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Chapter 11-2

Modeling and Simulating Dynamic Systems with Matlab

The Physical Model

The physical model treated in this example is described as follows.

Consider a simple mass-spring-dashpot mechanical system that can

be described by

wherey(t) is the instantaneous displacement of mass mfromequilibrium, and kand care proportionality constants that describe

the spring and dashpot properties.

This system can be put into state matrix form by letting x1 y and

x! dy"dt, gi#ing

where the matrices associated with the standard state

space format for $T% systems are gi#en by direct comparison with

the general equations,

Time Domain Simulation

Time domain simulations are possible using three state-space

&atlab functions, impulse, step, and lsim. 'or example, the

sequence for an arbitrary input function u(t) could be written as

sys = ss(!"!C!D#

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$%!T!&' = lsim(sys!!t!)o#

These commands can also be used with the frequency domain form

of a system where the $T% ob*ect is defined #ia sys = t*(num!den#.

+s a typical example, the impulseand stepfunctions are used in

&atlab file &$&.&to show the time domain beha#ior of

the mass-spring-dashpot system (see 'igs. /.! and /. below).

'ig. /.!. %mpulse response of typical second order system.

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'ig. /.. 0tep response of typical second order system.

Model Con+ersion

The frequency domain representation of $T% systems is gi#en by

where the system transfer function matrix is

erforming the indicated matrix operations for the 0%0

mechanical system described abo#e gi#es a single scalar transfer

function,

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This could also be easily deri#ed from the original !nd

orderdifferential equation.

3owe#er, multiple-input multiple-output (&%&) systems are

quite tedious to wor4 with algebraically, and the model con#ersion

functions in &atlab become quite useful. %n particular, the

con#ersions from state-space form to transfer function form (and

#ice #ersa) and from the transfer function form to 5ero-pole-gain

form are #ery useful. 6ith the new $T% ob*ect representation in

&atlab 7ersion / and higher, these con#ersions are quitestraightforward. The ss, t*, and ,pcommands easily con#ert

among the #arious forms. The &atlab functions to do the

con#ersions suggested abo#e are

sys1 = ss(!"!C!D#

sys2 = t*(sys1#

sys. = ,p(sys2#

ne can also extract the information stored within the data

structure associated with a particular $T% ob*ect with the ssdata,

t*data, and ,pdatacommands, or by direct structure-li4e

referencing. 'or example, the command,

$num!den' = t*data(sys2#

returns the numerators and denominators of the $T% ob*ect sys2.numand denare cell arrays with as many rows as outputs and as

many columns as inputs, and their i,* entries specify the transfer

function from input * to output i. ach cell array is a #ector that

contains the coefficients of the numerator or denominator

polynomial in s, in order of decreasing powers of s.

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+n alternate approach would be to extract the indi#idual cell arrays

from the data structure and then display it using &atlab9s celldisp

command and"or perform other manipulations as desired. The

sequence for simply displaying the data could be accomplished

with the following commands:

*ieldnames(sys2#

num1 = sys2/num! den1 = sys2/den

celldisp(num1#! celldisp(den1#

0imilar manipulations can be performed with the 5ero-pole-gain

representation. %n particular, the command,

$0!P!' = ,pdata(sys.#

returns the 5eros, poles, and gain for each % channel of the $T%

system sys.. The 0and Pcell arrays and the matrix ha#e as

many rows as outputs and as many columns as inputs, and their i,*

entries specify the 5eros, poles, and gain of the transfer function

from input * to output i.

