[37] | 1 | yalmip('clear') |
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| 2 | |
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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 | % Future state |
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| 13 | % Now for two different noises |
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| 14 | x1 = sdpvar(repmat(nx,1,N),repmat(1,1,N)); |
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| 15 | x2 = sdpvar(repmat(nx,1,N),repmat(1,1,N)); |
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| 16 | |
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| 17 | % Current state |
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| 18 | x = sdpvar(repmat(nx,1,N),repmat(1,1,N)); |
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| 19 | |
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| 20 | % Inputs u(k), ..., u(k+N) (last one not used) |
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| 21 | u = sdpvar(repmat(nu,1,N),repmat(1,1,N)); |
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| 22 | v = sdpvar(repmat(nu,1,N),repmat(1,1,N)); |
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| 23 | |
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| 24 | % Binary for PWA selection |
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| 25 | d = binvar(2,1); |
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| 26 | |
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| 27 | % Value functions |
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| 28 | J = cell(1,N); |
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| 29 | |
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| 30 | % Initialize value function at stage N |
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| 31 | J{N} = 0; |
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| 32 | J1{N} = pwa(norm(x1{N},1),set(-10<x1{N}(1)<10)); |
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| 33 | J2{N} = pwa(norm(x2{N},1),set(-10<x2{N}(1)<10)); |
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| 34 | |
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| 35 | t = sdpvar(nx+nu,1); |
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| 36 | bounds(t,0,600); |
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| 37 | k = N-1 |
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| 38 | for k = N-1:-1:1 |
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| 39 | |
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| 40 | bounds(x{k},-5,5); |
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| 41 | bounds(u{k},-1,1); |
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| 42 | bounds(x1{k+1},-5,5); |
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| 43 | bounds(x2{k+1},-5,5); |
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| 44 | |
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| 45 | % Feasible region |
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| 46 | F = set(-1 < u{k} < 1); |
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| 47 | F = set(-1 < u{k}+v{k} < 1); |
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| 48 | F = F + set(-1 < C*x{k} < 1); |
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| 49 | F = F + set(-5 < x{k} < 5); |
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| 50 | |
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| 51 | F = F + set(-1 < C*x1{k+1} < 1); |
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| 52 | F = F + set(-1 < C*x2{k+1} < 1); |
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| 53 | F = F + set(-5 < x1{k} < 5); |
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| 54 | F = F + set(-5 < x2{k} < 5); |
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| 55 | |
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| 56 | % Two possible extreme predictions |
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| 57 | F = F + set(x1{k+1} == A*x{k}+B*u{k}); |
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| 58 | F = F + set(x2{k+1} == pi*A*x{k}+B*(u{k}+v{k})); |
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| 59 | |
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| 60 | F = F + set(-t < [x{k};u{k}] < t) ; |
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| 61 | |
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| 62 | if k<N-1 |
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| 63 | % Create two value functions, minimize worst case |
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| 64 | J1{k+1} = pwf(mpsol{k+1},x1{k+1},'convex'); |
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| 65 | J2{k+1} = pwf(mpsol{k+1},x2{k+1},'convex'); |
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| 66 | sdpvar w |
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| 67 | F = F + set(J1{k+1} < w) + set(J2{k+1} < w); |
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| 68 | obj = sum(t) + w; |
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| 69 | else |
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| 70 | % J1{N} = 0; |
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| 71 | % J2{N} = 0; |
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| 72 | sdpvar w |
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| 73 | F = F + set(J1{k+1} < w) + set(J2{k+1} < w); |
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| 74 | obj = sum(t)+w; |
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| 75 | end |
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| 76 | [mpsol{k},sol{k},Uz{k},J{k}] = solvemp(F,obj,[],x{k},u{k}); |
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| 77 | end |
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| 78 | |
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| 79 | |
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| 80 | mpsol{k} = rmovlps(mpsol{k}) |
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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 | J{N} = pwa(norm(x{N},1),set(-10<x{N}(1)<10)); |
