[37] | 1 | echo on |
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| 2 | clc |
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| 3 | % This example shows how to (locally) solve a small BMI using |
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| 4 | % a simple linearization-based algorithm. |
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| 5 | % |
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| 6 | % The main motivation of this example is not to describe a |
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| 7 | % particularily efficient solver, but to describe how easily |
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| 8 | % a BMI solver can be implemented using high-level YALMIP code. |
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| 9 | % |
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| 10 | % The problem is to find a feedback u = Lx so that the |
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| 11 | % L2 gain from w to y is minimized, for the system |
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| 12 | % |
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| 13 | % x' = Ax+Bu+Gw |
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| 14 | % y = Cx |
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| 15 | % |
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| 16 | % This can be formulated as the BMI |
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| 17 | % |
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| 18 | % min t |
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| 19 | % s.t P > 0 |
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| 20 | % [(A+BL)P+P(A+BL)+C'C PG; G'P -tI] < 0 |
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| 21 | % |
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| 22 | pause |
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| 23 | clc |
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| 24 | |
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| 25 | % Create system data |
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| 26 | A = [-1 -1 -1; |
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| 27 | 1 0 0; |
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| 28 | 0 1 0]; |
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| 29 | B = [1;0;0]; |
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| 30 | C = [0 0 1]; |
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| 31 | G = [-1;-1;-1]; |
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| 32 | |
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| 33 | % Define decision variables |
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| 34 | P = sdpvar(3,3); |
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| 35 | L = sdpvar(1,3); |
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| 36 | t = sdpvar(1,1); |
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| 37 | pause |
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| 38 | |
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| 39 | % A reasonble initial guess is valuable |
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| 40 | [L0,P0]=lqr(A,B,eye(3),1); |
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| 41 | setsdpvar(P,P0); |
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| 42 | setsdpvar(L,-L0); |
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| 43 | setsdpvar(t,100); |
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| 44 | pause |
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| 45 | clc |
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| 46 | |
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| 47 | % Recover all involved decision variables |
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| 48 | % in one single variable (simplifies code later) |
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| 49 | x = recover(getvariables([P(:);L(:);t])) |
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| 50 | pause |
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| 51 | |
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| 52 | % Define the nonlinear matrix (simplifies the code) |
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| 53 | H = -[(A+B*L)'*P+P*(A+B*L)+C'*C P*G;G'*P -t]; |
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| 54 | pause |
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| 55 | |
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| 56 | % Save old iterates and old objective function |
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| 57 | x0 = double(x); |
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| 58 | t0 = double(t); |
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| 59 | pause |
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| 60 | |
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| 61 | % Linearized constraints |
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| 62 | F = set(linearize(H)>0) + set(P>0) |
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| 63 | pause |
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| 64 | |
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| 65 | % add a trust region |
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| 66 | F = F + set(cone(x-x0,0.5*norm(x0))); |
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| 67 | pause |
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| 68 | |
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| 69 | % Solve linearized problem |
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| 70 | solvesdp(F,t) |
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| 71 | pause |
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| 72 | |
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| 73 | % Optimal decision variables for linearized problem |
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| 74 | xnew = double(x); |
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| 75 | pause |
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| 76 | |
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| 77 | % Original problem is not guaranteed to be feasible |
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| 78 | % Line-search for feasible (and improving) solution |
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| 79 | pause |
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| 80 | alpha = 1; |
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| 81 | while (min(eig(double(H)))<0) | (min(eig(double(P)))<0) | (double(t)>t0*0.9999) |
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| 82 | alpha = alpha*0.5; |
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| 83 | setsdpvar(x,x0+alpha*(xnew-x0)); |
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| 84 | end |
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| 85 | |
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| 86 | % Current (squared) gain |
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| 87 | double(t) |
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| 88 | pause |
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| 89 | |
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| 90 | % repeat.... |
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| 91 | % |
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| 92 | % for i = 1:15 |
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| 93 | % |
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| 94 | % % Save old iterates and old objective function |
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| 95 | % x0 = double(x); |
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| 96 | % t0 = double(t); |
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| 97 | % |
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| 98 | % % Linearized constraints |
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| 99 | % F = set(linearize(H)>0) + set(P>0); |
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| 100 | % % add a trust region |
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| 101 | % F = F + set(cone(x-x0,0.25*norm(x0))); |
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| 102 | % |
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| 103 | % % Solve linearized problem |
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| 104 | % solvesdp(F,t,sdpsettings('verbose',0)); |
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| 105 | % |
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| 106 | % % Optimal decision variables for linearized problem |
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| 107 | % xnew = double(x); |
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| 108 | % |
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| 109 | % % Original problem is not guaranteed to be feasible |
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| 110 | % % Line-search for feasible (and improving) solution |
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| 111 | % alpha = 1; |
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| 112 | % while (min(eig(double(H)))<0) | (double(t)>t0*0.9999) |
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| 113 | % alpha = alpha*0.5; |
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| 114 | % setsdpvar(x,x0+alpha*(xnew-x0)); |
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| 115 | % end |
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| 116 | % double(t) |
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| 117 | % end |
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| 118 | pause |
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| 119 | |
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| 120 | echo off |
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| 121 | for i = 1:15 |
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| 122 | % Save old iterates and old objective function |
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| 123 | x0 = double(x); |
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| 124 | t0 = double(t); |
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| 125 | |
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| 126 | % Linearized constraints |
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| 127 | F = set(linearize(H)>0) + set(P>0); |
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| 128 | % add a trust region |
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| 129 | F = F + set(cone(x-x0,0.5*norm(x0))); |
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| 130 | |
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| 131 | % Solve linearized problem |
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| 132 | solvesdp(F,t,sdpsettings('verbose',0)); |
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| 133 | |
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| 134 | % Optimal decision variables for linearized problem |
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| 135 | xnew = double(x); |
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| 136 | |
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| 137 | % Original problem is not guaranteed to be feasible |
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| 138 | % Line-search for feasible (and improving) solution |
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| 139 | alpha = 1; |
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| 140 | while (min(eig(double(H)))<0) | (min(eig(double(P)))<0) | (double(t)>t0*0.9999) |
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| 141 | alpha = alpha*0.5; |
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| 142 | setsdpvar(x,x0+alpha*(xnew-x0)); |
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| 143 | end |
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| 144 | disp(['#' num2str(i) ' L2 gain : ' num2str(double(t))]) |
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| 145 | end |
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| 146 | clc |
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| 147 | echo on |
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| 148 | % An alternativ is to work with the complete set of |
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| 149 | % constraints by linearizing the LMI object |
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| 150 | % |
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| 151 | % Define the constraints |
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| 152 | |
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| 153 | F = set(H>0) + set(P>0) |
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| 154 | pause |
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| 155 | % Solve linearized problem |
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| 156 | solvesdp(linearize(F),t) |
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| 157 | while (min(eig(double(H)))<0) | (double(t)>t0*0.9999) |
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| 158 | alpha = alpha*0.5; |
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| 159 | setsdpvar(x,x0+alpha*(xnew-x0)); |
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| 160 | end |
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| 161 | double(t) |
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| 162 | |
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| 163 | pause |
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| 164 | % The rest is done similarily... |
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| 165 | pause |
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| 166 | clc |
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| 167 | |
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| 168 | % In fact, all this can be done automatically |
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| 169 | % by using the solver bmilin (coded using high-level YALMIP) |
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| 170 | pause |
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| 171 | solvesdp(F,t,sdpsettings('solver','bmilin','usex0',1)); |
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| 172 | pause |
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| 173 | echo off |
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| 174 | |
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| 175 | |
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