Differential equation to state space representation

Question

Suppose I have a dynamic system in the form: $$ a \, \dot{x}_1 + b \, \dot{x}_2 + c \,x_1 + d \, x_2 + e = u_1 \\ f \, \dot{x}_1 + g \, \dot{x}_2 + h \,x_1 + i \,x_2 + j = u_2 $$ How would I transform this system into a state space representation?

I would usually isolate for \$\dot{x}_1\$ or \$\dot{x}_2\$, but in this case, they are both a function of each other, and this is a MIMO system. Would one way be to rewrite these equations in matrix form and isolate for the \$[\dot{x}_1; \, \dot{x}_2]\$ vector (assuming matrix \$[a \; b; f \; g]\$ is not singular of course)?

Answer 1

You have

$$ a \, \dot{x}_1 + b \, \dot{x}_2 + c \,x_1 + d \, x_2 + e = u_1 \\ f \, \dot{x}_1 + g \, \dot{x}_2 + h \,x_1 + i \,x_2 + j = u_2 $$ That can be re written as

$$ a \, \dot{x}_1 + b \, \dot{x}_2 + c \,x_1 + d \, x_2 = u_1-e \\ f \, \dot{x}_1 + g \, \dot{x}_2 + h \,x_1 + i \,x_2 = u_2-j $$ And then $$ H = \begin{bmatrix} a & b \\f & g\end{bmatrix}, \quad -A = \begin{bmatrix} c & d \\h & i\end{bmatrix}, \quad \tilde{u} = \begin{bmatrix} u_1 -e \\ u_2 -j \end{bmatrix} \text{ and } x = \begin{bmatrix}x_1 \\ x_2\end{bmatrix}.\\ H \dot{x} - Ax = \tilde{u},\\ H \dot{x} = Ax + \tilde{u}. $$ That representation should be somewhat familiar to you except that it is a descriptor representation due to the \$H\$ matrix. If that matrix is invertible you will be able to find the usual \$\dot{x} = Ax+Bu\$. But if it is not invertible then you should dig into descriptor representation theory and find out when you can find a state-space representation to them. Or read about Differential-algebraic system of equations. I guess that \$H\$ having a left-inverse would be a must, but I don't know a thing about the descriptor representation.