Aircraft digital control design methods

eigenvalues found. These z-plane eigenvalues are transformed back into the s-plane -by the inverse relation s = #n(z)/T. A program to perform these .operations as well as certain synthesis methods is described in the appendix.

2.2.1. Pitch SAS The pitch SAS is simple pitch rate feedback to the elevator for better damping and higher bandwidth. A short period model of the airplane is used here: a block diagram of the continuous system is shown in Figure 1.

In this system it is assumed that the control law is calculated instanta- neously; there is no sampling interval delay at the output of the controller.

Also note that there is no washout on pitch rate in this model. Although a washout is common in most stability augmentation systems, it does not affect root locations or their variation since its break frequency is quite low. Without the washout, the digital controller has no states and no difference equation; there'fore, this example is somewhat degenerate but will serve as a basis for comparison with other design metnods in later sections. The root movement shown in Figure 2 is due entirely to the lag induced by the ZOH. This lag causes immediate degradation of the system damping. The real root is very slow and is not affected by sampling rate.

2.2.2 Pitch Attitude Control-Wheel Steering The aircraft model for the attitude control-wheel steering mode is the same as that shown in Figure 1 with the addition of a pitch angle state. In this mode pitch rate is fed back through a washout to eliminate any steady-state component. The output of the washout and difference between the pitch angle and the preselected pitch angle is fed back through a one cycle delay to simulate computational delay.

The existence of this delay depends on the particular mechanization and does not always exist. Its presence degrades the stability and per- formance of the system more than would be encountered without it.

A block diagram of this system is shown in Figure 3.

-7- The continuous washout for the TCV is: d(s) 16s c(s) 16s + 1 which yields a discrete transfer function (Table U) of -T/16 d(z) -] z - 1 = c(z) ~ -T/16 z - e and a difference equation d ' = C, d_ + C_(c - c ) n+1 in 2 n+1 n where the values of the coefficients are given in Table II for several sample rates.

elevator short period model s + 0.6 s + 10 (s + 0.54 ± 0.82J) Short Period model in state space form: w .6283 207.1 w .1408 -.4499 .01739 .0033 Figure 1 Pitch SAS Model -8- (r/sec) 8 cps _ 5 16 cps 4 cps 32 cps . 4 cont inuous . 3 _ 1 real root -3 -2 -1 -5 -4 CT(sec~ ) Figure 2 Pitch SAS roots, s-plane design -9- s + 0.6 1 (s + 0.54 + 0.82J) s digital d 2.16 washout r •^ 9desired Figure 3 Pitch Attitude Control Wheel Steering Model Table II Washout Coefficient Values -T/16 -T/16 1+e Sampling C 2 ' Rate 0. 999 0.998 32 cps 16 0.996 0. 998 0. 996 8 0.992 0.992 4 0.984 For the purposes of analysis and simulation, the input from the control wheel is assumed to be zero. The variation in dominant root locations versus sampling rate is shown in Figure 4. The transient response of pitch angle and pitch rate is shown in Figure 5 for an initial pitch error of 0.1 radian.

-10- Sampling at 4 cps causes instability and is not shown.

4 J03 h (r/sce) 8 cps _ 3 4 cps . 2 continuous _ 1 -3 -1 (sec' ) Figure 4 Dominant roots of the Control-wheel steering mode 2.2.3 Longitudinal Autoland The longitudinal autoland aircraft model is the same as that for the control-wheel steering model with the addition of a vertical position state given by: h = w - 207.16 The feedback control (Figure 6) was taken from drawings of the NASA Langley's TCW Boeing 737 and linearized. As witBi the control-wheel steering mode, a one cycle delay is assumed in the controller. The digital washout coefficients are the same as in 'Cable II while the difference equation for the digital integration is

•v,*!

-11- c •H I .01

-.01

o o> M •d 0) ,01 .

-.01 .

Figure 5 Pitch Angle/Rate, Control Wheel Steering -12- The variation in dominant root locations versus sampling rate is shown in Figure 7. Figure 8 shows the error in pitch rate for an initial altitude error of 100 feet off the glide slope. Again sampling at 4 cps results in instability.

In summary, digitized s-plane design is suitable for higher sampling rates or for systems with high relative stability. The addition of delays due to computation(which are cascaded in several multi-computer systems) has an increased effect with slower sampling.

-13- one cycle delay digital

c

integration elevator command Figure 6 Autoland Controller -14- ._ 5 JO) (r/sed)

•

/ / • 32 cps 16 cps • f < _ 4

Y /

i 8 cps continuous • 4 cps _ 2 - 1 \ i . i i i -5 -4 -3 -2 -1 , -1 ' a (sec ; Figure 7 Dominant autoland root locations -15- 0.1 Cent inuous 32, 16, cps t (sec) o o> CO -0.1 _t_ Figure 8 Pitch Rate, Autoland -16- 2.3 Digitized S-Plane Design with Hold Approximations A modification to the design method described in section 2.2 is to include the effect of the ZOH in the s-plane design. Since this effect is sample rate dependent, the s-plane design needs to be refined for any selected sample rate. This design technique will be evaluated for the Pitch SAS system which was discussed previously in section 2.2.1. In this case the s-plane model is represented as shown in Figure 9, where G (s) is nominally 1 for the continuous or extremely fast sampling case c and varied to cause an acceptable design as T increases.

Assuming that the loss of damping shown to occur in Figure 2 is due primarily to the hold characteristics, some compensation such as a classical lead: (A<B) will be required to retain sufficient damping when T increases. The lead parameters can be chosen to meet root location or frequency response criteria, both of which will be described.

( s + 1 0 ) ( s + . 5 4 ! . 8 2 j ) Figure 9 Modified s-Plane Design Model -17- 2.3.1 Root Location Design If the natural frequency and damping of the short period roots are preserved, the time response will be essentially unchanged.

For purposes of constructing s-plane root loci, we approximate the transcendental Laplace transform of the zero-order hold by a rational expression of polynomials in s. Fade approximates, , ~ ^ , x m a (-ST) + a (-sT) + — — + a m m-1 a ( - s T ) ' + n -sT of e are formed by matching the largest possible number of terms of the respective McLaurin series by suitable choice of the a.'s and b.'s.

The approximates thus formed are given in Table III (from ref. [ 1 7 ] ) .

m = 0 m= \ m => 2

1 1 + * + i**

1+*

I 1 1

1 1 + §* + \1*

1 + i*

a= I l — x

1-1* 1-1*

1 + J* l + iz + Az'

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I - i* + AX*

—-5

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T" ~i2T

Fig. 10 compares the amplitude and phase of all three representations of H (s) -18- .1 ..

.09 .

. 0 8 q -34 .03.

. 0 2 . . . .

Onn 1/100 1/30 1/10 Figure 10 Frequency Response Comparisons of the Hold and its Approximations.

-19- The m=n=l approximate is accurate for frequencies less than 1/10 the sample rate and the m=n=2 approximate is very good for 60 less than 1/3 the sample rate. For the short period closed loop poles at approximately 1 cps (Fig. 2), the 2nd order (m,n=2) approximate must be used to evaluate sample rates of 5 cps.

The compensation is designed by arbitrarily locating the lead zero at s=~8 and by successive trials determining the gain K and pole

q

location -B which restore the short period roots that have been perturbed by sampling. This is easily done since the transcendental nature of the lag is eliminated by replacing it by a Pade approximate. The results of these calculations are contained in Table IV.

T(sec.) Kq A B .05 4.15 8 8.85 .1 4.1 8 10.

.2 4.7 8 14.1 TABLE IV - Compensation Parameters Figure 11 shows the equivalent s-plane roots that result after converting the s-plane design in Table IV to discrete form using Tustin's (or Bilinear) approximation and the matched z-transform.

2 Z—1 The Tustin transform is formed by substituting — —- for s.

J.

AT/2 - 1 Kq s+A _ Kq (AT/2 + 1) z+ AT/2 + 1 w s+B (BT/2 + 1) BT/2 - 1 Z+ BT/2 +1 The matched z-transform was defined in the previous section. As also described in the previous section, the equivalent s-plane roots are obtained by performing an exact discrete analysis of the system and transforming the z-plane closed loop roots back into the s-plane by:

s = — S/n z

-20- o <1> w csi CM -H I <H •H . T) •H m m O to s, •H m I H Q> D N •p 0) o d o as K i-i Ctl I CO M H CO u Q> CO -21- The root migration in Figure 11 is caused primarily by the approximations in the matched z transform and Tustin digitization.

The figure shows that root migration at the lower sample rates is much reduced compared to the identical design (Figure 2) performed without the Fade approximate for the hold.

As a further aid in evaluating these designs, frequency response data are presented in Figures 12 and 13 for the Tustin digiti- zation. The open loop gain = 1 crossover frequency can be seen to be f • ' lower in the compensated discrete case (GHZ) at 00 = 5Hz than the s continuous case ( G ) . Furthermore, the closed loop bandwidth can be seen in Figure 13 to decrease as the sample rate decreases. These effects occur in spite of the fact that the short period root locations remain essentially unchanged; primarily because the additional root introduced by the compensation affects the frequency response.

2.3.2 Frequency Response Design An alternate method of choosing the lead compensation is to maintain certain characteristics of the open loop frequency response.

In other words, use the lead to restore the frequency response de- gradation due to sampling.

The closed loop transfer function of plant G with feedback compensation H is: GZ 1 + GHZ where the zero-order hold (Z) must be included for sampled systems.

At low frequency the ZOH and lead network have vanishingly G small effect, and the closed loop transfer function is -r—- as desired.

1+G At.high frequency where |GHZJ « 1 the transfer function is GZ instead of G, and the compensation H cannot counteract the zero order hold.

But this is well above the bandwidth of the system, the highest frequency of interest. The critical location and sharpness of the Band edge, (Closed loop magnitude of -3db) is heavily dependent on the frequency at which I GHZ I = 0 db and the corresponding phase angle.

