Hybrid computer solutions for optimal control of time- time-varying systems with parameter uncertainties*

by W. TRAUTWEIN and C. L. CONNOR Lockheed Missiles & Space Company

Huntsville, Alabama

SUMMARY

A hybrid computing scheme is described which eco-nomically solves optimal regulator problems for time-varying systems. The variational problem of deter-mining optimal time-varying controller gain schedules is reduced to a sequence of standard one-dimensional parameter searches and high-speed simulations by restricting the class of optimal gain schedules to be piecewise linear.

A major shortcoming of most optimization tech-niques-the high sensitivity to parameter uncertainties -is mitigated by making the performance index de-pendent upon two or more different simulations.

Choosing highly adverse operating conditions for per-formance evaluation largely reduces the sensitivity of the optimized system to off-nominal conditions. This is illustrated by an example, the design of a load-relief attitude controller for large launch vehicles. Other practical features of the hybrid optimization include the use of free-form, non-quadratic performance mea-sures to specify design goals in the most direct manner with only minor mathematical constraints concerning their functional form.

INTRODUCTION

Hybrid computers have long found wide use in param-eter optimization of time-invariant dynamic systems.

Fast-time, repetitive analog simulation of the system dynamics is combined with a functional minimization scheme programmed digitally. A performance index, formulated in accordance with the design goals, is minimized by iteratively updating the adjustable parameters between subsequent analog runs.¹

* Work supported by NASA-MSFC Contract No. NAS8-30515, Mod II, Task 1.

135

A basic advantage of hybrid optimization over linear optimal control theory is that performance measures can be freely selected to best reflect the design objectives, whereas the quadratic criteria required by linear con-trol theory are in gen~ral difficult to relate to the ~esign goals. System nonlinearities and constraints imposed upon certain state variables or control parameters can be accounted for in the hybrid approach without difficulty.

These practical features made it desirable to solve another important engineering problem, optimal con-troller design for time-varying systems, by a similar technique.· Optimal controller adjustments in this case become functions of time which in general require variational techniques for their solution. For regulator design without specified terminal conditions the varia-tional problem can readily be reduced to a sequence of parameter optimizations if the optimal gain schedules are restricted to piecewise linear functions of time. In another section the basic approach will be described for completely defined systems. An extension of the hybrid technique to drastically reduce the sensitivity of the optimized system to parameter uncertainties or other off-nominal conditions is given later in this paper. A more detailed discussion of the optimization method and its application to booster load relief controller design is given in Reference 2.

OPTIMAL REGULATOR DESIGN FOR TIME-VARYING SYSTEMS UNDER NOMINAL CONDITIONS

Formulation oJthe control problem

The dynamics of. the system (plant) be described by the first-order vector equation

x=](x, il, t) (1)

136 :Fall Joint Computer Conference, 1970

r

Figure I-Assumed polygonal form of optimal gain schedules

where i is the n-dimensional state vector,

J

an n-dimen-sional functional vector,

u

an m-dimensional control vector (m5:n) and (") =d/dt.

The control law be dictated by available sensors or other limitations in cost and complexity. For this dis-cussion we assume a linear control law

u(t) =K(t)i(t) (2)

where K is an m X n gain matrix. The design problem then is to determine the gain matrix K (t) so that the performance measure

J =J(x, u, t) (3)

;.

is minimized. J is a positive definite function of other-wise free form, selected to best reflect the design objectives.

Restriction to piecewise linear gain schedules

The computational load can be greatly reduced with little or no loss in optimality of the solutions if the mn gain schedules of K (t) are assumed to be piecewise linear functions of time as shown in Figure 1. Then the optimization problem is reduced to a parameter search

Figure 2-During optimization cycle at time t^{p ,} a search is performed in Ki parameter space of the mn gain slopes for

optimum performance

Time Varying _ _ _ - , Parameters

Disturbances

,----i

_I

I

I OPERATE) RESET I

I

1---,

COMPUTER LOGIC CONTROL

I I I L_

PLANT DYNAMICS

PERFORMANCE ANALYZER J = I(i, u, t)

DIGITAL Slopes 01 Controller . Parameter. I K

I I

, I

GRADIENT MINIMIZA TION

J -MIN

---=-

^~

-=-=-=- -=--=- ^--=-

-=-r:::-::-_-__

HOLD/COMPUTE I

I

Figure 3-Basic control system optimization scheme. Complete plant and control system dynamics are simulated repetitively on analog console of hybrid computer. Performance is analyzed after simulation in digital computer and optimized by iterative

changes in slopes of controller gain schedules.

