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| '''Controllability''' is an important property of a [[control system]], and the controllability property plays a crucial role in many control problems, such as stabilization of [[BIBO stability|unstable systems]] by feedback, or optimal control.
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| Controllability and [[observability]] are [[duality (mathematics)|dual]] aspects of the same problem.
| | Blogging seems regarding gaining appeal and losing it, depends upon who you are and what your expertise in. To be a very good writer for a blog has its perks but so does being very personable. Should your blog be considered driftless and inconsequential because you are not a good writer or a tech expert? Not quite.<br><br>Purchasing traffic is very popular these a number of days. When a marketer boasts hundreds of thousands in revenue every month, many bet the child is buying traffic 1 source or another.<br><br>Web space and bandwidth are item important things to consider. When you select a hosting provider, you should certainly consider web space and bandwidth certain. Most of the hosting providers offer unlimited bandwidth and unlimited web space. Prepared to select a hosting service agency which gives limited web space or limited information.<br><br>While web traffic is all fine and dandy, the traffic that can yield the most results is targeted web web traffic visitors. Why is this? Well among the rules of promoting is to get traffic in the neighborhood . targeted as part of your specific product or business. If you don't find targeted traffic, you may have a problem converting your traffic into sales and profit. Understand way to try to to this is simply by using a targeted targeted traffic spurning call campaign.<br><br>Now utilizing the proper just how an individual can contain a tool like Facebook and develop a successful business without spending a red cent. Besides social media, there are tools like YouTube and EzineArticles to achieve the word along with. While these tools require a fair amount of your energy and effort, they will get you great results without the costs of a standard campaign.<br><br>For the most part, can actually submit your site content to ezines that accept text format with some HTML included. You can use some HTML to make your article look the more organized and that's include links when authority. Further, be sure incorporate at least one link in the "resource" section (short bio) that frequently a separate paragraph range. Otherwise, you can place bio individuals of the number one body i have told.<br><br>So you need to start to surf the pattern by now, having your link into public view is a crucial element to get website road traffic. If you can get url viewed are able to boost to apply far beyond imagination. You're finding others within the identical niche market and targeting their appeals to. You may even learn something from others within your market that have acquired more experience than you have probably. You can generate traffic and increase sales with a link building techniques. Success is significantly as you, work with these tools you can create the wealth you will need.<br><br>If you adored this information and you would such as to obtain more facts concerning [http://www.trafficfaze.com/ premium traffic] kindly go to the web page. |
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| Roughly, the concept of controllability denotes the ability to move a system around in its entire configuration space using only certain admissible manipulations. The exact definition varies slightly within the framework or the type of models applied.
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| The following are examples of variations of controllability notions which have been introduced in the systems and control literature:
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| * State controllability
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| * Output controllability
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| * Controllability in the behavioural framework
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| == State controllability ==
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| The [[state space (controls)|state]] of a system, which is a collection of the system's variables values, completely describes the system at any given time. In particular, no information on the past of a system will help in predicting the future, if the states at the present time are known.
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| ''Complete state controllability'' (or simply ''controllability'' if no other context is given) describes the ability of an external input to move the internal state of a system from any initial state to any other final state in a finite time interval.<ref name="Ogata97">{{cite book|author=Katsuhiko Ogata|title=Modern Control Engineering|edition=3rd|year=1997|publisher=Prentice-Hall|location=Upper Saddle River, NJ|isbn=0-13-227307-1}}</ref>{{rp|737}}
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| == Continuous linear systems ==
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| Consider the [[continuous function|continuous]] [[linear]] [[time-variant system]] <ref group="note">A [[Linear time-invariant system]] behaves the same but with the coefficients are constant in time.</ref>
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| : <math>\dot{\mathbf{x}}(t) = A(t) \mathbf{x}(t) + B(t) \mathbf{u}(t)</math>
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| : <math>\mathbf{y}(t) = C(t) \mathbf{x}(t) + D(t) \mathbf{u}(t).</math>
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| There exists a control <math>u</math> from state <math>x_0</math> at time <math>t_0</math> to state <math>x_1</math> at time <math>t_1 > t_0</math> if and only if <math>x_1 - \phi(t_0,t_1)x_0</math> is in the [[column space]] of
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| : <math>W(t_0,t_1) = \int_{t_0}^{t_1} \phi(t_0,t)B(t)B(t)^{T}\phi(t_0,t)^{T} dt</math>
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| where <math>\phi</math> is the [[state-transition matrix]].
