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

`

`Eigenstructure Assignment for Design of
`Mu,timode Flight Control Systems
`Kenneth M. Sobel and Eliezer Y. Shapiro
`
`ABSTRACT: Advanced aircraft such as
`control configured vehicles (CCV) provide
`the capability for implementing multimode
`control laws, which allow the aircraft perfor(cid:173)
`mance to be tailored to match the character(cid:173)
`istics of a specific task or mission. This is
`accomplished by generating decoupled air(cid:173)
`craft motions, which can be used to improve
`aircraft effectiveness. In this article, we de(cid:173)
`scribe a task-tailored multimode flight con(cid:173)
`trol system, which was designed by using
`eigenstructure assignment.
`
`Introduction
`Advanced aircraft such as control con(cid:173)
`figured vehicles (CCV) provide the capabil(cid:173)
`ity to control the aircraft in unconventional
`ways. One such approach is to generate de(cid:173)
`coupled motions, which can be used to im(cid:173)
`prove tracking and accuracy. The decoupled
`motions are obtained by utilizing a task(cid:173)
`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(cid:173)
`tor form a set of redundant control surfaces
`capable of decoupling normal control forces
`and pitching moments. The decoupled mo(cid:173)
`tions include pitch pointing, vertical transla(cid:173)
`tion, and direct lift control. Pitch pointing is
`characterized by pitch attitude command
`without a change in flight path angle. Vert(cid:173)
`ical translation is characterized by flight
`path command without a change in pitch atti(cid:173)
`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(cid:173)
`figured vehicle, the vertical canard and rud(cid:173)
`der form a set of redundant surfaces that is
`capable of producing lateral forces and yaw(cid:173)
`ing moments independently. The decoupled
`
`This work was done while Kenneth M. Sobel
`and Eliezer Y. Shapiro were with Lockheed(cid:173)
`California Company, in Burbank, CA 9l520.
`Eliezer Y. Shapiro is now with HR Textron,
`Valencia, CA 91355.
`
`motions include yaw pointing, lateral trans(cid:173)
`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 sideforee is char(cid:173)
`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(cid:173)
`ment to conventional flight control design
`has been described by Shapiro et a!. in
`Ref. [1]. A design methodology that uses
`eigenstructure assignment to obtain decou(cid:173)
`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(cid:173)
`tem. The longitudinal design is illustrated by
`using the unstable dynamics of an advanced
`fighter aircraft, and the lateral design is illus(cid:173)
`trated by using the dynamics of the flight pro(cid:173)
`pulsion control coupling (FPCC) vehicle.
`
`Eigenstructure Assignment Basics
`Consider an aircraft modeled by the linear
`time-invariant matrix differential equation
`given by
`
`x =Ax + Bu
`y = Cx
`
`(1)
`
`(2)
`
`where x is the state vector (n x I), u is the
`control vector (m x !), andy 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(cid:173)
`sume that m, 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 y.
`
`u = -Fy
`
`The feedback problem can be stated as fol(cid:173)
`lows [I]: Given a set of desired eigenvalues,
`(A1), i = I, 2, ... , r and a corresponding
`set of desired eigenvectors, (vf), i
`I,
`2, ... , r, find a real m X r matrix F such
`that the eigenvalues of A - BFC contain
`(,\f) as a subset, and the corresponding
`
`0272-1708/85/0500-0009$01.00 © 1985 IEEE
`
`BFC are close to the
`eigenvectors of A
`respective members of the set (v1).
`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,J - A)- 1B.
`This subspace is of dimension m, which is
`the number of independent control variables.
`In general, a chosen or desired eigenvector
`v1 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 v1 onto the
`subspace spanned by the columns of
`(A 11 - A)- 18.
`We summarize with the following:
`
`• The matrix F will exactly assign r eigen(cid:173)
`values. It will also exactly assign each of
`the corresponding r eigenvectors to
`m-dimensional subspaces, which are con(cid:173)
`strained by Af, A, and B.
`• If more than m elements are specified for
`a particular eigenvector, then an achiev(cid:173)
`able eigenvector is computed by projecting
`the desired eigenvector onto the allowable
`subspace. This is the subspace spanned by
`the columns of (A;l A)- 1B.
`• If control over a larger number of eigen(cid:173)
`values is required, then additional inde(cid:173)
`pendent sensors must be added.
`• If improved eigenvector assignability is
`required, then additional independent con(cid:173)
`trol surfaces must be added.
`
`Now suppose that in addition to transient
`shaping, we desire the controlled (or tracked)
`aircraft variables y, to follow the command
`vector Uc with zero steady-state error where
`
`y, = Hx
`
`(3)
`
`The complete control law is derived by
`Broussard [6] and Davison [7]. If the com(cid:173)
`mand inputs Uc are constant, and if the track(cid:173)
`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 = (fizz + FCf!12)uc
`feedforward
`
`- Fv
`......_,...;:..
`feedback
`
`(4)
`
`May 7985
`
`9
`
`Ex. PGS 1029
`
`

