Kapalı çevrimli örneklenmiş-verili kontrol sistemlerinde optimum kontrol parametrelerinin tayini

Abstract

ÖZET Geri beslemeli kontrol sistemleri tasarımında II. Dünya Savaşı' ndan beri bir çok gelişme olmuştur. İlk zamanlarda sanayide fazla yaygın olarak kullanılmayan kontrol çevrimleri şu an vazgeçilmez hale gelmiştir. Özellikle seri üretim yapılan sektörlerde otomatik kontrol son derece önemlidir. İyi yönde yapılacak her değişiklik seri ve kitle üretiminde çok büyük ekonomik kazançlar sağlayacaktır. Bu nedenle, son yıllarda endüstrinin çeşitli kollarında yaygın olarak kullanılan proses kontrol sistemlerinde, optimum kontrolün sağlanması problemi çok önem verilen bir konu haline gelmiştir. Bu çalışma esas olarak, yukarıda sözü edilen problemin dijital simülasyon metodu ile çözümünü amaçlamaktadır. Örneklenmiş - verili kapalı çevrimli bir otomatik kontrol sisteminde, kontrol organı olarak PI ve PID' ye çeşitli performans kriterlerinin uygulanması ile optimum kontrol parametreleri bulunarak karşılaştırılmaları elde edilmiştir. Üzerinde çalışılan proses kontrol sisteminin şekli ve parametreleri uygulamadan (İngiltere' de ICI Billingham Endüstriyel Kuruluşu' nun kendi testleri ile bulduktan parametre gruplarından) alınmıştır. Çalışılan sistemin proses transfer fonksiyonu, iki zaman sabiti ve bir zaman gecikmesinden oluşmaktadır. Ayrıca geri besleme kolu üzerinde süreksiz ölçmeler bulunmaktadır. Ölçme için kullanılan kromatograf cihazı bir örnekleme zamanına ve bir tutucuya sahiptir. Cihazda cevap gecikmesi (analiz zaman) ve sistemde iletim gecikmesi bulunmaktadır. Kontrol organı olarak PI ve PID seçilmiştir. Sisteme referanstan birim basamak girişi uygulanarak çıkış alınmıştır. Bu çalışmada öncelikle kontrol organı ve optimum kontrol organı için kullanılan performans kriterleri tanıtılmıştır. Sistemin simülasyonu ve optimizasyonu Turbo C (Version 2.01) kullanılarak yapılmıştır. Sistem üzerinde yapılan çalışmada, başlangıç kontrol parametreleri (K, Ti ve Td) kullanılarak sisteme performans kriterleri uygulanmış ve sistem cevabı alınmıştır. Daha sonra bunların karşılaştırılması yapılmıştır. Ayrıca bozucuya birim basamak girişi uygulanarak sistem üzerindeki etkileri incelenmiştir. Son olarak farklı örnekleme zamanlan için optimum ayar değerleri bulunarak incelenen sistemin senkronize çalışıp, çalışmadığı tespit edilmiştir. XVU

SUMMARY When a performance criterion or a set of performance specifications is stipulated for a system and these conditions are not met, a control problem exists. Generally, in order to obtain the desired system performance additional equipment must be used in conjunction with the basic system. Either modern control or conventional design techniques are utilized to determine the configuration and parameters of the required additional equipment. By the nature of the modern control technique a specified performance criterion ( PI = J e2dt to be minimum) is met exactly, and a unique design is obtained. The performance of the system is therefore said to be optimal in terms of the defined performance criterion. In contrast, the conventional technique satisfies a required set of performance specifications. These requirements may be met by a number of different designs. In general, these designs do not simultaneously meet a defined optimal performance criterion. Therefore, these designs may be called suboptimal. The characteristics used to judge performance are as follows: 1. Maximum overshoot Mp 2. Time to reach the maximum overshoot (peak time) tp 3. Settling time (also called solution time), which is the time for the response to settle within a given percentage of the final value, t, 4. Frequency of oscillation of the transient Wd 5. Steady-state error for a given input ess The ready availabilility of the digital computer has led to the development and exploitation of modern control theory. This permits the achievement of an optimal system performance which meets some specified performance criterion. It involves minimizing (or maximizing) a performance index. This method, in contrast to the conventional design technique, relies on the extensive use of mathematical analysis. The selection of a performance index is often based on mathematical convenience, i.e., the selected performance index permits the mathematical design of the system. While the method yields a unique mathematical solution, the actual system performance may not have all the desired performance characteristics. xviuIn other words, the resulting optimal control satisfies the mathematical performance index but may not yield desired values of Mp, tp, ts, etc. A compromise must be made between specifying a performance index which includes all the desired system characteristics and a performance index which can be achieved with a reasonable amount of computation. It should further be noted that it is most difficult to analyze multiple-input multiple- output systems by conventional control theory, whereas modern control theory is quite adaptable to the analysis and design, with the use of the digital computer, of such systems. One of the major problems in the process control systems is the determination of the controller settings to obtain satisfactory transient performance. A number of useful methods for determining optimum settings of industrial controllers have been developed. The research presented here considers a closed-loop sampled-data system which consisted of a distillation column, a chromatograph as a measuring instrument, a chromatograph sampler and a controller. The mathematical model of the process consisted of two time- constant and pure time delay. In the practical application of process control systems, some methods for tuning and process identification are needed. The selection of controller modes depends on the process to be controlled. Proportional control is simple, but the response exhibits offset. The derivative action in PD control makes it possible to increase the controller gain with the result that the response has less offset and responds more quickly compared to proportional control. To eliminate offset, integral action must be present in the controller in the form of PI and PID control. PI control often causes the response to have large overshoot and a slow return to the set point especially for high- order processes. The presence of derivative action in a PID