`
`
`Ex. PGS 1029
`
`
`
`EX. PGS 1029
`
`
`
`
`
`

`

`Eigenstructure Assignment for Design of
`Multimode Flight Control Systems
`Kenneth M. Sobel and Eliezer Y. Shapiro
`
`to the
`
`eigenvectors of A - BFC are close
`respective members of the set (v?).
`The feedback gain matrix F will exactly
`assign r eigenvalues. It will also assign the
`corresponding eigenvectors,
`provided that
`they were chosen to be in the
`subspace
`spanned by the columns of ( A i l - A ) - ’ B .
`This subspace is of dimension m, which is
`the number of independent control variables.
`In general, a chosen or desired eigenvector
`v f will not reside in the prescribed subspace
`and, hence, cannot be achieved. Instead, a
`“best possible”
`choice for an achievable
`eigenvector is made. This “best possible”
`eigenvector is the projection of I,? onto the
`subspace spanned by the columns of
`( A i l - A ) - ’ B .
`We summarize with the following:
`
`The matrix F will exactly assign r eigen-
`values. It will also exactly assign each of
`r eigenvectors to
`the corresponding
`m -dimensional subspaces, which are con-
`strained by A:, A , and E .
`If more than 172 elements are specified for
`a particular eigenvector, then an achiev-
`able eigenvector is computed by projecting
`the desired eigenvector onto the allowable
`subspace. This is the subspace spanned by
`the columns of ( A i l - A)-’B.
`If control over a larger number of eigen-
`values is required, then additional inde-
`pendent sensors must be added.
`If improved eigenvector assignability is
`required, then additional independent con-
`trol surfaces must be added.
`
`ABSTRACT: Advanced aircraft such as
`control configured vehicles (CCV) provide
`the capability for implementing multimode
`control laws, which allow the aircraft perfor-
`mance to be tailored to match the character-
`istics of a specific task or mission. This is
`accomplished by generating decoupled air-
`craft motions, which can be used to improve
`aircraft effectiveness. In this article, we de-
`scribe a task-tailored multimode flight con-
`trol system, which was designed by using
`eigenstructure assignment.
`
`Introduction
`Advanced aircraft such as control con-
`figured vehicles (CCV) provide the capabil-
`ity to control the aircraft in unconventional
`ways. One such approach is to generate de-
`coupled motions, which can be used to im-
`prove tracking and accuracy. The decoupled
`motions are obtained by utilizing a task-
`tailored multimode flight control
`system,
`which implements feedback gains not only as
`a function of flight condition but also as a
`function of the mode selected. The aircraft
`performance can then be tailored to match
`the desired characteristics of a specific task
`or mission.
`For the longitudinal dynamics of a control
`configured vehicle, the flaperons and eleva-
`tor form a set of redundant control surfaces
`capable of decoupling normal control forces
`and pitching moments. The decoupled mo-
`tions include pitch pointing, vertical transla-
`tion, and direct lift control. Pitch pointing is
`characterized by pitch attitude command
`without a change in flight path angle. Vert-
`ical translation is characterized
`by flight
`path command without a change in pitch atti-
`tude. Direct lift control is characterized by
`normal acceleration command without
`a
`change in the angle of attack.
`For the lateral dynamics of a control con-
`figured vehicle, the vertical canard and rud-
`der form a set of redundant surfaces that is
`capable of producing lateral forces and yaw-
`ing moments independently. The decoupled
`
`This work was done while Kenneth M. Sobel
`and Eliezer Y . Shapiro were with Lockheed-
`91520.
`California Company, in Burbank, CA
`Eliezer Y . Shapiro is now with HR Textron,
`Valencia, CA 91355.
`
`motions include yaw pointing, lateral trans-
`lation, and direct sideforce. Yaw pointing is
`characterized by heading command without a
`change in lateral directional flight path angle.
`Lateral translation is characterized by lateral
`directional flight path command without a
`change in heading. Direct sideforce is char-
`acterized by lateral acceleration command
`without a change in sideslip angle. All three
`lateral modes also require that there be no
`change in bank angle.
