Project Report on

CONTROL SYSTEM DESIGN FOR SPEED CONTROL & DIVING AUTOPILOT FOR MAYA AUV July, 2006

Project Guide:

Prof. António M. Pascoal DSOR Lab & Dept. of Electrical Engg. Instituto Superior Técnico Lisbon Portugal

Submitted By:

Rakesh Kumar B.Tech (3rd Year) Dept. of Mechanical Engg. IIT Kharagpur India

Acknowledgement I would like to express my gratitude to Professor António M. Pascoal for his guidance, patience and motivation throughout the project work. His high level of technical competence and uncanny ability to teach provided me with a great understanding of the subject area. I would also like to thank Professor Maria I. Ribeiro, Director - ISR, for giving me this opportunity to undertake my project in ISR, Instituto Superior Técnico, Lisbon. I am also thankful to the students, researchers and professors in DSOR Lab who helped me throughout the project and made me feel at home. Finally, I would like to thank Dr. Elgar Desa at the National Institute of Oceanography, Dona Paula, Goa for having kindly granted access to the dynamic model of the MAYA -AUV.

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CONTENTS S. No. Title

1 2 2.1 2.2

3 3.1 3.2

4

4.1 4.2

5 5.1 5.2

6 7

Page No.

Abstract

3

Introduction AUV Dynamic Model

3

Kinematic Equations of Motion Dynamic Equations of Motion

Linearization about the operating conditions For Diving Plane For Longitudinal Motion

Control System Design and Implementation LQR Technique Gain Scheduling Speed Control Autopilot Diving Autopilot

MATLAB ® Codes and Simulink ® Models Speed Control Autopilot Diving Autopilot

Conclusion References

2

4 4 5 7 7 7 8 8 9 9 11 17 17 18 22 23

Abstract The report describes the design and simulation of a control system for MAYA AUV in the vertical plane for diving autopilot and for speed control in longitudinal direction. The methodology adopted here is non linear gain scheduling control where a set of linear controllers is obtained using LQR technique and then scheduled on the vehicle’s forward speed. The report summarizes the basic steps in modeling and control design for the vehicle. The simulated results are given at the end.

1

Introduction:

Small AUVs are increasingly appearing in different marine application areas particularly in Oceanography, naval applications of mine reconnaissance, and as effective tools for monitoring the coastal environment. This is not surprising as AUV technology has benefited from advances in control systems, local area networks, large memories, and high performance micro-controllers, and new technologies that are resulting in smaller size sensor payloads for marine applications. [1]. A small Autonomous Underwater Vehicle [AUV] called Maya, is now nearing development at the National Institute of Oceanography in Goa, India. Part of the research and development effort that led to MAYA was carried out in the scope of an on-going Indian-Portuguese collaboration programme that aims to build and test the joint operation of AUVs for marine science applications. The MAYA AUV has a streamlined torpedo-like body propelled by a single thruster. For vehicle maneuvering, two stern planes and a single stern rudder underneath the hull are used [2] . Figures 1 and 2 show the vehicle in Goa.

Fig 1. Maya AUV developed at NIO Goa

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Fig 2. Maya cruising underwater

2

AUV Dynamic Model:

Two coordinate frames are defined in order to derive the equations of motion. The earth frame {U} is an inertial reference frame and is composed of orthonormal axes { xG , yG , zG }. The body frame {B} is composed of orthonormal axes { xB , yB , z B } and it is attached to the vehicle. The following entities are defined [Fossen, 1994] adopting the SNAME notation:

υ = [u, v, w, p, q, r]T - Linear Velocity and Angular Velocity in {B} η = [ x, y, z, φ ,θ ,ψ ]T - Position and Attitude in {U} τ = [X, Y, Z, K, M, N]T - Acting Forces and Torques in {B}

• • •

Fig.1: Body reference frame

2.1

Kinematic Equations of Motion:

The velocity vector υ and position and attitude vector η are related through the Euler angle transformation,

