Practical Realization of Implicit Homogeneous Controllers for Linearized Systems

Abstract

This paper deals with the practical implementation of implicit homogeneous controllers (IHCs) for linearized mechanical systems. The control design includes the methodology to get gains of the IHC based on the linearized approximation of the system. If the approximation error enforced by the linearization is vanishing with the state, locally measurable and bounded, the IHC can lead the state to the origin in finite-time. This IHC control allows accelerating the convergence rate of the states. A semi-explicit algorithm is provided to exert the digital implementation of the controller. The application of the bisection method estimates the controller gain ensuring the finite-time convergence of the state to the origin. A complementary analysis provides a simplified algorithm with a reduced number of computation stages but equally efficient gain estimation. The proposed IHC is applied to a rotary inverted pendulum QUBE ™ Servo 2 platform of Quanser ® . The obtained results for the state convergence are compared with other classical feedback controllers to validate the effectiveness of the proposed scheme. The comparative analysis of the state convergence provides evidence of the faster convergence for the state trajectory to a zone centered on the origin with a smaller hypervolume than the one gotten with the classical controllers.

Nonlinear optimal control for the rotary double inverted pendulum

“…The article's nonlinear optimal (H‐infinity) control scheme achieves fast and accurate tracking of reference setpoints under moderate variations of the control inputs. The control problem of the rotary single and double inverted pendulum can be used as a benchmark for testing linear and nonlinear control algorithms, and the results of the present article confirm that the proposed nonlinear optimal control method is a meaningful contribution to the research area of nonlinear control 33‐38 . The use of the article's nonlinear optimal control method can be extended to more nonlinear underactuated dynamical systems which have been the subject of control systems research during the last years 39‐43 …”

“…The control problem of the rotary single and double inverted pendulum can be used as a benchmark for testing linear and nonlinear control algorithms, and the results of the present article confirm that the proposed nonlinear optimal control method is a meaningful contribution to the research area of nonlinear control. [33][34][35][36][37][38] The use of the article's nonlinear optimal control method can be extended to more nonlinear underactuated dynamical systems which have been the subject of control systems research during the last years. [39][40][41][42][43] A comparison of the nonlinear optimal (H-infinity) control method against other linear and nonlinear control schemes for complex dynamical systems has shown the following: (1) unlike Lie algebra-based control, the new nonlinear optimal control method does not rely on complicated transformations (diffeomorphisms) of the system's state variables.…”

On Generalized Homogeneous Leader-Following Consensus

“…This paper develops a generalized homogeneous control protocol for MAS, where each agent is modeled by a linear single input plant. We follow the idea of an "upgrade" of an existing linear control to a homogeneous one developed in [17,Chapter 9], [26], [27]. In this paper we showed that such an upgrade is possible under the assumption that MAS has a supervisor, which can observe whole system, but it cannot be utilized as a centralized controller due to communication constraints.…”