Robust backstepping ship autopilot design

ABSTRACT

A ship is always under the influence of matched as well mismatched uncertainties and external disturbances owing to its operational environment and, therefore, designing an autopilot that offers uniform performance in the presence of uncertainties and disturbances poses a challenging task. In this work, to address the issue, control design based on adaptive backstepping robustified by Uncertainty and Disturbance Estimator (UDE) is proposed. The design does not need knowledge of bounds of uncertainties and disturbances and is able to effectively deal with the uncertainties and disturbances. Simulation results show that the performance of the backstepping autopilot is enhanced by combining the backstepping design with the UDE approach.

1. Introduction

Marine surface vessels operate in highly uncertain environment that exists at sea and, therefore, effective autopilot on board that performs satisfactorily is of utmost importance as it helps the economy by reducing fuel consumption and manpower, improving safety and security, better machinery maintenance and comfort to the crew. Designing an efficient controller, however, is a challenging task owing to the ship's nonlinear dynamics, parametric variations, external disturbances due to waves, wind and underwater currents, varied sailing conditions, model mismatches, inaccuracies etc. The factors such as input and rate limitation constraints complicate the issue further.

Due to the advancements in automation and microprocessor-based modern control strategies, the urge to improve manoeuvrability performance of ships has caught attention of researchers, resulting in many significant and noteworthy results (McGookin et al. 2000; Sheng et al. 2006; Rigatos and Tzafestas 2006; Fossen and Perez 2009; Tao and Jin 2012; Li and Sun 2012; Lee et al. 2013; Xia et al. 2014; Yao et al. 2014; Das and Talole 2016a, 2016b; Das and Talole 2018). Backstepping, a well-known recursive methodology, represents one viable technique that can be used for the design of the ship autopilot controller (Fossen and Strand 1999b; Marquez 2003; Zhou and Wen 2008). The approach has received a lot of attention during the last few decades and the application of the technique in diverse areas including in the field of the marine vessel autopilot design have appeared in the literature. For example, in Witkowska and Smierzchalski (2009), nonlinear control structure obtained in the backstepping method is optimised by means of the genetic algorithm. Adaptive filtering backstepping for the ship steering problem is demonstrated in Xia et al. (2014) and its use in the presence of input saturation due to steering gear is presented in Witkowska et al. (2007) and Li et al. (2011). Adaptive filtering Backstepping Neural Network (ABNN) for uncertain nonlinear system is implemented for the autopilot design in order to solve the stability problem, convergence error and robustness in Meziou et al. (2011). In Fossen and Grovlen (1998), observer backstepping is used for designing nonlinear output feedback control for dynamic positioning of ship. In Do et al. (2002), the development of controller based on Lyapunov's direct method and the backstepping technique is demonstrated.

Numerous research can be cited over the last decade wherein conventional backstepping controllers have been combined with some other approaches. These techniques are aimed towards achieving better performance of the controller. The performance of the conventional backstepping controllers in the presence of uncertainty and disturbances has been found to improve by augmenting it with some appropriate method. One such example of a similar approach as mentioned above is the augmenting backstepping controller with the help of Uncertainty and Disturbances estimation. In Hwang and Tahk (2006), a study on the integrated backstepping design of missile guidance and control was demonstrated by combining it with the robust disturbance observer. Similarly, in Ji et al. (2012), a study on robust adaptive backstepping synchronisation for a class of uncertain chaotic systems was implemented by combining with the fuzzy disturbance observer. Extended State Observer (ESO) was used in conjunction with backstepping for nonlinear robust control of hydraulic systems in Ji et al. (2012). Control of the nonlinear system with multiple mismatched disturbances by using the composite technique of the backstepping controller with adaptive disturbance observer was used in Suna and Guoa (2014). In all of these attempts, it has been demonstrated in their respective work that the backstepping controller is robustified by integrating it with a suitable methodology.

