Depth control for a deep-sea self-holding intelligent buoy system based on active disturbance rejection control method

QIU Zu-rong，WANG Qiang,2，YANG Shao-bo,2，LI Hong-yu，LI Xing-fei，2

(1. State Key Laboratory of Precision Measurement Technology and Instruments，Tianjin University，Tianjin 300072, China；2. Qingdao Institute for Ocean Technology of Tianjin University，Qingdao 266235, China；3. School of Mechanical and Electrical Engineering，Shandong University of Science and Technology，Qingdao 266590, China)

Abstract： The net buoyancy of the deep-sea self-holding intelligent buoy (DSIB) will change with depth due to pressure hull deformation in the deep submergence process. The net buoyancy changes will affect the hovering performance of the DSIB. To make the DSIB have better resistance to the external disturbances caused by the net buoyancy and water resistance, a depth controller was designed to improve the depth positioning based on the active disturbance rejection control (ADRC). Firstly, a dynamic model was established based on the motion analysis of the DSIB. In addition, the extended state observer (ESO) and nonlinear state error feedback controller were designed based on the Lyapunov stability principle. Finally, semi-physical simulations for the depth control process were made by using the ADRC depth controller and traditional PID depth controller, respectively. The results of the semi-physical simulations indicate that the depth controller based on the ADRC can achieve the predefined depth control under the external disturbances. Compared with the traditional PID depth controller, the overshoot of the ADRC depth controller is 1.74%, and the depth error is within 0.5%. It not only has a better control capability to restrain the overshoot and shock caused by the external disturbances, but also can improve intelligence of the DSIB under the depth tracking task．

Key words： deep-sea self-holding intelligent buoy (DSIB)； active disturbance rejection control (ADRC)； depth control； buoyancy change； pressure hull deformation

0 Introduction

Recently, with the development of exploration in deep ocean, applications of the deep-sea self-holding intelligent buoy (DSIB) have experienced a substantial increase, which is mainly due to the tasks arising from various commercial and scientific needs. As an efficient mobile sensor platform, the DSIB has broad application prospects in ocean monitoring, exploring and detection, for both civilian and military purposes, such as scientific sampling, undersea searching and mining, observation of the ocean environment, hydrographic surveys and biological surveys, building of military affairs and national defense, etc.[1-3].

The DSIB has the virtues of extensive scope of activity, small volume, convenience in operation, good maneuverability, strong security, intelligence, etc, which can carry on various kinds of dangerous tasks instead of human. To achieve the aforementioned tasks, a control method for depth positioning of the DSIB is indispensable. Depth positioning is an important function of a float. However, the dynamics of the DSIB have nonlinear relation with the environmental disturbances. Several control methods for nonlinear depth positioning have been proposed, which, however, renders the depth positioning method difficult to design[4-5]. Many researchers have extensively investigated various linear control approaches as well, and their achievements have been acknowledged in Refs.[6-10].

A cascade PID controller was designed to conduct the double closed-loop depth control on the float[6]. The float reached a depth of 7 m with the depth error within 2.85%. However, this controller was expensive in terms of energy consumption. Other studies available in Ref.[7] is related to the state-space feedback control algorithm. The control algorithm proposed in Ref.[7] was used not only to increase the accuracy of the float dynamics model, but also to accomplish the control procedure for depth positioning and altitude trajectory. However, this method had to be combined with empirical motor efficiency data so that the tradeoffs between the efficiency and the performance could be studied. In addition, an intelligent control strategy based on the ocean model was used to design the depth positioning strategy of the DSIB[8]. Nevertheless, the algorithm presented in Ref.[8] was dependent on a large-scale ocean model. Considering the nonlinear and strong coupling characteristics of the DSIB motion model, a fuzzy PID controller based on a dual closed-loop was proposed to achieve the depth control[9]．By simulation, the overshoot of the fuzzy PID controller is 2.0%, and the depth error is within 1.0%. However, the implementation of this method needs to be further verified by the experiments. Based on the principle of trial-and-error method and the unique design of the characteristic curve function, an automatic bathymetric depth control method for a class of the DSIB without propellers was proposed[10]. The float hovered at a depth of 5 m with an average depth tracking error of less than 1 m. But the relevant dynamic model was not established in the application of this method, so it was difficult to optimize the bathymetric control method. A model-free feedback control method was used to achieve the depth control with the depth error within 5%[11]. Although the depth control was realized by the model-free feedback method, the optimal effect of the depth control process had not been achieved. This method needs to be further optimized.

