[37] | 1 | yalmip('clear') |
---|
| 2 | clc |
---|
| 3 | echo on |
---|
| 4 | |
---|
| 5 | %********************************************************* |
---|
| 6 | % |
---|
| 7 | % Sum-of-squares decomposition |
---|
| 8 | % |
---|
| 9 | %********************************************************* |
---|
| 10 | % |
---|
| 11 | % This example shows how to solve some simple SOS-problems. |
---|
| 12 | pause |
---|
| 13 | clc |
---|
| 14 | |
---|
| 15 | % Define variables |
---|
| 16 | x = sdpvar(1,1); |
---|
| 17 | y = sdpvar(1,1); |
---|
| 18 | z = sdpvar(1,1); |
---|
| 19 | pause |
---|
| 20 | |
---|
| 21 | % Define a polynomial to be analyzed |
---|
| 22 | p = 12+y^2-2*x^3*y+2*y*z^2+x^6-2*x^3*z^2+z^4+x^2*y^2; |
---|
| 23 | pause |
---|
| 24 | |
---|
| 25 | % and the corresponding constraint |
---|
| 26 | F = set(sos(p)); |
---|
| 27 | pause |
---|
| 28 | |
---|
| 29 | % Call solvesos to calculate SOS-decomposition. |
---|
| 30 | solvesos(F); |
---|
| 31 | pause |
---|
| 32 | |
---|
| 33 | clc |
---|
| 34 | % Extract decomposition (a lot of spurious terms so the display is messy...) |
---|
| 35 | h = sosd(F); |
---|
| 36 | sdisplay(h) |
---|
| 37 | pause |
---|
| 38 | |
---|
| 39 | % Cleaned display... |
---|
| 40 | |
---|
| 41 | sdisplay(clean(h,1e-4)) |
---|
| 42 | pause |
---|
| 43 | |
---|
| 44 | |
---|
| 45 | % To obtain more sparse solutions, it can sometimes be |
---|
| 46 | % beneficial to minimize the trace of the Gramian Q |
---|
| 47 | % in the decomposition p(x) = v(x)'Qv(x)=h'(x)h(x) |
---|
| 48 | % |
---|
| 49 | % This can be done by specifying sos.traceobj=1 |
---|
| 50 | pause |
---|
| 51 | |
---|
| 52 | solvesos(F,[],sdpsettings('sos.traceobj',1)); |
---|
| 53 | h = sosd(F); |
---|
| 54 | sdisplay(h) % Still a lot of small terms... |
---|
| 55 | pause |
---|
| 56 | |
---|
| 57 | sdisplay(clean(h,1e-4)) % cleaned... |
---|
| 58 | pause |
---|
| 59 | |
---|
| 60 | |
---|
| 61 | % To clean the decomposition from small monomials already in solvesos, |
---|
| 62 | % use the option 'sos.clean'. This will remove terms in chol(Q)v(x) with |
---|
| 63 | % coefficients smaller than sos.clean. |
---|
| 64 | % |
---|
| 65 | % NOTE, this means that the match between p and h'h will be detoriate, since we |
---|
| 66 | % clean h after the SOS computations are done. This should only be used if |
---|
| 67 | % you only want to display the polynomial. |
---|
| 68 | pause |
---|
| 69 | |
---|
| 70 | solvesos(F,[],sdpsettings('sos.traceobj',1,'sos.clean',1e-4)); |
---|
| 71 | h = sosd(F); |
---|
| 72 | sdisplay(h) |
---|
| 73 | pause |
---|
| 74 | |
---|
| 75 | % Is it really a SOS-decomposition |
---|
| 76 | % If so, p-h'*h=0 |
---|
| 77 | pause |
---|
| 78 | sdisplay(p-h'*h) |
---|
| 79 | pause |
---|
| 80 | clc |
---|
| 81 | |
---|
| 82 | % What! Failure!? No, numerical issues. |
---|
| 83 | % Remove all terms smaller than 1e-6 |
---|
| 84 | clean(p-h'*h,1e-6) |
---|
| 85 | pause |
---|
| 86 | |
---|
| 87 | % The largest coefficient in the polynomial |