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| In fact, if <math>\eta_0</math> is a solution to <math>W(t_0,t_1)\eta = x_1 - \phi(t_0,t_1)x_0</math> then a control given by <math>u(t) = -B(t)^{T}\phi(t_0,t)^{T}\eta_0</math> would make the desired transfer.
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| Note that the matrix <math>W</math> defined as above has the following properties:
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| * <math>W(t_0,t_1)</math> is [[symmetric matrix|symmetric]]
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| * <math>W(t_0,t_1)</math> is [[positive semidefinite matrix|positive semidefinite]] for <math>t_1 \geq t_0</math>
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| * <math>W(t_0,t_1)</math> satisfies the linear [[matrix differential equation]]
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| :: <math>\frac{d}{dt}W(t,t_1) = A(t)W(t,t_1)+W(t,t_1)A(t)^{T}-B(t)B(t)^{T}, \; W(t_1,t_1) = 0</math>
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| * <math>W(t_0,t_1)</math> satisfies the equation
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| :: <math>W(t_0,t_1) = W(t_0,t) + \phi(t_0,t)W(t,t_1)\phi(t_0,t)^{T}</math><ref>{{cite book|first=Roger W.|last=Brockett|title=Finite Dimensional Linear Systems|publisher=John Wiley & Sons|year=1970|isbn=978-0-471-10585-5}}</ref>
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| === Continuous linear time-invariant (LTI) systems ===
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| Consider the continuous linear [[time-invariant system]]
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| : <math>\dot{\mathbf{x}}(t) = A \mathbf{x}(t) + B \mathbf{u}(t)</math>
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| : <math>\mathbf{y}(t) = C \mathbf{x}(t) + D \mathbf{u}(t)</math>
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| where
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| : <math>\mathbf{x}</math> is the <math>n \times 1</math> "state vector",
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| : <math>\mathbf{y}</math> is the <math>m \times 1</math> "output vector",
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| : <math>\mathbf{u}</math> is the <math>r \times 1</math> "input (or control) vector",
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| : <math>A</math> is the <math>n \times n</math> "state matrix",
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| : <math>B</math> is the <math>n \times r</math> "input matrix",
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| : <math>C</math> is the <math>m \times n</math> "output matrix",
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| : <math>D</math> is the <math>m \times r</math> "feedthrough (or feedforward) matrix".
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| The <math>n \times nr</math> controllability matrix is given by
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| :<math>R = \begin{bmatrix}B & AB & A^{2}B & ...& A^{n-1}B\end{bmatrix}</math>
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| The system is controllable if the controllability matrix has full [[Rank (linear algebra)|rank]] (i.e. <math>\operatorname{rank}(R)=n</math>).
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| == Discrete linear time-invariant (LTI) systems ==
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| For a discrete-time linear state-space system (i.e. time variable <math>k\in\mathbb{Z}</math>) the state equation is
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| :<math>\textbf{x}(k+1) = A\textbf{x}(k) + B\textbf{u}(k)</math>
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| Where <math>A</math> is an <math>n \times n</math> matrix and <math>B</math> is a <math>n \times r</math> matrix (i.e. <math>\mathbf{u}</math> is <math>r</math> inputs collected in a <math>r \times 1</math> vector. The test for controllability is that the <math>n \times nr</math> matrix
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| :<math>\mathcal{C} = \begin{bmatrix}B & AB & A^{2}B & \cdots & A^{n-1}B\end{bmatrix}</math>
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| has full row [[Rank (linear algebra)|rank]] (i.e., <math>\operatorname{rank}(C) = n</math>). That is, if the system is controllable, <math>C</math> will have <math>n</math> columns that are [[linearly independent]]; if <math>n</math> columns of <math>C</math> are [[linearly independent]], each of the <math>n</math> states is reachable giving the system proper inputs through the variable <math>u(k)</math>.
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| ===Example===
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| For example, consider the case when <math>n=2</math> and <math>r=1</math> (i.e. only one control input). Thus, <math>B</math> and <math>A B</math> are <math>n \times 1</math> vectors. If <math>\begin{bmatrix}B & AB\end{bmatrix}</math> has rank 2 (full rank), and so <math>B</math> and <math>AB</math> are [[linearly independent]] and span the entire plane. If the rank is 1, then <math>B</math> and <math>AB</math> are [[Line (geometry)|collinear]] and do not span the plane.
