Who Should Read This Book

This book is intended for engineers and designers who seek to develop effective estimation and control algorithms for nonlinear systems. The reader is assumed to have some background in random processes and estimation (see, for example, [Pap91]) along with familiarity with concepts of feedback control phrased within the context of linear state space models (see, for example, [DB95] or [Wo194]). This background should include knowledge of

• random variables and processes,

• probability density and distribution functions for random variables, both continuous and discrete,

• moments and cross moments, including correlation functions,

• second-order properties of stationary processes, including power spectral densities,

• conditional expectations with respect to an observation process,

• fundamental properties of feedback systems, including stability and controllability of a system model.

Both time-continuous and time-discrete processes will be encountered. Some familiarity with mean-square estimation is useful. For example, in the development of the Kalman filter [BW92, Chapter 7], linear state space models are integrated with Gaussian white noise. The Kalman filter will form a basis of comparison for many of the estimators that follow.

Our treatment is applications oriented, but the reader will find that nonlinear systems require more detailed analysis than is necessary in the study of linear systems.

Introduction

Common problems in design require that an engineer devise a control or decision algorithm that converts measurements of system and environmental variables into signals that aid in system regulation. For example, a control node converts sensor outputs into an actuating signal that moves the system toward the desired operating point and keeps it there. At this foundational level, the engineer must formulate a mapping from the system observables into an action or report; for example, a feedback regulator converts the measured outputs of the system to be controlled (the plant) into an input that stabilizes the system.

Design is made difficult by disturbances internal to the system and by noise at its output. For example, there may be no sensors that measure those plant variables most useful for regulation, or, if measured, the variables may be masked by noise in the sensor-to-regulator link. Lacking omniscience, an engineer must process the available measurements to produce a good approximation to relevant but “hidden” variables. And this inference must be done on-line. The processing algorithm must not only be adapted to the incoming data stream, it must be of a form that can be implemented: An implementable estimation algorithm is an explicit mapping of the sensor output process (the measurements) into a (nearly) concurrent estimate of the required variables. In the applications studied here, the need for contemporaneous response limits consideration to finite-dimensional recursive algorithms; new observations are integrated into an estimate in an accretive manner.

Tracking an Agile Target

One of the most thoroughly studied applications of hybrid estimation arises in the synthesis of tracking algorithms for agile targets. These targets, sometimes intentionally and sometimes inadvertently, have motion paths that make them difficult to follow. One example is that of a piloted vehicle whose location must be tracked from a fixed sensor.

The operator of the vehicle exploits its maneuver capability to create a path that he hopes will cause the tracker to lose lock. Such paths have a familiar pattern, with nearly constant acceleration over intervals of unpredictable length followed by discrete maneuver mode changes. Even autonomous platforms can be designed to take such evasive maneuvers; for example, an antiship missile on approach to its objective may follow a preprogrammed jinking path. Such motions often have a constrained geometric structure. If the missile has limited speed control, it will focus its evasive efforts on turning motions, where the acceleration is perpendicular to the velocity.

Problem Statement

State estimation and control is made difficult in a hybrid system by the multiplicative nonlinearities in the equation of base-state evolution. The PME fuses complementary data streams using the dual path architecture shown in Figure 4.1 in a finite-dimensional algorithm for approximating the Gt-error moments useful in a broad range of applications.

In earlier chapters we have used the PME to estimate the base-state of a moving platform (called here the target). In these applications, a plausible motion model for the target was known a priori. There are situations in which this important information is lacking, for example in cases where the target must be identified while simultaneously tracking it and predicting its future motion. Identification in this context is called automatic target recognition (ATR). Prediction can take many forms, but we will focus on predicting the location where the target intersects a boundary in state space.

Uncertainties in target identification compound those already present in target location at time of detection and the accretion of disturbances along the path. Model-based path-following algorithms utilize a formal model to represent target evolution, and the selection of a tracking algorithm is based upon the target dynamics as articulated in the model. Because the tracking algorithm is tuned to a particular dynamic class, it is advantageous to know the proper class in advance, or if unknown, to identify it as soon as possible.

Introduction

At its basic level, an estimation algorithm is a causal mapping from a spatiotemporal observation (the measurement) to an approximation of a primary process (the signal or state, depending on the context). Estimators have become increasingly sophisticated as more versatile online processors have become available. If the state process has a suitable structure, improved estimation and prediction is achieved through model-based synthesis procedures. These approaches, as their name suggests, use a comprehensive analytical model to delineate both the state dynamics and the precise relationship between the state and its measurement. Using this model as an intermediary, a problem in optimal inference is posed. The solution to this optimization problem is then said to be the best estimation algorithm in the application. Through the model, the estimator is tuned to subtle patterns in the measurement, thus enabling good performance to be achieved in the presence of significant measurement ambiguity. Of course, the model is only an abridgement of reality, and to the degree that the model fails to adequately portray the salient features of the actual state processes, there is justifiable concern that the algorithm may be tuned improperly and may see things in the observation that are not actually there.

