Chapter 6

Extremal Plant Identification and Controller Designfor Multi-linear lTncertain Launch Vehicle

in the Atmospheric Phase6.1 Introduction

Most control design methods focus on the nominal plant. with some emphasis on the

possible perturbations. While it is not practical to design a controller meeting the

performance specifications if)r all possible plant perturbations, there is a possibility of

designing a controller meeting the performance specifications for a set of plants.

Parametric uncertain systems such as the atmospheric phase launch vehicles considered

in Chapter::; have lightly damped modes with large phase uncertainty, as well as large

aerodynamic uncertainties. with each parameter varying within specified bOllnds. (Imler

such conditions, it is possible to have a case-based design: i.e. to design a controller

meeting the performance specifications for a set of plants. If the plants selected f()r the

design are the worst cases in some sense, then the controller is robust with respect to the

set of cases identified. The major drawback of the case-based design is the heuristic

element involved. !-\ny new case identified later leads to a design modification if the case

falls outside the robustness boundary. The case-based design is employed in both

industry and aerospace applications. with classical design tuned for worst case systems.

The major dra\vback of the method is that it relies strongly on the experience of the

designer.

It is seen in Chapter 5 that the extended Kharitonov theorem along \A/ith the ivlapping

theorem define extremal systems for the launch vehicle plant \\7ith controller. which give

the guaranteed stability margins of the multi-linear uncertain system with negligibl)

small conservatism. At present, no systematic procedure exists for identification of the set

of extremal plants of systems with multi-linear parametric uncertainties, such as the

launch vehicle plant. The current chapter addresses this problem by developing a

systematic. Kharitol1ov-based approach for the design of robust controller for this class of

plants.

109

The following are the significant contributions of this chapter:

(i) The development of a systematic. Kharitonov-based approach for the c1ec.;ign of

robust controller for a class of multi-linear parametric uncertain systems. and the

application of the method for launch vehicle autopilot design. This method gives

pmgressive improvement in the stability margins obtained in successive

iterations, and can enable design for the maximum performance possible for given

parametric uncertain plant. The Kharitonov based approach. which is a robust

stability analysis tooL is thus converted into a robust design tool. To the best of

the author's knowledge. no such systematic method for determination of the worst

case plant templates for multi-linear parametric uncertain svstems has been

developed elsewhere.

(ii) With the systematic method evolved for definition of the plant templates. the

powerful Quantitative Feedback Theory (QFT) approach for robust stability

analysis of parametric uncertain systems has become a viable method for robust

control of launch vehicle autopilot. A robust low order controller is designed to

meet the system specifications, which are translated into QFT bounds.

The chapter is organized as follows:

The proposed design approach is described 111 Section 6.2. The first design iteration.

\vhere the extremal plants corresponding to the classical controller designed in Chapter S

are identified. is described in Section 6.3 Section 6.4 gives the second iteration. in which

LQG controller design is carried out to give satisfactory response for the identified

extremal set of systems. Section 6.5 evaluates the extremal plants correspondi ng, to the

LQG controller. while Section 6.6 gives the QFT based design. and extremal plant

evaluation Section 6.7 summarises the major results.

6.2 A Systerrlatic Approach for Case-Based Control SystemDesign

Since no systematic approach exists for extremal plant identification and robust controller

design for plants with multi-linear parametric uncertainty. a new approach is proposed in

this section. The proposed design approach is as follo\\s:

110

Consider a nominal plant P and extremal set SO. A controller CO designed for the plant P.

Assume that the design method used for generating CO is MO. For the pair (P,CO). the

method KO is used for the identification of the set of extremal plants S I. This is the first

iteration of the design. Now for the extremal plants S L assume a new method M] is used

for the design ofa modified controller C1 catering to the cases Sl. For CP, C1) there is a

set of extremal plants S2 determined using the algorithm K 1. This process can be

continued and the iterative process is defined as follows.

(1 ) For the extremal cases of the previous iteration. design controller.

(2) For the plant and the controller designed. identi f'y the new set of worst cases.

(3) Continue the procedure until the iteration process converges.

The convergence of the iterative process can happen in one of the two ways. The

sequence of the set of extrema] plants (SO, S1, S2, ... ) converges to S or the sequence of

controllers (C'eJ.Cl,C2, ... ) converges to C. The process is pictorially represented in Fig.

6.1.

Mn

KIKo

Fig. 6.1 Systematic Robust Controller Design Approach for Parametric Uncertain S~'stems

III

6.3 Extremal Systems with Classical Design

The fust design iteration is carried out with the classical controller. The classical design

considerations, the design specifications and the generation of the extremal plants are

described in this section.

