# Adaptive control for double-integrator class systems in the absence of velocity feedback

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## Abstract

This work considers formulation of new classes of adaptive controllers for double-integrator type systems where the underlying system parameters are uncertain and the complete state-vector is not available for feedback. Given the parameter uncertainty within the system model, a "separation principle" cannot generally be invoked towards an observer geared towards reconstruction of the full state vector using only measured variables.

In this report, controllers are designed for some important sub-classes of Euler-Lagrange type mechanical systems, where states are physically interpreted as position and velocity variables, and only the position part of the state vector is available as measured output. The typical approach to obtain velocity estimates using position interpolation (also known as dirty differentiation), is known to be strongly susceptible to measurement noise and therefore does not usually represent a robust option for feedback control implementation. The proposed control scheme achieves global asymptotic stability for system dynamics subject to the condition that velocity states appear within the governing dynamics in a linear fashion. This arguably restrictive condition is loosened for the special case of scalar system with friction non-linearity as is typical within hardware implementations. The objective is to study prototypical mechanical systems with non-linearity appearing in the velocity rate equations with the eventual applications envisioned towards the attitude control problem accounting for the gyroscopic non-linearity in the Euler rotational dynamics.

Based on classical certainty equivalence approaches for adaptive control, one cannot readily deal with cross terms associated with parameter estimates and unmeasured states in the Lyapunov function derivative in order to make the Lyapunov function negative definite or negative semi-definite. However, employing a new approach, this obstacle is shown in this report to be circumvented for scalar systems. In order to generalize the methodology for higher-order dynamics, a filtered state approach is used. Specifically, an auxiliary variable is introduced which plays an important role in determining restrictions on the control parameters and analyzing the stability. The proposed approach helps to overcome the uniform detectability obstacle. Additionally, this work can be applied to uncertain linear systems where independent control inputs are applied on each of the velocity state dynamics.

Lastly, the solution for the scalar is applied to the rotor speed control system and is extended to the case where Coulomb friction is considered in addition to viscous friction. Since a sign function can be approximated as a hyperbolic tangent, the tanh model is used for the Coulomb friction. A controller is developed with the assumption that the coefficients of these frictions are unknown. The proposed control is then verified with Educational Control Product Model 750 Control Moment Gyroscope, and the simulation and actual test results are compared.