Extreme learning‐based non‐linear model predictive controller for an autonomous underwater vehicle: simulation and experimental results

Abstract

In this study, an extreme learning‐based non‐linear model predictive controller (NMPC) is proposed for path following planning of an autonomous underwater vehicle (AUV) using horizontal way‐points. The proposed controller comprises a kinematic controller and a dynamic controller. The kinematic controller is designed by using back‐stepping approach whilst the dynamic controller is designed by employing the NMPC approach. The dynamics of the AUV is identified in real‐time by employing an extreme learning machine (ELM) structure. In view of achieving improved performance of the ELM structure, its hidden layer parameters are optimally determined by applying Jaya optimisation algorithm. The resulting ELM model is then used to design a NMPC considering the constraint on rudder planes. The tracking performance of the proposed controller is compared with that of two recently reported control algorithms namely, $H_{\mathrm{\infty }}$ state feedback controller and inverse optimal self‐tuning proportional–integral–derivative (PID) controller. The proposed controller is implemented using MATLAB and then in real‐time on a prototype AUV developed in the authors’ laboratory. From both the simulation and experimental results obtained, it is observed that the proposed controller exhibits superior tracking performance compared to both $H_{\mathrm{\infty }}$ state feedback controller and inverse optimal self‐tuning PID controller.

1 Introduction

Research on control of the autonomous underwater vehicle (AUV) has gained momentum since the last decade owing to its several interesting applications including defence, pipeline survey, and mine survey measure missions. Controlling an AUV is a challenging task owing to its uncertain dynamics and communication constraints in the acoustic medium. Thus, there is a need for identifying the dynamics of the AUV in real-time that could capture its uncertain dynamics to design suitable controllers. A single-hidden layer feed-forward neural network (SLFN) is a very popular model for identification of AUV dynamic owing to the fact that each basis function in the hidden layer of such a network is a non-linear mapping of the multi-variable inputs to scalar values [1–5]. However, training of such models encounters difficulties e.g. slow convergence rate and a very large number of iterations are required to reach the global optima [6]. To overcome the aforesaid problems, an extreme learning machine (ELM) model has been proposed in [7, 8]. It offers improved performance for solving a regression problem. In the ELM model, the input weights and hidden biases are randomly chosen while the weights connecting the hidden layer and the output are analytically determined by using Moore–Penrose generalised inverse operation of the hidden layer output matrix. Owing to the random assignment of the input weights and biases for the hidden layer, the ELM model may not always give a good performance. This is due to the reason that more number of hidden layer neurons is needed which in turn results in slow convergence of the ELM model [9–11]. A lot of efforts have been made in the past for overcoming these issues, e.g. in [9], a modified differential evolution algorithm is used to optimise the input weights and input biases of the ELM network and the concept of such a hybrid algorithm is called evolutionary ELM (EELM). The EELM structure not only provides faster convergence but also more accurate generalisation performance than the ELM structure. Many gradient-based algorithms based on Levenberg–Marquardt and backpropagation algorithms can be hybridised with the ELM. However, these hybrid models may exhibit very slow convergence. Furthermore, searching for the global optimum using gradient-based algorithms is not always guaranteed to result in premature convergence. Owing to the excellent global optimisation feature of Jaya algorithm with less computational time and faster convergence characteristics [12, 13], it is expected that the hybridisation of Jaya algorithm and ELM would be beneficial for training SLFN. Furthermore, Jaya algorithm does not need to depend on any optimisation specific parameters except for population size, number of design variable, and maximum number of iterations. The predicted EELM model is then used to carry out on-line predictions of the AUV dynamics using the non-linear model predictive controller (NMPC) [14–16].

Considering both fully-actuated and under-actuated AUVs, path following control algorithms achieve better smooth convergence to the path as compared to trajectory tracking control algorithms [17–19]. Moreover, in path following control, the actuation signal is less likely to undergo actuation saturation, so the path following algorithm is certainly a better choice for under-actuated AUV as compared to trajectory tracking algorithm. However, considering the suitability of the real-time application of the AUV in ocean environment such as any surveillance application, way-points tracking based on the line-of-sight (LoS) path is often used [19, 20] as it is less computationally expensive [21, 22]. In the LoS path following approach, two way-points are connected by a rectilinear path as shown in Fig. 1b and the guidance law objective is to track the yaw angle obtained from the LoS path so that the AUV can reach within the defined acceptance region of way-point [23, 24].

