Dear Rodrigo Preuss, You are wrong about ZOH. It is the simplest of all hold circuits and is infact a rudimentary way of Euler discetization (something which I am not satisfied about). What I am looking for is something that is analytically equivalent to a infinite-order-hold circuit. The solution is matrix exponential for LTI systems. My query was for the equivalent in nonlinear systems.

Use finite element method. I know it is somewhat complicated with nonlinear dynamical systems. But you can try...

Dear Nita Shah, I have a system which is lumped parameter nonlinear system described by nonlinear ordinary differential equations. Finite element methods are used for spatial discretization of distributed parameter systems which are described by partial differential equations.

Dear David Trebing,

I had come across the notion of "finitely discretizable nonlinear" systems, sometime ago. The reference is

A. Chelouah, M. Petitot, "FInitely Discretizable Nonlinear Systems : Concepts and definitions", in Proc. IEEE CDC, pp. 19-24, New Orleans, Dec. 1995.

However, the systems under consideration in the paper are all of type wherein the jacobian matrix is nilpotent.

A LTI parallel would be dx/dt = Ax+Bu, with A^n = 0 for some small n, with some additional restrictions.

In this case the LTI discretization would be a polynomial in A of 'finite' length.

I work in the area of multirate sampling, i.e., using more than one sampling rate in the same systems to an advantage and had worked on the concept applied to the aformentioned class of systems.

https://www.researchgate.net/publication/221645392_Multirate_Output_Feedback_Based_Discrete-Time_Sliding_Mode_Control_for_a_Class_of_Nonlinear_Systems?ev=prf_pub

Now, I am interested in generalizing it to systems wherein the jacobian is NOT nilpotent. Hence, the query.

Now, the generalization in case of LTI systems with A not being nilpotent is the matrix exponential.

There is no matrix exponential for nonlinear systems, but, there may be a concept that parallels it.

Conference Paper Multirate Output Feedback Based Discrete-Time Sliding Mode C...

There are some techniques like collocation method to discretize nonlinear systems; however, I am not sure does they meet the "exact" condition you mentioned. Please see "Nonlinear programming" by Biegler or "Nonlinear Model-based Prediction control" by Grune for more details.

There is a method of the construction of exact integrable non linear systems that could be useful for you. See the classical paper M. J. Ablowitz, J. F. Ladik, Nonlinear differential-difference equations. J.Math. Phys. 16 (1975), 598-603.

If yous discretization system let you have and the jacobian matrix of you nonlinear state system is norm bounded, you might easily discretization you nonlinear differential equation with finitely many sample. the idea is that you may locally linearize your nonlinear system and discretized your system locally with a exponential matrix approach. however i am not quite sure how can you bound the required sample rate but it shouldn't a big deal if you completely know you system and the theory of discretization for linear system.

Dear Behrad Mahoobi,

My aim is to discretize the system and find an exact discrete time nonlinear equivalent for the continuous time system. The purpose is not for getting a good enough model for simulation.

In general you it is not possible to discretized the system without approximation. One example of such systems are chaotic system. If you system isn't chaotic you my find the exact sampling rate by analyzing the first order and second order derivatives of your system update equation.

Have you studied chaotic system characteristics?

Dear Janardhanan,

if you have exact solutions for the two parts of your problem "dx/dt = Ax+Bu", i.e.,

1) dx/dt = Ax (matrix exponential)

2) dx/dt = Bu (ODE solution)

then you may combine both by using an splitting method, such as the second-order Strang splitting. Of course this will also introduce a numerical approximation error, but this will only depend on the time discretisation and not in the spatial one.

Dear Alfonso Bueno-Orovio,

My focus is specifically on nonlinear systems and that too of exact discretization. I already know of reliable methods of approximate discretization of nonlinear systems also. But, those are useful only for simulation purpose and not for proper analysis.

An example

dx1/dt = u1

dx2/dt = u2

dx3/dt = (x1+1)*x2

is a system with an exact discrete model

x1(k+1)=x1(k)+t*u1(k)

x2(k+1)=x2(k)+t*u2(k)

x3(k+1)=x3(k)+t*x1(k)*(x2(k)+1)+t^/2*(x2(k)*u1(k)+(1+x1(k))*u2(k))+t^3/3*(u1(k)*u2(k))

for ANY arbitrary sampling period t>0.

For a piecewise constant input u(k), both the continuous and discrete system responses will match at all sampling instants.

Dear Janardhanan, how are you sure your nonlinear system is not chaotic? for a chaotic system never finite sampling rate can assure accuracy!

I am not saying that 'all' nonlinear systems can be discretized. Probably, chaotic systems cannot be discretized accurately. My aim is to find the class of nonlinear systems which 'can be' discretised.

Hence, it may be that the answer that I seek is a sub-set of non-chaotic systems. But, that still is a very large set.

I am not sure even in the field of control engeering there is a boundry between chaotic and nonchaotic system! Hence it might not be possible to classify in that way !

I think the answer depends on what you mean by the word 'exact' and 'equivalent representation'. One reasonable requirement is this: Any mathematical result on the discrete representation can be transfered to the continuous system to get a (provably correct) result on the latter. For example, a controller designed for the discrete representation that enforces a prescribed specification should provably enforce the specification if connected to the continuous plant.

If this is what you are looking for, the topic to study is "discrete abstractions of continuous systems". Please have a look at my list of publications (2011 paper in ieee trans AC). Don't hesitate to contact me if this seems to be what you are looking for.

To my knowledge this argument has been discussed in the following article:

https://ieeexplore.ieee.org/abstract/document/6858697

Where Peano-Baker series are used for the scope.

Mirko Aveta

Hi Mirko,

I read the paper you introduced but does this method really hold for a time when we have inputs (non-Autonomous systems)? Because in the discretized system, the input part doesn't appear explicitly.

I mean if we write the nonlinear system as dx=Ax+Bu (where A and B are nonlinear), the Jacobian part doesn't explicitly consist of the " u " part.

Dear Nahdi, it also includes TV cases that are approximated by expanding the Peano-Baker series.

Mirko Aveta Yes, I know that. My question is about how it can deal with inputs, not time. Because the Jacobian that is mentioned in the paper is only the Jacobian of the states and not the inputs.

Mahdi Nabipour If your input is simple state feedback, for instance, you would have to properly discretize your system with this logic and then use the discretized system to define your controller.

Thanks, Mirko Aveta