Dear Ali, in my book: "Shaikhet L.. Lyapunov Functionals and Stability of Stochastic Difference Equations. Springer, 2011" there is the Chapter 10: "Difference Equations as Difference Analogues of Differential Equations". Look it please, maybe it will give you some answer on your question.

I do not think that it is possible in general. Counterexample: consider a very simple system x_dot = 0.5x (1). The discrete time version of (1) x(k+1) = 0.5 x(k) is easily proven asymptotically stable but the continuous form is unstable. The stability criteria between two versions are in general very different although some of the key ideas of this version can be used along another one.

We must be careful with discretization.

If the sampling time is T, the simple approximation is usually

xdot~(x(k+1)-x(k))/T

This gives :

(x(k+1)-x(k))/T=0.5x(k)

x(k+1)=x(k)+0.5Tx(k)

and finally

x(k+1)=(1+0.5T)x(k)

Here, it is clear that the x(k) in the discrete system keeps increasing and so, the discrete sytem is also unstable, like the original continuous system.

I would say that one should know the bandwidth of the continuous-time system and should use a sufficerntly small discretisation periodf T. In such a case, the discrete-time system is a sufficiently good representation of the continuous-time system and its analysis, including stability analysis, is also representatiive.

Stability Theorem is a very important issue in the study of dynamical systems.

There are also many theories to examine the stability of linear and nonlinear systems.

The below book can to help you.

https://www.researchgate.net/profile/Muhammad_Ahsan23/post/What_is_the_general_principle_of_designing_the_switching_control_in_sliding_mode_control/attachment/59d62f0dc49f478072e9f5e3/AS%3A272534484783104%401441988692243/download/H-K-Khalil-Nonlinear-Systems-3rd-Edition-2002.pdf

Dear Ali,

your question is interesting. In general, the discrete version will depend on two fundamental choices: the discretization scheme used (there are several) and the discretization time. Notice that it is typicaly not sufficient just to replace t for k and \dot{x(t)} with x(k+1). The stability of discretized models will usually depend on the discretization time. This feature is obviously beyond the continuous-time counterpart.

Dear Dr. Luis Antonio Aguirre and Dr. Itzhak Barkana

I think as you said , the discritization scheme is very important. now If we assume that our numerical scheme for discritization is convergent (i.e the solution of discritized system converges to the solution of continuous time system ) for example by taking very small time steps dt, now is it possible to establish the stability results for continuous system based on descritized system?

Dear Ali,

if the discretized version of the continuous-time system is valid, then it should share the same dynamical features, including stability. Remember that the parameters of a discretized model may have different interpretations, so be careful if you are trying to establish the stability of the continuous system as a function of a given parameter.

Yes, Luis Antonio is right. We must make sure that the sampling time is short enough, such that the discrete representation is indeed a good representation of the continuous-time system. This is true not only for stabilty analysis, but this is also how real-world continuous-time systems, such as machines, robots, planes, etc., actually work, being ruled by discrete-time controllers, which must be fast enough to control the continuous-time system.

Dear Dr. Itzhak Barkana

you mentioned that

"We must make sure that the sampling time is short enough, such that the discrete representation is indeed a good representation of the continuous-time system"

and Dr. Luis Antonio Aguirre

mentioned that

"if the discretized version of the continuous-time system is valid"

I somehow get confused.

how can we assure that our discritized system will be a good representation of the continuous system.

is there a criteria for that?

maybe if we are looking at ODEs the answer is obvious, because most of the time we know how the continuous time system behaves, so when our discritized system behaves like the continuous time system we can conclude that our disceritized system is a good representation of the continuous one.

but imagine you are given a time-dependent PDE and there is no way to solve it analytically, (not even its stationary solution), so you disceritize it and solve it, now the question is, if our descritized version of PDE is stable, does it imply stability of the original PDE?

In fact because we can simply establish stability results for system of ODEs

(like eigenvalue analysis for linear systes, or linearized nonlinear system), or Lyapunov theory, maybe my question is irrelevant for system of ODEs.

but for PDEs the stability results can not be established so easily for continuous time version of PDE, (for example the PDE operator should be self-adjoin) so I want to know is it possible to investigate the stability result of PDEs by analysis of their discritized version?(because when we descritize a PDE it simply turns to a system of algebric equation or it turns to a truncated system of high dimensional ODEs and for these systems stability results are easy to investigate)

Ali,

My feeling is that discretized versions of stable continuous-time models are closer to becoming unstable. So if a discretized version of a PDE is stable, I wiuld tend to imagine that the continuous-time is also stable. The oposite is certainly not true, unless the discretization time AND space (in case od PDEs) are sufficiently small.

Hi Ali, (and I am Itzhak for all, no need for Prof., Dr., Mr., or Sir, except for strictly official occasions.) I happened to deal quite a bit with stability analysis of systems, in particular with nonlinear systems. Nevertheless, although for this purpose I had to use and even keep developing quite a bit of high-level Math, I always have the safety of real-systems in my mind and I thought that this is what also was bothering you.

