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

Buck-Boost Converter

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, 0]);
\[\mathrm{\tt (\%o4) }\quad \begin{pmatrix}0 & 0\end{pmatrix}\]

state 2 matrices

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

basic computation; keep it as it is

(%i9) A: D0 * A1 + (1 - D0) * A2;
\[\mathrm{\tt (\%o9) }\quad \begin{pmatrix}0 & \frac{1-\mathit{D0}}{L}\cr -\frac{1-\mathit{D0}}{c} & 0\end{pmatrix}\]
(%i10) A: ratsimp(A);
\[\mathrm{\tt (\%o10) }\quad \begin{pmatrix}0 & -\frac{\mathit{D0}-1}{L}\cr \frac{\mathit{D0}-1}{c} & 0\end{pmatrix}\]
(%i11) B: D0 * B1 + (1 - D0) * B2;
\[\mathrm{\tt (\%o11) }\quad \begin{pmatrix}\frac{\mathit{D0}}{L} & 0\cr 0 & -\frac{\mathit{D0}}{c}-\frac{1-\mathit{D0}}{c}\end{pmatrix}\]
(%i12) B: ratsimp(B);
\[\mathrm{\tt (\%o12) }\quad \begin{pmatrix}\frac{\mathit{D0}}{L} & 0\cr 0 & -\frac{1}{c}\end{pmatrix}\]
(%i13) C: D0 * C1 + (1 - D0) * C2;
\[\mathrm{\tt (\%o13) }\quad \begin{pmatrix}0 & 1\end{pmatrix}\]
(%i14) C: ratsimp(C);
\[\mathrm{\tt (\%o14) }\quad \begin{pmatrix}0 & 1\end{pmatrix}\]
(%i15) D: D0 * D1 + (1 - D0) * D2;
\[\mathrm{\tt (\%o15) }\quad \begin{pmatrix}0 & 0\end{pmatrix}\]
(%i16) D: ratsimp(D);
\[\mathrm{\tt (\%o16) }\quad \begin{pmatrix}0 & 0\end{pmatrix}\]
(%i17) X0: matrix([Il], [Vc]);
\[\mathrm{\tt (\%o17) }\quad \begin{pmatrix}\mathit{Il}\cr \mathit{Vc}\end{pmatrix}\]
(%i18) U0: matrix([Vin], [Iout]);
\[\mathrm{\tt (\%o18) }\quad \begin{pmatrix}\mathit{Vin}\cr \mathit{Iout}\end{pmatrix}\]
(%i19) E: (A1 - A2) . X0 + (B1 - B2) . U0;
\[\mathrm{\tt (\%o19) }\quad \begin{pmatrix}\frac{\mathit{Vin}}{L}-\frac{\mathit{Vc}}{L}\cr \frac{\mathit{Il}}{c}\end{pmatrix}\]
(%i20) E: ratsimp(E);
\[\mathrm{\tt (\%o20) }\quad \begin{pmatrix}\frac{\mathit{Vin}-\mathit{Vc}}{L}\cr \frac{\mathit{Il}}{c}\end{pmatrix}\]
(%i21) F: (C1 - C2) . X0 + (D1 - D2) . U0;
\[\mathrm{\tt (\%o21) }\quad 0\]

merging E and F; keep it as it is

(%i22) B: addcol(B, E);
\[\mathrm{\tt (\%o22) }\quad \begin{pmatrix}\frac{\mathit{D0}}{L} & 0 & \frac{\mathit{Vin}-\mathit{Vc}}{L}\cr 0 & -\frac{1}{c} & \frac{\mathit{Il}}{c}\end{pmatrix}\]
(%i23) D: addcol(D, matrix([F]));
\[\mathrm{\tt (\%o23) }\quad \begin{pmatrix}0 & 0 & 0\end{pmatrix}\]

