30079_27a.pdf

CHAPTER 27

MATHEMATICAL MODELS OF

DYNAMIC PHYSICAL SYSTEMS

K. Preston White, Jr.

Department of Systems Engineering

University of Virginia

Charlottesville, Virginia

27.1 RATIONALE

795

27.5 APPROACHES TO LINEAR

SYSTEMS ANALYSIS

813

27.5.1 Transform Methods

813

27.2 IDEAL ELEMENTS 796

27.2.1 Physical Variables 796

27.2.2 Power and Energy 797

27.2.3 One-Port Element Laws 798

27.2.4 Multiport Elements

27.5.2 Transient Analysis Using

Transform Methods

818

27.5.3 Response to Periodic

Inputs Using Transform

Methods

799

827

27.3 SYSTEM STRUCTURE AND

INTERCONNECTION LAWS 802

27.3.1 A Simple Example 802

27.3.2 Structure and Graphs 804

27.3.3 System Relations

27.6 STATE-VARIABLE METHODS 829

27.6.1 Solution of the State

Equation

829

27.6.2 Eigenstructure

831

806

27.3.4 Analogs and Duals

807

27.7 SIMULATION

840

27.7.1 Simulation—Experimental

Analysis of Model

Behavior

27.4 STANDARD FORMS FOR

LINEAR MODELS

807

840

27.4.1 I/O Form

808

27.7.2 Digital Simulation

841

27.4.2 Deriving the I/O Form—

An Example

808

27.4.3 State-Variable Form

810

27.8 MODEL CLASSIFICATIONS 846

27.8.1 Stochastic Systems

27.4.4 Deriving the "Natural"

State Variables—A

Procedure

846

27.8.2 Distributed-Parameter

Models 850

27.8.3 Time-Varying Systems 851

27.8.4 Nonlinear Systems

811

27.4.5 Deriving the "Natural"

State Variables—An

Example

852

812

27.8.5 Discrete and Hybrid

Systems

27.4.6 Converting from I/O to

"Phase-Variable" Form 812

861

27.1 RATIONALE

The design of modern control systems relies on the formulation and analysis of mathematical models

of dynamic physical systems. This is simply because a model is more accessible to study than the

physical system the model represents. Models typically are less costly and less time consuming to

construct and test. Changes in the structure of a model are easier to implement, and changes in the

behavior of a model are easier to isolate and understand. A model often can be used to achieve

insight when the corresponding physical system cannot, because experimentation with the actual

system is too dangerous or too demanding. Indeed, a model can be used to answer "what if" questions

about a system that has not yet been realized or actually cannot be realized with current technologies.

Mechanical Engineers' Handbook, 2nd ed., Edited by Myer Kutz.

The type of model used by the control engineer depends upon the nature of the system the model

represents, the objectives of the engineer in developing the model, and the tools which the engineer

has at his or her disposal for developing and analyzing the model. A mathematical model is a

description of a system in terms of equations. Because the physical systems of primary interest to

the control engineer are dynamic in nature, the mathematical models used to represent these systems

most often incorporate difference or differential equations. Such equations, based on physical laws

and observations, are statements of the fundamental relationships among the important variables that

describe the system. Difference and differential equation models are expressions of the way in which

the current values assumed by the variables combine to determine the future values of these variables.

Mathematical models are particularly useful because of the large body of mathematical and com-

putational theory that exists for the study and solution of equations. Based on this theory, a wide

range of techniques has been developed specifically for the study of control systems. In recent years,

computer programs have been written that implement virtually all of these techniques. Computer

software packages are now widely available for both simulation and computational assistance in the

analysis and design of control systems.

It is important to understand that a variety of models can be realized for any given physical

system. The choice of a particular model always represents a tradeoff between the fidelity of the

model and the effort required in model formulation and analysis. This tradeoff is reflected in the

nature and extent of simplifying assumptions used to derive the model. In general, the more faithful

the model is as a description of the physical system modeled, the more difficult it is to obtain general

solutions. In the final analysis, the best engineering model is not necessarily the most accurate or

precise. It is, instead, the simplest model that yields the information needed to support a decision. A

classification of various types of models commonly encountered by control engineers is given in

Section 27.8.

A large and complicated model is justified if the underlying physical system is itself complex, if

the individual relationships among the system variables are well understood, if it is important to

understand the system with a great deal of accuracy and precision, and if time and budget exist to

support an extensive study. In this case, the assumptions necessary to formulate the model can be

minimized. Such complex models cannot be solved analytically, however. The model itself must be

studied experimentally, using the techniques of computer simulation. This approach to model analysis

is treated in Section 27.7.

