Goodwin Modelagem

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KAILATH T (1974) A fIT

. \leW 0 three decades of filtering theory JEEE T-20, 2 p. 146. JailS. Inf. Thearr

The advantages of digital control over analog control are discussed more fully in:

CASSELL D. A. (1983) Microcomputers and Modem .Englewood Cliffs, NJ. Control Engineering. Prentice-Hall,

For a brief introduction to delta operators, see:MIDDLETON, R. H . and G. C GOODWIN (1986) I .

tlCS in digital control using delta operators. J ~ p r o v e dfimte word length characteris-N. 11, pp. 1015-1021. . E Trans. AutomatIC COl/trol AC-31.

TSCHAUNER. J (1963) dlra uclion a a theorie des systemes echanrillonnes. Dunod, Paris.

6 Introduction Chap 1

2

ontinuous ime Models

2 1 INTRODUCTION

The main topic of this book will be the study of control and estimation theory amIhow this theory relates to physical processes. Before commencing a detailed study ofcontrol and estimation, it is therefore helpful to have an elementary appreciation of

the dynamics of physical systems. In particular, it is helpful to revic , , various

mathematical descriptions for dynamical systems and to see how these descriptionscan be arrived at from the laws of nature. This will be the topic of the current

chapter.In describing physical systems, it is useful to distinguish Four types of

variables: the inputs, the disturbances, the outputs, and the inter nal varil bles. We

will use the term input to describe those variables that influence the operation of asystem and which can be directly manipulated. The term disturbance will be used to

describe other variables that influence the operation of a system but that cannot bedirectly manipulated. The outpurs are those variables thaI are directly measured andwhose behavior is a result of the action of the inputs and disturbances. Finally. theterm infern l v ri ble will be llsed for any variable that is relevant to the behavior ofa system. A special form of internal variable is the system state. which allo s one to

summarize the effect of all past inputs and disturbances on the system. The concept

of system state forms the basis of a powerful modeling and analysis technique forboth linear and nonlinear systems. We will study these models since they are central

to much of modern control theory and have a close link with the fundamental

physics o the process under study.

7



Many models derived from physical laws will exhibit nonlinear behavior.However, an extremely important subclass of state space models are those thatexhibit linear behavior. We show how some nonlinear systems can be exactlycOlWerted to linear models, We also show how other nonlinear systems can beapproximated by a linear model about an equilibrium position.

State space models generally have the form of a coupled set of first-orderordinary differential equations. An alternative to thjs model format is to use a singlehigh-ord er differen tial equation relating the inputs directl y to t he ou tputs. We will

call this description an input-output model. We will briefly describe this class ofmodels and their relationship with state space descriptions.Finally, we show how certain simple types of disturbances can also be

described in state space and input-output form.

2.2 ST TE SP CE MODELING

The state of a system can be loosely defined as a (nonunique) set of variables suchthat knowledge of these variables at some time, together with the future inputs and

disturbances. is sufficient to allow determination of the system's future behavior.Thus the system state completely summarizes the influence of all past inputs anddisturbances on the system.

In many practical cases it turns out that the state can be adequately represented by a finite-dimensional vector. We will designate these systems as finite-dimensional systems. For this class of systems, we will denote the dimension by theinteger and the system state by xC ). which is an vector. We will also denote theinput to the system by u I), the disturbance by del), and the output by yr/). With

this notation, we can describe the relationship between these various variables by aSlale space model as follows:

~ X l )= / X I). U I). d I) .I) .2.2.1 )

and

Y I) = g .\ I), 11 1). d I) .I)22.2)

The modeling problem. in state space form, is then one of:

a) Determining an appropriate vector of state variables, x.

(b) Finding the functions / and g that describe the dynamic behavior of thesystem.

In many cases, IlOlUral state variables can be suggested. In electrical circuits.these would usually be inductor currents or capacitor voltages. In dynamic mechanisms. the natural states may be linear or angular positions and velocities. In fluidMow problems. the states may be tluid pressures. temperatures. and so on. In

8Continuous Time Models Chap. 2

chemical systems, the natural state variables are often temperatures or chemical

concentrations. ,To find the functions / and g, we can employ physical laws such as Newton s

law of motion, heat or mass balance equations, rate of reactIOn e ~ u a t l O n s ,andKirchhoff's law for electrical circuits, plus the electncal propertIes of mductors and

capacitors. . fi ld -h t ' l l t t eWe will present next several examples taken from dIfferent e s l a I us r

the derivation of state space models using the laws of nature.

Example 2.2.1. Electrical Circuit

Consider the electric circuit shown in Figure 2.2.L A natural. choice for the tatevariables in this case is the capacitor voltages. ApplicatIOn of Kirchhoff's lav\s olves

Iicc = un - vn ), R2

Using i = C dvjdl). we then have

(2.2.3)

(2.2.4)

. dLn , -- - - - v T Vi1 1 1 1 .

Vn = d r = - L n R l . Cl R2· Cl c 2 C l · R2 Cl · Rl

. n _ _ _ , __dv 1) 1 )vn = d r = vn R2· C2 u R2· C2

Thus if x = [un Vn]T and u = u . we can write

; =1

R2 · C2

_ 1 [ 1 1_ Cl . : 2 J Cl Rl ,u

R2· C2 J

y= [0 IJx

which is in state space form [ 2.2.1), 2.2.21].

