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(IAA Guidance, Navigation and;ontrol Conference and Exhibit4-17 August 2000 Denver, CO

AOO-37068 AIAA-2000-4151

NUMERICAL TECHNIQUE FOR SOLVING TIME- AND ROBUST TIME-OPTIMAL CONTROL PROBLEMS

Bassam A. Albassam*King Saud University, Riyadh, Saudi Arabia

AbstractWe present a numerical technique for solving time- androbust time-optimal control problems. The methodrelies on the special feature that the optimal controlstructure contains as many free parameters as there areinterior or boundary conditions. Two boundary valueproblems (BVPs) are formulated as a result of thenecessary conditions of optimality. The first BVPincludes only the state variables, whereas, the secondone includes only the costates. This separation reducesthe dimension of the original problem into half, thus,taking less computer time to solve. Then, a numericalalgorithm is developed, that is based on the shootingmethod, to solve the resulting BVPs. While satisfyingall the necessary conditions of optimality, the computedoptimal solution is verified by comparing the controlinput resulting from the solution of the first BVP withthe switching function from the second BVP. Thenumerical technique is modified, next, to solve robusttime-optimal control problem. The capability of theproposed method is demonstrated through numericalexamples, whose output optimal solution is shown to beidentical to those presented in the literature. Theseexamples include linear as well as nonlinear systems.Finally, the numerical technique is utilized to design atime-optimal control for a rest-to-rest maneuver of aflexible structure while el iminating the residualvibrations at the end of the maneuver.

1.0 IntroductionIn recent years there has been a considerable interest in

Assistant Professor, King Saud University, Meeh. Eng. Dept.,College of Eng., P.O. Box 800, Riyadh 11421, Saudi Arabia.Copyright © 2000 The American Institute of Aeronautics andAstronautics Inc. All rights reserved.

modeling and control of flexible structures. This is dueto the use of lightweight materials for the purposes ofbuilding systems with high speed and fuel efficiency.Furthermore, many applications, such as roboticmanipulators, disk drive heads and pointing systems inspace, are required to maneuver as quickly as possiblewithout having significant structural vibrations duringand/or after a maneuver.The time-optimal control for general maneuvers andgeneral flexible structures has posed a challengingproblem and is still an open area for research. Inparticular, the time-optimal control for rest-to-restslewing maneuvers of flexible structures has been anactive area of research, and only limited solutions havebeen reported in the literature. Solution to the time-optimal control problem of a general flexible system isfaced with two main obstacles. First, the number ofcontrol switching times is unknown a priori and mustbe guessed. Second, as the number of modes includedin the model is increased, the computer time requiredby these numerical techniques becomes prohibitive.In the recent literature, many researchers1"7 havedeveloped computational techniques that deal withsolving time-optimal control of flexible structures. Inall of these published works, the exact time-optimalcontrol input, which is of the bang-bnag type, iscalculated based on transforming the time-optimalcontrol problem into parameter optimization.This paper presents a new numerical technique to solvea general time-optimal control problem. The basic ideaof the approach is to transform the necessary conditionsof optimality into two boundary value problems(BVPs), which involve the states and costatesseparately. Consequently, a large saving in computertime is gained with the proposed approach. The methoddepends on the special feature that the optimal controlstructure contains as many free parameters as there areinterior or boundary conditions for the state variables8.

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The paper begins with the presentation of theformulation of the problem through a benchmarkexample. This is followed by introducing the necessaryconditions of optimality to set up the BVP's and thenumerical algorithm with which these BVP's aresolved. Then, the numerical technique is modified withthe purpose of computing a robust time-optimal controlinput. This is followed by applying the method to anonlinear system proving the method to be applicable tolinear as well as nonlinear systems. The design of atime-optimal control input for a flexible structure withone rigid body mode and ten flexible modes ispresented in the next section. This is followed by someconcluding remarks.