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| 87 | t = sdpvar(nx+nu,1); |
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| 88 | bounds(t,0,600); |
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| 89 | k = N-1 |
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| 90 | for k = N-1:-1:1 |
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| 91 | |
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| 92 | bounds(x{k},-5,5); |
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| 93 | bounds(u{k},-1,1); |
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| 94 | bounds(x{k+1},-5,5); |
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| 95 | % Feasible region |
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| 96 | F = set(-1 < u{k} < 1); |
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| 97 | F = F + set(-1 < C*x{k} < 1); |
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| 98 | F = F + set(-5 < x{k} < 5); |
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| 99 | % Two possible extreme predictions |
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| 100 | F = F + set(x{k+1} == A*x{k}+B*u{k}); |
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| 101 | F = F + set(-t < [x{k};u{k}] < t) ; |
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| 102 | obj = sum(t)+J{k+1}; |
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| 103 | [mpsol1{k}] = solvemp(F,obj,[],x{k},u{k}); |
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| 104 | end |
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| 105 | |
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| 106 | J{N} = pwa(norm(x{N},1),set(-10<x{N}(1)<10)); |
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| 107 | t = sdpvar(nx+nu,1); |
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| 108 | bounds(t,0,600); |
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| 109 | k = N-1 |
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| 110 | for k = N-1:-1:1 |
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| 111 | |
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| 112 | bounds(x{k},-5,5); |
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| 113 | bounds(u{k},-1,1); |
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| 114 | bounds(x{k+1},-5,5); |
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| 115 | % Feasible region |
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| 116 | F = set(-1 < u{k} < 1); |
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| 117 | F = F + set(-1 < C*x{k} < 1); |
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| 118 | F = F + set(-5 < x{k} < 5); |
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| 119 | % Two possible extreme predictions |
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| 120 | F = F + set(x{k+1} == pi*A*x{k}+B*u{k}); |
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| 121 | F = F + set(-t < [x{k};u{k}] < t) ; |
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| 122 | obj = sum(t)+J{k+1}; |
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| 123 | [mpsol2{k}] = solvemp(F,obj,[],x{k},u{k}); |
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| 124 | end |
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| 125 | |
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| 126 | |
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| 127 | |
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| 128 | |
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| 129 | |
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| 130 | |
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| 131 | J{N} = pwa(norm(x{N},1),set(-10<x{N}(1)<10)); |
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| 132 | bounds(x{k},-5,5); |
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| 133 | bounds(u{k},-1,1); |
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| 134 | bounds(x{k+1},-5,5); |
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| 135 | % Feasible region |
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| 136 | F = set(-1 < u{k} < 1); |
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| 137 | F = F + set(-1 < C*x{k} < 1); |
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| 138 | F = F + set(-5 < x{k} < 5); |
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| 139 | % Two possible extreme predictions |
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| 140 | F = F + set(x{k+1} == pi*A*x{k}+B*u{k}); |
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| 141 | F = F + set(-t < [x{k};u{k}] < t) ; |
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| 142 | |
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| 143 | obj = sum(t)+J{k+1}; |
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| 144 | [mpsol2{k}] = solvemp(F,obj,[],x{k},u{k}); |
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| 176 | |
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| 177 | break |
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| 178 | |
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| 179 | |
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| 180 | % Compare |
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| 181 | sysStruct.A{1} = A; |
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| 182 | sysStruct.B{1} = B; |
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| 183 | sysStruct.C{1} = C; |
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| 184 | sysStruct.D{1} = [0]; |
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| 185 | sysStruct.A{2} = A; |
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| 186 | sysStruct.B{2} = B*pi; |
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| 187 | sysStruct.C{2} = C; |
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| 188 | sysStruct.D{2} = [0]; |
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| 189 | sysStruct.guardX{1} = [-1 0]; |
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| 190 | sysStruct.guardU{1} = [0]; |
