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Fig. 1. Developed Architecture

FDI methods to HVAC systems has been proposed using

different methods [2]. Because HVAC systems are highly

nonlinear and complex systems, different strategies to

solve particular problems can be investigated.

In this work, a methodology is presented for fault de-

tection and isolation using multiple models. The scheme

is applied to a heating unit of an HVAC system. The

paper is organized as follows: the second section intro-

duces the fault tolerant control scheme that was appliedto the system; in the third section the results obtained

are presented; finally, in the fourth section, concluding

remarks are provided. For simplicity, the presentation is

restricted to single-input single-output (SISO) systems,

but it can easily be expanded to cover multiple-input

multiple-output (MIMO) systems.

II. THE FDI ARCHITECTURE

Fig. 1 shows the architecture developed to test the FDI

mechanism in the HVAC. The proposed architecture has

two different and important modules:

A bank of models.

FDI algorithms.

Each one of these modules are described below.

A. Bank of models

The bank of models is composed of several linear or

non linear models that represent the behavior of the pro-

cess at different operating regimes. Using a nonlinear state

space representation, the set ofNmodels is described as

follows:

Mi:

xi(k+ 1) =fi(k, x) +gi(k, u) +(k)i= 1, , N

yi(k) =hi(k, x) +(k)(1)

whereMi is the i-th model, xi ni represents the statevector of the i-th model,yirepresents the output of the i-th

model,and represent system and measurement noises,

respectively. The multiple model approach decomposes

the process in different operating regimes where a local

model is associated to each regime. It should be noted that

(i) not every operating regime needs a highly accurate

model; (ii) a complex model can represent the process

in a wider operating regime; (iii) process complexity is

not always the same in every regime [4]. The use of the

multiple model approach calls for the use of the following

preliminary steps: Decompose the domain of interest into several

regimes of operation.

Identify one model for each of the regimes.

Define a scheduling method to identify the operating

regime and choose the active model or models.

One of the main difficulties with the approach is the

choice of the number of models that are necessary to

describe a certain process. If all models represent input

and output variables in the same space, it is possible toquantify and measure the distance between models. In the

linear case, using theH norm [10], one can calculate adistance between the models:

Mi Mk f or all i, k = 1, 2, , N. (2)

Even with a distance measure it is difficult to decide the

number of models that should be used to represent the

plant. A practical rule can be applied: the models should

not be too close as it will make the decision between

models difficult, and they should not be too far apart since

numerical problems can occur [11].

The idea of dividing the system representation into

several models can be directly applied to FDI. One has

to think that a faulty process is a process in a different

operating regime. Thus, the bank of models represents

the process under different faults that can occur. Some

issues arise when thinking of modeling faults in the

process: not every kind of fault can be simulated for

the purposes of identification. However, using sate-space

models it is fairly easy to build models that can represent

total or partial, actuator or sensor failures (1). It may

be impossible to build models that can represent every

possible fault in the system. It also may be difficult to

decide how many models are necessary to represent the

behavior of an incipient fault.The models are identified for different regimes or for

different faults that can occur in the system. It is possible

for the process to operate between different regimes, and

in this way the nonzero weights will be applied to the

models near those regimes. In this way, a regime that is

not explicitly represented in the model set can potentially

be identified by the weights.

B. FDI Algorithms

The FDI block was built using two different ap-

proaches. On one hand, a cost function is minimized in or-

der to obtain the best model combination to approximate

the dynamic behavior of the system. On the other hand, a

probabilistic approach helps with the identification of the

correct mode of operation. The combination between the

two different approaches is obtained as follows,

i= i+iN

j j+ jf or i= 1, 2, , N (3)

where i represents the weight associated with each

fault mode, represents the result of the optimization

of the cost function (9), p represents the result of the

probabilistic approach, and represent the degree of

confidence in each one of the approaches.

The values of the weights help to isolate a fault, hencethis solution performs the FDI function. If a certain weight

is equal or very close to one (then all other weights are

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very close to zero), that indicates that the real process

is best represented by the model corresponding to that

weight. In such case, it can be said that a particular model

is being fired, which indicates that the plant is likely to

be working in a particular (faulty or normal) regime.

1) Optimization: In this module the operating regime

(including faulty regimes) is identified. Define a vector

Yp H with the last H samples of the plant output asfollows:

Yp =

y(k)y(k 1)

...

y(k H+ 1)

(4)

Define the following model data matrix:

Ym=

y1(k) . . . yN(k)y1(k 1) . . . yN(k 1)

.... . .