2.0 Problem FormulationTo introduce the proposed control design technique, weconsider the control of a two-mass-spring-dampersystem, shown in Fig. (1), which has one flexible modeand one rigid body mode. The equations of motion are

dH = \.A

m2x2 + c(x2 —xl -Jt) = 0where JC] and x2 represent the displacement of the firstand second masses respectively. The system in Eqs. (1)is to be transferred from the initial conditions

x,(0)=;c2(0)=0 , dx1(0)/dt=dx2(0)/dt=0 (2a)to the final conditions

*i(tf) = *2(tf)=l, d*i(ta/dt = dx2(tf)/dt = 0 (2b)and subject to the control magnitude constraints

-l<u(t)<l Vte[0,tf] (3)in minimum time. This system can be writtencompactly in state space form as

x = Ax + bu (4)where x = [vector, and

A=

00

klm2

X\ X2

00

-klm2

X3=d*

1

0

clm2

I/At X4

01

-c/m2

b=

00

\Jml

0

(5)

are the system matrix and input vector, respectively.

2.1 The Necessary Conditions of OptimalityFollowing Jacobson et al. , the necessary conditions ofoptimality can be derived by defining, first, theHamiltonian H as

where A^ i=1..4 are the costates (Lagrange multipliers).Then, the necessary conditions of optimality require thefollowing equations to be satisfied

1. The state equations_ dH

1 **\ i "" *• *

(7)

3. The boundary conditions, Eqns. (2).4. The transversality condition for the free end

time tf yieldsH(t) = H(X(t),A(t),u(t)) = 0, 0 < t < t f (8)

5. The time-optimal control u(t) minimizing theHamiltonian H is

"max =1

(9)-u =-\

2. The costate equations

where $t) = dH/du = A,3(t) is the switching function forthe control input u(t). It can be shown that the system inEqs. (4) is normal9'10, a property that eliminates theexistence of singular intervals and forces the necessaryconditions to also be sufficient. Therefore, the control isof the bang-bang type, as given by Eq. (9), which canbe parameterized by its switch times. Manyresearchers1"5 have noted that the time-optimal bang-bang solution for a spring-mass-damper dynamicalsystem with n degrees of freedom and with the type ofmaneuver described by Eqs. (2) (i.e., a rest-to-restmaneuver) has, in most cases, 2»-l switches.Consequently, for two degrees of freedom system, andassuming the control structure shown in Fig. (2), theabove necessary conditions of optimality can lead to setup the BVP of order twelve with

(i) Twelve variables x\, x2, jc3, jc4, KI, A,2, ^3,-̂4> tsi, ts2, tj3 tf.

(ii) Eight boundary conditions, Eqs. (2), onetransversality condition, Eq. (8) applied ateither the initial or final time, and thethree switching conditions A,3(t) = 0 at t =tsl, ta and t^.

Many researchers have developed numerical techniquesto solve the above BVP. Lastman11 and Li12 have usedthe shooting method to arrive at the tune-optimalsolution. Other researchers1"6 have transformed thetime-optimal control problem into constrainedparameter optimization problem. The basic idea for thistransformation is to assume a number for the controlswitch tunes, integrate the system equations, Eqs. (4),and imposing the boundary conditions, Eqs. (2). Theresulting nonlinear equations, which are functions ofthe control switch and final times, are used asconstraints in the parameter optimization problem.Therefore, the resulting constrained parameteroptimization problem, for the system in Eqs. (4), is todetermine the four parameters, ts!, t^, tg and tfi thatminimize the maneuver time t6 and subject to the fournonlinear equations resulting from applying the controlstructure, shown in Fig. (2), to the system in Eqs. (4).Since the formulation of the parameter optimization

