Introduction

When a mass–spring–dashpot is attached to any mechanical system, including flexible space structures, the damping of the system is almost always augmented regardless of the system size. The parameters of the mass–spring–dashpot are arbitrary, model independent, and thus insensitive to the system uncertainties. To satisfy the system performance requirements, we adjust the parameters by using the knowledge of the system model. The more the system is known, the better the parameters of the mass–spring–dashpot may be adjusted to meet the performance requirements. However, no matter what happens, the mass–spring–dashpot will not destabilize the system because it is an energy-dissipative device. The question arises as to whether there are any feedback control designs that use sensors and actuators that behave like the passive mass–spring–dashpot.

We discuss a robust controller design for flexible structures in this chapter by using a set of second-order dynamic equations similar to that describing the passive mass–spring–dashpot. Under certain realistic (practical) conditions, this method provides a stable system in the presence of system uncertainties. For better understanding, two major steps are involved in developing the formulation of the method. First, consider only the direct output feedback for simplicity, implying the absence of dynamics in the feedback controller. Conditions are identified in terms of the number and the type of sensors and their locations to make the system asymptotically stable. Second, assume that the feedback controller contains a set of second-order dynamic equations.

Introduction

In a state–space model, the relationship between the input(s) and the output(s) of a system is described by means of an intermediate variable called the state. Recall that in a state-feedback-control system, the state information is required for computing the control input. In Chap. 8, we assume that the state of the system can be directly measured and used in the state-feedback-control law. This is certainly possible for a simple system (if enough sensors are available to measure all elements of the state vector), but usually not the case in practice. Fortunately, we can use a state estimator, otherwise known as an observer, to estimate the state from input and output measurements. The estimated state is then used in the state-feedback-control law as if it is the true state. Obviously it is desirable that the estimated state provided by an observer is as close as possible to the true state. Indeed, under ideal conditions, the estimated state can be the same as the true state (in theory at least).

The first part of this chapter addresses the problem of state estimation by use of an observer (Ref. [1–2]). The issue of integrating a state estimator into a state-feedback-control system and the stability analysis of the overall system will then be discussed. Under certain conditions, the determination of a state-feedback-control law and the design of an observer to be used can be treated as separate problems.

Introduction

This chapter shows the reader how to rewrite the second-order equations of motion for a general multiple-degree-of-freedom system into the form of a first-order matrix differential equation. What we have learned from Chap. 1 about the mathematics of a first-order matrix differential equation then applies. The first-order matrix differential equation is known as the state–space model, which is a fundamental equation on which modern control theory is based.

In this chapter, we will also study a sampled-data (or discrete-time) representation of the continuous-time state–space model. Assuming a constant input at each sampling interval, it is possible to represent the continuous-time state–space model that is in the form of a first-order matrix differential equation by a discrete-time first-order matrix difference equation. At the sampling points, the corresponding discrete-time state–space model describes exactly the continuous-system with a constant input at each sampling interval without any kind of approximation (Ref. [1–3]). The corresponding discrete-time model is also of the first order. As a result, control methods that are originally developed in the continuous-time domain can be converted almost trivially to the discrete-time domain for digital control implementation (Ref. [2]). Digital control is flexible in that changing a control strategy amounts to writing a different program (software) rather than constructing a different analog control circuitry (hardware). For the same reason, different signal processing techniques such as filtering, identification (Ref. [1]), etc., can be incorporated much more conveniently into the discrete-time digital format.

Introduction

In this chapter, the response of linear dynamic systems is studied in more detail. We begin with a study of efficient techniques for solving first-order and second-order single-degree-of-freedom (SDOF) equations of motion. Taking advantage of the simplicity of the first-order equation, we show how to obtain analytical expressions for the system response to harmonic excitation and how to write the steady-state response in terms of the magnitude and phase difference. It leads to the definition of the frequency-response function (FRF) and its analytical expression in terms of a complex variable (Ref. [1]). We then show how two- or more degree-of-freedom equations of motion can be treated similarly. The focus will be on multiple DOF mass–spring–damper systems (lumped parameter systems). The last part of this chapter deals with the bending vibration of flexible beams, which are described by partial differential equations rather than by ordinary differential equations for mass–spring–damper systems (Ref. [2]). The partial differential equations are decoupled by the separation of variable method to yield two sets of ordinary differential equations. From these equations, the common features of flexible structures are introduced, including the natural modes, frequency equations, and modal amplitudes. A discussion of the orthogonality property of the natural modes and the expansion theorem is then presented.

Single Degree of Freedom

A system of considerable importance in vibrations is the mass–spring–damper system, shown in Fig. 5.1.

Introduction

In the previous chapter, we paid attention to the solution of a scalar ODE. A scalar ODE describes the dynamics of a single-input single output (SISO) dynamic system. In general, a system can have several inputs and several outputs. Such a multi-input multi-output (MIMO) dynamic system can be described by a set of coupled ODEs involving several input and output variables. We must solve these equations simultaneously to obtain the dynamic response of the system. We can find such a solution by using matrix theory, which conveniently rearranges the set of coupled equations in a compact form. Instead of having several coupled scalar ODEs, we now have one single matrix ODE. Matrix operations can be performed on the matrix differential equations, and the final solution can be expressed in a simple form. It is important to realize that applying such matrix operations is equivalent to operating on the scalar ODEs individually, although with the scalar approach it is very easy to miss the general picture. Thus, matrix theory is one fine example in which one gains by simply rewriting an old problem in a new form that can be analyzed more effectively. Another reason why the matrix formulations are so useful is that a set of coupled differential equations of any order can be rewritten as a single matrix differential equation of first order. The same statement applies for a single scalar ODE of any order as well.