The DSIB ascends or dives at a certain speed from the initial depth to the target depth. But the external disturbances caused by the pressure hull deformation and water resistance may be unavoidable in the depth control process. These external disturbances will affect the effect of the depth positioning, but they are not taken into account in the aforementioned Refs.[6-10]. Because ADRC method is based on poor dynamic quality of the tracking control caused by the internal and external uncertainties of controlled object[12], ADRC depth controller is suitable for solving the depth control problem under the external disturbances. Thus, based on the research of depth control in Ref.[9], a deep control strategy based on ADRC technology is proposed. Firstly, based on the pressure hull deformation, the dynamic model of the DSIB is modeled. Then, the nonlinear tracking differentiator (TD) is used to extract the differential signal for the transition process of the depth tracking, extended state observer (ESO) is designed to estimate the DSIB system state and integrate DSIB system disturbances and make compensation according to the external disturbances, and a nonlinear state error feedback controller (NLSEF) is designed to control the buoyancy regulating system of the DSIB. Finally, the simulation experiment and comparative analysis of the proposed ADRC depth control strategy are carried out by the semi-physical simulation system, which verifies the effectiveness of the control method in the depth control process.

1 DSIB motion modeling in vertical plane

1.1 Deformation of spherical pressure hull under external pressure

In our study, a spherical pressure hull is used in the pressure-proof structure of the DSIB. Only the volume changes induced by the deformation with increasing seawater pressure are considered. The deformation of the spherical pressure hull under seawater pressure condition slightly reduces the active buoyancy of the float. Thus, the deformation of the spherical pressure hull cannot be ignored in deep sea. The spherical pressure hull deformation at different pressures is expressed as[13]

whereris the radius of the spherical pressure hull (0.216 m),λis the Poisson’s ratio (0.2),Eis the modulus of elasticity on the spherical pressure hull (63 GPa),δis the thickness of the spherical pressure hull (0.013 5 m), andPzis the pressure at the target depthz(Pz=ρgz,g=9.8 m/s2). The material used for the spherical pressure hull is glass. Substituting the values for the above parameters in Eq.(1), we obtain

ΔV=qρz,

(2)

whereq=1.25×10-11.

1.2 Drag analysis

The drag force on the DISB is assumed to be a quadratic drag law and is expressed as[3]

(3)

whereRis the drag force,Ais the effective cross-sectional area of the float, andCdis the drag coefficient. The effective cross-sectional area can be obtained by the projection of the 3D model onto the vertical plane. In the current study,Ais calculated to be 0.301 m2by using Eq.(3). Through CFD simulation analysis, the drag coefficients of the DSIB in the ascending and diving processes are calculated to beCd-up=0.73 andCd-down=0.66, respectively.

Eq.(3) shows that the drag force is nonlinear, which can be approximated to a linear expression at the velocity equilibrium point. The velocity component of the drag term is expressed asv|v| instead ofv2. Considering the current velocityu, the drag can be linearized as

(4)

The relationship between the quadratic and linearized drags is shown in Fig.1. The horizontal axis is the motion velocity of the DSIB, and the vertical axis is the drag force. The drag law is approximately quadratic as indicated by the dotted line, and the linearized drag is represented by the solid line. Finally, the linearization velocity of the float (|v-u|=0.1 m/s) under the current disturbances, is represented by the gray point.