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Several control strategies have been reported in the literature, which are successful in the path following problem such as back-stepping controller [25], sliding mode controller [26, 27], $H_{\mathrm{\infty }}$ controller [28–30], proportional–integral–derivative (PID) controller [31, 32] etc. In [26, 27], the sliding mode controller is developed, which is based on the feedback linearisation method. Though it provides good tracking performances, issues such as robustness, internal stability, and system constraints were not taken into consideration while forming the control law. The development of $H_{\mathrm{\infty }}$ provides a robust control strategy for the non-linear system [28–30]. Recently, $H_{\mathrm{\infty }}$ controller based on the energy dissipation concept has been developed to obtain a static control law for steering control of the AUV [30]. In [30], the non-linear heading dynamics of the AUV are converted into the sum of Taylor's series. Furthermore, Taylor's series has been approximated till second degree for control law formulation. However, to improve the controller performance, if higher-order terms are also considered in Taylor's series, it will make the controller computationally complex and expensive. Though there are many developments in control strategy recently still PID controllers are extensively used for tracking performance owing to their simplicity and ease of implementation [31, 32]. However in these controller design [25–32], the system constraints have not been considered in deriving control law, thus a constrained optimal controller is necessary to address the issues owing to the above constraints. Furthermore, in most of the previous works [21, 32], a higher-dimensional model structure of the AUV is used for controlling the AUV, which makes the design of the controller computationally expensive. So, in this study, a reduced dimensional model of the AUV is considered for the design of the controller for the AUV. Furthermore, the model predictive controller is an efficient optimal control strategy for a non-linear system with actuator constraints [33–36] has been used in the present work to address the horizontal way-point tracking problem of the AUV. However, the accuracy and stability of the closed-loop system using NMPC depend on accurate modeling of the non-linear system. As discussed earlier, the ELM model is used for modelling AUV. So, in this study, a constrained NMPC waypoint tracking algorithm for an AUV is proposed by using the ELM model of the AUV.

The contributions of this paper are as follows:

• A reduced order EELM model of an AUV is derived by using the experimental data.

• By testing the above EELM model, a NMPC is designed for heading control of the AUV.

• The proposed NMPC controller is realised on a prototype AUV developed in the laboratory.

The rest of the paper is organised as follows. The AUV modelling and problem formulation are presented in Section 2. Section 3 presents the model identification of the AUV using the ELM model structure and Jaya optimisation algorithm. Based on the resulting ELM model of the AUV, the proposed control design problem is considered in Section 4. The modelling accuracy by the ELM model of the AUV is verified first followed by the evaluation of the efficacy of the proposed NMPC algorithm is presented in Section 5. In the same section, the performance of the proposed NMPC is compared with that of both $H_{\mathrm{\infty }}$ state feedback controller [30] and inverse optimal self-tuning PID controller [32]. Finally, the paper is concluded in Section 6.

2 Modelling of AUV and problem formulation

The structure of the AUV in three degree-of-freedom is shown in Fig. 1a, which is used for controlling the AUV in the horizontal plane. The AUV is represented by using a body fixed frame $(u^{\mathrm{s}},v,r)$ and inertial reference frame $(x,y,\psi )$, where $u^{\mathrm{s}},v,r$ denotes the surge velocity, sway velocity, and yaw rate. x and y are the inertial coordinates of the AUV and ${\psi }_k$ denotes the yaw orientation. The kinematic and dynamic equation of the AUV, adopted from [21], is given as follows.

The Euler forward form of kinematics equation for heading motion is given as 1 $\begin{array}{rl}{x}_k& =x_{k-1}+T\left(u_{k-1}^{\mathrm{s}}\cos {\psi }_{k-1}-v_{k-1}\sin {\psi }_{k-1}\right)\\ y_k& =y_{k-1}+T\left(u_{k-1}^{\mathrm{s}}\sin {\psi }_{k-1}+v_{k-1}\cos {\psi }_{k-1}\right)\\ {\psi }_k& ={\psi }_{k-1}+T{r}_{k-1}\end{array}$ where k denotes the sampling instant and T denotes the sampling time.

The Euler forward form of a dynamic model for heading motion of the AUV is given as 2 $\begin{array}{rl}{v}_k& =v_{k-1}+T\left(\frac{Y_{uv}}{m-Y_{\stackrel{.}{v}}}{u}_{k-1}^{\mathrm{s}}{v}_{k-1}+\frac{Y_{ur}}{m-Y_{\stackrel{.}{v}}}{u}_{k-1}^{\mathrm{s}}{r}_{k-1}\right)\\ & +T\left(\frac{Y_{vv}}{m-Y_{\stackrel{.}{v}}}\left|v_{k-1}\right|v_{k-1}+\frac{Y_{rr}}{m-Y_{\stackrel{.}{v}}}\left|r_{k-1}\right|r_{k-1}\right)\\ r_k& =r_{k-1}+T\left(\frac{N_{ur}}{I_{zz}-N_{\stackrel{.}{r}}}{u}_{k-1}{r}_{k-1}+\frac{N_{uv}}{I_{zz}-N_{\stackrel{.}{r}}}{u}_{k-1}\right)\\ & +T\left(\frac{N_{rr}}{I_{zz}-N_{\stackrel{.}{r}}}\left|r_{k-1}\right|r_{k-1}+\frac{N_{u{\delta }_r}{u}_{k-1}^2}{I_{zz}-N_{\stackrel{.}{r}}}{\delta }_{r_{k-1}}\right)\end{array}$ where ${\delta }_r$ is the input to rudder plane and $Y_{uv}$, $Y_{\stackrel{.}{v}}$, $Y_{ur}$, $Y_{vv}$, $Y_{rr}$, $N_{ur}$, $N_{\stackrel{.}{r}}$, $N_{uv}$, $N_{\stackrel{.}{r}}$, $N_{rr}$, and $N_{u{\delta }_r}$ are the hydrodynamic coefficients and are defined in [21].

