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7 AD-A 39 44 A PL IIIt'9 NG A D DEPT OF AERONAUT ICAL AND MEC - FG 17/7"FG-PEA ORMANCE T MAC . NG SYSTEMSTE4FA6flA C O
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AD A119440
AFWAL-TR-82-3032
DESIGN OF HIGH-PERFORMANCE TRACKING SYSTEMS
PROFESSOR B PORTER
DEPARTMENT OF AERONAUTICAL AND MECHANICAL ENGINEERING
UNIVERSITY OF SALFORD
SALFORD M5 4WT
ENGLAND
JULY 82
TECHNICAL REPORT FOR PERIOD 15 NOVEMBER 1980 - 30 SEPTEMBER 1981
APPROVED FOR PUBLIC RELEASE:DISTRIBUTION UNLIMITED
FLIGHT DYNAMICS LABORATORYAF WRIGHT AERONAUTICAL LABORATORIESAIR FORCE SYSTEMS COMMANDWRIGHT-PATTERSON AFB, OHIO 45433
0.. EUROPEAN OFFICE OF AEROSPACE RESEARCH AND DEVELOPMENT3 DEPARTMENT OF THE AIR FORCE
223-231 MARYLEBONE ROADLONDON NW1 5TH S SEP 21 982
82 09 21 056 A
NOTICE
When t.ove.nment drawings, specifications, or other data are used for any purposeother than in connection with a definitely related Government procurement operation,the United States Government thereby incurs no responsibility nor any obligationwhatsoever; and the fact that the government may have formulated, furnished, or inany way supplied the said drawings, specifications, or other data, is not to be re-garded by implication or otherwise as in any manner licensing the holder or anyother person or corporation, or conveying any rights or permission to manufactureuse,. or sell any patented invention that may in any way be related thereto.
This report has been reviewed by the Office of Public Affairs (ASD/PA) and isreleasable to the National Technical Information Service (NTIS). At NTIS, it willbe available to the general public, Including foreign nations.
This technical report has been reviewed and is approved for publication.
DUANE P. RUBURTUS, Chief EVARD H. FLN, ChiefControl Techniques Group Control Systems Development BranchControl Systm Development Branch Flight Control Division
FOR THE COMMANDER
ERNEST F. HDEREColonel, USAFChief, Flight Control Division
"If your address has .changed, if Vou wish to be removed from our mailing list, orif the addressee is no longer employed by your organization please notfyJ"J=,W-PAFB, OH 45433 to help us maintain a current mailing list*.
Copies of this report should not be returned unless return Is required bv securityconsiderations, contractual obligations, or notice on a specific document.
!m
REPORT DOCUMENTATION PAGE READ INSTRUCTIONSBEFORE COMPLETING FORM
1. Report Number 2. Govt Accession No. 3. Recipient's Catalqg Number
4. Title (and Subtitle) 5. Type of Report & Period CoveredScientific Report
DESIGN OF HIGH-PERFOPMANCE TRACKING 15 November 1980 -SYSTEMS 30 September 1981.
6. Performing Org. Report NumberUSAE/DC/120/81
7. Author(s) 8. Contract or Grant NumberProfessor Brian Porter F49620-81-C-0026
9. Performing Organization Name and Address 10. Program Element, Project, TaskDepartment of Aeronautical & Mechanical Area & Work Unit Numbers
EngineeringUniversity of Salford 62201FSalford M5 4WT, England 2304/02
11. Controlling Office Name and Address 12. Report DateAFWAL/FIGL 1 July 1982Wright-Patterson AFBOhio 45433, USA 13. Number of Pages
_ 16714. Nlonitoring Agehcy Name and Address 15.
EOARD/LNVDepartment of t Air Force223/231 Old Marylebone Road, London NW1 5
16. & 17. Distribution Statement
Approved for public release; distribution unlimited.
18. Supplementary Notes
19. Key WordsMultivariable control systemsp tracking systems; high-gain controllers;fast-sampling controllers; singular perturbation methods.
20. Abstract-7p Conceptually and computationally simple methods for the design of high-
performance tracking systems are described. These methods are equallyapplicable to the design of both analogue and digital controllers, andare based upon the exploitation of a solitary system-theoretic resultfrom the singular perturbation analysis of transfer function matrices.Illustrative examples involving the design of tracking systemsincorporating both high-qain analogue controllers and fast-sampling digitalcontrollers are presented.
FORK 14739. -. ,,
• m U m nl i Ul In ,.. [ ,1[--
PREFACE
This research was sponsored by the Air Force Flight Dynamics
Laboratory, (AFWAL/FIGL), Wright-Patterson AFB, United States
Air Force, under Contract F49620-81-C-0026 and was performed
in the Engineering Dynamics and Control Laboratory at the
University df Salford. In this work, the significant con-
ceptual contributions of Dr A Bradshaw and the computational
assistance of Dr A Hemami are gratefully acknowledged. This
report was expertly typed and collated by Mrs D Millward.
NT I "e-D7'_' ':' ;I
7-a4_
IU L . ..
Ii.
ATABLE OF CONTENTS
Page
CHAPTER I : INTRODUCTION . . . " . ...... 1
1,1 : INTRODUCTION ... 2... ........... 2
1.2 : REGULAR AND IRREGULAR PLANTS ...... 2
CHAPTER 2 : DESIGN OF TRACKING SYSTEMS INCORPORATINGHIGH GAIN ERROR ACTUATED CONTROLLERS ... 5
2.1 : INTRODUCTION ... ... ... ... .. .. 6
2.2 : ANALYSI.S .. s .............. 8
2.3 : SYNTHESIS ... ... ... .. . . . .. 11
2.4 : ILLUSTRATIVE EXAMPLE .... ... ... 12
2.5 : CONCLUSION ... ... ... ... .. . .. 14
CHAPTER 3 DESIGN OF TRACKING SYSTEMS INCORPORATINGFAST-SAMPLING ERROR-ACTUATED CONTROLLERS 33
3.1 : INTRODUCTION 35
3.2 : ANALYSIS .................. . .. 37
3.3 SYNTHESIS 39
3,4 : ILLUSTRATIVE EXAMPLE .. ... ... .. 40
3.5 : CONCLUSION .* ... ... ... ... ... 43
CHAPTER 4 : DESIGN OF TRACKING SYSTEMS INCORPORAT idGINNER-LOOP COMPENSATORS AND HIGH-GAINERROR-ACTUATED CONTROLLERS .. .. .... 62
4.1 : INTRODUCTION --- ** o ..... .... .. 63
4.2 : ANALYSIS 0. 00. ... ... *.. ... 66
4.3 : SYNTHESIS . 0 * ..... ...... 68
4.4 : ILLUSTRATIVE EXAMPLE .... .... .. 70
4.5 : CONCLUSION . .... ... ... .. 73
!, &A
it ~-7
Page
CHAPTER 5 : DESIGN OF TRACKING SYSTEMS INCORPORATINGINNER-LOOP COMPENSATORS AND FAST-SAMPLINGERROR-ACTUATED CONTROLLERS ... ... ... 111
5.1 : INTRODUCTION ... ... ... ... ... ... 112
5.2 : ANALYSIS ... . ... *..... ... 116
563 : SYNTHESIS ... ... ...... ... ... 118
5,4 : ILLUSTRATIVE EXAMPLE .... . . .. ... 120
5.5 : CONCLUSION ... *** . . .... ... 123
CHAPTER 6 : CONCLUSIONS AND RECOMMENDATIONS ... ... 161
6.1 : CONCLUSIONS ... ... ... ... ... ... 162
6.2 : RECOMMENDATIONS ..... .. ... ... 162
REFERENCES e.. ... ... ... ... ... ... 64
_,_....."_._i__,____ - - - . _ _- _ __ _ __._, .
ABBR E V I AT I ON S AND SY MB 0 L S
A plant matrix
B input matrix
C output matrix
F measurement matrix
K0 K1 controller matrices
M transducer matrix
T sampling periodcAk-l k-th Markov parameter
G(A) transfer function matrixr(A) asymptotic transfer function matrix
e error vector
f sampling frequency
g gain parameter
u control input vectorv command input vector
w measurement vector
x state vector
y output vectorz integral of error vector
C set of complex numbers
9set of real numbers
,pxq set of real pxq matricesT contlnuous-time set O,+-)TT discrete-time set (O,T,2T,...}
z t set of transmision zeros
Zs set of 'slow' modes
z fset of 'fast' modesv11
CHA PTE R
INTRODUCTION
1.1 INTRODUCTION
In spite of the extensive effort which has been expended
during the past twenty years, most currently available
techniques (Athans and Falb 1966, Rosenbrock 1974, MacFarlane
1980, Wonham 1974, Wolovich 1974, Davison and Ferguson 1981)
for the design of tracking systems incorporating linear
multivariable plants are both conceptually and computationally
rather complicated. Furthermore, such techniques are almost
exclusively concerned with the design of analogue controllers
and are therefore inapplicable to the much more important
practical task of designing digital controllers.
The design techniques described in this report are both
conceptually and computationally simple. In addition, these
techniques are equally applicable to the design of both
analogue and digital controllers. This universality and
simplicity derives from the fact that these techniques are
based upon the systematic exploitation of a solitary system-
theoretic result from the theory of singular perturbations
(Porter and Shenton 1975). This system-theoretic result
exhibits the distinctive asymptotic structure of the transfer
function matrices of linear multivariable systems with 'slow'
and 'fast' modes in a manner which is directly applicable to
the design of tracking systems incorporating either high-
gain analogue controllers or fast-sampling digital controllers.
162 REGULAR AND IRREGULAR PLANTS
In the context of the design of high-performance tracking
2
systems, it is convenient to classify linear multivariable
plants as follows:
(i) Regular plants: first Markov parameter of maximal
rank, minimum phase;
(ii) Irregular plants: first Markov parameter of non-
maximal rank, non-minimum phase.
The design of tracking systems incorporating regular plants
is considered in Chapters 2 and 3, whilst the design of
tracking systems incorporating irregular plants is considered
in Chapters 4 and 5. In the case of regular plants, arbitrarily
fast and non-interacting tracking behaviour is achievable by
appropriate selection of gain parameters or sampling periods
simply by implementing error-actuated analogue or digital
proportional-plus-integral controllers. However, in the case
of irregular plants, good tracking behaviour is achievable
only by the simultaneous implementation of inner-loop com-
pensators to provide appropriate extra measurements for
control purposes.
This result is crucially important since it makes explicit
the intuitively obvious fact that controller and transducer
designs are inseparable, and confirms the fact that the
irrelevance of much of 'modern' control theory to practical
engineering design derives from its failure to consider in-
tegrated controller/transducer system. The importance of
the difference between regular and irregular plants obviously
requires that the set of transmission zeros of linear multi-
variable plants (Porter and D'Azzo 1977) be readily computable
3
and very powerful software for this purpose has accordingly
been developed (Porter 1979).
I
. . . ... . i - i _ i _ - _ - _ . , , 4,.
- -
CHAPTER 2
DESIGN OF TRACKING SYSTEMS INCORPORATING
HIGH-GAIN ERROR-ACTUATED CONTROLLERS
4 -.
. 5
2.1 INTRODUCTION
In this chaptersingular perturbation methods are used
to exhibit the asymptotic structure of the transfer function
matrices of continuous-time tracking systems incorporating
linear multivariable plants which are amenable to high-gain
error-actuated analogue control (Porter and Bradshaw 1979a).
Such tracking systems consist of a linear multivariable plant
governed on the continuous-time set T = [O,+-) by state and
output equations of the respective forms (Porter and
Bradshaw 1979b)
x1 (t) F 11 , A121 Fx(t) FoF 1 = , I + u(t) (2.1)
L 2 (t) [A2 1 , A2 2J Lx2 (t)- 2
and
y(t) [c1 , c2 tIJ ( (2.2)
together with a high-gain error-actuated analogue controller
governed on T by a control-law equation of the form
u(t) = g{K 0e(t)+Klz(t)} (2.3)
which is required to generate the control input vector u(t)
so as to cause the output vector y(t) to track any constant
command input vector v(t) on T in the sense that the error
vector e(t) - v(t) - y(t) assumes the steady-state value
lim e(t) - lim{v(t)-y(t)} - 0 (2.4)
6
for arbitrary initial conditions. In equations (2.1),(2.2) ,(2.3),and (2.4) ,x (t) E: e-, x (t) E: RL, u (t) R' , y (t) E RL,
AlEl R(n-L)x(nL)# A1 E R(n-)xL A RLx(n- L) , A2 2 ERxk,B2 ER xL, CIERLx(n-L), c 2 ER'xL, rank C2B2 -L, e(t)IR t ,
v(t) E: RL, z(t) = z(o) + Lt edt ER1, KoC_X , K1 CRxL,
and g E R.
It is evident from equations (2.1),(2.2),(2.3) that
such continuous-time tracking systems are governed on T by
state and output equations of the respective forms
1 (t)- 1 0 , -C 1 C2z (t)1
c1 (t) =(0 A A11 A 12 x 1 (t)
x2 (t)j [B 2K1 , A2 1 -gB 2 KoC1 , A22-gB2KoC_ Lx2 (t)j
+ 0O v(t) (2.5)
and
y (t) = [O , C 1 , C2]X 1(t) .(2.6)
Lxc2 (t)J
The transfer function matrix relating the plant output vector
to the command input vector of the closed-loop continuous-
time tracking system governed by equations (2.5) and (2.6) is
clearly
71.
Xit #, C 2
G(,X)m[OC 1 1 C 21 0 , X1 n...LA 11 -A 120
L-gB 2K1, A2 1 +gB2 KoC1 , XI -A 2 2+gB2 KoC2.] LgB2KoJ
(2 7)
and the high-gain tracking characteristics of this sytem can
accordingly be elucidated by invoking the results of Porter
and Shenton (1975) from the singular perturbation analysis
of transfer function matrices.
These results yield the asymptotic form of G(X) as the
gain parameter g - - and thus greatly facilitate the deter-
mination of controller matrices K and K1 such that the
continuous-time tracking behaviour of the closed-loop system
becomes increasingly 'tight' and non-interacting as g is
increased. The frequency-response and step-response
characteristics of a continuous-time tracking system in-
corporating an open-loop unstable chemical reactor (MacFarlane
and Kouvaritakis 1977) are presented in order to illustrate
these general results.
2.2 ANALYSIS
It is evident from equation (2.7) thatby regarding
e - 1/g as the perturbation parameter, the asymptotic form
of the transfer function matrix G(A) of the continuous-time
tracking system as g - - can be determined by invoking the
results of Porter and Shenton (1975) from the singular
perturbation analysis of transfer function matrices. Indeed,
these results indicate that as g e the transfer function
matrix G(W) assumes the asymptotic form
r(A) = r(x) + r () (2.8)
where
r(X1 , Co(x)In-A o ) - I B ° (2.9)
r() - C2 (II gA4 ) gB2K° (2.10)
-K (2.11)0 = LAl 2 C2 KoI , Al _A1 2 cClc (2.11)
B o (2.12)12C2
C- [K- 1 K , o (2.13)
and
A4 - -B 2 KoC2 ( (2.14)
It follows from equations (2 .8),(2.9)and(2.11) thatthe 'slow'
modes Z. of the tracking system correspond as g - e to the
poles Z1U Z 2 of r(X)lwhere
z- 0 Cc'IxKo+K1 Imo} (2.15)
and
2 {1 1C:IXInt -A1 i+A 2 C2Cl ' 0} , (2.16)
and from equations(2.8) ,(2.10) ,and(2.14)that the 'fast'modes Zf of the tracking system correspond as g . - to I
p9
the poles Z3 of r(x) where
z ()iC:1AI +CB 2K~0 (2.17)z3 - I gC, c2Kol-01
Furthermore, it follows from equations (2.9),(2.11),(2.12),
and (2.13)that the 'slow' transfer function matrix
r( ) - 0 (2.18)
and from equation (2.10) and(2.14) that the ' fast' transfer
function matrix
-1.r)= (It+gC2 B2ro)' gC2B2Xo . (2.19)
Hence, in view of equations(2.18) and (2.19), it is
evident from equation (2.8) that as g - the transfer
function matrix G(M) of the continuous-time tracking system
assumes the asymptotic form
rF() - (xIt+gC2B2K0)- gC2B2Ko (2.20)
in consonance with the fact that only the 'fast' modes
corresponding to the poles Z3 remain both controllable and
observable as g - -. Indeed, the 'slow' transfer function
matrix i(A) vanishes precisely because the 'slow' modes
corresponding to the poles Z1 become asymptotically un-
controllable as g . - in view of the block structure of
the matrices Ao and B. in equations (2.11) and (2.12) whilst the
'slow' modes corresponding to the poles Z2 become asymptotically
unobservable as g - in view of the block structure of the
matrices Ao and CO in equations (2.11) and (2.13)
10
3. SYNTHESIS
It is evident from equations (2.5) and (2.6) that tracking
will occur in the sense of equation (4) provided only that
Zs U Zf CC (2.21)
where C" is the open left half-plane. In view of equations
(2.15),(2.16),and(2.17),the 'slow' and 'fast' modes will
satisfy the tracking requirement (2.21) for sufficiently
large gains if the controller matrices Ko and K1 are chosen
such that both ZICC- and Z3 C C in the case of minimum-
phase plants for which (Porter and D'Azzo 1977) the set of
transmission zeros
z- A c:kz n 9-A1+A2C21Cl1.0} CC- (2.22)
since it is then immediately obvious from equation (2.16)
that Z2 C C. Moreover, in such cases, tracking will become
increasingly 'tight' as g . in view of equation (2.20).
Furthermore, if K0 is chosen such that
C2B2Ko M diag{ala 2,...,O} (2.23)
where a ER + (j-1,2,...,L), it follows from equation (2.20)
that the transfer function matrix G(X) of the continuous-
time tracking system will assume the diagonal asymptotic
form
go gV2 gor, diag , ... , (2.24)
119o2
and therefore that increasingly non-interacting tracking
behaviour will occur as g . e.
2.4 ILLUSTRATIVE EXAMPLE
These general results can be conveniently illustrated
by designing a high-gain error-actuated analogue controller
for an open-loop unstable chemical reactor governed on T
by the respective state and output equations (MacFarlane
and Kouvaritakis 1977)
" 1(t) . 1.38 , -0.2077 , 6.715 ,-5.676' X1 (t),
;2(t) -0.5814 , -4.29 , 0 , 0.675 ix2()
3(t) 1.067 , 4.273 , -6.654 , j 3(t)Lc4 (t) J I .o.048 4.273 ,1.343,-2.104J JLx4( t ) -
5.679 , 0 ul(t)(+ ut)(2.25)11.136 , -3.146 [u2 (t)JL1.136 , 0
and
rx1 (t)ly 2(tlJ L, 1 , o , oJx3(t)J 1.6
t -"64 ( t)j
from which it can be readily verified that Zt - (-1.192,
-5.0391 C C" and that the first Narkov pareter
12
C 2B2 (2o.27)22 - .5.679 ..0 •(2.2
is of full rank. In case {q ,a2 } - {1,i} and K - 2Ko , itfollows from equations 2.3),t2.23)and(2.27)that the correspond-
ing high-gain error-actuated analogue controller is governed
on by the control-law equation
,ct 0 060. 76 1 e F (t) 0 .3522 iZ lt)
Lu2 (t .3179, 0 1Le2 (t)] +[0.6358, .o L 2'(-t)
.... (2.28)
and if is evident frdm eiquationis (2.17)1,(" 2.i)81, and f2.19)"that
z - {-2,-21},Z" {l.192-4.1039}, 'and Z- i {-g,-g). : It
is algo evident from: equation "(1. 2 4) that the asyptotic
transfer fuiction matrix assufis the'diagonal form
24g) 0 (2.29)r m.and therefore that the closed-loop continuous-time tracking
-"" ,l • . . :
sytmincorporating the chemical reactor will exhibitincreasingly tight' and non-interacting tracking behaviour
as g "*"when the control input vector Eul ( t ) U z(t)]T is
generated by the high-gain analogue controller governed on
T by equation (2.28).
The actual frequency-response loci G(iw) for w E(--,+ml are
shn in Figs 2.1, 2.2, and2.3 when g - 25, 50, and 100,
respectively and it is clear that the actual frequency-response
13
loci approach the asymptotic frequency-response loci r(iw)
as the gain parameter g is increased. The corresponding
step-response characteristics are shown in Fig 2.4 and Fig 2.5
when CVl(t) , v2 (t)JT - [I , T and Cv 1(t) , v2 (t)]T -
[i , 1] T, respectively, and it is evident that increasingly
'tight' and non-interacting tracking occurs as the gain
parameter g is increased.
2.5 CONCLUSION
Singular perturbation methods have been used to exhibit
the asymptotic structure of the transfer function matrices
of continuous-time tracking systems incorporating linear
multivaxiable plants which are amenable to high-gain error-
actuated analogue control. It has been shown that these
results greatly facilitate the determination of controller
matrices which ensure that the closed-loop behaviour of such
continuous-time tracking systems becomes increasingly 'tight'
and non-interacting as the gain parameter g is increased.
These general results have been illustrated by the presenta-
tion of the frequency-response and step-response character-
istics of a continuous-time tracking system incorporating
an open-loop unstable chemical reactor.
14
1.0
0.5
0.0
-0.5
-1.0 ~..-0.5 6.0 0.5 1. 1.5
Fig 2.1 (a): G 1 1 iw:q -25
V 15
1.0
0.5
-0.5
Fig 2.1I(b)~ G 12 w); 25
16
. ... . . .. . . . . . .
0.5
0.0
I
-0.5
-1.0 I I
-0.5 0.0 0.5 ie . 1.5Fig 2.1(c): G2 .(iw)., g , 25.
17 -
.. .. .. . . . . . .. . . . . . . . .
1.0
0.5
0.0
-0.5
-1.0 . ., . . . . .-0.5 0.0 0.5 1.8 1.5
Fig 2.1(d)-. G 22 (iw); g -25
0.5e
-0.0-
0.0.5.0 1.5Fig 2 .2(a): Gj 1 j) g' 50
19
1 . . . . . . . . .
Fig 2.2(b): G 1 2 (iw); g =50
20
1 .0 I I ..
1 .0 . . . . . ... . . .. .., .... .. .. . . .. . . .
0.5
-0.5 0.01 6.5 1.a 1.5Fig 2 .2 (c): G 2 1() g1 50
21
1.0
o.5
0.0
-8.5
-6.5 6.6 6.5 1.8 1.5-Fig 2.2 (d): G 22 (±w); g So5
22
0.5
0.0
-0.5-
-0.5 0.0 8.5 1.0 1.5rig .3 (a): G 1 (i-); g;m - lo
23
1.0
0.5
0.0
-0.5 -
- 1 .0 .. . I . . .
-0.,5 08. 0.5 1.0 1.5Fig 2.3(b): G12 (iw); g - 100
24
0.5
0.0
-0.5
-- 1.8 ' * . . I t | ,I | -
--. 5 8.8 0.5 1.0 1.5Fig 2 .3(a): G2 1 (iw); g - loo
25
1 .0
0.5
0.0
-0.5
-1'.3 * . .. .... S . . * I . , a
-0.5 a.0 0.5 1.0 1.5Fig 2.3(d): G2 2 (iw); g " 100
26
.,: l .. , 11 - r _ .. ..=, , . .m
* I * I * I * I * aN
4J
4.' 4J
'.4
b)
U,
'.4
II
0*'
- S
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*1*4r34
U,a
S
N CD6 0 6 0 S a a
27
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C~CD
co ODco(!
28,
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44)
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C% GO CO (N CD C
32
CHAPTER 3
DESIGN OF TRACKING SYSTEMS INCORPORATING
FAST-SAMPLING ERROR-ACTUATED CONTROLLERS
33
t.
3.1 INTRODUCTION
In this chapter,singular perturbation methods are used
to exhibit the asymptotic structure of the transfer function
matrices of discrete-time tracking systems incorporating
linear multivariable plants which are amenable to fast-
sampling error-actuated control (Bradshaw and Porter 198Oa).
Such tracking systems consist of a linear multivariable
plant governed on the continuous-time set T = CO,+Ga) by
state and output equations of the respective forms (Bradshaw
and Porter 1980b)
= + u(t) (3.1)
Lx2 tJ [A2 1 , A2 2 j x 2 (t)
and
y x ) = C (3.2)y~t) = [ I , 2 x(t)]
together with a fast-sampling error-actuated digital con-
troller governed on the discrete-time set TT = {O,T,2T,...}
by a control-law equation of the form
u(kT) = f{K 0e(kT)+K 1 z(kT)} (3.3)
which is required to generate the piecewise-constant control
input vector u(t) - u(kT), t F[kT,(k+l)T), kT ETT, so as
to cause the output vector y(t) to track any constant command
input vector v(t) on TT in the sense that the error vector
e(t) = v(t) - y(t) assumes the steady-state value
34
lim e(kT) - lim{v(kT)-y(kT)} - 0 (3.4)k-0 k-w
for arbitrary initial conditions where f - i/T and T is
the sampling period. In equations (3.1) , (3.2) , (3.3) , and (3.4),Xl(t) E R n-1, x(t)€ R, Lu(t) E R' , y(t) ( _R' , A '7l-R (n-U)x (n-£)
Al12 R(n-Z)xL A2 1 'Rx(n-1) , A22 , B2 FRlx,
Cl Rtxln-L), C CRLx , rank C2 B2 9 ., e(t) C R , v(t) ER' ,j-k-1 j-2X1
z(kT) - z(o) +T E e(jT) E Rt , Ko0 CRx KIR and
f ER + .
It is evident from equations( 3 .1),( 3 .2 ), and (3.3) that
such discrete-time tracking systems are governed on TT by
state and output equations of the respective forms
z {(k+l) T})
22{ (k+l) T}_
[I? -TC1 -TC2 z(kT)1
[fT T2 2 l-f2KoC I, '22- f' 2 KoC 2 J x2 (kT)J
TI
fy 1 K v(kT) (35)
and
z(kT)1
y(kT) - , C1 , C21 xI(kT) (3.6)
35
where
0 11 01" exp 1 , 37ad 21 ' 02 2J LA2 1 ' A2 2jf
and
"sj[A ,A t [B2 dt .(3.8)
2A21 ' 22J
The transfer function matrix relating the plant output
vector to the command input vector of the closed-loop
discrete-time tracking system governed by equations (3.5)
and (3.6) is clearly
G(A)
X I-I TC 1 TC 2 JlITI I= [0'CI' C21 I -flKl" In-k -0ll+f~iKoCl '- 12 +fT 1Ko0C2 fT 1f Ko0
-f f2K 1', -'21 +f'2KoCl ,XAI - 2 2 +fV2 K0 C2 Lf 2K
(3.9)
and the fast-sampling tracking characteristics of this
system can accordingly be elucidated by invoking the results
of Porter and Shenton (1975) from the singular perturbation
analysis of transfer function matrices.
These results yield the asymptotic form of G(X) as the
sampling frequency f - and thus greatly facilitate the
determination of controller matrices K0 and K1 such that the
discrete-time tracking behaviour of the closed-loop system
becomes increasingly 'tight' and non-interacting as f is
36 _________
increased. The frequency-response and step-response
characteristics of a discrete-time tracking system incorporat-
ing an open-loop unstable chemical reactor (MacFarlane and
Kouvaritakis 1977) are presented in order to illustrate these
general results.
3,2 ANALYSIS
It is evident from equation (3.9) that, by regarding
e - 1/f as the perturbation parameter, the asymptotic form
of the transfer function matrix GMA) of the discrete-time
tracking system as f - - can be determined by invoking the
results of Porter and Shenton (1975) from the singular
perturbation analysis of transfer function matrices. Indeed,
since it follows from equations (3.7) and (3.8) that
lim f l1In-£' 1 12 [ " 11 , A1 21 (3.10)
L 21 ' 22-I1j LA2 1 , A22 J
and
limf - (3.11)
these results indicate that as f - the transfer function
matrix G(A) assumes the asymptotic form
r(A) - r( ) + r( ) (3.12)
where
r(m) - C0 (An-I n-TAo)-I'm (3.13)
37
[A -o-0ol
o l 1 (3.15)
B - , (3.16)2 2
Co - 1KJo 1 , o] (3.17)
and
A = -B KoC . (3.18)4 20o2
It follows from equations (3.12),(3.13), and (3.15) that the
'slow' modes Z of the tracking system correspond as f *
to the poles Z1IU Z2 of r(x) where
z - {X CC: XI -I +TKoKIf=O} (3.19)
and
z2 - (X CC:II n_-ZIn_1-TA 1 1 +TA1 2 C 21 -C1 120 (3.20)
and from equations (3.12),(3.14), and (3.18) that the 'fast'
modes Z f of the tracking system correspond as f . - to the
poles Z3 of r(m) where
z - {(A CC:IAI-+B 2 K2Xo'0 . (3.21)
Furthermore, it follows from equations (3.13), (3.15) ,(3.16),
and(3.17) that the 'slow' transfer function matrix
)- 0 (3.22)
38
and from equations (3.14) and (3.18) that the 'fast' transfer
function matrix
( - (xi-I +C2B2,o) - 2 2Ko . (3.23)
Hence, in view of equations (3.22) and (3.23) , itis evident
from equation(3.12) thatas f - o.the transfer function
matrix GMd of the discrete-time tracking system assumes
the asymptotic form
r ( 1i -L ' BIl-- C B2BK (3-24)
in consonance with the fact that only the 'fast' modes
corresponding to the poles Z remain both controllable and
observable as f - m. Indeed, the 'slow' transfer function
matrix r(d vanishes precisely because the 'slow' modes
corresponding to the poles Z1 become asymptotically un-
controllable as f - - in view of the block structure of the
matrices A and B in equations (3.15) and (3.16) whilst the
'slow' modes corresponding to the poles Z2 become asymptotically
unobservable as f - in view of the block structure of the
matrices Ao and Co in equations (3.15) and (3..7).
33 SYNTHESIS
It is evident from equations (3.5) and (3.6) that tracking
will occur in the sense of equation (3.4) provided only that
Zs U Zf CP (3.25)
where 0- is the open unit disc. In viewof equations (3.19)
39
(3.20),and(3.21),the 'slow' and 'fast' modes will satisfy
the tracking requirement (3.25) for sufficiently fast sampling
frequencies if the controller matrices K° and K1 are chosen
such that both Z1C V for sufficiently small sampling
periods and Z3 C - in the case of minimum-phase plants for
which (Porter and D'Azzo 1977) the set of transmission zeros
Zt {X eC:Ixin -A +A 1 C (3.26)
where C is the open left half-plane since it is then
immediately obvious from equation (3.20)that Z2 C V- for
sufficiently small sampling periods. Moreover, in such cases,
tracking will become increasingly 'tight' as f - in view
of equation (3.24). Fufthermore, if K is chosen such that0
C2B2Ko - dUial{ala 2 ,...,at1 (3.27)
where 1-o LRflV (j-1,2,...,L), it follows from equation
(3.24)that the transfer function matrix G(X) of the discrete-time
tracking system will assume the diagonal asymptotic form
M) - dag a la 2 a ... (3.28)
and therefore that increasingly non-interacting discrete-
time tracking behaviour will occur as f . -.
3.4 ILLUSTRATIVE EXAMPLE
These general results can be conveniently illustrated
by designing a fast-sampling error-actuated digital controller
for an open-loop unstable chemical reactor governed on T
40
-- . .. --. . . • • : --
by the respective state and output equations (MacFarlane
and Kouvaritakis 1977)
" It)- 1.38 , -0.2077 , 6.715 , -5.676 "xl(t)1x 2 (t) m -0.5814 , -4.29 , 0 0 0.675 x 2 (t)
X3(t) ( 1.067 , 4.273 r -6.654 , 5.893 x3 (t)
4 (t). 0.048 , 4.273 , 1.343 , -2.104j Lx 4 (t),
0 015.679 , 0 Ul(t)+ ult](3.29)1.136 , -3.146 Lu2 (t)J
1.136 , 0
and
ol o 1 1 (It)1
y.(t) - 0 [ 1 x2 (t) (3.30), , 0, olx3(t)j
LX4 (t) J!!
from which it can bo readily verified that Zt - {-1.192,
-5.039} C C and that the first Markov parameter
0 -3.146"
C2B2 .679 : (3.31)
In case {l, 2 } - {1,l},and 11 - 2Ko , it follows from
equations (3.3) ,(3.27),and(2.38)that the corresponding fast-
saupling-error-actuated digital controller is governed on
TT by the control-law equation
41
M 1(T)1 [ 0 ,0.17611re1 (kT)1 1 ,o32i (kT)1mf +
U2 (kT)j 1C-O.3179, 0 e 2 (kT)" .0.6358, 0 Lz2 (kT)JJ
(3.32)
and it is evident from equations '(3.19),(3.20), and (3.21) that
Z1 {1-2T,1-2T}, Z2 - {1-1.192T,1-5.039T}, and Z = {0,O}.
It is also evident from equation (3.24) that the asymptotic
transfer function matrix assumes the diagonal form
rm - (3.33)
and therefore that the closed-loop discrete-time tracking
system incorporating the chemical reactor will exhibit
increasingly 'tight' and non-interacting tracking behaviour
as f * when the piecewise-constant control input vector
ul(t) , u2(t) ] T - ul(kT) , u2 (kT)]T, tC~kT , (k+l)T),
kT E TT, is generated by the fast-sampling digital con-
troller governed on TTby equation (3,32).
The actual frequency-response loci G(eiwT) for wT [0 , 2w]
are shown in Figs 3.1, 3.2, and3.3whenT-0.04, 0.02,and0.0l,
respectively, and it is clear that the actual frequency-
response loci approach the asymptotic frequency-response
loci r(eiWT) as the sampling frequency f is increased. The
corresponding step-response -charateriatics are shown in
Fig3.4 and Fig 3. 5whenl(t) , v 2 (t)] T _ [1 , O]T and [vl(t)
v2 (t) ] _[O , 1] T , respectively, and it is evident that in-
creasingly 'tight' and non-interacting tracking occurs as the
sampling frequency f is increased.
42
I t I I t u -- t t n I-.--
3,5 CONCLUSION
Singular perturbation methods have been used to exhibit
the asymptotic structure of the transfer function matrices
of discrete-time tracking systems incorporating linear multi-
variable plants which are amenable to fast-sampling error-
actuated digital ctzol. It has been -shown that these
results greatly facilitate the determination of controller
matrices which ensure that the closed-loop behaviour of
such discrete-time tracking system become increasingly
'tight' and non-interacting ai the sampling frequency f is
increased. These general results have been illustrated by
the presentation of the frequency-response and step-response
characteristics of a discrete-time tracking system incorporat-
ing an open-loop unstable chemical reactor.
41 •.
1 .5
6.5
-6.5
-1.0
-1 .5-1.5 -1.0 -6.5 6.6 6.5 1.0 1.5
Fig 3.1(a): G11 08 );T T - 0.04
44
1 .5
0.5
0.0
-0.5
-1.5 - 18 -0'.5 6.6 -0.5 1 .0 1.5
Fig 3.1(b): G 1 2 (e );T T'- 0.04
45I
1.
6.5
6.0
-6.5
-15 -1.0 -0.5 0.0 0.5 1.6 1.5
Fig 3.1(c):G1( w T - 0.04
46
1.5 * , , ,
1.0
0.5
0.0
-0.5
-1.0 L
-1 .5 * , I , I , I
-1 .5 -1.0 -0.5 0.0 0.5 1.0 1.5Fig 3.1(d): G2 2 (ewT; T -0.04
47
II
1.0
0.5
0.0
-0.5
-1.5 -1.0 -0 .5 0.0 0.5 1.0 1.5
Fig 3.2 (a):G(ew T =0.02
48
1.5 .
1.0-
0.5
0.0
-0.5
-1 .0
-1 5 I I
-1.5 -1.0 -0.5 0.6 0.5 1.0 1.5Fig 3.2(b): G12(iT); T - 0.02
49
1.5
0.5
0.0
-0.5
-1.5
-15 -1.0 -0.5 0.0 0.5 1.0 1.5Fig 3. 2 (c): G2 1 (e i UT T - 0.02
50
1.5'
1.0-
0.5
0.0
-0.5
-1.0
-1 .5
-1.5 -1.0 -6.5 0.0 0.5 1.0 1.5
Fig 3.2(d): G22 T - 0.02
51
1.5
10
-0.5
-1.0
-1.5 -1.0 -0.5 0.10 0.5 1.0 1.5Fig 3.3(a): G 1 1 (.±wT) T - 0.01
52
1.5| I i I *
1.0
0.5
0.0
-0.5
-1.0
-1 .5 ....-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
Fig 3.3(b): G1 2 (e ±wT T - 0.01
53
LL
1.5'
1.0
0.5
0.0
-0.5
-1.8
- 1 .5 ,AI I .
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
Fig 3.3(c): G2 1 (e~" 'r ); T - 0.01
54
1.0
0.5
0.0
-0.5
-1.0
-1.5 -1.6 -6.5 0.0 0.5 1.0 1.5Fig 3.3 (d): G 2 2 (eiWT); T - 0.01
,1 _I ll .- * i |1 I I J _
4-)
Lo
56I(~,4
0
I"
56
___ ___ __ - -~ _-~~ -- ~~-.-
E-4
U,I
0 .44 D
-4z
00
I|
In
57
-- I -II
CN
'-44
N*>1S
00 coN (24
580
44)
44
C1
0
E.4
r3
a; .D CD CD
59
- 1
0
6EA
N I --
60 • '
Lfl')
,,,--S
S
C~4 CO (S S S SS S S
9-E-,-
60 .
)*61
CHAPTER 4
DESIGN OF TRACKING SYSTEMS
INCORPORATING INNER-LOOP COMPENSATORS
AND HIGH-GAIN ERROR-ACTUATED CONTROLLERS
62
4.1 INTRODUCTION
In this chapter, singular perturbation methods are used
to exhibit the asymptotic structure of the transfer function
matrices of continuous-time tracking systems incorporating
linear multivariable plants which are amenable to high-gain
error-actuated analogue control (Porter and Bradshaw 1979)
only if extra plant output measurements are generated by
the introduction of appropriate transducers and processed
by inner-loop compensators. Such tracking systems consist
o0 a linear multivariable plant governed on the continuous-
time set T = [o,+-) by state, output, and measurement equations
of the respective forms
= [+u W A2 2 f 2] u (t) (4.1)
y(t) - c, C2 (4.2)
and
w(t) - Fl , F [] ( t ) (LX2 W_)
together with a high-gain error-actuated analogue controller
governed on T by a control-law equation of the form
u(t) - g{Koe(t)+KlZ(t)} (4.4)
which is required to generate the control input vector u(t)
so as to cause the output vector y(t) to track any constant
63
command input vector v(t) on T in the sense that
lim{v(t)-y(t)1 = 0 (4.5)
as a consequence of the fact that the error vector e(t) =
v(t) - w(t) assumes the steady-state value
lir e(t) = lim{v(t)-w(t)} 0 (4.6)
for arbitrary initial conditions. In equations (4.1), (4.2),
(4.3) ,and(4.4),xl(t) E Rn - 1, x2 (t) E R', u(t) E R', y(t) E RZ,
w(t) E R i , All(ER ( n - U~x ( n - Z) , A12 E R ( n - t ) x t , A 21 ER 1x ( n - E) ,
A2 2 ER"xL, B2 ER XL , C I E Rx(n- ), C2 ER xL, FIERtX(n-.E)
F2 ER 1XI , rank C2B2 < L, rank P2B2 = Z, e(t) R1, v(t)ZRL,
z(t) - z(o) + ot edt ER, K E RX, KI zRX, g(ER + , and
[F1 , F2] = [Cl+MAll , C2 +MA1 2] (4.7)
where M ERLx(n -L). It is evident from equations (4.2), (4.3), and
(4.7) that the vector
w(t) - y(t) = CMA 1 l , MAI. 2 ] (4.8)
of extra measurements is such that v(t) and y(t) satisfy
the tracking condition( 4.5) for any M ERx(n-L) if S(t)
satisfies the steady-state condition (4.6) since equation (4.1)
clearly implies that
lira[All AI1211 x l ( t ) 1 0(49
64p -
in any steady state. However, the condition that rank F2 B2
requires that C2 C R 1X and A1 2 E R(n - j ) x 1 are such that
M ERLx(n-L) can be chosen so that
rank F2 = rank(C2 +MAl2) - 9 . (4.10)
It is evident from equations (4.1),(4.2),(4.3), and (4.4)
that such continuous-time tracking systems are governed on
T by state and output equations of the respective forms
0(t IL ' F1 F - 2 ]Z(t)
i 1(t = , 1 1 A12 ()x2 (t) B2X1 , A21-gB2KoF1 , A2 2 -gB 2KoF 2J -x2 (t)J
+ O vlt) (4. 11)
and
[Z (t)1
Y (t) - C , C 1 1 C2 ] [xl(t)} (4. 12)
The transfer function matrix relating the plant output
vector to the command input vector of the closed-loop
continuous-time tracking system governed by equations (4.11)
and (4.12) is clearly
r~ ~ I , F1 , 2 -tI
G(A)-'E' lC 2] 1 0 X )I n--A 1 A1
-7gB2K11 -A214.qD2 Xo1 1, I9 -A22+gB2K0 2 g2O
65 (4.13)
and the high-gain tracking characteristics of this system
can accordingly be elucidated by invoking the results of
Porter and Shenton (1975) from the singular perturbation
analysis of transfer function matrices.
These results yield the asymptotic form of G(X) as
the gain parameter g * and thus greatly facilitate the
determination of controller and transducer matrices Ko, Ki ,
and M such that the continuous-time tracking behaviour of
the closed-loop system becomes increasingly non-interacting
as g is increased. The frequency-response and step-response
characteristics of a continuous-time flight-control system
for the longitudinal dynamics of an aircraft (Kouvaritakis,
Murray, and MacFarlane 1979) are presented in order to
illustrate these general results.
4.2 ANALYSIS
It is evident from equation(4.13)that, by regarding
e = 1/g as the perturbation parameter, the asymptotic form
of the transfer function matrix G(X) of the continuous-time
tracking system as g * = can be determined by invoking the
results of Porter and Shenton (1975) from the singular
perturbation analysis of transfer function matrices. Indeed,
these results indicate that as g * - the transfer function
matrix G(A) assumes the asymptotic form
r(m) - r(A) + r( (4. 14)
where
66
F(A) Co (AIn-Ao) -1B (4.15)
A -() - C2 (xIt-gA4 ) -gB 2K0 (4.16)
[ -o 1K1 , 0 1(A0 - El 2~u~1 I.K , A1 _A1 2 (4.17)
Bo - [A,2F-] (4.18)2 2
Co" [ -2 1 K0 1 C 1 - 2 F] , (4. 19)
and
A4 - -B 2KoF 2 (4. 20)
It follows from equations (4.14), (4.15), and (4.17) that the
'slow' modes Zs of the tracking system correspond as g -
to the poles Z 1 ,UZ 2 of r() where
zI - (AEc:IXK o+K1 10) (4. 21)
and
Z2 - {X CCIAIn_t-A11 .+A 1 2 1FI4-} (4. 22)
and from equations (4.14), (4.16), and (4.20) that the 'fast'
modes Z of the tracking system correspond as g * to the
poles Z3 of F(A) where
z 3 - (X EC:Ixl L+gF 2B 2 K0o0} . (4. 23)
Furthermore, it follows from equations (4.15), (4.17), (4.18),
and(4.19) that the 'slow' transfer function matrix
67
r(X) = (C1 -C 2 F21 F1 ) (XInk-A1 1 +A1 2 F2 F1 ) A1 2 F2
1 (4.24)
and from equations(4 .16) and(4.20) that the 'fast' transfer
function matrix
r -x) = C2F2 1 GI F2 B2 Ko ) gF2B2Ko (4.25)
Hence, in view of equations (4.24) and (4.25), it is
evident from equation(4.14)that as g - - the transfer
function matrix G(X) of the continuous-time tracking system
assumes the asymptotic form
( = (CI-CF 1 FI) (XI -A +A F-1 F)AIF2 1
1 2 21 n-2. 11 1 2F2 F1) 12 2
+ C2F2 1 (XI z+gF 2 B2 K0 ) 1gF2 B2 K0 (4.26)
in consonance with the fact that both the 'slow' modes
corresponding to the poles Z2 and the 'fast' modes correspond-
ing to the poles Z3 possibly remain both controlldble and
observable as g - . However, the 'slow' transfer function
matrix r(A) reduces to the form expressed by equation (4.24)
precisely because the 'slow' modes corresponding to the poles
Z1 definitely become asymptotically uncontrollable as g
in view of the block structure of the matrices A and B in0 0
equations ( 4 .1 7 ) and (4.18).
4,3 SYNTHESIS
It is evident from equations (4.7),(4.9),(4.11), and (4.12)
that tracking will occur in the sense of equation (4.5)
provided only that
68
zsU Zf Cc- (4.27)
where C is the open left half-plane. In view of equations(4.21),
(4.22), and (4.23), the 'slow' and 'fast' modes will
satisfy the tracking requirement(4.27)for sufficiently large
gains if the controller and transducer matrices K0 , K1 , andm are chosen such that Z1C C-, z 2 C C-, and Z 3C C" in
the case of plants for which the transducer matrix M can be
simultaneously chosen so as to satisfy the measurement con-
dition expressed by equation (4.10). Moreover, if K and M
are chosen so that both r(X) and r(X) are diagonal transfer
function matrices by requiring that
F2B2K° W (C2+MA12)B2KO - diag{a 1,a2,...,0 } (4.28)
where ao R+ (j=,2,...,Z) in the case of r(A), it follows
from equation(4.20)that the transfer function matrix G(X)
of the continuous-time tracking system will assume the
diagonal asymptotic form F(A) and therefore that increasingly
non-interacting tracking behaviour will occur as g .
Furthermore, such tracking behaviour will exhibit high
accuracy in the face of plant-parameter variations provided
that the steady-state conditions expressed by equation (4.9)
correspond to 'kinematic' relationships which hold between
the state variables as a consequence of the fundamental
dynamical structure of the plant.
However, increasingly 'tight' tracking behaviour will
not in general occur as g -P . in view of the possible presence
in the plant output vector of 'slow' modes corresponding t(
the poles Z2 of the 'slow' transfer function matrix r().
69
ilk
But the presence of any such 'slow' modes is the inevitable
consequence of the introduction of appropriate transducers
which generate extra plant output measurements so as to
ensure that rank F B2 = and thus render plants for which
rank C2B2 < I amenable to high-gain error-actuated control.
Indeed, increasingly 'tight' tracking behaviour is achievable
as g - only in the case of plants for which rank C2B2 -2
(Porter and Bradshaw 1979) and which are accordingly amenable
to high-gain error-actuated control without the necessity
for the generation of extra plant output measurements.
4,4 ILLUSTRATIVE EXAMPLE
These general results can be conveniently illustrated
by designing a high-gain error-actuated analogue flight
controller for the longitudinal dynamics of an aircraft
governed on T by the respective state and output equations
(Kouvaritakis, Murray, and MacFarlane 1979)
1() 0, 1.132 , 0 , 0 , 1 x 1(W
2t) 0 0 , 0 , 1 , 0 x2(t)
113 t) - 0 , -0.1712 , -0.0538 , 0 , 0.0705 x3 (t)
4(t) 0 , 0 , 0.0485 , -0.8536 , -1.013 x4 (t)
5 J , o , -0.2909 , 1.0532 , -0.6959 .xS(t)_
0 0' 0
0 , 0 ' O u1
+ -0.0012 , 1 , O .2 (t) (4.29)
0.4419 , 0 , -1.6646 Lu3 (t)
0.1575 , 0 , -0.0732J
70
and
Mx1 (t)
LY2 t J 0 0 1 , O J x (M (4.30)Y3(t)- , 1 0 , Q 4 4(t) I
x 5 (t
from which it can be readily verified that the first Markov
parameter
C2B2 O.0012 , 1 , (4. 31)
is rank defective. In case {a1 , 2 03 } - {1,1,1, K1 - Kof
and
M L O ] , (4. 32)0 0.25
it follows from equations (4.3), (4.4), and (4.28) that the
corresponding transducers and high-gain error-actuated
analogue controller are governed on T by the respective
measurement and control-law equations
"x1 (t) 7
f1 (t) 1 [1 , 0.283 ,0 , 0 ,0. 25] .2(t)
1w2(t) 0 , 0 ,1 ,0 ,0 x 3 (t) (4.33)S3(t)J . , 1 , , 0.25 , x4 (t)
x7(t)71"
and
F -28.97 0 -1.274
U2Mtg) -. 1 , -o.oo5 e2 (t)
3u3 L-7.690 , 0 , -2.740 Le3 (t),
F 28.97 ,0 ,-1.274 lzi(t)f+ -0.0348 ,1 ,-0.0015 2 (t) (4'34
L-7.690 , 0 , -2.740 -Z3 (t)i}
and it is evident from equations (4.21), (4.22), and (4.23)
that Z1 = {-1,-1,-i}, Z2 = {-4,-4}, and Z3 = {-g,-g,-g}.