Following a similar approach, in Das (2015), an Uncertainty and Disturbance Estimation UDE) (Zhong and Rees 2004) approach is used in conjunction with the backstepping controller to improve the performance of the backstepping controller designed for ship steering autopilot. In their work, the authors have shown that the UDE technique is quite effective in tackling the problem of uncertainties and disturbances without having any specific information about the same. The work, however, considers the ship model with servo dynamics in the phase variable form making the uncertainties and disturbances as matched one. In this work, the issue is addressed by using a more practical model wherein the disturbances are mismatched. For the mismatched dynamics, an adaptive backstepping controller is designed and the UDE technique is used to robustify the same. Comparison of the backstepping controller is carried out with the UDE augmented backstepping controller and it is shown that the UDE approach results in noticeable improved performance. The remaining paper is organised as follows: in Section 2, the mathematical model including the rudder steering system are presented. The design of the nonlinear controller based on e stepping approach is presented in Section 3 followed by the construction of a UDE-based backstepping controller in Section 4. Closed-loop stability analysis is presented in Section 5 whereas simulation results are presented in Section 6. Lastly, Section 7 concludes this work.

2. Mathematical model

The dynamic characteristic describing the ship's motion is given by Nomoto's first-order model as (Nomoto et al. 1957) (1) $\frac{r\left(s\right)}{\delta \left(s\right)}=\frac{K}{Ts+1}$(1) where r is the yaw rate, δ is the actual rudder deflection, and T and K are the time constant and gain parameters, respectively. Noting $r=\dot{\psi }$, the corresponding transfer function relating to the yaw angle, ψ, to rudder movement is given as (2) $\frac{\psi \left(s\right)}{\delta \left(s\right)}=\frac{K}{s(Ts+1)}$(2) which when written in the form of differential equation gives (3) $T\ddot{\psi }+\dot{\psi }=K\delta$(3) The model of Nomoto (3(3) $T\ddot{\psi }+\dot{\psi }=K\delta$(3) ) is suitable for a small rudder angle and does not account for nonlinearity. A nonlinear model as proposed by Norbin (H. 1963; Amerongen 1984) addressing the effect of course instability as well as a large rudder angle is widely used in the literature. Accordingly, the nominal model (3(3) $T\ddot{\psi }+\dot{\psi }=K\delta$(3) ) is modified wherein the heading angle rate, $\dot{\psi }$, is given by a nonlinear function $H\left(\dot{\psi }\right)$. Hence, the modified nonlinear model takes the form as (4) $T\ddot{\psi }+H\left(\dot{\psi }\right)=K\delta$(4) where $H\left(\dot{\psi }\right)$ describes the nonlinear manoeuvring characteristic produced by Norbin's reverse manoeuvre and the shape of the curve of $H\left(\dot{\psi }\right)$ is found by the spiral test. The expression is given by the following function (Amerongen 1982; Witkowska et al. 2007): (5) $H\left(\dot{\psi }\right)=b_3{\dot{\psi }}^3\left(t\right)+b_2{\dot{\psi }}^2\left(t\right)+b_1\dot{\psi }\left(t\right)+b_0$(5) For a course stable