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In this paper, we present a control strategy for an axially moving web system that regulates transverse vibration and ensures that the web speed tracks a desired trajectory. The control strategy is implemented using roller torque inputs applied at the boundary and a pair of control forces applied to the web via a mechanical guide. Given the hybrid model of the web system (i.e. the distributed parameter field equation coupled to discrete actuator equations), Lyapunov-type arguments are utilised to design a model-based control law that exponentially regulates the axial speed tracking error and the web displacement. The proposed control law is based on measurements of the web speed and displacement, slope, and slope rate at the mechanical guide. Experimental results demonstrate the speed tracking and vibration damping provided by the control strategy.

In this paper, we present the design of boundary controllers for a two-dimensional, spinning flexible rotor system. Specifically, we develop a model-based boundary controller which exponentially regulates the rotor's displacement and the angular velocity tracking error, and an adaptive boundary controller which asymptotically achieves the same control objective while compensating for parametric uncertainty. As opposed to previous boundary control work, which focused on the velocity setpoint problem and placed restrictions on the magnitude of the desired angular velocity setpoint, the proposed control architecture achieves angular velocity tracking with no restrictions on the magnitude of the desired velocity trajectory. Experimental results conducted on a flexible rotor tested are presented to illustrate the feasibility of implementing the proposed boundary control laws. Copyright © 2001 John Wiley & Sons, Ltd.

This paper introduces boundary controllers for a general class of non-linear string–actuator systems. The non-linear distributed-parameter model accounts for large amplitude displacement and the associated varying tension according to a general class of non-linear stress–strain relationships. A non-linear, model-based controller asymptotically drives the system energy to zero. Redesign of the controller using adaptation laws allows compensation for parametric uncertainty while asymptotically driving the system energy to zero. Experimental results verify the feasibility of implementing the proposed controllers and compare the performance with damper and linear controllers.

During continuous manufacture of axially moving materials such as fiber, paper, foil, and film, accurate speed and tension control are essential. In this paper, control torques applied to rollers at the boundaries of an axially moving system regulate the material speed and tension using speed and tension sensors for each roller. Given a distributed parameter model, Lyapunov techniques are used to develop a model-based boundary control system that exponentially stabilizes the material tension and speed at desired setpoints and stabilizes longitudinal vibration. Experimental results compare the tension and speed setpoint regulation provided by the proposed control strategy with proportional plus integral speed control and proportional tension feedback. [S0022-0434(00)00203-3]

In this paper, we propose a robust near output feedback (which refers to the fact that the control scheme uses dynamic feedback of the output but requires knowledge of the initial conditions of other states) link position tracking controller for a class of robotic systems, which includes the rigid-link flexible-joint (RLFJ) robot manipulator and the rigid-link electrically-driven (RLED) robot. The method uses a bank of dynamic filters, with the link position tracking error as the input, to generate surrogates for the unmeasurable system states. One of these generated filter states is then used in the feedback control strategy. The method is distinguished from previous results by the limited amount of modelling information required, the utilization of only link position measurements and knowledge of the initial conditions, and the overall simplicity of the controller. Even though the approach only requires link position measurements and knowledge of initial conditions, the controller exhibits good robustness properties while ensuring semiglobal uniformly ultimately bounded link position tracking.

We present a control strategy for an axially moving string system that achieves vibration regulation along the string and ensures that the string tracks a desired axial velocity trajectory. The control strategy is implemented by utilizing the roller torque inputs applied at the entry and exit points of the string and a pair of control forces applied to the string via an interstitial mechanical guide. Given the hybrid model of the string system (i.e., the distributed parameter field equation coupled to discrete actuator equations), Lyapunov-type arguments are utilized to design a model-based control law that exponentially regulates the axial velocity tracking error and the string displacement along the entire span. The proposed control law is based on measurements of the string displacement, velocity, slope, and slope rate at the mechanical guide.

In this paper, we present the design of boundary controllers for a two-dimensional, spinning flexible rotor system. Specifically, we develop a model-based boundary controller which exponentially regulates the rotor's displacement and the angular velocity tracking error, and an adaptive boundary controller which asymptotically achieves the same control objective while compensating for parametric uncertainty. As opposed to previous boundary control work, which focused on the velocity setpoint problem and placed restrictions on the magnitude of the desired angular velocity setpoint, the proposed control architecture achieves angular velocity tracking with no restrictions on the magnitude of the desired velocity trajectory.

In this paper, we design a modal controller for conservative flexible systems that uses distributed sensing. Spatial filtering of the distributed displacement and velocity measurements based on the system eigenfunctions prevents spillover instabilities in the closed-loop system. The proposed control is proven to stabilize a discrete set of controlled modes without destabilizing the remaining, residual modes. We then apply the theory to a single flexible link robot arm and experimentally demonstrate the feasibility of the proposed control strategy. The experiments use high speed video feedback with image processing to determine the spatial beam curve. The controller quickly damps the first modal response without causing instability in the remaining modes.

