This paper is considered with the robust stability problem for linear discrete-time systems with polytopic uncertainty and an interval time-varying delay in the state. On the basis of a novel Lyapunov-Krasovskii functional, new delay-range-dependent stability criteria are established by employing the free-weighting matrix approach and a Jensen-type sum inequality. It is shown that the newly proposed criteria can provide less conservative results than some existing ones. Numerical examples are given to illustrate the effectiveness of the proposed approach.

Servocontrol loops are an important component in the control architecture of machine tools. Such loops manage the axes based on requirements of velocity, position, and acceleration. The success of modeling and control techniques applied to this level sets the basis for production of parts with high quality as well as cycle time reduction. First, an introduction is presented for the three mayor control levels: adaptive control, interpolation, and servocontrol level. The drives and transmission components for traditional machine tools as well as for precision machines are discussed; including compareison between systems, such as linear versus rotational electric actuators. Then, relevant control techniques, such as Zero Phase Error Tracking Control (ZPETC) and Cross Coupling Control (CCC), are presented with experimental results. After that, modern control strategies to mitigate the effect of nonlinearities, model uncertainties, and perturbations are studied. Furthermore, the problem of High Speed Machining (HSM) is also reviewed. In addition, precision control for precision machining is examined. At the end, the future tendency of Open Architecture Systems (OAS) that allows the implementation of any control strategy in machine tools is presented. Finally, it is included a summary on: electric and mechanical components, control strategies, precision machining, high speed machining, and a discussion on why new control techniques are not available in commercially machine tools.