%n the case of the single-input single-output system (0%0) gi#en

here, the transfer function and 5ero-pole-gain form simplify to the

common representation gi#en by

The user is cautioned that model con#ersion for large systems is

tric4y because of potential numerical difficulties - so ma4e sure

that the results are reasonable;

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esidue or Partial 3raction 3orm

ne can also write the system in the partial fraction expansion

form. 'or example, =(s) for the 0%0 mechanical system can be

written as

where,

and r1and r!can be computed using a #ariety of methods >and 4(s)

?@. 'or problems of this type, &atlab has the residuefunction,

$r!p!' = residue("!#

where "and are row #ectors containing the polynomialcoefficients and rand pare column #ectors containing the residues

and poles, respecti#ely. (s#is only non-5ero if the order of "(s#is

greater than that of (s#/

3re4uency Domain Simulation

'requency domain simulations are conceptually straightforward,

but they can be computationally intensi#e. %n practice, one

e#aluates the expression,

for se#eral #alues of and then usually plots the data in one of

three ways - using Aode, Bichols, or Byquist plots.

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&atlab has se#eral functions for obtaining frequency domain

signatures. The first step is to generate a #ector of frequencies. The

points are usually logarithmically spaced between decades 1?d1and

1?d!with a total of n points,

w = logspace(d1!d2!n#

The bodeor ny4uistfunctions can then be used to e#aluate

at each ,

$M5!P6S7' = bode(sys!w#

or

$7!8M' = ny4uist(sys!w#

%f syshas BD inputs and BE outputs and $6 length(w), M5

and P6S7are BE-by-BD-by-$6 arrays and the response at the

frequency w(#is gi#en by M5(9!9!#and P6S7(9!9!#. +

similar description of the ny4uistoutput can be obtained from the

&atlab help files.

The frequency response for the 0%0 mass-spring-dashpot systemis computed as part of the &atlab sample session (see

&$&.&and 'igs. /. - /.

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'ig. /.. Aode magnitude plots for typical second order system.

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'ig. /./. Aode phase plots for typical second order system.

!!

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'ig. /.2. Bichols plots for typical second order system.

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'ig. /.8. 3omemade Byquist plots for typical second order system.

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'ig. /.

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:/ Modeling and Simulating Dynamic Systems with Matlab

"loc Diagrams;Model "uilding

&any real systems that are comprised of se#eral components and

feedbac4 loops are gi#en in bloc4 diagram form. %t can be difficult

in some cases to find equi#alent system representations and the

corresponding state space matrices. 3owe#er, we can use &atlab

to help build the state space representation from the bloc4 diagram

form. ne can do this in &atlab using a #ariety of commands

within the Control Toolbox or by using the 0imulin4 graphical

interface for modeling dynamic systems.

To illustrate these processes, consider the simple mechanical

system example gi#en abo#e. $etFs first put this system in bloc4

diagram form, and then we can automatically recreate the original

state space and o#erall transfer function forms as a test of the

model building capabilities in &atlab and 0imulin4.

xpanding the state space equations and ta4ing the $aplace

transform of the indi#idual equations gi#es

The $aplace transform equations gi#e the indi#idual bloc4s, as

follows:

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Bow connecting these bloc4s and specifying the desired output as

E(s) G1(s), gi#es

0ome minimal bloc4 diagram arithmetic gi#es the o#erall transfer

function between E(s) and D(s), where the resultant =(s) is the

same as before.

The

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1. efine the indi#idual 0%0 transfer functions for each bloc4

(numbered 1, !, , ..., nbloc4s). %n this example there are three

bloc4s where each transfer function is defined as the ratio

Bn(s)"n(s):

!. 6e then build an unconnected state-space model by repeated

calls to &atlabFs t*and appendcommands, as needed. 'or

example, we can append bloc4s 1, ! and to gi#e an

Hunconnected9 system with the following commands:

sys1 = t*(n1!d1#! sys2 = t*(n2!d2#! sys. = t*(n.!d.#

sysuct = append(sys1!sys2!sys.#! sysucs = ss(sysuct#

which gi#es the system

To simplify this process for the case of se#eral 0%0 bloc4s, the

appendcommand can ha#e as many systems in the input argument

list as needed.

. efine a matrix, >, that indicates how the indi#idual bloc4s are

interconnected. The matrix >has a row for each bloc4. The firstelement in each row is the bloc4 number and the subsequent

entries contain the bloc4 numbers where the current bloc4 gets its

summing inputs (a negati#e bloc4 number implies a negati#e

feedbac4 connection). 'or the abo#e example, the >matrix is

gi#en as

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> = $1 ? ?@ 2 1 -.@ . 2 ?'

