ผู้ใช้:Robosorne/กระบะทราย 2

ให้ ${\displaystyle f}$ เป็นฟังก์ชันต่อเนืองบน ${\displaystyle [a,b]}$ โดยที่ ${\displaystyle a,b}$ เป็นจำนวนจริง จะได้ว่า ${\displaystyle f(x)}$ มีค่าได้ทุกค่าที่อยู่ระหว่าง ${\displaystyle f(a)}$ และ ${\displaystyle f(b)}$

กรณี ${\displaystyle f(a)}$ และ ${\displaystyle f(b)}$ มีเครืองหมายต่างกัน กล่าวคือ ${\displaystyle f(a)f(b)<0}$ จะได้ว่ามี ${\displaystyle c\in (a,b)}$ ซึ่ง ${\displaystyle f(c)=0}$

สมมติให้ ${\displaystyle f(a)>0}$ และ ${\displaystyle f(b)<0}$ และให้ ${\displaystyle A={x\in [a,b]:f(x)\geq 0}}$ เนื่องจาก ${\displaystyle b\in A}$ ดังนั้น ${\displaystyle A\neq \phi }$ และกำหนดให้ ${\displaystyle c=\sup A}$ ซึ่งหมายความว่า ${\displaystyle a<c<b}$ สมมุติว่าถ้า ${\displaystyle f(c)\neq 0}$ เราจะแบ่งกรณีพิจารณาได้เป็นสองกรณีดังนี้

กรณีที่หนึ่ง ถ้า${\displaystyle f(c)>0}$

การแปลง Z ขั้นสูง

การแปลง Z ขั้นสูง Advanced Z-transform modified Z-transformเป็นการแปลง Z ที่ได้ผนวกผลของการหน่วง (delay) ที่ไม่ได้เป็นพหุคูณของอัตราการชักตัวอย่าง (sampling rate) บนโดเมนเวลาของสัญญาณ การแปลง Z ขั้นสูงถูกประยุกต์ใช้กันอย่างมากในการประมวลผลสัญญาณ (signal processing) และการควบคุมดิจิทัล (digital control) ตัวอย่างเช่น การสร้างแบบจำลองการประมวลผลสัญญาณที่รวมผลของการหน่วงเชิงเวลาแบบแม่นยำ เป็นต้น

การแปลง Z ขั้นสูง ถูกเสนอโดย จูรี่ (Eliahu Ibraham Jury)

การแปลง Z ขั้นสูง มีนิยามดังต่อไปนี้

${\displaystyle F(z,m)=\sum _{k=0}^{\infty }f(kT+m)z^{-k}}$

โดยที่

• T คาบของการชักตัวอย่าง (sampling period)
• m พารามิเตอร์การหน่วง (delay parameter) โดยที่ ${\displaystyle m\in [0,T).}$

${\displaystyle {\mathcal {Z}}\left\{\sum _{k=1}^{n}c_{k}f_{k}(t)\right\}=\sum _{k=1}^{n}c_{k}F(z,m).}$

${\displaystyle {\mathcal {Z}}\left\{u(t-nT)f(t-nT)\right\}=z^{-n}F(z,m).}$

${\displaystyle {\mathcal {Z}}\left\{f(t)e^{-a\,t}\right\}=e^{-a\,m}F(e^{a\,T}z,m).}$

${\displaystyle {\mathcal {Z}}\left\{t^{y}f(t)\right\}=\left(-Tz{\frac {d}{dz}}+m\right)^{y}F(z,m).}$

${\displaystyle \lim _{k=\infty }f(kT+m)=\lim _{z=1}(1-z^{-1})F(z,m).}$

หมายเหตุ: ในกรณีที่ พารามิเตอร์การหน่วงmเป็นคงคงที่ ในกรณีนี้คุณสมบัติของการแปลง Z แบบปรกติกับการแปลง Z ขั้นสูงจะเหมือนกันทั้งหมด

ในที่นี้เรากำหนดให้ ${\displaystyle f(t)=\cos(\omega t)}$

${\displaystyle {\begin{aligned}F(z,m)=&{\mathcal {Z}}\left\{\cos \left(\omega \left(kT+m\right)\right)\right\}\\=&{\mathcal {Z}}\left\{\cos(\omega kT)\cos(\omega m)-\sin(\omega kT)\sin(\omega m)\right\}\\=&\cos(\omega m){\mathcal {Z}}\left\{\cos(\omega kT)\right\}-\sin(\omega m){\mathcal {Z}}\left\{\sin(\omega kT)\right\}\\=&\cos(\omega m){\frac {z\left(z-\cos(\omega T)\right)}{z^{2}-2z\cos(\omega T)+1}}-\sin(\omega m){\frac {z\sin(\omega T)}{z^{2}-2z\cos(\omega T)+1}}\\=&{\frac {z^{2}\cos(\omega m)-z\cos(\omega (T-m))}{z^{2}-2z\cos(\omega T)+1}}\end{aligned}}}$.