η& = J (η )υ

(1)

where, ⎡cψ cθ ⎢ sψ cθ ⎢ ⎢ − sθ J (η ) = ⎢ ⎢ 0 ⎢ 0 ⎢ ⎣ 0

− sψ cφ + cψ sθ sφ

sψ sφ + cψ sθ cφ

0

0

cψ cφ + sψ sθ sφ

−cψ sφ + sψ sθ cφ

0

0

cθ sφ

cθ cφ

0

0

0 0

0 0

1 0

sφ tθ cφ

0

0

0 sφ / cθ

4

⎤ 0 ⎥⎥ 0 ⎥ ⎥ cφ tθ ⎥ − sφ ⎥ ⎥ cφ / cθ ⎦ 0

2.2

Dynamic Equations of Motion:

For underwater vehicles, the dynamic equations of motion are usually expressed in {B}. The dynamic equations for a generic rigid body are as follows:

M υ& + C (υ )υ + D(υ )υ + g (η ) = τ where M is a 6 × 6 inertia matrix including hydrodynamic added mass, C( υ ) is a matrix of Coriolis and centripetal terms (including added mass), D( υ ) is hydrodynamic damping matrix, g(η) is vector of gravitational forces and moments and τ is vector of control inputs. This equation of motions is mainly derived from first physics principles [Fossen, 1994]. The general model can be divided into three sub models. Each system consists of the state variables: 1. speed system state: u(t) 2. steering system state: v(t), r(t) and ψ (t) 3. diving system states: w(t),q(t), θ (t) and z(t) This can be done for those vehicles where the vehicles execute maneuvers that require only slight interaction between steering in the horizontal plane and diving in the vertical plane. In order to reach a model of the vehicle for maneuvers in the diving or vertical plane, some simplifying assumptions are made: • • •

The velocity of sway v should be zero so that the vehicle does not leave the diving plane. Angle of roll φ and angular velocity p are all zero. Angle of yawψ and angular velocity r are all zero.

The above simplifications lead to the following model: Surge Motion Equation:

mu& + mqw = −(W − B)sin θ + CX u 2 + CX u& u& + T

(2)

Heave Motion Equation:

mw& − mqu = (W − B)cosθ + CZw uw + CZq uq + u 2CZδ δ S + CZw& w& + CZq& q&

(3)

S

z& = −u sin θ + w cos θ

(4)

Pitch Motion Equation:

I y q& = BZCB sin θ + CM w uw + CM q uq + u 2CMδ δ S + CM w& w& + CM q& q&

(5)

θ& = q

(6)

S

5

Here C(.) are the simplified model coefficients which are related to non dimensional hydrodynamic coefficients as shown below: 1 2 1 = ρ Z w L2 2 1 = ρ Z w& L3 2 1 = ρ M w L3 2 1 = ρ M w& L4 2

1 2 1 = ρ Z q L3 2 1 = ρ Z q& L4 2 1 = ρ M q L4 2 1 = ρ M q& L5 2

C X = ρ X uu L2

C X u& = ρ X u& L3

CZ w

CZq

CZw& CM w CM w&

CZq& CM q CM q&

1 2

CZδ = ρ Zδ L2 S

1 2

S

CMδ = ρ M δ L3 S

S

Table 1: MAYA AUV simplified Model Coefficients

X uu = -0.12

X u& = -0.00062

Z w = -0.10272890

Z q = -0.01963362

Z w& = -0.02982800

Z q& = -0.00082247

Zδ S = -0.04028200 M w = -0.00832880

M q = -0.00641877

M w& = -0.00082247

M q& = -0.00190690

M δ S = -0.00786620 Table 2: MAYA AUV nondimensional hydrodynamic coefficients expressed in the body frame {B}

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3 3.1

Linearization about the operating conditions: For Longitudinal Motion:

Neglecting the interactions from other degrees of freedom, the linearized equation of motion for the longitudinal direction is given by: (m − C X u& )u& = 2C X u0u + T

3.2

(7)

For Diving Plane:

