ABSTRACT Optimal estimator synthesis algorithm for dynamic ship system control is considered. Estimator is constructed in the form of Kalman gain as steady-state Kalman filter. Estimator design is realised by MATLAB codes using account of wind gust noise with a set spectral intensity

Biya Motto

Frederic

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RESEARCH AND REVIEWS: JOURNAL OF ENGINEERING AND 
TECHNOLOGY 
 
 
Optimal Estimator Synthesis for Ship Dynamic Object Control. 
 
BIYA MOTTO Frederic* 
 
Department of Physics, Faculty of Sciences, University of Yaoundé I, P O Box 812, Cameroon. 
 
 
Research Article 
 
Received: 15/05/2014 
Revised: 16/06/2014 
Accepted: 20/06/2014 
 
*For Correspondence 
 
Department of Physics, 
Faculty of Sciences, 
University of Yaoundé I, P O 
Box 812, Cameroon. 
 
Keywords: Control, Optimal 
estimator, Kalman filter, Ship 
dynamic object, gust noise. 
 
ABSTRACT 
 
        Optimal estimator synthesis algorithm for dynamic ship system 
control is considered. Estimator is constructed in the form of 
Kalman gain as steady-state Kalman filter. Estimator design is 
realised by MATLAB codes using account of wind gust noise with a 
set spectral intensity. 
 
 
 
 
 
 
INTRODUCTION 
 
In no small way, the great technological progress in automatic control and communication systems 
during the past years has depended on advances and refinements in the mathematical study of such 
systems. Conversely, the growth of technology brought forth many new problems to challenge the ingenuity 
and competence of research workers concerned with theoretical questions. 
 
Obtaining good quality information on vector of state variables for dynamic system is very 
important when studying the evolution process and synthesis of control functions that insure oriented 
behaviour in time and space. However, phase coordinate vector is often difficult to get for measurement 
because of various reasons. In such a case, we can evaluate the coordinate vector through the output 
vector. Since measurements are done with errors and also considering noise at input of the system, we 
need indicators based on procedures of stochastic filtering. 
 
In systems with optimal control, we need full information for state vector since in this case the 
synthesis of regulator is simplified. In case of insufficient measurements, we should solve the problem of 
determining state vector from observation results through output expressions. When measurements and 
noise errors are significantly acting on the system, together with the observer, we need to synthesise a 
special filter while presenting the process either in continuous or discrete time. We consider the solution in 
continuous time. 
 
KALMAN FILTER SYNTHESIS 
 
Let us assume that the object is described by the matrix equations: 
 
ẋ(t) = Ao∙x(t) + Bo∙u(t) + Go∙w(t)        eq1 
y(t) = C∙x(t) + v(t)         eq2 
Where; 
x(t) is (nx1) state vector 
 
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RRJET | Volume 3 | Issue 3 | July - September, 2014                  24 
u(t) is (mx1) control vector 
 
y(t) is (px1) output vector 
 
Ao, Bo, Go, C are Matrices of corresponding dimensions. 
 
w(t) is the unknown noise acting on the object (such noise can be the influence of the wind or agitation of 
water surface, etc.) 
 
v(t) is the measurement noise (for example, it can be noise from amplifier) 
 
x(t)  Rn, u(t)  Rm, y(t)  Rp 
 
In case there is no information about the precise value of x(0), then we can use the mathematical 
expectation  ̅(0) and the co-variation matrix Po, then  
 
 ̃(o) = ( ̅, Po)         eq3 
 
Let us consider x(o), w(t) and v(t) as being orthogonal to each other. Much more, suppose that w(t) and v(t) 
are white noise, 
 
Therefore, 
 
Rw(τ) = E [w(t + τ))∙wT(t)] = Q∙δ(t)       eq4 
 
Rv(τ) = E [v(t + τ))∙vT(t)] = R∙δ(t)       eq5 
 
Where, 
 
Q and R are matrices of spectral densities, 
 
δ(t) is delta function of Dirac. 
 
Assume that matrices Q and R are known and that R is not a singular matrix, that is  
 
w(t) ~ (0,Q), Q ≥ 0        eq6 
 
R(t) ~ (0,R), R> 0        eq7 
 
We should calculate the observer for stochastic system at entrance of which two signals are acting:  
Control signal and output signal from object 
 
 (t) = A∙x(t) + B∙u(t) + L∙(y- ŷ)       eq8 
 
or 
 
 (t) = (A – L∙C)∙ ̂ - L∙y        eq9 
 
Vector  ̂(t) is state evaluator. 
 