ถ้า ${\displaystyle m=0}$ แล้ว ${\displaystyle F(z,m)}$ จะลดรูปกลายเป็นการแปลง Z แบบปรกติ

${\displaystyle F(z,0)={\frac {z^{2}-z\cos(\omega T)}{z^{2}-2z\cos(\omega T)+1}}}$

ซึ่งก็คือผลการแปลงการแปลง Z แบบปรกติของ${\displaystyle f(t).}$นั้นเอง

• en:Z-transform

• Eliahu Ibraham Jury, Theory and Application of the Z-Transform Method, Krieger Pub Co, 1973. ISBN 0-88275-122-0.

แม่แบบ:DSP

Eliahu I. Jury
รางวัล	รางวัล ริชาร์ด อี. เบลแมน (Richard E. Bellman Control Heritage Award)
อาชีพทางวิทยาศาสตร์
สาขา	ทฤษฎีระบบควบคุม

อีไลฮุ อิบาร์ฮัม จัวรี (Eliahu Ibraham Jury 23 พฤษภาคม ค.ศ. 1923 - ปัจจุบัน)^[1]) is an American engineer,^[2] born in Baghdad, Iraq. He received his Doctor of Engineering Science degree from Columbia University of New York in 1953. He was professor of Electrical Engineering at the University of California, Berkeley, and the University of Miami.

He developed the advanced Z-transform, used in digital control systems and signal processing.^[2]

He is a Life Fellow of the IEEE^{[ต้องการอ้างอิง]} and has received the Rufus Oldenburger Medal from the ASME,^[2] the First Education Award of IEEE Circuits and Systems Society,^[2] and the IEEE Millennium Medal.^[2]

• Theory and Application of the z-Transform Method, John Wiley and Sons, 1964.
• Inners and stability of dynamic systems, John Wiley & Sons, 1974

แม่แบบ:Richard E. Bellman Control Heritage Award 1979-2000 Laureates

[[วิกิพีเดีย:|ข้อมูลบุคคล]]
ชื่อ	Jury, Eliahu I.}
ชื่ออื่น
รายละเอียดโดยย่อ
วันเกิด	May 23, 1923
สถานที่เกิด	Baghdad, Iraq
วันตาย
สถานที่ตาย

แม่แบบ:Iraq-academic-bio-stub

Rudolf Emil Kálmán
เกิด	(1930-05-19) 19 พฤษภาคม ค.ศ. 1930 (94 ปี) Budapest, Kingdom of Hungary
สัญชาติ	Hungarian-born American citizen
ศิษย์เก่า	Massachusetts Institute of Technology; Columbia University
รางวัล	IEEE Medal of Honor; National Medal of Science; Charles Stark Draper Prize ; Kyoto Prize
อาชีพทางวิทยาศาสตร์
สาขา	Electrical Engineering; Mathematics; Applied Engineering Systems Theory
สถาบันที่ทำงาน	Stanford University; University of Florida; Swiss Federal Institute of Technology
อาจารย์ที่ปรึกษาในระดับปริญญาเอก	John Ragazzini

Rudolf (Rudy) Emil Kálmán^[1] (in Hungarian Kálmán Rudolf Emil; born May 19, 1930) is a Hungarian-American electrical engineer, mathematical system theorist, and college professor, who was educated in the United States, and has done most of his work there. He is currently a retired professor from three different institutes of technology and universities. He is most noted for his co-invention and development of the Kalman filter, a mathematical formulation that is widely used in control systems, avionics, and outer space manned and unmanned vehicles. For this work, U.S. President Barack Obama awarded Kálmán with the National Medal of Science on October 7, 2009.

Rudolf Kálmán was born in Budapest. After emigrating to the United States in 1943, he earned his bachelor's degree in 1953 and his master's degree in 1954, both from the Massachusetts Institute of Technology, in electrical engineering. Kálmán completed his doctorate in 1957 at Columbia University in New York City.

