Nonlinear 𝐻_∞ Optimal Control Scheme for an Underwater Vehicle with Regional Function Formulation

Abstract

A conventional region control technique cannot meet the demands for an accurate tracking performance in view of its inability to accommodate highly nonlinear system dynamics, imprecise hydrodynamic coefficients, and external disturbances. In this paper, a robust technique is presented for an Autonomous Underwater Vehicle (AUV) with region tracking function. Within this control scheme, nonlinear 𝐻_∞ and region based control schemes are used. A Lyapunov-like function is presented for stability analysis of the proposed control law. Numerical simulations are presented to demonstrate the performance of the proposed tracking control of the AUV. It is shown that the proposed control law is robust against parameter uncertainties, external disturbances, and nonlinearities and it leads to uniform ultimate boundedness of the region tracking error.

1. Introduction

A valuable robotic system for the ocean environment is known as an Autonomous Underwater Vehicle (AUV). It has been used for many years in the oil and gas industry to obtain detailed maps of the ocean floor as well as to supervise pipeline activities [1]. The ongoing research on AUVs has given attention to the improvement of navigation and tracking control schemes. The conventional control methodologies are not the most suitable choice and they cannot guarantee the required tracking performance since an underwater vehicle exhibits inherent highly nonlinear system dynamics, imprecise hydrodynamic coefficients, and external disturbances. On the other hand, sliding mode control, due to its robustness against modelling inaccuracies and external disturbances, has been demonstrated to be a very attractive approach to cope with these problems [2–6]. However, a well-known drawback of conventional sliding mode controllers is the chattering effect. Therefore, to overcome the undesired effects of the control chattering, the authors in [7, 8] proposed a saturation function rather than a sign function. This substitution can minimize or, when desired, even completely eliminate chattering, but the trajectory tracking error is uniformly ultimately bounded (UUB), which in fact means that a steady-state error will always remain. In order to enhance the tracking performance inside the boundary layer, some adaptive strategy should be used for uncertainty/disturbance compensation.

Recently, a nonlinear 𝐻_∞ optimal control scheme was adopted for an underwater robotic system as an external tracking control loop and a disturbance observer was used as an internal disturbance compensation loop [9, 10]. The resultant control obtained by combining these two controls is then derived. A brief review of 𝐻_∞ optimality control has been presented in [11]. Moreover, the disturbance observer [12] is chosen, so that the 𝐿₂-gain conditions of the nonlinear 𝐻_∞ optimal control are relaxed, the magnitude of extended disturbances is reduced, and the robustness of the resulting control is improved without increasing the control input beyond that of the nonlinear 𝐻_∞ optimal control alone. By using this control, the underwater robotic system can successfully follow the given trajectories, even when uncertainties and disturbances exist. An adaptive region tracking control was presented in [13] for an AUV where a region is used rather than a point due to minimizing the control effort to track the region. Note that the total potential energy of the desired region is a summation of the potential energy associated with each region. Inspired from [13], some related research works such as in [14, 15] have been carried out to ensure that the marine robotic systems can cope with the underwater conditions and missions.

In this paper, a nonlinear 𝐻_∞ optimal control with region tracking function is proposed for an underwater vehicle. The proposed dynamic region control, where it is formulated in task space, aims to reduce the energy consumed by vehicle thrusters. Within the region function formulation, the controller activates and sends commands to the thrusters only when the AUV is outside the desired region, and hence it significantly reduces energy consumption. However, the disturbances such as ocean currents may pull the underwater vehicle out of its desired region. This is likely to occur when the AUV navigates near to the boundary as illustrated in Figure 1. Hence, a nonlinear 𝐻_∞ optimal control is proposed in this paper to counteract this problem. The performance of conventional region tracking control and region function adopted with nonlinear 𝐻_∞ optimal control law can be observed with respect to the existence of unidirectional and bounded ocean current. The rest of the paper is organized as follows: Section 2 describes the kinematic and dynamic properties of an AUV. In Section 3, the nonlinear 𝐻_∞ optimal control with region function formulation is briefly explained. The stability analysis using a Lyapunov-like function is also given in this section. In Section 4, numerical simulation results are provided to demonstrate the performance of the proposed control. Finally, the paper is concluded with some remarks in Section 5.

