\( \DeclareMathOperator{\abs}{abs} \)

Boost Converter, esr

state 1 matrices

(%i1) A1: matrix([0, 0], [0, 0]);
\[\mathrm{\tt (\%o1) }\quad \begin{pmatrix}0 & 0\cr 0 & 0\end{pmatrix}\]
(%i2) B1: matrix([1/L, 0], [0, -1/c]);
\[\mathrm{\tt (\%o2) }\quad \begin{pmatrix}\frac{1}{L} & 0\cr 0 & -\frac{1}{c}\end{pmatrix}\]
(%i3) C1: matrix([0, 1]);
\[\mathrm{\tt (\%o3) }\quad \begin{pmatrix}0 & 1\end{pmatrix}\]
(%i4) D1: matrix([0, -R]);
\[\mathrm{\tt (\%o4) }\quad \begin{pmatrix}0 & -R\end{pmatrix}\]

state 2 matrices

(%i5) A2: matrix([-R/L, -1/L], [1/c, 0]);
\[\mathrm{\tt (\%o5) }\quad \begin{pmatrix}-\frac{R}{L} & -\frac{1}{L}\cr \frac{1}{c} & 0\end{pmatrix}\]
(%i6) B2: matrix([1/L, R/L], [0, -1/c]);
\[\mathrm{\tt (\%o6) }\quad \begin{pmatrix}\frac{1}{L} & \frac{R}{L}\cr 0 & -\frac{1}{c}\end{pmatrix}\]
(%i7) C2: matrix([R, 1]);
\[\mathrm{\tt (\%o7) }\quad \begin{pmatrix}R & 1\end{pmatrix}\]
(%i8) D2: matrix([0, -R]);
\[\mathrm{\tt (\%o8) }\quad \begin{pmatrix}0 & -R\end{pmatrix}\]

state variables, dc

(%i9) X0: matrix([Il], [Vc]);
\[\mathrm{\tt (\%o9) }\quad \begin{pmatrix}\mathit{Il}\cr \mathit{Vc}\end{pmatrix}\]

input variables, dc

(%i10) U0: matrix([Vin], [Iout]);
\[\mathrm{\tt (\%o10) }\quad \begin{pmatrix}\mathit{Vin}\cr \mathit{Iout}\end{pmatrix}\]

basic computation; keep it as it is

(%i11) A: D0 * A1 + (1 - D0) * A2;
\[\mathrm{\tt (\%o11) }\quad \begin{pmatrix}-\frac{\left( 1-\mathit{D0}\right) \cdot R}{L} & -\frac{1-\mathit{D0}}{L}\cr \frac{1-\mathit{D0}}{c} & 0\end{pmatrix}\]
(%i12) A: ratsimp(A);
\[\mathrm{\tt (\%o12) }\quad \begin{pmatrix}\frac{\left( \mathit{D0}-1\right) \cdot R}{L} & \frac{\mathit{D0}-1}{L}\cr -\frac{\mathit{D0}-1}{c} & 0\end{pmatrix}\]
(%i13) B: D0 * B1 + (1 - D0) * B2;
\[\mathrm{\tt (\%o13) }\quad \begin{pmatrix}\frac{\mathit{D0}}{L}+\frac{1-\mathit{D0}}{L} & \frac{\left( 1-\mathit{D0}\right) \cdot R}{L}\cr 0 & -\frac{\mathit{D0}}{c}-\frac{1-\mathit{D0}}{c}\end{pmatrix}\]
(%i14) B: ratsimp(B);
\[\mathrm{\tt (\%o14) }\quad \begin{pmatrix}\frac{1}{L} & -\frac{\left( \mathit{D0}-1\right) \cdot R}{L}\cr 0 & -\frac{1}{c}\end{pmatrix}\]
(%i15) C: D0 * C1 + (1 - D0) * C2;
\[\mathrm{\tt (\%o15) }\quad \begin{pmatrix}\left( 1-\mathit{D0}\right) \cdot R & 1\end{pmatrix}\]
(%i16) C: ratsimp(C);
\[\mathrm{\tt (\%o16) }\quad \begin{pmatrix}\left( 1-\mathit{D0}\right) \cdot R & 1\end{pmatrix}\]
(%i17) D: D0 * D1 + (1 - D0) * D2;
\[\mathrm{\tt (\%o17) }\quad \begin{pmatrix}0 & -\mathit{D0}\cdot R-\left( 1-\mathit{D0}\right) \cdot R\end{pmatrix}\]
(%i18) D: ratsimp(D);
\[\mathrm{\tt (\%o18) }\quad \begin{pmatrix}0 & -R\end{pmatrix}\]

