Reduced-Order Multiple Model Adaptive Controller for Flexible

0018-9251/9&$3.00 @ 1992 IEEE Adaptive fdter and control algorithms can be designed by multiple model adaptive estimation and control (MMAE [14] and MMAC [5-71) techniques. They have the structure of a bank of parallel Kalman filters or parallel LQG (linear system model, quadratic cost, and Gaussian noise models used for synthesis) controllers that are used to form a state estimate or control output as a probability-weighted sum. Each elemental filter or controller is based upon an assumed value of the uncertain parameter to which adaptation is to occur, and it is the characteristics of the residuals from each of these filters that allows the evaluation of the probability that each elemental filter or controller is in fact based upon the best assumed value of the parameter at the current time. These algorithms are ideally suited to distributed and/or multiprocessor computation, and they have some performance attributes that are far superior to adaptive algorithms without the multiple model structure [6-171. One important application area for multiple model algorithms is the adaptive control of large flexible spacecraft to allow for rapid pointing and efficient quelling of bending oscillations [18-261. For this case, the use of a “moving-bank” multiple model estimation and/or control algorithm has been investigated. Rather than implementing parallel filters or controllers based upon all viable discrete parameter values for all time, only a subset of these discrete values is used at any one time. There is a dynamic reallocation logic that decides which subset of values should be currently implemented. To motivate this concept, consider the potentially debilitating problem of multiple model algorithms, namely, the number of elemental filters or controllers in the parallel bank. For instance, if there are two uncertain parameters and each can assume 10 possible values, then lo2 = 100 separate filters (or controllers) must be implemented, even if the parameters are treated as unknown constants. To circumvent this problem, one can propose a “moving bank” of fewer estimators or controllers. In the preceding example, one might choose the three discrete values of each parameter that most closely surround the estimated value, requiring only 32 = 9 separate elemental filters (or controllers). This is depicted in Fig. 1. The particular choice of 9 filters could then depend on the most recent estimate of the parameters, generated in real time. Maintaining fewer elemental filters in the bank enhances the feasibility of multiple model algorithms, but it could aggravate the behavior observed earlier in fixed-bank algorithms of making inappropriate decisions when the “true model” is not included in the model set of the filter [5]. Some research has been directed at the information theoretic problem associated with this condition [27-291. Furthermore,

They have the structure of a bank of parallel Kalman filters or parallel LQG (linear system model, quadratic cost, and Gaussian noise models used for synthesis) controllers that are used to form a state estimate or control output as a probability-weighted sum.Each elemental filter or controller is based upon an assumed value of the uncertain parameter to which adaptation is to occur, and it is the characteristics of the residuals from each of these filters that allows the evaluation of the probability that each elemental filter or controller is in fact based upon the best assumed value of the parameter at the current time.These algorithms are ideally suited to distributed and/or multiprocessor computation, and they have some performance attributes that are far superior to adaptive algorithms without the multiple model structure [6-171.One important application area for multiple model algorithms is the adaptive control of large flexible spacecraft to allow for rapid pointing and efficient quelling of bending oscillations [18-261.For this case, the use of a "moving-bank" multiple model estimation and/or control algorithm has been investigated.Rather than implementing parallel filters or controllers based upon all viable discrete parameter values for all time, only a subset of these discrete values is used at any one time.There is a dynamic reallocation logic that decides which subset of values should be currently implemented.
To motivate this concept, consider the potentially debilitating problem of multiple model algorithms, namely, the number of elemental filters or controllers in the parallel bank.For instance, if there are two uncertain parameters and each can assume 10 possible values, then lo2 = 100 separate filters (or controllers) must be implemented, even if the parameters are treated as unknown constants.To circumvent this problem, one can propose a "moving bank" of fewer estimators or controllers.In the preceding example, one might choose the three discrete values of each parameter that most closely surround the estimated value, requiring only 32 = 9 separate elemental filters (or controllers).This is depicted in Fig. 1.The particular choice of 9 filters could then depend on the most recent estimate of the parameters, generated in real time.