ship, the value $b_1=1$ and for the course unstable ship $b_1$ takes the value of $-1$. The quantity $b_0$ is considered as an additional rudder offset and becomes non-zero only when the ship's hull is asymmetric or the ship is propelled by a single screw propeller. For the ship considered in this work, $b_0=0$. On a similar line, symmetry in the hull is considered for almost all the ship design and implies $b_2=0$. Therefore, $H\left(\dot{\psi }\right)$ can be expressed as (6) $H\left(\dot{\psi }\right)=b_3{\dot{\psi }}^3\left(t\right)+b_1\dot{\psi }\left(t\right)$(6) The movement of the rudder is controlled by the steering machine and the rate at which the rudder moves depends mainly on the speed of the steering pump and the opening of its discharge valve. The hydraulic fluid flow is controlled by the movement of the swash plate, governed by the telemotor system. Due to this limitation of the hydraulic steering pump, the rudder remains as a low bandwidth actuation device. In addition to rate saturation, it also has magnitude limitation. Attributable to these hard nonlinearities like saturation (SAT) limit and slew rate limit (SRL), rudder dynamics should be integrated in the controller design. In few of the earlier works (Cheng 2002, 2004; Lei and Guo 2015), the effect of steering dynamics is considered as time delay. In this work, the servo dynamics has been considered in the plant model used in controller design. The dynamics is given as (Amerongen 1984) (7) $\delta \left(s\right)=sat\left(\frac{1}{s{\tau }_{\delta }+1}\right){\delta }_c\left(s\right)$(7) where ${\delta }_c$ and ${\tau }_{\delta }$ represent the commanded rudder angle and steering machine time constant, respectively, whereas, sat(·)=sign(·)min(${\delta }_{max}$, $|\cdot |$). The typical values of saturation and slew rate limitations are ${\delta }_{max}={35}^0$ and ${\dot{\delta }}_{max}=2-7$ (deg/s), respectively. Since the gain of the transfer function as shown in rudder dynamics is of unity value, it follows that if ${\delta }_c\le {\delta }_{max}$, $t\ge 0$ then $\delta \le {\delta }_{max}$, $t\ge 0$. In view of this limitation, stalling of rudder, large control gain cannot solve the stability problem of a vessel due to inadequate rudder (Triantafyllou and Hover 2003). Accordingly, assuming that ${\delta }_c\le {\delta }_{max}$, rudder dynamics can be treated linear as (8) $\frac{\delta \left(s\right)}{{\delta }_c\left(s\right)}=H_a\left(s\right)=\left(\frac{1}{s{\tau }_{\delta }+1}\right)$(8) Defining the state variables as $x_1=\psi$, $x_2=\dot{\psi }$, $x_3=\delta$ and input $u={\delta }_c$, the plant model is obtained as (9) $\begin{array}{rl}{\dot{x}}_1& =x_2\\ {\dot{x}}_2& =-\omega x_2+K\omega x_3+d_1\\ {\dot{x}}_3& =-{\omega }_a{x}_3+{\omega }_{a}u+d_2\end{array}$(9) where $\omega =1/T$ and ${\omega }_a=1/T_a$ are the bandwidths corresponding to the ship and actuator, respectively. Further, $d_1$ represents the effect of parametric uncertainties, external disturbances, plant nonlinearities, and model inaccuracies, whereas $d_2$ represents the effect of uncertainties and disturbances appearing in the plant through the input channel. It is worth to note that the disturbance $d_1$ is mismatched.

