In this chapter we introduce many of the modeling and control concepts to be developed in this book through several examples. The examples in this chapter are extremely simple, but are intended to convey key concepts that can be generalized to more complex networks. We will return to each of these examples over the course of the book to illustrate various techniques.

A natural starting point is the single server queue.

Modeling the single server queue

The single server queue illustrated in Fig. 2.1 is a useful model for a range of very different physical systems. The most familiar example is the single-line queue at a bank: Customers arrive to the bank in a random manner, wait until they reach the head of the line, are served according to their needs and the abilities of the teller, and then exit the system. In the single server queue we assume that there is a single line, and only one bank teller. To understand how delays develop in this system we must look at average time requirements of customers and the rate of arrivals to the bank. Also, variability of service times or interarrivals times of customers has a detrimental effect on average delays.

Even in this very simple system there are control and design issues to consider. Is it in the best interest of the bank to reserve a teller to take care of customers with short time requirements?

This chapter develops extensions of the fluid and stochastic network models to capture a wider range of activities. As in the previous chapters we allow scheduling and routing. In the demand-driven models considered in this chapter we also permit “admission control” of raw material arriving to the network so that the total amount of material in the system can be regulated. Although manufacturing systems will motivate most of the discussion in this chapter, power distribution systems as described in Section 2.7 and some communication systems can be modeled as demand-driven networks.

Figure 7.1 illustrates a typical example of the class of models to be investigated. In this 16-buffer network there are two sources of exogenous demand, and the release of two different types of raw material into the system is controlled. At two of the five stations there are multiple buffers so that scheduling is required, and routing is controlled at the exit of Station 3.

This is the most complex example considered in any detail in this book, although it is far simpler than a typical semiconductor wafer fab as described in Section 1.1.1. The International Semiconductor Roadmap for Semiconductors (ITRS) provides an annual assessment of the challenges facing the semiconductor industry [279]. In recent years their reports have contained some recurring themes:

Contention for resources There may be dozens of different product flows in a single factory.

Chapter 4 touches on many of the techniques to be developed in this book for controlling large interconnected networks. The fluid model was highlighted precisely because control is most easily conceptualized when variability is disregarded. The infinitehorizon optimal control problem with objective function defined in (4.37) can be recast as an infinite-dimensional linear program when c is linear. In many examples, such as the simple re-entrant line introduced in Section 2.8, a solution is explicitly computable. The MaxWeight policy and its generalizations are universally stabilizing, in the sense that a single policy is stabilizing for any CRW scheduling model satisfying the load condition ρ• < 1 along with the second moment constraint E[∥A(1)∥2] < ∞.

What is missing at this stage is any intuition regarding the structure of “good policies” for a network with many stations and buffers. In this chapter we introduce one of the most important concepts in this book, the workload relaxation. Our main goal is to construct a model of reduced dimension to simplify computation of policies, and to better visualize network behavior.

In the theory of optimization, a relaxation of a given model is simply a new model obtained by removing constraints. In the case of networks there are several classes of constraints that complicate analysis:

1. (i) The integer constraint on buffer levels.

2. (ii) Constraints on the allocation sequence determined by the constituency matrix.

3. (iii) State space constraints, including positivity of buffer levels, as well as strict upper limits on available storage.

The chief motivation for performance evaluation is to compare candidate policies. For example, many of the policies described in Chapters 4 and 10 depend upon static or dynamic safety-stock parameters, and we would like to know how to choose the best parameter values in order to optimize performance.

We have seen in Chapter 8 that linear programming techniques can provide bounds on performance for the CRW model. This approach can be successfully applied in network models with many buffers and stations. However, linear programming techniques are not flexible with respect to the operating policy. For example, in order to distinguish similar policies with different safety-stock levels, constraints must be introduced in the LP specific to each safety-stock parameter. It is not clear how to introduce such constraints in the approaches that have been developed to date.

