Model Predictive Control - Stanford University

Model Predictive Control linear convex optimal Control finite horizon approximation Model Predictive Control fast MPC implementations supply chain managementProf. S. Boyd, EE364b, Stanford UniversityLinear time-invariant convex optimal controlminimizeJ= t=0 (x(t), u(t))subject tou(t) U, x(t) X, t= 0,1, ..x(t+ 1) =Ax(t) +Bu(t), t= 0,1, ..x(0) =z. variables: state and input trajectoriesx(0), x(1), .. Rn,u(0), u(1), .. Rm problem data: dynamics and input matricesA Rn n,B Rn m convex stage cost function :Rn Rm R, (0,0) = 0 convex state and input constraint setsX,U, with0 X,0 U initial statez XProf. S. Boyd, EE364b, Stanford University1 Greedy Control useu(t) = argminw{ (x(t), w)|w U, Ax(t) +Bw X} minimizes current stage cost only, ignoring effect ofu(t)on future,except forx(t+ 1) X typically works very poorly; can lead toJ= (when optimalugivesfiniteJ)Prof. S. Boyd, EE364b, Stanford University2 Solution via dynamic programming (Bellman)value functionV(z)is optimal value of Control problem as afunction of initial statez can showVis convex Vsatisfies Bellman or dynamic programming equationV(z) = inf{ (z, w) +V(Az+Bw)|w U, Az+Bw X} optimalugiven byu (t) =argminw U, Ax(t)+Bw X( (x(t), w) +V(Ax(t) +Bw))Prof.

• MPC problem is highly structured (see Convex Optimization, §10.3.4) – Hessian is block diagonal – equality constraint matrix is block banded • use block elimination to compute Newton step – Schur complement is block tridiagonal with n×n blocks • can solve in order T(n+m)3 flops using an interior point method

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  Control, Predictive, Predictive control

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