86.126.47.219: /* History */

2014-08-24T16:41:34Z

History

en>Lejarrag: /* Single particle systems */

2013-12-31T16:06:31Z

Single particle systems

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	~~Fascinated~~ to ~~meet you! My name~~ is [~~http~~://~~Thesaurus~~.~~com~~/~~browse~~/~~Eusebio~~+~~Ledbetter Eusebio Ledbetter~~]. It's not ~~a common place but what I want doing~~ is ~~bottle best collecting~~ and ~~now My hubby~~ and i ~~have time to take on new things~~. ~~Software developing is how My personal support my family~~. ~~My house is but~~ in ~~Vermont~~. I~~'ve been toiling on my website just~~ for ~~some time now~~. ~~Investigate it out here~~: http://~~circuspartypanama~~.com/<br><br>~~Also visit my website :~~: [http://~~circuspartypanama~~.com/ ~~how to hack clash~~ of ~~clans with cydia~~]		{{for\|the concept in quantum mechanics\|observable}}
			In [[control theory]], '''observability''' is a measure for how well internal states of a [[system]] can be inferred by [[knowledge]] of its external [[output]]s. The observability and [[controllability]] of a system are mathematical [[duality (mathematics)\|duals]]. The concept of observability was introduced by American-Hungarian scientist [[Rudolf E. Kalman]] for linear dynamic systems.<ref>Kalman R. E., "On the General Theory of Control Systems", Proc. 1st Int. Cong. of IFAC, Moscow 1960 1
			481, Butterworth, London 1961.</ref><ref>Kalman R. E., "Mathematical Description of Linear Dynamical Systems", SIAM J. Contr. 1963 1 152</ref>

			==Definition==
			Formally, a system is said to be '''observable''' if, for any possible sequence of state and control vectors, the current state can be determined in finite time using only the outputs (this definition is slanted towards the [[state space (controls)\|state space]] representation). Less formally, this means that from the system's outputs it is possible to determine the behaviour of the entire system. If a system is not observable, this means the current values of some of its states cannot be determined through output [[sensors]]. This implies that their value is unknown to the [[controller (control theory)\|controller]] (although they can be estimated through various means).

			For [[LTI system theory\|time-invariant linear system]]s in the state space representation, a convenient test to check if a system is observable exists. Consider a [[Single-Input and Single-Output\|SISO]] system with <math>n</math> states (see [[state space (controls)\|state space]] for details about [[MIMO]] systems), if the row [[Rank (linear algebra)\|rank]] of the following ''observability matrix''

			:<math>\mathcal{O}=\begin{bmatrix} C \\ CA \\ CA^2 \\ \vdots \\ CA^{n-1} \end{bmatrix}</math>

			is equal to <math>n</math>, then the system is observable. The rationale for this test is that if <math>n</math> rows are linearly independent, then each of the <math>n</math> states is viewable through linear combinations of the output variables <math>y(k)</math>.

			A module designed to estimate the state of a system from measurements of the outputs is called a [[state observer]] or simply an observer for that system.

			;Observability index
			The Observability index <math>v</math> of a linear time-invariant discrete system is the smallest natural number for which is satisfied that <math>rank{(O_v)} = rank{(O_{v+1})}</math>, where

			:<math> \mathcal{O}_v=\begin{bmatrix} C \\ CA \\ CA^2 \\ \vdots \\ CA^{v-1} \end{bmatrix}.</math>

			;Detectability
			A slightly weaker notion than observability is detectability. A system is detectable if and only if all of its unobservable modes are stable. Thus even though not all system modes are observable, the ones that are not observable do not require stabilization.

			== Continuous time-varying system ==

			Consider the [[continuous function\|continuous]] [[linear]] [[time-variant system]]

			: <math>\dot{\mathbf{x}}(t) = A(t) \mathbf{x}(t) + B(t) \mathbf{u}(t) \, </math>
			: <math>\mathbf{y}(t) = C(t) \mathbf{x}(t). \, </math>

			Suppose that the matrices <math>A,B, \text{ and } C</math> are given as well as inputs and outputs <math>u \text{ and } y</math> for all <math>t \in [t_0,t_1]</math> then it is possible to determine <math>x(t_0)</math> to within an additive constant vector which lies in the [[null space]] of <math>M(t_0,t_1)</math> defined by
			: <math>M(t_0,t_1) = \int_{t_0}^{t_1} \phi(t,t_0)^{T}C(t)^{T}C(t)\phi(t,t_0) dt</math>
			where <math>\phi</math> is the [[state-transition matrix]].

			It is possible to determine a unique <math>x(t_0)</math> if <math>M(t_0,t_1)</math> is [[nonsingular]]. In fact, it is not possible to distinguish the initial state for <math>x_1</math> from that of <math>x_2</math> if <math>x_1 - x_2</math> is in the null space of <math>M(t_0,t_1)</math>.

