w05-turningcontrol

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Design of Turning Con tro lfor aGeng G. Wang

Tracked Vehic leShih H. Wang and Cheng W. Chen

ABSTRACT: This article treats the designof a turning control system for an M113tracked vehicle, which is modeled as a two-input/two-output system. Since the vehiclemodel is nonlinear, time varying, and im-plicit, it is replaced by a set of linear time-invariant models, which is valid over certainoperating regions. The two inputs to the ve-hicle are throttle and steering. The two mea-sured output variables are the vehicles lon-gitudinal speed and heading rate. Using theQuantitative Feedback Theory QFT), a ro-bust control system is obtained. The controlloop for vehicle speed is designed to trackthe speed command and reject the distur-bance caused by steering. The control loopfor the vehicle heading rate is designed totrack only the rate command, since the dis-turbance caused by throttle is negligible. Thecontrol system design is tested by simulationand shows satisfactory results.

IntroductionA previous article [ I ] described an auto-

matic pilot that could drive an M113 ar-mored personnel carrier as an autonomousland vehicle. Field tests have demonstratedthe successful execution of road-following atspeeds up to 20 km/h. However, a more so-phisticated vehicle control system is requiredto respond faster and more accurately to avariety of commands including cruising,turning, and advancing a specified distance.

The design of the turning control systemfor a tracked vehicle is difficult because ofcomplexities and uncertainties in the model.In particular, in this model, some nonlinear-ities lack explicit expression and many re-lationships are determined from empiricalformulas. Since the internal models are socomplex, it seems reasonable to consider ex-ternal models or the system input-output re-lationships for the control design. The prob-lems are how to obtain a simple and sufficientexternal model and how to design a control-ler based on such a model. QuantitativeGeng G . Wang and Shih H. Wang are with theDepartment of Electrical Engineering and Com-puter Science, University of California, Davis, C A95616. Cheng W. Chen is with the Central En-gineering Laboratory, FMC Corporation, 1205Coleman Avenue, Santa Clara, CA 95052.

Feedback Theory [2]-[4] provides a system-atic procedure for solving these problems.According to the Quantitative Feedback

Theory, the plant is treated as a black box.Only input-output behavior is considered. Aset of linear time-invariant models, whichexhibits the same input-output behavior overcertain operating regions as the given non-linear system, is regarded as equivalent tothat nonlinear system. A robust controller isconstructed based on the set of linear time-invariant models.

However, the design of controllers for realsystems involving significant nonlinearitiesand parameter uncertainties is not thatstraightforward, especially with multi-inputmulti-output systems. This article treats thedesign of a turning controller for a vehiclemodel with two inputs and two outputs basedupon QFT. A speed controller for a single-input/single-output vehicle model has beenreported earl ier [ 5 ] .

The whole design procedure can be di-vided into four steps. First, the two-input/two-output nonlinear model is replaced by aset of two-by-two linear transfer functionmatrices. This set approximates the originalnonlinear model in a confined operating re-gion, because each linear transfer functionin the set is derived from an input-output pairof the original nonlinear model. The nonlin-earities of the original nonlinear model havebeen replaced by the parameter uncertaintiesof linear models. Second, this two-inputhwo-output linear control problem is convertedinto two single-loop design problems, whichtake into account both tracking and distur-bance rejection. Third, the single loops aredesigned using the Quantitative FeedbackTheory techniques. Finally, the controller isverified by simulation.

Conversion to Linear ModelThe vehicle system block diagram shown

in Fig. 1consists of three main parts, whichare the power train, the rigid body, and theground interaction. The behavior of thepower train is approximated by using a first-order differential system with a number ofnonlinear empirical formulas. The rigid bodyis described by a five-degree-of-freedomnonlinear differential system. The ground in-

teraction is modeled using Coulombs fric-tion law, and the complex interaction be-tween the tracks and the ground leads to animplicit description. The control inputs tothe vehicle are the throttle, the steering, andthe brake. The vehicle outputs are the lon-gitudinal speed, the transversal speed, head-ing rate, and left and right track speeds.