The objective of this study is to develop a control law using the reduced order dynamic model (2) such that the AUV tracks the desired way-points using the LoS guidance algorithm. As shown in Fig. 1b, the present waypoint $(x_i^{\mathrm{o}},y_i^{\mathrm{o}})$ and previous way-point $(x_{i-1}^{\mathrm{o}},y_{i-1}^{\mathrm{o}})$ are connected using a rectilinear path. Once, the AUV reaches within the defined acceptance region of a way-point then the next way-point is allotted for tracking based on the following criterion [19]: 3 $\begin{array}{cc}\mathrm{Step}1.& \mathrm{For}i=1,h=\sqrt{{(x_k-x_i^{\mathrm{o}})}^2+{(y_k-y_i^{\mathrm{o}})}^2}\\ \mathrm{Step}2.& \mathrm{if}(h\le \mathrm{acceptance}\mathrm{region})\\ & i=i+1\\ & \mathrm{else}\\ & \mathrm{go}\mathrm{to}\mathrm{Step}1\end{array}$ where $x_i^{\mathrm{o}}$ and $y_i^{\mathrm{o}}$ are the ith way-point positions.

3 Estimation of AUV dynamics using EELM network

Let ${\mathit{z}}_{\mathrm{in}}\in {\mathrm{\Re }}^{m+n}$ be the augmented input vector to the ELM model and is given by ${\mathit{z}}_{\mathrm{in}}=[z_k,u_k]$, where $z_k=[v_k,r_k]$, $u_k={\delta }_{r,k}$, m and n are the number of states and input of the AUV. The non-linear heading dynamic of the AUV (2) is identified using the ELM model as shown in Fig. 2 and is given by 4 ${\hat{z}}_{k+1}=f({\mathit{z}}_{\mathrm{in},k})$ where $f({\mathit{z}}_{\mathrm{in},k})=\sum _{i=1}^{n_{\mathrm{h}}}{\phi }_i\left(w_{j,i}^h{\mathit{z}}_{\mathrm{in}}+b_i\right)w_i={\mathbf{\Phi }}^{\ast }w$ where $\phi$ denotes the hidden layer activation function, ${\mathbf{\Phi }}^{\ast }$ is the output matrix of the hidden layer, $n_{\mathrm{h}}$ is the number of hidden nodes, w indicates the output parameters linking the hidden nodes and the output node, $w^h$ is the input parameters of the ELM model and j varies from 1 to number of samples N, $f\left(\cdot \right)$ denotes the non-linear Lipschitz function, which is continuously differentiable. The off-line identification of the non-linear function $f\left(\cdot \right)$ is modelled using the ELM structure as shown in Fig. 2.

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In the ELM learning algorithm, the hidden layer parameters, i.e. input parameters $w_h$ and biases b are not tuned rather are randomly assigned between [–1 1] [9–11] thus, the training steps of the ELM model involves around finding only the least square solution to output parameters w and is given by solving the following ridge regression-based optimisation problem 5 $\underset{w}{min}\left\{∥{\mathbf{\Phi }}^{\ast }w-{{\mathit{y}}_{\mathrm{o}}∥}^2+\eta \parallel w{\parallel }^2\right\}$ where $\eta$ denotes the regularisation coefficient and ${\mathit{y}}_{\mathrm{o}}$ is the desired output vector.

The penalty term $\eta \parallel w{\parallel }^2$ in (5) is used to penalise the weight estimates, thus increases the generalisation performances of the predicted model. The output optimum weights are obtained by solving (5) and is given as 6 $w^{\ast }={\left({\mathbf{\Phi }}^{\ast \mathrm{T}}{\mathbf{\Phi }}^{\ast }+\eta \mathit{I}\right)}^{-1}{\mathbf{\Phi }}^{\ast \mathrm{T}}{\mathit{y}}_{\mathrm{o}}$ where $\mathit{I}$ is the identity matrix.

3.1 Selection of hidden nodes of the ELM model of the AUV

Selection of a number of hidden nodes for the ELM model is a heuristic approach. However, in this study, the mean square error (MSE) criterion is used for the determination of an optimal number of hidden neurons and is given as 7 $\mathrm{MSE}=\sum _{k=1}^{\hat{N}}\frac{{\left({\hat{z}}_k-z_k\right)}^2}{\hat{N}}$ ${\hat{z}}_k$ and $z_k$ are the predicted value and actual value of the output and $\hat{N}$ is the number of samples. The MSE of the network is obtained for a different number of hidden neurons while training and validating the ELM model. The node where the MSE value is minima during training and validating the ELM model gives the optimal number of hidden neurons for the ELM network.

3.2 EELM model using Jaya algorithm)

The output weights (5) are obtained by randomly assigning the input weights and hidden biases between [–1 1] [9–11]. To increase the efficacy of the network ELM, the EELM concept is introduced where the hidden layer parameters (input weights and hidden biases) are optimised by applying an optimisation technique. Owing to the potency of Jaya algorithm as pointed out in the Introduction section, it is used to optimise the hidden layer parameters, i.e. input weights and hidden biases of the ELM model. The hybridisation of the ELM and Jaya algorithm (EELM) is briefly described in Algorithm 1 (see Fig. 3). Once the network parameters are determined, then the network is trained and validated through the experimental input–output data sets.

