ADP based output-feedback fault-tolerant tracking control for underactuated AUV with actuators faults

Abstract

In this work, the output-feedback fault-tolerant tacking control issue for underactuated autonomous underwater vehicle (AUV) with actuators faults is investigated. Firstly, an output-feedback error tacking system is constructed based on the theoretical model of underactuated AUV with actuators faults. Then, an adaptive dynamic programming (ADP) based fault-tolerant control controller is developed. In our proposed control scheme, a neural-network observer is designed to approximate the system states with actuators faults. An online policy iteration algorithm is designed with critic network and action network in order to improve the tracking accuracy. Based on Lyapunov stability theorem, the stability of the error tracking system is guaranteed by the proposed controller. At last, the simulation results show that the underactuated AUV achieves better tracking performance.

Keywords

Adaptive dynamic programming (ADP)fault-tolerant tracking control actuators faults neural network observer autonomous underwater vehicle (AUV)

1 Introduction

Tracking control is a complex motion control problem for underactuated autonomous underwater vehicle (AUV) in an unknown underwater environment [1, 2]. Traditionally, the tracking control problems of underactuated AUV without actuators faults have been solved through a variety of control schemes [3–7]. However, actuators are the very important parts of underactuated AUV. The actuators faults may lead to performance degradation of AUV [8, 9], which adds more difficulties in the process of trajectory-tracking control. These difficulties serve as the motivation of this work.

Table 1
Notations and variable used in this paper

‖· ‖ 2-norm of a vector

{O_e - X_eY_eZ_e} the universal coordinate system

{O_b - X_bY_bZ_b} the body-fixed coordinate system

η the position and attitude vector with respect to {O_e - X_eY_eZ_e}

χ the location along with x-axis

y the location along with y-axis

z the location along with z-axis

φ the roll angle

θ the pitch angle

ψ the yaw angle

J (η) the coordinate transformation matrix

ξ the position and attitude vector with respect to {O_b - X_bY_bZ_b}

u the surge velocity

v the sway velocity

w the heave velocity

p the roll angular velocity

q the pitch angular velocity

r the yaw angular velocity

M the inertia matrix

C (ξ) the Coriolis and centripetal matrix

D (ξ) the hydrodynamic damping matrix

τ the thrust force vector

g (η) the gravity and buoyancy forces vector

f the actuators faults

γ the discount factor

β, δ_i (i = 1, 2, . . . , 13) the positive constants

W₀, W_a, W_c the ideal weights

φ₀ (·) , φ_a (·) , φ_c (·) the activation functions

ɛ₀, ɛ_a, ɛ_c the approximation errors

I the identity matrix

K the diagonal matrix

Q, R, L the positive definite matrices

ϱ₀, ϱ₁, ϱ₂ the learning rates

‖· ‖	2-norm of a vector
{O_e - X_eY_eZ_e}	the universal coordinate system
{O_b - X_bY_bZ_b}	the body-fixed coordinate system
η	the position and attitude vector with respect to {O_e - X_eY_eZ_e}
χ	the location along with x-axis
y	the location along with y-axis
z	the location along with z-axis
φ	the roll angle
θ	the pitch angle
ψ	the yaw angle
J (η)	the coordinate transformation matrix
ξ	the position and attitude vector with respect to {O_b - X_bY_bZ_b}
u	the surge velocity
v	the sway velocity
w	the heave velocity
p	the roll angular velocity
q	the pitch angular velocity
r	the yaw angular velocity
M	the inertia matrix
C (ξ)	the Coriolis and centripetal matrix
D (ξ)	the hydrodynamic damping matrix
τ	the thrust force vector
g (η)	the gravity and buoyancy forces vector
f	the actuators faults
γ	the discount factor
β, δ_i (i = 1, 2, . . . , 13)	the positive constants
W₀, W_a, W_c	the ideal weights
φ₀ (·) , φ_a (·) , φ_c (·)	the activation functions
ɛ₀, ɛ_a, ɛ_c	the approximation errors
I	the identity matrix
K	the diagonal matrix
Q, R, L	the positive definite matrices
ϱ₀, ϱ₁, ϱ₂	the learning rates

In order to maintain the system stability and the acceptable tracking accuracy, many fault-tolerant control strategies have been developed for AUV with actuators faults, such as adaptive control method [10], robust control method [11], backstepping approach [12] and so on. In this work, the adaptive dynamic programming (ADP) is introduced to solve the output-feedback fault-tolerant tacking control problem for underactuated AUV with actuators faults.

Compared with above control methods [10–12], ADP algorithm has a better adaptive and self-learning ability. An actor-critic networks based constrained generalized policy iteration framework was proposed to solve the nonlinear non-affine optimal control problem in [13]. An event-triggered-based ADP control scheme was designed for distributed formation control of multi-UAV in [14]. An online ADP algorithm was proposed to solve the robust tracking control problem for uncertain nonlinear systems in [15]. A data-based policy iteration algorithm was designed to solve the output-feedback optimal control problem for uncertain linear systems in [16]. And an event-driven ADP scheme was proposed to solve the output tracking control problem for nonlinear systems in [17]. The time delays are considered and HDP is designed to solve the tracing control problem for a class of nonlinear systems in [18]. The ADP based tracking control scheme is designed for coal gasification system in [19]. The ADP algorithm is designed for tracking control with unknown system dynamics in [20, 21].

Motivated by the aforementioned discussion, an action-critic networks based ADP control scheme via neural network observer is proposed for the output-feedback fault-tolerant tracking control of underactuated AUV. The main contribution of this work can be summarized as follows:

Compared with existing output-feedback tracking control method for underactuated AUV with actuator faults, the proposed novel tracking control scheme employed the ADP schme, which tries to find a near-optimal control strategy in order to keep the higher stability and better tracking accuracy under the actuator faults.

