AIM-90-1 226-C.P A LINE-OF-SIGHT PERFORMANCE CRITERION FOR CONTROLLER DESIGN

OF A PROPOSED LABORATORY MODEL

Kyong B. Lim* and Lucas G. Hortat NASA Langley Research Center, MSl230

Hampton, VA 23665 680 ?

Abstract In the design of control systems, the control objective

must play a central role. In this paper, a line-of-sight per- formance criterion is derived for a proposed Controls Struc- tures lnteraction model and its many uses in the control design process for fine pointing control are illustrated. A lin- earized LOS criterion is used for direct controller design and as a performance measure to judge different control method- ologies. Numerical simulation results are shown where the three approaches; linear quadratic gaussian theory, robust eigensystem assignment, and local velocity feedback are used for vibration control. Results indicate that the LQG con- troller, which incorporates a linearized LOS weighting matrix directly, yields good performance without wasting energy to control motions that have no influence on the LOS

Nomenclature Cv, Sv cos(v), sin(v) e' LOS error vector

unit vector normal to mirror plane

& unit normal vector to target plane

A, Trn position, deformation vector of a point on mirror reference point on target plane

f unit vector of reflected laser beam 3 unit vector at laser source

z,, 6 position, deformation vector at laser source

5 position vector of laser point on mirror position vector of laser point on target plane

7 incident and reflected angle at mirror 8 rotation angles of mirror plane v rotation angles at laser source

, q physical and modal displacements

P parameter defining an equation of line in 3-0

CP direction cosine matrix 'I' mode shape matrix of structure (-),, ( s ) ~ retained and truncated modes ( * ) O undeformed configuration

* Research Associate, ODU Research Foundation, Norfolk, VA 23508-0369, Member AlAA t Aerospace Engineer, NASA Langley Research Center, Member AlAA

Copyright @ 1990 by the American Institute of Aeronautics and Astronautics, Inc. No copyright is asserted in the

United States under Title 17, U.S. Code. The U.S. Govern- ment has a royalty-free license to exercise all rights under the copyright claimed herein for Governmental purposes.

1. Introduction

Future NASA missions will require spacecraft which are large in size, flexible, and have stringent performance re- quirements. Because of the large size of such space- craft systems and the lightweight constraint imposed for cost effectiveness, sophisticated control methodologies are needed for their successful operation. Some of the require- ments for these systems are discussed in Ref. 1. For large flexible antennas, for example, the operating wavelength de- termines the surface accuracy to be maintained in the pres- ence of disturbances. Over the years rigid body attitude con- trol has been sufficient for control since the structures were sufficiently stiff. The new generation of large space struc- tures will include large solar panels, long appendages, and large antennas whose flexibility can no longer be ignored. For space applications where antennas are used for comu- nications or earth observation, accurate pointing in the pres- ence of structural flexibility must be maintained. To inves- tigate some of the generic problems arising for such large systems2 NASA under the Controls Structures Interaction (CSI) program have undertaken the analysis and design, ground testing and implementation of generic structures that would permit evaluation and validation of the existing tech- nology. One such system is currently being designed and built for testing at the NASA Langley Research Center as a ground experiment.

The proposed structure, described later in detail, is suspended by two cables which gives rise to the "pseudo rigid body" or pendulum modes in addition to the elastic vibratory motion. A laser beam, whose source is near one end of the structure, is reflected off a mirror, on the other end, onto a fixed target. The performance of the system is measured by the line-of-sight (LOS) accuracy in the presence of system disturbances. The novelty of the proposed structure is that it captures the heart of the (CSI) problem, i.e, the elastic modes must be controlled simultaneously with the "pseudo rigid body" modes because its structural flexibility prevents a simple uncoupling of the flexible and rigid body motion for control purposes.

The objective of this work is threefold. First, the kine- matic equations describing the three dimensional LOS mo- tion including structural flexibility are developed for use as performance criterion. Second, the importance of the LOS criterion not only for controller gain design but also in the determination of reduced order models and actuator/sensor locations is discussed. Third, results using three different control methodologies for vibration control are shown. The three approaches used are linear quadratic gaussian (LQG),

All other rights are reserved by the copyright iwner. 349

robust eigensystem assignment (REA), and local velocity feedback (LVF). Among these, LQG is the only approach that uses the LOS criterion directly in the determination of the control gains. In the other two approaches the criterion is only used to judge how well the design performs.