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4 Control law formulation

The structure of the proposed controller is shown in Fig. 4. It comprises the kinematic controller and dynamic controller. The purpose of the kinematic controller, adopted from [19], is to generate the desired yaw rate $(r_{\mathrm{d}})$ using the LoS path as shown in Fig. 1b, such that the AUV can track the desired path $(x_{\mathrm{d}}{y}_{\mathrm{d}})$. The desired yaw rate so obtained is then tracked by using the NMPC dynamic controller, which is based on the EELM model. $\hat{r}$ in Fig. 4 represents the model predicted output. The inputs to the identified EELM model (Fig. 4) are given via trapped delay line, which stores EELM inputs temporarily.

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4.1 Design of kinematics controller using the back-stepping approach

Let a rectilinear path connecting two way-points as shown in Fig. 1b is to be followed by an AUV with constant surge velocity. The hyper-plane equation representing the heading cross-track error $(h_{e,k})$, adopted from [19], is given as 8 $h_{e,k}=\mathit{A}{\left[\begin{array}{cc}{x}_k& y_k\end{array}\right]}^{\mathrm{T}}.$ where $\mathit{A}=\left[\begin{array}{cc}a& b\end{array}\right]$ represents the vector of path parameters.

Using expressions $x_k$ and $y_k$ from (1) in (8), $h_{e,k}$ is modified as 9 $h_{e,k}=h_{e,k-1}+T{u}_k^t{a}_1\sin \left({\psi }_{k-1}-{\delta }_h\right),$ where ${\delta }_h={\tan }^{-1}(b/a)$ denotes the slope of the LoS path, $u_k^t=\sqrt{{(u_k^{\mathrm{s}})}^2+v_k^2}$ denotes the net velocity and $a_1=\sqrt{a^2+b^2}$ is the resultant value.

A Lyapunov function $V_k$ of the following form is chosen so as to minimise $h_{e,k}$ 10 $V_k=\frac{1}{2}{h}_{e,k}^2$ Using (9) in (10), one obtains 11 $V_k-V_{k-1}=\frac{1}{2}[h_{e,k}^2-h_{e,k-1}^2]$ Expanding (12), one obtains 12 $\begin{array}{rl}{V}_k-V_{k-1}& =\frac{1}{2}[h_{e,k-1}^2+(T{u}_k^t{a}_1\sin ({\psi }_{k-1}-{\delta }_h)){}^2\\ & +2h_{e,k-1}T{u}_k^t{a}_1\sin ({\psi }_{k-1}-{\delta }_h)-h_{e,k-1}^2]\\ & =\frac{1}{2}(T{u}_k^{t}a{h}_{e,k-1}\sin ({\psi }_{k-1}-{\delta }_h)){}^2\\ & +T{u}_k^{t}a{h}_{e,k-1}\sin \left({\psi }_{k-1}-{\delta }_h\right)\end{array}$ Owing to the slow varying dynamic of the AUV, the first term in the right-hand side of (12) can be neglected by considering $T\ll 1$, so (12) can be rewritten as 13 $V_k-V_{k-1}=T{u}_k^t{a}_1{h}_{e,k-1}\sin \left({\psi }_{k-1}-{\delta }_h\right)$ In order to minimise $V_k$ at every sampling instant, the following condition should be satisfied, i.e. 14 $\begin{array}{rl}& V_k-V_{k-1}\le 0\\ & \Rightarrow u_k^t{a}_1{h}_{e,k-1}\sin \left({\psi }_{k-1}-{\delta }_h\right)\le 0\\ & \Rightarrow u_k^t{a}_1{h}_{e,k-1}\sin \left({\psi }_{d,k}\right)\le 0\end{array}$ where ${\psi }_{d,k}$ is the desired yaw orientation.

Equation (14) is valid if ${\psi }_{d,k}$ is given as 15 ${\psi }_{d,k}=-{\psi }_a\frac{{\mathrm{e}}^{2q_{\delta }{h}_{ek-1}}-1}{{\mathrm{e}}^{2q_{\delta }{h}_{ek-1}}+1}$ where $q_{\delta }$ is a positive constant and ${\psi }_a$ is the desired approaching angle.

Replacing $h_{e,k}$ by ${\psi }_{e,k}$ in (10) and solving (10)–(14) to minimise ${\psi }_{e,k}$ gives the desired yaw rate at the kth instant and is given by 16 $r_{d,k}=\frac{1}{T}\left(\hat{\epsilon }{\psi }_{e,k}+{\psi }_{d,k}-{\psi }_{k-1}+{\delta }_h\right)$ where $\hat{\epsilon }$ is a positive constant $\in (0,1)$, ${\psi }_{d,k}$ is the required approaching angle and ${\psi }_{e,k}$ is the orientation error between the vehicle and the LoS given as 17 ${\psi }_{e,k}={\psi }_k-{\delta }_h-{\psi }_{d,k}$ Using the identified AUV model from (4), a constrained NMPC is designed in the next section for tracking the desired yaw rate ($r_{d,k}$), so that the cross-track error will asymptotically be zero for tracking heading motion.

