F E A T U R EF E A T U R E

Fundamentals of the Analog Computer

Circuits, technology, and simulation

The analog computers created in theyears immediately following WorldWar II were based on electronicversions of the mechanical differen-tial analyzer, first conceived byLord Kelvin and implemented at

MIT during the 1930s by Vannevar Bush. Theheart of the analog computer is the operationalamplifier, which consists of a high-gain dcamplifier with a feedback impedance Zf andinput impedances Z1, Z2, and Z3, as shown inFigure 1. Operational amplifiers, configured asin Figure 1, can be combined to solve lineardifferential equations with constant coeffi-cients. Figure 2 shows how two integratingamplifiers and one summing amplifier can be

interconnected to solve a second-order mass-spring-damper system.

The dc operational amplifier was first devel-oped by Philbrick and researchers at Bell Laborato-

ries, but later improved by Goldberg at RCALaboratories with the addition of drift stabilization.

U.S. government-sponsored projects for the develop-ment of analog computers for real-time simulation of flight

equations included Project Cyclone at Reeves Instrument Cor-poration, Project Typhoon at RCA Laboratories, and the DACL

(Dynamic Analysis and Control Laboratory) at MIT, while ProjectWhirlwind funded the initial development of digital computers for real-

time flight simulation at MIT.To solve differential equations with specified initial values for the state variables, it is neces-

sary to include initial-condition relays with the integrating amplifiers. Figure 3 illustrates the cir-cuit for a single integrator, which includes both a reset relay and a hold relay. When the resetrelay is energized, the amplifier input is switched from the input computing resistors to thesumming junction of the two initial-condition resistors; the integrator output voltage thenassumes the value eo(0), the desired initial condition. When the reset relay is de-energized, the

By Robert M. Howe

© DIGITALVISION

June 2005 290272-1708/05/$20.00©2005IEEE

IEEE Control Systems Magazine

June 200530 IEEE Control Systems Magazine

amplifier input is reconnected to the input computingresistors, and the analog solution to the differential equa-tion proceeds. Figure 3 also shows a hold relay that, when

energized, disconnects the amplifier input from all inputresistors. This procedure then stops the integration andfreezes the integrator output voltage at its current value.

The hold mode offers a useful methodfor accurately reading all integratorvalues at predetermined times duringan analog solution.

To solve linear differential equa-tions with time-varying coefficients,the author utilized a stepping relay tovary the appropriate computing resis-tor of an operational amplifier at afixed sample rate in time. The resistorfor each time step was chosen so thatthe time integral of the resistor valuematched the time integral of the cor-responding continuous time function.Figure 4 shows a simple example, ananalog circuit for solving Bessel’sequation of order zero.

In [1], examples are given of ana-log solutions of the time-dependenttransient responses of one- and two-degree-of-freedom mechanical sys-tems. The transient response ofsimple feedback control systems,where the analog computer solutionsare recorded as a function of the inde-pendent variable time by using adirect-inking oscillograph, is exploredas well. Also shown are the solutions

of both Bessel’s and Legendre’sdifferential equations, which aresolved as second-order linearsystems with time-varying coef-ficients using the stepping-relayscheme described above (andshown in Figure 4) to approxi-mate the variable coefficients.Solutions for the static displace-ment of structural beams withvarious boundary conditionsand load distributions are alsosolved by allowing time on theanalog computer to correspondto distance along the beam. Thenormal-mode frequencies ofboth uniform and nonuniformbeams are solved as two-pointboundary-value problems, againby letting time on the analogcomputer represent distancealong the beam. Here the un-known ratio of the two nonzero

Figure 2. Analog circuit for simulating a mass-spring-damper system. Amplifier 1 sumsthe terms in the equation for d2x/dt2 and integrates the sum to obtain −dx/dt. Amplifier2 integrates −dx/dt to obtain x. Amplifier 3 simply inverts x to obtain −x. Note that allintegrators are inverting.