~ ~ ~ 4 0

igure 2 2 1 Example electflcal : y ~ l ~ m

Sec. 2.2 Slate Space Modeling

(2.2.5)

(2.2.6)

(2.2.7)

2.2.8)

9



m

Figure 2.2.2 Example mechanicalsystem

Example 2.2.2. Torque-driven Inverted Pendulum

Consider the torque-driven inverted pendulum shown in Figure 2.2.2. We may selectas state variables the angle B of the rod about a vertical axis and w, the angularvelocity corresponding to B. The input in this case is the torque applied to the massand rod. Using Newton s second law of motion (in angular form). we have that

Um/2 ,;, = T + m/g)sinB

Letting XI = J and x 2 = W we then have(2.2.9)

(2.2.10). 2 g ) T

Xl = / In XI - 22m/ (2.2.11)

and

y = j = [1 O x

which is in state space form.(2.2.12)

xample 2.2.3. Decalcification Plant

Consider the decalcification plant shown in Figure 2.2.3. The idea of the system is toreduce the concentration of calcium hydroxide present in the water by forming acalcium carbonate precipitate. Although not shown in Figure 2.2.3 we assume that

some means for removing the precipitate is included in the system. In modeling thissystem, we will make the following simplifying assumptions: that the tank iscompletely stirred and that all Ihe carbon dioxide injected dissolves in the water.The basic reaction in the system is

(2.2.13 )

We further assume that the reaction proceeds only in the forward direction,that all calcium carbonate formed precipitates, and that the amount of waterproduced in the reaction (2.2.13) is insignificant. We will use the following notation


I

II

Lime water ~ _ _ _ ,

\ CO gas injection

O control valve

CO2 gas tank

Figure 2.2.3 Exampl e chemical system

to descri be the system:

k

V

c

u

y

State variables representing the concentrations of calcium hydroxide and carbon dioxide (respectively) moles/lIter.IIl the tank.

Rate of inflow. in moles/second, of calcium hydroXIde.

Total tank volume in liters.

Rate of production of calcium carbonate in moles/second.

Constant (in liters 2/ mole . second» re lating r to XI and x 2

Input rate of flow of CO 2 in moles/second.

Calcium hydroxide concentration at the outlet moles/Ilter).

We hypothesize that the rate of reaction is proportional to the product of theconcentrations; that is.

2.2.14)= cX X

Note t h ~in practice, the constant of proportionality c is strongly e p e n ~ e ~ ~on the temper;ture of the reactants. The rate of change of concentratIOns caexpressed as

(2.2.15 )

Sec. 2 2 State Space Modeling 11



andrV

Equations (2.2.14) to (2.2.16) can be written in state space form as

. k cXI = V - V X I X 2

. u Cxl = V - VXIXl

andY = [ 1 J x

2.3 LINEAR STATE SPACE MODELS

(2.2.16)

(2.217)

(2.2.18)

(2.2.19)

In the prev'ious sections we hav'e given some examples of the development of state

space models for dynamic systems. Some of these models have the special property

that the functions f and g are linear in the variables: that is. f( X. II. I) = Ax + Bliand g(x. u. I) = Cx + Du. We will call these models linear Slole space models.These systems have many useful properties and thus we will give them specialemphasis in this book. for example, one key property satisfied by linear state space

models is superposition. This principle can be stated as follows. Let YI(I) denote theresponse with x(O) = 0, u ( i ) = III I J. h l J he response with x(O) = 0 and 11(1) =

u,U), and yoU) the response \\'ith x O) = -'0 and U(I) = O. Then the response to

xeD) = "'0·'1' 11(1) = ' lu l(I) - - "'l l l(l) is ,1'(1) = 0: 0 )'0(1) + (ttY (I) + (t2h(l) for

any . c a l a r ~'d. 'I. l,. This principle and other properties make l inear systems

partlcularly amenable to analysis. In subsequent chapters we will explore the richtheory available for this special class of svstems.

Unfortunately, however, not all s y s t ~ m sare linear, see, for example, Examples

2.2.2 and 2.2.3. These systems exhibit more complex behavior and are more difficultto a n a l ~ · z efor this reason. it is often desirable to transform a given nonlinear

system IIlto an associated linear system whenever this is feasible. We discuss next

several methods for linking particular nonlinear systems to associated linear models.

2.3.1 Compensating Nonlinearities

Compensation techniques rely on the presence of special structures in the nonlinear

ities exhibited in (2.2.1) and (2.2.2) and are thus applicable only for certain classesof systems. Several illustrative examples are discussed next. -

(a) Input-output nonlinearities

Consider the case where the system model (2.2.1) and (2.2.2) has the followingstructure:

d(fiX(I) = AX(I) + BfuCu)

Y(I) = g,( eX(I))

12Continuous Time Models

(2.3.1 )

(2.3.2)

Chap. 2

1 1

I l : p : ~ u t P u t : 1

I nocllnearit y_ Linear system ~ n [ } + - G t o n l l n e a r r t YIu .' u ' u Ax + Bli Y I y 1 1 '

I \;1111 fu 9 y I 9 y I y. Y = ex .

I I , I1 1 II ~ I1 O n i n e - = - : s = n ~ J

Auxiliary model

igure 2.3.1 D Jgr.:rm show ing the concept of input-omput nontmelrity compensation.

If f and gr are invertible functions over some operating range. then we define

auxiliary variables 1 / ' and y' hy the equations

(2.3.3)

and

(2.34)

Then. clearlv, over the range of values for which f and gr can be inverted. 1/' and

y can be reiated by the following linear model: .

d(frX(I) = AX(I) + Bu (I) (2.3.5)

.1 (1) = CX(I) (3.3.6 )

This technique of nonlinearity compensation is depicted in Figure 2.3.1.The following simple example illustrates how this technique may be used.