6.3.1 Design Considerations

In the atmospheric flight regime, the prime concern is to stabilize the vehicle while

maintaining the loads experienced by the structure at a minimum. The aerodynamic load

experienced by the vehicle is a function of the dynamic pressure. A typical dynamic

pressure profile during atmospheric ascent is given in Fig. 6.2. It is seen that the dynamic

pressure reaches its maximum value around 65s after lift off. The aerodynamic effects are

maximum here, and the design has been carried out with the data corresponding to this

flight instant.

15010050

-- ---_._-

V \/-----/ \f--------

/ \---_. / \

/ \'-------

- /----------------

--------- -i\

/ ! ~oo

0.5

1.5

X 104 Typical dynamic pressure profile4

2.5

3

3.5

Flight time in s

Fig 6.2 Typical Dynamic Pressure ProfIle during Atmospheric Ascent

The usual practice in atmospheric flight is to phase stabilize the dominant flexible mode

(which ensures that the bending mode signal that gets fed back into the system is out of

phase with the excitation), and thus to quickly damp out any oscillations that might be

excited by the command or disturbances. The critical stability margin points are the aero

112

margin (low frequency gain margin), the rigid body phase margin. rigid body gain margin

( strongly influenced by the bending mode due to proximity) and the phase lag and phase

lead margins of the phase stabilized bending mode. A typical frequency response plot in

the atmospheric phase is given in Fig 6.3.

(mag Axis

BM Lead Phase Mar~gl_'n_--r-~~

Aero Margin

,,,

BM Lag Phase Margin

Unity Circle

Real Axis

Bending Mode (BM)

Fig 6.3 Typical Frequency Response Plot in the Atmospheric Phase

The set of extremal system plants. hence consists of those plants which capttm: the

minimum aero margin. phase margin and gain margin. as well as milJlmum lag and lead

phase margins for the lightly damped structural mode, with the controller in loup. The

classical controller design is first carried out to meet the specifications for the nominal

plant as detailed in the next section.

6.3.2 Design Specifications

The autopilot has to track the guidance command of around 1 deg/s \Nith tracking error of

less than Ideg. With damping ratio of 0.7, this roughly corresponds to a minimum

bandwidth requirement of 1.4 rad/s. The upper limit of the bandwidth is dictated by the

proximity of higher order dynamics. The first bending mode frequency is beyond 10

rad/s.

113

To maintain a reasonable amount of separation between rigid body and ben.ding mode

(t~lctor of two to three)! and to obtain good high frequency noise rejection, it is desirable

to limit the rigid body bandwidth to around 3.5 rad/s.

The frequency domain specifications for the rigid body are

Phase margin> 35 deg

Gain margin> 6dB

Bending mode stability is to he ensured in all nominal and perturhation cases.

(),3.3 Generation of Extremal Plant Set

The baseline classical controller is the same as that used in Chapter 5 for robust stahility

analysis. This design is used to generate the extremal systems with plant and controller

using the Kharitonov-based procedure developed in Chapter 5.

The open loop transfer function for the system with classical controller is

The extremal plants are obtained by rearrangmg the system transfer functions and

isolating the extremal plant transfer function set, as follows.

Plant transfer function from the actuator input to the attitude angle:

Plant transfer function from the actuator input to the body rate:')

CVa - CTclc - Tc¢' CJ', )act rg

114

(6.2)

(6.3 )

The extremal plant set is derived in the same manner as in the earlier chapter. and

l-ls Bsconsists of pairs of -. - transfer functions.8 8

With the same notations as 111 Chapter 5 and defining the additional uncertain

polynomials P5, P6 and P7, where

2Ps = ¢COa Tc

P7 = CJ/ rg ,

the attitude and rate transfer functions can be represented as

CSPjS - P3PSP7 s=

Pj P3(s2 +C35+C4 )

The Kharitonov polynomials of Ps are

11 21, d. I 2TKs1 =¢! OJa c .K52 ='1-'/. (jJa caCI acf

The Kharitonov polynomials of P6 are

K· 11 K. 61 = CJ! ,. 62 -.': er/PK pg

(6.4)

K71= (J/rg

11, Kn = CJ/

pg

. . h f' f Os esc: fi .The extremal plant famIly consIsts of t e set 0 pairs 0 -, - tranSler unctions,. 88

() s_ = C 5 K Ii - K 3k K 5111 K 6/1

5 (.1'2 +C 1 S+C 4 }K li K 3k

=('5 K I; -K3kK5111K7/1

(s2 +C 3S+C 4 )K li K 3k

(6.5 )

for i == 1:4, k ==1 :2,111=1:2 and n =1:2

115

The next section give the second iteration. in which Linear Quadratic Gaussian (LQG)

controller design is carried out to give satisfactory response for this set of systems.