Different from the compared method [22, 24], the proposed novel control scheme introduced a discount coefficient into the performance index due to the nonlinearity and complexity of underactuated AUV. The critic-action neural networks are employed and online policy iteration algorithm and weight update law are designed.

The neural network observer is designed to approximate the actuators faults.

The rest of paper is organized as follows. The output-feedback error tracking system is constructed and problem formulation is described in Section 2. In Section 3, the fault-tolerant ADP tracking controller with neural network observer is designed. Simulation examples are provided to demonstrate the effectiveness of the proposed method in Section 4. The conclusion is drawn in Section 5.

2 Theoretical model of underactuated AUV and problem formulation

2.1 Theoretical model of underactuated AUV

Two coordinate systems are employed in the theoretical model of underactuated AUV as shown in Fig. 1. The theoretical model of underactuated AUV without actuators faults is shown as ${\begin{matrix} \dot{η} = J (η) ξ \\ M \dot{ξ} + C (ξ) ξ + D (ξ) ξ + g (η) = τ \end{matrix}$ (1)

Fig. 1

AUV coordinate systems

where η = [χ, y, z, φ, θ, ψ] ^T; ξ = [u, v, w, p, q, r] ^T.

The kinematics of underactuated AUV is given as ${\begin{matrix} ξ = J^{- 1} \dot{η} \\ \dot{ξ} = J^{- 1} \ddot{η} - J^{- 1} \dot{J} J^{- 1} \dot{η} \end{matrix}$ (2) where J = J (η).

Combining (1) with (2), we can get $\begin{matrix} \ddot{η} & = ({MJ}^{- 1})^{- 1} ({MJ}^{- 1} \dot{J} J^{- 1} - {CJ}^{- 1} \\ - {DJ}^{- 1}) \dot{η} - ({MJ}^{- 1})^{- 1} g + ({MJ}^{- 1})^{- 1} τ \end{matrix}$ (3) where C = C (ξ); D = D (ξ); g = g (η).

2.2 Problem formulation

The desired trajectory is given as $\begin{matrix} {\ddot{η}}_{d} & = (M_{d} J_{d}^{- 1})^{- 1} (M_{d} J_{d}^{- 1} \dot{J} J^{- 1} - C_{d} J_{d}^{- 1} \\ - D_{d} J_{d}^{- 1}) \dot{η_{d}} - (M_{d} J_{d}^{- 1})^{- 1} g_{d} \\ + (M_{d} J_{d}^{- 1})^{- 1} τ_{d} . \end{matrix}$ (4)

The error vectors are defined as ${\begin{matrix} e_{η} = η - η_{d} \\ μ = τ - τ_{d} . \end{matrix}$ (5)

Then substituting (4), (5) into (3), the output feedback based error tracking system without actuators faults is given $\begin{matrix} {\ddot{e}}_{η} & = ({MJ}^{- 1})^{- 1} ({MJ}^{- 1} \dot{J} J^{- 1} - C_{d} J_{d}^{- 1} \\ - D_{d} J_{d}^{- 1}) \dot{e_{η}} + Θ_{1} + ({MJ}^{- 1})^{- 1} μ \end{matrix}$ (6) where $Θ_{1} = ({MJ}^{- 1})^{- 1} (M_{d} J_{d}^{- 1} - {MJ}^{- 1}) {\ddot{η}}_{d} + ({MJ}^{- 1})^{- 1} ({MJ}^{- 1} \dot{J} J^{- 1} - {CJ}^{- 1} - {DJ}^{- 1} - M_{d} J_{d}^{- 1} {\dot{J}}_{d} J_{d}^{- 1} + C_{d} J_{d}^{- 1} + D_{d} J_{d}^{- 1}) {\dot{η}}_{d} + ({MJ}^{- 1})^{- 1} (g_{d} - g)$ .

We define error vector $x = [e_{η} {\dot{e}}_{η}]^{T}$ , then (6) can be transformed as follows $\dot{x} = [\begin{matrix} 0 & I \\ 0 & Θ_{2} \end{matrix}] x + [\begin{matrix} 0 \\ Θ_{1} \end{matrix}] + [\begin{matrix} 0 \\ ({MJ}^{- 1})^{- 1} \end{matrix}] μ$ (7) where $Θ_{2} = ({MJ}^{- 1})^{- 1} ({MJ}^{- 1} \dot{J} J^{- 1} - {CJ}^{- 1} - {DJ}^{- 1})$ .

The error tracking system with actuators faults is given as $\dot{x} = ϖ (x) + ρ (x) (μ - f)$ (8) where $ϖ (x) = [\begin{matrix} 0 & I \\ 0 & Θ_{2} \end{matrix}] x + [\begin{matrix} 0 \\ Θ_{1} \end{matrix}]$ , $ρ (x) = [\begin{matrix} 0 \\ ({MJ}^{- 1})^{- 1} \end{matrix}]$ .

Assumption 1. Because underactuated AUV does not have independent actuators in the sway and heave axes, the available controls are the surge force, pitch moment and the yaw moment. The actuators faults f satisfies that ∥f ∥ = ∥ Kμ ∥ ≤ ∥ μ ∥ ≤ δ₁. And the element k_ii of diagonal matrix K satisfies 0 ≤ k_ii < 1. ϖ (x) and ρ (x) are locally Lipchitz continuous.

The performance index function is defined as $\begin{matrix} V_{1} (x, μ) & = \int_{t}^{\infty} e^{γ (t - σ)} (β {\hat{f}}^{T} (σ) \hat{f} (σ) \\ + U (x (σ), μ (σ))) d σ \end{matrix}$ (9) where U (x, μ) = x^TQx + μ^TRμ and U (0, 0) =0; $\hat{f}$ is the approximate actuators failures f; 0 ≤ γ < 1.

