In this paper, the output feedback based direct model reference adaptive control of piecewise affine systems and its parameter convergence are investigated. Under the slow switching assumption, it is shown that all the closed-loop signals are bounded and the output tracking error is small in the mean square sense. Built upon this result, the estimation error of controller parameters is proved to converge to a residual set if the input signal is sufficiently rich. The relationship between the size of this residual set and the switching frequency is established. Moreover, the convergence of the estimated controller parameters to their nominal values can be achieved for a certain subsystem given that this subsystem is activated for infinitely long time. Simulation results validate the effectiveness of the proposed approach.
The humanoid robot David is equipped with a novel robotic neck based on an elastic continuum mechanism (ECM). To realize a model-based motion control, the six dimensional stiffness characteristics needs to be known. The paper presents an approach to experimentally identify the stiffness characteristic using a robot manipulator to deflect the ECM and measure the Cartesian wrenches and Cartesian poses with external sensors. A three-step process is proposed to establish Cartesian wrench and pose pairs experimentally. The process consists of a simulation step, to select a good model, a second step that extracts effective poses from workspace which are sampled experimentally and the third step, the pose sampling procedure in which the robot drives the ECM to these effective poses. A full cubic polynomial regression model is adopted based on simulation data to fit the stiffness characteristics. To extract the poses to be sampled in the experiments, two different approaches are evaluated and compared to ensure a well-posed identification. The identification process on the hardware is performed by using Cartesian impedance and inverse kinematics control in combination to comply with the physical constraints imposed by the ECM.
In this paper, we investigate the model reference adaptive control approach for uncertain piecewise affine systems with state tracking performance guarantees. The proposed approach ensures the error metric, defined as the weighted Euclidean norm of the state tracking error, to be confined within a user-defined time-varying performance bound. We introduce an auxiliary performance bound to construct a barrier Lyapunov function. This auxiliary performance bound is reset at each switching instant, which prevents the barrier transgression caused by the jumps of the error metric at switching instants. The dwell time constraints are derived such that the auxiliary performance bound resides within the user-defined performance bound. We prove that the Lyapunov function is nonincreasing at and in between the switching instants. Therefore, it does not impose extra dwell time constraints and ensures the error metric to fulfill the performance guarantees. Furthermore, we study the robust modification of the adaptive controller for the uncertain piecewise affine systems subject to unmatched disturbances. A numerical example validates the correctness of the proposed approach.
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