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This paper presents a novel fuzzy selftuning PID control scheme for regulating industrial processes. The essential idea of the scheme is to parameterize a ZieglerNicholslike tuning formula by a single parameter a, then to use an online fuzzy inference mechanism to selftune the parameter. The fuzzy tuning mechanism, with process output error and error rate as its inputs, adjusts c~ in such a way that it speeds up the convergence of the process output to a setpoint Yr, and slows down the divergence trend of the output from Yr. A comparative simulation study on various processes, including a secondorder process, processes with long deadtime and nonminimum phase processes, shows that the performance of the new scheme improves considerably, in terms of setpoint and load disturbance responses, over the PID controllers welltuned using both the classical ZieglerNichols formula and the more recent Refined ZieglerNichols formula.
Fuzzy Sets and Systems 56 (1993) 37-46 North-Holland 37 Fuzzy self-tuning of PID controllers Shi-Zhong He 1, Shaohua Tan Feng-Lan Xu: Department of Electrical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 0511 Pei-Zhuang Wang Institute of System Science, National University of Singapore, Heng Mui Keng Terrace, Kent Ridge, Singapore 0511 Received August 1992 Revised October 1992 Abstract." This paper presents a novel fuzzy self-tuning PID control scheme for regulating industrial processes The essential idea of the scheme is to parameterize a Ziegler-Nichols-like tuning formula by a single parameter a, then to use an on-line fuzzy inference mechanism to self-tune the parameter The fuzzy tuning mechanism, with process output error and error rate as its inputs, adjusts c~ in such a way that it speeds up the convergence of the process output to a set-point Yr, and slows down the divergence trend of the output from Yr A comparative simulation study on various processes, including a second-order process, processes with long dead-time and non-minimum phase processes, shows that the performance of the new scheme improves considerably, in terms of set-point and load disturbance responses, over the PID controllers well-tuned using both the classical Ziegler-Nichols formula and the more recent Refined Ziegler-Nichols formula Keywords: Fuzzy self-tuning; fuzzy control; adaptive control Introduction Despite the advent of many sophisticated control theories and techniques, the majority of industrial processes nowadays are still regulated by PID controllers This status quo not just indicates the cautious attitude of the practicing Correspondence to: Shaohua Tan, Department of Electrical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 0511 On leave from Department of Automation, Tsinghua University, Beijing 100084, China On leave from Department of Electrical Engineering, Tsinghua University, Beijing 100084, China On leave from Department of Mathematics, Beijing Normal University, Beijing 100088, China world towards the new invention, it does reveal the rich potential of this extremely simple (almost primitive, perhaps, in the eyes of some control theorists) control strategy for meeting various specifications for a vast variety of industrial processes A crucial issue in the PID control is the setting of the controller parameters, the so-called tuning problem The conventional way to the tuning is to study the mathematical models of processes, and try to come up with a simple tuning law that will establish a set of constant PID parameters based on the models It is not hard to show theoretically that the PID is adequate for the processes modelled perfectly by linear first or second order systems Tuning laws can easily be established in these cases Unfortunately, real industrial processes can never be modelled perfectly as simply as the linear first and the second order systems They may have such marked characteristics as high-order, dead-time, nonlinearity, etc., and may be affected by noise, load disturbance and other ambient conditions that cause parameter variation and sudden model structural change The existing theories can no longer provide systematic and robust tuning laws for these complex situations Thus, many of the PID tuning laws actually incorporate empirical evidences to compensate for the model complexity and variation As can be expected, these tuning laws are often ad hoc in nature, and may only be useful for a certain class of processes or under certain conditions A typical example of the model-based tuning laws is the famous Ziegler-Nichols tuning formula [15] Apart from the fact that it may completely fail to tune the processes with, for example, relatively large dead-time, its tuning will have to be supplemented with purely experience-based fine-tuning to meet the response requirements Along the line of empirical investigation and approximate analysis, the work on improving the PID tuning has been going on especially in the past decade There has been attempt to revise 0165-0114/93/$06.00 © 1993 Elsevier Science Publishers B.V All rights reserved 38 Shi-Zhong He et al / Fuzzy self-tuning of PID controllers the half-century-old Ziegler-Nichols formula to enhance its performance and applicability, resulting in the so-called Refined ZieglerNichols tuning formula [6] These refinements, useful as they may be in improving certain aspects of the responses for certain processes, may perform worse for certain other processes In a sense, they hit upon the delicate boundary of performance-process tradeoff, and the complexity may never allow a clear