Assessment and Future Directions of Nonlinear Model Predictive ...

298 L. Xie, P. Li, and G. WoznyDetailed discussion about the properties of the truncated normal distribution canbe found in [9] and it is easy to extend Definition 1 to the multivariate case. Inthe following, a truncated normally distributed ξ with mean µ, covariance matrixΣ and truncated points a 1 , a 2 , denoted as ξ ∼ TN(µ, Σ, a 1 , a 2 ), is considered.3.1 Inverse Mapping Approach to Compute the Probability andGradientIf the joint PDF of the output y(k+i|k) is available, the calculation of P{y min ≤y(k + i|k) ≤ y max } and its gradient to u can be cast as a standard multivariateintegration problem [8]. But unfortunately, depending on the form of g 2 ,theexplicit form of the output PDF is not always avaliable. To avoid directly usingthe output PDF, an inverse mapping method has been recently proposed forsituations in which the monotone relation exists between the output and one ofthe uncertain variables [5].Without loss of generality, let y = F (ξ S ) denotes the monotone relation betweena single output y and one of the uncertain variables ξ S in ξ=[ξ 1 , ξ 2, ...,ξ S ] T . Due to the monotony, a point between the interval of [y min , y max ]canbeinversely mapped to a unique ξ S through ξ S = F −1 (y):P{y min ≤ y ≤ y max }⇔P{ξ minS≤ ξ S ≤ ξ maxS } (5)It should be noted that the bounds ξSmin,ξmax S depends on the realization ofthe individual uncertain variables ξ i , (i =1, ··· ,S− 1) and the value of inputu, i.e.[ξSmin ,ξmax S ]=F −1 (ξ 1 , ··· ,ξ S−1 ,y min ,y max ,u) (6)and this leads to the following representationP{y min ≤ y ≤ y max } =∫ ∞−∞···∫ ∞−∞ξS∫maxρ(ξ 1 , ··· ,ξ S−1 ,ξ S )dξ S dξ S−1 ···dξ 1 (7)ξ minSFrom (6) and (7), u has the impact on the integration bound of ξ S .Thusthefollowing equation can be used to compute the gradient of P{y min ≤ y ≤ y max }with respect to the control variable u:∂P{y min≤y≤y max}∂u=−∞∞∫∞∫··· {ρ(ξ 1 , ··· ,ξ S−1 ,ξ max −S−∞(8)ρ(ξ 1 , ··· ,ξ S−1 ,ξ min ) ∂ξmin SS ∂u }dξ S−1 ···dξ 1A numerical integration of (7) is required when taking a joint distributionfunction of ξ into account. Note that the integration bound of the last variablein (7) is not fixed. A novel iterative method based on the orthogonal collocationon finite elements was proposed in ref [5] to accomplish the numerical integrationin the unfixed-bounded region.) ∂ξmax S∂uExtending inverse mapping to the dynamic case. If a monotone relationalso exists between y(k + i|k) andξ(k + i) fori =1,...,P in g 2 of Eq.(1), the