. %dentify the system inputs and outputs in row #ectors iuand iy,

respecti#ely, where the entries are simply the appropriate bloc4

numbers. %n our example, the iuand iy#ectors are identified as

iu = $1@,iy = $2'

/. The final step is to simply connect the bloc4s as described in the

>matrix into a full state matrix representation. The appropriate

statement is

syscs = connect(sysucs!>!iu!iy#

erforming these steps gi#es a state matrix formulation equi#alent

to the system described #ia the original bloc4 diagram.

The abo#e process may not gi#e a minimal state space reali5ation

of the system. This implies that there are probably some poles and

5eros that can be canceled. =enerally, this is not a problem, but

minimal systems are usually more efficient during time and

frequency domain simulations. &atlab has commands to remo#e

the extra states and find a minimal reali5ation (see minreal). 'orexample, for this system the proper command is

syscsm = minreal(syscs#

+t this point the user has an o#erall state space formulation of the

original system that was in the form of se#eral connected 0%0

bloc4s. The bloc4 diagram arithmetic that can often be a tedious

underta4ing has been completed automatically, resulting in a set of

state space matrices that capture the full dynamics of the originalsystem. ne can now wor4 with this system as desired. %n

&$&.&and &$&.T(an edited #ersion of the

diary file) we simply show that con#ersion bac4 into the transfer

function form gi#es the original transfer function that we started

with, thus successfully completing this modeling demonstration.

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The Aew ay

+s a final tas4 let9s build the same system as *ust described, but

this time we will use the 0imulin4 graphical interface. %n simple

terms, 0imulin4 represents a graphical interface whichautomatically performs the appendand connectoperations that we

illustrated abo#e. %n fact, howe#er, 0imulin4 has e#ol#ed into a

#ery powerful modeling capability that can perform sophisticated

analyses of coupled linear and nonlinear systems within a

relati#ely easy-to-use en#ironment.

6e will briefly explore some additional 0imulin4 capabilities at a

later time. 'or now the goal is to simply build the abo#e -bloc4

system within 0imulin4, extract the state space matrices from the

resultant model, and illustrate that it gi#es the expected dynamic

response for the simple mechanical spring-mass-dashpot system.

The process starts with the graphical representation of the system

in 0imulin4 (see class demo). ne simply drags the appropriate

bloc4s from the 0imulin4 bloc4 library to the wor4space area.

ragging the mouse between desired connections connects the

inputs and outputs of the bloc4s. 'inally, the indi#idual bloc4s arecustomi5ed to contain the desired parameters for the current

simulation. The resulting graphical representation is sa#ed as a

model file (with an mdl extension) for later use within &atlab or as

a standalone 0imulin4 model.

%n the current demo, the 0imulin4 model was sa#ed as

&&0$.&$. 0imply typing &&0$ at the &atlab

prompt displays the graphical representation of the system. +lso,

the last step in &$&.& gi#es an illustration of how to usethe graphical model embedded in &&0$.&$. %n this case

we simply extract the $T% state matrices using the linmod

command,

$!"!C!D' = linmod(Bmdemosl#

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and proceed to show that the system is identical to those generated

abo#e. Thus, this first 0imulin4 demo simply shows that the

0imulin4 graphical interface can be used for model construction as

an alternati#e to the append, connect, sequence from the

Control Toolbox library. 0imulin4 is often the preferred optionprimarily because of its flexibility and ease of use.

6e ha#e completed this brief demo illustrating some of &atlab9s

modeling capabilities. %n general, the reader is encouraged to

browse through the help files for many of &atlabFs interesting and

useful functions and to gi#e 0imulin4 a try. 6ith some experience,

you will find that the &atlab"0imulin4 combination is one of the

best tools a#ailable for the modeling and simulation of medium-si5ed dynamic systems. 'eel free to experiment with some of their

many features.

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