...

y1(k H+ 1) . . . yN(k H+ 1)

where k is an integer index that denotes present time,

the row dimension H represents the number of past

samples that are considered, and the column dimension

N is the number of models. Matrix Ym contains the last

H samples predicted by each of the N models in the

model bank. Notice that the elements of Ym are one-

step-ahead model predictions of the output, such that

yi(k j) =yi(k j|k j 1), j = 0, . . . , H 1.Define a vector of weighting factors =

[1, . . . , N]T which satisfy:

Ni=1

i = 10 i 1 i= 1, , N (5)

The purpose of these weighting factors is to quantify the

degree of activation of each model in the model bank.

Using the individual models from the model bank, and

consideringHpast output samples from all models, it is

possible to define a weighted historical output vector of

the model bank, Yw H that considers the weighted

outputs of the individual models:

Yw =

Ni=1

Ym,ii= Ym (6)

where Ym,i represents the past H samples of the i-thmodel output, and consists of the i-th column of matrix

Ym.

It is now possible to define the residuals H

between the weighted historical output of the model bank

Yw and the historical plant output Yp, as follows:

= Yp Yw =Yp Ym (7)

and, using these residuals, it is possible to define a cost

function to be minimized:

(Yp, Ym, ) =T= (Yp Ym)

T(Yp Ym) (8)

Given the plant data Yp and the model data Ym, thiscost function is minimized with respect to the weight

vector subject to the constraints given in Equation

(10). The solution of the optimization problem gives an

optimum vector of weighting factors , which indicate

what combination of the model set best represents the

real process under the current operating conditions. This

optimization problem can be solved at every sampling

instant using quadratic programming [12], with the model

data matrixYmand the plant data vectorYp being updated

with new values using the receding data window concept.

When experimenting this solution in a real process it was

clear that in some situations the fault diagnosis could

be improved. One solution proposed in [14] to solve

this problem is to change the cost function making sure

that the penalty is not only on the residuals described

by (7), but also on the weighted sum of the square of

the difference between the most recent plant output y(k)and the corresponding prediction yi(k) from each of themodels:

(Yp, Ym, ) = (Yp Ym)T(Yp Ym)

+Ni=1

ii(y(k) yi(k))2 (9)

where > 0 is a scalar parameter that defines theimportance of the second term of the cost function and

N defines a vector with the level of confidence ineach one of the models.

2) Probabilistic approach: In order to achieve a higher

level of confidence in the FDI mechanism, a probabilis-

tic approach was introduced. Markovian Jump Systems

(MJS) [15],[16], [17], are used as a switching mecha-

nism between a series of models that represent different

operating regimes of the system. One can consider a

system similar to the one represented in (1). Suppose now,that the system faults could be represented by a Markov

chain, taking values in: s(k) M ={1, 2, , N}. Theswitching can be calculated using initial and transition

probabilities,

P{si(0)} = i(0)

P{si(k)|si(k 1)} = P{si(k)|si(k 1), Yk1p }= ij

for all i, j = 1, 2, , N (10)

where i represents the probability of each model, ijrepresents the transition probability matrix and Yk1prepresents the measurement obtained from the system

until sample time (k-1). The probability transition matrix

represents the transition probability between different

operating regimes. The ij represents the probability of

transition from modeito modej. Using a minimum mean-

square error (MMSE) estimation it is possible to obtain

the optimal mode for a certain operating regime [17].

Because several numerical problems can occur (exponen-

tial computation and memory growth) using this optimal

estimation, a sub-optimal method known as the interacting

multiple model (IMM) has been proposed [16]. Although,

this sub-optimal method solves the numerical problems

has a major drawback, the fact that the transition matrix

has to be known in advance. In this sense the resultsachieved with this method are very dependent of the initial

guess of the transition matrix. In order to make this

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method more robust to the initial transition matrix, it has

been proposed that this matrix can be updated at every

sample time using the measurement information received

from the process. In this work two different algorithms are

tested to update the transition matrix. Both algorithms are

based on a simple idea:

At every sampling time the mode probability isupdated and the mode likelihood functions (just like

in IMM algorithm or generalized pseudo-Bayesian

algorithm [16]):

i(k) P{si(k)|(k 1), Y(k1)p }

Li(k) p[Yp(k)|si(k),(k 1), Y(k1)p ](11)

The transition probability matrix is updated using

(k 1) and using new information obtained fromthe process measurements.