The simplified rigid-body equations of motion in heave and pitch are given by Fossen, 1994: m( w& − uoq) = Z (8) Iyq& = M (9) where u0 is the constant surge speed. Linear modeling of hydrodynamic added mass, damping and effects of stern plane deflection yields: Z = CZw& w& + CZq& q& + CZwu0 w + CZq u0 q + CZδ s u0 2δ s

(10)

M = CMw& w& + CMq& q& + CMwu0 w + CMq u0 q + CM δ s u0 2δ s − mg ( zG − zB ) sin θ

= CMw& w& + CMq& q& + CMwu0 w + CMq u0 q + CM δ s u0 2δ s − W BGθ

(11)

In addition, the moment caused by the vertical distance between the center of gravity and the center of buoyancy, BGz = zG-zB, must also be modeled. In steady state we have θ 0 = q 0 = φ 0 = 0. This suggests the following relations (dropping the delta notation for simplicity):

θ0 = q z& = −θ uo + w

(12) (13)

The above equations can be written as

⎡ m − CZw& ⎢ −C Mw& ⎢ ⎢ 0 ⎢ ⎣ 0

−CZq& Iy − CMq& 0 0

0 0 ⎤ ⎡ w& ⎤ ⎡ −CZwu0 ⎢ 0 0 ⎥⎥ ⎢⎢ q& ⎥⎥ ⎢ −CMwu0 + 1 0 ⎥ ⎢θ& ⎥ ⎢ 0 ⎥⎢ ⎥ ⎢ 0 1 ⎦ ⎣ z& ⎦ ⎣⎢ −1

−(m + CZq )u0

0

−CMq u0

BGzW

−1 0

0 uo

0 ⎤ ⎡ w ⎤ ⎡ C Z δ s u0 2 ⎤ ⎥⎢ ⎥ ⎢ ⎥ 0 ⎥ ⎢ q ⎥ ⎢ C M δ s u0 2 ⎥ δs = 0 ⎥ ⎢θ ⎥ ⎢ 0 ⎥ ⎥⎢ ⎥ ⎢ ⎥ 0 ⎦⎥ ⎣ z ⎦ ⎣ 0 ⎦ (14)

As heave velocity w is small during diving, we can neglect it in the linear model. Hence the linear model reduces to:

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⎡ CMq u0 ⎢ ⎡ q& ⎤ ⎢ Iy − CMq& ⎢θ& ⎥ = ⎢ 1 ⎢ ⎥ ⎢ ⎢⎣ z& ⎥⎦ ⎢ 0 ⎢ ⎣

BGzW − Iy − CMq& 0 −uo

⎤ ⎡ C M δ s u0 2 ⎤ 0⎥ ⎢ ⎥ ⎥ ⎡ q ⎤ ⎢ Iy − CMq& ⎥ 0 ⎥ ⎢⎢θ ⎥⎥ + ⎢ 0 ⎥ δ s ⎥ ⎢ ⎥ 0 ⎥ ⎢⎣ z ⎥⎦ ⎢ 0 ⎥ ⎥ ⎢ ⎥ ⎣ ⎦ ⎦

(15)

4 Control System Design and Implementation: This section focuses on the design of a control system for the MAYA AUV in the longitudinal plane and diving plane, based on the model presented in the previous section. The feedback controllers are designed using the LQR technique for the linearized plant model at different operating conditions (vehicle’s forward speed u). Then the controller is implemented on the non-linear model based on Gain Scheduling technique. A brief overview is given below: LQR technique: A linear quadratic regulator (LQR) can be designed to regulate the outputs of the system to zero while ensuring that they exhibit desirable time-response characteristics. Consider the state-space model:

x& = Ax + Bu ,

The technique involves choosing a control law u = − K ( x) which stabilizes the origin (i.e., regulates x to zero) while minimizing the quadratic cost function ∞

J = ∫ x(t )T Qx(t ) + u (t )T R(u )tdt 0

where Q = QT ≥ 0 and R = RT ≥ 0 . The term “linear-quadratic” refers to the linear system dynamics and the quadratic cost function. Under suitable generic stabilizability and detectability conditions, a stead-state solution to the this problem is well defined and −1 T unique and is given by K = R B P , where P is found by solving the algebraic Riccati Equation

AT P + PA − PBR −1 BT P + Q = 0 The matrices Q and R are called the state and control penalty matrices, respectively. If the components of Q are chosen large relative to those of R, then deviations of x from zero will be penalized heavily relative to deviations of u from zero. On the other hand, if the components of R are large relative to those of Q, then control effort will be more costly and the state will not converge to zero as quickly.

