AIM-92-21 13-CP

MINIMUM TIME MANEUVER OF STRUCTURES USING PULSE RESPONSE BASED CONTROL

Jeffrey K. Bennighor and Murugan ~ u b r a m a n i a m t

The University of Texas at Aus t in Austin, Texas 78712

A b s t r a c t

The Pulse Response Based Control (PRBC) method is applied t o minimum time maneuver of flex- ible structures in this paper. When PRBC is applied to linear systems, an explicit model of the system is not needed, because the behavior of the system is rep- resented without modal truncation in terms of the measured response to pulses in control inputs. How- ever, for maneuver problems, nonlinear behavior asso- ciated with rapid large-angle rotation can be signifi- cant. These nonlinear effects are taken into account by explicitly modeling the response in a few of the lowest flexible modes, and using the results t o cor- rect the pulse response measurements upon which the minimum time control profile is based. A numerical example demonstrates that even for extremely rapid maneuver, nonlinear effects only need t o be considered for a very small number of flexible modes, so that re- quirements for identification and computation are very low.

I n t r o d u c t i o n

In the minimum time maneuver of flexible space- craft, a set of bounded control inputs are used to ac- complish a given change in rigid body modal states as quickly as possible, with minimal residual energy in flexible modes a t the end of the maneuver. Over the past decade or so, a number of different approaches have been taken to address problems in which nonlin- ear behavior is significant.

Turner and ~ u n k i n s ' represent flexible motion in terms of several assumed modes, and minimize a quadratic performance index with a single-actuator control, with specified final states and final time. A continuation method is used t o obtain a solution of the nonlinear two-point boundary value problem that results when kinematic nonlinearity is present in the formulation. Turner and Chun extend this approach for the case in which a number of actuators are dis- tributed throughout the structure.' Chun, Turner and Juang obtain a frequency-shaped open-loop control for the rigid body modes, using a continuation method to handle nonlinearity, and then design a feedback con- trol for the flexible motion by linearizing the flexible

*Assistant Professor, Aerospace Engineering & Engineering Mechanics. Member AIAA.

tGraduate Research Assistant.

response about several points in the open-loop rigid- body trajectory.3 In a later paper, they replace the solution of the open-loop problem for the rigid body modes with a programmed-motion/inverse dynamics approach, where the trajectories of the rigid body modes are simply specified as smooth functions and the required control torques are obt,ained from the non- linear rigid body equations of m ~ t i o n . ~ Meirovitch and Quinn5 develop a perturbation approach in which the rigid body maneuver is taken as a zero order problem and the flexible motion is treated as a first order per- turbation, for nonlinear flexible maneuver problems. Meirovitch and Sharony6 then use the perturbation approach and use a minimum time (bang-bang) con- trol to handle the rigid body problem, and a linear quadratic regulator with integral feedback and pre- scribed convergence rate to drive the flexible motion to zero a t the end of the maneuver. Bounds on con- trol inputs are only considered in the design of the rigid body portion of the control, so, a.s in all of the work mentioned thus far, a true minimum time control problem for the flexible maneuver is not solved. True minimum time control is investigated by Ben-Asher, Burns and CliffI7 in which the flexible behavior is mod- eled in terms of a small number of assumed modes, and switching times for the bang-bang control of this model are found for both linear and nonlinear problems us- ing a parameter optimization approach. More recently, Junkins, Rahman and Bang have pursued an approach in which near minimum time maneuver is achieved by smoothing a bang-bang control profile, and designing a stable feedback control t o suppress the flexible motion that is excited by this profile."

The difficulty of these problems is due t o several issues which are all related to modeling the flexible behavior of the spacecraft. First, there is the difficulty of obtaining an accurate model, i.e., system identifi- cation and realization. Second, since the systems t o be controlled are distributed parameter systems, they typically require high order models for representing all significant behavior, and this greatly increases compu- tational requirements associated with control, particu- larly if a nonlinear two point boundary problem must be solved. Third, a finite order model is guaranteed to have a bang-bang minimum time control, for which only the switching times must be determined, whereas the minimum time control for the distributed parame- ter system being rnodeled is not ordinarily bang-bangIg

so the optimal control profile is qualitatively quite dif- ferent from the bang-bang minimum time control of a finite order model of the system.

