An Optimal Consensus Tracking Control Algorithm for Autonomous Underwater Vehicles with Disturbances

7/31/2019 An Optimal Consensus Tracking Control Algorithm for Autonomous Underwater Vehicles with Disturbances

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International Journal of Instrumentation and Control Systems (IJICS) Vol.2, No.2, April 2012

DOI : 10.5121/ijics.2012.2204 45

AN OPTIMAL CONSENSUSTRACKING CONTROL

ALGORITHM FORAUTONOMOUS UNDERWATER

VEHICLES WITH DISTURBANCES

Jian Yuan1 Wen-Xia Zhang2 and Zhou-Hai Zhou1

1Institute of Oceanographic Instrumentation of Shandong Academy of Science, Qingdao,China

[email protected]

2 Mechanical & Electrical Engineering Department, Qingdao College, Qingdao,China

[email protected]

ABSTRACT

The optimal disturbance rejection control problem is considered for consensus tracking systems affected by

external persistent disturbances and noise. Optimal estimated values of system states are obtained by

recursive filtering for the multiple autonomous underwater vehicles modeled to multi-agent systems with

Kalman filter. Then the feedforward-feedback optimal control law is deduced by solving the Riccati

equations and matrix equations. The existence and uniqueness condition of feedforward-feedback optimal

control law is proposed and the optimal control law algorithm is carried out. Lastly, simulations show the

result is effectiveness with respect to external persistent disturbances and noise.

KEYWORDS

Autonomous Underwater Vehicles; Consensus Tracking; Optimal Disturbance Rejection

1.INTRODUCTION

The AUVs formation control is a typical problem of multi-robot coordination and cooperation.The coordinated control of multiple AUVs can significantly improve many applications includingocean sampling, imaging, and surveillance abilities. The large-scale multiple AUVs system ismodelled to multi-agent system to study the consensus problem on their spatial location. Wangand Xiao [1 ] proposed a finite-time formation control framework for large-scale multi -agentsystem. They divide the formation information into two types: global information and localinformation, in which the global information can decide the formation shape and only the leadercan obtain such the global information; followers can only get local information. This framework

can reduce the amount of communication between the agents. And then they design a nonlinearconsensus protocol, and apply it to the time-invariant, time-varying and trajectory-trackingcontrol. Wang and Hong [2] propose some types of consensus control algorithms for the first-orderdynamic system with variable coupling topology. They design a finite-time consensus protocoland give its non-smooth controller using time-invariant Lyapunov function and graph theory tool.Further, they propose a non-smooth time-invariant consensus algorithm for a second-orderdynamic systems. They demonstrate the existence of the finite-time control law using Lyapunovfunctions and graph theory in [3]. Wang and Chen [4] study the consensus problem of continuous-


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time multi-agent with communication time delay. They design a class of continuous but non-smooth finite-time controller which ensures that the multi-agent systems with time delay reach aconsistent state in finite time. The underwater environment is extremely complex, of strong noiseand disturbance, resulting in multi-agent status subjecting to these external disturbances.Reducing or overcoming the impact of these disturbances and noise on multi-agent formation has

important theoretical and practical significance. In this paper, the feedforward-feedback optimaltracking control problem for the multi-agent system with external disturbance and noise under agiven performance index based on the Kalman filter is studied. Optimal estimated values ofsystem states are obtained by recursive filtering for the multiple autonomous underwater vehiclesmodelled to multi-agent systems with Kalman filter. Then the feedforward-feedback optimalcontrol law is deduced by solving the Riccati equations and matrix equations. The existence anduniqueness condition of feedforward-feedback optimal control law is proposed and the optimalcontrol law algorithm is carried out. Simulations show the result is effectiveness with respect toexternal persistent disturbances and noise.

