问题提出
对于如图所示的电路而言，包括一个电阻 R R R，电感 L L L和电容 C C C，以及电压源 v ( t ) v(t) v(t)。
根据基尔霍夫电压定律(Kirchhoff’s Voltage Law, KVL)，串联电路中各元件的电压与电压源的电压相等，电流则处处相等
v ( t ) = R ⋅ i ( t ) + L d i ( t ) d t + v C ( t ) d i ( t ) d t = − R L i ( t ) − 1 L v C ( t ) + 1 L v ( t ) \begin{aligned} v(t)&=R\cdot i(t)+L\frac{\text di(t)}{\text dt}+v_C(t)\\ \frac{\text di(t)}{\text dt}&=-\frac{R}{L}i(t)-\frac{1}{L}v_C(t)+\frac{1}{L}v(t) \end{aligned} v(t)dtdi(t)=R⋅i(t)+Ldtdi(t)+vC(t)=−LRi(t)−L1vC(t)+L1v(t)
其中，电容中的电压由充放电过程决定，可以记作
v C ( t ) = 1 C ∫ i ( t ) d t d v C ( t ) d t = d v C ( t ) d t = i ( t ) C \begin{aligned} v_C(t)&=\frac{1}{C}\int i(t)\text dt\\ \frac{\text dv_C(t)}{\text dt}&=\frac{\text dv_C(t)}{\text dt}=\frac{i(t)}{C} \end{aligned} vC(t)dtdvC(t)=C1∫i(t)dt=dtdvC(t)=Ci(t)
状态空间表示
状态空间可用于表示线性控制系统，其形式为
X ′ ( t ) = A X ( t ) + B ∗ u ( t ) Y ( t ) = C X ( t ) + D ∗ u ( t ) X'(t)=AX(t)+B*u(t)\\ Y(t)=CX(t)+D*u(t) X′(t)=AX(t)+B∗u(t)Y(t)=CX(t)+D∗u(t)
其中， A , B , C , D A,B,C,D A,B,C,D均为状态空间矩阵。接下来，将RLC电路的方程改写为状态空间的形式。
由于此前已经构造了关于电流 i i i和电容中的电压 v C v_C vC的微分方程组，故而可将二者记作
X ( t ) = [ i ( t ) v C ( t ) ] X(t)=\begin{bmatrix}i(t)\\v_C(t)\end{bmatrix} X(t)=[i(t)vC(t)]
则微分方程组可以改写为
X ˙ ( t ) = [ − R L − 1 L 1 C 0 ] X ( t ) + [ 1 L 0 ] v ( t ) \dot X(t)=\begin{bmatrix} -\frac{R}{L}&-\frac{1}{L}\\ \frac{1}{C}&0\end{bmatrix} X(t)+ \begin{bmatrix}\frac{1}{L}\\0\end{bmatrix}v(t) X˙(t)=[−LRC1−L10]X(t)+[L10]v(t)
从而 A = [ − R L − 1 L 1 C 0 ] , B = [ 1 L 0 ] A=\begin{bmatrix}-\frac{R}{L}&-\frac{1}{L}\\\frac{1}{C}&0\end{bmatrix}, B=\begin{bmatrix}\frac{1}{L}\\0\end{bmatrix} A=[−LRC1−L10],B=[L10]，令 C = [ 0 1 ] , D = [ 0 ] C=\begin{bmatrix}0&1\end{bmatrix}, D=[0] C=[01],D=[0]，则 Y ( t ) Y(t) Y(t)可构造为
Y ( t ) = [ 0 1 ] X ( t ) + [ 0 ] v ( t ) Y(t)=\begin{bmatrix}0&1\end{bmatrix}X(t)+[0]v(t) Y(t)=[01]X(t)+[0]v(t)
sympy代码
【sympy】的【control】模块提供了状态空间类【StateSpace】，RLC电路的状态空间为
StateSpace ( [ − R L − 1 L 1 C 0 ] , [ 1 L 0 ] , [ 0 1 ] , [ 0 ] ) \operatorname{StateSpace}\left(\left[\begin{matrix}- \frac{R}{L} & - \frac{1}{L}\\\frac{1}{C} & 0\end{matrix}\right], \left[\begin{matrix}\frac{1}{L}\\0\end{matrix}\right], \left[\begin{matrix}0 & 1\end{matrix}\right], \left[\begin{matrix}0\end{matrix}\right]\right) StateSpace([−LRC1−L10],[L10],[01],[0])