While not a topic of this book, there are classes of networks for which the invariant measure π is known. The crucial property required is reversibility, from which it follows that π has a product form, π(x) = π1(x1) … πℓ(xℓ) for x ∈ X [502, 290]. Outside of this very special class of models the computation of π is essentially impossible in large networks. We are thus led to simulation techniques to evaluate performance.

The simulation techniques surveyed in this chapter all involve a Markov chain X on a state space X.

LEARNING OBJECTIVES FOR THIS CHAPTER

1. 14–1 To characterize the steady-state behavior of a system through analysis of the system transfer function.

2. 14–2 To evaluate the steady-state disturbance sensitivity of a system.

3. 14–3 To understand the trade-off between transient and steady-state performance specifications in control system design.

4. 14–4 To select gains of a proportional–integral–derivative controller based on open-loop system performance.

5. 14–5 To design an appropriate cascade compensator based on steady-state and transient performance specifications.

INTRODUCTION

In Chap. 1, it was pointed out that negative feedback is present in nearly all existing engineering systems. In control systems, which are introduced in this chapter, negative feedback is included intentionally as a means of obtaining a specified performance of the system.

Control is an action undertaken to obtain a desired behavior of a system, and it can be applied in an open-loop or a closed-loop configuration. In an open-loop system, shown schematically in Fig. 14.1, a process is controlled in a certain prescribed manner regardless of the actual state of the process. A washing machine performing a predefined sequence of operations without any information and “with no concern” regarding the results of its operation is an example of an open-loop control system.

In a closed-loop control system, shown in Fig. 14.2, the controller produces a control signal based on the difference between the desired and the actual process output. The washing machine previously considered as an open-loop system would operate in a closed-loop mode if it were equipped with a measuring device capable of generating a signal related to the degree of cleanness of the laundry being washed.

LEARNING OBJECTIVES FOR THIS CHAPTER

1. 13–1 To understand the concept of system stability and its implication for dynamic feedback systems.

2. 13–2 To apply algebraic stability criteria for linear dynamic models.

3. 13–3 To apply frequency-domain stability criteria for linear dynamic models.

4. 13–4 To assess relative stability of linear dynamic systems by using phase margin and gain margin.

5. 13–5 To plot and interpret the root locus of linear dynamic systems.

INTRODUCTION

Up to this point, the modeling and analysis in this text have dealt mainly with systems resulting from the straightforward interconnection of A-type, T-type, and D-type elements together with energy-converting transducers. The graphical representation of these system models by use of simulation block diagrams has revealed the widespread natural occurrence of closed loops containing one or more integrators, each loop of which involves feedback to a summing point. The techniques of analysis used so far with these passive systems have led to descriptions of their dynamic characteristics by means of sets of state-variable equations and/or input–output differential equations and transfer functions. These systems are considered passive because no attempts have been made intentionally to close additional loops with signal-amplifying or signal-modifying devices. They are simply collections of naturally occurring phenomena that, to be naturally occurring, must be inherently stable in order to survive.

LEARNING OBJECTIVES FOR THIS CHAPTER

1. 2–1 To apply constitutive equations for the fundamental translational and rotational mechanical components: lumped mass, springs, and dampers.

2. 2–2 To derive correct equations of motion for systems involving multiple instances of these fundamental components, some of which may be nonlinear.

3. 2–3 To apply a linearization procedure based on Taylor series expansion to approximate nonlinear systems with simplified linear models.

INTRODUCTION

As indicated in Chap. 1, three basic ideal elements are available for modeling elementary mechanical systems: masses, springs, and dampers. Although each of these elements is itself a system with all the attributes of a system (inputs, parameters, state variables, and outputs), the use of the term “system” usually implies a combination of interacting elements. In this chapter, systems composed of only mechanical elements are discussed. In addition to the translational elements (moving along a single axis) introduced in Chap. 1, a corresponding set of rotational elements (rotating about a single axis) is introduced to deal with rotational–mechanical systems and mixed (translational and rotational) systems.