`

`0 I
`
`where
`
`.n = [.nll I .nl2J = [¥]-! ~ H,O
`
`Further details of the eigenstructure assign(cid:173)
`ment algorithm may be found in [1].
`
`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(cid:173)
`trol actuator dynamics (elevator and flap(cid:173)
`erons). The equations of motion are de(cid:173)
`scribed by Eqs. (1) and (2) where the state x
`has five components and the control u has
`two components.
`
`(} -pitch attitude
`q -pitch rate
`x = a -angle of attack
`8, -elevator deflection
`Of -
`flaperon deflection
`
`u = [O•c] -elevator deflection command
`ofr - flaperon deflection command
`
`A=
`0
`0
`0 -0.8693 43.223 -17.251 -1.5766
`0
`0.9933 -1.341 -0.1689 -0.2518
`0
`0
`0
`-20
`0
`0
`0
`0
`0
`0
`-20
`
`B=l~ ~I
`
`0
`20
`0 20
`
`The eigenvalues of the open-loop system
`from matrix A are given by
`
`At = -7.662} unstable short
`period mode
`5.452
`Az =
`
`AJ = 0.0
`A4 = -20
`A5 = -20
`
`pitch attitude mode
`
`elevator actuator mode
`
`flaperon actuator mode
`
`The normal acceleration at the pilot's sta(cid:173)
`tion n,P is used as a controlled aircraft vari(cid:173)
`able for pitch pointing.
`
`nzp = [ -0.268, 47.76, -4.56, 4.45]
`
`q
`a
`a.
`of
`where nzp is in g 's and q, a, 8., and of are
`in radians or rad/sec. In what follows, we
`
`(5)
`
`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(cid:173)
`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
`where TAS is true airspeed. The surface de(cid:173)
`flections are measured by using linear vari(cid:173)
`able differential transformers (L VDT).
`We include 'Y as a state because this is the
`variable whose perturbation we require to re(cid:173)
`main zero. Thus, we replace (} 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
`
`(6)
`
`(7)
`
`(8)
`
`(9)
`
`x = [y,q,a,8.,8tY
`u = [o.co OfcY
`y = [q, nzp, y, 8., Of Y
`Our first step in the design is to compute
`the feedback matrix F. The desired short pe(cid:173)
`riod frequency and damping are chosen to be
`( = 0.8 and w" = 7 rad/sec. These values
`were chosen to meet MIL-F-8785C specifi(cid:173)
`cations for category A, level 1 flight. Cate(cid:173)
`gory A includes nonterrninal flight phases
`that require rapid maneuvering. precision
`tracking, or precise flight path control.
`Level I flying qualities are those that are
`clearly adequate for the mission objectives.
`We can arbitrarily place all five eigen(cid:173)
`values because we have five measurements.
`We can also arbitrarily assign two entries in
`each eigenvector because we have two in(cid:173)
`puts. Alternatively, we can specify more than
`tv.·o entries in a particular eigenvector, and
`then the algorithm will compute a corre(cid:173)
`sponding achievable eigenvector by taking
`
`the projection of the desired eigenvector onto
`the allowable subspace .
`We choose the desired eigenvectors to de(cid:173)
`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(cid:173)
`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(cid:173)
`vectors represent elements that are not speci(cid:173)
`fied ·because they are not directly related to
`the decoupling objective.
`We now compute the feedforward gains by
`using Eq. (4). For the pitch pointing problem
`y, = Hx = [e. 'Y Y
`Uc = [(}co 'YcY
`
`(lOb)
`
`(lOa)
`
`where
`(}c = pilot's pitch attitude command
`'Yc = pilot's flight path angle command
`I 0 0 OJ
`0 0 0 0 0
`
`H = [~ 0
`
`The feedforward gain matrix consists of
`four gains, which couple the commands (}c
`and 'Yc to the actuator inputs. The control law
`is described by
`
`When 'Yc = 0. we can command pitch atti(cid:173)
`tude without a change in flight path angle
`(pitch pointing). Alternatively, when (}c =
`0, we can command flight path angle without
`a change In pitch attitude (vertical trans(cid:173)
`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(cid:173)
`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 Pointing/Vertical Translation
`
`Desired Eigenvectors
`
`Achievable Eigenvectors
`
`l~- t~l t~~~ I~ I 1~1 ; I ~~= = = 1[~~~5 I I ~~~ I ~-~:~~571 ~-~:~~~~~
`
`X
`X
`X
`
`1 X
`X
`X
`X
`X
`
`X
`X
`a
`1 X
`8,
`1 of
`X
`
`-0.9286
`-5.13
`8.36
`
`1
`0.1286
`-5.16
`
`-1
`-2.80
`3.23
`
`-0.0508
`1
`0
`
`0.0106
`0
`1
`
`10
`
`IEEE Control Systems Magazine
`
`Ex. PGS 1029
`
`

