Phase-lock loop circuit with fuzzy control

A phase-lock loop circuit with fuzzy control, includes a phase comparator whose output is connected to a low-pass filter that drives a voltage-controlled oscillator. The phase comparator generates a signal that represents the phase difference between an input signal and a signal generated by the oscillator. The oscillator of the present invention is furthermore driven by a control signal generated by fuzzy control. The input of the fuzzy control is the signal that represents the phase difference.

BACKGROUND OF THE INVENTION 
1. Field of the Invention 
The present invention relates to a phase-lock loop circuit (PLL) with fuzzy 
control. 
2. Discussion of the Related Art 
As is known, the Phase Lock Loop (PLL) is the electronic system that allows 
to match and follow the phase 5 of an input signal with the phase of a 
locally generated signal. 
Simple circuits are normally used to provide this function, such as a phase 
comparator, a voltage-controlled oscillator (VCO), and a low-pass filter 
connected in a particular negative-feedback configuration that adapts to 
phase variations of the input signal and thus to its frequency variations. 
The structure of a conventional PLL is shown in FIG. 8. The conventional 
PLL is essentially composed of a phase comparator 2, a voltage-controlled 
oscillator 4, and a low-pass filter 3. 
The purpose of the phase comparator 2 is to provide information on the 
phase difference between the input signal x and the output signal y 
generated inside the PLL by the voltage-controlled oscillator 4. 
A typical example of transfer curve of the phase comparator 2 is shown in 
FIG. 9. A first fundamental characteristic of the phase comparator 2 is 
the periodic nature of the criterion h that relates the inputs of the 
phase comparator and the output, i.e.: 
EQU z=h(.DELTA..phi.) 
where z is the output signal of the comparator 2 and .DELTA..phi. is the 
phase difference between the two input signals x and y. 
A second fundamental characteristic of the phase comparator is the 
nonlinearity of h within each period. 
The voltage-controlled oscillator 4 performs the role of generating the 
output signal y(t) which is provided to the input of the phase comparator 
2. The frequency of the oscillator 4 is driven by a single input voltage. 
It should be noted that the relationship between the value of the input 
voltage that originates from the filter 3 and the output frequency of the 
oscillator 4 may be nonlinear. 
The low-pass filter 3 has the fundamental purpose of closing the feedback 
loop and of permanently setting the static and dynamic characteristics of 
the PLL. Operation of the PLL is in fact mainly dependent upon the order 
and type of the filter used. 
A simplified layout of the conventional lock circuit, designated by the 
reference numeral 1, is shown in FIG. 7. A typical input waveform to the 
PLL 1 is given by the function: 
EQU x(t)=g.sub.T (w.sub.f t+.phi..sub.g), 
and a typical output waveform is given by the function: 
EQU y(t)=f.sub.T (w.sub.f t+.phi..sub.f). 
The purpose of the conventional PLL circuit is to equalize the phase and/or 
frequency of the two signals. 
Different situations can arise according to the initial frequency 
difference of the two signals. 
If .vertline.w.sub.f -w.sub.g .vertline.&gt;.DELTA.w.sub.L, where 
.DELTA.w.sub.L is the locking range of the PLL, the circuit will tend to 
lock gradually, exhibiting cycle skipping. This condition generally has 
the drawback of being excessively long. If the loop does not converge, the 
condition becomes infinitely long. This situation must therefore be 
absolutely avoided in normal operating conditions. 
If, however, .vertline.w.sub.f -W.sub.g .vertline.&lt;.DELTA.w.sub.L, then the 
output signal of the PLL circuit correctly follows the phase and/or 
frequency variations in the input signal. This is the optimum operating 
condition. 
A particular factor of such phase-lock circuits is their lock-in range, 
i.e. the frequency range within which the PLL correctly operates to follow 
any phase and/or frequency variation of the input signal. It is usually 
desirable that this range be the widest possible. 
From this point of view the need to keep this range as wide as possible is 
evident. The goal of making the lock-in range as wide as possible can 
entail problems for other important factors of the PLL. A particularly 
critical problem is the behavior of the phase-locked loop with respect to 
noise. 
An object of the present invention is to provide a phase-lock circuit with 
fuzzy control that considerably extends its lock-in range compared with 
known phase-locked loop circuits. 
Another object of the present invention is to provide a circuit that allows 
a control that is independent of the type of phase comparator used. 
Another object of the present invention is to provide a circuit that can 
reduce locking times with respect to known circuits. 
