Ölü zamanlı kontrol sistemlerinde tasarım yöntemleri - Arıkan, Taner (Fen Bilimleri Enstitüsü, 1997)
Bu çalışmada, endüstride çok sık karşılaşılan ölü zamanlı sistemler ele alınmıştır. Kontrol edilen süreç olarak, endüstriyel ölü zamanlı bir sistemin birçok davranışım gerçekleyebilen bir çalışma seti kullanılmıştır, ölü zamanlı süreçler için, endüstriyel tasarım yöntemleri incelenmiş ve bu yöntemlere ilişkin kontrol algoritmaları oluşturulmuştur. Bu kontrol algoritmaları, kontrolör olarak kullanılan bir programlanabilir lojik kontrolörde programlanarak kontrol edilen sisteme uygulanmıştır. Kontrolörün ürettiği kontrol işaretinin sınırlı olması dolayısıyla sistemde oluşturduğu integral yığılması sorunu ele alınmış ve çözüm yöntemleri verilmiştir. Ayrıca, ölü zaman kompanzasyonu ile ilgili yöntemler incelenmiş, endüstride oransal- türevsel - integral (PID) kontrolörü kadar kolay bir biçimde uygulanabilen bir öngörüsel kontrol yöntemi sunulmuştur.

Many industrial processes have a model which is given by the transfer l + s.T -s.L (1) It is characterized by three parameters; the gain (Kp), the time constant (T) and the time delay (L). The step response of the system is shown in Figure 1. o i z 3 4 Figure 1 The step response of the system This system has a monoton step response and a time delay. The parameter L is called the apparent time delay. The PT326 Process Trainer which is used as a plant has the model which is given by the transfer function Equation 1 and the step response which is shown in Figure 1. The PT326 Process Trainer is a self-contained process and control equipment. It has the basic characteristics of a large plant, enabling distance/velocity lag, system response, proportional and two-step control etc. to be demonstrated. Due to its relatively fast responce, changes in set value and measured value can be displayed on an oscilloscope. In this equipment, air drawn from atmosphere by a centrifugal blower is driven past a heater grid and through a length of tubing to atmosphere again. The process consists of heating the air flowing in the tube to the desired temperature level and the purpose of the control equipment is to measure the air temperature, compare it with a value set by the operator and generate a control signal which determines the amount of electrical power supplied to a correcting element, in this case a heater mounted adjacent to the blower The elements which form the system are shown in Figure 2. XV The instrument contains integrated circuit operational amplifiers and self- contained power supplies. It can be coupled to an external control equipment to control the process. correcting element process motor element detecting element controlling ^ element^ comparing element measuring element set value Figure 2 Basic elements of closed loop process control system The response of the detector to a step change in heater power is affected by two time lags; distance/velocity lag, which has no effect on the form of the input signal, and transfer lag, which does affect the form of the signal. An alteration to the condition of a process affects the detecting element after a time interval which is dependent on the velocity of the process and the distance between the point of change and the detector. This time interval L is the distance/velocity or transport lag, as given by the equation: L = distance velocity input output ramp *L* signal. time - >. time Figure 3 The effect of distance/velocity lag It represents a pure lag, there being no change in the magnitude or form of the XVI The effect of distance/velocity lag on different forms of input signal is shown in Figure 3. In any stage of a thermal process where heat is transferred through thermal resistance to or away from a thermal capacity, the temperature rise following a step change of input, is exponential as shown in Figure 4 (a). It reaches 63.2 % of its final value Kp after time T, which is the exponential lag of that stage. In a complete process, several exponential lags are normally present, leading to a response curve of the form shown in Figure 4 (b), and producing a time lag which is referred to as the transfer lag of the process. output signal output signal "Kt^ time time (a) with single exponential lag (b) with more than one exponential J Figure 4 Response of system to a step change of input signal The shifting property of the Laplace transformcan be used to determine the response of a system with dead time. This occurs when an input results from a transporting fluid, for example. This is shown in Figure 5. A fluid with a temperature y flows through a pipe. The fluid velocity v is constant with time. The pipe length is D, so it takes a time L=D/v for the fluid to move from one end to the other. Let