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problem does not depend on the necessary conditions ofoptimality, the resulting solution needs to be verifiedfor optimality. This is accomplished by checking if theresulting solution satisfies the necessary conditions ofoptimality. Ben-Asher2 has proposed a numericaltechnique to verify the optimality of a solution resultingfrom a parameter optimization problem.The numerical technique introduced in this paper is toformulate a BVP involving only the state variables byusing the fact that optimal control problems contains asmany free parameters as there are interior or boundaryconditions. A second BVP which involves only thecostates is also formulated and solved to verify theoptimality of the state space solution, resulting fromsolving the first BVP. Consequently, the numericalcomplexity of the original time-optimal control problemis reduced considerably with this approach. Anumerical algorithm is outlined to solve both BVPs byusing the shooting method.The formulation of the BVP depends, first, onknowledge of the optimal control structure. For theabove example, if the control structure, shown in Fig.(2), is believed to be time-optimal then, the resultingBVP has the eight variables, x\, x2, x3, x$, tsj, t^, t^ andtf. The boundary conditions necessary to solve this BVPare the eight boundary conditions stated in Eqs. (2). Thesecond BVP contains the four costate variables, Xj, A,2,A/3, and 7^ as unknowns with the three interior switchingconditions , A,3(t) = 0 at t = t^, t^ and t^, and onetransversality condition, Eq. (8), applied at either t = 0or t = tf.

2.2 Numerical AlgorithmIn this paper, a simple numerical technique is proposedto solve the BVP formulated in Section (2.1). Themethod is based on the simple shooting method, and isapplicable to any general BVP. The numericalprocedure is illustrated in the following steps :Step 1. Assume starting values for the unknowns. Inthe example of Section (2.0), these are the controlswitch and final times, tsl, ts2, ts3, and tf.Step 2. Integrate the state equations, Eqs. (4), from t = 0to t = t f, using the initial conditions in Eqs. (2a) , thecontrol structure shown in Fig. (2) and the assumedvalues for the control switch and final times.Step 3. Form the vector of errors,

Step 4. Calculate the Jacobian matrix,

f = (10)

It is obvious, when all the components of the vector/equals to zeros, then all the boundary conditions aresatisfied and we hope that we arrive at the optimalsolution.

Sf

dt

5/2 dtf$3 $3 $3

5/4 5/4 S/, 5/4

dt

(ii)

'f _

where /) andg^ i = 1..4, are the H/z component ofthe vectors/and g. The vector g is defined by,-['„ .„ I, >,rThe above Jacobian matrix can be calculated eitheranalytically or numerically using, for example, finitedifference. Analytical derivatives can be calculated bysolving, symboliclly, the state equations, Eqs. (4), usingthe control structure, shown in Fig. (2), anddifferentiating the resulting equations with respect tothe control switch and final times.Step 5. Calculate the step direction using

= hr- >< / (12)

where Del is a 4 x 1 vector representing the stepdirection with which the control switch and final timesshould change in order for the components of the vector/ to reach zeros.Step 6. Update the unknown variables using

gt^gt-pxDeli i=l . .5 (13)where (.)j is the i-th component of (.), and p is aconstant representing the step size. As a general guide,the value of the constant p should be between zeroand one to avoid divergence.Step 7. Repeat Steps (1) through (6) until

norm(f) < tolerancewhere norm refers to the Euclidean norm of the vectorf, and tolerance is a specified small number affectingthe accuracy of the resulting solution.

2.3 Calculation of the CostatesOnce the optimal solution for the state variables and thecontrol switch times are calculated, the correspondingtime-optimal solution for the costates is computed next.A BVP involving the costates can be formulated withthe help of the necessary conditions of optimality. Itinvolves solving for the four unknownsIi(0),/l2(0),/l3(0),andl4(0) given the three interiorconditions,

= 0 (14a)

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and the transversality condition , Eq. (8), applied at thefinal time t&

-̂ - (14b)

where u^ is the maximum value for the control input,taken here to be one.The same numerical procedure outlined in Section (2.2)can be used to solve for the costates by replacing thevectors g and / with the corresponding vectors ofunknowns and errors, respectively.For nij = m2 = k = 1, and c = 0, the above numericalalgorithm is programmed in Matlab15, where thederivatives in Eqs. (11) is calculated using the finitedifference equations

The resulting time-optimal control switch times, inseconds, are presented in Table 1 .