---|
| 88 | % p-h'*h is displayed in checkset as the |
---|
| 89 | % primal residual for a SOS constraint |
---|
| 90 | checkset(F) |
---|
| 91 | |
---|
| 92 | pause |
---|
| 93 | clc |
---|
| 94 | |
---|
| 95 | clc |
---|
| 96 | % Let us take deeper look at the decomposition |
---|
| 97 | pause |
---|
| 98 | [sol,v,Q,res] = solvesos(F,[],sdpsettings('sos.traceobj',1)); |
---|
| 99 | pause |
---|
| 100 | |
---|
| 101 | % Gramian (Block diagonal due to the feature sos.congruence) |
---|
| 102 | Q{1} |
---|
| 103 | pause |
---|
| 104 | |
---|
| 105 | % Should be positive definite (may depend on your solver due to numerical precision) |
---|
| 106 | eig(Q{1}) |
---|
| 107 | pause |
---|
| 108 | |
---|
| 109 | % Monomials used in decomposition |
---|
| 110 | sdisplay(v{1}) |
---|
| 111 | pause |
---|
| 112 | |
---|
| 113 | % ...numerical mismatch |
---|
| 114 | sdisplay(p-v{1}'*Q{1}*v{1}); |
---|
| 115 | pause |
---|
| 116 | |
---|
| 117 | % Largest coefficient (in absolute value) in the error polynomial p-v'Qv |
---|
| 118 | mismatch = max(abs(getbase(p-v{1}'*Q{1}*v{1}))) |
---|
| 119 | pause |
---|
| 120 | |
---|
| 121 | % Should be the same as reported in checkset. |
---|
| 122 | % (May be different if Q not is positive semidefinite) |
---|
| 123 | checkset(F) |
---|
| 124 | pause |
---|
| 125 | |
---|
| 126 | % The numerical value can easily be extracted also |
---|
| 127 | % from CHECKSET. |
---|
| 128 | mismatch = checkset(F) |
---|
| 129 | pause |
---|
| 130 | |
---|
| 131 | % Also given as 4th output from solvesos |
---|
| 132 | res |
---|
| 133 | pause |
---|
| 134 | |
---|
| 135 | clc |
---|
| 136 | % By studying the diagonal of the Gramian, we see that |
---|
| 137 | % a lot of monomials are not used in the decomposition |
---|
| 138 | % (zero diagonal means that the corresponding row and |
---|
| 139 | % column is zero, hence the corresponding monomial is |
---|
| 140 | % only multiplied by 0) |
---|
| 141 | diag(Q{1}) |
---|
| 142 | pause |
---|
| 143 | |
---|
| 144 | % Let us re-solve the problem, but this time we manually |
---|
| 145 | % specify what monomials to use. |
---|
| 146 | % Since we know that the monomials generated by YALMIP |
---|
| 147 | % are guaranteed to be sufficient, but Q indicates that |
---|
| 148 | % some actually are redundant, let us re-use the old ones, |
---|
| 149 | % but skip those corresponding to small diagonals. |
---|
| 150 | % |
---|
| 151 | % User specified monomials is the fifth input. |
---|
| 152 | usethese = find(diag(Q{1})>1e-3); |
---|
| 153 | pause |
---|
| 154 | [sol,v,Q] = solvesos(F,[],sdpsettings('sos.traceobj',1),[],v{1}(usethese)); |
---|
| 155 | pause |
---|
| 156 | |
---|
| 157 | % The net result is a smaller decomposition |
---|
| 158 | sdisplay(sosd(F)) |
---|
| 159 | pause |
---|
| 160 | |
---|
| 161 | % The problem is better conditioned and leads to smaller residuals |
---|
| 162 | checkset(F) |
---|
| 163 | pause |
---|
| 164 | |
---|
| 165 | % The Gramian is of-course smaller now. |