dc computation, still without d hat, keep it as it is

(%i19) iA: invert(A);
\[\mathrm{\tt (\%o19) }\quad \begin{pmatrix}0 & -\frac{c}{\mathit{D0}-1}\cr \frac{L}{\mathit{D0}-1} & \frac{c\cdot R}{\mathit{D0}-1}\end{pmatrix}\]
(%i20) iA: ratsimp(iA);
\[\mathrm{\tt (\%o20) }\quad \begin{pmatrix}0 & -\frac{c}{\mathit{D0}-1}\cr \frac{L}{\mathit{D0}-1} & \frac{c\cdot R}{\mathit{D0}-1}\end{pmatrix}\]
(%i21) x0: -iA . B . U0;
\[\mathrm{\tt (\%o21) }\quad \begin{pmatrix}-\frac{\mathit{Iout}}{\mathit{D0}-1}\cr \frac{\mathit{Iout}\cdot R}{\mathit{D0}-1}-\frac{L\cdot \left( \frac{\mathit{Vin}}{L}-\frac{\mathit{Iout}\cdot \left( \mathit{D0}-1\right) \cdot R}{L}\right) }{\mathit{D0}-1}\end{pmatrix}\]
(%i22) x0: ratsimp(x0);
\[\mathrm{\tt (\%o22) }\quad \begin{pmatrix}-\frac{\mathit{Iout}}{\mathit{D0}-1}\cr \frac{\mathit{Iout}\cdot \mathit{D0}\cdot R-\mathit{Vin}}{\mathit{D0}-1}\end{pmatrix}\]
(%i23) y0: (D - C . iA . B) . U0;
\[\mathrm{\tt (\%o23) }\quad \mathit{Iout}\cdot \left( \frac{R}{\mathit{D0}-1}-\frac{\left( 1-\mathit{D0}\right) \cdot R}{\mathit{D0}-1}\right) -\frac{\mathit{Vin}}{\mathit{D0}-1}\]
(%i24) y0: ratsimp(y0);
\[\mathrm{\tt (\%o24) }\quad \frac{\mathit{Iout}\cdot \mathit{D0}\cdot R-\mathit{Vin}}{\mathit{D0}-1}\]

ac computation, keep it as it is

(%i25) E: (A1 - A2) . X0 + (B1 - B2) . U0;
\[\mathrm{\tt (\%o25) }\quad \begin{pmatrix}-\frac{\mathit{Iout}\cdot R}{L}+\frac{\mathit{Il}\cdot R}{L}+\frac{\mathit{Vc}}{L}\cr -\frac{\mathit{Il}}{c}\end{pmatrix}\]
(%i26) E: ratsimp(E);
\[\mathrm{\tt (\%o26) }\quad \begin{pmatrix}-\frac{\left( \mathit{Iout}-\mathit{Il}\right) \cdot R-\mathit{Vc}}{L}\cr -\frac{\mathit{Il}}{c}\end{pmatrix}\]
(%i27) F: (C1 - C2) . X0 + (D1 - D2) . U0;
\[\mathrm{\tt (\%o27) }\quad -\mathit{Il}\cdot R\]
(%i28) F: matrix([ratsimp(F)]);
\[\mathrm{\tt (\%o28) }\quad \begin{pmatrix}-\mathit{Il}\cdot R\end{pmatrix}\]

merging E and F; keep it as it is

(%i29) B: addcol(B, E);
\[\mathrm{\tt (\%o29) }\quad \begin{pmatrix}\frac{1}{L} & -\frac{\left( \mathit{D0}-1\right) \cdot R}{L} & -\frac{\left( \mathit{Iout}-\mathit{Il}\right) \cdot R-\mathit{Vc}}{L}\cr 0 & -\frac{1}{c} & -\frac{\mathit{Il}}{c}\end{pmatrix}\]
(%i30) D: addcol(D, F);
\[\mathrm{\tt (\%o30) }\quad \begin{pmatrix}0 & -R & -\mathit{Il}\cdot R\end{pmatrix}\]