2. Proposed CSZ Structure

The structure shown in Fig.1 is a prototype test bed being considered by NASA Langley Research Center as part of the CSI program. The main section of the truss is 62 bays long (each bay is 10 in. x 10 in. x 10 in.), 6 bays long along the short appendage, and 11 bays along the long appendage. Mounted on the short appendage is a reflector 16 ft. in diameter with a circular mirror attached to its center. The reflector flexibility is included in this study. The truss is supponed from two cables 65 ft. long, located to minimize the interaction between the suspension and the structural modes. The total structural weight is approximately 701 Ibs. without the actuators. The NASTRAN finite element model has 460 grid points for a total of 2755 degrees of freedom. A reduced order model consisting of 15 modes is obtained through a controllability and observability analysis. A subset of 6 mode shapes along with the frequencies are shown in Fig.2. The vibration analysis revealed the first four pendulum modes which range from 0.1 12 Hz to 0.962 Hz. The first two flexible modes are reflector modes and the first truss beam flexible mode starts at 1.62 Hz. For this initial study, eight force actuators are distributed on the structure along with eight noncollocated accelerometers and eight angular velocity sensors. The configuration is shown in Fig.3.

Using finite element modeling, the governing equation is

where 6 denotes the displacement vector at the finite el- ement degrees-of-freedom. The physical acceleration and angular velocity measurements are

The controller design is based on the following reduced modal state space equations

where

The subscripts "r" and "t" represent the "retalned" and "truncated" subsets respectively and wi and Ci denotes the frequency and damping ratios of the ith structural mode.

3. Geometry of the Line-of-Sight (LOS) As part of the experimental set-up described previously,

a laser beam is pointed at the mirror surface which is mounted on the reflector. The laser beam reflection from the mirror is then detected by a spatially fixed optical detector array off the structure. The optical detector array will be referred to in the following as the target plane. The rigid body motion will also affect the targeting accuracy and because the laser beam source and mirror are mounted on the structure, any structural flexibility will affect the location of the reflection on target plane.

Consider the undeformed configuration as shown in Fig. 4 with the position of the laser source, PUI the position vector of the impinging laser beam on the target plane, go a unit vector providing the pointing direction of the laser from its source, @O a unit vector in the direction of the reflected beam from the mirror surface, and A% and ti,,, be the unit vectors normal to the mirror and target planes respectively. The superscript 'O' refers to the undeformed configuration. The position vector of the intersection point of the laser beam and the mirror plane can be written as

where ph is the distance from the laser source to the mirror along the laser path. To determine this distance, and hence the position of the laser on the flat mirror, consider the mirror plane equation,

where Tm is the position vector of any known reference point on the mirror and "." denotes a dot product. The distance, pO,, is obtained by combining Eqs.(5) and (6),

At the mirror plane, the incidence angle of the incoming laser beam equals the angle of reflection. The incidence angle is

given by cos(7) = -3' . n& (8)

so that the orientation of the reflected beam can be obtained by

Po = so + 2cos(y)h& (9)

By using Eq.(9), the position vector of the reflected beam on the target plane is written as

where the only unknown is the distance from the mirror to the target plane, 4. The target plane equation is given by

where &, is the position vector of any reference point on the fixed target plane. The distance from the mirror to the target plane p t along the laser path is evaluated by substituting Eq.(lO) into (1 1) so that

Substituting Eq. (12) into (10) gives the location of the laser beam on the target plane for the undeformed configuration. To include the flexibility of the structure, the preceding analysis is easily modified as follows. Let the position of the laser source after deformation be given by

and the deformed position of the reference point on the mirror plane be given by

At any instant of time, the laser source and mirror will have translated and rotated. Let (vl , v2, e) and (81, 02, 03) denote the rotation angles at the laser source and mirror plane respectively. The rotated directions 3, 4 and .iL, is related to the initial directions go, f0 and n& and are written in their component forms

where @(01,f12, 03) is the direction cosine matrix parame- terized by 1-2-3 Euler angles so that

w 1 , 8 2 , w 2

Note that for the proposed CSI structure, the magnitudes of the rotation angles will be such that the kinematic singularity, inherent in an Euler angle parameterization, will not pose a problem.