Using this idea and after some mathematics manipulation

it is possible to prove that [17]:

p[1j ] = {1 +j(k)[1j 1j(k 1)]

Lj(k)}p[1j|Yk1p ] (12)

for j = 1, 2, , N (13)

where:

j(k) = i(k 1)

(k 1)(k 1)L(k) (14)

Using this approach the update of the transition prob-

ability matrix can be done on-line using two different

methods:

Quasi-Bayesian Algorithm (QBA)- in this algorithm

it is defined that:

ij(k) = 1

k+i(0)ij(k) (15)

where,

ij(k) =ij(k1)+ ij(k 1)gij(k)Nj=1ij(k 1)gij(k)

(16)

and:

gij(k) = 1 +i(k)[Li(k)

i(k 1)L(k)] (17)

Numerical-Integration Algorithm (NIA) - Using the

non decoupled probability density function versionof (12):

p[|Zk] = (k 1)L(k)

(k 1)(k 1)L(k)p[|Zk1]

(18)

Using tpm = 1, 2, , V transition probability ma-trixes it is possible to:

p(tpm)(k) = (k 1) (tpm)L(k)

(k 1)(k 1)L(k)p(tpm)(k 1)

(k) = 1

N

tpm=1

V

tpmptpm(k) (19)

See [17] in order to see a complete description of both

algorithms.

0 500 1000 1500 2000 2500 3000 3500 40001

2

3

4

5

0 500 1000 1500 2000 2500 3000 3500 40000

10

20

30

0 2000 4000 6000 8000 10000 12000 14000 16000 180000

1

2

3

4

5

Time (s)

Signal Fault

Fig. 2. Faults introduced in the system.

III. RESULTS

In order to test the proposed FDI mechanism an HVAC

system is used. Using models of the different system

components under different fault conditions, a simulation

is created to evaluate the performance of the FDI, before

the real experiments are performed. After the simulation

test the proposed method was applied to the real HVAC

system, using real faults introduced in the HVAC compo-

nents.

A. Simulation Results

The different fault regimes in the actuator are modelled

using neural network ARX models [13] and were built in

order to cover the operating regime of the heater bank

actuator fault. Sensor faults were modelled using four

different operating regimes and a nominal model. The four

different regimes were computed to cover the domain of

the output of the temperature sensor of the heater bank.

The following simulation tests were conducted:

Actuator Fault - in this case an incipient fault was

introduced in the system (see Fig. 2 a)). The fault

was introduced in the 1000 second of the simulation.

Fig. 3 presents three different results with three

different strategies in the FDI mechanism. The first

result presented is obtained with an FDI mechanism

that uses only (9) without the probabilistic approach.

Fig. 3 b) and 3 c) represent the results obtained using

the cost function (9), algorithm QBA and algorithm

NIA, respectively. The sample time used in this

simulations was 10 seconds and the data window

used in the optimization of the cost function is of 40

samples.

Sensor Fault - the sensor fault introduced was also

a incipient fault, and was introduced in the 1000

second of the simulation (Fig. 2 b)). Also, in thiscase three different approaches were used to test

the different algorithms. In Fig. 4 is presented the

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0 500 1000 1500 2000 2500 3000 3500 40000

0.5

1FDI without Probability Analysis

Weights

Time(s)

0 500 1000 1500 2000 2500 3000 3500 40000

0.5

1

FDI with Probability Analysis QBA

Time(s)

Weights

0 500 1000 1500 2000 2500 3000 3500 40000

0.5

1FDI with Probability Analysis NIA

Time(s)

Weights

Process Model Fault Model 1 Fault Model 2 Fault Model 3

Fig. 3. Actuator incipient faults in simulation

0 500 1000 1500 2000 2500 3000 3500 40000

0.5

1FDI without Probability Analysis

Weights

Time(s)

0 500 1000 1500 2000 2500 3000 3500 40000

0.5

1FDI with Probability Analysis QBA

Time(s)

W

eights

0 500 1000 1500 2000 2500 3000 3500 40000

0.5

1FDI with Probability Analysis NIA

Time(s)

Weights

P. Model F. Model 1 F. Model 2 F. Model 3 F. Model 4

Fig. 4. Sensor incipient faults in simulation

different results, with and without the probabilistic

approach. In this case the sample time is also of 10

seconds and the cost function is calculated using 400

seconds of data.