6.4 LQG Controller Design for Extremal Plants

LQG design, which minimizes quadratic performance indices subject to linear state

space dynamics driven by Gaussian white noise. has been used in the atmospheric phase

in the Ariane series of vehicles as well as in the Apollo and Space Shuttle. In this section.

an LQG controller is designed for the extremal plants identified in the previous section.

6.4.1 Tuning of LQG Controller Weights to Achieve Performance and

Robustness

An approach similar to that used in Chapter 3 is used to achieve performance as well as

robustness to parameter perturbation through appropriate tuning of the LQG weighting

matrix Q. by making use of the reduced set of extremal plants. The states corresponding

to the rigid hody response (attitude and body rates) and those of the bending mode have

been treated separately. The tuning of the weights is done in two steps: in the tirst step.

weights assigned to the rigid body states are tuned to achieve acceptable rigid hody

response based on transient response results. Once these are fixed. the weights on Ihe

bending mode parameters have been tuned with a unique approach. hased on the

frequency domain analysis over the extremal set of plants. Thus the main emphasis for

the choice of the bending mode \veights has shifted from optimization to rohustness. The

overall int1uence of different weightages in the optimized performance index on the

control performance is assessed in terms of transient response as well as robustness to

parameter perturbation. The choice of weights that gives the best relative performance is

used for the tInal controller design. The LQG contro11er tuning procedure is given in Fig.

6.4.

116

Formulate extremalplant set

AssignQ=[I]R=[I]

DesignLQGController

Design LQGcontroller

Tunediagonalflexiblemode stateweights No

Yes

TransientResponseSatisfactory?

FrequencyResponsesatisfactory?

NoTuneRigidbodyweightsQ(I,I)Q(2,2)

Fig 6.4 LQG Design Tuning Procedure

The rigid body weights are first tuned to get acceptable transient response. while

maintaining the bending mode weights unity. The nominal system response after tuning

is given in Fig 6.5 and Fig 6.6. The initial undershoot in the system response ( Fig 6.5) is

due to the non-minimum phase nature of the plant. Since the launch vehicle is highly

aerodynamically unstable. the achievable aeromargin is limited. The aeromargin strnngl v

influences the transient response- the lower the aeromargin. the larger the peak overshoot.

In this case. the achievable peak overshoot is around 38(~'o. The rise time is less than half

a second and the system settles in around 5 seconds. The steady state error of around 4U/o

is due to the aerodynamic disturbance. The maximum control actuator command (Fig 6.6)

is around 6 deg 'vvhich is well within the actuator maximum capability of 8deg.

117

Slep respo~se willl LOG controller, hlgll dynamic prossure regime

2 :'i Body r~le'lnd~9/S--'-' -,- . .. -1

1,S !.. \.>.:::y-.: : ! ! -.1" ·····[···-···1-··-····:-······1I V i'·..... ~ AlIit~de angle in de~ i ! ~

i ,Ii it~ 1,,;~':.:\;;;::~=h~-'~-=j=~-d~~=

~ o.s-J.:.\ : -.. .l.. ! L. : :!:\ ..,: \

0,.1 .. L.. \. ......,......-"~.\/ \-..; ;//--r- ~ ~",--" ....,.-;.

,. .~O.5 t L..._.... _; ~_. l.. i. -.l-_.__.:.. . t__..L,--_L-.

o 1 2 3 4 S 6 7 8 9 10lime in s

Fig 6.5 - Transient Response with LQG Controller - Attitude and Body Rate

Step response with LOG controller· high dynamic pressure regime

81----~--·~--· : ' ._,6/1\': :'''' ,.. -..r··.·,· ..·.. ·~· ...... ~ ..·.. ··!..-.... :· ......·

9.' 41"'A:ctualbr\:6mhi~rid"''':''''''':'''' .. ". ''''~''-''i''''

~ , \: : : : : , , :iE

2\ ::, .. ; : -:-- -..: -, .. - : -..: '.- .. -..

i o .. i.· : .: .l. \ J. .. L :.:_._ ..:.......!2\ /<-=-T~ ·~--~~r--~j

I li:/ ' : : . : . : :-\'\Y", : ... :.......:. l' ....r ....:-... :......l ::::: '/:

-6 ----..--.-.. ;".--- ---j-------- L ._ :__ .L 1.. ~....-....J_. i .l__~__

a 1 2 3 4 S 6 7 8 9 10lime In s

Fig 6.6 - Transient Response with LQG Controller - Command to Actuator

The frequency response analysis is canied out next to determine the bending mode

weights, The response of the extremal plant set is evaluated with the LQG controller. The

system response with unity weights on the bending mode states is given in Fig 6.7,