Resistors in megohms, capacitors in microfarads

m/c

m/k

m

− x− xx

− —dxdt

− —dxdt

f(t)

111

1121 3

f(t)

m

k

x

c

m—— + c — + kx = f(t), or —— = — f(t) − — — − —x .d2xdt2

d2xdt2

dxdt

dxdt

1m

cm

km

Figure 1. Operational amplifier. When the feedback and input impedances areresistors, the output voltage is proportional to the sum of the input voltages. Whena capacitor C is used as a feedback impedance, Zf = 1/(C s) and the output volt-age is proportional to the time integral of the sum of the input voltages.

InputVoltages

OutputVoltage

dc AmplifierGain = −

Z1 Zf

eo

ifi1

i2

i3

Z2

Z3

e1

e'

e2

e3

Zf

Z1

Zf

Z2

Zf

Z3— e1+ — e2+ — e3eo ≅ − .

µ

For the dc amplifier gain >> 1 + — + — + — ,Zf

Z1

Zf

Z2

Zf

Z3µ

Letting i1+i2+i3 = if and e' = − —1 eo, it follows from Ohm's Law thatµ

Zf

Z1

Zf

Z2

Zf

Z3— e1+ — e2+ — e3

Zf

Z1

Zf

Z2

Zf

Z3— + — + —

1+1–

eo = ————————— .

µ 1+

June 2005 31IEEE Control Systems Magazine

initial state variables, together with the unknown eigenval-ue for each mode, are varied by trial and error until thetwo specified end conditions are met.

Stability of Operational AmplifiersBecause the operational amplifier represents a feedbacksystem with a wide choice of feedback and input imped-ances, it is important to consider the dynamic require-ments on the amplifier gain µ(s) to ensure closed-loopamplifier stability. These requirements can be understoodby breaking the feedback loop, as shown in Figure 5, withthe input impedance Zi grounded to represent zero inputvoltages. The overall open-loop transfer function is thengiven by

eo

e′ (s) = − µZi

Zi + Zf,

also known as the return ratio. For the closed-loop systemto be stable, the phase shift of the open-loop system mustbe more positive than −π at the frequency of unity open-loop gain, defined as the crossover frequency. To be con-servative, we usually require the phase shift of theopen-loop system to be equal to −π/2 at the crossoverfrequency. Clearly

|µ| =∣∣∣∣

Zf + Zi

Zi

∣∣∣∣

at the crossover frequency. If the feedback and inputimpedances are resistors, as in the case of a summingamplifier, then the crossover frequency occurs when

|µ| = Rf + Ri

Ri.

If the feedback impedance is a capacitor, then |Zf | � |Zi|and the crossover frequency occurs when |µ| = 1. Toensure that the open-loop phase shift is equal to −π/2 atthe crossover frequency for all possible summer and inte-grator configurations, it follows that the phase shift of theamplifier gain µ(s) should be −π/2 or greater at all frequen-cies for which |µ| > 1. When the dc operational amplifier isa stable minimum-phase system (no zeros or poles in theright-half plane), which is indeed the case for both vacuum-tube and transistor amplifiers, a phase shift of −π/2 orgreater is realized when the slope of the log |µ| versuslog |ω| plot is greater than or equal to −6 dB/octave. Figure5 shows the resulting open-loop frequency response.

An example of a more complex configuration of feedbackand input impedances is the single amplifier circuit for sim-ulating a second-order linear system, as shown in Figure 6.To examine the stability of this feedback system, weground the input terminal −ei, break the feedback loop asshown in the figure, and let the open-loop input be e′

o. Thisone-amplifier circuit for simulating a second-order systemcan be used in the feedback loop of a high-gain amplifier to

Figure 3. Circuit for integrator mode control. The resetrelay, when energized, establishes the initial condition forthe integrator. The hold relay, when energized, stops theintegration and freezes the output voltage eo.

Figure 4. Analog circuit for solving Bessel’s equation of order zero. The variable coefficient is approximated with the stair-case function using a stepping relay.

