ijms 40(2) 200-206ff

Indian Journal of Geo-Marine Sciences

Vol. 40(2), April 2011, pp. 200-206

Depth and pitch control of USM underwater glider: performance comparison

PID vs. LQR

Maziyah Mat Noh, Mohd Rizal Arshad & Rosmiwati Mohd Mokhtar USM Robotics Research Group, School of Electrical and Electronic Engineering ,Universiti Sains Malaysia,

Engineering Campus, 14300 Nibong Tebal, Seberang Perai Selatan, Pulau Pinang, Malaysia

[E-mail: [email protected], [email protected],[email protected]]

Received 23 March 20011; revised 28 April 2011

Present study consists the design of an optimal state feedback controller for our USM Underwater Glider (USM-

Glider). The glider mathematical model for motion control is obtained using MATLAB System Identification Toolbox,

where ballast pumping rate is an input signal to the system. Different parameters were observed independently, the pitching

angle, and the depth. Two different models were obtained for the respective observations that relates ballast pumping rate to

pitching angle and depth. From the transfer obtained, we apply a Linear Quadratic Regulator (LQR) and PID control

schemes to observe the performance of the controllers over the pitching angle, and the depth. The optimal performances are

obtained via tuning of Qs and Rs matrices of the LQR and gain of Kp, Ki and Kd of the PID controller. The results show both

controllers provide satisfactory performance.

[Keywords: Underwater Glider, Model Order Reduction, System Identification, PID, LQR]

Introduction

An underwater glider is a buoyancy-propelled and

fixed-winged glider that shifts internal actuators to

control attitude. Underwater glider has greater range

than other underwater vehicle, and does not need

expensive support vessels. This idea was inspired by

Stommel in1 and has been realized after almost

10 years. The first generation gliders have been

developed and tested in ocean including SLOCUM

glider2, Seaglider glider

3 and Spray glider

4. Currently

many laboratory-scale gliders progressively have been

developed and tested including ALBAC (University

of Tokyo), ROGUE (University of Princetone),

STERNE (Ecole Nationale Superieure D’Ingenieurs –

INSIETA) and ALEX (Osaka Perfecture University).

After developing the glider, controller is needed to

ensure the glider designed is gliding steadily.

Feedback control is the most commonly used method

in designing controller for glider motion5-9

. In [6]

combination of feedforward and feedback control was

used to get a better result compare to feedback control

alone. Among feedback control PID is the most

popular used due to its simple architecture and

controller tuning parameters as used in6-7

. The optimal

control so called Linear Quadratic Regulator (LQR) is

another approach has been used in5,8-9

, where in this

method only two tuning knobs (Q and R) need to be

varied in order to obtain an optimal gain that

minimizing the cost function and be a solution for the

Ricatti equation. Another approach is Hierarchical

Supervisory Control (HSC)10

, and Decentralized

Supervisory Control (DSC)11

both are based on RW

(Ramadge & Wonham) supervisory control theory of

Discrete Event Dynamic System (DEDS),where the

control implementation has been divided into layers

which mission, task, and behavior layers , and motion

control is fall under behaviour layer. Detail

implementation is explained in10-11

. The most recent

approach used is Model Predictive Control (MPC)12

.

In this approach, the authors divided the dynamic into

two-level hierarchical controller by decompose the

problems into two simple control problems that are

attitude control loop as high level controller to control

ballast and internal sliding mass position, and low-

level control for internal configuration control loop to

check the control inputs have matches the high-level

references.

This paper presents assessment of PID and LQR

controllers on a nonlinear glider. The Simulation

works of the nonlinear control design of glider system

for pitching angle, and depth control that having one

control input are performed in Matlab software and

the results of the responses are presented in time

domain. The outline of this paper is as follows,

Section Method discusses on methods which include

the glider system, modelling of glider system, model

NOH et. al: DEPTH AND PITCH CONTROL OF USM UNDERWATER GLIDER

201

identification, and controller design. Section Results

and Discussion discusses the controller performance

on glider system through simulation works. Final

section is conclusion where we reiterates the main

contribution of the work and highlights some of the

possible future improvement.

Material and Methods

Glider System

All the symbols used throughout this paper are

shown in Table 1. The general kinematic and dynamic

equations of the glider system designed is based on

Leonard and Graver13

; is shown in Figure 114

.

Reference frames are established that is inertial frame

and body frame. We consider inertial frame is the

non-rotating reference frame of i, j and k. Let i and j

inertial axes lie in the horizontal perpendicular to

gravity. The k axis lies in the direction of the gravity

vector and is positive downwards. The inertial value

for k=0 coincides with the water’s surface, in which

case k is depth.