Simpler models frequently can be justified, particularly during the initial stages of a control system

study. In particular, systems that can be described by linear difference or differential equations permit

the use of powerful analysis and design techniques. These include the transform methods of classical

control theory and the state-variable methods of modern control theory. Descriptions of these standard

forms for linear systems analysis are presented in Sections 27.4, 27.5, and 27.6.

During the past several decades, a unified approach for developing lumped-parameter models of

physical systems has emerged. This approach is based on the idea of idealized system elements,

which store, dissipate, or transform energy. Ideal elements apply equally well to the many kinds of

physical systems encountered by control engineers. Indeed, because control engineers most frequently

deal with systems that are part mechanical, part electrical, part fluid, and/or part thermal, a unified

approach to these various physical systems is especially useful and economic. The modeling of

physical systems using ideal elements is discussed further in Sections 27.2, 27.3, and 27.4.

Frequently, more than one model is used in the course of a control system study. Simple models

that can be solved analytically are used to gain insight into the behavior of the system and to suggest

candidate designs for controllers. These designs are then verified and refined in more complex models,

using computer simulation. If physical components are developed during the course of a study, it is

often practical to incorporate these components directly into the simulation, replacing the correspond-

ing model components. An iterative, evolutionary approach to control systems analysis and design

is depicted in Fig. 27.1.

27.2 IDEAL ELEMENTS

Differential equations describing the dynamic behavior of a physical system are derived by applying

the appropriate physical laws. These laws reflect the ways in which energy can be stored and trans-

ferred within the system. Because of the common physical basis provided by the concept of energy,

a general approach to deriving differential equation models is possible. This approach applies equally

well to mechanical, electrical, fluid, and thermal systems and is particularly useful for systems that

are combinations of these physical types.

27.2.1 Physical Variables

An idealized two-terminal or one-port element is shown in Fig. 27.2. Two primary physical variables

are associated with the element: a through variable f(t) and an across variable v(t). Through variables

represent quantities that are transmitted through the element, such as the force transmitted through a

spring, the current transmitted through a resistor, or the flow of fluid through a pipe. Through variables

have the same value at both ends or terminals of the element. Across variables represent the difference

Define the system, its

components, and its

performance objectives

and measures

Formulate a lumped- ^

parameter model

_________ Formulate a

mathematical model

...... .

Translate the model

Simplify/lmeanze ^

into an appropriate -«

the model

computer code

Analyze the model

Simulate the model

*• and test alternative •*

and test alternative •*

designs

Examine solutions

and assumptions

Design control

Implement control

systems

system designs

Fig. 27.1 An iterative approach to control system design, showing the use of mathematical

analysis and computer simulation.

in state between the terminals of the element, such as the velocity difference across the ends of a

spring, the voltage drop across a resistor, or the pressure drop across the ends of a pipe. Secondary

physical variables are the integrated through variable h(t) and the integrated across variable x(t).

These represent the accumulation of quantities within an element as a result of the integration of the

associated through and across variables. For example, the momentum of a mass is an integrated

through variable, representing the effect of forces on the mass integrated or accumulated over time.

Table 27.1 defines the primary and secondary physical variables for various physical systems.

27.2.2 Power and Energy

The flow of power P(t) into an element through the terminals 1 and 2 is the product of the through

variable f(t) and the difference between the across variables v2(t) and v^t). Suppressing the notation

for time dependence, this may be written as

P = №2 - ^1) = fv2i

A negative value of power indicates that power flows out of the element. The energy E(ta, tb) trans-

ferred to the element during the time interval from ta to tb is the integral of power, that is,

ftb ftb

E= \ P dt = fv21 dt

Jta

Fig. 27.2 A two-terminal or one-port element, showing through and across variables.1

A negative value of energy indicates a net transfer of energy out of the element during the corre-

sponding time interval.

Thermal systems are an exception to these generalized energy relationships. For a thermal system,

power is identically the through variable q(i), heat flow. Energy is the integrated through variable

3G(fa, tb), the amount of heat transferred.

By the first law of thermodynamics, the net energy stored within a system at any given instant

must equal the difference between all energy supplied to the system and all energy dissipated by the

system. The generalized classification of elements given in the following sections is based on whether

the element stores or dissipates energy within the system, supplies energy to the system, or transforms

energy between parts of the system.

27.2.3 One-Port Element Laws

Physical devices are represented by idealized system elements, or by combinations of these elements.

A physical device that exchanges energy with its environment through one pair of across and through

variables is called a one-port or two-terminal element. The behavior of a one-port element expresses

the relationship between the physical variables for that element. This behavior is defined mathemat-

ically by a constitutive relationship. Constitutive relationships are derived empirically, by experi-

mentation, rather than from any more fundamental principles. The element law, derived from the

corresponding constitutive relationship, describes the behavior of an element in terms of across and

through variables and is the form most commonly used to derive mathematical models.