Figure 1 

An illustration of an underwater robotic system which performs tracking task in a spherical region.

2. Kinematic and Dynamic Model of an AUV

2.1. Kinematic Model

The relationship between inertial and body-fixed vehicle velocity can be described using the Jacobian matrix 𝐽(𝜂₂) in the following form: 𝜂̇₁𝜂̇₂=𝐽₁𝜂₂0_3×30_3×3𝐽₂𝜂₂𝑣₁𝑣₂⟺𝜂̇=𝐽𝜂₂𝑣, (1) where 𝜂₁=[^𝑥𝑦𝑧]^𝑇∈ℝ³  and 𝜂₂=[^𝜙𝜃𝜓]^𝑇∈ℝ³ denote the position and the orientation of the vehicle, respectively, expressed in the inertial fixed frame. 𝐽₁ and 𝐽₂ are the transformation matrices expressed in terms of the Euler angles. The linear and angular velocity vectors, 𝑣₁=[^𝑢𝑣𝑤]^𝑇∈ℝ³ and 𝑣₂=[^𝑝𝑞𝑟]^𝑇∈ℝ³, respectively, are described in terms of the body-fixed frame.

2.2. Dynamic Model

Let the velocity state vector with respect to the body-fixed frame be defined by 𝑣∈ℝ⁶; and the underwater vehicle dynamic equation can be expressed in closed form as [16] 𝑀𝑣̇+𝐶(𝑣)𝑣+𝑔𝜂+𝐹_ext=𝜏, (2) where 𝑀 and 𝐶(𝑣) represent the inertia matrix and the Coriolis and centripetal forces matrix including the effects of added mass and hydrodynamic damping by body motion and 𝑔(𝜂) is the restoring force. 𝐹_ext contains the effects of external disturbances and the effects of added mass and hydrodynamic damping by body motion in static water. The dynamic (2) preserves the following properties [16, 17].

Property 1. The inertia matrix 𝑀 is symmetric and positive definite such that 𝑀=𝑀^𝑇>0 and 𝛾𝐼≤𝑀≤Υ𝐼.

Property 2. 𝐶(𝑣) is the skew-symmetric matrix such that 𝐶(𝑣)=−𝐶^𝑇(𝑣).
In the Property 1, 𝛾 and Υ denote the minimum and maximum eigenvalues of the inertia matrix, respectively. The matrix 𝐼 is the identity matrix that has suitable dimension.

3. Nonlinear 𝐻_∞ Optimal Control Law with Region Formulation

In the region-based control framework, the desired moving target is specified by a region at the desired trajectory. A robust nonlinear 𝐻_∞ optimal control for AUV proposed in this paper is formulated as follows.

First, the vehicle needs to converge into a region with specific shape. The objective function for this region is defined by the following: 𝑓𝛿𝜂_𝐵≤0, (3) where 𝛿𝜂_𝐵=𝐵(𝜂−𝜂_𝑑)∈ℝ⁶ are the continuous first partial derivatives of the dynamic region; 𝜂_𝑑(𝑡) is the time-varying reference point inside the geometric shape and 𝐵(𝑡) is a time-varying and nonsingular  scaling factor. It is assumed that 𝜂_𝑑(𝑡) and 𝐵(𝑡) are bounded functions of time. To achieve the scaling formation, that is, if the scaling factor increases, then the size of a desired region also increases, a nonsingular matrix is defined as follows: 𝐵𝐵=¹00𝐵₂, (4) where 𝐵₁ is the scaling matrix of 𝜂₁ and 𝐵₂ is the scaling matrix of 𝜂₂. This function is useful when the AUV needs to adapt the moving region, depending on the situation and environment.

