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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