Another object of the present invention is to provide a circuit that is 
highly reliable, relatively easy to manufacture, and low cost. 
SUMMARY OF THE INVENTION 
These and other objects and advantages are achieved by providing a 
phase-lock circuit with fuzzy logic control, which includes a phase 
comparator whose output is connected to a low-pass filter, said low-pass 
filter being suitable to drive a voltage-controlled oscillator. The phase 
comparator is suitable to generate a signal that represents the phase 
difference between an input signal and a signal generated by the 
oscillator. The oscillator is furthermore driven by fuzzy logic control 
which receives the signal representing the phase difference.

DETAILED DESCRIPTION 
In order to better explain the inventive concept of the present invention, 
some introductory principles of fuzzy-logic control, as used in the 
circuit and the process according to the present invention, are described 
below. 
Fuzzy logic, as compared with classical logic, attempts to model reasoning 
processes that are typical of the human mind, allowing machines to make 
rational decisions in uncertain and inaccurate environments. 
Fuzzy logic provides a set of rules for handling non-exact facts. These 
rules are expressed by means of the semantics of a linguistic method. 
The basic concepts of fuzzy logic are linguistic variables and fuzzy sets, 
the latter being characterized by membership functions. 
Fuzzy logic operates using linguistic descriptions of reality. This means 
that a problem is not characterized exactly (as with a mathematical model) 
but rather is provided as a linguistic representation of the algorithms. A 
particular class of variables, known as linguistic variables, is used to 
represent information that becomes available during the linguistic 
description step. Linguistic variables are characterized based on the type 
of values that can be assigned to them. The type of value can include 
words or sentences in any natural or artificial language. 
Accordingly, linguistic variables contain the semantic meaning of the 
sentences used in modeling the problem. Syntactically speaking, a set of 
values that depends on the selected variable can be found for each 
linguistic variable. This set can assume different meanings according to 
the context in which it is used. 
For each linguistic variable it is possible to provide a table that 
summarizes all the values that the linguistic variable can assume. These 
values can generally be obtained by applying appropriate modifiers to a 
primary term, which represents the variable, or to its opposite. The 
following table gives an idea of this. 
______________________________________ 
Linguistic variable 
name: TEMPERATURE 
______________________________________ 
Primary term COLD 
opposite WARM 
Modifiers NOT, VERY, MORE, or LESS 
______________________________________ 
Fuzzy sets and the associated membership functions are closely linked to 
the above-mentioned linguistic variables. Each value assigned to a 
linguistic variable is in fact represented by a fuzzy set. 
A fuzzy set can be considered as a distribution of possibilities that links 
a particular value of a linguistic variable to a definition domain (the 
universe of discourse). If a fuzzy set is plotted on a chart, the degrees 
of membership (or truths) are plotted on the ordinate, whereas the 
universe of discourse, i.e., the definition domain of the fuzzy variable 
(in this case, the temperature and the related fuzzy set), is plotted on 
the abscissa. 
This domain can be a continuous space {x} or a discrete representation {x1 
. . . x2}. For example, if X is a temperature, {x} represents its range of 
variability, whereas {x1 . . . x2} represents the discrete values that 
characterize it. 
A membership function .mu.(x) is a function that identifies a fuzzy set in 
the universe of discourse that is characteristic of a linguistic variable, 
and that associates a degree of membership of a given value to the fuzzy 
set for each point of the definition domain (universe of discourse), 
accordingly mapping the universe of discourse in the interval 0,1!. 
A membership value .mu.(x)=0 indicates that point x is not a member of the 
fuzzy set being considered, which is identified by the function .mu., 
whereas a membership value .mu.(x)=1 indicates that the value x is 
certainly a member of the fuzzy set. 
Membership functions are entities on which fuzzy calculus is performed. 
This calculus is performed by means of appropriate operations on the sets 
represented by the membership functions. 
The collection of all the fuzzy sets of a linguistic variable is known as a 
"term set". FIG. 2 summarizes the definitions given earlier. FIG. 2, for 
the sake of graphic simplicity, plots triangular membership functions 
.mu..sub.cold, .mu..sub.medium and .mu..sub.warm which can generally be 
represented by any linear or non-linear function. 
The adoption of a particular computational model is one of the factors that 
affect the performance of the device. However, the fuzzy control process 
for phase-lock circuits according to the present invention can be 
implemented with any fuzzy computational model. Examples of these 
computational models will be described hereinafter. 
