Development of Algorithms for Control of Motor Boat as Multidimensional Nonlinear Object

In this paper authors develop and research system for motor boat control, that allows to move along the stated paths with the given speed. It is assumed, that boat is equipped by the measuring system that provides current coordinates, linear and angular velocities. Control system is based upon the mathematical model, presented earlier (see references). In order to analytically find the necessary controls, all equations were transformed to Jordan controllable form. Besides solution this transformation also allows to handle model nonlinearities and get required quality of movement along the stated paths. Control system includes algorithms for control of longtitudal velocity and boat course. Research of the proposed control system according to boat design limitations for the values of control variables was performed by simulation in MATLAB. Results of two experiments, different in value of the required velocity are discussed.


Introduction
One of the major tasks in creation of fast motor boats is development of control system and control algorithms providing boat movement along the stated path with the necessary speed [1,2,3].It is known, that solution of this task requires precise mathematical model describing the boat movement.In this paper, a refined mathematical model of surface boat "Neptune", introduced in [4] is used.According to this model, motor boat is a multidimensional controllable object.It has two controls and two controlled variables.Thus, boat control system can be synthesized as multidimensional system, based on its model in Jordan controllable form (JCF).This allows both handling model nonlinearities and getting required quality of movement along the stated paths by selection of control laws and algorithms.Developed nonlinear algorithm will be implemented in onboard processing unit and required input will be collected by continuous sensors.Research of closed-loop control system properties was performed by computer simulation.

Mathematical model of the motor boat
Considering taken coordinate system (Fig. 1.a), refined mathematical model of the motor boat [2,4] can be described by the following equations: where ‫̇ݔ‬, ‫̇ݖ‬ -velocities of changes of boat centre of gravity in fixed coordinate system; V x , V z -projections of velocity vector to the axes of body coordinate system XZY; φ -current boat course, where ω y =߮̇ is an angular velocity of the boat relating to its vertical axis OY.J ymoment of inertia relating to axis OY; c x , c y , c zhydrodynamic coefficients of forces and moments of resistance forces applied to the boat; m x , m y -total boat mass including additional water mass relating to OX and OZ axes.F T -total force of all engines and propellers in the boat (Fig. 1b); α -rotational angle of movable frame with engines and propellers; k -distance from moving force application point to spinning axis of the boat, equal to 0.4 meter; y 1 , y 2 -controlled variables of the boat: course and travel speed.pitching, rolling and heaving motions of the boat are not considered.It is assumed that boat moves in still water [3].Eq. ( 1) are kinematic ones as they link boat velocities with its coordinates and course.Current coordinates x g (t), z g (t) and course φ(t) could be defined by integrating Eq.
In this case we intend that required boat path is setup by the following laws: change of travel speed V * x (t) and change of boat course -angle φ * = φ * (t).Here V * x (t), φ * (t) and ω * y = dφ * /dt are given continues functions of time.
Thus, movement of the boat with the given travelling speed ܸ ௫ * ‫)ݐ(‬ is characterized by corresponding changes of course φ * (t), angular velocity ω y (t) and lateral offset V z (t).Specified boat design includes two independent controls: u 1 = F T cosα and u 2 = F T sinα.Control u 1 allows to change travelling speed V x .Control u 2 is used for course, angular velocity and lateral shift.Actually this gives and opportunity to control only two variables: angular velocity and lateral shift of the boat having course as noncontrolled or course and angular velocity having lateral shift non-controlled.In this paper the last option is taken.
Considering synthesis of the system on the base of Jordan controllable form, we will define boat equations in deviations.Let's introduce state variables -deviations: As in our case, lateral shift is non-controlled, then in steady state Eq. ( 8) follows from Eq. ( 4) ~ (7).
Consequently, in steady state, control values depend on velocity and acceleration of the boat on the desired path.This conclusion corresponds to physical laws of body movement in viscous fluids one of which is water.
So, kinematics equations (1) and dynamic equations (4) ~ ( 7) describe movement of the boat according to the desired path.Equations ( 4) ~ ( 6) can be used to synthesize the necessary control.Let's suppose that variables φ * = φ * (t), ω y = ω y (t), V z = V z (t) and V x = V x (t) are measures be special sensors.As φ * (t) and V * (t) are given functions of time, then state variables ‫ݔ‬ = 0, ݅ = 1, ݊ in Eq. ( 4) ~ ( 7) can be treated as measurable and can be used in synthesis of the system and control algorithms.