			Note that the matrix <math>M</math> defined as above has the following properties:
			* <math>M(t_0,t_1)</math> is [[symmetric matrix\|symmetric]]
			* <math>M(t_0,t_1)</math> is [[positive semidefinite matrix\|positive semidefinite]] for <math>t_1 \geq t_0</math>
			* <math>M(t_0,t_1)</math> satisfies the linear [[matrix differential equation]]
			:: <math>\frac{d}{dt}M(t,t_1) = -A(t)^{T}M(t,t_1)-M(t,t_1)A(t)-C(t)^{T}C(t), \; M(t_1,t_1) = 0</math>
			* <math>M(t_0,t_1)</math> satisfies the equation
			:: <math>M(t_0,t_1) = M(t_0,t) + \phi(t,t_0)^T M(t,t_1)\phi(t,t_0)</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>

			== Nonlinear case ==
			Given the system <math>\dot{x} = f(x) + \sum_{j=1}^mg_j(x)u_j </math>, <math>y_i = h_i(x), i \in p</math>. Where <math>x \in \mathbb{R}^n</math> the state vector, <math>u \in \mathbb{R}^m</math> the input vector and <math>y \in \mathbb{R}^p</math> the output vector. <math>f,g,h</math> are to be smooth vectorfields.

			Now define the observation space <math>\mathcal{O}_s</math> to be the space containing all repeated [[Lie derivative]]s. Now the system is observable in <math>x_0</math> if and only if <math>\textrm{dim}(d\mathcal{O}_s(x_0)) = n</math>.

			Note: <math>d\mathcal{O}_s(x_0) = \mathrm{span}(dh_1(x_0), \ldots , dh_p(x_0), dL_{v_i}L_{v_{i-1}}, \ldots , L_{v_1}h_j(x_0)),\ j\in p, k=1,2,\ldots.</math><ref>Lecture notes for Nonlinear Systems Theory by prof. dr. D.Jeltsema, prof dr. J.M.A.Scherpen and prof dr. A.J.van der Schaft.</ref>

			Early criteria for observability in nonlinear dynamic systems were discovered by Griffith and Kumar,<ref>Griffith E. W. and Kumar K. S. P., "On the Observability of Nonlinear Systems I, J. Math. Anal. Appl. 1971
			35 135</ref> Kou, Elliot and Tarn,<ref>Kou S. R., Elliott D. L. and Tarn T. J., Inf. Contr. 1973 22 89</ref> and Singh.<ref>Singh S.N., "Observability in Non-linear Systems with immeasurable Inputs, Int. J. Syst. Sci., 6 723, 1975</ref>

			== Static systems and general topological spaces ==

			Observability may also be characterized for steady state systems (systems typically defined in terms of algebraic equations and inequalities), or more generally, for sets in <math>\mathbb{R}^n</math>
			,.<ref>[http://gregstanleyandassociates.com/whitepapers/DataRec/CES-1981a-ObservabilityRedundancy.pdf Stanley G.M. and Mah, R.S.H., "Observability and Redundancy in Process Data Estimation, Chem. Engng. Sci. 36, 259 (1981)]</ref><ref>[http://gregstanleyandassociates.com/whitepapers/DataRec/CES-1981b-ObservabilityRedundancyProcessNetworks.pdf Stanley G.M., and Mah R.S.H., "Observability and Redundancy Classification in Process Networks", Chem. Engng. Sci. 36, 1941 (1981) ]</ref> Just as observability criteria are used to predict the behavior of [[Kalman filter]]s or other observers in the dynamic system case, observability criteria for sets in <math>\mathbb{R}^n</math> are used to predict the behavior of [[data validation and reconciliation\|data reconciliation]] and other static estimators. In the nonlinear case, observability can be characterized for individual variables, and also for local estimator behavior rather than just global behavior.

			== See also ==
			* [[Controllability]]
			* [[State observer]]
			* [[State space (controls)]]

			==References==
			{{reflist}}

			==External links==
			*{{planetmath reference\|id=6074\|title=Observability}}
			* [http://www.mathworks.com/help/toolbox/control/ref/obsv.html MATLAB function for checking observability of a system]
			* [http://reference.wolfram.com/mathematica/ref/ObservableModelQ.html Mathematica function for checking observability of a system]

			[[Category:Control theory]]

			[[fr:Représentation d'état#Observabilité et détectabilité]]

en>Cesaranieto: /* First quantization */

2012-09-01T14:08:54Z

First quantization

New page

Fascinated to meet you! My name is [http://Thesaurus.com/browse/Eusebio+Ledbetter Eusebio Ledbetter]. It's not a common place but what I want doing is bottle best collecting and now My hubby and i have time to take on new things. Software developing is how My personal support my family. My house is but in Vermont. I've been toiling on my website just for some time now. Investigate it out here: http://circuspartypanama.com/<br><br>Also visit my website :: [http://circuspartypanama.com/ how to hack clash of clans with cydia]

Canonical quantization - Revision history

86.126.47.219: /* History */

en>Lejarrag: /* Single particle systems */

en>Cesaranieto: /* First quantization */