Adaptive Control Strategies for Vibration Suppression in Flexible Structures Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2653 A.M. Annaswamy Department of Aerospace and Mechanical EngineeringBoston University Boston, MA 02215 ana D.J. Clancy Missile Systems Division Raytheon Company Tewksbury, MA 01876 study if the adaptive system which is stable in the absence of these perturbations, will behave satisfactorily in their presence. The results obtained using this approach have been grouped under the heading of Tobust adaptive c o n t ~ o P - ~Parallel ~. results have also been obtained for systems with multi-inputs and multio u t p ~ t s ' ~ -as ~~ well , as for stochastic systems16. In all cases, the aim has been to develop conditions for global boundedness of the underlying adaptive system. Requiring that the order of the system be known is quite a restrictive condition, especially in the context of flexible structures. Even if this number is known, often, it tends to be quite large. As a result, it is not feasible to implement the requisite controller, since the order of the controller depends directly on the order of the system. Considerable effort has been directed towards establishing satisfactory behavior of adaptive systems using low-order controllers. Despite this, the existing results still prove to be Active control of large flexible structures is an important topic inadequate for flexible structures. that has been studied by researchers over the past two d e ~ a d e s l - ~ . The control of flexible structures is a highly researched area Due to the large size and light weight of these structures, proband has received considerable attention during the past decade. lems related to their control assume paramount importance, and In fact, optimal control theory has served as the cornerstone for control issues such as vibration suppression, and shaping need to designing control systems for flexible space structures17. Modal be adequately dealt with. Many of these structures possess indecoupling procedures have demonstrated robustness to paramherently low damping, thereby necessitating control procedures eter uncertainties, along with eliminating spillover effects from that will introduce damping using active means. residual modes, but require that each controlled mode have arL In addition to being poorly damped, the elastic modes of flexindividual actuator associated with it'8. The use of positivity ible structures are quite often large, and possess uncertainties. It concepts is another common approach used for flexible structures. is well known that modal models of large structures derived using It is well known that the use of colocated velocity output feedfinite-element methods contain about 10% errors in both modal back results in a stable design without requiring high order truth frequencies and mode shapes4. Active control methods based on models, modal truncation, or accurate knowledge of the modal finite-dimensional techniques unavoidably introduce modal trunfrequencies or modal However, in such cases, since cations at some point. Due to these reasons, not only are the the underlying requirement is the positive realness of the sysmodal frequencies and the modal shapes unknown, but since diftem's transfer function, it implies that the controlled outputs ferent modes may be excited during different operating condimust be from either velocity sensors, or velocity with scaled potions, the total number of dominant modes in any given problem sition sensors, which makes position following impossible. LQGmay also be unknown. Structural modifications, failure of system based controllers have demonstrated superior performance over components, changes in the operating environment, and actuatorcompensators using colocated actuators with position and velocand sensor-dynamics are other sources of significant uncertainties. ity sensors, along with requiring fewer control inputs to meet Since large deformations are invariably present in these problems, performance requirementsz1. Control systems synthesis in the the effect of nonlinearities cannot be ignored. The use of adaptive presence of modeling uncertainties and parameter errors has been controllers, which can cope with the uncertainties and deliver the accomplished using LQG/LTR loop shaping methods. LQG/LTR required performance, is therefore qf considerable interest and controllers are excellent a t alleviating spillover effects common to importance for the control of flexible structures. flexible space structures, but can be overly conservative when The field of adaptive control has evolved over the past thirty satisfying robustness constraints, which tends to produce infeyears and grew out of attempts to control systems that are parrior performancez2. Modifications to the LQG/LTR robustness tially known. A significant part of this field has addressed dytests have been proposed which have resulted in less conservanamic systems that have parametric uncertainties. The adaptive, higher performance controllers. Unfortunately, LQG and tive control of linear time-invariant plants with unknown transLQG/LTR controllers designed using reduced order space strucfer functions has been studied at length and i s currently well ture models do not automatically guarantee closed-loop stability understood6. The major landmark in this field is the proof of of the actual space structure. In addition, fixed and LQG-based global stability of a class of adaptive systems that arise while controllers are extremely dependent on the accuracy of the plant controlling such plants, which was vstablished in 1980. This remodel, i.e., modal frequencies and mode shapes, along with the sult holds under certain conditions regarding the plant transfer structure and order of the reduced order plant model used to defunction, one of which is that its order must be known. Also, the velop the compensatorz3. All of the above indicates that control order of the controller that *stabilizes the plant is almost twice methodologies involving fixed compensators may not be adequate as large as that of the plant. Since 1980, several extensions to for the control of flexible space structures, and that new and adaptive control theory have been attempted to include timeinnovative strategies, which utilize adaptive compensation with variations, unmodeled dynamics, and nonlinearities in the plant. time-varying parameters, may be warranted. The P-rnach used was to model these as perturbations, and to Flexible structural systems are high dimensional and lightly damped, and invariably contain significant uncertainties in their dynamic behavior. Adaptive controllers, which are capable of overcoming such uncertainties and delivering high performance by providing a time-varying compensation on-line, are therefore desirable for such systems. In this paper, we present a new adaptive controller which can globally stabilize a class of flexible structures. This controller is applicable whether position measurements, rate measurements, or combinations thereof are available, as well as for colocated and non-colocated actuator-sensor pairs. The superior performance generated using such controllers is demonstrated using two practical structural systems. Copyright @ 1991 American Institute of Aeronautics and Astronautics, Inc. All rights reserved. 