Output evaluator  ̂ is obtained by formula 
 
ŷ=E{C∙x +v} = C∙ ̂         eq10 
 
Suppose that the couple(C, A) is observable and (A, G√ ) is feasible. Therefore matrix algebraic Riccati 
equation is the unique positive determined solution of P: 
 
A∙P + P∙AT+ G∙Q∙GT– P∙CT∙R-1∙C∙P = 0       eq11 
 
Thus, we can synthesize the evaluator L which can assure the best evaluation of error in the presence of 
noises w(t) and v(t). 
 ̂ 
 ̂ 
 
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 If error of the system  ̃ = x(t) – ̂(t), then we can define it as a solution of matrix differential 
equation; 
 
= (A - L∙C)∙ ̃ + G∙w – L∙v = A3∙ ̃ + G∙w - L∙v        
          eq12 
 
Where we use the matrix of constant coefficients of amplification (Matrix of Kalman) 
 
L = P.CT.R-1         eq13 
 
The Solution of equation eq12 is stable. 
 
 The evaluator eq13 can be called static filter of kalman. It minimizes the error value (co-variation 
of error) at the completion of transient processes. 
 
The static filter of kalman is the best evaluator that contains constant coefficients. Dynamic 
equations of the system can contain a model used for transforming white noise into a signal with a given 
spectral density. 
 
Let us study this point; 
 
Energy spectrum for most of the processes can be represented in the form of square module for a 
certain function H(jw): 
 
Φ(ω) = |H(jω)|2         eq14 
 
It is assumed that a random process with energy spectrum Φ (ω) can be composed of white noise 
with an energy spectrum equal to that in the frequency diapason  -      if it passes though linear 
filter with frequency characteristic H(jω): 
 
Φ(ω) = |H(jω)|2∙Nƒ(ω) = |H(jω)|2 = H(jω)H*(jω)     eq15 
 
Where H*(jω) – conjugate frequency characteristic. 
 
The energy spectrum Φ(ω) can be approximated by rational frequency function, that is a ratio of 
polynomial function of ω. 
  
Since the energy spectrum is real, even number and not a negative function of ω, then such 
polynomial should have real coefficients and have only even number degrees of ω: 
 
Φ(ω) =                                 =                             eq16 
 
 
 
From the condition of real coefficients in eq16, poles of energy spectrum can be either real 
numbers or in pairs of complex conjugates (symmetrical relatively to the real axis). 
 
Impulse and frequency characteristics of the filter are related by the Fourrier transformation. If all 
the poles of frequency characteristics are lying on the left hand plan then the impulse function is feasible. 
In that case, the filter happens to be physically realizable. From the complex transfer coefficient, we can 
establish the transfer function and then synthesize dynamic of obtaining H(s).The canonic form of the 
system can be defined from the well-known procedure equation [1]. 
 
xw = Aw∙xw + Bw∙n      
W = Cwxw + Dw∙n      
 
Where n(t) is white noise, and w(t) is output random signal with given properties. 
 
Equations eq1, eq2 and eq17 are used for the obtention of enlarged dynamic systems with the following 
structure: 
eq1
7 
 
 ̃ 
M(ω) 
N(ω) 
 
bO + b1ω
2 + … + bmω
2m
ao + a1ω
2 + … + anω
2n 
 
 
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[
 
  
]  = [
  
 
     
  
]∙[
 
  
] + [
 
 
] ∙ u + [
    
  
]∙n     eq18 
 
y = [    ]∙[
 
  
] + v        eq19 
 
We should observe that on input of enlarged system, apart from the control signal u(t) we also 
have the action from the measurements of noise w(t) and white noise n(t). 
 
Using obtained relations, we will calculate the evaluator – Kalman filter for ship object under the 
influence of turbulent air flux. 
 
SHIP DYNAMIC OBJECT 
 
Due to the development and utilization of offshore petroleum resources, the continuous 
development of marine transport and marine science and technology, ship and ocean engineering 
technology started rising rapidly. Ship and platform often need to locate their particular position in a site 
but the traditional way to do so by the anchor method is restricted by the following factors: operating water 
depth, working time and accuracy demands. 
 
To solve this problem of ship position and maintaining its course, a ship control technology called 
the dynamic positioning [4,6] at a particular time in history is bringing considerable progress to the 
development of correlated techniques using the electronic computer, sensors and the control theory. 
 
Dynamic properties of the object are described by equations eq1 and eq2 with the following 
matrices [2,3]: 
 
Ao= [
       
      
      
       
], Bo = [
       
       
],  Go = [
      
       
] 
 
On the object is acting noise with spectral density 
 
Φ(ω) = 2L σ2     , 
 
 
Where σ =10; L = 3.49 
 
Matrix C in equation eq2 has the following coefficients values: 
 
C =  [
                                 
                                             
] 
 
For Kalman filter calculation, assuring minimal errors with noise actions and in Riccati equation eq11 we 
chose weight matrices Q and R: 
 
Q = 1;  R = [
     
     
] 
 
Factorizing of Φ(ω) in the form  
 
Φ(ω) = H(jω)∙H(-jω) enables us to obtain the realizable part of the filter: 
 
 
H(s) =                   =   
 
 
 
Transformation of H(s) into canonic form of model in states space will be done by function tf2sss, 
contained in Matlab pack. We then obtained model eq17: 
 
Aw = R = [
              
          
], Bw = [
 
 
],  
1 + 3L
2
ω
2
 
1 + L
2
ω
2
 
 
13.1118·s + 2.1691 
s
2
 + 0.5731s + 0.0821 
M(s) 
N(s) 
 
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Cw =[            ], Dw = 0. 
 