Definition 1. A control law μ is defined as an admissible control policy for (8) with f = 0, if μ is continuous on a set $Ω \subset R^{12}$ and can stabilize the error tracking system (8) with f = 0, μ (0) =0 and V₁ (x₀, 0) is finite for all x₀ ∈ Ω.

Based on the optimal control theory, the performance index function (9) is a Lyapunov function and satisfies as $\begin{matrix} 0 & = β {\hat{f}}^{T} \hat{f} + U (x, μ) + (\nabla V_{1} (x, μ))^{T} (ϖ (x) \\ + ρ (x) μ) - γ V_{1} (x, μ) \end{matrix}$ (10) where V₁ (0, 0) =0 and ∇V₁ (x, μ) is the partial derivative of V₁ (x, μ) with respect to x, $\nabla V_{1} (x, μ) = \frac{\partial V_{1} (x, μ)}{\partial x}$ .

Then, the Hamiltonian function is defined as $\begin{matrix} H (x, μ, \nabla V_{1} (x, μ)) & = β {\hat{f}}^{T} \hat{f} + U (x, μ) \\ + (\nabla V_{1} (x, μ))^{T} (ϖ (x) \\ + ρ (x) μ) - γ V_{1} (x, μ) . \end{matrix}$ (11)

The optimal cost function is defined as $\begin{matrix} V_{1}^{*} (x, μ) & = min_{μ \in Ψ (Ω)} \int_{t}^{\infty} e^{γ (t - σ)} (β {\hat{f}}^{T} \hat{f} \\ + U (x (σ), μ (σ))) d σ \\ \leq δ_{2} . \end{matrix}$ (12)

The optimal cost function (12) satisfies the HJB equation, then $0 = min_{μ} H (x, μ, \nabla V_{1}^{*} (x, μ)) .$ (13)

The optimal control is expressed as $μ^{*} (x) = - \frac{1}{2} R^{- 1} ρ^{T} (x) \nabla V_{1}^{*} (x, μ) .$ (14)

The PI scheme is designed as shown in Algorithm 1.

Algorithm1 Online PI

Step1: Select an initial admissible control policy μ⁽⁰⁾ and a

positive constant ϵ and an initial performance index

function

\nabla V_{1}^{(0)} (x, μ^{(0)}) = 0

;

Step2: Solve

V_{1}^{(i)}

according to

0 = β {\hat{f}}^{T} \hat{f} + U (x, μ^{i}) + (\nabla

V_{1}^{(i)} (x, μ^{(i)}))^{T} (ϖ (x)

+ ρ (x) μ^{(i)}) - γ V_{1}^{(i - 1)} (x, μ^{(i - 1)})

;

Step3: Update the control policy with

μ^{(i + 1)} = - \frac{1}{2} R^{- 1} ρ^{T} (x) \nabla V_{1}^{(i)} (x, μ^{i})

;

Step4: if

∥ V_{1}^{(i + 1)} (x, μ^{(i + 1)}) - V_{1}^{(i)} (x, μ^{(i)}) ∥ \leq ϵ

stop the iterations;else return to Step2.

3 Fault-tolerant ADP tracking controller design via neural network observer

3.1 Problem transformation

The structural diagram of neural network observer based fault-tolerant ADP control scheme is shown in Fig. 2.

Fig. 2

Structural diagram of neural network observer based fault-tolerant ADP control scheme.

Assumption 2. The approximate error of actuators faults $e_{f} = f - \hat{f}$ satisfies that ∥e_f ∥ ≤ δ₃.

Lemma 1. ([23, 24]) With Assumption 1,2 and the control policy (14) for error tracking system (8) with f = 0, the continuously differentiable function $V_{1}^{*} (x, μ)$ is a Lyapunov function if the conditions $β \geq γ δ_{1}^{- 2} δ_{2} λ_{max} (R^{- 1}) + λ_{max} (R)$ and $∥ x ∥ \geq \sqrt{\frac{(γ δ_{1}^{- 2} δ_{2} λ_{max} (R^{- 1}) + λ_{max} (R)) (2 δ_{1} + δ_{3})}{λ_{min} (Q)}}$ hold. So, the optimal control law (14) is a solution to the error tracking system (8) with f ≠ 0 and error tracking system (8) with f ≠ 0 is UUB.

The Proof was given in [23, 24].

3.2 Design of neural-network observer

For the error tracking system (8), we developed a radial basis function (RBF) neural network to approximate the actuators faults. $f = - (W_{0} φ_{0} (x) + ɛ_{0}) .$ (15)

Substituting equation (15) into error tracking system (8), we can get $\dot{x} = ϖ (x) + ρ (x) (μ + W_{0} φ_{0} (x) + ɛ_{0}) .$ (16)

Then the neural-network faults observer is designed as $\dot{\hat{x}} = ϖ (\hat{x}) + ρ (\hat{x}) (μ + {\hat{W}}_{0} φ_{0} (\hat{x})) + L (x - \hat{x})$ (17) where $\hat{x}$ the approximation of x; ${\hat{W}}_{0}$ is the approximation of W₀.

The weight vector ${\hat{W}}_{0}$ should be updated as ${\dot{\hat{W}}}_{0} = - ϱ_{0} ρ^{T} (\hat{x}) e_{x} φ_{0}^{T} (\hat{x})$ (18) where $e_{x} = \hat{x} - x$ is the approximation error of x, and ϱ₀ > 0.

