LIST OF DYNAMIC SYSTEMS 2008

8/13/2019 LIST OF DYNAMIC SYSTEMS 2008

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LIST OF DYNAMIC SYSTEMS 2008

The following are suggested systems for assignment 1 enclosed recommended references.

You can collect more information from different sources and select appropriate numerical

values of system parameters for your system.

1. Torsional Mechanical Feedback Control System (Dorf and Bishop, 2005)

A torsional mechanical system is shown in Figure 1(a). The torque due to the twisting of the

shaft is k ; the damping torque due to the braking device is b ; the disturbance torque is

d(t); the input torque is r(t); and the moment of inertia of the mechanical system is J. The

transfer function of the torsional mechanical system is:

( )( )2

1/ JG s

s b / J s k / J=

+ +

A closed-loop control system for the system is shown in Figure 1(b). Suppose the desired

angle od 0 = , k = 5, b = 0.9 and J = 1.

(a)Determine the open-loop response, ( )t , of the system for a unit step disturbance,d(t), using MATLAB (set r(t) = 0)

(b)With the controller gain K0= 50, determine the closed-loop response, ( )t , to a unitstep disturbance, d(t), using MATLAB.

(c)Co-plot the open-loop versus the closed-loop response to the disturbance input.Discuss your results and make an argument for using closed-loop feedback control to

improve the disturbance rejection properties of the system.

Braking

devicer(t), input

torque

Elastic d(t), Disturbance

torque

(a)


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( )( )2

1/ JG s

s b / J s k / J=

+ +

Figure 1 A torsional mechanical system

2. DC Motor Level Control System (Kuo and Golnaraghi, 2003) (pp. 208-209)

The following schematic diagram represents a control system whose purpose is to hold the

level of the liquid in the tank at a desired level. The liquid level is controlled by a float whose

position h(t) is monitored. The input signal of the open-loop system is e(t). The system

parameters and equations are as follows:

Motor resistance: Ra= 10 Ohms

Torque constant Ki= 10 oz-in/A

Back-emf constant Kb= 0.0706 V/rad/sec

Load inertia JL= 10 oz.-in-sec2

Amplifier gain Ka= 50

Motor inductance La= 0 H

Rotor inertia Jm= 0.005 oz-in.-sec2

Gear ratio n = N1/N2= 1/100

Load and motor friction = negligible

Area of tank A = 50 ft2

Figure 2 Liquid level system with a DC motor

K0

d(t)

r(t)

(t)d(t)

(b)


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( ) ( ) ( )a a a b me t R i t K t= +

( ) ( )m

m

d tt

dt

=

( ) ( ) ( ) ( )n2

m i a m L

d tT t K i t J n J

dt

= = +

( ) ( )y mt n t =

The number of valves connected to the tank from the reservoir is N = 10. All the valves have the same

characters are and controlled simultaneously by y . The equations that govern the volume of flow are

as follows:

( ) ( )i I yq t K N t= KI= 10 ft3

/sec-rad

( ) ( )o oq t K h t= Ko= 50 ft3/sec

( ) ( ) ( )o ivolume of tank 1

h t q t q t dtare of tank A

= =

Figure 3 Liquid level control system

Ninlet

valves


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3. Permanent-magnet DC-motor control system (Kuo and Golnaraghi, 2003)

Figure 4 Permanent-magnet DC motor control system

(See attached)

4. Process Control System (Michael L Luyben and Willim L. Luyben, Essentials of

Process Control)

Figure 5 Process control system


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5. DC motor speed control system (Kuo and Golnaraghi, 2003)

The following are parameters of an armature-controlled DC motor system:

Jm = 0.0004 kg-m^2

Bm = 0.001 Nm/rad/sec

Ra = 2 Ohms

La = 0.008 H

Km = 0.1 Nm/A

Kb = 0.1 V/rad/sec

Kt = speed sensor sensitivity (tachometer or encoder)

Simulate a DC motor speed control system with PID control law using the above motor.

6. DC motor feedback control system (Kuo and Golnaraghi, 2003)

Simulate a DC motor feedback control system with PID control law using a DC motor as in

Exercise 8 (pp. 122-123 Lecture Notes and also see Kuo and Golnaraghi, 2003).