The corresponding potential energy function for the desired region described in (3) can be specified as 𝑃𝛿𝜂_𝐵=0,𝑓𝛿𝜂_𝐵𝑘≤0_𝑝2𝑓²𝛿𝜂_𝐵,𝑓𝛿𝜂_𝐵>0, (5) such that 𝑃𝛿𝜂_𝐵=𝑘_𝑝2max0,𝑓𝛿𝜂_𝐵², (6) where 𝑘_𝑝 is a positive scalar. Differentiating (6) with respect to 𝛿𝜂_𝐵 gives 𝜕𝑃𝛿𝜂_𝐵𝜕𝜂_𝐵^𝑇=𝑘_𝑝max0,𝑓𝛿𝜂_𝐵𝜕𝑓𝛿𝜂_𝐵𝜕𝜂_𝐵^𝑇. (7) Now, let (7) be represented as the region error 𝑒_𝐵̃ in the following form: 𝑒_𝐵̃=max0,𝑓𝛿𝜂_𝐵𝜕𝑓𝛿𝜂_𝐵𝜕𝜂_𝐵^𝑇. (8) If 𝐵 is set to an identity matrix, then a useful vector 𝑣_𝑟 is defined as 𝑣_𝑟=𝐽⁻¹𝜂̇_𝑑−𝛼𝐽⁻¹𝑒_𝐵̃−𝛽𝐽⁻¹𝑒_𝐵̃𝑑𝑡, (9) where 𝛼 and 𝛽 are arbitrary positive constants. The matrix 𝐽⁻¹ represents the inverse of the Jacobian matrix. From the arguments of trigonometric functions, this matrix is bounded. Based on the structure of (8) and (9) and the subsequent stability analysis, a filtered tracking error vector for an underwater vehicle is defined as 𝑟(𝑡)=𝑣−𝐽⁻¹𝜂̇_𝑑+𝛼𝐽⁻¹𝑒_𝐵̃+𝛽𝐽⁻¹𝑒_𝐵̃𝑑𝑡. (10) From the definition of 𝑟 in (10), the control law for an AUV can be proposed in the following form: 𝜏=−𝐾_𝑣𝑟+𝑀𝑣̇_𝑟+𝐶(𝑣)𝑣_𝑟+𝑔𝜂, (11) where 𝐾_𝑣=𝐾+(1/𝜅²)𝐼; 𝐾 and 𝐼 are the positive definite matrix and identity matrix, respectively. 𝑀, 𝐶(𝑣), and 𝑔(𝜂) are the nominal matrices and vectors of 𝑀, 𝐶(𝑣), and 𝑔(𝜂), respectively. The derivative of 𝑣_𝑟 in (9) is given as 𝑣̇_𝑟=𝐽̇⁻¹𝜂̇_𝑑+𝐽⁻¹𝜂̈_𝑑−𝛼𝐽̇⁻¹𝑒_𝐵̃−𝛼𝐽⁻¹̇𝑒_𝐵̃−𝛽𝐽̇⁻¹𝑒_𝐵̃𝑑𝑡−𝛽𝐽⁻¹𝑒_𝐵̃, (12) where 𝜂̇_𝑑(𝑡), 𝜂̈_𝑑(𝑡), and 𝐽̇⁻¹(𝑡) are all assumed to be bounded functions of time. Substituting (11) into (2) produces a closed-loop dynamic equation for 𝑟(𝑡) as follows: 𝑀𝑟̇+𝐶(𝑣)𝑟+𝐾_𝑣𝑟−𝜔=0, (13) where 𝜔 is the extended disturbance vector which is defined in the following form: 𝜔=𝑀𝑣̇_𝑟+𝐶(𝑣)𝑣_𝑟+𝑔𝜂−𝐹_ext, (14) where (⋅)=(⋅)−(⋅)  denotes the parameter estimation error. The modeling error acts as a disturbance in (14) when the AUV is in motion. Note that 𝜅 is the  𝐿₂ gain for disturbance attenuation satisfying the following condition: ^𝑇₀𝑧^𝑇𝑧𝑑𝑡=^𝑇₀𝜔^𝑇𝜔𝑑𝑡, (15) where 𝑧^𝑇𝑧 is defined as the weighted sum of the quadratic forms of the error states and the control input. Since nonlinear 𝐻_∞ optimal control scheme is based on feedback tracking errors, 𝑧^𝑇𝑧 can be approximated up to magnitude of these errors.