`

`Table 2
`Pitch Pointing/Vertical Translation Control Law
`
`Desired
`Eigenvalues
`
`Feedforward
`Gains
`
`A1.z = -5.6 ± j4.2
`A~=-1.0
`A~= -19.0
`A~= -19.5
`
`-2.88 -0.367]
`2.02
`4.08
`
`[
`
`[
`
`Feedback Gains
`oJ
`y
`o.
`nz
`q
`0.747]
`-0.931 -0.149 -3.25 -0.153
`0.537 -1.04
`0.954
`0.210
`6.10
`
`(.!J
`u.J
`CJ
`
`F
`
`CD
`
`0.0
`
`1.0
`
`2.0
`
`3.0
`
`4.0
`
`5.0
`
`(A) PITCH POINTING RESPONSE
`
`1.0
`
`0.5
`
`0.0
`
`B
`u.J
`CJ
`?=.::
`CD
`
`e
`
`-0.5L------L------L------L----~------~
`5.0
`4.0
`1.0
`0.0
`2.0
`3.0
`
`(B) VERTICAl TRANSlATION RESPONSE
`
`Fig. 1. Longitudinal decoupled responses.
`
`Table 3
`Direct Lift Control Summary
`
`Mode 3: Direct Lift Control
`The objective in direct lift control (DLC)
`is to command normal acceleration (or equi v(cid:173)
`alently flight path angle rate) without a
`change in angle of attack. To achieve acceler(cid:173)
`ation command following, we include inte(cid:173)
`grated normal acceleration in the state vector.
`Thus, for the DLC problem, we choose the
`state vector to be
`
`where
`
`nz1 =
`
`{
`
`integral of normal acceleration
`at the pilot's station
`
`The measurement vector is chosen to be
`
`The desired eigenvalues and desired eigen(cid:173)
`vectors are shown in Table 3. Observe that
`the zeros in the desired eigenvectors are
`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 n,P and commanded
`nzp. The control law is described by
`
`u = [-jll -!12 -!14 -!15] a
`
`q
`
`-j21 -j22 -Jz4 -j25 Oe
`OJ
`
`+ [ =~:] J e(t) dt
`
`(12)
`
`where
`
`e(t) = llzp -
`
`(n,p)command
`
`The DLC responses to a lg normal accelera(cid:173)
`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
`
`mode decoupling
`
`Au= -5.6 ± j4.2
`A3 = -5.6
`A4=-19.0
`As= -19.5
`
`[ -0.722 -~.70 -0.220 0.468 0.0587]
`-0.301
`).51
`0.994 0.223 0.187
`
`q
`a
`Oe
`OJ
`llzJ
`
`Wliv ~ ~ m I
`
`llz1 Oe
`
`OJ
`
`a
`q
`short
`period
`
`May 1985
`
`11
`
`Ex. PGS 1029
`
`