computing transfer functions; keep it as it is

(%i24) S0: s * diagmatrix(2, 1) - A;
\[\mathrm{\tt (\%o24) }\quad \begin{pmatrix}s & \frac{\mathit{D0}-1}{L}\cr -\frac{\mathit{D0}-1}{c} & s\end{pmatrix}\]
(%i25) S: invert(S0);
\[\mathrm{\tt (\%o25) }\quad \begin{pmatrix}\frac{s}{\frac{{{\left( \mathit{D0}-1\right) }^{2}}}{c\cdot L}+{{s}^{2}}} & -\frac{\mathit{D0}-1}{\left( \frac{{{\left( \mathit{D0}-1\right) }^{2}}}{c\cdot L}+{{s}^{2}}\right) \cdot L}\cr \frac{\mathit{D0}-1}{c\cdot \left( \frac{{{\left( \mathit{D0}-1\right) }^{2}}}{c\cdot L}+{{s}^{2}}\right) } & \frac{s}{\frac{{{\left( \mathit{D0}-1\right) }^{2}}}{c\cdot L}+{{s}^{2}}}\end{pmatrix}\]
(%i26) S: ratsimp(S);
\[\mathrm{\tt (\%o26) }\quad \begin{pmatrix}\frac{c\cdot s\cdot L}{c\cdot {{s}^{2}}\cdot L+{{\mathit{D0}}^{2}}-2\cdot \mathit{D0}+1} & -\frac{c\cdot \mathit{D0}-c}{c\cdot {{s}^{2}}\cdot L+{{\mathit{D0}}^{2}}-2\cdot \mathit{D0}+1}\cr \frac{\left( \mathit{D0}-1\right) \cdot L}{c\cdot {{s}^{2}}\cdot L+{{\mathit{D0}}^{2}}-2\cdot \mathit{D0}+1} & \frac{c\cdot s\cdot L}{c\cdot {{s}^{2}}\cdot L+{{\mathit{D0}}^{2}}-2\cdot \mathit{D0}+1}\end{pmatrix}\]
(%i27) tox: S . B;
\[\mathrm{\tt (\%o27) }\quad \begin{pmatrix}\frac{c\cdot s\cdot \mathit{D0}}{c\cdot {{s}^{2}}\cdot L+{{\mathit{D0}}^{2}}-2\cdot \mathit{D0}+1} & \frac{c\cdot \mathit{D0}-c}{c\cdot \left( c\cdot {{s}^{2}}\cdot L+{{\mathit{D0}}^{2}}-2\cdot \mathit{D0}+1\right) } & \frac{c\cdot s\cdot \left( \mathit{Vin}-\mathit{Vc}\right) }{c\cdot {{s}^{2}}\cdot L+{{\mathit{D0}}^{2}}-2\cdot \mathit{D0}+1}-\frac{\mathit{Il}\cdot \left( c\cdot \mathit{D0}-c\right) }{c\cdot \left( c\cdot {{s}^{2}}\cdot L+{{\mathit{D0}}^{2}}-2\cdot \mathit{D0}+1\right) }\cr \frac{\left( \mathit{D0}-1\right) \cdot \mathit{D0}}{c\cdot {{s}^{2}}\cdot L+{{\mathit{D0}}^{2}}-2\cdot \mathit{D0}+1} & -\frac{s\cdot L}{c\cdot {{s}^{2}}\cdot L+{{\mathit{D0}}^{2}}-2\cdot \mathit{D0}+1} & \frac{\mathit{Il}\cdot s\cdot L}{c\cdot {{s}^{2}}\cdot L+{{\mathit{D0}}^{2}}-2\cdot \mathit{D0}+1}+\frac{\left( \mathit{Vin}-\mathit{Vc}\right) \cdot \left( \mathit{D0}-1\right) }{c\cdot {{s}^{2}}\cdot L+{{\mathit{D0}}^{2}}-2\cdot \mathit{D0}+1}\end{pmatrix}\]
(%i28) tox: ratsimp(tox);
\[\mathrm{\tt (\%o28) }\quad \begin{pmatrix}\frac{c\cdot s\cdot \mathit{D0}}{c\cdot {{s}^{2}}\cdot L+{{\mathit{D0}}^{2}}-2\cdot \mathit{D0}+1} & \frac{\mathit{D0}-1}{c\cdot {{s}^{2}}\cdot L+{{\mathit{D0}}^{2}}-2\cdot \mathit{D0}+1} & -\frac{-\mathit{Il}+c\cdot s\cdot \mathit{Vc}-c\cdot s\cdot \mathit{Vin}+\mathit{Il}\cdot \mathit{D0}}{c\cdot {{s}^{2}}\cdot L+{{\mathit{D0}}^{2}}-2\cdot \mathit{D0}+1}\cr \frac{{{\mathit{D0}}^{2}}-\mathit{D0}}{c\cdot {{s}^{2}}\cdot L+{{\mathit{D0}}^{2}}-2\cdot \mathit{D0}+1} & -\frac{s\cdot L}{c\cdot {{s}^{2}}\cdot L+{{\mathit{D0}}^{2}}-2\cdot \mathit{D0}+1} & \frac{\mathit{Vc}-\mathit{Vin}+\left( \mathit{Vin}-\mathit{Vc}\right) \cdot \mathit{D0}+\mathit{Il}\cdot s\cdot L}{c\cdot {{s}^{2}}\cdot L+{{\mathit{D0}}^{2}}-2\cdot \mathit{D0}+1}\end{pmatrix}\]
(%i29) toy: C . S . B + D;
\[\mathrm{\tt (\%o29) }\quad \begin{pmatrix}\frac{\left( \mathit{D0}-1\right) \cdot \mathit{D0}}{c\cdot {{s}^{2}}\cdot L+{{\mathit{D0}}^{2}}-2\cdot \mathit{D0}+1} & -\frac{s\cdot L}{c\cdot {{s}^{2}}\cdot L+{{\mathit{D0}}^{2}}-2\cdot \mathit{D0}+1} & \frac{\mathit{Il}\cdot s\cdot L}{c\cdot {{s}^{2}}\cdot L+{{\mathit{D0}}^{2}}-2\cdot \mathit{D0}+1}+\frac{\left( \mathit{Vin}-\mathit{Vc}\right) \cdot \left( \mathit{D0}-1\right) }{c\cdot {{s}^{2}}\cdot L+{{\mathit{D0}}^{2}}-2\cdot \mathit{D0}+1}\end{pmatrix}\]
(%i30) toy: ratsimp(toy);
\[\mathrm{\tt (\%o30) }\quad \begin{pmatrix}\frac{{{\mathit{D0}}^{2}}-\mathit{D0}}{c\cdot {{s}^{2}}\cdot L+{{\mathit{D0}}^{2}}-2\cdot \mathit{D0}+1} & -\frac{s\cdot L}{c\cdot {{s}^{2}}\cdot L+{{\mathit{D0}}^{2}}-2\cdot \mathit{D0}+1} & \frac{\mathit{Vc}-\mathit{Vin}+\left( \mathit{Vin}-\mathit{Vc}\right) \cdot \mathit{D0}+\mathit{Il}\cdot s\cdot L}{c\cdot {{s}^{2}}\cdot L+{{\mathit{D0}}^{2}}-2\cdot \mathit{D0}+1}\end{pmatrix}\]

