School of Aeronautics ad Asronauti. Purdu Univerity. West Lafayette, TN whe. of yi(.) be. Fmid the gain. bjec. where. given dist (1.

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1 FA3 1:15 AN ITERATiVE ALGORIT1HM COMBINING MODEL REDUCION AND CONTlOL DESIGN AbWrad A design sragy which irgrates mode reduction by modal co alysis ax a muld-objective contoer design is sopmpod a Te ciary modeli}ng d on! aorhinms ae easly programmed in Malabstand sotwa. Hence, th medhod is very prncal for contr design f lre a p strtm"eurs 1designddgorithm also presents a ulution for the very imp rtt rbm of "tung muliple lop controle MMO). In stedoftwefslkgin clgs that is ued in tanda Row Loomsnd gain ai p e r thofis, the new nwthod tn multiple loop lr(n n"lw-o obhigh" gain in a sysematic way i the designp. 'is deign regy is applied to the NASA's Mini-Ms systm at Langley Research Cater. 1Ịn It is well known that finding a good model for control desig is a difficuk priben, in only becae the parels of the system ar uncerin, but became the model that is a e for contro deign depends on the contrt fcs (magniude ad 9 spectal cflrt). Cosquenty, mdlngi ax! contirol prblms remain an itratve proces See the exaples in [11. Here we iragrate a model r cio tm e ad a coroller design metbod to sy size a iominal for NASA's Mni-Mast syn. lhe model reducio te upe used he is the Modal Cost Analysis (MCA) which cal es each modal niutio Vi to a weighted quadratic cost fucton V=EyTQy=% V. where E. A lim E is the expeaton operor and N is hel number of modes in the model. The smalest modal contribuiom (snallest Vs) indicate the modes to be dekled in the reduced model. An importan feature of this MCA is tm there is a closed forn for each Vi. In this closed form shion only multicaon and additon are neded, no big Lyapunov equation needs to be solved, and no matrix manipulaton (such as inverse, SVD and eigen value computio) is involved. As a rsult, there is no limit on the system size, ax! m exact soluto can be obtaied. For fiurter information about MCA and its other cations to Mini-Mast see (1,2]. The emphasis of this paper is on Output Variance Constraint (OVC) controller design algorithm ad its integratio with MCA model reduction Basically, the pardcular OVC algorithm which incorporates with MCA provides an LQG controler with a specially tuned output weighting matrix. To be complete, we also discuss the general measurement feedback case, the fixed order and full order controller. In the OVC controler design the following linear timeinvariant stabilizable and detectable discrete system is given, x(k+l) = Ax(k) + Bu(k) + Dw(k) y(k) = Cx(k) zk) = Mx(k) u(k) = Gz(k) C. Hsit, J. H. Kimt, G. Zhjut, K Liut, R. E Skelton" School of Aeronautics ad Asronauti Purdu Univerity West Lafayette, TN 4797 (1.1) where xern, ye R", ze R, uelrc, we l" are vectors of state, t Research Assistant tt Professor output,ni aedierince, e dy. The definition o(the OVC ca be addessd in two differen ways - s cally ad- detemkisdtcly. - Stochsic Problm Outjxx Varisce Corstrint Control Conder system (1.1), where w(.) is a zer mean white noise with covaiac E(w(l)wTC)) = W. Fldx theiwo x gain G to minimri Js =E. utru sujc to the nequalty vilawe coisrain E.. y2j:9 q2, j = 1,.. Ny (1.2) where R is a sym ic poditve weighting matix. DeterminisicProblem - Output L. Coraint Conr Prolmn Cosder system (1.1), where w(k) is a derministic exogernms siga. Lette12 normofw() be whe of yi(.) be Fmid the gain bjec where IkU 2 A k= wt(k)ww(k), (1.3) W is a symmetic positive definite mtix, and the L. norm to given dist IYi(&.2 A V ;(@,. (1.4) W w() 1 11w12 P). minimize JD ri fsug,lkiudi2) r,>o oyi 41 r. e?, i = 1,... my 'Vwe w Ikk The significance of tihse two prblems is obmvious: (1.5) under a enviromet we seek a controler to satisfy the output constaints (either RMS value or maximum value) with minimum mtol effort (as determined by Js Md JD). A continuous verion of this problem has been studied in [3], which also contains a review of oter wlated workl Even hugh the problems stated above am to achieve some prescibed specificatons (a and c), the algorithm itself will automatically create a sequence of controlls with different contol gains. If we treat the specification also as a design paramneter, then this is very helpful for doing on site laboratory tests. Frm the safety point of view, because of the inevitable error of the model, it is usually more ap Dpie test a lower gain controler first, and then tune up the gan gradually. For the SISO system, this can be done by multiplying the controller with a scale factor, using a small factor first and gradually ineasing the scale factor if stability is maintained. However, for the MIMO 212