Yes, in LTI systems, poles and zeros define the properties of the system, including stability.

However, if stability analysis of continuous-time systems is what bothers you, I would try to read and understand the literature on stability analysis of continuous-time systems. This is what all books and papers do. If PDE is your object, I don’t think that discretization can replace understanding the analysis of the actual continuous-time system.

These days, writing “Stability of PDE systems” on Google, gives you quite a bit of literature, such as

math.bu.edu/people/mabeck/lin_stab_minicourse_2012.pdf

(which could be a good start) and many others. Have Fun! Itzhak

Dear Dr. Luis Antonio Aguirre

intuitively , I agree with you, but it seems there is no theorem or prove for this

" if the descritized version of PDE is stable then the continuous-time is also stable "

Dear Itzhak Barkana

I hope there was a way to investigate the stability of PDE by descritized version of that, in this case, we simply descritized PDE with method of lines and again we could use our tools (such as eigenvalues or Lyapunov theory) for the truncated ODE system,

I thinks this might be a very big question that has no answer today

Is there any descritization scheme that preserve the stability properties of the original system?

I typically start searching for inequalities, that produce upper or lower bounds for an unknown function of interest, when attempting to answer questions like you asked. For example, if some function is known to be concave (because the sign of the second derivative is known), then the value of this function at some point between two other points is less than or equal to a linear interpolation between those two other points. I know that this suggestion is a long way away from answering your question, because there will be a lot of details to work through that are unique to your application, but I think that an investigation of bounds derived from inequalities is a good approach.

Dear Ali, Although I have had to deal a lot with stability, it was mostly with nonlinear (in particular adaptive control) systems, yet not PDE. If the system is continuous, I don’t even know why you cannot directly address its stability using some Lyapunov-style method, and why one would have to analyze its stability by first discretizing it. However, you seem to have a reason and, as no one knows all, I can only hope that you will ultimately find the right person(s) who can answer your questions.

Good Luck and Have Fun, Itzhak.

"Is it possible to prove the stability of a continuous time system from its corresponding discrete version?"

Thus posed the question, if it refers to any system in continuous time and to any of its possible discretizations, it is known, as stressed by people above, the answer is, in general, no.

If "its corresponding discrete version" alludes to a certain discrete representation in particular, the answer depends on the continuous system and that particular representation.

Sometimes a continuous system that behaves as a discrete system is sought. The latter is the starting point and the center of interest. The corresponding continuous system, for some reason easier to analyze, should agree on the most important characteristics of the original discrete system, such as those related to stability.

Going back to your original equation written as

x_dot=f(x). (1)

This equation can be solved by separation of variables so I assume that the function f makes this procedure too complicated to do (it can get complicated when the solution t as a function of x is not an invertible function), which motivates your interest in the discrete time version that you wrote as

x(k+1)=F(x(k)) . (2)

Unfortunately, unless the function f has certain properties (which might more directly lead to conclusions about stability), I don't think you can reach guaranteed conclusions about anything (including stability) from an investigation of the discrete time version unless the latter investigation uses either multiple choices or a special choice for the discrete points x1, x2, ... . If one and only one definite and arbitrary choice is made for these points, they could miss important properties of the function f (i.e., they might be outside some very important interval within the domain of f). Some kind of strategy regarding the selection of points would be needed to answer your question.

A small note: Lyapunov methodology is meant exactly for that: to be able to decide about the stability of the system without having to actually solve the nonlinear differential equation.

My apologies. I missed the statement "finite-dimensional" in the original question and assumed we were talking about scalar functions. "Finite dimensional" makes things more difficult.

However, if a scalar version of the problem has difficulties (explained in my previous post) then a more general version (finite-dimensional) will have at least those difficulties. I don't think you can reach guaranteed conclusions about anything (including stability) from an investigation of the discrete time version unless the latter investigation uses either multiple choices or a special choice for the discrete points x1, x2, ... , because one definite and arbitrary choice can miss important properties of the function f. Perhaps Itzhak Barakana gave the best advice. Instead of trying to reach conclusions about the continuous version from a study of some specific discrete representation, it might be easier to reach conclusions about the continuous version via Itzhak's suggestion.

I think what Dr Luis Antonio Aguirre said might be correct

when the discrete system is unstable we can not tell anything about the stability of continuous time system, but

when the discrete system is stable we can conclude that the continuous system was stable

otherwise we should say that some numerical scheme exist that can stabilized the unstable system.

have you ever seen that your unstable continuous-times systems in your computer simulations (where you had to discritized your continuous system) shows stable behavior, i.e the descritization scheme stabilized the system?

as an example for ODE systems:

consider dx/dt=2x (1)

this is an unstable continuous time system:

we use firs order Euler discretization with Time-step=h

x(k+1)=(2h+1)x(k) (2)

for any h>0, system (2) is also unstable.

can our discritization scheme stabilize the unstable system?

h is the sampling time and it can not be negative, it is trivial that always h>0

so our first order Euler method can not stabilized our unstable system.

could you find any counter example for that? i.e a numerical scheme stabilizes an unstable system?

