{j} - {\hat{ψ}}_{i} Θ_{i})

(19)

where $\hat{ψ}$ is the disturbance estimation, which will be determined in the next subsection, and $Θ_{i} = [1, θ_{i}]$ .

Step n. In the last step, we choose the Lyapunov function as

\begin{matrix} V_{n} = V_{n - 1} + \frac{1}{2} {z_{n}}^{T} z_{n} + \frac{1}{2} {\tilde{w}}_{n}^{T} Γ_{w_{n}}^{- 1} {\tilde{w}}_{n} \\ + \frac{1}{2} {\tilde{w}}_{n, f}^{T} Γ_{w_{n, f}}^{- 1} {\tilde{w}}_{n, f} \end{matrix}

(20)

with

φ_{n - 1} ({\bar{z}}_{n - 1}, {\bar{\hat{w}}}_{r - 1}, y_{d}^{(r)}) = {\overset{\cdot}{α}}_{n - 1}

(21)

f_{n} ({\bar{x}}_{n}) - φ_{n - 1} ({\bar{z}}_{n - 1}, {\bar{\hat{w}}}_{n - 1}, y_{d}^{(n)}) = w_{n}^{T} ϕ_{n} + ϵ_{n}

(22)

{\overset{\cdot}{z}}_{n} = w_{n}^{T} ϕ_{n} + ϵ_{n} ({\bar{x}}_{r}) + G_{r} ({\bar{x}}_{j}) u (t)

(23)

where $∥ ϵ_{n} ∥ \leq ψ_{n, 1} + ψ_{n, 2} θ_{n}$ as in Assumption 2, ${\tilde{w}}_{n} = w_{n} - {\hat{w}}_{n}$ and ${\tilde{w}}_{n, f} = w_{n} - {\hat{w}}_{n, f}$ . $Γ_{w_{n}} \in R^{n \times n}$ and $Γ_{w_{n, f}} \in R^{n \times n}$ , which are symmetric positive matrices. The adaptive laws are

\begin{matrix} {\overset{\cdot}{\hat{w}}}_{n} = Γ_{w_{n}} [ϕ_{r} z_{n} - μ ({\hat{w}}_{n} - {\hat{w}}_{n, f}) - k_{n} {\hat{w}}_{n}] \\ {\overset{\cdot}{\hat{w}}}_{n, f} = Γ_{w_{n, f}} [{\hat{w}}_{n} - {\hat{w}}_{n, f} - k_{n, f} {\hat{w}}_{n, f}] \end{matrix}

(24)

where $k_{n}$ and $k_{n, f}$ are positive constants. The control of the adaptive neural network controller $u$ is designed as

u (t) = α_{r} = - G_{n}^{- 1} (G_{n - 1} z_{n - 1} + z_{n} + {\hat{w}}_{n}^{T} ϕ_{n} - {\hat{ψ}}_{n} Θ_{n})

(25)

where ${\hat{ψ}}_{n}$ is the disturbance estimation, which will be determined in the next subsection, and $Θ_{n} = [1, θ_{n}]$ . With (20) and (24) we have

{\overset{\cdot}{V}}_{n} \leq - β_{n} V_{n} + \sum_{i = 1}^{n} {\tilde{ψ}}_{i} Θ_{i}

(26)

Now we have finished designing the adaptive neural control for uncertain nonlinear systems with neural network approximation assumption. A disturbance observer for eliminating the disturbance item $\sum_{i = 1}^{n} {\tilde{ψ}}_{i} Θ_{i}$ caused by the neural network approximation error is proposed in the following subsection.

Nonlinear disturbance observer

Using the same step-by-step method as in the previous section, we now design a disturbance observer.

Step 1.