8

Gain-scheduling: It is a divide and conquer approach for the design of nonlinear control systems which has been applied in fields ranging from aerospace to process control. The conventional gain-scheduling design approach typically involves (Rugh & Shamma, 2000; Silvestre et al., 2002):

Step 1. Linearize the plant about a finite number of representative operating points. Step 2. Design linear controllers for the plant linearizations at each operating point. Step 3. Interpolate the parameters of the linear controllers of Step 2 to achieve adequate performance of the linearized close loop system at all points where the plant is expected to operate. Step 4. Implement the above gain scheduled controller on the non linear plant. The implementation of the non linear gain scheduled controller is done using the Dmethodology described in Kaminer et al. (1995). Here the control system is so designed that some of the outputs have to be differentiated and an integral action is provided at the input to the plant. This ensures that no feed forward is required and that the transfer is “bump less”. Other than the cases where the derivatives have physical meaning and are available from dedicated sensors, differentiation of the plant’s output cannot be done in practice. Hence the differentiation operation is replaced by a causal transfer function s /(ε s + 1) with ε > 0 so that the linearization property is recovered asymptotically as ε → 0 .

4.1

Speed Control Autopilot:

The linearized equation for longitudinal motion is obtained as shown in the previous sections. An integrator is added after the plant and the augmented State Space matrices take the following form: ⎡ 2C X uo ⎡ u& ⎤ ⎢ ⎢ξ& ⎥ = ⎢ (m − C X u& ) ⎣ ⎦ ⎢ 1 ⎣

1 ⎤ ⎡ ⎤ 0⎥ ⎡u ⎤ ⎢ (m − C X u& ) ⎥ T ⎥ ⎢ξ ⎥ + ⎢ ⎥ ⎣ ⎦ ⎢ ⎥ ⎥⎦ 0 0⎦ ⎣

(16)

⎡u ⎤ y = [1 0] ⎢ ⎥ ⎣ξ ⎦

(17)

For the speed control autopilot, we design the controllers by pole placement technique as the closed loop system is fairly simple. We choose ξ =0.7 and ωn =1 rad/s as our design specifications. The feedback gains thus obtained are: K1 = 1.4(m − C X u& ) + 2C X uo K 2 = m − C X u&

9

The closed loop eigenvalues obtained for the system are • -0.7000 + 0.7141i • -0.7000 - 0.7141i

Fig. 2: Step Response

Fig. 3: Bode Plot

10

Non linear control implementation Since the controllers obtained are functions of u0, hence we can implement them directly on the non linear plant as the gain scheduled controllers with the scheduling variable as u0. Simulation result obtained with gain scheduled nonlinear controller is given below:

Fig. 4: Desired and Obtained speed for non linear plant.

4.2

Diving Autopilot:

Open loop system analysis: Linearization of the non linear plant about equilibrium point determined by (q, θ , z )T = (0, 0, 0)T , u0 =1.5 m/s and δ S = 0 leads to linear model with state space and input matrices

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⎡ −1.8247 −0.1266 0 ⎤ A = ⎢⎢ 1.0 0 0 ⎥⎥ ⎢⎣ 0 −1.5 0 ⎥⎦ ⎡ −1.8635⎤ B = ⎢⎢ 0 ⎥⎥ ⎢⎣ 0 ⎥⎦ The resultant linearized model gives 3 real eigenvalues: two stable eigenvalues at -1.7525 rad/s and -0.0722 rad/s and one eigenvalues at zero that corresponds to the pure integrator from q to θ . The control system for the linearized plant is designed using LQR method. The weighing matrix C (Q = C '× C ) is chosen as [1 10 0.5 0.1]. The Control matrix (K)and eigenvalues (E) for u0 = 1.5 m/s are thus given by: • •