for the mn gain slopes

Kj

of the gain matrix K (t) as shown in Figure 2 which can be carried out based on standard hybrid techniques. In this regulator problem the performance index does not contain the terminal state. Thus, forward integrations are sufficient to deter-mine the optimum. The basic computing scheme is shown in Figure 3. The analog simulation serves a dual purpose. Performance is repetitively evaluated during an optimization cycle in fast time. Once the optimum set of gain slopes It( j = 1, 2, ... mn) is determined the optimum parameters are used during a real time simulation from one update time (t,,) to the next (t,,+l) using the same analog circuit (Figures 4 and 5).

Fast Time Predictions / ' Optimization Cycle"

Optimization -Cycle "+ I 'r--~----~~----~~~~~~--~----t

t"

Figure 4-Series of fast time predictions at update time tp leads to optimum gain slopes Xi ^opt(j = 1 ... mn). Subsequent real time simulation uses Iti opt from tv to next update time t,,+l

LOgi~L

Control

Input '3 T S

el eZ

Track 8. Store Unit Behavior

Real Time Mode Fast Time Mode

o

fl .... _ ...

^O~

__

^Machine

t/l_ l t" t" t"+l Time Track 8. Store

Unit Output ,...

(To Integrator IC) -e3

Ie ....

~

.... --' .

r---

^Machine

Integrator Output eZ

~ II" ·

^Time

~I:: :"-+1- ^M~_

Recorded Output e Fast Runs . . . Time

(Recorder is stopped Z

kfl

Real Tune Slmulatlon during fast runs)

-+-..L...JL...J.... _ _ _ _ _ _ _ _ ... Real Time t"-lt,, ~l

Figure 5-Simple analog circuit for real time simulation and fast time predictions using same integrators

Minimization method

In order to distinguish between local ffilmma and absolute minima the search in the Xj parameter space is performed in two phases:

1. Systematic Grid Search

All possible parameter combinations within a region of specified limits and a grid of specified fineness are evaluated for their performance J.

Such a complete survey is feasible as long as the parameter space is of low dimension as in present applications (Figure 6).

2. Gradient Search

The Fletcher-Powell-Davidon gradient method³ uses the grid search minimum as a starting point to precisely locate the minimum.

Example: Booster load relief controller design

The practical features of the optimization scheme are best illustrated using the following example. The peak bending loads at two critical stations, Xl and X2 (Figure 7) of a Saturn V launch vehicle shall be minimized during powered first stage flight while the vehicle is subjected to severe winds with a superimposed gust (Figure 8, top strip) and to an engineout failure at the time of maximum dynamic pressure.

The major variables are:

a vehicle angle of attack control engines gimbal angle

Hybrid Computer Solutions 137

Parameter Combinations Evaluated During Grid Search

Grid Search Minimum

o

-0.1 C>=---O---C>--~-~----(l_-o

-0.1 0 Kl 0.1

Gradient Search Minimum

Figure 6--Parameter optimization performed in two phases:

(1) Systematic grid search (0) for complete survey of parameter space; Grid point of minimum J ( 0) serves as starting point for (2) gradient search which locates the minimum more precisely (D). From grid search contour plots (Lines of J = Const) can be

displayed at CRT for better insight into J-topology.

1/ generalized displacement of bending mode at nose of vehicle

cP vehicle pitch attitude error

X commanded pitch attitude relative to vertical at time of launch

z drift, normal to the reference trajectory

~ slosh mass displacement, normal to refer-ence traj ectory

Y (x), Y' (x) bending mode normalized amplitude and slope at station x

Disturbance terms due to engine failure:

cPF pitch acceleration due to engine failure ZF lateral acceleration due to engine failure

1/F bending mode acceleration due to engine failure

Control law

The attitude controller to be optimally adjusted has feedback of attitude error, error rate and lateral acceleration:

(4)

138 Fall Joint Computer Conference, 1970

Figure 7-Minimax load relief performance criterion to minimize peak bending loads at two critical locations, Xl, X2:

j

^tv+To

~ =max/MBi(t) /MBi /

+q

()2RG(t) dt--+min

~=1,2 tv

tv~t<tv+To

Normalized Peak Bending Loads

System equations

Mean Square of Rate Gyro Output to Ensure Trajectory Stability

(6)

