An analytical fuzzy-based approach to -gain optimal control of input-affine nonlinear systems using Newton-type algorithm

Abstract

This paper is concerned with ${\mathcal{L}}_2$-gain optimisation of input-affine nonlinear systems controlled by analytic fuzzy logic system. Unlike the conventional fuzzy-based strategies, the non-conventional analytic fuzzy control method does not require an explicit fuzzy rule base. As the first contribution of this paper, we prove, by using the Stone–Weierstrass theorem, that the proposed fuzzy system without rule base is universal approximator. The second contribution of this paper is an algorithm for solving a finite-horizon minimax problem for ${\mathcal{L}}_2$-gain optimisation. The proposed algorithm consists of recursive chain rule for first- and second-order derivatives, Newton’s method, multi-step Adams method and automatic differentiation. Finally, the results of this paper are evaluated on a second-order nonlinear system.

Keywords:

• analytic fuzzy control
• ${\mathcal{L}}_2$-gain optimisation
• Newton’s method
• input-affine nonlinear systems

Notes

1. We use term ‘${\mathcal{L}}_2$-gain control’ as in Van Der Schaft (1992), Van Der Schaft (1996) because it is more correct than ‘nonlinear ${\mathcal{H}}_{\infty }$ control,’ since in the time domain ${\mathcal{H}}_{\infty }$ norm is nothing else than an induced ${\mathcal{L}}_2$-norm of the nonlinear systems.

2. According to the usual convention, for some function z = f (v) the Hessian is defined by $$\begin{array}{c}\hfill \frac{{\partial }^2\mathbf{z}}{\partial \mathbf{v}\partial {\mathbf{v}}^{\text{T}}}=\frac{\partial }{\partial \mathbf{v}}\left[\mathrm{vec}\left(\frac{\partial \mathbf{z}}{\partial \mathbf{v}}\right)\right].\end{array}$$

Additional information

Notes on contributors

Vladimir Milic

Vladimir Milic received his BE and ME in Mechanical Engineering from the University of Zagreb in 2007 and 2008, respectively. He is a PhD candidate in the Department of Robotics and Automation of Manufacturing Systems at Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Croatia. His current research interests include numerical algorithms for optimal and robust control of nonlinear systems.

Josip Kasac

Josip Kasac is an Associate Professor in the Department of Robotics and Automation of Manufacturing Systems at Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Croatia. Dr. Kasac received his PhD in Mechanical Engineering from the University of Zagreb in 2005. His current interests are control of nonlinear mechanical systems, repetitive control systems and optimal control.

Branko Novakovic

Branko Novakovic is a Professor emeritus in the Department of Robotics and Automation of Manufacturing Systems at Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Croatia. Prof. Novakovic received his PhD in Mechanical Engineering from the University of Zagreb in 1978. His research interests include control systems, robotics, neural networks, and fuzzy control. He is author of two books, Control Systems (1985) and Control Methods in Robotics, Flexible Manufacturing Systems and Processes (1990), and co-author of a book Artificial Neural Networks (1998).