-22- 00 (rad./sec. ) .3 .4 .5 .6 .7.8. .91.

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3. 4. 5. 6.7. 8.9. 10.

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.1 .2 5. 6. 7.&9.

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. 1 W (rad/sec) Frequency 3-Plane Design Closed Loop 13 S- Figure Response -24- O _L. A Thus in the frequency response design, H(s) = K q s + B is picked so that the 0 db or cross-over frequency and phase margin of G is maintained for GHZ as shown in Figure 14 for 5 cps.

The lead compensation is designed by arbitrarily locating the zero at s = -8sec~ and choosing K , and B to meet the phase margin and gain criterion of 0 db at 0) = 4.1 radians/second, the cross-over frequency of the continuous plant. The parameters thus obtained are contained in Table V.

Table V Lead Compensation by Frequency Response Design K A B T. (Sec. )

q

8 8.42 .01 4.48 8 8.88 .02 4.68 10.54 .05 5..41 8 14.90 7.44 8 .1 .2 31.8 64.53 In Figure 15 the equivalent s-plane poles of the discrete system using the compensation of Table V are shown.

The approximate discrete compensation is formed using both Tustin (or bilinear) and matched z-transform methods giving rise to two different s-plane loci.

Also shown in the figure are the s-plane root locations before digitization .The root movement for the higher order Fade approximates is due to the preservation of frequency response characteristics.

Furthermore, comparison of the various s-plane designs with different Fade approximates indicates that the 2nd order Fade yields sufficient accuracy for this design although for the higher bandwidth roots at T =.2 seconds there is a noticeable difference between the 2nd and 3rd approximates.

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i "V CM O bo (H -c cj a C C •H i-< 0 •H . 03 -27- The discretization changes the predicted s-plane roots significantly, especially if the compensation is fast relative to the sample rate. For our case where A = 8, the Bilinear transform is un- acceptably inaccurate at moderate sample rates. The z-transform is somewhat more accurate and should be used. The bilinear and matched z-transform and Fade approximate curves coalesce for higher sampling and slower compensation. An example of this is Figure 16 which resulted from a design using the same frequency response techniques but fixing the pole at -9.5 sec and varying the zero for proper compen- sation. The compensation is relatively slow, but clearly the curves are more consistent in this case. However, if the lead compensation pole is too slow, closed loop frequency response will be adversely affected.

The migration of the short period root that occurs in this design (Figure 16) does not adversely affect the closed loop response because the resonant peak from the reduction in damping of the short period is offset by the lag from the real root originating from the compensation pole and zero. Furthermore, the adverse phase lag of the zero-order hold is counteracted by the increased bandwidth of the short period poles.

The closed loop frequency response characteristics of the designs with lead parameters given in Table V and root locations given in Figure 15 are shown in Figure 17. Comparing these results to the designs based on short period pole fixing ( Figure 13), we find that the magnitude matches better at low frequencies and the phase is slightly better over all frequencies.

The designs with slower compensation yielded less discretization errors (note correspondence between dotted curves and solid curves in Figure 16) because of the slower compensation to be digitized; however, the slower compensation yielded a poorer match to the continuous system at the lower frequencies than either the fast compensation frequency response design on the root location design.

Either frequency response design criteria or dominant root location design criteria appear to be feasible methods for adjusting -28- .-. .'

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The frequency response method, however, appears to offer a more reliable method in general since movement of all roots is accounted for in the design criteria. For flight control modes with higher order dynamics, it may be more difficult to pick the dominant poles and adjust the compensation without introducing some new dominant pole.

-31- 2.4 Direct Digital Design Direct digital design refers to the design of a controller in the z-plane (or the w-plane, with an appropriate transformation).

It differs from digitized s-plane design in that the effects of the hold circuit are known exactly at all sampling rates. In the z-plane, techniques such as root locus and pole placement may be used exactly, as they are in the s-plane.

Direct digital design is most widely used in the construction of estimators or of controllers with state variable feedback inasmuch as optimal control techniques may be applied with little change to the discrete system. However, the starting point for most digital systems is a continuous mechanization, for which engineers still have the most insight as well as an arsenal of design tools. The object, therefore, is not to completely redesign the system, but instead, to have changes to account for the discrete characteristics and the lag of the hold circuit. Toward this aim, two of the systems discussed in the section on digitized s-plane design (2.2) were re-analyzed with the addition of a lead network at the input to the hold circuit. The lead ratio and gain of this network was chosen in order to place the dominant poles of the closed-loop system at the location chosen in the continuous design. This process can, in general, be done by trial and error; but for this effort,a systematic procedure was developed and implemented in a computer program (called LEAD, See Appendix A). It calculates the lead required from the pole/zero constellation in the z-plane.

It is important to note that this design was performed in the z-plane, which is an exact representation of the digital system. This FORTRAN computer program has been expanded so that it contains options to do the analysis of the s-plane design method as well as calculating the lead network.

2.4.1 Pitch SAS To the SAS system shown in Figure 1 is added a digital lead network which performs the equivalent of the continuous lead: -32- T s using the matched z-transforra method, the digitized version of this becomes: f(z) 1 - e z - ™ The constants for this lead network were determined so as to match the equivalent s-plane complex roots of the discrete system with the original complex roots at s =-5.15 + 4.5j. Table VI shows the lead network characteristics determined and the resulting equivalent s-plane root locations. Except for the 4 cps case, the complex roots were matched exactly while the position of the additional real root added by the network varied. The inability to match precisely the 4 cps complex TABLE VI Pitch SAS Lead Characteristics Roots Sampling K 1/T, l/T Rate 0.917 32 cps 8.0 8.46 -0.77, -5,15+4.5, -7.75 16 8.0 9.03 0.841 -0.79, -5.15 + 4.5, -7.45 10.7 8 8.0 0.714 -0.82, -5.15+4.5, -6.75 4 8.0 75. 0.45 -0.97, -5.14 + 3.12,-5.08 roots was due to the arbitrary design procedure of fixing the lead zero at 8 r/sec and varying the lead pole to obtain the desired roots. A zero placement closer to the s-plane origin would have allowed matching at 4 cps also.

The additional real root arising from the lead compensation dynamics is sufficiently fast to have a relatively small effect on the system response. However, for progressively slower sampling the root becomes progressively slower and more important in the response. In subsequent designs, the detrimental effects of the additional real root on the transient response will be evaluated.

-33- 2.4.2 Pitch Attitude Control-Wheel Steering Just as for the SAS described above, an appropriate compensation for the control-wheel steering mode is a lead network .

at the output of the digital controller (see Figure 18). LEAD was used to match the dominant poles at -1.34 + 2.13j. With decreasing sampling rate, however, one of the faster roots moves toward the origin to in- fluence the system response. Basically, there is no way to keep all the system roots constant with a decreasing sampling rate with only one compensation. However, there is a great improvement in performance i.e. the response is good down to 4 cps, whereas the system with a digitized s-plane design goes unstable. Table VII presents the para- meters of the digital lead network employed, and Figure 19 indicates that there is no movement of the root at -1.34 + 2.13j at all versus sampling rate. That figure also shows the position of the next faster root at 4 cps versus the lead network zero (sampling faster than 4 cps puts this root on the negative real axis and much faster than the dominant root). Since these two roots have bandwidths of the same order of magnitude, they both must be taken into consider- ation in determining performance. Figures 20 and 21 show the pitch error and pitch rate transient response for an initial pitch error of 0.1 radian with lead compensation (they may be compared directly to Figure 5). Although there is some degradation in transient per- formance, it is considerably better than that of the system without compensation.

-34- — (s + 0.6) (s + 0.54 '+ 0.82j: digital d washout 2.16

•+

?•

' 4.32 6 desired Figure 18 Control Wheel Steering (Discrete Design) Table VII Control Wheel Steering Lead Characteristics Sampling K 1/T 1/T rate 2.0 2.23 0.87 32 cps 0.77 16 2.0 2.49 0.61 2.0 3.31 0.45 4 2.0 6.03 -35- 0 ° 3 <D •r-> M CO <N r-l ( 0 •P tn o Q.

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o -(-> •H 0, Figure 20 Pitch angle/rate, Control Wheel Steering (Discrete Design) -37- 2.5 Discrete Design Using Optimal Techniques An alternate method to the design techniques discussed so far is to use modern control's "optimal" estimator/controller approach.

This entails a somewhat arbitrary selection of weighting factors in a cost function; however, once these are selected, the discretization process is routine.

: An estimator/controller implementation can be exactly the same as a classical compensator, the difference being solely in the methods used to arrive at the design. In many continuous cases, however, the estimator/controller design results in a higher order compensation than would result using classical methods; for discrete systems, the difference in order is less pronounced.

Design of compensation using modern methods is made up of two parts; the controller and the estimator (sometimes called an observer).

The controller is determined by minimizing the quadratic performance index: °° T T 1 r r J = - \ x Ax + u Bu dt = - \ \ x c 2 L ^o and the estimator by minimizing:

i r r T-I T T ~i

J = (z Hx) R dt o 2 \ I ~ (z - Hx) + w Q w J where x is the state vector x = estimated state vector u = control vector z = measurement vector H = measurement state distribution matrix A = state weighting matrix B = control weighting matrix W = process noise vector Q = white process noise power spectral density matrix -38- R = white measurement noise power spectral density matrix v = z - Hx = measurement error vector It can be shown, (ref. [18] [19]) that the solution of the optimal control and estimation problem consists of linear state feed- back to the control u and linear measurement error feedback to the estimator: u = Cx x = (F + GC)x + K(z - Hx) Using eigenvector decomposition methods [19], [20] the steady state controller gains, C, and estimator gains, K, can be efficiently determined. To accomplish this, the A, B matrices and the Q, R values of the noise model are chosen to give a "satisfactory" system design. The judgement in determining when the system is satis- factory is based on the same factors that one would use for a classically designed system. In this particular design exercise, the goal is to obtain a modern design which has all the same root locations as the previous design methods; thus enabling a direct comparison with the other methods. Therefore, A, B, Q, and R were selected to yield identical closed loop short period roots while maintaining the additional roots which occur with the introduction of the estimator relatively fast.