3. Backstepping controller

The objective of good autopilot controller design for marine vessel lies in its ability to maintain and change the ship's course by controlling the movement of the rudder without compromising mechanical limitations imposed by the actuating mechanism in the presence of all the environmental disturbances (Triantafyllou and Hover 2003). The backstepping control strategy represents one of the most researched techniques that can be used to deal with mismatched uncertainties. Hence, its usage in the design of the ship autopilot controller has been explored numerous times facilitated by combining the backstepping controller with other design strategies. Backstepping is a recursive methodology which designs the controller in a systematic stepwise manner by breaking the actual control into smaller parts called virtual controls. It breaks down the problem into a number of lower-order control problems and a control law is recursively constructed. The conventional backstepping method involves the use of a positive definite function to construct the virtual and actual controls. In this work, the adaptive backstepping (ABS) design is employed wherein the control laws are recursively designed using input–output linearisation (IOL) theory (Slotine and Li 1991).

3.1. Controller design

The controller is designed using the nominal plant of (9(9) $\begin{array}{rl}{\dot{x}}_1& =x_2\\ {\dot{x}}_2& =-\omega x_2+K\omega x_3+d_1\\ {\dot{x}}_3& =-{\omega }_a{x}_3+{\omega }_{a}u+d_2\end{array}$(9) ); however, in simulations, the actual plant composed of (4(4) $T\ddot{\psi }+H\left(\dot{\psi }\right)=K\delta$(4) ) and (7(7) $\delta \left(s\right)=sat\left(\frac{1}{s{\tau }_{\delta }+1}\right){\delta }_c\left(s\right)$(7) ) with all the limitations imposed by the actuator is used. As mentioned in the previous section, the controller is an IOL-based ABS controller, i.e. it amalgamates the concept of the IOL-based feedback law in designing each step of the control law design of ABS as described below.

1. Considering the outermost loop variable $x_1$, let the error variable pertaining to this loop be $z_1=x_1-x_{1d}$ where $x_{1d}$ is the desired trajectory. The virtual control in order to make this loop stable is obtained as (10) ${\overline{x}}_2=-k_1{z}_1+{\dot{x}}_{1d}$(10)

2. Now considering the loop corresponding to variable $x_2$, the error variable is defined as $z_2=x_2-{\overline{x}}_2$ and the virtual control ${\overline{x}}_3$ is designed as (11) ${\overline{x}}_3=\frac{1}{k\omega }[\omega x_2-k_2{z}_2]+{\dot{\overline{x}}}_2$(11)

3. Lastly, arriving at the innermost loop, the error variable corresponding to this loop is $z_3=x_3-{\overline{x}}_3$. The actual control is calculated as (12) $u=\frac{1}{{\omega }_a}[{\omega }_a{x}_3-k_3{z}_3+{\dot{\overline{x}}}_3]$(12)

where, $k_1$, $k_2$, and $k_3$ are the controller gains. Upon implementing the designed controller, the closed-loop system becomes (13) $\begin{array}{rl}{\dot{z}}_1& =-k_1{z}_1+z_2\\ {\dot{z}}_2& =-k_2{z}_2+K\omega z_3\\ {\dot{z}}_3& =-k_3{z}_3\end{array}$(13)

4. UDE-based backstepping

As stated earlier, the backstepping design is augmented by using the UDE technique for designing robust control law for ship autopilot. The UDE enables the estimation of the effect of uncertainties and disturbances acting on the system and the estimate is used to augment the backstepping controller designed for a nominal system.

Concept of UDE: Consider a first-order dynamical system as (14) $\dot{x}\left(t\right)=f(x,t)+\mathrm{\Delta }f(x,t)+d^{\prime }\left(t\right)+u\left(t\right)$(14) where $\mathrm{\Delta }f(x,t,u(t\left)\right)$ represents the parametric uncertainty, $u\left(t\right)$ is the control input and $d^{\prime }\left(t\right)$ represents the unknown external disturbance acting on the system. Defining the total disturbance as $d(x,t)=\mathrm{\Delta }f(x,t)+d^{\prime }\left(t\right)$, the control law is designed as $u_c\left(t\right)=v_f(x,t)-\hat{d}(x,t)$, where $\hat{d}(x,t)$ is the estimate of d. By following the UDE theory (Zhong and Rees 2004; Kuperman et al. 2010), the disturbance is estimated as (15) $\hat{d}\left(t\right)=G_f\left(s\right)d\left(t\right)=G_f\left(s\right)\left[\dot{x}\right(t)-f(x,t\left)\right]$(15) where the term $v_f(x,t)$ represents control law designed for the nominal system, i.e. for the plant without any uncertainties or disturbances and $G_f\left(s\right)$ is an appropriately chosen low pass filter of steady state gain of unity and wide enough bandwidth. In this work, the low pass filter as $G_f\left(s\right)=1/1+s\tau$ with time constant of τ is used for disturbance estimation.

