Transcription of Model Predictive Control - Stanford University
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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.
Model Predictive Control • linear convex optimal control • finite horizon approximation • model predictive control • fast MPC implementations ... • under some conditions, can give performance guarantees for MPC Prof. S. Boyd, EE364b, Stanford University 13. Variations on MPC
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