2 system, how to safely change the gain of a given controller is not clear because of the muliple loops. Fortunately, the OVC algorithm can solve this prblem. By testing the contollers cresed during the design procedure one by one (from low gain, of course, and no scale factor is needed for any of them), we can find an "appropriate" controller for the system. Therefore, this algorithm can also be eted as a self-tuming mechanism. As a byprduct of the OVC contrl design algorithm, the relative imporance of each output is reflected by the weighting matrix produced by the algorithm. This weighting matrix Q can then be used for MCA, in which Q is the only design parameter, and hence a more appropriate reduced model can be obtained using the new updated weight This is the essence of the philosophy to integrate MCA model rducon and OVC controller design. This design saegy tums out to be very useful for Large Space Stncus. The original idea was first applied to a 56-state model of NASA's Pinhole Occulter Experime on the space station. In this paper we modify the strategy and apply it to the NASA's Mini-Mast system at Langley Research Center sponsored by the Guest-Investigator (GD) Program. In section 2 some facts about sochasic systems and determinisfic systems wil be stated. With the help of these facts both problems can be simplified to one nonlinear programming prblem. Tle necessary conditions and design algorithm for this programming problem are given in sctim 3. In section 4, we show dte fixed order and full order controller design case and redesign the algorithm to integrate the full order controller design and model reduction. In section 5, we apply this integrated design algorithm to NASA's Mini-Mast system, folowed by some conclusions. 2. Fundamental Facts In this section some fundamel fac ar given for both the stochastic and detenninistic problems With the help of these facts we can conbine ese two problems into one problan. Consider (1.1) with a zero mean whte noise w(.) with covariance W. It is well known hat the suady tat covariance X is the solution of the following discrete Lyapov eqaion (4]. X = (A+BGM)X(A+BGM)T + DWDT (2.1) and the output and input variances can be exprssed as E y2 = [CXCT1. =qxct E,.uI = [GMXMTGTIh whe (.k stands for the ith diagonal elemert of (1 and is the it row of C. fe control effort Js can ten be written as Js = E.uTRu = trrgmxmtgt Hnce, the stchastic OVC problem is equivalent to the followmig nonlinear programming problem. Let Q A inssible set of (X,G) (A+BGM)X(A+BGM)T+DWT-X = -= (X,G) [CXCT]L S of, i=1,2,..., ny Find (X,G)e to minimize J = trrgm X>O XMTGT Now consider (1.1) again with w(k) being a deterministic exogenous signal and we W. Fron [5-71 we know that for the closed loop systen (2.2) and sm llylii = picxctju sium 111, = pigmxmtg (2.3) where X is the solution of (2.1) X = (A+BGM)X(ABGMT + DWDT Hence, (2.1) JDJo - ;ri 2~Jujui. ( SUP vi i=1 = ri P[GMXM GT1 = trrgmxmwt where R = diag (ri, r2..., rt,). As a result, by replacing ca? by 4 i = 1,..., ny, the equivalent non-linear programming problem for the L. inequality constraint problem is the same as that of the stochastic OVC problem. It is obvious now that both the deterministic and stochastic problems are mathematically exacdy the same. Furthermore, if wew then with the following w(k) and kf large enough we can make iyiell. -* PCXCT]U. See reference (6]. where Y(kf)= 4 WID T (A )T CTy-11(k) ; C k gcf(24 Wk, (k) (A";O.sk1(2.4 k> kf CAk DWDTCT. Here we assume Y(kQ) is nonzero. Since W is positive definite, if Y(kf) is zero, the ith outu will remain zem (if initially zero) for any w(s). (This is the cmplee disturbance rejection property.) The usefiuness of wk4(k) given in (2.4) is two-fold. Firt, the existence of such a w(k) proves that ti inequality the best possile bound. Second:ly, for lab Ijy,(e)fl.CKXCT is tests the disturbance wk,(k) can be applied to the systemn as a worst case check It is noted t as vjfe-, w4(k) is in fact the inverse impulse response of the following system = ATI(k) + CTU y =WIJT4 withi i() = and f() = (.p -)1. Studies [CXCT of k other worst case distubances can be found in (8]. 3. Necessary Conditions and the OVC Algorithm In the equivalent nonlinear programming problem, it is noted that the third condition in the admissible set, i.e., X >, is considered for the purpose of stability. Since X > is not a binding consai, this condition does not contribute to tie derivation of necessary conditions. Itiswellknownthatif(XXG)Elis aregular poire and is a klo minimizer for J, then there exist qkc ri,..., n,,) an! KJ1 = K!(ij 1,..., n.) such that (X*, G J q, Ki) is the solution to [aiax =o (3.1a) JJ/aKJi =O forall i,j (3.1b) aii-og= (3.1c) L Oand q&ktk-a=2)= k=l,..., ny (3.1d) where J reprsts t unostrained cost, J = trrgmxmtgt+ 5 K,, A Now since 5 K j = trka,(3.2) becones + E(T-oh ; = trrgmxmitgt + trka + trq[cxct -P] = J+trKA +trcxct -P3 (3.2) (3.3) 2121