Table 1 Computed and exact control switch times.

NumericalExact Values1

td1.00261.0026

ts2

2.10862.1089

ts3

3.21483.2152

tf

4.21754.2178

The corresponding numerical values for the parametersused and the number of iterations with which thealgorithm is able to reach the time-optimal solution areshown in Table 2.

Table 2 Parameter values used in the algorithm andnumber of iterations.

Tolerance

PNo. of iterationsStarting Values

lxl(T3

0.510

ts,=l ts2=2 tfi=3 tp4

Table 1 shows that the resulting tune-optimal controlprofile is very close to those listed in the literature. Thenumerical simulation for the states, and control inputu(t) and switching function X3(t) are shown in Figs. (3)and (4), respectively.

3.0 Robust Time-Optimal ControlIt is well known13, that the resulting time-optimalsolution, calculated in Section (2.0), is very sensitive toplant modeling uncertainty. If the resulting time-optimal control, from the preceding example, is appliedto the system of Eqs. (4) while changing the value ofthe stiffness from k = 1 to k = 1.1, 1.2 and 1.3, then theresulting system trajectories is shown in Fig. (5), whereit is seen that small changes in the value of k(10%~30%) result in a large increase in the vibrational

amplitude of the flexible mode. Therefore, there is aneed to design a robust time-optimal control input thatreduces the sensitivity of the optimal solution tomodeling errors. In particular, we would like tominimize the sensitivity of the final states of a dynamicsystem, to errors in the estimated parameters of thesystem. Following Liu et al.13, we consider a generalsystem represented by the state equation

x = f(x,p,u) (16),where x is the state vector of the system, p a parametervector and u the control input. To determine thesensitivity of the final states of the system to a changein the parameter vector p, we could use the finitedifference approximation for the derivatives as

c dx(p) x(pi+Api)-x(pi)•>, = —;— = ————-————— U',)

dpt AP,-where the subscript / denotes the fh parameter ofinterest. Evaluating Spi at the final time, provides uswith an estimate of the sensitivity of the final states to achange in the parameter pt. Alternatively, and moreaccurately13, the state sensitivities could also beevaluated by the direct differentiation of the stateequation as

=^+y.^^ 08)dp,J -

Integrating Eqs. (16) along with Eq. (18) leads to thesensitivity of the states to the parameter pt.Now, let us consider designing a robust time-optimalcontrol input for the system in Eqs. (1). Applying Eq.(18) to the system defined by Eqs. (4) for thesensitivities of x\ and x2 to the parameter k leads to

- x - - X(19)

*\>-*\*2ii ~*\s>~^*2s ~A\s>

where *is = dx\ldk and x2s = dx2/dk are the sensitivitiesof Xi and x2 to a change in the parameter k. From Eqs.(19), for mi = m2 = m , then

xls = ~x2s => xls ~ ~x2s => xls ~ ~x2s (20)

and, therefore, one of the sensitivities can be eliminatedto give

mxls = -(*! -x2)-2kxla-2cxla (21)Then, the augmented state equations are the systemequations, Eqs. (4), and the state space form of Eq.(21). The augmented boundary conditions are theoriginal boundary conditions, Eqs. (2), and

It is clear that the boundary conditions in Eq. (22) forcethe sensitivities of the states jci and x2 at the final timeto zero.Now the objective is to design a control input u(t) thatminimizes the final time t6 subject to the stateequations, Eqs. (1) and (21), the control constraints, Eq.(3), and the boundary conditions, Eqs. (2) and (22).

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Since we have added the robustness constraint,represented by Eq. (21), in the formulation of the BVP,the number of control switches must be increased tomatch the number of constraint equations. Li Refs. (1)and (3), it has been shown that adding the robustnessconstraint, Eq. (21), increases the number of time-optimal control switches by two.Therefore, the resulting BVP of order twelve has

(i) Twelve variables x\, x2, x3, x4, XH, dxis/dt,tsi, ts2, ts, t-4, t^, and tf

(ii) Eight boundary conditions, Eqs. (2) andthe four boundary conditions, Eqs. (22).

It is noted that the number of control switch times hasincreased by two1'3, emphasizing the feet that thenumber of variables should match the number ofboundary and interior conditions.The numerical algorithm, outlined in Section (2.2), isapplied to solve for the robust time-optimal controlproblem, with , m1=m2=k=l and c=0. The resultingcontrol switch times, in seconds, are presented in Table3. In addition, the numerical simulation for thedisplacement x\ is shown in Fig. (6) for the nominalvalue of k=l and k=l.l, 1.2 and 1.3. Comparing Figs.(5) and (6), it can be seen that the application of therobust time-optimal control to the system in Eqs. (1)has reduced the sensitivity of the state x\ to changes inthe stiflhess value k and, consequently, reduced theresidual vibration significantly.

Table 3 Robust time-optimal control switch times.tsl

0.7124ts2

1.6563ts3

2.9330ts4

4.2096tsS

5.1536tf

5.8660

4.0 Time-Optimal Control for NonlinearSystems

This section demonstrates the capability of theproposed numerical algorithm to solve for the time-optimal control of nonlinear systems. We consider thetwo-mass-spring-damper system, with nonlinear springelement representing hard spring characteristics13. Theequations of motion are

-x2)3 = "(23)

x2 +ki(x2 - -k2(x2 - = 0

For a rest-to-rest maneuver, we assume the sameboundary conditions as those given by Eqs. (2).It is desired to perform a minimum time maneuver forthe system of Eqs. (23), subject to the controlconstraints, Eq. (3), and satisfying the boundaryconditions, Eqs. (2). The necessary conditions ofoptimality can be derived easily following Section(2.1).

Assuming the number of time-optimal control switchesto be three, the resulting BVP has

(i) The eight variable, jci, x2, x3, x4, tsl, t^, t^,andtf

(ii) The eight boundary conditions, Eqs. (2).

Similarly, the BVP for the costates contains(i) The four variables, A.J, A.2, A,3, A,4(ii) The three switching conditions

X3(t-) = 0 /=1..3 (24)and the transversality condition applied atthe final time

A9(td = m, (25)The expressions for the costate equations can bederived using Eq. (7).The numerical algorithm, outlined in Section (2.2) isapplied to this example, with kl=k2=ml=m2=l andc=0, to calculate the time-optimal control input. Thecontrol switch times are shown in Table 4. The systemresponse to the time-optimal control input is shown inFig. (7), whereas, the control input and switchingfunction A.3(t) are shown in Fig. (8). Similarly, it isstraightforward to calculate the robust time-optimalcontrol input with respect to changes in either k] or k2using similar procedure as for the linear exampletreated in Section (3.0).

Table 4 Computed control switch times.tsl

1.0013ts2

2.0755ts3

3.1498tf

4.1512

5.0 Numerical ExampleAs an example to illustrate the numerical procedure andshow its effectiveness, we consider the time-optimalsingle-axis maneuver problem for a system consistingof a rigid hub with two uniform elastic appendagesattached to it. A single actuator that exerts an externaltorque on the rigid hub controls the motion of thesystem. This system is schematically shown in Fig. (9).It is desired to rotate the flexible system by 90 degreesas quickly as possible while suppressing the vibration atthe end of the maneuver to achieve good pointingaccuracy.For comparison purposes, we use the same material andmaneuver specifications that are used in Ref. (4), andare shown in Table 5.

Table 5 System dimensions, appendage material,and maneuver specifications.