7. Roll stabilization system


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8. Diesel-electrical control system (Dorf and Bishop, 2005)


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9. Pitch Control System (Dorf and Bishop, 2005)


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10. Additional Topics:

10.1 Ship Autopilot System using PID control law

Nonlinear model of vessel: Guidance and Control of Ocean Vehicles (Fossen, 1994)

Training Vessel, Container Vessel, Mariner, Tanker and Underwater Vehicle

(See attached Container Vessel)

10.2 Liquid level control system with a hydraulic controller or a DC motor (see Exercise

3, Exercise 5 in Chapter 2 of Lecture Notes 2007).

10.3 A dynamic system related to your final year research project(discuss with and get

approved from lecturer).


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Attachment 1 (for 3. Permanent-magnet DC-motor control system)


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Extracted from Fossen (1994), Guidance and Control of Ocean Vehicles, Wiley and Sons Inc.

122

Appendix 3.5 Mathematical Models of Marine Vehicles

In order to verify a good control design it is useful to simulate the control law against a realistic

model of the vessel. The following motion parameters will be used to describe the differentmathematical models:

uuu 0 ppp 0 0 vvv 0 rrr 0 0 (A3.5.1)

For instance, the definition implies that u is a small perturbation from a nominal (constant)surge velocity u0 while u denotes the total surge velocity. The total speed of the vessel is defined

according to:

22 vuU = 220 vuu (A3.5.2)3. Container Ship

A mathematical model for single screw high speed container ship in surge, sway, roll and yawhave been presented by Son and Nomoto (1981, 1982). The main results of this work are

presented below as three mathematical models, all describing the couplings in sway, roll and yaw.

The models are:

1. Nonlinear equations of motion in surge, sway, roll and yaw.

2. Nonlinear course-keeping equations of motion (sway, roll and yaw)3. Linearised course-keeping equations of motion (sway, roll and yaw)

The container ship is given by the following set of data:

Container Ship

Length (L) 175.00 (m)

Breadth (B) 25.40 (m)

Draft fore (dF) 8.00 (m)

aft (dA) 9.00 (m)

mean (d) 8.50 (m)

Displacement volume ( ) 21,222.00 (m3)Height from keel to transverse metacentre (KM) 10.39 (m)

Height from keel to centre of buoyancy (KB) 4.6154 (m)

Block coefficient (CB) 0.559 (-)Rudder area (AR) 33.0376 (m

2)

Aspect ratio ( ) 1.8219 (-)Propeller diameter 6.533 (m)


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3.1 Nonlinear Equations of Motion (Surge, Sway, Roll and Yaw)

Xrvmmumm yx Yplmrmrummvmm yyyyxy

x x y y x xI J p m l v m l u r W GM K = (A3.5.5)z z y y GI J r m v N Y x =

Here xm , ym , zJ and xJ denote the added mass and added moment of inertia in the x and ydirections and about the z and x axes, respectively. Furthermore, y denotes the x-coordinates ofthe centre of ym , and xI and yI the z-coordinates of the centres of xm and ym , respectively.The hydrodynamic forces and moment are:

X = ( ) ( ) ( ) 2vvvr vXrvXJTt1uX +++ + ++ sinFcXrX NRX

22

rr

K = 3rrr3vvvprv rKvKKpKrKvK + 2v

2

vv

2

vrr

2

vvr vKvKrvKrvK + 2 2rr r H R NK r K r 1 a z F cos

Y = 3rrr3vvvprv rYvYYpYrYvY (A3.5.6)+ 2v

2

vv

2

vrr

2

vvr vYvYrvYrvY + 2 2rr r H NY r Y r 1 a F cos

N = 3rrr3vvvprv rNvNNpNrNvN + 2v

2

vv

2

vrr

2

vvr vNvNrvNrvN + ( ) +++ cosFxaxrNrN NHHR

2

r

2

rr

where ( )uX is a velocity dependent damping function, e.g. ( ) uuXuXuu

= . The rudder force

NF can be resolved as

( ) R2R2R2R

N sinvuL

A

25.2

13.6F +

+

=

+=

R

R1

Ru

vtan

( )2T

PRJ

kK81uu

+= (A3.5.7)

vrcrcrcvv 2Rrrv3

RrrrRrR +++=


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where

nD

UuJ P=

( ) ( ){ }rcvcrxvw1vcosu prpv2ppP ++++= (A3.5.8)

The different parameters in the model are given below.

