Robust adaptive finite-time course tracking control of vessel under actuator attacks

1. INTRODUCTION

With the continuous development of marine economy, unmanned surface vessels (USV) have become the most economical and effective means of marine transportation and have received special attention in the field of marine engineering ^{[1, 2]}. At the same time, the course tracking control of USV is a classic basic research topic, and many researchers have published important research results in this field. The goal of course tracking control is to overcome various internal and external disturbances and realize the high-precision tracking target course of USV ^{[3, 4]}. In order to complete the control task, many control methods, such as neural networks (NNs), sliding modes, self-adaptation, event-triggered control (ETC), nonlinear feedback, and nonlinear decoration, are applied in the design of the control scheme ^[5].

In course tracking control, PID has been widely used in engineering because of its simple control structure and good control effect ^[6]. Witkowska et al.^[7] combined backstepping and genetic algorithm to propose a course tracking control scheme. However, this scheme does not consider the disturbance of the environment. In addition, PID is in the face of disturbances, such as wind, waves, and currents. Its robustness also fails to meet further demands. In order to further solve the problem of external interference, Le et al.^[8] combined PID and fuzzy logic control to develop an automatic driving scheme for surface vessels and verified the feasibility of the control scheme through simulation. Annamalai et al.^[9] developed a robust USV adaptive course keeping control scheme combined with the gradient descent algorithm. Yang et al.^[10] proposed a robust adaptive nonlinear control algorithm for ship steering based on the projection method and Lyapunov stability theory, which simultaneously solved the uncertain dynamics inside and outside the system. Li et al.^[11] and Zhang et al.^[12] combine Radial Basis Function NNs (RBFNNS) and dynamic surface control (DSC) technology to further discuss the problem of "differential explosion" in backstepping.

In practice, there is often the challenge of controlling signal transmission, which can lead to channel overload ^[13]. This problem is more prominent in many control systems, especially when long-distance transmission is required or when operating under harsh environmental conditions^[14]. Factors such as bandwidth limitations, signal delays, and others all bring great challenges to the reliability of the control scheme. ETC adopts an event-driven approach, which triggers the controller to send a control signal only when the system state reaches a certain condition^[15]. This means that signals are only sent when adjustments or corrective control actions are required, saving communication bandwidth. Zhang et al.^[16] combined ETC technology and proposed a heading tracking fault-tolerant control (FTC) scheme. This scheme effectively improves bandwidth efficiency and saves computing resources.

In theoretical research, nonlinear feedback^[17] and nonlinear decoration^[18] have also been widely used in course tracking control. In Ref.^[19], a course tracking control algorithm is designed by establishing the error driving function to address unknown time-varying disturbance, uncertain ship model parameters, and input saturation. Zhang et al.^[20] introduced a nonlinear function of heading error in the feedback loop to replace the heading error itself and designed an improved compact backstepping controller based on the Lyapunov candidate function. Zhang et al.^[21] adopted PID technology, introduced a bipolar sigmoid function, and designed a nonlinear feedback algorithm. Finally, the effect of the algorithm is analyzed using the theory of closed-loop gain shaping. Cao et al.^[22] proposed an active disturbance rejection control algorithm based on nonlinear feedback to solve the problems of external disturbance, internal model uncertainty, and rudder angle energy input in the process of ship course keeping.

Robustness is an important performance index of the ship control system, which plays a vital role in ensuring the safe and effective operation of the ship ^[23]. FTC technology is widely used in the field of ship control because of its good control effect. Compared with traditional control methods, FTC techniques focus on achieving the control objectives of the system within a predetermined finite time. Through the research of the above reference, this paper designs a USV robust adaptive finite-time course tracking control scheme under actuator attacks. The main contributions of this paper are as follows:

(1) Using the features of adaptive online approximation and high reconstruction accuracy of nonlinear finite-time disturbance observer (NFTDO), a robust adaptive tracking control scheme based on depth information robust adaptive method is developed. The scheme not only overcomes internal and external uncertainties but also effectively resists the impact of actuator attacks.