Remark 1. Equation (13) can be represented in state space such that the nonlinear 𝐻_∞ optimality satisfies [18] ^𝑇₀𝑥^𝑇𝑄𝑥+𝑢^𝑇𝑅𝑢𝑑𝑡=𝜅^2𝑇₀𝜔^𝑇𝜔𝑑𝑡 (16) with 𝛼²>2𝛽. 𝑥 and 𝑢 in (16) denote the state and input variables, respectively. Meanwhile, the matrices 𝑄 and 𝑅 are state weighting and input weighting matrices, respectively, and they are determined by inverse optimal problem with respect to specific 𝐿₂ attenuation gain, 𝜅.

Theorem 2. Let the filtered tracking error vector 𝑟 be upper bounded as the following form: ‖𝑟‖≤Υ𝛾𝜅²2𝑘_𝑚𝜅²‖𝜔‖+1^∞, (17) where 𝑘_𝑚 is scalar constant and ‖𝜔‖_∞ denotes an infinity norm of 𝜔 for a given time interval. Then, the control law (11) above is continuous and the closed-loop system is uniformly ultimately bounded (u.u.b) as defined in [11].

Proof. The following nonnegative function is introduced to analyze the stability of the proposed control law: 1𝑉=2𝑟^𝑇𝑀𝑟. (18) Differentiating 𝑉 with respect to time and utilizing (10) and (14), a closed-loop dynamic (13) yields 𝑉̇=−𝑟^𝑇𝐶(𝑣)𝑟−𝑟^𝑇𝐾_𝑣𝑟+𝑟^𝑇𝜔. (19) Simplifying (19) leads to 𝑉̇=−𝑟^𝑇𝐾_𝑣𝑟+𝑟^𝑇𝜔=−𝑟^𝑇1𝐾+𝜅²𝐼𝑟+𝑟^𝑇𝜔=−𝑟^𝑇1𝐾+2𝜅²𝐼𝑟−𝜅²21𝜅²𝑟−𝜔²+𝜅²2‖𝜔‖²_∞, (20) where Property 2 is used. Let 𝑘_𝑚 be the minimum diagonal element of gain matrix 𝐾 utilizing the worst case disturbance [11] to yield the following inequality: 𝑉̇≤−𝑘_𝑚+12𝜅²‖𝑟‖²+𝜅²2‖𝜔‖²_∞. (21) From (21), it is necessary to choose sufficiently large value of 𝑘_𝑚 to ensure the negative definiteness of 𝑉̇. Therefore, implying the results and terminology of [19], the ultimate boundedness of ‖𝑟‖ can be obtained as in (17).

Remark 3. It is assumed that the norm of extended disturbance which includes tracking errors is not deviated largely, when a control input (11) is used in (2). Thus, the control gain can be changed according to (17), so that the satisfactory performance of proposed control law with region function formulation can be achieved.

