Is it possible to convert a linear, time-variant (LTV) dynamical system into a time invariant system (LTI)?

Question

Lets consider a dynamical system, whose state transition matrix is defined by $A(t)$, where size of A is $2$ x $2$ matrix. Can't we change $A(t)$ into $\hat{A}$, where $\hat{A}$ is a $3$x$3$ constant matrix, by introducing new state that incorporates time?

Answer 1

It is not possible if you mean a system of the form $$ \dot{x}=Ax+Bu $$ with arbitrary initial condition. However if you are ok with a constant offset like $$ \dot{x}=Ax+Bu+d $$ then it is possible in some very special cases. For example $$ \dot{x}=-x+u+t $$ can be converted by $$ \begin{align} \dot{x}&=-x+y+u\\ \dot{y}&=1 \end{align} $$ Strictly speaking this is not linear but affine. It can even be made linear by introducing another state and fixing initial conditions: $$ \begin{align} \dot{x}&=-x+y+u\\ \dot{y}&=z\\ \dot{z}&=0\\ y(0)&=0\\ z(0)&=1 \end{align} $$ and $x(0)$ arbitrary. This can be extended to more complicated situations. For example $$ \dot{x}=-x+u+\sin(t) $$ can be converted to $$ \begin{align} \dot{x}&=-x+y+u\\ \dot{y}&=z\\ \dot{z}&=-y\\ y(0)&=0\\ z(0)&=1 \end{align} $$ and $x(0)$ arbitrary. If this is useful or not depends on your specific problem.