501 . I Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2653 Adaptive control of flexible structures has been considered elsewhere in [24-261. The approach that has been used in these papers is based on what is termed as the Command Generator Tracking (CGT) For adaptive control based on the CGT method to result in global stability, several assumptions have to be satisfied". Of these, the most restrictive one, is that the outputs used for feedback must include rate measurements. As in the case of the fixed compensators, this arises because an underlying transfer matrix in the system is required to be strictly positive real. Other conditions that have to be met which also prove restrictive are that the plant has to be (i) invertible, (ii) stabilizable using output feedback, and (iii) minimum phase, (iv) that no transmission-zero of the plant can be equal to an eigenvalue of the reference model, and (vi) that the model outputs must be generated from the solution of homogeneous differential equations. Some attempts have been made to relax these conditions, but invariably they have led to other assumptions which are either more restrictive, or unverifiable prior to the application in a practical problem. While this theory has been applied to flexible structures, many of the assumptions mentioned above have either not been shown to be satisfied in the structure, or clearly invalid. For instance, when rigid body modes are present, pure velocity feedback cannot be used due to assumptions (i), (iv), and (v). In [ 2 5 ] ,this difficulty was avoided by introducing an inner-loop augmentation, i.e., an inner loop control gain matrix is added to alter the modal characteristics of the plant and to convert the rigid body modes into fbite frequency modes. It was claimed that this matrix can be implemented without having accurate knowledge of the plant. Clearly, this approach ignores the destabilizing interaction that has been known to occur in many instances between rigid body modes and the flexible modes of a structural system. If the outputs from velocity -I- scaled position sensors are used for feedback, even though the rigid body modes are accomodated, position following cannot be ensured. Most importantly, in almost all results related to the CGT method, velocity sensing is a must, and has even been claimed to be a necessity. In this paper, we develop new adaptive control strategies for vibration suppression as well as shape control in flexible structures with unknown mode shapes, modal frequencies, and number of dominant modes. These adaptive controllers do not require prior knowledge of the order of the system, are of low order, have a simple structure, and guarantee stability as well as satisfactory performance. The main result in this paper is that regulation and tracking can be achieved using these new adaptive controllers for flexible structures whether (i) position measurements, (ii) rate measurements, or (iii) a combination of both, are available. The result also implies that satisfactory performance can be achieved not only with colocated actuator-sensor pairs, but also with a class of non-colocated actuator-sensor pairs. In section 2, we discuss the underlying dynamics of a flexible structure. We consider a linear finite-dimensional model, whose parameters correspond to the modal frequencies and the mode shapes. We then derive the input-output representations for different cases that depend upon the number, the locations, and the types of actuators and sensors. In section 3, we present the new adaptive controller, and state the main result of this paper in Theorem 1, where the global boundedness of all signals in the system is established. In section 4, the application of the new controller to flexible structures is considered. It is shown that the main result in Theorem 1 leads to vibration suppression as well as shape control in flexible structures. Due to space limitations, all results are stated without proofs. For the latter, we refer the reader to [30]. Finally, in section 5 , the performance of these controllers is evaluated in the context two flexible structures, an experimental control facility a t JPL, which has 30 modes whose frequencies lie between 0.1 and 5 Hz., and a flexible space station with 2 rigid body modes and 4 flexible modes31. The Dynamic Model 2. For a flexible space structure with small displacements, the linear dynamic equations describing the system are given by f -I- diag (2(iwi) i. + diag (w!) r = Bau + Bdv [ Y = [;I c:a (1) where T is the vector of n modal coordinates, and wi and (; are the natural frequency and damping ratio of the ith mode, respectively. u E IRna is a vector of actuator inputs, v E End is a vector is a vector of poof external disturbances, and y E I R n p + n c + n w sition, velocity plus scaled position, and velocity sensor outputs. The input matrices are given by [ Bo = b:], Bd b, = [ '1 dn where bi E ELna,d ; E I R n d , i = 1,. .. ,n,are the mode shapes evaluated a t the no actuator locations, and n d disturbance locations, respectively. The output matrices are defined as where hp; E E.'+, hc; E lR."c, h,; E IRnV,i = 1,...,n,are the mode shapes a t the np position sensor locations, n, scaled position velocity sensor locations, with Q > 0 as the scaling factor, and nu velocity sensor locations, respectively. It is assumed that the system parameters in (1) as well as the number of modes n are unknown, and that an arbitrarily small, usually smaller than 0.005 in flexible structures1, but nonzero amount of modal damping ( i is present. Depending on the number, the locations, and the types of actuators and sensors used, numerous input-output representations of the system in (1)c a n be derived. Por instance, if the actuators and sensors are colocated', it follows that + h,; V i = 1, ...,n = bi where the subscript z = p , c , or v , depends on whether position, scaled position velocity, or velocity measurements are used. On the other hand, if the actuators and sensors are non-colocated but sufficiently close, the mode shapes at these different locations are similar. In such a case, the relation +- h,; = (k+q)bi where k+ci>O V i = l , ...,n ( 3 ) may be satisfied. In such a case, i.e., if the locations are such that Eq. (3) is satisfied, we shall define the actuators and sensors to be prozimally located. The results developed in this paper are applicable for both colocated and proximally located actuator-sensor pairs. We now derive the underlying input-output representations and their properties. 502 A single-input single-output model: Assuming that there is a single actuator-sensor pair {ui,y i } , and a scalar disturbance vi a t the actuator location, and ni modes are controllable and observable, the input-output representation of the model in (1) is given by + = Wpi(B)ui(t) W d i ( s ) v i ( t ) * 'As in Ref. 1321,by colocated aensora and actuolorr, we mean that sensors and actuators are placed not only at the same physical positions but also along or about the name axis. Throughout this paper, the variable 8 will be used to denote the differentiation operator d / d t . The following conclusions can. be drawn regarding Wpi(s) and W d i ( s ) . v -K< fk,. (5) Kri Thus, when a position controlled dynamic system with rigid body modes satisfies Eq. ( 5 ) , the underlying transfer function has relative degree two, with zeros inC-. A similar property can also be derived when scaled position velocity sensors are used for a system containing rigid body modes. However, the use of velocity sensors in the presence of rigid body modes leads to undesirable pole-zero cancellations and, hence, is avoided. For a given flexible structure, the smallness of (Kfi/K,.i)depends on the sum total of the contributions from all flexible modes relative to that of the rigid body mode a t the i t h location. If this is small, which is the case for most flexible structures, the above discussion indicates that the underlying transfer function is minimum phase. For the class of flexible structures considered in this paper, we shall assume that if rigid body modes are present, they satisfy Eq. ( 5 ) . (4) + We note that W p i ( s ) is of order ni 5 n, and has relative degree (= no. of poles - no. of zeros) two. If a transfer function W ( s )= x y ! l ( k i / q i ( s ) ) ,where k; > 0 and q i ( s ) = s 2 + 2 ( i w ; s + w ~ , it can be shown that30 the zeros of W ( s )are in C-, the open left half of the complex plane. For both colocated as well as proximally located actuator-sensor pairs, from Eqs. ( 2 ) and ( 3 ) , it follows that the numerator gain hpkbk > 0 for all k = 1,. . . ,n i . Hence, it follows that in both cases, all zeros of W p i ( s )are inC-. Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2653 > 0 exists the zeros of Wpi(s) has roots inC- Case 1. Position MeasuremenkWhen y; corresponds to a position measurement, the transfer function Wpi(s)is given by + tively. This implies that an arbitrarily small constant k such that Case 2. Scaled Position Velocity Measurements:When yi corresponds to a scaled position velocity measurement, with a scale factor Q > 0, the transfer function Wpi(s)is given by + Remark 2.1: In all the above cases, except when the output corresponds to a pure velocity measurement, W , ( g ) is a minimum phase transfer function. This property follows since (i) eachmode is assumed to have nonzero damping, (E) the numerator gain of the i t h mode h,;bi is positive, and (iii) the contributions from any existing rigid body modes are relatively large. All flexible structures have a certain amount 01 passive damping which justifies (i). Colocated actuator-sensor pairs automatically satisfy (ii), whereas non-colocated actuator-sensor pairs satisfy (ii) provided that the locations are sufieiently close, i.e., the pair is prozimally located. This is quantified in Eq. ( 3 ) . Theoretically, as the number of modelled modes for any system represented by two non-identical actuator-sensor locations becomes sufficiently large, W p ; ( s ) can become nonminimum phase, which is pointed out in [ 3 3 ] . However, for a given number of modes, ni, a finite, nonzero number of non-colocated actuator-sensor locations can be found for which the input-output transfer functions retain the minimum phase property. Once again, W p i ( s )is of order ni, but has a relative degree unity. As in case 1, if the input-output pair is colocated or proximally located, we have that the numerator gain h&bk > 0. Since a > 0, the zeros of Wp;(s)are [email protected] Case 3. Velocity Measurements:When yi corresponds to a velocity measurement, the transfer function Wpi(s)becomes As in the previous case, Wpi(s)is of order n;,and has relative degree unity. Even though h,,kbk > 0 for colocated and proximally located pairs, this only implies that all but one of the zeros of Wpi(s) are inC-, with one zero at s = 0. In all the three cases, the transfer function Wdi(s) between the disturbance and the output is given by A multi-input multi-output model: We now consider m actuator-sensor pairs {ui,y i } , i = 1,. ,m, which include np position sensors, n, scaled position velocity sensors, and nu velocity sensors, with m = np n, t np. The MIMO model can be expressed as + + where the subscript 2 = p p , or v , and p ( s ) = l , ( s a ) , or s depending on whether position, scaled position velocity, or velocity measurements are used. Therefore, the order and the relative degree of Wdi(s) are equal to those of Wpi(s).Since the disturbance can be located anywhere on the structure, the zeros of W d i ( s ) need not be inC-. For the dynamic model in Eq. (l),no rigid body modes were included, i.e., ( i , wi > 0 for all i = 1,. ,n. Obviously, the presence of rigid body modes leads to a modification in Eq. (1) and likewise in W p i . In this case, when a position sensor is used, the transfer functions Wr; and Wfi represent the contributions due to rigid body modes and flexible modes, respectively, a t the ith output. Eq. (4)is modified to be Wpi(s) = Wri(s) W f i ( s ) where W,i(s) = ( K p ; / s 2 ) ,and + + .. where y ( t ) = [ y l , ..., y,,,IT, u ( t ) = [ u l ,...,urn]*,and v ( t ) = [VI,. ,v,lT is a disturbance vector. We assume that the locations of the actuators, sensors, and disturbances are such that .. .. + The transfer function Wpi(s) of the ith subsystem between u; and yi will therefore satisfy all the properties discussed in the SISO case depending on the type of sensor a t the i t h location, for all i = 1 , . ,m. The same holds for W,i(s) as well. Such a decoupled system c a n be achieved by choosing the m locations in such a way that the n; modes present in each Wpi(s)are controllable and observable only through the corresponding input-output pair .. { u i ,V i } * Hence In summary, the class of flexible structures that is considered in this paper is of the form of Eq. (6), where W p ( s )and Wd(s) satisfy Eq. (7) and the following properties hold. From case 1, it follows that K f ; > 0, and D j ( s ) and N f ( s ) are polynomials with roots in C-, of degrees 2 n and 2 n - 2, respec- ( P a ) The relative degree of Wpi(s) is equal to (Pl) The order n; 5 n. 503 i well as proximally located actuator-sensor pairs with either a pure velocjty measurement, or a velocity scaled position measurement. + (P3) The relative degree of W ~ ( sis)equal to two for both colocated and proximally located pairs when the output measurement corresponds to a pure position measurement. (P4) In cases 1 and 2, all zeros are in(C-; in case 3, one zero is at s = 0 and the remaining are in(C-. (P5) If a rigid body mode is present, it is assumed that its contribution is large relative to those of the flexible modes, i.e., Eq. (5) is satisfied. (P6) I f a disturbance is present, the relative degree of ( W d i ( s ) ) 2 Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2653 relative degree of ( W p ; ( s ) ) . S t a t e m e n t of the problem: With the model of the flexible structure given by Eqs. (6) and (7), the problem is to design a control input u such that when a disturbance v is present, or if there is an initial deflection on the structure, the displacements at various points on the structure settle down to zero as quickly as possible, Le., limt+m y ( t ) = 0. We also consider the problem of static shape control where it is required that the position response at different points on the structure are displaced by a f i t e amount34. As discussed in section 4, this problem can be posed so that the output y follows a desired trajectory ym, where the latter is specified as the solution of a homogeneous differential equation whose coefficients and initial conditions are appropriately determined. If the output error is defmed as e , where e = y, - y m , our aim is to choose the control input u so that limt+m e ( t ) = 0. When the assumptions (Al)-(A5) hold, we shall establish that an adaptive controller exists which ensures global boundedness of all signals in the system, and leads to command following as well as disturbance rejection. Without loss of generality, we assume that k, > 0. We present two error models, Error Model A and Error Model B, which illustrate the principle behind the new controller that is being proposed. Using these error models, we derive two lemmas, which in turn can be used to establish the main stability result of this paper. Error m o d e l A: In all adaptive control systems, there are two kinds of errors present, the output error e l , and the parameter error #. While the output error el(t) can be measured at every instant of time, the parameter error # ( t ) is unknown but can be adjusted a t every instant. If the adaptive system is such that the relation between these two errors is of the form where w ( t ) is a vector-signal present in the system that is accessible for measurement, v ( t ) is a scalar signal that arises due to disturbances, and W ( s )is a strictly positive real transfer function, we refer to Eq. (10) as Error Model .A. Lemma A summarizes the various results that can be derived for this error model. L e m m a A: In Error Model A given by Eq. eter # b e adjusted according to the rule (lo), let the pararn- r = rT> 0. i ( t ) = -rel(t)w(t) ff the disturbance has a finite energy, 3. A New Adaptive Controller v E L2 In this section, we consider a single-input single-output linear + e l ( t ) and # ( t ) are bounded V t t-m lim e1 E f?. e ( t ) = 0, t+m lirn e l ( t ) = 0. Error m o d e l B: In the context of flexible striictures, a special case needs to be addressed, which arises when only velocity neasurements are available. In this case, the underlying error :quation is once again of the form where u is the control input, yp is the measured output, and V I is a bounded external disturbance. The transfer functions between the input and the output, and the disturbance and the output, are respectively W,(s) = h," ( S I - AP)-' bp = k,H and W ~ ( S = )h,' (SI- A P ) - l d p .The desired trajectory that the output y p must follow is specified as the scalar output ym of a homogeneous differential equation referred to as Error Model B, where all the variables are as defined in Error Model A with the exception that the transfer function W ( s )is positive real. Lemma B summarizes the results obtainable from Error Model B. (9) where Am is any stable matrix in lR."mx"m.In the absence of disturbances, the control objective is to ensure that the error el between the plant output and the model output defined as el(t) = y p ( t )- y,(t) must satisfy the condition e l ( t ) = 0 and if disturbances are present in the plant, then it is only required that lel(t)l must remain bounded for t 2 t o . Our main result requires the following assumptions regarding the plant transfer function: L e m m a B: Let the state-variable description corresponding to Error Model B be given as ; = Ae + b(CTw + v ) el = hTe (13) where W ( s ) = hT(sI - A ) - l b is positive real. If a matrix P = PT > 0 exists such that it satisfies the equation A T P + P A = -hhT-Mo ( A l ) The relative degree ' n of Wp(s)must be known, and ' n 5 2. (A2) The zeros of W,(s) must lie i n C . Pb = h where MO,= MOT 2 0, and the parameter q5 is adjusted as (A3) The sign of kp must be known. i ( t ) = -rel(t)w(t) We also make the following assumptions regarding the disturbance: r = rT > o (11) then v E L2implies that e l ( t ) and + ( t ) are bounded for all t 2 t o and el E L2.If, in addition, w ( t ) and v ( t ) are bounded, then limt+m q ( t ) = 0. (A4) vi E L2. 