Radius of the rigid central hub, R

Length of one appendage, L

Appendage material stiffness, El

LOOM

40.00 m

1 500.00 N.m2

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Appendage material density, p

Mass of the rigid central hub

Maximum torque available, u

Command slew angle, xjf

Total rotational inertia, J

0.04096 Kg/m

400.00 Kg

1 50.00 N.m

90.00 deg

2081. 97547 Kg.m2

The Euler-Bernouli beam assumption is made to obtainthe equations of motion for the system. It is assumedfurther that the beam is inextensible, in the sense thatthe stretch of the neutral axis is negligible, and nostructural damping is present. The assumed modemethod based on modal expansion is applied to the twoflexible appendages, where the rigid body mode and thefirst 10 vibrational modes are retained in the evaluationmodel. The normalized mode shapes for a fixed-freecantilever beam are used as the assumed mode shapes.A detailed derivation of the equations of motion for thissystem can be found in Ref. (4). The discretizedequations of motion, in matrix form, are

Mx + Kx = bu (26)where x, and b are 11 x 1 vectors and M and K are 11 x11 matrices. Using the matrix of eignevectors,normalized with respect to the mass matrix, the systemin Eq. (26) is transformed to

q + Aq = J3u (27)using the transformation

x = <bq (28)where A is an 11 x 11 diagonal matrix with thediagonals being o>i2 , i = 0 .. 10, and <D is 11 x 11matrix of eigenvectors. Table 6 shows the naturalfrequencies Oi and the modal participation vectorcomponents p\ ( f} = ®Tb).

Table 6 Modal quantitiesi<BjPi

00

0.0219

11.15070.0543

23.0098-0.0353

37.5342-0.0143

414.5654-0.0077

524.0232-0.0049

i<Bi

ft

635.8722-0.0034

750.2238-0.0026

866.8855-0.0020

986.6068-0.0015

10108.1406-0.0013

Using the numerical technique outlined hi Section (2.2),we first design a time-optimal control input for the rigidbody mode. Then the number of modes in the time-optimal control calculations is increased until theflexible system is rotated with the smallest error whileminimizing the structural vibrations at the end of themaneuver. The time-optimal control switches and finaltimes for the rigid body mode, rigid body mode and oneflexible mode, and rigid body mode and two flexible

modes are presented in Table 7. The rigid hub attituderesponse for the three time-optimal control inputs,designated by Ui, u2 and u3, are shown in Fig. (10). Asseen from the figure, the control input designed for therigid body mode and two flexible modes is able toperform the maneuver in minimum time, whileeliminating the residual energy at the end of themaneuver.

Table 7 Time-optimal control switch and finaltimes.

Ul

U2

U3

4.6710

4.21693.9689

9.3424.72874.1833

5.23844.7456

9.45325.3190 5.5190 9.4600

6.0 Summary and ConclusionsThis paper outlined a new numerical algorithm tocalculate both time- and robust time-optimal controlproblems for linear as well as nonlinear systems. Thealgorithm relies on formulating and solving a BVP,derived from the necessary conditions of optimality,that contains the state variables and costates, separately,thus, resulting in smaller dimensional problem. Anumerical procedure is outlined to solve, based on theshooting method, the resulting BVP. The algorithm isfirst tested by solving a benchmark problem (mass-spring-damper system) and the resulting time- androbust time-optimal controls are shown to be very closeto those presented in the literature. Then, the numericaltechnique is used to solve for the time-optimal controldesign for a nonlinear system. Finally, the algorithm isutilized to design a minimum time control input for aflexible structure while reducing the residual vibrationsat the end of the maneuver.

References'Singh, T., and Vadali, S., "Robust Time-OptimalControl : Frequency Domain Approach," Journal ofGuidance, Control, and Dynamics, Vol. 17, No. 2,1994, pp. 346-353.

2Ben-Asher, J., Burns, J., and Cliff, E.," Time-OptimalSlewing of Flexible Spacecraft," Journal of Guidance,Control, and Dynamics, Vol. 15, No. 2, 1992, pp. 360-367.