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| Assume that the initial state is zero.
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| At time <math>k=0</math>: <math>x(1) = A\textbf{x}(0) + B\textbf{u}(0) = B\textbf{u}(0)</math>
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| At time <math>k=1</math>: <math>x(2) = A\textbf{x}(1) + B\textbf{u}(1) = AB\textbf{u}(0) + B\textbf{u}(1)</math>
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| At time <math>k=0</math> all of the reachable states are on the line formed by the vector <math>B</math>.
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| At time <math>k=1</math> all of the reachable states are linear combinations of <math>AB</math> and <math>B</math>.
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| If the system is controllable then these two vectors can span the entire plane and can be done so for time <math>k=2</math>.
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| The assumption made that the initial state is zero is merely for convenience.
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| Clearly if all states can be reached from the origin then any state can be reached from another state (merely a shift in coordinates).
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| This example holds for all positive <math>n</math>, but the case of <math>n=2</math> is easier to visualize.
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| ===Analogy for example of ''n'' = 2===
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| Consider an [[Car analogy|analogy]] to the previous example system.
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| You are sitting in your car on an infinite, flat plane and facing north.
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| The goal is to reach any point in the plane by driving a distance in a straight line, come to a full stop, turn, and driving another distance, again, in a straight line.
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| If your car has no steering then you can only drive straight, which means you can only drive on a line (in this case the north-south line since you started facing north). | |
| The lack of steering case would be analogous to when the rank of <math>C</math> is 1 (the two distances you drove are on the same line).
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| Now, if your car did have steering then you could easily drive to any point in the plane and this would be the analogous case to when the rank of <math>C</math> is 2.
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| If you change this example to <math>n=3</math> then the analogy would be flying in space to reach any position in 3D space (ignoring the [[Orientation (rigid body)|orientation]] of the [[aircraft]]).
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| You are allowed to:
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| *fly in a straight line
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| *turn left or right by any amount ([[Yaw (flight)|Yaw]])
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| *direct the plane upwards or downwards by any amount ([[Pitch (flight)|Pitch]])
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| Although the 3-dimensional case is harder to visualize, the concept of controllability is still analogous.
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| == Nonlinear systems ==
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| Nonlinear systems in the control-affine form
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| : <math>\dot{\mathbf{x}} = \mathbf{f(x)} + \sum_{i=1}^m \mathbf{g}_i(\mathbf{x})u_i</math>
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| is locally accessible about <math>x_0</math> if the accessibility distribution <math>R</math> spans <math>n</math> space, when <math>n</math> equals the rank of <math>x</math> and R is given by:<ref>Isidori, Alberto (1989). ''Nonlinear Control Systems'', p. 92–3. Springer-Verlag, London. ISBN 3-540-19916-0.</ref>
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| :<math>R = \begin{bmatrix} \mathbf{g}_1 & \cdots & \mathbf{g}_m & [\mathrm{ad}^k_{\mathbf{g}_i}\mathbf{\mathbf{g}_j}] & \cdots & [\mathrm{ad}^k_{\mathbf{f}}\mathbf{\mathbf{g}_i}] \end{bmatrix}.</math>
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| Here, <math>[\mathrm{ad}^k_{\mathbf{f}}\mathbf{\mathbf{g}}]</math> is the repeated [[Lie bracket of vector fields|Lie bracket]] operation defined by
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| : <math>[\mathrm{ad}^k_{\mathbf{f}}\mathbf{\mathbf{g}}] = \begin{bmatrix} \mathbf{f} & \cdots & j & \cdots & \mathbf{[\mathbf{f}, \mathbf{g}]} \end{bmatrix}. </math>
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| The controllability matrix for linear systems in the previous section can in fact be derived from this equation.
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| == Output controllability ==
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| ''Output controllability'' is the related notion for the output of the system; the output controllability describes the ability of an external input to move the output from any initial condition to any final condition in a finite time interval. It is not necessary that there is any relationship between state controllability and output controllability. In particular:
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| * A controllable system is not necessarily output controllable. For example, if matrix ''D'' = 0 and matrix ''C'' does not have full row rank, then some positions of the output are masked by the limiting structure of the output matrix. Moreover, even though the system can be moved to any state in finite time, there may be some outputs that are inaccessible by all states. A trivial numerical example uses ''D''=0 and a ''C'' matrix with at least one row of zeros; thus, the system is not able to produce a non-zero output along that dimension.