Recently, a new approach t o minimum time con- trol of flexible spacecraft known as Pulse Response Based Control (PRBC) has been developed for linear problems.10 For linear problems, no explicit modeling of the system is required. Pulses in control inputs are applied to the system, and sensor measurements of the response t o these pulses are recorded. Then sen- sor measurements of response to a train of pulses in control inputs are available from a convolution sum. A true minimum time problem is solved by finding a train of pulses that results in measurements consis- tent with the desired final state of the system, with a minimal number of time steps. The two-point bound- ary value problem encountered in a standard optimal control approach is replaced with a numerical opti- mization problem, for which an efficient algorithm has been developed." Pulse response measurements can be obtained much more quickly and easily than the ex- plicit model data required for conventional approaches. There is no modal truncation because all modes are represented in the pulse response measurements. The precision with which the final state of the linear system can be specified is limited only by the observability of the system with the given outputs. The control history obtained is not bang-bang, and the only known exact solution of a minimum time control problem for a flexi- ble structure, which happens to be piecewise constant, is easily obtained using PRBC.

In this paper the Pulse Response Based Control method is modified so that the nonlinear behavior as- sociated with large-angle rotation in maneuver prob- lems can be handled. This requires explicit modeling of those flexible rnodes in which nonlinear behavior is significant, which is ordinarily a very srnall num- ber of modes. There is still no modal truncation or explicit modeling required for linear behavior, so the method has substantial advantages over previous methods. The amount of computation required for nonlinear problems is close to the amount required for linear problems. In the second section of this paper, the PRBC method is briefly presented. The extension of the method to nonlinear problems is discussed in the third section. In the fourth section, a numerical example is presented. The final section contains con- clusions.

P u l s e R e s p o n s e Based C o n t r o l

Suppose a linear dynamic system is controlled by a single input u(t) , with no other excitations, and its response is measured in terms of m outputs that con- stitute a vector y(t). If the system is initially a t rest and a square pulse in u(t) , of unit amplitude and dura- tlon A t , is applied, and the outputs y(t) are sampled

every A t for a total time n A t , an m x n pulse response matrix H is obtained, of the form

If the system is initially at rest and a control profile u(t) that is constant over each time step At between t = 0 and t = nAt is applied to the system, so that the control input is equal to ui over the i th time step, the output vector a t nAt can be found by superposi- tion, in terms of a convolution sum. Each value ui is multiplied by a vector in the pulse response matrix H given above, so that the output vector takes the form

where u = [ U I u2 . . . u, I T . If there are p inputs, the vectors h, must be re-

placed by m x p submatrices H, , and the entries u; in the vector u are replaced by the vectors u , = [ul , , . . . up,, I T . The pulse response matrix H must then be filled by applying pulses in the control inputs one a t a time, and sampling the outputs resulting from each pulse every At for 0 < t 5 nAt . The outputs re- sulting from a pulse in the first control input will fill the first vector of each submatrix H, in H, and each of the other p - 1 columns in a submatrix Hz will be obtained from response to a pulse in another control input.

If a control task that consists of driving the system from rest a t t = 0 t o a desired state a t a time tf = nAt is successfully accomplished by a piecewise constant control profile of the type described above, the set of linear equations

~ ( t f ) = H u (3)

must be satisfied by the control profile u , where the vector y ( t f ) contains outputs consistent with the de- sired final state of the system. Also, if there is no ex- citation of the system after t f , the outputs a t a time tf + j A t must be given by the equations

where the submatrix HI-^ contains outputs measured n + j time steps after applying test pulses in control inputs, for example, and where the outputs y(tf + j A t ) must be consistent with the desired state of the system at t f , with free response after t f .

These observations suggest the following approach to a minimum time control problem for the system when the control inputs are bounded: Find the small- est integer n such that there exists a control history vector u -- [uT . . . uZlT satisfying the equations

and the input bounds, typically of the form

B: 5 ujIi 5 By, j = I , . . , i = l , . . , , (6)

in which Bj and BjU are lower and upper bounds for the j th control input. The vector gj contains outputs a t tJ and at 1 time steps afterward:

and the matrix TI is given by

Satisfaction of Eq. (5) is a necessary but not sufficient condition for accomplishing the control task exactly. It has been shown that the precision with which the final state of the system can be specified is limited only by the observability of the system with the given out- puts. The effect of measurement noise on the accuracy of the outputs at the end of the control task has been investigated.14 With the assumptions that error due to noise in the pulse response data is zero-mean and uncorrelated in time, and that measurement hardware is selected appropriately for pulse response signal lev- els, so that mean square error can be expected t o be proportional to the mean square value of the signal, it is found that the expected values of the predicted out- puts are equal to the correct values, and the covariance matrix for the output vector a t the end of the control history is inversely proportional to the number of time steps in the control history. This reflects the averaging of error that takes place in the convolution sum. The error in pulse response data can be reduced further by repeatedly generating pulse response measurements and averaging the results, in which case the covariance rnatrix for a column of H is inversely proportional to the number of samples averaged.

