Generalized Kernel Approximation Approach to $\mathcal{L}_{1}$ Optimal Control of Sampled-Data Systems

This paper introduces a new approach to the
$\mathcal{L}_{1}$
optimal control of sampled-data systems, in which the problem of minimizing their
$\mathcal{L}_{\infty}$
-induced norm is concerned with. After establishing an operator-based representation of sampled-data systems via the lifting approach, the hold function of the output operator and the kernel function of the input operator are approximated by piecewise constant functions. More precisely, we divide the sampling interval
$[0,\ h)$
into
$M$
subintervals with an equal width, and take the freedom to select a point for each subinterval, at which Taylor expansions of the relevant functions are derived. This developed approach is regarded as a generalized version of the conventional kernel approximation (KA) approach, in which Taylor expansions are considered only at the beginning point for each subinterval. Based on the generalization, we derive a discretization procedure of the continuous-time plant in terms of the
$\mathcal{L}_{1}$
optimal control. We then show that the discrete-time
$l_{1}$
optimal controller with respect to the discretized plant also approximately minimizes the
$\mathcal{L}_{\infty}$
-induced norm of the original sampled-data system with the associated convergence rate of
$1/M$
. Finally, it is shown that taking the central point for each subinterval minimizes a relevant performance deterioration occurring from employing the generalized KA approach when the approximation parameter
$M$
is fixed.