Maintaining fewer elemental filters in the bank enhances the feasibility of multiple model algorithms, but it could aggravate the behavior observed earlier in fixed-bank algorithms of making inappropriate decisions when the "true model" is not included in the model set of the filter [5].Some research has been directed at the information theoretic problem associated with this condition [27-291.Furthermore, recent research [18-251 has portrayed the performance capabilities of various proposed decision logics for "moving" (and changing the sue of) the bank of currently used values around the parameter space.
issues of such an adaptive estimator/controller. How well does the adaptation work in the '.ice of unmodeled effects in the real world?A final implementable controller will necessarily be based upon reduced-order, simplified models in order to maintain reasonable computational loading.Will this cause graceful degradation in the controlled system characteristics, or will the adaptation process and/or the bank-moving logic become totally confounded by the unmodeled effects?This investigation quantifies the impact of unmodeled higher order bending modes, and it thereby indicates fundamental limits on model order reduction for synthesis of multiple model adaptive controllers for flexible spacestructures.Subsequent sections of the paper present the structure of a multiple model adaptive estimator and controller, and then develop some possible logics for moving the bank.The model of the two-bay truss structure of interest is then described, the simulation characteristics are depicted, and the robustness of MMAE and Mh4AC algorithms to unmodeled higher order bending mode effects is assessed.

II. MULTIPLE MODEL ADAPTIVE ESTIMATION
Let a denote the vector of uncertain parameters in a given linear stochastic state model for a dynamic system.These parameters can affect the matrices defining the structure of the model or depicting the statistics of the noises entering it.In order to make simultaneous estimation of states and parameters tractable, the continuous range of a values is discretized into K representative values.If we define the hypothesis conditional probability pk(t;) as the probability that a assumes the value ak (for k = 1,2,. . ., K), conditioned on the observed measurement history to time t;: pk(t;) = Pr{a = ak I Z(t;) = Zi} (1) then it can be shown [14] that pk(t;) can be evaluated recursively for all k via the iteration: f<ii )la,Z(rj-,) (zi I ak 7 Zi-1) .pk (ti-1) C~=lfi(ri)l~,z(rj-I)(Zi I ajyzi-1) 'pj(ti-1) pk(ti) = (2) in terms of the previous values of pl(f;-l), . . ., p ~( t ; -l ) , and conditional densities for the current measurement z(t;) to be defined explicitly in (12).Notationally, the measurement history random vector Z(t;) is made up of partitions z(tl), . . ., z(t;) that are the measurement vectors available at the sample times f1,. . ., t;; similarly, the realization Z; of the measurement history vector has partitions z1,. . .,z;.Furthermore, the Bayesian minimum mean square error estimate of the state is the probability-weighted average: where ?k ( t + ) is the state estimate generated by a Kalman filter based on the assumption that the parameter vector equals ak.More explicitly, let the model corresponding to ak be described by an "equivalent discrete-time model" 14, 7] for a continuous-time system with sampled data measurements: where xk is the state, U is a control input, W k is discrete-time zero-mean white Gaussian dynamics noise of covariance Q k ( t ; ) at each t;, z is the measurement vector, and Vk is discrete-time zero-mean white Gaussian measurement noise of covariance & ( t i ) at ti, assumed independent of W k ; x(t0) is modeled as Gaussian, with mean 4 0 and covariance P ~o and is assumed independent of W k and vk.Based on this model, the Kalman fdter is specified by the measurement update: is calculated in the kth Kalman filter as in (6).
The denominator in (2) is simply the sum of all the computed numerator terms and thus is the scale factor required to ensure that the p k ( t i ) s sum to one.
One expects the residuals of the Kalman filter based upon the "best" model to have mean squared value most in consonance with its own computed A k (ti), while "mismatched" filters will have larger residuals than anticipated through A&).Therefore, (2), (3), and ( 6)-( 12) will most heavily weight the filter based upon the most correct assumed parameter value.However, the performance of the algorithm depends on there being significant differences in the characteristics of residuals in "correct" versus "mismatched" fdters.Each filter should be tuned for best performance when the "true" values of the uncertain parameters are identical to its assumed value for these parameters.One should specifically avoid the "conservative" philosophy of adding considerable dynamics pseudonoise, often used to open the bandwidth of a single Kalman filter to guard against divergence, since this tends to mask the differences between good and bad models.
estimator is identical to the full-bank estimator just described, except that K does not correspond to the total number of possible parameter vector values.Instead, it is the smaller number of elemental Kalman filters maintained within the bank.Which particular K fdters are in the bank at a given time is determined by one of the decision mechanisms of Section IV.