computing transfer functions; keep it as it is

(%i31) S0: s * diagmatrix(2, 1) - A;
\[\mathrm{\tt (\%o31) }\quad \begin{pmatrix}s-\frac{\left( \mathit{D0}-1\right) \cdot R}{L} & -\frac{\mathit{D0}-1}{L}\cr \frac{\mathit{D0}-1}{c} & s\end{pmatrix}\]
(%i32) S: invert(S0);
\[\mathrm{\tt (\%o32) }\quad \begin{pmatrix}\frac{s}{s\cdot \left( s-\frac{\left( \mathit{D0}-1\right) \cdot R}{L}\right) +\frac{{{\left( \mathit{D0}-1\right) }^{2}}}{c\cdot L}} & \frac{\mathit{D0}-1}{L\cdot \left( s\cdot \left( s-\frac{\left( \mathit{D0}-1\right) \cdot R}{L}\right) +\frac{{{\left( \mathit{D0}-1\right) }^{2}}}{c\cdot L}\right) }\cr -\frac{\mathit{D0}-1}{c\cdot \left( s\cdot \left( s-\frac{\left( \mathit{D0}-1\right) \cdot R}{L}\right) +\frac{{{\left( \mathit{D0}-1\right) }^{2}}}{c\cdot L}\right) } & \frac{s-\frac{\left( \mathit{D0}-1\right) \cdot R}{L}}{s\cdot \left( s-\frac{\left( \mathit{D0}-1\right) \cdot R}{L}\right) +\frac{{{\left( \mathit{D0}-1\right) }^{2}}}{c\cdot L}}\end{pmatrix}\]
(%i33) S: ratsimp(S);
\[\mathrm{\tt (\%o33) }\quad \begin{pmatrix}-\frac{c\cdot s\cdot L}{\left( c\cdot s\cdot \mathit{D0}-c\cdot s\right) \cdot R-c\cdot {{s}^{2}}\cdot L-{{\mathit{D0}}^{2}}+2\cdot \mathit{D0}-1} & -\frac{c\cdot \mathit{D0}-c}{\left( c\cdot s\cdot \mathit{D0}-c\cdot s\right) \cdot R-c\cdot {{s}^{2}}\cdot L-{{\mathit{D0}}^{2}}+2\cdot \mathit{D0}-1}\cr \frac{\left( \mathit{D0}-1\right) \cdot L}{\left( c\cdot s\cdot \mathit{D0}-c\cdot s\right) \cdot R-c\cdot {{s}^{2}}\cdot L-{{\mathit{D0}}^{2}}+2\cdot \mathit{D0}-1} & \frac{\left( c\cdot \mathit{D0}-c\right) \cdot R-c\cdot s\cdot L}{\left( c\cdot s\cdot \mathit{D0}-c\cdot s\right) \cdot R-c\cdot {{s}^{2}}\cdot L-{{\mathit{D0}}^{2}}+2\cdot \mathit{D0}-1}\end{pmatrix}\]
(%i34) S: facsum(S, s);
\[\mathrm{\tt (\%o34) }\quad \begin{pmatrix}\frac{c\cdot s\cdot L}{-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}} & \frac{c\cdot \left( \mathit{D0}-1\right) }{-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}}\cr -\frac{\left( \mathit{D0}-1\right) \cdot L}{-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}} & \frac{c\cdot s\cdot L-c\cdot \left( \mathit{D0}-1\right) \cdot R}{-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}}\end{pmatrix}\]
(%i35) tox: S . B;
\[\mathrm{\tt (\%o35) }\quad \begin{pmatrix}\frac{c\cdot s}{-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}} & -\frac{c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R}{-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}}-\frac{\mathit{D0}-1}{-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}} & -\frac{c\cdot s\cdot \left( \left( \mathit{Iout}-\mathit{Il}\right) \cdot R-\mathit{Vc}\right) }{-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}}-\frac{\mathit{Il}\cdot \left( \mathit{D0}-1\right) }{-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}}\cr -\frac{\mathit{D0}-1}{-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}} & \frac{{{\left( \mathit{D0}-1\right) }^{2}}\cdot R}{-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}}-\frac{c\cdot s\cdot L-c\cdot \left( \mathit{D0}-1\right) \cdot R}{c\cdot \left( -c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}\right) } & \frac{\left( \mathit{D0}-1\right) \cdot \left( \left( \mathit{Iout}-\mathit{Il}\right) \cdot R-\mathit{Vc}\right) }{-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}}-\frac{\mathit{Il}\cdot \left( c\cdot s\cdot L-c\cdot \left( \mathit{D0}-1\right) \cdot R\right) }{c\cdot \left( -c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}\right) }\end{pmatrix}\]
(%i36) tox: ratsimp(tox);