Fig.1 Relationship between quadratic drag and linearized drag

1.3 Kinematic equation of deep-sea intelligent float

The dynamic model of the float in disturbing currents is considered to be nonlinear and coupled in the vertical plane. However, a linearized dynamic model can be used in this study with linear techniques. Therefore, simplifying them at the operating point based on special assumptions is necessary as follows.

1) The total mass of the DSIB is constant. The center of mass and the center of buoyancy are axially collinear. The float is regarded as a spherical entity in the vertical plane, and the direction of its movement is also vertical.

2) The current is not negligible in the vertical plane, and the drag force is considered to be linear.

Based on these special assumptions, the dynamic model of the DSIB is established under the current disturbances in the vertical plane. The DSIB ascends or descends at a velocityvfrom the initial depth to the target depth. In the above process, the relevant forces and relevant velocity vectors acting on the float include buoyancyF, gravityG, dragR, and motion velocity of the floatv, as illustrated in Fig.2. The floating (ascending) process is represented by the region marked solid line, and the submerging (diving) process is represented by the region dotted line.

Fig.2 Significant forces acting on DSIB during floating and submerging processes

(5)

(6)

Table 1 presents the non-dynamic parameters of the DSIB.

Table 1 Non-dynamic parameters of DSIB

According to Eqs.(5) and (6), the state-space equation of the DSIB system can be obtained as

(7)

(8)

wherebis control gains,b=R4；f(x1,x2) is the unknown nonlinear dynamics of the DISB,f(x1,x2)=R1x2+R2x1； Δ1is unknown nonlinear disturbance of DISB, Δ1=R5.

2 Design of active disturbance rejection depth controller

Active disturbance rejection controller (ADRC) is an applied control to improve the shortage of PID control in many aspects and has good disturbance rejection ability and robust ability[14]. The schematic diagram of the ADRC is shown in Fig.3. TD is the tracking differentiator for the expected response； ESO is an extended state observer for estimating general disturbances and object output；NLSEF is a kind of non-linear state error feedback which uses the error and its derivatives to achieve the good control performance in a non-linear way[15].

Fig.3 Schematic diagram of ADRC depth controller

2.1 Design of tracking differentiator

The noise amplification effect of the classical differentiator is overcome by the steepest tracker for the tracking differentiator. A transition process is arranged for a given signal so that the given signal is tracked without overshooting. The DSIB system is controlled by the error with the feedback amount so as to avoid the DSIB system overshoot caused by the excessive initial control force. Leti=w1-w0, the second-order tracking differentiator is designed as[16]

(9)

wherew(t) is a given signal；w1andw2are the tracking signal and differential tracking signal, respectively;ris a fast factor;his the simulation step;fhan(·) is the fastest control synthesis function, and its specific expression is[15]

(10)

2.2 Design of extended state observer

The extended state observer can be used to estimate the nonlinear part of the DSIB system. The nonlinear disturbance part is continuously differentiable and bounded. Then the extended state system of the DSIB system is defined as

(11)

whereλis the derivative ofx3;ηis the derivative of Δ1;λandηare bounded in practice. Defininge1=z1-x1，According to Eq.(11)，the extended state observer of the system is represented as

(12)

wherez1，z2andz3are observations ofw1，w2andw3, respectively；α1，α2andα3are observer gains；σ1=0.5 andσ2=0.25; the role of the saturation functionfal(e,σ,δ) is to suppress signal chattering, and it is defined as[15]

(13)

Considering systems (11) and (12), we get the following error system as

(14)

Theorem1: For error system(14)，there exist appropriate α1, α2 and α3 guaranteeing error system variables to converge 0. That is, the extended state observer (12) estimates the state variables of system (11) effectively.