At the high level, a fuzzy program is a set of rules of the IF-THEN type. 
The following example shows a set of three rules with two inputs (A and B) 
and two outputs (C and D). The various terms A1, A2 . . . D3 represent the 
knowledge of the system, obtained from expert technicians or in other 
ways, in the form of membership functions: 
rule 1: IF (A is A1)! AND (B is B1)! THEN (C1 is C.sub.1) AND (D1 is 
D.sub.1)! 
rule 2: IF (A is A2)! AND (B is B2)! THEN (C2 is C.sub.2) AND (D2 is 
D.sub.2)! 
rule 3: IF (A is A3)! AND (B is B3)! THEN (C3 is C.sub.3) AND (D3 is 
D.sub.3)! 
The part of each rule that precedes THEN is commonly 25 termed the "left 
part" or "antecedent", whereas the part that follows THEN is termed 
"consequent" or "right part". 
The inputs A and B, after being appropriately fuzzified, i.e., converted 
into membership functions, are sent to the rules to be compared with the 
premises stored in the memory of the control device (the IF parts). 
Multiple rules are combined simply by means of a fuzzy union operation on 
the membership functions that are the result of each rule. 
Conceptually, the better the equalization of the inputs with the membership 
function of a stored rule, the higher the influence of said rule in 
overall computation. 
In order to determine this equalization, weight functions which identify 
some particularly indicative values are determined. One of these weight 
functions is the function .alpha., which indicates the extent to which the 
input propositions (A1, B1) match the stored premises (A, B). In the above 
example of rules, the function .alpha. is given as: 
EQU .alpha..sub.i A=max(min(A1,A.sub.i)) 
EQU .alpha..sub.i B=max(min(B1,B.sub.i)) 
for i=1, 2, 3 (number of rules). 
The second weight function is .OMEGA..sub.i, which indicates the extent of 
the "general resemblance" of the IF part of a rule. For the above example, 
the function .OMEGA..sub.i is calculated as: 
EQU .OMEGA..sub.i =min( .alpha..sub.i A, .alpha..sub.i B, . . . ) 
for i equal to the number of rules and with as many items inside the 
parentheses as there are propositions (the IF part) of each rule. As an 
alternative to the above membership function, a weight function equal to 
the product of the individual membership values is usually used: 
EQU .OMEGA..sub.i=.alpha..sub.i A.times..alpha..sub.i B 
These values, which in practice define the activation value of the 
antecedent of the fuzzy inference, are used subsequently to calculate the 
activation value of the consequent (i.e., the right part). 
As far as this subject is concerned, two different inference methods are 
generally considered: MAX/DOT and MAX/MIN. Essentially, both methods act 
by modifying the membership functions of the consequent by means of a 
threshold value supplied by the antecedent. 
The MAX/MIN method acts by clipping the membership functions related to the 
consequent in the manner shown in FIG. 2. The rule of the fuzzy inference 
of FIG. 2 is as follows: 
IF alpha IS low AND delta IS high THEN gamma IS medium 
As regards the values "alpha" and "delta" in input, one uses the related 
lower (threshold) membership value with which the membership function of 
the output "gamma" is clipped. In practice, the membership function in 
output will have no value higher than the threshold value. 
The MAX/DOT method instead acts by modifying the membership functions of 
the right part (the consequent), so that the membership function of the 
output is "compressed", while trying to maintain its original shape as 
much as possible. The MAX/DOT method for the same rule as above is shown 
in FIG. 3. 
In the case of fuzzy control, it is possible to simplify the calculation of 
the weights .alpha.. It is in fact possible to considerably reduce the 
amount of calculation by assuming that one is dealing with a degenerate 
case of fuzzy calculus in which the input variables are not fuzzy sets 
(ambiguous values) but are variables which generally originate from 
sensors and are thus definite numeric values. The input data are not fuzzy 
sets but crisp values. 
In order to represent these values within a fuzzy system, they must be 
converted into crisp membership functions, i.e., into particular 
membership functions which have an activation value of 1 ("TRUE") at the 
point which corresponds to the value provided in input. Equally, these 
crisp values have a zero ("FALSE") value in the remaining part of the 
definition range. This concept is shown in FIG. 4. 
In order to convert a physical value, provided for example by an external 
sensor, into a fuzzy value, it is thus sufficient to assign the maximum 
truth value that is characteristic of the system to the point of the 
definition range that is identified by the measured value. With reference 
to computation, this means that the case shown in FIG. 5 always occurs. 