All the position tracking controllers presented in Chapter 3 required fullstate feedback (FSFB). That is, the control implementation requires the measurement of the position and velocity of the mechanical system. Since the cost of implementing a FSFB controller for achieving position tracking would typically include the cost of motion sensors, this chapter addresses the problem of position tracking under the constraint of minimizing the sensor count (i.e., elimination of velocity measurements). Hence, we are motivated to investigate means of constructing velocity signal surrogates for use in closed-loop, position tracking control strategies, i.e., output feedback (OFB) controllers. A standard approach for removing velocity measurements is to apply the so-called backwards difference algorithm to the position measurements. Even though this method of eliminating velocity may provide reasonable performance, the use of this discrete-time velocity approximation is not satisfying from a theoretical viewpoint since the dynamics of the backwards difference algorithm are normally not included during the closed-loop stability analysis.

This chapter will concentrate on the full-state feedback (FSFB) (i.e., position and velocity are available for feedback) control problem for general, nonlinear, MIMO, rigid mechanical systems with parametric uncertainty. Specifically, we will emphasize the adaptive control solution to this problem. First (as in Chapter 2), we will present, owing to its popularity and relative simplicity, the standard adaptive controller of Slotine and Li [27] in its original MIMO version. The idea is for this adaptive controller to serve as a benchmark for the subsequent MIMO, FSFB controllers of this chapter and the output feedback controllers of Chapter 4.

In this chapter, we present boundary control strategies for reducing vibration in three different engineering applications of flexible mechanical systems. Two of these applications will share the interesting characteristic of having flexible and rigid subsystems, which leads to hybrid dynamic models containing coupled PDEs and ODEs. The design of boundary controllers and the closed-loop stability analysis will be built upon the arguments set forth in Chapters 5 and 6.

In the previous chapters, controllers were designed for mechanical systems that are modeled by nonlinear ODEs. For the remainder of this book, we will focus our attention on the development of control algorithms for mechanical systems that are assumed to be modeled by PDEs.

Friction is a natural phenomenon that affects almost all motion. It has been the subject of extensive studies for centuries, with the main objectives being the design of effective lubricating processes and the understanding of the mechanisms of wear. Whereas friction effects at moderate velocities are somewhat predictable, it is the effect of friction at low velocities that is very difficult to model. The facts that friction changes sign with velocity, is asymmetric about the velocity axis, has evolutionary characteristics, and exhibits the stick-slip phenomenon, etc., aggravates the problem. Although friction effects have been well understood qualitatively, researchers have often relied on experimental data to formulate various mathematical models. A heuristic model for friction was first proposed by Leonardo da Vinci [11] in 1519; however, the model failed to capture the low-velocity friction effects such as the Stribeck effect, presliding displacement, rising static friction, etc., which play a major role in high-precision position/velocity tracking applications. In recent years, several dynamic models have been introduced to describe this highly nonlinear behavior exhibited by friction. For example, Dahl [12] proposed a dynamic model to capture the spring-like behavior during stiction. Canudas et al. [9] proposed a dynamic state-variable model to capture friction effects such as the Stribeck effect, hysteresis, spring-like behavior of stiction, and varying breakaway force.

In this paper, the tension and speed of an axially moving material system are regulated using control torques applied to rollers at each end of a controlled span. Given a distributed parameter model, Lyapunov-type arguments produce a model-based boundary control law that exponentially stabilizes the material tension and speed at the desired setpoints. Dynamic simulation results compare the tension and speed setpoint regulation provided by the proposed control strategy with standard PID approaches.

We design a control strategy for a spinning rotor with an unbalanced disk attached to its free end. The control strategy is composed of a boundary torque applied to the clamped end, and two boundary forces and two boundary torques applied to the free-end. At the clamped-end, the boundary torque ensures that the rotor tracks a desired angular velocity trajectory, while at the free-end, the boundary forces and torques ensure that the rotor displacement is regulated at every point along the length of the rotor. Under the assumption of exact model knowledge, we first develop a model-based control law which exponentially achieves the control objectives. We then illustrate how the model-based control law can be redesigned as an adaptive controller which asymptotically achieves the same control objectives while compensating for parametric uncertainty associated with unbalanced operation.

In this paper, we develop a boundary controller to actively cancel acoustic disturbances in an enclosed duct of finite length using a loudspeaker. Given a distributed parameter acoustic model and the actuator electrical dynamics at the boundary, Lyapunov-type arguments are used to design a model-based backstepping boundary control law that asymptotically regulates the noise within the duct. Dynamic simulation results demonstrate the improved performance achieved by the proposed active noise cancellation algorithm.

We study the control of an undamped, nonlinear string model with actuator dynamics at the boundary. Specifically, we develop a boundary controller which asymptotically stabilizes the out-of-plane displacement. The performance of the controller is illustrated via dynamic simulation.

We develop boundary controllers for a flexible cable with actuator dynamics at the boundary. Specifically, we develop an exact model knowledge controller which exponentially stabilizes the position of the cable and an adaptive controller which asymptotically stabilizes the position of the cable while compensating for parametric uncertainty. The performance of the controllers is illustrated via experimental results.