构造过程如下
from sympy import Matrix, symbols, print_latex
from sympy.physics.control import *
R, L, C = symbols('R L C')
A = Matrix([[-R/L, -1/L], [1/C, 0]])
B = Matrix([[1/L], [0]])
C,D = Matrix([[0, 1]]), Matrix([[0]])
ss = StateSpace(A, B, C, D)
print_latex(ss)
【dsolve】是状态空间的求解方法，若想求出RLC电路的状态方程，则需给出 X ( t ) , v ( t ) X(t), v(t) X(t),v(t)的初值，
U = symbols('U')
i = Matrix([0, 0])
u0 = Matrix([U])
res = ss.dsolve(i, u0).simplify()
print_latex(res)
结果如下，非常抽象
[ { U ( − C 4 R 7 e t ( R 2 + C ( C R 2 − 4 L ) C ) L + C 4 R 7 e t ( R + C ( C R 2 − 4 L ) 2 C ) L + C 3 L R 5 e R t 2 L + 8 C 3 L R 5 e t ( R 2 + C ( C R 2 − 4 L ) C ) L − 9 C 3 L R 5 e t ( R + C ( C R 2 − 4 L ) 2 C ) L − C 3 R 6 C ( C R 2 − 4 L ) e t ( R 2 + C ( C R 2 − 4 L ) C ) L + C 3 R 6 C ( C R 2 − 4 L ) e t ( R + C ( C R 2 − 4 L ) 2 C ) L − 6 C 2 L 2 R 3 e R t 2 L − 19 C 2 L 2 R 3 e t ( R 2 + C ( C R 2 − 4 L ) C ) L + 25 C 2 L 2 R 3 e t ( R + C ( C R 2 − 4 L ) 2 C ) L + C 2 L R 4 C ( C R 2 − 4 L ) e R t 2 L + 6 C 2 L R 4 C ( C R 2 − 4 L ) e t ( R 2 + C ( C R 2 − 4 L ) C ) L − 7 C 2 L R 4 C ( C R 2 − 4 L ) e t ( R + C ( C R 2 − 4 L ) 2 C ) L + 8 C L 3 R e R t 2 L + 12 C L 3 R e t ( R 2 + C ( C R 2 − 4 L ) C ) L − 20 C L 3 R e t ( R + C ( C R 2 − 4 L ) 2 C ) L − 4 C L 2 R 2 C ( C R 2 − 4 L ) e R t 2 L − 9 C L 2 R 2 C ( C R 2 − 4 L ) e t ( R 2 + C ( C R 2 − 4 L ) C ) L + 13 C L 2 R 2 C ( C R 2 − 4 L ) e t ( R + C ( C R 2 − 4 L ) 2 C ) L + 2 L 3 C ( C R 2 − 4 L ) e R t 2 L + 2 L 3 C ( C R 2 − 4 L ) e t ( R 2 + C ( C R 2 − 4 L ) C ) L − 4 L 3 C ( C R 2 − 4 L ) e t ( R + C ( C R 2 − 4 L ) 2 C ) L ) e − t ( R + C ( C R 2 − 4 L ) 2 C ) L C 4 R 7 − 9 C 3 L R 5 + C 3 R 6 C ( C R 2 − 4 L ) + 25 C 2 L 2 R 3 − 7 C 2 L R 4 C ( C R 2 − 4 L ) − 20 C L 3 R + 13 C L 2 R 2 C ( C R 2 − 4 L ) − 4 L 3 C ( C R 2 − 4 L ) for − C R 2 + 4 L + R C ( C R 2 − 4 L ) ≠ 0 ∧ C R 2 − 4 L + R C ( C R 2 − 4 L ) ≠ 0 U ( C L R e t ( R + C ( C R 2 − 4 L ) 2 C ) L − C L R e t ( 3 R 2 + C ( C R 2 − 4 L ) C ) L + C R 2 t e t ( R + 3 C ( C R 2 − 4 L ) 2 C ) L − 2 L t e t ( R + 3 C ( C R 2 − 4 L ) 2 C ) L + L C ( C R 2 − 4 L ) e t ( R + C ( C R 2 − 4 L ) 2 C ) L − L C ( C R 2 − 4 L ) e t ( 3 R 2 + C ( C R 2 − 4 L ) C ) L + R t C ( C R 2 − 4 L ) e t ( R + 3 C ( C R 2 − 4 L ) 2 C ) L ) e − t ( 3 R 2 + C ( C R 2 − 4 L ) C ) L C 2 R 3 − 4 C L R + C R 2 C ( C R 2 − 4 L ) − 2 L C ( C R 2 − 4 L ) for C R 2 − 4 L + R C ( C R 2 − 4 L ) ≠ 0 U ( − C 4 R 7 e t ( R 2 + C ( C R 2 − 4 L ) C ) L + C 4 R 7 e t ( R + C ( C R 2 − 4 L ) 2 C ) L + 8 C 3 L R 5 e t ( R 2 + C ( C R 2 − 4 L ) C ) L − 8 C 3 L R 5 e t ( R + C ( C R 2 − 4 L ) 2 C ) L − C 3 R 6 t e R t 2 L − C 3 R 6 C ( C R 2 − 4 L ) e t ( R 2 + C ( C R 2 − 4 L ) C ) L + C 3 R 6 C ( C R 2 − 4 L ) e t ( R + C ( C R 2 − 4 L ) 2 C ) L − 19 C 2 L 2 R 3 e t ( R 2 + C ( C R 2 − 4 L ) C ) L + 19 C 2 L 2 R 3 e t ( R + C ( C R 2 − 4 L ) 2 C ) L + 7 C 2 L R 4 t e R t 2 L + 6 C 2 L R 4 C ( C R 2 − 4 L ) e t ( R 2 + C ( C R 2 − 4 L ) C ) L − 6 C 2 L R 4 C ( C R 2 − 4 L ) e t ( R + C ( C R 2 − 4 L ) 2 C ) L − C 2 R 5 t C ( C R 2 − 4 L ) e R t 2 L + 12 C L 3 R e t ( R 2 + C ( C R 2 − 4 L ) C ) L − 12 C L 3 R e t ( R + C ( C R 2 − 4 L ) 2 C ) L − 13 C L 2 R 2 t e R t 2 L − 9 C L 2 R 2 C ( C R 2 − 4 L ) e t ( R 2 + C ( C R 2 − 4 L ) C ) L + 9 C L 2 R 2 C ( C R 2 − 4 L ) e t ( R + C ( C R 2 − 4 L ) 2 C ) L + 5 C L R 3 t C ( C R 2 − 4 L ) e R t 2 L + 4 L 3 t e R t 2 L + 2 L 3 C ( C R 2 − 4 L ) e t ( R 2 + C ( C R 2 − 4 L ) C ) L − 2 L 3 C ( C R 2 − 4 L ) e t ( R + C ( C R 2 − 4 L ) 2 C ) L − 5 L 2 R t C ( C R 2 − 4 L ) e R t 2 L ) e − t ( R + C ( C R 2 − 4 L ) 2 C ) L C 4 R 7 − 9 C 3 L R 5 + C 3 R 6 C ( C R 2 − 4 L ) + 25 C 2 L 2 R 3 − 7 C 2 L R 4 C ( C R 2 − 4 L ) − 20 C L 3 R + 13 C L 2 R 2 C ( C R 2 − 4 L ) − 4 L 3 C ( C R 2 − 4 L ) for − C R 2 + 4 L + R C ( C R 2 − 4 L ) ≠ 0 U t C ( C R 2 − 4 L ) ( e t C ( C R 2 − 4 L ) C L − 1 ) e − t ( R + C ( C R 2 − 4 L ) C ) 2 L C ( C R 2 − 4 L ) otherwise ] \left[\begin{matrix}\begin{cases} \frac{U \left(- C^{4} R^{7} e^{\frac{t \left(\frac{R}{2} + \frac{\sqrt{C \left(C R^{2} - 4 L\right)}}{C}\right)}{L}} + C^{4} R^{7} e^{\frac{t \left(R + \frac{\sqrt{C \left(C R^{2} - 4 L\right)}}{2 C}\right)}{L}} + C^{3} L R^{5} e^{\frac{R t}{2 L}} + 8 C^{3} L R^{5} e^{\frac{t \left(\frac{R}{2} + \frac{\sqrt{C \left(C R^{2} - 4 L\right)}}{C}\right)}{L}} - 9 C^{3} L R^{5} e^{\frac{t \left(R + \frac{\sqrt{C \left(C R^{2} - 4 L\right)}}{2 C}\right)}{L}} - C^{3} R^{6} \sqrt{C \left(C R^{2} - 4 L\right)} e^{\frac{t \left(\frac{R}{2} + \frac{\sqrt{C \left(C R^{2} - 4 L\right)}}{C}\right)}{L}} + C^{3} R^{6} \sqrt{C \left(C R^{2} - 4 L\right)} e^{\frac{t \left(R + \frac{\sqrt{C \left(C R^{2} - 4 L\right)}}{2 C}\right)}{L}} - 6 C^{2} L^{2} R^{3} e^{\frac{R t}{2 L}} - 19 C^{2} L^{2} R^{3} e^{\frac{t \left(\frac{R}{2} + \frac{\sqrt{C \left(C R^{2} - 4 L\right)}}{C}\right)}{L}} + 25 C^{2} L^{2} R^{3} e^{\frac{t \left(R + \frac{\sqrt{C \left(C R^{2} - 4 L\right)}}{2 C}\right)}{L}} + C^{2} L R^{4} \sqrt{C \left(C R^{2} - 4 L\right)} e^{\frac{R t}{2 L}} + 6 C^{2} L R^{4} \sqrt{C \left(C R^{2} - 4 L\right)} e^{\frac{t \left(\frac{R}{2} + \frac{\sqrt{C \left(C R^{2} - 4 L\right)}}{C}\right)}{L}} - 7 C^{2} L R^{4} \sqrt{C \left(C R^{2} - 4 L\right)} e^{\frac{t \left(R + \frac{\sqrt{C \left(C R^{2} - 4 L\right)}}{2 C}\right)}{L}} + 8 C L^{3} R e^{\frac{R t}{2 L}} + 12 C L^{3} R e^{\frac{t \left(\frac{R}{2} + \frac{\sqrt{C \left(C R^{2} - 4 L\right)}}{C}\right)}{L}} - 20 C L^{3} R e^{\frac{t \left(R + \frac{\sqrt{C \left(C R^{2} - 4 L\right)}}{2 C}\right)}{L}} - 4 C L^{2} R^{2} \sqrt{C \left(C R^{2} - 4 L\right)} e^{\frac{R t}{2 L}} - 9 C L^{2} R^{2} \sqrt{C \left(C R^{2} - 4 L\right)} e^{\frac{t \left(\frac{R}{2} + \frac{\sqrt{C \left(C R^{2} - 4 L\right)}}{C}\right)}{L}} + 13 C L^{2} R^{2} \sqrt{C \left(C R^{2} - 4 L\right)} e^{\frac{t \left(R + \frac{\sqrt{C \left(C R^{2} - 4 L\right)}}{2 C}\right)}{L}} + 2 L^{3} \sqrt{C \left(C R^{2} - 4 L\right)} e^{\frac{R t}{2 L}} + 2 L^{3} \sqrt{C \left(C R^{2} - 4 L\right)} e^{\frac{t \left(\frac{R}{2} + \frac{\sqrt{C \left(C R^{2} - 4 L\right)}}{C}\right)}{L}} - 4 L^{3} \sqrt{C \left(C R^{2} - 4 L\right)} e^{\frac{t \left(R + \frac{\sqrt{C \left(C R^{2} - 4 L\right)}}{2 C}\right)}{L}}\right) e^{- \frac{t \left(R + \frac{\sqrt{C \left(C R^{2} - 4 L\right)}}{2 C}\right)}{L}}}{C^{4} R^{7} - 9 C^{3} L R^{5} + C^{3} R^{6} \sqrt{C \left(C R^{2} - 4 L\right)} + 25 C^{2} L^{2} R^{3} - 7 C^{2} L R^{4} \sqrt{C \left(C R^{2} - 4 L\right)} - 20 C L^{3} R + 13 C L^{2} R^{2} \sqrt{C \left(C R^{2} - 4 L\right)} - 4 L^{3} \sqrt{C \left(C R^{2} - 4 L\right)}} & \text{for}\: - C R^{2} + 4 L + R \sqrt{C \left(C R^{2} - 4 L\right)} \neq 0 \wedge C R^{2} - 4 L + R \sqrt{C \left(C R^{2} - 4 L\right)} \neq 0 \\\frac{U \left(C L R e^{\frac{t \left(R + \frac{\sqrt{C \left(C R^{2} - 4 L\right)}}{2 C}\right)}{L}} - C L R e^{\frac{t \left(\frac{3 R}{2} + \frac{\sqrt{C \left(C R^{2} - 4 L\right)}}{C}\right)}{L}} + C R^{2} t e^{\frac{t \left(R + \frac{3 \sqrt{C \left(C R^{2} - 4 L\right)}}{2 C}\right)}{L}} - 2 L t e^{\frac{t \left(R + \frac{3 \sqrt{C \left(C R^{2} - 4 L\right)}}{2 C}\right)}{L}} + L \sqrt{C \left(C R^{2} - 4 L\right)} e^{\frac{t \left(R + \frac{\sqrt{C \left(C R^{2} - 4 L\right)}}{2 C}\right)}{L}} - L \sqrt{C \left(C R^{2} - 4 L\right)} e^{\frac{t \left(\frac{3 R}{2} + \frac{\sqrt{C \left(C R^{2} - 4 L\right)}}{C}\right)}{L}} + R t \sqrt{C \left(C R^{2} - 4 L\right)} e^{\frac{t \left(R + \frac{3 \sqrt{C \left(C R^{2} - 4 L\right)}}{2 C}\right)}{L}}\right) e^{- \frac{t \left(\frac{3 R}{2} + \frac{\sqrt{C \left(C R^{2} - 4 L\right)}}{C}\right)}{L}}}{C^{2} R^{3} - 4 C L R + C R^{2} \sqrt{C \left(C R^{2} - 4 L\right)} - 2 L \sqrt{C \left(C R^{2} - 4 L\right)}} & \text{for}\: C R^{2} - 4 L + R \sqrt{C \left(C R^{2} - 4 L\right)} \neq 0 \\\frac{U \left(- C^{4} R^{7} e^{\frac{t \left(\frac{R}{2} + \frac{\sqrt{C \left(C R^{2} - 4 L\right)}}{C}\right)}{L}} + C^{4} R^{7} e^{\frac{t \left(R + \frac{\sqrt{C \left(C R^{2} - 4 L\right)}}{2 C}\right)}{L}} + 8 C^{3} L R^{5} e^{\frac{t \left(\frac{R}{2} + \frac{\sqrt{C \left(C R^{2} - 4 L\right)}}{C}\right)}{L}} - 8 C^{3} L R^{5} e^{\frac{t \left(R + \frac{\sqrt{C \left(C R^{2} - 4 L\right)}}{2 C}\right)}{L}} - C^{3} R^{6} t e^{\frac{R t}{2 L}} - C^{3} R^{6} \sqrt{C \left(C R^{2} - 4 L\right)} e^{\frac{t \left(\frac{R}{2} + \frac{\sqrt{C \left(C R^{2} - 4 