Tamás Gábor Molnár

On the robust stabilizability of unstable systems with feedback delay by finite spectrum assignment

An application of the finite spectrum assignment (FSA) control technique is presented for unstable systems with feedback delay. The FSA controller predicts the actual state of the system over the delay period using an internal model of the real system. If the internal model is perfectly accurate then the feedback delay can be compensated. However, parameter mismatches of the internal model or implementation inaccuracies of the control law may result in an unstable control process. In this paper, the stabilizability of an undamped second-order system is analyzed for different system and delay parameter mismatches. Theoretical stability and robustness to implementation inaccuracies of the control law are discussed. It is shown that, for small parameter uncertainties, the FSA controller allows stabilization for significantly larger feedback delays than conventional delayed proportional-derivative-acceleration controllers do.

Estimation of the Bistable Zone for Machining Operations for the Case of a Distributed Cutting-Force Model

Regenerative machine tool chatter is investigated for a single-degree-of-freedom model of turning processes. The cutting force is modeled as the resultant of a force system distributed along the rake face of the tool, whose magnitude is a nonlinear function of the chip thickness. Thus, the process is described by a nonlinear delay-differential equation, where a short distributed delay is superimposed on the regenerative point delay. The corresponding stability lobe diagrams are computed and are shown numerically that a subcritical Hopf bifurcation occurs along the stability boundaries for realistic cutting-force distributions. Therefore, a bistable region exists near the stability boundaries, where large-amplitude vibrations (chatter) may arise for large perturbations. Analytical formulas are obtained to estimate the size of the bistable region based on center manifold reduction and normal form calculations for the governing distributed-delay equation. The locally and globally stable parameter regions are computed numerically as well using the continuation algorithm implemented in dde-biftool. The results can be considered as an extension of the bifurcation analysis of machining operations with point delay.

Investigating Multiscale Phenomena in Machining: The Effect of Cutting-Force Distribution Along the Tool’s Rake Face on Process Stability

Regenerative machine tool chatter is investigated in a nonlinear single-degree-of-freedom model of turning processes. The nonlinearity arises from the dependence of the cutting-force magnitude on the chip thickness. The cutting-force is modeled as the resultant of a force system distributed along the rake face of the tool. It introduces a distributed delay in the governing equations of the system in addition to the well-known regenerative delay, which is often referred to as the short regenerative effect. The corresponding stability lobe diagrams are depicted, and it is shown that a subcritical Hopf bifurcation occurs along the stability limits in the case of realistic cutting-force distributions. Due to the subcriticality a so-called unsafe zone exists near the stability limits, where the linearly stable cutting process becomes unstable to large perturbations. Based on center-manifold reduction and normal form calculations analytic formulas are obtained to estimate the size of the unsafe zone.Copyright

On the analysis of the double Hopf bifurcation in machining processes via centre manifold reduction

The single-degree-of-freedom model of orthogonal cutting is investigated to study machine tool vibrations in the vicinity of a double Hopf bifurcation point. Centre manifold reduction and normal form calculations are performed to investigate the long-term dynamics of the cutting process. The normal form of the four-dimensional centre subsystem is derived analytically, and the possible topologies in the infinite-dimensional phase space of the system are revealed. It is shown that bistable parameter regions exist where unstable periodic and, in certain cases, unstable quasi-periodic motions coexist with the equilibrium. Taking into account the non-smoothness caused by loss of contact between the tool and the workpiece, the boundary of the bistable region is also derived analytically. The results are verified by numerical continuation. The possibility of (transient) chaotic motions in the global non-smooth dynamics is shown.

On the Stabilizability of the Delayed Inverted Pendulum Controlled by Finite Spectrum Assignment in Case of Parameter Uncertainties

An application of the Finite Spectrum Assignment (FSA) control technique is presented for an inverted pendulum with feedback delay. The FSA controller predicts the actual state of the system over the delay period using an internal model of the real system. If the internal model is perfectly accurate then the feedback delay can be compensated. However, slight parameter mismatch of the internal model may result in an unstable control process. In this paper, stabilizability of the inverted pendulum for different system and delay parameter mismatches are analyzed. It is shown that, for the same parameter uncertainties, the FSA controller allows stabilization for significantly larger feedback delays than a conventional delayed proportional-derivative controller does. In the analysis, it is assumed that the controller input is piecewise constant (sampled), this way the destabilizing effect of the difference part of the governing neutral functional differential equation is eliminated. The relation of the FSA controller to the Smith predictor is also described in time domain.Copyright

Application of Predictor Feedback to Compensate Time Delays in Connected Cruise Control

In this paper, we investigate a vehicular string traveling on a single lane, where vehicles use connected cruise control to regulate their longitudinal motion based on data received from other vehicles via wireless vehicle-to-vehicle communication. Assuming digital controllers, the sample-and-hold units introduce time-periodic time delays in the control loops and the delays increase when data packets are lost. We investigate the effect of packet losses on plant and string stability while varying the control gains and determine the minimum achievable time gap below which stability cannot be achieved. We propose two predictor feedback control strategies that overcome the destabilizing effect of the time delay caused by the sample-and-hold unit and packet losses.