`

`ANGLE OF ATTACK, a (DEGREES!
`
`-0.086°
`
`FLIGHT PATH ANGLE, 'Y (DEGREES!
`
`13.0°
`
`0
`
`-0.05
`
`-0.10
`
`'Y
`
`10
`
`5
`
`0.0
`
`0.5
`
`-1.0
`
`1.5
`
`2.0
`
`2.5
`
`3.0
`
`3.5
`
`4.0
`
`4.5
`
`5.0
`
`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 y. The eight state variables are
`sideslip angle ({3), bank angle (c/>), roll rate
`(p), yaw rate (r), lateral directional flight
`path (y = 1/J + {3), rudder deflection (D,),
`aileron deflection (Da), and canard deflec(cid:173)
`tion (De).
`The three control variables are rudder
`command (D,c), aileron command (Dac), and
`canard command (Dec). The five measure(cid:173)
`ment variables are r, {3, p, r/J, y. Because of
`space limitations, the detailed numerical re(cid:173)
`sults are not presented, but the general ap(cid:173)
`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(cid:173)
`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
`of the flight path mode. For design #I,
`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(cid:173)
`actly those that were desired.
`
`12
`
`We compute the feedforward gains by
`using Eq. ( 4). The tracked variables are
`given by
`
`y, = [1/J,y.c/>Y
`
`and the pilot commands are given by
`lie = [we. Yc· wX
`
`where
`1/J,. = commanded heading
`Yc = commanded lateral directional
`flight path
`rPc = 0
`Since bank angle is commanded to be zero,
`we need not implement the gains that
`multiply We· It is included in the numerical
`computations only to avoid the need for a
`pseudo-inv'ei-sion.
`The time histories for design #I are not
`shown; however, the yaw pointing responses
`to a unit step heading command are such that
`IYI ~ 0.0004 degree and lc/> 1 ~ 0.0031
`degree. The lateral translation responses
`to a unit step lateral flight path command
`are such that 11/Ji ~ 0.008 degree and
`:r~J; ~ 0.004 degree.
`Design #2 is characterized by an addi(cid:173)
`tional specification that seven of the feed(cid:173)
`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(cid:173)
`tical point of view. Furthermore, by reducing
`the number of gains, we have increased the
`reliability of the control system.
`
`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.
`
`Mode 6: Direct Sideforce Control
`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
`
`where
`
`= {integral of lateral acceleration
`at pilot's station
`
`n,.P,
`
`The measurement vector is chosen to be
`y = [r, {3, p, <b, n,.p,y
`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
`I
`0 0
`0 0
`X
`X X
`X .x
`X X
`
`0 0
`X
`I X
`0 0
`X X
`X X
`
`;oi
`{3
`:o:
`c/>
`:o: p
`~0~
`r
`X
`1/1+{3
`X D,
`X
`Da
`X
`De
`1
`
`nYPl
`
`dutch roll roll mode ace mode
`mode de<;oupling
`
`To obtain lateral acceleration command
`following, we feed back the integral of the
`error between measured n,P and commanded
`n,.P. This approach is similar to that used for
`direct lift control.
`Several designs are investigated. Design
`#I is characterized by an output feedback
`
`IEEE Control Systems Magazine
`
`Ex. PGS 1029
`
`