The bias estimate state (which performs the same function as the classical washout) is an exception to this; its root was maintained at the washout break point.

A continuous controller/estimator for the Pitch SAS with a washout was determined for the TCV vehicle. The state space model for the aircraft was given in section 2.2.1. Combining this with the actuator model and augmenting the state to account for a possible bias in the pitch rate measurement (the equivalent of a washout), we obtain the system model: -39-,

r

0 0 0 0 0

V

s

* 0 -.628 207 -.14 w 0 w 6e i . \ -3 N.-2 N X 0 -3.28 -.45 -1.74 0

q q

6e 0 0 0 -10. 10

;6e

fl y = Hx + v where: H = [1010]; q, = pitch rate measurement bias b q = pitch rate w = z-component of aircraft velocity 6 = elevator angle in degress e 6 = commanded elevator angle e c y = measurement The values of A and B selected for use in the cost function were: 0 0 0 B = 1 A = 0 .33 0 0 0 7.33X 0 0 -.86

o

which yielded: 8 9 C = [0, -1.32^~ , 247.23, 9.cN~ ] the values of Q and R selected are: 22.877 0. 0. 0.

0. 0. 0.

5.03\ R = 1 Q =• 0. 0. 33.92 0.

0. 0. . o: i.ioe^

which yielded: 4.783 -1.908^ K = 8.816 -314.5 .- the notation 3.1> means 3.1 x 10 -40- For comparison purposes, the poles and zeros of the resulting design's open loop transfer function were computed . Figure (2l) shows the' selected closed loop root locations and a root locus vs the open loop gain. This design yields three roots more than a classical design with a washout. If compensating the system using classical methods to overcome effects of sampling, one or two extra roots may be added, a step that is not required when discretizing a controller/estimator design.

By formulating the system equations in a discrete fashion but using an identical cost function and weighting values to the continuous design, an optimal discrete controller/estimator results which closely matches the performance of the continuous design.

In this case, the discrete system representation is as stated in section 2.2 and results in a discrete controller: u. = Cx.

i i z . = Hx . +v .

i 1 1 or - x, ^ = «K + 1U _ + KHSXx. - x ) + ± J ± The actual cost function used in the minimization was discrete, i.e. :

i °° r* T T ~i

Controller: min J = —Z.J I x . A x . + u . B u . I u. 2 L i Di i DiJ 00 f^ 1 -y I T —T AT \ Estimator: min J = - Li \ w.Q w. + (z. - H x . ) R (z. - Hx. ) [ 2 0 ] however, the values of A and B are computed based on A and B from the continuous design. The values of Q and R are the same as the Q and R used in the continuous system optimization. Eigenvector [19,201 decomposition methods yield a very efficient tool in solving for the optimal gains for either the continuous or discrete problem.

Figure (22) shows the equivalent s-plane root locations for the discrete system designed as a continuous system in Figure (21). The roots show very little migration due to sampling. Figure (23) shows the closed loop -41- (rad/sec) JOJ

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-200 .02 sec.

Figure 23 r Closed loop Frequency Response versus Sample Period for Design Using Quadratic Synthesis -44- frequency response characteristics for the various sample rates. To demonstrate the modification to the compensator that is being brought about by this digitization procedure, we have included Figure ( 2 4 ) .

Note the very large changes that occur in some gains vs. sample rate.

The digitization procedure is essentially routine and no design iteration is required for any of the sample rates evaluated.

The design iteration is carried out on the continuous system with the end product being a set of controller weighting matrices, A & B, and estimator noise matrices Q & R. 'These matrices are easily turned into discrete control laws (or compensators) for any reasonable sample period in a non-iterative manner.

The most difficult part of the design method is obtaining the continuous optimal design. For systems being designed for the first time, the iterations required are on the same order as those required for a classical design. For systems whose continuous design exists and was obtained using classical means, the iterations required * to obtain a matching optimal design is somewhat cumbersome, but the requirement to "match" only exists in a study such as this where it is important to make consistent comparisons.

* • . . . • actually can't be matched exactly because the order of the continuous compensation is higher for the optimal method.

-45- o 0> to CO CO w d •H cd O ol •H •P snonut^uoo T3 cd a!

CO •p c eg o en •Q 'SUTB3 -46- 2.6 Comparisons of Design Methods Five different design methods have been evaluated for various modes of the TCV flight control system. For a common comparison of the methods, the fastest loop (inner pitch stabilization) has been evaluated by all of the methods.

Figure 25 shows the equivalent s-plane short period root locations of the discrete systems for four of the design methods and Figures 26 and 27 show the closed loop frequency response character- istics for 5 of the methods. The methods are: 1) Uncompensated s-plane design (section 2.2) 2) Modified s-plane design based on pole fixing with the matched z-transform digitization (section 2.3.1) 3) Modified s-plane design based on frequency response with the matched z-transform digitization(section 2.3.2) 4) Discrete design based on pole fixing (section 2.4) 5) Optimal discrete design (section 2.5) Figure 25 shows the most severe degradation in stability for the 'uncompensated" design method (as expected), a conclusion that is amplified by Figures 26 and 27. For this flight control design problem, the uncompensated design method should not be used with sample rates much slower than 20 cps. Since the system dominant poles are at approximately 1 cps, this represents sampling at 20 times the dominant closed loop roots.

Further comparison of the designs shows a large root movement of the frequency response design but no corresponding change in the frequency response. This arises because the short period root movement is being counteracted by shifts in the other real roots which are not shown and demonstrates the usefulness of the frequency response information in performing a total design evaluation.

With the exception of the uncompensated design, all other design methods would probably be considered acceptable; however, most show some loss of bandwidth at the slow sampling rate (5 cps).

-47- ju (r/sec) <N en C O •H CM -P I CO o CO -P o O o o 0) <D O o w W CO CM O T3 I •o • • 0) o O o h •H - -H II II CO (D o O o 0) CQ CO <u a r-l OH m I Kl i - l CO to •H I cr in t~ I bfi •H -48- u c a) a

•8

ii H

s

CO o m

a

ts J= S a,

I"' g fi

to I N '-49- to (rad./sec. ) r desired continuous design i modern optimal design -100 i frequency response design ! modified S-Plane pole fixing design ! discrete pole fixing design O _ , !._ uncompensated design -150 -200 _ Figure 27 Comparison of Closed Loop Phase Frequency Response by Design for T = .2 sec.

-50- Since the frequency response design was based on retaining the same gain and phase at crossover (4.1 r/sec) it's interesting to note that the closed loop frequency response information does not match the continuous system at 4.1 r/sec. This is due to the error brought about by the digitization procedure (the matched z-transform), an error that is present for any design method that is based on s-plane results, regardless of whether the hold characteristics are, accounted for or not.

Note also however, that accounting correctly for the digitization process is not nearly sufficient in itself to obtain a good design. This is demonstrated by the discrete pole fixing design in Figures 26 and 27, which, except for the uncompensated design, is the worst match to the continuous of all the designs. This design was not shown on the root locus in Figure 25 because - it was based on maintaining the z-plane roots at the location which produced no move- ment of the equivalent s-plane roots. The shift in the frequency response characteristics is due to shifting real root locations which were not taken into account when compensating the system.

Factors other than the accuracy of the method must also be taken into account when evaluating design procedures. Table VIII is included to illustrate these differences. The first column refers to the quality of match to the continuous closed loop bandwidth as evidenced by Figure 26. Intuitive design refers primarily to whether the design iterations are performed in the s-plane or not, which most designers prefer; however, the modern discrete is given a qualified yes because many designers do not consider the weighting matrix selection an intuitive process. The third column refers to whether the digitization effects are evaluated in the z-plane or not. For the s-plane methods, the magnitude of the error is dependent on the frequency placement of the compensation — the higher the frequencies, the worse the error. This error can always be computed by an analysis such as that performed by this study, but this is not con- sidered to be part of the design method. The fourth column refers to whether any design iterations are required when changing sample rate, -51- M b4 .

S O M § 3 o o o in m N. iH iH H S>3 | fc, i o w 03 W 55 H Pi O •i CN eg CM •<» § O fi 0 0 •* w S TO GO 01 ^ ^ H *-* H 11 a «« s: Oj o M S § W § bo •H g O O CK fl gS O > 01 0) O t) o **^ CD Q) O O o >« >H ^ fc S ID • . X CO •H 1 g ^ « CO

§

«

M g 2 M do •H * o 5 S5 >« >• tx 2 Jx H * S g 9 • a I Q.

1 • CO ^ e H g -H O •' 0. | S c o o > •o -H S 6 a -H O •<!

m o to v M o a •H -P — P S § 8 X 2 " 0 fa < o o £ P O .J CO Q) 0} i-t « *> P e o •O O • O O bo C O CX, CK h ti •rl 4-> O O n >.

M « <0 o a OJ -H -H •o e a -a Q Q a -H c P o •0 <M Oj O fe B O -H iH. O 0) o u a, h to C O 1 -H O 3 E W Q S -52- a feature that is attractive if part of the design exercise is to determine the minimum practical sample rate; hence designs at many sample rates being required. The order of the compensation refers to the complexity of the control mechanization. For all but the modern design, considerable freedom exists in the order of the compensation, although the order tends to increase with slower sampling. For the modern design the compensation order is typically equal to the plant order but can be made equal to the plant minus the number of measurements when using a reduced order estimator. The last column is a somewhat arbitrary attempt to indicate at what sample rates the method can be used. For example the uncompensated design method should not be used at sample rates below 20 times the dominant closed loop poles. These estimates are based primarily on the quality of the closed loop bandwidth match as indicated in Figure 26 and in the more detailed design data in the individual sections. They are valid for the example used in the designs, the 737 TCV pitch stabilization mode, and should be used with some caution for other design problems.

