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.

  • $\begingroup$ 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. $\endgroup$
    – fedja
    Sep 16 '13 at 8:50
  • $\begingroup$ @fedja, Dear Professor, I have edited the question and added an example to clarify my question. Thank you! $\endgroup$ Sep 16 '13 at 9:21

Controllability is an algebraic yes-or-no question: if your linear time-invariant system is controllable, then it is controllable in any finite interval, no matter how small. Its study doesn't require any extra geometric structure of the state space besides what is needed to define differential equations on it.

If you want to discus the compromise between convergence and size of the controls, you need extra structure: a metric on how far the states and the controls are from a desired value. This is done in optimal control theory. Linear-quadratic optimal control for example uses quadratic distances (costs). That is to say, it introduces an Euclidean geometry to the state space. Other geometries and costs are possible. Quadratic costs are simply the easiest ones to deal with. Pick you favorite cost, and search the extensive literature - someone is bound to have obtained useful results.


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