IntegratorOutputInput

ComputingResistors

C

eo

−eo(0)

1

1

1 ResetRelay

HoldRelay

0.1

0.1

0.1

0.1

1− x

− xx− —dx

dt− —dx

dt

11 1

1121 3

—— + – — + x = 0, or —— = −– — −x .d2xdt2

d2xdt2

dxdt

dxdt

1t

1t 1

0.75

0.5

0.25

00 0.25 0.5 0.75 1.0

t

ttˆ

tˆ

June 200532 IEEE Control Systems Magazine

obtain a representation of the reciprocal of a second-ordersystem. The author utilized such a scheme to implementan electronic drive circuit for the two-channel direct-inkingoscillograph used for recording analog solutions. In thiscase, the recorder by itself was well represented by thedynamics of a second-order system with an undamped nat-ural frequency of approximately 30 Hz. When combinedwith the electronic drive circuitry described earlier, thebandwidth of the overall recording system was extendedto more than 100 Hz.

The Use of Potentiometers to SetCoefficient Values in Analog SimulationsAll of the circuits described thus far for the analog solutionof differential equations have utilized resistors to setsumming and integrating amplifier gains. However, a muchmore practical method for implementing the precision set-ting of arbitrary coefficients in analog simulations utilizespotentiometers. For example, the potentiometer shown atthe top of Figure 7 with input voltage x provides an output

voltage cx, where c is acoefficient that can beset between 0 and 1. Thediagram of the analogcircuit for simulating themass-spring-damper sys-tem, shown earlier in Fig-ure 2, is repeated inFigure 7 using poten-tiometers to representthe fixed coefficients inthe problem. The feed-back impedances are notshown in the operationalamplifiers in Figure 7.Instead, only the fixed

input gains (nominally one or ten)are shown, with a triangle repre-senting a summing amplifier and atriangle attached to a vertical barrepresenting an integrating amplifi-er. Thus, in Figure 7, amplifiers 1and 2 are integrators, whereasamplifier 3 is an inverter. The inputresistors m/c and m in Figure 2 arereplaced in Figure 7 with coefficientpots 1 and 2 set at c/m and 1/m,respectively, in both cases drivingunity gain inputs. The input resis-tor m/k in Figure 2 is replaced bycoefficient pot 4, set at 0.1 c/m anddriving a gain-of-ten input. Pot 4 isused in Figure 7 to set the initialcondition x(0) on the mass

displacement, whereas there is no indication of initial-condition circuitry in Figure 2.

It should be noted that the setting of the output wiperarm on the coefficient pot must take into account the inputresistor driven by the pot, since that input resistor is inparallel with the pot resistance from wiper arm to ground.In general-purpose analog computers, the coefficient potsare normally set in a pot-set mode, with the pot input xreplaced by +1 reference (that is, +100 V for analog com-puters based on 100-V reference, and +10 V for analogcomputers based on 10-V reference). The pot is then set tothe desired value, either by reading a digital voltmeter orby means of a null meter connected between the pot out-put and a reference pot with a precision dial. Note that thecoefficient pots are not set by reading the pot dials, eventhough the coefficient pots are usually ten-turn helical potswith precision, counter-type dials.

It is apparent that the variables used in analog simula-tions must be properly scaled both to avoid over-rangingamplifier outputs and to ensure that amplifier outputs

Figure 5. Operational amplifier frequency response. This open-loop frequency response isneeded to provide 90° of phase margin for any ratio of feedback to input impedances.

= Open-Loop Transfer Function

eo'

eo'

eo−

e'

Zf

Zi

Return Ratio = —eo

| (j )|(dB)

log 0

6 dB/octZf + Zi

Zi

µ

µ ω

ω

Figure 6. Single-amplifier circuit for simulating a second-order system when e′o = eo.

For the closed-loop system to be stable with a phase margin of π/2, the open-loop fre-quency response should exhibit a slope of −6 dB/octave at and around the frequencyof unity (0 dB) open-loop gain.

