Re: Lyapunov stability [Archive]

Dave Langers

Nov14-04, 06:26 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>> 1) what could be an apropriate candidate for a Lyapunov function, V(x), so\n> that one can determine the stability of a simple mass (m) -spring\n> (k) -damper (c)system\n>\n> the model equation is: mx(ddot)+cx(dot)+kx = 0\n>\n> V(x) must be positive definite & d/dt{V(x)} must be negative definite\n\nMust V(x) be a function of x only? I don\'t think that can be done in\ngeneral (if you require d/dt{V(x)} to be negative). However, if x(dot)\ncan be used as a variable as well, it seems feasible. The system moves\nin a spiral pattern in state space (in a coordiante system with axes x\nand x(dot) I mean), so choose the equi-V contours to be ellipses.\nI would suggest\nV(x,x(dot)) = A * x^2 + B * x(dot)^2\nwith A and B yet unknown.\nNow\nd/dt{V(x)} = 2*A*x*x(dot) + 2*B*x(dot)*x(ddot)\n= 2*x(dot) * (A*x + B*x(ddot))\nIf you take A = k and B = m, then this equals\nd/dt{V(x)} = 2*x(dot) * -c*x(dot) = -2c*x(dot)^2 < 0\nif c > 0, with the obvious exception of x(dot) = x = 0.\n\n> 2) Also how does one prove the above system stable using LaSalle\'s invariant\n> principle?\n\nI am not familiar with LaSalle\'s principle...\n\n--\nM.vr.gr.\nDave\n("d-dot-langers-at-wxs-dot-nl")\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>> 1) what could be an apropriate candidate for a Lyapunov function, V(x), so
> that one can determine the stability of a simple mass (m) -spring
> (k) -damper (c)system
>
> the model equation is: mx(ddot)+cx(dot)+kx =
>
> V(x) must be positive definite & d/dt{V(x)} must be negative definite

Must V(x) be a function of x only? I don't think that can be done in
general (if you require d/dt{V(x)} to be negative). However, if x(dot)
can be used as a variable as well, it seems feasible. The system moves
in a spiral pattern in state space (in a coordiante system with axes x
and x(dot) I mean), so choose the equi-V contours to be ellipses.
I would suggest
V(x,x(dot)) = A * x^2 + B * x(dot)^2
with A and B yet unknown.
Now
d/dt{V(x)} = 2*A*x*x(dot) + 2*B*x(dot)*x(ddot)= 2*x(dot) * (A*x + B*x(ddot))
If you take A = k and B = m, then this equals
d/dt{V(x)} = 2*x(dot) * -c*x(dot) = -2c*x(dot)^2 <
if c > 0, with the obvious exception of x(dot) = x = .

> 2) Also how does one prove the above system stable using LaSalle's invariant
> principle?

I am not familiar with LaSalle's principle...

--
M.vr.gr.
Dave
("d-dot-langers-at-wxs-dot-nl")

Robert Israel

Nov14-04, 06:27 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>In article <cn0a3g\\$eue\\$1@nunki.unm.edu>, GRad <dchristo@unm.edu> wrote:\n\n>1) what could be an apropriate candidate for a Lyapunov function, V(x), so\n>that one can determine the stability of a simple mass (m) -spring\n>(k) -damper (c)system\n\n>the model equation is: mx(ddot)+cx(dot)+kx = 0\n\n>V(x) must be positive definite & d/dt{V(x)} must be negative definite\n\nIt\'s not a function of x, it\'s a function of x and x(dot).\nHint: since you\'re cross-posting to physics groups, think about energy.\n\nRobert Israel israel@math.ubc.ca\nDepartment of Mathematics http://www.math.ubc.ca/~israel\nUniversity of British Columbia Vancouver, BC, Canada\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>In article <cn0a3g$eue$1@nunki.unm.edu>, GRad <dchristo@unm.edu> wrote:

>1) what could be an apropriate candidate for a Lyapunov function, V(x), so
>that one can determine the stability of a simple mass (m) -spring
>(k) -damper (c)system

>the model equation is: mx(ddot)+cx(dot)+kx =

>V(x) must be positive definite & d/dt{V(x)} must be negative definite

It's not a function of x, it's a function of x and x(dot).
Hint: since you're cross-posting to physics groups, think about energy.

Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

Arnold Neumaier

Nov14-04, 07:09 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>GRad wrote:\n>\n> i wish to ask\n>\n> 1) what could be an apropriate candidate for a Lyapunov function, V(x), so\n> that one can determine the stability of a simple mass (m) -spring\n> (k) -damper (c)system\n>\n> the model equation is: mx(ddot)+cx(dot)+kx = 0\n>\n> V(x) must be positive definite & d/dt{V(x)} must be negative definite\n>\n> 2) Also how does one prove the above system stable using LaSalle\'s invariant\n> principle?\n\n\ns.p.r is not the right forum for getting your exercises done.\nHave you ever seen an athlete win a medal who had others do the\nphysical exercises for him? Science is not really different!\n\nYou can find background for the above in most introductions to\ntheoretical mechanics. Read it from several different perspectives\n(i.e., authors) if you can\'t get the insight from the first one.\n\n\nArnold Neumaier\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>GRad wrote:
>
> i wish to ask
>
> 1) what could be an apropriate candidate for a Lyapunov function, V(x), so
> that one can determine the stability of a simple mass (m) -spring
> (k) -damper (c)system
>
> the model equation is: mx(ddot)+cx(dot)+kx =
>
> V(x) must be positive definite & d/dt{V(x)} must be negative definite
>
> 2) Also how does one prove the above system stable using LaSalle's invariant
> principle?

s.p.r is not the right forum for getting your exercises done.
Have you ever seen an athlete win a medal who had others do the
physical exercises for him? Science is not really different!

You can find background for the above in most introductions to
theoretical mechanics. Read it from several different perspectives
(i.e., authors) if you can't get the insight from the first one.

Arnold Neumaier

Igor Khavkine

Nov14-04, 11:59 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\n\nOn Fri, 12 Nov 2004 19:11:15 +0000, GRad wrote:\n\n> good afternoon\n> i wish to ask\n>\n> 1) what could be an apropriate candidate for a Lyapunov function, V(x), so\n> that one can determine the stability of a simple mass (m) -spring (k)\n> -damper (c)system\n>\n> the model equation is: mx(ddot)+cx(dot)+kx = 0\n>\n> V(x) must be positive definite & d/dt{V(x)} must be negative definite\n\nEnergy E(xdot,x) = (1/2)(m xdot^2 + k x^2). It\'s positive definite and\nintuitively we know that it is only going to decrease since the system is\ndissipative. This is also very simple to demonstrate mathematically.\n\n> 2) Also how does one prove the above system stable using LaSalle\'s\n> invariant principle?\n\nKnowing not what LaSalle\'s invariant principle is, I cannot say. But I\'d\nbe interested in the answer if someone knows.\n\nIgor\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Fri, 12 Nov 2004 19:11:15 +0000, GRad wrote:

> good afternoon
> i wish to ask
>
> 1) what could be an apropriate candidate for a Lyapunov function, V(x), so
> that one can determine the stability of a simple mass (m) -spring (k)
> -damper (c)system
>
> the model equation is: mx(ddot)+cx(dot)+kx =
>
> V(x) must be positive definite & d/dt{V(x)} must be negative definite

Energy E(xdot,x) = (1/2)(m xdot^2 + k x^2). It's positive definite and
intuitively we know that it is only going to decrease since the system is
dissipative. This is also very simple to demonstrate mathematically.

> 2) Also how does one prove the above system stable using LaSalle's
> invariant principle?

Knowing not what LaSalle's invariant principle is, I cannot say. But I'd
be interested in the answer if someone knows.

Igor