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)?