2 and tf, in addition, w ( t ) and v ( t ) are bounded, then time-invariant plant whose order as well as parameters are unknown. The plant is represented by a differential equation (A5) Relative degree of W d ( s ) 2 to, n*. R e m a r k 3.1: In the adaptive control literature, the concept of error models has been extensively used. The use of error models 504 no longer holds, since Lemma A is no longer applicable. In fact, bounded disturbances can be found for which the controller in Eq. (16) leads to unbounded solutions [ 5 , Chapter 81. To ensure boundedness in such a case, modifications to the adaptive law have to be introduced. For instance, the adaptive laws in Eq. (16) have to be modified as enables one to develop an intuitive understanding of the stability questions that may arise, without a detailed analysis of the overall system dynamics. These error models have been developed thus far mainly for the disturbance-free case. In this paper, we have developed two error models, both of which include a disturbance as well as a dynamical system which is either strictly positive real or positive real. &(t)= -ueo(t)-el(t)y,(t), Remark 3.2: The results of Lemma A and Lemma B hold provided the disturbance v E L 2 . Disturbances due to measure- Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2653 (Al)-(A5), and the model is given by Eq. (9), then a control input u(.) can be found such that all the signals in the loop are globally bounded, and (15) Remark 3.8: If additional prior information is available regarding the poles and zeros of Wp(s), the controller in Eq. (17) can be simplified further. In the context of flexible structures, often the poles (pi's) and zeros ( x i ' s ) of W p ( s )satisfy the relation The input is given by [Fig. 11 + u(t) = e ~ ( t ) ~ p ( t )C(t)zrn(t) e O ( t ) = -7Oel(t)Y~(~) 70 > 0 em(t) = -rmei(t)zm(t) rm >o (16) k(pi)-F j=l( x j ) < o i=l when n' = 1, and by [Fig. 21 This in turn enables us to simplify the control input as + 4 ( t ) = -xcw1(t) u(t) u ( t ) = Bo(t)~p(t) p(t)wl(t) e$(t)zm(t) - el(t) ( i ~ : ( t ) ~ : ( t ) zz(t)zrn(t)) e O ( t ) = -7Oel(t)Vp(t) 70 > 0 P ( t ) = -7pel(t)vl(t) 7P > 0 ern(t) = -rmel(t)zm(t) rm> o + + + + + Remark 3.3: In contrast to the standard adaptive controller structure used in the literature, we note that the controller in Eq. (16) is simple, consists of n, + 1 adjustable parameters, and, most importantly, does not require the knowledge of the order n of the plant. Remark 3.4: The control parameters in Eq. (16) can be modified to include a proportional adjustment as "(t) = eo(t)Yp(t) +e W m ( t ) + [email protected]) = eop(t) eop(t) = -el(t)yp(t) where p, = x, (17) when 71' = 2, where a , ( t ) = ( l / ( s + a ) ) w i ( t ) , Vp(t)(l/(s+a))yp(t), zm(t)(l/(s ~ ) ) z , ( t ) , and a , ~>, 0. To facilitate the understanding of the main result in Theorem 1, we provide below a few qualitative remarks. While all comments are made in reference to the controller in Eq. (16) for n* = 1, they are equally applicable to the controller in Eq. (17) which is used when n* = 2. - >o Remark 3.7: It should be pointed out that the assumption (A2), which requires all zeros of the plant to lie inC-, limits the applicability of this controller. In the context of flexible structures, it is well known that nonminimum phase transfer functions are quite common, especially in the case of non-colocated actuator-sensor pairs33. Extensions of the approach presented here to plants with arbitrary zeros are currently under investigation. Theorem 1: When the plant in Eq. (8) satisfies assumptions c'. (T Remark 3.6: Elsewhere in the literature, boundedness has been established for adaptive systems with relative degree unity. We are however establishing for the first time that boundedness follows even when the underlying system has relative degree two, and therefore represents a significant breakthrough in this area. We now state the main stability result in Theorem 1. E -cre,(t)-el(t)z,(t), : - which can be shown to result in boundedness of all solutions. A similar modification to the adaptive laws in Eq. (17) will assure boundedness when n* = 2 as well. ment noise, or actuator errors invariably contain finite energy, and hence, E L2. However, if disturbances are present due to other sources of uncertainties such as nonlinear dynamics, such an L2 assumption may not hold. If these uncertainties are such that v E Cm, the results of Lemma A and Lemma B are no longer valid. The adaptive law for adjusting the parameter error 4 has to be appropriately modified to assure boundedness of the errors e and 4. el e,(t) - &(t) = Qmp(t) t ern(t) OmP(t) = -ei(t)zm(t) with 00 and 0, adjusted as in Eq. (16). The proof of stability can once again be established along very similar lines to that in Theorem 1. Remark 3.5: The global stability in Theorem 1 follows provided the disturbance u E L2.If on the other hand, u E Coo,this result + 6, and 6 is an arbitrary positive constant3'. Remark 3.9: All our discussions in this section pertain to SISO plants. While the result in Theorem 1 can be extended simply to totally decoupled MIMO plants and with relative ease to diagonally dominant MIMO plants, the problem proves to be nontrivial for the case when the MIMO plant is symmetric and strongly coupled. It is the latter case that is of interest in a flexible structure. Work is under progress to determine the least restrictive set of assumptions on the plant transfer matrix under which a stabilizing controller exists. 4. Adaptive Control Strategies for a Flexible Structure In section 2, the dynamic models that describe the behavior of a flexible structure were discussed at length. It was seen that these structures have high order, low damping, unknown moaal mequencies and/or modal shapes, and even an indeterminate number of dominant modes. The control of these structures has to therefore be carried out in the presence of such uncertainties. The new adaptive controller developed in the previous section is ideal for realizing the control objectives commonly encountered in these structures such as vibration suppression and shape control. In this section, we show that indeed the results of Theorem 1 are applicable to the control of a flexible structure, and consider these two control objectives in sections 4.1 and 4.2. In each case, we show that satisfactory control is achievable whether (i) only position measurements are available, (ii) a combination of position and velocity measurements are available, or (Gi)I only 505 uniform settling time, it follows that the real parts of the (2n-2) zeros are identical to that of 2n- 2 poles which once again leads to the relation in Eq. (20). In such cases, as pointed out in Remark 3.8, the parameter p ( t ) no longer needs to be adaptively adjusted and can be fixed as p ( t ) E p , = zc 6,6 > 0. This simplifies the control input as velocity measurements are available. Also, we show that these objectives are attainable using colocated as well as proximally located actuator-sensor pairs. 