---|
| 166 | % No zero diagonal, hence we have no simple way to |
---|
| 167 | % obtain a smaller decomposition. |
---|
| 168 | Q{1} |
---|
| 169 | pause |
---|
| 170 | |
---|
| 171 | % However, rather annoyingly there are a lot of terms |
---|
| 172 | % in Q that are almost 0, but not quite (once again, this |
---|
| 173 | % depends on what solver you have, how succesful YALMIPs |
---|
| 174 | % post-processing is etc.) |
---|
| 175 | % |
---|
| 176 | % It is possible to tell YALMIP to analyse the sparsity |
---|
| 177 | % of the Gramian after it has computed it, and re-solve |
---|
| 178 | % the problem, but this time forcing the elements to be 0. |
---|
| 179 | % |
---|
| 180 | % Note, we are not cleaning the Gramian a-posteriori, but |
---|
| 181 | % resolving the problem. Hence, the decomposition is correct. |
---|
| 182 | % |
---|
| 183 | % This can be obtained with the option sos.impsparse |
---|
| 184 | pause |
---|
| 185 | [sol,v,Q] = solvesos(F,[],sdpsettings('sos.traceobj',1,'sos.impsparse',1),[],v{1}); |
---|
| 186 | pause |
---|
| 187 | |
---|
| 188 | % Sweet... |
---|
| 189 | Q{1} |
---|
| 190 | pause |
---|
| 191 | |
---|
| 192 | clc |
---|
| 193 | % Alternatively, we can tell YALMIP to study the almost-zero pattern |
---|
| 194 | % of the Grmamians, and derive a block-diagonalization based on this. |
---|
| 195 | % By setting the options sos.numblkdg to a number larger than zero, |
---|
| 196 | % YALMIP will declare number smaller than this tolerance as zero, and |
---|
| 197 | % detect any hidden block-structure, re-solve the problem with this |
---|
| 198 | % more economic parameterization, and repeat until the block-structure |
---|
| 199 | % not changes any more. Note that this not only gives "nicer" solutions, |
---|
| 200 | % but also improves the numerical conditioning of the problem. |
---|
| 201 | % |
---|
| 202 | % This is the prefered way to post-process the solution. |
---|
| 203 | pause |
---|
| 204 | [sol,v,Q] = solvesos(F,[],sdpsettings('sos.traceobj',1,'sos.numblkdg',1e-5)); |
---|
| 205 | pause |
---|
| 206 | |
---|
| 207 | % Sweet... |
---|
| 208 | Q{1} |
---|
| 209 | pause |
---|
| 210 | |
---|
| 211 | |
---|
| 212 | |
---|
| 213 | clc |
---|
| 214 | % As a second example, we solve a somewhat larger problem. |
---|
| 215 | % |
---|
| 216 | % We want to show that the following matrix is co-positive |
---|
| 217 | J = [1 -1 1 1 -1; |
---|
| 218 | -1 1 -1 1 1; |
---|
| 219 | 1 -1 1 -1 1; |
---|
| 220 | 1 1 -1 1 -1; |
---|
| 221 | -1 1 1 -1 1]; |
---|
| 222 | pause |
---|
| 223 | |
---|
| 224 | % It is clearly co-positive if this polynomial |
---|
| 225 | % is a sum-of-squares |
---|
| 226 | x1 = sdpvar(1,1);x2 = sdpvar(1,1);x3 = sdpvar(1,1);x4 = sdpvar(1,1);x5 = sdpvar(1,1); |
---|
| 227 | z = [x1^2 x2^2 x3^2 x4^2 x5^2]'; |
---|
| 228 | p = z'*J*z; |
---|
| 229 | pause |
---|
| 230 | |
---|
| 231 | solvesos(set(sos(p))) |
---|
| 232 | |
---|
| 233 | % Hmm, failure... |
---|
| 234 | % |
---|
