N. van de Wouw

Convergent dynamics, a tribute to Boris Pavlovich Demidovich

We review and pay tribute to a result on convergent systems by the Russian mathematician Boris Pavlovich Demidovich. In a sense, Demidovichs approach forms a prelude to a field which is now called incremental stability of dynamical systems. Developments on incremental stability are reviewed from a historical perspective.

Robust Stability of Networked Control Systems with Time-varying Network-induced Delays

In this paper, the stability of a networked control system (NCS) with time-varying delays is analyzed. A discrete-time state-space model is used to analyze the dynamics of the NCS. The delay is introduced by the network itself and is assumed to be upperbounded by a fraction of the sample-time. A typical motion control example is presented in which the time-variation of the delay results in an unstable system, although for each fixed delay the system is stable. Conditions in terms of LMIs are presented guaranteeing the robust asymptotic stability of the discrete-time system, given bounds on the uncertain time-varying delay. Moreover, it is shown that the robust stability conditions also guarantee asymptotic stability of the intersample behavior. Additionally, LMIs are presented to synthesize a feedback controller that stabilizes the system for the uncertain time-varying delay. The results are illustrated on an example concerning a mechanical model of a motor driving a roller in a printer

On convergence properties of piecewise affine systems

In this paper convergence properties of piecewise affine (PWA) systems are studied. In general, a system is called convergent if all its solutions converge to some bounded globally asymptotically stable steady-state solution. The notions of exponential, uniform and quadratic convergence are introduced and studied. It is shown that for non-linear systems with discontinuous right-hand sides, quadratic convergence, i.e., convergence with a quadratic Lyapunov function, implies exponential convergence. For PWA systems with continuous right-hand sides it is shown that quadratic convergence is equivalent to the existence of a common quadratic Lyapunov function for the linear parts of the system dynamics in every mode. For discontinuous bimodal PWA systems it is proved that quadratic convergence is equivalent to the requirements that the system has some special structure and that certain passivity-like condition is satisfied. For a general multimodal PWA system these conditions become sufficient for quadratic convergence. An example illustrating the application of the obtained results to a mechanical system with a one-sided restoring characteristic, which is equivalent to an electric circuit with a switching capacitor, is provided. The obtained results facilitate bifurcation analysis of PWA systems excited by periodic inputs, substantiate numerical methods for computing the corresponding periodic responses and help in controller design for PWA systems.

Robust Active Chatter Control in the High-Speed Milling Process

Chatter is an instability phenomenon in machining processes which limits productivity and results in inferior workpiece quality, noise and rapid tool wear. The increasing demand for productivity in the manufacturing community motivates the development of an active control strategy to shape the chatter stability boundary of manufacturing processes. In this work a control methodology for the high-speed milling process is developed that alters the chatter stability boundary such that the area of chatter-free operating points is increased and a higher productivity can be attained. The methodology developed in this paper is based on a robust control approach using -synthesis. Hereto, the most important process parameters (depth of cut and spindle speed) are treated as uncertainties to guarantee the robust stability (i.e., no chatter) in an a priori specified range of these process parameters. Effectiveness of the proposed methodology is demonstrated by means of illustrative examples.

Tracking Control for Hybrid Systems With State-Triggered Jumps

This paper addresses the tracking problem in which the controller should stabilize time-varying reference trajectories of hybrid systems. Despite the fact that discrete events (or jumps) in hybrid systems can often not be controlled directly, as, e.g., is the case in impacting mechanical systems, the controller should still stabilize the desired trajectory. A major complication in the analysis of this hybrid tracking problem is that, in general, the jump times of the plant do not coincide with those of the reference trajectory. Consequently, the conventional Euclidean tracking error does not converge to zero, even if trajectories converge to the reference trajectory in between jumps, and the jump times converge to those of the reference trajectory. Hence, standard control approaches can not be applied. We propose a novel definition of the tracking error that overcomes this problem and formulate Lyapunov-based conditions for the global asymptotic stability of the hybrid reference trajectory. Using these conditions, we design hysteresis-based controllers that solve the hybrid tracking problem for two exemplary systems, including the well-known bouncing ball problem.

