It’s not hard to show that the (linearized) state equationscorresponding to Figure 4 are

Taking Laplace transforms of (1) and (2) and solving for Ta(s), which is the output of interest, gives the following open-loop model of the thermal system:

where K is a constant and D(s) is a second-order polynomial。K, tz, and the coefficients of D(s) are functions of the variousparameters appearing in (1) and (2)。Of course the various parameters in (1) and (2) are completely unknown, but it’s not hard to show that, regardless of their values, D(s) has two real zeros。 Therefore the main transfer function of interest (which is the one from Q(s), since we’ll assume constant ambient temperature) can be written

Moreover, it’s not too hard to show that 1=tp1 <1=tz <1=tp2, i。e。, that the zero lies between the two poles。 Both of these are excellent exercises for the student, and the result is the openloop pole-zero diagram of Figure 5。

Obtaining a complete thermal model, then, is reduced to identifying the constant K and the three unknown time constants in (3)。 Four unknown parameters is quite a few, but simple experiments show that 1=tp1 _ 1=tz;1=tp2 so that tz;tp2 _ 0 are good approximations。 Thus the open-loop system is essentially first-order and can therefore be written

 (where the subscript p1 has been dropped)。

Simple open-loop step response experiments show that,for a wide range of initial temperatures and heat inputs, K _0:14 _=W and t _ 295 s。1

4。2 Control System Design

Using the first-order model of (4) for the open-loop transfer function Gaq(s) and assuming for the moment that linear control of the heater power output q(t) is possible, the block diagram of Figure 6 represents the closed-loop system。 Td(s) is the desired, or set-point, temperature,C(s) is the compensator transfer function, and Q(s) is the heater output in watts。

Given this simple situation, introductory linear control design tools such as the root locus method can be used to arrive at a C(s) which meets the step response requirements on rise time, steady-state error, and overshoot specified in Table 1。 The upshot, of course, is that a proportional controller with sufficient gain can meet all specifications。 Overshoot is impossible, and increasing gains decreases both steady-state error and rise time。

Unfortunately, sufficient gain to meet the specifications may require larger heat outputs than the heater is capable of producing。 This was indeed the case for this system, and the result is that the rise time specification cannot be met。 It is quite revealing to the student how useful such an oversimplified model, carefully arrived at, can be in determining overall performance limitations。

4。3 Simulation Model

Gross performance and its limitations can be determined using the simplified model of Figure 6, but there are a number of other aspects of the closed-loop system whose effects on performance are not so simply modeled。 Chief among these are

·quantization error in analog-to-digital conversion of the measured temperature and

· the use of PWM to control the heater。

Both of these are nonlinear and time-varying effects, and the only practical way to study them is through simulation (or experiment, of course)。

Figure 7 shows a SimulinkTM block diagram of the closed-loop system which incorporates these effects。 A/D converter quantization and saturation are modeled using standard Simulink quantizer and saturation blocks。 Modeling PWM is more complicated and requires a custom S-function to represent it。

This simulation model has proven particularly useful in gauging the effects of varying the basic PWM parameters and hence selecting them appropriately。 (I。e。, the longer the period, the larger the temperature error PWM introduces。 On the other hand, a long period is desirable to avoid excessive relay “chatter,” among other things。) PWM is often difficult for students to grasp, and the simulation model allows an exploration of its operation and effects which is quite revealing。