cut Taking, as an example, the processes with long dead-time, if they are controlled by the PID plus the Ziegler-Nichols tuning, the first overshoot in the set-point response will be excessively high, which is considered unacceptable for many applications The Refined Ziegler-Nichols formula can be employed in this case to reduce the overshoot In doing so, however, the response time will be slowed down, sometimes considerably Further, in the case of mild nonlinearities, it is hard to tell for a particular process if the original formula or its refinement should be used All these may be contributed to the single fact of requiring the PID parameters to be fixed throughout the control In other words, with fixed parameters, the controllers of a simple structure, such as the PID, cannot go beyond a certain limit in handling the model complexity and uncertainty If we still insist on using the same controller structure, the controller parameter adaptation with time seems to be the only way to extend beyond the limit Carrying on with the thought, a natural step ahead is to consider self-tuning PID control, which tunes the PID parameters on-line to adjust the controller actions for meeting the real-time need This is precisely the direction pursued by a number of researchers [3, 10, 1] The idea of the existing PID self-tuning schemes essentially follows that of the conventional self-tuning controllers, i.e., the tuning at any time instance is based on a structurally-fixed parameter-evolving process model produced by an on-line identification procedure The momentary tuning itself will still have to be done by using some design formula, or just the Ziegler-Nichols formula Thus, these schemes can be seen as trying to deal with the model complexity and uncertainty problem by localizing (on the time scale) the conventional tuning methods Aside from the often-cited problem of high computational demand, a major difficiency for the schemes seems to be that it does not change the model-based nature of non-adaptive PID controllers By assuming a model, the consequent robustness issue still needs to be settled (even at localized time instances), which proves to be difficult Further, because an a priori assumed model will have to be necessarily simple, it often cannot accommodate the structural disturbance, such as load disturbance on processes If such a disturbance happens, the identified model will be highly inaccurate, leading to the momentarily degraded controller performances [7] Against this backdrop, we propose to use a fuzzy inference based self-tuning scheme for PID controllers The essence of the scheme is that at every time instance, the controller evaluates the trend of the controlled process output to detect the possible deviation from a prescribed course If a deviation is found, an appropriate control action according to the nature of the deviation will be generated instantaneously to correct it Compared to the existing model based selftuning schemes, our scheme is empirical based, and acts more like what we when we, for example, try to steer a controller manually to keep the output of a process on a fixed course Our own experience tells that a precise description of the process (in the form of a dynamical model) is often irrelevant for our steering actions What is more important is the instant observations of the error, and subsequent rational actions for bringing it back to course There are two key ideas in our scheme First, the Ziegler-Nichols formula is parameterized by a single parameter a This a is arranged so that its increase (decrease) will lead to the increase (decrease) in the proportional term and the decrease (increase) in both the integral and differential terms in the PID controller Such an arrangement is intended to divert the trend of the process output using the knowledge of the qualitative relationship between the proportions of the PID feedbacks and the profiles of the process output Secondly, the on-line tuning formula for a is a discrete dynamical equation driven by a fuzzy inference procedure A simple fuzzy map is formed in such a way that it updates a in accordance with the current regulation error and error rate Specifically, it speeds up the convergence of the process output to a set-point Shi-ZhongHe et al / Fuzzyself-tuningof PID controllers Yr, and slows down the divergence trend of the output from Yr This is, in fact, the fuzzy adaptation mechanism we have used successfully in one of our early works on adaptive fuzzy controller [9] Many forms of adaptive fuzzy control schemes exist, see [12,13,9] Virtually all of these schemes are genuine fuzzy control schemes in the sense that the controllers are actually fuzzy, although the adaptation mechanisms may sometimes employ non-fuzzy tuning laws The proposed fuzzy auto-tuning PID is different in nature from all these schemes in that it is a non-fuzzy controller tuned with a fuzzy inference mechanisms Further, because of its connection with the Ziegler-Nichols formula, the proposed control scheme is not completely model-free A simple initialization procedure will have to be used to obtain the ultimate gain and ultimate period of the process to be controlled in order to start the fuzzy adaptation This limited degree of model dependence reflects the consideration that if certain information on the process can be acquired easily and directly, the control scheme should be able to make use of it This is in sharp contrast to a genuine fuzzy controller, which are completely model-free and all the control rules are supposed to come directly from the experiences