# Control of DC Motors to Guide Unmanned Underwater Vehicles
## 1 Introduction  
[PERSON] is often credited with the development of the least squares method [1], which can be done recursively, as well-described by [PERSON] [2]. Various instantiations of recursive least squares estimation algorithms have often been combined with modeling assumptions to produce adaptive decision and control methods; for example, the self-tuning regulators introduced by [PERSON] [3]. Not long afterward, two similar algorithms based on least squares and minimum variance control emerged. The first was presented by [PERSON] in Prague at an International Federation on Automatic Control conference [4], followed shortly after by [PERSON] and [PERSON]'s publication of an adaptive control using real time identification [5]. This lineage established the paradigm that exists today: adaption in two parts--modeling and estimation. From this auspicious inception, self-tuning regulators remain foundational, being first analyzed thoroughly and presented by [PERSON] and [PERSON] [6], and later revised [7], at which point the term self-tuning regulator was adopted. [PERSON] and [PERSON] later produced a seminal text [8] elaborating on self-tuning regulators, with several variations on modeling assumptions and estimation algorithms in addition to model-reference adaptive control topologies. This text included sections on persistent excitation and parameter convergence--as did [PERSON] and [PERSON]'s text [9], which also included analytic methods for proving stability of the resulting time-varying nonlinear systems. More recently, published comparative analyses fixed all factors except the modeling approach and/or the chosen estimation method, to produce advice regarding whichmethods to use to control direct current (DC) motors. While [10] illustrated quick methods to decide model degree based on minimizing estimation error, [11] compared indirect and direct self-tuning models with various degrees of pole excess, with and without plant cancelling. Subsequently, several authors have published research incorporating learning methods into self-tuning control algorithms. [PERSON] and [PERSON] [12] tackled the problem of speed control of a DC motor using feedforward artificial neural networks to capture and emulate detailed nonlinear mappings and implement a full nonlinear control law. [PERSON] et al. [13] addressed the problem of DC motor speed control using a constant gain feedback controller (which failed to counter dynamic load torques)--using artificial neural networks to identify and control online, and therefore modulate control rate. [PERSON] [14] used artificial neural networks (ANNs) in estimating and controlling speed for a separately excited DC motor, (one in which the rotor speed can be made to follow an arbitrarily selected trajectory to achieve accurate trajectory control of the speed, especially when the motor and load parameters are unknown). [PERSON] et al. [15] proposed neural network-based self-tuning proportional, integral, and derivative (PID) control for underwater vehicles. Due to difficulties designing based on cost functions, the controllers were trained with imitation learning [16] on optimized trajectories (which is sometimes referred to as a form of explicit model predictive control [17]). Model reference adaption remains an active area of research as well--e.g., [PERSON]'s integration [18] of a control model with PID components into model reference adaptive control of robots. [PERSON] [19] incorporated indirect self-tuning PID control with a radial basis function neural network (RBFNN), where its parameters were adjusted depending on linear adaptive neural network (LANN) parameters using system identification for DC servo motor speed control. Just last year [PERSON] et al. [20] used adaptive kinematic modeling and feedforward artificial neural networks to control surgical robots. [PERSON] et al. [21] developed a feedback control of a mobile inverted pendulum using a form of a parametric function to both stabilize the system and perform target tracking. Others include [PERSON] et al.