In our design, the measured outputs arethe longitudinal speedy, and the heading ratey 2 , and the inputs to the vehicle are the throt-tle u I and the steering u 2 . The closed-loopstep responses for the longitudinal speed andthe heading rate are required to be withinspecified bounds. Under normal operatingconditions, the longitudinal speeds vary be-tween 3.5 and 8 m/sec, and the heading ratesvary between 0.1 and 1 radh ec. The linearmodels are derived from input-output pairsof the nonlinear model. Each output of thepair is within the specified bounds. Since thesystem lacks an explicit description, the de-sired input-output pairs cannot be generatedby inverting the system equations. Instead,various combinations of the throttle, steer-ing, and turning times are tried in order togenerate a family of satisfactory input-outputpairs. From experience, the throttle u I andthe steering u2 are assumed to be in the fol-lowing form where r is the time to start tum-ing:

u I t ) = U , + 1 ~ , ) e x p - a t ) ,r O

& ( t ) = 0, < 7

Different input-output pairs of the nonlinearmodel are generated with various combina-tions of u l , u2 , a , @ and r. System identi-fication techniques are used to find a two-by-two transfer function matrix for eachinput-output pair. A first-order form is as-sumed for each of the four elements of thetransfer function matrix P s ) so that p o s )= a,,/ . + b,,), i , j = 1 , 2 ) . Since thethrottle input does not affect the heading rate,one of the four elements of the transfer func-tion matrix is zero-i.e., p 2 s )= 0. Sam-

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P o w eTrain

riglil l r ack speed

le11 l rack speedFig. 1. Vehicle system block diagram.TableSample Parameter Values

long speedlrans speedyaw raterlglil l rack speedtell track w e e d

ll bil a 2 bl a22

domain specifications. The correspondingfrequency-domain values are plotted on aBode diagram; their envelope forms the fre-quency-domain specification for h , s) asshown in Fig. 4.Similar bounds are derivedfor h I 2 ( s ) nd h Z 2 s ) .

Design ProcedureThere are many published papers onQuantitative Feedback Theory [2]-[4]. The

compensator design for h22 the heading rateloop) is a single-input/single-output esignproblem. The compensator design for h , andh I 2 the longitudinal speed loop) is a littlebe constructed to satisfy both the tracking

~ more complex since the compensator g , mustb22 -

4 , 106.223 4.6744.209 0.7143 17.98 10.804.22 0.7206 21.57 12.55 1.3974.484 0.9828 2.705 0.71 1 0.7544 1.610

and the disturbance rejection requirements.In either design, the main idea is as follows:Consider the set of linear transfer functionsP, s ) describing an uncertain single-inputand single-output system. The closed-loopsystem transfer function isple values of the parameters of the linearmodels are given in the Table. p, SI G s ) F ( s )

Conversion to Single Loops Fig. 2. Closed-loop system configuration. 1 + Pl (S) G( s)H , s )The closed-loop control configuration of i = 1 , . . , n )the linear model P is shown in Fig. 2, where

F and G are diagonal two-by-two compen-sator gain matrices. The transfer functionmatrix H of the closed-loop system from in-put r to output y is determined directly.

H = I + P C ) - P G F= P - I + G ) - G F

The inverse of the plant P is particularly sim-ple because pZ1s zero.

Because the matrices G and F are diagonal,their product is diagonal. Hence, after takingthe inverse of (P-I + G ) , he componentsh, of H can be determined directly.

hll = Pl lg l f i / l + P l l g l )h22 = P 2 2 g2h/ 1 + P22 gz)

wheredl2 = hZ*P12/(PllP22)

The quantity dI2 orresponds to the distur-bance on y l caused by input r,, as shown inthe block diagrams in Fig. 3.

Fig. 3. Single-loop control systems.