Example 2.3.1. Electric Water Heater

Let us suppose that the heat capacity of the water in the tank in Figure 2.3.2 is Cand that. if amhient (outside) temperature is To heat is lost to the outside at a rateequal to k( T - ~ . As our state variable. let x = T - To be the v ater temperature

abol'e ambient. Then we can show that an appropriate state space model is

(2.3.7)

and

y = x (2.3.8)

Clearly, this model is linear except for the term u 2 in (2.3.7), and so, by defining an

Sec. 2.3 Linear State Space Models 13



~ S i r r e r

Water tank Water temperature f11easurement, T

Heating element resistance, R

L . =--=--=--=-=tVoltage, U

Figure 2.3.2 Electric water heater diagram for Example 1.31.

auxiliary input variable p = u 2 , we can obtain the following linear model:

k 1x = - e + CRP 2.3.9)

Y = x (2.3.10)

~ o t ethat this auxiliary model is only valid oVer the operating range p > o. VV V

(b) Output feedback nonlinearities

We now consider another compensation technique for obtaining a linear modelwhen the nonlinear model has the following structure:

~ x ( r )= Ax r) + B [ f ) y ) + u(r)]

d r ) = Cx r)

(2.3.11)

(2.3.12)

The system is linear, save for the term ro We can then define the auxiliary input

u = u + f r ( y ) (2.3.13)

which leads to the following linear auxiliary state space form. illustrated in Figure2.3.3:

ddix r) = Ax r) + Bu'(r) (2.3.14)

y t) = Cx t) (2.3.15)

This technique is illustrated by the following example:

14 Conti nuous Time Models Chap. 2

1l

I 1 1

, 1+ 1 ; ;= Ax + u r - . ~ - - \ - - - t _ - ,......U \ Ir 1 v = ex

I I + 1

I1 \

\ I ~ : , : ,~ ~I fy (.)

1 inear auxiliary S y s t ~_ _ _ _ _ _

. r . t' of a feedback nonlinearitv.igure 2.3.3 Block dia;ral11 illustratmg meanla IOn

Example 2.3,2. Inverted Pendulum (continu ed)

f E I 2 ' ) ' ) Define an

C-·d r aOa·l·l tile torque-driven inverted pendulum 0 xamp e ~ ~

o n ~ e ~ , .auxIliar\ in-put torque, T , bv the eljuat,OIl(2.3.16)

T =T+mlgs ineI f ' 11 0 Imear auxiliarv_ . 2' )]0)-.(') ,12), \\e can obtam t1e a \\ I b ' . .Then, from Equal10ns ( ._. _._.

model:

~ X l = X 2 (2.3.17)T

,-( 2 = T,:;;2(2.3.18)

l ~ e = [ I 01x

. . . deoree of freedom robot. The same lineariza-Example 2.3.2 IS II1 fact. a one . \> deoree s of freedom. in which case the

tion method can be applied to multIp c Co

l11e1l1OU IS known as computed torque control.

(c) Linearization by change of states

Consider a general state space model of the form(23.191

i )- . \ = ( ( X . l iit

.= v( X ) such

. .. . j nlinear functIOn o[ the states. sa) ,I I

Somel1mes It IS pOSSible to nne a no _ f . ..tl espect to time are indep enden t ofthat v and the lirst II - 1 ucnvatl\e, 0 .\ \ \ 1 1 r

15

Sec. 23 l inear State Space Models



11. In this case we can define a new state vector as- 7 6 . I

X = y .I ..... , y -I)

Using this new state vector, the corresponding state equations becomed _ _f j X I = x 2

(2.3.20)

(2.3.21)

: i ; ~ ~ I Y , ~ f;he function Ii(.\ , iJ) is invertible .\vith respect to 11, then \\e can solve( ) u for U I I I terms of u and x, Ieadmg to the following linear svstem

d .( f j I=X 2

This method is illustrated by the following (;xampJe.

Example 2.3.3

Consider the following nonlinear system:

:1 = Xl + .yi U

)22 = x j + u

We define .t I = Y as

·:rl ,g, y g Xl - X2

Then

Also.(independent of u)

i = 3.\ 5. ,

= 1 ~ ( X I+ u)

Hence. defining - \ £ , ) , .\: 2 g .1 , \\:e have

.Xl = x2

~ , =< t / l ( t + t I / l ) , -2 /1- -. 2 - 1 . 2 T : l X 2 - II

16Continuous Time MOdels

(2.3.22)

(2.3.23)

(2.3.24)

(2.3.25)

(2.3.26)

(2.3.27)

(2.3.28)

(1.3.29)

Chap. 2

Finally, putting

u = tu .t=;2/3 - (.\\ + X1/3)

we obtain the following linear model:

XI = ~ 2

x = 1.1

(2.330)

(2.3.31)

(2332)

The techniques just described can be combined in various obvious ways.

However. there still remain some difficult nonlinear systems that either cannot beexactly linearized or are difficult to exactly linearize (for example, the simple

decalcification plant of Example 2.2.3 has a model that cannot be linearized using

either technique a or b and that can be linearized by method c only if the output isdifferentiated and the process constants C. k, and V are known). In such cases the

use of linear approximations to nonlinear systems may be a useful way of linking

the nonlinear behavior to linear behavior.

2.3.2 Linear Approximation 01 Nonlinear Systems

Consider the case where the nonlinear system (2.2.1) and (2.2.2) is operating with

small deviations about a fixed operating point (which is assumed to be an equilib-

rium point for the system). Let

and

x t) = X f ~ x t )

u(t ) = U -:- J.u(t)

y / ) = y -:- , ;y(/)

where X. U. and Yare the constants de:1ning the operating point.

The assumption of equilibria ensures that

f X U ) = O

Y = g ( ~ \ : , U )

(2.3.33)

(2.3.34)

(2.3.15)

(2.3.36)

(2.3 .37)

Then. using a Taylor s series approximation. we have the following approximatemodel for the S\ stem (where we assume that f and g do not depend explicitly on

time).

~ . 1 x ( / )= ~ x ( / )= f X + . 1X(I).U+ .ell/(I))

=0 [ ; ~ l l : : : . ' 1 X ( t )+ r ~ ] I ; : : : . l u ( / ) (2.3.38)

and

.1 Y( I ) ~ Ii X + . 1x ( ). U + , ;l/ ( t )) - y

== [ ~ ~ ] l x _ x , ' ; X ( t )+ [ ~ ~ l p x , ' ; U ( I )1 /= l; . 11= L

(2.3.39)

Sec. 2.3 Linear State Space Models 17



J b Note that this technique of linearization requires the computation of variousaco Jan matflces that afIse when we differentiate a vector function with respect to

a vector (for e' I Dfl8 .N xamp e, . i X IS the 1 X 11 matrix whose i, j t h element is 8 j j 8 x ), ate also that, by lettmg A = [8//8x)l, (and so on for B C i D ) ) .