A nonlinear disturbance observer is designed to estimate the unknown disturbances $ψ_{1} = (ψ_{1, 1}, ψ_{1, 2}$ ) as follows:

\begin{matrix} {\overset{\cdot}{D}}_{1} = - l_{1} (x) Θ_{1} (x) D - l_{1} (x) [Θ_{i} (x) p_{1} (x) \\ + {\hat{w}}_{1}^{T} ϕ_{1} + G_{1} (x) x_{2}] \end{matrix}

(27)

{\hat{ψ}}_{i} = D_{1} + p_{1} (x) - Θ_{1} (x)

(28)

where ${\hat{ψ}}_{i}$ denotes the disturbance estimation vector, $l (x)$ is the nonlinear gain function, $D$ is the internal state of the nonlinear observer, and $p (x)$ is the nonlinear function to be designed. The nonlinear disturbance observer gain $l_{1} (x)$ is determined by

l_{1} (x) = \frac{\partial (p_{1} (x))}{\partial (x_{1})}

(29)

Then we have

\begin{matrix} {\overset{\cdot}{\tilde{ψ}}}_{1} = {\overset{\cdot}{\hat{ψ}}}_{1} - Θ_{1} (x) - {\overset{\cdot}{ψ}}_{1} \\ = {\overset{\cdot}{D}}_{1} + \frac{\partial (p (x))}{\partial (x)} \overset{\cdot}{x} - Θ_{1} (x) - {\overset{\cdot}{ψ}}_{1} \\ = - l_{1} (x) Θ_{1} (x) D - l_{1} (x) [Θ_{1} (x) p (x) + {\hat{w}}_{1}^{T} ϕ_{1} \\ + G_{1} (x) x_{2}] + l_{1} (x) [F_{1} ({\bar{x}}_{1}) + G_{1} x_{2}] - Θ_{1} (x) - {\overset{\cdot}{ψ}}_{1} \\ = - l_{1} (x) Θ_{1} (x) D - l_{1} (x) [Θ_{1} (x) p (x) + {\hat{w}}_{1}^{T} ϕ_{1} \\ + G_{1} (x) x_{2}] + l_{1} (x) [{\hat{w}}_{1}^{T} ϕ_{1} + ψ_{1} Θ_{1} + G_{1} x_{2}] - Θ_{1} (x) - {\overset{\cdot}{ψ}}_{1} \\ = - l_{1} (x) Θ_{1} (x) {\hat{ψ}}_{1} + l (x) Θ_{1} (x) ψ_{i} - Θ_{1} (x) - {\overset{\cdot}{ψ}}_{1} \\ = - l_{1} (x) Θ_{1} (x) {\tilde{ψ}}_{1} - Θ_{1} (x) - {\overset{\cdot}{ψ}}_{1} \end{matrix}

(30)

Consider a new Lyapunov function candidate

V_{1}^{*} = V_{1} + \frac{1}{2} {\tilde{ψ}}_{1}^{T} {\tilde{ψ}}_{1}

(31)

Then the time derivative of $V_{1}^{*}$ along the state trajectory is

\begin{matrix} {\overset{\cdot}{V}}_{1}^{*} = - β_{1} V_{1} + z_{1} G_{1} z_{2} + {\tilde{ψ}}_{1} Θ_{1} + {\tilde{ψ}}_{1} {\overset{\cdot}{\tilde{ψ}}}_{1} \\ = - β_{1} V_{1} + z_{1} G_{1} z_{2} + {\tilde{ψ}}_{1} Θ_{1} - l_{1} (x) Θ_{1} (x) {\tilde{ψ}}_{1}^{2} \\ - {\tilde{ψ}}_{1} Θ_{1} (x) - {\tilde{ψ}}_{1} {\overset{\cdot}{ψ}}_{1} \\ \leq - β_{1} V_{1} + z_{1} G_{1} z_{2} + \frac{1}{2} {\tilde{ψ}}_{1}^{2} + \frac{1}{2} {\overset{\cdot}{ψ}}_{1}^{2} - l_{1} (x) Θ_{1} (x) {\tilde{ψ}}_{1}^{2} \\ = - β_{1} V_{1} + z_{1} G_{1} z_{2} + \frac{1}{2} \nabla_{1}^{2} - [l_{1} (x) Θ_{1} (x) - \frac{1}{2}] {\tilde{ψ}}_{1}^{2} \end{matrix}

(32)

where $l (x)$ is a designed function which can be chosen to make $δ_{1} = l_{1} (x) Θ_{1} (x) - \frac{1}{2}$ positive. Then

{\dot{V}}_{1}^{*} = - β_{1}^{*} V_{1}^{*} + z_{1} G_{1} z_{2} + \frac{1}{2} \nabla_{11}^{2 *}

(33)

where $β_{1}^{*}$ is a positive constant.

