Assessment and Future Directions of Nonlinear Model Predictive ...

506 R. De Keyser and J. Donald III⎧⎪⎨ e(t) =white noise (uncorrelated noise with zero mean value)C(q −1 )=1+c 1 q −1 + ...+ c nc q −nc⎪⎩D(q −1 )=1+d 1 q −1 + ...+ d nd q −n dThe filter C(q −1 )/D(q −1 )isthedisturbance model. It is specified as a designfilter, mainly affecting the robustness of the control loop against non-measurabledisturbances and against modelling errors [4].RecipeNEPSACLampsREACTORWafer4 SCR4 TCFig. 2. Basic control structureThe model output x(t) represents the effect (as described by a process model)of the process input u(t) on the process output y(t) and is also a non-measurablesignal. The relationship between u(t) andx(t) is given by the generic dynamicmodel:x(t) =f[x(t − 1),x(t − 2), ...; u(t − 1),u(t − 2)...] (3)where f[.] represents a known function (process model), which can be a linearor a nonlinear function of x(.),u(.).The nonlinear multivariable model used in the RTCVD predictive controllerconsisted of 4 sub-models (one for each output i =1..4):x i (t) =c i x i (t − 1) 4 + a i 1 x i(t − 1) ...+ a i 3 x i(t − 3) + b i11 u 1(t − 1) + ...+b i13 u 1(t − 3) + ...+ b i41 u 4(t − 1) + ...+ b i43 u (4)4(t − 3) + d iFrom a theoretical standpoint, the model (4) should include other nonlinearterms; indeed, the radiative heat flux is based on the temperature differencebetween heat source and wafer surface raised to the 4 th power. However, froman engineering standpoint, a simplified yet accurate model was obtained duringthe identification phase. This simplified model (4) still contains 68 parameters.The parameters have been identified successfully from a multiple of real-lifeexperiments on a pilot-plant RTCVD-reactor. Some model responses can beobserved in Fig. 3, clearly illustrating the interactive multivariable and nonlinearcharacter of the process.3.2 EPSACThe fundamental step in MPC methodology consists in prediction of the processoutput y(t + k) attimeinstantt, indicated by y(t + k|t),k =1..N 2 over theprediction horizon N 2 and based on: