songchaikul metin
TRANSCRIPT
i Nonlinear C o n t r o l S y s t e m Des ign
u s i n g a Gain Schedu l i ng Techn ique
A T h e s i s P resen ted t o
The F a c u l t y o f t h e Co l lege o f Eng ineer ing and Techno logy
Ohio U n i v e r s i t y
I n P a r t i a l F u l f i l l m e n t
o f t h e R e q u i r e m e n t s f o r t h e Degree
M a s t e r o f Sc ience
b y
M e t i n - Songcha i ku l ) ,/
March, 1993
iii
Table of contents
Page
... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements ..... i v
........... ..................................................................................... Abstract ... v
Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Chapter 2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Basics of Lyapunov S t a b i l i t y Theory 3
... , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 . 2 Linear Contro l ler Design .... 13
.... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 . 3 Gain Scheduling .... 29
... . . . . . . . . . ..... Chapter 3 Missile Flight Control Problem .. 3 2
........... 3 .1 Mathematical Descr ip t ion of M i s s i l e Mode1 3 3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 . 2 Design Object ives 3 6
Chapter 4 Design Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 8
4.1 Equi l ibr ium Point Select ion .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
......... 4 . 2 L inear izat ion around each Equ i l ib r ium Point 41
4.3 Linear Contro l ler Designs .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.4 Scheduling the Set o f Linear Cont ro l le rs .. . . . . . . . . . . . . . . 60
Chapter 5 Summary and Conclusions ................................ 77
References ........... ....... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Appendix A Computer programs .................................................. A1
Acknow ledaement
1 w ishes t o express s incere appreciat ion t o my advisor Dr.
Douglas Lawrence, f o r h i s support and guidance throughout the
course of t h i s study, w i t h o u t wh ich the complet ion o f t h i s thes is
would not have been possible.
Apprec ia t ion i s extended to the member o f the examinat ion
commi t tee f o r sac r i f i c i ng the i r f ree t ime, and g iv ing t h e i r valuable
c r i t i c i s m s : Dr. Aysin Yeltekin , Dr. Dennis I r w i n and Dr. Bra in Fabien.
Special appreciat ion i s extended to my fami l y : Pramot, Arunee
and Nutharin Songchaikul, f o r t he i r encouragement and support which
I have been able t o depend on.
Also, thanks t o Sr ikasem's f a m i l y who a lways help and care
throughout the years of my academic i n Ohio Univers i ty .
F ina l ly , deepest thanks to my special f r i end Nuthaya f o r her
help i n typing.
Abst rac t
Real i s t i c models o f engineering sys tems of ten are nonl inear .
As a consequence, t he dynamical behav io r o f a s y s t e m t o be
cont ro l led changes w i t h the operat ing region. In recent years, one
design methodology t o cont ro l t h i s e f f e c t , ca l l ed Gain Scheduling,
has proven t o be successful . The basic idea o f gain scheduling i s t o
break the cont ro l design process i n to t w o steps. The f i r s t s tep i s t o
l inear ize the model about one or more operat ing po in ts . Then l inear
design methods are applied t o the l inear ized model a t each operat ing
p o i n t i n order t o ob ta in the s t a b i l i z e d sys tem w i t h i n the design
object ives. The f i na l step, the actual gain scheduling, i s obtained by
schedul ing o r i n te rpo la t i ng the gains of the loca l operat ing po in ts
designs i n o rder to handle the non l inear aspec ts o f t he design
problem.
In t h i s thes is , a nonl inear cont ro l le r w i l l be designed using a
gain schedul ing technique f o r a hypo the t i ca l m i s s i l e model. The
m i s s i l e considered here i s the same as discussed i n recent papers on
ga in schedul ing. Here, a nonl inear au top i l o t i s designed us ing
c lass ica l servomechanism theory and s t a t e f e e d b a c k k t a t e observer
based techniques.
C h a ~ t e r 1
Introduction
I t i s a we l l - known f a c t t h a t a r e a l i s t i c sys tem model f o r
eng ineer ing app l i ca t i ons i s non l inear . As a consequence, t he
dynamical behavior of a system t o be con t ro l l ed changes w i t h the
operat ing region. One design method t o handle t h i s e f f e c t i s ca l led "
Gain Scheduling " . Recently, several papers have descr ibed studies
of gain schedul ing i n cont ro l sys tem design both i n l inear sys tem
aspects and nonl inear sys tem aspects, and of those studies, many
have focused on the appl icat ion of gain scheduling i n f l i g h t cont ro l
problems as publ ished i n [5] , [9], [ I 01, [ I 11, [ I 21, and [ 131. These
s tud ies have demonstrated tha t ga in scheduling can be a successful
design methodology f o r many appl icat ions o f engineering. The design
process f o r gain scheduling involves 2 basic steps.
1 . The sys tem t o be c o n t r o l l e d i s l i n e a r i z e d a t severa l
e q u i l i b r i u m po in ts (The equi 1 i b r i u m po in ts should be se lec ted t o
cover the desired operat ing range). Then, f o r each 1 inear ized plant, a
l inear t ime- inva r ian t design technique i s appl ied t o c rea te a loca l
c o n t r o l l e r w h i c h s a t i s f i e s the design o b j e c t i v e s f o r t h e sys tem
when operat ing su f f i c i en t l y close t o the given equ i l i b r i um po in t .
2 . The ac tua l gain schedul ing i s obta ined by "schedu l ing" or
2 in terpo la t ing the gains of the cont ro l le rs i n step 1 between the
equil ibrium points. In th is way, a nonlinear control ler f o r a nonlinear
system w i l l be obtained.
Despite i t s populari ty, the gain scheduling method s t i l l has
some rest r ic t ions. For example, the operating condition i s normally
specif ied by the value of one or more exogenous variables and the
scheduled gain depends on the instantaneous values o f these
variables. The studies described in [ I 31 show that current gain
schedul ing i s necessar i ly l i m i t e d t o s low va r i a t i ons i n the
scheduling variables. Previously, th i s l im i t a t i on was jus t i f i ed only
through implementation and simulation, but [ l 11, [13], and a recent
paper 151 shows a mathematical formula to j us t i f y th is res t r ic t ion.
In t h i s thesis, the author w i l l use the gain scheduling
technique to design a nonlinear control ler fo r a hypothetical m iss i le
model, The miss i le considered here i s the same as discussed in [9-
101, but, instead of using an H- inf in i ty contro l ler as done in 191, the
author w i l l employ type- 1 servo system as the control ler.
Chapter 2 explains the theory that w i l l be used throughout th is
thesis. Chapter 3 describes the miss i le model. The design process
and s imulat ion resu l ts w i l l be discussed in Chapter 4. Chapter 5
w i l l set fo r th the conclusions of th is design work.
C h a ~ t e r 2
Theory
In th is chapter the theory that related to th is thesis w i l l be
reviewed. Fi rst , the basics of the Lyapunov s tab i l i t y theory w i l l be
given. The detai l i n th is section w i l l include the theory of nonlinear
system, equi l ibr ium point, s tab i l i t y i n the sense of Lyapunov and
l inearizat ion. Next, the theory of l inear control design, type- 1 servo
system w i l l be described. Last, the technique fo r gain scheduling
w i 1 1 be considered.
2.1 Basics of Lva~unov Stabi l i tv Theory
For a given contro l system, the f i r s t and most important
aspect t o be determined i s i t s s tab i l i t y . A system i s described as
stable i f when we s t a r t the system somewhere around a desired
operating point, the system w i l l operate around t h i s point fo r all
fu tu re t ime. I f the system i s l inear and t ime- invar iant , many
c r i t e r i a are available fo r determining s tab i l i t y such as the Nyquist
S tab i l i t y C r i t e r i a and Routh's S tab i l i t y Cr i ter ia. I f the system i s
nonlinear, or t ime-varying, one cannot apply these c r i te r ia , The
most useful theory fo r determining the s t a b i l i t y of a nonlinear
4 and/or t ime-varying system i s Lyapunov S tab i l i t y Theory.
Lyapunov's work, The General Problem o f Plot ion Stabil ity.,
together w i t h t w o methods fo r s tab i l i t y analysis (the l inearizat ion
method and the di rect method) was published i n 1892 by the Russian
mathematician Alex Mi khai louich Lyapunov. However, th i s theory did
not rece ive much a t t en t i on u n t i l the ear ly 1960 's when the
publ icat ion of the work of Lure's and a book by La Sal le and
Lefschetz brought Lyapunov's work t o the fore f ront o f the control
engineering community. Today, Lyapunov's 1 inearizat ion method has
come t o represent the theoret ical j u s t i f icat ior l of l inear control,
wh i le Lyapunov's d i rect method has become the most important tool
for nonlinear system analysis and design [ 141.
The Theory of Lyapunov also plays an important ro le i n the
design o f a contro l ler fo r a nonlinear system. In order t o provide
foundational informat ion regarding Lyapunov Theory, the fo l low ing
terms are defined and explained:
- Nonlinear System
A nonlinear dynamic system can be represented by a se t of
nonlinear d i f ferent ia l equation of the form
x = f(x,u,t)
where
X - 1 i. ,n x 1 state vector
U = , m 1 input or control vector
In the case where the system does not e x p l i c i t l y contain
control input variables, the system i s described by the fo l lowing
equation.
x = f(x,t)
The number of states n i s cal led the order of the system. A
so lu t ion x ( t ) of Equation (2.1-2) i s re fe r red t o as the system
, p x 1 output or measurement vector Y =
Y1 -
Y2
... - YP -
t ra jec tory in the state space for t 2 0
- Equilibrium Point
Def in i t ion: A s t a t e x u i s an equi l ibr ium s ta te (or
~ q u i l i b r i u m point) of the system i f once x ( t l is equal to x u , i t
remains equal to xu for a i l future time.
Mathemat ical ly , t h i s de f in i t i on means that an equi l ib r ium
point of the system i s a t r i p l e (xo,uo,yo of a constant state, input
and output such that
f(~g,uo,t) " 0
Many s t a b i l i t y problems are na tu ra l l y f o rmu la ted w i t h
respected to equi l ibr ium points.
- Stabi l i ty in the sense of Lyapunov
Definition: The Equil ibr ium s ta te xo = 0 i s said to be
stable i f , for any R>U, there exist r>U,, such that i f I/ x(UI //< r,
then /' x f t l I/< R for a l l t>U. Otherwise, the equil ibrium point i s
8
system t ra jec tor ies , which s ta r t closely t o the equi l i br ium point,
actual ly converge to the equi l ibr ium point as t ime goes to in f in i t y .
Also, i t i s necessary t o know how fast the system tra jector ies w i l l
converge t o the equi l ibr ium point. The fo l lowing def in i t ions address
these concepts.
Definition: An equil ibr ium point xu = U i s as,ymptotical/y
stable i f i t i s stable,, and i f in addition there ~ . x i s t s some r > 0
such that If x(UI If < r implies that x(tl 7' U as t 7' a,.
Definition: .An equil ibrium point xu = 0 i s e.xponential1,~
stable i f there exist two s t r i c t l y positive numbers a and A such
that
V t > 0 , 11 x(t) 11 I a Ilx(0) lle-At
for a1 1 x(Ul in some ha l l Br around the origin.