Step j. $(2 \leq j < n)$

As is done in the backstepping process, the Lyapunov function is chosen as

V_{i}^{*} = V_{i} + \frac{1}{2} {\tilde{ψ}}_{i}^{T} {\tilde{ψ}}_{i}

(34)

with disturbance observers

\begin{matrix} {\overset{\cdot}{D}}_{i} = - l_{i} (x) Θ_{i} (x) D_{i} - l_{i} (x) [Θ_{i} (x) p_{i} (x) \\ + {\hat{w}}_{i}^{T} ϕ_{i} + G_{i} (x) x (i + 1)] \end{matrix}

(35)

{\hat{ψ}}_{i} = D_{i} + p_{i} (x) - Θ_{i} (x)

(36)

By selecting the appropriate $l_{i} (x), i = 2, . . ., n - 1$ ,

l_{i} = \frac{\partial (p_{i})}{\partial x}

(37)

with

\begin{matrix} {\overset{\cdot}{\tilde{ψ}}}_{i} = {\overset{\cdot}{\hat{ψ}}}_{i} - Θ_{i} - {\overset{\cdot}{ψ}}_{i} \\ = - l_{i} (x) Θ_{i} (x) {\tilde{ψ}}_{i} - Θ_{i} - {\overset{\cdot}{ψ}}_{i} \end{matrix}

(38)

we can get

{\overset{\cdot}{V}}_{i}^{*} \leq - β_{i}^{*} V_{i}^{*} + z_{i} G_{i} z_{i + 1} + \sum_{j = 1}^{i} \frac{1}{2} \nabla_{i}^{2}

(39)

Step $n$

Consider the following Lyapunov function candidate

V_{n}^{*} = V_{n} + \frac{1}{2} {\tilde{ψ}}_{n}^{T} {\tilde{ψ}}_{n}

(40)

and disturbance observers

\begin{matrix} {\overset{\cdot}{D}}_{n} = - l_{n} (x) Θ_{n} (x) D_{n} - l_{n} (x) [Θ_{n} (x) p_{n} (x) \\ + {\hat{w}}_{n}^{T} ϕ_{n} + G_{n} (x) u] \end{matrix}

(41)

{\hat{ψ}}_{i} = D_{n} + p_{n} (x) - Θ_{n} (x)

(42)

Choose $l_{n}$ as

l_{n} = \frac{\partial (p_{n})}{\partial x}

(43)

The time derivative of V3 is

{\overset{\cdot}{V}}_{n}^{*} \leq - β_{n}^{*} V_{n}^{*} + B^{*}

(44)

where $B^{*} = \sum_{j = 1}^{n} \frac{1}{2} \nabla_{i}^{2}$ . We have the signals $z_{i}$ , $w_{i}$ , and $ψ$ , which are uniformly bounded. Therefore, all the internal signals of the closed-loop system are globally uniformly bounded.

To summarize, the following theorem for disturbance-observer-based controller can be obtained,

Theorem 1. Considering the unknown nonlinear system (1) which fulfills Assumptions (1) and (2), with the adaptive neural control law (25), and neural network parameters updating laws (10)(17)(24) and disturbance observers (28)(36)(42). The asymptotic tracking of the closed loop system is achieved and all the system states and the designed parameters are bounded.

Remark 2 : In practice, a neural network can be developed to explore the dynamic information of an unknown nonlinear system and used in controller design. In traditional neural network control design, the approximation error is assumed to be constant, which is difficult to guarantee without optimal neural network approximation. In this paper, by contrast, the NN approximation error is assumed to be time-varying and state-based, which is treated as a disturbance to be estimated using a nonlinear disturbance observer. To the best of our knowledge, this is the first time that the neural network approximation error has been considered as a state-based structure in the control design for uncertain nonlinear systems.