Combining (16) with (17), we can get $\begin{matrix} {\dot{e}}_{x} & = \dot{\hat{x}} - \dot{x} \\ = ϖ (\hat{x}) + ρ (\hat{x}) (μ + {\hat{W}}_{0} φ_{0} (\hat{x})) - {Le}_{x} \\ - (ϖ (x) + ρ (x) (μ + W_{0} φ_{0} (x) + ɛ_{0})) \\ = - {Le}_{x} + (ϖ (\hat{x}) - ϖ (x)) + (ρ (\hat{x}) \\ - ρ (x)) μ + ρ (\hat{x}) {\hat{W}}_{0} φ_{0} (\hat{x}) \\ - ρ (x) W_{0} φ_{0} (x) - ρ (x) ɛ_{0} \\ = - {Le}_{x} + \tilde{ϖ} + \tilde{ρ} μ + ρ (\hat{x}) {\hat{W}}_{0} φ_{0} (\hat{x}) \\ - ρ (\hat{x}) W_{0} φ_{0} (\hat{x}) + ρ (\hat{x}) W_{0} φ_{0} (\hat{x}) \\ - ρ (x) W_{0} φ_{0} (x) - ρ (x) ɛ_{0} \\ = - {Le}_{x} + \tilde{ϖ} + \tilde{ρ} μ + ρ (\hat{x}) {\tilde{W}}_{0} φ_{0} (\hat{x}) \\ + ρ (\hat{x}) W_{0} φ_{0} (\hat{x}) - ρ (x) W_{0} φ_{0} (\hat{x}) \\ + ρ (x) W_{0} φ_{0} (\hat{x}) - ρ (x) W_{0} φ_{0} (x) \\ - ρ (x) ɛ_{0} \\ = - {Le}_{x} + \tilde{ϖ} + \tilde{ρ} μ + ρ (\hat{x}) {\tilde{W}}_{0} φ_{0} (\hat{x}) \\ + \tilde{ρ} W_{0} φ_{0} (\hat{x}) + ρ (x) W_{0} {\tilde{φ}}_{0} - ρ (x) ɛ_{0} \end{matrix}$ (19) where $\tilde{ϖ} = ϖ (\hat{x}) - ϖ (x)$ ; $\tilde{ρ} = ρ (\hat{x}) - ρ (x)$ ; ${\tilde{φ}}_{0} = φ_{0} (\hat{x}) - φ_{0} (x)$ ; ${\tilde{W}}_{0} = {\hat{W}}_{0} - W_{0}$ and ${\dot{\tilde{W}}}_{0} = {\dot{W}}_{0}$ .

Assumption 3. $\tilde{ϖ}$ , $\tilde{ρ} μ$ , $\tilde{ρ} W_{0} φ_{0} (\hat{x})$ , $ρ (x) W_{0} {\tilde{φ}}_{0}$ and ρ (x) ɛ₀ are norm-bounded as $∥ \tilde{ϖ} ∥ \leq δ_{4}$ , $∥ \tilde{ρ} μ ∥ \leq δ_{5}$ , $∥ \tilde{ρ} W_{0} φ_{0} (\hat{x}) ∥ \leq δ_{6}$ , $∥ ρ (x) W_{0} {\tilde{φ}}_{0} ∥ \leq δ_{7}$ and ∥ρ (x) ɛ₀ ∥ ≤ δ₈.

Theorem 1. With Assumptions 1,3, the updating law (18) of weight vector (17) can guarantee e_x to be UUB based on the neural network observer.

Proof 1. Select an Lypunov function as $V_{2} = \frac{1}{2} e_{x}^{T} e_{x} + \frac{1}{2 ϱ_{0}} tr [{\tilde{W}}_{0}^{T} {\tilde{W}}_{0}] .$ (20)

Substituting (19) into the time derivative of (20), we can get $\begin{matrix} {\dot{V}}_{2} & = e_{x}^{T} {\dot{e}}_{x} + \frac{1}{ϱ_{0}} tr [{\dot{\tilde{W}}}_{0}^{T} {\tilde{W}}_{0}] \\ = e_{x}^{T} (- {Le}_{x} + \tilde{ϖ} + \tilde{ρ} μ + ρ (\hat{x}) {\tilde{W}}_{0} φ_{0} (\hat{x}) \\ + \tilde{ρ} W_{0} φ_{0} (\hat{x}) + ρ (x) W_{0} {\tilde{φ}}_{0} - ρ (x) ɛ_{0}) \\ - tr [(ρ^{T} (\hat{x}) e_{x} φ_{0}^{T} (\hat{x}))^{T} {\tilde{W}}_{0}] \\ = e_{x}^{T} (- {Le}_{x} + \tilde{ϖ} + \tilde{ρ} μ + ρ (\hat{x}) {\tilde{W}}_{0} φ_{0} (\hat{x}) \\ + \tilde{ρ} W_{0} φ_{0} (\hat{x}) + ρ (x) W_{0} {\tilde{φ}}_{0} - ρ (x) ɛ_{0}) \\ - tr [φ_{0} (\hat{x}) e_{x}^{T} ρ (\hat{x}) {\tilde{W}}_{0}] \\ = e_{x}^{T} (- {Le}_{x} + \tilde{ϖ} + \tilde{ρ} μ + \tilde{ρ} W_{0} φ_{0} (\hat{x}) \\ + ρ (x) W_{0} {\tilde{φ}}_{0} - ρ (x) ɛ_{0}) \\ \leq - λ_{min} (L) e_{x}^{T} e_{x} + e_{x}^{T} (\tilde{ϖ} + \tilde{ρ} μ \\ + \tilde{ρ} W_{0} φ_{0} (\hat{x}) + ρ (x) W_{0} {\tilde{φ}}_{0} - ρ (x) ɛ_{0}) \\ \leq - \frac{11 λ_{min} (L)}{16} e_{x}^{T} e_{x} + \frac{4}{λ_{min}} ({\tilde{ϖ}}^{T} \tilde{ϖ} \\ + (\tilde{ρ} μ)^{T} (\tilde{ρ} μ) + (\tilde{ρ} W_{0} φ_{0} (\hat{x}))^{T} (\tilde{ρ} W_{0} φ_{0} (\hat{x})) \\ + (ρ (x) W_{0} {\tilde{φ}}_{0})^{T} (ρ (x) W_{0} {\tilde{φ}}_{0}) \\ + (ρ (x) ɛ_{0})^{T} (ρ (x) ɛ_{0})) \\ \leq - \frac{11 λ_{min} (L)}{16} ∥ e_{x} ∥^{2} + \frac{4}{λ_{min}} (∥ \tilde{ϖ} ∥^{2} \\ + ∥ \tilde{ρ} μ ∥^{2} + ∥ \tilde{ρ} W_{0} φ_{0} (\hat{x}) ∥^{2} \\ + ∥ ρ (x) W_{0} {\tilde{φ}}_{0} ∥^{2} + ∥ ρ (x) ɛ_{0} ∥^{2}) \\ \leq - \frac{11 λ_{min} (L)}{16} ∥ e_{x} ∥^{2} + \frac{4}{λ_{min}} (δ_{4}^{2} + δ_{5}^{2} + δ_{6}^{2} \\ + δ_{7}^{2} + δ_{8}^{2}) . \end{matrix}$ (21)