These s tab i l i t y def in i t ions are formulated to characterize the
loca l behavior of systems when the system operates near an
equi l ibr ium point. Local properties do not describe the behavior of
the system when the i n i t i a l s tate i s some distance away f rom the
equil ibr ium point. The fo l lowing def in i t ion describes a concept o f
s tab i l i t y i n th is case.
Definition: I f asymptotic (or e.xponential) s tab i l i t y holds
f o r any i n i t i a l state,, the equi l ibr ium point i s said to be
asymptotical ly (or e.xponentially) stable in the large. I t is also
called globally asymptotically (or e.xponentiall,~) stable.
- Linearization
The las t s tab i l i t y theorem that w i l l be given i s the important
theorem fo r th is gain scheduling technique. This theorem give the
idea of the s tab i l i t y f o r nonlinear systems w i t h s l ow ly varying
inputs. I t i s used to guarantee nonlocal performance of the nonlinear
system. The t3eorem and the detai ls of the proof are discussed in
[S]. Here, the theorem in [51 w i l l be given again as;
For the system described as
~ ( t ) = f(x(t), u(t)) , x(b) =xo , t 2 to
assume
( H I ) f: R* x ~ m - + i s t w i c e continuously d i f ferent iable
(H2) there i s bounded, open set r c R~ and a continuously
10
d i f f e ren t i ab le func t ion x:;- R" such tha t f o r each constant
input value u E r, ~(x(u), U) =O,
(H3) there i s a A > Osuch tha t f o r each u E r, the eigenvalues
of (~V~X)(X(U),U) have real pa r t s no greater than -A .
Theorem: Suppose the sys tem ( 1 . 1 ) s a t i s f i e s ( t i 1 ), (H2), and (H3) .
Then there i s a p* > o such tha t given any p E (0, p*] and T > 0, there
e x i s t 6,(p), B2(p, T)> 0 f o r wh ich the f o l l o w i n g proper ty holds. I f a
cont inuously d i f f e ren t i ab le input u ( t ) s a t i s f i e s 11 xo - x(u(to)) 11 <a1 f o r
~ ( t ) E r, t 2 to and
then the corresponding so lut ion o f the system given above s a t i s f i e s
II x(t> - x(u(t)) II < P , t 2 to.
Now the l i nea r i za t i on method w i 1 1 be discussed. Lyapunov's
l i n e a r i z a t i o n method i s concerned w i t h the l oca l s t a b i l i t y o f a
11
nonlinear system. The idea of th is approach comes f rom the we l l -
known fac t that a nonlinear system, when operated in a suf f ic ient ly
small neighborhood of an equil ibrium point, may behave much l ike a
l inear system. This method involves l inearizing the given system in
the neighborhood of an equi l ib r ium point and determin ing the
behavior of the nonlinear system's t r a j ec to r i es by studying t h i s
1 inearized system using 1 inear system techniques.
Mathematically, the idea i s to expand the nonlinear funct ions
in to a Taylor series around the equil ibrium point and re ta in only the
l inear term, neglecting the higher-order terms provided they are
small compared to the l inear term. Consider the nonlinear dynamics
system described below;
Recal l an equ i l i b r ium po in t of the system i s a t r i p l e
( x ~ ~ u ~ ~ Y o 1 of constant state, input, and output such that
f(xo,uo,t) = 0 for all t 2 to
Define Jacobian matr ices
By Taylor Series Expansion of f and g, the funct ions f and g
can be expanded around the equil ibrium state and input (xo,uo) as
f (x ,~ , t ) = f(x0,u0,t) + A(x0,uO,t) (x - xo)+ B(xO,uO,t) (u - uo) + hot's (2.1-6)
g(x,u,t) = g(q,uo,t) + C(xo,uO,t) (X - xo)+ D(%,u0,t) (U - uo) + hot's (2.1-7)
where hot's means higher-order terms
Assuming that the higher-order terms are smal l enough t o be
neglected one can approximate these functions as
f(x,u,t) = A(xg,uo,t) (x - xo)+ B(xo,~o,t) (u - uo)
then def ine deviat ion var iables
x6 = x - xo u6 = u - uo Ya = Y - Yo,
Since x( t ) i s a constant vec tor
Using the l inear approximat ion of f and g around equ i l i b r i um
s t a t e and inpu t one can then descr ibe a l i n e a r s y s t e m t h a t
approximates the behavior of the nonl inear system (2.1-3) near the
equ i l ib r ium po in t as
y6(t) = C(xo,uO,t) xa(t) + D(xo,uo,t>uij(t) (2.1-8)
Based on the l inear ized sys tem (2.1-8>, one can apply l inear
design techniques t o guarantee the s t a b i l i t y of tP~is sys tem.
2.2 Linear Controller Desian
The design o f the cont ro l le r , wh ich regulates the given m i s s i l e
i n t h i s thes is problem, employs a type- l servo sys tem based on the
pole placement approach and s t a t e observers . Theorems re la ted t o
t h i s des ign inc lude po le p lacement des ign and t h e des ign o f
observers.
- P o l e P l a c e m e n t
Consider SISO system
x = A x + B u
where x = s ta te vector (n x 1 vec tor )
Y = output s ignal (scalar)
u = contro l signal (sca la r )
A = n x n constant m a t r i x
B = n x 1 constant m a t r i x
C = 1 x n constant m a t r i x
The cont ro l s ignal w i l l be
u s - K x .
The 1 x n vector K i s ca l led the s ta te feedback gain vec tor .
Subs t i t u t i ng Equation (2.2-3) i n to Equation (2.2-1 ), w e obta in
The so lu t ion o f Equation (2.2-4) i s given by
The s tab i l i t y of th is system i s determined by the eigenvalues
of the matr ix (A-BK). By choosing a proper K, one can construct the
mat r i x (A-BK) such that i t i s asymptot ical ly stable. This problem
of placing the closed-loop poles a t the desired location i s called the
pole placement problem. The technique used to solve t h i s problem
cons t ruc t s an asymp to t i ca l l y s tab le c losed- loop sys tem by
speci fy ing the desired locat ions f o r the closed-.loop poles. By
assuming the contro l law t o be u = -Kx, one can determine the
feedback gain vector K such that the closed-loop system as shown i n
FIGURE 2.2- 1 w ill have a desired characterist ic equation.
FIGURE 2.2-1 Block diagram w i t h u = -Kx
16 When using t h i s technique, one must meet the necessary and
s u f f i c i e n t c o n d i t i o n t h a t t h e s y s t e m s t a t e i s c o m p l e t e l y
contro l lab le. Algebra ica l ly , t h i s i s equivalent t o nonsingular i ty of
t h e n x n c o n t r o l l a b i l i t y m a t r i x C ( A P B ) = [ B A B . - - A " - ~ B ] - Note, a l l
s t a t e var iab les are assumed to be avai lab le and measurable f o r
feedback. De ta i l ed i n f o r m a t i o n concern ing t h i s technique i s
explained i n [61. An approach f o r the determinat ion o f the s t a t e
feedback ga in m a t r i x K presented n e x t w a s developed by
J.E.Ackermann. This approach i s known as Ackermann's formula.
Ackermann's formula
The s ta te equation f o r th i s system i s given by
x = A x + B u .
Assume that the system i s completely s ta te contro l lab le .
Ackermann's formula i s given as
K = [0 0 ... 0 11 [B f AB f A*B f . . . i B ] - ' a ( ~ ) (2.2-6)
and
a (s) = ( s - ~ i ) ( s - ~ 2 ) . . . ( s - c I ~ )
= sn+ a 1 s n - l + . . . +a,.ls + an
where p 1 .p2, ... .pn = the desired closed-loop poles.
- Design of State Observers
I n the pole placement approach, one assumes t h a t a l l s t a t e
var iables are avai lab le f o r feedback. For a f i r -s t o r second order
sys tem, f u l l s t a t e feedback i s not an unreasonable expecta t ion .
However, f o r most high order systems, a l l s t a t e var iab les are not
avai lable f o r feedback; t o implement pole-placement design i n these
sys tems, i t i s necessary t o e s t i m a t e these unava i lab le s t a t e
var iab les f r o m the measurements tha t can be made on the system.
The method used to es t ima te the unavai lable s t a t e s i s commonly
ca l l ed a s t a t e observer A s t a t e observer e s t i m a t e s t h e s t a t e
va r iab les based on the measurements o f the output and con t ro l
var iables w i t h a r b i t r a r i l y speci f ied er ror dynamics, bu t can only do
so on the condi t ion that the system i s completely observable. Also,
[6] has provided proof of t h i s fac t
A fu l l -o rder s t a t e observer i s one tha t es t ima tes a l l s ta tes
va r iab les o f the sys tem regardless o f whe the r o r n o t they are
d i r e c t l y measurable. A minimum-order s t a t e observer i s def ined as
an observer t h a t e s t i m a t e s only the m i n i m u m number of s t a t e
var iables.
Th is thesis w i l l only consider the f u l l order s t a t e observer t
Use ? t o designate the observed s ta te vector f o r
Assume that state x i s to be approximated by the state ? of
the dynamic model as show in FIGURE 2.2-2
From FI GURE (2.2-2),
+ X
; = A ? + B U + L ( ~ - G ) (2.2-9)
which represents the state observer w i t h y and u as input and - x as output.
1/s
A
b Y + 2
X -
U
-
C B => q +
-
: l/s ,
C A ,
2
B u u -
3
+ +
X, > + - - 4 +
L 4
FIGURE 2.2-2 Block diagram o f system w i t h ful l-order s ta te observer
19 To obta in the observer e r ro r equation, subt rac t Equation (2 .2 -
9) f r o m Equation (2 .2 -7)
x - 2 = Ax+Bu - Ax?-Bu-L(Cx-Cs = (A-LC) (x - x) (2.2- 10)
Define the d i f fe rence between x and ii as the e r ro r vec to r or
e = X-ii and Equation (2.2- 10) becomes
e = (A-LC) e . (2.2- 1 1)
Th is i l l u s t r a t e s t h a t t he eigenvalues of t h e m a t r i x A-LC
determine the dynamic behavior of the e r ro r vec tor .
I f the eigenvalues of m a t r i x A-LC are chosen i n such a way
t h a t t he e r r o r s y s t e m 2 2 - I i s exponen t ia l l y s t a b l e w i t h
acceptable r a t e o f decay, then any e r r o r vec to r w i l l tend t o zero
w i t h adequate speed.
Since the problem o f designing a fu l l -o rder observer requi res
t h a t the observer gain m a t r i x L be such t h a t A-LC has desi red
eigenvalues, t h i s problem resembles the pole p lacement problem.
Thus, using the Pr inc ip le of Dual i ty, l e t
z = A*Z + C*V (2.2- 12)
and assume the contro l s ignal y t o be
v = -L*z
20
L e t I J ~ . I J ~ . .Pn be the des i red eigenvalues o f t he s t a t e
observer m a t r i x , and assume the dual sys tem i s complete ly s t a t e
con t ro l lable. Fur thermore, tak ing the same p i ' s as the desi red
eigenvalues o f the s ta te feedback gain mat r ix , one can w r i t e :
Is1 - (A* - C8L*)I = (s - pl)(s- pZ) ...( s - p,)
* * Since (A*-c L 1 has the same eigenvalues as (A-LC), one
can determine the observer gain L by f i r s t de termin ing L* i n the
pole placement approach
A c k e r m a n n ' s F o r m u l a
Consider Equation (2.2- 12) and Equation (2.2- 13)
The Ackermann Formula f o r pole placement can be w r i t t e n as
L* = [O 0 ... 0 11 [C* A'C* I . . . (A*)"-~c*]" a(A)
Taking transposes, one w i l l obta in Ackermann's Formula f o r
the s t a t e observer gain as
and
where P ~ . P ~ . ... .Pn = the desired eigenvalues of observer e r ro r
dynamics.
I n the pole placement design process, w e assumed t h a t the
actual s t a t e x ( t ) was avai lab le f o r feedback; however, the actual
s t a t e x ( t ) may not be measurable. Therefore, w e need t o design an
observer and use the observed s t a t e G(t ) f o r feedback. Thus, the
design step involves a two-stages, f i r s t determine the feedback gain
m a t r i x K t o y i e l d the desi red closed-loop c h a r a c t e r i s t i c equat ion
assuming s t a t e feedback and second determine the observer gain
m a t r i x L t o y i e l d the desired observer cha rac te r i s t i c equat ion. The
e f f e c t o f using g(t) instead of the actual s t a t e x(t) on the closed-
loop contro l system i s discussed i n [ 6 ] . Thus, only the conclusion o f
t h i s e f f e c t w i l l be mentioned.
22
Since the character is t ic equation that described the dynamics
of the observed-state feedback control system i s given as
I sI-A+BK I 1 sI-A+LC I = 0
Obviously, i t shows that the closed-loop poles of the combined
observer-state feedback system comprise the poles due to the pole
placement design together w i t h the poles due to the observer design.
This means that the pole placement design and the observer design
can be done separately and combined together to form the observer-
state feedback control system.
- Servo system
FIGURE 2.2-3 Block diagram of Type-1 servp system
In the discussion of pole placement and the design of a state
observer, only a closed-loop system wh ich has no input was
considered. The purpose of such a design Is t o re tu rn a l l s ta te
variables f rom the i r i n i t i a l values t o values of zero when the states
23 have been perturbed. Such a system i s ca l led a regulator . However,
many con t ro l sys tems, inc lud ing the con t ro l sys tem discussed i n
t h i s thes is , requi re the system output t o t rack an external reference
inpu t . I n such cases, t h i s necess i ta tes m o d i f y i n g t h e design
equation of the pole placement and the s t a t e observer. These types
o f s y s t e m s are known as servo sys tems and are i l l u s t r a t e d i n
FI GURE 2.2-3.
Servo system design involves cons t ruc t ing compensators and
feedback l a w s tha t y i e l d a s table (BIB0 and /o r asympto t ic ) closed-
loop system able t o t rack a speci f ied c lass of re ference signals. In
FIGURE 2.2-3, the in tegra tor , together w i t h s t a t e feedback scheme,
i s used t o s t a b i l i z e the sys tem and a s y m p t o t i c a l l y t r a c k s tep
reference inputs w i t h zero steady-state e r ro r .
Since the g iven p lan t ( m i s s i l e problem) does no t invo lve an
in tegrator , t h i s thes is w i l l consider only the design theory o f a type
1 servo sys tem where the p lan t has no i n teg ra to r . As ment ioned
e a r l i e r , i n m o s t cases, no t a l l s t a t e va r iab les can be d i r e c t l y
measured, theref ore t h i s considerat ion of servo sys tem design w i l l
a lso inc lude a d iscuss ion o f the s t a t e observer. A type- 1 servo
sys tem where the p lant has no in tegra tor i s shown i n FIGURE 2.2-4.
FIGURE 2.2-4 Block dlagram o f type- 1 servo system w i t h s t a t e observer
F r o m this f igure, we have
x(t) = Ax(t) + Bu(t)
The c o n t r o l l aw i s descr ibed as
where u( t ) = contro l s ignal (sca la r )
y ( t ) = p lant output signal (scalar)
r(t) = reference input signal
c ( t ) = output of in tegra tor ( s ta te var iab le o f t he system)
I t w i l l be assumed tha t :
1 . The p lant i s contro l lab le and observable
2. The plant has no pole a t s=O
3. The p lant has no zero a t s=O
Assume t h a t the re fe rence input (r(t) = s t e p func t i on ) i s
appl ied a t t = 0. As a consequence of the e f f e c t of the add i t ion of
the observer on a closed-loop system, the pole placement design and
the observer design can be design separately and combined together
t o fo rm the observer-state feedback system. Thus, f r o m FIGURE 2.2-
4, we w i l l use the pole placement approach t o design gain K and K i
t o s tab i l i ze the system. Then the observer design f o r gain L w i l l be
app 1 i ed.
Assuming the actual s t a t e s x ( t ) are ava i lab le f o r feedback,
one can fo rm the dynamic equation of type- 1 servo sys tem as
An asympto t ica l l y s table system w i l l be designed such tha t f o r
r(t) = r, t 2 0 as t -> m, x(t), f(t), and u(t) approach cons tan t
values, denoted xss, fss, and us, respect ive ly . Further, f(t) -> 0 and
A t steady state, one has
Since r(t) i s a s tep input , thus r(t) = r (cons tan t ) . By
subtract ing Equation (2.2-22) f r om Equation (2.2-21) and def in ing
x(t) - xss = xe(t>
E(t> - ESS = Ee(t)
w e have
where ue(t> = -Q(t) + KiEe(t>
Define a new (n+ 1 ) th-order error vector e( t ) by
then Equation (2.2-23) becomes
$(t) = &t) + Bue(t)
where
The control signal u,(t) becomes
where K = [ K I -Ki]
The idea of f i r s t design stage i n type-1 servo system i s to
design a stable (n+ l ) th-order regular system that w i l l b r ing the
new error c(t) to zero. And the s ta te error equation of t h i s system
can be found by put t ing Equation (2.2-25) into Equation (2.2-24)
$(t) = (i - BK) a t )
Therefore, i f the desired eigenvalues o f ma t r i x X - B K are
28
specif ied as pl , p2 , ..., pn + I in order to have the zero steady state
er ror , the s ta te feedback gain m a t r i x K and the in tegra l gain
constant K i can be determined by the pole placement approach.
Now consider the s ta te observer t o design the gain L. To
obtain the observer er ror equation, subtract ing Equation (2 .2-20)
from Equation (2.2- 16), we have
x - i? = Ax+Bu - Ax--Bu-L(Cx-C%) = (A-LC) (x - 2) (2.2-26)
Define the difference between x and ii as the error vector e or
e = (A-LC)e (2.2-27)
From Equation (2.2-27), we see that the dynamic behavior of
the er ror vector i s determined 5y the eigenvalues of mat r i x A-LC.
I f the eigenvalues o f mat r ix A-LC are chosen in such a way that the
dynamic behavior of the error vector i s asymptot ical ly stable and i s
adequately fast , then any error vector w i l l tend t o zero w i t h an
adequate speed. Since we assumed that th is system i s completely
observable, the gain L of state observer approach can be chosen by
- speci f icat ion of the desired eigenvalues F1 , F2 , .... pn of the matr ix
A-LC. At th is point, one can f ind the gain K, Ki and L which makes
th i s type-1 servo system have z e r o steady state error.
Next the closed-loop s ta te equation o f t h i s type-1 servo
29
system in FIGURE 2.2-4 w i l l be developed for future reference i n the
design steps.
Consider Equation (2.2- 1 6 ) - Equation (2.2-20);
Put Equati on(2.2- 1 9) into Equation (2.2- 16) t o obtain
~ ( t ) = Ax(t) + B ( - E ( t ) + Kic(t) )
Put Equation (2.2- 17) into Equation (2.2- 18) t o obtain
Putting Equation (2.2- 17), (2.2- 19)) and (2.2-28) into Equation (2.2-
20) yields f inal ly,
Thus, the combination of Equation (2.2-28), (2.2-29)) and (2.2-
30) gives the closed-loop system as
A -BK BKi LC A-BK-LC 13Ki ] - C 0 0
? - x(t)
%t)
- SO) , + [H ] d t )
2-3 Gain Schedulinq
Rea l i s t i c models of engineering systems are t y p i c a l l y
nonlinear. In studying control system design, an important e f fec t of
th i s kind o f system emerges: the dynamic behavior of a system t o be
cont ro l led changes w i t h the operat ing region. An approach to
handling th is e f fec t i s called "Gain Scheduling".
As f i r s t noted, current gain scheduling pract ice i s l im i t ed to
s low var ia t ion of exogenous scheduling var iables. Thus i n the
considered model, the operating condition had to be arranged so that
i t would be spec i f ied by the value of one o r more exogenous
var iables, then the gains w i l l be scheduled according t o the
instantaneous values of the exogenous variable. The model of the
system w i l l resemble as shown in FIGURE 2.3-1.
w( t),exogenous (scheduling) variables
FIGURE 2.3- 1 System for applying gain scheduling
r( t ) r
The application of gain scheduling t o the contro l ler design i s
divided into 4 steps:
1 , select a set of equil ibrium points to cover desired operating
range
2. l inearize the plant around each equil ibrium point
3. design a l inear control ler for each l inearizat ion
4. schedule the set of l inear control lers
To determine an equil ibrium point (step I ) , set f(x,u,t)=O and
Nonlinear u ( t ) Nonlinear Control ler plant
The l inearizat ion about an equi l ibr ium point i n step 2 involves
expanding f and g i n a Taylor series at the equi l ibr ium point and
neglecting the higher order terms.
32 The type 1 servo system w i l l be designed fo r each l inear
contro l ler in step 3 as previously described.
Scheduling or interpolat ing the set of l inear control lers in the
las t step has the basic idea to interpolate the l inear cont ro l le r a t
intermediate operating conditions. That is, a scheme i s devised fo r
changing the gains i n the con t ro l le rs based on the operat ing
condit ion of the system. The detai ls of scheduling techniques used
i n t h i s thesis w i l l be discussed in chapter 4.4.
C h a ~ t e r 3
Missi le Fl iaht Control Problem
Consider m i s s i l e -a i r f rame con t ro l problem i l l u s t r a t e d i n
FI GURE 3- 1 .
F in defect ion Ve loc i t y vec to r
attack
FIGURE 3- 1 M iss i l e Fl ight contro l problem
When the vehic le i s f l y i n g w i t h an angle of a t tack ( a ) , l i f t i s
developed. T h i s l i f t may be represented as a c t i n g a t a cen t ra l
loca t ion (center o f pressure). The vehic le w i l l be s t a t i c a l l y s tab le
o r uns tab le ( w i t h o u t co r rec t i ve t a i 1 de fec t ions) depending on the
loca t ioh of the center of pressure re la t i ve t o the center of mass [21.
The problem focused on i n t h i s t hes i s i s t h a t o f c o n t r o l l i n g t h i s
vehic le t o t rack commanded normal accelerat ion by generat ing a t a i l
f i n defect ion angle. The au top i lo t wh ich needs t o be designed w i l l
34 accept a normal accelerat ion command f rom some ou te r guidance
system. The f i r s t par t of t h i s chapter w i l l introduce the descr ip t ion
of a hypothet ica l m i s s i l e model tha t w i l l be used i n the f o l l o w i n g
design discussion. Some of the m i s s i l e ' s var iab le are measured by
gyros and accelerometers. The las t par t of t h i s chapter w i l l show
the requirements of the autopi lo t design. The process o f t h i s design
w i l l be discussed i n the next chapter.
3.1 Mathematical D e s c r i ~ t i o n of Missile Model
The m i s s i l e f l i g h t contro l problem used i n the thes is design i s
shown i n FIGURE 3.1-1.
FIGURE 3.1 - 1 The block diagram of m i s s i l e model
M ( t )
Air f rame Dynamics
h(t> = &M(t)G[a(t>,6(t),M(t)lcos(a(t>>+q(t)
6 b actuator
6 b a i r f r ame
-b accel erome t o r -b p i t c h r a t e
Actuator Dynamics
Output
Var iables
a(t) = angle of attack, range -20'1 a 5 20'
M(t) = Mach number, range 2 M 4
q(t) = pitch rate.
GC(t) = commanded tail fin deflection angle
6(t) = actual tail fin deflection angle.
qc(t) = commanded normal acceleration.
qz(t) = actual normal acceleration.
note : The angles a re measured i n degrees.
The acce le ra t i on i s measured i n gees
Simulation Variable
36 F o r simulation purposes, a state equation for Mach number i s
defined as
Aerodynamic Coefficients
Constants
K, = (0.7) PoS/mv,
Ax = (0.7) PoSCa/m
where Po = stat ic pressure at 20,000 f t = 973.3 lbs / f t2
s = surface area = 0.44 f t 2
m = mass = 13.98 slugs
37 vs = speed of sound at 20,000 f t = 1036.4 f t /sec
d = diameter =0.75 f t
IY = pitch moment of inert ia = 182.5 slug-ft2
Ca = drag coefficient = -1.5
=0.7
3.2 Desian Obiectives
The requirements of the design are as fol lows:
( 1 ) Obtain robust s tab i l i t y over the operating range. The
operating range i s specif ied by the angle of at tack a and Mach
number M and consists of those points (a,M) such that -200sas200
and 2 s M s ( alt itude = 20,000 f t . ) .
38 ( 2 ) Track step normal accelerat ion commands w i t h t i m e
constants of approximately 0.25 second or less.
( 3 ) Maintain greater than 30 dB attenuation at 300 rad/sec for
the open-loop l inearized t ransfer funct ion w i t h the loop broken at
the actuator input. This requirement seeks t o avoid exc i t ing the
unmodelled structural dynamics.
Chapter 4
Gain Schedulina Desian
I n t h i s chapter t he non l inear sys tem design us ing a ga in
schedul ing technique w i l l be discussed. An a u t o p i l o t w i l l be
designed i n o rder t o c o n t r o l the m i s s i l e p rob lem discussed i n
chapter 3. By using the gain schedul ing technique the a u t o p i l o t
design i s div ided i n t o 4 steps as
1 , equ i l ib r ium point se lect ion
2. l inear iza t ion around each equ i l ib r ium po in t
3, l i near contro l l e r designs
4. schedclling the set o f l inear cont ro l le rs .
Each o f these des'ign s teps w i l l be d iscussed n e x t A f t e r
obta in ing the au top i lo t f rom the design method, the local s t a b i l i t y
of the m i s s i l e w i l l be checked. And a t the end o f t h i s chapter the
s i m u l a t i o n by SIMULAB w i l l be appl ied t o the m i s s i l e i n order t o
check the m i s s i 1 e 's performance.
4.1 Eauilibrium Point Selection
From the mathemat ica l descr ip t ion i n Chapter 3, the m i s s i l e
model i s w r i t t e n as a set o f nonlinear d i f f e r e n t i a l equations as
where
x = a 4 X 1 s tate vector so that
Thus,
= K,M(t)G[a(t),G(t),M(t)lcos (a(t)) + q(t) =, f l(x(t),u(t),w(t))
To determine the equi l ib r ium po in ts of t h i s system, by
def ini t ion,we set f(x,u,y) = 0. Thus the set of equi l ibr ium poirlts
i s calculated and shown as fo l lows:
6(t) = - Isgn(a(t))laJa(t)13 + bmla(t)12 + G,($M(~) -7)la(t)ll , and dm
f~(x(t),u(t),w(t)) = 0, imp ly ing tha t
a n d q(t) = KaM(t)C,[a(t),6(t),M(t)lcos (a(t)) = 0:
q(a,M) = -K,MCJa,G(a,M),Mlcos ( a )
and the constant operating point o f the output func t ion i s ca lcu lated
as
g(x(t),u(t> ,w(t>) = y
and rl,(t) = KzM2(t)G[a(t),G(t),M(t)l:
rlz(a,M) = KzM2G[a,G(a,M),Ml
4.2 Linearization around each eauilibrium point.
The nonlinear plant i s given as
where
To l inearize the nonlinear system, we use the Taylor series
expansion of f and g around an equi l ibr ium s ta te and neglecting the
higher-order term of order greater than 2, which are assumed t o be
small, we have
where
Z(t) = x(t) - x(a,M)
G(t) = 1Z(t) - rlz(a,M)
The Jacoblan mat r ices A(a,M), B(a,M), C(a,M)are calculated as
I-
44
The coe f f i c i en t ma t r i ces of the 1 inear i zed p lan t are ca lcu la ted
v i a PROGRAMftl shown i n appendix f o r any e q u i l i b r i u m p o i n t
spec i f ied by (a,M).
4.3 Linear Controller Desians
As ment ioned ear l ier , i n order t o design a con t ro l l e r t o cont ro l
nonl inear plants, i t i s necessary t o break the cont ro l design process
i n t o t w o steps. F i rs t , one must design local l inear con t ro l l e rs based
on 1 i n e a r i z a t i o n o f t he non l inear p l a n t s a t seve ra l d i f f e r e n t
operat ing condi t ions. Second, one must in te rpo la te the gains of the
local designs. The process o f a l inear cont ro l le r design i s described
be low.
In designing these cont ro l le rs , the p lan t t h a t w e consider i s
t h e l i n e a r i z e d p l a n t c a l c u l a t e d f rorn S e c t i o n 4.2, s i n c e i t i s
necessary t o design a cont ro l le r f o r the l inear ized p lan t a t several
d i f f e r e n t opera t ing p o i n t s . Here, consider 3 d i f f e r e n t opera t ing
po in ts a t a ; 1 2 , 3 and (1;,4). I t was prev iously observed i n
[9] t ha t the so lut ions are a f fec ted by var ia t ions i n Mach numbers and
on ly weak ly a f f e c t e d by changes i n angle o f a t tack ; t he re fo re , 0
se lec t ing an angle o f a t tack a t 10 f o r each o f th ree Mach numbers
represents a reasonable compromise i n tha t t h i s va lue represents
the m i d po in t of the desired operat ing range. The f i r s t ob jec t ive f o r
45 t h i s s tep is , f o r f i xed Mach number, the con t ro l l e r mus t s tab i l i ze a l l
p lan t l inear iza t ions corresponding t o a l l values of angle o f a t tack
be tween -200 < a < 200 By s y m m e t r y p r o p e r t i e s o f the p l a n t
descr ip t ion, one need consider only $ < a < 2$
The type 1 servo system based on pole placement i s u t i l i z e d i n
order t o design the desired cont ro l le r . Since some o f s t a t e var iab les
are not avai lable f o r measurement, the s t a t e observer i s placed in to
t h i s servo system. De ta i l s about t h i s type-1 servo sys tem were
given i n Sect ion 2.2 f rom wh ich the closed -loop system i s described
In order t o meet the design ob jec t ive , the open-loop t r a n s f e r
f unc t i on of the l inear ized system requi res the loop t o be opened a t
the input t o the ac tua tor . Before going f u r t h e r t o t h e c o n t r o l l e r
46 design step, the open-loop system i s developed here. FIGURE 4.3- 1
shows a blclck diagram of the open-loop system which i s developed
f rom FIGURE 2.2-4.
FIGURE 4 3 - 1 B l o c k d 1 2 c r w - g f o a e n - l o g p 5ys:en
The dynamics o f the open-loop system i s described as
r
A 0 0 LC A-LC 0 -C 0 O J
- 1
d t i 1
%(ti +[E3]ul(t)
- f(t) J
I n Chapter 2.2, a method t o f i n d the ga in K o f the po le
placement design and the gain L of the s t a t e observer was given by
using Ackermann's formula. However, i n t h i s cont ro l le r design step
these can be found d i r e c t l y by using the "PLACE" command i n the
Control System Toolbox of MATLAB.
The f i r s t design i s calculated i n PROGRAM #2. The p lan t used
i s the l i nea r i zed p l a n t f r o m PROGRAMXl. I n de te rm in ing the
s u i t a b l e ga in v e c t o r K t h a t g i ves the bes t o v e r a l l sys tem
performance, the several d i f f e r e n t ma t r i ces K are examined v i a
computer s imu la t i on t o ob ta in the response charac te r i s t i cs o f the
s y s t e m The m a t r i x K i s based on the se lec t ion o f the eigenvalues
wh ich give the desired charac ter is t i c equation. I n our design, a f t e r
several s imu la t ions to check the system charac ter is t i cs , i t i s found
t h a t the e i genvalues a t [ -36 .3 , - 3 6 . 9 6 + 0 . 6 6 i , - 3 6 . 9 6 - 0 . 6 6 i , -
37.62+1.32 i , -37.62- 1 .32 i l g ive the bes t gain m a t r i x K t h a t i s
su i table f o r the design object ives.
The observer gain m a t r i c e s L are considered i n the same
48 manner as the gain K . The best gain matr ix L which gives the
suitable response for the design objectives is defined by placing the
eigenvalues for states observer technique at [-6 16+ 1 1 i, -61 6-1 1 i, -
627+22i, -627-2213.
In addition, PROGRAM "2 simulates the closed-loop system
step response and the frequency response of the open-loop system
From PROGRAM #2, the gain K, K i and L for the controller a t 0 0 0
the constant equi l ibr ium point (a,M); ( 10,2), ( 1 0,3), and ( 10,4) are
given respectively as
at f ixed Mach number 2:
KK2 = [ -5.7 137e+00, -4.56 13e-0 1 , -4.0079e-0 1 , - 1 . 1 296e-031
Ki2 = 6.3689e+O 1
KL2'= [-7.7941 e+02, -7.6420e+04, 2.0806e+03, -2.5828e+051
at f ixed Mach number 3:
KK3 = [-2.2529e+00, - 1.7727e-0 1 , -4.0539e-0 1 , - 1 . 1 405e-031
Ki3 = 1.4099e+01
KL3'= [-4.0547e+02, -3.9609e+04, 9.19 1 8e+02, - 1.1657e+05]
at f ixed Mach Number 4:
KK4 = [ - 1.3009e+00, -9.5867e-02, -4.0555e-0 1 , - 1 . 1 457e-031
Ki4 = 5.0734e+00
where KK2=the gain K design a t f i xed Mach number 2
Ki2=the gain K i design a t f i xed Mach number 2
KL2=the gain L design a t f i xed Mach number 2
and KK3, Ki3, KL3, KK4, K i 4 and KL4 are defined i n the same but a t
the f i xed Mach number 3 and 4 respect ive ly .
The step response of the closed-loop l i nea r sys tem a t th ree
cons tan t ope ra t i ng p o i n t s a re p i c t u r e d i n FIGURE4.3-2, w h i c h
i l l u s t r a t e s t h a t the ou tpu t t racks the s tep command w i t h t i m e
constants less than 0 .25 sec. FIGURE 4.3-3 descr ibes the frequency
response of the open-lcop system a t those equ i l ib r ium points; w i t h a
frequency of 3 0 0 rad/sec, a l l the magnitude are less than -30 dB.
Both graphs demonstrate t h a t the prev iously spec i f i ed eigenvalues
y i e l d gains K, K i and L wh ich achieve the design object ives.
0 FIGURE 4.3-2 Step Response of closed-loop system at M=2, 3 , and 4 ,alfa= 10
Frequency (radlsec) 0
FIGURE 4.3-3 Frequency Response of open loop system at M=2, 3, and 4 ,alfa= 1 0
51 As prev iously noted, a con t ro l l e r a t a f i xed Mach number must
s a t i s f y the design ob jec t ives over the range o f an angle of a t tack
between 6 t o 200 Thus PROGRAM "3 i s created in order t o check
the s t a b i l i t y o f the sys tem as the angle o f a t t a c k va r ies . Th i s
program uses the constant gain K, K i and L found e a r l i e r a t each
f i x e d cons tan t ope ra t i ng po in ts . The same p l a n t c o e f f i c i e n t s
ma t r i ces a t those points are use t o be the c o e f f i c i e n t s ma t r i ces of
the s t a t e observer i n the considered system.
For convenience i n represent ing the design s tep and output, the
author now w i l l f i r s t consider the c o n t r o l l e r design s tep a t the
f i x e d Mach number 2. The f i x e d Mach number 3 and 4 w i l l be
addressed la te r .
The s i m u l a t i o n are app l ied t o the c o n t r o l l e r w h i c h are
designed i n PROGRAMs3 t o check the s t a b i l i t y of the sys tem as the
angle of a t t a c k vary between t o 26 We s imu la ted PROGRAMz3
w i t h a l l the angle of a t tack i n the range of i n te res t . Around the
0
constant operat ing design po in t (a,M) = (10,2), the c o n t r o l l e r can
s a t i s f y a l l the design object ives, but when the angles of a t tack are
changed, the con t ro l l e r performance i s degraded. I t means t h a t the
c o n t r o l l e r i s able t o s t a b i l i z e the sys tem only a t t h e va lues o f
0
at tack wh ich do not d i f f e r much f r o m the design point , a = l o . When
the angle o f a t t a c k changes s i g n i f i c a n t l y , the c o n t r o l l e r cannot
52 s t a b i l i z e t h e s y s t e m . A t t e m p t s t o s e l e c t d i f f e r e n t des ign p o i n t s
cor responding t o d i f f e r e n t angle o f a t t ack y i e l d comparab le r e s u l t s .
The l i n e a r s i m u l a t i o n s a t d i f f e r e n t ang le o f a t t a c k a r e shown
i n FIGURE 4 . 3 - 4 and FIGURE 4 .3 -5 . FIGURE 4 . 3 - 4 s h o w s t he s t e p
responses o f the c losed- loop sys tem a t the cons tan t Mach number 2 0 0 0
and t h e angle o f a t t a c k 0 , 10, and 20 . The f requency responses o f
the open-loop sys tem a t the same design p o i n t s va lues a re d isp layed
i n FIGURE 4.3-5. The p rob lem o f t he c o n t r o l l e r w h i c h ment ioned a re
c l e a r l y shown f r o m these graphs
Now consider the design s tep a t t he f i x e d Mach number 3 and 4.
We a l so s i m u l a t e the sys tem a t these f i x e d Mach number w i t h a l l the 0 0
angle o f a t t a c k f r o m 0 t o 20. The p rob lem encountered a t the f i x e d
Mach numbers 3 and 4 resemble as shown t h a t faced a t f i x e d Mach
number 2 . W i t h s i g n i f i c a n t changes o f t h e ang le o f a t t a c k , t he
c o n t r o l l e r per formance i s degraded. FIGURE 4 .3 -6 - FIGURE 4 .3 -9 are
man ipu la ted as t he same as FIGURE 4.3-4, and FIGURE 4.3-5. FIGURE
4 .3 -6 , and FIGURE 4.3-8 show s t e p response o f t h e c losed- loop
s y s t e m a t f i x e d Mach number 3 and 4 r e s p e c t i v e l y . I n t hese graph 0 0 0
the d i s t i n c t angle o f a t t a c k a t 0, 10 and 20 a re chosen. FIGURE 4.3-
7, and FIGURE 4 . 3 - 9 i l l u s t r a t e the open- loop 's f requency responses
a t the same angle o f a t t a c k va lues f o r f i x e d Mach number 3 and 4.
Frequency (radlsec)
FIGURE 4.3-5 Frequency Response of open loop system at M=2.
L
1.5
1
......... : ado
- : a-10' - ++++ : a -20'
-
- - a +++** n. - a 0
-
-
-0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
time(sec)
FIGURE 4.3-4 Step Response of closed-loop system at M=2.
time(sec)
FIGURE 4.3-6 Step Response of closed-loop system at M=3.
Frequency (radiscc)
FIGURE 4.3-7 Frequency Response of open loop system a t N=3.
time(sec)
FIGURE 4.3-8 Step Response of closed-loop system at M=4.
Frequency (radlsec)
FIGURE 4.3-9 Frequency Response of open loop system a t W-4.
56 We redesign the con t ro l l e r again as ca lcu late i n PROGRAM #4.
Once again, f i r s t consider the design s tep a t the f i x e d Mach number 2
and then a t the f i xed Mach numbers 3 and 4.
In the new design, use the same gain K, Ki , and L but l e t the
l inear ized p lant coe f f i c i en t mat r ices used i n the s t a t e observer vary
w i t h angle o f a t tack as ind ica ted i n Sec t ion 4.2. By l e t t i n g the
coe f f i c i en t ma t r i ces of the observer depend on angle o f a t tack, step
response and frequency response o f the 1 i near ized m i s s i le model
w i t h type- 1 servo system a t f i xed Mach number 2 meet the design 0 0
ob jec t ives a t a l l angle o f a t tack between 0 t o 20. Three d i f f e r e n t 0
angle of a t tack 6 1; and 2 0 are chosen t o v e r i f y these r e s u l t s as
shown i n FIGURE 4.3- 10 and FIGURE 4.3- 1 1 .
Next, consider the con t ro l l e r design a t the f i xed Mach numbers
3 and 4 by us ing the same considerat ion bu t changing the gains (K,
K i , and L). The s imu la t ion resu l t s are also the same as w e discuss
i n design a t f i xed Mach number 2 f o r a l l the angle o f a t tack between 0 0
0 t o 20. The closed-loop s tep responses and open-loop frequency
responses are shown i n FIGURE 4.3- 12 and FIGURE 4.3-1 3 f o r f i xed
Mach number 3 and i n FIGURE 4 .3 -14 and FIGURE 4 .3-15 f o r f i x e d
Mach number 4 , respec t ive ly . The s i m u l a t i o n response a t t h ree
d i f f e r e n t angles o f a t tack are shown in these graphs.
Thus w e can conclude t h a t t he l i n e a r c o n t r o l l e r w h i c h i s
57 designed by the chosen eigenvalues g iven e a r l i e r and l e t t i n g the
c o e f f i c i e n t s m a t r i c e s of the observer p a r t of the con t ro l l e r depend
on angle o f a t tack as w e l l as Mach number, the l inear ized closed-
loop sys tem f o r f i xed Mach number (M=2,3,4) and a l l angle of a t tack
i s s table and meets the design speci f icat ions.
time(sec)
FIGURE 4.3- 10 Step Response of closed-loop system at M=2.
tirne(sec)
FIGURE 4.3- 1 2 Step Response of closed-loop system at M=3.
Frequency (radlsec)
FIGURE 4.3- 13 Frequency Response of open loop system at M=3.
FIGURE 4.3- 1 5 Frequency Response of open loop system at M=4.
4.4 Schedulina the Set of Linear Controllers
Gain schedul ing i s broken i n t o t w o s teps . The f i r s t s tep
involves designing a local cont ro l l e r based on 1 i near iza t ion of the
nonl inear p lan t a t several d i f f e r e n t equ i l i b r i um po in ts . Th i s was
accompl ished i n the prev ious sec t ion . The loca l c o n t r o l l e r was 0 0
designed a t three d i f f e r e n t equi l i b r i u m po in ts (( a,M)=( 10,2), ( 10,3), 0
1 0 4 Each equ i l ib r ium point gives a spec i f ied gain wh ich makes
the con t ro l l e r capable of sa t i s f y ing the system requi rements loca l l y
around each design point .
The second step, t o be discussed i n t h i s chapter, requ i res
in te rpo la t ing , or "schedul ing" , the qains of the l i nea r designs t o
ob ta in a nonl inear con t ro l l e r . The three spec i f i ed gains f r o m the
local con t ro l l e r are r e w r i t t e n as fo l l ows :
A t f i xed Mach number 2:
A t f i xed Mach number 3 :
KK3 = [-2.2529e+00, - 1.7727e-0 1 , -4 .0539e-0 1 , - 1.1 405e-031
K i 3 = 1.4099e+O 1
At f ixed Mach Number 4:
KK4 = [ - 1.3009e+00, -9.5867e-02, -4.0555e-0 1 , - 1 . 1 457e-031
K i4 = 5.0734e+00
KL4'= [-2.7506e+02, -2.68 1 3e+04, 5.1287e=02, -6.6569e+041
It i s known from the linear control ler analysis that the gains
at each f ixed Mach number can stabi l ize the l inearized plant at the
f i xed number and over the ent i re range of the angle of at tack 0 0
between 0 to 20. In considering th is problem, an attempt i s made to
schedule a l l these gains when the Mach nurriber i s d i f ferent from the
f ixed point. This means that the gain i s defined when the value of
the Mach number l i es between the Mach numbers 2 and 3 or the
Mach numbers 3 and 4, or outside the range 2 5 M i 4.
Fo r the gain scheduling method, K i i s considered f i r s t . At the
f ixed Mach numbers 2, 3 and 4, the values of the gain K i are known.
To fac i l i ta te an understanding of the discussion that fo l lows, these
3 values a r e shown in the FIGURE 4.4- 1 .
From th is figure, one can see that a t each f ixed Mach number
2, 3, and 4, the gain K i i s set a t the known values. Thus, when the
Mach number i n the considered system l i es a t one of these f ixed
63 points, the appropriate gain K i w i l l be used in the control ler.