Having obtained the quantities after deformation, i.e., &',, F?, f&, 3, 4, pm and pw, the position vector, Sw, which grves the location of the laser beam on the target plane after deformation, can be computed from the same equations as the undeformed case just described.

Finally, the LOS error is defined by

Equation (17) gives the instantaneous LOS error. There are 12 independent degrees-of-freedom that determines the LOS motion, namely, 3 displacements and 3 rotations at the laser source and mirror.

4. Line-of-Sight Performance Index The integral LOS error between times to and tl is given

where the LOS error in component form is

and xw(t) is the coordinate of the impinging LOS beam h

on target plane at time t. Note that Q = IsXs denotes a radial distance measure. In general, the above coordinate depends nonlinearly on the instantaneous physical displace- ments and rotations at the laser source point and the mirror (see Section 3) in addition to the geometrical layout of the laser sourcelmirror planeltarget plane, i.e.,

For small physical displacements and rotations, the above physical displacement and rotations can be wriiten in terms of the modal state, so that

where IEp is a subset of the structural mode shape matrix corresponding to the above critical degrees of freedom.

In order to obtain an integral LOS error in terms of modal states, an hence a LOS dependent quadratic weight matrix, Eq.(20) is linearized about the undeformed physical displacement, 6;

1

so that if we let 20, = f(l$), then

where

In this study the gradient matrix. $!-In, is computed by a P

finitedifference. The integral in Eq.(18) can now be approximated as

where the line-of-sight weight matrix is given by

As a measure of the degree of participation of structural modes in the LOS motion, we choose the free response due to modal initial condition from time to to tl = oo is chosen. The participation of mode4 is given by the magnitude of the time integral of Eq.(26) due to the initial condition response of

x(t) = h ( t ) + Fw(t); %(to) = vi = ith eigenvector (281

For asymptotically stable systems, the steady-state integral can be computed conveniently since

where Pw satisfies

In addition to the degree of structural mode participation in the LOS integral, the effect of input noise on the LOS can be obtained. From Eqs.(28) and (24), the steady-state variance matrix of LOS due to white noise with zero mean and constant intensity, V can be written asS

where the steady state variance of the state vector, Qz, satisfies

A&, + Q , A ~ + F V ~ = o (32)

Hence, trace@,], or the maximum singular value, @[&,I, could be used as a measure of the sensitivity of the LOS due to random noise at the input.

For the class of structural control problems whose pri- mary goal is to control or supress LOS disturbances, the

above LOS quadratic measures are helpful in various as- pects of the design such as: (1) obtaining models for con- troller design or simulation studies, (2) deciding optimal ac- tuatorlsensor locations, and (3) controller design. Clearly, for line-of-sight control, the control design model should in- clude, at least, all contributing modes. This should also be the case for simulation models. Having obtained a struc- tural model that includes all significant modes, i.e., modes corresponding to large costs as defined by Eq. (29), the actuators and sensors should be placed to maximize the controllability and observablity. Finally, the control gains for controlling LOS motion can be designed by using the previ- ous structural model with the actuators and sensors located strategically to detect and control LOS motion. In the next section, the use of LOS weight matrix to obtain control gains for optimal performance is outlined.

5. Constant Gain Feedback Control In this section, three candidate control configurations are

examined for the vibration supression of the proposed CSI model. The control laws are briefly summarized and the advantages and the disadvantages of the controller design are discussed.

5.1 LQG with UIS Cost Briefly, the steady state Linear Quadratic Gaussian

(LQG) problem can be defined as followss:

Minimize $ eTOe + uT RU dt (33)

subject to x(t) = Ax(t) + Bu(t)

~ ( t ) = Cx(t) (34) e(t) = T x ( ~ )

The output matrix, 3, is given by Eqs. (24)-(25). The optimal LQG control is given by

where Z is the optimal state estimate and P satisfies the algebraic Riccati equation

In the above equation, represents the LOS error weight- ing matrix. For the control input weight matrix R an 8 X 8 identity matrix is chosen to reflect eight identical actuators. To represent a radial distance mcasure of the LOS error, a 3 x 3 identity matrix is used for Q.

