Low power membership function generator for interval type-2 fuzzy system

Abstract

An analog type-2 fuzzy membership function generator is analyzed, while its low power features are demonstrated. Developed and designed to work as part of an energy-efficient type-2 fuzzy system (i.e. controller, neural network), it operates in current-mode, allowing multiple settings, such as: variable shape, position, slope and size of the Footprint of Uncertainty. All these parameters can be updated/adjusted during actual operation, being suitable for real-time applications, even with online learning. The achieved robustness and flexibility of the architecture allow operation with different values of supply voltage and maximum degree of membership current. Simulations based on CADENCE software were performed to assess the functionalities. The circuit was prototyped in TSMC CMOS 0.18μm technology and experimental results show a power consumption of 72.56μW, 7.86μW, and 478.1nW when operating with power supplies of 1.2V, 1V, and even with 0.6V, respectively, much less than the nominal supply voltage of the TSMC CMOS 0.18μm technology (1.8V). These results attest the circuit is robust, low-power, functional and suitable for systems that require energy efficiency, presenting a high potential for practical implementations including those that require real-time dedicated hardware.

Keywords

Analog circuits low-power fuzzifier interval type-2 fuzzy logic

1. Introduction

The aptitude of fuzzy logic to represent complex systems, and to operate as a controller, in many different situations, has been thoroughly demonstrated [1]. This capacity, however, has not yet been translated into a large number of practical, low-power and embedded applications, such as portable devices, biological implants, wearables, and other small System-on-Chip (SoC) devices. There are some exceptions in: systems implementing wireless body sensors [2], real-time object recognition [3], and even a maximum power point tracker for a solar energy harvester in a wearable sensor device [4], but most fuzzy hardware implementations still do not aim at small, low-power devices [5]. In recent years, studies have concentrated on the benefits of type-2 fuzzy logic, with promising results in intelligent control [6, 7] and learning systems applications [8, 9]. It has even been demonstrated that a type-2 fuzzy system – whether Mamdani or Sugeno – is a universal approximator [10]. In type-2 fuzzy logic, the membership function maps each point of the universe of discourse (UOD) to a type-1 fuzzy set, which adds another degree of uncertainty. This allows the system to better deal with the uncertainty of data and, therefore, to present a potential for higher performance [6, 11]. Also, actual systems can be described using type-2 fuzzy logic with a smaller rule base when compared with a type-1 implementation; therefore, fewer fuzzifier circuits are needed, saving area, power, and increasing performance. Type-2 fuzzy logic can also address more complex relationships, meaning that it can represent systems that an equivalent (same rule base) type-1 fuzzy inference system is not able to [6].

The associated cost of using this kind of architecture is an increase in the number of computational operations required to achieve the desired output [11]. But, to operate in real-time applications, embedded systems, or even in implantable devices, it is critical that the hardware presents high performance, small area, and, mainly, low power consumption, what leads to a potential incompatibility between those applications and a type-2 fuzzy system, unless a novel architecture overcomes those main drawbacks.

One of the main reasons that prevent the implementation of small embedded systems exploring the advantages of fuzzy logic is the power consumption. Portable applications, or implants, usually require batteries or power-harvesting, which severely limit the power budget available for the IC. Also, traditional digital approaches usually need analog-digital converters, which may compromise even further the feasibility of the system.

Following this complex scenario, this paper analyzes novel operation modes – and respective power consumptions – of an analog low power membership function generator circuit, previously proposed by the authors in [12]. It is capable of creating an interval type-2 fuzzy set with variable shape, position, inclination, and size of the footprint of uncertainty (FOU). This versatility allows it to be used as a fuzzifier in a fuzzy controller, or as part of a membership layer in a fuzzy neural network, for instance, while being updated, online and in real-time. It is demonstrated here that it can operate with a maximum degree of membership represented by currents of 10μA, 1μA, and even 100 nA, presenting average power consumptions of 72.56μW, 7.86μW, and 478.1 nW, respectively. Just a few analog hardware implementations of interval type-2 fuzzy membership function generators [13 –16] can be found in the literature, but this architecture makes it possible to operate with a power reduction of up to 1190 times when compared with [16], and a reduction of 150 times even when compared to the most energy-efficient type-1 membership function generator [17].