(2) The FTC technology is introduced to further improve the control performance of the course tracking system so that the errors of the system can be converged within a finite time. Compared with traditional control schemes, the steady-state performance and transient response of the system are improved.

2. PROBLEM FORMULATION AND PRELIMINARIES

The nonlinear ship course tracking mathematical model can be expressed in the following form^[24]:

where $$ T $$ and $$ K $$ are the maneuverability index of the ship, respectively. $$ \xi $$ is the unknown environmental disturbances. $$ \partial $$ is the actuator attack signal. $$ F(\dot \psi ) = a\dot \psi + b{\dot \psi ^3} $$ is a nonlinear function of $$ \dot \psi $$, where $$ a $$ and $$ b $$ are constants.

Let $$ {x_1} = \psi $$, $$ {x_2} = \dot \psi = r $$, $$ u = \delta $$, and then it can be obtained from Eq. (1)

where $$ y \in R $$ is the output of the system, $$ u $$ is the control input of the system, $$ \omega = \frac{K}{T} $$, $$ f({x_2}) = {[ - {x_2}, - x_2^3]^T} $$, $$ \theta = {\left[ {\frac{a}{T}, \frac{b}{T}} \right]^T} $$.

Assumption 1 The external environment disturbances $$ \xi $$ and the actuator attack signal $$ \partial $$ are unknown and bounded; that is, there is a constant a greater than $$ 0 $$, s satisfying $$ \left| \xi \right| \le {\xi _d} $$, $$ \left| \partial \right| \le {\partial _d} $$.

Assumption 2 The reference course $$ {y_d} $$ is smooth guideable, and $$ {\dot y_d} $$ and $$ {\ddot y_d} $$ are available.

Assumption 3 Both model parameters $$ \theta $$ and $$ \omega $$ are unknown.

Lemma 1^[25] For system $$ x = {u_c} + \xi $$, where $$ {u_c} $$ is the control input, $$ \xi $$ is the external unknown and bounded disturbances, satisfying $$ \left| {\dot \xi } \right| \le {\xi _d} $$, and $$ {\xi _d} $$ is a positive definite constant. If there are parameters $$ {\lambda _1} $$, $$ {\lambda _2} $$, $$ {\lambda _3} $$ satisfying $$ 0 < {\lambda _3} < 1 $$, $$ {\lambda _1}, {\lambda _2}, {\lambda _3} > 0 $$, and then the disturbance observer shown in Eq. (4) can converge in a finite time.

where $$ \hat x $$ and $$ \hat \xi $$ are estimates of $$ x $$ and $$ \xi $$, $$ \pi = x - \hat x $$, $$ si{g^{{\lambda _3}}}\left( \pi \right) = {\left| \pi \right|^{{\lambda _3}}}{\mathop{\rm sgn}} \left( \pi \right) $$.

Lemma 2^[26] Assuming that there is a positive definite Lyapunov function $$ V\left( x \right) $$: $$ {\Omega _0} \to R $$ and any scalars $$ a > 0 $$, $$ b > 0 $$, and $$ 0 < \kappa < 1 $$, such that the inequality $$ \dot V(x) + aV(x) + b{V^\kappa }(x) \le 0 $$ holds, then the system is finite-time stable, and its adjustment time satisfies:

where $$ V\left( {{x_0}} \right) $$ is the initial value of $$ V\left( x \right) $$.

3. CONTROL DESIGN AND STABILITY ANALYSIS

The designed procedure of control law is shown in Figure 1. First, define the error variable

Figure 1. The designed procedure of control law.

where $$ {\psi _d} $$ is the reference course, and $$ {r_d} $$ is the virtual control variable.

Define the following virtual control variables

where $$ {\gamma _{11}} $$, $$ {\gamma _{12}} $$, $$ {\gamma _{13}} $$, and $$ {\gamma _{14}} $$ are positive definite parameters.

Then, using the following DSC technique, we can obtain the derivative of the virtual control

where $$ {\gamma _r} $$ is a positive definite parameter.