The vehicle equations of motion as simulated on the analog console are:

Rotation:

Translation:

Bending Mode:

Sloshing:

Position Gyro Output:

cJ>i = cJ>+ Y' (xPG)7J

Rate Gyro Output:

(Ji=

cb+

Y' (XRG) 7j

Accelerometer Output:

Ti=Z+Al4>-A2cJ>-A31i+A47J

Gimbaled Engine Dynamics:

Shaping Filters:

Attitude Error Filter: cJ>s=</li(Pos+1)/P1s+1) Rate Error Filter:

Accelerometer Filter: Ts=TiRo/(S2+R1S+Ro) Angular Relationship:

a=cJ>-i/V+aw

aw= tan-1(Vw cosx/V - Vw sinx)

Wind

Disturbance

aw:o~

Failure Mode

Optimized Gain Schedules

Pitch Angle

Engine Deflection

Bending Moments at Station Z

At Station I

Leeward Engine Failure

. ,outF

^~

Engme No. I lIn I ~

al I

r==

^'¥

(sec) O~

I:

~gz

~r

m/sec

~

:r

(deg)

+5.0 [ (d~g) 0 MB2 +0.:

r

MBZ

MBI +0.:[

MBI

60

I I :

I I

I~~

I I I

r:;::

Peak Moment for

I / ' - Constant Gain Case

~ 53% higher than peaks I L""";= of optimized system

I

I

; , / \ _ Peak Moment for

~ Constant Gain Case

50% higher than peaks

I~ of optimized system

70 80 90

Figure 8-Typical Saturn V S-IC optimization results. Three gain schedules are optimally adjusted to minimize the peak bending loads among two stations (1541 and 3764 inches) for the disturbance and failure history of the top charts. Peak loads are substantially reduced compared with nominal Saturn V performance (without accelerometer feedback). Weighting factor q=O.05; floating optirilization time interval To=20 sec in

performance criterion (6)

Engine Failure Switching Function:

Complete numerical data for all coefficients used in the simulation are compiled in Reference 4.

Selection of performance index

In earlier studies⁴ quadratic performance criteria such as

(5) were-used. They allow a straightforward physical inter-pretation and at the same time can still be loosely related to linear optimal control theory. Neglecting external disturbances (aw ~ 0), Equation (5) can be re-written in the more familiar form

where a is an n-dimensional coefficient vector, q3, q4 are constants depending upon ql, q2, M a' and M / and superscript T denotes transpose.

The results from optimizations where Equation (5) was minimized were disappointing insofar as peak bending loads were reduced by a few percent only, whereas the major reductions were in the RMS values of the bending loads.

Since peak loads are of major concern to the designer, a more direct approach was made to reduce peak loads by using the minimax criterion (6) of Figure 7. During each run bending load peaks at both critical stations were sampled and compared. Only the greater of the two peaks was included in J. This peak amplitude normalized with respect to the structural limit M B was the major term in J. The only additional term, the mean square of measured error rate, was included to ensure smooth time histories and trajectory stability. This performance criterion reduced the number of weighting factors to be empirically adjusted to one, whereas n such factors must be selected in linear optimal control theory for an nth order system.

Hybrid Computer Solutions 139

Wind for assumed windward engine failure under otherwise identical flight conditions. Optimal gain schedules are strongly dependent

upon assumed failure condition

Results

A typical optimization result is shown in Figure 8.

Drastic reductions in bending moment peaks result from the minimax criterion compared with the constant gain case. It should be noted, however, that perfect predictions 20 seconds ahead were used in the optimiza-tion including the anticipated failure.

In Figure 9 the results of a similar case ar.e shown.

All flight conditions are identical to the previous case except for the failure mode: a leeward engine fails in Figure 8,a windward engine in Figure 9. Again, peak bending loads are substantially reduced in magnitude compared with a case with nominal constant adjust~

ments of the controller. However, two of the three opti-mal gain schedules (ao(t) and g2(t)) differ drastically for the two failure modes. In view of the lack of any a priori knowledge about time and location of a possible engine failure no useful information can therefore be gained from the two optimizations concerning load relief controller design. This basic shortcoming of all strictly deterministic optimization schemes must be relieved before the method can be applied to practical engineering design problems characterized by parameter or failure uncertainties.