-53- 3.0 Roughness Since the output of a ZOH is constant for an entire sampling period, the control is a series of steps which may be perceived by a pilot through his motion or visual cues as jerkiness or roughness.

These effects will be most noticeable in the high frequency modes of the aircraft and become increasingly noticeable as the sample rate decreases.

To evaluate and quantify this effect of digital flight control for varying sample rates, the following designs have been selected for implementation on the Langley motion simulator. The autoland and control wheel steering modes have been selected for this investigation. The autoland mode essentially places the pilot in a monitoring type of function and may produce different reactions compared to the control wheel steering where he will be required to execute various tasks.

The holds to be considered are (Fig. 28) the standard zero order hold (ZOH) for a baseline and the integral hold.

X U U i , COMPUTER ZOH PLANT

U

u u .

1 1

t _, TJT A T^P OOMPIITFR 7 OH \ • s ' •- • — - Integral Hold Figure 28 Hold Definition -54- The integral hold (IGH) is motivated by the desire to eliminate the step changes in u characteristic of the ZOH but to retain its hardware simplicity. Its mechanization simply requires the addition of an anolog integrator between the standard ZOH and the actuators in the aircraft.

Figure 29 combines the different elements of the evaluation and indicates where the different portions of the computations are to be carried out.

The algorithms to be implemented for the hold are: ZOH: u(t) = u.;t. < t < t.

t f IGH: u(t) =A u^t The aircraft model and autopilot for the control wheel steering mode for use with the ZOH is given in Figure 18 with the washout coefficients given in Table II and lead coefficients in Table VII for the sample rates to be used on the CDC 6600.

The aircraft model for the autoland mode is given by Figures 1 and 3 and the additional position state relationship: h= w - 207.1 6 The autoland autopilot for use with a ZOH is given in Figure 6. The digital washout coefficients are again given in Table II and the integration by: T b . = b + — (a , + a ) n+1 n 2 n+1 n For adequate stability with decreasing sample rates, the overall loop gain shown to be 0.85 in Figure 6 was adjusted according to Table IX.

TABLE IX Autoland Gains Sampling Rate Gain 32 cps 0.60 16 0.51 8 0.34 4 0.21 -55- Analog Computers (EAI 231R's) airerait moael and u(t) u.

motion simulator transfor- simulator mations and washouts q-lT autopilot tO =4,8,16,32cps 'Pilot commai ds -=»- s Roughness function evaluation Digital Computer (CDC 6600) Figure 29 Roughness Simulation Schematic -56- For the integral hold, designs are preliminary and need further refinement for good response characteristics. Because of the extra 90 phase lag of the integrator and the additional feed- back from the integrator, the design is more difficult than that for the ZOH. One preliminary design was performed using a modified s-plane approach where an approximation to the delay is contained • in the system model. The design was obtained for 6 feedback only so did not represent the full control wheel steering mode. Figure 30 shows the block diagram and defines portions of the compensation.

| .2s (J Dlan1- +20 s Hold Delay!

s+3 '• —S-T ^ ' s+9.5 | r digital analog Figure 30 Integral Hold Design The transient response of this controller obtained from a hybrid simulation with a 10 cps sample rate in the digital portion is shown in Figure 31. It shows that the performance can be made to match that of the system without the extra integrator even though the controller input to the actuator is ramped instead of stepped.

Furthermore, the magnitude closed loop frequency response is shown in Figure 32 to match very closely as well. Further design efforts are -57- ZOH

e

t(sec) IGH t(sec) Figure 31 Transient Response of Pitch Stabilization Mode from Hybrid Simulation underway to determine compensation for the control wheel steering and autoland modes.

-58- CO (rad/sec) Figure 32 Closed loop Frequency Response Comparison of the Integral and Conventional Holds Analytical evaluations of roughness have been proposed [20] and can be used for a quantitative comparison with pilot evaluations; they are: N RF, = Au

N

Au RF 2 ' RF = 1^2 AU[ 0 .

-59- where the terms are defined in Figure 33. These functions are to be evaluated in the digital computer during the simulations.

The exact nature of the evaluations with regards to initial conditions for the autoland, test maneuvers for the control wheel steering, and form of pilot evaluations, have not been determined.

ZERO ORDER HOLD Au 2T INTEGRAL HOLD 2T Figure 33 Roughness Function Parameter Definitions -60- CONCLUSIONS Discrete Design methods have been compared at various sample rates for several different flight control modes for the Langley TCV Boeing 737. It has been shown that digitization of continuous auto- pilot design leads to a substantial degradation in performance for sample rates slower than 20 Hz (or 20 times the closed loop bandwidth).

Modifications to the s-plane design to account for the lag due to sampling improves performance considerably and can be used with confidence at sample rates of 10 Hz (or 10 times the closed loop bandwidth). For sample rates much lower than 10 Hz, a discrete design procedure is advised to eliminate the digitization error present in any s-plane design procedure. A new discrete design procedure has been evaluated which is based on optimal control and estimation techniques. It entails performing the design iterations on the continuous system, then using the resulting weighting matrices in a discrete optimal control procedure. The resulting discrete designs match the continuously designed system better than any other method evaluated and requires no iteration during the discretization procedure, a feature that no other acceptably accurate method possesses.

A test of roughness has been defined and control algorithms determined for the Langley simulator facility with sample rates of 32, 16, 8 & 4 Hz. Preliminary designs are included for the integral hold, a hold which requires the addition of an integrator between the ZOH and the actuator.

-61-

Appendix A: Program LEAD Program LEAD calculates the eigenvalues in the z-plane of a continuous system with a ZOH, actuator dynamics, and a digital controller mechanized in difference equations. An option of the program allows the user to calculate the lead network required at the input of the ZOH to produce given s-plane roots.

Derivation: A system may be written * / / x = A x + B u Al where X — " ' , a state vector partitioned into the original states x , from x = Ax + Bu, and x , the s a actuator states

B!

A' —

[

o c J ,

B A2

, ' . H

-.L» i

L J

Thus u is the control signal to the actuators whose differential equations are of the form x = Ax + Bx A3 s a If the transition matrix is calculated for equation Al x = PX + Qu .„ n iz+1 n n A4 where A'T . / A T e B ' d r J o To equation A4 must be added the difference equations of the digital controller. The control is of the form • u = C,x + C^x + CLx A5

n In 2 en 3 sn

-62-

where x is the controller state variables.

c x is the derivative of original state variables such as S acceleration, (and x = A x) then

' = [P + Q(C + C A')] x + QC x A6

n+1 1 3 n 2 e n The difference equations can be written • • ' x , = M x • + MX , + MX + MX ., + M x_ + MX A7 cn+1 1 en 2 cn+1 3 n 4 n+1 5 5n 6 sn+1 The presence of x , . x , , x is due to the existance of these cn+1 n+l' sn+1 types of terms in most matched 3-trans forms.

A8 Now x . , = AX sn+1 n+1 which can be rewritten using A6 A9 * n^i = - A ' [ P + Q(C +C A ' ) ] x + A'QC x C sn+1 13 n 2 en and finally x _ = (I-M.)' I M + (M. + M J V ' ) [ P + Q(C + C A ' ) ] + M _ A ' \ x n Q cn+1 Z ^ 3 4 6 1 . 3 o j n Thus, given the state definition A2, A3, the sampling interval T, the control law A5 , and the difference equations A7 , LEAD solves for the eigenvalues of the matrix j (I-Mj)" I « + (M^ + M A ' ) [ P '] J 3 g To calculate the lead network required to fix a specific root in the s-plane, the program first computes the poles and zeroes of the system in the z-plane. It then transforms the desired root into the -63- z-plane, calculate the lead angle deficiency, and places the pole of the lead network from this. The gain is computed by standard root- locus methods.

The user must program 3 FORTRAN subroutines: size, diffeq, and plant. They define respectively the size of various arrays, the difference equation matrices, and the state equations of the plant and actuators. A listing follows of the main program and relevant subroutines.

-64- I M P L I C I T REAL ( K , L , M ) R £ A L > 8 F, ICNT, INT ,GN , CS , ACL R E A L * B IC«T<20),INT<20),GN<10,U) ,09Uv»,CliMlu),C.wniO» f T I U V t 10,10) ,ACL(1C, 10)

Kvecua.io) .vFCPf (io,u» i v£cin(io,io), VECRC 10,10)