2 3 4 Mach number
FIGURE 4.4- 1 Scheduling the gain Ki
Now consider the points between the f i xed Mach numbers 2
and 3 and draw a l ine between these two points. An equation can
then be created fo r t h i s l ine. The same can be done between the
f ixed Mach numbers 3 and 4.
Recall now, the equation of the l ine jo in ing between t w o
poi n t s P(xi,y 1) and P(x2,yz):
Using th i s t w o equations, one can obtain the gain K i as a
funct ion of Mach number, w r i t t e n Ki(M). Therefore, when the Mach
number i s given between either the two f ixed points 2 and 3 or the
64 t w o f ixed points 3 and 4, one can f ind the gain K i re lated to the
corresponding Mach number.
One can schedule the gains K and the gain L, i n the same
manner t o obtain K(M) and L(M).
Gain "scheduling" i s the most important component of the
nonlinear contro l ler design and w i 1 1 be u t i l i zed f o r the remainder of
t h i s thesis. The M- f i le f rom MATLAB i s w r i t t e n to calculate th is.
The M-f i le i s named "schedu1ing.m" and i s described i n PROGRAM
#5.
Using PROGRAM l f5 the gain scheduling i s created f o r the
nonl inear control ler. With the gain scheduling, i t i s expected that
our nonlinear contro l ler w i 11 s tabi l ize the nonlinear p lant over the
ent i re operating range. In order t o conf i rm t h i s expectation, the
local s tab i l i t y around any equil ibrium point i n the desired range w i l l
be examined f i r s t and then the overall performance of the nonlinear
system w i t h the designed contro l ler w i l l be examined by means of
s imulat ion using SIMULAB.
- Checking the local stabil i ty of the linearized system.
For the discussion of local s tab i l i t y , consider the nonlinear
control ler, the "Autopi lot ". F i rst , connect the autopi lot to the plant
65 and close t h i s loop. The block diagram i s i l l u s t r a t e d i n FIGURE 4.4-
FIGURE 4.4- 2 Block diagram of the missi le model w i t h nonlinear contro l ler
n c nz
The closed-loop dynamic system i s given as
where
-
t = n , - n z
6, = -K(M)Z + Ki(M)t
where f a n d g a r e t h e n o n l i n e a r f u n c t i o n o f t h e m i s s i l e m o d e l
miss i le /actuator Au top i lo t . )
66
A t the cons tan t opera t ing po in t fc, (x,:,f,h,~) = o w h i c h i m p l i e s t h a t
Thus, a t any c losed- loop constant opera t ing po in t , w e have
?(a,M) = x(a,M)
so t h a t Sz(a,M) = nz(a,M)
f (a ,M) = &[Wa,M) - K(M)x(a,M)I .
Next , 1 i nea r i za t i on o f the non l inear c losed- loop s y s t e m around
t h i s s e t o f cons tan t ope ra t i ng p o i n t s y i e l d s a s e t o f l i n e a r i z e d
s y s t e m s whose p rope r t i es w i 1 1 be analyzed. The Jacob ian m a t r i c e s
f o r t h i s l i n e a r i z a t i o n method are g iven as f o l l o w s .
F rom t h e c losed- loop sys tem:
De f ine
G = %a3+bna2+cn (2-$M)a, G, = 3%a2+2ba+c , (2-$M)
The Jacobian m a t r i c e s t h a t descr ibe the l i n e a r i z a t i o n of the
non l i nea r c losed- loop s y s t e m i n FIGURE 4.4-2 a re c r e a t e d i n
PROGRAMlt6, Inc luded i n t h i s p rogram i s t h e s i m u l a t e d s t e p
response of the l inear ized closed-loop sys tem i n order t o check the
local s t a b i l i t y of the system. For the l oca l s t a b i l i t y , t h e l inear ized
sys tems w h i c h are s i m u l a t e d a t any cons tan t ope ra t i ng p o i n t s
should provide s table response i n the en t i re range o f in te res t .
As f i r s t men t ion f o r the l o c a l s t a b i l i t y check, m o r e than
hundred po in ts o f the f i x e d Mach number and the angle o f a t tack i n
68 the interested range are simulated and checked. Each step response
f rom those simulat ions show the ab i l i t y to track the step command
which indicated the local s tab i l i t y of the l inearized system. Also,
these step responses sat is fy the design object ives of t ime constant
less than 0.25 second.
To ver i f y the conclusion, the simulat ion a t the angle of attack 0 00 and 2 0 w i t h the f ixed Mach number 2.3, 2.3, 3.3, and 3.7 are
examined. The step response which provided the s t a b i l i t y of the
l inear ized system w i t h t i m e constant ly than 0 .25 second are
i l l u s t r a t e d i n FIGURE 4.4-3 and FIGURE 4.4-5. The frequency
response of the open-loop system which less than -30 dB a t 300
radian/second are shown in FIGURE 4.4-4 and FIGURE 4.4-6.
time(sec) 0
FIGURE 4.4-3 Step Response of the linearized system at alfa=O.
Frequency (radlsec) 0
FIGURE 4.4-4 Frequency Response of open loop linearized system at alfa=O.
FIGURE 4.4-5 Step Response of the l inearized system at alfa.26
Frequency (rad/sec) 0
FIGURE 4.4- 6 Frequency Response of open loop l inear ized system a t a l fa=20.
71 - Checking Performance of the Nonlinear Controller
Previously, a nonl inear au top i lo t i s created by scheduling the
gains of the 1 inear con t ro l l e rs designed a t 3 d i f f e r e n t equ i l i b r i um
points . A t the conclusion o f t ha t chapter, local s t a b i l i t y around any
e q u i l i b r i u m p o i n t o f t h e s y s t e m w a s checked. The r e s u l t s
demonstrated tha t the autopi l o t can s t a b i l i z e the nonl inear system
loca l l y around any operating point i n the desired range.
Since w e r e q u i r e t h a t t he a u t o p i l o t s t a b i l i z e the sys tem
throughout the e n t i r e operat ing range, a program f o r s i m u l a t i n g
nonl inear dynamic systems, SIMULAB, w i 1 1 be employed i n order t o
v e r i f y t h i s s t a b i l i t y .
For s i m u l a t i o n purpose, t h e a u t o p i l o t i s connected t o the
m i s s i l e as shown i n FIGURE 4 .4 -7 . The va r iab les i n t h i s b lock
diagram tha t are fedback are the actual v e r t i c a l acce le ra t ion (e ta )
and the Mach number (M). Note; the Mach number generated in t h i s
program i s not p roper ly p a r t o f t h e p l a n t bu t i t i s inc luded f o r
s imu la t i on purposes. PROGRAMZ7 and *8 conta in the M-f i l e s o f the
s - func t ions i l l u s t r a t e i n FIGURE 4.4-7. "Au top i 1ot .m" i s an M - f i l e
wh ich descr ibes the autopi l o t , the designed con t ro l l e r . " M iss i le .mn
also, i s an M- f i l e wh ich i s used t o describe the m i s s i l e and ac tua tor
dynamics.
73
To obtain the response of th is system over the ent i re range of
the Mach number 2 and 4, the Mach number are consider as 4 range;
4.0-3.5, 3.5-3.0, 3.0-2.5, and 2.5-2.0. The simulat ions are designed
t o s imulate each of these range. The f i r s t s imulat ion output are
shown in FIGURE 4.4-8. The graph shows the response of the system
compared to the step command. I t show that the response of the
system can track the step command w i t h t ime constant less than
0.25 second which i s the requirement of the design objectives. Also,
in FIGURE 4.4-8 , the Mach number pro f i le (s tar t ing a t Mach number
= 4) i s shown.
In FIGURE 4.4-9 - FIGURE 4.4-1 1 i l l u s t r a t e the s imula t ion
output as the same as in FIGURE 44-8 . The difference i n these graph
i s the range of the Mach number. From these graphs, one obviously
sees tha t no ma t te r what the range of the Mach number are
considered, the system i s able to track the step command w i t h t ime
constant less than 0.25 second. Thus , w i t h these data, i t i s clearly
show that the dynamic contro l ler y ie lds stable and well-behaved
response which sat is f ies the design objectives.
time(sec)
FIGURE 4.4-8 Step Response o f the miss i le model s tar t ing at M=4.
40
.----- step command - acceleration
time(sec)
Mach number fo r simulat ion i n FIGURE 4.4-8,
i ..
I
I ,
-
-
-
-10
-20
- -
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
FIGURE 4.4-9 Step Response of the miss i le model s tar t ing at M=3.5
- - - - - - step command - - acceleration
-
-
-
time(sec)
Mach number fo r simulat ion i n FIGURE 4.4-9.
-10 - -
-20. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
time(sec)
FIGURE 4.4- 10 Step Response of the miss i le model s tar t ing at M=3.
time(scc)
Mach number for simulation in FIGURE 4.4- 10.
time(sec)
FIGURE 4.4- 1 1 Step Response of the miss i le model s tar t ing at M=2.5.
time(sec)
Mach number fo r simulat ion in FIGURE 4.4- 1 1 .
Chapter 5
Summary and Conclusions
In t h i s thesis, we study the design o f an au top i lo t by applying
gain schedul ing, new cont ro l l e r design technique, t o the example of
non l inear sys tem. As w e ment ioned e a r l i e r , a l though t h e gain
s c h e d u l i n g i s a s u c c e s s f u l techn ique i n many e n g i n e e r i n g
appl icat ion, i t has a r e s t r i c t i o n on the exogenous var iab le wh ich had
t o vary s l o w l y . I n the example problem o r m i s s i l e f i g h t con t ro l
p rob lem, the Mach number i s considered t o be t h i s exogenous
var iable.
In Chapter 3, a l l tne matheniat ics desc r ip t i on of the m i s s i l e
problem together w i t h a s t a t e equat ion o f the Mach number were
def ined ( A s t a t e equation f o r the Mach number was no t a proper pa r t
o f t h e m i s s i l e sys tem but t h i s equat ion w a s necessary i n the
s imu la t ion p a r t f o r checking the performance o f the autopi lo t . ) .
The de ta i l s o f the design procedure were discussed i n Chapter
4. F i r s t , the l inear cont ro l le rs were designed f r o m l inear ized p lant
data a t 3 d i f f e r e n t constant operat ing po in ts . Since the cont ro l le rs
designed there used 1 inear t ime- invar iant technique, our con t ro l l e rs
can guarantee only local performance and nominal s t a b i l i t y o f the
m i s s i l e . The open-loop frequency response of the 1 inear ized system
79 shows t h a t a t 300 radian/second the magnitude was l ess than -30
dB. That i s one o f the designed requi rements t h a t seeks t o avoid
e x c i t i n g the unmodel led s t r u c t u r a l dynamics. Last , the gains o f
those con t ro l l e r are scheduled, and the autopi lo t w i t h the scheduled
gains was s i m u l a t e d t o check f o r the m i s s i l e per formance. The
r e s u l t f r o m t h e s i m u l a t i o n show the a b i l i t y o f the a u t o p i l o t t o 0 0
s tab i l i ze the system w i t h i n the operat ing range (-20 5 a_< 20 and 2 5
M 3 4 ). Th is resu l t i s also sa t i s f i es the design object ives.