K = [-2.5677 -10.8192 1.7421 0.1000] E = [-3.1768+2.9248i -3.1768 - 2.9248i

-0.1654

Step response and Bode plots for the closed loop system are:

12

-0.0906]

The values of K and E obtained for different longitudinal speeds are: For u0=1.0 m/s • K = [-3.8715 -10.9127 2.1126 0.1000] • E = [-2.1085+1.9596i -2.1085-1.9596i

-0.1280

-0.0781]

For u0=2.0 m/s • K = [-1.9181 -10.7447 1.5259 0.1000] • E = [-4.2439+3.8920i -4.2439-3.8920i -0.1998

-0.1000]

For u0=2.5 m/s • K = [-1.5303 -10.6889 1.3810 0.1000] • E = [-5.3114+4.8592i -5.3114 - 4.8592i -0.2322

-0.1076]

Taking u0 as the scheduling variable, we implement the controller on the nonlinear plant. Anti windup scheme is also used in the above control system. The simulation results obtained for different speeds are given below:

13

14

15

From the above simulations we can see the vehicle is diving smoothly. The heave velocity remains small as assumed while designing the controllers for the linear plant. The pitch angle of the vehicle also remains such that the vehicle doesn’t turn by awkward angles during diving. Also the stern angle remains small enough so that the diving is smooth. For the implementation of controllers on the non linear plant we also make use of pre-filter and anti windup scheme: Pre-filter: Usage of pre-filter enables us to smoothen the reference command (desired depth, desired surge velocity etc.) given to the control system. Its effect can be seen in the simulation results. Anti windup scheme: The windup effect occurs in all control systems where an integrator is used in the controller. The problem is obviously that the controller continues to integrate though the manipulated variable has already reached its bound. As the controller output further grows unnecessarily, this is called the windup effect. The goal of an anti-windup measure is to counteract the integration of the controller when such a case arises. This can be performed by feeding back the difference u − uC to the controller. This scheme has been applied in the model in order to counteract the windup effect when stern angle saturates at ±30° .

Fig.: Step response for a closed loop system with and without anti windup scheme

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5 5.1

MATLAB ® Codes and Simulink ® Models: Speed Control Autopilot

Simulink

®

Model

Fig.: Simulink ® model for speed control

MATLAB ® code for initialization: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Speed Control Autopilot %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% g=9.8; m=53; W=m*g; B=53.5*g; Zg=0.52e-2; Zb=-0.172e-2; Iy=9.921; L=1.8; rho=1025; Uo=1.5;

%mass in kg %weight in N %buoyancy in N %in Kg m^2 %density of sea water in Kg/m^3 %forward speed

X_uu=-0.12; X_udot=-0.00062;

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Cx=.5*rho*X_uu*L^2; Cxudot=.5*rho*X_udot*L^3; e=.001; A11=[ 2*Cx*Uo/(m-Cxudot) 0; 1 0]; B11=[1/(m-Cxudot); 0]; C11=[1 0]; D11=0; K1=1.4*(m-Cxudot)+2*Cx*Uo; K2=m-Cxudot; K=[K1 K2];