Design SpecificationsAccording to QFT, the design specifica-tions are given by the time-domain step re-sponse of the closed-loop system. For ex-ample, the output y I due to a step input rl isrequired to be within specified bounds. Thistime-domain specification is then convertedinto a frequency-domain specificationfor thetransfer function h l l by the following pro-cedure: The desired closed-loop step re-sponse is assumed to be that of a second-order system. We consider h, s) n the fol-lowing two forms:

h l l ( s )= w ; / s + 25w,,s + w ; )= 1 + S Z - I ) , ;

+ s2+ 2 . 5 4 s + U t )We choose different combinations of a,,,5 and z so that the step responses of these

systems fall within the bounds of the time-

According to the frequency-domain spec-ifications, the closed-loop transfer functionis required to be inside the bounds a U ) and/ 3 w ) at a set of specified frequencieswlr w 2 ,. . . , w,, i.e.,P w ) < 20 log I f f j w )

< . U ) , f or i E { 1 , , n } 1)This implies that

20 1 H , j w ) 20 1 H k ; w )a w ) @ U ) , for all i, k

An equivalent expression ismax { 20 log N , ; U ) 20 log N~ w }

1 . k )

. a) P w ) 2)where

N, j w ) = P , j w ) Me1+ 11 + P, j a )Me j l

M = IC JU)lJ = ~ G ( j w )

Notice that we have cancelled the termF ( j w ) . f we assume that P, U ) # 0 for alli and w , then for any w and 4, we can alwaysfind a large enough M to make both log func-tions close to zero. Hence, the above in-equality is satisfied. Let B w , 6 e a lowerbound on such M. For a set of frequencies

Apni 7990 123


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200- -20

3 -40' E%E: -60

80

3

10-1 100 101 102Frequency radsec)

Frequency-domain specijication fo r h ,ig. 4 .

w , i = 1, 2, . , m ) , a set of boundsB w , , 4 ) is plotted on the Nichols chart. Byproper choice of poles and zeros, a compen-sator G s ) s found such that G jw,) 2B w , , G j w , ) ) , i = 1, 2 , , m ) .Hence, the condition in 2) is satisfied. Inthe actual construction of G s ) , care mustbe taken to ensure that the system has properphase margin, gain margin, and bandwidth.This process is known as loop shaping [2]-[4]. For any w belonging to { w , , . . , U,,,} ,the feedback compensation G s ) as brought

20 1 H, j w ) IP , j w ) G j w )W w )

1 + P , j w ) G j w )= 20 logi 1, . . ., n )The above description only sketches thebasic concepts in QFT. Interested readers

should consult many articles on this subjectfor further details [2]-[4].In our case, the compensators obtained areas follows:

S- 10.5f i S ) =

0.43 (& + 1 )

-0.3579 (A + I ) ( ; + l ) ( + 1)~ Simulationsthe set of numbers P , j w ) G j w ) / [ +P I j w ) G j w ) ] i = 0, 1, , n ) closeto each other in order to satisfy 2 ) . Lastly,we need to design a feedforward compen-sator F s ) o bring the whole set of numberswithin the bounds a w ) and p ~ ) ,E { w , ,

, w,} , n order to satisfy 1) .

The last step is to perform the closed-loopsimulations. The discretized versions of G s )and F s ) at a IO-Hz sampling rate are usedwith the nonlinear vehicle model in the sim-ulations. Limiters are also included for theinputs to the vehicle model. Figure 5 shows

00 5 10IIE

Time sec)

Time sec)loo

0 5 10 15Time sec)

Time sec)Fig. 5 . Closed- loop sys tem s imulation,

one typical result of the closed-loop systemsimulation. The response time and overshootsatisfy the specifications except for somesteady-state errors in the tracking of speed.

In order to eliminate the steady-state errorin speed, we add an integrator to g , . Theintegrator is used only when the tracking er-ror is small. In this way, we can both elim-inate the steady-state error and improve thedynamic response. The switching formulaand the integrator gain are selected throughsimulations.