Ordinarily the inertial properties of spacecraft are known very accurately, in comparison with modal properties, in which case it is straightforward to de- termine response in rigid body modes to pulses in the control inputs so that rigid body modal states a t tf can be included in the vector of specified outputs ?If.

An approximate solution of a minimum time con- trol problem is found using PRBC by determining the minimum number of time steps for which a control history exists that satisfies Eq. (5) and inequality con- straints corresponding t o input bounds. As the time step size is reduced, the accuracy of the approximation improves, but the number of unknowns and the com- putational burden increase. An efficient algorithm has

been developed which synthesizes I1 for wide pulses from narrow pulse data and uses inexpensive solutions with large At as initial guesses for small At, thereby reducing computation significantly."

PRBC for Nonlinear Systems

The PRBC approach described in the previous sec- tion is appropriate for distributed parameter systems whose behavior is linear. However, for flexible struc- tures undergoing rapid large-angle rotations, the non- linear behavior arising from rotational motion will be significant, so i t must be taken into account. The pulse response measurements essentially represent a linearization about the state of the system when the pulses are applied, so nonlinear behavior will result in a deviation from this linearization. The basic ap- proach for dealing with the nonlinearity is t o use the pulse response measurements to obtain an initial ap- proximation of the minimum time control profile and the minimum time trajectory for the system, and then to correct the pulse response measurements so that they represent a linearization about the trajectory, a t least in those modes that exhibit significant nonlinear behavior. Then the minimum time problem can be solved again with the corrected pulse response mea- surements. If the new minimum time trajectory differs significantly from the one obtained based on the orig- inal pulse response measurements, the measurements can be corrected again so a more accurate control pro- file can be found. In this section, the procedure for correcting the pulse response measurements so that they represent a linearization about a given trajectory is presented.

For a deformable body which can undergo large rigid body displacements and small elastic deforma- tions, the instantaneous position vector of any point on the body is given by

where Ro locates the origin of a reference frame and is expressed in terms of an inertial frame, + is the position vector from this origin to the point in the undeformed position, and d is the elastic displacement. If the vec- tors r and d are expressed in terms of a frame that rotates with angular velocity w, the kinetic energy of the system is given by

[W X ( r + d) ] . [w x ( r + d ) ] dm -+

where M is the total mass of the structure. It is ad- vantageous to use a floating reference frame for the flexible structure t o simplify the equations of motion. Using the Tisserand frame," which is subject to the constraints

the expression for the kinetic energy becomes, after some manipulation,

The origin of the reference frame is now located a t the instantaneous center of mass of the structure, which has R, as its position vector.

In this paper, PRBC is applied t o an example prob- lem in which a uniform Bernoulli-Euler beam is to be rotated through a large angle in minimum time. For this bearn, the transverse elastic displacement is rep- resented as ~ ( x , t ) , and the vectors T , w and d are written in terms of unit vectors i , j and h as T = x i , w = 0b and

~ ' ( < , t ) ~ d < i , (13)

where only up to quadratic terms in v ( x , t ) and its derivatives are retained. The kinetic and potential en- ergies for the system become

and 1 L/2 v = - LLI2 EIvl ' dx, (15)

where m is the mass per unit length, EI is the flexural rigidity, and L is the length of the beam. The beam is controlled by transverse force inputs a t its ends, so the non-conservative virtual work, linearized in v(x, t ) and its derivatives, is

Applying the extended IIamilton's principle results in ordinary differential equations for translational and ro- tational motion and a partial differential equation gov- e r n i ~ g the flexible response. For a rotational maneuver

the translational equations of motion can be ignored and the control input,s are related by

Due t o the choice of the Tisserand frame the equation governing the rotation angle 6 is uncoupled from the flexible motion and is given by

mL3 .. 12

-0 = Lu.

Once the minimum time control problem associated with a given H matrix has been solved for a control profile u(t) , this equation can be integrated to obtain & t ) , which appears in the equation governing the flex- ible response. The angular velocity O(t) is then used to correct the pulse response measurements so that they represent a linearization about the controlled trajec- tory, so a more accurate solution of the minimum time maneuver problem can be found. The flexible response is governed by the partial differential equation

with boundary conditions

The terms involving i2 in Eq. (19) represent the non- linear effects. The control inputs are subject to the bounds Iu(t)l < B and the control task is to carry out a rotational maneuver in minimum time, with no resid- ual energy a t the end of the task. Hence the initial and final conditions become

and

The modes of a free-free uniform beam can be used to discretize the partial differential equation govern- ing the flexible motion. The symmetric modes need not be considered due t o the anti-symmetric nature of the problem. The mass-normalized anti-symmetric flexible modes are given by

where the &'s are roots of the characteristic equa- tion cos,O,.L cosh,B,L = l associated with the anti- symmetric modes. The displacement v(x, 2 ) can now

be written as to @(( i - 1)At) over the i th time step. In fact, i t is convenient to parametrize them in terms of 0 so that

Note that the choice of the Tisserand reference frame makes the flexible displacement v (x , t ) orthogonal to rigid body motion due to Eqs. (11).