Ill. MULTIPLE MODEL ADAPTIVE CONTROL
Using the system model of ( 4) and (9, and an appropriately chosen quadratic cost function, a standard LQ full-state feedback regulator can be designed for each a k value as uk(ti) = -G:(ti;ak)Xk(ri) (13) where the optimal controller gain is generated by solving a backward Riccati difference equation [7].
The gain of this control law and the state estimate of the corresponding elemental Kalman filter can then be combined via "certainty equivalence" to produce the elemental LQG control law: In MMAC, a separate elemental control law of this form is associated with each ak value and thus with each elemental filter in the bank of K filters.Thus, K parallel LQG controllers are formed, and the adaptive control is then generated as the probability-weighted average:

IV. MOVING THE BANK
When the "true" parameter point lies within the region of parameter space bracketted by the moving bank, the moving-bank estimator behaves much as the standard full-bank multiple model adaptive estimator.If the "true" parameter value should lie outside that region, or even inside the region but near its boundary, we must fmst detect this condition and then take some appropriate action, as to move or expand the bank in some fashion.Previous research investigated the behavior of four different decision logics for moving the bank plus an additional logic to allow expansion and contraction of the bank size [lS-251.Based on these analyses, the bank size was fixed at the size corresponding to the finest parameter space discretization, and two bank-moving decision logics were considered: residual monitoring and parameter position estimate monitoring.
Residual Monitoring: Let the likelihood quotient Lk(t;) be the quadratic form that appears in (12): In the case of scalar measurements, this is the current residual squared, divided by the filtercomputed variance for the residual.When the true parameter value does not lie within the moving-bank region, all K likelihood quotients can be expected to exceed a threshold level T, the numerical value of which is set in an ad hoc manner during performance evaluations.Thus, a possible detection logic would indicate that the bank should be moved at time t; if: Moreover, the elemental filter based on Bk nearest to the true parameter value should have the smallest likelihood quotient, thereby giving an indication of the direction to move the bank.While this logic would respond effectively to a real need to move the bank, it would also be prone to false alarms induced by single large samples of measurement noise.
Parameter Position Estimate Monitoring: Another means of keeping the true parameter value within the region bracketted by the moving bank is to keep the bank centered (as closely as possible, in view of the discrete values a is allowed to assume) on the current estimate of the parameter.This estimate is [4]: If the distance from the parameter value associated with the center of the bank to &(ti) becomes larger than some chosen threshold, a move of the bank in that direction is indicated.Since b(ti) depends on a history of measurements, it is less prone to the false alarms discussed in the previous paragraph.
Allowing for precomputability, each filter is an implementation of (8) and (lo), with the appropriate values for @k(t;+l,ti), &(ti), and &(ti) stored for each ak.When the bank is moved, any filters corresponding to newly declared ak locations must be initialized with &(ti) and pk(t;) values.A reasonable choice for the ik(t;)s is the current moving-bank estimate k(t:).For the pk (ti)s, many proposed techniques have proven to be computationally intense.Since there appears to be no loss of performance from the simple equal redistribution of the discarded filters' probabilities among the new filters [18-251, it is the method chosen for this investigation.hub with an appendage extending from the structure [21-251.The mass of the hub is large relative to the mass of the appendage, and the hub can be rotated to point the appendage in a commanded direction.In fact, this is an outgrowth of a fixed two-bay truss that was originally developed to study the effects of Structural optimization on optimal control design [MI.
A similar model was used to research active control laws for vibration damping [31].
The structure consists of 13 rods which are assumed to be constructed of aluminum, having a modulus of elasticity of lo7 lb/in2 and weight density of 0.1 l b h 3 [30].The two rods connected to the hub are of much greater cross-section than the rest; the resulting greater stiffness contributes some high frequency modes into the system.Nonstructural masses with a mass of 1.3 lb-s2/in are located at positions 1, 2, 3, and 4, as shown in Fig. 3.The nonstructural mass is very large compared with the structural mass so as to achieve the low frequencies associated with large space structures [31].
