I. Introduction
There exist many and various nonlinear control methods, such as backstepping [1], fuzzy logic [2], model predictive control [3], sliding-mode control [4], and more in literature [5]. In particular, recently the state-dependent Riccati equation (SDRE) scheme for nonlinear control systems has been attracting attention, which is also known or abbreviated as “SDR equation” [6] and “frozen Riccati equation” [7]. Although the scheme exhibits practical performance advantages over other methods [8], it still demands more theoretical fundamentals to support such potentials [9], for instance, tunable tradeoff between state and control responses, robustness to disturbance and modeling error/uncertainty, and exploitation of system's nonlinearities [10], [11]. Therefore, to grow more confidence in the SDRE scheme, this article analytically focuses on a nonlinear benchmark problem [12]; that is, the motion control of a translational oscillator by a rotational actuator. Along this way of research, Mracek and Cloutier [12] pioneer the literature and reveals many practical potentials using the SDRE scheme, including stability, suboptimality, robustness, and input constraint—with significant impacts in diverse fields ([8], [13], to name a few). Moving forward, this article investigates the computational performance that is motivated by the recent literature [14]. As a matter of fact, it has been the most critical concern among the mechatronics/control community for decades and quite limits the application spectrum in general [15], [16], at least with respect to an up-to-date control design whose system dimension is four [14]. To sum up, the main contributions are as follows.
The applicability of the SDRE scheme that is in terms of the pointwise solvability is completely analyzed, which precedes any further performance consideration. The analytical results are formulated in a lower dimensional state space, which significantly alleviates the associated dominant computational burden while enlarges the domain of interest as confined by numerical confidence [12].
The analysis is designed in a way that generalizes as follows.
the widely adopted -parameterization technique as built on heuristics [10], [17] (note that a rigorous proof for its (in)effectiveness has been being expected [18]);
the parameter values of this benchmark problem [19];
a spectrum of nonlinear control system designs—both within and beyond the SDRE framework.