\ = X , a n u , we can

rewrite (2.3.38) and (2.3.39) in the stand;r:tlinear state space form:

ddi( :1x) ' A( :1x) - BPII)

and

( :11) '" C( :1x) - D :111)

This technique is illustrated in the following example.

Example 2.3,11. Decalcification Plant

(2.3.40)

(2.3.41)

Consider the simple decalcification plant of Example 2 ) 3 with a . I .point, u = U = k d . _ v _ T • . - . • nomJl1a operatlllg

) ~ an J - ~ ' l - L TIl en, uSlllg EquatIOns (2.2.17)-(2.2.19) and(_.3._61, \ve can show that the nomma opera tino con d't ' f . ,

to I IOn or x 2 IS

(2.3.42)

Then. rewritmg (2.2.17). we have

Similarlv.

( :1'-(2) '" - -pX2 :1XI - fX j :1x: + ~ : 1 I 1

and thus an approximate linear model for the system is

dJi j \ =

k 1

-17 Xlk 1

'V Xl

y = [1 OJ.::.x

(2.3.43)

(2.3.44)

(2.3.45)

(2.3 .46)

In the next section. we consider an alternative to Ibased on input- output models. t le state space approach


2.4 INPUT - OUTPUT MODELS FOR DYNAMIC SYSTEMS

In an input-output model. we write the differential equations that describe a systemoS a higher,order differential equation relating input to output. rather than as afirst-order vector dillerential equation involving system states. This type of descrip'tion is sometimes more natural and bears close connection to the transfer functionfor linear systems to be discussed later in Chapter 5.

We introduce the symbol p to represent the differential operator. p dldl. Inthe general nonlinear case, an input-output model has the form

h(p y . . . . y; P 'u . . . . , II. r) = 0 (2.4.1 )

In the case of linear time invariant systems, (2.4.1) simplifies to

Allplly + An_1pll-I) ' + . . . +Ao.v = BmP l11 u + Bm_1pll l -IU + .,. +BOLI

(2.4.2)

That is.

A(p ) ) = B(p)u (2.4.3)

where A(·) and B ( ' I are matrix polynomials in the operator p.Note that linear input-output models and linear state space models give an

equivalent description of the input-output properties of the syskm; that is. given

any state space model, there exists an equivalent operator model relating y and 11,

and vice \·ersa. For example, given a single-input. single-output model of the form(2.4.2) (where a = I and m < for simplicity). the following state space modelcan he readily shown to be equivalent to (2.4.2):

-Ql l -1 1 0 0 bll

_ 1

- ( I n - 2 0 0 0 b,,_2

dJix = 0

x (2.4.4)

1

- a o 0 0 bo

Y = [I 0 0 OJx (2.4.5)

where we have used lowercase notation since A" Bj are scalars. To demonstrate thisequivalence. we compute py. p y . . . . p y from (2.4.4) and (2.4.5). ~ n then sho\\that

( ) a p - I • . + a ) \ = (b p - I + ... +bo )T 11--1 I 0 11-1 (2.4.6)

further details may be found in Section 8.8.3 of Chapter 8.The converse result, that any linear state space system can be written in

input-output form. is also true and is left as an exercise for the reader (Problem 8).The following examples illustrate how input-output models may be directly

determined from physical principles.

Sec. 2.4 Input- Output Models for Dynamic Systems 19



Example 2.4.1

Consider again the torque-driven inverted pendulum of ExampleNewton's second law of motion, we have directly that

~ m I 2 J i- mig sin 8 = Tor

{IJ} 2 g . e 2p- - T SIn = m i l T

which is in a nonlinear input-output form.

Example 2.4.2

As a further example, consider the problem of modeling an armature voltage-controlled dc motor. Assuming a constant field current, and neglecting saturation,armature reaction, and the armature time constant, we have the following equationsfor the dynamic behavior of the dc motor:

v = R1 + E(2.4.9)

E = kO(2.4.10)and

Tdec = Ie' = kI(2.4.11 )

where V is the terminal voltage (input). 1" is the armature current, R is thearmature resistance, e s the motor shaft position, Ea is the back emf generated inthe motor, k is the motor torque constant, I is the shaft inertia. and elec is thetorque produced by the motor. Equations (2.4.9)-(2.4.11) can be rearranged as

(2A.12)

2.5 MODELS FOR DISTURBANCES

The previous descriptions of system models have neglected the effects of theenvironment within which the system operates. To obtain a more complete descrip

tion. it is often helpful to adjoin an additional model that describes the influence ofthe environment.

A number of disturbance models are possible depending on the nature of theenvironment. A key ingredient in disturbance modeling is the extent to which thedisturbance can be predicted based on past values. We wil briefly discuss severalclasses of disturhances. These models can be combined with the system model togive a composite model for the system and its environment.

An important class of disturbances are those called deterministic. A determin_istic disturbance is one that can be perfectly predicted into the future. Examples of


. ~ re dc offsets drift at constant rate. and periodic disturbancessuch disturbances a T h' I 'nd ,f disturbance can usually be modeled byI d' asonal components IS ~ I L _ Iinc u mg se '. t t ut model Examples of these mode s area homogeneous state space or mpu -ou p .

(a) DC offset: d = constant; that is, the model is

pd = O. d(O) = constant (2.5.1)

(b) Drift at constant rate: d = ext + /3; that is. the model is

p d = 0, d O) = /3, pd O) = a (2.5 .2)

c) Sinusoidal disturbances: d = C cos( - 01+ ¢); that is,

) d(O) = C cos( ¢), pd(O) = 'oC sine 9)(p2 + wi) d = 0, (25.3)

These models have the general form

S p ) d = O 25A)

where

S p) = p -'- S,,_IP,,-I + +so

. . I b b t led by the techniques\n equivalent state space descnptlon can a so e 0 all.. . I d I ( J " 4) can also be wntten asf the previous sectIon. Thus, t le mo e _.".

[

-S;,_I

px = .