`The application of eigenstructure assign-
`ment to conventional flight control design
`has been described by Shapiro et al. in
`Ref. [l]. A design methodology that uses
`eigenstructure assignment to obtain decou-
`pled aircraft motions has been described by
`Sobel et al. in Refs. [2]-[5]. In this article,
`we use eigenstructure assignment to design
`a task-tailored multimode flight control sys-
`tem. The longitudinal design is illustrated by
`using the unstable dynamics of an advanced
`fighter aircraft, and the lateral design is illus-
`trated by using the dynamics of the flight pro-
`pulsion control coupling (FPCC) vehicle.
`
`Eigenstructure Assignment Basics
`Consider an aircraft modeled by the linear
`time-invariant matrix differential equation
`given by
`
`P = A x + B u
`
`(1)
`(2)
`y = CX
`where x is the state vector (n x l), u is the
`control vector (rn x l), and y is the output
`vector (r x 1). Without loss of generality,
`we assume that the m inputs are independent
`and the r outputs are independent. Also, as is
`usually the case in aircraft problems, we as-
`sume that rn, the number of inputs, is less
`than r, the number of outputs. If there are no
`pilot commands, the control vector u equals
`a matrix times the output vector J.
`= -Fy
`
`The feedback problem can be stated as fol-
`lows [l]: Given a set of desired eigenvalues,
`i = 1,2, . . . , r and a corresponding
`(A?),
`set of desired eigenvectors, (vf), i = 1,
`2 , . . . , r , find a real rn x r matrix F such
`that the eigenvalues of A - BFC contain
`(A:)
`as a subset, and the corresponding
`
`0272-1708/85/0500-0009$0l .OO 0 1985 IEEE
`
`Now suppose that in addition to transient
`shaping, we desire the controlled (or tracked)
`aircraft variables yr to follow the command
`vector u, with zero steady-state error where
`v, = Hx
`(3)
`The complete control law is derived by
`Broussard [6] and Davison [7]. If the com-
`mand inputs u, are constant, and if the track-
`ing objective is to have the aircraft variables
`y, approach the command inputs in the limit,
`then the control input vector is given by
`u = ( f l z + FC&)uc
`feedforward
`
`- Fv - -
`
`feedback
`
`(4)
`
`May 7985
`
`9
`
`Ex. PGS 1029
`
`

`

`where
`
`Further details of the eigenstructure assign-
`ment algorithm may be found in [ 11.
`
`Longitudinal Multimode Flight
`Control Design
`The model of the advanced fighter aircraft
`[8] will be described by the short period
`approximation equations augmented by con-
`trol actuator dynamics (elevator and flap-
`erons). The equations of motion are de-
`scribed by Eqs. (1) and (2) where the state x
`has five components and the control u has
`two components.
`-pitch attitude
`-pitch rate
`-angle of attack
`-elevator deflection
`-flaperon
`deflection
`-elevator
`deflection command
`- flaperon deflection command
`
`x =
`
`-
`A =
`0
`0
`
`1
`0
`0 -0.8693 43.223 -17.251 -1.5766
`0
`0.9933 -1.341 -0.1689
`-0.2518
`0
`0
`
`0
`0
`-20
`0
`0
`0
`-20
`-0
`
`0
`
`
`
`0
`
`B = [ ;
`
`j]
`I
`
`The eigenvalues of the open-loop system
`from matrix A are given by
`
`A1 = -7.662
`hz = 5.452
`
`unstable short
`period mode
`
`pitch attitude mode
`h 3 = 0.0
`actuator mode
`h4 = -20 elevator
`h5 = -20 flaperon actuator
`mode
`The normal acceleration at the pilot's sta-
`tion nrp is used as a controlled aircraft vari-
`able for pitch pointing.
`
`implement modes 1 and 2, using the same
`gain matrix.
`
`Pitch Pointing (Mode 1) and Vertical
`Translation (Mode 2)
`The objective in pitch pointing control is
`to command the pitch attitude while main-
`taining zero perturbation in the flight path
`angle. The measurements are chosen to be
`pitch rate, normal acceleration, altitude rate.
`and control surface deflections. The altitude
`rate is obtained from the air data computer,
`and it is used to obtain the flight path angle
`via the relationship
`y = h/TAS
`
`(6)
`where TAS is true airspeed. The surface de-
`flections are measured by using linear vari-
`able differential transformers (LVDT) .
`We include y as a state because this is the
`variable whose perturbation we require to re-
`main zero. Thus, we replace 0 by y + a in
`the state equations and obtain an equation for
`y. The resulting state-space model is given
`by Eqs. (1) and (2) with
`x = [Y. 9, a, s e , Sf]'
`(7)
`24 = [ L , 6 , J
`(8)
`y = [q. nvP, y, 6,, Sf]'
`(9)
`Our fiist step in the design is to compute
`the feedback matrix F. The desired short pe-
`riod frequency and damping are chosen to be
`6 = 0.8 and on = 7 rad/sec. These values
`were chosen to meet MIL-F-8785C specifi-
`cations for category A, level 1 flight. Cate-
`gory A includes nonterminal flight phases
`that require rapid maneuvering. precision
`tracking, or precise flight path control.
`Level 1 flying qualities are those that are
`clearly adequate for the mission objectives.
`We can arbitrarily place all five eigen-
`values because we have five measurements.
`We can also arbitrarily assign two entries in
`each eigenvector because we have two in-
`puts. Alternatively, we can specify more than
`two entries in a particular eigenvector, and
`then the algorithm will compute a corre-
`sponding achievable eigenvector by taking
`
`the projection of the desired eigenvector onto
`the allowable subspace.
`We choose the desired eigenvectors to de-
`couple pitch rate and flight path angle. Such
`a choice should prevent an attitude command
`from causing significant flight path change.
`The desired eigenvectors and achievable ei-
`genvectors are shown in Table 1, from which
`we observe that we have achieved an exact
`decoupling between pitch rate and flight path
`angle. The "X" elements in the desired eigen-
`vectors represent elements that are not speci-
`fiedkcause they are not directly related to
`the decoupling objective.
`We now compute the feedforward gains by
`using Eq. (4). For the pitch pointing problem
`J, = H x = [e, y]'
`U, = [e,, yclr
`
`(lob)
`
`(loa)
`
`where
`0, = pilot's pitch attitude command
`yc = pilot's flight path angle command
`H = [ 1 0 1 0 0 0
`1
`0
`0
`0
`0
`0
`
` 1
`
`The feedforward gain matrix consists of
`four gains, which couple the commands 0,
`and yc to the actuator inputs. The control law
`is described by
`
`When yc = 0. we can command pitch atti-
`tude without a change in flight path angle
`(pitch pointing). Alternatively, when 8, =
`0, we can command flight path angle without
`a change in pitch attitude (vertical
`trans-
`lation).' The feedback and feedforward gains
`are shown in Table 2. The pitch pointing and
`vertical translation responses are shown in
`Fig. 1. We observe that both responses ex-
`hibit excellent decoupling between pitch
`attitude and flight path angle. An additional
`feature of the design
`is that the aircraft
`is stable with good handling qualities
`in
`the event of a flaperon failure. Of course,
`decoupled mode control would no longer
`be possible.