3 whiere Q=diag[qlctoi p t *1 [1 ai... As a reut (3.la-c) is euwale to...s2.. vq 6, 2 (A+BGTK (A+BGM) + MTGTRGM +CT9C - K = (3.4) (A+BGMX(A+BGT+ DWT X = (3.5) [RGMbTK(ABGMIT =O (3.6) fle c don X > cmbined with the sm the cosed lop syn is d ae k to the sability of A + BGM. Funhemon, if A + BGM is ab, then ie soltin K to the Ly ov equatio (3.4) is at a postve nmidelkie From the above dsus we c u the fmowing terem. Thon 1 (fin ordeniecessy canditio) Jf(X,G)eeisaregulrpoh* mdisalocalmiminrfor J. then there ei a wno-gative denit diagonal matrix q..] ad a sm-negatv debit matrix KE R4XE, - ssisfy the conditions(3.4) (36) mad (3.1d). Equatku - (3.4) (3.6) are e sae a due for th discrte optimum measureme feedba r m: d to miize E. (utr + yt%y) with sm dional cmpu matix Q. With this obsvsk,n& we have the folowing comp algorithm for the gp varlae consa'i (OVC) control OVC Algoithm Givi data (A.B. C.D, a, W,.R,Q(O)yr>. n>o) 1. Start with Q(O) >-. 2. Solve the Lpimum measument feedbac problem (3.4) - (3.6) with Q(k) and R as de outu and input weighting matrices, rpevely. TIis yields X(k), G(k). 3. If IrCX(k)cTd-cX(k-l)CTk I <yforaltistop. 4. Updaie Q(k) with aid go to step 2. Q (k+)=k) There is no guarntee of convergence of this algonrthm. However, from numerous lage scale examples it always converges if n is small enough. Further discussion of this algorthm and equations (3.4) - (3.6) wil be given later. Fimt, le's look at the dynamic controller. 4. Dynamic, Controle Desin and Integratin of OVC and MCA Case 1: Fixed Order (ui_ Srin) Dynamic Controler with Feed Forward Considering system (1.1). we desire a dynamic controler of ordern, nc S n., xc(k+l) = Acxc(k) + B,z(k) u(k)c=cxc(k)+dz(k) tosolve the OVC problem. By defining * See [131 for e definition ofa mgwarpoint ** If (X,G) obtnied from (3.4) - (3.6) urn out a to be a nguls point, it is still a cu,dkae for a minimizer. oa Bo YMD N GA [Dc CcAC [Bc Ac] DB [o C-oIC Ol amd ACL -A AO + Bo GMOt, theod loop sysem cm be exprse in the wn fom [9,lJ im=- AcaLx + IOw [x] Let X, L x&ocx. 'k) x k n. n X al Sifi XCL = (AO + BoGMo)XWL(Ao + BoGMo)T + DoWDI md J = trrgmoxclmiot when R= R ] lll n variate mnhfts beom i=l,z...,n, [CoXa.CIJasG?, h is flred ta iall nw m vwuabes ae coiwakied in and XaL. Thereo the prom b. die e form as in the measreme feedbak cm, ad can be solved in the same mar as shwn in section 3. Case 2: Full order dynamic controller (nc = n) with no feed forward To design e fll toder controler fbr a system with white noise of covariae V in fdie measumet one can prve by using tce i-epraon c Near pgrkg tory that the OVC controller is in fac the stadard LQG control with some diagona output weighikg matx sasfying (3.1d). Hece, the alouith to solve ftis problem is the sme as the stdard OVC algoithm except thk the 2nd stp is relacd by saxdad LQG design and the estimati error is inclued in calclatng the steady stat closed loop covarse. It is noted ta the state esimation error covarimme P, which satisfies the following disce Riccati equato is a lower boun for the dosed loop state covariance P = APAT -APT ( + V-' MPAT + DWDT Hence, befoe entering th iteratin gorithm we may check yprt. jf any [IZPTk>>of, the specificadon is not achievable fbr this. It is noted dt duing the design itraton prcedure the outu weighting matrix Q is adjused so t if a particular output specification oj is not acheved (E.yf >cv,) then the corrspondig Qn wilbe inreased (step 4 of the OVC algoriihm) according to the disrepamy between the curent E.y2 and. Conquently, tose "difficult" outputs with hard-to-achieve specifications will end up with large Q's, and the with easy to achieve specificatis will have smalier QJ 's. In fact, for thdse "easiest" outputs with variances mmalier dtan the correpnding of's the final converged Q's wiu be ze (ind g E.yf c). This implies tha these ouu csraints are not important and can be disregarded during