[37] | 1 | yalmip('clear') |
---|
| 2 | %clear all |
---|
| 3 | |
---|
| 4 | % Data |
---|
| 5 | A = [2 -1;1 0];nx = 2; |
---|
| 6 | B = [1;0];nu = 1; |
---|
| 7 | C = [0.5 0.5]; |
---|
| 8 | |
---|
| 9 | % Prediction horizon |
---|
| 10 | N = 3; |
---|
| 11 | |
---|
| 12 | % States x(k), ..., x(k+N) |
---|
| 13 | x = sdpvar(repmat(nx,1,N),repmat(1,1,N)); |
---|
| 14 | |
---|
| 15 | % Inputs u(k), ..., u(k+N) (last one not used) |
---|
| 16 | u = sdpvar(repmat(nu,1,N),repmat(1,1,N)); |
---|
| 17 | |
---|
| 18 | % Value functions |
---|
| 19 | J = cell(1,N); |
---|
| 20 | |
---|
| 21 | % Initialize value function at stage N |
---|
| 22 | J{N} = 0; |
---|
| 23 | U{N} = 0; |
---|
| 24 | for k = N-1:-1:1 |
---|
| 25 | |
---|
| 26 | % Feasible region |
---|
| 27 | bounds(x{k},-5,5); |
---|
| 28 | bounds(u{k},-1,1); |
---|
| 29 | bounds(x{k+1},-5,5); |
---|
| 30 | |
---|
| 31 | F = set(-1 < u{k} < 1); |
---|
| 32 | F = F + set(-1 < C*x{k} < 1); |
---|
| 33 | F = F + set(-5 < x{k} < 5); |
---|
| 34 | F = F + set(-1 < C*x{k+1} < 1); |
---|
| 35 | F = F + set(-5 < x{k+1} < 5); |
---|
| 36 | |
---|
| 37 | % LTI Dynamics |
---|
| 38 | F = F + set(x{k+1} == A*x{k}+B*u{k}); |
---|
| 39 | |
---|
| 40 | obj = norm(x{k},inf) + norm(u{k},inf); |
---|
| 41 | |
---|
| 42 | % Compute value function for one step backwards |
---|
| 43 | % [mpsol{k},sol{k},Uz{k},J{k},U{k}] = solvemp(F,max(obj,J{k+1}),[],x{k},u{k}); |
---|
| 44 | [mpsol{k},sol{k},Uz{k},J{k},U{k}] = solvemp(F,obj+J{k+1},[],x{k},u{k}); |
---|
| 45 | end |
---|
| 46 | |
---|
| 47 | |
---|
| 48 | % MPT implementation |
---|
| 49 | sysStruct.A= A; |
---|
| 50 | sysStruct.B= B; |
---|
| 51 | sysStruct.C= C; |
---|
| 52 | sysStruct.D= [0]; |
---|
| 53 | |
---|
| 54 | %set constraints on output |
---|
| 55 | sysStruct.ymin = -1; |
---|
| 56 | sysStruct.ymax = 1; |
---|
| 57 | |
---|
| 58 | %set constraints on input |
---|
| 59 | sysStruct.umin = -1; |
---|
| 60 | sysStruct.umax = 1; |
---|
| 61 | |
---|
| 62 | sysStruct.xmin = [-5;-5]; |
---|
| 63 | sysStruct.xmax = [5;5]; |
---|
| 64 | |
---|
| 65 | probStruct.norm=inf; |
---|
| 66 | probStruct.Q=eye(2); |
---|
| 67 | probStruct.R=1; |
---|
| 68 | probStruct.N=N-1; |
---|
| 69 | probStruct.P_N=zeros(2); |
---|
| 70 | probStruct.subopt_lev=0; |
---|
| 71 | probStruct.y0bounds=1; |
---|
| 72 | probStruct.Tconstraint=0; |
---|
| 73 | ctrl=mpt_control(sysStruct,probStruct) |
---|
| 74 | mpt_isPWAbigger(mpsol{1}{1},ctrl) |
---|
| 75 | |
---|
| 76 | |
---|
| 77 | |
---|
| 78 | |
---|
| 79 | |
---|
| 80 | break |
---|
| 81 | |
---|
| 82 | |
---|
| 83 | |
---|
| 84 | |
---|
| 85 | |
---|
| 86 | % Prediction horizon |
---|
| 87 | N = 5; |
---|
| 88 | |
---|
| 89 | % States x(k), ..., x(k+N) |
---|
| 90 | x = sdpvar(repmat(nx,1,N),repmat(1,1,N)); |
---|
| 91 | |
---|
| 92 | % Inputs u(k), ..., u(k+N) (last one not used) |
---|
| 93 | u = sdpvar(repmat(nu,1,N),repmat(1,1,N)); |
---|
| 94 | |
---|
| 95 | % Value functions |
---|
| 96 | J = cell(1,N); |
---|
| 97 | |
---|
| 98 | % Initialize value function at stage N |
---|
| 99 | J{N} = 0; |
---|
| 100 | F = set([]); |
---|
| 101 | obj = 0; |
---|
| 102 | for k = N-1:-1:1 |
---|
| 103 | % Feasible region |
---|
| 104 | F = F + set(-1 < u{k} < 1); |
---|
| 105 | F = F + set(-1 < C*x{k} < 1); |
---|
| 106 | F = F + set(-5 < x{k} < 5); |
---|
| 107 | F = F + set(-1 < C*x{k+1} < 1); |
---|
| 108 | F = F + set(-5 < x{k+1} < 5); |
---|
| 109 | |
---|
| 110 | % LTI Dynamics |
---|
| 111 | F = F + set(x{k+1} == A*x{k}+B*u{k}); |
---|
| 112 | |
---|
| 113 | % Compute value function for one step backwards |
---|
| 114 | obj = max(obj,norm([x{k};u{k}],inf)); |
---|
| 115 | |
---|
| 116 | end |
---|
| 117 | %F = F + set(norm(x{k+1},1) < norm(x{k},1)) |
---|
| 118 | solvesdp(F+set(x{1}==[0.5;-1]),obj) |
---|
| 119 | |
---|
| 120 | [mpsol{k},xx,cc,vv,U{k}] = solvemp(F,obj,[],x{k},u{k}); |
---|
| 121 | |
---|
| 122 | |
---|
| 123 | |
---|
| 124 | |
---|
| 125 | assign(x{k},[-0.5;-1]); |
---|
| 126 | xx = [-0.5;-1]; |
---|
| 127 | for i = 1:10 |
---|
| 128 | xx = A*xx + B*double(U{k}) |
---|
| 129 | assign(x{k},xx); |
---|
| 130 | double(x{k}) |
---|
| 131 | plot(xx(1),xx(2),'*k');drawnow |
---|
| 132 | end |
---|
| 133 | |
---|
| 134 | |
---|
| 135 | |
---|
| 136 | |
---|