`
`Table 1
`Eigenvectors for Pitch PointinglVertical Translation
`
`
`
`
`
`
`
` Desired Eigenvectors Achievable Eigenvectors
`
`
`
`
`
`
`
`- 9 -
`a
`6,
`-Sf-
`where nrp is in g's and 9. a, &, and 8, are
`in radians or rad/sec. In what follows, we
`
`n, = [-0.268,47.76, -4.56,4.45]
`
`(5)
`
`10
`
`/E€€ Control Systems Magazine
`
`~
`
`~~
`
`Ex. PGS 1029
`
`

`

`Table 2
`Pitch PointingNertical Translation Control Law
`
`9
`
`n,
`
`Y
`
`6,
`
`Sf
`
`
`
` 0.210 6.10 0.537 -1.04
`
` Desired Feedforward
`
`
`
`Eigenvalues
`Gains
`
` -2.88 -0.367 -0.931 -0.149 -3.25 -0.153 0.747
`
`A;'.2 = -5.6 5 j4.2
`A: = -1.0
`
`
`
`
`
`
`A i = -19.0
` 2.02
`A < = -19.5
`
`1.5
`
`4.08 ] [ 0.954
`
`
`
`
`
`1
`
`Mode 3: Direct Lift Control
`The objective in direct lift control (DLC)
`is to command normal acceleration (or equiv-
`alently flight path angle rate) without
`a
`change in angle of attack. To achieve acceler-
`ation command following, we include inte-
`grated normal acceleration in the state vector.
`Thus, for the DLC problem, we choose the
`state vector to be
`x = [q. a, a,, Sf, 4'
`n d = { integral of normal acceleration
`
`at the pilot's station
`
`where
`
`The measurement vector is chosen to be
`Y = [q, a , nd: 6,, %I'
`The desired eigenvalues and desired eigen-
`vectors are shown in Table 3. Observe that
`are
`the zeros in the desired eigenvectors
`chosen to decouple the short.period motion
`from the normal acceleration. The feedback
`-
`gain matrix is also shown in Table 3.
`To obtain normal acceleration command
`following, we feedback the integral of the
`error between measured nzp and commanded
`nip. The control law is described by
`
`.=[
`
`-f11
`-f21
`
`-f12
`
`-f,4
`
`- h 2
`
`-f24
`
`-f15
`
`a
`6,
`6f-
`
`-fis 1 - 9 -
`
`"
`g +.
`m
`
`w
`
`I
`
`1.0
`
`0.5
`
`-I
`
`0.0
`0.0
`
`1 .o
`(A) PITCH POINTING RESPONSE
`
`3.0
`
`2.0
`
`4.0
`
`5.0
`
`- 0.5
`0.0
`
`1 .o
`
`2.0
`
`3.0
`
`4.0
`
`5.0
`
`(6) VERTICAL TRANSLATION RESPONSE
`Fig. 1. Longitudinal decoupled responses.
`
`Table 3
`Direct Lift Control Summary
`
`where
`
` = rlq - (nrp)command
`)
`4
`The DLC responses to a lg normal accelera-
`tion command are shown in Fig. 2. Observe
`that we achieve a large change in flight path
`angle with an insignificant deviation in angle
`of attack. Thus, the aircraft is climbing with
`almost no change in angle of attack.
`
`Desired Eigenvalues
`
`Desired Eigenvectors
`
`Feedback Gains
`
`I
`
`0.468 0.0587
`0.223 0.187
`
`-0.722 -6.70 -0.220
`
`
`-0.301 5.51 0.994
`
`
`
`a
`4
`short
`period
`
`May 1985
`
`1 1
`
`Ex. PGS 1029
`
`

`

`The couplings that we wanted to be zero
`are now greater than they were for design # 1.
`Also, the eigenvalues are not quite where
`we had specified that they should be. The
`responses for design #2 are shown in Figs. 3
`and 4. Figure 3 shows the
`lateral pointing
`response to a unit step heading command.
`The change in flight path angle is less than
`0.01 degree, and the change in bank angle is
`less than 0.25 degree. Figure 4 shows the
`lateral translation response
`to a unit
`step
`flight path command. The change in heading
`is less than 0.012 degree, and the change in
`bank angle is less than 0.14 degree. Both
`designs are considered to achieve acceptable
`performance.
