[37] | 1 | yalmip('clear') |
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| 2 | %clear all |
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| 3 | |
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| 4 | % Data |
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| 5 | A = [2 -1;1 0];nx = 2; |
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| 6 | B = [1;0];nu = 1; |
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| 7 | C = [0.5 0.5]; |
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| 8 | |
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| 9 | % Prediction horizon |
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| 10 | N = 3; |
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| 11 | |
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| 12 | % States x(k), ..., x(k+N) |
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| 13 | x = sdpvar(repmat(nx,1,N),repmat(1,1,N)); |
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| 14 | |
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| 15 | % Inputs u(k), ..., u(k+N) (last one not used) |
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| 16 | u = sdpvar(repmat(nu,1,N),repmat(1,1,N)); |
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| 17 | |
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| 18 | % Value functions |
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| 19 | J = cell(1,N); |
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| 20 | |
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| 21 | % Initialize value function at stage N |
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| 22 | J{N} = 0; |
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| 23 | U{N} = 0; |
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| 24 | k=N-1; |
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| 25 | |
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| 26 | for k = N-1:-1:1 |
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| 27 | |
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| 28 | % Feasible region |
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| 29 | bounds(x{k},-5,5); |
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| 30 | bounds(u{k},-1,1); |
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| 31 | bounds(x{k+1},-5,5); |
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| 32 | |
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| 33 | F = set(-1 < u{k} < 1); |
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| 34 | F = F + set(-1 < C*x{k} < 1); |
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| 35 | F = F + set(-5 < x{k} < 5); |
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| 36 | F = F + set(-1 < C*x{k+1} < 1); |
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| 37 | F = F + set(-5 < x{k+1} < 5); |
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| 38 | |
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| 39 | % LTI Dynamics |
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| 40 | F = F + set(x{k+1} == A*x{k}+B*u{k}); |
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| 41 | |
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| 42 | obj = norm([x{k};u{k}],1); |
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| 43 | |
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| 44 | % Compute value function for one step backwards |
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| 45 | [mpsol{k},sol{k},Uz{k},J{k}] = solvemp(F,obj + J{k+1},[],x{k},u{k}); |
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| 46 | end |
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| 47 | |
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| 48 | % MPT implementation |
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| 49 | sysStruct.A= A; |
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| 50 | sysStruct.B= B; |
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| 51 | sysStruct.C= C; |
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| 52 | sysStruct.D= [0]; |
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| 53 | |
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| 54 | %set constraints on output |
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| 55 | sysStruct.ymin = -1; |
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| 56 | sysStruct.ymax = 1; |
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| 57 | |
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| 58 | %set constraints on input |
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| 59 | sysStruct.umin = -1; |
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| 60 | sysStruct.umax = 1; |
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| 61 | |
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| 62 | sysStruct.xmin = [-5;-5]; |
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| 63 | sysStruct.xmax = [5;5]; |
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| 64 | |
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| 65 | probStruct.norm=1; |
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| 66 | probStruct.Q=eye(2); |
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| 67 | probStruct.R=1; |
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| 68 | probStruct.N=N-1; |
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| 69 | probStruct.P_N=zeros(2); |
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| 70 | probStruct.subopt_lev=0; |
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| 71 | probStruct.y0bounds=1; |
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| 72 | probStruct.Tconstraint=0; |
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| 73 | ctrl=mpt_control(sysStruct,probStruct) |
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| 74 | mpt_isPWAbigger(mpsol{1}{1},ctrl) |
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| 75 | |
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| 76 | |
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| 77 | |
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| 78 | |
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| 79 | |
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| 80 | break |
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| 81 | |
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| 82 | |
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| 83 | |
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| 84 | |
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| 85 | |
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| 86 | % States x(k), ..., x(k+N) |
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| 87 | x = sdpvar(repmat(nx,1,N),repmat(1,1,N)); |
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| 88 | |
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| 89 | % Inputs u(k), ..., u(k+N) (last one not used) |
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| 90 | u = sdpvar(repmat(nu,1,N),repmat(1,1,N)); |
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| 91 | |
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| 92 | % Value functions |
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| 93 | J = cell(1,N); |
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| 94 | |
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| 95 | % Initialize value function at stage N |
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| 96 | J{N} = 0; |
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| 97 | F = set([]); |
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| 98 | obj = 0; |
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| 99 | for k = N-1:-1:1 |
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| 100 | % Feasible region |
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| 101 | F = F + set(-1 < u{k} < 1); |
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| 102 | F = F + set(-1 < C*x{k} < 1); |
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| 103 | F = F + set(-5 < x{k} < 5); |
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| 104 | F = F + set(-1 < C*x{k+1} < 1); |
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| 105 | F = F + set(-5 < x{k+1} < 5); |
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| 106 | |
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| 107 | % LTI Dynamics |
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| 108 | F = F + set(x{k+1} == A*x{k}+B*u{k}); |
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| 109 | |
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| 110 | % Compute value function for one step backwards |
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| 111 | % obj = obj + norm([x{k};u{k}],1); |
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| 112 | obj = obj + x{k}'*x{k}+ u{k}'*u{k};%;u{k}],1); |
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| 113 | |
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| 114 | end |
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| 115 | [mpsol2{k},sol2{k},Uz2{k},J2{k},U2{k}] = solvemp(F,obj,[],x{k},u{k}); |
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| 116 | |
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| 117 | mpt_isPWAbigger(mpsol{1}{1},mpsol2{1}{1}) |
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| 118 | |
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| 119 | assign(x{k},[1;0.5]) |
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| 120 | double(J{k}) |
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| 121 | solvesdp(F+set(x{1} == [1;0.5]),obj); |
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| 122 | |
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| 123 | |
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| 124 | |
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| 125 | |
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| 126 | |
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