Owing to the fact that the desired LoS path is independent of any time constraint and the surge velocity does not affect the tracking performance of the vehicle, the surge velocity $u^{\mathrm{s}}$ is considered to be constant in heading dynamics of the AUV.

4.2 Design of dynamic controller using NMPC

With a given reference trajectory $(r_k)$, it is intended to determine a predictive optimal law such that the predicted output will follow the reference trajectory. To minimise the error between the reference trajectory and the predicted values of the EELM model (${\hat{y}}_k$) over the prediction horizon, a cost function is defined as follows: $\underset{\mathrm{\Delta }{u}_k}{min}J=\sum _{j=1}^{N_p}{e}_{k+j|k}^{\mathrm{T}}{W}_e{e}_{k+j|k}+\sum _{j=0}^{N_u-1}\mathrm{\Delta }{u}_{k+j|k}^{\mathrm{T}}{W}_{\mathrm{\Delta }u}\mathrm{\Delta }{u}_{k+j|k}$ subjected to 18 $\begin{array}{rl}{u}_{min}& \le u_{k+j|k}\le u_{max}\\ \mathrm{\Delta }{u}_{min}& \le \mathrm{\Delta }{u}_{k+j|k}\le \mathrm{\Delta }{u}_{max}\\ {\hat{y}}_{min}& \le {\hat{y}}_{k+j|k}\le {\hat{y}}_{max}\end{array}$ where $e_{k+j|k}=r_{k+j|k}-{\hat{y}}_{k+j|k}$ $\mathrm{\Delta }{u}_{k+j|k}=u_{k+j|k}-u_{k+j-1|k}$

The output of EELM network (4) introduces non-linearities. Solving non-linear optimisation for (18) may require a lot of time to identify whether the problem has a global solution or not and generally leads to a non-convex optimisation problem.

With Remark 2, (18) is reformulated as a convex problem by linearising the non-linear EELM model around the given operating point $(z_k^{\mathrm{o}},u_k^{\mathrm{o}})$. The linear parameters A and B vectors are thus extracted at every sampling instant by finding the Jacobian $\mathrm{\partial }Y/\mathrm{\partial }{x}_{\mathrm{in}}$ around the current states and current input and is given as 19 $\frac{\mathrm{\partial }Y}{\mathrm{\partial }{x}_{\mathrm{in}}}=\left[\mathit{A}\left|\mathit{B}\right.\right]$ Thus, (4) can now be presented as 20 ${\hat{z}}_{k+1}=\mathit{A}{\hat{z}}_k+\mathit{B}{u}_k+d_{1,k}$ 21 $y_k=C{\hat{z}}_k$ where $d_{1,k}=f(z_k^{\mathrm{o}},u_k^{\mathrm{o}})-\mathit{A}{z}_k^{\mathrm{o}}-\mathit{B}{u}_k^{\mathrm{o}}+\mathrm{higher}\mathrm{order}\mathrm{terms}$.