`'OI 5
`Mode 6: Direct Sideforce Control
`5.0
`0.0
`0.5
`4.5
`-1.0
`1.5
`2.0
`2.5
`3.0
`3.5
`4.0
`The objective in direct sideforce control
`(DSC) is to command lateral acceleration
`(or equivalently lateral directional flight
`path) without a change in sideslip angle. To
`achieve acceleration
`command following,
`we include integrated lateral acceleration in
`the state vector. Thus, for the DSC problem,
`we choose the state vector to be
`.x = M , 4 , p - r, Y. 6,, 6.. S,, n,,,lT
`where
`
`[ integral of lateral acceleration
`
`at pilot's station
`
`n p , =
`
`The measurement vector is chosen to be
`.v = [r, P , p . 6. n,,lr
`The desired eigenvectors
`were chosen
`such that the lateral acceleration mode would
`be decoupled from both the dutch roll mode
`and the roll mode. This choice yields the
`following:
`- _ - _
`x 1
`0 0
`0 0
`1 x
`x x
`X J X
`x x
`x x
`x x
`x x
`:y,o,--
`0 0
`- _ -
`x 1
`-
`-
`dutch roll roll mode
`
`-
`
`I
`
`x
`x
`1
`- _
`acc mode
`mode deGoupling
`To obtain lateral acceleration command
`following. we feed back the integral of the
`error between measured n,, and commanded
`nyp. This approach is similar to that used for
`direct lift control.
`Several designs are investigated. Design
`#1 is characterized by an output feedback
`
`ANGLE OF ATTACK, a (DEGREES1
`
`-0.086"
`
`cy
`I
`0.0 0.5
`
`I
`1.0
`
`I
`1.5
`
`I
`2.0
`
`I
`2.5
`
`I
`3.0
`
`I
`3.5
`
`I
`4.0
`
`I
`
`4.5 5.0
`
`I
`
`'I
`
`FLIGHT PATH ANGLE, 7 (DEGREES)
`
`13.0"
`
`TIME (SECONDS)
`Fig. 2. DLC responses.
`
`Lateral Multimode Control
`Law Design
`The model of the flight propulsion control
`coupling (FPCC) lateral dynamics is
`described by eight state variables x , three
`control variables u, and five measurement
`variables J. The eight state variables are
`sideslip angle (p), bank angle (4). roll rate
`( p ) , yaw rate (r), lateral directional flight
`path ( y = $ + p), rudder deflection ( c S r ) .
`aileron deflection (ijO), and canard deflec-
`tion (6<).
`rudder
`The three control variables are
`command (6rc), aileron command (60c.), and
`canard command ( t j C c ) . The five measure-
`ment variables are r-. p , p . &, y. Because of
`space limitations, the detailed numerical re-
`sults are not presented, but the general ap-
`proach is outlined. Further details may be
`found in [9]. In what follows, we implement
`modes 4 and 5 using the same gain matrix.
`
`Yaw Pointing (Mode 4) and Lateral
`Translation (Mode 5)
`We desire to decouple the lateral direc-
`tional flight path response from
`the bank
`angle, roll rate, and yaw rate responses. Thus,
`the desired eigenvectors are chosen such that
`the flight path mode will not affect the bank
`angle, roll rate, or yaw rate responses and so
`that the flight path response will consist only
`#I,
`of the flight path mode. For design
`which corresponds to the full feedback gain
`matrix. the achievable eigenvectors are very
`close to those that were desired. The control
`law gives achievable eigenvalues almost ex-
`actly those that were desired.
`
`12
`
`~
`
`by
`We compute the feedforward gains
`using Eq. (4). The tracked
`variables are
`given by
`?'I = [4, y. $IT
`and the pilot commands are given by
`I(, = [&. y<., &IT
`
`where
`&. = commanded heading
`yc = commanded lateral directional
`flight path
`$< = 0
`Since bank angle is commanded to be zero.
`we need not implement the gains that
`multiply dc. It is
`included in the numerical
`computations only to avoid the need for a
`pseudo-inxrekion.