It is also evident from equation(4.26) that the asymptotic
transfer function matrix assumes the diagonal form
4~A 0 ,0
_o 9 o (4. 35)r( ) - , +g ,
[a0 0 '
and therefore that the closed-loop continuous-time flight-
control system for the longitudinal dynamics of the aircraft
will exhibit increasingly non-interacting tracking behaviour
as g * - when the control input vector Cul(t) , u2 (t) , u3 (t)]T
is generated by the high-gain analogue controller governed
on T by equation (4.34). However, it is apparent from equation
(4.35)that increasingly 'tight' tracking behaviour will be
achieved only in the case of Y2 (t) in view of the presence
Ln yl(t) and Y3 (t) of .he 'slow' modes corresponding to the
poles Z2.
72
J.i
The actual frequency-response loci G(iw) for w
are shown in Figs4.1,4.2,and 4.3wheng -1P,20, and 50,
respectively, and it is clear that the actual frequency-
response loci approach the asymptotic frequency-response
loci r(iw) as the gain parameter g is increased. The
corresponding step-response characteristics are shown in
Fig 4.4,Fig4.5,andFig4.6whe[vl(t) , v2 (t) , v3 (t)]T =
[1 , o , 0]T, Cvp (t) I v2 (t) v 3 (t)IT =[O , 1 , O]T, and
[Vl(t) , v2 (t) , v3 (t)]T _ [O , O , 1]T, respectively, and
it is evident that increasingly non-interacting tracking
occurs as the gain parameter q is increased but that in-
creasingly 'tight' tracking occurs only in the case of Y2 (t).
4.5 CONcujsxoN
Singular perturbation methods have been used to exhibit
the asymptotic structure of the transfer function matrices
of continuous-time tracking systems incorporating linear
multivariable plants which are amenable to high-gain error-
actuated analogue control only if extra plant output measure-
ments are generated by the introduction of appropriate trans-
ducers and processed by inner-loop compensators. It has been
shown that these results greatly facilitate the determination
of controller and transducer matrices which ensure that the
closed-loop behaviour of such continuous-time tracking systems
becomes increasingly non-interacting as the gain parameter g
is increased. These general results have been illustrated by
the presentation of the frequency-response and step-response
73
characteristics of a continuous-time flight-control system
for the longitudinal dynamics of an aircraft.
74
0.5
0.0
-0.5
-0.5 0.0 0.5 1.0 1.5Fig 4.1(a): G 1 1 (iw); g-10
75
1 .0
0.5
0.0
-0.5
- 1 , , I I , ,0, i ,-0.5 0.0 0.5 1.0 1.5
Fig 4.1(b): G 1 2 (ii); g = 10
76
1.0 . .
0.5
0.0
-0.5
-1 .0-0.5 0.0 0.5 I .0 1 .5
Fig 4.1(c) G 13 (iw) g = 10
77
1 .0 I. I ~ I
0.5
0.0
-0.5
-1.0 p
-0.5 0.0 0.5 1.0 1 .5Fig 4.1(d): G 2 1 (iw); g =10
78
1.0
0.5
0.8
-0.5
-1 .0 . f . .
-0.5 0.0 0.5 1.8 1.5
Fig.4.1(e). G 2 2 (iw); g 10
79
1.0 .1 1 1
0.5
0.0
-0.5
-0.5 0.0 0.5 1.0 1.5Fig 4.1(f): G2 3 (iw); g = 10
80
___,___23
1 ... ...
0.5
0.0r)
-0.5
"- 1 .0 . . . . . ' .. .. . , p
-0.5 0.0 6.5 1.0 1.5Fig 4.1(g): G3 1 (iw); g 10
81
dA
1 .
-0.5
-1 .8-0.5 0.0 8.5 1.0 1.5
Fig 4 . 1(h): 3 2 ('W) g 10
82
1.5
0.5
-0.5
-o.5 0.0 0.5 1.0 1.5Fig 41(i)-. G 3 3 (iw); g-10
83
0.5
0.0
-0 .5
-o.5 0.0 0.5 1.0 1.5Fig 4.2(a): Gljjiw); g =20
84
1.60 .
0.5
0.0
-0.5
-0.5 0.6 0.5 1.0 1.5rig 4.2 (b): G 1 2 (iw) g -20
85
1 .0 . . .
8.5
0.0
-0 .5
-1
.0
k
-0.5 0.0 0. 1.0Fig 4.2(c): G1 3 (iw); g =20
86
1.0
0.5
0.0
-0.5 0.0 0.5 1.0 1.5Fig 4.2 (d): G 2 1 (iw); g -20
87
1.0 ,
0.5
0.0
-0.5
*-1 .0 p p I
-0.5 0.0 0.5 1.0 1.5
Fig 4 . 2 (e): G 2 2 (iw); g =20
88
1.0 . .
0.5
0.0
-0.5
- 1 .0I . . . . . . p I I .
-0.5 0.0 0.5 1.0 1 5
Fig 4.2(f): G2 3 (iw); g - 20
7AD-A119 4.0 SALFOflO UNIV IENGLANI DEPT OF AERONAUTICAL AND MEC-EC Pie 17/7DESIGN OFm P MISW.PgNFCE TRACKING SYSTEMS. 4W
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-0.5 0.0 0.5 1.0 1.5Fig 4.2(g) G3 1 (iw); g = 20
f90
0.5
8.0
-0.5
-0.5 8.0 0.5 1.8 1.5
Fig 4.2(h): G 3 2 (iw); g -20
_ _ _ _ _ _91
1 .0 *
0.5
0.0
-0.5
-0.5 0.0 0.5 1.0 1.5Fig 4.2 (1): G 3 3 (iw); g =20
92
1.0 . .. . . . ."
0.5
0.0
I ~ -0.5
-1 .0 . . . . . . . ,. . ,A 2 .1..
-0.5 0.0 0.5 1.0 1.5Fig 4 .3(a): Gl1(iw); g - 50
93
1 0 1 T I I
0.5
-0.5
-1 0 .
-9.5 0.0 8 .5 1.0 1.5Fig 4.3 (b): G 1 2 (iw);t 50
94
1.0
0.0
-05 .0 1. .a 1.5Fig 4 3 (c): G (w); g -50
134
_ _-~~ <5 __
1.0
0.0
Il -0.5
-1.0 a a a
-0~. 0.0 0.5 1.0 1.5Fig 4.3(d): G 2 1 (iw); g So5
96
0.5
6.0
-1.1
-o.5 0.0 3.5 1.3 1.5Fig 4.3(e): G 2 2 (iw); g 50S
97 _ _ _ _ _ _ _
1.0
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-0.5
-1 .e , , , _., . . . I , , , . I . . . . .
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98
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-0.5
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99
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-0.5
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-085 0. .5 .. 0 1Fig 4.3 (h): G 3 2 (iw); g 50
100
1.0
0.5
-0 0.0 0.5 1.0 1.5Fig 4.3(1): G 3 3 (iw); g -50
101
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110
CHAPTER 5
DESIGN OF TRACKING SYSTEMS
INCORPORATING INNER-LOOP COMPENSATORS
AND FAST-SAMPLING ERROR-ACTUATED CONTROLLERS
111
5.1 INTRODUCTION
In this chapter singular perturbation methods are used
to exhibit the asymptotic structure of the transfer function
matrices of discrete-time tracking systems incorporating
linear multivariable plants which are amenable to fast-
sampling error-actuated digital control (Bradshaw and Porter
1980) only if extra plant output measurements are generated
by the introduction of appropriate transducers and processed
by inner-loop compensators. Such tracking systems consist
of a linear multivariable plant governed on the continuous-
time set T = CO,+-) by state, output, and measurement
equations of the respective forms
-I+ [B u(t) (5.1)Lx 2 ( t LA21 , 22 x2(2)J
Fx(t 1= i y(t)=[C 1 1 C] LxtJ (5.2)2 (5.2)
and
w(t) = [F1 , F2] 1 (.3)
together with a fast-sampling error-actuated digital con-
troller governed on the discrete-time set TT = {0,T,2T,...}
by a control-law equation of the form
u(kT) - f{Koe(kT)+Klz(kT)} (5.4)
which is required to generate the control input vector
112
u(t) - u(kT), t ECkT , (k+l)T), kT CTT, so as to cause the
output vector y(t) to track any constant command input
vector v(t) on TT in the sense that
lim{v(kT)-y(kT)} = 0 (5.5)
as a consequence of the fact that the error vector e(t) =
v(t) - w(t) assumes the steady-state value
lim e(kT) = lim{v(kT)-w(kT)} = 0 (5.6)k-b- k-w
for arbitrary initial conditions. In equations (5.1), (5.2),
(5.3),and(5.4) ,xz(t) CR -1 x 2 (t) CE , u(t) CR , y(t) (Rz
w(t) C RI, All R(n- i)x(n- L), AI12 R(n - t)xL, A21 uRlx(n - 9),
A2 2 CR xt , B 2 E RIXI, C 1RIx(n-1), C2 ER x t , F ECRLx(n - ),
F2 ERXI, rank C B < Z,rank F 2B2 = L, e(t)C R', v(t)( R
z(kT) - z(o) + T Z e(jT) CR , KoCRXL, KC RLX, f eR ,
J-Oand
[F 1 , F2] = [Cl+MAI1, C2+MA1 2] (5.7)
where M CRLx(nh). It is evident from equations (5.2), (5.3),
and (5.7) that the vector
w(t) - y(t) - [MAll , MA1 2 ] [xt] (5.8)X2()
of extra measurements is such that v(kT) and y(kT) satisfy
the tracking condition(5.5)for any M CRLx(n - ) if e(kT)
satisfies the steady-state condition (5.6) since equation (5.1)
clearly implies that
113
lim[Azl , AI2 0o (5.9)t ®W £x2 L =
in any steady state. However, the condition that rank
F2B2 = £ requires that C2ER x z and A1 2 ER(n-I)xk are
such that M ER x(n- k) can be chosen so that
rank F 2 = rank(C 2+MA1 2 ) = z . (5.10)
It is evident from equations (5.1),(5.2),(5.3), and (5.4)
that such discrete-time tracking systems are governed on TT
by state and output equations of the respective forms
z{ (k+l)T}1x (k+1)T}
2{ (k+l) T}j
9 , -TF 1 , -TF 2 1 rz(kT) 1= T K 1 ,1 1 1 -f! 1 K0 F 1 , 12 fI*KoF 2 xl(kT)
f L 2 KI , 121-fT2 KoF1, 2 2 -fI 2 KoF 2J Lx2 (kT)J
_f~;jk (5.1K1)
and
Fz(kT)1ykT)= [O , C1 , C2] Lxl(kT) (5.12)
2C2 (kT)_
where
114
" 1 0 expi1l A1 jT (5. 13)t21 02 J LA21 ' A22JJ
and
~TfexA11 Al2]ltlFdt . (5. 14)J 1LA21 A 2 [
The transfer function matrix relating the plant output
vector to the command input vector of the closed-loop
discrete-time tracking system governed by equations (5.11)
and (5.12) is clearly
G(X) =
X1I-I , TF I TF2 iTI[0C1' C2 ] {- f 1EK l , IIn - '' l + f ' 1E O F I ' - (' 1 2 + f ' 1E O F 2 fT 1f Ko0
L-f2K1, -021+f'2KoF1 , AIX-0 222+fY2KoF2 f 2 K 0K
(5.15)
and the fast-samling tracking characteristics of this system
can accordingly be elucidated by invoking the results of
Porter and Shenton (1975) from the singular perturbation
analysis of transfer function matrices.