118

Im(s}

Nyquist pial of extremal systems wilh LOG controller- unity weight for BM

8[ '\ ", \ ' "";,

i \ Nominal ' ""':61, , '

4\I

2\I

:\Ii

-41-I

Ra(s}

ll'ig 6.7 - Nyquist Plot of Extremal Systems with LQG Controller - Unity Weight onBending Mode States

It is seen that t11e lead phase margin of the bending mode is lost in some of the extremal

cases (there is an encirclement of the (-1,0) point). The nominal response, which is

ovcrplotted, shows good stability margins, and perfect phase stabilization, with 'lear1y

equal phase lag and phase lead margins. The bending mode weights are iteratively tuned

to ensure stability for the extremal set of systems. Weights of 50 on the bei " I

slate and 10 on its derivative are seen to give stable response for the set ofsYSl: "ig

6.8). Vhth higher weightage on bending mode stabilization- to ensure stability the

extremal set it is seen that the overall attenuation on the bending mode has incrc8sccl,

and the mode magnitudes have shrunk to a maximum amplitude of around 4 as against

nearly 8 viith unity bending mode weights. The nominal response also has shrunk

considerably and the mode is no longer phase stabilized, but almost totally attenuated in

the nominal case. It is seen that the low frequency response is not significantly affected.

The comparison of the nominal response for the two sets of weights is given in Fig 6.9.

119

8 10

-'~.

642oRe(s)

·2 "

: /:;'J "

-4-6-8

G

-8 _.._-i_.__:.._.._. __L_--,-_

-10

-6 ..... ,. ~.. . .. ;

Nyquist plOI or OXll'OllIfll systoms with LOG controllor- with Incronsod 8M1 slnlo wolghls8 .-'

-4···" " .

4

o

-2

Im(s)

Fig 6.8 - Nyquist Plot of Extremal Systems with Increased Weight on BM States

Nominal response with low and high 8M weights2 .-_. ~··T\-··-T'-"---".'----l ....····~~··!-----:-·----l----;-··----r 7-'-'--"'-

15 I.-.L..! ..~ )<~r>~, .. ~ .. \ \ : " : : : \' :r ; : :: :

I 0.: ~· •••••k••·.l ••••••J"···¥,J•• /'hC'YBMI~,g,,,.·-0.5 l ~'\1<~j('.. ·····i.·::~~ll ..: + + ..

, I , , , I

I , I , I I ,

R1 .. -..... -.. ,,:, ... -- ... -r' -_. -... ·~i~--~·-··· ·1-····· ~ .. ~[.._.. ·-··-1~~·--·- ---t ---.- ...., , , ,. , . ,

4

. .,-1 .5 .. ", .. " ~ : ~ ~ ; ~ : .

, " I, I". '"I "I, , , , , , ,

-2 ...,....._._L... "...L•.._.__.L-.....__.......L"__L._.__l. ...._.........-J ...._

-4 -3 -2 ·1 0 1 2 3Re(s)

Fig 6.9 -Comparison of Response with Low and High Weights on BendingMode States

120

The final set of chosen weights arc

r1000 0 0o 100 0

o 0 200

Q= 0 0 0

000

000

000

o 0 00

o 0 00

o 0 00

500 00

o 10 0 0

o 0 10

o 0 01

R =[1]

The Bode plot of the LQ } controller is given in Fig 6.10.

" Diagrams 01 LOG conlroller, ql=1000,q2=100,q3=200,q4=50,q5=10

~

Ir~- - ._-

-- l, -- --)

/

ll~~--

.-1 --

- I --

Frequency (rad/s)

Fig 6.10LQG Controller Bode Plot

The LQG controller is of seventh order, and has a wide notch structure around the

bending mode frequency. The LQG controller is given in Appendix 7.

6.5 Evaluation of Extrernal Plants Y"ith LQG Design

The extremal systems consisting of the LQG controller along with the set of extremal

plants is constructed, This family of extremal systems is now evaluated to determine

whether the extremal plant set for the earlier controller design (in this case, a classical

controller) captures the extreme system margins with the new controller. This is done by

comparing the response of the set of systems consisting of the extremal plant with LQG

121

controller against the response of the set of systems obtained by gridding the uncertain

system parameters, with the same controller.

6.5.1 Frequency Domain Evaluation

The gridded set response is given in Fig 6.11 and the two set of responses are overplotted

in Fig 6.12. It is seen that the aeromargin ranges from 8dB to nearly 10dB over the set of

systems. The nominal aeromargin is of the order of 9.5dB.Rigid body phase margin of

around 30deg and gain margin of around 6dB is achieved for all the systems. The

minimum lag margin for the bending mode is around 30deg and the lead margin, around

10deg. The robustness of the set of systems, which was marginally stable with the

classical controller, has improved significantly with the LQG controller.

