Asymptotic stabilization of underactuated surface vehicles with actuator saturation

This paper investigates the problem of global asymptotic stabilization of underactuated surface vessels (USVs) with input saturation. A novel input transformation is presented, so that the USV system can be transformed to a cascade structure. For the obtained system, the improved fractional power control laws are proposed to ensure input signals do not exceed actuator constraints and enhance convergence rates. Finally, stabilization and parameter optimization algorithm of USVs are proposed. Simulations are given to demonstrate the effectiveness of the presented method.


INTRODUCTION
Underactuated surface vehicles (USV) stabilization has many practical uses in engineering practice, such as monitoring, rescue, and fishing. Generally, the USV has no side thruster considered, whose reduction always causes by an actuator failure or a deliberate decision. For example, the limitation of the actuator number due to, e.g., cost and weight considerations. This paper establishes a feedback control law to stabilize three states containing both position and orientation with only two available control inputs.
The reference (Pettersen & Egeland, 1996) shows that USVs have no asymptotic stability on desired equilibrium points with time-invariant continuous control laws. The reason is that the dynamic models of USVs have nonholonomic constraints, so they do not meet Brockett's necessary condition (Brockett & Millman, 1983). Furthermore, the existing control approaches designed for conventional nonholonomic systems cannot stabilize USVs directly since the dynamics of a USV are not drift-less (Oriolo & Nakamura, 1991). For these reasons, significant interest exist in stabilization of USVs as evidenced by various control schemes presented for such a problem (Reyhanoglu, 1996;Pettersen & Egeland, 1996;Ghommam et al., 2006;Ma, 2009;Pettersen & Nijmeijer, 2000;Zhang & Wu, 2014;Mazenc, Pettersen & Nijmeijer, 2002;Dong & Yi, 2005;Ma & Xie, 2013;Zhang & Wu, 2015;Xie & Ma, 2015). To name some, the references (Reyhanoglu, 1996;Pettersen & Egeland, 1996) proposed control laws that can stabilize the system to a small neighborhood of origin. Pettersen & Nijmeijer (2000) proposed a control law guaranteeing the semiglobal asymptotic stability of USVs. Ghommam et al. (2006), Ma (2009), Zhang & Wu (2014), Mazenc, Pettersen & Nijmeijer (2002) and Dong & Yi (2005) proposed control methods that can globally asymptotically stabilize USVs. Other related researches on the relative to the geographic north. The kinematics and of the USV with mismatched and matched disturbances are given as follows: where x, y denotes the coordinate of surface vessel mass center in the earth-fixed frame, ψ is the orientation of vessel. Variables u, v, r are the velocities in surge, sway and yaw respectively. Parameters m 11 , m 22 , m 33 are the inertia coefficients, d 11 , d 22 , d 33 are the damping coefficients of the vessel. τ u is the force of surge, τ r is the torques of yaw. Y v , N r are the coefficients from Taylor series, which are described in Fossen (1994). The saturation function is defined as follows: (2) where β and z denote the saturation constant and arbitrary variable. In terms of Eqs. (1a) and (1b), we can then define that τ umax and τ rmax are, respectively, the maximum of input variables τ u and τ r .

The objective
The objective of this paper is to design the control inputs τ u and τ r that can globally asymptotically stabilize the USV system as modeled in (1a) and (1b) subject to input saturation, namely, the following holds true for any initial conditions lim t!þ1 ½xðtÞ; yðtÞ; wðtÞ; uðtÞ; vðtÞ; rðtÞ ¼ 0: The saturation restricts the controllbility of USV systems, thus some references (such as Reyhanoglu, 1996, Pettersen & Egeland, 1996, Ghommam et al., 2006, Ma, 2009, Pettersen & Nijmeijer, 2000, Mazenc, Pettersen & Nijmeijer, 2002, Dong & Yi, 2005 are unable to be followed in the presence of saturation making it quite challenging to stabilize system (1a) and (1b) to the origin. As a compensatory research, this paper emphasize the input saturation problem instead of disturbances, input dead-zones and output constraints. This is because the methods in  can be directly applied to solve these problems with some saturation constraints. The only work that deals with stabilization of USVs with yaw constraints is Li & Yan (2016). However, it is based on MPC and relies on singular state transformations to ensure iterative feasibility.

