Block Backstepping Design of Nonlinear State Feedback Control Law for Underactuated Mechanical Systems

Abstract

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Self-generated limit cycle tracking of the underactuated inertia wheel inverted pendulum under IDA-PBC

“…We recall that x r is the state of the reference model (16), which generates the desired period-1 stable limit cycle, whereas x is the state of the IWIP, as the physical system modeled via the nonlinear dynamics (15), under the tracking control law v (73).…”

“…We recall that the dynamics of the IWIP under parametric uncertainties in the friction coefficients δ 1 and δ 2 and subject to the external disturbing torque ζ is given by (17). The reference model generating the desired period-1 stable limit cycle is always the nominal one given by the dynamics (16). Moreover, in order to show how well behaved the closed-loop system is to the uncertainties in the two friction parameters δ 1 and δ 2 and the external disturbance ζ, we will realize several simulations.…”

“…Thus, our goal is to design the new control law v such that the constrained nonlinear system (15) of the controlled IWIP tracks the reference model (16). The tracking error, e = x − x r , between the nonlinear dynamics (15) and the reference model (16) is defined as:…”

“…Recently, the interest in the control problem of underactuated mechanical systems has increased due to their complexity and applications. Several kinds of underactuated mechanical systems exist in literature such as the double inverted pendulum, the rotational inverted pendulum, the Furuta pendulum, the cartpendulum system, the wheeled inverted pendulum, the flywheel inverted pendulum, among others [11][12][13][14][15][16][17][18]. In general, inverted pendulums are very suitable to illustrate many ideas in automatic control of nonlinear systems.…”

“…A special underactuated mechanical system is the Inertia Wheel Inverted Pendulum (IWIP), which has two degrees of freedom with only one actuator [19,20]. Such underactuated mechanical system has attracted the attention of several researchers in robotics and control system engineering where several control strategies have been designed (see for example [11][12][13][14][15][16][17] and references therein). The most recognized works are those based on the energy point-of-view.…”