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| * An output controllable system is not necessarily state controllable. For example, if the dimension of the state space is greater than the dimension of the output, then there will be a set of possible state configurations for each individual output. That is, the system can have significant [[zero dynamics]], which are trajectories of the system that are not observable from the output. Consequently, being able to drive an output to a particular position in finite time says nothing about the state configuration of the system.
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| For a linear continuous-time system, like the example above, described by matrices <math>A</math>, <math>B</math>, <math>C</math>, and <math>D</math>, the <math>m \times (n+1)r</math> ''output controllability matrix''
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| :<math>\begin{bmatrix} CB & CAB & CA^2B & \cdots & CA^{n-1}B & D\end{bmatrix}</math>
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| must have full row rank (i.e. rank <math>m</math>) if and only if the system is output controllable.<ref name="Ogata97" />{{rp|742}} This result is known as Kalman's criteria of controllability.{{Citation needed|date=February 2012}}
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| == Controllability under input constraints ==
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| In systems with limited control authority, it is often no longer possible to move any initial state to any final state inside the controllable subspace. This phenomenon is caused by constraints on the input that could be inherent to the system (e.g. due to saturating actuator) or imposed on the system for other reasons (e.g. due to safety-related concerns). The controllability of systems with input and state constraints is studied in the context of [[reachability (controls)|reachability]]<ref>{{cite journal | author = Claire J. Tomlin, Ian Mitchell, Alexandre M. Bayen and Meeko Oishi | title = Computational Techniques for the Verification of Hybrid Systems | journal = Proceedings of the IEEE | version = | publisher = | year = 2003 | url = http://www.cs.ubc.ca/~mitchell/Papers/publishedIEEEproc03.pdf | accessdate = 2012-03-04
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| }}</ref> and [[viability theory]].<ref>{{cite book|author=Jean-Pierre Aubin |title= Viability Theory |edition= |year=1991 |publisher=Birkhauser |location= |isbn=0-8176-3571-8 }}</ref>
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| == Controllability in the behavioural framework ==
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| In the so-called [[behavioral system theoretic approach]] due to Willems (see [[people in systems and control]]), models considered do not directly define an input–output structure. In this framework systems are described by admissible trajectories of a collection of variables, some of which might be interpreted as inputs or outputs.
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| A system is then defined to be controllable in this setting, if any past part of a behavior (trajectory of the external veriables) can be concatenated with any future trajectory of the behavior in such a way that the concatenation is contained in the behavior, i.e. is part of the admissible system behavior.<ref name="Polderman98">{{cite book|author=Jan Polderman, Jan Willems|title=Introduction to Mathematical Systems Theory: A Behavioral Approach|edition=1st|year=1998|publisher=Springer Verlag|location=New York|isbn=0-387-98266-3}}</ref>{{rp|151}}
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| == Stabilizability ==
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| A slightly weaker notion than controllability is that of '''stabilizability'''. A system is determined to be stabilizable when all uncontrollable states have stable dynamics. Thus, even though some of the states cannot be controlled (as determined by the controllability test above) all the states will still remain bounded during the system's behavior.<ref name="Anderson+Moore/book:1989">{{cite book|author1=Brian D.O. Anderson|author2=John B. Moore|title=Optimal Control: Linear Quadratic Methods|year=1990|publisher=Prentice Hall|location=Englewood Cliffs, NJ|isbn=978-0-13-638560-8}}</ref>
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| == See also ==
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| * [[Observability]]
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| * [[State observer]]
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| ==Notes==
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| {{Reflist|group=note}}
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| ==References==
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| {{reflist}}
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| ==External links==
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| *{{planetmath reference|id=6073|title=Controllability}}
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| * [http://www.mathworks.com/help/toolbox/control/ref/ctrb.html MATLAB function for checking controllability of a system]
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| * [http://reference.wolfram.com/mathematica/ref/ControllableModelQ.html Mathematica function for checking controllability of a system]
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| [[Category:Control theory]]
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| [[fr:Représentation d'état#Systèmes linéaires]]
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