We can conclude that ${\dot{V}}_{2} < 0$ if e_x satisfies $∥ e_{x} ∥ > \frac{8}{λ_{min}} \sqrt{\frac{δ_{4}^{2} + δ_{5}^{2} + δ_{6}^{2} + δ_{7}^{2} + δ_{8}^{2}}{11}}$ . Based on the Lyapunov stability theorem, e_x is guaranteed to be UUB. This completes the proof.

3.3 Design of critic neural network

The ADP controller consists of critic neural network and action neural network. The critic neural network is utilized to approximate $V_{1}^{*} (x, μ^{*})$ . $V_{3} (x, μ) = W_{c} φ_{c} (x, μ) + ɛ_{c} .$ (22)

The derivative of the cost function V₃ (x, μ) is given as $\nabla V_{3} (x, μ) = (\nabla φ_{c} (x, μ))^{T} W_{c}^{T} + \nabla ɛ_{c}$ (23) where $\nabla φ_{c} (x, μ) = \frac{\partial φ_{c} (x, μ)}{\partial x}$ and $\nabla ɛ_{c} = \frac{\partial ɛ_{c}}{\partial x}$ .

Substituting (23) into (10), we can obtain $\begin{matrix} 0 & = β {\hat{f}}^{T} \hat{f} + U (x, μ) + (W_{c} \nabla φ_{c} (x, μ) \\ + \nabla ɛ_{c}) (ϖ (x) + ρ (x) (μ - \hat{f}) \\ - γ (W_{c} φ_{c} (x, μ) + ɛ_{c}) . \end{matrix}$ (24)

Then the Hamiltonian function can be expressed as $\begin{matrix} H (x, μ, W_{c}) & = β {\hat{f}}^{T} \hat{f} + U (x, μ) - γ W_{c} φ_{c} (x, μ) \\ + W_{c} \nabla φ_{c} (x, μ) (ϖ (x) + ρ (x) (μ \\ - \hat{f})) \\ = e_{c} \end{matrix}$ (25) where e_c is the residual error.

Then, V₃ (x, μ) is approximated as ${\hat{V}}_{3} (x, μ) = {\hat{W}}_{c} φ_{c} (x, μ)$ (26) where ${\hat{W}}_{c}$ is the approximation of W_c.

The derivative of ${\hat{V}}_{3} (Z)$ can be expressed as $\nabla {\hat{V}}_{3} (x, μ) = (\nabla φ_{c} (x, μ))^{T} {\hat{W}}_{c}^{T} .$ (27)

Then, the approximate Hamiltonian function can be expressed as $\begin{matrix} \hat{H} (x, μ, {\hat{W}}_{c}) & = β {\hat{f}}^{T} \hat{f} + U (x, μ) - γ {\hat{W}}_{c} φ_{c} (x, μ) \\ + {\hat{W}}_{c} \nabla φ_{c} (x, μ) (ϖ (x) + ρ (x) (μ \\ - \hat{f})) \\ = {\hat{e}}_{c} . \end{matrix}$ (28)

Given any admissible control policy μ, it is desired to select ${\hat{W}}_{c}$ to minimize the squared residual error $E_{c} ({\hat{W}}_{c})$ as $E_{c} ({\hat{W}}_{c}) = \frac{1}{2} {\hat{e}}_{c}^{T} {\hat{e}}_{c} .$ (29)

The weight update law for the critic neural network is given as ${\dot{\hat{W}}}_{c} = - \frac{ϱ_{1} {\hat{e}}_{c} ς_{1}^{T}}{(1 + ς_{1}^{T} ς_{1})^{2}}$ (30) where ϱ₁ satisfies that ϱ₁ > 0; $ς_{1} = \nabla φ_{c} (x, μ) (ϖ (x) + ρ (x) (μ - \hat{f})) - γ φ_{c} (x, μ)$ and $ς_{1} \in R^{l_{1}}$ .