J

140 Fall Joint Computer Conference, 1970

-Upper Bound J of Performance for Failure A ~ B

Failure A

B

K K

optA or B optB

Figure lO-Optimum adjustment of scalar control parameter K considering possible occurrence of failure A or B

OPTIMAL REGULATOR DESIGN INSENSITIVE TO FAILURE UNCERTAINTIES

Previous work to reduce parameter sensitivity has centered around inclusion of sensitivity terms

aJo/aK

in the performance index to be minimized, where J ⁰ denotes performance under nominal conditions and K is the uncertain parameter vector.5 Substantial addi-tional computaaddi-tional load would arise if such an ap-proach were implemented on the hybrid computer.

Moreover, in the case of possible failures the uncertain parameters may vary discontinuously from one discrete value to another like the engine failure switching func-tion in the preceding example:

{

I for Failure A 0= 0 for Nominal Case

-1 for Failure B

Another approach is therefore chosen: The hybrid optimization method is extended to account for such failure uncertainties even if no partial derivatives exist.

Consider the case of two possible failure conditions,

A or B. The performance index evaluated for each failure may be of the form of Figure 10. Neither KoptA

nor KoPtB would be optimal in view of the uncertainty concerning the failure mode. Performance might be unacceptably poor at the level

I

^Au if Failure A occurred and the control parameter were adjusted at the opti-mum for Failure B. The best tradeoff in view of the failure uncertainty is the minimum of the upper bound of J ^A and J ^B (solid line in Figure 10). In the example of Figure 10 this optimum which is the least sensitive to the type of failure lies at the lower intersection of the two curves.

Extension of the optimum-seeking computing scheme The most direct way to locate the minimum of the upper performance bound is to simulate all possible failure modes for a given set of control parameters in order to determine the upper bound

I.

A gradient de-pendent minimization technique can again be applied to seek the minimum. One might expect convergence difficulties around the corners of these I-functions.

However, only minor modifications were necessary to the basic Fletcher-Powell-Davidon gradient scheme and to the preceding grid search to locate comer-type minima. The changes included a relaxing of the gradient convergence test

(I

VJ

I

~~, where }; is a small specified number). If all other convergence tests are passed, then ~ is doubled in subsequent iterations. In

q1MB1

:T") -:IJ

~j.B4\1J

Flight Condition A

I

Flight Condition B (Windward Engine Out) (Leeward Engine Out)

MAxlMBil MBi

Figure ll-Generalized load relief performance criterion to minimize upper performance bound J for two possible

operating conditions

j

^t,,+To

} = max I MBi/MBd

+

82RG(t) dt~min t"

CaseA,CaseB i=l,2 t,,<t<t,,+To

order to keep the number of analog simulations low, only one failure case associated with the local upper bound is simulated for approximate gradient computa-tions based on

J

(K

+

llK). During the grid search much computing time can be saved by checking if the upper performance bound

J

is much larger than the performance index J of the less critical case. Then evaluation of only one failure case is necessary for the neighboring grid points.

A pplication to the booster load relief problem

In its extended form the optimization technique is ideally suited to minimize the effects of possible failures in the booster attitude- control problem. The design goal is to minimize peak bending loads for the worst of several possible failure conditions. To this end the two most adverse flight and failure conditions are simulated to determine the upper performance bound

J

of both cases for each set of parameters as shown in Figure 11.

The performance index J, now a function of both ad-verse cases, is then minimized following much the same scheme as in the basic approach.

In Figure 12 results obtained by considering a single failure mode only (solid lines) as in Figures 8 and 9 are compared with results gained from the extended op-timization scheme which considers both adverse failure conditions to minimize the upper performance bound.

The former gain schedules (solid lines) are closely tuned to the specific flight and failure condition and therefore, differ drastically for the two adverse failure conditions and cannot be used for controller design.

Simultaneous consideration of both adverse failure modes (dotted lines) leads to a single set of controller

Flight COI1ditiOll "A": Flight Condition "B'I:

LEEWARD ENGINE OUT WINDWARD ENGINE OUT

j.'~'.b#¢···1 ""3J-"~i-,pl

.---+ ^j

¹

.-E-i

lJ ':",1 ~~i.J ... :9

!l ~:+ ?§f J~'-E .?"1iJ

bO eec 100 eec 60 etc IO!l.p(.