1 , V 5 C I ( 1C.1CJ) , V R V ( 1 0 ) , V I V ( 1 0 ) ,VR*V (10) , V * I V 4 1 0 ) D'IMtHSICN APR I M^ ( 10 10 ) ,DF.P IME ( 1O ,i U ) ,P( 10,10) ,g(10,10) 1 ,C1(10,10> , C 2 < 10,10),C3UO,10),MiUd,10) ,M2( 10,10) 2 , ( " 3 ( 1 0 , 1 0 ) , M4( 10,10) , P 5 ( 1 0 , l J ) . » r t o { l J , 10) 3 ,A( 10,10), 8(10, 10 ),C (10, 10), 0(10,1 d ) , JNuMllO, lu ) , T c K 5 B A { 10 ,10) 4 ,TEMA(10,10) , T E M 2 A / S ( 1 0 , 1 C ) ,Tfcf-!BAUj,lJ) , T c M A 6 ( 1 0 , 1 0 ) 2 , V 1 ( 2 0 ) , V 2 « 2 C ) , T £ M 3 E / H 1 C , 10> , TtKJB( 10 , 10) , T^.'-l^BB ( 10 , 10) 6 , T E K 2 A E ( 1 0 , 1 0 ) ,FUO,10) , T EH2BAI 10 , 10) , TcK'tBA ( 10, 10) O A T A A P r t I K E / 1 0 0 " 0 . / , B P H I M £ / l O O * 0 . / , P/1JO*0./ ,u/iOO*0./ 3 ,Cl/10C*0./,C2/10C«C./,C3/10G*0./,Mi/iOJ*0./, M<;/100*0./ 2 , V I 3 / 1 0 0 > ' 0 . / , ^ K / 1 0 C » 0 . / , ^ 5 / 1 0 0 * 0 . / C ^ T A A / 1 C O O . / , 8/100*0. /,C/100*0./ C A L L S I Z E < ^ P L A ^ ' T , ^ O I G , ^ C , ^ A C T ) N S T C T = N S + N D I G NSTOT2 = 2*,NSTOT • CALL CaiT(NPLAN'T,NDIG,NC,NACT,Nd,^STQT,U5TaT2, 2 VrCI ,VRV,VIV,VRP.V, VK I V , APR IMS , BPRI Me , P , Q , Ci ,C 2 C3 , Mi , M2,M3 f 3 ^5,H6,A,B,C,D,TF.N58A,TEMA/!,T£M2AA, Ti.;4bA , TtMAb , Vi. , V2, 4 T E V 3 e A , T E M 8 B , T E M 2 3 B , 1 Ef 2 AG ,F , FEM26 A, TEMHbA, D.SUM) RETURN fc.ND S L E R G U T I N E OOIT ( NPLANT ,ND I GtMC ,NACT ,NS, NST JT ,i\i STUT2, 1 ICNT, INT, GlJ,D9»CV>P. ,C V« I ,T I NV ,ACL , RVtC » V5Ck,<, VtCIH,Vi:CR, 2 Vf;CI,yFV,V!V,VRRV,VRlV,APSIMEfbPf»IMc,P,W,CifC2,C3,,Ml,M2,M3 3 V5,W6,A,E,C,D,TEV50A,TEMA*,TEM2AA, i iiMdA ,T£MAb , V 1 , V2, I M P L I C I T Pb'AL ( K , L , W ) R E A L * f i F, ICNT, INT, G N t C S , ACL.

r P.. .AL*8 T SAMP, C P G L E S . C Z E PCS C I M E N S I C N Z t R . a ( 2 , 8 ) , P C L F . ( 2 , 8 ) O I M t f V S I L N I C N T ( N S T O T 2 J , I M T { N S T C T 2 ) , G i i ( . \ j T O r , » c > r j T I , 0 9 { N S T O T 2 ) , 1 C W R J N S T G T l ,CWI (MSTCT ), TI NV (MC1G.NJ i«) , ACt (NorOT,NS TJ T ) , : 2 RV'EC ( N S T C T , f l S T G I » , V t f C l N ( i \ S T O T J S T T ) , V c C l N t n S T u T , > I S T C T ) , f ) J 1 3 V E C R d \ S T O T , N S T C T ) , VEC f ( N S T C T , NST JT • , VK V< ro TCH , VI V ( .'iSTOT I , H V K F , V ( . \ S T C r > , V R ! V ( N S T C T ) , iF SI KC l,<5 , i^S ) , BPKlMi.< NS ,NC i t f 5 P < r . $ , N S ) , 0 ( N > » » C > fCl ( i N C , N S - ) , C 2 ( N C , i , J i G ) f C 3 ( i - i C ,:<):>) v 3iN;r.'5iCN i«i(f:n!r '!orr; ) W i ( N o r G , r - i j i G ) , - i 3 { ! j j i G , . t M S » , M 4 ( N O i G , N S « , M ( 1 M5 ( NC I G ,NS ) , M6( ,'!CI G ^ S » , f ( NPLAiiT , ;;Pi.A/,T ) , 3J rJP LANT ,NC ) , 2 ' C I M A C T , N A C T ) , D ( N A C T , N C ) , T E i " 5 B A | N D i ' 0 , / » a ) , l c r t A A l , - J S » : 4 S ) , 3 T c K 2 / 5 A ( N S , N S S ., TEI"P.A( NC IG , N S ) ^TfcMAti (Ni, NulG) » V it i-»ST OT 2 I , •', V2 ! ,'•; STi:T2 ) > Tt."3P A (i'JB IG ,.N'S ) ,TS/-!3a( ,,0 I L-, ,'*0 J G ) ,T oi2ii8 ( :iU I 0,^0 IG ) , i> T t T ' ' 2 A f c ( . N S , i i 4 D I G ) , r- ( M S T C T , ^ S T U T ) ,T M i3AtrlUliJ, nS ) , Tc •*.•*-> A ( NOIG ,NS ) i: 6 i K N C M i . A C T ^ N C ) T A L L l ' L / i N T ( N - P L A N T , N O i G , N C , K A C T , ( i ' i j r ' < o r u r , H . S r j T c , ?

X I C « T , I N T , G r i , D < 2 , C V , : ; , C v . I T I N V , A C L , s V t - C , V S C K ! - 4 , V c C l n , v e C ! \ » l !

^ V'JC I , V;. V, Vi V. V f - . ^ V , ,/••••. .IV ,.".JT IMF, bP.- i r-i^ ,P ,Q,Ci , C 2 , C j , M i ,M x 3 '•:• - ?' t , A P. ,C , U, Tt:":>t: i . f f. '< '? .1' j *ci'-:/:AM • 'i ;"''i;<lA, T :-.,-(/.;.>, V i V^it t ( !

M. r :f-OL:.A ,T t'-luci . T u f ' ? 0 i : , 'n":."2^-L ,F,T:::i2L».-;, T c M - V d A , u;iUMJ t 10M'? C A L L c - K C . V F T f ' / . c : A G ? ' , £ , 1 . 0 !

-65- CCCCC R E A C C C N F E N S A T I O N ZERC (PCSITIVt SIGN) REACUO,*) CZEROS 1 C C N T I N U E CALL FrtOFTCINS? ',5,10) CCCCC CCCCC R E A D INSTRUCTIONS CCCCC INS = I , 2 i 3 t OF. 4 (INTEGER) CCCCC T S A M P = S A M P L I N G I N T E R V A L (SECONDS!

CCCCC THE T H I R D AND FOURTH V A R I A B L E S READ HAVE McArtlNGi OcPtiNOENT CCCCC CN THE V A L U E OF INS CCCCC INS = 1; < ? C C T S IN THE I P L A N E ANJ S PLANc W I L L Be FOUND FOR CCCCC THE S Y S T c f mTh AN INPUTTED GAIN A,,D COMPcroATION CCCCC P C L E AMD Z E K O . TH2: THIRD V A K I M L 3 E 13 Trie G A H . IT CCCCC M U L T I P L I E S ALL THE ELEMENTS JF THE UNuM M A T R I X . THE CCCCC FCURTH V A R I A B L E IS The CuMPiNSAT I urt PULc ( P O S I T I V E CCCCC SIGi^i). IF NC C C K P E K S A T I O N IS O c S I R E J , ScT THc POLE CCCCC L O C A T I O N AT THE Z E F C L C C A T I O U . Trie KGJTo WILL THEN CCCCC INCLUDE AN UNCCUPLFC ONc COftR^SPOUUlNG TO THt CCCCC C C f P E N S A T I C N o CCCCC INS = 2; THE GAIN AND P C L E LCCAT10N W I L L B^. C A L C U L A T E D THAT CCCCC UILL G U A R A N T E E P A R T I C U L A R K J U T S Iw THt S P L A N E . THE CCCCC DESIRED S P L A N E HOCT IS IUPJTTEO Ai THc T H i R O Ai^D FOUPTH CCCCC V A R I A B L E S (VJAL AND I M A G I N A R Y ) . AS A CHcCK, THE CCCCC FiCCTS A R E C A L C L L A T E C W I T H THIS CJMPUTcU CUnPtNSATIOM A N D CCCCC G A I M . T H I S CAN BE LSED TO CHLCN UU The L O C A T I O N OF C T H E K R O O T S W h C S E LCCATIUNi WERt uuT S P c C I F i t D .

CCCCC CCCCC T H I S OPTION W I L L W C R K C U R R t U T L Y JULY F O R 1 A C T U A T O R CCCCC T H A T INFLU^ICtiS THE S T A T c S THiXJUGH ' Jws uJU-ZERO ELFMEMT CCCCC CF QNOM. ANY C T h E H FORMULATION W I L L Gi Vc U ^PKcD ICTA BLE CCCCC R E S U L T S I. N T H I S V E R S I O N .

CCCCC INS=3: THIS PERFORMS THE S A M E CuMPJTATION «S FuR INS=2.

CCCCC T H I S CAN Bt U S E C OM.V AFT£ i liiz-2 HAS BtcN j P £ C I F I E D ; CCCCC FCR A P A ? . T I C U L / 6 « Z = RO L O C A T I O N ANO iAMPLluG I.-iTSRVAL.

CCCCC T H I S O P T I C M E X I S T S T C E L I M i N A T d SOME C O M P U T A T I O N THAT CCCCC UAS PEliFQ^FO F P - E V I C U S L Y .

CCCCC ^-15 = 4; RETURNS TO A .°FC*PT FOR A NtW ZcKO. The SECJND, THIRD CCCCC AND FOURTH V A R I A B L E S MUjT Bt I N P U T T c D t BUT AR£ IGNORED.

CCCCC R E A C d O , * ) I N S , T S A M F , £ R F N T , S I P N T GO TO ( 9 5 , ^ 4 , 166, 1009), INS C G A I N = S R F N T CPGLES^SIFNT GO TO 199 GAIN=1.