The advantage of the optimal LQG control design is that it allows the combined minimization of state error and control effort while always giving stable state feedback gains. On the other hand, the significant arbitrariness of the Q and R

weight matrices usually requires some form of adjustment or optimization to obtain satisfactory performance. However, the LOS weight matrix can be used to weigh the states and only R remains arbitrary. Full state or its estimate is necessary for implementing this control law and so a Kalman-Bucy filter based on an assumed set of input and measurement noise statistics Is designed and implemented.

52 Robust Eigenstructure Assignment (REA)

In this controller design4, the closed-loop transient per- formance is satisfied by assigning eigenvalues at desired locations. The remaining design freedom, beyond assigning eigenvalues, is used to tailor the closed loop eigenvectors, as well.

From the equation of motion given by Eq.(34), the cor- responding eigensolution can be written as

where Xk denotes the kth desired closed loop eigenvalue and p is the number of eigenvalues to be assigned. The bases for assignable subspace, Vok, can be computed by Singular Value or QR decomposition, i.e.,

The set of assignable eigenvectors are defined by

and in the robust eigensystem assignment strategy, the remaining design freedom, (cl . . . cp), is used to make the closed loop eigenvectors as orthogonal as possible.

The advantage of this method is that it allows arbitrary placement of poles and shape eigenvectors. The shaping of the closed loop eigenvectors gives robustness to the sys- tem. It is expected that this method will work well if the designer knows the disturbance environment a priori. A word of caution when using this procedure is that certain eigenvalue/eigenvector configurations may required unreal- istic control actions. If a desired pole configuration is known, for example from LQG design using LOS weights (as de- scribed earlier in section 5.1), then it can be used for place- ment using this procedure. However, the degree of feasibil- ity and the efficiency of this method can only be determined by simulation studies for a particular problem. This method also requires a full state feedback.

5.3 Local Velocity Feedback (LVF)

In this control strategy, the actuators and sensors are collocated so that

where the physical measurements are given by

The total closed loop energy and the rate of change of total energy is

From Lyapunov's second theorem, it is clear that semi- definiteness of E together with the fact that E f 0 along any trajectory guarantee asymptotic stability. Observe from Eq.(42) that the the feedback gains corresponding to dis- placement feedback do not affect the energy dissipation rate, i.e., the addition of springs do not affect energy dis- sipation. Hence, only velocity measurements are used ih the sequel.

For further simplification, if only the velocity measure- ments at a particular point are fed back to the collocated actuator, i.e. "local feedback", then the gain matrix is con- strained to the diagonal form

Clearly, the main advantage of this controller is its sim- plicity in the controller design. The uncoupled nature of the feedback structure substantially reduces the real time com- putational requirements. This is an output feedback con- troller and requires no real-time state estimation. Further- more, this controller structure is theoretically robust with re- spect to actuator andlor sensor failures. On the other hand, the main drawback in this controller design is that it requires the sensors and actuators to be collocated and obviously the same number of sensors and actuators are required. A practical limitation may be that accurate translational ve- locity measurements are not readily available as compared to say acceleration transducers. In addition, the relatively small number of control gains available for tuning may limit system performance. However, the degree and the nature of this limitation predicted through qualitative analysis and simulation studies remains to be verifiedldemonstrated in the laboratory.

6. Simulation & Discussion of Results

In this section, the importance of the LOS index for model reduction and actuator/sensor placement is dis- cussed. Using the proposed model, three controller designs are obtained and compared in terms of closed-bop eigen- value locations, energy removal rate, rms control effort, and LOS error trajectory.