Development of control for the longtitudal velocity
Control equation ‫ݑ‬ ଵ = ‫ܨ‬ ் ‫,ߙݏܿ‬ defines modification of lateral component of the boat speed.It is included only in one equation (4).Thus, this equation can be considered as equation in JCF and control u 1 can be defined with respect to ‫ݔ‬ ଵ = ܸ ௫ − ܸ ௫ * and ‫̇ݔ‬ଵ = −ߣ ଵ ‫ݔ‬ ଵ by the following expression: (9) But this control equation for u 1 , does not provide astatism of the control system by velocity V x , and even in steady state and at constant speed it will give some error.
To zero the error, let's add integrator to the control device and temporary assign its equation to the model of the considered channel.For this u 1 should be equal to x 5 and ‫̇ݔ‬ହ = ‫ݑ‬ ଵ,௩௦ , where u 1,vsp is an assistant control.Having so, equation of channel, controlling V x velocity becomes: As ߲߶ ଵ ‫ݔ(‬ ଵ , ‫ݔ‬ ହ ‫ݔ߲/)‬ ହ = ݉ ௫ ିଵ , then Eq. ( 12) are in JCF when ߶ ଵ ‫ݔ(‬ ଵ , ‫ݔ‬ ହ ) = 1.This allows to find the necessary equation using method of analytical synthesis for nonlinear control systems using JCF.Method is described in details in [5].According to it, let's introduce new variables ‫ݓ‬ ଵ = ‫ݔ‬ ଵ , ‫ݓ‬ ଶ = ‫̇ݓ‬ଵ + ߣ ଵ ‫ݓ‬ ଵ .In this case: According to [5], in order to define control, it is required to get derivative with respect to time for variable w2.As this function directly depends on set of variables that are functions of time, then preliminary it is necessary to get partial derivative for these variables.In particular, w2 depends on function ߶ ଵ ‫ݔ(‬ ଵ , ‫ݔ‬ ଶ ) (Eq. ( 12)).And the latter depends on variable Function ߶ ଵ ‫ݔ(‬ ଵ , ‫ݔ‬ ଶ ) has multiplication |ζ|ζ that includes function modulus.Taking this into account, development of formula for derivative of this multiplication with ζ in explicit form will be: Besides, left and right parts of the Eq.(10) depend not only upon state variables, x1, x2, but also ܸ ௫ * and ܸ ̇௫ * .If ܸ ௫ * ≠ ‫ݐݏ݊ܿ‬ , then in general case ܸ ௫ * = ܸ ௫ * ‫)ݐ(‬ and ܸ ̇௫ * = ܸ ̇௫ * ‫)ݐ(‬ .So, following [5] control u1,vsp can be found using condition ‫̇ݓ‬ଶ = −ߣ ଶ ‫ݓ‬ ଶ .Taking derivative 04005-p.2ICMME 2015 with respect to time for Eq. ( 11) with respect to Eq. ( 12), ( 13) and (10) we get And with regard to Eq. ( 11): As u 1 =x 5 and ‫ݔ‬ ଵ = ܸ ௫ − ܸ ̇௫ * , then from Eq. ( 10) and ( 14) the following formulas for the astatic boat control device and be produced: In such a way, boat velocity control device is described by Eq. ( 15).Its output values gives F T and α as u 1 =F T cosα.Positive constants λ 1 and λ 2 in Eq. ( 15) define length and character of the transient process for boat velocity.When choosing these constants it is necessary to account the limitation for control modulus u 1 .Important to note that increase of constants λ 1 and λ 2 leads to shortening of the transition process but also increase the necessary |u 1 |.

Development of course control
As it was shown above, changes in boat course are described by Eq. ( 5) and (6). where (18) In this case ݀߶ ෨ ଵ ‫ݔ݀/‬ ଷ = 1 ≠ 0, so Eq. ( 16) and ( 17) are in JCF allowing to find control u 2 .In order to provide astatism by the course, let's introduce integrator into the control device.If u 2 = x 6 , then union of integrator equation ‫̇ݔ‬ = ‫ݑ‬ ଶ,௩௦ with equations ( 16) and (17) would give the following course control equation (with respect to (17): Eq. ( 19) ~ (21) are in JCF and in order to get control equations the following additional variables are required: Derivation of variable ‫ݓ‬ ଷ with respect to time and substitution of ‫ݓ(‬ ଷ ) ̇= −ߣ ‫ݓ‬ ଷ 3 with the account to Eq. ( 17) ~ (21) will produce: Character of the transition processes by variables ω y and φ depends on values of λ 5 , λ 6 , and λ 7 constants.
So, with account to Eq. ( 21) and Eq. ( 23), controlling device while calculating course and angular speed can be described by the following equation: Equations ( 15) and (24) allows to find values u 1 and u 2 .Necessary values of the real control actions -total force F T and rotational angle α can be found by the following equations: Details of the implementation of FT and α with provision for design constraints are discussed in paper [4].Presence of these limitations results in limitations in boat movement stability area.Analytical definition of this area is challenging, so in this paper authors research it by simulation in MATLAB.

Simulation of developed algorithms
The following boat values were used for the experiment:

Conclusions
This paper was devoted to the development of the automatic control system for the motor boat.It is assumed, that boat is equipped by the measuring system that provides current coordinates, linear and angular velocities.Control system was created upon the boat mathematical model, described in paper [4].To solve the problem, equation of control channels were transformed to Jordan manageable form that allowed to analytically find necessary controls.Research of the proposed control system according to boat design limitations for the values of control variables was performed by simulation in MATLAB.Now the next task is to perform the experiment with a real boat.It will be necessary to develop hardware of the control system, specify measuring system and setup the equipment.The early prototype is planned for the spring 2016.

Figure 1 .
Figure 1.Coordinate system and force projection.Eq.(1) ~ (3) do not take in to account aerodynamic forces and moments acting on the boat as they are far less then hydrodynamic forces.Flexion of the sea surface,

Figure 2 .
Figure 2. Results of the experiment 1: a) modification of boat velocity; b) rotational angle c) total force; d) modification rotational angle of movable frame.

Figure 3 .
Figure 3. Results of the experiment 2: a) modification of boat velocity; b) rotational angle c) total force; d) modification rotational angle of movable frameCharts at Fig.3show that boat did not met the required velocity because of limitation for F T .Requirement for the angle α were met correctly.Results for these two experiments prove efficiency of the proposed algorithms.