Is anyone here familiar with the Barbalat lemma and it's application in control engineering?

Sundarapandian Vaidyanathan

Barbalat's Lemma is detailed well in "Nonlinear Systems" (Hassan K. Khalil, 3rd edition, page 323).

Lemma 8.2: Let \phi: R --> R be a uniformly continuous function on [0, \infty). Suppose that

\lim_{t --> \infty) \int\limits_0^t \phi(\tau) d\tau

exists and is finite. Then \phi(t) --> 0 as t --> \infty.

[Khalil proves this lemma by a simple contradiction argument].

[Barbalat's lemma has many applications in adaptive control systems].

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Timm Faulwasser

Technische Universität Dortmund

Barbalat's Lemma is also used to prove asymptotic convergence of continuous-time sampled-data nonlinear model predictive control, cf.

Fontes, F. A General Framework to Design Stabilizing Nonlinear Model Predictive Controllers. Sys. Contr. Lett., 2001, 42, pp. 127-143.

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Itzhak Barkana

BARKANA Consulting

Sorry if I am writing a long story that may also surprise some people.

It all started when people “discovered” that Lyapunov's need for a negative definite derivative is not satisfied in many other than class-room examples.

Because what most people know about LaSalle or Krasovsky-LaSalle Invariance principle only deals with autonomous (I prefer this term rather than time-invariant) systems of the form xdot=f(x) (where time does not explicitly appear) and not with real control systems, which are nonautonomous of the form xdot=f(x,t) (where time does explicitly appear), people adopted Barbalat’s Lemma from function theory.

The Lemma says that if V(t) is known to have a finite limit and if its derivative Vdot(t) is uniformly continuous, then Vdot tends to 0 as t tends to ∞.

So, if you can find an appropriate Lyapunov function and if can prove that its derivative is uniformly continuous, you may reach the conclusion that Vdot tends to 0. Note that, although a lot of effort is invested in proving uniform continuity, there is another issue of the finite limit of V, which people may forget. How do you know that V reaches a finite limit unless Vdot tends to zero and why would Vdot reach zero unless V has a finite limit and…?

Also, note that, while Lyapunov and LaSalle deal with trajectories, Barbalat deals with functions and why would a function decide to go to zero.

The real issue here is that most people only know the pre-historical LaSalle (or Krasovsky-LaSalle) of 1950-1960, while the REAL contribution of LaSalle of 1976-1980, which extended the Invariance Principle to nonautonomous systems, have remained strangely unknown. I happened to have written a paper where at first I only wanted to tell people about those ignored works of LaSalle and followers, yet because of questions and comments from reviewers and other readers, I saw that, actually, I could go beyond LaSalle and simplify even his already mitigated conditions.

Bottom line, if you can find an appropriate Lyapunov function and its derivative is negative semidefinite, all you have to check is that the system cannot pass an infinite distance in finite time along bounded trajectories. In this case, trajectories tend to ultimately reach Vdot=0.

So, if f(x,t) is bounded for bounded x, such as xdot=x^5+sint, it clearly satisfies the assumption.

However, even if f(x,t) is not bounded, such as in xdot=x^5+δ(t), the integral of f(x,t) is still bounded for any finite time-interval.

The appropriate references and more you can find in my recent paper:

I. Barkana: "Defending the Beauty of the Invariance Principle," International Journal of Control, Vol. 87, No. 1, pp. 186-206, 2014

(Published On-Line 06 Sep 2013), DOI:10.1080/00207179.2013.826385.

Any questions/comments/objections are most welcome

All the Best,

Itzhak

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Raja Muhammad Imran

Aalborg University

Barbalat's lemma is a useful tool in stability analysis of nonlinear systems and also used to check the asymptotic convergence of adaptive control.

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Alireza Modirrousta

Sazeh Consultant

from theorical aspect you can refer to Applied Nonlinear Control (Slotine.Li), Nonlinear systems (Khalil) and Nonlinear Dynamical systems and control (M. Haddad, Chellaboina)

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Sundarapandian Vaidyanathan

Vel Tech - Technical University

Dear A. Modirrousta - Thanks for good references for Barbalat's Lemma.. Friends may read its proof also! Thanks..

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Hengameh Nouraei.ch

Shahid Bahonar University of Kerman

Thank you all for your useful answers.

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Ankit K Shah

L. D. College of Engineering

It is mainly used for time varying system to prove the stability of the system in order to design robust control or optimal control law.

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Samir Ladaci

National Polytechnic School of Algiers

An essential tool for demonstrating stability of adaptive control systems.

Many thanks for this illustrative introduction to Barbalat' Lemma.

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MohammadAli Daneshpajouh

Shahrood University of Technology

in time variable systems when derivative of lyapunov function is negetive the close loop system is not asymptotically stable, if barbalar lemma is set then we can say that the close loop system is asymptotically stable.

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Itzhak Barkana

BARKANA Consulting

Sorry again, yet there is some confusion here.

Lyapunov always works if the derivative is negative DEFINITE. The result is asymptotic stability because the negative DEFINITE derivative is zero if and only if x=0.