6.1 Applications of LOS index As an illustrative example on some uses of the LOS

index, consider the lowest 49 structural modes of the struc- ture shown in Fig. 1. Fig. 5 shows the results of the relative importance or participation of each structural mode (in as- cending order of frequency) on the LOS index. A larger LOS index implies a larger contribution from a particular mode. The varying parameter, 'distance', is the distance from the target plane to the structure. It is clear from the figures that the relative importance of the modes depends on the dis- tance to the target. In particular, for a nearby target (300 in), the higher frequency modes are relatively insignificant with respect to the lower frequency modes. On the other hand for a distant target (200 miles), higher frequency modes play a more significant role in the LOS index. Physically, the LOS error becomes more sensitive to angular motion as the target distance increases. Since the higher frequency nor- malized mode shapes have more curvature than the lower frequency modes, their contribution to the angular motion is larger. This curvature effect is counteracted by the faster decay rate. Based on Fig. 5, the designer can select, at the very least, the significant modes for inclusion in the model to be used in control design and simulation.

Ideally, the structural modes that most influence the LOS motion should be made maximally controllable and observable. If some of the significant modes are either not sufficiently controllable andlor observable, the designer must reconfigure the actuators andlor sensors locations. For example, it turns out that for 'distance=300 in.' case, mode 1 is the most significant in terms of LOS index (Fig. 5) and is very controllable but unfortunately it is the least observable. In this paper, the degree of controllability of each mode is defined as the minimum control effort required to drive a zero state to the corresponding eigenvector state in a given period of time. The degree of observability of each mode is taken to be the time integral of the output signal of the free response using a particular eigenvector initial condition.

6.2 Comparison of Controller Designs Using the control methodologies discussed in the pre-

vious section, three controller designs are obtained for the proposed CSI structure. Starting from zero initial state, the structure is excited for 5 seconds using all 8 thrusters with a uniform random disturbance force of 2 Ibs. peak ampli- tude. The simulation is for 30 seconds and the controllers are assumed to be turned on from time zero.

Figure 6 shows the resulting pole locations for the un- controlled case, LQG (estimator poles are not shown) with LOS performance criterion, REA, and LVF. A damping ratio

of 0.2% is assumed for the uncontrolled structure, hence, all roots lie very close to the imaginary axes (Fig. 6a).

For the LQG controller (Fig. 6b), the pendulum modes are well damped whereas a selected few modes are left almost undamped. Notice that mode 7, which is the first significant beam bending mode, is damped most heavily. This design also added significantly large damping ratios to all pendulum modes.

The LOS weights used were based on the target dis- tance of 300 inches so that the pendulum modes are more significant to LOS motion (see Fig. 5a). For the REA case (Fig. 6c), the poles were selected such that the state tran- sient response is supressed to 5 % of the initial state in 10 seconds for all modes while the frequencies are kept fixed to open loop values. The design freedom remaining is used to maximize robustness via closed loop eigenvector shaping. Note that for the REA case, the pole locations of the LQG controller could be used as the desired locations to efficiently control LOS motion while maximizing system robustness.

For LVF case, the gains were chosen by analyzing multiparameter root-locus plots. A set of sub-optimal gains were chosen such that (1) all LOS modes are damped as much as possible without significant overdarnping (which may produce slow response), and (2) the control amplitudes do not saturate. Note that the LVF and the REA controllers have no knowledge of the LOS criterion, rather, it simply dissipates the energy in the system.

Figure 7 show logarithmic plots of the total energy in the system as a function of time. The slopes of the curves represent the energy removal rate for each of the three designs. The uncontrolled case shows negligible energy removal because of the low damping level. The REA controller yields the highest energy removal rate of all three methods. On the other hand, the LQG controller have the largest residual energy remaining after 30 seconds. If a large energy level remains after 30 seconds but the LOS error is minimized, the energy in the system must be distributed among modes that do not affect the LOS. This highlights the fact that LQG control damps only those modes associated with LOS motion.

Figure 8 show time histories of the X-Y motion of the impinging laser point on the target plane. The uncontrolled case is shown in Fig. 8a for comparison. Out of all three designs, LQG produces the smallest deviation from the nom- inal. Although the LQG case had the most energy remaining after 30 seconds (see Fig. 7b), its LOS performance is the best thus indicating that its control effort is focused to main- tain good pointing accuracy without wasting control energy. The LOS performance for the LVF and REA is about equal. To examine the control effort for each of the designs, Fig. 9 shows the total rms control force, using all 8 actuators, as a function of time. All three controllers require forces below 4 Ibs. which is within the limitations of the proposed thrusters. Note that LVF requires a smaller peak RMS control force whereas REA control requires the largest.