Nyquist plot of systems with g1idding

64oRe(s)

-2-4'-_-"-__L--_-"-__L--_-'-_--'

-6

Fig 6.11 -Frequency Response of Systems with Gridding

Comparison of Nyquist plot of systems with griddlng, and extremal systems8 r--.--..,..---r--r-.------....---r__.__-,---.-....,----..,...----,

61---t--HI

41---t--\l\--tt-I· 1-t--r"'f=-orl--I---t------JI---1

2

Im(s)0

-2

-4

-6

..a-10 ..a -6 -4 -2 0 2 4 6 8 10

Re(s)

Fif! 6. t 2 -Evaluation of Extremal Svstems vis a vis Gridded Svstems

122

It is seen that the extremal systems (plotted in black) form almost the exact boundary of

the set of gridded responses( plotted in blue). Thus it is established that the extremal

plants pulled out from the extremal systems with PD controller design remain the

extremal plants for the LQG design also.

The pole zero set of the set of extremal systems is given in Fig 6.13.

Pole-zero map of extremal uncertain system- attitude transfer function

Act alar poles

Bending me~e pole

Rigid body P- es

- ,/

" ~BelllfuJg:/

moduefQL1---

• Bending mo e poles

--Act alar poles

20

15

10

5Ul

~ 0Ol

'"E-5

-10

-15

-20-15 -10 -5

Real Axiso 5

Fig 6.13(a) Pole Zero Map of Extremal Plants

Pole-zero map of extremal uncertain system- expanded plot

2

1.5

.~ 0.5q;g>E 0

-0.5

-1

-1.5

-

I---L-1----

----- - - -I- _.-

d

- - -C

-I-- --b a a b

c

~ igid-15OOY pole pairs

d

-1 -0.5 oReal Axis

0.5

Fig 6.13(b) Pole Zero Map of Extremal Plants- Expanded Plot

123

It is seen that the bending mode zeros form a pair on the real axis, with one zero on the

right and Jine,u movement symmetric with respect to the imaginary axis over the

uncertain set. 'The rigid body poles move from a symmetric pair on the real axis to a

symmetric pair on the imaginary axis, over the uncertainty set ( i.e. the system moves

from aerodynamic instability to aerodynamic stability). It is interesting to note that the

number ofpoles in the right halfplane is not constant over the set.

6.5.2 Titnc Domain Evaluation

The transient response of the entire set of systems obtained by gridding the uncertain

parameter set, with the controller designed based on the extremal plants, is plotted in Fig"

6.14. It is seen that the peak overshoot ranges from 30% to 55% over the set of systems,

corresponding to the range of aeromargins from 8dB to nearly 10dB, as seen in the

frequency i·cspOl1se. Also, while the nominal response is smooth, indicating perfect

stabilization of the bending mode, the bending mode oscillations are seen overriding 011

the rigid body response in those systems with low bending mode margins.

Sel 01 transient responses wilh gridding· LOG coniroller

2 ..1.5 .

0.5 '

-0.5a

, , ; Allilucie angi~ indeg:." . ' ".,."", .. ' ,. ":

m"li'~~. ' : : : : : :\.'t<-I1f' ' , , , , , , ,~',," W : : : : : :

__ u~ .~ •••_~~,~: Sle? cal I: :: ~ V:' :' :

, ' , " ,I , , " ,

:~ ::::',:,.•• -i~ • - -:- •. - - .• ,- _••••• ;. _••..• -: •• ~ ••• -:••.•••. \ ••.••••:•.••~ .I':.: - •.••••:

j~ jdym'J'""~' ;.~: ~ : ' , : ::' '" : " l

_ ...L..-----L.__, I I ---I----J

2 3 5 6 7 8 9 10llme In s

Fig 6.14(a) Transient Response with LQG Controller

124

Normalised bending co-ordinate- lor lull sel 01 systems

-6a 5

nme in s

Fig 6.14(b) Bending Mode Response with LQG Controller

Thus it is seen that the extremal plant set for the classical design remain the extremal

plant set with the LOG design also~ the tuned LOC; design has resulted in signi [icantly

improved stability margins for the worst case plant template compared to the baseline

classical design. In the next section, the controller design is carried out using the

Quantitative Feedback Method.