| 235 | % Note that the error-message displayed can be somewhat mis-guiding. |
---|
| 236 | % Depending on the solver and the option sos.model, infeasible problems can |
---|
| 237 | % be declared as unbounded, and vice versa. In most cases (using SeDuMi, |
---|
| 238 | % PENSDP, SDPT3,... and kernel representation model) infeasibility is |
---|
| 239 | % reported as unbounded and an unbounded objective is reported as |
---|
| 240 | % infeasible (the reason is that YALMIP solves a problem related to the |
---|
| 241 | % dual of the SOS problem when the kernel representation model is used.) |
---|
| 242 | % |
---|
| 243 | % Let's multiply the polynomial with the positive definite function |
---|
| 244 | % sum(x_i^2). If this new polynomial is SOS, then so is the original. |
---|
| 245 | |
---|
| 246 | p_new = p*(x1^2+x2^2+x3^2+x4^2+x5^2); |
---|
| 247 | |
---|
| 248 | pause |
---|
| 249 | F = set(sos(p_new)); |
---|
| 250 | [sol,v,Q] = solvesos(F); |
---|
| 251 | checkset(F) |
---|
| 252 | % We found a decomposition, hence p is SOS and J is co-positive |
---|
| 253 | pause |
---|
| 254 | |
---|
| 255 | % Just for fun, let us solve the problem again, this time |
---|
| 256 | % removing some redundant terms |
---|
| 257 | pause |
---|
| 258 | usethese = find(diag(Q{1})>1e-3); |
---|
| 259 | [sol,v,Q] = solvesos(F,[],[],[],v{1}(usethese)); |
---|
| 260 | pause |
---|
| 261 | |
---|
| 262 | clc |
---|
| 263 | % The basic idea in sum of squares readily |
---|
| 264 | % extend also to the matrix valued case. |
---|
| 265 | % |
---|
| 266 | % In other words, find a decomposition of |
---|
| 267 | % the symmetric polynomial matrix P(x), |
---|
| 268 | % hence proving global postive definiteness. |
---|
| 269 | pause |
---|
| 270 | |
---|
| 271 | % Define a symmetric polynomail matrix |
---|
| 272 | sdpvar x1 x2 |
---|
| 273 | P = [1+x1^2 -x1+x2+x1^2;-x1+x2+x1^2 2*x1^2-2*x1*x2+x2^2]; |
---|
| 274 | pause |
---|
| 275 | |
---|
| 276 | % Call SOLVESOS |
---|
| 277 | [sol,v,Q] = solvesos(set(sos(P))); |
---|
| 278 | |
---|
| 279 | % The basis is now matrix valued |
---|
| 280 | sdisplay(v{1}) |
---|
| 281 | pause |
---|
| 282 | |
---|
| 283 | % Check that the polynomials match |
---|
| 284 | clean(v{1}'*Q{1}*v{1}-P,1e-8) |
---|
| 285 | pause |
---|
| 286 | |
---|
| 287 | clc |
---|
| 288 | % Let us now solve a parameterized SOS-problem. |
---|
| 289 | % |
---|
| 290 | % A typical application is to find a lower bound |
---|
| 291 | % on the global minima. |
---|
| 292 | % |
---|
| 293 | % This is done by solving |
---|
| 294 | % |
---|
| 295 | % max t |
---|
| 296 | % subject to p(x)-t is SOS |
---|
| 297 | pause |
---|
| 298 | |
---|
| 299 | % Define p and t |
---|
| 300 | x = sdpvar(1,1); |
---|
| 301 | y = sdpvar(1,1); |
---|
| 302 | z = sdpvar(1,1); |
---|
| 303 | |
---|
| 304 | p = 12+y^2-2*x^3*y+2*y*z^2+x^6-2*x^3*z^2+z^4+x^2*y^2; |
---|
| 305 | t = sdpvar(1,1) |
---|
| 306 | pause |
---|
| 307 | |
---|
| 308 | % maximize t subject to SOS-constraint |
---|
| 309 | % |
---|