-so

(2.5.5)

o(2.5.6)[ l 0 OJx .

Tl d I for disturbances can be combined with a system Illodel to obtamlese mo e s . I . Ex' mple 2 5 1a composite description. This is illustrated for a sImp e case

Example 2,5,1. Water Heater

E I? 3 1 In the previous derivation ,,'eConsider again the water heater of ~ ' , 1 1 l Pe _ t t A better model could be toh h b'ent temperature J was cons an .

assumed t at t e a m I i 0 d l and is thus describable as 3I h b'ent temperature I uctuates al y .aSSllme t lat t e a m I . d t tl temperature above mean ambIentsinusoidal disturbance. Lettmg XI eno e le .temperature. then the following model IS appropnatc

k 1 k (2.5.7)P x = - - x - P - - d. 1 C 1 ( C

) ;.....; l

Sec. 2.5 Modets for Disturbances 21



'here d is modeled as a sine wave and satisfies

p [ : ] = [ - °W ~}[ l(2.5.8)

d = [I 0 J :::] (2.5.9)

Combining (2.5.7)-(2.5.9) leads to the following composite model in statespace form:

[ ) [ ~k

l )[1]p X 2 = 0 0 ~ ~ + pJ 0 w5(2.5.10)

Y = XI

(2.5.11)

Equa tions (2.5.10) and (2.5.11) are also equivalent to tIle f II .model: 0 OWIng inpu t -ou tpu t

( e 2)( k ( ,)1p + wo. p + c y = p + wii - p. c (2.512)

~ o t ethat it is not possible to cancel the term (p2 + w J from both sides of (2.5.12).mce, If thIS IS done, the effect of the disturbance is lost. (More will be said aboutthiS In Chapter 8.)

. In the preceding discussion, we have restricted attentil>n to simple determinis-tIC dIsturbances. ThIS covers manv practical cases l>f l'llterest Ha I " . oweve r, ml>re~ e n e rdIsturbance models are pl>ssible. Fl>r example. in Chapter 10, we willcl>n'lder randl>m dIsturbances that have the prl>perty that they are nl>t perfectlvpredlctahle from theH pas . "

2.6 SUMMARY

The key points cl>vered in this chapter were:

• The concept of system Slate. which SU111marl'zes th fl f II- e e ect l> a past inputs anddisturbances on future system behavior.

• Nonlinear state space models of the form

ddix(l) = f(x(I) , U I), d(I), I),

(22.1 )Y(I) = g(X(I), U I), d(I), I)

• \ special class of state space ml>dels, linear models, described by(2.2.2)

dd i X = A x B u

) = ex + Du


• The idea that many nonlinear systems can be com'erted to exact linear systems byvarious compensation techniques (Section 2.3.1).

• The idea that nonlinear state space models can be approximated by linear systemsabout an equilibrium point (Section 2.3.2).

• A discussion l>f input-l>utput models and their properties (Section 2.4).• A discussion of the equivalence between l inear state space models and lmear

input-l>utput ml>dels.

• A descriptil>n of deterministic disturbances, including steps, ramps, and sinewaves, and a discussion of their corresponding models.

Thus, Chapter 2 cl>vered various aspects of the modeling of phy,ical processes

by continuous time models. In Chapter 3, we will take the next step of introducing

the notion of s:J.l1lpling and hl>w this affects the modeling process.

2,7 REFERENCES

Further discussion of physical models may be found in:

DOEBELJN. E. O. (1985) Confrol Systems: Princ:ples Gnd Design. Wiley, New York.

Good mtroductioJls to linear state space models are contained in:

CHEN, C. T. (1984) Linear Syslem Theory and DeSign. Holt, Rinehart and Winston, NewYork.

FORH1AN ', T. E., and K. L. HITZ (1977) n inrroduClIoll1O Lilleur COlllrol 5, ,SI<II1S. }"farceDekker, )Ie\\, York.

KAILATH, T. (1980) Lmear Svslems Prentice-HalL Englewood Cliffs, N.J.

Additional information on input-output based models is given in:

MIl.bTH, T. (1980) Linfilr Sysfems. Prentice-Hall, Englewood Cliffs, N.J.

WOlOVICH. W. A. (1974) Linear Mulfluariable Sysfems. Springer-Verlag, Ne\\' York

2,8 PROBLEMS

L Extend the model of Example 2.2.1 to include nonzero sOUrce resistance and nonzeroload conductance (requires some electrical background).

2, Extend the model of Example 2.2.2 to include friction in the Joint and the mass of thelink (requires some mechanical background).

3. Extend the model of Example 2.2.3 to include reaction limiting by product inhibition(re4uires some chemical engineering background).

4. Consider the level control problem in Figure 2.S.P4. Assuming that f elY and that thecross-sectional area of the tank is A, deri\c a nonlinear model for the system.

Sec. 2.8 Problems 23



Inlet f low

Lj

~ - - - - - - - - - - ~utle t f luw

igure 2.8 ~

5. Linearize the model in Problem 4 llsing both feedback linearization [Section 2J.l(bl) andbnear approxlmatlOn [SectJOn 2.3.2).

6. Extend Problems 4 and 5 (1f possible) to the two coupled tanks shown in Figure 2.8 P6.~ -

I I ~ c ~

I · ~ ( · F

Figure 2.8. P6

7. a) Show by successively ell1ninating states that (2.4.5) and (2.4.4) can be expressed in theform of 2.4.21.

b) Show that a model of the form 2.4.2) can be e.'pressed in the form of (2. 1.5) and12.4.4) by appropnately defining x in terms of y. u and their Jerivatives.

8. Gi\ en a gene ral s i n g l e ~ i n p u t s i n g ] e ~ o u t p L 1 t]jnear state ~ p a c emodel o c t j m e n ~ i o n/1. show;hat t ~ l re.X1sts an equivalent lth·order input-output model. HIIlI: Find an >.pressionor Pl ' 1Il terms of x.u . . . . p , - l using the staie space model. Then use the

C a ~ · l e v . - H a m l l t o ntheorem to eliminate x hy adding the aerivati"e, o f ) with appropriatewelghtlngs. .