4.1 + Vibration Suppression In section 2, it was shown that the input-output representation can be expressed as y = Wp(s)u + Wd(S)Y (6) where y , u , v : lR+ -+ IR", and that the locations of the m actuator-sensor pairs can be chosen in such a way that the transfer matrices Wp(s)and Wd(a) are diagonal. It was also seen satisfies properties in section 2 that each diagonal entry W+(s) (Pl)-(P5) for i = 1,.. ,m. The control objective is to choose an input u in Eq. (6) so that the displacements in various locations on the structure settle down to zero as quickly as possible, even and Wd(s) are unknown. In Theorem 2, we describe when W,(s) the adaptive controller that generates the control input which ensures global stability as well as vibration suppression in all the modes that can be controlled through the m actuators in Eq. (6). Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2653 . Theorem 2: Let the dynamic model in Eq. ( 6 ) denote the inputoutput relation of a flexible structure. Let the m sensors consist of np position sensors, n, velocity scaled position sensors, and nu velocity sensors. The transfer matrix Wp(s) can be arranged in such a way that the sensor outputs yi, i = 1,.. . ,np, correspond to position measurements, y;, i = (n, l), ..,(n, t n,) correspond scaled position sensors, and y;, i = (np n, to velocity 11,. . ,m, correspond to velocity sensors, where m = n,+n, tn,. Let u = [q,. ..,umIT and y = [yl, . ,ymIT. If + + . + . + + .. -zcWi(t) t u;(t) 2, > 0 Boi(t)yi(t) +pi(t)wi(t) - yi(t) ($(t) t @ ( t ) ) -7i~i(t)jii(t) ri > 0 -rpi~i(t)gi(t) rpi > 0 (18) % ( t ) = (Wo(s))~;(t) gi(t) = ( W o ( s ) ) w ; ( t ) 1 W.(S) = a>O a+a hi(t) = ~ ( t = ) $i(t) = ?ii(t) = ..,np,and for i = 1,. ui(t) = eoi(t)yi(t) = -7i~"t) &i(t) ri > 0 (19) + . for i = np 1,. .,m, global boundedness of all signals follows and l i ~ - , - y ( t ) = 0. Remark 4.1: The adaptive controller suggested in this paper is applicable to the control of all flexible structures which can be described by decoupled, linear, finite-dimensional, MIMO models, including rigid body modes as well as lightly damped flexible modes. Most importantly, the results of Theorem 2 imply that stable control of flexible structures can be achieved using only position measurements, with either colocated actuator-sensor pairs, or sufficiently close non-colocated pairs. Without loss of generality, this input can also be expressed as In simulation studies, it is seen that such a control input leads to good performance with reasonable values of BO;. It is obvious that extensions to strongly coupled MIMO plants as well as to plants with arbitrary zeros need to be established before the approach presented in this paper can be considered as a viable practical methodology. We strongly believe that the controller suggested here will result in boundedness even for these extensions. In fact, in section 5 , we provide simulation studies of two practical flexible structures that have been studied extensively in the literature, which illustrate that superior performance can be obtained using this controller not only for decoupled MIMO plants but also for strongly coupled ones. 4.2 Shape Control We now consider the problem of shape control where it is required that the displacements at various points on the structure have to achieve certain steady-state values so as to obtain a static shape. The desired values for the various points can then be represented as ym, the output of a homogeneous differential equation, as in Eq. (8). The problem is then posed as the choice of u in Eq. (6) so that y, follows ym asymptotically. The results of Theorem 1 once again enable us to determine such an input which is only a slight modification of that in Theorem 2. An additional term OZz, is added to the control input, where z, is the state of the model in Eq. (8), and 0, is a time-varying parameter that is adaptively adjusted. This is stated in Theorem 3. Once again, the disturbance Y is assumed to have a finite energy. For ease of exposition, we assume that Eq. (%a) is satisfied by the flexible structure. Theorem 3: Let the dynamic model in Eq. (6) denote the inputoutput relation of a flexible structure, and Eq. (8) specify the desired response. Let the output y consist of n, position sensors and nc velocity t scaled position sensors, so that y;, i = 1 , . . . ,n,, correspond t o position measurements, yi, i = (n, I ) , . . .,(np ne) correspond to velocity + scaled position measurements. We define 0, = [Bo;, BZilT, and W; = [yi(t), z f ( t ) l T . If the various components ui of the control input u are chosen as + A Special Case: Often in many flexible structures, the damping characteristics are satisfy the relation such that the poles and zeros of W+(s) . . ,n,, for i = 1,. where pi and zj are the real parts of the ith pole and the j t h zero. For instance, when there is uniform modal damping, using simple extensions of the result in [32], it can be shown that Pi < zj < P;+I v i , j = 1,...,n - 1 and u;(t) = (20) B?(t)w;(t) ei(t) = -r;e;(t)yi(t) Bmi(t) = -I'miei(t)zm(t) + + . ri > o rmi> o (2%) for i = np 1,. . ,m, global boundedness of all signals follows, , ~ = 0. and l i ~ - ei(t) which in turn implies Eq. (20). Similarly, when all modes have a 506 Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2653 In this section, simulation studies of two popular flexible structures are presented. The f i s t is an experimental facility called the Large Spacecraft Control Laboratory developed jointly by JPL and AFAL, which replicates the main properties of a flexible space structure that are most relevant when implementing active control methods. The second is a flexible space station with a two-panel configuration. In both cases, we discuss the dynamic model, the number of dominant modes, the modal frequencies and the mode shapes. We then choose the actuator-sensor locations, their number, and the type of sensors. We incorporate the new controller proposed in this paper, and present the resulting system performance. In each case, we discuss whether the assumptions under which the proposed controller is stable are indeed satisfied by the structures. 