# More than controlability: Speed of controllability!

Consider the continuous linear time-invariant system $$\begin{array}{l} \dot{\mathbf{x}}(t) = A \mathbf{x}(t) + B \mathbf{u}(t)\\ \mathbf{y}(t) = C \mathbf{x}(t) + D \mathbf{u}(t) \end{array}$$ where $\mathbf{x}$ is the $n \times 1$ state vector, $\mathbf{y}$ is the $m \times 1$ output vector, $\mathbf{u}$ is the $r \times 1$ input (or control) vector, $A$ is the $n \times n$ state matrix, $B$ is the $n \times r$ input matrix, $C$ is the $m \times n$ output matrix, $D$ is the $m \times r$ feedthrough (or feedforward) matrix.

For this system, we say it is controllable iff the controllability matrix $$R = \begin{bmatrix}B & AB & A^{2}B & ...& A^{n-1}B\end{bmatrix}$$ is full rank, i.e. for any $\mathbf{x}_0,\mathbf{x}_1\in {\mathbb R}^n$ and $T>0$, there exist a control $\mathbf{u}$, such that for the system $$\begin{array}{l} \dot{\mathbf{x}}(t) = A \mathbf{x}(t) + B \mathbf{u}(t)\\ \mathbf{x}(0) = \mathbf{x}_0 \end{array}$$ we have $\mathbf{x}(T)=\mathbf{x}_1$.

Beside this, I am looking for a condition on the speed of reaching vs boundedness of $\mathbf{u}$, i.e. for a given speed value $V$ and a bound $u_\max$, what is the condition on $A$ and $B$, such that for any given $\mathbf{x}_0,\mathbf{x}_1\in {\mathbb R}^n$, there exist a control input $\mathbf{u}$ and a final time $T$, where $||\mathbf{u}||_\infty\le u_\max$ and $T\le \frac{||\mathbf{x}_1-\mathbf{x}_0||}{V}$, such that for the system $$\begin{array}{l} \dot{\mathbf{x}}(t) = A \mathbf{x}(t) + B \mathbf{u}(t)\\ \mathbf{x}(0) = \mathbf{x}_0 \end{array}$$ we have $\mathbf{x}(T)=\mathbf{x}_1$.

A special case: Let $A = 0$ and $B = \frac{V}{u_\max} I_n$. So the resulting system is $$\dot{\mathbf{x}}(t) = \frac{V}{u_\max}\mathbf{u}(t).$$ This system satisfies the requested property. The question is a generalization for systems with similar property.

It seems to be a standard question which might be found in the texts, so in the case of any references please mention the reference, either books or papers.

• Let's try to get rid of typos first: $||\mathbf{x}||_\infty\le u_\max$ should really be $\|\mathbf{u}\|_\infty\le u_\max$, right? Also, if $x_0$ or $x_1$ are large, $u$ has to be large too (or the control may not be even noticeable on the background of the main dynamics), so you, probably, want to change the order of quantifiers to avoid stupid counterexamples. Sep 16 '13 at 8:50
• @fedja, Dear Professor, I have edited the question and added an example to clarify my question. Thank you! Sep 16 '13 at 9:21