According to Linear Technology’s Jim Williams, “The unfortunate relationship between servo systems and oscillators is very apparent in thermal control systems.” (Linear Applications Handbook, 1990).
High-performance temperature control is certainly one of those topics that look easy enough in theory, but prove to be anything but simple in practice. To their chagrin, thermal-control system designers discover that heater-load thermal time constants can conspire shamelessly with heater-sensor response delays. Together, they produce wild oscillatory instability where precision thermostasis was expected.
Over the years, a long laundry list of feedback techniques and control strategies has been devised to tame the dynamic-stability gremlins that inhabit temperature-control servo loops. Some of these ideas incorporate integration of the temperature-control error term (TS − T) to force the control loop error to converge toward zero (e.g., “Hybrid Digital-Analog Proportional-Integral Temperature Controller,” Electronic Design, Aug. 3, 1998).
In fact, one tempting alternative scheme would be to make the heater power proportional to the integrated temperature error alone. In such a simplistic “straight-integration” algorithm, the temperature (T) would be sampled and subtracted from the setpoint (TS). Then, the difference would be multiplied by the loop gain (F) and added as a cumulative adjustment to the heater power setting H, once each cycle through the loop. Consequently, H = H + F(TS − T).
Many desirable properties, including simplicity, noise rejection, easy, one-parameter (F) “tuning,” and zero steady state error would be offered by the resulting servo loop (whether implemented by software or analog integration). Unfortunately, it would also have one other rather undesirable property—virtually guaranteed oscillation that would never allow final convergence to TS (Fig. 1).
Fatal oscillation can be expected since, by the time system temperature has corrected from a deviation and struggled back to T = TS, the heater power will inevitably have been grossly overcorrected. In fact, the resulting overshoot of H is likely to be about as large as the original perturbation. Later, H’s opposite undershoot will be as large as the initial overshoot, and so on—and on.
Intuitively, the problem might be fixed by adopting a better estimate of H whenever the system crosses the setpoint (T = TS). The Take-Back-Half (TBH) method is a way of doing exactly that. It refers to taking deliberate advantage of the (approximate) equality of straight-integration’s undamped over/undershoots.
To do so, variable HO is added and the modified servo run as before, except for what happens when the sampled temperature (T) passes through setpoint (T = TS). Whenever such a setpoint crossing occurs, both H and HO are replaced by the bisecting value: (H + HO)/2. As a result, at each setpoint crossing, H and HO are set midway between the values corresponding to the current (H) and previous (HO) crossings. This action amounts to “taking back half” of the adjustment applied to the heater setting between crossings —hence, TBH. Figure 2 shows the ability of (simulated) TBH to force quick (half-cycle) convergence.
Experience gained from applying TBH reveals that the algorithm’s stability is robust, given a reasonable choice for F. In general, TBH systems have a natural cycle time proportional to the square root of the ratio of the thermal time constant to F. Simulation and experimentation agree that convergence is assured if this system period is at least eight times longer than the heater-sensor time delay. Therefore, convergence can always be achieved if F is set low enough. Meanwhile, steady-state (TS − T) error remains equal to zero.