2. PROBLEM DESCRIPTION

The consensus algorithm with external disturbance is described as

0 0( ) ( ( ) ( )) ( ) ( ( ) ( )) ( )t

j i

i ij j i vi i i i iv N

t a t t a t a t t t

= - + + - +&x x x v x x m (1)

where ( )i

tv denotes external disturbance and satisfies ( ) ( ( ))i i

t f t=&v v , 0 ( )tx is the desired

trajectory and satisfies 0 0( ) ( ( ))t f t=&x x , ( )i tm denotes Gaussian white noise where

0 0, 0i via a> > . without loss of generality, the individual AUV dynamics of multi-AUV systemswith disturbances described as follows:

1 2

0

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )(0)

i ii i ij j i i i i ij i

i i i i

i i

t t t t t t

t t t

= + + + +

= +=

&x A x A x B u B w m

y C x n

x x

, i N (2)

where ( )i tx denotes the states of AUV i and ( )p

i t Rx . ( )j tx denotes the states of AUVj

surrounding AUV i . ( ) si t Ry denotes the system output. ( )q

i t Ru denotes the input of

AUV i .

( ) ri t Rw denotes the disturbances, ( )p

i t Rm denotes the process noise, ( )p

i t Rn denotes

the measurement noise. p pii

R A , p pij

R A , 1p q

iR B , 2

p r

iR B and s p

iR C are

constant matrice of appropriate dimensions.ij

A denotes the corresponding matrix between

AUV i and AUVj .

The output trajectory-trackingi

%y of virtual leader which the system outputi

y tracks is described

as


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( ) ( ) ( )

( ) ( ) ( )i i i zi

i i i zi

t t t

t t t

= +

= +

&

%

z F z m

y H z n(3)

where li

Rz , si

R%y , l li

R F and s li

R H are constant matrice of appropriate dimensions.

And ( , )i iF H is observable. ( ) lzi t m denotes the process noise of the external system,

( ) szi

t Rn is the measurement noise of the external system. So the equation (2) is rewritten as

1 2

0

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

(0)

t t t t t

t t t

= + + +

= +

=

&x Ax B u B w m

y Cx n

x x

(4)

where

TT T T1 2( )

Np

Nt R = Lx x x x ,

TT T T1 2( )

Np

Nt R = Lm m m m

TT T T1 2( )

Nq

Nt R = Lu u u u ,

TT T T1 2( )

Nr

Nt R = Lw w w w

TT T T1 2( )

Ns

Nt R = Ly y y y ,

TT T T1 2( )

Ns

Nt R = Ln n n n

11 1

1

N

Np Np

N NN

R

=

L

M O M

L

A A

A

A A

,

11

1

1

Np Nq

N

R

=

O

B

B

B

21

2

2

Np Nr

N

R =

O

BB

B

,

1Ns Np

N

R =

O

CC

C

We define the disturbance ( )tw as

( ) ( ) ( )wt K t t = +&w w m (5)

The output trajectory-tracking % of virtual leader which the system output tracks is describedas

( ) ( ) ( )( ) ( ) ( )

z

z

t t t

t t t= += +

&

%

z Fz m

y Hz n(6)

where NlRz , ( ) Nlz t Rm ,Ns

R%y , ( ) Nsz t Rn ,Nl Nl

RF and Ns NlR H are constant

matrices of appropriate dimensions. And ( , )F H is observable.

The filtered system by optimal state filtering is described as


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1 2

0

( ) ( ) ( ) ( )

( ) ( )

(0)

t t t t

t t

= + +

=

=

&x Ax B u B w

y Cx

x x

(7)

whereTT T T

1 2 ( )Np

Nt R = Lx x x x ,

TT T T1 2 ( )

Nq

Nt R = Lu u u u

TT T T1 2

( ) NrN

t R = Lw w w w ,TT T T

1 2 ( ) Ns

Nt R = Ly y y y

The external disturbance system is described as

( ) ( )t K t=&w w (8)

The external trajectory-tracking system is described as

( ) ( ) ( ) ( )

t t

t t==

&

%

z Fz

y Hz(9)

where NlRz , NsR%y . Choose Infinite horizon quadratic performance index as

( )

( )

T T

0

T T

0

( ) ( ) ( ) ( ) d

( ) ( ) ( ) ( ) d

J E t t t t t

t t t t t

= +

= +

e Qe u Ru

e Qe u Ru

(10)

where Q and R are positive definite matrix of appropriate dimensions. ( )te and ( )te are output

error and estimated value of the output error, described as( ) ( ) ( )

( ) [ ( )] ( ) ( )

t t t

t E t t t

= -

= = -

%

%

e y y

e e y y(11)

The optimal formation control is to solve the optimal tracking control law * ( )tu to make Jobtain the minimal value.