R1

Output

Input−ei

C2

C1

eo'

1

R3

R2

From Kirchhoff's and Ohm's Laws:

= —– [R2 + R3(1 + R2/R1)C2]ωn

2

k = –— ,R3

R1n = —————— ,

1

R3C1R2C2√

ζ

Differential equation: –— —— + —– —— + eo = kei1 ζ2d2eo

dt2

deo

dt ωn

ωeo

ωn2

June 2005 33IEEE Control Systems Magazine

range over a reasonable fraction offull-scale during any given solution.A similar problem occurs whenusing fixed-point digital computa-tion. The simplest solution is torepresent each variable as a scaledfraction, the variable divided by itsmaximum value. Then each scaled-fraction variable will range over±1, which corresponds to ±100 Vin the case of 100-V analog comput-ers and ±10 V in the case of 10-Vanalog computers. Actually, manyengineers (including the author)have found that the process ofscaling problem variables can lendconsiderable insight into the prob-lem itself, including the identifica-tion of dominant terms.

Servomultipliers forAnalog ComputersAll of the example analog simula-tions described thus far haveinvolved the solution of linear dif-ferential equations, including dif-ferential equations with time-varying coefficients. Figure 8shows a schematic of a typical ser-vomultiplier. The three gangedpotentiometers at the right side ofthe figure are driven by a two-phase induction motor with an n:1gear reduction (not shown in thefigure). The servomotor is in turndriven by a magnetic amplifier,which receives its dc input from ananalog controller unit. The error inservo output angle is given bye = Xin − Xout, where Xin is the command input angle forthe servo and Xout is the measured output angle, asobtained from the wiper arm output −Xout of the poten-tiometer with −100 V and +100 V connected to the highand low side, respectively. With ±Y and ±Z connectedacross the second and third potentiometers, the pot out-puts represent XY and X Z , respectively. Note that the sta-tic accuracy of the servomultiplier depends on thelinearity of each ganged pot as well as the static nullingerror of the servo.

Typical pots used in servomultipliers exhibit a maxi-mum linearity error ranging between 0.02% for ten-turnwire-wound, ganged potentiometers to 0.1% for one-turnganged potentiometers, either wire wound or film type.The static nulling error of the servomultiplier depends on

the overall design of the servo, the static friction due toboth pot friction and gear-train friction, and the static reso-lution of the ganged potentiometers. In the case of wire-wound potentiometers, the static resolution is equal to thereciprocal of the number of wire turns, typically 3,000 forone-turn pots and more than 10,000 for ten-turn pots. Thestatic resolution error associated with conducting-filmpots can approach zero. To preserve static accuracy in theservomultiplier, it should also be noted that it is importantfor the multiplier pots with outputs XY and X Z in Figure 8to drive input resistors that are identical with the resistorthat loads the output −Xout of the reference pot (that is, 1megohm in the example shown here).

The transfer function of a typical servomotor plus themagnetic amplifier driver is given by

Figure 7. Using potentiometers to set coefficient values in an analog simulation cir-cuit. Note that pot 3 is set at 0.1 c/m because of the gain-of-10 input. To speed up thesetting of coefficients in large problems, servo-set potentiometers were widely used instate-of-the-art analog computers.

Figure 8. A servomultiplier that uses a 60-Hz, two-phase servo motor driven by amagnetic amplifier. The analog controller circuit is used to provide additional damp-ing. Typical multiplier accuracy was 0.1% or better. Because of the limited servobandwidth, the servo input Xin should be assigned to the slower of the two variablesin computing a product.

1/m

c/m

0.1k/m

2

1

3

1 1 11 2

4

3

1

10

Coefficient Pot

f (t) x

–x(0)

–x

cxcx

cc

x

x

–1(ref)

m—— + c — + kx = f(t), ord2xdt2

dxdt

d2xdt2

dxdt

1m

cm

km

dxdt

Xin

−Z

XY XZ

+Z+Y−100

−Y+100

−Xout

−e u1

1

1

AnalogController

Circuit

MagneticAmplifier

ServoMotor

60-Hz Ref.60-Hz Ref.