Table 27.1 Primary and Secondary Physical Variables for Various Systems1

Integrated

Through

Variable h

Translational

momentum p

Angular

momentum h

Charge q

Through

Variable f

Force F

Across

Variable v

Velocity

difference u21

Angular velocity

difference H2i

Voltage

difference u21

Pressure

difference P2l

Temperature

difference 021

Integrated Across

Variable x

Displacement

difference x2l

Angular displacement

difference @2i

Flux linkage A21

System

Mechanical-

translational

Mechanical-

rotational

Electrical

Torque T

Current i

Fluid

Fluid flow Q

Volume V

Pressure-momentum

r21

Not used in general

Thermal

Heat flow q

Heat energy X

Table 27.2 summarizes the element laws and constitutive relationships for the one-port elements.

Passive elements are classified into three types. T-type or inductive storage elements are defined by

a single-valued constitutive relationship between the through variable f(t) and the integrated across-

variable difference x2l(f). Differentiating the constitutive relationship yields the element law. For a

linear (or ideal) T-type element, the element law states that the across-variable difference is propor-

tional to the rate of change of the through variable. Pure translational and rotational compliance

(springs), pure electrical inductance, and pure fluid inertance are examples of T-type storage elements.

There is no corresponding thermal element.

A-type or capacitive storage elements are defined by a single-valued constitutive relationship

between the across-variable difference v2l(t) and the integrated through variable h(f). These elements

store energy by virtue of the across variable. Differentiating the constitutive relationship yields the

element law. For a linear A-type element, the element law states that the through variable is propor-

tional to the derivative of the across-variable difference. Pure translational and rotational inertia

(masses), and pure electrical, fluid, and thermal capacitance are examples.

It is important to note that when a nonelectrical capacitance is represented by an A-type element,

one terminal of the element must have a constant (reference) across variable, usually assumed to be

zero. In a mechanical system, for example, this requirement expresses the fact that the velocity of a

mass must be measured relative to a noninertial (nonaccelerating) reference frame. The constant

velocity terminal of a pure mass may be thought of as being attached in this sense to the reference

frame.

D-type or resistive elements are defined by a single-valued constitutive relationship between the

across and the through variables. These elements dissipate energy, generally by converting energy

into heat. For this reason, power always flows into a D-type element. The element law for a D-type

energy dissipator is the same as the constitutive relationship. For a linear dissipator, the through

variable is proportional to the across-variable difference. Pure translational and rotational friction

(dampers or dashpots), and pure electrical, fluid, and thermal resistance are examples.

Energy-storage and energy-dissipating elements are called passive elements, because such ele-

ments do not supply outside energy to the system. The fourth set of one-port elements are source

elements, which are examples of active or power-supply ing elements. Ideal sources describe inter-

actions between the system and its environment. A pure A-type source imposes an across-variable

difference between its terminals, which is a prescribed function of time, regardless of the values

assumed by the through variable. Similarly, a pure T-type source imposes a through-variable flow

through the source element, which is a prescribed function of time, regardless of the corresponding

across variable.

Pure system elements are used to represent physical devices. Such models are called lumped-

element models. The derivation of lumped-element models typically requires some degree of approx-

imation, since (1) there rarely is a one-to-one correspondence between a physical device and a set

of pure elements and (2) there always is a desire to express an element law as simply as possible.

For example, a coil spring has both mass and compliance. Depending on the context, the physical

spring might be represented by a pure translational mass, or by a pure translational spring, or by

some combination of pure springs and masses. In addition, the physical spring undoubtedly will have

a nonlinear constitutive relationship over its full range of extension and compression. The compliance

of the coil spring may well be represented by an ideal translational spring, however, if the physical

spring is approximately linear over the range of extension and compression of concern.

27.2.4 Multiport Elements

A physical device that exchanges energy with its environment through two or more pairs of through

and across variables is called a multiport element. The simplest of these, the idealized four-terminal

or two-port element, is shown in Fig. 27.3. Two-port elements provide for transformations between

the physical variables at different energy ports, while maintaining instantaneous continuity of power.

In other words, net power flow into a two-port element is always identically zero:

P = faVa + fbVb = 0

The particulars of the transformation between the variables define different categories of two-port

elements.

A pure transformer is defined by a single-valued constitutive relationship between the integrated

across variables or between the integrated through variables at each port:

xb = f(Xa) or hb = f(ha)

For a linear (or ideal) transformer, the relationship is proportional, implying the following relation-

ships between the primary variables:

vb = nva, fb = —fa

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