4.1. UDE-based controller

In order to design the controller using UDE, the steps are similar to those discussed in Section 3.1 except that the disturbance estimate terms, ${\hat{d}}_i$, are taken into consideration. This leads to the following control strategy development (16) $\begin{array}{rl}{\overline{x}}_2& =-k_1{z}_1+{\dot{x}}_{1d}\\ {\overline{x}}_3& =\frac{1}{K\omega }[\omega x_2+{\dot{\overline{x}}}_2-k_2{z}_2-{\hat{d}}_1]\\ u& =\frac{1}{{\omega }_a}[{\omega }_a{x}_3+{\dot{\overline{x}}}_3-k_3{z}_3-{\hat{d}}_2]\end{array}$(16) where ${\hat{d}}_1$ and ${\hat{d}}_2$ are the estimates of the actual disturbances $d_1$ and $d_2$, respectively, appearing in the plant dynamics (9(9) $\begin{array}{rl}{\dot{x}}_1& =x_2\\ {\dot{x}}_2& =-\omega x_2+K\omega x_3+d_1\\ {\dot{x}}_3& =-{\omega }_a{x}_3+{\omega }_{a}u+d_2\end{array}$(9) ). The resulting closed-loop dynamics can be obtained as (17) $\begin{array}{rl}{\dot{z}}_1& =-k_1{z}_1+z_2\\ {\dot{z}}_2& =-k_2{z}_2+K\omega z_3+(d_1-{\hat{d}}_1)\\ {\dot{z}}_3& =-k_3{z}_3+(d_2-{\hat{d}}_2)\end{array}$(17) Now using (15(15) $\hat{d}\left(t\right)=G_f\left(s\right)d\left(t\right)=G_f\left(s\right)\left[\dot{x}\right(t)-f(x,t\left)\right]$(15) ), (16(16) $\begin{array}{rl}{\overline{x}}_2& =-k_1{z}_1+{\dot{x}}_{1d}\\ {\overline{x}}_3& =\frac{1}{K\omega }[\omega x_2+{\dot{\overline{x}}}_2-k_2{z}_2-{\hat{d}}_1]\\ u& =\frac{1}{{\omega }_a}[{\omega }_a{x}_3+{\dot{\overline{x}}}_3-k_3{z}_3-{\hat{d}}_2]\end{array}$(16) ) and (17(17) $\begin{array}{rl}{\dot{z}}_1& =-k_1{z}_1+z_2\\ {\dot{z}}_2& =-k_2{z}_2+K\omega z_3+(d_1-{\hat{d}}_1)\\ {\dot{z}}_3& =-k_3{z}_3+(d_2-{\hat{d}}_2)\end{array}$(17) ), the disturbance estimates are obtained as (18) $\begin{array}{rl}{\hat{d}}_1& =\frac{1}{\tau }[z_2+k_2{\int }_0^t{z}_2\mathrm{d}t-K\omega {\int }_0^t{z}_3\mathrm{d}t]\\ {\hat{d}}_2& =\frac{1}{\tau }[z_3+k_3{\int }_0^t{z}_3\mathrm{d}t]\end{array}$(18)

The schematic diagram of the UDE based controller is shown in Figure 1

Figure 1. Schematic block diagram of UDE-based backstepping controller.