\[\mathrm{\tt (\%o36) }\quad \begin{pmatrix}-\frac{c\cdot s}{\left( c\cdot s\cdot \mathit{D0}-c\cdot s\right) \cdot R-c\cdot {{s}^{2}}\cdot L-{{\mathit{D0}}^{2}}+2\cdot \mathit{D0}-1} & \frac{-1+\mathit{D0}+\left( c\cdot s\cdot \mathit{D0}-c\cdot s\right) \cdot R}{\left( c\cdot s\cdot \mathit{D0}-c\cdot s\right) \cdot R-c\cdot {{s}^{2}}\cdot L-{{\mathit{D0}}^{2}}+2\cdot \mathit{D0}-1} & \frac{-\mathit{Il}-c\cdot s\cdot \mathit{Vc}+\mathit{Il}\cdot \mathit{D0}+\left( c\cdot \mathit{Iout}-c\cdot \mathit{Il}\right) \cdot s\cdot R}{\left( c\cdot s\cdot \mathit{D0}-c\cdot s\right) \cdot R-c\cdot {{s}^{2}}\cdot L-{{\mathit{D0}}^{2}}+2\cdot \mathit{D0}-1}\cr \frac{\mathit{D0}-1}{\left( c\cdot s\cdot \mathit{D0}-c\cdot s\right) \cdot R-c\cdot {{s}^{2}}\cdot L-{{\mathit{D0}}^{2}}+2\cdot \mathit{D0}-1} & -\frac{\left( {{\mathit{D0}}^{2}}-\mathit{D0}\right) \cdot R-s\cdot L}{\left( c\cdot s\cdot \mathit{D0}-c\cdot s\right) \cdot R-c\cdot {{s}^{2}}\cdot L-{{\mathit{D0}}^{2}}+2\cdot \mathit{D0}-1} & -\frac{\mathit{Vc}-\mathit{Vc}\cdot \mathit{D0}-\mathit{Il}\cdot s\cdot L+\left( \mathit{Iout}\cdot \mathit{D0}-\mathit{Iout}\right) \cdot R}{\left( c\cdot s\cdot \mathit{D0}-c\cdot s\right) \cdot R-c\cdot {{s}^{2}}\cdot L-{{\mathit{D0}}^{2}}+2\cdot \mathit{D0}-1}\end{pmatrix}\]
(%i37) tox: facsum(tox, s);
\[\mathrm{\tt (\%o37) }\quad \begin{pmatrix}\frac{c\cdot s}{-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}} & \frac{1-\mathit{D0}-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R}{-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}} & \frac{-\mathit{Il}\cdot \left( \mathit{D0}-1\right) -c\cdot s\cdot \left( -\mathit{Vc}-\mathit{Il}\cdot R+\mathit{Iout}\cdot R\right) }{-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}}\cr \frac{1-\mathit{D0}}{-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}} & \frac{\left( \mathit{D0}-1\right) \cdot \mathit{D0}\cdot R-s\cdot L}{-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}} & \frac{\left( \mathit{D0}-1\right) \cdot \left( \mathit{Iout}\cdot R-\mathit{Vc}\right) -\mathit{Il}\cdot s\cdot L}{-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}}\end{pmatrix}\]
(%i38) tox: ev(tox, Il = x0[1, 1], Vc = x0[2, 1]);
\[\mathrm{\tt (\%o38) }\quad \begin{pmatrix}\frac{c\cdot s}{-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}} & \frac{1-\mathit{D0}-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R}{-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}} & \frac{\mathit{Iout}-c\cdot s\cdot \left( \mathit{Iout}\cdot R+\frac{\mathit{Iout}\cdot R}{\mathit{D0}-1}-\frac{\mathit{Iout}\cdot \mathit{D0}\cdot R-\mathit{Vin}}{\mathit{D0}-1}\right) }{-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}}\cr \frac{1-\mathit{D0}}{-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}} & \frac{\left( \mathit{D0}-1\right) \cdot \mathit{D0}\cdot R-s\cdot L}{-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}} & \frac{\frac{\mathit{Iout}\cdot s\cdot L}{\mathit{D0}-1}+\left( \mathit{D0}-1\right) \cdot \left( \mathit{Iout}\cdot R-\frac{\mathit{Iout}\cdot \mathit{D0}\cdot R-\mathit{Vin}}{\mathit{D0}-1}\right) }{-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}}\end{pmatrix}\]
(%i39) tox: ratsimp(tox);
\[\mathrm{\tt (\%o39) }\quad \begin{pmatrix}-\frac{c\cdot s}{\left( c\cdot s\cdot \mathit{D0}-c\cdot s\right) \cdot R-c\cdot {{s}^{2}}\cdot L-{{\mathit{D0}}^{2}}+2\cdot \mathit{D0}-1} & \frac{-1+\mathit{D0}+\left( c\cdot s\cdot \mathit{D0}-c\cdot s\right) \cdot R}{\left( c\cdot s\cdot \mathit{D0}-c\cdot s\right) \cdot R-c\cdot {{s}^{2}}\cdot L-{{\mathit{D0}}^{2}}+2\cdot \mathit{D0}-1} & -\frac{-\mathit{Iout}-c\cdot s\cdot \mathit{Vin}+\mathit{Iout}\cdot \mathit{D0}}{\left( c\cdot s-2\cdot c\cdot s\cdot \mathit{D0}+c\cdot s\cdot {{\mathit{D0}}^{2}}\right) \cdot R+\left( c\cdot {{s}^{2}}-c\cdot {{s}^{2}}\cdot \mathit{D0}\right) \cdot L-{{\mathit{D0}}^{3}}+3\cdot {{\mathit{D0}}^{2}}-3\cdot \mathit{D0}+1}\cr \frac{\mathit{D0}-1}{\left( c\cdot s\cdot \mathit{D0}-c\cdot s\right) \cdot R-c\cdot {{s}^{2}}\cdot L-{{\mathit{D0}}^{2}}+2\cdot \mathit{D0}-1} & -\frac{\left( {{\mathit{D0}}^{2}}-\mathit{D0}\right) \cdot R-s\cdot L}{\left( c\cdot s\cdot \mathit{D0}-c\cdot s\right) \cdot R-c\cdot {{s}^{2}}\cdot L-{{\mathit{D0}}^{2}}+2\cdot \mathit{D0}-1} & \frac{\mathit{Vin}-\mathit{Vin}\cdot \mathit{D0}-\mathit{Iout}\cdot s\cdot L+\left( \mathit{Iout}\cdot \mathit{D0}-\mathit{Iout}\right) \cdot R}{\left( c\cdot s-2\cdot c\cdot s\cdot \mathit{D0}+c\cdot s\cdot {{\mathit{D0}}^{2}}\right) \cdot R+\left( c\cdot {{s}^{2}}-c\cdot {{s}^{2}}\cdot \mathit{D0}\right) \cdot L-{{\mathit{D0}}^{3}}+3\cdot {{\mathit{D0}}^{2}}-3\cdot \mathit{D0}+1}\end{pmatrix}\]