C A L L C W A T ( D , O N O « , K A C T , N C , G A I N » C A L L C r ; T R L < N P L A N T , r . ! n i C , N C , M C T , N S , . < S T U T , N S T O T < ! , 1 I C N T , I N T G M , r ) 9 , C ^ P v » C U l »T I NV, ACL » K V Q C , VcCKiSt V r . C I N , V £ C R t t 2 ' V l i C I V ? . V , V l V , V i - i < V i V > . r v , A F r ( I M E , i > P . - \ I , ' - - . = i f , a , C 4 . , ^ t 3 M 3 , M 6 f A , f c t C « D T E V 5 B A , T E M A * ,TCM2AA, TcMBA T tMAb Vi f f t CO 1403 J = 1,NS'TOT P = CWJ\ ( J ) * » 2 * C W I ( IF (f< . G T . 1. ) GC TC 1402 X R = f , W R ( J ) Y I = C W I ( J > GO TO 1409 14-03. C C N T I M . E CALL c ^ A T i D f D N C ? » N A C T I I S C , G A I N ) C A L L C N T f ' . L l N P L A i - a , r ! t ; i G , r i C » f . A C T f h i , . J S T o T , i » S r j T 2 f -66- 1 I C N T , I N T , G N , n 9 , C V i P , C M t T I N V , A C L , * V L C V t C / < N f V c C I N i V c C R t f 2 V E C I t V R V , V l V , V F R V , V R I V , A P R I ' ' F , B P V l M c , P » U , O i i L ; : , C : » t M l , M 2 , M 3 M 4 i f . 4 SF=1.

CO 404 I=1,NSTQT S F * S F / O S C R T ( ( X P . - C k R ( I ) ) * * 2 * ( Y l - C W l ( l ) ) * * 2 » P C L F U t I)=CWR.Ul 4C4 P O L E ( 2 , I ) = C W I { I) GAIN=1.E10 C A L L C M A T ( 0 , D M ^ , N A C T , N C , C M N ) > CALL C N T F , L < f s P L A N T , j C I G , N C f N A C T , N i , N b T U T , f > i : i T O T 2 r i 1 I C N T , I M , G N O q , C ^ « i C U I i T I N V , A C i . , K V « . C f V t : C R i « t y t C I N f y/cCRi f 2 V 5 C ! . V R V . V I V . V R R V . V r a v . A F R I H E . S H K l M d . P . C . C i . C i t C j f Mi. , M2» M3 3 ( > ' 5 X 6 f A a , C f D , T e N 5 B A T £ H A A t T ^ M i : A A i Tei-iBA, TcMAti »\/i t V 2 » > t t NZ=0 •CO 5 0 5 I=1,NSTCT IF ( C k R ( I ) * * . 2 + C M I ( I ) * * 2 » G T . 2) GO TJ 505 Z E R C ( l f K Z » = C W R ( I 1 Z E f ? 0 ( 2 N Z ) = C W K I ) f 505 CCKTIM.E C-J 609 I = liNZ 6 0 9 S F = S F * S C R T < { X R - Z E R Q ( l f I ) ) * * 2 * ( Y I - Z e R O ( 2 i I ) ) * * 2 » C Z e R O Z = e X P ( - T S A V P * C Z f c T « O S ) = o C A L L L ^ A D t Z R P N T f Z1PNT , f \ S T C T , POLL, M^. » ZcKD Cl»A IN CPOLtZ , PHI t t 1 , C Z E S J Z f S F » C P a L E S = - A L O G ( C P O L E Z ) / T S ^ H F W R I T E ( 6 ' , 5 A < ' O J C P Q L E S . C G A I N F O R P A T C POLE « , G 1 5 . 6 , « GAIN *,G15.6) CCMTINUE C A L L C v i A T < 0 , O N G M , K A C T , r , C , C G A I N ) CALL C N T R L U P L A N T , M O I G f N C f N A C T , N j , , N o T u T , H o T j r 2 , 1 l C N T , I N T G N , D 9 , C k R , C U I f T I N V f A C L f R V n C , V £ i C k U f V i - C I M f V C C R f t 2 V E C I f V K V f V I V , V H K V , V ^ I V , A P R I M E , B P r < I M d i P t & i C i , C t > C . 3 f M i t M 2 » M 3 f M 4 , 3 V . 5 , M 6 t A t E , C » D , T E M t i B A , T F M A A , T C M 2 A A , T i i M a A , T t M A t i , V i t V 2 , 4 T C M 3 B A t T f M B B f T E f 2 B a , T E f 2 A E F , T c M 2 U A r TcM4B f 8 F C R M A T < » ' , G 1 5 . 6 , L 5 X , G 1 5 . O 10 F C R M A T C S 4 G 1 5 . 6 ) CC 960 J - ' l t N S T O T Si<E = D L U G { C S Q R T ( C U R ( J ) * * 2 * C V , I(J ) * * 2 J )/TiAMP SlM = C A T 4 r , ( C W I ( J ) / C W ? . ( Jl J / 1 S A M P

IF < c m ( j ) .EQo o.oo) cc TC 497-*

H R I T E ( 6 , 1 C » C W R { J l , C ^ I ( J ) » S R E t S I M GC TO 960 4 9 7 4 W P . I T E ( 6 t £ ) C W S ( J ) , S R E 960 C O N T I N U E GC TC 1 46 RETURN END f.

C c C c c c c -67-

c

c

. . S L B R O U T I N E CNTRL ( K P L A K T ,NO IG NC,Ni: T ,^S , NSTUTf NS I JT2 » ( f 1 ICKT I N T f G ; 4 » 0 5 , C V » R . C V » I » T I/\V A C L , K V C C i V c C « N , VcCIrt , VECR » f f 2 VPC I , V R V f V l V . V P . i W , V R I V , A F P I ' - I E , b P K i M : £ , P , Q , i . i r C < L , C 3 , M l , M 2 , M 3 , y 4 f 3 M S f W c t A , E , C , 0 , T E " 5 B A , T E M A ; , T E M 2 H A , T = MbA, TiiMAb ,Vi , V2» <t T E M 3 e A , T E M B B , T E M 2 B B , T t P 2 A S , F f TEt4«.6A TiiM^BAf UNOM) f I M P L I C I T R E A L (K,LfM)

F, ICNT,INT,GN,C9,ACL

SH:,SIM,CUI,CWR

R£AL*8 OLCG,DSORT,DAT*N

CCMKCN/T/TSAMP,CFCLES,CZEFCS

DIMENSICN ICNT(N5T3T2),INT(NSTOT2),GN(,\STL)r,HSTjTJ,D9(NSTOT2), 1 C W S ( r ; 5 T O T » , C W I ( N S T O T ) f T I N V ( N O I G f . - i i 3 I t i » f ACL (N jf uT n^iFJT I , 2 R V £ C ( N S T C T t N S T O T ) i V f c C R N ( r . S T O T , U S r j T ) , V t C l N ( KoT J T / . 4 S T U T 1 3 V t C R I N S T C T . M S T G T J , VEC I ( N S T G T ,NSTGT ) , V-.V < iJSTar ) , VI V {.tiTOT ) , 4 V r t K V ( N S T C T ) , V R I V ( N S T C T ) , £ F R I M c t r f S » N S ) t B P k I M t < H S t U C ) » 5 P ( N S , N S ) , G ( N S , N C ) , C l ( ^ C , ^ S ) , C 2 ^ l ^ i L , l ^ , O I G » , C 3 ( ^ C , i > ^ S ) D I M E N S I O N M l ( N D I G » H D I G J f M 2 ( N D I G f \ i ) i t » ) f M 3 ( i i D i i i t M i , ) , M < » l N O I G t N S l f 1 M 5 ( r l C I G , h S ) t M 6 ( ^ C I G , ^ S ) , / S ( ^ P L A l ^ i r , • ^ P L A l ; T J , ^ ( : ^ , P L A H T , f ^ C I, !

2 C ( N A C T , N / i C T ) , D ( N A C T h C ) , T r f / 5 R A ( N D I G , i 4 S » , T e M A A ( M S , N S ) t f 3 TEMZAA(NSfNS) T E M B A ( N C I G t N S ) f f A V 2 ( N S T G T 2 ) i T E M 3 B A ( N D I G t N S » » T E M 8 B U D I G f r 5 T E H 2 A 6 ( N S , N O I G > F (NST01 , NS'TQT) tTtM^iiAiiJLiIo.fii ) , T t S ^ b A l MDIG f CALL C I F F E G ( N P L A N T , N O I G t N C t N A C f » N S , N S r j T , N j T i j T 2 , 1 1C NT, l N T , G N , 0 9 , C h R f C K I t T I N V t A C L R V L C , V c C K U f Vr.CIi«t V t C R » f 2 V E C I , V R V , V l V , V R ^ V , V K l V , i \ F K I M E | B P ; \ l M e , P t ( , C l f o ^ , C 3 , « l f M 2 , M 3 f M 4 i t 3 M 5 i M 6 , A , E , C , 0 , T i £ M 5 B A , T E M y i i | T ^ M ^ A A , TuM3A TtMAb , Vi » V2 t f ^ T F M 3 E A , 7 E M B B , T E M 2 B 3 , T t N 2 A e , F , T t M < i £ i A , T c M f 3 A f OMuMI CO 91 1= l.NPLANT DC 90 J= 1,INPLANT .90 A P R I M E ( I , J ) = A(I,J) CC 91 J= 1,NC 91 iPR I M E ( I , J * N P L A N T > = 8(1, J) CO 94 1= 1,NACT

cc 9^ J=I,NACT

94 A P P I M £ ( I + N P L A N T , J+NPLANT)= C(I,J) DC 96 I=1,NACT 00 1b J=1,NC 96 BPKIME(I«f,PLAMT, J)=0( It J) CALL FGPC(APRIM6,HPRIf£,P»C,NS,NC,TSAMP) CALL MMULT(C3,APflIME,KI*5EAfHCfK;i,U:>) : C A L L N A C C (TLM5fiA,Cl,TE"5i )A,KC,NS,U) CALL NfULT(C,TrM5eA,T£^AA,NS,NC,NS) C A L L VAtT(TF/-1AA,P,TKM.t.4,fiS jhSfO!