`The time histories for design #I are not
`shown; however. the yaw pointing responses
`to a unit step heading command are such that
`I y ( 2 0.0004 degree and 14, 5 0.0031
`degree. The lateral translation
`responses
`to a unit step lateral flight path command
`14; 5 0.008 degree and
`are such that
`:b, 5 0.004 degree.
`Design #2 is characterized by an addi-
`tional specification that seven of the feed.-
`back gains be constrained to be zero. The
`zero elements are chosen based upon the
`physical insight that the roll autopilot should
`be able to operate somewhat independently
`of the lateral directional control system. Of
`course. some degradation will result, but the
`responses will still be acceptable from a prac-
`tical point of view. Furthermore, by reducing
`the number of gains, we have increased the
`reliability of the control system.
`
`0 -:"
`
`0
`x 1
`
`10:
`
`10; 10;
`
`F E E Control Systems Mogozine
`
`Ex. PGS 1029
`
`

`

`1.20 r
`
`0.00
`
`1 .oo
`
`3.00
`2.00
`HEAOING
`
`4.00
`
`5.00
`
`0.08 r
`
`/
`
`0.00
`
`-0.04
`
`-0.08
`
`-0.12 I
`0.00
`2.00
`
`I
`1.00
`
`
`
`1.20 r
`
`I
`
`I
`3.00
`HEADING
`
`I
`4.00
`
`I
`5.00
`
`0.00
`
`-0.04 I
`
`0.00
`
`I
`1.00
`
`I
`2.00
`
`I
`3.00
`
`I
`4.00
`
`I
`5.00
`
`LATERAL FLIGHT PATH
`
`0.00
`
`
`
`1.00 3.00
`
`
`2.00
`LATERAL FLIGHT PATH
`
`4.00
`
`5.00
`
`0.00
`
`1 .oo
`
`3.00
`2.00
`BANK ANGLE
`Fig. 3. Yaw pointing response.
`
`4.00
`
`5.00
`
`0.00
`
`1 .oo
`
`5.00
`
`3.00
`2.00
`BANK ANGLE
`Fig. 4. Lateral translation response.
`
`
`
`4.00
`
`gain matrix without any gain constraints.
`The time histories (not shown) exhibit a lat-
`eral acceleration, which achieves its
`lg
`commanded value while the steady-state
`sideslip angle and bank angle are described
`by Ip,.I = 0.44 degree and 145rl =
`0.014 degree, respectively.
`Design #2 is characterized by the con-
`straints that bank angle and roll rate shall not
`be fed back to either the rudder or canard. In
`this design, the number of gains is reduced
`by more than 25 percent as compared to
`
`May 7985
`
`design #l. The time histories (not shown)
`exhibit almost the same behavior as
`in
`design #l.
`Design #21 is characterized by feeding
`back proportional plus integral sideslip angle
`described by
`
`The time histories for this design are
`shown in Fig. 5. Sideslip angle attains
`
`a
`
`maximum of 0.42 degree, but it approaches
`zero as the time into the maneuver increases.
`The bank angle attains a maximum value of
`0.18 degree, which is acceptable although it
`is larger than for designs #1 and #2. Fur-
`ther, we observe that both heading and lateral
`flight path have achieved a change in excess
`of 17 degrees during the 10-sec maneuver.
`We conclude that design #21 is acceptable
`because it achieves heading and flight path
`changes with insignificant variations in side-
`slip angle and bank angle.
`
`73
`
`Ex. PGS 1029
`
`

`

`1.20 r
`
`o.*ot--r
`
`-0.120
`
`-0.178
`
`0.69 r
`0.46 c
`
`LATERAL ACC AT PILOT (G'sl
`
`BANK ANGLE IDEGI
`
`I
`I
`0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00
`
`I
`I
`
`TIME (SECONDS)
`Fig. 5. Direct sideforce responses.