The body frame is coordinate frame fixed to the

glider body with its origin at the glider center of

buoyancy (CB) as shown in Figure 114

.

In this paper, the vertical plane dynamics is

controlled using movable mass and variable ballast

mass is considered. Since the fix rudder is used to

stabilize the glider straight motion in vertical plane,

therefore the lateral dynamics can be ignored, hence

all the lateral components such as V2 are equal to zero

except for the pitching component, Ω2. This is reason

why the study for the vertical plane dynamics and

representative of actual glider operation are useful.

The following vertical motion equations suggested by

Leonard and Grave5 is used in

14 to validate the

estimation model during the system identification

process. All the symbols used through this paper are

shown in Table 1.

θθ sincos 31 vvx += … (1)

θθ cossin 31 vvz +−= … (2)

2Ω=θ … (3)

Table 1Description of the symbols

Symbol Description

α Angle of attack

θ Pitching angle

i, j, k Inertial reference frame

b =[x, y, z]T Body position vector from

inertial frame

D Drag force

J = Jh + Jf =diag(J1, J2, J3) Total Inertia Matrix

L Lift force

M = msI +Mf = diag(m1,m2,m3) Sum of body and added mass M

DL Viscous moment

m Mass of displaced fluid

mb Variable ballast mass

m Internal movable mass

m0 = mv - m Excess mass

Ω=[Ω1, Ω2, Ω3]T Angular velocity

PP =[PP1, PP2, PP3]

T

Linear momentum of m

R Rotation matrix

rP =[rP1, rP2, rP3]T Position of the internal

movable mass

v=[v1, v2 ,v3]T Translational velocity

Fig. 1Glider dynamics14

INDIAN J. MAR. SCI., VOL. 40, NO. 2, APRIL 2011

202

( )

2 3 1 1 3 1 3 3 2

2

1 3 3 1 1 3

1[( ) ( )

cos sin )]DL

m m v v rPp rp Pp mgJ

rp rp M rp u rp uθ θ

Ω = − − + Ω −

+ + − +

… (4)

( )10232331

1 cossinsin1

uDLgmPpvmm

v −−+−Ω−Ω−= ααθ

… (5)

( )30212113

3 sincoscos1

uDLgmPpvmm

v −−+−Ω−Ω−= ααθ

… (6)

( )23111

1Ω−−= PPP rvP

mr

… (7)

( )21333

1Ω−−= PPP rvP

mr

… (8)

11 uPP =

… (9)

33 uPP =

… (10)

4umb = … (11)

where α is the angle of attack, D is drag, L is lift, MDL

is viscous moment, Ѳ is pitching angle, Ω is angular

velocity, rp is position of movable mass, Pp is linear

momentum, x and z are components of vehicle

position, and v1 and v3 are velocity with respect to

fixed body cord as shown in Figure 1 and appears in

equations 1 - 11. The estimation of hydrodynamics

forces acting on the underwater glider are obtained

using CFD analysis as explained in15-16

. The

simulation of motion is also explained in15-16

.

Model Identification

The System Identification (System Id) method has

been used to observe the ballast system (pumping

rate) of the USM-Glider causes changes in glider

motion behaviour. For this purpose, the input signal

(pumping rate/ballast rate) are positive and negative

pulses to command the ballast to pull and push the

water from the ballast tank. Then this input will

manipulate the mass distribution in the vehicle , hence

the vehicle dynamics are observed. The pitching angle

and the depth have been observed independently. The

detail implementation is explained in 14

. From 14

, we

have the following mathematical models that relates

the input (ballast rate) to the outputs. Using the

models obtained, we construct the controller

accordingly.

05.033.02

37.0

369.1

417.1

5083.1

695.1

7

56.92

7.21

384.16

594.21

61.10

)(

+−−

+−+−

+−

++

=

zz

zzzzz

zz

zzz

zPitch

G

… (12)

977.0934.12

063.03

02.24

665.02

327.13

662.0)(

−++−

−+−=

zzzz

zzzzDepthG

… (13)

All the transfer functions obtained during the

identification process14

are converted into continuous

time using ZOH approximation with sampling time Ts

= 0.5 seconds. This conversion is done using Matlab

simulation software. As a result the following transfer

functions are obtained.

