$L_{1}$ Discretization for Sampled-Data Controller Synthesis via Piecewise Linear Approximation

This paper develops a new discretization method with piecewise linear approximation for the $L_{1}$ optimal controller synthesis problem of sampled-data systems, which is the problem of minimizing the $L_{\infty}$-induced norm of sampled-data systems. We apply fast-lifting on the top of the lifting technique, by which the sampling interval $[0,h)$ is divided into $M$ subintervals with an equal width. The signals on each subinterval are then approximated by linear functions by introducing two types of ‘linearizing operators’ for input and output, which leads to piecewise linear approximation of sampled-data systems. By using the arguments of preadjoint operators, we provide an important inequality that forms a theoretical basis for tackling the $L_{1}$ optimal controller synthesis problem of sampled-data systems more efficiently than the conventional method. More precisely, a mathematical basis for the piecewise linear approximation method associated with the convergence rate is shown through this inequality, and this suggests that the piecewise linear approximation method may drastically outperform the conventional method in the $L_{1}$ optimal controller synthesis problem of sampled-data systems. We then provide a discretization procedure of sampled-data systems by which the $L_{1}$ optimal controller synthesis problem is converted to the discrete-time $l_{1}$ optimal controller synthesis problem. Finally, effectiveness of the proposed method is demonstrated through a numerical example.