`

`1.20
`
`0.80
`
`0.40
`
`0.00
`0.00
`
`1.00
`
`3.00
`
`2.00
`HEADING
`
`4.00
`
`5.00
`
`0.08
`
`0.04
`
`0.00
`
`I
`0
`
`-0.04
`0.00
`
`0.22
`
`0.14
`
`1.00
`
`2.00
`
`3.00
`
`4.00
`
`5.00
`
`LATERAL FLIGHT PATH
`
`-0.04
`
`0
`
`-0.08
`
`-0.12
`0.00
`
`1.20
`
`0.80
`
`0.40
`
`0.00
`0.00
`
`-0.02
`
`-0.06
`
`-0.10
`
`1.00
`
`2.00
`
`3.00
`
`4.00
`
`5.00
`
`HEADING
`
`1.00
`
`2.00
`
`3.00
`
`4.00
`
`5.00
`
`LATERAL FLIGHT PATH
`
`BANK ANGLE
`
`Fig. 3. Yaw pointing response.
`
`BANK ANGLE
`
`Fig. 4. Lateral translation response.
`
`- 0.14 ...._ __ __,_ ___ _,_ ___ ......._ ___ ,__ __ ---~._
`3.00
`5.00
`0.00
`1.00
`2.00
`4.00
`
`gain matrix without any gain constraints.
`The time histories (not shown) exhibit a lat(cid:173)
`eral acceleration, which achieves its 1g
`commanded value while the steady-state
`sideslip angle and bank angle are described
`by I.Bssl = 0.44 degree and lcf>ssl =
`0.014 degree, respectively.
`Design #2 is characterized by the con(cid:173)
`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
`
`design #1. The time histories (not shown)
`exhibit almost the same behavior as in
`design #1.
`Design #21 is characterized by feeding
`back proportional plus integral sideslip angle
`described by
`
`Yz = ,B - 0.1 J ,B dt
`
`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(cid:173)
`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(cid:173)
`slip angle and bank angle.
`
`May 7985
`
`73
`
`Ex. PGS 1029
`
`