The approximate weight error of critic neural network is defined as ${\tilde{W}}_{c} = {\hat{W}}_{c} - W_{c}$ . Then, (30) can be transformed as $\begin{matrix} {\dot{\tilde{W}}}_{c} & = - \frac{ϱ_{1}}{(1 + ς_{1}^{T} ς_{1})^{2}} ({\tilde{W}}_{c} ς_{1} - \nabla ɛ_{c} (ϖ (x) \\ + ρ (x) (μ - \hat{f})) - γ ɛ_{c}) ς_{1}^{T} . \end{matrix}$ (31)

Assumption 4. $∥ \nabla ɛ_{c} (ϖ (x) + ρ (x) (μ - \hat{f})) + γ ɛ_{c} ∥ \leq δ_{9}$ and ς_min ≤ ∥ ς₁ ∥ ≤ ς_max, where ς_min and ς_max are positive constants.

Theorem 2. The approximate weight error is UUB, if the weight of the critic neural network is updated by (31).

Proof 2. Select an Lyapunov function as $V_{4} = \frac{(1 + ς_{1}^{T} ς_{1})^{2}}{2 ϱ_{1}} {\tilde{W}}_{c} {\tilde{W}}_{c}^{T} .$ (32)

Then, the time derivative of V₄ is $\begin{matrix} {\dot{V}}_{4} & = \frac{(1 + ς_{1}^{T} ς_{1})^{2}}{2 ϱ_{1}} {\dot{\tilde{W}}}_{c} {\tilde{W}}_{c}^{T} \\ = - ({\tilde{W}}_{c} ς_{1} - \nabla ɛ_{c} (ϖ (x) + ρ (x) (μ - \hat{f})) \\ - γ ɛ_{c}) ς_{1}^{T} {\tilde{W}}_{c}^{T} \\ = - {\tilde{W}}_{c} ς_{1} ς_{1}^{T} {\tilde{W}}_{c}^{T} + (\nabla ɛ_{c} (ϖ (x) + ρ (x) (μ \\ - \hat{f})) + γ ɛ_{c}) ς_{1}^{T} {\tilde{W}}_{c}^{T} \\ \leq - \frac{1}{2} ∥ {\tilde{W}}_{c} ς_{1} ∥^{2} + \frac{1}{2} ∥ \nabla ɛ_{c} (ϖ (x) + ρ (x) (μ \\ - \hat{f})) + γ ɛ_{c} ∥^{2} \\ \leq - \frac{1}{2} ∥ {\tilde{W}}_{c} ς_{1} ∥^{2} + \frac{1}{2} ∥ δ_{9} ∥^{2} . \end{matrix}$ (33)

Hence, ${\dot{V}}_{4} < 0$ if $∥ {\tilde{W}}_{c} ∥ > ∥ \frac{δ_{9}}{ς_{min}} ∥$ . The approximate weight error is UUB, according to the Lyapunov stability theorem. This completes the proof.

3.4 Design of action neural network

The optimal control μ^* is approximated by the action neural network as $μ = W_{a} φ_{a} (x) + ɛ_{a} .$ (34)

Because the ideal weight $W_{a}^{T}$ is unknown, μ^* is approximated as $\hat{μ} = {\hat{W}}_{a} φ_{a} (x)$ (35) where ${\hat{W}}_{a}$ is the estimate of W_a.

The approximate feedback error used for training action neural network is defined as the difference between the feedback control input applied to the error tracking system (8) and the optimal control μ^* as ${\hat{e}}_{a} = {\hat{W}}_{a} φ_{a} (x) + \frac{1}{2} R^{- 1} ρ^{T} (x) (\nabla φ_{c} (x, μ))^{T} {\hat{W}}_{c}^{T} .$ (36)

The action neural network is defined to minimize the objective function as $E_{a} ({\hat{W}}_{a}) = \frac{1}{2} {\hat{e}}_{a}^{T} {\hat{e}}_{a} .$ (37)

The weight updating law for the action neural network is given as follows ${\dot{\hat{W}}}_{a} = - ϱ_{2} {\hat{e}}_{a} φ_{a}^{T} (x)$ (38) where ϱ₂ > 0.

According to (14), (23) and (34), we have $\begin{matrix} 0 & = W_{a} φ_{a} (x) + ɛ_{a} + \frac{1}{2} R^{- 1} ρ^{T} (x) ((\nabla φ_{c} (x, μ))^{T} W_{c}^{T} \\ + \nabla ɛ_{c}) . \end{matrix}$ (39)

The approximate weight error of action neural network is defined as ${\tilde{W}}_{a} = {\hat{W}}_{a} - W_{a}$ . Then, (38) can be transformed as $\begin{matrix} {\dot{\tilde{W}}}_{a} & = - ϱ_{2} ({\tilde{W}}_{a} φ_{a} (x) - ɛ_{a} - \frac{1}{2} R^{- 1} ρ^{T} (x) \nabla ɛ_{c} \\ + \frac{1}{2} R^{- 1} ρ^{T} (x) (\nabla φ_{c} (x, μ))^{T} {\tilde{W}}_{c}^{T}) φ_{a}^{T} (x) . \end{matrix}$ (40)

Assumption 5. ∥R^-1ρ^T (x) (∇ φ_c (x, μ)) ^T ∥ ≤ δ₁₀, $∥ ɛ_{a} + \frac{1}{2} R^{- 1} ρ (x)^{T} \nabla ɛ_{c} ∥ \leq δ_{11}$ and φ_a,min ≤ ∥ φ_a (x) ∥ ≤ φ_a,max, where φ_a,min and φ_a,max are positive constants.

Theorem 3. The approximate weight error is UUB, if the weight of the action neural network is updated by (40).

