###### Abstract  
This paper proposes a method for active vibration control to a two-link flexible robot arm in the presence of time delay, by means of robust pole placement. The issue is of practical and theoretical interest as time delay in vibration control can cause instability if not properly taken into account in the controller design. The controller design is performed through the receptance method to exactly assign a pair of pole and to achieve a given stability margin for ensuring robustness to uncertainty. The desired stability margin is achieved by solving an optimization problem based on the Nyquist stability criterion. The method is applied on a laboratory testbed that mimic a typical flexible robotic system employed for pick-and-place applications. The linearization assumption about an equilibrium configuration leads to the identification of the local receptances, holding for infinitesimal displacements about it, and hence applying the proposed control design technique. Nonlinear terms, due to the finite displacements, uncertainty, disturbances, and the coarse encoder quantization, are effectively handled by embedding the robustness requirement into the design. The experimental results, and the consistence with the numerical expectations, demonstrate the method effectiveness and ease of application. recptance method; pole placement; robust control; active vibration control; flexible systems; robotic arm; vibration suppression; Nyquist criterion; genetic algorithm +
Footnote †: journal: Applied Sciences  
control [7], Smith predictor [5; 8; 9], communication disturbance observer [10; 11], sliding mode control [12], and switching control [13]. A technique attracting an ever-growing interest for active vibration control of vibrating linear systems with time delay is pole placement, borrowed by the traditional approaches for systems without time delay [14; 15]. The seminal receptance method for pole placement [16] has been extended in [17] accounting also for time delay. The extension to partial pole placement requires that a subset of the system poles is assigned while the remaining unassigned poles are kept unchanged with respect to the open-loop configuration. This problem has been tackled in [18]. The same problem has been solved using the system matrices instead of the measured receptances in [19] for the single input case and later extended for multi-input control in [20]. The papers previously quoted require evaluating a posteriori the stability of the closed-loop system. Recently, a two-stage method that embeds an a priori stability condition has been developed by [PERSON] and [PERSON] in [21]. It relies on the powerful theory of Linear Matrix Inequalities (LMI) and ensures the placement of the dominant poles of interest while imposing stability of the remaining unassigned poles, either those due to the mechanical resonances and those induced by the time delay. This method uses both the measured receptances to assign the dominant poles, and the system matrices, that are required by the LMIs. Inspired by the controller parametrization proposed in the paper of [PERSON] and [PERSON], a method that only exploits the measured receptances has been proposed by [PERSON], [PERSON], and [PERSON] in [22]. The state feedback control gains are computed to assign the dominant poles and simultaneously impose the closed-loop system stability and robustness through the generalized Nyquist criterion. Robustness is achieved using the sensitivity function of the loop gain as an index and the problem is solved through a genetic algorithm. In this paper, such a method is experimentally applied to control a flexible robot arm that mimics a typical system for pick-and-place applications. The arm flexibility is due to the passive joint torsional spring, that is an approach commonly used to represent flexibility of robots through a lumped model (see, e.g., the milestone paper in [23]). Time delays in this kind of system usually arise due to the instrumentation employed for real-time control.