The general second-order differential equations which describe the forced vibration of a large space structure with active controls and n frequency modes can be written in physical coordinates as [M, 311:

-M-'b h-by+
It is assumed that the noises can be represented as inputs that enter the system at the same location as the actuators (B = G).Measurements are assumed available from position and velocity sensors that are colocated for simplicity (as, integrated outputs of accelerometers) and located at nodes 1 and 2 in Fig. 3, and from gyros located at the hub.It is further assumed that the measurements are noise corrupted due to deficiencies in the model of the sensor and/or some actual external measurement noise.Thus, the sampleddata measurements are modeled as where m is the number of scalar measurements, v is an uncertain measurement disturbance of dimension m and modeled as a discrete-time white noise of covariance R (assumed diagonal for simplicity), H is the position measurement matrix, and H' is the velocity measurement matrix.In this application, the position and velocity measurement matrices are identical because of the colocation of the position and velocity sensors.
system into a set of decoupled modal equations [30][31][32][33].In order to achieve decoupling, the damping matrix is assumed to be a linear combination of the mass and stiffness matrices [31].Modal coordinates are related to physical coordinates by Modal Analysis: Modal analysis transforms the where r is as defined previously and r' represents the modal coordinates.Substituting (24) into (20) yields x'(t) = F'x'(t) + B'u(t) + G'w(t) (3) where x' = [rlTrrTIT and the open-loop plant matrix F', the control input matrix B' and the noise input matrix

G' are
The F' matrix is also of the form [30, 311: where each of the four partitions are n-by-n and diagonal.The measurements become: It is assumed that uniform damping exists throughout the structure.Furthermore, the damping coefficient of C = 0.005 is chosen for implementation because it is characteristic of damping associated with large space structures [31].Parameter Variation: A 10-by-10 point discretization in parameter space is created by considering two physically motivated parameter variations, a scalar multiplier on the four nonstructural masses and another on the entire stiffness matrix.Physically, the mass variation can be related to fuel being depleted or shifted to a different section of the space structure.The change in the stiffness matrix can be associated with structural fatigue in the rods or a failure of a member within the structure itself.the rms error of the state variable estimates from a nonadaptive filter operating in a simulated "truth model" environment, as the true parameter and the filter-assumed parameter are moved apart.Discretization was established at points yielding a certain percentage amount of degradation in performance, thereby producing elemental filters for the multiple model adaptive estimator that are readily distinguishable from each other on the basis of residual characteristics.In the f i a l discretization level, the nonstructural masses vary -50% to +50% from nominal in a nonlinear fashion, while the entire stiffness matrix varies -50% to +40% in a nonlinear manner [23-251.This parameter space discretization enhanced algorithm performance substantially beyond what was achievable with a simple linear discretization.
Order Reduction: The method of order reduction based on singular perturbations [7 : 219, 31, 341 is used to reduce the system model from 24 to 6 states.(This choice was based on the natural frequencies associated with the 12 modes, averaged over the 100 discrete models: 0, 1. 4, 2.9, 5.0, 5.4, 7.1, 9.1, 9.9, 150, 1400, 1800, and3100 Hz.)This method assumes that faster modes reach steady state essentially instantaneously.Letting x1 be the states to be retained and x2 be The discretization was accomplished by monitoring the states to be ignored, the deterministic system is reformulated as H21X (30) where F11 and F z are square matrices.The high frequency modes are eliminated by assuming steady state is reached instantaneously in these modes (x2 = 0).It can be shown that, for the state equations developed previously as ( 20)-(B), the reduced-order system model is where F, = Fll, B, = B1, H, = HI, and D, = (-H*F;'Bz).
The direct feedthrough matrix D, is the only term in (31) that is dependent upon terms associated with the modes assumed to reach steady state instantaneously.In contrast, the other reduced-order matrices are calculated simply by truncating those states associated with x2.Numerical problems in the computation of D, can be avoided by a method described in [21-251.

VI. PERFORMANCE ANALYSES
A ten-run Monte Carlo analysis was used to evaluate the performance and robustness of a 6-state moving-bank multiple model adaptive estimator and controller against a 24-state "truth model" of the structure.As suggested in Section I, this algorithm implemented nine elemental Filters, corresponding to a 3-by-3 array of points within the full 10-by-10 grid of discrete parameter points.The "true" parameter was represented as constant and equal to one of the discretized values for the purposes of this study.For all algorithms, only steady state constant-gain elemental filter (and controller) gains were considered.The residual monitoring decision logic described in Section IV was used to move the bank for the results depicted in this section; the alternative logic produced very similar results.