These results yield the asymptotic form of G(M) as the
sampling frequency f . and thus greatly facilitate the
determination of controller and transducer matrices Ko, KI,
and M such that the discrete-time tracking behaviour of the
closed-loop system becomes increasingly non-interacting as
f is increased. The frequency-response and step-response
115
characteristics of a discrete-time flight-control system
for the longitudinal dynamics of an aircraft (Kouvaritakis,
Murray, and MacFarlane 1979) are presented in order to
illustrate these general results.
5.2 ANALYSIS
It is evident from equation (5.15) that, by regarding
E = 1/f as the perturbation parameter, the asymptotic form
of the transfer function matrix G(X) of the discrete-time
tracking system as f - can be determined by invoking the
results of Porter and Shenton (1975) from the ingular
perturbation analysis of transfer function matrices. Indeed,
since it follows from equations (5.13) and (5.14) that
lim f ll-In-9 , 12 1 = 12I[Al A1 2] (5.16)
f c L (21 ' 22- 1 21' A2 2]
andI (
lim f = , (5.17)
these results indicate that as f ® the transfer function
matrix G(A) assumes the asymptotic form
r(x) = r() + r(x) (5.18)
where
r(A) = C0 ( Bn-I-TA0o , (5.19)
116
rm c 2 (XI CI A 4) B E2 K0 ,(5.20)
[Al22KI1 AII-A1 2 F2 F] (5.21)
B = , (5.22)0 [A2F 21
= EC2 F 2 QK 1 , C -C F21 F1] (5.23)
and
A 4 = -B 2 KoF 2 (5.24)
It follows from equations (5.18), (5.19), and (5.21) that the
'slow' modes Zs of the tracking system correspond as f -
to the poles Z1 U Z2 of F(A) where
z = {X C:lI-I +K-X 1KI=0} (5.25)
and
Z= { X EC:IXInt-I nZ-TAI+TAI 2 F 21 FI =O} , (5.26)
and from equations (5.8), (5.20), and (5.24)that the 'fast'
modes Z f of the tracking system correspond as f * to
the poles Z3 of r(m) where
Z3 - {X eC:IXI I z+F 2 B2 K0 I=0} (5.27)
Furthermore, it follows from equations (5.19),(5.21),(5.22),
and(5.23)that the 'slow' transfer function matrix
= (Ct-CF21F) (XIn_-In_- TAI+TA F21 F )-TA F2112 2 1 - 1 12 2 1 12 2
(5.28)117
and from equations (5.20) and(5.24)that the 'fast' transfer
function matrix
^-1 ( I£I£FBK )-1FBKfl() = C2F 2 (XIzI z+F2B2K0) F 2 . (5.29)
Hence, in view of equations (5.28) and (5.29), it is
evident from equation(5.18)that as f - - the transfer
function matrix G(X) of the discrete-time tracking system
assumes the asymptotic form
r(A1 -( CFl )X TA +TA F-1F' -1 TAF 1F (X) = (CI-C 2 F2 IF I ) (x~n_£-In_£Tl+A2;11 IA2;1 21 n- nk 11 12 2 F1) 12 2
+ C2F21 (XII-I +F2 B2 Ko ) -1F2B2Ko (5.30)
in consonance with the fact that both the 'slow' modes
corresponding to the poles Z2 and the 'fast' modes correspond-
ing to the poles Z3 possibly remain both controllable and
observable as f - =. However, the 'slow' transfer function
matrix r(X) reduces to the form expressed by equation (5.28)
precisely because the 'slow' modes corresponding to the
poles Z1 definitely become asymptotically uncontrollable as
f - - in view of the block structure of the matrices A and0
B in equations (5.21) and (5.22).
5,3 SYNTHESIS
It is evident from equations (5.7),(5.9),(5.11), and (5.12)
that tracking will occur in the sense of equation (5.5)
provided only that
ZsU Zf CD (5.31)
118
where V- is the open unit disc. In view of equations (5.25),
(5.26)iand(5.27),the 'slow' and 'fast' modes will satisfy
the tracking requirement(5. 31)for sufficiently small sampling
periods if the controller and transducer matrices K0 , Ki,
and M are chosen such that both Z1CV _ and Z2 CD- for suf-
ficiently small sampling periods and Z3 C D- in the case of
plants for which the transducer matrix M can be simultaneously
chosen so as to satisfy the measurement condition expressed
by equation(5.10). Moreover, if Ko and M are chosen so that
both r(X) and r(X) are diagonal transfer function matrices
by requiring that
F2B2K o = (C2+MA12)B2Ko = diag{al1a2 ,...,a2, (5.32)
where l-a(:Rnv- (j=l,2,...,Z) in the case of r(X), it
follows from equation (5.24)that the transfer function matrix
G(X) of the discrete-time tracking system will assume the
diagonal asymptotic form r(A) and therefore that increasingly
non-interacting tracking behaviour will occur as f - -.
Furthermore, such tracking behaviour will exhibit high accuracy
in the face of plant-parameter variations provided that the
steady-state conditions expressed by equation(5.9)correspond
to 'kinematic' relationships which hold between the state
variables as a consequence of the fundamental dynamical
structure of the plant.
However, increasingly 'tight' tracking behaviour will
not in general occur as f w in view of the possible presence
in the plant output vector of 'slow' modes corresponding to
the poles Z2 of the 'slow' transfer function matrix r(A).
But the presence of any such 'slow' modes is the inevitable119
consequence of the introduction of appropriate transducers
which generate extra plant output measurements so as to
ensure that rank F2B2 = Z and thus render plants for which
rank C2B2 < Z amenable to fast-sampling error-actuated control.
Indeed, increasingly 'tight' tracking behaviour is achievable
as f - only in the case of plants for which rank C2B2 =
(Bradshaw and Porter 1980) and which are accordingly amenable
to fast-sampling error-actuated control without the necessity
for the generation of extra plant output measurements.
5,4 ILLUSTRATIVE EXAMPLE
These general results can be conveniently illustrated
by designing a fast-sampling error-actuated digital flight
controller for the longitudinal dynamics of an aircraft
governed on T by the respective state and output equations
(Kouvaritakis, Murray, and MacFarlane 1979)
" (t) 7 0 , 1.132 , 0 , 0 , 1 -x (t)
2 () 0, 0 0 0 , 1 , 0 x2 (t)
x3 (t) = 0 , -0.1712 , -O.0538 , 0 , 0.0705 x 3 (t)
4(t) , 0 , 0.0485 , -0.8536 , -1.013 x 4 (t)
x 5 (t)] 0 , 0 , -0.2909 , 1.0532 , -0.6859 x 5 (t)
0 , 0
0 10 0 Fut 1+ -0.0012 , 1 , 0 u2 (t) (5.33)
0.4419 , 0 ,-1.6646 u3 (t)J
0.1575 , 0 , -0.0732
120
and
"xl(t)]L yit 1 10 0 0,0 x2(tM
2 (t) , 0 , 1 , 0 , x3 (t) (5.34)
3(t)],0 , , 0 , x4(t)X4(t)
from which it can be readily verified that the first Markov
parameter
00
C2B2 = -0.012 , 1 , (5.35)
is rank defective. In case {a 1, 2 ,a3} - {1,1,1}, K1 = K0 ,
and
M [0 , (5.36)
0 0.25
it follows from equations (5.3), (5.4), and (5.32) that the
corresponding transducers and fast-sampling error-actuated
digital controller are governed on T and TT by the respective
measurement and control-law equations
x1 (t)
Mw1t) [1 , 0.283 , 0 ,0 ,0.251 x 2t)W2(t) 0 , 0 , 1, 0 , (3t) 5.37)
3 (t) , 1 , 0 , 0.25 , x4(t)
x5 (t).
121
and E 1 (kT)1 f[28.97 , 0 , -1.274 l e 1(kT)1U2 (kT) f -0.0348 , 1 r -0.0015 e2 (kT)
u3 (kT)" -7.690 , 0 , -2.740 j Le3 (kT)I
-28.97 , , -1.274 - zl(kT)i
S-.038 ,1 ,--01 z 2(kT) (5.38)
.690 , 0 , -2.740 ilz 3 (kT)J
and it is evident from equations (5.25),(5.26),and(5.27) that
z 1 = {l-T,1-T,1-T}, Z2 = {1-4T,l-4T,1-4T}, and 73 = {0,0,0}.
