9.3 Fuzzy controller for the Location Update procedure 164
9.3.1 Defining the interval ranges and membership functions
The range of possible EI values must be defined before starting to train the controller, thus minimum and maximum values. When an EI is out of min-max value, it cannot be predicted. The predicted value will be at the extreme of the domain, but still within it.
Increasing the difference between the defined domain and real values increases the prediction error. On the other hand, unnecessarily large intervals require more membership sets to achieve sufficient precision of the controller. More membership sets require more Fuzzy rules to cover all possible value combinations. More rules means more CPU and memory resources at the device. To recapitulate, the value range cannot be increased to infinite. An interval between 0 and 24 hours is defined in M-VPN, which seems to be a good value. The domain (range) should not cover all Event Intervals, but it should cover the majority of them. The output and input domain are the same in M-LU because the input and the output are the same variable types.
Each domain must be divided into regions of the membership functions. The number of regions defines the precision of the prediction. The higher the precision, the more regions are needed. The Maximum Disconnection Interval (MDI) defines the precision necessary to some degree. The membership region is proportional of the maximum disconnection. The membership functions overlap by 50% on the left and right side. An overlapping is very important since it covers directly not considered values combinations. The total number of regions in the input-output domain is:
size region
size domain regions
of number
R
N 2.D
= ,
where Ddomainsizeis the domains size, Rregionsize is the region size proportional to the maximum Disconnection Interval. The region size is set 3 times the MDI in this simulation. This value is chosen trough multiple simulation for delivering good results. It is an optimization task to find optimal values and it is out of scope in this thesis.
The antecedent membership functions are triangular. The advantages of the symmetric triangular form are the ease of calculation and good performance.
9.3.2 Creation of rule base
The expert rules provide raw information on how the controller should act. For example:
they can be created using the assumption that the M-LU is implemented in the device used by an employee. As generally known, there are two maximums in business days: in the morning and after lunch. The maximums are higher on Monday than on Friday. There is a reduced level of activity at the weekend. Typical diagrams for day and week activity can be seen in Figure 9.5. Since there is no real data real data of mobile usage available, the data has been gathered from the Internet exchange point - PARIX. The Mobile Client movements depend on the usage, like smart phone, train etc.
Abstracting rules can generally describe the day and week activity and can be involved in covering the main extremes. They could consist of two rules, for example:
IF 7:oo h < time AND time < 20:oo h THEN Location Update Interval IS small IF Monday < day AND day < Friday THEN Location Update Interval IS small In the next step, the rules are created trough training as described in chapter 9.2.2.1.
9.3 Fuzzy controller for the Location Update procedure 165
Figure 9.5: Day and week activity
9.3.3 Number of inputs of the Fuzzy controller
Before starting the OP (One Pass) method, the number of inputs of the Fuzzy controller must be set, i.e. how many past values must be considered in the prediction. Increasing the number of inputs improves the prediction precision. This can be compared with the interpolation of curves using more given points. The interpolation becomes more realistic when involving more points.
Unfortunately, increasing the number of inputs requires more rules to cover transition cases. The exact number of rules cannot be determined, since the form of the interval transition plays a major role. It is factorial incensement of the number of inputs according to the authors subjective experience. In the simulation, about 100 rules were required for domain 0 to 700 sec with 3 input variables. Using more than 10 past input values requires many thousands of rules.
The input values (Measured Intervals) are typical sequentially following values, thus values with index k, k-1, k-2...k-n where k is the current index and n is the number of inputs.
Reducing the number of inputs reduces the past time period considered in the controller.
Reducing the considered past values decreases the prediction ability. To overcome this problem, not sequential following sequential values can be involved. For example series such as k, k-1, k-3, k-5, k-10,…. The considered period increases and the precision decreases. It is a trade off between the precision and the considered past values. The use of not sequential following past values is only mentioned and it is part of future work. M-LU uses monotone following values.
After the rule base is created, the centres of the consequent membership function must be updated according the RLS optimisation method. The implementation is straightforward from the theoretical part, see 9.2.2.2.
9.3.4 Distribution of the Update Time Points
The Fuzzy controller predicts the coming Event Interval using past Measured Intervals (MIs). The predicted Event Interval (EI) shows the highest probability where the event (PoA change) could happen. The Fuzzy controller does not deliver the PDF of the EI but only a single value in contrast to the Particle filter described in chapter 8. The PDF cannot be constructed from this single value. This is a general issue of the Fuzzy method. The requirement of the chapter 7 cannot be met directly, i.e. the Update Intervals(UIs) to be distributes as the real PDF. This shortcoming is solved with the assumption that the probability of EI is in approximately Normal distributed with the standard deviation proportional to the Update Interval(UI). This principle is also applied in the extended Kalman Filter for M-LU presented and derived in 10.3.