3Liu, Q., and Wie, B., "Robust Tune-Optimal Controlof Uncertain Flexible Spacecraft," Journal ofGuidance, Control, and Dynamics, Vol. 15, No. 3,1992, pp. 597-604.

4Singh, G., Kabamba, P., and McClamroch, N., "Planar,Time-Optimal, Rest-to-Rest Slewing Maneuvers of

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Flexible Spacecraft," Journal of Guidance, Control,and Dynamics, Vol. 12, No. 1, 1989, pp. 71-81.

5Pao, L., "Minimum-Time Control Characteristics ofFlexible Structures," Journal of Guidance, Control,and Dynamics, Vol. 19, No. 1, 1996, pp. 123-129.

6Meier, E., and Bryson, A., "An Efficient Algorithm forTime-Optimal Control of a Two-Link Manipulator,"Journal of Guidance, Control, and Dynamics, Vol.13, No. 5, 1990, pp. 859-866.

7Scrivener, S., and Thompson, R., "Survey of Time-Optimal Attitude Maneuvers," Journal of Guidance,Control, and Dynamics, Vol. 17, No. 2, 1994, pp. 225-233.

8Maurer, H., and Wiegand, M., "Numerical Solution ofa Drug Displacement Problem with Bounded StateVariables," Optimal Control Applications andMethods, Vol. 13, 1992, pp. 43-55.

9Kirk, D., Optimal Control Theory, Prentice Hall, NewJersey, 1970.

10Ryan, E., Optimal Relay and Saturating ControlSystem Synthesis, Peter Peregrinus, London, 1982.

uLastman, G. J., "Shooting Method for Solving Two-Point Boundary Value Problems, Arising from Non-Singular Bang-Bang Optimal Control Problems,"International Journal of Control, Vol. 27, No. 4,1978, pp. 513-524.

12Li, F., and Bainum, P. M., "An Improved ShootingMethod for Solving Minimum-Time ManeuverProblems," American Society of MechanicalEngineering, Winter Annual Meeting, Dallas, TX,Nov. 26-30, 1990.

13Liu, S., and Singh, T., "Robust Time-Optimal Controlof Flexible Spacecraft with Structured Uncertainties,"AIAA Guidance, Navigation, and ControlConference, San Diego, CA, July 29-31, 1996.14Jacobson, D., Lele, M., and Speyer, J., "NewNecessary Conditions of Optimality for ControlProblems with State-Variables Inequality Constraints,"Journal of Mathematical Analysis and Applications,Vol.35, 1971, pp. 255-284.

15Matlab User's guide, 1999, The MathWorks Inc.,South Natick, MA.

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Xj X2

^ mi \-S\Sv\__W2

OT) (TOFigure (1) Mass-spring-damper system.

u(t)

tsl ts2 ts3

Figure (2) Time-optimal control structure.

0.5 1 1.5 2 2.5 3Time (s)

3.5 4 4.5 5

Figure (3) Response of 'a spring-mass-damper system to a time-optimal control input

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-1.50 0.5 1 1.5

Figure (4) Time-optimal control input uft) and switching function

4 5 6Time (s)

8 9 10

Figure (5) Response to time-optimal control input for k = l, LI, 1.2 and 1.3.

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1.4

1.2 -

1

0.8

0.6

0.4

0.2

k=1.2

k=1.1

0 1 2 3 4 5 6 7 8 9 1 0Time (s)

Figure (6) Response to robust time-optimal control input

0.5 1 1.5 2 2.5 3Time (s)

3.5 4.5

Figure (7) Nonlinear system response to time-optimal control

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2 -1.5^ 0 0.5 1 1.5oO

2 2.5 3Time (s)

Figure (8) Time-optimal control input and switching function for the nonlinear system.

Figure (9) Flexible structure.

10 15 20Time (s)

25 30

Figure (10) Rigid hub attitude resulting from the application of the three time-optimal control inputs.

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