Another rationale behind our scheme is that it is often hard to directly acquire the knowledge or possess direct human experiences for controlling a complex process However, if the controller structure is fixed to be the PID control, then the experiences are narrowed down to more specialized experiences of choosing a few PID parameters This latter problem has been under scrutiny for so long a time that there have been a great amount of knowledge and experiences accumulated on the subject In this context, it seems more meaningful to keep the PID structure and let the self-tuning part be handled by the fuzzy logic approach The main objective of the present paper is to propose this new type of fuzzy self-tuning control scheme, provide the details of the design procedure, and conduct a simulation analysis to compare the scheme with two tuning schemes, namely, the Ziegler-Nichols tuning and the Refined Ziegler-Nichols tuning The general conclusion of the simulation analysis is that the 39 new fuzzy self-tuning PID controller outperforms the PID controllers tuned by the two fixed tuning laws The paper is organized as follows In Section 2, the new fuzzy self-tuning PID controller is described in detail, the exposition covers the basic structure of the controller, fuzzy tuner, as well as the initialization of the controller The simulation analysis is carried out in Section followed by Section 4, which contains further discussions and conclusions The controller and the fuzzy adaptation Basic structure To begin with, we assume that the process to be controlled has single input u(t) and single output y(t), and the control objective is to bring the process output y(t) to a prescribed set-point Yr The scheme can actually be extended to the tracking problems where y~ is a time-varying target output However, this extension will not be discussed here to keep our exposition concise As mentioned in the previous section, the fuzzy self-tuning PID controller consists of a standard PID controller and a fuzzy tuning mechanism used for the on-line adaptation of the PID parameters (Figure 1) The PID controller, which generates a control u(t) based on the closed-loop error e(t)= Yr- y(t), has the following standard form u(t)= Kc[e(t)+ Tdde~t) +~ fe(t)dt ] , (1) where Ko Td, T~ are, respectively, the proportional gain, the derivative time and the integral time of the controller, which are to be adjusted on-line One of the key ideas of the control scheme is to parametrize the three PID parameters by a single parameter c~ as shown below Kc 1.2c~ku, T~= 0.75 ~-a t°, Ta = 0.25T~, (2) where ku, tu are, respectively, the ultimate gain and the ultimate period of the underlying process, which will be determined shortly Shi-Zhong He et al / Fuzzy self-tuning of PID controllers 40 f- Fuzzy Self-tuning Mechanism I I I I ]FuzzyAdaptatio~ -~ DPasrgm~oerrimZelda I [- m _ _1 - Yr Controller Process Fig The basic structure of the fuzzy self-tuning PID controller The form of the parameterization is inspired by the Ziegler-Nichols formula, and in fact reduces to it when a = ½ Thus we can think of the controller as compensating the basic control of the Ziegler-Nichols by biasing all the parameters on-line in order to adjust the process output to a prescribed course In what follows, both the fuzzy adaptation and the initialization will be discussed in detail Table The fuzzy map from E and R to H H E R -3 -2 -1 -3 -3 -2 -2 -3 -2 -2 -1 -2 -2 -I -1 -2 -1 -1 -1 -1 -1 1 1 1 -1 1 2 -1 1 2 2 3 -3 -2 -1 0 Fuzzy adaptation As shown in Figure 1, the fuzzy self-tuning mechanism will generate an a(t) given the instant values of e(t) and O(t) at time t It is composed of two parts: a fuzzy core and a conditional updating formula for a The fuzzy core starts with fuzzifying e(t) and O(t) into two fuzzy variables E, respectively, R To ensure a speedy fuzzy inference, both the range of interest for e(t) and that for ~(t) are covered by seven different fuzzy sets as shown below E = {NL, NM, NS, ZO, PS, PM, PL} R = {NL, NM, NS, ZO, PS, PM, PL}, (3) where, as usual, the meanings of the acronyms used in (3) are, respectively, PL for positive large, PM for positive medium, PS for positive small, Z O for zero, NS for negative small, NM for negative medium, and N L for negative large For ease of the notation, we shall assign the integers for the fuzzy sets as PL=3, NS=-I, PM=2, NM=-2, PS=I, ZP=0, NL=-3, and denote the fuzzy sets by their corresponding numbers The second part of the fuzzy core is the fuzzy mapping from E and R to H, where H is another fuzzy variable whose defuzzified version will be used for the later updating equation of a The range of H similarly consists of seven fuzzy sets H = { - , - , - , 0, 1, 2, 3}, (4) and it is linked to E and R by a fuzzy map (a particular example that we shall use is given in Table 1) The fuzzy inference is the standard CRI procedure, hence all the fuzzy rules, such as those shown in Table 1, will be involved in every single inference We also assume that the membership functions for all fuzzy sets have the following standard form re(x) = e ( ~ ), (5) where xi, o- denote, respectively, the centre and the spread of each fuzzy set, and their choices are somehow subjective 41 Shi-Zhong He et al / Fuzzy self-tuning of PID controllers The last part of the fuzzy core is the defuzzification of H into a real variable h(t) For smooth opeation, we choose to use the centre of gravity method After h(t) is obtained, it is used in the following recursive equation to update a c~(t + 1) = {a(t) + yh(t)(1 - a(t)) ( a ( t ) + vh(t)ce(t) for ol(t) >0.5, for c~(t) ~