Taylor–Lie formulation based discretization of nonlinear systems

Abstract

This paper proposes a discretization technique based on Taylor–Lie formulation. It also presents an overview of various techniques that can be applied for the discretization of a class of nonlinear systems. An in-depth analysis and comparison of the proposed technique with existing techniques has also been done. The efficiency of these different techniques have been compared on the basis of approximation error and discrete model complexity.

Appendix

Fourth derivative for the discrete time version of (26) is,

$$\begin{aligned} \begin{aligned}&D^{(4)}=d_{0}^{(4)}(x)+d_{1}^{(4)}(x)u+d_{2}^{(4)}(x)u^2+d_{3}^{(4)}(x)u^3\\&\quad \quad \quad \;+d_{4}^{(4)}(x)u^4 \\&\text {where }d_{0}^{(4)}(x)= \frac{\partial d_0^{(3)}}{\partial x}(x) \cdot f(x)={\mathcal {L}}_{d_0^{(3)}}f=-105x^{9},\\&d_{1}^{(4)}(x)=\frac{\partial d_1^{(3)}}{\partial x}(x) \cdot f(x)+\frac{\partial d_0^{(3)}}{\partial x}(x) \cdot g(x)\\&\quad \quad \quad \quad ={\mathcal {L}}_{d_1^{(3)}}f+{\mathcal {L}}_{d_0^{(3)}}g = -240x^{7},\\&d_{2}^{(4)}(x)=\frac{\partial d_2^{(3)}}{\partial x}(x) \cdot f(x)+\frac{\partial d_1^{(3)}}{\partial x}(x) \cdot g(x)\\&\quad \quad \quad \quad ={\mathcal {L}}_{d_2^{(3)}}f+{\mathcal {L}}_{d_1^{(3)}}g = 174x^{5},~~~~~~~~~~\\&d_{3}^{(4)}(x)=\frac{\partial d_3^{(3)}}{\partial x}(x) \cdot f(x)+\frac{\partial d_2^{(3)}}{\partial x}(x) \cdot g(x)\\&\quad \quad \quad \quad ={\mathcal {L}}_{d_3^{(3)}}f+{\mathcal {L}}_{d_2^{(3)}}g = -40x^{3},~~~~~~~~~~\\&d_{4}^{(4)}(x)=\frac{\partial d_3^{(3)}}{\partial x}(x) \cdot g(x)={\mathcal {L}}_{d_3^{(3)}}g = -x,~~~~~~~~~~\\&D^{(4)}={\mathcal {L}}_{d_0^{(3)}}f+({\mathcal {L}}_{d_1^{(3)}}f+{\mathcal {L}}_{d_0^{(3)}}g)u+({\mathcal {L}}_{d_2^{(3)}}f +{\mathcal {L}}_{d_1^{(3)}}g)u^2\\&\quad \quad \quad +({\mathcal {L}}_{d_3^{(3)}}f+{\mathcal {L}}_{d_2^{(3)}}g)u^3+({\mathcal {L}}_{d_3^{(3)}}g)u^4 \\&D^{(4)}=-105x^{9}-240x^{7}+174x^{5}-40x^{3}-x \end{aligned} \end{aligned}$$

and fifth derivative for the discrete time version of (26) is,

$$\begin{aligned}&D^{(5)}=d_{0}^{(5)}(x)+d_{1}^{(5)}(x)u+d_{2}^{(5)}(x)u^2+d_{3}^{(5)}(x)u^3\nonumber \\&\quad \quad \quad \quad +d_{4}^{(5)}(x)u^4 +d_{5}^{(5)}(x)u^5\nonumber \\&\text {where }d_{0}^{(5)}(x)= \frac{\partial d_0^{(4)}}{\partial x}(x) \cdot f(x)={\mathcal {L}}_{d_0^{(4)}}f= -945 x^{11},\nonumber \\&d_{1}^{(5)}(x)=\frac{\partial d_1^{(4)}}{\partial x}(x) \cdot f(x)+\frac{\partial d_0^{(4)}}{\partial x}(x) \cdot g(x)\nonumber \\&\quad \quad \quad \quad ={\mathcal {L}}_{d_1^{(4)}}f+{\mathcal {L}}_{d_0^{(4)}}g = 2625x^{9} ,\nonumber \\&d_{2}^{(5)}(x)=\frac{\partial d_2^{(4)}}{\partial x}(x) \cdot f(x)+\frac{\partial d_1^{(4)}}{\partial x}(x) \cdot g(x)\nonumber \\&\quad \quad \quad \quad ={\mathcal {L}}_{d_2^{(4)}}f+{\mathcal {L}}_{d_1^{(4)}}g = -2550x^{7} ,~~~~~~~~~~\nonumber \\&d_{3}^{(5)}(x)=\frac{\partial d_3^{(4)}}{\partial x}(x) \cdot f(x)+\frac{\partial d_2^{(4)}}{\partial x}(x) \cdot g(x)\nonumber \\&\quad \quad \quad \quad ={\mathcal {L}}_{d_3^{(4)}}f+{\mathcal {L}}_{d_2^{(4)}}g = 990x^{5},~~~~~~~~~~\nonumber \\&d_{4}^{(5)}(x)=\frac{\partial d_4^{(4)}}{\partial x}(x) \cdot f(x)+\frac{\partial d_3^{(4)}}{\partial x}(x) \cdot g(x)\nonumber \\&\quad \quad \quad \quad ={\mathcal {L}}_{d_4^{(4)}}f+{\mathcal {L}}_{d_3^{(4)}}g = -121x^{3},~~~~~~~~~~\nonumber \\&d_{5}^{(5)}(x)=\frac{\partial d_4^{(4)}}{\partial x}(x) \cdot g(x)={\mathcal {L}}_{d_4^{(4)}}g = x,~~~~~~~~~~\nonumber \\&D^{(5)}={\mathcal {L}}_{d_0^{(4)}}f+{\mathcal {L}}_{d_1^{(4)}}f+({\mathcal {L}}_{d_0^{(4)}}g)u+ ({\mathcal {L}}_{d_2^{(4)}}f+{\mathcal {L}}_{d_1^{(4)}}g) u^{2}\nonumber \\&\quad \quad \quad \quad +({\mathcal {L}}_{d_3^{(4)}}f+{\mathcal {L}}_{d_2^{(4)}}g)u^{3}+ ({\mathcal {L}}_{d_4^{(4)}}f+{\mathcal {L}}_{d_3^{(4)}}g)u^{4}+ ({\mathcal {L}}_{d_4^{(4)}}g)u^{5}\nonumber \\&D^{(5)}= -945 x^{11}+2625x^{9}u-2550x^{7}u^{2}\nonumber \\&\quad \quad \quad \quad +990x^{5}u^{3}-121x^{3}u^{4}+xu^{5} \end{aligned}$$

(29)