7. Concluding Remarks The kinematic equations governing the line-of-sight mo-

tion have been developed including the flexibility of the truss structure. By formulating the control problem in terms of the line-of-sight (LOS) and using this criterion directly in the de- sign, the control effort is concentrated on controlling only those structural modes which have the most affect on the LOS.

Among the three controllers compared, the LQG formu- lation produced the most effective controller for LOS control. Although the LVF configuration is the easiest to implement (assuming collocated translational velocity measurements are available) and perhaps the most robust, its performance is comparable to the other two controllers which required a full state estimator.

It is found that the relative importance of the modes depend on the LOS geometry and the distance to the target. Higher frequency modes have more curvature so that it generally produces more angular motion except when the laser source and the mirror are located very cbse to zero- slope points of mode shapes. Consequently, the higher frequency modes generally become more significant as the distance to the target increases.

Finally, for the class of structural control problems whose primary goal is to control or supress LOS disturbance, the LOS performance criierion is helpful in various aspects of the design such as: (1) obtaining models for controller

design or simulation studies, (2) deciding optimal actua- torlsensor locations, and (3) controller design.

Acknowledgements The authors would like to thank Mr. James Bailey of the

Spacecraft Dynamics Branch for providing the NASTRAN model. We would like to acknowledge Dr. Jerry Housner of the Spacecraft Dynamics Branch and Dr. Suresh Joshi of the Spacecraft Controls Branch for reviewing the draft.

References Strunce, R.R., Jr. and Turner, J.D., "Enabling Technolo- gies for Large Precision Space Systems," Large Space Antenna Systems Technology - 1982, NASA CP-2269, Part 2. "SSTAC Ad Hoc Subcommittee on Controls/Structures Interaction," Chairman: J.R. Garibotti, Final Report, June 8, 1983. I

Kwakernaak, H., and Sivan, R., tinear Optimal Contrd Systems, John Wiley & Sons, Inc., 1972.

Juang, J-N., Lim, K.B., and Junkins, J.L., "Robust Eigen- system Assignment For Flexible Structures," Journal of Guidance, Control, and Dynamics, V01.12. No.3, May- June 1989, pp.381387.

LLJ Fig. 1 Finite element model

MODE 1 o = 0.111 Hz.

MODE 3 61 = 0.145 Hz.

LY

MODE 5 o = 1.37 Hz.

MODE 2 o = 0.115 Hz.

MODE 4 = 0.962 Hz.

MODE 7 o = 1.62 Hz.

Fig. 2 Six structural mode shapes for the proposed CSI model.

356

Accelerometers Angular Rate Sensors

Fig. 3 Actuators and sensors locations

Fixed

Truss Structure

Fig. 4 Geometry of line-of-sight

LOS Index (log)

3

LOS Index 2 (log)

1

0

Mode Number

4 , z

Distance = 200 miles

0 5 10 15 20 25 30 35 40 45 50

-

-

-

Mode Number

Fig. 5 Relative importance of modes on LOS index for varying distance to target.

- - Distance = 300 in. - - - -

- -

- n ,,

Fig. 6 Pole locations for the different controller designs.

50

40-

30 Im(V

20-

10

0

lo3 -, 3 Total

Energy 100 r (lb-in)

- a) Uncontrolled

Total Energy (Ib-in)

a) Uncontrolled 0 -

- 0

0 0

-

- c) REA

- 1

10-3 I

0 10 20 30 T i e (sec)

0

50

40-

30 MA)

20

10-3 1 I 0 10 20 30

Time (sec)

-

b) LQG 0

0 - 0

- 0

- 0

0 0

1 10

Fig. 7 Time histories of total energy in the system.

-1.5 -0.5 0 -1.5 -1 -0.5 0

Re(W Re(U

-0 0 0 0-

0

0 n n

Control Force

RMS (lbs)

a) Uncontrolled b) LQG -10

10 20 -20 -10 0 10

x (in) x (in)

Fig. 8 Target plane laser beam motion for different controllers.

Control Force

RMS (lbs)

Time (sec) Time (sec)

Control Force 2 1

RMS (lbs)

C) LVF I

Fig. 9 Total control force RMS value for different controllers.