PID vs Fuzzy Control

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Temperature control based on traditional PID versus fuzzy controllersIndustrial temperature-control applications demand speed and precision. Improved temperature controllers can meet these demands by adding features not found in traditional PID controllers but increase system complexity. Fuzzy temperature controllers give you another option.By Peter Galan, Control Software Designer, Nortel Networks

Common perception holds that temperature control is a mature and largely static area of technology. Some industrial applicationsfor example, injection-molding processesstill desire not only precise temperature control but also a faster warm-up phase and a quicker response to disturbances with minimal overshoot and undershoot when the set point changes. Traditional PID (proportional-integral-derivative) control techniques cannot meet these extra challenges. Figure 1 shows a continuous-time (analog) PID controller, typical for most closedloop control systems. Its output, the actuating value y(t), is a function of the regulation error e(t):

y (t ) = K P e(t ) + K I e(t )dt + K D

de(t ) . dt

The use of a standard PID controller is fully adequate for some applications. Such applications commonly feature low to moderate control-quality (timing, precision) constraints and well-defined and stable dynamic system behavior. Precision industrialtemperature control, however, does not belong among these standard applications. For example, injection-molding processes require fast changingreadjusting of controlled temperatures with minimal overshoot. In addition, heating processes do not exhibit stable dynamic behavior, because heating and cooling rates are different at each temperature set point. In addition, coupling between the zones of a multizone heating system makes dynamic behavior very unpredictable.

Heat-transfer-specific problemsFigure 2's simplified thermal model of a typical heating system ignores heattransport delays. Assume that when you switch on a constant power source, W, it powers an electrical heating element with a heat capacity, Ce. The temperature of the heating element, Te(t), rises with time, t. The heat continuously propagates, by direct conduction, to the heated system. Res represents a thermal resistance between the electrical element and the heated system, and Cs represents the heated systems thermal capacity. The system temperature, Ts(t), also rises with time. Rsa thus represents a thermal resistance between the heated system and surrounding environment (with ambient temperature Ta(t)), which tends to cool the system. Ideally, without any cooling from the outside (that is, Rsa ), both temperatures, Te(t) and Ts(t), would rise forever. In practice, however, natural cooling prevents this occurrence. So, after a certain time period, both temperatures stabilize at certain constant values. Figure 2's model represents a second-order system with the following transfer function:

G p (s ) =

K . 1 + sT1 + s 2T2

The PID controller would be an ideal controller, because its transfer function with two nulls could, at least theoretically, cancel both poles of Gp. Figure 2 demonstrates that two cascaded first-order systems can replace this second-order system. Therefore, in practice, system response to a step function will always be aperiodical, and if Res is relatively small, the output temperature will follow a simple exponential curve when rising or falling. For such systems, even PI controllers (without the derivative member) are fully adequate. An interesting feature of a typical heated system, which represents the first difficulty for any temperature controller, is that increasing the temperature a couple of degrees from the surrounding temperature takes substantially less time than does cooling down to the surrounding temperature. In contrast, increasing the temperature of the same system by a couple of degrees when it is close to its maximum temperature takes substantially longer than does bringing the temperature back down by a couple of degrees. At one temperature set point, the heating rate (the speed of heating up) and the cooling rate (the speed of cooling down) are equal. This set point is at the temperature that requires the application of exactly 50% of the maximum applicable power to the heater. What is that balanced temperature? Theoretically, from Figure 2's thermal model, the dependence between power (W) and temperature (Ts) should be linear. That is, the stabilized temperature corresponding to 50% of maximum power should be exactly in the middle between the minimum and maximum temperature. The assumption here is that the maximum reachable temperature requires 100% power, and the minimum (ambient) temperature requires you to apply 0% power to the heating element for an unlimited time. Figure 3 shows the above-described feature for three set points: One is close to the ambient temperature, one is at the balanced temperature, and one is close to the maximum temperature. Temperatures are rising and decreasing exponentially with different rates (time constants) for heating and cooling. The ratio of the rate of heating and cooling processes at any stage depends only on the value of the set point.

Enhancing the traditional PID controllerThe PID constants basically depend on the gain and time constant of the controlled system. If you select them properly, they will cancel poles of the controlled systemtransfer function. Different rates of the heating and cooling processes affect the optimal values of the PID constants, making their estimation very difficult. If you want to use autotuning, be aware that all autotuning methods provide only one set of PID constants, which will at best suit only the set points close to the balanced temperature. A different situation exists when the temperature set point is different from the balanced temperature. The farther the set point is from the balanced temperature, the farther the PID constants will be from their optimal values. Just how far depends on how far the set point is from the balanced temperature and whether the current temperature is below or above the set point. To make the PID controller suitable for temperature control at any set point, you need some automatic adaptation of the PID constants. Generally, all three PID constants, K P , K I , and K D , are inversely proportional to the rate of temperature change. Autotuning algorithms (known in control theory as the Ziegler-Nichols rules) also adhere to this rule. The rate of temperature change is not a linear function of the temperature set point (the derivative of exponential is also exponential), but a straight line is a rea-

sonable approximation (Figure 4). As Figure 4 shows, the rate (speed) of heating at the balanced temperature is 1/2, where is a time constant of the exponential (shown previously in Figure 3). At the ambient temperature, 1/ is the rate of heating. The rate of cooling at the ambient temperature is infinitesimally low. The opposite situation exists at the other end of the exponential curve, close to the maximum temperature. Here, the rate of heating is infinitesimally low, and the rate of cooling is very high. To compensate for varying heating and cooling rates, the PID constants, K P , K I ,

and K D , must also vary, requiring modification of the basic PID controller. Figure 5 shows the addition of two new blocks: an adapter and a heater model. The heater model must at first determine a value of the balanced temperature. For most applications, applying 50% of the maximum output power to the heating element and waiting until the temperature settles down is infeasible. Therefore, the control system can first assume that the balanced temperature equals one half of the maximum temperature (a value for which the controller has been designed). You can determine a more precise value of the balanced temperature during the control process. The adapter block (knowing the balanced and set-point temperatures) then calculates two correction coefficients for the modification of the PID constants. This calculation is based on a linear dependence of the temperature-change rate as shown in Figure 4. You apply the first correction coefficient, kh, (it multiplies the PID constants) during the heating and the other coefficient, kc, during cooling: kh TMAX/2(TMAX TSP), [1] and kc TMAX/2TSP, [2] where TMAX is the maximum controllable and TSP is the set point temperature.

Introducing a feedforward memberOnce you have created the heater model (which is actually an inverse static characteristic of the heating process), you can use its output variable power level as a feedforward contribution to the actuating variable. Because the heating process is linear (the output temperature is proportional to the applied power), a simple line with the slope km can approximate the heating model characteristic. You can then calculate an initial value of km as: km MAX_POWER / MAX_TEMPERATURE. You can express the MAX_POWER value in the percentage of the output pulse width if you use the PWM method to drive the heater elements. If you knew in advance what power value youd need for the set-point temperature, you could immediately replace the contribution of the controller's integral member with the output of the feedforward member. However, at the beginning of the control process, you would not have the correct heater model (the correct km value). Fortunately, the integral member can provide this information to the adapter block. So, this approach provides the adapter block with a new task. Once the temperature stabilizes at the set-point value, the adapter block can easily calculate the required output power to maintain this temperature. It is a sum of the integral member and the feedforward member outputs. Because the temperature is stabilized at the set point, the regulation error is zero, and the contributions of the propo