1. The LSCL: The structure is a large, 20 foot diameter, 1 2 rib circular antenna-like flexible structure with a gimballed central hub and a long flexible feed-boom assembly (Fig. 3). The ribs are very flexible in the vertical, out-of-plane direction, and are coupled to one another by tensioned wires which dynamically simulate the coupling effect of a mesh on a real antenna. A 10 degree-of-freedom finite-element model of this structure yields 30 flexible modes, all of which are below 5 Hz and no rigid body modes. All modes are assummed to have a uniform modal damping of 0.001. A total of 6 actuators can be placed at the locations H1, H10 (hub torquers), Rl,R4,R7, and R10 (rib root torquers). 30 sensors can be placed throughout the structure, including the 6 actuator locations and 24 locations on the 12 ribs. The disturbance input is a single, 0.20 Hz pulse with an amplitude of 1.5, which is introduced a t the hub torquers. The control objectives are to minimize the the displacements at various locations on the structure, while returning the structure to equilibrium as quickly as possible. Table 1 summarizes the 2 cases examined for the LSCL. Case Transfer Sensor Adaptive Functions Type Gains from to pposition, u-velocity 1 H1-H1 P 71 = l E l l R1-R1 u+p 72 = l E l l __ 2 U R1-R1 R4-R4 R7-R7 R10-R10 P ~ + U 71 = 1E6 p 72=1E6 73 = 1E6 7 4 = 1E6 a decoupled 2x2 system and hence is a direct application of Theorem 2. All but 8 of the 30 modes were controllable. In this case with position measurements and with velocity+ scaled position measurements, shown in Figures 4 and 5 , as the adaptive gains 71 and 72 increased, the settling times of the position responses decreased, but the magnitude of the actuator inputs increased as well. Case 2 leads to a strongly coupled 4 x 4 plant transfer matrix, and hence the theory developed in this paper is not directly applicable. However, when we used the same controller structure as in Theorem 2, we obtained very satisfactory performance. Similar observations and conclusions to those in case 1 can be drawn. The responses using position sensors and velocity -t scaled position sensors are presented in Figs. 6 and 7. The amplitudes of the actuator inputs and the steady-state feedback gains 01 and 82 were in fact slightly lower than those for case 1, since the burden of control was spread over 4 actuators, decreasing the requirements on any one actuator. These results give credence to OUT belief that the approach reported here can be extended to general multivariable systems that occur in flexible structures. Next, case 1was modified to include the same f i s t colocated actuator-position sensor pair and a proximally located actuatorposition sensor pair at R1-LI1, with the resulting 2x2 system being totally decoupled. With 71 = 1E6 and 72 = 1E12, the responses were excellent only at the locations controlled by actuator H1, and at position sensor location, LI1, which was controlled by actuator R1. However, the responses at all other sensor locations affected by actuator R1 were unsatisfactory. Similar observations were made when 4 proximally located actuator-position sensor pairs, R1-LI1, R4-L14, R7-LI7,and R10-LI10, were used. This indicates that while the theory developed in this paper is applicable to non-colocated actuator-sensor pairs, more work remains to be done before realizing results that are practically reasonable. 2. The Flexible Space Station: Various configurations have been developed by NASA for the proposed space station. One of them has a two-panel planar configuration, whose dynamic model consists of 2 rigid modes, and 4 flexible modes between 0.04 Hz and 0.3947 Hz. The problem is to determine a control strategy to contain the effect due to initial condition deflections. The initial conditions placed on the individual nodes of the space station were the same as those used in [25]. In addition to being strongly coupled, the underlying system also has rigid body modes. With the same adaptive controller structure as in Theorem 2, we obtained uniformly superior performance (see Figures 8 and 9) with a simple structure utilizing 4 adjustable parameters when position measurements are utilized and 1adjustable parameter when velocity scaled position measurements are used. A comparison of Figures 8 and 9 clearly demonstates the advantages of using pure position measurements over velocity scaled position responses. Figure 9 further clarifies the fact that a satisfactory velocity + scaled position response doesn’t always result in a satisfactory response a t the corresponding position sensor. The benefits of using pure position measurements with the new adaptive controller over the approach used in [25] are obvious. + In Table 1, The second column indicates the input-output pairs and their locations, the third column indicates the types of sensors used, and the fourth gives the values of the adaptive gains used. All initial conditions are set to zero. The closed-loop responses obtained a t the various sensor locations along the structure, a t the actuator inputs, and at selected signals in the feedback loop, for the 2 cases presented in Table 1, are shown in Figs. 4-7. The adaptive controller for all cases is turned on after 5 seconds, providing a direct comparison between the amplitudes of the open-loop and closed-loop responses. In addition, since the open-loop settling times would be extremely slow due to assuming a uniform modal damping of 0.001, a direct comparison between the settling times is demonstrated as well. The units for the hub sensor responses (Hl,H10), for the rib sensor responses (Rl, R4, R7, R10, LI1 - LI12, and LO1 - LOlZ), and the actuator inputs are in radians, meters, and Newton-meters, respectively. Since the velocity with scaled position responses were exactly the same as the pure velocity responses, only measurements from the velocity with scaled position sensors are presented. Case l leads to + 6. Conclusions In this paper, a new adaptive controller is proposed to achieve the objectives of vibration suppression as well as shape control in flexible structures. The controller is simple, requires no knowledge of the order of the underlying dynamic model, and achieves global boundedness as well as the desired objectives. By making model, it is shown that satisfactory use of the underlving- dynamic performance can be realized using only position measurements. Such a result, to the authors’ knowledge, is totally new and is being reported for the fist time in the literature. The same results I 507 I can also be derived using either only velocity measurements, or velocity scaled position measurements. Both colocated as well as proximally located actuator-sensor pairs can be used. This paper represents the f i s t step in realizing a practically viable adaptive controller for flexible structures. The main result reported in this paper is based on the assumptions that underlying dynamic model is decoupled, and minimum-phase. These assumptions have to be relaxed before significant practical results can be derived using the suggested approach. Work is currently in progress in these directions. + Acknowledgments Meirovitch, L. and Baruh, H.,”On the Robustness of the Independent Modal Space Control Method,” Journal of Guidance, Control and Dynamics, Vol. 6, No. 1, Jan.