C A L L ^ y L L T ( o , c 2 , T t M A B ^ 5 , ^ c , N D I G )

r CALL C A T I M T I ; W , N C I G , 1) . .

. C A L L M A C C « T I N V , M 2 , T I N V , N O I C , N D I G » i ) C A L L M N V ( N D I G * N O i r , , T I N V , M : l G , U i , V i , V t ) C A L L M M U L T 1 M 6 , / P !MS , T £,'-2P.-l ND IG,,'io , f!i» r s C A L L N . A C C < T : f , y 2 B A , f ' f , T ( : f ' * B A , < ! Q I G , r ; } , 0 ) C A L L N V U l . T ( r e V 2 P A , T e M A / s , r r N 3 ^ A , i < D K . , . \ S , N S ) v/ C A L L MMUL T ( M 5 , Ai^RIM;.-, T £, .t A ,iiDIG,r;:> ,,'lb.J C A L L V A D n ( T t M 4 P A f T ~ c 3 e £ , T E C 3 H A , r<i)i ! i ,i-<S,0) C A L L MAC D < T ' : M 3 ' 5 A ,!•*:•» ,T i>' ';A ,i\OlGiN-M 0) C A L L M . / U L T ( i ; M V , T ' : : M 3 i ; X , Ii-rM:A,NjIi,iNuIG,.-4i) : C A L L M M U L T ( 0 , C 2 , T e " 2 A L ^ 5 , , V C , r - U l o } -68- C A L L M M U L 7 ( T E M 2 B * t T E M 2 ^ B , T £ f ' 2 8 r i , N J I G , N S , N O i G ) C A L L V A C C { T c M 2 P 3 t ^ l » T E f ' 2 9 e , N O I G t N i J l G . U ) C A L L f - M J L T { T I N V , T E M 2 e e , T E | i ' E B , l S L > l G , f M O l G N l > l G ) f DC 125 1=1, NS CO 125 J=1,NS F( I, J ) = T £ M A A ( I, J) CO 76 l=liNDIG CO 76 J=1,NS F( J, I + N S ) = T f P A 8 ( J , I ) 76 F ( I + N S f - J ) = T E M B A ( I , J | DO 77 IM.NDIG CC 77 J=1,NDIG 77 F( H-NSt J + f ^ S ) = T G M R B ( I ,J) «tv:«44*,#***4 ****** ******** 4 »*,<***# ****** C A L L E A L i l s C ( N S T C T . N S T C T , F, ILOW , IrUGH,D9J C A L L E L M H E S ( N S T C T , NS1CT , ILCW , IHIGH ,F , INT ) CALL HCR2 (NSTQT ,NST(J T , I LCU . I HIGH, F Crt.AiC Wl , AC Lt iCrtT, t ^6 rtFTURN END C C C '

c • - . - ' . •

C C C

c

c

c

SUBRGLTIN5L LEAO( X« , Y I ,NPCLES, POLt ,NZcKJS , ZcKU, GA IN, C JMPP.PHI , 1 C C M P Z . S F ) C I M E h S I C N POLL ( 2 , 6 ) , Z F R C ( 2 , 8 I ANGLE=0.

GAIN=1.

I F t N P C L E S . F U . O ) GC TO 15 CC 10 I=1,NPQLE5 G A I N = G A l i > * S U K T ( ( XR-PULE ( 1 , I ) ) **2* I Y I-PJLt U,IJ T t M P = / . T A M t Y I - P i D L E < 2 , I ) ) / ( . X F - P U L t U t i m I F { POL £ ( 1 , ! ) . GT . XR ) T E NP= 3 . 14 1 592 *T uMP : 15 IF(NZE. ftCS.EO.O) GC TO 25 CO 20 I.= l,NZtROS G A I N = G A i r / ( S O R T ( ( XR-Z EPC ( I , I ) )**2*< Y I -Zcff^J ( 2 , 1 ))**2) J (Yl-ZERr(2, D.GT.XH) TEf 20 A,NGLt = AnGLE-T£MP 25 PHI = A?vGLr*3. 141592 FH!Z = A T A N 1 Y I / ( X R - C C M P Z ) I F ( C C M P Z . G T . X R ) C C M P P = XP. - Y I / T A . N ( F H Z - F H ) S Q f t T ( ( X^'.-CCA'PP ) *-*2*( VI ) / (i.-CCWPP) KETUrtH END C C C

c

-69-

c

c

c

c

c

c

S U B R O U T I N E D M A T ( C , D I S C ^ f f v A C T t N C , G A I r t l

Di^.cNSiafv ONGM<NACT,NO ,O(NACT,NU

cc 10 i=i NACT

f DO 10 J=1,NC 10 C(I , J ) = D N C M ( I , J ) * G A I N END -70- S U B R O U T I N E S I Z F . ( N P L A N T , N r > I G , ; : C , V A C T ) CCCCC CCCCC S I Z E IS A UF-FR-VR rTTF>: P P O I 7 R A P "F'A" DFTF.PWT TF.S THE SIZE OP CCCCC 7HF S Y S T F " . TT S H O U L D PE P F O G 3 A KKF. D A? IN TUTS E X A M P L E .

1 1

ccccc M P L A N T = T ?* snnp-t? OF DiFFF.RFt'TirL EQUATIONS OP Til* PLANT

CCCCC ?.'DTG = THE N'tlMnFP O ^ D T F F F P F N C F F O L I A T I O N T!. THE CONTROL CCCCC NC = TflF N U f B F " OF C O N T R O L S > CCCCC N A C T = T H E Vn»1PEr O F A C T U A T O R D I F F F P F N T A L E g U A T T O V S .

CCCCC N O T E : T H I S PFAJ.'S T H A T N P L A K T DO?S K O T I N C L U D E A N Y CCCCC A C T ' J A T O P D Y N A M I C S : CCCCC T H E P P O G P / V M SHOVN ' SRE I S F O F T H E L O N G I T U D I N A L A T T I T U D E CCCCC C O N T R O L - K H E L L S T O R I N G PROBI.F!' PISCOSSFD TN THE BODY OF CCCCC R E P O R T CCCCC N P L A N T = 3 N C I G = 3 N A C T = 1 NC=1 R E T U R N E N C C C c .

c

c

c

c

c

c

c

S CP.ROOTIMF- P L A H T ( J ^ P L A N T , N D I G , N C , >?ACT, N5 , V'STOT ,H STOT2 , 1 T C N T r l N " , G", D ^ f C K P j C W T , T I N V ,7\CL ,? VEC , V F G P ? ? , V F C T H , V'CP, « T EH 3 E A , 7 E" BB , T 2" 2BB , ? FM 2A 3 , F , TF M2^ ». , TEK« n A , DN' I H P L I C I ? P R A t ( K . , L , M ) P E A L * R ? , . I C N T , I N T , G M , D 9 , A C L r - B K , S T H , C W I , C W R OIKFNSTON ZrFO(2,P) ,POT.E(2,fl) 1 n DT"ENriON IC'I '' (VST072) ,IKT (NP70T2) ,GM {'ISTO' ,MSTOT) , n^ (N3TOT2) , 1 CW° ( K S T O ? ) ,C MI ( N S T O T ) ^ I N V f N D I G , N r > T G ) , ACL ( N ? T O T , 5TSTOT) f n T 2 E VEC ( N S T O T , K S T T ) , V F C ? N ( N S T O T , f'FTOT) , V F C I K (KSTOT, f.'STOT) , n "? V E C P ( K S ^ O T / V S T O T ) , VFCI f'!S" "', VSTO") , VRV (I'S'OT). , VT V (NSTOT) , C U VP? V (KSTO") , V T V (T!STOT) , A PP.I VT ( KS , NS) , B P F T M E (NS , KC) , 5 P ( ? ; S , K S ) , 0 ( M 5 V C ) ,C1 ( V C , ' ( S ) ,C2 (KC, V D I G ) , C? ( f ' C , N / m 1 ? C ( N A C T , ? ; AC > , n (V AC "''ICJ ,•".•"• SPA ( N ' P T G , NS) , "? "*. A (!IS , ".!

(4 V2 ( N ^ ^ n T ? ) ,T7.'M'BA (!.T>rn , N 3 ) ,Tr."!Pr< ( N D I R , ?'i>TG) , T F»l r T ' TFfl? A P ( * I ? , f.'OTG) , F('r"'" )'-.r;TTO'') , "F'^BA (!'DTG , V C ) ,"?.*U W. f K . T T G , WS) 6 , D K O M ( N r > C 7 , N C ) CCCCC 171 I 7 J ; 1 5 1 C'*fCC P T i'' '•'."" ~~ i ti^ ^ - w~ CT^ '' P' OH f P'1 "J's " r) ! ^'^ l<*1 r " E? T FI F r r r CCCCC D T F r E ^ E N ' T T M . F C V ' s T l " " '•? T!! P ' . A M T . X 0 ? ' ^ P C T FIT AT f Y TT' ! > p ccccc rm'i-FT 7nr. i*.?~Mc-r, .•. , ,c, AI:;I D N O « . TFT' "v:r IMF " ':;,LI.so. .

i t? v ^ f ccccc oner r'F.7 F I / N , ANT) 7.'!"?:-T- o " ;-AY co.'^rAf? R?.'/•• p r:7^~" r^• :? A'?. n CCCCC THE LIK ?,.

-71- CCCCC T H F P P P H P A M S H O W " !I»»E I S F O P T H E L O N G I T U D I N A L A L T I T U D E CCCCC C C N T R O L - W H E L L S T F F P T N G P H O B I - F K D I S C U S S E D I N THF. n o n ? O F T H E CCCCC P.EPOHT CCCCC A (1, 1) = -6. 2328751-1 A (2,1) = - 3 . 2 7 9 Vi?.- 3 A ( 1 , 2 ) = 2 0 7 , 0 6 t t A (2,2) = - U . a 9 9 S U F . - 1 A ( 3 , 2 ) = 1 .

