Assessment and Future Directions of Nonlinear Model Predictive ...

Application of the NEPSAC Nonlinear Predictive Control Strategy 509The controller output is then the result of minimizing the cost function:V (U) =∑N 2k=N 1[r(t + k|t) − y(t + k|t)] 2 (10)with r(t + k|t) the desired reference trajectory (called recipe in semiconductorterminology) and the horizons N 1 , N 2 being design parameters. It is nowstraightforward to derive the (unconstrained) EPSAC solution:U ∗ =[G T G] −1 [G T (R − Y)] (11)Only the first element δu ∗ (t|t) inU ∗ is required in order to compute the actualcontrol action applied to the process. At the next sampling instant t+1, the wholeprocedure is repeated, taking into account the new measurement informationy(t + 1); this is called the principle of receding horizon control. As well-knownin current MPC-practice, the cost index (10) can be extended with constraints,leading to a quadratic programming problem. This has been the approach totackle input saturation constraints in the RTCVD application.3.3 NEPSACThe calculation of the predicted output with (6) involves the superposition principle.When a nonlinear system model f[.] isusedin(3),abovestrategyisonlyvalid - from a practical point of view - if the term y optimize (t+k|t) in (6) is smallenough compared to the term y base (t + k|t). When this term would be zero, thesuperposition principle would no longer be involved. The term y optimize (t + k|t)will be small if δu(t + k|t) is small, see (7). Referring to Fig. 4, δu(t + k|t) willbe small if u base (t + k|t) isclose to the optimal u ∗ (t + k|t) .This can be realized iteratively, by executing the following steps at each controllersampling instant:1. Initialize u base (t + k|t) as:u 1 base (t + k|t) =u∗ (t + k|t − 1) , i.e. the optimalcontrol sequence as computed during the previous sampling instant; in otherwords: u ∗ (t + k|t − 1) is used as a 1 st estimate for u ∗ (t + k|t)2. Calculate δu 1 (t + k|t) using the linear EPSAC algorithm3. Calculate the corresponding yoptimize 1 (t + k|t) with (7) and compare it toybase 1 (t + k|t) , which is the result of u1 base (t + k|t)4. • In case yoptimize 1 (t + k|t) is NOT small enough compared to y1 base(t + k|t):re-define u base (t + k|t) asu 2 base (t + k|t) =u1 base (t + k|t)+δu1 (t + k|t) andgo to 2. The underlying idea is that u 1 base (t + k|t)+δu1 (t + k|t) -whichis the optimal u ∗ (t + k|t) for a linear system - can act as a 2 nd estimatefor the optimal u ∗ (t + k|t) in case of a nonlinear system• In case yoptimize i (t + k|t) is small enough compared to yi base(t + k|t): useu(t) =u i base (t|t)+δui (t|t) as the resulting control action of the currentsampling instant (notice that i =1, 2,..., according to the number ofiterations).