2. Type-2 Fuzzy membership function generator

There are two main possible applications that may require a membership function generator circuit: a type-2 fuzzy inference system or a type-2 neuro-fuzzy system. In the first case, the fuzzy inference system consists of five main blocks: the Fuzzifier, the Inference Engine, the Rule Base, the Type-reducer, and the Defuzzifier [6]. In short, the fuzzifier takes crisp input values and outputs an interval type-2 fuzzy set. Then, the inference engine, responsible for applying the rules, defined by the rule base, receives a certain number of fuzzy sets from the fuzzifier, corresponding to each one of the membership functions used in the system, outputting a new type-2 fuzzy set. The type-reducer converts the type-2 fuzzy set into a type-1 fuzzy set and, finally, the defuzzifier converts a type-1 fuzzy set into a crisp output value.

In the second case, a neural network can be organized in a way that represents an inference system, similar to the one described previously, usually with layers directly associated with fuzzification, inference, type-reduction and defuzzification [18 –21]. The advantage of this architecture is that both the structure and the parameters can be updated during the learning process.

In both cases, the membership function generator – whether for the fuzzifier block or for the membership function layer – must be able to output a membership function of an interval type-2 fuzzy set. A general type-2 fuzzy set is defined by (1), in which $0 < μ_{\tilde{A}} (x, u) < 1$ . $\tilde{A} = {\begin{matrix} ((x, u), μ_{\tilde{A}} (x, u)) | \forall x \in X, \\ \forall u \in [0, 1] \end{matrix}}$ (1) while x is the input variable, X is the UOD, u is the primary membership function and $μ_{\tilde{A}}$ is the secondary membership function [6].

If $μ_{\tilde{A}}$ is always equal to 1 within the FOU, then $\tilde{A}$ is determined only by the upper membership function (UMF) and by the lower membership function (LMF), which defines an interval type-2 fuzzy set. For a circuit to be used as a membership function generator, it must be able to create both the upper and lower functions, while being able to alter shape, inclination, position and size of the FOU during operation. The simplest way to represent the degrees of membership in an analog circuit is based on electric currents, since they make the processing easier (currents can easily be added, multiplied or processed using MAX or MIN operators).

Type-2 analog membership function generators usually work with combinations of type-1 fuzzifiers to create the type-2 fuzzy set, but this is not an optimal solution. Considering that many membership generator circuits are needed in the membership layer, the power consumption becomes paramount in making it feasible for use in a real-world application that presents severe energy constraints.

To overcome the main existing drawbacks, the analyzed circuit uses a current steering mirror to facilitate the simultaneous creation of both membership functions, while allowing an online update of all its defining parameters [12]. Another distinguishing feature of this circuit is that the used wide-swing current mirrors allow a higher dynamic range and, consequently, a reduction in the power supply. Furthermore, this assists in the power consumption reduction, since the current representing the maximum membership function value can also be decreased to a value as low as 100 nA. Previous current mode membership function circuits found in the literature usually work with 10μA as maximum current, increasing the power consumed by the circuit, when compared to our solution. The low power feature achieved and demonstrated in this work – consuming only 478.1 nW – improves the performance previously demonstrated in [12], making this circuit suitable not only for real-time dedicated applications, but also for those that require energy-efficient hardware.

3. Circuit architecture

To work as described in the previous section, a membership function generator must have several configuration inputs, which set the desired shape and position of the membership function. The architecture analyzed, and briefly reviewed in this section, operates in current mode, being able to generate the most common shapes of membership functions, such as trapezoidal, triangular, Z and S [12].

The input parameters are defined for a general trapezoidal function, from which the other shapes can be obtained. Figure 1 shows the basic function, where I_MAX represents the maximum value of the membership function, I_D and I_S set the shape and position. The inclinations are determined by a 2-bit input, therefore having four different possible values. The last configuration parameter is the input that sets the size of the FOU, by means of a differential voltage input ΔV_c. All the described parameters can be updated/modified during the operation of the circuit, as required in a system with online learning features.

Fig. 1

Upper and lower membership functions of a type-2 fuzzy set, as functions of the input current.

The block diagram representing the circuit’s operation is presented in Fig. 2. Current I_IN is the input, and currents I_U and I_L are the UMF and LMF, respectively. The two main blocks in this architecture are: i. the Current Steering Mirror (CSM); and ii. the Programmable Current Mirror (PCM).

Fig. 2

Diagram showing the components of the circuit: the CSM, the PCM and the path of all the currents involved.

The CSM is responsible for the voltage-dependent FOU. Each CSM has two control voltage inputs, which set the size of the FOU, also working as bias voltages. It works with a differential pair, as current steering element [22], and can be seen in Fig. 3.

Fig. 3

The circuit of the current steering mirror (CSM).

The circuit works by making a copy of the input I_CMS current and dividing it into two parts. By doubling the transistors in the output branch, the sum I₁ + I₂ is always twice the input current. Therefore, when Vcp= Vcn, or ΔVc = Vcp - Vcn= 0, the current gain is 1 for both outputs. As ΔVc increases, I₁ becomes larger than I₂, corresponding to an increase in the range of the FOU.

By construction (using wide-swing current mirrors to perform bounded difference operations), while if I_IN < I_D, the block CSM1 mirrors the difference I_D - I_IN. When I_IN > I_D, CSM1 cuts off. For the same reasons, if I_IN <I_S, CSM2 is also in a cut-off state. Once I_IN > I_S, CSM2 mirrors the difference I_IN–I_S, and creates the other part of the general trapezoidal membership function. To form the complete upper and lower functions, the currents are added at the input of the PCM, as seen in Fig. 2, making I_P1 = I_A1 + I_A2 and I_P2 = I_B1 + I_B2. The CSM creates the FOU by outputting modified copies of the input current, but the average inclination is still the same such that another circuit is needed in the form of a PCM, a digitally programmable current mirror, allowing different inclinations of the membership function. Also, there are two control voltages (V₁andV₂) that change the number of copies made by the current mirror. Consequently, there are four possible gains.

The topology in Fig. 4 shows the configuration inputs, V₁ andV₂, and the bias voltage, Vb, needed in this wide-swing configuration. After adjusting the inclination, the output currents are then subtracted from I_MAX, resulting in the final type-2 fuzzy set, as the upper and lower membership functions I_U andI_L.

Fig. 4

The circuit of the digitally Programmable Current Mirror (PCM) and its control voltages V₁ and V₂.

4. Simulation results

4.1. Monte carlo

To ensure the proper operation of the circuit under different circumstances, and to be sure of its ability to create the desired membership function independent of process and mismatch variations, a Monte Carlo simulation was performed for 1500 different points. Figure 5 shows the result for the general trapezoidal function, obtained for a power supply of 1.2V. The current I_MAX was set as 10μA, while the other configuration currents were I_D = 4μ Aand I_S = 6μ A and I_s = 4μ Aand I_S = 6μ A the bias voltages Vcn and Vb were set as 800mV and 400mV, respectively; Vcp was set as 810mV, so that the size of the FOU was determined by ΔVc = 10mV, and the inclination was set at the maximum value by V₁=V₂=0V. The detail shown in Fig. 5(b), close to the input current of 2μ A, depicts that the variation around the typical case is minimal, guaranteeing the robustness and functionality of the circuit as a membership function generator.

Fig. 5

Output currents of the Monte Carlo Simulation. (a) Monte Carlo analysis on trapezoidal function for 1500 points (b) Detail of the variation for different membership function results.

4.2. Power analysis

The robustness of the circuit allows a correct operation with different values of power supply voltage, bias voltages, and even the maximum membership function current – and therefore the input current range as well. Different simulations were performed to evaluate the power consumption under each one of three different settings. The first results were obtained for the configuration parameters shown in Table 1, as a function of the input current varying from 0 to 10μ A.

Table 1
Configuration parameters using maximum of 10μA universe of discourse (input current)

Parameter Value

V _DD 1.2 V

V _cn 800 mV

V _cp 810 mV

V _b 400 mV

I _MAX 10μA

Parameter	Value
V _DD	1.2 V
V _cn	800 mV
V _cp	810 mV
V _b	400 mV
I _MAX	10μA

Figure 6 shows the S-shaped function, obtained with I_D = 9μA and I_S = 11μA, and its power consumption as a function of the input current. The average power consumption, in this case, is equal to 76.48μW.

Fig. 6

S-shape membership function and its power consumption as a function of the input current varying between 0 and 10μA.

Figure 7 shows the Z-shaped function, obtained with I_D = 0μA and I_S = 1μA, and its power consumption, as a function of the input current. The average power consumption, in this case, is equal to 65.45μW.

Fig. 7

Z-shape membership function and its power consumption as a function of the input current varying between 0 and 10μA.

Also, Fig. 8 shows the triangular function, obtained with I_D = 5μA and I_S = 5μA, and its power consumption as a function of the input current. The average power consumption is equal to 67.46μW.