9. Convert Ihe model in Example 2.4.2 to state space form as in Equation 2.4.4).

10. Can the nonlinear element in Figure 2.S.PlO be locally linearized Give reasons foranswer. Jour

Output

l I- - - t - - - - + - -

I~ - - - - +-----. Input

Figure 2.8.PlO


II Suggest a suitable deterministic rlisturbance model to add wind efrecls on the inverted

pendulum model. Discuss some of the shortcomings of deterministic models in thisinstance,

12. Develop an input-output model for a disturbance of the form

d I) ~ (;1 sine w,1 "' 9 , ) ' (;, cos w, l + ¢,)

13. Suppose you are required to model the dynamic behavior of an automobile (driven in

high gear) so that an automatic cruise control may be added. Suppose that the wind

resistance of the vehicle (Fw in kN) is given by

F = - 7 X to-'s'where 5 is the car speed in meters/second. When in high gear, the engine force is (in kN)

F = 0 3';( 1 - 0.15 + 0.7{ i )I . 1 ..,. 0.15

. .where T is the Ihrotl1e (that is, accelerator) position in percent. (T = 100 is accelerator

fully depressed. and T ~ 0 is fully released.) 11 the car has an effective mass M of 1 tonne

tOoo kg), answer the following questions:(a) Find a nonlinea.r state spacc model for the system. (T is the input. 2nd 5 is the

measured outpUL) Hill : Newton's laws of motion give that S = total force.(b) Find an approximate linear model for the system operatmg near a steady-state

condition 5 = 30 m/s.e) Show how a linear auxiliary system may be formed by defining a new input. II. Give

an explicit expression for T in terms of V.(d) Describe qualitativel:' the effect On Ihe model in part (a) of including the efl'ect of

gravity on the car, if the road is not flat.

14. Suppose for x(O) . ~ O that is, zero initial conditions. the response of the linear time

varyjng system

. ~ ( [ )= A(I)X(I) + B(I)U(I)

Y I) = c(1)x(r) + D(I) U l)

for input u(l) = ",(I) is x(I) = X,(I) and .1·(1) - 1(1). and for u ( t ) = , ( I ) . theresponse is x(1) ~ x,(1) and \, 1) = .1',(1). Show that the system obeys superposition:

that is. for any constants ", . "" the response to u(t) = "1",(1) + , u , l I ) is X I) =

,x,(1) -- , x , ( I ) , Ylr) = ,YI(I) + a,v,(r) . Hint: Find (d /d l )x ( r ) and X(O).

IS. Show that the following system is nonlinear (that is, does nol obey superposition):

r x

Hin1: For x O) = 0 and u(t) = k (constant). x(1) = k' 1 - e- I); then consider u,ltJ =

+ l . u , ( I ) = - l .

16. (al Show that the folluwing system is linear (from u to y I although it appears otherwise:

,t = - 3 x + (3X 2, 3 )u

Hilll : For x(O) ~ O. show that X(l) { f , : e - u - u ( , ) d ,} l .(b) Show that the system has a nonlinear initial condition response. HIIlI: For uI11 = 0,

0 , (1 ) = e- 3Ix 0). Thus find y( t ) for U(I) = O

Sec. 2.8 Problems 25



+

17. o m i d ~ rthe mechanical system in Figure 2.8.P17. If the output l' is i I , =i

.I ., [I

( I n I u = (dldl)d" , = (dld )"j ' and a. = (<i/JI)[' which of thcfollowin Q setsof variables are valid states: - - <0 •

(i) XI = .1, x, = I

(ii) XI = d l X = d,. x, = [ j , X 4 = I ' ,

(iii) XI = y. x. = v - ['1

(iv) I =)''. x, = exp( , - I)

l ~ f o r c e ~Ml M2

V.d,

d 2

Figure 2.8.PI7

Frictionless surface

18. COIlSider the electromagnetic system in Figure 2.8.P18, If tbe total reluctance of the~ 1 a g n c t l cIfcmt is R + k\lY and the \ \ i n d i n g ~are assumed to have zero resi tancefwd a state space model for the system. Hill : '

(i) Try using L .v. aJld I as stales.(ii) The i n d l l c ~ demf is u = N(d?/dl),

(iii) The fiux is ? = NI/R.(h) The magnetic force is 1N' i ' (d /d l ' ) ( l /R) ,

(

Figure 2,s'PI8

19. Using states l 1

Figure 28,P19.and x, = [> find a linear state 'pace model for the system in

t ~ H1 V-+i

c 1 c v

_ _ _ _ T ~ 2 _ r 2Figure 2.8.P19

26 Continuous Time Models Chap, 2

3

Discrete Time Models

3 1 INTRODUCTION

Wesa\\' in Chapter 2 how continuous time physical systems can be modeled b\' sctsof ordin:H\ ditTerential equatIons. Here we turn to the question of the interaction ofa physlcal system with a computer. In particular. we wish to describe the model asseen b\' the computer. Thus, our objective is to relate sampled \'alues of the outputresponse to sampled \'alues of the input. The interface between a Cl1mputer and acOlllinuous process is typically achieved with an analog to digital (A/D) COl1\ erterand a digital to analog (D/A) converter, as shown in figure 3 1 1

The process of A/D conversion is basically one of sampling. However, \\ e willsee that it is also important, in general. that the signal be preconditioned prior tosampling, This preconditioning usually takes the form of a special loss-pass Illler(called an antialiasing filter) that remoyes high-frequency Cl1mponents frOlll the

Continuous Input

Ana',og process

BDiscrete

Computer

Output

F i g l l r ~~ . I . l CnlllpulU intcrfJ.cc to n

;I lalng. r t - 0 L ~ I ' . '

27



1 2 2 Proress s y s t e m • analy. ' a n d con fro/

P H O B L E M SHI.l :\. pneumatic PJ ('oEtroll"r has an olltput pressure uf ] l psi II'hen the set point