-Feb. 1983, pp. 20-25. Arbel, A. and Gupta, N. K.,”Robust Colocated Control for Large Flexible Space Structures,” Journal of Guidance, Control and Dynamics, Vol. 4, No. 5, Sept.-Oct. 1981, pp. 480-486. Auburn, J. N., ”Theory ofthe Control ofstructures by Low-Authority Controllers,” Journal of Guidance, Control and Dynamics, Vol. 3, No. 5, Sept.-Oct. 1980, pp. 444-451. Sundararajan, N., Joshi, S. M., and Armstrong, E. S.,”Robust Controller Synthesis for a Large Flexible Space Antenna,” Journal of Guidance, Control and Dynamics, Vol. 10, No. 2, March-April 1987, pp. 201-208. ” The work reported here was carried out under NSF Grant No. ECS 8915276. We would also like thank Dr. Aziz Ahmed a t JPL for providing us with the data for the LSCL structure. Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2653 l7 Martin, G. D. and Bryson, A. E.,”Attitude Control of a Flexible Spacecraft,” Journal of Guidance and Control, Vol. 3, No. 1, Jan.-F&-. 1980, pp. 37-41. a1 Blelloch, P. A. and Mingori, D. L.,”Robust Linear Quadratic Gaussian Control for Flexible Structures,” Journal of Guidance, Control and Dynamics, Vol. 13, No. 1,Jan.-Feb. 1990, pp. 66-72. References ” Clancy, ‘Balas, M. J.,“Trends in Large Space Structure Control Theory: Fondest Hopes,Wildest Dreams,” IEEE Transactions on Automatic Control, Vol. AC-27, No. 3, June 1982, pp. 522-535. ’Meirovitch, L., Baruh, H., and Os, H., “A Comparison of Control Techniques for Large Flexible Systems,” Journal of Guidance, Control and Dynamics, Vol. 6, No. 4, July-August 1983, pp. 514-526. SNurre, G. S., Ryan, R. S., Scofield, H. N., and Sims, J. L., “Dynamics and Control of Large Space Structures,” Journal of Guidance, Control and Dynamics, Vol. 7, No. 5, Sept.-Oct. 1984, pp. 514-526. 4Benhabib, R. J., Iwens, R. P., and Jackson, R. L.,“Stability of Large Space Structure Control Systems Using Positivity Concepts,“ Journal of Guidance ,Control, and Dynamics, Vol. 4, No. 5, Sept.-Oct. 1981, pp. 487-494. 6Narendra, K. S., and Annaswamy, A. M., Stable Adaptive Systems, Prentice- Hall Inc., Englewood Cliffs, N.J., 1989. ‘Peterson, B. B., and Narendra, K. S., “Bounded Error Adaptive Control,” IEEE Transactions on Automatic Control, Vol. AC-27, No. 6, Dec. 1982, pp. 1161-1168. 7Kreisselmeier, G., and Narendra, K. S., “Stable Model Reference Adaptive Control in the Presence of Bounded Disturbances,” IEEE Transactions on Automatic Control, Vol. AC-27, No. 6, Dec. 1982, pp. 1169-1175. D. J.,”Vibration Suppression in a Flexible Space Structure,“ M.S. Thesis, Boston University, Jan. 1991. Bar-Kanna, I., Kaufman, H., and Balas, M.,”Model Reference A d a p tive Control of Large Structural Systems,” Journal of Guidance, Control and Dynamics, Vol. 6, No. 2, March-April 1983, pp. 112-118. ” Ih, C. C., Wang, S. J., and Leondes, C. 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Annaswamy, A.M. and Clancy, D.J., “Adaptive Control Strategies for Vibration Suppression in Flexible Structures,” Submitted to the AIAA Journal of Guidance, Control, and Dynamics, 1991. ’’ Vivian, H. C., Blaire, P. E., Eldred, D. B., Fleischer, G. E., Ih, C. ‘Ioannou, P. A., and Kokotovic, P. V., Adaptive Systems with Reduced Models, Springer-Verlag, Berlin, 1983. C., Nerheim, N. M., Scheid, R. E., and Wen, J. T.,”Flexible Structure Control Laboratory Development and Technology Demonstration,” Final Report, J P L Publication 88-29, Oct. 1987. DNarendra, K. S., and Annaswamy, A. M., “A New Adaptive Law for Robust Adaptive Control Without Persistent Excitation,” IEEE Transactions on Automatic Control, Vol. AC-32, No. 2, Feb. 1987, pp. 134-145. ‘ONarendra, K. S., and Annaswamy, A. M., “Robust Adaptive Control in the Presence of Bounded Disturbances,” IEEE Transactions on Automatic Control, Vol. AC-31, No. 4, April 1986, pp. 306-315. “Kosut, R. L., and Johnson Jr., C. 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A.,”Parameter Adaptive Control of Linear Multivariable Systems,” IEEE Transactions on Automatic Control, Vol. AC-27, No. 2, April 1982, pp. 340-352. Goodwin, G. C. and Long, R. S.,”Generalization of Results on Multivariable Adaptive Control,” IEEE Tranactions on Automatic Control, Vol. AC-25, No. 6, Dec. 1980, pp. 1241-1245. ’’Singh, R. P., and Narendra, K. S.,”Prior Information in the Design of Multivariable Adaptive Controllers,” IEEE Transactions on Automatic Control, Vol. AC-29, No. 12, Dee. 1984, pp. 1108-1111. l e Goodwin, G. C. and Sin, K. S., Adaptive Filtering Prediction and Control, Prentice-Hall, Englewood Cliffs, N.J., 1984. 508 Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2653 Figure 1: The new adaptive controller (n*= 1). Figure 3: A plan view of the LSCL structure with actuators and Figure 2: The new adaptive controller (n*= 2). A PY 75 1~ f : 5~ r 0 H 5 - . I : ' 1 1 ' 5 2 t L5 ~ " ,. " , " , " " " '- Figure 4: Adaptive control for a decoupled MIMO flexible structure with two colocated actuator-sensor pairs using position feedback. 509 Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2653 A 1.1 : 2.5 , ? Figure 5: Adaptive control for a decoupled MIMO flexible structure with two colocated actuator-sensor pairs using velocity scaled position feedback. + A P II R 0 A 0 L I 1 P A I P n A 0 I G 0 L 0 L I 0 Figure 6: Ad,aptive control for a strongly coupled MIMO flexible structure with four colocated actuator-sensor pairs using position feedback. 5 10 Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2653 Figure 7: Adaptive control for a strongly coupled MIMO flexible structure with four colocated actuator-sensor pairs using velocity scaled position feedback. + " P P I I 2 I Y P P ; I 1 I Figure 8: Adaptive control for a space station, a strongly coupled MIMO flexible structure with four colocated actuator-sensor pairs using position feedback. Figure 9: Adaptive control for a space station, a strongly coupled MIMO flexible structure wit four colocated actuator-sensor pairs using velocity scaled position feedback. + 511

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