Proof. In order to prove the convergence of the extended state observer, the Lyapunov function need to be defined as

(15)

The derivative ofV1is given as

e2[e3-α2fal(e1,0.5,δ)]+

e3[-α3fal(e1,0.25,δ)-η]=

e3α3fal(e1,0.25,δ)-e3η,

2.3 Design of nonlinear error feedback controller

The system uncertainty and total disturbance are measured through the extended state observer. It is the core of ADRC. Error signals between the tracking differentiator and the extended-state observer are defined as

(16)

wherew1andw2are the given position and velocity, respectively;z1andz2are the observations of the corresponding output. The nonlinear state error feedback controlleruis given as

(17)

whereβ1andβ2are two gains of the nonlinear controller, andz3/b0is compensation for DSIB system (8). In order to analyze the stability of the DSIB closed-loop system, the errors between the given and the output are presented as

(18)

Furthermore, the error system can be defined as

(19)

wherew3is the derivative ofw2, and it is continuous and bounded. From Eqs.(16) and (18), it is easy to get the following expression as

(20)

Moreover, we have

bβ2fal(σ2,γ2,δ2)+e3-Δ1.

(21)

Theorem2: Considering the closed-loop system (19) with the error feedback controller (17), choosing appropriate controller gainsβ1andβ2in controller (17), the closed-loop system (19) is stable. That is, outputsx1andx2converge to inputsw1andw2, respectively.

Proof. Constructing a Lyapunov function as

(22)

(θ1-e1+θ2-e2){θ2-e2+α1e1+w3-

bβ1fal(σ1,γ1,δ1)-bβ2fal(σ2,γ2,δ2)+

e3-Δ1-[e3-α2fal(e1,0.5,δ)]}=

(θ1-e1+θ2-e2)[θ2-e2+α1e1+w3-

bβ1fal(σ1,γ1,δ1)-bβ2fal(σ2,γ2,δ2)-

Δ1+α2fal(e1,0.5,δ)]=(σ1+σ2)×

(σ2+α1e1+w3-Δ1+α2fal(e1,0.5,δ))+

(σ1+σ2)[-bβ1fal(σ1,γ1,δ1)-bβ2fal(σ2,γ2,δ2)],

letβ1=β2=β≥0 for simplified analysis, we have

(σ1+σ2)(σ2+α1e1+w3-Δ1+α2fal(e1,0.5,δ)).

According to Ref.[9], the functionfal(·) is an odd function that is monotonically increasing. Thus,σ1fal(σ1,γ1,δ1)≥0 andσ2fal(σ2,γ2,δ2)≥0. To get the sign of expression (σ1+σ2)[fal(σ1,γ1,δ1)+fal(σ2,γ2,δ2)], letγ1=γ2=γandδ1=δ2=δ, the following analysis is given.

1) Ifσ1< 0,σ2< 0, we getfal(σ1,γ,δ)<0,fal(σ2,γ,δ)<0. Then (σ1+σ2)[fal(σ1,γ,δ)+fal(σ2,γ,δ)]>0.

2) Ifσ1≥0,σ2≥0, we getfal(σ1,γ,δ)≥0,fal(σ2,γ,δ)≥0. Then (σ1+σ2)[fal(σ1,γ,δ)+fal(σ2,γ,δ)]≥0.

3) Ifσ1<0≤σ2,σ2≥|σ1|, we getfal(σ2,γ,δ)≥0>fal(σ1,γ,δ),fal(σ2,γ,δ)≥|fal(σ1,γ,δ)|.

Then (σ1+σ2)[fal(σ1,γ,δ)+fal(σ2,γ,δ)]≥0.

4) Ifσ1<0≤σ2,σ2<|σ1|, we getfal(σ2,γ,δ)≥0>fal(σ1,γ,δ),fal(σ2,γ,δ)<|fal(σ1,γ,δ)|.

Then (σ1+σ2)[fal(σ1,γ,δ)+fal(σ2,γ,δ)]>0.

In a similar way, ifσ1≥0>σ2,

(σ1+σ2)[fal(σ1,γ,δ)+fal(σ2,γ,δ)]>0.

In conclusion, we obtain

(σ1+σ2)[fal(σ1,γ,δ)+fal(σ2,γ,δ)]≥0.

(23)

Therefore, we have

Thus, the designed nonlinear error feedback controller (17) guarantees the stability of the closed-loop system (19) by selecting appropriate controller parametersβ1andβ2.