82.1992.788.2762714

5.94229923.75978.7

10335.11086.1

10172.110256.2

67591.184108713.7

)(

23

45678

55

2534

4568

++++

++++

++

++

++−−

=

sss

sssss

xsx

sxsx

ssss

sG pitch

… (14)

03994.0016.01263.0

48.3909031.0

04793.02351.044.26606.66765.0

)(

2

345

234

+−

++

−+−−−

=

ss

sss

ssss

sGdepth

… (15)

Controller Design

In this section, a PID and LQR control schemes are

proposed and described in detail. The design

objectives to examine the performance of both

controllers in terms of system overshoot (OS%),

settling time (Ts) and the steady-state error (ess). The

gain of PID controller and the Qs and Rs values were

tuned heuristically.

PID Controller

The design methods for PID controllers are

typically based on a time-domain or frequency-

domain performance criterion. However, the


203

relationships between the dynamic behavior of the

closed-loop system and these performance indices

are not straightforward. A proportional–integral–

derivative controller (PID controller) is a generic

control loop feedback mechanism (controller) widely

used in industrial control systems. This is a type of

feedback controller whose output, a control variable

(CV), is generally based on the error (e) between

some user-defined set point (SP) and some measured

process variable (PV). The PID controller calculation

(algorithm) involves three separate parameters, and is

accordingly sometimes called three-term control: the

proportional, the integral and derivative values,

denoted Kp, Ki, and Kd. Each element of the PID

controller refers to a particular action taken on the

error.

• Proportional: error multiplied by a gain, Kp.

This is an adjustable amplifier. In many

systems Kp is responsible for process stability:

too low and the PV can drift away; too high

and the PV can oscillate.

• Integral: the integral of error multiplied by a

gain, Ki. In many systems Ki is responsible for

driving error to zero, but to set Ki too high is to

invite oscillation or instability.

• Derivative: the rate of change of error

multiplied by a gain, Kd. In many systems Kd is

responsible for system response, if set too high

then the PV will oscillate; if set too low then

the PV will respond sluggishly.

The structure of the PID control scheme is shown

in Figure 2.

In this study the proportional, integral and

derivative gains are obtained through manual tuning

process. We did not use Ziegler-Nichols or Cohen-

Coon tuning methods due to very oscillatory open

loop response as shown in Figures 4 and 5. Through

manual tuning process of the proportional, integral

and derivative gains shows that the optimum response

of PID controller for controlling pitching angle is

achieved by setting Kp = 0.00003, Ki = 1.5, and Kd =

82. For depth control the optimum response is

achieved by setting Kp = 0.001, Ki = 0.001, and Kd =

0.01.

Linear Quadratic Regulator (LQR)

LQR is a method in modern control theory that

uses state-space approach to analyze such a system.

This the standard optimal control design which

produces a stabilizing control law that minimizes a

Fig. 2The PID controller structure

Fig. 3The LQR structure

Fig. 4Pitching angle

Fig. 5Depth


204

cost function, J that is weighted of sum of squares of

the states and input variables. By determines the

feedback gain matrix that minimizes J, we can

establish the trade-off between the use of control

effort, the magnitude, and the speed of response that

will guarantee a stable system. Assume that all the

states are available for feedback. The cost function is

to be minimized is defined as

dttuRtutQxtxJTT

)()()()(

0

∫∞

+=

… (16)

Where Q is symmetric positive semi-definite state

penalty matrix and R is symmetric positive semi-

definite control penalty matrix. Choosing Q relatively

large than 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 R is relatively large

than those of Q, then control effort will be more

costly and the state will not converge to zero as

quickly as we wish. The tracking performance of the

LQR applied to the underwater glider was

investigated by setting the value of vector K and N

which determines the feedback control law and for

elimination of steady state error capability

respectively. The LQR structure is shown in Figure 3.

The transfer functions are converted into state

space form to enable LQR control to be constructed

over the system. After conversion the following

matrices are obtained.

[ ]

[ ]

0

60.3239.1158.334.041.004.011.012.0

000000064

025.0000000

005.000000

0005.00000

00004000

00000400

00000040

000000016

31.031.054.0650.268.367.4702.498.7

=

−−=

=

−−−−−−−−

=

pitch

pitch

Tpitch

pitch

D

C

B

A

… (17)

[ ]

[ ]

0

05.006.065.121.017.0

00004

025.0000

0025.000

00050.00

00008

16.002.003.093.409.0

=

−−−−=

=

−−−−

=

depth

depth

Tdepth

depth

D

C

B

A

… (18)

Q and R matrices are tuned to obtain the

satisfactory results. The value Q and R are set to be

diag(500,50,10,10,0.5,0.5,0.5,5), and 0.005 for

pitching control and diag(80000,0.05,0.8,0.1,0.01),

and 0.01 for depth control. By setting such Qs and Rs

values we obtain the following Kpitch and Kdepth.