Fig. 4 depicts the simulation used to test the MMAE In this diagram, X is a scalar multiplier that is applied to the columns of the truth model measurement matrix H, that correspond to all but the six states modeled within the MMAE.It can be varied from zero, the case in which the truth model is of the same order as the filter model, to one, the case in which the full effects of all the higher order mode states of the truth model are being incorporated.Thus, the effect of unmodeled states can be gradually increased as desired.Once the state estimation (and later, control) capability of the algorithm is established with no unmodeled effects, the robustness of this performance to the nine unmodeled modes can thus be evaluated.The ability of the moving-bank adaptive algorithm to provide adequate state estimation accuracy (and adequate state control later) is the primary criterion of performance: the main objective is to design a good adaptive state estimator (and controller), not a parameter identifier.Thus, the most important output in Fig. 4 is the state estimation error ex(t;) = Hlx,(t;) -HfB(f+).

+--
(32) This is a six-dimensional vector, composed of errors in the estimated positions and velocities of the structure at nodes 1 and 2 and at the hub (node 3) of Fig. 3; it portrays how well the algorithm estimates the true physical shape and velocity of the space structure.As shown in Fig. 4, when no feedback control was being applied to the system, a square wave dither signal (one that alternates its value every sample time and is thus of minimum period = 0.1 s = twice the measurement sample period of 0.05 s; and of magnitude 10, determined by trial and error) was applied to excite the system and enhance parameter identification; the filter is informed of the dither signal.
a 10 s simulation) of mean and standard deviation of the position and velocity errors, respectively, for a nonadaptive single filter benchmark.The first three entries for each node show the effect of varying X for a single filter that is artificially informed of the correct parameter value, (7,6), i.e., the mass parameter at its seventh discrete value out of ten and the stiffness matrix at its sixth discrete location.The last entry for each node, indicated by the "-" in the X column, corresponds to a nonadaptive filter that incorrectly assumes the parameter value is ($5); this data in the table corresponds to the case of X equal to 0. The table indicates that the effects of unmodeled higher order modes are not very significant on position estimates at the midpoint and end of the truss (especially relative to the effect of incorrectly assumed parameter value, as indicated by the corresponding last table entry for each node, designated with "-" in the X column), but they do seriously impact the standard deviation in velocity estimates at these two points and in position estimates at the hub (again, particularly Ebles I and I1 present the temporal averages (over  Fig. 5 presents the mean k one standard deviation time histories for the position estimate errors at node 1 (center of the truss) for a moving-bank MMAE initially centered at the ( 5 5 ) location with all nine pk(ro) values set to 1/9, whereas the true parameter is again at (7,6).The lower bound on computed probabilities was set to 0.05, and the movement logic threshold T of ( 17) was set to 0.25 through repeated performance analyses.Fig. 5a corresponds to the case of A = 0 and plot b to A = 0.5, and no serious degradation is seen from the introduction of unmodeled higher mode effects.(Note the different scales in plots a and b.)The other trends seen in Thbles I and I1 were repeated in this case as well.
For the controller evaluation, the setup was analogous to that depicted in Fig. 4, except that the "Estimator" block was replaced with the MMAC controller that provided feedback to the "Buth Model" block.For the first 0.5 s, no feedback was provided but the dither signal was used to excite the system; thereafter, the dither was turned off and the MMAC output was fed back to the system to quell the oscillations.Here the most meaningful system outputs are the actual system positions and velocities at the three nodes, H,x,, and the commanded controls, as well as the estimation errors of (32).