6.6 Quantitative Feedback Theory Based Robust ControllerDesign

The most appropriate controller design method for the atmospheric phase launch \ehicle

plant. with its large phase uncertainty and the need to phase stabilize the dominant mode.

is one that accounts for the parametric uncertainty nature of the plant. Robust uncertainty

norm bounding approaches such as the 1-1" approach are not directl:' suitable for this

system since in such approaches the uncertainty is assumed to be bounded onh in

magnitude. and the phase information is not considered. In the Quantitative Feedback

Theory (OF'f)7 framework. the uncertain plant family can be considered at the design

stage itself. avoiding the trial and error of classical design, The design fully exploits the

intuitiveness of classical frequency domain loop shaping concepts to cope simultaneously

with performance specifications and plant uncertainties. The design is not conservative

and controller orders are low. Hence the QFT method is chosen for the next iteration,

125

6.6.1 ControHcr Design

The d i rfcrcn t steps in the QFT design procedure are as follows:

(i) Selection of the transfer function set

The set of worst case plant transfer functions defined in Section 6.3 hold good for this

design aho, since the previous design iteration with LQG design has not resulted in any

modification in the set of worst case plants.

(ii) Selection of frequency array

Low frequcncy point of 0.1 rad/s, high frequency point of 100 rad/s and intermediate

points around the rigid body bandwidth as well as a large number of points around the

lightly damped bending mode frequency have been chosen in the frequency array. 'The

extremal plant templates are given in Fig 6.15.

Fig G.IS(b) Plant Templates withExtremal Kharitonov

Planl Templates- worst case (exlremal) systems

~ I 0 ~ , ---~--~--·-1

10.: j , : ~ ; '1

o .: ~ , : .<.>.0,1;... ·"1

!

'1:1:lT T U LC :•..• ·.1·70 .; -j-- ~~1oo j : j'.'"~.L.~J

·350 ·300 -250 -200 ·150 ·100 -SO 0Phase (degrees)

I:Lj9ioEJ~I ••'i:: ;::; -40·:···· .. ··~·· ~..... .. ..

Fig 6.15(a) IJlant Templates with ExtremalKharitonov Plants, around ~endingmode

As expected, the Nichols plots show that the extremal Kharitonov plants define the

.bounds of the response at each frequency.

(iii) Computation of QFT Bounds

The two conditions for robust stability are (l) Stability of the nominal system and (2) the

Nichols envelope should not intersect the critical pClnt (-180deg,OdB) point in the

Nichols chart or the (-1 + j 0) point in the complex plane. QFT translates the robust

margins constraint into required conditions on the phase and magnitude of the controller.

126

These form the robust stability bounds. Based on the phase margin and gain margin

specifications,

PUw) G(jw)

--------------------- < I.l ,where I.l =1.2 for all PEp ,0) E(O,a,)

1 + PUw) GUw)

has been chosen as the robust stability specification, where p is the parametric family of

transfer functions describing the uncertain plant and G is the controller transfer function.

The value of J.l has been arrived at based on formulae l1G to satisfy the phase and gain

margin specifications. Specifications are also assigned on the transfer function between

the output and the disturbance input in terms of a constant upper bound. This f0rI11s the

robust perJorrnance bound. The individual bounds are given in Fig 6.16(a) to Fig 6.16(t).

The worst case bound of all the stability and performance bounds are then computed. Thc

intersection of bounds is given in Fig 6.16(d).

Robust Margins Bounds

Fig. 6.16(a) Robust Margins Bounds

127

·20 .~- •. -.-~-:l..:.- .. - ..:---.---:-. . :1:.. : \: :: I I

-30 .~-'_. -. - -.;_.. .:-- -. ",. I

-40 ) ---.~ --~- ..~.~~.:-~-~!.~:~.=.:-':,"-.. '-, ', , I ,. , , ,1---,------...J... ! --1 .. 1_ ~ •• _

-350 -300 ·250 -200 ·150 -100 -50Phase (degrees)

Fig 6.16(b) Robust Output DisturbanceRejection Bounds

rloIHJ~;1 hll'lll DlslurtJnnr:1) nolocl~Jfl UOU!1,lr,

-',.g:.1....I'...1 Inlomccllon 01 Bounds

Ii.. I')06b 50: ..

..'~7'.li ,tOk ....:;.. 1

] I :1:1i~~4t}~f=I_\[~~lJ.45~ ·350 ·300 ·250 ·200 ·'50 ·100 ·50 0

~.?J~~ Ph0lJ6 (do{Jroos)

, . -_.._--'- ----- -.- ~_.- .; ... -.,--~- ;r'~-~:' _-.: -;,-:--,

..;...

·250

Fig 6.16(c) Robust Input DisturbanceRejection Bounds

Fig 6.16 (d) - Intersection of Bounds

(iv) Contl'oller Design

The uncompensated response of the arbitrarily chosen nominal plant is given in Fig 6,17.