5.2

Diving Autopilot

Simulink

®

Model

Fig.: Simulink ® model for depth control 18

MATLAB ® code for initialization: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Diving Autopilot Initialization %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% g=9.8; m=53; %mass in kg W=m*g; %weight in N B=53.5*g; %buoyancy in N Zg=0.52e-2; Zb=-0.172e-2; Iy=9.921; %Kg m^2 L=1.8; rho=1025; %density of sea water in Kg/m^3 Ka=1; %Gain for anti windup X_uu=-0.12; X_udot=-0.00062; Z_q=-0.01963362; Z_qdot=-0.00082247; Z_w=-0.10272890; Z_wdot=-0.02982800; Z_deltas=-0.04028200; M_w=-0.00832880; M_wdot=-0.00082247; M_q=-0.00641877; M_qdot=-0.00190690; M_deltas=-0.00786620; Cx=.5*rho*X_uu*L^2; Cxudot=.5*rho*X_udot*L^3; Czq=.5*rho*Z_q*L^3; Czqdot=.5*rho*Z_qdot*L^4; Czw=.5*rho*Z_w*L^2; Czwdot=.5*rho*Z_wdot*L^3; Czdeltas=.5*rho*Z_deltas*L^3; C_Mw=.5*rho*M_w*L^3; C_Mwdot=.5*rho*M_wdot*L^4; C_Mq=.5*rho*M_q*L^4; C_Mqdot=.5*rho*M_qdot*L^5; C_Mdeltas=.5*rho*M_deltas*L^3; B_zCB=-(Zg-Zb)*W; Uo=1.5; A11=[(C_Mq*Uo)/(Iy-C_Mqdot) B_zCB/(Iy-C_Mqdot) 0; 1 0 0; 0 -Uo 0]; B11=[C_Mdeltas*(Uo^2)/(Iy-C_Mqdot); 0; 0]; C11=[0 0 1];

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D11=0; [numl,denl]=ss2tf(A11,B11,C11,D11); roots(denl) %Applying LQR technique for finding Control Matrix K%% Anew=[A11 zeros(3,1); 0 0 1 0]; %Adding integrator Bnew=[B11;0]; %after plant Cnew=[C11 0]; Dnew=0; e=.001; wq=0.1; wtheta=10; wz=0.5; wIntz=0.1; Cbar=[wq wtheta wz wIntz];% Cbar=[wq wtheta wz wIntz]; Q=Cbar'*Cbar; rhoo=1; [K,S,E]=lqr(Anew,Bnew,Q,rhoo); K1=K(1); K2=K(2); K3=K(3); K4=K(4); Kx=K(1:3); Kzdot=K(4); Ac=[A11-B11*Kx B11*Kzdot -Cll 0 ]; Bc=[0;0;0;1]; Cc=[0 0 1 0]; Dc=0; Clsys=ss(Ac,Bc,Cc,Dc); eig(Clsys) figure, subplot(211) bode(Clsys); title(sprintf('Cut Off Frequency= %g(rad/s)\nwq=%g wTheta=%g wz=%g wIntz=%g',bandwidth(Clsys),wq,wtheta,wz,wIntz));,grid on; subplot(212) step(Clsys),grid on;

S-Function for Non Linear Model of AUV %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% S-Function for Non Linear Model %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [sys,x0]=AUV_NL_sfunc(t,state,input,flag) %state = [w, q, theta, z]

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%input = [u, delta_s] %output = [w, q, theta, z] %% Coefficients for MAYA %% g=9.8; m=53; %mass in kg W=m*g; %weight in N B=53.5*g; %buoyancy in N Zg=0.52e-2; Zb=-0.172e-2; Iy=9.921; %Kg m^2 L=1.8; rho=1025; %density of sea water in Kg/m^3 X_uu=-0.12; X_udot=-0.00062; Z_q=-0.01963362; Z_qdot=-0.00082247; Z_w=-0.10272890; Z_wdot=-0.02982800; Z_deltas=-0.04028200; M_w=-0.00832880; M_wdot=-0.00082247; M_q=-0.00641877; M_qdot=-0.00190690; M_deltas=-0.00786620; Cx=.5*rho*X_uu*L^2; Cxudot=.5*rho*X_udot*L^3; Czq=.5*rho*Z_q*L^3; Czqdot=.5*rho*Z_qdot*L^4; Czw=.5*rho*Z_w*L^2; Czwdot=.5*rho*Z_wdot*L^3; Czdeltas=.5*rho*Z_deltas*L^3; C_Mw=.5*rho*M_w*L^3; C_Mwdot=.5*rho*M_wdot*L^4; C_Mq=.5*rho*M_q*L^4; C_Mqdot=.5*rho*M_qdot*L^5; C_Mdeltas=.5*rho*M_deltas*L^3; B_zCB=(Zg-Zb)*W;

if flag == 0 %return system dimensions and initial state %[cont. state, discr. state, outputs, inputs., disc. roots, feedforward] sys=[4, 0, 4, 2, 0, 0]; x0 = [0,0,0,0]; %[w0, q0, theta0, z0] elseif flag == 1 %return state derivatives w = state(1);