established as a single nonadaptive LQG controller with artificial knowledge of the true parameter value, (7,6).Fig. 6 shows the mean f one standard deviation in estimating the position at node 1, for the cases of X = 0 and X = 1.The rms position error is held at about 0.002 in, and the only appreciable effect of the unmodeled higher order modes is during the control acquisition transient for about a half-second once the feedback control is turned on at 0.5 s into the simulation.Fig. 7 shows the corresponding actual position mean f one standard deviation at both nodes 1 and 2, for the case of X = 1 (with full higher order modes); the effectiveness of the controller is obvious from the instant it is turned on at 0.5 s.The misinformed nonadaptive controller based upon an assumed parameter value of (5,5) yielded an unstable As for the estimator, a benchmark controller was closed-loop system: adaptivity is seen to be critical for control of this system.The moving-bank MMAC was initialized with its center at the (55) location that led to instability in the nonadaptive controller case; all other initial conditions and design parameters were also identical to the Mh4AE evaluation.Fig. 8 presents the mean f one standard deviation time histories for the node 1 position estimate errors, like those of Fig. 5 for the open-loop case, but for X = 0 and X = 1.In both cases, the rms errors are kept at about 0.002 in, essentially the same as for the artificially informed nonadaptive controller case.Fig. 9 exhibits the associated mean time histories for both nodes 1 and 2, for X = 0 and 1, respectively; the additional f o n e standard deviation plots were not included as in Fig. 7 because they were small and not very informative.The moving-bank MMAC was able to perform as well as the artificially informed nonadaptive single controller, and performance was not degraded (but in fact improved) by unmodeled mode effects.It was also determined that the moving-bank MMAC algorithm can maintain good control of the structure in the face of both unmodeled higher order effects and an unanticipated external disturbance.At 4 s into the simulation, the dither was turned on again (with a magnitude of 50 rather than 10 as for the initial half-second) but closed-loop control was maintained.
Although the velocities were affected directly, the disturbance effects were completely quelled in about 1.5 s after the disturbance was removed, and the node 1 and 2 positions and position estimate errors displayed virtually no disturbance effects throughout the entire simulation time.

VII. SUMMARY
The robustness capabilities of moving-bank MMAE and MMAC algorithms have been investigated for application to a flexible spacestructure in which parameter uncertainties dictate a need for online adaptivity.For both estimation and control, the algorithms are sufficiently robust to allow a 6-state dynamics design model to yield excellent performance against a 24-state "truth model" of the actual system.Particularly for control, where nonadaptive controllers were shown to lead readily to unstable closed-loop systems, the moving-bank MMAC provided control that was essentially equivalent to that of an artificially informed benchmark controller.
The values of the measurement noise covariance play an important role in the performance potential of the moving-bank algorithms.A range of admissible measurement precisions exist, beyond which the effective movement of the bank in parameter space is seriously impaired.Similarly, appropriate discretization of the parameter space as used in this study is essential, in order to ensure that the various elemental filters in the multiple model adaptive structure can be distinguished from each other on the basis of their residual characteristics.The less precise the measurements are, the coarser this discretization must be.Modifications to bank-moving logic or to the computation of the p k ( f i ) probabilities and/or the state estimate or control outputs can enhance algorithm capabilities.Current research is exploring the sensitivity of performance and robustness to these and other important factors.
of Xchnology The robustness of a moving-bank multiple nmdel adaptive estimatodcontroller (0 order reduction in the controller design model is examined.It is shown that the adaptive mechanism and bank-mving logic are not confounded by the effects of unmodeled higher order d e s of a large flexible spacestructure.Control characteristics are achieved that are essentially equivalent to those of an artificially informed benchmark controller.Manuscript received January 21, 1991; revised September 21, and October 21, 1991.IEEE Log No. 9107195.Authors' current addresses: P. S. Maybeck, Dept. of Electrical and Computer Engineering, Air Force Institute of Xchnology, Wright-Patterson Air Force Base, OH 45433-6583; M. R. Schore, Ballistic Systems Division, Norton Air Force Base, CA 92409.U.S. Government work.Not protected by U.S. copyright.
n-vector representing physical controlled variables.Constant n-by-n mass matrix.Constant n-by-n damping matrix.Constant n-by-n stiffness matrix.Vector of length r representing actuator n-by-r matrix identifying position and relationship between actuators and controlled variables.Vector of length s representing dynamics driving noise, where s is the number of scalar white noise inputs.n-by-s matrix identifying position and relationships between dynamics driving noise and controlled variables.input.The mass and stiffness matrices are obtained from finite element analysis [32].For this application, the control system is assumed to consist of a set of discrete actuators, and the external disturbances and unmodeled control inputs are represented by white noise.The state representation of (19) can be written as (20) x(t) = Fx(t) + Bu(t) + Gw(t) where x = [rTiTIT and the open-loop plant matrix F, the control input matrix B, and the noise input matrix G are given by