Dear S. A. Nahian,

I suggest for you a links and attached files in topics.

-Industrial Applications of Neural Networks

https://books.google.dz/books?isbn=0849398029

-The Navier-Stokes Equations: Theory and Numerical Methods

https://books.google.dz/books?isbn=0203908503

-Stability - Scholarpedia

www.scholarpedia.org/article/Stability 

Best regards

Hi Nahian,

can you explain your simulation please?

what is the system, which adaptive law did you take?

what is the lyapunov function?

Regards

Thank you Mr. Lafifi and Harmouche.

@Mr Harmouche: I am trying to control an Electrohydraulic actuator cylinder. The top graph is its reference and cylinder position. and the bottom graph is the derivative of lyap_function.

The adaptation law I used like BELBIC controller,

e=tracking error

s=60*e;

rew=-40*s*sd-20*e^2;

Learning:

-----------

v_dot=la*max(rew-s*v,0);

w_dot=lo*(u-rew);

Control_output:

---------------------

u=s*v-s*w;

Lyapunov function:

--------------------------

V=.5*(e/v)^2+.5*(w)^2

for all time v>0 and bounded up to certain value v_max.

I am quite new in this field, please let me know if you need any other infos.

my another question: Is it true, if any controller is asymptotically stable, then there must be existing such lypunov_candidate function which satisfies V_dot

Hi Nahian,

The convergence imply the existance of Lyapunov function. Otherwise it's not necessary that you can find the Lyapunov function itself, you can refer to the work of Baht&Bernstein 2005 on géometric homogeneity.

Back to your system:

I find a paper that probably can help you.

http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=4602180

In this paper you can find the lyapunov function and the adaptive controller. I can see that the lyapunov function is a bit different from yours.

Regards.

Hi Mr Harmouche, Thanks for your reply and paper. I will try with that manner.

Regards

Nahian

Dear S. A. Nahian,

In the enclosed pdf file rg3.pdf I prepared a simple example which explains the behaviour

you have described.

Regards,

Piotr Grabowski

Hi Prof. Piotr,

Thanks for your kind effort on the problem, I will try to understand the situation as you described. I would let you know if I have any quarry. your pdf is very much appreciated.

Regards

Nahian

Hi Nahian,

First, although pretty many people may still tell you the opposite, the answer to your question is yes. Your Lyapunov derivative may contain a non-negative term and still end with stability conclusion.

I must worn you though: while in linear time invariant (LTI) systems poles tell you all about stability, dealing with nonlinear systems is not as simple and require diligence. During the years, people have managed to obtain various partial results that solved one aspect or other and then, same people or others managed to move on and get more results.

Indeed, the original Lyapunov requires the derivative of the positive definite V(x)  to be negative definite or, in other words, the derivative Vdot(x) must be negative for any nonzero x and zero only for x=0. In such a case, all trajectories are bounded and end at Vdot(x)=0 which implies x=0. For an example, if V(x)=x1*4+x2^4 and the derivative is Vdot(x)=-(x1*2+x2^2), then the trajectories must end at  Vdot(x)=0, which is equivalent with x=0.

However, in many cases other than class-room examples, the derivative is hardly negative semidefinite. Many developers have tried to get some results even in such a case. Some required some tough prior conditions in order to end with the conclusion that trajectories still end with Vdot(x)=0. However, this does not necessarily imply x=0. For an example, if V(x)=x1*4+x2^4 and the derivative is Vdot(x)=-x1*2, then the trajectories still end at Vdot(x)=0, yet this is not equivalent with x=0 any more. 

In your case, you may get Vdot(x)=-x1*2+x1*exp(-t). This is a particular case of Vdot(x)=W1(x)+W2(x), where W1(x) is negative definite or semidefinite and W2(x) is some non-negative term. Because of the not necessarily negative transient term, this derivative is not even negative semidefinite, so for some time, the analysis seems to be finished. However, new results show that, if W2(x) is bounded on bounded trajectories, W2(x) does not affect the ultimate behavior of bounded trajectories and they still end at W1(x)=0.

Instead of making this into a dissertation, you may look for the Technical Report

Technical Report: Can Stability Analysis be really simplified? (Revisiting Lyapunov, Barbalat, LaSalle and all that)

which you can get from Research Gate under my name and which may give a better review of results from Lyapunov to these very days.

Please let me know if you got it and also feel free to come back with any questions or comment and also don't be shy if you feel the need to come back with objections.

Best regards,

Itzhak.

Dear @Nahian, as I do follow @Itzhak, I am very pleased with his answer. It is fine report that you may download, link follows. 

Technical Report Can Stability Analysis be really simplified? (Revisiting Lya...

Dear Dr, Itzhak Barkana, Thank you for your excellent answer and it is perfectly match to my case what I understood. Some of my lyapunov term has x*(...) . I will work on it.

I would try to understand your technical report and get back to you for further quarries.. Thank you once again for your kind effort and answers.

@Dr. Ljubomir: Thank you sir for giving me the link of the report.. 

Regards 

Nahian

Dear Nahian,

(As I use to tell everybody, I am just Itzhak or all, no need for Prof., Dr., Mr., or Sir, except for strictly official occasions, of course.)

Glad you liked it. As I wrote, please feel free to come back any time and as many times you need.

Best,

Itzhak

for all, of course :-)

Dear @Itzhak, dear @Nahian, let me bring some more resources which are coauthored by my former academic adviser Lj. Grujic. His adviser was Prof. D. Siljak.

https://www.crcpress.com/Stability-Domains/Gruyitch-Richard-Borne-Gentina/p/book/9780415308489

http://www.springer.com/us/book/9783540183006

Thanks Ljubomir.

 in the past I used to refer the works of Prof. D. Siljak and  vice-versa, yet the last years took me on a different path.

All my "purely abstract:" works on stability of last years came only because of my worry as an Engineer that my customers may get scared with my Simple Adaptive Control just because they may read that in nonlinear systems one has no guarantee of stability unless one can guarantee uniform continuity, etc, or, like in our specific case, because the Lyapunov derivative lost its negativity even for a moment.

They shouldn't.