ENGG7302 - UQ Machine Health Monitoring System using Markov Chains

This report provides an analysis of the University of Queensland's (UQ) machine health monitoring system using Markov Chains, as part of the ENGG7302 Advanced Computational Techniques in Engineering course. The model assesses the health of machines across various schools within UQ, treating the arrival of new machines as independent Bernoulli processes. The health of each machine is represented as a Markov Chain, where states indicate health levels at the end of each month, and transitions are modeled as independent Binomial distributions. The report includes a methodology detailing the number of schools, arrival process parameters, the number of states representing machine health, and the transition matrix. The implementation uses MATLAB to simulate the Markov Chain, display state transitions, and analyze posterior distributions related to machine health. The study also references various sources on condition monitoring, diagnostics, and the application of Markov models in similar contexts.

UNIVERSITY AFFILIATION
FACULTY OR DEPARTMENT
COURSE ID & NAME
TITLE: ADVANCED COMPUTATIONAL TECHNIQUES IN ENGINEERING
STUDENT NAME
STUDENT REGISTRATION NUMBER
PROFESSOR (TUTOR)
DATE OF SUBMISSION
2018

REPORT
A researcher and scientist, Andrei Andreevich Markov, introduced a model into the
statistical and logistics field that would provide a simple generalization of the probability model
of independent trials. The expected outcomes of the successive trials are based on previously
done trials whose data is fed into the model. The random variables based on time form the
stochastic process. These processes are also known as Markov processes when they include the
property states. The attributes follow that the probability of a system having a transition from one
state to another is highly based on the current state and not on previous states. The system tends
to exhibit the memory-less property. The case study of the UQ equipment maintenance uses the
continuous-time Markov chain. The homogenous mode allows for easy calculation of the state
probabilities by using a transition matrix. It has transition rates matrix Q where the elements in
the matrix transition from state i→ state j such that,
qii=−∑
i ≠ j
qij
To compute the steady-state probabilities using the Gauss-Jordan elimination method,
{Πe P=Π e
∑
kϵs
πk
e=1
The steady-state distribution is obtained as,
Πe=e . ( 1+ E−P )−1
πi=E ( hi ) . πi
e
∑ π k
e E ( hi )
The semi-Markov process are much more suitable in the design of aging processes and
maintenance of equipment. The model introduces the sojourn time to enable modeling of
complex stochastic processes. Its accuracy impacts the entire model’s accuracy.

METHODOLOGY
1. The number of schools in the University of Queensland, n
2. Vector of n elements, with each element indicating the parameter for the arrival process
of the machines in each school.
3. Number of states in the Markov chain that represent the health of machines in the
different schools, 1… n.
4. Transition matrix for the Markov chain
The transition rate is defined as,
The state probabilities through the transition rate is given as,

When the transition rate is leaving state i, it is given as,
MATLAB IMPLEMENTATION (r2018a)
%% ENGG7302 advanced computational techniques in Engineering
n=input ('Enter number of schools in UQ');
% To model the arrival of the new machines in each school
% Independent Bernoulli process- Component form
p=n/10;
for k=1:n,
bernp2=rand<=p;
end
%% creating a Markov chain
mc = dtmc(P);
% to display the number of states in the Markov chain
numstates=mc.NumStates
figure (1) % observe the states as they transition
graphplot(mc)
title('Markov Chain on Health of Machines in UQ')
grid on
figure (2)
D=[0.0137,6.1162];
alpha=0.7; %shape parameter for gamma process
alpha0=0.05; %shape parameter for conjugate prior
beta0=25; %inverse scale parameter of conjugate prior
MEAN=alpha0/beta0;
beta=(0:0.00001:0.03); %generate beta values for posterior pdf plot
countj=0;

sumD=0;
for j=1:size(D,2),
alpha0=alpha+alpha0; %alpha0 is updated with each new observation of D
beta0=beta0+D(j); %beta0 is update with each new observation of D
sumD=sumD+D(j);
MEAN(j)=alpha0/beta0;
countj=countj+1;
Arrayalpha0(countj)=alpha0;
Arraybeta0(countj)=beta0;
ArraysumD(countj)=sumD;
counti=0;
for i=1:size(beta,2), %evaluate the posterior distribution for each beta
%value and store in an array
Posterior=((beta(i)^(alpha0-1))*(exp(-
(beta0)*beta(i)))*((beta0)^alpha0))/gamma(alpha0);
if Posterior < 1.e-40
Posterior=0;
end
counti=counti+1;
ArrayPosterior(counti,j)=Posterior;
end
temp=ArrayPosterior(1:counti,j);
plot(beta,temp,'k-')
hold on
end
xlabel('\beta')
ylabel('f(\beta)')
hold off

0 0.005 0.01 0.015 0.02 0.025 0.03
0
20
40
60
80
100
120
140
160
180
f( )
REFERENCES
AFNOR, “Condition monitoring and diagnostics of machines—prognostics—part 1: general
guidelines,” Tech. Rep. NF ISO 13381-1, 2005.
A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via
the EM algorithm,” Journal of The Royal Statistical Society, Series B, vol. 39, no. 1, pp. 1–38,
n.d.
D. A. Tobon-Mejia, K. Medjaher, N. Zerhouni, and G. Tripot, “A mixture of gaussians hidden
markov model for failure diagnostic and prognostic,” in Proceedings of the IEEE International
Conference on Automation Science and Engineering (CASE '10), pp. 338–343, Toronto, Canada,
August 2010.
M. Azimi, P. Nasiopoulos, and R. K. Ward, “Offline and online identification of hidden semi-
Markov models,” IEEE Transactions on Signal Processing, vol. 53, no. 8, pp. 2658–2663, 2005.

T. Honkanen, Modelling industrial maintenance systems and the effects of automatic condition
monitoring [Ph.D. dissertation], Helsinki University of Technology, Information and Computer
Systems in Automation, 2004.