Since the m i s s i l e ' s performance m e t a l l design ob jec t ives , i t
i s concluded tha t the autopi lo t , the cont ro l le r , designed by us ing a
gain scheduling i s achieved.
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12, no. 3, pp. 10 1 - 107,1992.
[ 141. S lo t i ne , J.-J., E., " A p p l i e d Non l inear Con t ro l " . P ren t i ce -Ha l l ,
Inc . ,New Jersey, 199 1
[ 151. V i dyasagar, M., "Nonl i near S y s t e m A n a l y s i s " . P r e n t i ce -Ha l l ,
Inc . , New Jersey, 1978.
% PROGRAM # 1 % Linearization of nonlinear system
m=input('The value of Mach number (m)=') alfa=input('The value of the angle of attack (alfa) = ' )
% Airframe and actuator constants % Kalfa=0.02069; Kq=l. 23196; Kz121.4432; Ax=32.1648; damp=0.7; Wa=150;
% Some constants that change from degree to radian % alfan= alfa*pi/l80; Kalfan= 1.18587; Kzn= 0.6661697; Kqn=70.586;
% Aerodynamic coefficients % an=0.000103; bn=-0.00945; cn=-0.1696; dn=-0.034; am-0.000215; bm=-0.0195; cm=0.051; dm=-0.206; Cn=an*alfaA3+bn*alfaA2+cnf(2-m/3)*alfa; ~m=am+alfa~3+bm*alfa^2+cm*(8*m/3-7)*alfa;
% Differential values % cnd=3*an*alfaA2+2*bn*alfa+cnt(2-m/3); ~md=3*am*alfa~2+2*bm+alfa+cmt(8fm/3-7); delta=-cm/dm; .
% Jacobian matrices 8 all=~alfan*m*(Cnd*cos(alfan)-(Cn+dnfdelta)*sin(alfan)*pi/l80); a12=1; al3=Kalfan*m*dn*cos(alfan); a14=0; a21=Kqn*mA2*Cmd; a22=O; a23=Kqn*mA2*dm; a24=0; a31=0; a32=0; a33=0; a34=1; a4 1-0; a42=O; a4 3=-WaA2; a44=-2 *damp*Wa;
% The the linearized system % aa=[all,al2,a13,a14;a21,aZZ,a22Ia23,a24;a31la32,a33la34;a4lla42la43la44] bb=[bll;b21;b31;b41] cc=[cll,c12,~13,~14]
% PROGRAM # 2 % ~inearization of nonlinear system A2
m=input ('The value of Mach number (m) = ' ) alfa=input('The value of the angle of attack (alfa) = ' I % % Airframe and actuator constant
Kalfa=0.02069; Kq=1.23196; Kz=21.4432; Ax=32.1648; damp=O. 7; Wa=150; % % Some constant that change from degree to radian
alfan= alfa*pi/l80; Kalfan= 1.18587; Kzn= 0.6661697; Kqn=70.586; % % Aerodynamic coefficients
% % Some differential value
% % Jacobian matrix
ades=[all,al2,al3,al4;a21,a22,a23,a24;a3l,a32,a33,a34;a4l,a42,a43,a44]; bdes=[bll;b21;b31;b41] ; cdes=[cll,cl2,cl3,cl4]; % sk=pole locatin
% % Controller design step % % Finding the gain K for poles placement % sk=the desired eigenvalues
q=[0;0;0;01; sk=[-36.3,-36.96+0.66*i,-36.96-0.66*i,-37.62+1.32*i,-37.62-1.32*il;
A3 abar=[ades,q;-cdes,OI; bbar= [bdes; 01 ; kk=place(abar,bbar,sk);
% Finding the gain L for states observer % sl=the desired eigenvalues
% Finging the step response for the closed-loop system
aclose=[ades -bdesek bdes*ki;l*cdes ades-l*cdes-bdes*k bdes*ki;-cdes q' 01; bclose= [q;q; 11 ; cclose= [cdes q' 0 I ; dclose=O;
t=linspace(O, . 5 ) ; [yclose,xclose] =step(aclose,h~1ose,cclose,dclose, 1, t) ; plot it, yclose) ,grid title(' GRAPH-? Step response for closed-loop system ' ) xlabel('time(sec) '),ylabel('output') pause
% Finding the frequency response of the open-loop system
qs=[s#q.q#sl; aopen=[ades,qq,q;l*cdes,ades,ades-l*cdes,q;-cdes,q',O]; bopen=[bdes;bdes;Ol; copen= [q' , -k, ki j ; dopen=O; [mag, phase, w] =bode (aopen, bopen, copen, dopen, 1) ; semilogx(w, 20*log (mag) ) ,grid title('GFAPH-? Frequency response of the open-loop system') xlabel('Frequency (rad/sec) ');ylabel('Magnitude (dB) ' ) ;
% PROGRAM # 3 % Design the Type 1 servo system(use the coefficients matrices 8 of the state observer the same as the plant coefficients % matrices at design point. % m= 2 alfa=input('The value of the angle of attack (alfa) = I )
% Airframe and actuator constants % Kalfa=0.02069; Kqi1.23196; Kz=21.4432; Ax=32.1648; damp=O. 7 ; Wa-150;
% Some constants that change from degree to radian % alfans alfa*pi/l80; Kalfan= 1.18587; Kzn= 0.6661697; Kqnz70.586;
% Aerodynamic coefficients % an=0.000103; bn=-0.00945; cn=-0.1696; &I=-0. 034; am=O.000215; bm=-0.0195; cm=O. 051; dm=-0.206; ~n=an*alfa^3+bn*alfa*Z+cn+(2-m73)+alfa; ~m=am*alfa~3+brn*alfa~2+cm*(8*m/3-7)Palfa;
% Some differential values % Cnd=3*an*alfaA2+2+bn*alfa+cn+(2-m/3); Cmd=3*am*alfaA2+2*bm*a1fa+cm*(8*m/3-7); delta=-cm/dm;
% Jacobian matrices coefficients % all=~alfan*m*(Cnd*cos(alfan)-(Cn+dn+delta)*sin(alfan)*pi/l80); a1231; al3=Kalfan*m*dn*cos(alfan); a14=0; a21=Kqn*mA2+Cmd; a22-0; a23=Kqn*mA2*dm; a24=O; a3 1=0; a32=0; a33=0; a34=1; a41=0; a4 2=O ; a43=-WaA2; a44=-2 *damp*Wa;
% Controller's design step % % The gain K from poles placement % k=[-5.7137e+00,-4.5613e-01,-4.0079e-01,-1.1296e-03]; ki=6.3689e+01:
% The gain L from states observer % 1=[-7.7941e+02;-7.6420e+04;2.0806e+03;-2.5828e+05];
% 0 8 % The gains are changed for the difference constant m as 8 % at m=3; k=[-2.2529e+00,-1.7727e-01,-4.0539e-01,-1.1405e-03] % % ki=1.4099e+01 % % 1= [ -4 .0547e+02, -3 .9609e+04I9 .1918e+02, -1 . l657e+O5] % % % % at m=4; k=[-1.3009e+00,-9.5867e+02-4.0555e-01,-1.1457e-O3] % % ki=5.0734e+00 % % 1=[-2.7506e+02,-2.6813e+04,5.1287e+02,-6.6596e+O4] % %
% Finging the step response for the closed-loop system % aclose=[aa -bb*k bb*ki;l*cc aa-l*cc-bb*k bb*ki;-cc q f 01; bclose=[q;q; 11 ; cclose=[cc q' 01; dclose=O;
t=linspace(0,.5); [yclose,xclose]=step(acloseIbclosetcclosetdclosetltt); plot(t,yclose),grid title(' GRAPH-? Step response for closed-loop system ' ) xlabel('time(sec)'),ylabel('output(y(t))') pause
% Finding the frequency response of the open-loop system 8 qqr[q!qlqlql; aopen=[aa,qq,q;ltcc,aa-l*cc,q;-ccIq'IO]; bopen=[bb;bb;O]; copen=[qt, -k,ki]; dopen-0 ; [mag,phase,w]=bode(aopenIbopen,copen~dopenll); semilogx(wt20*log(mag)),grid title('GRAPH-? Frequency response of the open-loop system') xlabel('Frequency (rad/sec)');ylabel('Magnitude (dB)');
o PROGRAM # 4 % Design Type 1 servo System (let the plant coefficient used % in the state observer vary with angle of attack and Mach % number)
m= 2 alfa=input('The value of the angle of attack (alfa) = ' )
0 Airframe and actuator constants % Kalfa-0.02069; Kq=1.23196; Kz=21.4432; kx=32.1648; damp=O .7 ; Wa=150;
0 Some constants that change from degree to radian % alfan=alfa+pi/l80; Kalfan= 1.18587; Kzn= 0.6661697; Kqn=70.586;
% Aerodynamic coefficients % an=0.000103; bn=-0.00945; cn=-0.1696; dn=-0.034; arn=0.000215; bm=-0.0195; cm=O. 051; dm=-0.206; Cn=an*alfaA3+bn+alfa^2+cn*(2-m/3)talfa; Cm=am+alfaA3+bm+alfa^2+cmt(8*m/3-7)talfa;
% Some differential values % Cnd=3*an*alfaA2+2+bn+alfa+cn*(2-m/3); Cmd=3*am*alfaA2+2*bm+alfa+cm+(8fm/3-7); delta=-cm/dm;
% Jacobian matrices coefficient % all=Kalfan+m*(Cndfcos(alfan)-(Cn+dn*delta)*sin(alfan)*pi/l8O); a12=1; al3=Kalfan*m+dn*cos(alfan); a14=0; a21=Kqn+mA2*Cmd; a22=0; a23=Kqn+mA2*dm; a24=0; a3 1-0; a32=0; a33=0; a34=1; a41=0; a42=0; a43=-Waa2; a44=-2*damp+Wa;
% Controller's design step % % The gain K from poles placement % k=[-5.7137e+00,-4.5613e-01,-4.0079e-01,-1.1296e-03]; ki=6.3689e+Ol;
% The gain L from states observer % 1=[-7.7941e+02;-7.6420e+04;2.0806e+03;-2.5828e+05];
% % 8 % The gains are changed for the difference constant m as 8% at m=3; k=[-2.2529e+00,-1.7727e-01,-4.0539e-01,-1.1405e-03] % % ki=1.4099e+01 % % 1= [ -4 .0547e+02, -3 .9609e+04I9 .1918e+02, - l . l 657e+O5] % 0 % % at m=4; k=[-1.3009e+00,-9.5867e+O2-4.0555e-01,-1.1457e-O3] % % ki=5.0734e+00 % % 1 = [ - 2 . 7 5 0 6 e + 0 2 , - 2 . 6 8 1 3 e + O 4 , 5 . 1 2 8 7 e + O 2 I - 6 . 6 5 9 6 e + O 4 ] % %
% Finging the step response for the closed-loop system % aclose=[aa -bb*k bb*ki;l*cc ades-ltcdes-bdes*k bdes*ki;-cc q r 01; bclose=[q;q;l]; cclose=[cc q' 0] ; dclose=O ;
t=linspace(0,.5); [yclose,xclose]=step(acloseIb~loserccloseldcloserllt); plot(t,yclose),grid title(' GRAPH-? Step response for closed-loop system ' ) xlabel('time(sec)'),ylabel('output(y(t))') pause
% Finding the frequency response of the open-loop system 8 ~¶~[SrSrSr¶l; aopen=[aa,qq,q;l*ccIades-l*~deslq;-~~Iq'rO]; bopen=[bb;bdes;O]; copen=[q',-k,ki]; dopens0 ; [mag,phase,w]=bode(aopenIbopen,dopen,l); sem~logx(w,20*log(mag)),grid title('GRAPH-? Frequency response of the open-loop system') xlabel('Frequency (rad/sec)');ylabel('Magnitude (dB)');
% Program # 5 % Scheduling the gains for nonlinear controller
% Gains from the constant operating points % kk2=[-5.7137e+00,-4.5613e-01,-4.0079e-01,-1.1296e-03]; ki2=-6.3689e+01; k12=[-7.7941e+02,-7.6402e+04I2.0806e+03,-2.5828e+05]; kk3=[-2.2529e+00,-1.7727e-01,-4.0539e-01,-1.1405e-03]; ki3=-1.4099e+01; k13=[-4.0547e+02,-3.9609e+04I9.1918e+02,-1.1657e+05]; kk4=[-1.3009e+00,-9.5867e-02,-4.0555e-01,-1.1457e-03]; ki4=-5.0734e+00; k14=[-2.7506e+02,-2.6813e+04,5.1287e+02,-6.6596e+04]; kdata=[kk2;kk3;kk4] ldata=[kl2;kl3;kl4] kidata=[ ki2, ki3, ki4] .