June 200534 IEEE Control Systems Magazine

M(s) = Km/ Is2 + s/Tm

, (1)

where I is the total inertia of the servomultiplier(including servomotor, gear train, and ganged poten-tiometers referred to the servo output shaft) and Tm isthe motor time constant. If the analog controller circuitin Figure 8 consists of a pure gain K , then the overallservo transfer function is approximately represented bya second-order system with an undamped natural fre-

quency given by ωn = √

K Km/ I and damping ratio givenby ζ = (1/2)

√

I/K Km/Tm . To obtain satisfactory nullingerrors, the gain constant K for typical servomultipliersmust be sufficiently large that the resulting damping ratioζ is too small for satisfactory transient performance. Thus,it is necessary to add rate control to the proportional con-trol in the analog controller circuit. It also may be desir-able to add integral control to ensure low nulling error(that is, good servo tracking) for a slowly varying input Xin.Figure 9 shows an operational amplifier circuit that pro-vides the necessary controller transfer function, where the

rate control term Ces is bandwidth-limited by the transfer function1/(τs + 1).

The overall servomultiplier trans-fer function for sinusoidal inputspredicts the dynamic performanceof the closed-loop system based onlinear models of the subsystems.However, the servomotor has twoimportant characteristics that arenot modeled by the motor transferfunction (1). These characteristicsinclude the acceleration limit|X ′′|max and the velocity limit |X ′|max

for large motor controlinputs. The experiencegained in designing servo-multipliers for analog com-puters led to the realizationthat the design of an electro-mechanical control system,including the choice of com-ponents, is dominated bynonlinear requirements suchas the acceleration andvelocity limits (large-motionnonlinearities) and thenulling error or static resolu-tion (small-motion nonlinear-ities). The role of lineardesign methods is generallylimited to the choice of con-troller parameters needed toproduce satisfactory systemstability margin and transientresponse [2].

Other NonlinearAnalog-ComputerComponentsWhile the servomultipliersdescribed above were widelyutilized in analog computers

Figure 10. Diode circuit for approximating ((X + Y)/2)2 for X + Y > 0. The same circuitis repeated with inputs −X and −Y to approximate ((X + Y)/2)2 for X + Y < 0. A similarpair of circuits with inputs X and −Y, and −X and Y, with diodes reversed and +100 biasvoltages, is used to approximate −((X − Y)/2)2 . A single operational amplifier is used tosum the outputs of all four circuits to produce the product XY.

Figure 9. One-amplifier analog controller circuit for proportional, bandwidth-limit-ed rate and integral control. Alternatively, a circuit that requires two conventionalintegrating amplifiers and two summing amplifiers can be used.

Where K =

Input−e

Output

R2

R1C1

Rf

Zf

Zi

C2

ue

K== 1+Ces

τs + 1

τ = R1C1.

(1+ Cis−1)

Rf

R2

, Ce = (R1 + R2) C1 , Ci =1

R2 C2

,

−u1

••••

••••

2R1a

2R2a

2R3a

2R3a

2R2a

2R1a

R1b

R2b

R3b

Y

X

−100

X+Y

Y

X

−100

Y

X

−100D3

D2

D1

i1

i2

X+Y X+Y

i

i2

i=i1+i2+i3+ ...

i3

i3

i1

≡ −2

2

X+Y

X+Y

0.1

during the 1950s, several alternative, all-electronic deviceswere developed to improve the overall dynamics of analogmultipliers. The first of these was the time-division multi-plier, initially developed by Goldberg [3]. In this scheme,the voltage X is converted to a pulse-width modulated sig-nal used to drive an electronic switch, which modulatesthe voltage Y . The area under each cycle of the switchedsignal is proportional to the product XY . The switched sig-nal drives a low-pass filter to produce a smoothed outputXY . Although time-division multipliers achieved dynamicerrors that were typically two orders of magnitude smallerthan those associated with servomultipliers, these multi-pliers often exhibited undesirable drift in voltage offsetwith time. Another device was the quarter-square multipli-er, which was based on the equation

