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Lyapunov-Based Control of Mechanical Systems

verfasst von: Marcio S. de Queiroz, Darren M. Dawson, Siddharth P. Nagarkatti, Fumin Zhang

Verlag: Birkhäuser Boston

Buchreihe : Control Engineering

Enthalten in: Professional Book Archive

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The design of nonlinear controllers for mechanical systems has been an ex tremely active area of research in the last two decades. From a theoretical point of view, this attention can be attributed to their interesting dynamic behavior, which makes them suitable benchmarks for nonlinear control the oreticians. On the other hand, recent technological advances have produced many real-world engineering applications that require the automatic con trol of mechanical systems. the mechanism for de Often, Lyapunov-based techniques are utilized as veloping different nonlinear control structures for mechanical systems. The allure of the Lyapunov-based framework for mechanical system control de sign can most likely be assigned to the fact that Lyapunov function candi dates can often be crafted from physical insight into the mechanics of the system. That is, despite the nonlinearities, couplings, and/or the flexible effects associated with the system, Lyapunov-based techniques can often be used to analyze the stability of the closed-loop system by using an energy like function as the Lyapunov function candidate. In practice, the design procedure often tends to be an iterative process that results in the death of many trees. That is, the controller and energy-like function are often constructed in concert to foster an advantageous stability property and/or robustness property. Fortunately, over the last 15 years, many system the ory and control researchers have labored in this area to produce various design tools that can be applied in a variety of situations.

Inhaltsverzeichnis

Frontmatter

1. Introduction

Abstract

The synthesis of controllers basically involves two steps: (i) the design step, where the goal is to construct a control algorithm for a given system to satisfy certain performance specifications; and (ii) the analysis step; where the goal is to verify the closed-loop system behavior (e.g., stability properties) once the controller has been designed. For a nonlinear system, these two steps become interdependent in the sense that the design and analysis are the result of an iterative procedure. The nuances of this iterative procedure become explicitly apparent if one exploits Lyapunov’s stability theory ¹ (i.e., the so-called Direct Method) for the control design and analysis. Roughly speaking, Lyapunov’s Direct Method is a mathematical interpretation of the physical property that if a system’stotal energy is dissipating, then the states of the system will ultimately travel to an equilibrium point[6]. Simply stated, this property can be explored by constructing a scalar, energy-related function for the system (e.g., V (t), where this function usually contains the closed-loop system states²) and then that investigating its time variation denoted by V(t). V(t) ≤ 0, then we know that V (t) is a decreasing or constant function of time (i.e., the energy is being dissipated or held at a constant level) and will eventually reach a constant; hence, the closed-loop system is considered to be stable in some sense.

Marcio S. de Queiroz, Darren M. Dawson, Siddharth P. Nagarkatti, Fumin Zhang

2. Control Techniques for Friction Compensation

Abstract

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.

Marcio S. de Queiroz, Darren M. Dawson, Siddharth P. Nagarkatti, Fumin Zhang

3. Full-State Feedback Tracking Controllers

Abstract

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.

Marcio S. de Queiroz, Darren M. Dawson, Siddharth P. Nagarkatti, Fumin Zhang

4. Output Feedback Tracking Controllers

Abstract

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.

Marcio S. de Queiroz, Darren M. Dawson, Siddharth P. Nagarkatti, Fumin Zhang

5. Strings and Cables

Abstract

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.

Marcio S. de Queiroz, Darren M. Dawson, Siddharth P. Nagarkatti, Fumin Zhang

6. Cantilevered Beams

Abstract

In many distributed parameter mechanical systems, the flexible element can be modeled as a beam-type structure (e.g., space structures, flexible link robots, helicopter rotor/blades, turbine blades, etc.). The most commonly used beam model is based on the classical Euler-Bernoulli theory, which provides a good description of the beam’s dynamic behavior when the beam’scross-sectional dimensions are small in comparison to its length (i.e., this model neglects the rotary inertia of the beam). A more accurate beam model is provided by the Timoshenko theory, which takes into account not only the rotary inertial energy but also the beam’s deformation owing to shear. As discussed in [15], the Timoshenko beam model has been shown to have a broader applicability than the Euler-Bernoulli model. In [1], Aldraihem et al. compared the accuracy and validity of these two beam models by simulating a cantilevered beam under distributed piezoelectric sensoring/actuation. The results provided in [1] indicated that the Timoshenko model is superior to the Euler-Bernoulli model in predicting the beam’s response. While the Timoshenko model may be more accurate at predicting the beam’s response in comparison to the Euler-Bernoulli model, the Timoshenko model is more difficult to utilize for control design purposes owing its higher order. For this reason, the design of boundary controllers for flexible-beam-type structures has been based mainly on the Euler-Bernoulli model.

Marcio S. de Queiroz, Darren M. Dawson, Siddharth P. Nagarkatti, Fumin Zhang

7. Boundary Control Applications

Abstract

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.

Marcio S. de Queiroz, Darren M. Dawson, Siddharth P. Nagarkatti, Fumin Zhang

Backmatter

Titel: Lyapunov-Based Control of Mechanical Systems
verfasst von: Marcio S. de Queiroz
Darren M. Dawson
Siddharth P. Nagarkatti
Fumin Zhang
Verlag: Birkhäuser Boston
Electronic ISBN: 978-1-4612-1352-9
Print ISBN: 978-1-4612-7108-6
DOI: https://doi.org/10.1007/978-1-4612-1352-9

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Über dieses Buch

Inhaltsverzeichnis

Frontmatter

1. Introduction

2. Control Techniques for Friction Compensation

3. Full-State Feedback Tracking Controllers

4. Output Feedback Tracking Controllers

5. Strings and Cables

6. Cantilevered Beams

7. Boundary Control Applications

Backmatter