substituting values

(%i31) tox1: ev(tox, D0 = 0.5, Vin = 10, Iout = -1, Il = 2, Vc = -10, L = 100e-6, c = 1e-3);
\[\mathrm{\tt (\%o31) }\quad \begin{pmatrix}\frac{5.0\cdot {{10}^{-4}}\cdot s}{1.0\cdot {{10}^{-7}}\cdot {{s}^{2}}+0.25} & -\frac{0.5}{1.0\cdot {{10}^{-7}}\cdot {{s}^{2}}+0.25} & -\frac{-1.0-0.02\cdot s}{1.0\cdot {{10}^{-7}}\cdot {{s}^{2}}+0.25}\cr -\frac{0.25}{1.0\cdot {{10}^{-7}}\cdot {{s}^{2}}+0.25} & -\frac{1.0\cdot {{10}^{-4}}\cdot s}{1.0\cdot {{10}^{-7}}\cdot {{s}^{2}}+0.25} & \frac{2.0\cdot {{10}^{-4}}\cdot s-10.0}{1.0\cdot {{10}^{-7}}\cdot {{s}^{2}}+0.25}\end{pmatrix}\]
(%i32) tox1: ratsimp(tox1);
\[\mbox{}\\\mbox{rat: replaced 5.0E-4 by 1/2000 = 5.0E-4}\mbox{}\\\mbox{rat: replaced 0.25 by 1/4 = 0.25}\mbox{}\\\mbox{rat: replaced 1.0E-7 by 1/10000000 = 1.0E-7}\mbox{}\\\mbox{rat: replaced -0.5 by -1/2 = -0.5}\mbox{}\\\mbox{rat: replaced 0.25 by 1/4 = 0.25}\mbox{}\\\mbox{rat: replaced 1.0E-7 by 1/10000000 = 1.0E-7}\mbox{}\\\mbox{rat: replaced -1.0 by -1/1 = -1.0}\mbox{}\\\mbox{rat: replaced -0.02 by -1/50 = -0.02}\mbox{}\\\mbox{rat: replaced 0.25 by 1/4 = 0.25}\mbox{}\\\mbox{rat: replaced 1.0E-7 by 1/10000000 = 1.0E-7}\mbox{}\\\mbox{rat: replaced -0.25 by -1/4 = -0.25}\mbox{}\\\mbox{rat: replaced 0.25 by 1/4 = 0.25}\mbox{}\\\mbox{rat: replaced 1.0E-7 by 1/10000000 = 1.0E-7}\mbox{}\\\mbox{rat: replaced -1.0E-4 by -1/10000 = -1.0E-4}\mbox{}\\\mbox{rat: replaced 0.25 by 1/4 = 0.25}\mbox{}\\\mbox{rat: replaced 1.0E-7 by 1/10000000 = 1.0E-7}\mbox{}\\\mbox{rat: replaced -10.0 by -10/1 = -10.0}\mbox{}\\\mbox{rat: replaced 2.0E-4 by 1/5000 = 2.0E-4}\mbox{}\\\mbox{rat: replaced 0.25 by 1/4 = 0.25}\mbox{}\\\mbox{rat: replaced 1.0E-7 by 1/10000000 = 1.0E-7}\]\[\mathrm{\tt (\%o32) }\quad \begin{pmatrix}\frac{5000\cdot s}{{{s}^{2}}+2500000} & -\frac{5000000}{{{s}^{2}}+2500000} & \frac{10000000+200000\cdot s}{{{s}^{2}}+2500000}\cr -\frac{2500000}{{{s}^{2}}+2500000} & -\frac{1000\cdot s}{{{s}^{2}}+2500000} & \frac{2000\cdot s-100000000}{{{s}^{2}}+2500000}\end{pmatrix}\]
(%i33) tox1: factor(tox1);
