Dear friends,
Is it a correct way to use a linear system with uncertainties instead of the main nonlinear system?why?
I recommend transforming the system into LTV system. I tried that.you can transform a nonlinear system into a linear time varying one. The uncertainty will be covered by the state and input matrixes.
Hi, in case your physical problem is modelled by a nonlinear dynamics, you can deal with uncertainties on certain papameters but I think you system is not equivalent to a linear system with uncertainties. Could you tell us more about your problem ?
Hey Navid,
In 1932, Carleman T. published a paper in Acta Mathematica, showing that a finite dimensional system of nonlinear differential equations can be embedded into an infinite system of linear differential equations. Today, the Carleman embedding technique is known as Carleman linearization.
The Carleman Linearization tool in Mathematica 10 is a new powerful tool in the research of nonlinear systems to obtain a linear representation of a generic nonlinear system. An analogy between the Carleman linearization and the Taylor series (a representation of a function as an infinite sum of the function's derivatives at a single point) can be drawn.
Like Taylor series, the infinite linear system needs to be truncated after an arbitrary order yielding an approximation at a desired accuracy. Are you trying to refer the truncation error of the Carleman linearization as an uncertainty of modeling error?
If you are interested, consider getting the book "Nonlinear Dynamical Systems and Carleman Linearization" by K Kowalski and W-H Steeb.
http://www.worldscientific.com/worldscibooks/10.1142/1347
https://www.wolfram.com/mathematica/new-in-10/nonlinear-control-systems/carleman-linearization.html
http://reference.wolfram.com/language/ref/CarlemanLinearize.html
I recommend the answer of Prof.Yew. Moreover, through Carleman Linearization it is possible to design interesting control scheme and to justify the stability performance using Lyapunov theory. It is a very helpful tool which can bring remedy for several classical problematics in control theory.
Thank you all.
Dear Richard Epenoy ,
The considered system is a linearization of a nonlinear system with interval uncertainties; i.e. (y=[y0-eps, y0+eps])
I would like to thank Prof. Yew-Chung for the advertising my joint book together with Prof. Steeb. I also recommend my monograph: K. Kowalski, Methods of Hilbert Spaces in the Theory of Nonlinear Dynamical Systems, World Scientific, Singapore, 1994, on the method discovered by me relying on reduction of nonlinear differential equations to the linear, abstract Schrodinger-like equation in Hilbert space.
For Nolinear feedback register, which gives us binary sequences or sequences in the finite field Fp , we can find feedback register which gives us the same sequences but with more difficulty achieving them because the length of the linear feedback register would be very large. will be have it
I recommend transforming the system into LTV system. I tried that.you can transform a nonlinear system into a linear time varying one. The uncertainty will be covered by the state and input matrixes.
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