Convergent piecewise affine systems: analysis and design Part II: discontinuous case

In this paper convergence properties of piecewise affine (PWA) systems with discontinuous right-hand sides are studied. It is shown that for discontinuous PWA systems existence of a common quadratic Lyapunov function is not sufficient for convergence. For discontinuous bimodal PWA systems necessary and sufficient conditions for quadratic convergence, i.e. convergence with a quadratic Lyapunov function, are derived.

Performance of convergence-based variable-gain control of optical storage drives

In this paper, a method for the performance assessment of a variable-gain control design for optical storage drives is proposed. The variable-gain strategy is used to overcome well-known linear control design trade-offs between low-frequency tracking properties and high-frequency noise sensitivity. A convergence-based control design is proposed that guarantees stability of the closed-loop system and a unique bounded steady-state response for any bounded disturbance. These favourable properties, guaranteed by virtue of convergence, allow for a unique performance evaluation of the control system. Moreover, technical conditions for convergence are derived for the variable-gain controlled system and a quantitative performance measure, taking into account both low-frequency tracking properties and high-frequency measurement noise sensitivity, is proposed. The convergence conditions together with the performance measure jointly constitute a design tool for tuning the parameters of the variable-gain controller. The resulting design is shown to outperform linear control designs.

Stability analysis of nonlinear networked control systems with asynchronous communication: A small-gain approach

In this paper, we study the stability of decentralized networked control systems (NCSs) in which the sensors, controllers and actuators communicate through a finite number of local networks. These local networks accommodate the communication between local (decentralized) controllers at uncertain transmission times and operate asynchronously and independently of each other. In addition, each of the local networks exhibits communication constraints that require the presence of a protocol that decides which of the (local) network nodes is allowed to transmit its corresponding information at which transmission time. Due to the asynchronous nature of the networks, most existing works on the stability analysis of NCSs are not applicable as their stability characterizations assume that there is only one global communication network, or at least one global coordinator (or clock). Therefore, we present a novel approach that leads to maximal allowable transmission intervals for each of the individual local networks that guarantee the global asymptotic stability of the overall closed-loop system. The approach combines ideas from emulation-based stability analysis for NCSs and techniques from the stability of large-scale systems.

On dropout modelling for stability analysis of networked control systems

This paper presents three discrete-time modelling approaches for networked control systems (NCSs) that incorporate time-varying sampling intervals, time-varying delays and dropouts. The focus of this work is on the extension of two existing techniques to describe dropouts, namely (i) dropouts modelled as prolongation of the delay and (ii) dropouts modelled as prolongation of the sampling interval, and the presentation of a new approach (iii) based on explicit dropout modelling using automata. Based on polytopic overapproximations of the resulting discrete-time NCS models, we provide LMI-based stability conditions for all three approaches. Herewith, we compare the extensions of the existing approaches and the newly proposed method in terms of modelling accuracy, conservatism and complexity of the stability analysis. Using an illustrative example, we provide a thorough numerical comparison of the alternative modelling approaches.

A piecewise linear approach towards sheet control in a printer paper path

In this paper an approach towards sheet control in a printer paper path is presented. To make the control problem feasible, the complex overall control question is formulated in a hierarchical control set-up with a low level motor control part and a high level sheet control part. To understand the essence of the sheet control problem we consider a basic paper path in which industrial constraints and requirements are relaxed. Furthermore, the motor control part is assumed to be ideal and the sheet dynamics are captured in the piecewise linear modeling formalism. Based on the model of the sheet dynamics, the controller synthesis is carried out. Both state and output feedback control designs are presented and stability and tracking performance are analyzed. The effectiveness of the control design approaches is demonstrated via simulations