Matrices of enlarged system look as follows. 
 
A=
[
 
 
 
 
                   
                    
          
             
                 
                  
                        
                        
                                             
                                   ]
 
 
 
 
 
 
B=[          ]; G = [        ] 
 
State matrix F of Kalman filter is equal to: 
 
F = (A – L.C) =  
[
 
 
 
 
                   
                    
                       
             
               
                  
                       
  
                             
                        ]
 
 
 
 
 
 
Kalman transform coefficient is calculated by the aid of Ɩqe function: 
 
L=
[
 
 
 
 
             
             
             
             
  ]
 
 
 
 
 
 
Kalman filter represents dynamic system: 
 
= F. ̂ + B.u + L∙Y         eq20 
 
 
Where, the 5th state (the last line of matrix L) is equal to zero because noise does not act directly 
on the drive of the rudder. 
 
NUMERICAL CALCULATIONS AND CONCLUSION 
 
The calculation of filter is done with a program, represented by File sah175a.m, whose content is 
precised in the text by commentaries. 
 
% File Sah175a.m 
 
% Calculation of stationary Kalman filter. 
 
% Dynamic of the system: 
 
A0= [                             ]; 
B0 = [              ]’; 
G0 = [             ]’; 
 
% Determination of the model in state space through 
% function of spectral density of external noise signal 
 
sig = 10; L = 3.49; 
v = Sig*( (6/L)^0.5); 
 
% Numerator and denominator of complex transform coefficient: 
 
num[1 1/(L*(3^0.5))].*v; 
den = [1 2/L 1/(L^2)]; 
 ̂ 
 
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% Determination of states space model with % white noise influence: 
 
[Aw, Bw, Cw, Dw] = tƒ2ss (num, den); 
 
% Formation of generalized dynamic system: 
 
A = [A0 G0*Cw B0;[0 0;0 0;0 0] [Aw;[0 0]] [0 0 -20.2]’]; 
B = [0 0 0 0 20.2]’; 
G = [0 0 1 0 0]’; 
C = [15.8788 1.4811 0 0 0; 0 1 0 0 0]; 
R = [1/20 0; 0 1/60]; 
Q = [1]; 
% The filter is synthesis with “ℓqe∙m”. 
% The matrix of Kalman amplifier coefficients: 
L = ℓqe (A, G, C, Q, R)  
 
Pause, 
 
% Kalman filter: 
  
  
 = F*x + B*u + L*y; 
F = A – L*C 
 
Pause, 
 
% Proper values of the optimal evaluator matrix: eig(F) 
 
The program is ended by calculation of values of matrix F, represented in the form of vector 
elements: 
 
eig(F) = [-1.1661 -0.3759 -1.5224 -1.1734 -20.2000]T 
 
Finally, through expressions eq17 – eq20 in a Matlab environment, we synthesized the Kalman 
filter while assuring stability of dynamic process. 
 
REFERENCES 
 
1. Grinkevich YM, Sakharov VV. 2001. Observer and state evaluator in ship control systems, Saint-
Petersburg, 193p (in Russian). 
2. Lewis FL, D Vralie, VL Syrmos – 3er ed. – N.Y : John Willy and sons, 2012 – 544p. 
3. Model parametric identification of flexible mechanical structures in mechatronic system/ C. K. 
Pang [et al.]; Trans Inst Measurement Control. 2010;32(2):137 – 154. 
4. Girard AR, Bourges De Souza J, Hedrick JK. Dynamic positioning concepts and strategies for the 
mobile offshore base. IEEE Conference on Intelligent Transportation Systems, Proceedings, ITSC, 
2001, pp 1095 – 1101.  
5. Strensen Asgei J. A survey of dynamic positioning control of ships and ocean structures, Norwegian 
university of science and technology, No - 7491 Trondheim, Norway, vol. 35, No. 1, April 2011, pp 
123 – 136. 
6. Smallwood, David A, Whitcomb, Louis L. Model – Based Dynamic positioning of Underwater 
Robotic vehicles: theory and experiment. IEEE J Oceanic Eng. 2004;29(1):169 – 186. 
 
 
 


Optimal Estimator Synthesis for Ship Dynamic Object Control

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Optimal Estimator Synthesis for Ship Dynamic Object Control

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