3.DESIGN ON OPTIMAL FORMATION CONTROL

In this section we focus on the designing on the optimal formation control law of AUVs with

disturbance and noise effects. First, we give the following theorem:Theorem 1 Considered the optimal tracking control problem of disturbed multiple AUVs systemwith equation (4) and (6) under the performance indicators (10), the optimal formation controllaw only exists and is represented by the following formula:

( )* 1

1 1 2 ( ) ( ) ( ) ( )Tt t t t -= - + +u R B Px P z P w (12)


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where P is the unique semi-positive definite solution of the following matrix algebraicequation:

T T+ - + = 0A P PA PSP C QC (13)

where 1P is the unique solution of the following matrix algebraic equation:

T1 1 1- - + = 0

TP F PSP C QH A P (14)

where 2P is the unique solution of the following matrix algebraic equation:

T2 2 2 2 0- + =A P + P K PSP PB (15)

where 1 T1 1-=S B R B .

Proof: A necessary condition of the optimal formation control under performance indicators (10)for system (4) and (6)for leads to solving the following two point boundary value:

T T T

2 0

0

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ,

( )

(0)

t t t t

t t t t t t

l- = - +

= - + < <

=

=

0

&

&

C QCx C QHz A

x Ax S B w

x x

(16)

Its optimal formation control law is described as

1 T1( ) ( )t t

-= -u R B (17)

For solving the two point boundary value(16), we define

1 2 ( ) ( ) ( ) ( )t t t t = + +Px P z P w (18)

Take the derivative of both sides of(17), and take the second equation of (16) and (5) intoaccount, we obtain

1 1 2 2 ( ) ( ) ( ) ( ) ( ) ( ) ( )t t t t t t t = + + + + +& &&& & & &Px Px P z P z P w P w (19)

Compare (19) with the first equation of(16), we obtain:

( ) ( )( )

T T T1 1

T2 2 2 2

( ) ( )

( )

t t

t

+ - + + - - +

+ - + = 0

TA P PA PSP C QC x P F PSP C QH A P z

A P + P K PSP PB w(20)

For all ( )tx , ( )tz and ( )tw , (19) establishes all the time. So we obtain the Ricatti matrixalgebraic equation (13), (14) and (15)For equation (13) is the matrix algebraic equation about P


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with unique semi-positive definite solution, we take it into matrix algebraic equation(14)and(15)to solve the unique 1P and 2P .

When the uniqueP , 1P and 2P are obtained, ( )t is solved. Furthermore, by (18) we determine

the optimal formation control law(12).For the optimal tracking control problem described by (4) and(6), the designing procedure of theabove algorithm is as follows:

Algorithm: (the optimal consensus tracking control algorithm)

Solving the expected output ( )t%y through(9);

Solving ( )tP , 1( )tP and 2 ( )tP through equations(13), (14) and (15), respectively;

Calculating ( )tx through(16);

Calculating ( )tu through(12);

Calculating ( )te through(11);

Calculating J through(10).

According to the above system description and optimal consensus tracking control algorithm, weobtain the control block of the optimal consensus tracking control illustrated in Fig.1.

