Assessment and Future Directions of Nonlinear Model Predictive ...

510 R. De Keyser and J. Donald IIIThis algorithm results after convergence to the optimal solution for the underlyingnonlinear predictive control problem. A convergence proof is not available;however, simulation results and practical experience both look very promising.The number of required iterations depends on how far the optimal u ∗ (t + k|t)is away from the optimal u ∗ (t + k|t − 1). In quasi-steady-state situations, thenumber of iterations is low (1 ...2). On the other hand, during transients thenumber of iterations might raise to 10. As the NEPSAC algorithm consists of arepetitive use of the basic linear EPSAC algorithm and as EPSAC requires a lowcomputational effort, this is acceptable in practice. In the RTCVD application,10 iterations require about 100 ms, which is a small fraction of the controllersampling period (2 s).4 Experimental ResultsHundreds of test runs on different types of real-life RTCVD-reactors have shownthe excellent performance of the MPC strategy compared to the traditional(commercial) PID approach. During the comparison, the PID controllers wereconfigured and tuned by experienced staff of the company, according to theirexpert skill of many years. Figure 5 presents typical PID results during a recipe104010401020102010001000Temperatures (°C)980960940920Temperatures (°C)980960940920900900880880860100 150 200 250 300 350 400Samples (2 seconds)860100 150 200 250 300 350 400Samples (2 seconds)Fig. 5. PIDwithramprate6 o C/s (center/front/side/rear wafer surface temperatures)(left) and NEPSAC with ramp rate 15 o C/s (center/front/side/rear wafer surface temperatures)(right)with ramp rates of about 6 o C/s. This is the maximum ramp rate that is feasibleon this kind of equipment under PID control (note that there is a trendtowards higher ramp rates in order to reduce the process cycle time; however thehigher the ramp rate, the more difficult it is to keep control of the temperatureuniformityover the wafer surface). Figure 5 presents also typical NEPSAC-MPCresults during a recipe with ramp rates of about 15 o C/s. Although the ramprate is much higher, it is clear that tight control of the temperature uniformityis still possible.