controller gives less overshoot and a faster return to the set point, compared to the response for PI control. Proportional (P) control action. For a controller with proportional control action, the relationship between the output of the controller m(t) and the actuating error signal e(t) is m(t) = K.e(t) or, in Laplace-transformed quantities, M(s)/E(s)=K where K is termed the proportional sensitivity or the gain. XIXIntegral (T) control action. In a controller with integral control action, the value of the controller output m(t) is changed at a rate proportional to the actuating error signal e(t). The transfer function of the integral controller is M(s)/E(s)=l/TiS where Ti represents the integral time. Tjis an adjustable constant. If the value of e(t) is doubled, then the value of m(t) varies twice as fast. For zero actuating error, the value of m(t) remains stationary. The integral control action is sometimes called reset control. Proportional-plus-integral (PI) control action. The transfer function of the controller is M(s)/E(s) = K.(l + l/Tis) where K represents the proportional sensitivity or gain, and Tj represents the integral time. Both K and Tj are adjustable. The integral time adjusts the integral control action, while a change in the value of K affects both the proportional and integral parts of the control action. The integral time Tj is called reset control. Proportional-plus-derivative (PD) control action. The transfer function of the controller is M(s)/E(s) = K.(l+Tds) where K represents the proportional sensitivity and Ta represents the derivative time. Both K and Ta are adjustable. The derivative control action, sometimes called rate control, is where the magnitude of the controller output is proportional to the rate of change of the actuating error signal. The derivative time Ta is the time interval by which the rate action advances the effect of the proportional control action. Note that derivative control action can never be used alone because this control action is effective only during transient periods. Proportional-plus-integral-plus-derivative (PH>) control action.The combination of proportional control action, integral control action, and derivative control action is termed proportional-plus-integral-derivative control action. This combined action has the advantages of each of the three individual control actions. The transfer function of the controller is M(s) / E(s) = K.(l + 1 / TiS + T`s) where K represents the proportional sensitivity, Ta represents the derivative time, and Ti represents the integral time. Errors analysis. Errors in a control system can be attributed to many factors. Changes in the reference input will cause unavoidable errors during transient periods and may also cause steady-state errors. Imperfections in the system components, such XXas static friction, backlash, and amplifier drift, as well as a ging or deterioration, will cause errors at steady state. The steady-state performance of a stable control system is generally judged by the steady-state error due to step, ramp, or acceleration inputs. Any physical control system inherently suffers steady-state error in response to certain types of inputs. A system may have no steady-state error to a step input, but the same system may exhibit nonzero steady-state error to a ramp input. (The only way we may be able to eliminate this error is to modify the system structure.) Whether or not a given system will exhibit steady-state error for a given type of input depends upon the type of open-loop transfer function of the system, to be discussed in what follows. Error criteria. In the design of a control system, it is important that the system meet given performance specifications. Since control systems are dynamic, the performance specifications may be given in terms of the transient-response behavior to specific inputs, such as step inputs, ramp inputs or the specifications may be given in terms of a performance index. Performance indexes. A performance index is a number which indicates the ` goodness ` of system performance. A control system is considered optimal if the values of the parameters are choosen so that the selected performance index is minimum or maximum. The optimal values of the parameters depend directly upon the performance index selected. Requirements of performance indexes. A performance index must offer selectively; that is, an optimal adjustment of the parameters must clearly distinguish nonoptimal adjustments of the parameters. In addition, a performance index must yield a single positive number or zero, the latter being obtained if and only if the measure of the deviation is identically zero. To be useful, a performance index must be a function of the parameters of the system, and it must exhibit a maximum or minimum. Finally, to be practical, a performance index must be easily computed, analytically or experimentally. Error performance indexes. In what follows, we shall discuss several error criteria in which the corresponding performance indexes are integrals of some function or weighted function of the deviation of the actual system output from the desired output. Since the values of the integrals can be obtained as functions of the system parameters, once a performance index is specified, the optimal system can be designed by adjusting the parameters to yield, say, the smallest value of the integral. To compare the quality of control on a numerical basis, several criteria that integrate some function of the error with respect to time have been proposed. Various error performance indexes have been proposed in the literature. We shall discuss the following six in this work. These performance indexes are as follows: 1. ISE (Integral of squared error) criterion 2. IAE (Integral of absolute value of error) criterion 3. ITSE (Integral of time weighted by squared error) criterion XXI4. ITAE (Integral of time weighted by absolute value of error) criterion 5. ISTSE (Integral of squared time weighted by squared error) criterion 6. ISTAE (Integral of squared time weighted by absolute value of error) criterion Integral of squared error (ISE) criterion. According to the integral of squared error (ISE) criterion, the quality of system performance is evaluated by the J e2(t)dt integral. Where the upper limit oo may be replaced by T which is chosen sufficiently