Also, this chapter deals with only so-called lumped-parameter models of real mechanical systems. In certain situations, such as modeling a real spring having both mass and stiffness uniformly distributed from one end to the other, suitable lumped-parameter models can be conceived that will adequately describe the system under at least limited conditions of operation.

LEARNING OBJECTIVES FOR THIS CHAPTER

1. 11–1 To drive system transfer functions as a compact representation of relationships between inputs and outputs in linear systems.

2. 11–2To predict time-domain system behavior by using the system transfer function.

3. 11–3To apply the concept of the transfer function to systems with multiple inputs, multiple outputs, or both.

4. 11–4To develop, manipulate, and interpret system transfer function block diagrams.

INTRODUCTION

A significant body of theory has been developed for analyzing linear dynamic systems without having to go through the classical methods of solving the input–output differential equations for the system. This body of theory involves the use of the complex variable s = σ + jω, sometimes known as a complex frequency variable. This variable is essentially the same as the Laplace transformation variable s in many respects, but its use does not need to involve the complex (in both senses of the word) transformation problems of ensuring convergence of integrals having limits approaching infinity, nor does it involve the need to carry out the tedious process of inverse transformation by means of partial fraction expansion and the use of a table of transformation pairs. Here the variable s is simply considered to be the coefficient in the exponential input function est, and the transfer function emerges from solving for the particular or forced part of the response to this input. The notion of transfer function is then combined with the use of simple input–output block diagrams to symbolically express the input–output characteristics of the dynamic behavior of the system.

MATLAB OVERVIEW

MATLAB, an interactive and powerful engineering package from The MathWorks, Inc., is one of the most common software tools used in the analysis and design of dynamic systems and the control systems implemented on them. The basic package has many powerful numerical analysis functions as well as the ability to write scripts and functions to aid in repetitive analysis and to allow the user to augment the environment with custom capabilities.

Although it is likely that the reader will have a working knowledge of MATLAB, this tutorial is included to bring the uninitiated reader up to speed and to expose the casual user to several important features that are pertinent to this text. In this tutorial we cover the basic running environment of MATLAB (with its default windows), the fundamental data structure (the matrix), writing scripts and functions, and plotting data. We also explore some capabilities of the Control Systems Toolbox, which has many tools designed to aid in the analysis of dynamic systems. Appendix 4 contains a tutorial for Simulink, a MATLAB add-on, which allows for the rapid development of system simulations.

Launching MATLAB

Find MATLAB from the START menu on your computer and launch the application. Figure A3.1 shows the default frame with the three standard windows. The “current directory” window has an alternative window that you can access by clicking on the tab. Figure A3.2 shows the alternative window and describes its function.

From its beginnings in the middle of the 20th century, the field of systems dynamics and feedback control has rapidly become both a core science for mathematicians and engineers and a remarkably mature field of study. As early as 20 years ago, textbooks (and professors) could be found that purported astoundingly different and widely varying approaches and tools for this field. From block diagrams to signal flow graphs and bond graphs, the diversity of approaches, and the passion with which they were defended (or attacked), made any meeting of systems and control professionals a lively event.

Although the various tools of the field still exist, there appears to be a consensus forming that the tools are secondary to the insight they provide. The field of system dynamics is nothing short of a unique, useful, and utterly different way of looking at natural and manmade systems. With this in mind, this text takes a rather neutral approach to the tools of the field, instead emphasizing insight into the underlying physics and the similarity of those physical effects across the various domains.

This book has its roots as lecture notes from Lowen Shearer's senior-level mechanical engineering course at Penn State in the 1970s with additions from Bohdan Kulakowski's and John Gardner's experiences since the 1980s. As such, it reveals those roots by beginning with lumped-parameter mechanical systems, engaging the student on familiar ground.