It is also evident from equation (5.30) that the asymptotic
transfer function matrix assumes the diagonal form
1(x) 0 [ 0 (5.39)
4T
and therefore that the closed-loop discrete-time flight-control
system for the longitudinal dynamics of the aircraft will
exhibit increasingly non-interacting tracking behaviour as
f * when the piecewise-constant control input vector
[ul(t) , u2 (t) , u3 (t)3T = [ul(kT) , u2 (kT) , u 3 (kT)] T ,
t E[kT , (k+l)T), kT ETT, is generated by the fast-sampling
digital controller governed on TT by equation (5.38). However,
it is apparent from equation (5.35)that increasingly 'tight'
tracking behaviour will be achieved only in the case of Y2 (t)
in view of the presence in yl(t) and Y3 (t) of the 'slow'
modes corresponding to the poles Z2 .
122
The actual frequency-response loci G(ei T ) for WT C[O , 2r]
are shown in Figs 5.1, 5.2,and 5.3whenT=0.i0, 0.05, and 0.02,
respectively, and it is clear that the actual frequency-response
loci approach the asymptotic frequency-response loci r(eiWT )
as the sampling frequency f is increased. The corresponding
step-response characteristics ara shown in Fig 5.4, Fig 5.5, and
Fig5.6when[vl(t) , v2(t) , v3(t)]T = [ , 0 , 0]T, [Vljt) ,
v2(t) , v 3 (t)]T = Co , 1 o] T , and [VC1(t) , v2(t) , v 3 (t)] T =
[O , O , 1]T, respectively, and it is evident that increasingly
non-interacting tracking occurs as the sampling frequency f
is increased but that increasingly 'tight' tracking occurs
only in the case of Y2 (t)"
5.5 CONCLUSION
Singular perturbation methods have been used to exhibit
the asymptotic structure of the transfer function matrices
of discrete-time tracking systems incorporating linear
multivariable plants which are amenable to fast-sampling
error-actua!:ad digital control only if extra plant output
measurement& are generated by the introduction of appropriate
transducers and processed by inner-loop compensators. It
has been shown that these results greatly facilitate the
determination of controller and transducer matrices which
ensure that the closed-loop behaviour of such discrete-time
tracking systems becomes increasingly non-interacting as the
sampling frequency f is increased. These general results
have been illustrated by the presentation of the frequency-
123
response and step-response characteristics of a discrete-
time flight-control system for the longitudinal dynamics of
an aircraft.
124
1 .5 *
1 .0
0.5
0.0
-0.5
-1.0
-1.5 * I * t ** I *
-1.5 -1.0 -0.5 0.0 0.5 1.0. 1.5
Fig S. I(a): G (e iwT T 0.10
125
1.5 * * I '
1.0
0.5
0.0
-0.5
-1 .0
-1.5 I , I , , I I
-1 .5 -1 .0 -0.5 0.0 0.5 1.0 1.5
Fig 5.1(b): G1 2 (e ; T 0.10
126
1.5 *
1 .0
0.5
6.0
-0.5
-1.0
-1.5 I
-1.5 -1.6 -6.5 0.0 0.5 1.0 1.5
Fig 5.1(C): G 1 3 (e );T T - 0. 10
127
1.5 *
1 .0
0.5
0.0
-0.5
-1 .0
-1 .5- 1.5 -1.0 -0.5 0.0 0.5 1 .0 1 .5
Fig 5.1(d): G2 eiT T=01
128
1 .0
0.5.
0.0
-0.5
-1.0
-1.5 -1.0 -0.5 0.0 6.5 1.0 1.5
Fig 5.1(6): G 2 2 (ei); T =0. 10
129
1.5
1 .0
0.5
0.0
-0.5
-1 .0
-1.5 I
-1.5 -1.0 -0.5 0.0 0.5 I.0 1.5
Fig 5.1(f): G2 3 (e i ) T ) 0.10
130
1.5 *
1 .0
0.5
0.0
-0.5
-1.0
-1.5 I
-1.5 -1.0 -0.5 0.0 0.5 1.8 1.5Fig 5.1 (g): G 3 1 (e jtT) T = 0. 10
1.31
1.5 *
1 .0
0.5
0.0
-0.5
-1 .0
-1.5 I i _ _ _ _ _ _ _ _ _ _ _ _
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
Fig 5.1 (h) : G 32e iwT) T =0. 10
132
1.5 *
1 .6
0.5
0.0
-6.5
-1 .5 I
-1.5 -1.0 -0.5 6.0 6.5 1.0 1.5
Fig 5.1(1): G 3 3 (e );T T - 0.10
133
1.5
1 .0
0.5
0.0
-0.5
-1 .0
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
Fig 5.2(a): G 1 2 (e );T T =0.05
134
1.5 * I I *
1.0-
0.5
0.0
-0.5
-1.0
- 1.5 I , I , , ! , I
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
Fig 5.2(b): G1 2 (e i T); T = 0.05
135
1 .5
1 .0
0.5
0.0
-0.5
-1 .0
-1.5 *
-1 .5 -1 .0 -0.5 0.0 0.5 1.0 1.5Fig 5.2 (c): G1 3 (eiwT T=00
136
1.5 ,
1.0
0.5
0.0
-0.5
-1 .0
-1.5 I * I * p I I
-1 .5 -1.0 -0.5 0.0 0.5 1.0 1 .5
Fig 5.2(d): G2 1 (eiT); T 0 0.05
137
1 .0
0.5
0.0
-1 .5 -1.0 -0.5 0.0 0.5 -1.0 1.5
Fig 5.2(e): G 2 2 (e );T T =0.05
138
1.5 ! * I
1.0
0.5
0.0
-0.5
-1.0
-1.5 I * I * * I * 1-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
Fig 5.2(f): G2 3(e ); T = 0.0.5
139
1.5 *
1 .0
0.5
0.0
-0.5
-1.0
-1.5 I
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
Fig 5.2(g): G 31(e iwLT) T = 0.05
140
1.5 *
1 .0
0.5
0.0
-0.5
-1 .0
-1.5 I a
-1 .5 -1.0 -0.5 0.0 0.5 1.0 1.5
Fig 5.2(h): G 3 2 (e );T T - 0.05
141
1.5 *
1 .0
0.5
0.0
* 1 -0.5
-1 .0
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
Fig 5.2(1): G 3 3 (e ) ; T -0.05
142
1.5 *
1 .0
0.5
0.0
-0.5
-1.0
-1.5 I 1
-1.5 -1.0 -0.5 0.0 0.5 1.0 1 .5
Fig S.3(a): G 1 e );T T = 0.02
143
1 .5
1 .0
0.5
0.0
-0.5
-1 .0
-1 .5-1.5 -1.0 -0.5 0.0 0.5 1 .0 1.5
Fig 5.3 (b): G 1 2 (e );T T 0.02
144
1.5 *
1 .0
0.5
0.0
-0.5
-1 .0
-1.5 I
-1 .5 -1.0 -0.5 0.0 0.5 1.0 1.5
pig 5.3(c): G 1 3 (e );T T =0.02
145
1 .5 I
1 .0
0.5
-0.5
-1 .0
-1.5-1.5 -1.0 -0.5 0.0 0.5 1.0 1 .5
Fig 5.3(d): G 2 1 (e );T T =0.02
146
1 .0
0.5
0.0
-0 .5
-1 .0
-1 .5 -1.0 -0.5 0.0 0.5 1 .0 1.5Fig 5.3(e): 0 2 2 (e );T T =0.02
147
1 .5 s
1 .0
0.5
0.0
-0.5
-1 .0
-1.5 I-1 .5 -1 .0 -0.5 0.0 0.5 1.0 1.5
Fig 5 .3 (f): G 23(e iwT) T =0.02
148
1.5 *
1 .0
0.5
0.0
-6.5
-1 .0
-1.5 I
-1.5 -1.6 -6.5 0.0 6.5 1.6 1.5i wT
rig 5.3(g): G 3 1 (e );T =0.02
1494
1.5
1 .0
0.5
0.0
-0.5
-1 .0
-1.5 p
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
Fig 5.3(h): C 3 2 (eiwT T =0.02
150
1.5 I
1.0
0.5
0.0
-0.5
-1 .0
-1.5 a
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
Fig 5.3(1): G 3 3 (e );T T =0.02
151
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CONCLUS IONS AND REcoMENDATIONS
1L61
L____
6.1 CONCLUSIONS
The conceptual and computational simplicity of singular
perturbation methods (Porter and Shenton 1975) in the design
of both analogue and digital tracking systems incorporating
error-actuated controllers has been demonstrated. This
demonstration has been effected by designing such controllers
for plants which have previously been considered by other
methods (MacFarlane and Kouvaritakis 1977, Kouvaritakis,
Murray, and MacFarlane 1979) so that detailed comparisons
between alternative design techniques can readily be made,
at least in the case of error-actuated analogue controllers.
Moreover, further demonstration of the power of the
design techniques described in this report has been provided
by the design of direct digital flight-mode controllers for
the F-16 aircraft (Porter and B' adshaw 1981). In addition,
the fact that high-gain and fast-sampling error-actuated
controllers remain highly effective in the presence of gross
actuator nonlinearities has been conclusively demonstrated
(Porter 1981a, Porter 1981b, Porter 1981c, Porter 1981d).
Thus, the same basic design techniques have been shown to be
applicable to both linear and nonlinear multivariable plants.
6.2 RECOMMENDATIONS
The controllers for the F-16 aircraft designed by
singular perturbation methods (Porter and Bradshaw 1981) are
extremely robust in the sense that the same controller
remains effective throughout a wide range of different flight
162
conditions. However, in case even higher performance
characteristics are required, the use of adaptive controllers
becomes unavoidable and the study of fast-sampling adaptive
digital controllers is therefore recommended.
Finally, although fast-sampling digital controllers for
the F-16 aircraft have been successfully implemented in the
Engineering Dynamics and Control Laboratory at the University
of Salford using microprocessor hardware (Garis 1981), the
implementation of such controllers in actual aircraft and sub-
sequent flight testing is recommended. Such practical im-
plementation would obviously entail close liaison with the
aerospace industry, based initially upon the wider dissemination
of the design methodologies described in this report.
163
116
goof
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167
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