B. Experimental Setup

A heater bank (part of the HVAC system) was used to

test the FDI algorithm. The heater bank is an electric

heater with a control action between 0 to 4 V. The

temperature is measured before and after the heater. The

air supply flow of the air handling unit passes through

the heater and is delivered to three different rooms. Theheater has a complex nonlinear behavior if the air flow in

the inlet is changed. A supply fan is used to distribute

Fig. 5. Experimental setup of the developed architecture.

the hot air flow to the rooms, and a terminal unit is

used to control the flow entering the room. Fig. 5 shows

a schematic of the process used to test the algorithms.

The fan works normally at a constant speed, but it is

possible to change it by means of a 0 to 10 V signal. In

this work, two temperature sensors reading the inlet air

temperature and outlet air temperature in the heater are

used. The instrumentation used in the system is connected

by a LON Network and all the signals are acquired by

a Personal Computer with Microsoft Windows R and

are available to the operating system by a DynamicData Exchange (DDE) Server. Matlab R, SimulinkR, andthe Optimisation Toolbox is used to implement the FDI

mechanism.

Two types of faults were simulated. A combination of

incipient fault and abrupt faults were tested. The fault

introduced represent usual problems in heaters, such as

material settling in the heater surface. The faults are

simulated a power losses in the heater. The multiple

model approach was used using three different models

to cover the domain of the fault plus one model for

the normal conditions. Neural network ARX models [13]

were employed. (see Table I). The faults were introduced

in the system during a period of 17000 seconds and

within that period three faults were tested: two of them

incipient faults and one abrupt fault. The structure of the

faults was built in order to simulate the degradation of

this kind of systems. The degree of confidence in each

one of the methods was = 1 = 0.8. Regardingthe probabilistic approach and NIA algorithm the initial

number of transition probability matrix were 50, and the

initial guess of the transition matrix for algorithm QBA

was,

0.9 0.08 0.01 0.01

0.02 0.9 0.04 0.04

0.01 0.03 0.9 0.060.01 0.01 0.08 0.9

(20)

and the initial probability vector was:

0.9 0.05 0.0025 0.0025

(21)

Fig. 6 shows the results obtained with the different

approaches. Observing Fig. 6 it is possible to see that in

different situations the probabilistic approach can improve

the detection and isolation mechanism. One of the aspects

that is more relevant is the less variability in the models

identified by the weights. This will introduce less false

alarms in the system and will make control reconfigu-

ration more accurate. In the period between 12000 and16000 seconds the fault is constant and it is clear that the

methods with the probabilistic approach are more accurate

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TABLE I

NEURALN ETWORK MODEL.

Weights

Neuron 1 Input to Hidden Layer 0.0404 -0.08210.0000 -0.9310

Hidden to Output Layer 35.2373

Neuron 2 Input to Hidden Layer 0.0404 -0.08210.0000 -0.9310

Hidden to Output Layer 35.2373

0 2000 4000 6000 8000 10000 12000 14000 16000 180000

0.5

1FDI without Probability Analysis

Weights

Time(s)

0 2000 4000 6000 8000 10000 12000 14000 16000 18000

0

0.5

1FDI with Probability Analysis QBA

Time(s)

Weights

0 2000 4000 6000 8000 10000 12000 14000 16000 180000

0.5

1FDI with Probability Analysis NIA

Time(s)

Weights

Process Model Fault Model Fault Model 2 Fault Model 3

Fig. 6. FDI results in a heater bank of a HVAC system

in the isolation of the fault. With the abrupt changeintroduced at 10000 seconds, the probabilistic approach

(QBA) achieves a fast fault identification. Nevertheless,

the probabilistic approach can result in worst results when

the fault is not clearly firing a certain mode of operation.

Between 4000 and 6000 seconds the fault introduced in

the system is between two different fault models, and in

this case the probabilistic approach has some difficulties

to obtain a suitable combination of weights. This kind

of behavior can be modified by changing the degree of

confidence of the two techniques, or even by making some

small changes in the initial transition probability matrix.

IV. CONCLUSIONS

A hybrid fault diagnosis and isolation scheme has

been proposed and applied to a heating unit of a Heat-

ing, Ventilation Air Conditioning system. The hybrid

technique combines probabilistic and optimisation based

methods. The results indicate that the introduction of the

probabilistic approach leads to better fault identification.

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