検出性

Wikipedia preview

出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2015/01/01 18:10:22」(JST)

wiki en

In control theory, observability is a measure for how well internal states of a system can be inferred by knowledge of its external outputs. The observability and controllability of a system are mathematical duals. The concept of observability was introduced by American-Hungarian scientist Rudolf E. Kalman for linear dynamic systems.^[1]^[2]

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 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 (although they can be estimated through various means).

For time-invariant linear systems in the state space representation, there is a convenient test to check if a system is observable. Consider a SISO system with states (see state space for details about MIMO systems), if the row rank of the following observability matrix

is equal to , then the system is observable. The rationale for this test is that if rows are linearly independent, then each of the states is viewable through linear combinations of the output variables .

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 of a linear time-invariant discrete system is the smallest natural number for which is satisfied that , where

Detectability

A slightly weaker notion than observability is detectability. A system is detectable if and only if all of its unobservable modes are asymptotically 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 linear time-variant system

Suppose that the matrices are given as well as inputs and outputs for all then it is possible to determine to within an additive constant vector which lies in the null space of defined by

where is the state-transition matrix.

It is possible to determine a unique if is nonsingular. In fact, it is not possible to distinguish the initial state for from that of if is in the null space of .

Note that the matrix defined as above has the following properties:

is symmetric
is positive semidefinite for
satisfies the linear matrix differential equation

satisfies the equation

^[3]

Nonlinear case

Given the system , . Where the state vector, the input vector and the output vector. are to be smooth vectorfields.

Now define the observation space to be the space containing all repeated Lie derivatives. Now the system is observable in if and only if .

Note: ^[4]

Early criteria for observability in nonlinear dynamic systems were discovered by Griffith and Kumar,^[5] Kou, Elliot and Tarn,^[6] and Singh.^[7]

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 ,.^[8]^[9] Just as observability criteria are used to predict the behavior of Kalman filters or other observers in the dynamic system case, observability criteria for sets in are used to predict the behavior of 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.

References

^ Kalman R. E., "On the General Theory of Control Systems", Proc. 1st Int. Cong. of IFAC, Moscow 1960 1 481, Butterworth, London 1961.
^ Kalman R. E., "Mathematical Description of Linear Dynamical Systems", SIAM J. Contr. 1963 1 152
^ Brockett, Roger W. (1970). Finite Dimensional Linear Systems. John Wiley & Sons. ISBN 978-0-471-10585-5.
^ 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.
^ Griffith E. W. and Kumar K. S. P., "On the Observability of Nonlinear Systems I, J. Math. Anal. Appl. 1971 35 135
^ Kou S. R., Elliott D. L. and Tarn T. J., Inf. Contr. 1973 22 89
^ Singh S.N., "Observability in Non-linear Systems with immeasurable Inputs, Int. J. Syst. Sci., 6 723, 1975
^ Stanley G.M. and Mah, R.S.H., "Observability and Redundancy in Process Data Estimation, Chem. Engng. Sci. 36, 259 (1981)
^ Stanley G.M., and Mah R.S.H., "Observability and Redundancy Classification in Process Networks", Chem. Engng. Sci. 36, 1941 (1981)

External links

Observability at PlanetMath.org.
MATLAB function for checking observability of a system
Mathematica function for checking observability of a system

UpToDate Contents

全文を閲覧するには購読必要です。 To read the full text you will need to subscribe.

1. 定量的冠動脈造影：技術的な問題 quantitative coronary angiography technical issues
2. 特発性細菌性腹膜炎の亜種 spontaneous bacterial peritonitis variants
3. 前立腺癌に対する局所療法に伴う血清PSAの上昇：定義、自然経過、およびリスク
層別化 rising serum psa following local therapy for prostate cancer definition natural history and risk stratification
4. 虚血性心疾患における加算平均心電図の利用 use of the signal averaged electrocardiogram in ischemic heart disease
5. 陥没乳頭 nipple inversion

English Journal

Increasing motorcycles attention and search conspicuity by using Alternating-Blinking Lights System (ABLS).

Gershon P, Shinar D.SourceDepartment of Industrial Engineering and Management, Ben-Gurion University of the Negev, Beer Sheva, Israel. Electronic address: pninage@bgu.ac.il.
Accident; analysis and prevention.Accid Anal Prev.2013 Jan;50:801-10. doi: 10.1016/j.aap.2012.07.005. Epub 2012 Aug 9.
Poor conspicuity of Powered Two-Wheelers (PTW) is one of the main factors contributing to their involvement in accidents. The current study examined the influence of different rider's outfits on PTWs' detectability, and evaluated the potential of a unique Alternating-Blinking Lights System (ABLS) to
PMID 22884378

Enhanced gene detection assays for fumarate-adding enzymes allow uncovering of anaerobic hydrocarbon degraders in terrestrial and marine systems.

von Netzer F, Pilloni G, Kleindienst S, Krüger M, Knittel K, Gründger F, Lueders T.SourceInstitute of Groundwater Ecology, Helmholtz Zentrum München-German Research Center for Environmental Health, Neuherberg, Germany.
Applied and environmental microbiology.Appl Environ Microbiol.2013 Jan;79(2):543-52. doi: 10.1128/AEM.02362-12. Epub 2012 Nov 2.
The detection of anaerobic hydrocarbon degrader populations via catabolic gene markers is important for the understanding of processes at contaminated sites. Fumarate-adding enzymes (FAEs; i.e., benzylsuccinate and alkylsuccinate synthases) have already been established as specific functional marker
PMID 23124238

Grey-scale inversion improves detection of lung nodules.

Robinson JW, Ryan JT, McEntee MF, Lewis SJ, Evanoff MG, Rainford LA, Brennan PC.SourceDiscipline of Medical Radiation Sciences, Faculty of Health Science, University of Sydney, Sydney, NSW, Australia.
The British journal of radiology.Br J Radiol.2013 Jan;86(1021):20110812. doi: 10.1259/bjr.20110812.
Objective The current study aims to establish whether detection of solitary pulmonary nodules can be improved by inverting the grey scale of posteroanterior (PA) chests. Methods 30 PA chest images were presented on 2 occasions to 16 senior radiologists on either primary or secondary class displays
PMID 23239692

Japanese Journal

ある多数決ルールを用いた不正者全員の特定が可能な(k,n)しきい値法 (情報理論)

島崎憲明,古賀弘樹
電子情報通信学会技術研究報告 = IEICE technical report : 信学技報 114(224), 19-24, 2014-09-19
NAID 40020225157

気筒間の空燃比変動の検出性を向上させた積層A/Fセンサの開発

林下剛,青木圭一郎,川井将司 [他]
自動車技術会論文集 45(3), 509-514, 2014-05
NAID 40020125231

Necessity for Reassessment of Patients with Serogroup 2 Hepatitis C Virus (HCV) and Undetectable Serum HCV RNA

Moritou Yuki,Ikeda Fusao,Takeuchi Yasuto,Seki Hiroyuki,Nanba Shintaro,Iwasaki Yoshiaki,Yamamoto Kazuhide
Journal of Clinical Microbiology 52(2), 544-548, 2014-02
… Discrepancies in the detectability of serum HCV RNA were investigated among 891 consecutive patients who were positive for anti-HCV. …
NAID 120005517856