Controlability of LTI Continuous-Time Systems

The state of a linear time-invariant system $$\dot{x}(\tau )=Ax(\tau )+Bu(\tau ),x(\tau )\in {\mathbb{R}}^n,u(\tau )\in {\mathbb{R}}^m$$ at time $t$, starting from the initial state $x(0)$ and influenced by a continuous input $u(\tau )$, can be expressed as $$x(t)=e^{At}x(0)+{\int }_0^t{e}^{A(t-\tau )}Bu(\tau )d\tau .\text{\hspace{1em}(1)}$$ For square matrices that satisfy their own characteristic equation the Cayley-Hamilton theorem has a consequence that the inifinite series $$e^{At}=I+At+\frac{(At{)}^2}{2!}+\dots$$ can be expressed using a finite number of terms $$e^{At}={\varphi }_0(t)I+{\varphi }_1(t)A+\cdots +{\varphi }_{n-1}(t)A^{n-1}.$$ After substituting it back into the second term of (1) we attain $$x(t)=e^{At}x(0)+{\int }_0^t({\varphi }_0(t-\tau )I+{\varphi }_1(t-\tau )A+\cdots +{\varphi }_{n-1}(t-\tau )A^{n-1})Bu(\tau )d\tau$$ which can be further manipulated into the form $$x(t)=e^{At}x(0)+R{\int }_0^t\left[\begin{array}{c}{\varphi }_0(t-\tau )u(\tau )\\ {\varphi }_1(t-\tau )u(\tau )\\ ⋮\\ {\varphi }_{n-1}(t-\tau )u(\tau )\end{array}\right]d\tau ,$$ where $$R=\left[\begin{array}{cccc}B& AB& \dots & A^{n-1}B\end{array}\right]$$ is the controllability matrix. If $R$ is rank deficient, some directions in the state-space cannot be effected by the inputs and therefore the system is uncontrollable.