4. Simulation Results

In this section, simulation studies are carried out to assess the effectiveness of the proposed nonlinear 𝐻_∞ optimal control law with region function formulation for an underwater vehicle. The performance of conventional tracking control and the proposed technique is observed concerning two cases: the first case is the conventional region tracking control and the second case is where the region function is adopted with nonlinear 𝐻_∞ optimal control law. Both control laws are observed with respect to the existence of random disturbances and bounded ocean current. The ODIN AUV [20, 21] that is known as a near-spherical omnidirectional vehicle equipped with four horizontal thrusters and four vertical thrusters is chosen as the Autonomous Underwater Vehicle model in these numerical simulations. The following inequality function is defined as 𝑓𝛿𝜂_𝐵=𝑠_𝑥𝑥−𝑥₀²+𝑠_𝑦𝑦−𝑦₀²+𝑠_𝑧𝑧−𝑧₀²+𝑠_𝜙𝜙−𝜙₀²+𝑠_𝜃𝜃−𝜃₀²+𝑠_𝜓𝜓−𝜓₀²≤𝜅²_𝑟, (22) where the element of {𝑠_𝑥,𝑠_𝑦,𝑠_𝑧,𝑠_𝜙,𝑠_𝜃,𝑠_𝜓} is the component of the time-varying scaling matrix 𝐵 and 𝜅_𝑟 is a scalar tolerance. In these simulations, the matrix 𝐵 is defined as the identity matrix and 𝜅_𝑟 is set to 0.25. Note that (22) can also be represented as the root mean square error for all axes. In Table 1, the norm values of required forces for four vertical thrusters are presented. The total control input is included to signify the overall energy needed for the system to maintain at depth −1.2 meter. Notice that when the proposed controller is utilized, the energy requirement is reduced as compared with set-point tracking method.

Table 1 

Simulation result: vehicle forces for four vertical thrusters (N).

The underwater vehicle is required to track a predefined trajectory as illustrated in Figure 2 where the green (cross-section) path is the horizontal basis position initialized at the position [^1.50−1.2]^𝑇  m. Moreover, the vehicle is initialized at the same position 𝜂₁(0)=[^1.50−1.2]^𝑇  m while its attitude is kept constant during simulation and the initial values are 𝜂₂(0)=[⁰⁰⁰]^𝑇 degrees. From Figures 3, 4, 5, and 6, it has been shown that the proposed control scheme exhibited a more robust tracking performance than the conventional region control, when parameter uncertainties, current effects, and disturbances exist. In Figure 4, the acute fluctuations in the early stages of the simulation were mainly caused by parameter uncertainties in the restoring force and moment. However, as long as the AUV is inside the desired region, the control input is turned off, and when the disturbances pull the vehicle out, the control input is applied to navigate the AUV back into the region.

Figure 2 

The desired lawnmower trajectory where “×” marks the initial position of the AUV.

Figure 3 

Three-dimensional view for conventional region tracking control.

Figure 4 

Planar view for conventional region tracking control.

Figure 5 

Three-dimensional view for the nonlinear 𝐻∞ optimal control law with region function formulation.

Figure 6 

Planar view for nonlinear 𝐻∞ optimal control law with region function formulation.

5. Conclusion

A new nonlinear 𝐻_∞ optimal control law with region function formulation for a hovering underwater vehicle with four horizontal and four vertical thrusters has been presented in this paper. Two cases have been considered: the first case is the conventional region tracking control and the second case is where the region function is adopted with the nonlinear 𝐻_∞ optimal control law. Both control laws are observed with respect to the existence of unidirectional and bounded ocean currents. Although the underwater disturbances exist during task execution, the AUV is still able to track a desired moving region. A Lyapunov-like function has been proposed for stability analysis. Simulation results have been presented to demonstrate the performance of the proposed controller.

Appendix

An omnidirectional intelligent navigator (ODIN) is a near-spherical AUV designed in the University of Hawaii. The dynamic model of ODIN is given by [20, 21] 𝑀_RB+𝑀_A𝑣̇+𝐶_RB(𝑣)+𝐶_A(𝑣)𝑣+𝐷(𝑣)𝑣+𝑔𝜂=𝜏, (A.1) where the subscripts RB and A represent the rigid body and added mass terms of the relevant parameters, respectively. The numerical values for the matrices of the vehicle dynamic equation (A.1) are given as 𝑀_RB=𝑚000𝑚𝑧_𝐺00𝑚0−𝑚𝑧_𝐺0000𝑚0000−𝑚𝑧_𝐺0𝐼_𝑥𝑥00𝑚𝑧_𝐺000𝐼_𝑦𝑦000000𝐼_𝑧𝑧, (A.2) where 𝐼_𝑥𝑥=𝐼_𝑦𝑦=𝐼_𝑧𝑧=𝐼=(8/15)𝜋𝜌_𝑣𝑟⁵_ODINare the moments of inertia about the principle axes.