L\right)}}{C}\right)}{L}} + C^{3} R^{6} \sqrt{C \left(C R^{2} - 4 L\right)} e^{\frac{t \left(R + \frac{\sqrt{C \left(C R^{2} - 4 L\right)}}{2 C}\right)}{L}} - 19 C^{2} L^{2} R^{3} e^{\frac{t \left(\frac{R}{2} + \frac{\sqrt{C \left(C R^{2} - 4 L\right)}}{C}\right)}{L}} + 19 C^{2} L^{2} R^{3} e^{\frac{t \left(R + \frac{\sqrt{C \left(C R^{2} - 4 L\right)}}{2 C}\right)}{L}} + 7 C^{2} L R^{4} t e^{\frac{R t}{2 L}} + 6 C^{2} L R^{4} \sqrt{C \left(C R^{2} - 4 L\right)} e^{\frac{t \left(\frac{R}{2} + \frac{\sqrt{C \left(C R^{2} - 4 L\right)}}{C}\right)}{L}} - 6 C^{2} L R^{4} \sqrt{C \left(C R^{2} - 4 L\right)} e^{\frac{t \left(R + \frac{\sqrt{C \left(C R^{2} - 4 L\right)}}{2 C}\right)}{L}} - C^{2} R^{5} t \sqrt{C \left(C R^{2} - 4 L\right)} e^{\frac{R t}{2 L}} + 12 C L^{3} R e^{\frac{t \left(\frac{R}{2} + \frac{\sqrt{C \left(C R^{2} - 4 L\right)}}{C}\right)}{L}} - 12 C L^{3} R e^{\frac{t \left(R + \frac{\sqrt{C \left(C R^{2} - 4 L\right)}}{2 C}\right)}{L}} - 13 C L^{2} R^{2} t e^{\frac{R t}{2 L}} - 9 C L^{2} R^{2} \sqrt{C \left(C R^{2} - 4 L\right)} e^{\frac{t \left(\frac{R}{2} + \frac{\sqrt{C \left(C R^{2} - 4 L\right)}}{C}\right)}{L}} + 9 C L^{2} R^{2} \sqrt{C \left(C R^{2} - 4 L\right)} e^{\frac{t \left(R + \frac{\sqrt{C \left(C R^{2} - 4 L\right)}}{2 C}\right)}{L}} + 5 C L R^{3} t \sqrt{C \left(C R^{2} - 4 L\right)} e^{\frac{R t}{2 L}} + 4 L^{3} t e^{\frac{R t}{2 L}} + 2 L^{3} \sqrt{C \left(C R^{2} - 4 L\right)} e^{\frac{t \left(\frac{R}{2} + \frac{\sqrt{C \left(C R^{2} - 4 L\right)}}{C}\right)}{L}} - 2 L^{3} \sqrt{C \left(C R^{2} - 4 L\right)} e^{\frac{t \left(R + \frac{\sqrt{C \left(C R^{2} - 4 L\right)}}{2 C}\right)}{L}} - 5 L^{2} R t \sqrt{C \left(C R^{2} - 4 L\right)} e^{\frac{R t}{2 L}}\right) e^{- \frac{t \left(R + \frac{\sqrt{C \left(C R^{2} - 4 L\right)}}{2 C}\right)}{L}}}{C^{4} R^{7} - 9 C^{3} L R^{5} + C^{3} R^{6} \sqrt{C \left(C R^{2} - 4 L\right)} + 25 C^{2} L^{2} R^{3} - 7 C^{2} L R^{4} \sqrt{C \left(C R^{2} - 4 L\right)} - 20 C L^{3} R + 13 C L^{2} R^{2} \sqrt{C \left(C R^{2} - 4 L\right)} - 4 L^{3} \sqrt{C \left(C R^{2} - 4 L\right)}} & \text{for}\: - C R^{2} + 4 L + R \sqrt{C \left(C R^{2} - 4 L\right)} \neq 0 \\\frac{U t \sqrt{C \left(C R^{2} - 4 L\right)} \left(e^{\frac{t \sqrt{C \left(C R^{2} - 4 L\right)}}{C L}} - 1\right) e^{- \frac{t \left(R + \frac{\sqrt{C \left(C R^{2} - 4 L\right)}}{C}\right)}{2 L}}}{C \left(C R^{2} - 4 L\right)} & \text{otherwise} \end{cases}\end{matrix}\right] ⎩ ⎨ ⎧C4R7−9C3LR5+C3R6C(CR2−4L)+25C2L2R3−7C2LR4C(CR2−4L)−20CL3R+13CL2R2C(CR2−4L)−4L3C(CR2−4L)U −C4R7eLt(2R+CC(CR2−4L))+C4R7eLt(R+2CC(CR2−4L))+C3LR5e2LRt+8C3LR5eLt(2R+CC(CR2−4L))−9C3LR5eLt(R+2CC(CR2−4L))−C3R6C(CR2−4L)eLt(2R+CC(CR2−4L))+C3R6C(CR2−4L)eLt(R+2CC(CR2−4L))−6C2L2R3e2LRt−19C2L2R3eLt(2R+CC(CR2−4L))+25C2L2R3eLt(R+2CC(CR2−4L))+C2LR4C(CR2−4L)e2LRt+6C2LR4C(CR2−4L)eLt(2R+CC(CR2−4L))−7C2LR4C(CR2−4L)eLt(R+2CC(CR2−4L))+8CL3Re2LRt+12CL3ReLt(2R+CC(CR2−4L))−20CL3ReLt(R+2CC(CR2−4L))−4CL2R2C(CR2−4L)e2LRt−9CL2R2C(CR2−4L)eLt(2R+CC(CR2−4L))+13CL2R2C(CR2−4L)eLt(R+2CC(CR2−4L))+2L3C(CR2−4L)e2LRt+2L3C(CR2−4L)eLt(2R+CC(CR2−4L))−4L3C(CR2−4L)eLt(R+2CC(CR2−4L)) e−Lt(R+2CC(CR2−4L))C2R3−4CLR+CR2C(CR2−4L)−2LC(CR2−4L)U CLReLt(R+2CC(CR2−4L))−CLReLt(23R+CC(CR2−4L))+CR2teLt(R+2C3C(CR2−4L))−2LteLt(R+2C3C(CR2−4L))+LC(CR2−4L)eLt(R+2CC(CR2−4L))−LC(CR2−4L)eLt(23R+CC(CR2−4L))+RtC(CR2−4L)eLt(R+2C3C(CR2−4L)) e−Lt(23R+CC(CR2−4L))C4R7−9C3LR5+C3R6C(CR2−4L)+25C2L2R3−7C2LR4C(CR2−4L)−20CL3R+13CL2R2C(CR2−4L)−4L3C(CR2−4L)U −C4R7eLt(2R+CC(CR2−4L))+C4R7eLt(R+2CC(CR2−4L))+8C3LR5eLt(2R+CC(CR2−4L))−8C3LR5eLt(R+2CC(CR2−4L))−C3R6te2LRt−C3R6C(CR2−4L)eLt(2R+CC(CR2−4L))+C3R6C(CR2−4L)eLt(R+2CC(CR2−4L))−19C2L2R3eLt(2R+CC(CR2−4L))+19C2L2R3eLt(R+2CC(CR2−4L))+7C2LR4te2LRt+6C2LR4C(CR2−4L)eLt(2R+CC(CR2−4L))−6C2LR4C(CR2−4L)eLt(R+2CC(CR2−4L))−C2R5tC(CR2−4L)e2LRt+12CL3ReLt(2R+CC(CR2−4L))−12CL3ReLt(R+2CC(CR2−4L))−13CL2R2te2LRt−9CL2R2C(CR2−4L)eLt(2R+CC(CR2−4L))+9CL2R2C(CR2−4L)eLt(R+2CC(CR2−4L))+5CLR3tC(CR2−4L)e2LRt+4L3te2LRt+2L3C(CR2−4L)eLt(2R+CC(CR2−4L))−2L3C(CR2−4L)eLt(R+2CC(CR2−4L))−5L2RtC(CR2−4L)e2LRt e−Lt(R+2CC(CR2−4L))C(CR2−4L)UtC(CR2−4L)(eCLtC(CR2−4L)−1)e−2Lt(R+CC(CR2−4L))for−CR2+4L+RC(CR2−4L)=0∧CR2−4L+RC(CR2−4L)=0forCR2−4L+RC(CR2−4L)=0for−CR2+4L+RC(CR2−4L)=0otherwise
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