yi (t) denote the incoming fluit temperature and y2 (t) the temperature of the fluid leaving the pipe. ^(t) >v y2(t) K D- A y(t) = y(t-L) A t L (a) Time plots of the input and the response (b) Block diagram Figures Process with dead time L=D/v Now, suppose that the temperature of the incoming fluid suddenly increases. If this is modeled as a step function, the result is shown in Figure 5. If no heat energy is XVU lost, then y2 (t), the temperature at the output, is yi (t-L), where yt (t) is the temperature at the input. Thus, a time L later, the output temperature suddenly increases. A similar effect occurs for any change in yi (t); in general, we can write y2(0=yl(t-L) From the shifting teorem, r2(*) = e-^W (2) (3) This result is shown in block diagram form in Figure 5 (b). Figure 6 shows the general case of an element with dead time. In this case, Y(s) = esL G(s) R(s) (4) Figure 6 System diagram with dead time The presence of dead time means that the system does not have a characteristic equation of finite order. In feet, there are an infinite number of characteristic roots for a system with dead time. This can be seen by noting that the term e"aL can be expanded in an infinite series as es.L = "s.L 1 l+s.L + (s2.L2/2\)+... (5) Although, there are some approximations as shown in (5) to express irratianol functions as rational, one of the most useful for dead time approximation in control systems is the Pad? approximation. There are some control methods for processes with dead time. The difficulty with system delays is that necessary information comes too late and creates stability problems. The problem of controlling systems with time delays was solved as early as 1957 by Otto Smith. He suggested a controller where a model of the system is included (Figure 7) and the controller consequently is called a Smith predictor. Such controllers are also called dead time compensating controllers. For processes with long dead time, the control performance obtained with a proportional-integral-derivative (PID) controller is limited. Predictive control is required to control a process with a long dead time efficiently. The derivative part of the PID controller can be interpreted as a prediction mechanism. Unfortunately, prediction through derivation of the measurement signal in not appropriatewhen the process contains long dead times. Therefore, ifa PID controller is applied on this kind of problems, the derivative part is mostly switched off, and only a PI controller without prediction is used. XVlll Figure 7 The Smith Predictor A Smith Predictor using the process model which is given Equation (1) combined with a PI controller requires five parameters to be determined, namely the PI controller parameters K and Tj and the process parameters Kp,T and L. The five parameters of the Smith Predictor are very difficult to tune manually without a systematic process identification experiment. There is a new model based predictive PI controller (PIP) with only three adjustable parameters. This dead time compensating controller can be tuned manually in the same way as a PID controller. The structure of the PIP controller is the same as the Smith Predictor, but with the exception mat two of the process model parameters are determined automatically based on the PI parameters. In the PIP controller, the parameters K, Tj and L determined by the operator. Parameters Kp and T are calculated as functions of the K and Tj. In a system with PI or PID controller, it is possible to occur windup. Although many aspects of a control system can be understood based on linear theory, some non-linear effects must be accounted for. All actuators have limitations: a motor has limited speed, a valve cannot be more than fully open or fully closed, etc. When a control system operates over a wide range of operating conditions, it may happen that the control variable reaches the actuator limits. When mis happens the feedback loop is effectively broken because the actuator will remain at its limit independently of the process output. If a regulator with integrating action is used, the error will continue to be integrated. This means that the integral term may become very large or, colloquially it "winds up". It is then required that the error change sign for a long period before things return to normal. The consequence is that any controller with integral action may give large transients when the actuator saturates. There are several ways to avoid integral windup.
Design methods for control systems with dead time
Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Sosyal Bilimler Enstitüsü, 1997
Keywords: Denetim sistemleri; Matematiksel modelleme; PID denetleme; Control systems; Mathematical modelling; PID control