(%i40) tox: facsum(tox, s);
\[\mathrm{\tt (\%o40) }\quad \begin{pmatrix}\frac{c\cdot s}{-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}} & \frac{1-\mathit{D0}-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R}{-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}} & \frac{\mathit{Iout}\cdot \left( \mathit{D0}-1\right) -c\cdot s\cdot \mathit{Vin}}{-c\cdot s\cdot {{\left( \mathit{D0}-1\right) }^{2}}\cdot R+c\cdot {{s}^{2}}\cdot \left( \mathit{D0}-1\right) \cdot L+{{\left( \mathit{D0}-1\right) }^{3}}}\cr \frac{1-\mathit{D0}}{-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}} & \frac{\left( \mathit{D0}-1\right) \cdot \mathit{D0}\cdot R-s\cdot L}{-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}} & \frac{\mathit{Iout}\cdot s\cdot L-\left( \mathit{D0}-1\right) \cdot \left( \mathit{Iout}\cdot R-\mathit{Vin}\right) }{-c\cdot s\cdot {{\left( \mathit{D0}-1\right) }^{2}}\cdot R+c\cdot {{s}^{2}}\cdot \left( \mathit{D0}-1\right) \cdot L+{{\left( \mathit{D0}-1\right) }^{3}}}\end{pmatrix}\]
(%i41) toy: C . S . B + D;
\[\mathrm{\tt (\%o41) }\quad \begin{pmatrix}\frac{c\cdot s\cdot \left( 1-\mathit{D0}\right) \cdot R}{-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}}-\frac{\mathit{D0}-1}{-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}} & \left( 1-\mathit{D0}\right) \cdot R\cdot \left( -\frac{\mathit{D0}-1}{-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}}-\frac{c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R}{-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}}\right) -\frac{c\cdot s\cdot L-c\cdot \left( \mathit{D0}-1\right) \cdot R}{c\cdot \left( -c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}\right) }+\frac{{{\left( \mathit{D0}-1\right) }^{2}}\cdot R}{-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}}-R & \left( 1-\mathit{D0}\right) \cdot R\cdot \left( -\frac{\mathit{Il}\cdot \left( \mathit{D0}-1\right) }{-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}}-\frac{c\cdot s\cdot \left( \left( \mathit{Iout}-\mathit{Il}\right) \cdot R-\mathit{Vc}\right) }{-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}}\right) -\frac{\mathit{Il}\cdot \left( c\cdot s\cdot L-c\cdot \left( \mathit{D0}-1\right) \cdot R\right) }{c\cdot \left( -c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}\right) }+\frac{\left( \mathit{D0}-1\right) \cdot \left( \left( \mathit{Iout}-\mathit{Il}\right) \cdot R-\mathit{Vc}\right) }{-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}}-\mathit{Il}\cdot R\end{pmatrix}\]
(%i42) toy: ratsimp(toy);
\[\mathrm{\tt (\%o42) }\quad \begin{pmatrix}\frac{-1+\mathit{D0}+\left( c\cdot s\cdot \mathit{D0}-c\cdot s\right) \cdot R}{\left( c\cdot s\cdot \mathit{D0}-c\cdot s\right) \cdot R-c\cdot {{s}^{2}}\cdot L-{{\mathit{D0}}^{2}}+2\cdot \mathit{D0}-1} & -\frac{-s\cdot L+\left( -c\cdot {{s}^{2}}\cdot L+{{\mathit{D0}}^{2}}-\mathit{D0}\right) \cdot R+\left( c\cdot s\cdot {{\mathit{D0}}^{2}}-c\cdot s\cdot \mathit{D0}\right) \cdot {{R}^{2}}}{\left( c\cdot s\cdot \mathit{D0}-c\cdot s\right) \cdot R-c\cdot {{s}^{2}}\cdot L-{{\mathit{D0}}^{2}}+2\cdot \mathit{D0}-1} & -\frac{\mathit{Vc}-\mathit{Vc}\cdot \mathit{D0}-\mathit{Il}\cdot s\cdot L+\left( -c\cdot \mathit{Il}\cdot {{s}^{2}}\cdot L+\left( \mathit{Iout}-c\cdot s\cdot \mathit{Vc}\right) \cdot \mathit{D0}+c\cdot s\cdot \mathit{Vc}-\mathit{Iout}\right) \cdot R+\left( c\cdot \mathit{Iout}\cdot s\cdot \mathit{D0}-c\cdot \mathit{Iout}\cdot s\right) \cdot {{R}^{2}}}{\left( c\cdot s\cdot \mathit{D0}-c\cdot s\right) \cdot R-c\cdot {{s}^{2}}\cdot L-{{\mathit{D0}}^{2}}+2\cdot \mathit{D0}-1}\end{pmatrix}\]
(%i43) toy: facsum(toy, s);
