Mertens' theorems: Difference between revisions

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In [[control theory]], the '''discrete Lyapunov equation''' is of the form
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:<math>A X A^{H} - X + Q = 0</math>
where <math>Q</math> is a [[Hermitian matrix]] and <math>A^H</math> is the [[conjugate transpose]] of <math>A</math>. The '''continuous Lyapunov equation''' is of form
:<math>AX + XA^H + Q = 0</math>.
 
The Lyapunov equation occurs in many branches of control theory, such as [[Lyapunov stability|stability analysis]] and [[optimal control]]. This and related equations are named after the Russian mathematician [[Aleksandr Lyapunov]].
 
==Application to stability==
In the following theorems <math>A, P, Q \in \mathbb{R}^{n \times n}</math>, and <math>P</math> and <math>Q</math> are symmetric. The notation <math>P>0</math> means that the matrix <math>P</math> is [[Positive-definite matrix|positive definite]].
 
'''Theorem''' (continuous time version). Given any <math>Q>0</math>, there exists a unique <math>P>0</math> satisfying <math>A^T P + P A + Q = 0</math> if and only if the linear system <math>\dot{x}=A x</math> is globally asymptotically stable. The quadratic function <math>V(z)=z^T P z</math> is a [[Lyapunov function]] that can be used to verify stability.
 
'''Theorem''' (discrete time version). Given any <math>Q>0</math>, there exists a unique <math>P>0</math> satisfying <math>A^T P A -P + Q = 0</math> if and only if the linear system <math>x(t+1)=A x(t)</math> is globally asymptotically stable. As before, <math>z^T P z</math> is a Lyapunov function.
 
==Computational aspects of solution==
 
The discrete Lyapunov equations can, by using [[Schur complement]]s, be written as
:<math>\begin{bmatrix}
X^{-1} & A \\ A^H & X-Q
\end{bmatrix}=0</math>
or equivalently as
:<math>\begin{bmatrix}
X & XA \\ A^HX & X-Q
\end{bmatrix}=0</math>.
 
Specialized software is available for solving Lyapunov equations. For the discrete case, the Schur method of Kitagawa is often used.<ref>{{cite journal |last=Kitagawa |first=G. |title=An Algorithm for Solving the Matrix Equation X = F X F' + S |journal=International Journal of Control |volume=25 |issue=5 |pages=745–753 |year=1977 |doi=10.1080/00207177708922266 }}</ref>  For the continuous Lyapunov equation the method of Bartels and Stewart can be used.<ref>{{cite journal |first=R. H. |last=Bartels |first2=G. W. |last2=Stewart |title=Algorithm 432: Solution of the matrix equation AX + XB = C |journal=Comm. ACM |volume=15 |year=1972 |issue=9 |pages=820–826 |doi=10.1145/361573.361582 }}</ref>
 
==Analytic Solution==
Defining the <math>\operatorname{vec} (A)</math> operator as stacking the columns of a matrix <math>A</math> and <math>A \otimes B</math> as the [[Kronecker product]] of <math>A</math> and <math>B</math>, the continuous time and discrete time Lyapunov equations can be expressed as solutions of a matrix equation. Furthermore, if the matrix <math>A</math> is stable, the solution can also be expressed as an integral (continuous time case) or as an infinite sum (discrete time case).
 
===Discrete time===
Using the result that <math> \operatorname{vec}(ABC)=(C^{T} \otimes A)\operatorname{vec}(B) </math>, one has
:<math> (I-A \otimes A)\operatorname{vec}(X) = \operatorname{vec}(Q) </math>
where <math>I</math> is a [[conformable]] identity matrix.<ref>{{cite book |first=J. |last=Hamilton |year=1994 |title=Time Series Analysis |at=Equations 10.2.13 and 10.2.18 |location= |publisher=Princeton University Press |isbn=0-691-04289-6 }}</ref> One may then solve for <math>\operatorname{vec}(X)</math> by inverting or solving the linear equations.  To get <math>X</math>, one must just reshape <math>\operatorname{vec} (X)</math> appropriately.
 
Moreover, if <math>A</math> is stable, the solution <math>X</math> can also be written as
:<math> X = \sum_{k=0}^{\infty} A^{k} Q (A^{H})^k </math>.
 
===Continuous time===
Using again the Kronecker product notation and the vectorization operator, one has the matrix equation
:<math> (I_n \otimes A +  \bar{A} \otimes I_n) \operatorname{vec}X = -\operatorname{vec}Q, </math>
where <math>\bar{A}</math> denotes the matrix obtained by complex conjugating the entries of <math>A</math>.
 
Likewise the discrete-time case, if <math>A</math> is stable, the solution <math>X</math> can also be written as
:<math> X = \int\limits_0^\infty e^{A \tau} Q e^{A^H \tau} d\tau  </math>.
 
==See also==
* [[Sylvester equation]]
* [[Algebraic Riccati equation]]
 
==References==
{{reflist}}
 
==External links==
* [http://calculator-fx.com/calculator/linear-algebra/solve-lyapunov-eqution Online solver for arbitrary sized matrices.]
 
{{DEFAULTSORT:Lyapunov Equation}}
[[Category:Control theory]]

Latest revision as of 16:47, 15 May 2014

If you have the desire to process settings instantly, loading files fast, but a body is logy plus torpid, what would we do? If you are a giant "switchboard" that is deficiency of effective administration program and effective housekeeper, what would we do? If you have send the exact commands to your notice, however the body cannot do properly, what would we do? Yes! You need a full-featured repair registry!

Firstly, you need to utilize the Antivirus or security tool and run a scan on a computer. It can be done that a computer is afflicted with virus or malware which slows down the computer. If there is nothing found inside the scanning report, it could be your RAM that cause the issue.

With RegCure you are able to easily improve the begin up plus shut down of the computer. The program shows the scan progress plus we shouldn't worry where it happens to be working at which time. It shows we what occurs. Dynamic link library section of the registry could result severe application failures. RegCure restores and repairs the registry plus keeps we out of DLL. RegCure can create individual corrections, thus it can functions for your requirements.

Windows errors is caused by any number of reasons, yet there's almost constantly one cause. There's a hidden part of the program that is responsible for making 90% of all Windows mistakes, plus it's called the 'registry'. This really is the central database for a system plus is where your computer shops all its program files plus settings. It's a important piece of Windows, that is requirements to be capable to function. However, it's furthermore among the largest causes of problems on the PC.

There are many tuneup utilities s available now. How do you know that you to choose? Well, whenever we bought the car you did several research on it, didn't we? You didn't just go out plus purchase the first red convertible we saw. The same thing works with registry cleaners. On any search engine, type in "registry cleaner reviews" plus they will receive posted for we to read about.

Your program is made plus built for the purpose of helping we accomplish tasks and not be pestered by windows XP error messages. When there are errors, what do we do? Some people pull their hair plus cry, while those sane ones have their PC repaired, while those certainly wise ones analysis to have the mistakes fixed themselves. No, these mistakes were not also designed to rob you off your cash and time. There are points which you can do to actually prevent this from happening.

By restoring the state of the program to an earlier date, error 1721 can not appear inside Windows 7, Vista plus XP. There is a tool called System Restore that you have to utilize in this procedure.

There are numerous businesses that offer the service of troubleshooting the PC each time you call them, all you need to do is sign up with them plus for a small fee, you have the machine constantly working well plus serve you greater.

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