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RussTedrake merged 1 commit into from
Sep 10, 2026
Merged

Avoid symbolic expansion when building approximate DP SOS programs #608

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64 changes: 28 additions & 36 deletions book/lyapunov/approximate_dp.ipynb
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Original file line number Diff line number Diff line change
Expand Up @@ -311,36 +311,32 @@
" J_dot = J_expr.Jacobian(z).dot(f(z, u))\n",
" LHS = J_dot + l(z, u)\n",
"\n",
" # Keep the multipliers as Polynomials to avoid expanding their symbolic\n",
" # coefficients when constructing the SOS constraints.\n",
" # S procedure for s^2 + c^2 = 1.\n",
" lam_r = prog.NewFreePolynomial(Variables(z), deg).ToExpression()\n",
" S_r = lam_r * (z[0] ** 2 + z[1] ** 2 - 1)\n",
" lam_r = prog.NewFreePolynomial(Variables(z), deg)\n",
" S_r = lam_r * Polynomial(z[0] ** 2 + z[1] ** 2 - 1)\n",
" S_Jdot = 0\n",
" for i in range(nz):\n",
" lam = prog.NewSosPolynomial(Variables(z), int(np.ceil(deg / 2) * 2))[\n",
" 0\n",
" ].ToExpression()\n",
" S_Jdot += lam * (z[i] - z_max[i]) * (z[i] - z_min[i])\n",
" lam = prog.NewSosPolynomial(Variables(z), int(np.ceil(deg / 2) * 2))[0]\n",
" S_Jdot += lam * Polynomial((z[i] - z_max[i]) * (z[i] - z_min[i]))\n",
"\n",
" # Enforce Input constraint\n",
" u_min = -u_max\n",
" for i in range(nu):\n",
" lam = prog.NewSosPolynomial(Variables(z), int(np.ceil(deg / 2) * 2))[\n",
" 0\n",
" ].ToExpression()\n",
" S_Jdot += lam * (u[i] - u_max[i]) * (u[i] - u_min[i])\n",
" lam = prog.NewSosPolynomial(Variables(z), int(np.ceil(deg / 2) * 2))[0]\n",
" S_Jdot += lam * Polynomial((u[i] - u_max[i]) * (u[i] - u_min[i]))\n",
" # Enforce Bellman inequality.\n",
" prog.AddSosConstraint(LHS + S_r + S_Jdot)\n",
" prog.AddSosConstraint(Polynomial(LHS, prog.indeterminates()) + S_r + S_Jdot)\n",
"\n",
" lam_r = prog.NewFreePolynomial(Variables(z), deg).ToExpression()\n",
" S_r = lam_r * (z[0] ** 2 + z[1] ** 2 - 1)\n",
" lam_r = prog.NewFreePolynomial(Variables(z), deg)\n",
" S_r = lam_r * Polynomial(z[0] ** 2 + z[1] ** 2 - 1)\n",
" S_J = 0\n",
" for i in range(nz):\n",
" lam = prog.NewSosPolynomial(Variables(z), int(np.ceil(deg / 2) * 2))[\n",
" 0\n",
" ].ToExpression()\n",
" S_J += lam * (z[i] - z_max[i]) * (z[i] - z_min[i])\n",
" lam = prog.NewSosPolynomial(Variables(z), int(np.ceil(deg / 2) * 2))[0]\n",
" S_J += lam * Polynomial((z[i] - z_max[i]) * (z[i] - z_min[i]))\n",
" # Enforce that value function is PD\n",
" prog.AddSosConstraint(J_expr + S_r + S_J)\n",
" prog.AddSosConstraint(J + S_r + S_J)\n",
"\n",
" # J(z0) = 0.\n",
" J0 = J_expr.EvaluatePartial(dict(zip(z, z0)))\n",
Expand Down Expand Up @@ -619,37 +615,33 @@
" LHS = J_dot + l_cost(z, u) * denominator\n",
"\n",
" lam_deg = Polynomial(LHS).TotalDegree()\n",
" # Keep the multipliers as Polynomials to avoid expanding their symbolic\n",
" # coefficients when constructing the SOS constraints.\n",
" # S procedure for s^2 + c^2 = 1.\n",
" lam = prog.NewFreePolynomial(Variables(z), lam_deg).ToExpression()\n",
" S_procedure = lam * (z[1] ** 2 + z[2] ** 2 - 1)\n",
" lam = prog.NewFreePolynomial(Variables(z), lam_deg)\n",
" S_procedure = lam * Polynomial(z[1] ** 2 + z[2] ** 2 - 1)\n",
" S_Jdot = 0\n",
" for i in np.arange(nz):\n",
" lam = prog.NewSosPolynomial(Variables(z), int(np.ceil(lam_deg / 2) * 2))[\n",
" 0\n",
" ].ToExpression()\n",
" S_Jdot += lam * (z[i] - z_max[i]) * (z[i] - z_min[i])\n",
" lam = prog.NewSosPolynomial(Variables(z), int(np.ceil(lam_deg / 2) * 2))[0]\n",
" S_Jdot += lam * Polynomial((z[i] - z_max[i]) * (z[i] - z_min[i]))\n",
"\n",
" # Enforce Input constraint\n",
" u_min = -u_max\n",
" for i in range(nu):\n",
" lam = prog.NewSosPolynomial(Variables(z), int(np.ceil(deg / 2) * 2))[\n",
" 0\n",
" ].ToExpression()\n",
" S_Jdot += lam * (u[i] - u_max[i]) * (u[i] - u_min[i])\n",
" lam = prog.NewSosPolynomial(Variables(z), int(np.ceil(deg / 2) * 2))[0]\n",
" S_Jdot += lam * Polynomial((u[i] - u_max[i]) * (u[i] - u_min[i]))\n",
"\n",
" prog.AddSosConstraint(LHS + S_procedure + S_Jdot)\n",
" prog.AddSosConstraint(Polynomial(LHS, prog.indeterminates()) + S_procedure + S_Jdot)\n",
"\n",
" # Enforce that value function is PD\n",
" S_J = 0\n",
" lam_r = prog.NewFreePolynomial(Variables(z), deg).ToExpression()\n",
" S_r = lam_r * (z[1] ** 2 + z[2] ** 2 - 1)\n",
" lam_r = prog.NewFreePolynomial(Variables(z), deg)\n",
" S_r = lam_r * Polynomial(z[1] ** 2 + z[2] ** 2 - 1)\n",
" for i in np.arange(nz):\n",
" lam = prog.NewSosPolynomial(Variables(z), int(np.ceil(deg / 2) * 2))[\n",
" 0\n",
" ].ToExpression()\n",
" S_J += lam * (z[i] - z_max[i]) * (z[i] - z_min[i])\n",
" lam = prog.NewSosPolynomial(Variables(z), int(np.ceil(deg / 2) * 2))[0]\n",
" S_J += lam * Polynomial((z[i] - z_max[i]) * (z[i] - z_min[i]))\n",
" # Enforce that value function is PD\n",
" prog.AddSosConstraint(J_expr + S_J + S_r)\n",
" prog.AddSosConstraint(J + S_J + S_r)\n",
"\n",
" # J(z0) = 0.\n",
" J0 = J_expr.EvaluatePartial(dict(zip(z, z0)))\n",
Expand Down
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