3 Semi-physical simulation experiment for depth control process of DSIB

3.1 Semi-physical simulation test platform

In order to verify the effectiveness and robustness of the designed ADRC depth controller, the DSIB system was selected as the research object in this study, and the simulation test was performed under the semi-physical simulation condition, as shown in Fig.4. The semi-physical simulation test platform was used to simulate the marine environment more realistically and provide a reliable reference value for the sea trial of the DSIB.

Fig.4 Schematic of semi-physical simulation test platform for depth control process of DSIB

Without considering the effects of the changes in salinity, temperature, and gravity acceleration, the seawater pressure can be approximately considered as 40 MPa at the depth of 4 000 m in this study. Therefore, the pressure change of the piston cylinder in the semi-physical simulation platform was used to simulate the depth change in the actual sea conditions. The experimental research on the depth tracking of the DSIB during the ascending and diving process was realized.

The floating and diving processes of the DSIB were performed by the buoyancy regulating system of the DSIB. The oil draining and oil returning stages were performed in the simulation experiment. When the DSIB ascends from 4 000 m underwater to the sea surface, the seawater pressure gradually decreases from 40 MPa to 0 MPa in the ascending process. When the pressure on the side of the rodless chamber for the 60 MPa hydraulic cylinder was reduced in the experimental platform and hydraulic oil was discharged from the external bladder by the buoyancy regulating system, in this case, the DSIB is in the floating process. When the float dives from the sea surface to 4 000 m, the seawater pressure gradually increases from 0 MPa to 40 MPa in this process. Hydraulic oil is allowed to flow from the external bladder back into the buoyancy regulating system by the external seawater pressure. In the test platform, when the pressure of the 60 MPa hydraulic cylinder without the rod cavity was increased, in this case, the float is in the diving process. If the DSIB hovers at a target depth, the pressure on the side of rodless chamber of 60 MPa hydraulic cylinder was remained constant at this time. The 60 MPa high-pressure accumulator was regarded as the external seawater pressure in the deep-sea environment for pressure-holding test.

The non-dynamic parameters of the DSIB for simulation were shown in Table 1. For the ADRC depth controller, the desirable parameters were as follows.

1) TD parameters:r=1,d=0.1;

2) ESO parameters:α1=200,α2=100,α3=200,δ=0.05;

3) NLSEF parameters：γ1=0.85,γ2=1.5,β1=1.75,β2=10,b=1.07,δ1=δ2=5.

Relevant simulations have been conducted to validate the proposed depth control method compared to that of a standard PID controller. For the PID depth controller, the desirable parameters were as follows:Kp=5,Ki=2,Kd=25.

In order to fully verify the control performance of the ADRC depth controller for the DSIB in the South China Sea, the test is divided into shallow water area and deep water area with the depth of 2 000 m as the demarcation point. The DSIB was always affected by the net buoyancy change caused by the pressure hull deformation and water resistance.

3.2 Experimental verification

The simulation scenario of the shallow water area was as follows: Assumeing that the DSIB ascended at a certain initial speed from 800 m deep water and hovers at a target depth of 600 m, the depth control results for the ADRC controller and PID controller were illustrated in Fig.5.

Fig.5 Depth control curves under different controllers in relatively shallow water

It can be seen that the DSIB is affected by the net buoyancy change caused by the pressure hull deformation and water resistance during the floating process. Under such circumstances, depth control is achieved by both the ADRC controller and PID controller. For the ascending process, the settling time of ADRC algorithm is minimal, only 346 s, and overshoot is less than 1.67%, which is much smaller than that of the PID algorithm(settling time is more than 696 s and overshoot exceeds 5.3%). The depth errors of the PID controller is ±5 m, which is much larger than the ADRC controller’s ±1 m, therefore the control effects are also worse than that of the ADRC controller. Fig.6 shows the depth tracking curves of the PID depth controller and the ADRC depth controller when the DSIB ascends to 600 m and 400 m, respectively.

