Assessment and Future Directions of Nonlinear Model Predictive ...

264 M. Cannon, P. Couchmann, and B. KouvaritakisTheorem 3. If (23) and (24) are replaced in the MPC online optimization (22)by∑j−1(κ 2 cT2 A j−1−l Z 2 (k + l +1|k + l)A j−1−lT ) 1/2c 2 ≤ Y2 − c T 2 ¯x(k + j|k)l=0∑j−2(κ 2 vTi A j−1−l Z 2 (k + l +1|k + l)A j−1−lT ) 1/2v il=0(26a)( )+r j vT 1/2i Z 2 (k + N|k + j − 1)v i ≤ 1 − vTi ¯x(k + N|k) (26b)for j = 2,...,N, where r j is defined by Pr(χ 2 ((N +1− j)L) ≤ r j ) =p 2 /p j−1Ω, then feasibility of (22) at time k implies feasibility at time k +1 withprobability p 2 .Proof. Condition (26a) ensures that: (i) Pr(y(k + j|k) ≤ Y 2 ) ≥ p 2 for j =1,...,N; (ii) Pr(y(k + j|k +1)≤ Y 2 ) ≥ p 2 , j =2,...,N,isfeasibleatk +1with probability p 2 ; and (iii) the implied constraints are likewise feasible withprobability p 2 when invoked at k + 1. Here (iii) is achieved by requiring that theconstraints Pr(y(k + l|k + j) ≤ Y 2 ) ≥ p 2 be feasible with probability p 2 wheninvoked at k + j, j =2,...,N − 1. Condition (26b) ensures recursive feasibilityof (22d) with probability p 2 through the constraint that Pr(x(k + N|k + j) ∈Ω) ≥ p 2 /p j Ω , j =0,...,N− 1 (and hence also Pr(x(k + N + j|k + j) ∈ Ω) ≥ p 2)should be feasible with probability p 2 .Incorporating (26a,b) into the receding horizon optimization leads to a convexonline optimization, which can be formulated as a SOCP. However (26) and theconstraint that Ω should be invariant with probability p Ω >p 1/(N−1)2 ≥ p 2 aremore restrictive than (22c,d), implying a more cautious control law.Remark 3. The method of computing terminal constraints and penalty termsdescribed in sections 4 and 5 is unchanged in the case that A contains random(normally distributed) parameters. However in this case state predictions are notlinear in the uncertain parameters, so that the online optimization (22) could nolonger be formulated as a SOCP. Instead computationally intensive numericaloptimization routines (such as the approach of [7]) would be required.Remark 4. It is possible to extend the approach of sections 4 and 5 to nonlineardynamics, for example using linear difference inclusion (LDI) models. In thecase that uncertainty is restricted to the linear output map, y j (k) =C j (k)x(k)predictions then remain normally distributed, so that the online optimization,though nonconvex in the predicted input sequence, would retain some aspectsof the computational convenience of (22).7 Numerical ExamplesThis section uses two simulation examples to compare the stochastic MPC algorithmdeveloped above with a generic robust MPC algorithm and the stochasticMPC approach of [12].