:llld pell point are t.ogethcl'. The set point and pCll point nr{' suddenl)' displaced byo.r ill. (i.{'" " .'t('j> dw,, ;c in errol' is introduced) and till' following ua ta are obtained:

i m e . r psi

o- W

0+ 820 7GO 590 3.5

DdcrlllilH' the Ilctul l ~ l i l l(psi IlI'f inch displacement) nnd the integnt time.]IJ.: A 1ln;l-sl "]> changl' in error is introduced into a PJ] ) ('olltrolkr. If Ii ,

TI = I, I l l ld 71. = iI.: >, plol til ' reRpons ' of tbc controller, 1>(1)

/0,;; An ideul PI) controller hll.' the trnnsfer function

As 0110 \ \ ' ' ' in Chul" ~ ~ un /lctuai PO controller hns thl' transfer function

lO,

\\'hl'rl' (l i s" inrI'' ' "onstanl ill nIl industrial controller.If u unit-step chunge in error is introducer into II controiler haying the second

r ~ l l s f c rfune: ioJl, sholl' t ha t

PIt) = I{ , l + -ic-su'l>i

where :1 is a function of f3 which you arc to determine . For fJ = 5 find J\ , = 0.5, plot1 (1) vcrsus I/T1> As 6 "' . show tha t the unit-step respoDse .opproaciIes tha t for the

icion controllerlOA A PI]) controller is at , tead, ' state with an ou tpu t pressure of Dpsi,.;. The Bet

point and pen point are initially togl ' lhcr. At t ime I = 0, the set point is moved away

from the pen point at l mte of 0.5 in./min. The motlOn of the set point is in the direction

of /OU'( ' r readings. If the knob s e t t m g ~rete

1\, = psi/il;. of pen tr,we]

7/ = 1.25 min

71' = 0.4 m i l l

plot the ou tpu t pressure versus t ime.

Blocl . D i a g r a m o a Cilemica l -react.@r Con t ro l Sys t c ln

To tie together the principles developed thus far and further to illusL _ ~ " ,I , nr()(:edul'e for reduction of a physical control system to a blockdiagralll, we eOllf;idr., j J ,."is '::'aptc,. t I l " t.wo-tank chemical-reactor controlsystem of Fig. 11.1. This entire chapter is an example and may be ollJittedby the reader with no loss ill c()lltinuity.

DCf'icription of System ,\ liquid stream enters tank 1 at a volumetric.flo\\' raU, F dIll and contains reactant A at a concentration of en moles A / f VReactant dC'composes in the tanks according: to the irreversible chemicalreaction

The reaction is first-order and proceeds at a rate

r = kc

where r

cIe

moles A decomposing/ (ft3)(time)concentration of A, moles A /f t 3

velocity constant, a function of temperature123



24 Process sys tems ana lys ; . and cont ro l

F -.

Co tI

Controller

Composition

/measurin g

element

Fig. 11.1 Control o f 8t i r red- tank chemicu l reac tor.

The reaction is to be carried out in a series of two stirred tanks. Thetallks arc muintained at different temperatures. The temperature in tank 2is to be g-reater than the temperature in tank 1, with the result that k2

the velocity constant in tank 2, is greater than that in tank 1, J :1. We shalllJeglcct any changes ill physical propcrties due to chemical reaction.

The purpose of the control system is to maintain Cz, the conecntration

of A leaving tank 2, at some desired value in spite of variation in inletconcentmtion Co This will be accomplished by adding a stream of pure A

to tank 1 through a control valve.Reac to r Trans fe r Func t ions 1,Ve bcgin the analysis by making It

material balance on A around t ank] ; thus

- = Co - ,+ - e1 - le e m- dc 1 F (F m \ TT +dt PAl

(11.1 )

where ' Ii = molar flOil rate of pure A through the valve, b moles/min

PA = density of pure A, Ib moles/ft 3

7 = holdup volume of tank, a constant, ftt is assumed that the volumetric flow of A through the valve m/ PA is much

less than the inlet flow rate F with the result that Eq. (11.1) can be written

v d; / + (F + k1V)e1 = FeD + m (11.2)

This last equation may be written in the form

dc] 1 1T l d + C1 = 1 + k1V/F

G + F l + klV/F) m(11.3)

vT1 = F + 1:1 V

At steady state, dCl/dl = 0, and Eq. (11.3) becomes

1 1Cl, - 1 + 7C; V/F co , + ( I - - + - k - 1 l ~ , - / F ~ )m,

where s refers to steady state.

Block d iagram of a chemica l - reac to r "on t ro s y s t e m I ~ i

Subtracting Eq. (11.4) from (11.3) and introducing the deviation

variables

C 1 = c, - Cl,

Co = Co - Co,

M = m - m,

(lL 5)

Taking the transform of Eq. (11.5) yields thc transfer function of the first

reactor:

Cl s) = 1/(1 + ~ l l / F )Co s) + l/[F(l + kiF/F)] M(s) (11.6)T]S -r 1 TIS + 1

A material balance on A around tank 2 gives

(11.7)

As with tank I, this la St equation can be written in terms of deviationvariables and arranged to give

dC 2 172 dt + C, = 1 + k 2 V/F C l

where

vT2 = F + k,V

C, = C2 - C2,

(11.8)

Taking the transform of Eq. (11.8) gives the transfer function for the second

reactor:

C, s) = 1/ 1 + ~ / F CI S)T,S (11.9)

To obtain some numerical results, we shall assume the following data

to apply to the system:

Molecular weight of A = 100 lb/ lb molePA = 0.8 Ib mole/ftCo, = 0.1 lb mole A/f t lF = 100 cfm

m, = 1.0 lb mole/min; = 7i min- I

k, = % min- l

V = 300 ft8



26 Proces •.• · t e m sanal .o; • a n d con trol

Substituting these constants into the parameters of the problem yields the

following values:

71 = :? min7, = 1 min

( 'I, = 0.0730 Ib Jl lOlc . 11ftC = O.024.J Ib lIlo]e A /ft'

111 /p = 1.25 cfm

Control Valve Assllllle that the control valve selected for the processhas til( following charactc>ristics: Thr flo\\ of A through the valve vurieslinearly frolll zero to 2 cfm as the valve-top pressure varies from 3 to 15 psig.The tilllc constant 7 of the valve is so smail compal ed with the other timeconstants in the system that its dynamics can l)( neglected.

