A LQR controller for an AUV depth control.

Burlacu, Paul ; Dobref, Vasile ; Badara, Nicolae 等

Abstract: This paper presents a design for the control of depth changing system for an autonomous underwater vehicle. This design is also based on a linearized model of the underwater vehicle, because the general model, of the course is nonlinear. The simulation on the vertical trajectory implies a lot of characteristics, characteristics which are obtained from direct observations and measurements.

Keywords: underwater vehicle, linearization, LQR controller

1. INTRODUCTION

This article presents a development of the control systems for depth changing, system which is considered to be decoupled from the entire system of the underwater vehicle control.

The motion equations linearization will be made around the equilibrium point, where we can approximate the entire behavior of the vehicle. For the close loop scheme we need to synthesize a controller. We have already chosen an LQR controller because this one fits our underwater vehicle needs.

2. THE AUTONOMOUS UNDERWATER VEHICLES EQUATIONS

To obtain diving equations of motion we must begin with the general motion equations. The matrix form (Fossen, 1994) of the nonlinear motion equations has the form:

[M.sub.RB][??] + [C.sub.RB](v)v = [[tau].sub.RB] (1)

where: [M.sub.RB]--inertial matrix;

[C.sub.RB]--coriolis forces and centripetal forces of rigid body matrix; [[tau].sub.RB]--forces and moments vectors of rigid body.

Further, the other form of the equations (1) is:

M[??] + C(v)v + D(v)v + g([eta]) = [tau] (2)

where

M = [M.sub.RB] + [M.sub.A]; C(v) = [C.sub.RB](V) + [C.sub.A](v) (3)

[M.sub.A]--adherent masses forces matrix;

[C.sub.A]--coriolis forces and centripetal forces of adherent masses forces matrix;

D(v)--total matrix of damping forces;

[tau]--propulsion forces and moments.

The linearized forms for these equations (Fossen, 1994) are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

[??] = A(t)x + B(t)u (5)

The diving equations of motion for the underwater vehicle should include the heave velocity w, the angular velocity in pitch q, the pitch angle [theta], the depth z and the stern plane deflection [delta]s. Assuming that the forward speed is constant and the sway and yaw modes can be neglected, the pitch and heave kinematics can be perturbed.

3. THE REPRESENTATION OF THE OPEN LOOP SIMULATION FOR THE NONLINEAR SYSTEM

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

In the beginning it is necessary to start with the simulation for the nonlinear model in the open loop scheme. The structure of our underwater vehicle is presented in fig. 1.

For this simulation it has been considered that the extern perturbations zero and the fins angles having the same value, zero. The open loop scheme is presented in fig. 2.

After studying the diagram from the figure 3 we can conclude that, the fixed part of the system, which is the vehicle body, has an unstable dynamic behavior for the both planes, horizontal and vertical.

4. THE REPRESENTATION OF THE CLOSED LOOP SIMULATION FOR THE LINEAR SYSTEM

If we consider the necessity of the underwater vehicles we should synthesize some controllers for changing and maintaining the depth and for the steering mode of an underwater vehicle.

Further, it will consider just the system for the depth changing. Starting with the linearised form of movement equations, we consider that, the command vector [delta] is formed by the angles from the four fins and the external forces and moments.

[??] = A(t)x + B(t)[B.sup.+][[[delta] [u.sub.i].sup.T]] (6)

where:

[delta] = [[[delta.sub.1] [delta.sub.2] [delta.sub.3] [delta.sub.4].sup.T]]--the fins angles vector;

[u.sub.1] = [[X Y Z K M N].sup.T]--the external forces and moments vector.

For the simulation, the linearized model will be around the equilibrium points where we have the values for the hydrodynamic coefficients:

--nominal forward velocity--3 m/s;

--damping force--416 N;

--nominal depths--0; 7 and 14 meters. The equilibrium point is:

[x.sub.0] = [[0 0 0 0 0 3 0 0 0 0 0 0 0].sup.T] with the entrances

[u.sub.0] = [[0 0 0 0 0 416 0 0 0 0 0].sup.T]

The model of the system will be:

[??] Ax x + = Bu (7)

with the matrixes:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII](9)

For the 7 and 14 meters depths the linearised models will have the same structure because the depth doesn't count for the system dynamic. After the decoupling the control systems, the equation for depth changing has the form:

[??] = ax + b[delta] (10)

where: x = [[z [theta] u w q].sup.T] (11)

[delta] = [[[delta.sub.1] [delta.sub.2] [delta.sub.3] [delta.sub.4].sup.T]] (12)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

The system has the eigenvalues:

e=0, -4.6172, 0.3433+0.33171*i, 0.3433-0.33171*i,-0.1852

It is obvious that the decoupled system for depth changing is at the limit of the stability. We will fit an optimal controller for the system. The closed loop system is shown in figure 4. Two simulations have been made:

1. First simulation: initial depth 0 to depth 7 (fig.5).

2. Second simulation: initial depth 7 to depth 14 (fig. 6).

5. CONCLUSION

The system is not total decoupled so this was the reason for using an optimal controller. Analyzing those two simulations in the present paper, we can conclude that the dynamic system doesn't depend on the depth.

6. REFERENCES

Fossen, T. I. (1994) Guidance and Control of Ocean Vehicles, Ed. John Wiley&Sons Ltd., Balfins Lane, Chichester, West Sussex, England, 1994.

Burlacu, P. (2006) Contributions in control systems with applications to underwater weapon techniques Ph. D. Thesis Tehnicall University Cluj Napoca, 2006.