MODEL TRANSFORMATION
Consider the system (1a) and (1b), similar to the previous literatures about stabilization of USVs, the state transformation is still utilized. However, in terms of the input saturation, existing input transformed methods can not be applied directly. Hence a novel input transformation is proposed in this section under the following assumption.
Assumption 1: The parameters d 11 , d 22 and d 33 satisfy the following condition: This assumption is relatively strong, and its specific meaning refers to a surface vehicle with a relatively small volume and a slight difference between the horizontal and vertical directions. To facilitate the control design, first the state transformation is cited from Pettersen & Egeland (1996). Define the vector ϑ = [ϑ 1 , ϑ 2 , ϑ 3 , ϑ 4 , ϑ 5 , ϑ 6 ] T with According to the system (1a) and (1b), one has 33 # 6 À ðm 11 À m 22 Þm À1 33 uv þ m À1 33 s s rmax ðs r Þ: For the transformed system, there are following results cited from previous works: Lemma 1 Stabilization of system (1a) and (1b) is equivalent to that of system (4a) and (4b).
According to the Lemmas 1 and 2, the global asymptotic stability of the USV system in (1a) and (1b) can be achieved by designing a global asymptotical control law for subsystem (4b). In addition, for the transformed system (4b), input transformations was usually introduced to further simplify calculations. However, for USVs with saturations, input transformations in previous works (such as Reyhanoglu, 1996;Pettersen & Egeland, 1996;Ghommam et al., 2006;Ma, 2009;Pettersen & Nijmeijer, 2000;Mazenc, Pettersen & Nijmeijer, 2002;Dong & Yi, 2005) cannot be used. Hence, in this paper we propose a novel input transformation for system (4b) to deal with input saturations.
Remark 1 By the proposed input transformation, the system (4b) who has input saturations can be further simplified as (13). This is one innovation point of this paper.
Remark 2 According to the inequalities (6) and (7), the values of β u and β r depend on the terms c 3 c 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi m 11 m 33 d 11 d 33 1 3 s , Δ 1 , Δ 2 , m 33 d 33 c 3 3 c 2 þ e and γ 1 . According to definitions of these parameters, we can have : A large 2 1 is useful to the large Δ 2 and contradictory with the requirement of similar m 11 m À1 22 1 and m 22 m À1 11 2 . Therefore, the difficulty lies that there exists a trade-off in selecting these parameters λ 1 , λ 2 and λ 3 . The details are stated in Section Optimization.

STABILIZATION OF USVS
Although the system is simplified as (13), there still be input saturation problem in the transformed system implying that the existing methodologies for (13) are not applicable here. To overcome the challenge, in this section, we propose a novel control law by combining the control law in Section "Stabilization of USVS" with the proposed input transformation in the Model Transformation section to solve the stabilization problem of the underactuated surface vessel.
We now have the following theorem, whose proof can be done along similar lines of Lemmas 1, 2 and 5.
Next, to facilitate practical application and summarize the main results, we provide the following algorithms to implement the proposed methods and results in USV.

Parameter optimization
In this subsection, the optimization algorithm of parameters λ 1 , λ 2 and λ 3 for the stabilization of the USV subject to input constraints are introduced.
To facilitate the calculation, first, define positive constant s as λ 1 = sλ 2 satisfying 0 < s < b 1 b 2 . Then we can have According to Eqs. (5)-(7), we have and T 4 ¼ m 33 d 33 c 3 ðm 11 Àm 22 Þ are constants. Therefore, the partial derivatives of Δ 0 , Δ 1 and Δ 2 about s can be @D 0 ffi ffi s p Àðb 1 Àb 2 sÞs rmax p . This means that Δ 0 and Δ 1 are increasing about s and the sign of @D 2 @s is determined by the values of s and τ rmax . Next, relations of @D 2 @s and the values of s and τ rmax will be analyzed as different cases. According to the formula (23), it is easy to know that , then we can have Δ 4 > 0 and @D 2 @s > 0 meaning Δ 0 , Δ 1 and Δ 2 are increasing. Therefore, in order to obtain a large β u , we choose s ¼ b 1 b 2 À j, where κ is the small positive constant.
Case 2: If s rmax < T 4 2 ffiffiffiffiffiffi b 1 b 2 p , according to the value of s, the case can be divided into two intervals.
In Figs. 2 and 3, it is shown that states x, y, ψ, u, v and r can be converged to zero. Fig. 4 shows the torque τ u and τ r . It is obvious that the maximum value of |τ u | and |τ r | are less than τ umax and τ rmax respectively.
As is known, there must be some trade-off in the input and convergence of states. In this paper, we consider the input saturation of USVs, meaning that the performance may be reduced. To investigate the performance of our controller with other methods, to be specific, consider the following performance index (25) Figure 5 shows that compared with the method in Zhang & Guo (2018), the convergence time is increased in this paper, which is caused by the input saturation term.
However, in Figs. 6 and 7, the maximum of τ u and τ r in this paper are far less than them in paper Zhang & Guo (2018). This implies that the anti-saturation control law in this We show the energy cost in Fig. 8, from which we can see that the energy consumption is far less than that of Zhang & Guo (2018).
To further illustrate the advantages of the method in this paper, we give a comparison result with the MPC casadi-windows-matlabR2016a-v3.4.5 method in Figs. 9-11.
The figure shows that although the MPC method can obtain faster convergence performance than our method. However, its control input oscillation frequency is Full-size  DOI: 10.7717/peerj-cs.793/ fig-7 relatively high, which requires relatively high actuators. Thus, we do not emphasize that our approach is superior to the MPC method, but proposes a particular strategy for USV stabilization control to avoid the problems of high oscillation frequency, difficulty in solving, and increased requirements for computing power in the MPC method. asymptotic stability of the USV with input saturation, and a parameter optimization method is given. The output limitation caused by the environment and the task may have a meaningful impact on the transient behavior and even the system's stability. The stability of the output-constrained USV is still an open issue. The energy consumption optimization and stability control of USV is another interesting problem for future research. At the same time, due to the extreme randomness of the environment faced by the sea task, some control methods for stochastic nonlinear systems, such as Zhang & Wang (2021), Yin et al. (2019), Zhang et al. (2016), Zhang, Hu & Gow (2020), should also be considered for application in the USV stabilization control in the future.