Perhaps the most widely studied model-based algorithm is the Kalman filter and its lineal variants. As described in Chapter 1, the dynamic features of the plant are represented by a base-state process {xt}.

Change of Measure

In earlier chapters of this book, we studied state estimation and regulation with an emphasis on time-continuous hybrid plants with a mix of time-continuous and time-discrete observations. The plant input/state-output representation is such that each state category is associated with an observation process and filtration: {yt} and {Yt} for the base-state, and {zt} and {t} for the modal-state. Low level data fusion generates the filtration Gt = Yt∨ t. An engineer seeks practical algorithms for approximating the Gt-regime probabilities along with those Gt-moments (including cross moments) important in the application. The primary tool in those chapters was the polymorphic estimator in its sundry realizations. The PME generates serviceable approximations to relevant Gt-conditional moments. Although t is the vector of conditional regime probabilities, the PME does not provide the Gt-distribution function of the zygostate.

The PME is premised on the assumption that the modal measurement is a good one; at least {zt} is the best measurement available as regards the regime. The modal measurement is not perfect to be sure, but it is good enough that base-state × modal-state cross moments depend incrementally upon {zt} alone. This hierarchical processing structure has its rationale in the way in which the engineer assembles the sensor suite. For example, the initial design of the panel temperature regulator for the solar central receiver (see Chapter 5) did not include insolation sensors.

Abstract

Distributed sensing and control of flexible shells and continua using distributed transducers has posted challenging issues for decades. This chapter focuses on distributed sensing and control of a generic double-curvature elastic shell and its derived geometries laminated with distributed piezoelectric transducers. Generic distributed orthogonal sensing and actuation of shells and continua are proposed. Spatially distributed orthogonal sensors/actuators and self-sensing actuators are presented. Collocated independent modal control with self-sensing orthogonal actuators is demonstrated and its control effectiveness evaluated. Spatially distributed orthogonal piezoelectric sensors/actuators for circular ring shells are designed and their modal sensing and control are investigated. Membrane and bending contributions in sensing and control responses are studied.

Introduction

Control of distributed parameter systems has posted many challenging problems and issues stimulating sophisticated research for decades (Balas, 1988; Brichkin et al., 1973; Butkovskii, 1962; Lions, 1968; Meirovitch, 1988; Oz and Meirovitch, 1983; Robinson, 1971; Sakawa, 1966; Tzafestas, 1970; Tzou, 1988, 1991, 1993; Vidyasagar, 1988; Wang, 1966; Zimmerman, Inman, and Juang, 1988). However, implementing distributed control of elastic continua, e.g., shells, plates, etc., using distributed devices has continuously been hampered by the practical availability of distributed sensing/actuation devices. Recent development of smart structures and intelligent structural systems (or structronic (structure-electronic) systems, in a new generic term) using active electromechanical materials has revealed the missing link of distributed transducers.

Distributed (parameter) systems (DPSs) are the most natural and generic systems existing today. Dynamic characteristics of most natural structures, manufacturing processes, fluids, heat transfer, control, etc., all fall into this DPS category. In general, their dynamics or responses are functions of spatial and time variables, and the systems are usually modeled by partial differential equations. Common practice often discretizes these distributed systems, and their lumped approximations (discrete systems modeled by ordinary differential equations) are then analyzed and evaluated. Original (distributed) behavior can only be observed at these discrete reference locations.

The dynamics and control of distributed systems have traditionally posed many challenging issues investigated by researchers and scientists for decades. However, new R&D activities and findings on DPSs have not been systematically reported for a long while. This book aims to document recent progress on the subject and to bring these technical advances to the engineering community.

Distributed structures are often coupled to external discrete components in engineering applications (e.g., disk heads and tape drives). A new transient analysis technique is developed to investigate the dynamics of coupled distributed-discrete systems in Chapter 1. Transient responses of time-varying systems and constrained translating strings are investigated. Transient behaviors of cables transporting dynamic payloads and translating (e.g., magnetic tape-head systems) are thoroughly studied.

Abstract

A model for a 3-D structural acoustic system, currently being used for parameter estimation and control experiments in the Acoustics Division, NASA Langley Research Center, is presented. This system consists of a hard-walled cylinder with a flexible circular plate at one end. An exterior noise source causes vibrations in the plate which in turn lead to unwanted noise inside the cylinder. Control is implemented through the excitation of piezoceramic patches bonded to the plate which generate in-plane forces and/or bending moments in response to an input voltage.

The plate and interior acoustic wave dynamics are approximated with expansions involving Fourier components in the circular direction and spline and spectral elements in the radial and axial directions. To guarantee uniqueness and differentiability at the coordinate singularity as well as to ensure stability and the expected convergence rate, the radial basis functions for both the wave and plate components are constructed in a manner that incorporates the Bessel or analytic behavior near the singularity while retaining sufficient generality so as to provide an approximation technique for discretizing complex coupled systems involving circular geometries.