\[\mathrm{\tt (\%o43) }\quad \begin{pmatrix}\frac{1-\mathit{D0}-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R}{-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}} & \frac{\left( \mathit{D0}-1\right) \cdot \mathit{D0}\cdot R-c\cdot {{s}^{2}}\cdot L\cdot R+s\cdot \left( c\cdot {{\mathit{D0}}^{2}}\cdot {{R}^{2}}-c\cdot \mathit{D0}\cdot {{R}^{2}}-L\right) }{-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}} & \frac{-c\cdot \mathit{Il}\cdot {{s}^{2}}\cdot L\cdot R+\left( \mathit{D0}-1\right) \cdot \left( \mathit{Iout}\cdot R-\mathit{Vc}\right) +s\cdot \left( c\cdot \mathit{Iout}\cdot \mathit{D0}\cdot {{R}^{2}}-c\cdot \mathit{Iout}\cdot {{R}^{2}}-c\cdot \mathit{Vc}\cdot \mathit{D0}\cdot R+c\cdot \mathit{Vc}\cdot R-\mathit{Il}\cdot L\right) }{-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}}\end{pmatrix}\]
(%i44) toy: ev(toy, Il = x0[1, 1], Vc = x0[2, 1]);
\[\mathrm{\tt (\%o44) }\quad \begin{pmatrix}\frac{1-\mathit{D0}-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R}{-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}} & \frac{\left( \mathit{D0}-1\right) \cdot \mathit{D0}\cdot R-c\cdot {{s}^{2}}\cdot L\cdot R+s\cdot \left( c\cdot {{\mathit{D0}}^{2}}\cdot {{R}^{2}}-c\cdot \mathit{D0}\cdot {{R}^{2}}-L\right) }{-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}} & \frac{\frac{c\cdot \mathit{Iout}\cdot {{s}^{2}}\cdot L\cdot R}{\mathit{D0}-1}+\left( \mathit{D0}-1\right) \cdot \left( \mathit{Iout}\cdot R-\frac{\mathit{Iout}\cdot \mathit{D0}\cdot R-\mathit{Vin}}{\mathit{D0}-1}\right) +s\cdot \left( c\cdot \mathit{Iout}\cdot \mathit{D0}\cdot {{R}^{2}}-c\cdot \mathit{Iout}\cdot {{R}^{2}}-\frac{c\cdot \mathit{D0}\cdot R\cdot \left( \mathit{Iout}\cdot \mathit{D0}\cdot R-\mathit{Vin}\right) }{\mathit{D0}-1}+\frac{c\cdot R\cdot \left( \mathit{Iout}\cdot \mathit{D0}\cdot R-\mathit{Vin}\right) }{\mathit{D0}-1}+\frac{\mathit{Iout}\cdot L}{\mathit{D0}-1}\right) }{-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}}\end{pmatrix}\]
(%i45) toy: ratsimp(toy);
\[\mathrm{\tt (\%o45) }\quad \begin{pmatrix}\frac{-1+\mathit{D0}+\left( c\cdot s\cdot \mathit{D0}-c\cdot s\right) \cdot R}{\left( c\cdot s\cdot \mathit{D0}-c\cdot s\right) \cdot R-c\cdot {{s}^{2}}\cdot L-{{\mathit{D0}}^{2}}+2\cdot \mathit{D0}-1} & -\frac{-s\cdot L+\left( -c\cdot {{s}^{2}}\cdot L+{{\mathit{D0}}^{2}}-\mathit{D0}\right) \cdot R+\left( c\cdot s\cdot {{\mathit{D0}}^{2}}-c\cdot s\cdot \mathit{D0}\right) \cdot {{R}^{2}}}{\left( c\cdot s\cdot \mathit{D0}-c\cdot s\right) \cdot R-c\cdot {{s}^{2}}\cdot L-{{\mathit{D0}}^{2}}+2\cdot \mathit{D0}-1} & \frac{\mathit{Vin}-\mathit{Vin}\cdot \mathit{D0}-\mathit{Iout}\cdot s\cdot L+\left( -c\cdot \mathit{Iout}\cdot {{s}^{2}}\cdot L+\left( \mathit{Iout}-c\cdot s\cdot \mathit{Vin}\right) \cdot \mathit{D0}+c\cdot s\cdot \mathit{Vin}-\mathit{Iout}\right) \cdot R+\left( c\cdot \mathit{Iout}\cdot s\cdot \mathit{D0}-c\cdot \mathit{Iout}\cdot s\right) \cdot {{R}^{2}}}{\left( c\cdot s-2\cdot c\cdot s\cdot \mathit{D0}+c\cdot s\cdot {{\mathit{D0}}^{2}}\right) \cdot R+\left( c\cdot {{s}^{2}}-c\cdot {{s}^{2}}\cdot \mathit{D0}\right) \cdot L-{{\mathit{D0}}^{3}}+3\cdot {{\mathit{D0}}^{2}}-3\cdot \mathit{D0}+1}\end{pmatrix}\]
(%i46) toy: facsum(toy, s);
\[\mathrm{\tt (\%o46) }\quad \begin{pmatrix}\frac{1-\mathit{D0}-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R}{-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}} & \frac{\left( \mathit{D0}-1\right) \cdot \mathit{D0}\cdot R-c\cdot {{s}^{2}}\cdot L\cdot R+s\cdot \left( c\cdot {{\mathit{D0}}^{2}}\cdot {{R}^{2}}-c\cdot \mathit{D0}\cdot {{R}^{2}}-L\right) }{-c\cdot s\cdot \left( \mathit{D0}-1\right) \cdot R+c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}} & \frac{c\cdot \mathit{Iout}\cdot {{s}^{2}}\cdot L\cdot R-\left( \mathit{D0}-1\right) \cdot \left( \mathit{Iout}\cdot R-\mathit{Vin}\right) -s\cdot \left( -\mathit{Iout}\cdot L+c\cdot \mathit{Vin}\cdot R-c\cdot \mathit{Vin}\cdot \mathit{D0}\cdot R-c\cdot \mathit{Iout}\cdot {{R}^{2}}+c\cdot \mathit{Iout}\cdot \mathit{D0}\cdot {{R}^{2}}\right) }{-c\cdot s\cdot {{\left( \mathit{D0}-1\right) }^{2}}\cdot R+c\cdot {{s}^{2}}\cdot \left( \mathit{D0}-1\right) \cdot L+{{\left( \mathit{D0}-1\right) }^{3}}}\end{pmatrix}\]

