Doğrusal kontrol sistemlerinin optimizasyonu

Abstract

Endüstrinin çeşitli kollarında, otomatik kontrol sistemleri yaygın olarak kullanılmaktadır. Otomatik kontrollü endüstri tesislerinde, sistem performansı doğrudan kontrol parametrelerine bağlıdır. Bu yüzden, bu parametrelerin optimum ayarlarının sağlanması problemi, üzerinde durulması gereken bir konudur. Proses kontrol sistemlerinde genellikle kontrolü yavaşlatan büyük zaman sabiti elemanları ve kontrolü zorlaştıran büyük ölçü zaman elemanları bulunmaktadır. Buna karşılık uygun kontrol sisteminin sağlanmasıyla elde edilecek imalat kalitesindeki iyileşme çok büyük miktarlarda imalat yapılması nedeniyle önemli ekonomik tasarruflar kazandırır. Bu bakımdan optimum kontrolün sağlanması proses kontrol sistemlerinde de çok önemlidir. Bu çalışmanın amacı; bir endüstriyel prosesin kapalı çevriminde kontrol organı olarak seçilen PI ve PID' nin çeşitli optimizasyon metodlanyla ayarlanması ve sistemin cevabının elde edilmesidir. Ayrıca optimum kontrol parametrelerinin elde edilmesinde kullanılan performans kriterlerinin karşılaştırılması da verilmiştir. Proses kontrol sisteminin şekli ve parametreleri uygulamadan alınmıştır. Üzerinde çalışılan sistem iki zaman sabiti ve bir zaman gecikmesinden oluşan geri beslemeli bir sistemdir. Ayrıca geri besleme kolunda ölçme elemanı olarak bir nakil gecikmesi vardır. Kontrol sistemine bir basamak girişi uygulanarak çıkış alınmıştır. Bu çalışmada öncelikle kontrol organı ve optimum kontrol organı için kullanılan performans kriterleri tanıtılmıştır. Sistemin optimizasyonu ve simülasyonu için Turbo C dilinde bir program 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 uygulanarak sistem üzerindeki etkileri incelenmiştir.

The selection of a controller type (P, PL PID) and its parameters (K, Ti, Td) is intimately related to the modal of the process to be controlled. The adjustment of the controller parameters to achieve satisfactory control is called tuning. The selection of the controller parameters is essentially an optimization problem in which the designer of the control system attempts to satisfy some criterion of optimality, the result of which is often referred to as " good " control. The process of tuning can vary form a trial-and-error attempt to find suitable control parameters for "good" control to elaborate optimization calculation based on a modal of the process and a specific criterion of optimal control. In many applications, there is no modal of the proses and the criterion for the good control is only vaguelly defined. A typical criterion for good control is that the response of the system to a step change in set point or load should have minimum overshoot and one-quarter decay ratio. Other criteria may include minimum rise time and minimum settling time. In this study, some of the widely using tuning rules for continuous controllers will be presented. Controller Tuning : Before presenting tuning rules, some discussion of the effect of each mode in a PID controller on the transient response of a controlled process will be instructive. Selection of Controller Modes : Consider a typical loop as shown in Fig.l in which the process is second order and the measuring element is a transport lag. Load responses for this process for four types of controllers (P, PD, PI, PID). For each response curve, the process was subjected to a unit step change in load (U=l/s) and the controller parameters were selected by the tuning rules to be presented later. The nature of the response for each type controller will now be described. Proportional Control : Proportional control procudes an overshood followed by an oscillatory response, which levels out at a value that does not equel the set point; this ultimate displacesment from the set point is the offset. Proportional - Derivative Control : For this case the reponse exhibits a smaller overshoot and a smaller period of oscillation compared to the response for proportional control. The offset that still remains is less than for proportional control. xxvi Proportional - Integral Control : In this case, the reponse has about the same overshoot as proportional control, but the period is larger; however, the response returns to the set point (offset=0) after a relatively long settling time. The most beneficial influence of the interal action in the controller is the elimination of offset. Proportional - integral - Derivative Control : As one might expect, the use of PID control combines the beneficial features of PD and PI control. The response has lower overshoot and returns to the set point more quickly than the responses for the other types of controllers. From the nature of responses just described, we can make following generalizations. Integral action, which is present in PI and PID controllers, eliminates offset. The addition of derivative action speeds up the response by contributing to the controller output a component of the signal that is proportional to the rate of change of the prosess variable. For simple, low order (first or second order) processes that can tolerate some offset, P or PD controlleris satisfactory. For processes that can not tolerate offset and are low order, PI control is required. For procosses that are of high-order (those whith transport lag or many first-order lags in series), PID control is needed to prevent large overshoot and long settling time. Before the avaibility of microprocessor-based controllers, it was customary to select a controller based on price. Pneumatic and electronic controllers with proportinal action were the least expensive and those with PID action were the most expensive. It was cosidered uneconomical to purchase a controller with more control actions than needed by the process. Today this price incentive no longer exists in the selection of the type of controller, for the modern microprocessor-based controller comes with all three actions, as well es other functions such as lead-lag and transport lag. There is probably little justification to select P or PD controller for most processes. The PI controller is often the choice because it eliminates offset and requires only two parameter adjustments. Tuning a PID controller is more diffucult because three parameters must be adjusted. The presence of derivative action can also cause the controller output to be very jittery if there is much noise in the signals. We now turn our attention to some of criteria for good control that are used to judge whether or not a control system is well tuned. Criteria for Good Control: Before we can be satisfied with the response of a control system for a choice of control parameters, we must have some concept of what we want as an ideal reponse. Most operators of processes know what they want in the form of a response to a change in set point or load. For example, a response that gives minimum overshoot and Va decey ratio is often considered as a satisfactory response. In many cases, tuning is done by trial and error until such a response is obtained. In order to compare different responses that use different sets of controller parameters, a criterion that reduces the entire response to a single number, or a figure of merit, is desireable. XXVI 1 Errors in a control system can be attributed to many factors. Changes in the reference input will cause unavoidable errors during transient periods and many also cause steady-state errors. Imperfections in the system companents, such as static friction, backlash, and amplifier drift, as well as aging or deteriotaration, will cause errors at steady state. The steady-state performance of 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 eliminanate 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. 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 givens in terms of the transient-response behaviour to specific inputs, such as step inputs, ramp inputs or the specifications may be given in terms of a performance index. 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. A performance index must offer selectivelly; 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, andit must exhibit a maximum or minimum. Finally, to be practical, a performance index must be easily computed, analitically or experimentally. 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 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. Various error performance indexes have been proposed in the literature. We shall discuss the following six in this section J e2(t)dt, j te2(t)dt, J |e(t)|dt, J t|e(t)|dt, JtV(t)dt,Jt2|e(t)|dt. xxviii According to the integral square-error (ISE) criterion, the quality of the system performance is evaluated by the J e2(t)dt integral. Where the upper limit qo may be replaced by T which is chosen sufficiently large so that e(t) for T