XY = (X + Y)2 − (X − Y)2

4. (2)

The circuit shown in Figure 10 is used to generate a seg-mented approximation to ((X + Y)/2)2 for X + Y > 0.When X + Y becomes positive enough, the diode D1 startsconducting and the current i1starts to increase linearly forfurther increases in X + Y . For alarger value of X + Y , diode D2

starts to conduct, and the cur-rent i2 starts to increase linearlywith increasing X + Y . Theoverall circuit in Figure 10 con-sists of n diode segments. Theamplifier output in Figure 10then produces an n-segmentapproximation to −((X + Y)/2)2

for X + Y > 0. With idealdiodes, the segmented approxi-mation to a quadratic functionthen exhibits a maximum frac-tional error of 1/(8n2). With thesilicon-junction diodes normallyused for quarter-square multi-pliers, the nonideal diode cur-rent-voltage characteristicproduces additional rounding ofeach segment knee, furtherreducing the quadratic-approxi-mation error.

It should be noted that (2) forthe quarter-square multiplier canbe replaced with the formula

XY = |X + Y |2 − |X − Y |24

.

By replacing (X + Y)2 with |X + Y |2 and replacing (X − Y)2

with |X − Y |2, we can eliminate the need for two of the foursegmented quadratic approximation circuits similar to theone shown in Figure 10. Furthermore, the calculation of|X + Y | and |X − Y | can be combined with the diode circuitfor the segmented quadratic approximation by using the cir-cuit shown in Figure 11. Here the voltage at the junction ofdiodes Da1 and Da2 is proportional to |X + Y |/2, and thevoltage at the junction of diodes Da3 and Da4 is proportionalto −|X − Y |/2, as indicated in Figure 11. These diode-junc-tion voltages then drive biased-diode circuits similar to theone in Figure 10 to produce segmented approximations tothe quadratic function. Because of the loading effect of thebiased-diode circuits on the passive absolute-value circuit inFigure 11, the effect of the nonideal diode characteristics onthe rounding of the knees of the segmented approximation isincreased. This rounding in turn causes error in the segment-ed approximation to the quadratic function to be consider-ably less than the value of 1/(8n2) for ideal diodes, where nis the number of diode segments used for the quadraticapproximation. Thus, the completely passive quarter-squaremultiplier circuit of Figure 11, in addition to reducing by

June 2005 35IEEE Control Systems Magazine

Figure 11. A completely passive quarter-square multiplier circuit terminated in a singleoperational amplifier. The circuit is based on XY = (|X + Y |/2)2 − (|X − Y |/2)2 . Thecircuit output is connected to the summing junction of an operational amplifier with a0.1 m� feedback resistor.

••••

••••

R

R

R

R

R

R

R

R

R1a

R1b

R2a

R2b

R1aDa1

Da2

D1

D2

−Vb1

−Vb1

+Vb1

+Vb1

R1b

R2a

R2b

X

≡XY

Y

−X

−Y

X

−Y

−X

Y

••••

••••

0.1

∝− X−Y 2

∝ X−Y 2

more than a factor of two the number of required compo-nents, also is more accurate than the circuit in Figure 10 for agiven number n of diode segments. The author developed aten-segment version of this circuit in 1959.

It should be noted that analog division of Y by X can beaccomplished by utilizing a multiplier in the feedback loopof an operational amplifier. With the amplifier input con-nected to −Y , the multiplier input Xin connected to X , themultiplier input Yin connected to the amplifier output, andthe multiplier output XinYin connected to the amplifierfeedback resistor, the amplifier output is the quotient Y/X .