`
`Conclusion
`A task-tailored multimode flight control
`system was designed by using eigenstructure
`can choose between
`assignment. The pilot
`six different modes in order to match the
`aircraft performance to a specific task or mis-
`sion. The design methodologies for the dif-
`ferent modes have been described including
`an explanation of the choices for the desired
`eigenvectors. Aircraft responses were shown
`to demonstrate the effectiveness of the flight
`control system.
`
`References
`A. N. .4ndq. E. Y. Shapiro, and J. C. Chung.
`"Eigenstructure Assignment for Linear Sys-
`tems." IEEE Trans. Aerosp. Electron. S w . ,
`vol. AES-19, pp. 711-729, Sept. 1983.
`K. M . Sobel and E. Y. Shapiro. "A Design
`Methodology for Pitch Pointing Flight Con-
`trol Systems,"J. Guid. Contr.. and Dpnam.,
`vol. 8, no. 2. Mar.-Apr.
`1985.
`K . M . Sobel. E. Y. Shapiro. and R. H.
`Rooney. "Synthesis of Direct Lift Control
`Laws Via Eigenstructure Assignment." Pro-
`
`ceedings of the 1984 Sational Aerospace ar7d
`Electronics Conference, Dayton. Ohio.
`pp. 570-575. May 1984.
`141 K. M . Sobel and E. Y. Shapiro. "Application
`of Eigensystem Assignment to Lateral Trans-
`lation and Yaw Pointing Flight Control.''
`Proceedings of rite '3rd IEEE Conference on
`Decision and Control. Las Vegas. Nevada,
`pp. 143-1428. Dec. 1984.
`[ 5 ] K. M . Sobel. E. Y. Shapiro. and A . N.
`Xndq. "Flat Turn Control Law Design Using
`Eigenstructure Assignment." Proceedings of
`7th International S~rnposiunl 017
`the Marhe-
`rnatical Theon of .vCht'OrkS and S.vsteins.
`Stockholm. Sweden. June 3985.
`[6] M. J. O'Brien and J. R. Broussard. "Feed
`Forward Control to Track the Output of a
`Forced Model." Proceedings of the 17th
`IEEE Conference on Decision and Control.
`San Diego, California. Jan. 1979.
`J. Davison. "The Steadystate Inverti-
`[7] E.
`bility and Feedforward Control
`of Linear
`Time-Invariant Systems." IEEE Trans. Auto.
`Contr.. vol. AC-?I. no. 4. pp. 529-534.
`Aug. 1976.
`[8] D. B. Ridgely. J. T. Silverthorn. and S. S.
`Banda. "Design and Analysis of a Multi-
`variable Control System for a
`CCV Type
`
`Fighter Aircraft," Proceedings of the AIAA
`9th Atmospheric Flight Mechanics Confer-
`ence. San Diego. California. Aug. 1982.
`[9] K. M . Sobel. "Application of Eigenstructure
`Assignment to Task-Tailored Multimode
`Flight Control System Design." Lockheed
`California Company Report LR-30852, Feb.
`1985.
`
`Kenneth M. Sobel was
`born in Brooklyn, New
`York. in 1954. He re-
`ceived the B.S.E.E.
`degree from the City Col-
`lege of New York in 1976
`andtheM.Eng. andPh.D.
`degrees in 1978 and 1980,
`respectively, from Rens-
`selaer Polytechnic Insti-
`tute. Troy, New York.
`From 1976 to 1980, he
`was a Research Assistant with the Department of
`Electrical and Systems Engineering. Rensselaer
`Polytechnic Institute. There he performed research
`in the areas of reduced state optimal stochastic
`feedback control and model reference adaptive
`control for multi-input multi-output systems. Since
`1980. he has been with Lockheed California Com-
`pany where he has published numerous papers on
`the application of modern control theory to aero-
`space problems. His wsork on adaptive control
`has been chosen to appear in Academic Press'
`Advances in Control a17d Dyamic Sxstems.
`Dr. Sobel has taught courses in lumped pa-
`rameter systems, feedback systems. and digital
`systems at both Rensselaer Polytechnic Institute
`and California State University, Northridge. He
`currently holds the position of Adjunct Assistant
`Professor at the University of Southern California,
`\vhere he
`teaches courses in the Electrical Engi-
`neering Department.