\[\mathrm{\tt (\%o33) }\quad \begin{pmatrix}\frac{5000\cdot s}{{{s}^{2}}+2500000} & -\frac{5000000}{{{s}^{2}}+2500000} & \frac{200000\cdot \left( 50+s\right) }{{{s}^{2}}+2500000}\cr -\frac{2500000}{{{s}^{2}}+2500000} & -\frac{1000\cdot s}{{{s}^{2}}+2500000} & \frac{2000\cdot \left( s-50000\right) }{{{s}^{2}}+2500000}\end{pmatrix}\]
(%i34) tox2: ev(tox, D0 = 0.5, Vin = 10, Iout = -1, Il = 2, Vc = -10, L = 100e-6, c = 1e-3);
\[\mathrm{\tt (\%o34) }\quad \begin{pmatrix}\frac{5.0\cdot {{10}^{-4}}\cdot s}{1.0\cdot {{10}^{-7}}\cdot {{s}^{2}}+0.25} & -\frac{0.5}{1.0\cdot {{10}^{-7}}\cdot {{s}^{2}}+0.25} & -\frac{-1.0-0.02\cdot s}{1.0\cdot {{10}^{-7}}\cdot {{s}^{2}}+0.25}\cr -\frac{0.25}{1.0\cdot {{10}^{-7}}\cdot {{s}^{2}}+0.25} & -\frac{1.0\cdot {{10}^{-4}}\cdot s}{1.0\cdot {{10}^{-7}}\cdot {{s}^{2}}+0.25} & \frac{2.0\cdot {{10}^{-4}}\cdot s-10.0}{1.0\cdot {{10}^{-7}}\cdot {{s}^{2}}+0.25}\end{pmatrix}\]
(%i35) toy1: ev(toy, D0 = 0.5, Vin = 10, Iout = -1, Il = 2, Vc = -10, L = 100e-6, c = 1e-3);
\[\mathrm{\tt (\%o35) }\quad \begin{pmatrix}-\frac{0.25}{1.0\cdot {{10}^{-7}}\cdot {{s}^{2}}+0.25} & -\frac{1.0\cdot {{10}^{-4}}\cdot s}{1.0\cdot {{10}^{-7}}\cdot {{s}^{2}}+0.25} & \frac{2.0\cdot {{10}^{-4}}\cdot s-10.0}{1.0\cdot {{10}^{-7}}\cdot {{s}^{2}}+0.25}\end{pmatrix}\]
(%i36) toy1: ratsimp(toy1);
\[\mbox{}\\\mbox{rat: replaced -0.25 by -1/4 = -0.25}\mbox{}\\\mbox{rat: replaced 0.25 by 1/4 = 0.25}\mbox{}\\\mbox{rat: replaced 1.0E-7 by 1/10000000 = 1.0E-7}\mbox{}\\\mbox{rat: replaced -1.0E-4 by -1/10000 = -1.0E-4}\mbox{}\\\mbox{rat: replaced 0.25 by 1/4 = 0.25}\mbox{}\\\mbox{rat: replaced 1.0E-7 by 1/10000000 = 1.0E-7}\mbox{}\\\mbox{rat: replaced -10.0 by -10/1 = -10.0}\mbox{}\\\mbox{rat: replaced 2.0E-4 by 1/5000 = 2.0E-4}\mbox{}\\\mbox{rat: replaced 0.25 by 1/4 = 0.25}\mbox{}\\\mbox{rat: replaced 1.0E-7 by 1/10000000 = 1.0E-7}\]\[\mathrm{\tt (\%o36) }\quad \begin{pmatrix}-\frac{2500000}{{{s}^{2}}+2500000} & -\frac{1000\cdot s}{{{s}^{2}}+2500000} & \frac{2000\cdot s-100000000}{{{s}^{2}}+2500000}\end{pmatrix}\]
(%i37) toy1: factor(toy1);
\[\mathrm{\tt (\%o37) }\quad \begin{pmatrix}-\frac{2500000}{{{s}^{2}}+2500000} & -\frac{1000\cdot s}{{{s}^{2}}+2500000} & \frac{2000\cdot \left( s-50000\right) }{{{s}^{2}}+2500000}\end{pmatrix}\]
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