Proof 3. Select an Lyapunov function as $V_{5} = \frac{1}{2 ϱ_{2}} {\tilde{W}}_{a}^{T} {\tilde{W}}_{a}^{T} .$ (41)

Then, the time derivative of V₅ is $\begin{matrix} {\dot{V}}_{5} & = \frac{1}{2 ϱ_{2}} {\dot{\tilde{W}}}_{a} {\tilde{W}}_{a}^{T} \\ = - {\tilde{W}}_{a} φ_{a} (x) φ_{a}^{T} (x) {\tilde{W}}_{a}^{T} \\ - \frac{1}{2} R^{- 1} ρ^{T} (x) (\nabla φ_{c} (x, μ))^{T} {\tilde{W}}_{c}^{T} φ_{a}^{T} (x) {\tilde{W}}_{a}^{T} \\ + (ɛ_{a} + \frac{1}{2} R^{- 1} ρ^{T} (x) \nabla ɛ_{c}) φ_{a}^{T} (x) {\tilde{W}}_{a}^{T} \\ \leq - \frac{1}{4} ∥ {\tilde{W}}_{a} φ_{a} (x) ∥^{2} + \frac{1}{2} ∥ ɛ_{a} + \frac{1}{2} R^{- 1} ρ^{T} (x) \nabla ɛ_{c} ∥^{2} \\ + \frac{1}{4} ∥ R^{- 1} ρ^{T} (x) (\nabla φ_{c} (x, μ))^{T} {\tilde{W}}_{c}^{T} ∥^{2} \\ \leq - \frac{1}{4} ∥ {\tilde{W}}_{a} φ_{a} (x) ∥^{2} + \frac{1}{4} δ_{10}^{2} ∥ {\tilde{W}}_{c} ∥^{2} + \frac{1}{2} δ_{11}^{2} . \end{matrix}$ (42)

Hence, ${\dot{V}}_{5} < 0$ if $∥ {\tilde{W}}_{a} ∥ > \frac{\sqrt{δ_{10}^{2} ∥ {\tilde{W}}_{c} ∥^{2} + 2 δ_{11}^{2}}}{φ_{a, min}}$ . The weight approximation error is UUB, according to the Lyapunov stability theorem. This completes the proof.

3.5 Stability analysis

Assumption 6. ∥ϖ (x) ∥ ≤ δ₁₂ and ∥ρ (x) ∥ ≤ δ₁₃.

Theorem 4. With the performance index function (9), the error tracking system (8) can be guaranteed to be UUB by the approximate fault-tolerant tracking control policy (35).

Proof 4. Select an Lypunov function as $V_{6} = \frac{1}{2} x^{T} x + V_{1}^{*} .$ (43)

Then, the time derivative of V₆ is $\begin{matrix} {\dot{V}}_{6} & = x^{T} \dot{x} + (\nabla V_{1}^{*})^{T} \dot{x} \\ = x^{T} (ϖ (x) + ρ (x) (μ - f)) \\ + (\nabla V_{1}^{*})^{T} (ϖ (x) + ρ (x) (μ - f)) \\ = x^{T} ϖ (x) + x^{T} ρ (x) μ - x^{T} ρ (x) f \\ - (\nabla V_{1}^{*})^{T} ρ (x) f + (\nabla V_{1}^{*})^{T} (ϖ (x) \\ + ρ (x) μ) . \end{matrix}$ (44)

According to (11), (13), (44) can be transformed as

$\begin{matrix} {\dot{V}}_{6} & = x^{T} ϖ (x) + x^{T} ρ (x) μ - x^{T} ρ (x) f - x^{T} Qx \\ - μ^{T} R μ - (\nabla V_{1}^{*})^{T} ρ (x) f - β {\hat{f}}^{T} \hat{f} + γ V_{1}^{*} \\ = x^{T} ϖ (x) + x^{T} ρ (x) μ - x^{T} ρ (x) f + 2 μ^{T} Rf \\ - x^{T} Qx - μ^{T} R μ - β {\hat{f}}^{T} \hat{f} + γ V_{1}^{*} \\ \leq \frac{3}{2} x^{T} x + \frac{1}{2} ϖ^{T} (x) ϖ (x) + \frac{1}{2} μ^{T} ρ^{T} (x) ρ (x) μ \\ + \frac{1}{2} f^{T} ρ^{T} (x) ρ (x) f + μ^{T} μ + f^{T} R^{T} Rf \\ - x^{T} Qx - μ^{T} R μ - β {\hat{f}}^{T} \hat{f} + γ δ_{2} \\ \leq - (λ_{min} (Q) - \frac{3}{2}) ∥ x ∥^{2} - (λ_{min} (Q) - \frac{1}{2} δ_{13}^{2} \\ - 1) ∥ μ ∥^{2} - (β - \frac{1}{2} δ_{13}^{2} - λ_{max} (R^{'})) ∥ \hat{f} ∥^{2} \\ + (\frac{1}{2} δ_{13}^{2} + λ_{max} (R^{'})) (2 δ_{1} + δ_{3}) δ_{3} \\ + \frac{1}{2} δ_{12}^{2} + γ δ_{2} \end{matrix}$ (45) where R′ = R^TR.

Hence, ${\dot{V}}_{6} < 0$ if $(λ_{min} (Q) - \frac{3}{2}) > 0$ , $(λ_{min} (Q) - \frac{1}{2} δ_{13}^{2} - 1) \geq 0$ , $(β - \frac{1}{2} δ_{13}^{2} - λ_{max} (R^{'})) \geq 0$ and $∥ x ∥ > \sqrt{\frac{(\frac{1}{2} δ_{13}^{2} + λ_{max} (R^{'})) (2 δ_{1} + δ_{3}) δ_{3} + + \frac{1}{2} δ_{12}^{2} + γ δ_{2}}{λ_{min} (Q) - \frac{3}{2}}}$ . The error tracking system (8) is UUB, according to the Lyapunov stability theorem. This completes the proof.