Simulation examples

In this section, the proposed adaptive neural network controller with a disturbance observer is used for tracking control of rigid robots. As covered in Lewis et al.,²³ the rigid robot model can be defined as

\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = D^{- 1} x_{3} - D^{- 1} N \sin x_{1} - D^{- 1} B * x_{2} \\ {\overset{\cdot}{x}}_{3} = - M^{- 1} K_{m} * x_{2} - M^{- 1} H x_{3} + M^{- 1} u \end{matrix}

(45)

where $D = 1$ , $N = 10$ , $B = 1$ , $M = 0.05$ , $H = 0.5$ , and $K_{m} = 10$ are simulation parameters of the robot system which are unknown for the designed controller. The initial states of the system are chosen to be $x (0) = [0.3, 3, 0]$ . We choose $γ_{ω_{2}} = 5$ , $γ_{ω_{2, f}} = 6$ , $γ_{ω_{3}} = 1$ , and $γ_{ω_{3, f}} = 40$ . The initial neural network activation function $ϕ_{i}$ is initialized with random parameters. Without loss of generality, the neural network initial weights ${\hat{w}}_{i} (0)$ are zero and $θ_{i} = ∥ {\bar{x}}_{i} ∥ + ∥ {\bar{x}}_{i}^{2} ∥$ .

The desired trajectory is $y_{d} = \sin (2 π t)$ . Figures 1 and 2 show the tracking trajectories and errors with the proposed disturbance observer-based control method. The nonlinear disturbance observer is designed according to (28) and (36) and the disturbance observer-based adaptive neural network control in (25) is verified. It can be seen that, for the time-varying desired tracking signal, the tracking errors of the system output are small and the desired tracking control performance is achieved with all the signals bounded. From Figure 3, we see that the control input is bounded within the input limit. These simulation results suggest that the proposed method delivers satisfactory attitude tracking control performance and is effective for uncertain nonlinear systems approximated by neural networks with state-based assumption of reconstruction errors in Assumption 2.

Figure 1.

Tracking trajectories.

Figure 2.

Tracking errors.

Figure 3.

Tracking trajectory.

In order to show the robustness performance of the proposed controller and observer, we can add an disturbance term to the system (46) as follows:

\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = D^{- 1} x_{3} - D^{- 1} N \sin x_{1} - D^{- 1} B * x_{2} + Ξ \\ {\overset{\cdot}{x}}_{2} = D^{- 1} x_{3} - D^{- 1} N \sin x_{1} - D^{- 1} B * x_{2} + Ξ \end{matrix}

(46)

where $Ξ = 20 * \sin (10 t)$ is the unknown external disturbance and all the other parameters keep remain unchanged as in (46). The tracking trajectories and errors can be shown in Figures 4 and 5 It can be found from the simulation that thanks to the proposed disturbance observer and neural network controller, although there is unknown disturbance in the system, the whole system can still remain relatively stable.

Figure 4.

Tracking trajectory.

Figure 5.

Tracking trajectory.

Conclusions

In this paper, we propose a disturbance observer-based adaptive neural network control for unknown nonlinear systems. A disturbance observer is introduced to deal with neural network approximation errors which are traditionally assumed to be bounded. The compact set assumption is released and replaced by system state-based error assumption. An adaptive neural network with updated parameters is designed to control the system and the disturbance observer is used to compensate the approximation error. It is demonstrated in the simulation section that the proposed method can guarantee system stability and save communication resources of the system. Although we have verified the effectiveness of the proposed method through simulation, we still face many problems in complex and changeable practical application which lead us to apply the algorithm to practical objects.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Meirong Zheng

References

Song

Yang

, et al. Design and analysis of hexapod bionic robot control system. In: Proceedings of IncoME-V and CEPE Net-2020, 2021.

Parizad

Mohamadian

Iranian

, et al. Power system real-time emulation: a practical virtual instrumentation to complete electric power system modeling. IEEE Trans Ind Inform 2019; 15(2): 889–900.