In our case, C is taken as $[01]{}^T$ as the output is yaw rate only. For finite horizon problem [37], (18) is written as 22 $\underset{\mathrm{\Delta }{U}_k}{min}\left(\sum _{j=1}^{N_p}{E}_k^{\mathrm{T}}{W}_E{E}_k+\sum _{j=0}^{N_u-1}\mathrm{\Delta }{U}_k^{\mathrm{T}}{W}_{\mathrm{\Delta }U}\mathrm{\Delta }{U}_k\right)$ where $E_k=R_k-Y_k$ $R_k=[r_{k+1|k}{r}_{k+2|k}\cdots r_{k+N_p|k}]{}^{\mathrm{T}}$ ${\hat{Y}}_k=[{\hat{y}}_{k+1|k}{\hat{y}}_{k+2|k}\cdots {\hat{y}}_{k+N_p|k}]{}^{\mathrm{T}},$ $\mathrm{\Delta }{U}_k=[\mathrm{\Delta }{U}_{k|k}\mathrm{\Delta }{U}_{k+1|k}\mathrm{\Delta }{U}_{k+2|k}\cdots \mathrm{\Delta }{U}_{k+N_u-1|k}]{}^{\mathrm{T}},$ $W_E\in {\mathrm{\Re }}^{N_p\times N_p}\mathrm{and}{W}_{\mathrm{\Delta }U}\in {\mathrm{\Re }}^{N_{u}n\times N_{u}n}$ The optimisation problem defined in (22) is a quadratic programming problem [37, 38] and is thus can be written in its equivalent form as follows: 23 $\begin{array}{rl}& \underset{\mathrm{\Delta }{U}_k}{min}\left(\frac{1}{2}\mathrm{\Delta }{U}_k^{\mathrm{T}}{\mathit{Q}}_1\mathrm{\Delta }{U}_k+Q_2\mathrm{\Delta }{U}_k\right)\\ & \mathrm{subjected}\mathrm{to}:\\ & H_k\mathrm{\Delta }{U}_k-G_k\le 0\end{array}$ where ${\mathit{Q}}_1=2\left[{\mathrm{\Gamma }}_{\mathrm{\Delta }U}^{\mathrm{T}}{W}_E{\mathrm{\Gamma }}_{\mathrm{\Delta }U}+W_{\mathrm{\Delta }U}\right],$ $Q_2=-2{\mathrm{\Gamma }}_{\mathrm{\Delta }U}^{\mathrm{T}}{W}_E\left[R_k-{\varphi }_k{z}_k-{\mathrm{\Gamma }}_u{u}_{k-1}-D_{1,k}{d}_{1,k}\right],$ ${\mathit{H}}_k={\left[\begin{array}{cccccc}{I}_{N_{u}m}& -I_{N_{u}m}& \mathrm{{\rm Y}}& -\mathrm{{\rm Y}}& {\mathrm{\Gamma }}_{\mathrm{\Delta }U}& -{\mathrm{\Gamma }}_{\mathrm{\Delta }U}\end{array}\right]}^{\mathrm{T}},$ ${\mathit{G}}_k={\left[\begin{array}{cccccc}\mathrm{\Delta }{U}_{max}& -\mathrm{\Delta }{U}_{max}& g_{1,k}& -g_{1,k}& g_{1,k}& -g_{1,k}\end{array}\right]}^{\mathrm{T}},$ $g_{1,k}=U_{max}-U_{k-1}$ $g_{2,k}={\hat{Y}}_{max}-{\hat{Y}}_k+{\mathrm{\Gamma }}_{\mathrm{\Delta }U}\mathrm{\Delta }{U}_k,$ $\begin{array}{rl}{\mathbf{\Gamma }}_u& ={\left[\begin{array}{ccc}CB& \cdots & CB+C{A}^{N_y-1}B\end{array}\right]}^{\mathrm{T}},\\ \mathit{\varphi }& ={\left[\begin{array}{ccc}CA& \cdots & C{A}^{N_y}\end{array}\right]}^{\mathrm{T}},\end{array}$ ${\mathit{D}}_{1,k}={\left[\begin{array}{ccc}C& \cdots & C+C{A}^{N_y-1}\end{array}\right]}^{\mathrm{T}}$ ${\mathbf{\Gamma }}_{\mathrm{\Delta }U}=\left[\begin{array}{ccc}CB& \cdots & 0\\ ⋮& ⋮& ⋮\\ CB+\cdots +C{A}^{N_u-1}B& \cdots & CB\\ ⋮& ⋮& ⋮\\ CB+\cdots +C{A}^{N_y-1}B& \cdots & CB+\cdots +C{A}^{N_y-N_u}B\end{array}\right]$

The objective function given by (23) is strictly convex owing to the fact that ${\mathit{Q}}_1$ is a symmetric positive definite matrix.

To solve the above optimisation problem effectively, (23) is formulated in terms of the Lagrangian dual problem form [38] and is given as 24 $\begin{array}{rl}& \underset{\alpha }{inf}\left(\frac{1}{2}\mathrm{\Delta }{U}_k^T{\mathit{Q}}_1\mathrm{\Delta }{U}_k+Q_2\mathrm{\Delta }{U}_k+\alpha \left(H_k\mathrm{\Delta }{U}_k-G_k\right)\right)\\ & \mathrm{subjected}\mathrm{to}:\alpha \ge 0\end{array}$ The objective function in (24) is strictly convex for given $\mathrm{\Delta }U$. Thus, the minimum value is obtained by finding the derivation of (24) with respect to $\mathrm{\Delta }U$ and is given by 25 ${\mathit{Q}}_1\mathrm{\Delta }{U}_k+Q_2+H^{\mathrm{T}}\alpha =0$ A unique solution to (25) is thus given as 26 $\mathrm{\Delta }{U}_k=-{\mathit{Q}}_1^{-1}(Q_2+H^{\mathrm{T}}\alpha )$ Using the expression $\mathrm{\Delta }U$ from (26) in (24), one obtains 27 $\underset{\alpha }{max}\left(\frac{1}{2}{\alpha }^{\mathrm{T}}{\lambda }_1\alpha +\alpha {\lambda }_2-\frac{1}{2}{Q}_2^{\mathrm{T}}{\mathit{Q}}_1^{-1}{Q}_2\right)$ 28 $\mathrm{subjected}\mathrm{to}:\alpha \ge 0$ where ${\lambda }_1=-H_k^{\mathrm{T}}{\mathit{Q}}_1^{-1}{H}_k$ and ${\lambda }_2=-G_k-H_k^{\mathrm{T}}{\mathit{Q}}_1{Q}_2$.

The optimal solution of $\alpha$ that satisfies (27) can be obtained using the steepest ascent method [37] and is given by the following adaptive law 29 ${\alpha }_{k+1}={\alpha }_k+\eta ({\lambda }_1{\alpha }_k+{\lambda }_2)$ where $\eta$ is the corresponding step size.

In order to satisfy (28), (29) is modified by using the max function in MATLAB and is given as 30 ${\alpha }_{k+1}=max({\alpha }_{k+1},0)$ Thus, the optimal control move is given by direct substituting of (29) into (26).