Furthermore, different speed commandssuch as sinusoidal functions are considered.To track such speed commands, a brake in-put of constant value is applied whenever thespeed exceeds the command value. Furtherdetails of the design can be found in [6].

DiscussionsThis article has investigated the applica-tion of the Quantitative Feedback Theory to

the design of a robust turning controller for

124 IEEE ontrol Sys rems Magazine


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the tracked vehicle M I 13. Since we are deal- G. G. Wang, 1. Horowitz, S . H. Wang, and Shih H. Wang receiveding with a real-world design problem, certain C. W . Chen, A Control Design for a his B.S.E .E. from the Na-engineering sense and judgment are needed. Tracked Vehicle with Implicit Nonlinearities tional Taiwan UniversityFor example, one needs to choose the ap- Using Quantitative Feedback Theory, Proc. in 1967, and his M.S . andIEEE Conk Decision and Contr., vol. 3, pp. Ph. D. degree s in electri-propriate set of input-output pairs in order to 2416-2418, Dec. 1988. cal engineering and com-[6] G. . Wang and S. H. Wang , Report on puter science from theerive the set of linear models to replace the

the Robust Controller Design for an Auton- University of California,onlinear model. The most crucial step inomous Land Vehic le, Technical Report, Berkeley, in 1970 andhe design is the loop shaping process, whichrequires some designer experience. Hence, Department of Electrical Engineering and 1971, respectively. He

it would be very useful to derive efficient Computer Science, University of California, has worked with the Uni-algorithms for computer implementation of Davis , Aug 9, 1988. versity of Toron to, NASAthis design approach. Langley Research Center, U niversity of Colorado,University of Maryland, and the Office of NavalResearch. Since 1984, he has been a Professor ofElectrical and Computer Engineering at the Uni-

[ 5 ]

AcknowledgmentThe authors would like t o thank Lou

McTamaney, Project Manager of the FMCAutonomous Vehicle Test Bed Project, for

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support and encouragement.

ReferencesJ J Nitao and A M. Parodi , A Real-TimeReflexive Pilot for an Autonomous Land Ve-hicle, IEEE Contr. Sysr. Mug. , vol . 6, no.1, pp. 14-23, 1986.I Horowitz and M. Breiner, QuantitativeSynthesis of Feedback Systems with Uncer-tain Non-Linear Multivariable Plants, Intl.J. Sysr. Sci.. vol . 12, no. 5 , pp. 539-563,1981.I. Horowitz, Feedback Systems with Non-linear Uncertain Plants, Intl. J . Contr.,vol.36. no. 1. DD. 155-171. 1982.

Geng G. Wang was bornin Wuhan, China, onOctober 12, 1947 . Hereceived the B.Sc. andM.Sc. degrees in elec-trical engineering fromthe Wuhan Universityof Technology, Wuhan,China , in 1982 and 19 84,respectively. He is nowworking toward hisPh.D. at the Department

versity of California at Davis. His research inter-ests are in control theory, robotics, and com-puters.

Cheng W Chen is amember of the technicalstaff in the Systems andControl Group of theFMC Corporate Technol-ogy Center, where he hasworked for over 3 years.Previously, he was anAssociate Professor ofcontrol engineering at Na-tional Chiao-Tung Uni-versity. Taiwan. His re-

search interests include control systems design,estimation and tracking, and stochastic systemsmodeling. Mr. Chen received his B.S. degree incontrol engineering from National Chiao-Tung

I , .

[4] 0 . Yaniv and I Horowitz, A QuantitativeDesign Method for MIMO Linear FeedbackSystems Having Uncertain Plants, Inrl. J.Contr., vol . 43, no. 2, pp. 401-421, 1986.

of Electrical Engineering and Computer Science,University of Califor nia, Davis. His current re-search interests are linear and nonlinear robustcontrol theory and applications.

University, Taiwan, in 1973, and his M.S. andD.Sc. degrees in systems science and mathematicsfrom the Washington University, St. Louis, in1978 and 19 82.

Apri l 1990

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