% Calculate the line equation between two fixed Mach numbers % skil=kidata(2)-kidata(1); bkil=kidata(l)-skil*2; kil=skil*m+bkil; skim=kidata(3)-kidata(2); bkim=kidata( 2 ) -skirn+3; kim=skim*rn+bkim:
% "scheduling" gains % if (m<3),
ki=kil; k=kl; 1=ll1;
else ki=kim; k = h ; 1=lmt;
end
% Program # 6 % Linearization of the system with nonlinear controller
m=input('The value of m=') alfa=input('The value of alfa ='I
% Using the gain'Schedulingm % scheduling
% Airframe and actuator constants % Kalfa=0.02069; Kq=1.23196; Kz=21.4432; danp=O. 7 ; Wa=150;
% Some constants that change from degree to radian % alfan=alfatpi/180; Kalfan=1.18587; Kzrk0.6661697; Kqn=70.585;
% Aerodynamic coefficients % an=0.000103; bn=-0.00945; cn=-0.1696; dn=-0.034; am=O. 000215; bm=-0.0195; cm=0.051; dm=-0.206; ~n=an*alfa~3+bn*alfa^2+cnf(2-m/3I*alfa; ~m=am*alfa~3+bm*alfa"2+cm*(8*m/3-7)'alfa;
% Some differential values % Cnd=3*an*alfaA2+2*bn*alfa+cn*(2-m/3); Cmd=3*am*alfaA2+2*bm'alfa+cm* (8*m/3-7) ; delta=-cm/dm;
% Jacobion matrices at an equilibrium points
$ Finding the step response of this linearized system % t=linspace(0,.6); [y,x] =step(aln,bln, cln, O , l , t) ; plot(t,y) ,grid title('GRAPH-? Step Response for the linearized system') xlabel('time(sec)'),ylabel('output(y(t))')
% PROGRAM 1 7 % SIMULAB M-file = autopil0t.m % This SIMULAB M-file describes the nonlinear controller % which used gain scheduling technique. % function (sys,xOl=autopilot (t,x,u,flag) % % Input: (1) u(t) (2) eta (3) M % outputs: (1) delta-c % States: (1) aiphaB (2) qB (3) deltaB (4) delta-dotB ( 5 ) integral % if abs(flag)==l, % %The gain ( k ,ki and 1 ) for gain scheduling % kk2=[-5.7137e+00,-4.5613e-01,-4.0079e-01,-1.1296e-031; k12=[-7.7941e+02,-7.6420e+04,2.0806e+03,-2.5828e+05]; ki2=6.3689e+Ol; kk3=[-2.2529e+00,-1.7727e-01,-4.0539e-01,-1.1405e-031; k13=[-4.0547e+02,-3.9609+04,9.1918e+02,-1.1657e+05]; ki3=1.4099e+Ol; kk4=[-1.3009e+00,-9.5867e-02,-4.0555e-01,-1.1457e-031; k14=[-2.7506e+02,-2.6813e+04,5.1287e+02,-6.6596e+04]; ki4=5.0734e+00;
kdata=[kk2;kk3;kk4]; ldata= [k12; k13; k141; kidata=[ki2,ki3,ki4]; % % Line equation between two constant operating points % skil=kidata(2)-kidata(1); bkil=kidata(l)-skil*2; kil=skil*u(3)+bkil; skim=kidata (3) -kidata (2) ; bkim=kidata(2) -skim*3; kim=skim*u (3 ) +bkim;
if (u(3)<3), ki=-kil; k=kl; l=1lt ;
else ki=-kim; k=km; l=lml ; end
% % Airframe and actuator constants pi=3.14159; Kalpha=1.18587; Kc~70.586;
Kz=0.6661697; Wa=150; damp=O . 7; % % ~erodynamic coefficient constants an=.000103; bn=-. 00945; cn=-.1696; dn=-. 034; am=.000215; bm=-,0195; cm=.051; dm=- .206; % % Definitions M=u(3); ar=x(l)*(pi/l80); aar =abs (ar ; aa=abs (x ( 1 ) ) ; % % Aerodynamic coefficients (afac and dfac are used for the % perturbation analysis) afac=l; df ac=l; Cn=sign ( ~ ( 1 ) ) (an*aa^3+bn*aa%cnV2-M/3) *aa) +dn*x(3) ; Cm=sign(x(l) ) *afac* (am*aaa3+bm*aaA2+cm* (-7+ (813) *M) *aa) +dfac*dm*x(3) ; deltac=-k(1) *x(l)-k(2) * ~ ( 2 ) - k ( 3 ) * ~ ( 3 ) - k ( 4 ) * ~ ( 4 ) -ki*.x(S); % % Plant state derivatives % sys(l)=(~alpha*~*Cn*cos(aar)+x(2) )+(1(1)*(u(2)-Kz*MA2*Cn) ) ; sys (2) = (~q*M^2*Cm) + (1 (2) '(~(2) - K z * M W n ) ) ; sys(3)=(~(4))+(1(3)*(~(2)-Kz*M^2*Cn)); s y s ( 4 ) = ( - ~ a ~ 2 * x ( 3 ) - ~ 2 * d a m p * W a * x ( 4 ) + W a ^ 2 f d e l t a c ~ + ~ l ~ 4 ~ * ~ ~ ~ 2 ~ - K z * M ~ 2 * C n ~ ) ; sys(5)=u(l)-u(2); % elseif flag==3 % % The gain (K,Ki and L) for gain scheduling % kk2=[-5.7137e+00,-4.5613e-01,-4.0079e-01,-1.1296e-03]; k12=[-7.7941e+02,-7.6420e+04,2.0806e+03,-2.5828e+05]; ki2=6.3689e+01; kk3=[-2.2529e+00,-1.7727e-01,-4.0539e-01,-1.1405e-03]; k13=[-4.0547e+02,-3.9609+04,9.1918e+02,-1.1657e+051; ki3=1.4099e+01; kk4=[-1.3009e+00,-9.5867e-02,-4.0555e-01,-1.1457e-03]; k14=[-2.7506e+02,-2.6813e+04,5.1287e+02,-6.6596e+04]; ki4=5.0734e+00; kdata=[kk2;kk3;kk4]; ldata= [ k12 ; k13 ; k14 I ; kidata=[ki2,ki3,ki4]; % % line equation between two constant operating points % skil=kidata(2)-kidata(1); bkil=kidata(l)-skil*2; kil=skil*u(3)+bkil; skim=kidata(3)-kidata(2); bkim=kidata(2)-skiin*3; kim=skimtu (3 ) +bkim;
if (u(3)<3), ki=-kil; k=kl; 1=ll1 ;
else ki=-kim; k=km; l=lm' ;
end % % Autopilot outputs % sys (l)=-k(l) *x(l)-k(2) * x ( 2 ) - k ( 3 ) * ~ ( 3 . ) - k ( 4 ) * ~ ( 4 ) -kltx(5); % elseif flag==O % sys=[5;0;1;3;0;0]; x0=[0;0;0;0;01; else sys=[l; % end %
% PROGRAM 11 8 % SIMULAB M-file = missi1e.m % This SIMULAB M-file describes the nonlinear missile. % % For simulation purposes the Mach number is generated here % though it is not properly part of the missile. % function [sys,xO]=missile(t,x,u,flag) % % Input: (1) delta-c % Outputs: (1) eta (2) M % States: (1) alpha (2) q (3) delta (4) delta-c (5) M % if abs (flag) ==l, % % Airframe and actuator constants pi=3.14159; Kalpha=1.18587; Kq=70.586; Kz=0.6661697; Wa=150; damp=O. 7; % % Aerodynamic coefficient constants an=.000103; bn=-. 00945; cn=-.1696; dn=-. 034; am=.000215; bm=-.0195; cm=.051; dm=-. 206; % % Definitions M=x(5) ; ar=x(l) (pi/l80) ; aar=abs (ar) ; aa=abs ( ~ ( 1 ) ) ; % % Aerodynamic coefficients (afac and dfac are used for the % perturbation analysis) afac=l; df ac=l ; Cn=sign(x(l) ) * (an*a-bn*aa%cn* (2-M/3) *aa) +dn*x(3) ;
% plant state derivatives sys (1) =KalphatM*Cn*cos (aar) +x(2) ;
' SYS (2 ) =Kq*MA2 *Cm; sys(3)=x(4); sys ( 4 ) = - ~ a ~ 2 * ~ ( 3 ) - 2 * d a m p * W a * x ( 4 ) + W a * 2 * ~ ( 1 ) ; % % Mach state derivative used for simulation purposes sys (5) =-0. 0207*MA2*abs (Cn) *sin(aar) -0. 0062*MA2*cos (aar) ; % % elseif flag==3 % % Airframe constant Kz=0.6661697; % % Aerodynamic coefficient constants an=.000103; bn=-. 00945; cn=-.1696; dn=-,034;
% % Definitions M=x(5); aa=abs ( x ( 1) ) ; % % Aerodynamic coefficient Cn=sign(x(l))*(an*aaA3+bn*aa^2+cn*(2-M/3)*aa)+~*x(3); % % Plant outputs SYS (1) =Kz*MA2 *Cn; % % Mach output used for simulation purposes % sys (2 ) =M; % elseif flag==O % sys=[5;0;2;1;0;01; x0=[0;0;0;0;2.51; else sys=[I; % end % %