large so that e(t) for T<t is negligible. The optimal system is the one which minimizes this integral. This performance index has been used extensively for both deterministic inputs (such as step inputs) and statistical inputs because of the ease of computing the integral both analytically and experimentally. A characteristic of this performance index is that is weighs large errors heavily and small errors lightly. This criterion is not very selective since, for following second order systemA system designed by this criterion tends to show a rapid decrease in a large initial error. Hence the response is fast and oscillatory. Thus the system has poor relative stability. Note, however, that the integral of squared error criterion is often of practical significance because the minimization of the performance index results in the minimization of power consumption for some systems, such a spacecraft systems. Integral of absolute value of error (IAE) criterion. The performance index defined by the integral of absolute value of error (IAE) criterion is j e(t)dt. This is one of the most easily applied performance indexes. If this criterion is used, both highly underdamped and highly overdamped systems cannot be made optimum. An optimum system based on this criterion is a system which has reasonable damping and a satisfactory transient-response characteristic; however, the selectivity of this performance index is not too good. Although this performance index cannot easily be evaluated by analytical means. Integral of time weighted by squared error (ITSE) criterion. The performance index based on the integral of time weighted by squared error (ITSE) criterion is J te2(t)dt. The optimal system is the one which minimizes this integral. This criterion has a characteristic that in the unit-step response of the system a large initial error is weighted lightly, while errors occurring late in the transient response are penalized heavily. This criterion has a better selectivity than the integral of squared error criterion. Integral of time weighted by absolute value of error (ITAE) criterion. According to this criterion, the ITAE criterion, the optimum system is the one which niinimizes the following performance index J te(t)dt. As in the preceding criteria, a large inital error in a unit-step response is weighted lightly, and errors occurring late in the transient response are penalized heavily. XXUA system designed by use of this criterion has a characteristic that the overshoot in the transient response is small and oscillations are well damped. This criterion possesses good selectivity and is an improvement over the integral of absolute value of error criterion. It is, however, very difficult to evaluate analytically, although it can be easily measured experimentally. Integral of squared time weighted by squared error (ISTSE) criterion. The performance index based on the integral of squared time weighted by squared error (ISTSE) criterion is Jt2e2(t)dt The optimal system is the one which minimizes this integral. This criterion has a characteristic that in the unit-step response of the system a large initial error is weighted lightly, while errors occurring late in the transient response are penalized heavily. Integral of squared time weighted by absolute value of error (ISTAE) criterion. The performance index based on the integral of squared time weighted by absolute value of error (ISTAE) criterion is Jt2e(t)dt The optimal system is the one which minimizes this integral. This criterion has a characteristic that in the unit-step response of the system a large initial error is weighted lightly, while errors occurring late in the transient response are penalized heavily. This criterion is not mathematically attractive, but it is used because of the availability of tables which have been derived from extensive simulation studies. Control systems are designed to perform specific tasks. The requirements imposed upon the control system are usually spelled out as performance specifications. They generally relate to accuracy relative stability and speed response. For routine design problems, the performance specifications may be given in terms of precise numerical values. In other cases they may be given partially in terms of precise numerical values and partially in terms of qualitive statements. In the latter case the specifications may be have to modified during the course of design since the given specifications may never be satisfied or may lead to a very expensive system. In this study, the performance criterions are applied to a closed-loop sampled-data process control system which consisted of a binary distillation column, a chromatograph as a measuring instrument, a chromatograph sampler and an industrial controller. The mathematical models of the distillation column and the chromatograph, representing a real plant (ICI Billingham Industrial Institution / U.K.). The closed-loop sampled-data system was digitally simulated K, Ti and Ta initial control parameter values for this system were determined by applying the simulated performance criterions, and the controller settings were computed according to relationships listed above. These settings and the corresponding time histories of the XXUloutput of the distillation process obtained from the simulation study are compared with, ISE, IAE, ITSE, ITAE, ISTSE and ISTAE criterion. Li this study, we shall solve the Flexible Simplex Optimization Method in which we minimize certain error performance indexes. A method is described for the minimization of a function of (n) variables, which depends on the comparison of function values at the (n+1) vertices of a general simplex, follwed by the replacement of the vertex with the highest value by another point. The simplex adapts itself to the local landscape, and contracts on to the final minimum. The method is shown to be effective and computationally compact. A procedure is given for the estimation of the Hessian Matrix in the neighbourhood of the minimum, needed in statistical estimation problems. The general block diagram represention of the process control system is shown in Fig.l. Disturbance r(t) e(t) b(t) Controller <*) '-*-&-{ Process Transportation Delay Retention. Zero Order Hold Sampler Figure 1. The general block diagram of a closed-loop sampled-data process control system. XXIV