In modeling an optimization problem, the easiest and most common mistake is to leave something out. This chapter shows how to reduce such omissions by systematically checking the model before trying to compute with it. Such a check can detect formulation errors, prevent wasteful computations, and avoid wrong answers. As a perhaps unexpected bonus, such a preliminary study may lead to a simpler and more clearly understandable model with fewer variables and constraints than the original one.

The methods of this chapter, informally referred to as boundedness checking, should be regarded as a model reduction and verification process to be carried out routinely before attempting any numerical optimization procedure. At the same time, one should be cautious about the limitations of boundedness arguments because they are based on necessary conditions, namely mathematical truths that hold assuming an optimal solution exists. Such existence, derived from sufficient conditions, is not always easy to prove. The complete optimality theory in Chapters 4 and 5 provides important additional tools to those presented in this chapter.

The chapter begins with the fundamental definitions of bounds and optima, allowing a precise definition of a well-bounded model. Since poor model boundedness is often a result of extensive monotonicities in the model functions, the boundedness theory presented here has become known as Monotonicity Analysis.

In the previous two chapters we set out the theory necessary for locating local optima for unconstrained and constrained problems. From the optimality conditions we derived some basic search procedures that can be used to reach optima through numerical iterations. In Chapter 7 we will look at local search algorithms more closely. But before we do that we need to pick up the thread of inquiry for two earlier issues: the role of parameters and the presence of discrete variables.

One of the common themes in this book is to advocate “model reduction” whenever possible, that is, rigorously cutting down the number of constraint combinations and variables that could lead to the optimum – before too much numerical computation is done. This reduction has two motivations. The first is to seek a particular solution, that is, a numerical answer for a given set of parameter values. The second is to construct a specific parametric optimization procedure, which for any set of parameter values would directly generate the globally optimal solution with a minimum of iteration or searching.

If the variables are continuous, the most practical way to find a particular optimum for a single set of parameter values is simply to employ a numerical optimization code. When discrete variables are present, the resulting combinatorial problem can only be solved if we examine all possible solutions, directly or indirectly.

Introduction to Situation Assessment

An abridged version of the polymorphic estimator was presented in Chapter 2. The PME is a finite-dimensional algorithm for generating the Gt-conditional mean of the zygostate along with several moments that are interesting in their own right. The plant dynamics of the hybrid system are partitioned into a base-state, modal-state pair, and the observation has a compatible partition. In this chapter we will focus on the modal-state measurement subsystem in which the measurements are discrete.

The specific application to be considered is the study of situation assessment by human decision makers. Many geographically distributed systems include both human decision makers and a diverse collection of sophisticated hardware/software subsystems. With the unavoidable errors and distortions in data accumulation, transfer, and presentation, it is difficult to determine the appropriate human role, or even if the decision maker is properly filling the role given him. Athans referred to decision-making systems as “event driven” and observed that “the state variables…are both continuous and discrete” [Ath87]. In Athans's partitioning of the comprehensive state space, the discrete states represent global (or meta) states that modulate the local (or micro) aspects of the task environment. The decision maker's reaction to local phenomena tends to have a reflexive quality. It is in this reaction to macroevents that particularly human idiosyncrasies are manifest.

A primary task for a decision maker is to identify changing circumstances in an environment characterized by noise and clutter.

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}.