Fig. 8

Triangular membership function and its power consumption as a function of the input current varying between 0 and 10μA.

Finally, completing all main shapes, Fig. 9 shows the trapezoidal function, obtained with I_D = 4μA and I_S = 6μA, and its power consumption as a function of the input current. The average power consumption is equal to 65.65μW.

Fig. 9

Trapezoidal membership function and its power consumption as a function of the input current varying between 0 and 10μA.

The average consumption of all the evaluated membership functions is equal to 69.02μW. This is only 12% of the membership function generator circuit proposed in [16]. But the robustness and the versatility of the architecture allows the circuit to be operated even with lower power supplies and currents. The next set of results was obtained for the configuration parameters shown in Table 2, as a function of the input current, varying from 0 to 1μA.

Table 2

Configuration parameters using maximum of 1μA universe of discourse (input current)

Parameter	Value
V _DD	1 V
V _cn	600 mV
V _cp	610 mV
V _b	400 mV
I _MAX	1μA

For the sake of space in this paper, and to simplify the analysis, the power consumption results for all four different membership functions were plotted in the same figure, as a function of the input current varying from 0 to 1μA. The results are shown in Fig. 10.

Fig. 10

Power consumption for all four different membership functions as a function of the input current varying between 0 and 1μA.

The S-shaped function was obtained with I_D 900nA and I_S = 1.1μA, while the average power consumption is equal to 8.29μW. The Z-shaped function was obtained with I_D = 0μA and I_S = 100nA, and the average power consumption is equal to 6.36μW. The triangular function was obtained with I_D=500nA and I_S = 500nA, while the average power consumption is equal to 7.01μW. Finally, the trapezoidal function was obtained with I_D =400nA and I_S = 600nA, and the average power consumption is equal to 7.22μW. The average consumption of all the evaluated membership functions is equal to 7.22μW.

The circuit is capable of correctly operating with an even lower level of power supply and currents, consequently leading to a lower power consumption. The third set of results was obtained for the configuration parameters shown in Table 3, as a function of the input current, varying from 0 to 100 nA.

Table 3

Configuration parameters using maximum of 0.1μA universe of discourse (input current)

Parameter	Value
V _DD	0.6 V
V _cn	500 mV
V _cp	510 mV
V _b	100 mV
I _MAX	100 nA

The power consumption results for all four different membership functions were depicted in Fig. 11, as a function of the input current varying from 0 to 100 nA.

Fig. 11

Power consumption for all four different membership functions as a function of the input current varying between 0 and 100 nA.

The S-shaped function was obtained with I_D = 90nA and I_S = 110nA, and the average power consumption is equal to 497.4 nW. The Z-shaped function was obtained with I_D = 0μA and I_S = 10nA, and the average power consumption is equal to 381.6 nW. The triangular function was obtained with I_D = 50nA and I_S = 50nA, and the average power consumption is equal to 420.6 nW. And finally, the trapezoidal function was obtained with I_D = 40nA and I_S = 60nA, and the average power consumption is equal to 433.5 nW. The average consumption of all the membership functions evaluated is equal to only 433.3 nW

5. Experimental results

Figure 12 shows the layout of the membership function generator circuit – created in CADENCE software – and the micrograph of the testing chip, manufactured using TSMC 0.18μm technology in cooperation with IMEC, in a multi-project wafer (MPW) run, which is part of the mini@sic program of Europractice-IC. The layout contains all circuits of the two CSM and of the two PCM blocks, and additional wide-swing current mirrors, at both the input and output to perform the bounded difference operations. The PMOS and NMOS transistors are separated by guard-rings to prevent latch-up and minimize possible external interferences. The total silicon area occupied by the design, in this technology, is equal to 0.0013mm².

Fig. 12

Micrograph of the testing chip and layout of the type-2 membership function generator circuit.

The power consumption of the prototyped circuit was measured for all three different parameter configurations to verify the simulation results in Section 4.2. The power measurements for the first set of results – using the same parameters from Table 1 – were plotted in Fig. 13 for all four different membership functions (Trapezoidal, Z-shape, S-shape and Triangular). The average power consumption was equal to 72.56μW.

Fig. 13

Experimental power consumption for all four different membership functions as a function of the input current varying between 0 and 10μA.

The power measurements for the second set of results – using the parameters from Table 2 – were plotted in Fig. 14 for all four membership functions. The average power consumption was equal to 7.82μW.