5 Results and discussion

5.1 Simulation results

This section presents the implementation of the proposed EELM network-based NMPC for heading tracking of the AUV and then it is extended for way-point tracking of the AUV using the LoS guidance to facilitate point-to-point navigation, which is the basis of any patrolling mission. The performance of the proposed control algorithm has been verified by conducting simulations in MATLAB environment using the parameters of INFANTE AUV [39]. The reference trajectory is taken to be constant during the prediction horizon. All the initial states of the AUV $[x,y,\psi ,v,r]$ are set to zero. $N_p$ and $N_u$ were selected as 5 and 2.

5.2 EELM network parameters

The input weights and hidden biases were initially taken randomly between $\left[-11\right]$. A hyperbolic tangent is considered as the activation function. Using the parameters defined in [39], input–output data sets are collected by simulation. The sampling time is taken as 0.5 s. During the training and validating of the ELM model, it is observed that the MSE during training and validation remains almost constant after the sixth hidden node. Based on the MSE values, the optimal number of hidden nodes is thus chosen as 6 in the ELM model. The hidden layer parameters of the ELM model are then optimised through Jaya optimisation algorithm by setting 100 as the maximum iteration, which is also the termination criterion. The number of design variables are set equal to the number of hidden layer parameters. The number of potential candidates used in Jaya algorithm to find global optima was considered as 6.

5.2.1 Heading controller

The proposed controller is implemented using the predicted EELM model of AUV. The desired approaching angle is set as 0.45 radian. $W_E$ was chosen to be $100I_{5\times 5}$ while $W_{\mathrm{\Delta }U}$ was kept to be as $10I_{6\times 6}$. For the implementation of the heading controller, the desired heading trajectory is in accordance to (31) and the surge velocity is kept constant, i.e. 1 m/s. The control input moves are subjected to the constraints given in (32) 31 ${\psi }_d=\left\{\begin{array}{rl}& 0,0\le t<50\\ & 1.57,50\le t<100\\ & 0,100\le t<150\\ & -1.57,150\le t<200\end{array}\right.$ To ensure the efficacy of the proposed controller in heading tracking, its performance is compared with that of the recently developed advanced controller algorithms, i.e. $H_{\mathrm{\infty }}$ state feedback controller [30] and inverse optimal self-tuning PID controller [32]. In Figs. 5a and c, it is observed that the proposed controller tracks the reference trajectory with less settling time as compared to the controllers in [30, 32].However, using the inverse optimal self-tuning PID controller [32], the heading trajectory reaches the reference trajectory faster than the proposed controller and $H_{\mathrm{\infty }}$ state feedback controller [30], but the output in [32] exhibits overshoots with oscillations before tracking the desired heading trajectory (Fig. 5a), which is not permissible for AUV application. Moreover, it is also seen from Fig. 5b that the control signal in the proposed controller lies well within the actuator (rudder plane) saturation region $\pm 45\text{°}(\pm 0.7856)$ radian, whereas the control signals in [30, 32] cross the saturation region and require very high input values, i.e. $\pm 9.62$ radian [32] and $\pm 1.53$ radian [30], which is not permissible in AUV applications and may cause actuator failure.

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In such a scenario, hard-limiters are generally used for these controllers [30, 32], which clips the control signal if it exceeds the actuator saturation limits as shown in Fig. 6b. However, due to the clipping of the control signals in [30, 32], it is observed from Fig. 6a that the tracking performances of the controller [30, 32] deteriorate further in terms of settling time as compared to the proposed controller. On comparing the error plot with and without hard-limiter, i.e. Figs. 5c and 6c, it is shown that in the presence of a hard-limiter for the controllers in [30, 32], the cross-track heading error using the proposed controller approaches zero faster as compared to [30, 32]. Hence, it may be concluded that the proposed controller not only resolves the problem of handling the constraints successfully but also drives the AUV to follow the desired heading reference with improved performance as compared to the controllers in [30, 32]. 32 $\begin{array}{rl}& -0.7856\le u(k+j\left|k\right.)\le 0.7856\\ & -2\text{\%}\le \mathrm{\Delta }U\le 2\text{\%}\mathrm{of}\mathrm{the}\mathrm{full}\mathrm{input}\mathrm{range}\\ & -0.5<\hat{Y}<0.5\end{array}$

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5.2.2 Robustness of the controller

In order to evaluate the robustness of the controller to disturbances, the proposed controller is reformulated to include disturbance that may arise from ocean waves or ocean currents. The disturbance regulated with zero mean white noise with a variance of 0.011 and is injected at the output. From Fig. 7a, it is evident that the proposed controller using the EELM network is efficient in tracking the desired heading orientation despite disturbances.Furthermore, from Fig. 7b, it is seen that the corresponding control input lies well within the actuator saturation, thus making it suitable for way-point tracking.