Figure 12-Deterministic optimization for one failure case only (solid lines) compared with optimization subject to failure uncertainty (dotted lines) with consideration of two possible

failure modes

Hybrid Computer Solutions 141

Failure B Station 1 50 Failure B

l_35~

Station Z

r:f '- 40 IQ

-L

III

I'Z3%

"0

..

₀

~ 30

tlO~

'::"0 ... 4)

'g .!l

4 )

-~

..

.lo: 6

Failur~

J

:

~ zo

o.e

Station 1

I I

I

Constant Gains

I I

Optimized Gains

ao=O. 9, a1=0. 67, ao(t)

gz = 0 gZ(t), a1 = const. = 0.67

Figure 13-Peak bending loads for Saturn V S-IC powered flight compared for constant gain case and for case optimized for failure of leeward or windward engine. Wind profile as shown

in Figure 8; Assumed engine failure time 76 sec

gain schedules. The reduction in peak load is smaller than before for Condition A, whereas the milder Con-dition B even shows some increase in bending load. A closer inspection of the resulting loads in Figure 13 reveals that an ideal tradeoff between failure effects A and B was achieved. All peak loads are brought to a common level of about 33 percent of the structural limit

M

^B, whereas the constant gain controller adjust-ment exhibits loads up to 52 percent of M ^B.

COMPUTATIONAL ASPECTS

Experiments were made on the EAI 8900 computer to determine if acceptable accuracy and repeatability is obtained for a time scale of 1000 times real time. Re-peatability was found to be within 2 percent for the time scale 1000 and within 0.5 percent for slower time scales using a circuit and interface similar to the opti-mization scheme. Two percent repeatability was con-sidered sufficient. Therefore, a time scale of 1000 times real time is used during fast time predictions. In the booster control problem the optimization of two gain schedules takes about 5.5 minutes hybrid time.

Up to 40 analog simulations per second are presently achieved for analog run times of typically 20 milli-seconds (Figure 14). Using nine grid points for each parameter during the grid search locates the minimum sufficiently close in the example presented. After 3

142 Fall Joint Computer Conference, 1970

Control Gain ao

Control Gain

Bending Moment MBI MBI

o z

z o

.5

0

-... "r

•

^II

I.

• • _•

--

l

l'

Fast Time Predictions

I "'_It-f--

^{I I r I}

t-~ Grid se,arch , ==R ^Gradient

1"'""- Search

... I ... C;i. ~ii:,f---' ^...

,. r. ,.

..

Real Time Simulations

..

ⁱ

• _•

• • •

. r

•

Machine Time

....

IT, ~

~r-I~

'.

••

t 1

;Cd '1

.-

Pi ...

I I I I

T

Figure 14-Typical computing speed is 20 to 40 analog simula-tions per second on EAI 8900 computer including AID transfer of performance index, D I A transfer of new parameters between simulations. Time scale used during fast runs: 1000 times real

time; Typical analog run time 20 millisec. per run

to 5 iterations of the following Fletcher-Powell-Davidon gradient search the minimum is found with sufficient accuracy.

ACKNOWLEDGMENT

The authors wish to acknowledge the contributions of Messrs. Roger Lin (now EAI, NASA Ames) and S. Lo in hybrid programming and computations.

REFERENCES

1 G A BEKEY W J KARPLUS Hybrid computation

John Wiley & Sons Inc Section 9 New York 1968 2 C L CONNOR W TRAUTWEIN

Control system optimization for Saturn V launch vehicles using gradient techniques

Final Report Contract NAS8-30515 Mod II Task 1 LMSC/HREC D162122-I & III Huntsville Alabama February 1970

3R FLETCHER M J D POWELL

A rapidly convergent descent method for minimization Computer J Vol61963

4 W TRAUTWEIN J G TUCK

Control system optimization for Saturn V launch vehicles using gradient techniques

Final Report Contract NAS8-:21335 LMSC/HREC A791836 Lockheed Missiles & Space Company Huntsville Alabama October 1968

5 P DORATO

On sensitivity in optimal control systems IEEE Trans Vol AC-8 pp 256-257 July 1963

The role of computer specialists in contracting for

Im Dokument FALL JOINT (Seite 146-154)