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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174 | |
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175 | |
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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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