`Dr. Sobel is a Senior Member of LEEE and a
`member of Eta Kappa Nu and Sigma Xi.
`
`Eliezer Y. Shapiro
`received the B.Sc. and
`M . S c . d e g r e e s f r o m
`Technion-Israel
`in 1962
`and 3965, respectively,
`the Sc.D. degree from
`Columbia University,
`New York. in 1972, and
`the MBA degree from
`UCLA in 1984.
`From 1962 to 1966, he was with the Armament
`Development .4uthority, Israeli Ministry of De-
`fense, where he was responsible for the analysis,
`design. and evaluation of high performance analog
`and digital systems. From 1968 to 1972,
`he
`developed computer-controlled typesetting equip-
`ment for Harris Intertype Corporation and was
`a Consultant to the Varityper Division of
`Addressograph-Multigraph Corporation. During
`1973-1974. he was a Senior Staff Engineer for
`PRD Electronics. Inc. In 1974, he joined Lock-
`heed California Company as an Advanced Systems
`Engineer involved in applying modern control
`theory to the SR-71 and other advanced aircraft.
`
`IEEE Control Systems Mogozine
`
`Ex. PGS 1029
`
`

`

`He was responsible for applying observer theory to
`the design, development, and successful flight
`test on a Lockheed L-1011 aircraft of a state re-
`constructor that generated
`a yaw rate signal
`through analytical means. Subsequently, he
`formed a team of highly qualified specialists who
`focus on developing new techniques for control of
`high performance aircraft and applying them in
`actual hardware. This team has been formalized
`into the Flight Control Research Department,
`
`which, under Dr. Shapiro’s direction, is respons-
`ible for company-wide research and development
`of advanced flight control systems analysis, de-
`sign, and evaluation methodology. Since 1985, Dr.
`Shapiro has been General Manager of the Special
`Products Division at HR Textron, Valencia, Cali-
`fornia. He has published over 70 papers on the
`application of modem control theory to aerospace
`problems.
`Dr. Shapiro is cocreator of a course entitled,
`
`“Analysis and Design of Flight Control Systems
`Using Modem Control Theory,” which he teaches
`at the University of California, Los Angeles, and
`the University of Maryland. Through this course,
`he shares his experience in using modem control
`theory to successfully design advanced flight con-
`trol systems.
`Dr. Shapiro is an Associate Fellow of AIAA,
`IEEE, and a member of
`Senior Member of
`Sigma Xi.
`
`1985 American Control
`Conference
`
`The Fourth Annual American Control
`Conference will take place in the historic and
`dynamic setting of the Boston Maniott Hotel
`Copley Place, Boston, Massachusetts, from
`June 19 to 21, 1985. A total of 40 invited
`sessions and 18 contributed sessions will be
`presented on all aspects of control theory and
`practice. The scheduled plenary speakers
`
`include Dr. Nam P. Suh, Assistant Director
`for Engineering at the National Science
`Vice
`Foundation; Dr. Donald C. Fraser,
`President of the Charles Stark Draper Labo-
`ratory; and Dr. Arthur Gelb, Founder and
`President of The Analytic Sciences Cor-
`poration (TASC). The General Conference
`Chairman is Professor Yaakov Bar-Shalom.
`
`For questions regarding the technical pro-
`gam, contact the Program Chairman: Pro-
`fessor David Wormley, MIT, Department
`of Mechanical Engineering, Cambridge, MA
`02139, phone: (617) 253-2246. See the Feb-
`ruary 1985 issue of IEEE Control Systems
`Magazine (p. 46) for the Hotel Reservation
`Form and the Advance Registration Form.
`
`Paul Revere Statue
`
`U . S . S . Constitution “Old Ironsides”
`
`Minuteman Statue in Lexington
`
`May 1985
`
`15
`
`Ex. PGS 1029
`
`