5. Closed-loop stability

For closed-loop stability analysis, new tracking error variables for states are defined as $e_i=x_i-{\overline{x}}_i$ with ${\overline{x}}_1$ denoting the reference $x_{1d}$ and the disturbance estimation error is defined as $e_{di}=d_i-{\hat{d}}_i$. Now, from the UDE filter transfer function, one has (Talole et al. 2007, 2011; Kolhe et al. 2013) (19) ${\dot{e}}_{di}=-\frac{1}{\tau }{e}_{di}+{\dot{d}}_i$(19) With reference to (16(16) $\begin{array}{rl}{\overline{x}}_2& =-k_1{z}_1+{\dot{x}}_{1d}\\ {\overline{x}}_3& =\frac{1}{K\omega }[\omega x_2+{\dot{\overline{x}}}_2-k_2{z}_2-{\hat{d}}_1]\\ u& =\frac{1}{{\omega }_a}[{\omega }_a{x}_3+{\dot{\overline{x}}}_3-k_3{z}_3-{\hat{d}}_2]\end{array}$(16) ), (17(17) $\begin{array}{rl}{\dot{z}}_1& =-k_1{z}_1+z_2\\ {\dot{z}}_2& =-k_2{z}_2+K\omega z_3+(d_1-{\hat{d}}_1)\\ {\dot{z}}_3& =-k_3{z}_3+(d_2-{\hat{d}}_2)\end{array}$(17) ) and (19(19) ${\dot{e}}_{di}=-\frac{1}{\tau }{e}_{di}+{\dot{d}}_i$(19) ), the closed-loop error dynamics can be obtained as (20) $\left[\begin{array}{c}\dot{e}\\ \dot{e_d}\end{array}\right]=\left[\begin{array}{cc}{A}_k& A_D\\ 0_{2\times 3}& A_{\tau }\end{array}\right]\left[\begin{array}{c}e\\ e_d\end{array}\right]+\left[\begin{array}{c}{0}_{3\times 2}\\ I_{2\times 2}\end{array}\right]\dot{d}$(20) where $e=[e_1{e}_2{e}_3{]}^T,e_d=[e_{d1}{e}_{d2}{]}^T,\dot{d}=[{\dot{d}}_1\dot{\phantom{a}}{d}_2{]}^T,A_k=\left[\begin{array}{ccc}-k_1& 1& 0\\ 0& -k_2& K\omega \\ 0& 0& -k_3\end{array}\right],A_D=\left[\begin{array}{cc}0& 0\\ 1& 0\\ 0& 1\end{array}\right]$ and $A_{\tau }=\left[\begin{array}{cc}-\frac{1}{\tau }& 0\\ 0& -\frac{1}{\tau }\end{array}\right]$ From (20(20) $\left[\begin{array}{c}\dot{e}\\ \dot{e_d}\end{array}\right]=\left[\begin{array}{cc}{A}_k& A_D\\ 0_{2\times 3}& A_{\tau }\end{array}\right]\left[\begin{array}{c}e\\ e_d\end{array}\right]+\left[\begin{array}{c}{0}_{3\times 2}\\ I_{2\times 2}\end{array}\right]\dot{d}$(20) ), it can be observed that the system matrix of the error dynamics is in a block-triangular form; thus, giving the characteristic equation as (21) $|sI-A_k||sI-A_{\tau }|=0$(21) Clearly, stability for the closed-loop dynamics can be ensured through a proper choice of the controller gains and filter time constant. It can also be observed that the ABS controller gain and the UDE filter constants can be independently designed to suffice the design requirements. Another aspect is related to the rate of change of actual disturbance, $\dot{d}$, acting on the system which can be neglected if the disturbance is slow varying with respect to the plant dynamics. However, for highly dynamic disturbances, a higher order filter can be used to mitigate its effect on the system performance (Talole and Phadke 2009).