For nonlinear systems the matrix H contains mea- surement,~ of the system response t o pulse inputs and represents the response linearized about the state of the system when the pulses are applied. In order for H to represent response linearized about a maneuver trajectory, it must be modified to account for nonlin- ear effects. This modification can be approximated by considering the nonlinear response of a finite number of modes. Modal equations of motion are obtained by inserting the representation of Eq. (24) in the partial differential equation of motion and exploiting the or- thogonality properties of the modes. The modal equa- tions of motion are given by

where p is the number of modes used to model the nonlinear effects, and the constants krs are given by

Using the state vector z = iV1 . state equations can be written as

in which

and

If the control input is piecewise constant, the solution of Eq. (27) is given by

x ( iAt ) = @ ( iAt , ( i - 1)At) x ( ( i - 1)At) + r ( iAt , ( i - 1)At) u,

= @ i x i - l + r i u i , (30)

wherc the state transition matrix cP and the input matrix r have the usual definitions. From the state equations, it is evident that @ and r depend on the angular velocity 0, which varies slowly SF that these matrises can be approximated by taking 0 t o be equal

a, il a ( ( ( i - ) A t ) ) , r, il r (@((i - 1 ) ~ t ) ) .

(31) Denoting the contribution t o the output vector y from the first p anti-symmetric flexible modes by yp = C x , the linear relationship between u, and yp(nAt) is ob- tained using

so that

With this in mind, a vector hrL taking nonlinear be- havior into account and representing a linearization about the trajectory in 9 is available from

where hk is from the original pulse response mea- surements, and rL is the input matrix evaluated with the angular velocity equal to that of the system when pulses in control inputs were applied t o generate pulse response measurements. Since

the vector hrkl is given by

where @L is the state transition matrix evaluated with the angular velocity equal t o that of the system when pulses in control inputs were applied. In general, the

N L . vector hn-i is given by

(37) Generalization of these results for the multi-input case, in which the vector hr f , is replaced by the submatrix ~f;'?, , is straightforward.

Once the H matrix is corrected, the minimum time problem can be solved again for a more accurate con- trol profile u(t). Then Eq. (18) can be used to obtain 6(t), and if this angular velocity history differs signifi- cantly from the one obtained previously, the correction process can be repeated.

Numerical Example

In this section, the solution of the problem de- scribed in the preceding section, which is the minimum time maneuver of a slender beam through a large angle by means of transverse force inputs a t the ends of the

beam, is obtained using PRBC. The exact solution of this minimum time control problem is unknown. The angle of rotation is taken to be $0 = 90' in this exam- ple.

The ratio of the input bound to the angle of rota- tion, B/Bo, must be specified to complete the state- ment of the problem. This ratio is chosen based on the time that would be required to carry out the de- sired maneuver on a rigid system with the same in- ertial properties. For the rigid system, the minimum time control is bang-bang, so the final time t fR,,,, can be related to the ratio B/Oo by considering the angle at mid-maneuver, which must be

The rapidity of the maneuver can be related to the flex- ibility of the system by specifying the ratio between tfRICID and the period T of the lowest (symmetric) flexible mode. For this example, the ratio BIBo is cho- sen so that tjR,,,, is equal to T/2, which means that a considerable amount of flexible response will be excited by the control history, and the system will exhibit sig- nificant nonlinear behavior. Also, the minimum time control history will not be a small perturbation of the bang-bang control history for the corresponding rigid system. Note that the minimum time for control of a uniform second-order one-dimensional system with unbounded inputs a t its two ends is equal to one half of the period of the first flexible mode.g

The s ta te transition matrix of the system whose state equations appear in Eq. (27) can be obtained us- ing Kinariwala's procedure.13 If the rotational effects are not very strong, A2 can be regarded as a pertur- bation of A1 and the state transition matrix can be written as

where = eAIAt and cP2 can be obtained using

If the control input is piecewise constant the solution of Eq. (27) can be approximated by

where the matrices r l and r2 are available from

t+At t+At

= @ l ( ~ , t ) B d r , T z = 1 0 2 ( r , t ) B d r

(42) The accuracy of the final state of the system is

measured in terms of the residual energy a t the end of the control task, given by the modal sum

which is normalized against the maximum energy in the corresponding rigid system in its minimum time control, which occurs a t mid-maneuver and is given

by

The vector of specified outputs ?If contains the final anti-symmetric rigid body modal states in all cases, and displacement and velocity a t one end of the beam at various time steps a t the end of the control task, so that displacement and velocity sensors are taken to be located a t one end of the beam. Because only anti- symmetric modes are excited by the anti-symmetric control, only one end of the beam needs to be consid- ered.