substituting values

(%i47) tox0: ev(tox, R = 0);
\[\mathrm{\tt (\%o47) }\quad \begin{pmatrix}\frac{c\cdot s}{c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}} & \frac{1-\mathit{D0}}{c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}} & \frac{\mathit{Iout}\cdot \left( \mathit{D0}-1\right) -c\cdot s\cdot \mathit{Vin}}{c\cdot {{s}^{2}}\cdot \left( \mathit{D0}-1\right) \cdot L+{{\left( \mathit{D0}-1\right) }^{3}}}\cr \frac{1-\mathit{D0}}{c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}} & -\frac{s\cdot L}{c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}} & \frac{\mathit{Vin}\cdot \left( \mathit{D0}-1\right) +\mathit{Iout}\cdot s\cdot L}{c\cdot {{s}^{2}}\cdot \left( \mathit{D0}-1\right) \cdot L+{{\left( \mathit{D0}-1\right) }^{3}}}\end{pmatrix}\]
(%i48) tox0: facsum(tox0, s);
\[\mathrm{\tt (\%o48) }\quad \begin{pmatrix}\frac{c\cdot s}{c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}} & \frac{1-\mathit{D0}}{c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}} & \frac{\mathit{Iout}\cdot \left( \mathit{D0}-1\right) -c\cdot s\cdot \mathit{Vin}}{c\cdot {{s}^{2}}\cdot \left( \mathit{D0}-1\right) \cdot L+{{\left( \mathit{D0}-1\right) }^{3}}}\cr \frac{1-\mathit{D0}}{c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}} & -\frac{s\cdot L}{c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}} & \frac{\mathit{Vin}\cdot \left( \mathit{D0}-1\right) +\mathit{Iout}\cdot s\cdot L}{c\cdot {{s}^{2}}\cdot \left( \mathit{D0}-1\right) \cdot L+{{\left( \mathit{D0}-1\right) }^{3}}}\end{pmatrix}\]
(%i49) toy0: ev(toy, R = 0);
\[\mathrm{\tt (\%o49) }\quad \begin{pmatrix}\frac{1-\mathit{D0}}{c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}} & -\frac{s\cdot L}{c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}} & \frac{\mathit{Vin}\cdot \left( \mathit{D0}-1\right) +\mathit{Iout}\cdot s\cdot L}{c\cdot {{s}^{2}}\cdot \left( \mathit{D0}-1\right) \cdot L+{{\left( \mathit{D0}-1\right) }^{3}}}\end{pmatrix}\]
(%i50) toy0: facsum(toy0, s);
\[\mathrm{\tt (\%o50) }\quad \begin{pmatrix}\frac{1-\mathit{D0}}{c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}} & -\frac{s\cdot L}{c\cdot {{s}^{2}}\cdot L+{{\left( \mathit{D0}-1\right) }^{2}}} & \frac{\mathit{Vin}\cdot \left( \mathit{D0}-1\right) +\mathit{Iout}\cdot s\cdot L}{c\cdot {{s}^{2}}\cdot \left( \mathit{D0}-1\right) \cdot L+{{\left( \mathit{D0}-1\right) }^{3}}}\end{pmatrix}\]
(%i51) tox1: ev(tox, D0 = 0.5, Vin = 10, Iout = 0.5, L = 50e-6, c = 0.5e-3, R = 0);
\[\mathrm{\tt (\%o51) }\quad \begin{pmatrix}\frac{5.0\cdot {{10}^{-4}}\cdot s}{2.5\cdot {{10}^{-8}}\cdot {{s}^{2}}+0.25} & \frac{0.5}{2.5\cdot {{10}^{-8}}\cdot {{s}^{2}}+0.25} & \frac{-0.25-0.005\cdot s}{-1.25\cdot {{10}^{-8}}\cdot {{s}^{2}}-0.125}\cr \frac{0.5}{2.5\cdot {{10}^{-8}}\cdot {{s}^{2}}+0.25} & -\frac{5.0\cdot {{10}^{-5}}\cdot s}{2.5\cdot {{10}^{-8}}\cdot {{s}^{2}}+0.25} & \frac{2.5\cdot {{10}^{-5}}\cdot s-5.0}{-1.25\cdot {{10}^{-8}}\cdot {{s}^{2}}-0.125}\end{pmatrix}\]
(%i52) tox2: ev(tox, D0 = 0.5, Vin = 10, Iout = 0.5, L = 50e-6, c = 0.5e-3, R = 0.1);
\[\mathrm{\tt (\%o52) }\quad \begin{pmatrix}\frac{5.0\cdot {{10}^{-4}}\cdot s}{2.5\cdot {{10}^{-8}}\cdot {{s}^{2}}+2.5\cdot {{10}^{-5}}\cdot s+0.25} & \frac{0.5+2.5\cdot {{10}^{-5}}\cdot s}{2.5\cdot {{10}^{-8}}\cdot {{s}^{2}}+2.5\cdot {{10}^{-5}}\cdot s+0.25} & \frac{-0.25-0.005\cdot s}{-1.25\cdot {{10}^{-8}}\cdot {{s}^{2}}-1.25\cdot {{10}^{-5}}\cdot s-0.125}\cr \frac{0.5}{2.5\cdot {{10}^{-8}}\cdot {{s}^{2}}+2.5\cdot {{10}^{-5}}\cdot s+0.25} & \frac{-0.025-5.0\cdot {{10}^{-5}}\cdot s}{2.5\cdot {{10}^{-8}}\cdot {{s}^{2}}+2.5\cdot {{10}^{-5}}\cdot s+0.25} & \frac{2.5\cdot {{10}^{-5}}\cdot s-4.975}{-1.25\cdot {{10}^{-8}}\cdot {{s}^{2}}-1.25\cdot {{10}^{-5}}\cdot s-0.125}\end{pmatrix}\]