4 Simulation results

In order to show the effectiveness of the proposed fault-tolerant tracking control based on ADP, two simulation examples are performed compared with the single critic network based ADP in this section [23, 24]. According to the kinematic and dynamic model of underactuated AUV (1) with the conditions that are η (4) =0 and ξ (4) =0, the matrices M, C (ξ), D (ξ) and g (η),J are given as follows.

$M = [\begin{matrix} 215 & 0 & 0 & 0 & 0 & 0 \\ 0 & 265 & 0 & 0 & 0 & 0 \\ 0 & 0 & 265 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 80 & 0 \\ 0 & 0 & 0 & 0 & 0 & 80 \end{matrix}] .$ (46)

$C (ξ) = [\begin{matrix} 0 & 0 & 0 & 0 & 265 w & - 265 v \\ 0 & 0 & 0 & 0 & 0 & 215 u \\ 0 & 0 & 0 & 0 & - 215 u & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ - 50 w & 0 & 0 & 0 & 0 & 0 \\ 50 v & 0 & 0 & 0 & 0 & 0 \end{matrix}] .$ (47) $D (ξ) = [\begin{matrix} D_{1} & 0 & 0 & 0 & 0 & 0 \\ 0 & D_{2} & 0 & 0 & 0 & 0 \\ 0 & 0 & D_{3} & 0 & 0 & 0 \\ 0 & 0 & 0 & D_{4} & 0 & 0 \\ 0 & 0 & 0 & 0 & D_{5} & 0 \\ 0 & 0 & 0 & 0 & 0 & D_{6} \end{matrix}]$ (48) where D₁ = 70 + 100|u|; D₂ = 100 + 200|v|; D₃ = 100 + 200|w|; D₄ = 0; D₅ = 50 + 100|q|; and D₆ = 50 + 100|r|. $\begin{matrix} g (η) & = [0, 0, - (1822.25 - G) cos (θ), 0, \\ (18.22225 - 0.01 G) sin (θ), 0]^{T} \end{matrix}$ (49) where G is the gravity. $J = [\begin{matrix} c θ c ψ & - s ψ & s θ c ψ & 0 & 0 & 0 \\ c θ s ψ & c ψ & s θ s ψ & 0 & 0 & 0 \\ - s θ & 0 & c θ & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & sec θ \end{matrix}]$ (50) where cθ = cos θ; sθ = sin θ; sψ = sin ψ; cψ = cos ψ.

4.1 Example one without actuators faults

Given f = 0, ϱ₀ = 0.1, ϱ₁ = 0.02, ϱ₂ = 0.04, γ = 0.3, B = 1822.25, β = 0.15 and τ_d = [500, 0, 0, 0, 200, 10] ^T, the simulation results are given as follows compared with the existing method [23, 24].

Figures 3 and 4 show the tracking error of desired position and attitude and the tracking error of desired velocity compared with the existing method [23, 24] respectively. The tracking trajectory is shown in Fig. 5. The method proposed in this work has received almost the same results with the existing method [23, 24]. From Figs. 3 and 4, we can know that the error tacking system (8) is bounded stable. The absolute values of the tracking error of desired position and attitude are no more than the threshold value 0.2. The absolute values of the tracking error of desired velocity are no more than the threshold value 0.05. From Fig. 5, we know that the value of the error trajectory between the desired trajectory and the simulation trajectory with the method proposed in this work is no bigger than 0.1m.

Fig. 3

Tracking error of desired position and attitude.

Fig. 4

Tracking error of desired velocity.

Fig. 5

AUV trajectory.

4.2 Example two with actuators faults

In this simulation example, we used the parameters values of example one except for f = 0.1μ. The simulation results compared with the existing method [23, 24] are given as follows.

Figures 6 and 7 show the tracking error of desired position and attitude and the tracking error of desired velocity. From Fig. 7, we know that the jitter happens with the existing method [23, 24] from 0s to 15s. The method proposed in this work has received better results and reduce the jitter effectively.

Fig. 6

Tracking error of desired position and attitude.

Fig. 7

Tracking error of desired velocity.

Figures 8 and 9 give the estimated actuators faults based on the RBF neural network when the values of f = 0.1μ. With the actuators faults, the jitter happened in the estimation of actuators faults with existing method [23, 24]. When the actuators faults became bigger, the jitter became bigger.

Fig. 8

Estimated actuators faults based on RBF neural network with the method propose in this work.

Fig. 9

Estimated actuators faults based on RBF neural network with the existing method [23, 24].

Figure 10 shows the tracking trajectories with f = 0.1μ. From the simulation results, we know that the value of the error trajectory between the desired trajectory and the simulation trajectory with the method proposed in this work is no more than 0.3m. The trajectory with the method proposed in this work is more close to the desired trajectory.

Fig. 10

AUV trajectory.

5 Conclusion

In this work, in order to apply action-critic networks based ADP to solve output-feedback fault-tolerant tracking control problems for underactuated AUV with actuators faults, the error tracking system with actuators faults (8) has been customized. Furthermore, the online policy iteration algorithm has been designed to improve the tracking accuracy, which reduces the impact of jitters effectively. The stability of error tracking system of underactuated AUV (8) is guaranteed under the Lyapunov stability theory. Finally, simulations have been performed. When the actuators faults happened, the jitter happened with the existing method [22, 24]. Simulation results have shown the better performance compared with the existing method [23, 24].

Future researches will concentrate on improving tracking accuracy and stability for full-coupled nonaffine AUVs with complex disturbances limitations. Therefore, online deep reinforcement learning will be taken into account in future study.

Footnotes

Declarations

Funding:There is no funding to support this work.

Conflicts of interest:The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this manuscript.

Authors’ contributions:G. Che designs the control method, does the simulation experiments and writes the manuscript and Z. Yu designs the control method and analyzes the stability of system.

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