Talk:Controllability

Controllability vs Reachability
What historically has been named 'controllability' is ambiguous. Let $$x_0$$ be the equilibrium state (usually taken to be 0) of a system in the absence of an input then.


 * 1) Reachability of an arbitrary state, $$ x_f $$, from an arbitrary state $$ x_i $$ is the ability to transfer from the initial state  $$ x_i $$ to the final state $$ x_f$$  in some time by applying a suitable input.
 * 2) Controllability of an arbitrary state, $$ x $$, is the ability to transfer from this state $$ x $$ to the equilibrium state $$ x_0 $$ in some time by applying a suitable input.

Obviously reachability implies controllability. For linear stystems in continuous time, both concepts are equivalent. However, a discrete time system my be controllable without being reachabable. A trivial example is: $$ x_{k+1} = A x_k+0 \,u_k $$ where $$A$$ is nilpotent.

Mastlab 21:29, 10 September 2006 (UTC)


 * Removed the line referring to not being able to stay in a state once you've reached it, because the ability to stay in a state is implied in the ability to reach any state (including the one you're in) from here. What a state I'm in.  —Preceding unsigned comment added by 76.64.141.149 (talk) 03:17, 27 November 2008 (UTC)


 * Controllability as it is defined in the subsequent math in this article, and in many places in the literature, is simply the ability to reach, but not necessarily maintain, any desired state. So I'm restoring the deleted passage. Loraof (talk) 14:13, 7 May 2015 (UTC)

Page needs attention/improvement
$$ A, B$$ are matrices, but they are treated as vectors. For example, see "n columns of $$ C$$ ", "$$B$$ and $$AB$$ are independent and span the entire plan", etc. --kris (talk) 08:00, 1 June 2009 (UTC)
 * See recent changes. I think dimensionality is now properly handled for the most general case.&mdash;TedPavlic (talk) 12:17, 2 June 2009 (UTC)

This mathematical gobblygook. Save it for the end. Not exactly laymans terms. Controllability in the context of highly complex systems (whether discrete or not, like integrated circuits) is the measure of (abilty) an input stimulus to change the state of internal state (regardless of whether it can be observed - that's observability).--71.245.164.83 (talk) 00:40, 12 September 2010 (UTC)

Seconding the above comment. The formal definition and derivation provided do not provide useful insight into what makes a system controllable nor do they indicate how the choice of A,B,C,D can influence the controllability of a system. Also, key theorems used in practice (e.g. picking specific operating points to judge controllability of the underlying system) would greatly improve the utility of this page. Adding a stab at the former. — Preceding unsigned comment added by 73.154.179.203 (talk) 01:06, 18 March 2021 (UTC)

Output controllability
"A controllable system is not necessarily output controllable"

How is this possible? Is it because D can cancel the effect of C? Is there an example in Katsuhiko Ogata book "Modern Control Engineering"? I think, we should definitely add an example. I guess I'm not the only one who has never thought about that. —Preceding unsigned comment added by FreedomM (talk • contribs) 22:12, 17 April 2010 (UTC)


 * You get a trivial example when $$ C=0,$$ $$ D=0 $$. More generally, The system is not output controllable if $$ D=0 $$ and $$ C $$ does not have full row rank. By the way, output controllablity is not really a very interesting property. &mdash;Malo Hautus 03:44, 16 June 2011 (UTC)


 * I agree that output controllability is not really a very interesting property. Nevertheless, comments like these illustrate why there may be some need to explore the differences between the two. I've added two broad examples that hopefully convince most readers that the two notions of controllability are distinct. &mdash;TedPavlic (talk/contrib/@) 15:34, 16 June 2011 (UTC)

Hautus condition for Controllability
Is there any reason why the Hautus conditions for controllability (and observability) are not mentioned on the respective pages? I cannot speak to the relevance of the lemma outside of my controls class, but homework-wise, this is a very nice way to check for controllability and stability. Kenblu24 (talk) 02:33, 25 October 2022 (UTC)