It is seen that the response violates the worst case bounds. A suitable controller was

designed for this system using the toolbox to satisfy the robust stability and performance

bounds. It may be noted that the order of the controller is low (second order filter). The

compensated system response is given in Fig 6.18.

I i 2 ,LiL mel 212: FiEF:

.,It n •• ,·<\ :~,~.t1I

(I(d"':')"_\I

O~.. \·I"''''

tl(lu'."""'1'

:-.--~::-:::-:'::::- "'I::L.-:..:..~

. Ol}·,

5,9 !li)6:b!19.L~:l1.1(l~:~J11.05-,~

lIi;;\n'l!,~!;t1.,~~J11"b~j,}11

;1~~!if~:~12.1.11

:i~1fJl?:~"j12.i.c 1

1.?~?,~..,;!},U1J.1S.g' , ;.---: )sc ·xc 1~ I',j:.'l ~Sil 10.; ~:l

g:~,bil.-- ---"-""",,'''~'''''''''''''''''''''''~' -'"y.............., ....""""."". _

.387.1"t1fg.11J.1~IlU

·1.21tll!U,·O.ltIlD

Opf'll.l(lop:

CIUH(I·1oOI·:

Fltqu,n.:y:. ---'-r- ------- --"- -1-'-'-- --.- ---'-r'-" '--··----1------·-

Fig 6.17 Uncompensated Response­Nominal Plant

Fig 6.18 Compensated Response ­Nominal Plant

128

6.6.2 Evaluation of CoutroHer: Performance

'The Nyquist plots of the extrema! systems with compensator are given in Fi~ 6.19. It is

seen thai t11e bending modes are phase stabilised, with good margins. A reprcsentali vc

Bode plot is given in Fig 6.20. The minimum stability margins for the CompenSated

system arc summarised below.

Rigid body

Minimum phase margin > 28deg

Minimum gain margin > 5.8 dB

Gain crossover fi'equency (indicative of system bandwidth):> 3.5radls

Bending mmk

Stable response is obtained for all the cases.

Minimum bending mode phase margin (lead): 33 deg

Minimum bending mode plJi1se margin (lag): 24.1 deg

Thus the bending mode is stabilized with good margins even for the worst case.

,4 . . .,'_ ~ ..• w": .•.• ..:- ••.. . .. . ,, ., ,'::

.,~TT~~J-.JlOuJT-L--J5 10

Fig 6.19 Nyquist Response - QFT Design

129

°i~-. ;~~-:f7~::-;-~;:;::-~n::::::-~:-~Tl

f: I· .. ... 'llj(:[!1.1'I~nJI~'f~\ j Ir"I'1~ <00, ··'H>:iH',:; "~h::>""!':"

~800 1. . _ ..• _ ,_ .. _.'--'_ ...d ._•.• _'-- ....... L_._'-_'-'-t._._•. _ ..._._ ...._J.-J_....L....l...t_~. .'- __ -'--l._!"'L_'_1.lo

10.2 10" 10° 10' 102

Frequency In radls

Fig 6.20 Representative Bode plot

Representative transient response plots are given in Fig 6.21.

Pepresentatl-.e transient response plots wth OFT controller .

, "

,i ... ty;~:::J:-I~~:-~-I: ..[i 1\ 11'/1' . : : .

0.81 If;fKP. . ;.... .... . ";"ill' i rtf. ,U ii U6/ I' ....•..... ,' ... 'fTT~ 0.41 .,.~~ "(····r····"!""···· : -:-- -:-- ; .« I : : : : :

0.21·· .,... . ..... ~ ..... j.....+ ... :.......~ ......:.......I : :: i

°ll\i!~' . ..:r······f······!······ i···· : "f" ! .. ':: ;

-0.2 1- ...--'---.----.1..- I. I 'o 1 2 3 4 5 6 7 8 9 10

orne in 5

Fig 6.21 (a) Attitude Angleswith QFT Controller

Body rate wlh OFTcontroller- representali>e p1o11.4

1

1 --:.----:-.;-.:-:-; --I ~, , . , , ., , • , , I, , , , , ,

1.2 .... ·r ......:......:......r..··(..r1.... - . ~ - .\ . -. - .' - .- _.l .• _. _ L •••••• L

" ." ,, I , , • •i 0.8· .. ·j·· ....i....·(- ..i..·..·(·.. i\

c: ::::;:j 0.6 ~ .... --: ~· .. · ..~ ...... i...... ~ ..

! 0.4 ~ ~ .. •· .. ~ f..·..·f ..·.. ·f .. ····~• , I I "'J\.. : : : :

0.2 ·r \;,: .. r· .... ·r .... ··i ......r ..,o --- ; ~ .. -\..~F·~,:<':'r· .._:!--+ -_._-.-;_ ..