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| 191 | sysStruct.guardC{1} = [0]; |
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| 192 | sysStruct.guardX{2} = [1 0]; |
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| 193 | sysStruct.guardU{2} = [0]; |
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| 194 | sysStruct.guardC{2} = [0]; |
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| 195 | |
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| 196 | %set constraints on output |
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| 197 | sysStruct.ymin = -1; |
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| 198 | sysStruct.ymax = 1; |
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| 199 | |
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| 200 | %set constraints on input |
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| 201 | sysStruct.umin = -1; |
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| 202 | sysStruct.umax = 1; |
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| 203 | |
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| 204 | sysStruct.xmin = [-5;-5]; |
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| 205 | sysStruct.xmax = [5;5]; |
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| 206 | |
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| 207 | probStruct.norm=1; |
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| 208 | probStruct.Q=eye(2); |
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| 209 | probStruct.R=1; |
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| 210 | probStruct.N=N-1; |
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| 211 | probStruct.P_N=zeros(2); |
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| 212 | probStruct.subopt_lev=0; |
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| 213 | probStruct.y0bounds=1; |
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| 214 | probStruct.Tconstraint=0; |
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| 215 | ctrl=mpt_control(sysStruct,probStruct) |
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| 216 | |
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| 217 | mpt_isPWAbigger(ctrl,mpsol{1}) |
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| 218 | break |
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| 219 | %[ii,jj] = isinside(ctrl.Pn,[1.2;0.8]); |
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| 220 | %ctrl.Bi{jj}*[1.2;0.8]+ctrl.Ci{jj} |
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| 221 | |
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| 222 | % |
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| 223 | % |
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| 224 | % % Online |
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| 225 | % obj = 0; |
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| 226 | % F = set([]); |
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| 227 | % dd = []; |
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| 228 | % for k = N-1:-1:1 |
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| 229 | % |
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| 230 | % bounds(x{k},-5,5); |
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| 231 | % bounds(u{k},-1,1); |
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| 232 | % bounds(x{k+1},-5,5); |
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| 233 | % |
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| 234 | % % Feasible region |
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| 235 | % F = F + set(-1 < u{k} < 1); |
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| 236 | % F = F + set(-1 < C*x{k} < 1); |
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| 237 | % F = F + set(-5 < x{k} < 5); |
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| 238 | % F = F + set(-1 < C*x{k+1} < 1); |
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| 239 | % F = F + set(-5 < x{k+1} < 5); |
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| 240 | % |
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| 241 | % % PWA Dynamics |
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| 242 | % d = binvar(2,1);dd = [dd;d]; |
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| 243 | % F = F + set(implies(d(1),x{k+1} == (A*x{k}+B*u{k}))); |
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| 244 | % F = F + set(implies(d(2),x{k+1} == (A*x{k}+pi*B*u{k}))); |
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| 245 | % F = F + set(implies(d(1),x{k}(1) > 0)); |
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| 246 | % F = F + set(implies(d(2),x{k}(1) < 0)); |
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| 247 | % F = F + set(sum(d) == 1); |
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| 248 | % % F = F + set(-0.1 < u{k}-u{k+1} < 0.1); |
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| 249 | % obj = obj + norm([x{k};u{k}],1); |
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| 250 | % %obj = obj + x{k}'*x{k}+u{k}'*u{k};%norm([x{k};u{k}],1); |
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| 251 | % % Compute value function for one step backwards |
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| 252 | % end |
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| 253 | % [mpsol2{k},sol{k},Uz{k},J2{k},U{k}] = solvemp(F,obj,[],x{k},u{k}); |
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| 254 | % solvesdp(F+set(x{k}==[0.5;0.5]),obj) |
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| 255 | % solvesdp(F+set(x{k}==[1.2;0.8]),obj) |
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| 256 | % mpsol{k} = solvemp(F,obj,[],x{k},u); |
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| 257 | % mpsol{1} = rmovlps(mpsol{1}); |
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| 258 | % |
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| 259 | % |
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