Introduction

A growing area of research in the structural acoustics community concerns the problem of reducing structure-borne noise levels within an acoustic cavity.

Abstract

We present two methodologies for studying periodic oscillations and wave transmission in periodic continuous systems with strongly nonlinear supporting or coupling stiffnesses. The first methodology is based on a nonsmooth transformation of the spatial variable and eliminates the singularities (generalized functions) from the governing partial differential equation of motion. The resulting smooth partial differential equation is then analyzed asymptotically. This method is used to study localized nonlinear normal modes (NNMs) of an infinite linear string supported by a periodic array of nonlinear stiffnesses. A second methodology is developed to study primary pulse transmission in a periodic system composed of linear layers coupled by means of strongly nonlinear stiffnesses. A piecewise transformation of the time variable is introduced, and the scattering of the primary pulse at the nonlinear stiffnesses is reduced to solving a set of strongly nonlinear first-order ordinary differential equations. Approximate analytical and exact numerical solutions of this set are presented, and the methodology is employed to study primary pulse transmission in a system with clearance nonlinearities.

Introduction

Flexible periodic structures such as truss aerospace systems, periodically supported or stiffened shells and plates, and turbine cyclic assemblies are very common in engineering applications. Periodic structures are composed of a number of identical (or near identical) coupled substructures.

Abstract

The problem of an elastic distributed system coupled with a moving oscillator, often referred to as the “moving-oscillator” problem, is studied in this chapter. The problem is formulated using a “relative displacement” model. It is shown that, in the limiting case, the moving-mass problem is recovered. The coupled equations of motion are recast into an integral equation, which is amenable to solution by both iterative and direct numerical procedures. The response of a string with a moving oscillator is studied using the direct numerical method.

Introduction

Perspective

The prediction of the dynamic response of a distributed elastic system that supports one or more translating elastic subsystems has been a fundamental problem of interest for well over a century. Interest in this problem originates in structural engineering for the design of railroad tracks, railroad bridges, and highway bridges, wherein the accurate calculation of loads is essential for reliable design and accurate life prediction (Stokes, 1883; Ting and Yener, 1983; Tan and Shore, 1968). It has been observed that, as a structure is subjected to moving loads, the dynamic deflection and stresses can be significantly higher than those observed in the static case. Hence, strict design criteria are now required as structural engineers become more aggressive in the use of long, flexible spans in cable-stayed and suspension bridges and compliant bearings in highway bridges to accommodate environmental loads.

Abstract

A new analytical method is presented for modeling and analysis of stepped cylindrical shells and cylindrical shells stiffened by circumferential rings. Through use of the distributed transfer functions of the structural systems, various static and dynamic problems of cylindrical shells are systematically formulated. With this transfer function formulation, the static and dynamic response, natural frequencies and mode shapes, and buckling loads of general stiffened cylindrical shells under arbitrary external excitations and boundary conditions can be determined in exact and closed form. The proposed method is illustrated on a Donnell–Mushtari shell and compared with the finite element method and other modeling techniques.

Introduction

Cylindrical shells are the basic element in many structures and machines and therefore have been extensively studied in the past; for instance, see Donnell (1933), Soedel (1981), Irie et al. (1984), Yamada et al. (1984), Sheinman and Weissman (1987), Koga (1988), Thangaratnam et al. (1990), Huang and Hsu (1992), Heyliger and Jilani (1993), Birman (1993), and Miyazaki and Hagihara (1993). The static and dynamic problems of cylindrical shells are often complicated by engineering design in which a cylindrical shell is composed of a finite number of serially connected shell segments and/or stiffened by circumferential rings. For such complex structural systems, numerical methods are usually adopted.

Abstract

Distributed structures are often coupled to external components in engineering applications: The recording head in disk and tape drives, the guide bearing in circular and band saws, and the payload in cable transport systems are a few examples. Because the transient response of a distributed structure is characterized by complex multiple wave scattering at an external component, this behavior has rarely been explored. Classical transient analyses by finite difference, finite element, and modal expansion approaches use spatial discretization. As a result, some discontinuities in the interaction force between a distributed structure and an external component, a characteristic of multiple wave scattering, are not predicted in the normal application of these methods. A new transient analysis is developed in this chapter for the response of distributed structures interacting with discrete components. The transient response of a time-varying, cable transport system is analyzed first. The transient and steady-state responses of constrained translating strings are obtained next. Transient behaviors of cables transporting dynamic payloads and translating, magnetic tape-head systems are thoroughly investigated. Application of the method to the classical piano string response under a hammer strike avoids key limitations of the existing standing and traveling wave methods.

Introduction

Distributed structures are usually coupled to external dynamic components in engineering applications.