I • I II • , I

, • • I, ,.I ,""..0,2 I' I I , L~ .L .. __,__ -'..

o 2 3 4 5 6 7 8 9 10llme In s

Fig 6.21 (b) Body Rates with QFTControIlzr

130

.... '.. ".,'

Represelltali'.'9 Actuator command- transient response' OFt design

':'" :1

5­l'

4:1':i' ,:1 \,

3iii " ",I . t "."l 21 \': ....:-- .... j."".':...... ;''.''.j..... ''(..... j.. '' ..

.5 1 1-. ..1- ...: -_··~··_··_·~······_j·_·_·_·1·······~··_···i····-:·~·· ....

i.+L/'><:(••,;=:1L··.[: •• :·•••••.'tti 1 \: \: : : : : : : :

~ -2 i if; ..:·".. ·: .. ,,·)···.. ·1"···+"···1····"·:"·,,·I \j' , , , ' ,-3 \.; ;' ... ;.. · .... :"· .."t ..".. 1· .. ·!·,,··-r....·r.. ·

-4 j \: I • • ' .••• •• : _ ...... ;" - ••••• : •••• ~ ••• -"~"""': ..••••

-51 __ ~ n ~ I !

o 1 2 3 4 5 6 7 8 9 101lme In s

Fig 6.21 c Actuator Command with QFT Controller

The controller transfer function is given below(s+19.51) 100

2.24 * *( s+ 5.923) ( s+ 100)

6.7 Sumlnary

A systematic method for robust autopilot design, for a class of multi-linear parametric

uncertain systems, which includes the launch vehicle plant, has been suc~essfully

developed. The extremal set of plant transfer functions for the aerodynamically unstable

launch vehicle plant has been formulated from the set of ex.tremal systems designed using

the classical method. A controller has been designed using the LQG design methodology

with the weights in the state weighting matrix tuned to impart robustness for this set of

worst case set of plants. The response of the extremal systems with LQG controller is

compared with that obtained by griclding the parameter interval. It is seen that the

extremal Kharitonov plants as identified from the extremal set of systems consis::ng of

plant along with a classical controller, form the extremal plants for the LQG controller

structure also. It is also seen that the extremal systems not only define the worst case set

of the system response, but they do so with negligible conservatism. There is a signifcant

improvement in the stability margins obtained for the extremal systems with the LQG

controller, as compared to the classical controller. While the classical controller just

J31

stabilizes the set of extremal plants. the LQG controller produces a minimum margin of

lOdeg. The transient response results for the set of uncertain systems are also seen to

corroborate the frequency domain analysis.

With the plant templates from the previous iteration. a robust low order QFT controller

has been designed to meet the system specifications. which have been translated into

QFT bounds. With the systematic definition of the plant template. the Quantitive

Feedback Theory approach has been shown to be viable for launch vehicle autopilot

design. giving low order controllers meeting the design specifications.The stability

margins are further improved compared to the earlier design. with the minimum bending

mode margin improved from 10 deg to 22deg. A second order controller is seen to be

sufficient to stabilize the sixth order plant, for nominal as well as perturbed data. as

compared to the seventh order LQG controller. Thus there is a progressive improvement

in the- stability margins obtained in successive- iterations.

For this class of systems. the method works even if the plant is unstable. and has \'arying

number of right half plane poles over the uncertainty set. and is hence a powerful design

tool. To the best of the author's knowledge. no such method has be-en developed

elsewhere which enables the worst ease set of plants to be identified for robust desif'.11 of

such parametric uncertain systems with multiple phase and gain margins.

The significant contributions of this chapter are the following

(i) /-\. systematic. Kharjtonuv-ba~:ed approach It)]' the design of robusl controllerlc)! ,I

das'; ,)1' multi-linear pann11t~tric uncertain systems is developed. and applied Ill!'

launch vehicle autopilot design. The successive design iterations involw the

classical. LQG and QFT controller design methodologies. There is a progressive

improvement in the stability margins obtained in successive iterations. The method

hence provides a powerful tool for robust controller design. without sacrificing the

phase information. The Kharitonov based approach. which is a robust stability

analysis tool. is thus converted into a robust design tool. To the best of the author's

knowledge. no such systematic method for determination of the worst case plant

I "'. 'J.c.

templates for multi-linear parametric uncertain systems has been developed

elsewhere.

(ii) One of the major inadequacies of QFT has been the inability to define exact plant

templates, which has been addressed in this thesis. With the systematic method

evolved for definition of the plant templates. QFT has become a viable method for

robust control of launch vehicle autopilot. A robust low order controller is designed

to meet the system specifications, which have been translated into QFT bounds.