The problem started because, in other than class-room examples, the derivative is at most negative SEMIdefinite, Here, all Hell broke loose.

First, theory only covered autonomous systems xdot=f(x) (that some would call time-invariant) and says that the trajectories tend to Vdot(x)=0. In specific cases, this may imply asymptotic stability, yet not necessarily. You may have a system with 20 variables and Vdot=-x1^2. only says that the end is x1=0, with no knowledge about the rest.

Then, Barbalat’s Lemma gave some extension for nonautonomous (or time-varying) systems of the form xdot=f(x,t) . At the time, it was a help. However, it unnecessarily requires uniform continuity of signals and led to people ignoring the REAL LaSalle’s Invariance Principle and further developments which lead to stability proofs with much lesser requirements and, ultimately, without even mentioning continuity.

Hope it helps. The rest is explained in my previous answer

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11 Recommendations

Leigh C. Becker

Christian Brothers University

I used Barbalat's lemma and Liapunov functionals to investigating asymptotic stability of Volterra integro-differential equations in the paper "Uniformly Continuous L1 Solutions of Volterra Equations and Global Asymptotic Stability" published in CUBO A Mathematical Journal Vol 11, No. 8 Vol 3 2009. I also give references to Barbalat's lemma in that paper that may be useful to you.

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Hengameh Nouraei.ch

Shahid Bahonar University of Kerman

Mr.Becker

Thanks for your attention and useful answer.

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Itzhak Barkana

BARKANA Consulting

@Professor Becker

...and yet, wouldn't you be interested to know that same stability conclusions could be obtained without the need for any continuity ?

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Dinesh D Dhadekar

Defence Institute of Advanced Technology

Mr. Itzhak Barkana

Thanks for useful concept !

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1 Recommendation

Itzhak Barkana

BARKANA Consulting

As I wrote in my two previous messages, Barbalat's Lemma was very useful when nothing else was available. The fact that relevant works of LaSalle dealing with nonautonomous systems have remained unknown for more than 40 years is one of the most amazing enigmas of this world.

Anyway, Barbalat’s Lemma is a nice and correct mathematical result related to the theory of functions. In practical terms, its interpretation in relation to stability requires strong conditions of uniform continuity of practically all signals involved. In other words, your “dangerous” nonlinear robot may blow up just because you decided to use some “threatening” discontinuous pulse input command signal in order to move it from A to B and then back to A.

OK, following along the lines of LaSalle’s extension of the Invariance Principle to nonautonomous systems, the stability conditions can be considerably simplified and mitigated.

Then, because LaSalle's works (as said, written some 40 years ago) still kept some conditions that at the time seemed to be needed, new works simplify the conditions even more. One of them I listed in my previous messages.

I. Barkana: "Defending the Beauty of the Invariance Principle," International Journal of Control, Vol. 87, No. 1, pp. 186-206, 2014 (Published On-Line 06 Sep 2013), DOI:10.1080/00207179.2013.826385.

It also contains the due references to the most relevant works of LaSalle and others. Still, even this new work kept some conditions that at the time seemed to be needed, so a new work simplify the conditions even more, as its title actually says:

I. Barkana: “The new Theorem of Stability – Direct extension of Lyapunov Theorem,” Mathematics in Engineering, Science and Aerospace (MESA), Vol. 6, no. 3, pp. 519-535, 2015.

I never had a problem with Barbalat’s Lemma itself, thinking that various people may use various approaches that fit their taste. However, although some people gladly received the new stuff, some others came up with counterexamples that were supposed to demonstrate the need for continuity and that were supposed to make the new stuff a “mumbo-jumbo theory.” Instead, they forced me to review all those well-established examples and, to my own shocking surprise, I found out that they all were using the right mathematical formulas… only in the wrong places. Bottom line, they are all wrong. This led to the recent

I. Barkana: “Barbalat’s Lemma and Stability – Misuse of a correct mathematical result?” Mathematics in Engineering, Science and Aerospace (MESA), Vol. 7, No. 1, pp. 197-218, 2016

People who are really interested in nonlinear systems stability analysis may find some interest in this stuff.

Best regards,

Itzhak Barkana

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6 Recommendations

Itzhak Barkana

BARKANA Consulting

Because my message above may sound too serious, I thought to add a somewhat lighter message:

No matter what I or some others may write now, or even what LaSalle already wrote 40 years ago on nonautonomous systems, the best books on Nonlinear Control and most people simply ignore them all and only use Barbalat. Nothing is wrong with Barbalat's lemma as a mathematical result, yet in the context of stability it requires uniform continuity of the Lyapunov derivative,

So, the optimists work hard to prove and guarantee uniform continuity of all signals, the pessimists work even harder to invent "solutions" for the "problems" of discontinuity (an entire Empire of Impulsive Control , etc.)

Instead, I dare to ask "Who on Earth needs continuity of derivative?" while others (such as Anthony Michel) even directly ask "Who on Earth needs derivative?" :-)

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Md Mashfique Reza

Intelsat US LLC

thanks

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