1

S1 T

R B 1B

A

2B

P

2P

1

S

F

&x xu

& %yH

Cy

1P

1

S

K

&w w

Figure 1. System block of optimal consensus tracking control system with disturbance and noise

4.SIMULATIONS

Considering the multiple AUVs system with noise and disturbance described by equations(4),(5) and (6) as follows:


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[ ]T1 0 3 2 0.1 0

( ) ( ) ( ) ( ) ( ), (0) 0 00 1 4 5 0 1

1 2( ) ( ) ( )

3 4

t t t t t

t t t

- - + = -

= +

&x = x + u + w m x

y x n

[ ]T0.1 1

( ) ( ) ( ), (0) 0.4 0.51 0.2

0.2 4( ) ( ) ( )

2 5

z

z

t t t

t t t

- + = - -

= +

&

%

z = z m z

y z n

[ ]T5 1

( ) ( ) ( ), (0) 0.3 0.41 9 w

t t t - + = - -

&w = w m w

The covariance mQ and nQ of ( )tm and ( )tn are 2n= =mQ Q , respectively. The covariancemQ and nQ of ( )z tm and ( )z tn are 2z zn= =mQ Q , respectively. And the covariance wQ of

( )w

tm is 2w

=Q . The total simulation time is 300(s)T= .we compare the Kalman filterbased feedforward-feedback tracking control law to the classical feedforward-feedbacktracking control law. The result comparison is showed in Fig.2 to Fig.7.

0 50 100 150 200 250 300-100

-80

-60

-40

-20

0

20

40

60

80

100

Error1

Time (Sec)

without filter

with kalman filter

0 50 100 150 200 250 300-150

-100

-50

0

50

100

150

Error2

Time (Sec)

without filter

with kalman filter

Figure 2. Comparison on error-1 Figure 3. Comparison on error-2

0 50 100 150 200 250 300-25

-20

-15

-10

-5

0

5

10

15

20

x1

Time (Sec)

without filter

with kalman filter

0 50 100 150 200 250 300-40

-30

-20

-10

0

10

20

30

40

x2

Time (Sec)

without filter

with kalman filter

Figure 4. Comparison on state-1 Figure 5. Comparison on stater-2


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0 50 100 150 200 250 300-80

-60

-40

-20

0

20

40

60

80

u1

Time (Sec)

without filter

with kalman filter

0 50 100 150 200 250 300-40

-30

-20

-10

0

10

20

30

40

u2

Time (Sec)

without filter

with kalman filter

Figure 6. Comparison on control-1 Figure 7. Comparison on control-2

From the simulation result comparison, it is ensured that the Kalman filter based optimalconsensus tracking control algorithm is effectiveness. The system tracks the expected the externalsystem(3) in higher precision under the noise and disturbance. The control law is of better noise

rejection and the tracking error is smaller than the classic feedforward-feedback tracking controllaw.

5. CONCLUSIONS

Because the system states of consensus protocol is polluted by the noise, we use Kalman filter tofilter the noised controlled system to obtain the optimal estimated values of each AUV states, inorder to achieve coordination control of multi-AUVs in noisy environment. Considering thecontrolled system affected by environmental noise and external disturbances, we design a Kalmanfilter-based feedforward and feedback optimal consensus tracking protocol.

REFERENCES

[1] F. Xiao, L. Wang, J. Chen, Y. Gao, (2009) Finite-time formation control for multi-agent systems,Automatica, Vol. 45, No. 11, pp 2605-2611.

[2] X. Wang, Y. Hong, (2010) Distributed finite-time -consensus algorithms for multi-agent systemswith variable coupling topology,Journal of Systems Science and Complexity, Vol.23, No.3, pp 209-218.

[3] X. Wang, Y. Hong, (2008) Finite-Time Consensus for Multi-Agent Networks with Second-OrderAgent Dynamics, Proceedings of the 17th World Congress of the International Federation ofAutomatic Control, Seoul, Korea, July 6-11.

[4] L. Wang, Z. Chen, Z. Liu, and Z. Yuan, (2009) Finite-time agreement protocol design of multi-agentsystems with communication delays,Asian Journal of Control, Vol.11, No.3, pp 281-286.

Authors

Dr. Yuan is Associate Professor, Institute of Oceanographic Instrumentation of

Shandong Academy of Science, Qingdao, China. His current research areas are:Multiple Agents System based control, Networked Control System and NonlinearFiltering. He has published over 20 papers in international journals and internationalconferences.

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An Optimal Consensus Tracking Control Algorithm for Autonomous Underwater Vehicles with Disturbances