9.3 Fuzzy controller for the Location Update procedure 166
Normal distribution of EI is continuous and not zero for all values. Implying this would mean that there is a nonzero probability at all time points, even in the past. This is certainly not the case. To overcome this controversy, we define the first condition: the probability for ETP is Normal distributed except for the past points. Using subjective knowledge, it is defined that the distribution must have very low values at the start of the filter cycle (zero time). The standard deviation is three times of the predicted EI as described in chapter 10.3.
The second condition is that the size of the Update Interval(UI) must also not exceed the Maximum Disconnection Interval (MDI) defined by the application.
An abstract curve of UI satisfying both conditions is shown in Figure 9.6.
Setting the UI proportional to the Normal distribution (PDF of EI) and not exceeding MDI is derived in chapter 10.5. The main idea is to create a transformation function for converting constant Update Intervals in a non-linear interval proportional to the PDF of EI.
The transformation function has an input the time point of the constant updates relative to the filter cycle. It depends on the parameter: the prior estimated EI (predicted EI) and the Maximal Disconnection Interval (MDI). The output of the function is Update Time Point considering the PDF of EI.
The transformation function T() derived in chapter 10.6 is:
+
−
+ = = −
= − − k
erf k erf
P e
C k e
erf u
k P 2 (0.95)
) 95 . 0 ( 2 1
1 1
. 2 95 . 0 2 ,
), 95 . 0 (
π π
( ) ( ) [ ]
∞
∪
−
−∞
∈ +
+
−
= ∈
′ ( , ) (, )
, /
.
2 x l l
C ax
l l x k
x erf x u
T
Then the inverse function is:
( ) ( ) [ ]
∞
′+
∪
′−
−∞
∈
−
′+
′−
= ∈
′− −
) ), ( ( )) ( , (
) ( ), ( /
.
2 1 1
l T l T x C x
l T l T x u
x erf x k
T
, where
( )
x =∫
xe−t dterf
0
2 2
π
,
Figure 9.6: Location update interval and the probability of PoA change
ETP Time relative to estimated ETP
Probability / Interval lenght
Probability of event time point and update interval
maximal user defined disconnetion interval
PDF of ETP Update interval lenght
9.3 Fuzzy controller for the Location Update procedure 167
The Pk+ is the prior estimated Event Interval (PoA change interval) and x time point of the constant updates. An example of the function is shown in Figure 9.7. The detailed derivation is presented in 10.3.
9.3.5 Implementation procedures and structures
An overview of the implementation is provided in this chapter. The main blocks of real implementations are presented in Figure 9.8.
The user defines the Maximum Disconnection Interval (MDI), the training period, user profile etc, see Figure 9.8 step 1. The MDI is different for different types of applications such as VoIP, FTP etc. The user profile contains the raw Fuzzy rules. For example: working employee, travelling sales man, night shift employee. The profile can be omitted if there is no exact idea of the behaviour. The rules can only be created using the self-learning OP methods. The training period must be sufficiently large, so that all possible situations can be observed.
During the second step (Figure 9.8, step 2) the Fuzzy rules are loaded from the profile.
The training data is collected during step 3 (Figure 9.8). The OP rules are created in step 4.
The centres of the output membership functions are optimised according to the RLS method in step 5. The Fuzzy controller is ready for use in the LU procedure.
Let n be the number of past values considered in the Fuzzy controller. The last n Measured Intervals (MI) of the training data are used as an input for the newly designed Fuzzy controller, step 6 on Figure 9.8. The result of the Fuzzy controller - predicted Event Interval is a parameter of the transformation function. Additional parameters of the transformation function are set as described in 9.3.4. The input of the transformation function is the time point of monotone constant updates equal to the MDI. The result of the transformation function is the Update Time Point. During step 7, the Location Update procedure is executed. The result of the LU procedure is Boolean - true or false, dependent on whether an event (PoA change) has occurred within the interval or not. If an event has occurred then the middle of UI is the Measured Interval(MI). The prediction is repeated again, where the last n MIs are the input of the Fuzzy controller at step 6. If there is no event (PoA change) in step 7, then the next LU Update Interval is calculated and LU executed at the time point. Step 7 is repeated until the PoA change event occurs. The implementation runs infinitely and it interrupts when the user stops the execution.
Figure 9.7: Transformation function
update time point, relative to estimated ETP [sec]
constant update time point [sec]
Transformation function
-10 -8 -6 -4 -2 0 2 4 6 8 10
-10 -8 -6 -4 -2 0 2 4 6 8 10
9.4 Simulation of M-LU with adaptive Fuzzy Logic