Fig. 14

Experimental power consumption for all four different membership functions as a function of the input current varying between 0 and 1μA.

The power measurements for the third set of results – using the parameters from Table 3 – were plotted in Fig. 15 for all four membership functions. The average power consumption was equal to 478.1 nW. The small difference between simulated and experimental results (3.54μW, 0.6μW, and 44.8 nW, for each case, respectively) is due to the slightly different configuration currents as a result of the calibration process to obtain the correct shape and position [12].

Fig. 15

Experimental power consumption for all four different membership functions as a function of the input current varying between 0 and 100 nA.

The measurement plotted in Fig. 16 shows that indeed, even in the least power consuming setting (more restrictive environment), the circuit is capable of generating a valid type-2 fuzzy membership function – in this case the general trapezoidal form.

Fig. 16

Experimental Trapezoidal membership function measured using parameters from Table 3.

The experimental results undoubtedly validate the capability of the circuit to work as a type-2 membership function generator with enough versatility to operate in different voltage and current settings, and with low power consumption. Table 4 presents a comparison between different membership function generators, both type-1 and type-2 fuzzy logic.

Table 4

Specifications and Power Consumption Comparison of Membership Function Generator Circuits

	Technology	Topology	Fuzzy Type	Area	Supply Voltage	Power
[23]	0.5μm	Transconductance	Type-1	–	3V	0.3 mW
[17]	0.18μm	Current-mode/ Transconductance	Type-1	–	1.8V	72μW–108 μW
[13]	0.35μm	Current-mode	Type-2	0.02 mm²	3.3V	1.29 mW
[14]	0.18μm	Transconductance	Type-2	0.006 mm²	3.3V	2.64 mW
[15]	0.35μm	Transconductance	Type-2	0.01 mm²	3.3V	657μW
[16]	0.18μm	Current-mode	Type-2	0.0183 mm²	1.8V	570μW
					1.2V	72.56μW
This work	0.18μm	Current-mode	Type-2	0.001 mm²	1V	7.82μW
					0.6V	478.1 nW

Even in the most power consuming configuration (worst results) of the evaluated circuit, with a 1.2V supply voltage, it presents a reduction of 87% when compared with the best (most energy-efficient) type-2 membership generator presented in the literature [16], being also in the same order of magnitude of the best result (in terms of power consumption) of a type-1 membership function generator [17]. Our intermediate result (improved in relation to our first one), with a supply voltage of 1V, represents a reduction of 72 times, considering the power presented by [16] and a reduction of 9 times of the power of [17]. Finally, in the most energy-efficient setting (our best results), with 0.6V supply voltage, the low power feature of the circuit becomes evident, given the reduction of 1190 times in relation to the power of [16] and a reduction of 150 times the power of [17].

6. Conclusion

A type-2 fuzzy membership function generator hardware implementation, capable of operating in small, embedded real-world applications with online learning, must have low power consumption while being able to update its parameters during operation. This paper demonstrated the low power feature of an analog solution for a fuzzy type-2 membership function generator that can be used as a fuzzifier in a fuzzy inference system, or as part of the membership layer in a type-2 neural network. It also confirmed its capability of generating all most common function shapes (trapezoidal, triangular, S-shape and Z-shape), with a variable FOU size and having all function parameters controllable, making it suitable for online updates.

The power consumption is a function of the voltage and current settings, being equal to 72.56μW with 1.2V supply voltage, 7.82μW with 1V supply voltage, and 478.1 nW with 0.6V supply voltage. This last result represents a major improvement when compared with previous results found in the literature (a reduction of 1190 times in relation to the power of [16] and a reduction of 150 times considering the power of [17]), and may prove to be very significant in the development of real-world low power hardware applications for fuzzy systems.

Footnotes

Acknowledgments

This study was financed by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.

The IC fabrication was made possible by IMEC as part of the mini@sic program.

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Low power membership function generator for interval type-2 fuzzy system

Abstract

Keywords

1. Introduction

2. Type-2 Fuzzy membership function generator

4.1. Monte carlo

Table 1 Configuration parameters using maximum of 10μA universe of discourse (input current) Parameter Value V DD 1.2 V V cn 800 mV V cp 810 mV V b 400 mV I MAX 10μA

Footnotes

Acknowledgments

References

Table 1
Configuration parameters using maximum of 10μA universe of discourse (input current)

Parameter Value

V _DD 1.2 V

V _cn 800 mV

V _cp 810 mV

V _b 400 mV

I _MAX 10μA