The pulse response matrix H of the system is gen- erated about the state e(t) = 0, by means of a modal simulation in which as many as one hundred anti- symmetric modes are included in Eq. (24) so that the effects of modal truncation will be negligible. The lin- ear problem associated with this pulse response matrix is solved using the algorithm described in Ref. 11. The smallest time step used is At = t~,,,,,/80. The min- imum time for this first linearized problem is found t o be t j = 0.54375T with outputs specified for twenty- two time steps after t j . Figure 1 shows how the nor- malized residual energy a t the end of the maneuver for the linearized system and the actual system vary as a function of the number of outputs specified. The bang-bang minimum time control for the correspond- ing rigid system results in residual energy of the lin- earized system equal t o about 57% of the energy in the rigid system at mid-maneuver. As more outputs are specified, this residual energy is quickly driven close to zero, according to a linear model. However, the actual residual energy of the nonlinear system remains much

Numkr of T i m Steps with Outputs Spxified

Figure 1: Residual energy variation before nonlinear correction

trr

Figure 2: Angular velocity of the beam

higher. Figure 2 shows the normalized angular velocity j ( t)

obtained by solving the initial linear problem. As de- scribed in the preceding section, the pulse response measurements are corrected so that they represent a linearization about this trajectory g(t). In this ex- ample, nonlinear behavior in only two modes is con- sidered to make this correction. The minimum time problem associated with the corrected H matrix is then solved. The minimum time of control is found t o be t f = 0 .5250T with outputs specified for seventeen time steps after t J . The minimum time t J is slightly less than the minimum time without accounting for nonlinear behavior.

Figure 3 shows the normalized residual energy a t the end of the maneuver. According to a model lin- earized about the angular velocity profile of Fig. 2, the residual energy is driven to less than 0.005% of the mid-maneuver energy for the rigid system. When non-

Nurnkr of Tim Stcp. with Outputs Spccificd

Figure 4: Control profile for maneuver

linear behavior is modeled, the residual energy of the beam is driven to less than 0.15% of the mid-maneuver energy for the rigid system. If nonlinear behavior in oniy one mode is taken into account in making the cor- rection, the residual energy is found t o decrease t o less than 0.7% of the mid-maneuver energy for the rigid system. The minimum time control profile is shown in Fig. 4. The control inputs are equal t o an upper or lower bound for most but not all of the control his- tory, which would ordinarily be the case for the exact minimum time profile. Considering the low residual energy a t the end of the maneuver and the fact that the algorithm used t o solve the minimum time prob- lems ensures that a control profile satisfying the speci- fied constraints with fewer time steps does not exist, it can be concluded that a very accurate solution of the minimum time maneuver problem has been found.

Conc lus ions

A method for applying Pulse Response Based Con- trol (PRBC) to maneuver problems involving nonlin- ear behavior associated with rapid large angle rotation is presented. Advantages of the PRBC approach are that linear behavior is represented in terms of pulse response measurements without modal truncation or a need for explicit modeling, and i t is not necessary to solve a two-point boundary value problem. For maneu- ver problems, the nonlinearity is handled by correcting the pulse response measurements t o account for non- linear behavior in a few flexible modes. The example presented in this paper demonstrates that the number of modes for which nonlinear behavior must be con- sidered to obtain an accurate solution of the problem is very small, even for an extremely rapid maneuver. This means that identification requirements are very low if this method is used.

Figure 3: Residual energy variation after nonlinear correction

A c k n o w l e d g e m e n t

This research is sponsored by the Air Force Of- fice of Scientific Research, Air Force Systems Com- mand, USAF, under grant number AFOSR-90-0297. The U.S. Government is authorized t o reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. This manuscript is submitted for publication with the un- derstanding tha t the U.S. Government is authorized to reproduce and distribute reprints for Governmen- tal purposes. The views and conclusions contained in this document are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either express or implied, of the Air Force Ofice of Scientific Research or the U.S. Government.

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