The generation of a function f(X ) can be accomplished inanalog computation by using a segmented approximation tof(X ). The segmented approximation can be generated usinga diode function generator, which consists of biased-diodecircuitry similar to that shown in Figure 10, with potentiome-ters used to set the slope and breakpoint of each diode seg-ment. Alternatively, a servomultiplier can be used, with ntaps on one of the servo-driven pots. Padding resistors con-nected between the taps can then be adjusted to obtain thepot output Yf(X ), with f(X ) represented by an approxima-tion with n + 1 straight-line segments. Similarly, the func-tions sin(X ) and cos(X ) are generated either by diodefunction generators or servo-driven sine-cosine pots. For adescription of these and other analog components, see [4].One of the first implementations of time-optimal control wasaccomplished using biased-diode analog circuitry to repre-sent the required nonlinear control law [5]. Also, the outputsaturation voltage characteristics of unstabilized operationalamplifiers were used to represent simple nonlinearities, suchas relay-type bang-bang controllers and Coulomb friction [6].

Current Availabilityof Analog ComponentsIt should be noted that many of the components utilized inthe early analog computers are currently available in inte-grated-circuit form. For example, operational amplifierswith bandwidths and drift characteristics equal to or bet-ter than the drift-stabilized amplifiers of two or threedecades ago are available at a fraction of the cost, with upto four amplifiers located on a single chip. Nevertheless,with the current and projected speed of digital-processorchips, it seems unlikely that analog computers will everagain match their prominent former role as a general-pur-pose device for the simulation of dynamic systems.

References[1] D.W. Hagelbarger, C.E. Howe, and R.M. Howe, “Investigation of theutility of an electronic analog computer in engineering problems,Aeronautical Research Center, Engineering Research Institute, Univ. Michigan, Ann Arbor, Michigan, UMM-28, Apr. 1, 1949.[2] E.O. Gilbert, “The design of position and velocity servos for multi-plying and function generation,’’ IRE Trans. Electron. Comput., vol. EC-8,no. 3, pp. 391–399, Sept. 1959.[3] E.A. Goldberg, “A high-accuracy time-division multiplier,’’ RCA Rev.,vol. 13, no. 3, pp. 265–274, Sept. 1952.

[4] R.M. Howe, Design Fundamentals of Analog Computer Components.Princeton, NJ: D. van Nostrand, 1961.[5] R.M. Howe and L.L. Rauch, “Physical implementation of a torque-saturated linear servo with optimum step response,’” in Proc. JointAutomatic Control Conf., Cambridge, Mass. Sept. 1960.[6] R.M. Howe, “Representation of nonlinear functions by means ofoperational amplifiers,’’ I.R.E. Trans. Electron. Comput., vol. EC-5, pp.203–206, Dec. 1956.

Robert M. Howe (bobhowe@umich.edu) is Professor Emeri-tus of Aerospace Engineering at the University of Michigan,where he served on the faculty for 41 years, including 15years as department chair. He received a B.S. in electricalengineering from Caltech. He received an A.B. from OberlinCollege, an M.S. from the University of Michigan, and a Ph.D.from MIT, all in physics. His interests include real-time simula-tion, as well as flight dynamics and control. He is the authorof over 100 technical papers and one book on analog comput-ers. He served as the first national chair of Simulation Coun-cils, Inc., the predecessor of the Society for ComputerSimulation. His many honors include the 1983 AIAA de FlorezTraining Award for Flight Simulation and the 1978 Award forMeritorious Civilian Service for “outstanding contributions inguidance and control, and flight simulation” while a memberof the USAF Scientific Advisory Board. He has served as atechnical consultant to many companies. His technical soci-ety memberships include AIAA (Associate Fellow), SCS, andIEEE (Fellow). He was a founder of Applied Dynamics Interna-tional, an Ann Arbor computer company for which he contin-ues to consult since his retirement in 1991 from theUniversity of Michigan. He can be contacted at 485 RockCreek Dr., Ann Arbor, MI 48104 USA.

June 200536 IEEE Control Systems Magazine

Robert M. Howe, one of the founders of Applied DynamicsInternational (ADI) and Professor Emeritus at the University ofMichigan, where he taught for 41 years. This image showsone of the latest ADI products, a distributed rtX system utiliz-ing multiple PCs for real-time hardware-in-the-loop simulation.