6. Simulations and results

Simulations are carried out by using the UDE-based backstepping controller (16(16) $\begin{array}{rl}{\overline{x}}_2& =-k_1{z}_1+{\dot{x}}_{1d}\\ {\overline{x}}_3& =\frac{1}{K\omega }[\omega x_2+{\dot{\overline{x}}}_2-k_2{z}_2-{\hat{d}}_1]\\ u& =\frac{1}{{\omega }_a}[{\omega }_a{x}_3+{\dot{\overline{x}}}_3-k_3{z}_3-{\hat{d}}_2]\end{array}$(16) ) with the disturbance estimates given by (18(18) $\begin{array}{rl}{\hat{d}}_1& =\frac{1}{\tau }[z_2+k_2{\int }_0^t{z}_2\mathrm{d}t-K\omega {\int }_0^t{z}_3\mathrm{d}t]\\ {\hat{d}}_2& =\frac{1}{\tau }[z_3+k_3{\int }_0^t{z}_3\mathrm{d}t]\end{array}$(18) ) on the actual plant consisting of (4(4) $T\ddot{\psi }+H\left(\dot{\psi }\right)=K\delta$(4) ) and (7(7) $\delta \left(s\right)=sat\left(\frac{1}{s{\tau }_{\delta }+1}\right){\delta }_c\left(s\right)$(7) ) and the yaw attitude profile tracking performance is evaluated in the presence of external disturbances. The simulation work has been experimented with the help of Matlab software. The performance of the UDE-based backstepping controller is compared with the backstepping controller of (12(12) $u=\frac{1}{{\omega }_a}[{\omega }_a{x}_3-k_3{z}_3+{\dot{\overline{x}}}_3]$(12) ) to verify the benefits of UDE. The parameters of the ship steering model are taken from Fossen (1994) wherein the time constant, T, and gain, K, are taken as 29 and 0.35, respectively. The values of $b_1$ and $b_3$ are taken as 1 and 0.2378, respectively, which correspond to the dynamics of a ship travelling at 16 knots, i.e. 8.2 m/s (Amerongen 1982). Simulations are carried out by considering a step input wherein the reference yaw angle is taken as ${\psi }^{\star }={10}^o$. The reference command is passed through a first-order filter of time constant 0.01 for smoothening the reference (Theodoulis and Duc 2009). In the present work, to demonstrate the effectiveness of the proposed design, two types of disturbances are considered (Li and Sun 2012). The disturbance $d_1$ is taken as a sinusoidal function given by $0.001\sin \left(0.08t\right)$ which mimics the first-order and second-order wave disturbances and $d_2$ is taken as $-0.0015$. It may be noted that $d_1$ represents the mismatched uncertainty, whereas $d_2$ is the matched uncertainty. The heading of a vessel is measured by a gyro compass, the accuracy of which is typically of 0.1 degree. Accordingly, the yaw angle measurement is assumed to be corrupted by zero mean Gaussian white noise (Fossen and Grovlen 1998; Fossen and Strand 1999a; Rigatos and Tzafestas 2006; Fossen and Perez 2009).

Simulations are carried out using only backstepping controller ($C_1$) and the UDE-based backstepping controller ($C_1+UDE$) and their performances are compared. In Figure 2, the simulation results of the controllers without the effect of any external disturbances are presented and it can be seen that both the controllers offer satisfactory results in the absence of disturbances and uncertainties. Next, simulations are carried out using the controllers in the presence of the disturbances and the results are shown in Figure 2. From the Figure 2(a), it is seen that the performance of the UDE-based backstepping controller is superior in tracking the yaw angle and the controller is able to track the desired reference accurately notwithstanding the disturbances and uncertainties.

Figure 2. Performance comparison of the backstepping controller (C1) and the UDE-based backstepping controller (C1+UDE) in tracking reference signal (ref) without disturbances and parametric uncertainty. (a) Yaw angle (deg), (b) yaw angle rate (deg/s) and (c) rudder angle (deg).

Figure 3. Performance comparison of the backstepping controller (C1) and the UDE-based backstepping controller (C1+UDE) in tracking reference signal (ref) in the presence of disturbances and parametric uncertainty. (a) Yaw angle (deg), (b) yaw angle rate (deg/s) and (c) rudder angle (deg).

The UDE is able to treat the effect of external environmental factors, model uncertainties, parametric variation, nonlinearities, etc., as a total disturbance and estimates them either in the form of matched or mismatched condition. When UDE is augmented with the controller, the estimated value of disturbances is used in the controller designed for the nominal system to compensate for the effects of uncertainty and disturbances. Therefore, in the presence of uncertainty and disturbances, the heading response of the proposed UDE-based controller is effective with the imperfect plant model.

7. Conclusions

In this work, an UDE approach is used to robustify the adaptive backstepping controller design in the presence of matched as well as mismatched disturbances. Closed loop stability of the proposed formulation is established. Numerical simulation results clearly show that the proposed formulation succeeds in robustification of the backstepping controller in spite of significant disturbances.

Disclosure statement

No potential conflict of interest was reported by the authors.