Introduction

This chapter describes several computational algorithms to compute the predictive control law that has some feature of adaptive control. All algorithms make use of the multi-step-ahead output prediction as derived in Chap. 10 based on the finite-difference model. The generalized predictive control (GPC) algorithm (Ref. [1–4]) is based on system output predictions over a finite horizon known as the prediction horizon. In determining the future control inputs, it is assumed that control is applied only over a finite horizon known as the control horizon. The GPC is computed with the Toeplitz matrix formed from the step-response time history of the system in conjunction with a cost function with weighted input and output. The control input is obtained by minimization of the cost function. There are three design parameters involved: the control weight, the prediction horizon, and the control horizon. A proper combination of these parameters is required in order to guarantee stability of the predictive control law.

In contrast to the GPC approach, another approach is the deadbeat predictive control (DPC) (Ref. [5–9]). The DPC feedback law is supposed to bring the output response to rest after a few specific time steps. Similar to GPC, DPC has a control design parameter and an identification parameter related to the order of the system. The control design parameter, which is similar to the GPC control horizon, gives the number of time steps for the system to become deadbeat (rest). The DPC guarantees closed-loop stability for a controllable system.

Introduction

In the analysis of continuous systems, the formulations describing the system response are governed by partial differential equations, as presented in the last chapter. The exact solution of the partial differential equations satisfying all boundary conditions is possible for only relatively simple systems such as a uniform beam. Numerical procedures must be used to approximate the partial differential equations and predict the system response.

The finite-element method is a very popular technique for the numerical solution of complex problems in engineering. It is a technique for solving partial differential equations that represent a physical system by discretizing them in their space dimensions. The discretization is performed locally over small regions of simple but arbitrary shape, i.e., finite elements. For example, in structural engineering, a structure is typically represented as an assemblage of discrete truss and beam elements. The discretization process converts the partial differential equations into matrix equations relating the input at specified points in the elements to the output at these same points. To solve equations over large regions, the matrix equations for the smaller subregions are summed node by node to yield global matrix equations.

Our objective in this chapter is to present the fundamental principles of the finite-element method. It is not our goal to summarize all the finite-element formulations available, but rather to establish only the basic and general principles that provide the foundation for a preliminary understanding of the finite-element method.

This book is based on a series of lecture notes developed by the authors. The first author has used part of the notes for two graduate-level classes in System Identification and Control of Large Aerospace Systems at the Joint Institute for Advancement of Flight Sciences, George Washington University at NASA Langley Research Center for the past 10 years. The second author has used part of the notes for senior and first year graduate-level courses in Dynamics and Control of Mechanical Systems and System Identification at Princeton University and Dartmouth College since 1995. There are many reasons that motivated the writing of this book; some of them are outlined below.

First, the lecture notes received overwhelming response from the students taking these courses, with many urging us to turn these materials into a textbook. When developing the notes, we tried to place emphasis on the fundamentals and clarity of presentation. Second, the subject matter is important in practice, but it is challenging both for students to learn and for us to teach because it is an integration of several disciplines: structural dynamics, vibration analysis, modern control, and system identification. The primary goal is for students to learn what these tools are without having to take a separate course for each subject and how they are brought together to solve a vibration control problem.

Introduction

The objective of a control system is to influence the dynamic system to make it behave in a desirable manner (Ref. [1–9]). Typical objectives of a control system are regulation and tracking. In a regulation problem, the system is controlled so that its output is maintained at a certain set point. In a tracking problem, the system is controlled so that its output follows a particular desired trajectory. A special case of the regulation problem is the stabilization problem, in which a control system is designed to bring the system to rest from any nonzero initial conditions (i.e., the desirable set point is zero). For a flexible structure that may be subjected to unwanted vibration, this is usually the most important goal of a controlled system. Stabilization is the focus of this chapter. In particular, we consider a special but very important class of control systems, namely state-feedback control, in which the control input is some function of the system states. For the moment we assume that there are enough sensors to measure the state of the system at any point in time to be used in computing the control input. If the state of the system cannot be measured directly, then a state observer is needed to estimate the system state from the measurements. The estimated state is then used in a state-feedback-control law. This subject of state estimation will be dealt with in the next chapter.

Designing is a complex human process that has resisted comprehensive description and understanding. All artifacts surrounding us are the results of designing. Creating these artifacts involves making a great many decisions, which suggests that designing can be viewed as a decision-making process. In the decision-making paradigm of the design process we examine the intended artifact in order to identify possible alternatives and select the most suitable one. An abstract description of the artifact using mathematical expressions of relevant natural laws, experience, and geometry is the mathematical model of the artifact. This mathematical model may contain many alternative designs, and so criteria for comparing these alternatives must be introduced in the model. Within the limitations of such a model, the best, or optimum, design can be identified with the aid of mathematical methods.

In this first chapter we define the design optimization problem and describe most of the properties and issues that occupy the rest of the book. We outline the limitations of our approach and caution that an “optimum” design should be perceived as such only within the scope of the mathematical model describing it and the inevitable subjective judgment of the modeler.

Mathematical Modeling

Although this book is concerned with design, almost all the concepts and results described can be generalized by replacing the word design by the word system.