[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 | B1 = [1;0]; |
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| 7 | B2 = pi*B1; |
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| 8 | nu = 1; |
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| 9 | C = [0.5 0.5]; |
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| 10 | |
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| 11 | % Prediction horizon |
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| 12 | N = 4; |
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| 13 | |
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| 14 | % States x(k), ..., x(k+N) |
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| 15 | x = sdpvar(repmat(nx,1,N),repmat(1,1,N)); |
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| 16 | |
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| 17 | % Inputs u(k), ..., u(k+N) (last one not used) |
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| 18 | u = sdpvar(repmat(nu,1,N),repmat(1,1,N)); |
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| 19 | |
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| 20 | % Binary for PWA selection |
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| 21 | d = binvar(repmat(2,1,N),repmat(1,1,N)); |
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| 22 | |
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| 23 | % Value functions |
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| 24 | J = cell(1,N); |
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| 25 | |
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| 26 | % Initialize value function at stage N |
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| 27 | J{N} = 0; |
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| 28 | |
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| 29 | % DP iteration from final state |
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| 30 | for k = N-1:-1:1 |
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| 31 | |
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| 32 | % Control constraints |
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| 33 | F = set(-1 < u{k} < 1); |
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| 34 | |
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| 35 | % We are in a feasible region... |
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| 36 | F = F + set(-1 < C*x{k} < 1); |
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| 37 | F = F + set(-5 < x{k} < 5); |
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| 38 | |
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| 39 | % ...and go to a feasible region |
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| 40 | F = F + set(-1 < C*x{k+1} < 1); |
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| 41 | F = F + set(-5 < x{k+1} < 5); |
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| 42 | |
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| 43 | % PWA Dynamics |
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| 44 | F = F + set(implies(d{k}(1),[x{k+1} == A*x{k}+B1*u{k}, x{k}(1) > 0])); |
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| 45 | F = F + set(implies(d{k}(2),[x{k+1} == A*x{k}+B2*u{k}, x{k}(1) < 0])); |
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| 46 | F = F + set(sum(d{k}) == 1); |
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| 47 | |
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| 48 | % L1 objective |
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| 49 | obj = norm(x{k},1) + norm(u{k},1); |
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| 50 | |
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| 51 | [mpsol{k},sol{k},aux{k},J{k},U{k}] = solvemp(F,obj + J{k+1},[],x{k},u{k}); |
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| 52 | |
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| 53 | end |
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| 54 | |
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| 55 | |
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| 56 | |
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| 57 | |
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| 58 | |
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| 59 | break |
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| 60 | |
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| 61 | |
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| 62 | |
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| 63 | break |
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| 64 | mpsol{1} = mpt_removeOverlaps(mpsol{1}) |
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| 65 | |
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| 66 | % Compare |
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| 67 | sysStruct.A{1} = A; |
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| 68 | sysStruct.B{1} = B1; |
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| 69 | sysStruct.C{1} = C; |
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| 70 | sysStruct.D{1} = [0]; |
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| 71 | sysStruct.A{2} = A; |
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| 72 | sysStruct.B{2} = B2; |
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| 73 | sysStruct.C{2} = C; |
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| 74 | sysStruct.D{2} = [0]; |
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| 75 | sysStruct.guardX{1} = [-1 0]; |
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| 76 | sysStruct.guardU{1} = [0]; |
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| 77 | sysStruct.guardC{1} = [0]; |
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| 78 | sysStruct.guardX{2} = [1 0]; |
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| 79 | sysStruct.guardU{2} = [0]; |
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| 80 | sysStruct.guardC{2} = [0]; |
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| 81 | |
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| 82 | %set constraints on output |
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| 83 | sysStruct.ymin = -1; |
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| 84 | sysStruct.ymax = 1; |
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| 85 | |
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| 86 | %set constraints on input |
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| 87 | sysStruct.umin = -1; |
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| 88 | sysStruct.umax = 1; |
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| 89 | |
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| 90 | sysStruct.xmin = [-5;-5]; |
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| 91 | sysStruct.xmax = [5;5]; |
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| 92 | |
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| 93 | probStruct.norm=1; |
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| 94 | probStruct.Q=eye(2); |
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| 95 | probStruct.R=1; |
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| 96 | probStruct.N=N-1; |
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| 97 | probStruct.P_N=zeros(2); |
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| 98 | probStruct.subopt_lev=0; |
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| 99 | probStruct.y0bounds=1; |
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| 100 | probStruct.Tconstraint=0; |
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| 101 | ctrl=mpt_control(sysStruct,probStruct) |
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| 102 | |
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| 103 | mpt_isPWAbigger(ctrl,mpsol{1}) |
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| 104 | break |
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| 105 | %[ii,jj] = isinside(ctrl.Pn,[1.2;0.8]); |
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| 106 | %ctrl.Bi{jj}*[1.2;0.8]+ctrl.Ci{jj} |
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| 107 | |
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| 108 | % |
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| 109 | % |
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| 110 | % Online |
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| 111 | obj = 0; |
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| 112 | F = set([]); |
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| 113 | dd = []; |
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| 114 | for k = N-1:-1:1 |
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| 115 | |
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| 116 | % Feasible region |
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| 117 | F = F + set(-1 < u{k} < 1); |
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| 118 | F = F + set(-1 < C*x{k} < 1); |
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| 119 | F = F + set(-5 < x{k} < 5); |
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| 120 | F = F + set(-1 < C*x{k+1} < 1); |
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| 121 | F = F + set(-5 < x{k+1} < 5); |
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| 122 | |
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| 123 | % PWA Dynamics |
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| 124 | d = binvar(2,1);dd = [dd;d]; |
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| 125 | F = F + set(implies(d(1),x{k+1} == (A*x{k}+B1*u{k}))); |
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| 126 | F = F + set(implies(d(2),x{k+1} == (A*x{k}+B2*u{k}))); |
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| 127 | F = F + set(implies(d(1),x{k}(1) > 0)); |
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| 128 | F = F + set(implies(d(2),x{k}(1) < 0)); |
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| 129 | F = F + set(sum(d) == 1); |
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| 130 | |
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| 131 | F1 = set(x{k+1} == (A*x{k}+B1*u{k})) + set(x{k}(1) > 0); |
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| 132 | F2 = set(x{k+1} == (A*x{k}+B2*u{k})) + set(x{k}(1) < 0); |
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| 133 | |
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| 134 | %F = F + hull(F1,F2); |
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| 135 | |
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| 136 | % obj = obj + norm([x{k};u{k}],1); |
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| 137 | obj = obj + norm([x{k};u{k}],1); |
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| 138 | % obj = obj + norm([x{k};u{k}],1); |
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| 139 | % Compute value function for one step backwards |
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| 140 | end |
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| 141 | mpsol2{k} = solvemp(F,obj,[],x{k},u{k}); |
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| 142 | mpsol3{k} = dua_test(F,obj,[],x{k}); |
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| 143 | solvesdp(F+set(x{k}==[0.5;0.5]),obj) |
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| 144 | solvesdp(F+set(x{k}==[1.2;0.8]),obj) |
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| 145 | mpsol{k} = solvemp(F,obj,[],x{k},u); |
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| 146 | mpsol2{1} = rmovlps(mpsol2{1}); |
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| 147 | |
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| 148 | mpsol1{k} = solvemp(F,obj,[],x{k},u{k}); |
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| 149 | % |
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