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q = state(2); theta = state(3); z = state(4); u = input(1); delta_s = input(2); nu_1 = [w; q]; A11 = [(m-Czwdot), -Czqdot; -C_Mwdot, (Iy-C_Mqdot)]; A11_inv = inv(A11); B11 = [Czw*u, (m+Czq)*u; C_Mw*u, C_Mq*u]; C11 = [Czdeltas*u*u*delta_s; C_Mdeltas*u*u*delta_s]; D11 = [(W-B)*cos(theta); B_zCB*sin(theta)]; dot_nu_1 = A11_inv*B11*nu_1 + A11_inv*C11 + A11_inv*D11; dot_theta = q; dot_z = -u*sin(theta) + w*cos(theta); sys=[dot_nu_1', dot_theta, dot_z]'; elseif flag == 3 %return system outputs output = state; sys = output; else %do nothing sys = []; end

6 Conclusion: This report described the design and simulation of the control system for speed control autopilot and diving autopilot for MAYA AUV. The general setup adopted for control design was non linear gain scheduling control where the feedback controllers were designed using LQR technique. The gain scheduling technique was implemented using the D methodology the advantage of which has already been stated above. The assumption that heave is small is also justified from the results that we obtain during the non linear implementation of the controller for diving autopilot. The simulation results show that the control system is working efficiently for speed ranges between 1 m/s and 2.5 m/s.

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7

References: [1] Mechanical design and development aspects of a small AUV – Maya, R. Madhan, Elgar Desa, S. Prabhudesai, Ehrlich Desa, A. Mascarenhas, Pramod Maurya, G. Navelkar, S. Afzulpurkar, S. Khalap, L.Sebastiao, to appear in the Proceedings of the MCMC´2006 Conference, Lisbon, Portugal, September 2006. [2] Control of the MAYA AUV in the Vertical and Horizontal Planes: Theory and Practical Results, P. Maurya, E. Desa, A. Pascoal, G.Navelkar, R Madhan, A. Mascarenhas , S. Prabhudesai, S. Afzulpurkar, A. Gouveia, S. Naroji, L. Sebastiao, to appear in the Proceedings of the MCMC´2006 Conference, Lisbon, Portugal, September 2006. [3] T. I. Fossen. Guidance and Control of Ocean Vehicles. John Wiley & Sons, New York, USA, 1994. [4] N. S. Nise. Control Systems Engineering. John Wiley & Sons, New York, USA, 2000. [5] G. F. Franklin, J. D. Powell, A. Emami-Naeini. Feedback Control of Dynamic Systems. Prentice Hall, New Jersey, 2002 [6] M. A. Abkowitz. Stability and Motion Control of Ocean Vehicles. The MIT Press, Cambridge, Massachusetts, USA, 1969 [7] I. Kaminer, A. M. Pascoal, P. P. Khargonekar, E. E. Coleman (1995). A velocity Algorithm for the Implementation of Gain Scheduled Controllers. Automatica, Vol. 31, No. 8, pp. 1185-1191 [8] C. Silvestre, A. M. Pascoal (2004). Control of the INFANTE AUV using gain scheduled static output feedback. Control Engineering Practice 12, 1501-1509 [9] B. Jalving. The NDRE-AUV Flight Control System. IEEE Journal of Oceanic Engineering, Vol. 19, No. 4, October 1994 [10] Healey, A.J. Underwater Vehicle Dynamics and Control. Unpublished Notes, Naval Postgraduate School, Monterey, CA. 93943 [11] C. Silvestre, A. M. Pascoal. Infante AUV Dynamic Model. DSORL-ISR Technical Report, Instituto Superior Técnico, Lisbon, Portugal, 2001.

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