`

`FLIGHT PATH ~ ,+ JIDEGI
`
`LATERAL ACC AT PILOT IG'sl
`0.00 L-------------------
`
`TIME !SECONDS)
`
`Fig. 5. Direct sideforce responses.
`
`Conclusion
`
`A task-tailored multimode flight control
`system was designed by using eigenstructure
`assignment. The pilot can choose between
`six different modes in order to match the
`aircraft performance to a specific task or mis(cid:173)
`sion. The design methodologies for the dif(cid:173)
`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
`
`[I] A.~- Andry. E. Y. Shapiro, andJ. C. Chung.
`"Eigenstructure Assignment for Linear Sys(cid:173)
`tems." IEEE Trans. Aerosp. Electron. S\'St.,
`vol. AES-19. pp. 711-729, Sept. 1983.
`[2] K. M. Sobel and E. Y. Shapiro. ''A Design
`Methodology for Pitch Pointing Flight Con(cid:173)
`trol Systems,'' J. Guid. Contr .. and Dynam.,
`vol. 8, no. 2, Mar.-Apr. 1985.
`[3] K. M. Sobel, E. Y. Shapiro. and R. H.
`Rooney. "Synthesis of Direct Lift Control
`Laws Via Eigenstructure Assignment," Pro-
`
`14
`
`ceedings of the 1984 l\'ational Aerospace and
`Electronics Conference, Dayton, Ohio.
`pp. 570-575, \<1ay 1984.
`[4] K. M. Sobel and E. Y. Shapiro. "Application
`of Eigensystem Assignment to Lateral Trans(cid:173)
`lation and Yaw Pointing Flight Control."
`Proceedings of the 23rd IEEE Conference on
`Decision and Control. Las Vegas. Nevada,
`pp. 1423-1428. Dec. 1984.
`[5] K. !VI. Sobel. E. Y. Shapiro. and A. N.
`Andry. "Flat Tum Control Law Design Using
`Eigenstructure Assignment." Proceedings of
`7th International Symposium on the .'vtathe(cid:173)
`matical Theon of .'Vefl.-orks and Systems,
`Stockholm, Sweden, June 1985.
`[6] M. J. O'Brien and J. R. Broussard, "Feed
`Forv-;ard Control to Track the Output of a
`Forced Model." Proceedings of the 17th
`IEEE Conference on Decision and Colltrol.
`San Diego. California. Jan. 1979.
`[7] E. J. Davison. "The Steady-State Inverti(cid:173)
`bility and Feedforward Control of Linear
`Time-Invariant Systems," IEEE Trans. Auto.
`Contr., vol. AC-21, no. 4, pp. 529-534,
`Aug. 1976.
`[8] D. B. Ridgely. J. T. Silverthorn. and S. S.
`Banda. "Design and Analysis of a Multi(cid:173)
`variable Control System for a CCV Type
`
`Fighter Aircraft," Proceedings of the AIM
`9th Atmospheric Flight Mechanics Confer(cid:173)
`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(cid:173)
`ceived the B.S.E.E.
`degree from the City Col(cid:173)
`lege of New York in 1976
`and the M.Eng. and Ph.D.
`degrees in 1978 and 1980,
`respectively. from Rens(cid:173)
`selaer Polytechnic Insti(cid:173)
`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(cid:173)
`pany where he has published numerous papers on
`the application of modern control theory to aero(cid:173)
`space problems. His work on adaptive control
`has been chosen to appear in Academic Press'
`Advances in Control and Dynamic Systems.
`Dr. Sobel has taught courses in lumped pa(cid:173)
`rameter systems, feedback systems, and digital
`systems at both Rensselaer Polytechnic Institute
`and California State L:niversity, Northridge. He
`currently holds the position of Adjunct Assistant
`Professor at the University of Southern California,
`where he teaches courses in the Electrical Engi(cid:173)
`neering Department.
`Dr. Sobel is a Senior Member of IEEE and a
`member of Eta Kappa 1\u and Sigma Xi.
`
`Eliezer Y. Shapiro
`received the B.Sc. and
`M.Sc. degrees from
`Technion-Israel in 1962
`and 1965, respectively,
`the Sc.D. degree from
`Columbia University,
`New York. in 1972, and
`the MBA degree from
`CCLA in 1984.
`From 1962 to 1966, he was with the Armament
`Development Authority, Israeli Ministry of De(cid:173)
`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(cid:173)
`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(cid:173)
`heed California Company as an Advanced Systems
`Engineer involved in applying modem control
`theory to the SR-71 and other advanced aircraft.
`
`IEEE Control Systems Magazine
`
`- - - - - · - - - - ----------
`
`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(cid:173)
`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(cid:173)
`ible for company-wide research and development
`of advanced flight control systems analysis, de(cid:173)
`sign, and evaluation methodology. Since 1985, Dr.
`Shapiro has been General Manager of the Special
`Products Division at HR Textron, Valencia, Cali(cid:173)
`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(cid:173)
`trol systems.
`Dr. Shapiro is an Associate Fellow of AIAA,
`Senior Member of IEEE, and a 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 Marriott 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
`Foundation; Dr. Donald C. Fraser, Vice
`President of the Charles Stark Draper Labo(cid:173)
`ratory; and Dr. Arthur Gelb, Founder and
`President of The Analytic Sciences Cor(cid:173)
`poration (T AS C). The General Conference
`Chairman is Professor Yaakov Bar-Shalom.
`
`For questions regarding the technical pro(cid:173)
`gram, contact the Program Chairman: Pro(cid:173)
`fessor David Wormley, MIT, Department
`of Mechanical Engineering, Cambridge, MA
`02139, phone: (617) 253-2246. See the Feb(cid:173)
`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
`
`