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5.2.3 Way-point tracking using LoS guidance law

A desired spine path defined in (33) is generated apriori using waypoints (0, 0), (245, 32), (420, 135), and (650, 195). Using NMPC, these waypoints are tracked as shown in Fig. 8a with a radius of the acceptance region set as 6 m for each waypoint by following the desired yaw rate (Fig. 8b).The corresponding control input is shown in Fig. 8c 33 $\begin{array}{rl}& x\left(k\right)=26k\\ & y\left(k\right)=2000\tanh \left(2\times {10}^{-3}{k}^3\right)\end{array}$

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5.2.4 Application of AUV for surveillance purpose

A surveillance application of an AUV is considered where an AUV is deployed to survey a given region represented through way-points, not known priori. Assuming the desired way-points are known and are given as (60, 2), (160, 105), (265, 5), (168, −110) and (270, −210). The radius of the acceptance region for each way-point is set as 6 m. Once AUV reaches within the acceptance region of a way-point, then a next way-point is assigned as given in the steps discussed in (3). Fig. 9a shows the way-point tracking performance of the proposed NMPC controller and Fig. 9b shows the control signal profile generated by applying the proposed control law.

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5.3 Experimental results

From the simulation results, it is confirmed that the system identification technique and the developed control algorithm is suitable for controlling the AUV. Thus the effectiveness of the developed control algorithm is verified in an experimental environment. The prototype AUV, a torpedo-shaped mono-hull structured consists of a single board computer along with Arduino microcontroller. The single board computer is installed with Ubuntu 14.04 Linux operating system and robot operating system (ROS) [40]. The Arduino microcontroller is used to generate the actuation signals for the AUV. The ROS nodes of the identification and control algorithm accept the data from the AUV sensor nodes and generate the control signal to the AUV actuation nodes. The experimentation is conducted in the institute swimming pool with a dimension of $10\mathrm{m}\times 5\mathrm{m}\times 1.6\mathrm{m}$.

5.4 Optimisation of EELM network parameters

Input–output data sets are collected experimentally by randomly giving the inputs to prototype AUV with a sampling time of 0.1 s. During the training and validating of the ELM model, it is observed from Fig. 10a that the MSE during the training and validation remains almost constant after the third hidden node.Based on the MSE values, the optimal number of hidden nodes is thus chosen as 3 in the ELM model. The hidden layer parameters of the ELM model are then optimised through Jaya optimisation algorithm by setting 30 as the maximum iteration. The number of potential candidates used in Jaya algorithm is set as 6. Jaya algorithm is compared with conventional particle swarm optimisation (PSO) algorithm and it is seen from Fig. 10b that the Jaya algorithm converges faster with less iteration as compared to convention PSO algorithm. Thus, shows the efficacy of the Jaya algorithm in estimating optimal hidden layer parameters of the ELM model. With optimised hidden layer parameters, the output weights connecting the hidden layer and the output were found analytically using (5). A comparison of the experimental model output with that of the measured output for the validation data set is presented in Fig. 11a and the corresponding prediction error is shown in Fig. 11b. From the autocorrelation analysis (Fig. 11c), it is observed that all the lags are within the confidence interval. Thus, it is envisaged that the identified EELM model can predict the non-linear heading dynamic of the AUV accurately and can be used for controlling the AUV.

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5.4.1 Tracking of yaw orientation

For the implementation of the heading controller, the desired yaw orientation is in accordance with (34) and the surge velocity is kept constant, i.e. 1 m/s. The initial states of the AUV $[v,r,\psi ]$ are set to zero. The proposed controller is implemented using the one step ahead predicted EELM model of the AUV. $W_E$ was chosen as $12I$ while $W_{\mathrm{\Delta }U}$ was selected as $I_{3\times 3}$. The control input moves are subject to the constraints given in (32) 34 ${\psi }_d=\left\{\begin{array}{rl}& 0,0\le t<5\\ & 1.57,5\le t<20\\ & 0,20\le t<60\end{array}\right.$ It is observed from Fig. 12 that the proposed controller tracks the reference heading trajectory as shown in Fig. 12a successfully by tracking the desired yaw rate (Fig. 13a) given by the kinematic controller.The settling time is nearly equal to 6 s with steady-state error to be nearly equal to 3% as seen in Figs. 12a and b. Moreover, it is also seen in Fig. 13b that the control signal in the proposed controller lies well within the actuator (rudder plane) saturation region $\pm 45\text{°}(\pm 0.7856)$ radian. The corresponding predicted output, i.e. the yaw rate is also well within the saturation limits $(\pm 0.5)$ (Fig. 13a).

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6 Conclusion

We presented the design of a new control scheme of an AUV by identifying its dynamic with an EELM model structure. The hidden layer parameters of the neural network model are optimised by using Jaya algorithm. A back-stepping approach is employed to design the kinematic controller followed by the design of a model predictive dynamic controller. The proposed controller is developed for trajectory tracking by considering the disturbances and the constraint on the rudder planes. Using the LoS path, the proposed controller was then used for way-point tracking. The proposed control scheme is simulated in MATLAB first and then its real-time implementation was pursued on a prototype AUV in the laboratory. From the obtained simulation and experimental results, it is observed that the proposed controller not only predicts the heading dynamics of the AUV accurately but also keeps the control input within the actuator saturation limits. Moreover, from the comparison of the proposed controller with that of recently developed algorithms namely, $H_{\mathrm{\infty }}$ state feedback controller and inverse optimal self-tuning PID controller, it is observed that the tracking performance of the proposed controller is satisfactory whilst the actuator constraints are taken care of.

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