Model Parameters

(a) Hull only

m 0.00792pY 0.0 vvN 0.019058

x

m 0.000238

Y 0.000063

v

N 0.0053766

ym 0.007049 vvvY 0.109 rrN 0.0038592

xI 0.0000176 rrrY 0.00177 rN 0.0024195

xJ 0.0000034 rvvY 0.0214 vK 0.0003026

zI 0.000456 rrvY 0.0405 rK 0.0003026

zJ 0.000419 vvY 0.04605 0.1 ( 1Fn )

y 0.05 vY 0.00304 0.2 ( 2.0Fn )

xl 0.0313 rrY 0.009325 Fn (0.1 < Fn< 0.2)

yl 0.0313 rY 0.001368 K 0.000021

KT 0.527 0.455J vN 0.0038545 pK 0.0000075

uuX 0.0004226 rN 0.00222 vvvK 0.002843

vrX 0.00311 pN 0.000213 rrrK 0.0000462

vvX 0.00386 N 0.0001424 rvvK 0.000558

rrX 0.00020 vvvN 0.00192 rrvK 0.0010565

X 0.00020 rrrN 0.00229 vvK 0.0012012

vY 0.0116 rvvN 0.0424 vK 0.0000793

rY 0.00242 rrvN 0.00156 rrK 0.000243

rK 0.00003569

(b) Propeller and rudder

Np(rpm) 79.10 (Fn0.2)118.64 (Fn0.3)

158.19 (Fn0.4)

aH

Hx cRX

0.2370.48

0.71

k

0.9210.631

0.088 ( v > 0)0.193 ( 0v )(1 t) 0.825

Rz 0.033(1 wp) 0.816 cpv 0.0 Rrc 0.156


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Rx 0.5 cpr 0.0 cRrrr 0.275

px 0.526 1.09 cRrrv 1.96

3.2 Nonlinear Course-keeping Equations of Motion (Sway, Roll and Yaw)

Consider a ship sailing nearly straight with an automatic course-keeping device in operation.

Hence, we can assume constant forward speed ( u = 1) which implies that the above equations ofmotion can be approximated by:

( )

+

+

+

zz

xxyy

yyy

JI00

0JIlm

0lmmm

r

p

v

=

N

K

Y

(A3.5.9)

where

Y = 2 2v x r p vv vY v m m Y r Y p Y Y v Y v + 2 2rr rY r Y r Y

K = ( ) ( ) ++++ 2vvrxxvp vKrKlmvKKMGWpK (A3.5.10)+ 2v vK +

2 2

rr rK r K r K N = +++ NpNvNrN pvr + 2v2vv vNvN + + 2rrN r

+ 2rN r N with p = = ( )U/L . The non-dimensional hydrodynamic derivatives for the course-keepingmodel with KG = 10.99 m and GM = 0.3 m are given below:

( )ymm + 0.01497 pN 0.000213( )zz JI + 0.000875 N 0.0001468

( )xx JI + 0.000021 vvN 0.018191

yym 0.0003525 vN 0.005299

yylm

0.0002205

rrN

0.003684

yY 0.012035 rN 0.0023843

( )rx Ymm + 0.00522 N 0.00126

pY 0.0 0.2

Y 0.000074 K 0.000021

vvY 0.046364 vK 0.000314

vY 0.003005 ( )rxx Klm + 0.0000692

rrY 0.0093887 vvK 0.0012094


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rY 0.0013523 vK 0.0000784

Y 0.002578 rrK 0.0002449

rN 0.00243 rK 0.00003528

vN 0.0038436 K 0.0000855

3.3 Linearised Course-Keeping Equations of Motion (Sway, Roll and Yaw)

The linearised course-keeping equations of motion are:

10000m00

00mm

00mm

33

2221

1211

r

p

v

+

11 12 13 14

21 22 23 24

31 32 33 34UL

d d d d

d d d d

d d d d

0 0 0

r

p

v

=

0b

b

b

3

2

1

where

11m = ymm +

yy12 lmm =

1221 mm =

xx22 JIm +=

zz33 JIm += 2

11 v vv 0 0 v 0d Y 2Y v Y p12 Yd =

2

13 x r rr 0 0 r 0d m m Y 2Y r Y 00r

2

rr00v

2

0vv14 rY2rYvY2vYYd = 2

21 v vv 0 0 v 0d K 2K v K p22 Kd =

2

23 x x r rr 0 0 r 0d m l K 2K r K 2 2

24 vv 0 v 0 0 rr 0 r 0 0d W GM K K v 2K v K r 2K r 2

31 v vv 0 0 v 0d N 2N v N p32 Nd =

2

33 r rr 0 0 r 0d N 2N r N 2 2

34 vv 0 v 0 0 rr 0 r 0 0d N N v 2N v N r 2N r = Yb1

= Kb2

= Nb3

LIST OF DYNAMIC SYSTEMS 2008

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