(%i53) toy1: ev(toy, D0 = 0.5, Vin = 10, Iout = 0.5, L = 50e-6, c = 0.5e-3, R = 0);
\[\mathrm{\tt (\%o53) }\quad \begin{pmatrix}\frac{0.5}{2.5\cdot {{10}^{-8}}\cdot {{s}^{2}}+0.25} & -\frac{5.0\cdot {{10}^{-5}}\cdot s}{2.5\cdot {{10}^{-8}}\cdot {{s}^{2}}+0.25} & \frac{2.5\cdot {{10}^{-5}}\cdot s-5.0}{-1.25\cdot {{10}^{-8}}\cdot {{s}^{2}}-0.125}\end{pmatrix}\]
(%i54) factor(toy1);
\[\mbox{}\\\mbox{rat: replaced 0.5 by 1/2 = 0.5}\mbox{}\\\mbox{rat: replaced 0.25 by 1/4 = 0.25}\mbox{}\\\mbox{rat: replaced 2.5E-8 by 1/40000000 = 2.5E-8}\mbox{}\\\mbox{rat: replaced -5.0E-5 by -1/20000 = -5.0E-5}\mbox{}\\\mbox{rat: replaced 0.25 by 1/4 = 0.25}\mbox{}\\\mbox{rat: replaced 2.5E-8 by 1/40000000 = 2.5E-8}\mbox{}\\\mbox{rat: replaced -5.0 by -5/1 = -5.0}\mbox{}\\\mbox{rat: replaced 2.5E-5 by 1/40000 = 2.5E-5}\mbox{}\\\mbox{rat: replaced -0.125 by -1/8 = -0.125}\mbox{}\\\mbox{rat: replaced -1.25E-8 by -1/80000000 = -1.25E-8}\]\[\mathrm{\tt (\%o54) }\quad \begin{pmatrix}\frac{20000000}{{{s}^{2}}+10000000} & -\frac{2000\cdot s}{{{s}^{2}}+10000000} & -\frac{2000\cdot \left( s-200000\right) }{{{s}^{2}}+10000000}\end{pmatrix}\]
(%i55) toy2: ev(toy, D0 = 0.5, Vin = 10, Iout = 0.5, L = 50e-6, c = 0.5e-3, R = 0.1);
\[\mathrm{\tt (\%o55) }\quad \begin{pmatrix}\frac{0.5+2.5\cdot {{10}^{-5}}\cdot s}{2.5\cdot {{10}^{-8}}\cdot {{s}^{2}}+2.5\cdot {{10}^{-5}}\cdot s+0.25} & \frac{-0.025-5.125\cdot {{10}^{-5}}\cdot s-2.5\cdot {{10}^{-9}}\cdot {{s}^{2}}}{2.5\cdot {{10}^{-8}}\cdot {{s}^{2}}+2.5\cdot {{10}^{-5}}\cdot s+0.25} & \frac{-4.975-2.2375\cdot {{10}^{-4}}\cdot s+1.25\cdot {{10}^{-9}}\cdot {{s}^{2}}}{-1.25\cdot {{10}^{-8}}\cdot {{s}^{2}}-1.25\cdot {{10}^{-5}}\cdot s-0.125}\end{pmatrix}\]
(%i56) factor(toy2);
\[\mbox{}\\\mbox{rat: replaced 0.5 by 1/2 = 0.5}\mbox{}\\\mbox{rat: replaced 2.5E-5 by 1/40000 = 2.5E-5}\mbox{}\\\mbox{rat: replaced 0.25 by 1/4 = 0.25}\mbox{}\\\mbox{rat: replaced 2.5E-5 by 1/40000 = 2.5E-5}\mbox{}\\\mbox{rat: replaced 2.5E-8 by 1/40000000 = 2.5E-8}\mbox{}\\\mbox{rat: replaced -0.025 by -1/40 = -0.025}\mbox{}\\\mbox{rat: replaced -5.125E-5 by -41/800000 = -5.125E-5}\mbox{}\\\mbox{rat: replaced -2.5E-9 by -1/400000000 = -2.5E-9}\mbox{}\\\mbox{rat: replaced 0.25 by 1/4 = 0.25}\mbox{}\\\mbox{rat: replaced 2.5E-5 by 1/40000 = 2.5E-5}\mbox{}\\\mbox{rat: replaced 2.5E-8 by 1/40000000 = 2.5E-8}\mbox{}\\\mbox{rat: replaced -0.125 by -1/8 = -0.125}\mbox{}\\\mbox{rat: replaced -1.25E-5 by -1/80000 = -1.25E-5}\mbox{}\\\mbox{rat: replaced -1.25E-8 by -1/80000000 = -1.25E-8}\mbox{}\\\mbox{rat: replaced -4.975 by -199/40 = -4.975}\mbox{}\\\mbox{rat: replaced -2.2375E-4 by -179/800000 = -2.2375E-4}\mbox{}\\\mbox{rat: replaced 1.25E-9 by 1/800000000 = 1.25E-9}\]\[\mathrm{\tt (\%o56) }\quad \begin{pmatrix}\frac{1000\cdot \left( 20000+s\right) }{{{s}^{2}}+1000\cdot s+10000000} & -\frac{\left( 500+s\right) \cdot \left( 20000+s\right) }{10\cdot \left( {{s}^{2}}+1000\cdot s+10000000\right) } & -\frac{\left( s-199000\right) \cdot \left( 20000+s\right) }{10\cdot \left( {{s}^{2}}+1000\cdot s+10000000\right) }\end{pmatrix}\]
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