| 310 | % SOLVESOS will automatically categorize t as a |
---|
| 311 | % parametric variable since it is part of the objective |
---|
| 312 | solvesos(set(sos(p-t)),-t); |
---|
| 313 | pause |
---|
| 314 | |
---|
| 315 | % Lower bound |
---|
| 316 | double(t) |
---|
| 317 | pause |
---|
| 318 | clc |
---|
| 319 | |
---|
| 320 | |
---|
| 321 | % Ok, now for some more advanced coding |
---|
| 322 | % |
---|
| 323 | % Given the nonlinear system dxdt=f(x), |
---|
| 324 | % we will try to prove that z'Pz is a lyapunov function |
---|
| 325 | % where z = [x1;x2;x1x2;x1^2;x2^2] and P positive definite |
---|
| 326 | pause |
---|
| 327 | |
---|
| 328 | % Define state-variables and system |
---|
| 329 | x = sdpvar(2,1); |
---|
| 330 | f = [-x(1)-x(1)^3+x(2);-x(2)]; |
---|
| 331 | |
---|
| 332 | pause |
---|
| 333 | |
---|
| 334 | % Define z, P, Lyapunov function and derivatives |
---|
| 335 | z = [x(1);x(2);x(1)^2;x(1)*x(2);x(2)^2]; |
---|
| 336 | P = sdpvar(length(z),length(z)); |
---|
| 337 | V = z'*P*z; |
---|
| 338 | dVdx = jacobian(V,x); |
---|
| 339 | dVdt = dVdx*f; |
---|
| 340 | pause |
---|
| 341 | |
---|
| 342 | % Try to prove that dVdt<0, while minimizing |
---|
| 343 | % trace(P), subject to P>0 |
---|
| 344 | % |
---|
| 345 | % All parametric variables (i.e. P) are constrained |
---|
| 346 | % hence SOLVESOS will find them automatically. |
---|
| 347 | % |
---|
| 348 | % Alternative |
---|
| 349 | % solvesos(F,trace(P),[],P(:)); |
---|
| 350 | % |
---|
| 351 | F = set(sos(-dVdt)) + set(P>eye(5)); |
---|
| 352 | [sol,v,Q] = solvesos(F,trace(P)); |
---|
| 353 | pause |
---|
| 354 | clc |
---|
| 355 | |
---|
| 356 | % Checking the validity of the SOS-decomposition is easily done |
---|
| 357 | % (checks SOS-decomposition at the current value of P) |
---|
| 358 | checkset(F) |
---|
| 359 | pause |
---|
| 360 | |
---|
| 361 | % So, according to the theory, we should have -dVdt==v'Qv |
---|
| 362 | clean(v{1}'*Q{1}*v{1}-(-dVdt),1e-2) |
---|
| 363 | pause |
---|
| 364 | |
---|
| 365 | % No! v{1}'*Q{1}*v{1} is a decomposition of -dVdt |
---|
| 366 | % when P is *fixed* to the computed optimal value. |
---|
| 367 | % |
---|
| 368 | % v{1}'*Q{1}*v{1} is a polynomial in x only, while |
---|
| 369 | % dVdt is a polynomial in x and P. |
---|
| 370 | pause |
---|
| 371 | |
---|
| 372 | % To check the decomposition manually, we need to |
---|
| 373 | % define the polynomials with the computed value |
---|
| 374 | % of P |
---|
| 375 | % |
---|
| 376 | % (Of-course, in practise it is most often more convenient |
---|
| 377 | % to use CHECKSET where this is done automatically) |
---|
| 378 | V = z'*double(P)*z; |
---|
| 379 | dVdx = jacobian(V,x); |
---|
| 380 | dVdt = dVdx*f; |
---|
| 381 | clean(-dVdt-v{1}'*Q{1}*v{1},1e-10) |
---|
| 382 | pause |
---|
| 383 | |
---|
| 384 | clc |
---|
| 385 | % Finally, make sure to check out the help in the HTML manual for more information. |
---|
| 386 | pause |
---|
| 387 | |
---|
| 388 | echo off |
---|
| 389 | |
---|
| 390 | |
---|