Consider𝑀_A=𝑋_𝑢𝑑000000𝑌_𝑣𝑑000000𝑍_𝑤𝑑000,𝐶000000000000000000_RB(𝑣)=⁰⁰⁰_{−𝑚𝑧𝐺𝑢6−𝑚𝑢3𝑚𝑢2}⁰⁰⁰_{𝑚𝑢3−𝑚𝑧𝐺𝑢6−𝑚𝑢1}⁰⁰⁰_{𝑢−𝑚2−𝑧𝐺𝑢4𝑚𝑢1+𝑧𝐺𝑢50}^{𝑚𝑧𝐺𝑢6−𝑚𝑢3𝑚𝑢2−𝑧𝐺𝑢4}_{0−𝐼𝑢6𝐼𝑢5}^{𝑚𝑢3𝑚𝑧𝐺𝑢6𝑢−𝑚1+𝑧𝐺𝑢5}_{𝐼𝑢60−𝐼𝑢4}^{−𝑚𝑢2𝑚𝑢10}_{−𝐼𝑢5𝐼𝑢40},𝐶_A(𝑣)=0000−𝑚𝑢₃𝑚𝑢₂000𝑚𝑢₃0−𝑚𝑢₁000−𝑚𝑢₂𝑚𝑢₁00−𝑚𝑢₃𝑚𝑢₂000𝑚𝑢₃0−𝑚𝑢₁000−𝑚𝑢₂𝑚𝑢₁0000,𝐷(𝑣)=−𝑑_𝑡1𝑢₁000000−𝑑_𝑡1𝑢₂000000−𝑑_𝑡1𝑢₃000000−𝑑_𝑟1𝑢₄−𝑑_𝑟2000000−𝑑_𝑟1𝑢₅−𝑑_𝑟2000000−𝑑_𝑟1𝑢₆−𝑑_𝑟2,𝑔𝜂=4𝑚𝑔−3𝜋𝑟³(𝜃)−4𝜌𝑔sin𝑚𝑔−3𝜋𝑟³(𝜃)𝜙−4𝜌𝑔cossin𝑚𝑔−3𝜋𝑟³(𝜃)𝜙𝑧𝜌𝑔coscos_𝐺(𝜃)𝜙𝑧𝑚𝑔cossin_𝐺(𝜃)0.𝑚𝑔sin (A.3)

Provided that 𝑟_ODIN=0.31m is the radius of ODIN, 𝑚=125.0kg is the mass of ODIN, 𝑧_𝐺=0.05m is the distance of the center of gravity from the geometric center, 𝜌_𝑣=965kg/m³ is the average density of the ODIN AUV, 𝜌=1000kg/m³ is the density of fresh water, and 𝑔=9.81m/s². The hydrodynamic derivatives are given by 𝑋_𝑢𝑑=𝑌_𝑣𝑑=𝑍_𝑤𝑑=(2/3)𝜋𝜌𝑟³_ODIN, the translational quadratic damping factor 𝑑_𝑡1=−248N(s/m)², the angular quadratic damping factor 𝑑_𝑟1=−280Ns²/m, and the angular linear damping factor 𝑑_𝑟2=−230Ns²/m.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank Offshore Geo-Surveys (OGS) Sdn. Bhd. for invaluable help and advice on the underwater vehicle. This work was supported in part by the Ministry of Higher Education, Malaysia, and Universiti Teknologi Malaysia under Grant no. R.J130000.7823.4L049.