Kálmán worked as a Research Mathematician at the Research Institute for Advanced Studies in Baltimore, Maryland from 1958 until 1964. He was a professor at Stanford University from 1964 until 1971, and then a Graduate Research Professor and the Director of the Center for Mathematical System Theory, at the University of Florida from 1971 until 1992. Starting in 1973, he also held the chair of Mathematical System Theory at the Swiss Federal Institute of Technology in Zürich, Switzerland.

Kálmán is a member of the U.S. National Academy of Sciences, the American National Academy of Engineering, and the American Academy of Arts and Sciences. He is a foreign member of the Hungarian, French, and Russian Academies of Science. He has been awarded many honorary doctorates from other universities.

Kálmán received the IEEE Medal of Honor in 1974, the IEEE Centennial Medal in 1984, the Inamori foundation's Kyoto Prize in High Technology in 1985, the Steele Prize of the American Mathematical Society in 1987, the Richard E. Bellman Control Heritage Award in 1997, and the National Academy of Engineering's Charles Stark Draper Prize in 2008.

Kálmán is an electrical engineer by his undergraduate and graduate education at M.I.T. and Columbia University, and he is noted for his co-invention of the Kalman filter (or Kalman-Bucy Filter), which is a mathematical technique widely used in the digital computers of control systems, navigation systems, avionics, and outer-space vehicles to extract a signal from a long sequence of noisy and/or incomplete technical measurements, usually those done by electronic and gyroscopic systems.

Kálmán's ideas on filtering were initially met with vast skepticism, so much so that he was forced to do the first publication of his results in mechanical engineering, rather than in electrical engineering or systems engineering. Kálmán had more success in presenting his ideas, however, while visiting Stanley F. Schmidt at the NASA Ames Research Center in 1960. This led to the use of Kálmán filters during the Apollo program, and furthermore, in the NASA Space Shuttle, in Navy submarines, and in unmanned aerospace vehicles and weapons, such as cruise missiles.

• Kalman, R.E. (1960). "A new approach to linear filtering and prediction problems" (PDF). Journal of Basic Engineering. 82 (1): 35–45. สืบค้นเมื่อ 2008-05-03.

• Kalman, R.E. (1961). "New Results in Linear Filtering and Prediction Theory". สืบค้นเมื่อ 2008-05-03. {{cite journal}}: Cite journal ต้องการ |journal= (help); ไม่รู้จักพารามิเตอร์ |coauthors= ถูกละเว้น แนะนำ (|author=) (help)

• The Kalman Filter website
• Kyoto Prize
• For Kálmán's PhD students see Rudolf Emil Kálmán on the Mathematics Genealogy Project page.
• Biography of Kalman from the IEEE

แม่แบบ:IEEE Medal of Honor Laureates 1951-1975 แม่แบบ:Richard E. Bellman Control Heritage Award 1979-2000 Laureates

[[วิกิพีเดีย:|ข้อมูลบุคคล]]
ชื่อ	Kalman, Rudolf Emil}
ชื่ออื่น
รายละเอียดโดยย่อ
วันเกิด	May 19, 1930
สถานที่เกิด	Budapest, Hungary
วันตาย
สถานที่ตาย

คำเตือน: หลักเรียงลำดับปริยาย "Kalman, Rudolf Emil" ได้ลบล้างหลักเรียงลำดับปริยาย "Jury, Eliahu I." ที่มีอยู่ก่อนหน้า

In mathematics, Lyapunov functions are functions which can be used to prove the stability of a certain fixed point in a dynamical system or autonomous differential equation. Named after the Russian mathematician Aleksandr Mikhailovich Lyapunov, Lyapunov functions are important to stability theory and control theory. A similar concept appears in the theory of general state space Markov Chains, usually under the name Lyapunov-Foster functions.

Functions which might prove the stability of some equilibrium are called Lyapunov-candidate-functions. There is no general method to construct or find a Lyapunov-candidate-function which proves the stability of an equilibrium, and the inability to find a Lyapunov function is inconclusive with respect to stability, which means, that not finding a Lyapunov function doesn't mean that the system is unstable. For dynamical systems (e.g. physical systems), conservation laws can often be used to construct a Lyapunov-candidate-function.

The basic Lyapunov theorems for autonomous systems which are directly related to Lyapunov (candidate) functions are a useful tool to prove the stability of an equilibrium of an autonomous dynamical system.