Fig.6 Multi-depth hovering curves under different controllers in relatively shallow water

It can be found that the aforementioned two depth controllers can achieve tracking effect on the different depths of the relatively shallow water area. Compared with PID controller, ADRC controller has a smaller overshoot and better tracking effect in the depth control process. In order to verify the stability of the PID depth controller and ADRC depth controller, when hovering at a predefined depth, external disturbance is added to the established simulation model at a certain moment, so as to observe the influence of external disturbance on the hovering motion of the DSIB. As shown in Fig.7, when the DSIB hovers at a depth of 600 m, an external disturbance of 50 N is added at 970 s, the settling time of the ADRC algorithm is less than that of the PID algorithm(585 s). Therefore, compared to the PID depth controller, the designed ADRC depth controller is better able to suppress the overshoot caused by the transient interference.

Fig.7 Anti-interference curves under different controllers in relatively shallow water

The simulation scenario of the deep water area was as follows: Assuming that the DSIB dived at a certain initial speed from 2 500 m deep water and hovered at a target depth of 2 800 m. Fig.8 shows the depth control results for the ADRC controller and PID controller.

Fig.8 Depth control curves under different controllers in relatively deep water

The results are similar to the floating process. The DSDC-ADRC has a shorter settling time at 685 s and a smaller overshoot at 1.74%. In contrast, the PID’s settling time is 1 030 s, and overshoot is more than 5.37%. Analogously, the depth errors of ADRC ARE less than THAT of PID, and the corresponding comparison results are ±1.5 m vs. ±6.5 m.

Fig.9 shows the depth tracking curves of the PID and ADRC depth controllers when the float doves to 2 800 m and 3 000 m, respectively. It can be seen that the ADRC algorithm has good control precision and low overshoot with less settling time for the different depths of deep water area.

Fig.9 Multi-depth hovering curves under different controllers in relatively deep water

According to the anti-interference depth tracking curve in Fig.10, when the DSIB hovers at a target depth of 2 800 m, a transient external disturbance of 50 N is added at 980 s, and the PID depth controller and ADRC depth controller are used to suppress the external disturbance. The ADRC controller shows a good performance of 516 s settling time while 605 s of the PID controller. Thus, for the same external disturbance, ADRC depth controller has a better anti-interference ability and stability than the PID depth controller.

In conclusion, by comparing the two methods, it can be found that the ADRC has smaller adjustment time, overshoot and depth error than the PID control method under the same external disturbance conditions. The ADRC performs better than the standard PID controller in the depth control process. For the dual closed-loop fuzzy PID depth control method[9], the overshoot of the fuzzy PID controller is 2.0%, and the depth error is within 1.0%. Compared to the fuzzy PID depth control method[9], the overshoot of the ADRC depth controller is 1.74%, and the depth error is within 0.5%. Thus, the depth positioning of the DSIB can be achieved by the ADRC method with high accuracy and fast response. The experimental results verified the effectiveness of the proposed depth controller.

Fig.10 Anti-interference curves under different controllers in relatively deep water

4 Conclusion

In this study, a buoyancy-driven deep-sea intelligent float was the object of research interest. Based on the pressure hull deformation of the DSIB, the dynamic model of the DSIB was modeled. In order to realize depth control process from the initial depth to the target depth, a depth control strategy based on ADRC method was proposed. The stability of the designed ADRC depth controller was proved by Lyapunov stability principle. Finally, the designed depth controller was verified by the semi-physical test platform. The depth controller based on ADRC method has better response ability and anti-jamming effect in comparison to the PID algorithm. The designed depth controller provides a stable operation condition of the marine sensors carried by the DSIB, and ensures the high-precision data acquisition of the DSIB on the ocean environment observation of a target depth. Since ADRC depth controller has various parameters and complicated setting process, the optimization method of ADRC needs to be further studied. In addition, the designed depth controller can be applied in the actual experiments to further verify the effectiveness of the method.