Frolll tlH d a b givell, the valYI sensitivit.y is computed as

. - 0 J f / .J E-- ':::-:: = () C Il l , pSI

Since /I/,/P , 1.2;, e l l l l . the norlllal operating pressure 011 the valvl is

.. 1.2;, , _ _ .p, = . + . .... l l . ) - .,) = l(l . ; , pSI

The equatioll for the \ a]v( is thercfon:

71i = i1.2;; + J{, ( l - 10.;;1lR.,

]1: tel lllS of dl I iatioll nuiablcs. this call he written

M = l\,·P.·J

where

M = /1/ - 1.25PAP = p - 10 . 5

Taking the t-ansform of Eq. ( l1 . l2) g ives

M(s)P(s) = K,p \

as the valve traIlsfer function.

(11.]0)

(11.11)

(1112)

(11.1:\)

Measur ing Ele.ment For illustration, assume that the pen on the

controller moves full scale (0 to 4.00 in.) as the concentration of A varies

from 0.01 to 0.05 Ib mole A/ft3. Vi c shall assume that the concentrationmeasuring device is linear and has negligible lag. The sensitivity for the

measuring device is therefore

K 4.00 O' 1/m = 0.0.5 _ 0.01 = ] a m. pen t ra ye (lb mole/ft

3)

Block d iagram o f a chemica l - reac to r c o n t r o l s y s t e m 1 2 7

Since C2, is 0.0244 Ib mole/ft ' , the normal pen reading is

0.0244 - 0.01 (4 00) = 1 44 .0.05 _ 0 0 1 · . I l l

The equation for the measuring device is therefore

b = 1.44 + Km(cz - 0.0244) ( l1. l4)

where b is the pen reading in iIlches of pen displacement. In terms ofdeviation variables, Eq. (11.14) becomes

where

B = b - 1.44

The transfer fUllctioll fol tIll measuring device is therefore

JJ(s) r

C z(s) = A. n, ( lUG)

A measuring device which changes the units between input and outputsignals is called a transducer. In the present case, the concentration signal

is trll.:lsduced to iL pen readillg.Cont ro l l e r For convenience, we shall assume the controller to have

proportional action, in which casr th.c.xclaticmship. betll:£.eIl. controller output

pressure and error is

p = p, + IC(c;. - b = p, + K f

where CR = desired pen reading or set point, in.K, = controller sensitivity, psi/in.

f = error = CR - bIn terms of deviation variables, Eq. (11.17) becomes

P = K,f

(1l.J7)

(I 1.18)

The transform of this equation gives the transfer function of the controller

(11.19)

Assuming the set point and the pen reading to be the same when the

system is at steady state under normal conditions, we have for the reference

value of the set point

CR = b = 1.44 in.

The corresponding deviation variable for the set point is

eR = eR - CR,



F i r l l 2 BJuck d h l ~ r n l nfor c h c l n i c u - r c u c l o r c o n t r o l M ~ H t e n l a

Transpor t a t ion L a ~ :\ portioll of the liquid le[wing tank 2 is COll

tinuously withdntwH through a salllple lint' containing n conccntmtioll

nleasurin;:; eleJllent, M 11 rute of 0.1 cfrn. The measuring element must /wrcmotely locaLed frail the process, because rigid ambicnt cOllditions IlIust 1)('

maintaincd for accumtl - cOHcPlltmti oll mcaSUrel lll'nts. TIl<' sampk linf'l l l ~ It,ugth of :)0 ft., and th(' cros,;-scctionaJ areu of tll(, lin ' is C .OOI ft".

Thl' s a l l ~ p i l11Il ' call Iw represcnted by u transportatiol\ luI I\'itlparameter

volu III ' (;,;0) (0.00]1

fio\ , I"a\.( ().1

Th · transfcr fUnciloll for th(, sumpl ' lill ' is, therefore,

Block i a ~ r a m\V · ha \ E no\'.' completed the a n a l ~ s i sof eaeh COlll

ponent of the control system and have obtained u transfer fUllction for each.

These transfer functions can no\\' be combined so t ha t the overall syst.em isrepresented by the block diagram ill Fig, J 1.2, All equivalent diagram IS

shown in Fig, ] I,:) in which some of the blocks have been combined.

Fig . 11 .3 E q u i a i e n t

block d i a g r a m for ac h e n l i c a i ~ r e a c t o rcon

t ro l s y s t e m , C is nowin coneen. t ra t ion uni ta.)

Blo('/; d;o{ ram o f chemica l - reac to r con t ro l .•y s t e m 120

2\ umericai q u a n t i t i e ~for the pU[D.mcters in the t ransfer functions are

givcn in Fig. 11.3, It should be emphasizcd that the blocl: dlUgral11 IS

writt.cn for dc\'iatioll vuriaolcs, The true steady-state values, \\'llIch are

HOi. givcn by the diagram, must. be obtained from t h l ~anal.\'sis of tlJ(' problem.

The exulllp\l' llnalyzed in this chapter will LH' ust,d kiter III diSCUSSIOn

of ('ol1troi s\'stem design, The design problel l will be tu select l.l vaiu(': I:, \\'llIci: gin: , satisf:.lc\.of\' control or the '.Olllj>ositioll C, desp ik the

mtliel lOll" tnUlsport.ntlOl1 l n ~ ill\'olv 'r\ in J, :l'l.t,illf; inforJllatioli to til(' COll

troller. l ~ nddit.ioll, \\,(, slmll wallt to (;ollsidcr possibll' Us(' of other ll\odes

of eontrol for this syst.elli.

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Goodwin Modelagem