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Interval type2 fuzzy PI controllers: Why they are more robust[C]//Granular Computing (GrC), 2010 IEEE International Conference on. IEEE, 2010: 802807.附 錄一、英文原文：Speed Control of a DC Brush Motor with Conventional PIDand Fuzzy PI controllersAbstractThis work presents the implementation of a conventional PID and a fuzzy PI control schemes in a 16 bits microcontroller. These schemes are designed for the speed control of a CD brush motor. This article makes emphasis in the handling of the floating point for the first case. For the fuzzy control, the data is obtained directly from the digital analogical converter of the system.1. IntroductionFeedback pensation is a widely used strategy when controlling a system using a PID controller. The implementation of such control using a microcontroller is an engineering area which is continuously growing. 90% of the industrial systems are based on this system because they are reliable at a reasonable cost [1]. In this case, the analog signals are converted into discrete samples before pensation. Sampling time depends on the characteristics of the control loop. Accordingly, calculations must be pleted before the next sampling cycle takes place [2,3]. Moreover, this calculation is affected by the variation of the PID parameters, being these the mon limitations of discrete systems. On the other hand, fuzzy control is an adaptive approach to intelligent control which can deal with vague information. Its advantages over the conventional systems are the following: (1) In order to control a system or to develop a new control scheme, it is not required to know the model of the system。 and (2) it is not affected by the variation of the parameters. The purpose of this article is to pare the performance of a conventional PID and a fuzzy based controller for the speed control of a DC brush motor. Implementing both strategies using a in a 16 bits microcontroller.2. PID conventionalIn order to implement the PID algorithm, the pensator has to be represented in the discrete domain. The flow chart, showed in figure 1, illustrates the general outline of such algorithm [5].The set point of the process is identified with Gd 。 G(t) is the value measured at certain time t. , Error is evaluated with both parameters, equation (1).The PID output correction x(t) is evaluated with the following equation (2).where Kp, Ki, Kd are the proportional, integral, and derivative constants.Rewriting equation (2) x(t) can be expressed as:In the discrete domain t=KT, where K = 1, 2, 3… being T the sampling period. In this way, t0= (K1) T.The limits of the integral are (k1)T and KT, and is evaluated according to the trapezoidal rule, as established in the NewtonCotes formulation. When only the error is considered in the evaluation, noise is introduced in the results if a single period is taken into account. Therefore, a fourth central point approach is used. The intervals of the differentials involved have the same length. Thus, the following equation is used:The IEEE 754 standard is used in most of the C pilers, which is also used in microprocessors and microcontrollers. In this case, real numbers are written in scientific notation, as a result, small and big numbers are written with a reduced number of digits [4].. Floating point formatThis algorithm is implemented following the 754 IEEE standard, considering the floating point format. It has four basic ponents: the sign S, the mantissa M, the base, which is always 2 and it is not represented, and the exponent E. The first bit of a word corresponds to the sign S. It can be positive or negative when a value of o or 1 is assigned respectively.This standard proposes several formats to represent numbers with different degrees of precision。 nonetheless, the microcontrollers of 8 bits use almost exclusively the simple precision format. Figure 2 illustrates the simple precision (32 bits) and double precision (64 bits) formats considered in the 754 IEEE standard.754 IEEE assumes that each number is standardized. This means that the mantissa range is between 1 and 2. In this form, the most significant bit of the mantissa is 1 and it is not needed to be written in an explicit form. A mantissa of 24 (or 53) can be represented with only 23 (or 52) bits.One mantissa of 24 bits provides 7 decimal digits of precision (224 is approximately 107). Besides, one mantissa of 53 bits can represent a precision of 15 decimal digits (253 is approximately 91015). The only case, in which a mantissa does not begin with 1, is when there is a 0。 this condition represents a special case that takes place when all the bits of the mantissa and the exponent are zero.The microcontroller, used in this work, is formatted with a 32 bits floating point. The acquisition data system is made with a converter that generates values of 8 bits. In this case, the range is between 0 and 255. In order to use the values within this range using the floating point format, it is necessary to use the following code in assembler.LDD $3F00S