Calculation of the weights a in the particular case of FIG. 5, where there 
are crisp values such as occur in the case of machines that control 
physical values, becomes merely a matter of finding the intersection 
.alpha.B and .alpha.A of the input variables with the membership functions 
imposed by the term sets A and B. 
The weights thus calculated are then used for computation on the consequent 
of the fuzzy inference (i.e., on the fuzzy rules). 
It should be noted that for control systems as in the case of the present 
invention, the output of the fuzzy regulator must be a definite physical 
value of the control criterion. Generally, once the inference has been 
performed on the right part of the fuzzy rules, a fuzzy set is obtained. 
It is accordingly necessary to defuzzify, i.e., to extract a definite 
numeric value from the calculated fuzzy set. There are various 
defuzzification methods, such as, for example, the centroid method, the 
maximum height method, etc. In practice, for reasons related to numeric 
precision, the most widely used method is the centroid method, according 
to which: 
##EQU1## 
where n is the number of rules and C represents the centroids (centers of 
gravity) of the membership functions of the consequents of each rule, 
appropriately modified by using the MAX/MIN or MAX/DOT method. The 
functions .OMEGA. are determined as described earlier, using either the 
minimum among the functions a or the product thereof. This computational 
model is referenced as the MAMDANI computational model. As an alternative, 
it is also possible to use another alternative fuzzy computational model, 
referenced as SUGENO model, in which defuzzification is performed simply 
by means of the following rule: 
##EQU2## 
In the above equation, .OMEGA..sub.0 is always equal to 1. In practice, the 
defuzzified value is determined by a linear combination of the activation 
values of each individual rule. 
With reference to FIG. 6, in the case of a fuzzy controller, the input 
values are numeric values (input 1-n) which originate from sensors. In 
this case it is necessary to fuzzify these values to obtain fuzzy values 
.alpha., apply the fuzzy inference (the rules) to obtain the weight 
functions a of said fuzzy values, and finally defuzzify these weight 
functions .OMEGA. so as to obtain a definite numeric value y in output. 
The simplified block diagram of the phase-lock loop circuit is shown in 
FIG. 10. A fuzzy controller 5 is added to the structure of the 
conventional PLL 1 and receives the phase error z. 
The output of the fuzzy controller 5 is a signal which is termed injection 
signal V.sub.inj. 
The detailed structure of the circuit according to the present invention is 
shown in FIG. 11. The circuit comprises the conventional elements of a 
PLL, i.e., the phase comparator 2, the low-pass filter 3, and the 
voltage-controlled oscillator 4. 
The input of the fuzzy controller 5 is connected to the output of the 
comparator 2, so as to receive as input the phase error signal z. The 
output of the fuzzy controller 5 is instead connected to an adder node 6 
which is interposed between the low-pass filter 3 and the 
voltage-controlled oscillator 4. In this manner, the output signal of the 
fuzzy controller 5 is added to the output of the low-pass filter 3. 
In this manner, the input of the voltage-controlled oscillator 4 receives a 
signal that is the sum of the injection voltage generated by the fuzzy 
controller 5 and of the voltage in output from the low-pass filter 3. 
The fuzzy controller 5, by accepting as input the voltage z which is 
proportional to the phase difference produced by the phase comparator 2, 
generates an injection signal V.sub.inj which is added to the signal that 
originates from the filter 3. The sum of the two signals drives the 
voltage-controlled oscillator 4, acting to compensate for large and thus 
dangerous variations in the phase error z that accordingly tend to cause 
the unlocking of the PLL. 
The fuzzy controller 5, whose relation V.sub.inj -z is typically nonlinear, 
acts by using signal processing based on fuzzy rules according to the 
method described earlier. 
Since there is a single input of the fuzzy controller, i.e., the phase 
error z, and a single output constituted by the injection voltage 
V.sub.inj, which is added to the output voltage from the low-pass filter 
3, two fuzzy variables are introduced. 
Examples of membership functions are shown in FIG. 13. For the input signal 
z there are three trapezoidal membership functions: NORMAL, ABNORMAL, and 
CRITICAL, which represent the various states of the phase difference z. 
Three triangular membership functions, LOW, MEDIUM, and HIGH, have instead 
been developed for the voltage injection signal V.sub.inj. 
Said membership functions are stored in the fuzzification means of the 
fuzzy controller 5. 