Taking the derivative of Eq. (6), we can get

where $$ F({x_2}) = - \frac{a}{T}{x_2} - \frac{b}{T}x_2^3 $$, $$ {\xi _\partial } = \xi + \partial $$.

According to Eq. (9), it can be further obtained

where $$ \hat F({x_2}) $$ and $$ {\hat \xi _\partial } $$ are estimated values of $$ F({x_2}) $$ and $$ { \xi _\partial } $$, respectively.

Define a new variable $$ \beta = {e_r} - {\hat e_r} $$, and according to Lemma 1, we can get

where $$ {\gamma _{31}} $$, $$ {\gamma _{32}} $$, $$ {\gamma _{33}} $$, and $$ {\gamma _{34}} $$ are positive definite parameters, and $$ 0 < {\gamma _{33}} < 1 $$.

The design control law is as follows

where $$ {\gamma _{21}} $$, $$ {\gamma _{22}} $$, $$ {\gamma _{23}} $$, $$ {\gamma _{24}} $$, $$ {\varepsilon _1} $$, and $$ {\varepsilon _2} $$ are positive definite parameters.

Construct the Lyapunov function as follows

where $$ \tilde F({x_2}) = \bar F({x_2}) - \hat F({x_2}) $$. $$ \tilde F({x_2}) $$, $$ \hat F({x_2}) $$, and $$ \bar F({x_2}) $$ are the estimated error, estimated value, and upper bound of $$ F({x_2}) $$, respectively.

Deriving Eq. (13) and substituting Eqs. (7)-(12), one can get

where $$ {\alpha _e} = {r_d} - \alpha $$. There exists a constant $$ {\lambda _\alpha } $$ greater than zero, which satisfies $$ \left| {{\alpha _e}} \right| \le {\lambda _\alpha } $$^[27]. According to Assumption 2 and Lemma 1, we can get A=B, where $$ {\lambda _{{\xi _\partial }}} $$ is a constant greater than zero. Then

Substituting Eq. (15) into Eq. (14), one can get

Using Young's inequality, we can get $$ {\varepsilon _2}\tilde F({x_2})F({x_2}) \le \frac{1}{4}{\varepsilon _2}{\tilde F^2}({x_2}) + {\varepsilon _2}{F^2}({x_2}) $$, $$ \frac{1}{4}{\varepsilon _2}\left| {\tilde F({x_2})} \right| \le \frac{1}{4}{\varepsilon _2}{\tilde F^2}({x_2}) + \frac{{{\varepsilon _2}}}{{16}} $$. In addition, for arbitrary variables $$ {\delta _m} $$, $$ {\delta _n} $$, and arbitrary real numbers$$ {\ell _1} $$, $$ {\ell _2} $$$$ {\ell _3} $$, satisfy

where $$ {\vartheta _1} = \min \left\{ {\left( {2{\gamma _{11}} - \frac{1}{2}} \right), 2{\gamma _{21}}, $$$$\left( {2{\gamma _{31}} - \frac{1}{2}} \right), \frac{1}{2}{\varepsilon _2}} \right\} $$, $$ {\vartheta _{ 2}} = {2^{\frac{{\gamma + 1}}{2}}}\min \left\{ {{\gamma _{12}}, {\gamma _{22}}, {\gamma _{32}}, \frac{{{\varepsilon _2}}}{{2\left( {{\varepsilon _2} + 1} \right)}}} \right\} $$, $$ \Xi = {\left| {{\lambda _{{\xi _\partial }}}} \right|^2} + \frac{1}{2}{\varepsilon _2}{F^2}({x_2}) $$$$+ {\lambda _\alpha }^2 + \frac{{{\varepsilon _2}\left( {1 - {\varepsilon _2}} \right)}}{{4\left( {{\varepsilon _2} + 1} \right)}} $$.

According to Eq. (17), it can be obtained

where $$ 0 < \iota < 1 $$. If $$ V > \frac{\Xi }{{\iota {\vartheta _1}}} $$, then

According to Lemma 2, $$ V $$ falls in the residual set $$ {\Omega _V} = \left\{ {V:V \le \frac{\Xi }{{\iota {\vartheta _1}}}} \right\} $$, and the stabilization time is

where $$ V_{(0)} $$ is the initial value of $$ V $$.