P, (1,1) =-1 .«08flOF--» B (2, 1) = - 1 . 7 3 9 2 P E - 2 C ( 1 , 1 ) = - 1 0 * DN01 (1,1) =10.

R F T U P H ' EN P C C C ' .

C C C

c

; C c c S U B r O U T I K ? DIFF^O ( N P L A NT , I!DIG , N C , K ACT , NF, NSTOT, NSTOT2, V 1 I C N T , T N ? , G , n ° , C V R , C W I , T T V V , ACL , * V F C , V E C F N , V F C I N , V^CP , T 2 V F . C T , V ? V , V I 7 , V F 7 V , V P I V , A PP^ MF, R P F >!F ,P ,9 ,C 1 , C2 ,C3 , M 1 ,M2 , W3 r L 1 3 M 5 , H 6 , S , B , C , D , ' r : ! l 5 3 A , r ? l J A A , T S « 2 A A T F M B A , T R H A B , V 1 , V 2 , f M < " T E V 3 P A , ^ E P P , r r M 2 ^ ' ? , T F , W 2 A D , ? T F P 2 B A , T P I K t t B A , D N O M ) / I M P I . I C I T F F A L ( f , L , M ) F E A!.*? F, TCNT , It-"? , T,M , D9, ACL PEA!.* 8 DLOG,!5SOPT, D A T A H M S T O M T C N T ( N S T 0 7 2 ) , " N ? (P5TOT2) , GK f P S ^ O T ; S'STOT) , P 9 ( K S T O T 2 ) , 1 CW? ( M ? T O T ) , C W I P'STOT) ,TI>?V ('TIG , JJDT(') , SCI. ( V R T O T ,HSTO?) , T

2 Rv?;r (?j??OT,!:sTO'r) ,VFCP.N (t'STOT, KS OT) ,VTCIM (KSTn

3 VFC? ( N S . T O T , K S T O T ) , V ? C T ( I . ' ? T O T » ! S ? O T ) , V ? V ( K S T O T ) , r U VP.P.'V ( K S T O ' J , V R I 7 (N'STOT) , A P P I t « E ( t ! S , T J ? ) , B P R I M F (115 ,NC) , r; MS ION ^1 rip-';, N n r o ) , M? (Korfi, NDIG) ,K? ( K D I G , M S ) ,.i« (NI>IG,VS) , c V : 1 1 5 ( ! J E 1 G , N 3 ) , f - 6 (N D I'.? ,M ) ,A ( J J P I A N T , NFI A T ) ,B (! PL A»T ,KC) , 2 C ( ? - ? A C T , M P . C T ) ,0 (NS C T , N C ) ,rFPl5B A f N n t G , NS) , T E K A A ( N ^ , N.9) , M ? T F 2 A A ( V S , N S ) , T ! ' M B A (NDT, , N 3 ) , T F P A 1 (KS,».*DIG) , u v 2 ( v ? T O T 2 ) ^ I M B P A (v DIG , M s) , Trr«n n CJOTG, >:ntr,) , 5 TF?12?." (KS,f!Prr,) ,"(HST07,NSTOT) , TF K2U A (HDIG , NS) ,T EMU EA (NDIG, 6 , DNOM (N ACT, VC) recce l CCCCC n i F F S O IS A » S ^ - W P I T 7 F N P P C G F A N T!IA" O E T F R M I VF.S T ' F O T F F E P F N C F CCCCC F Q U A T I O S S US'.r* IN THE D i r i T T A L C O N ^ P C L . " O P E ?P EC! f I C I M L Y CCCCC I I D E F I ^ r S THE M A T R I C E S : H 1, M2, f!3, 1U, PS, *ft, A"D C 1 , C 2 , C 3 CCCCC I V ^ E P « S O F T S A f P A N O A K Y C O N T P O T CO'^S" A'JTS. T T I S C A L L R T > C CCCCC S E V F ' A L T I M F S T'? G E T T I N G TH*" r O K P E f l S A " " O M ( I V S = 2) 0 COVTPOL T CCCCC P A £,",,* FT ErS F I I O ' f L T ) MOT H" R T : A f l IH TH.5 S'lP'OirTI V? , RHT .T rO'!LD V T CCCCC I N F T " A P P E K E A ? A T ""?E B ~ G I > T » ! G O F T i l " P P O ^ P A I A V P Pf-Sf.F.D t - 5 v ir T CCCCC TH f - T F F 0 TH A T..R y LED CO^:, !nN. T! E C ' ! f t T O K OF "*'!* 7 T r > < CCCCC Cf^".?" "? A T T O " t'''FT ' . ? n o n P A ' M F O H E T F , ?.VP "'HHS, N D T G , "!!?.

i i 71 ccccc ::"> '. r'.^ OF ciKF-'Ji-^vcir F O H \ " T O ^ S ^u?. T N C I K D / : TH" CCTIPFNSATTON ccccc r o i i ' ^ r n v .

r CCCCC ilK " F O G P A M S H O V N 'CiRH ~'.l FOR THF T O V n i r i j r r f j . A T T T T f l D K A r -72- 'CCCCC C O N T P O L - W H E I . L STF.FRI'JO P F C 3 L F W DISCUSSED IN TKF BODY OF THE CCCCC P K P O R T

ccccc

K01£ = ?XP (-TS»MP/16. ) E 0 1 6 F 3 = S 7 . 2 ° 5 7 7 « S 1 * (1. t F 0 1 6 ) / 2 .

C 2 ( 1 , 3 ) = 1 .

C W R < ? H O « T K1 (1, 1) = E016 !f 1(1,2) =-F016F.l*2. 16 , 1 U ( 1 , 2 ) =E016F3*2.16 C C O M P R M S / V T I O N « 1 ( 2 , 2 ) = T T 2 r G L L = (1.-' T2 M1 (2, 1) M 3 (2, H) =-r,LL*TT1*2U7. 53 "14 (2,3) =GLL*2«7.53 C DELAY » 1 ( 1 , 2 ) = 1 .

FETOP.H EfJD -73- The utility routines are: MMULT matrix multiplication MADD matrix addition MATIN matrix initialization (0 or I) MATEQ equate two matrices SCAMAT matrix-sealer multiplication FGPQ calculates <p given F MINV matrix inverse CDIV complex number division HQR2 eigenvalue solver (QR algorithm) BALANC preprocesses matrix for HQR2 ELMHES postprocesses matrix from HQR2

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References

1. Deets, D. A., and Szalai, K. J., "Design and Flight Experience with

a. Digital Fly-by-Wire Control System Using Apollo Guidance System

Hardware on an F-8 Aircraft," AIAA Paper No. 72-881, Aug. '72.

2. Mathews, Mark A., "SAAB Digital Flight Control," AIAA Paper No. 74-26, Feb. '74.

3. Edwards, J. W., " A Comparison of Digital Design Techniques for Digital

Flight Control Synthesis," JACC June '73.

4. Lee, J. F. L., "A Digital Flight Control System Design Approach for a

Space Shuttle Booster Type Vehicle," AIAA Paper No. 74-20, Aerospace

Sciences Meeting, Feb. 1974.

5. Stubbs, G. S., Penchuk, A., Schlundt, R. W., "Digital Autopilots for

Thrust Vector Control of the Apollo CSM/LM Vehicles," AIAA Paper No.

64-847, Aug. 1969.

6. Simmons, J. C. and Hall, B. M., "Digital Control System Development for

the Delta Launch Vehicle," AIAA Paper No. 73-847, Aug. 1973.

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Design Problem," AIAA Paper No. 73-848, Aug. 1973.

8. Johnson, J. M., "The Application of Digital Filters Using Observers to

the Design of an ICBM Flight Control System," AIAA Paper No. 73-845, Aug. 1973.

9. Powell, J. D. , and Katz, P., "Sample Rate Selection for Aircraft

Digital Control," AIAA J, August 1975.

10. Borow, M. S., et al., "Navy Digital Flight Control System Development,"

Honeywell Document No. 21857-FR, Honeywell, Inc., Minneapolis, Minn., Dec. 1972.

11. Sutton, M. L., "Feasibility Study for an Advanced Digital Flight Control

System," LSI Tech. Report No. ADR-773, Lear-Siegler, Inc., Santa Monica, CA., Oct. 1972.

12. Konar, A. F. , et al., "Digital Flight Control Systems for Tactical

Fighters," Report No. AFFDL-TR-73-119, by Honeywell, Inc. for the

A.F. Flight Dynamics Laboratory, Dec. '73.

13. Rader, C.M., Rabiner, L. R. , Digital Signal Processing IEEE Press, 1972.

14. Huelsman, L. P., Active Filters, McGraw-Hill, 1970.

15. Edwards, J. W., "A Comparison of Digital Design Techniques for Digital

Flight Control Synthesis", tutorial, NASA Flight Research Center, Memorandum.

-75- 16. Slater, G. L., "A Unified Approach to Digital Flight Control Algorithms", AIAA Paper No. 74-884, August 1974.

17. Tuttle Jr., David F., Network Synthesis, Vol. 1 John Wiley & Sons, 1958.

18. Bryson, Jr. , A.E.,"Control Theory for Random Systems" SUDAAR No. 447, September 1972.

19. Bryson, Jr., A. E., Hall, W. E.,"Optimal Control and Filter Synthesis by Eigenvector Decomposition" SUDAAR No. 436, November 1971.

20. Katz, Paul and Powell, J. D., "Selection of Sampling Rate for Digital Control of Aircraft". SUDAAR No. 486, September 1974.

Stanford University, Aero/Astro Dept. , -76-