Kpitch = [316.3 215.6 230.9 133.9 40.7 55.5 43.1 31.6]

Kdepth = 1000 *[2.8 0.1 0.03 0.01 0.001]

In order to reduce the steady state error of the

system output, a value of constant gain, Nbar should

be added after the reference. With a full-state

feedback controller all the states are fed back. The

steady-state value of the states should be computed,

multiply that by the chosen gain K, and use a new

value as the reference for computing the input. The

Nbar can be found using the user-defined function

which can be used in m-file code. The value of

constant gains, pitching is 0.97 and depth is -20.88.

Results and Discussion

In this section, the proposed control scheme is

implemented and tested within the simulation

environment of the glider system and the

corresponding results are presented. The system

responses namely pitching angle, and depth, are

observed. The performances of the control schemes

are assessed in terms of ballast pumping rate input

tracking capability and time response specifications.

Figures 4, and 5 shows open loop response, Figure 6

and 8 shows the closed-loop response for various

values of gain Kp and lastly Figures 7, and 9 shows

the closed loop response of pitching angle, and depth

for both controllers designed. The summarized result


205

is shown in Tables 2 and 3. Based on Table 1(pitching

angle), LQR demonstrates faster response which has

settling time of 12.2 seconds as compared to PID

which has much slower settling time of 88.1 seconds.

LQR also yields the maximum overshoot

considerably much lower than PID which is 4.13% as

compared to PID which is 22.5%. However both

controllers converge to the desired value with zero

steady-state error. Based on Figure 9, again LQR

demonstrates a good control over the system as

compared to PID which unable to converge to the

desired value, however the oscillation is reduced as

shown in Figure 10. Table 3 shows that LQR able the

reach settling time at 79.3 seconds, maximum

overshoot of 0.7%, and zero steady-state error.

An effective and considerable amount of research

work has been reported in the past on the tuning of PID

controllers. Some of them are Ziegler-Nichols step

response, Ziegler-Nichols ultimate cycling, Cohen-

Coon, internal model control, and error-integral criteria.

However, these tuning methods use only a small amount

of information about the systems, and often do not

provide good tuning for higher-order systems. Therefore

in his work, the controller parameters are obtained using

heuristic tuning approach.

The models for pitching and depth control are

obtained through system identification method using

data from simulation of the motion equations14

. The

restoring forces (buoyancy and gravity) are included

in the model derivation13

. The glider is modeled in

such that the external disturbances (water current and

wave) can be neglected. The PID controller provides an

Fig. 6Pitching angle-PID Controller with various values of

Kp Gain

Fig. 7Pitching angle LQR vs. PID

Fig. 8Depth- PID Controller with various values of Kp Gain

Fig. 9Depth- LQR vs. PID


206

overall control over the system not local. This is because

we only use ballast pumping rate as a control input to

observe the performance of overall glider system.

Conclusion

In this paper, two controllers: LQR and PID are

successfully designed. Based on the results and the

analyses, a conclusion has been made that both of the

control methods, modern controller (LQR) and

conventional controller (PID) are capable of controlling

the pitching angle. However for depth control, PID

control is unable to converge to the desired value as the

LQR does. All the successfully designed controllers

were compared. The responses of each controller were

plotted in one window and are summarized in Table 1

and Table 2. Simulation results show that LQR

controller has better performance compared to PID

controller in controlling the glider system. Further

improvement need to be done for LQR controller. LQR

controller should be improved so that the faster

response is obtained. In real system, not all states are

available to be fed back, therefore in future; observer-

based controller may be the option and Sliding Mode

Control (SMC) also a good option for control

technique to be used to control motion of the

underwater glider. In future online tuning also can be

considered to obtain optimal tuning over the system.

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Table 2Pitching angle

PID LQR

Maximum overshoot (OS%) 22.5 4.13

Settling time (Ts,secs) 88.1 12.2

Steady-state error (ess) 0 0

Table 3Depth

PID LQR

Maximum overshoot (OS%) unidentified 0.7

Settling time (Ts,secs) unidentified 79.3

Steady-state error(ess) unidentified 0

Fig. 10Depth – PID vs. Open loop

ijms 40(2) 200-206ff

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