Quote from Ali Namadchian: could you find any counter example for that? i.e a numerical scheme stabilizes an unstable system?

Consider the scalar system

\dot{x} = \lambda x

and discretize it via the implicit Euler scheme with time step h. Then we obtain

x(k+1) = (1-h\lambda)^{-1} x(k)

There are of course values for \lambda and h such that the continuous system is unstable but the discrete system is stable, for instance \lambda = 5 and h = 1.

I think the right question to ask is whether there is a numerical scheme that stabilizes an unstable system for any step size? And here also the answer is true, by simply picking the numerical scheme x(k+1) = 0 (which is an extremely efficient - though completely useless numerical scheme).

Dear Benjamin, the difference scheme has a sense only if it preserves the properties of the original differential equation. In this case should be x(k+1)=(1+\lambda h)x(k).

Yes Leonid. In order to see what we get after discretization, in my very first answer I mentioned that, in a differential equation, one must first make sure that one uses a proper representation of the derivative.

If the sampling time is T, then using xdot~(x(k+1)-x(k))/T leads to proper discrete-time representation, like you also show. So, at least for those simple first-order examples, if the continuous-time system is unstable, the discrete-time representation also comes out unstable.

Still, for a general answer, l would be very careful before reaching conclusions based on a too simple example.

Dear Itzhak, I prefer to write \dot x~(x(k+1)-x(k))/h, where h is the step of discretisation. And I think that discrete analogue needs save all properties of his original. In another way it is not an analogue but an absolutely another object.

Dear Dr. Roussel

It seems that shadowing lemma states that the error between the states of the discrete time system and continuous-time system is bounded, But does it help me to find my answer?

here is the interpretation of shadowing lemma in Wiki

Informally, the theory (shadowing lemma) states that every pseudo-orbit (which one can think of as a numerically computed trajectory with rounding errors on every step[1]) stays uniformly close to some true trajectory (with slightly altered initial position)—in other words, a pseudo-trajectory is "shadowed" by a true one.

I have some question about shadowing lemma, I really appreciate if you could answer them

1-Imagine we have descretized a continuous-time system, is there any tool to see if our descrite system (pseudo trajectories) shadowed by true one (continuous time-system) i.e a tool for examination of shadowing lemma

2- in numerical analysis we have a convergence definition

A numerical method is said to be convergent if the numerical solution approaches the exact solution as the step size h goes to 0.

what is the difference between shadowing lemma and convergence of numerical solution? Dont you think that convergence is stronger than shadowing lemma?

3- under shadowing lemma, does asymptotic stability of descrite-time system implies asymptotic stability of continuous-time system?

When we obtain discrete time system from continuous system, the step time (sampling time) is very important. The sampling time (T) must as follow

sampling frequency>2*biggest frequency in the first signal

If value of T selected be smallest the limitations of stability will be big.

Also for evaluation the stability in Continuous and Discrete system must we use below notation

for continuous system:

If Xd=A*x+B*u

and roots as below

Lambda=Beta(+-) j*Alfa

if Beta0 then the system is unstable

for discrete system:

X(k+1)=Ad*x(k)+Bd*u(k)

Lambda=Beta(+-) j*Alfa

|lambda|0 then the system is unstable

Ad=exp(A*t)

Bd=inv(A)*(Ad-I)*B

You see that the selection of T is very important for converting continuous system to discrete system.

Hamze Ahmadi Jeyed

Yes, you are right that it is important to understand that T must be small enough.

However, when you write

sampling frequency>2*biggest frequency in the first signal,

this is right when you assume you have all samples and want to reproduce a signal.

However, try to sample the simple sine(wt) at twice or three times its frequency and see what you get as digital signal. So, practically, you need much more than twice, I would say at least 8 times to get a good digital representation..

I also want to know more about this.

Exponential integration methods provide a good method for estimating the stability of a continuous system using a discrete counterpart. We start from the calculation of the Jacobian for the autonomous system

dx/dt=f(x) (1)

After performing a dynamic linearization with respect to the continuously evolving reference state, we can write (1) in a form

dx/dt=J x(t) + g(x(t)) (1 b)

where g(x(t))=f(x(t))-J x(t)

The solution of (1 b) is provided by the Volterra equation and can be used to derive a family of exponential integration schemes for (1) (Hochbruck and Ostermann, 2010). The main interesting property of these schemes is that they are exact for the linearized system regardless of the time step. Because of this, we can avoid many problems with schemes such as RK4.

The exponential method works just as well for a single ODE, sets of ODEs as for large sets of ODEs obtained after discretization of space derivatives in PDEs or even in the systems of PDEs.

Reference:

Hochbruck and Ostermann (2010): Exponential integrators, Acta Numerica, vol. 19, pp 209–286