Assessment and Future Directions of Nonlinear Model Predictive ...

Chance Constrained Nonlinear Model Predictive Control 297∆u(k + i|k) =u(k + i|k) − u(k + i − 1|k)u min ≤ u(k + i|k) ≤ u max ,i=0,...,M − 1.∆u min ≤ ∆u(k + i|k) ≤ ∆u max ,i=0,...,M − 1.P{y min ≤ y(k + i|k) ≤ y max }≥α, i =1,...,P.where P and M are the length of prediction and control horizon, ξ representsthe uncertain variables with known PDF, P{·} represents the probability tosatisfy the constraint y min ≤ y(k + i|k) ≤ y max and 0 ≤ α ≤ 1 is the predefinedconfidence level. States x, outputs y and controls u are all doubly indexed toindicate values at time k + i given information up to and including time k. Q i ,R i ,andS i are weighting matrices in the objective function. E and D are theoperators of expectation and variation, respectively.Since the outputs have been confined in the chance constraints, the objectivefunction f in Eq.(1) may exclude the quadratic terms on outputs for the sake ofsimplicity [10]. The simplified CNMPC objective function can be described asfollows:Min J =M−1∑i=1{‖u(k + i|k) − u ref ‖ Ri+ ‖∆u(k + i|k)‖ Si} (2)This problem can be solved by using a nonlinear programming algorithm. Thekey obstacle towards solving the CNMPC problem is how to compute P{·} andits gradient with respect to the controls. In the next section, the computationalaspects of CNMPC to address this problem as well as the feasibility analysis willbe discussed.3 Computational Aspects of CNMPCIn process engineering practice, uncertain variables are usually assumed to benormally distributed due to the central limit theory. However, a normal distributionmeans that the uncertain variable is boundless, which is not true forsome parameters with physical meanings, e.g. the molar concentration in a flowshould be in the range of [0, 1]. In order to describe the physical limits of theuncertainty parameters, it is preferable to employ truncated normal distributionwhich has been used extensively in the fields of economic theory [9]. The basicdefinition of truncated normal distribution is given as follows:Definition 1. Let z be a normally distributed random variable with the followingPDF:ρ(z) = 1σ √ − µ)2exp{−(z2π 2σ 2 } (3)Then the PDF of ξ, the truncated version of z on [a 1 , a 2 ] is given by:{ρ(ξ) =1σ √ (ξ−µ)2exp{−2π(Φ(a 2)−Φ(a 1)) 2σ},a 2 1 ≤ ξ ≤ a 20,ξ ≤ a 1 or a 2 ≤ ξwhere Φ(·)is the cumulative distribution function of z.(4)