Reza

Chen

Crandall

DJ.

Deep neural network based detection and verification of microelectronic images. J Hardware Syst Secur 2020; 4: 44–54.

Narendra

Mukhopadhyay

Adaptive control using neural networks and approximate models. IEEE Trans Neural Netw 1997; 8(3): 475–485.

Yang

Wang

Cheng

, et al. Neural-learning-based telerobot control with guaranteed performance. IEEE Trans Cybern 2017; 47(10): 3148–3159.

Zhang

Dong

Ouyang

, et al. Adaptive neural control for robotic manipulators with output constraints and uncertainties. IEEE Trans Neural Netw Learn Syst 2018; 29: 5554–5564.

Huang

Wang

Meng

, et al. Impact dynamic modeling and adaptive target capturing control for tethered space robots with uncertainties. IEEE/ASME Trans Mechatron 2016; 21(5): 2260–2271.

Chen

Yin

Adaptive neural network control of an uncertain robot with full-state constraints. IEEE Trans Cybern 2017; 46(3): 620–629.

Wang

Gao

Qiu

A combined adaptive neural network and nonlinear model predictive control for multirate networked industrial process control. IEEE Trans Neural Netw Learn Syst 2017; 27(2): 416–425.

10.

Chen

Sun

Qin

, et al. Global adaptive neural network control for a class of uncertain non-linear systems. IET Control Theory Appl 2011; 5(5): 655–662.

11.

Eggert

Neural network control. Hku Theses Online 2003; 75(2): 487495.

12.

Herbst

Practical active disturbance rejection control: Bumpless transfer, rate limitation and incremental algorithm. IEEE Trans Ind Electron 2016; 63(3): 1754–1762.

13.

Sira-Ramirez

Linares-Flores

Garcia-Rodriguez

, et al. On the control of the permanent magnet synchronous motor: an active disturbance rejection control approach. IEEE Trans Control Syst Technol 2014; 22(5): 2056–2063.

14.

Wei

Zhang

Guo

Composite disturbance-observer-based control and terminal sliding mode control for uncertain structural systems. Int J Syst Sci 2009; 40(10): 1009–1017.

15.

Chen

SS.

Direct adaptive neural control for a class of uncertain nonaffine nonlinear systems based on disturbance observer. IEEE Trans Cybern 2013; 43(4): 1213–1225.

16.

Yang

Sliding-mode control for systems with mismatched uncertainties via a disturbance observer. IEEE Trans Ind Electron 2013; 60(1): 160–169.

17.

Komada

Machii

Hori

Control of redundant manipulators considering order of disturbance observer. IEEE Trans Ind Electron 2000; 47(2): 413–420.

18.

Xiao

Nalakath

Sun

, et al. Complex-coefficient adaptive disturbance observer for position estimation of IPMSMS with robustness to dc errors. IEEE Trans Ind Electron 2020; 67(7): 5924–5935.

19.

Zhang

, et al. Disturbance observer-based finite-time control for three-phase AC–DC converter. IEEE Trans Ind Electron 2022; 69(6): 5637–5647.

20.

Girosi

Poggio

Networks and the best approximation property. Biol Cybern 1990; 63(3): 169–176.

21.

Xiang

Wen

Stable robust control of nonlinear systems with neural networks. In: Proceedings of the 2000 American control conference, Chicago, IL, 28–30 June 2000, vol. 5, pp.3439–3443. New York, NY: IEEE.

22.

Yucelen

Haddad

WM.

Low-frequency learning and fast adaptation in model reference adaptive control. IEEE Trans Automat Contr 2013; 58(4): 1080–1085.

23.

Lewis

Abdallah

Dawson

DM.

Control of robot manipulators. Upper Saddle River, NJ: Prentice Hall PTR, 1993.

Disturbance observer-based control with adaptive neural network for unknown nonlinear system

Abstract

Keywords

Introduction

System description and problem formulation

Problem formulation

Neural network function approximation

Adaptive neural network controller with disturbance observer

Adaptive neural network controller

Nonlinear disturbance observer

Simulation examples

Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References