Remark 1 The basic theory of control design is backstepping. As shown in Figure 1, both virtual control and controller design phases introduce finite-time techniques. At the same time, a finite-time disturbance observer is further introduced to compensate for actuator attacks and external disturbances. This ensures that the response speed and steady state of the system are improved compared to the traditional adaptive scheme.

4. SIMULATION

This paper takes the Dalian Maritime University practice ship "Yulong" as the test object for simulation research^[19]. The main parameters are shown in Table 1.

The relevant parameters used for simulation are $$ K = 0.478 $$, $$ T=216 $$, $$ a=1 $$, $$ b=30 $$, $$ {\gamma _{11}} = 0.15 $$, $$ {\gamma _{12}} = 0.08 $$, $$ {\gamma _{13}} = 0.5 $$, $$ {\gamma _{14}} = 0.003 $$, $$ {\gamma _{21}} = 0.1 $$, $$ {\gamma _{22}} = 0.05 $$, $$ {\gamma _{23}} = 0.5 $$, $$ {\gamma _{24}} = 0.001 $$, $$ {\gamma _r} = 0.01 $$, $$ {\varepsilon _1} = 0.1 $$, and $$ {\varepsilon _2} = 0.01 $$. The time-varying disturbances and actuator attack signal are set as $$ \xi = \left[ {{\rm{1 + 0}}{\rm{.3sin(0}}{\rm{.25t) + 0}}{\rm{.15cos(0}}{\rm{.6t)}}} \right] $$, $$ \partial = {\rm{0}}{\rm{.15 + 0}}{\rm{.1sin(0}}{\rm{.2t)}} $$.

Figure 2, Figure 3, Figure 4 and Figure 5 show the USV under the influence of time-varying disturbances and actuator attacks, the NFTDO control scheme designed in this paper, the scheme in Ref.^[19], and the ship course tracking effect of the traditional Backstepping control scheme.

Figure 2. Research methodology and main design steps.

Figure 3. Course-keeping.

Figure 4. Course-keeping error.

Figure 5. Rudder angle.

The control scheme in Ref.^[19] is as follows:

where $$ k_1 $$, $$ k_2 $$, $$ c_2 $$, and $$ \sigma $$ are positive definite parameters.

It can be seen from Figure 3 that both the NFTDO control scheme designed in this paper and the comparative scheme have completed the heading tracking task, but the dynamic adjustment performance of the scheme in Ref.^[19] and the traditional Backstepping control scheme is not as good as that of the NFTDO scheme. It can be seen from the tracking error duration curve shown in Figure 4 that the tracking accuracy under the NFTDO control scheme is higher than that of the comparison scheme. Figure 5 shows the control input response for the three control schemes. The control inputs of the three control schemes tend to be stable over time. Figure 6 shows the reconstruction effect of the NFTDO scheme on the composite uncertain term composed of time-varying interference and actuator attack signals. Figure 7 is the change curve of uncertain items over time. In summary, compared with the control scheme designed in this paper, the tracking effect has been greatly improved.

Figure 6. The reconstruction of $$ {\xi _\partial } $$.

Figure 7. Time evolution of $$ {\xi _\partial } $$, $$ \xi $$, $$ \partial $$.

5. CONCLUSIONS

In this paper, by combining FTC and disturbance compensation technology, the problems of actuator attacks, external unknown disturbances, and dynamic uncertainty in USV course tracking control are effectively solved. Without any prior knowledge, a finite-time course tracking control scheme is designed. Finally, the effectiveness is verified by simulation. The simulation results show that the steady-state performance and transient response of the USV are improved under the control scheme designed in this paper. In addition, since ships have unstable situations caused by uncertainties, such as mooring forces during offshore operations, it is necessary to systematically deal with such uncertainties and ensure stability. We will explore this area further in future research.

DECLARATIONS