Whenever a system is stable in one of the senses given below, Lyapunov functions can always be used to prove such stability. When a system is locally exponentially stable, one can prove local exponential stability by solving the Lyapunov equation for the linearized system.

As the areas of equal stability often follow lines in 2D, the computer generated images of Lyapunov exponents are visually appealing and very popular.

Let

${\displaystyle V:\mathbb {R} ^{n}\to \mathbb {R} }$

be a continuous scalar function.
${\displaystyle V}$ is a Lyapunov-candidate-function if it is a locally positive-definite function, i.e.

${\displaystyle V(0)=0\,}$

${\displaystyle V(x)>0\quad \forall x\in U\setminus \{0\}}$

With ${\displaystyle U}$ being a neighborhood region around ${\displaystyle x=0}$

Let

${\displaystyle g:\mathbb {R} ^{n}\to \mathbb {R} ^{n}}$

${\displaystyle {\dot {y}}=g(y)\,}$

be an arbitrary autonomous dynamical system with equilibrium point ${\displaystyle y^{*}\,}$:

${\displaystyle 0=g(y^{*})\,}$

There always exists a coordinate transformation ${\displaystyle x=y-y^{*}\,}$, such that:

${\displaystyle {\dot {x}}=g(x+y^{*})=f(x)\,}$

${\displaystyle f(0)=0\,}$

So the new system ${\displaystyle f(x)}$ has an equilibrium point at the origin.

บทความหลัก: Lyapunov stability

Let

${\displaystyle x^{*}=0\,}$

be an equilibrium of the autonomous system

${\displaystyle {\dot {x}}=f(x)\,}$

And let

${\displaystyle {\dot {V}}(x)={\frac {\partial V}{\partial x}}\cdot {\frac {dx}{dt}}=\nabla V\cdot {\dot {x}}=\nabla V\cdot f(x)}$

be the time derivative of the Lyapunov-candidate-function ${\displaystyle V}$.

If the Lyapunov-candidate-function ${\displaystyle V}$ is locally positive definite and the time derivative of the Lyapunov-candidate-function is locally negative semidefinite:

${\displaystyle {\dot {V}}(x)\leq 0\quad \forall x\in {\mathcal {B}}\setminus \{0\}}$

for some neighborhood ${\displaystyle {\mathcal {B}}}$ of ${\displaystyle 0}$, then the equilibrium is proven to be stable.

If the Lyapunov-candidate-function ${\displaystyle V}$ is locally positive definite and the time derivative of the Lyapunov-candidate-function is locally negative definite:

${\displaystyle {\dot {V}}(x)<0\quad \forall x\in {\mathcal {B}}\setminus \{0\}}$

for some neighborhood ${\displaystyle {\mathcal {B}}}$ of ${\displaystyle 0}$, then the equilibrium is proven to be locally asymptotically stable.

If the Lyapunov-candidate-function ${\displaystyle V}$ is globally positive definite, radially unbounded and the time derivative of the Lyapunov-candidate-function is globally negative definite:

${\displaystyle {\dot {V}}(x)<0\quad \forall x\in \mathbb {R} ^{n}\setminus \{0\},}$

then the equilibrium is proven to be globally asymptotically stable.

The Lyapunov-candidate function ${\displaystyle V(x)}$ is radially unbounded if

${\displaystyle \|x\|\to \infty \Rightarrow V(x)\to \infty }$.

(This is also referred to as norm-coercivity.)

Consider the following differential equation with solution x on ${\displaystyle \mathbb {R} }$:

${\displaystyle {\dot {x}}=-x.}$

Considering that |x| is always positive around the origin it is a natural candidate to be a Lyapunov function to help us study x. So let ${\displaystyle V(x)=|x|}$ on ${\displaystyle \mathbb {R} \setminus \{0\}}$. Then,

${\displaystyle {\dot {V}}(x)=V'(x)f(x)=\mathrm {sign} (x)\cdot (-x)=-|x|<0.}$

This correctly shows that the above differential equation, x, is asymptotically stable about the origin.

• Ordinary differential equations
• Control-Lyapunov function
• Foster's theorem

• เอริก ดับเบิลยู. ไวส์สไตน์, "Lyapunov Function" จากแมทเวิลด์.
• Khalil, H.K. (1996). Nonlinear systems. Prentice Hall Upper Saddle River, NJ.
• แม่แบบ:Planetmath

• Example of determining the stability of the equilibrium solution of a system of ODEs with a Lyapunov function
• Some Lyapunov diagrams