A series of fuzzy rules that are characteristic of the 30 behavior of the 
circuit to be obtained is then developed. 
An example of these rules is the following: 
rule 1: IF phase error IS NORMAL THEN voltage injection IS LOW 
rule 2: IF phase error IS ABNORMAL THEN voltage injection IS MEDIUM 
rule 3: IF phase error IS CRITICAL THEN voltage injection IS HIGH 
These rules are stored in the fuzzy inference unit of the fuzzy controller 
5. 
The fuzzy inference unit applies to the fuzzy rules, in, the above 
described manner, the measured values of the phase difference z and the 
corresponding membership functions so as to obtain a consequent of the 
fuzzy rules. 
The weights .OMEGA. of the antecedents are calculated, in the above 
described manner, by a weight calculation unit of the fuzzy controller 5. 
Defuzzification means of the fuzzy controller 5 extract crisp values of the 
voltage injection signal V.sub.inj on the basis of the calculated weights 
.OMEGA. and of the results of the rules. 
The defuzzification method can be any one among those described earlier 
(MAX/MIN, MAX/DOT). 
The crisp values can be calculated by means of the centroid method 
described earlier. 
The circuit executed according to the present invention considerably 
improves the lock-in range .DELTA.w.sub.L. Practical tests that have been 
conducted have furthermore demonstrated that locking times are shorter 
than those of the conventional structure described earlier. 
In order to better understand the operating principle of the new structure, 
reference should be made to FIG. 12, which plots the frequency spectra of 
a conventional PLL and of a PLL according to the present invention. 
FIGS. 12a, 12b, and 12c are related to a conventional PLL structure, 
whereas FIGS. 12d, 12e, and 12f relate to the circuit according to the 
present invention. 
While the locking situations shown in FIGS. 12a and 12b coincide in both 
cases, since they indeed correspond to the optimum locking situation, it 
is evident in FIG. 12e that a change in the input frequency w.sub.g 
entails a shift of the lock-in range .DELTA.w.sub.L in the case of the 
circuit according to the present invention. This occurs because the input 
frequency is kept within the lock-in range. This does not occur in a 
conventional PLL, where the lock-in range remains anchored to the original 
position, as shown in FIG. 12b. 
The substantial improvement becomes evident by comparing FIGS. 12c and 12f. 
In the case of FIG. 12c, the input frequency w.sub.g is outside the 
lock-in range .DELTA.w.sub.L. Accordingly, the subsequent locking occurs 
(if it can) with cycle skips and in a relatively long time. In the case of 
FIG. 12f, by using the circuit according to the present invention locking 
occurs every time and very quickly, since the input frequency lies always 
within the proposed lock-in range. 
The present invention accordingly fully achieves the intended aim and 
objects. 
The circuit according to the present invention is able to produce a 
considerable increase in the lock-in range, particularly at very high 
values, with respect to a conventional PLL. 
The introduction of fuzzy logic allows not only easy hardware 
implementation of the control surface, but also the possibility to 
optimize the control according to the type of phase comparator that is 
used. This structure is furthermore equally effective even in the presence 
of noise. 
Easy implementation is linked to the consideration that the provision of 
nonlinear curves, suitable to produce specific control surfaces, incurs in 
the difficulty of implementing desired curves. This occurs because if said 
curves are obtained with analog methods it is difficult to find components 
that perform the desired function over a sufficiently wide range. If 
instead the digital approach is used, one encounters the circuital 
complexity in the necessary A/D and D/A conversion and the consequent 
reduction in intervention speed, which might be insufficient for the very 
fast applications in which most PLLs must operate (telecommunications). 
These problems are avoided with the circuit and the related fuzzy control 
according to the present invention. 
The invention thus conceived is susceptible of numerous modifications and 
variations, all of which are within the scope of the inventive concept. 
Finally, all the details may be replaced with other technically equivalent 
ones. 
In practice, the materials employed, as well as the shapes and dimensions, 
may be any according to the requirements without thereby abandoning the 
protective scope of the appended claims. 
Having thus described at least one illustrative embodiment of the 
invention, various alterations, modifications, and improvements will 
readily occur to those skilled in the art. Such alterations, 
modifications, and improvements are intended to be within the spirit and 
scope of the invention. Accordingly, the foregoing description is by way 
of example only and is not intended as limiting. The invention is limited 
only as defined in the following claims and the equivalents thereto.