design. However, at the begiming we do not know which constaint is tial. As a result, the coverged Q appropriately reflects the impornce of each output with respect to the given spcfication. (Notc that each output may have different physical meaning and units, so we carmt justify the importance of each outu just by looking at the seification values.) This property is very helpful for model reduction, using Modal Cost Analysis, since MCA calculates the contribution of each mode to a weighted outu cost E_(yTQy) and deletes the least impotant modes accordingly. Therefore, if ft weighting matrix Q can appropriately rcflect the importance of each output, then the mduced model, obtained by MCA can keep the information which is most imporant to the controlled 2122

4 performance. It is noted that in the static measurement feedback case or fixed order dynamic contrller case (with feed forward term) no model reduction is used since the codntrer order is specified a priori. However, as shown in the design algorithn an optimum measurement feedback problemn needs to be solved (step 2) in each iteration. Unforunaely, just as in the continuous case, there is no efficient way (i.e., closed forn) to solve this sub-prblem and another iterating loop is needae, which usually induces a large computaion kad, limiting the prctical system size. On th other hand, the full order dynamic controller case (case 2) is quite simple in computation. Only a standard LQG problem needs to be solved in each itatio. The software to solve the LQG problen is quitc well developed. There are standard functons in the contol software packages such as Matlab, Matrixx and Control-C. However, the order of the controller is the same as ta of the sysen, as in the LQG prblem. This makes the contrller size too big without model reduction. A good reduced model can be obtained from MCA with the output weighting matrix provided by the OVC control design algorithn itself. Based on this, we propose an integration of MCA modeling and OVC controller design for Large Space Structures. (See Figur 1.) In this design procedure all softwae is well developed, and in each Q-iterain the calculations for MCA and LQG are easy compard with the optimum feedback problem with fixed conitroller order. Furthermore, the model reduction mtcnique is built into the design procedue. Thus, the proposed algorithm is a practical tool for controller design for LSS system. There is no guarantee for the convrgence of the iterating loop. However, in the examples we tried it converges very well. It should be pointed out ta in the standard OVC problem the specification a and noise covati e W are prespecified, and the purpose of OVC is to find the controler to satisfy these performnance requirements under the given nois environment. However, in the practcal situation the specification a and noise environment W are quite ad hoc in most cases. Therefore, frm the design poit of view a and W are also design parameters. The specification a influences the controller gain (the smaller the a the larger the cnrller gain) and puts weight on the modes. If a partcular output y is dominated by several modes, then by specifying small enough ai we will ed up with luer Q, and the control effort will have more influence on hes corresponding modes. In fact, these modes wil be included (by the MCA model reduction procedur) in the design model during the next Q- iteration. On the other hand, the noise covane W influences the estimator gain and weights on the input channels. In this design stegy, there are two loops to iterate Q: one is the global loop (labeled as Q-iteration in Figure 1), another is the local loop contained in the OVC design. It is noted that in the OVC design loop, the sarting Q(O) may not be the final Q from the last Q-iteration loop (which is used for MCA in the current loop). In fact, we prefer to start with the same Q() for each Q- iteration loop for the following sn. As explained earlier, a and w are quite ad hoc, and in most cases the purpose of design is to find an "aropriate" conrller. herefore, if a sequence of controlers can be created and compared (i.e., controller tuning), then it will be easier to find te approrate one. This can easily be achieved by using small sting Q(), say Q() = a! with a small number a. With this sartng point, the OVC algorithm will automatically adjust this Q (note that it is still in the same Q- iteration), and then a sequence of controllers from low gain to high gain are produced. This is a very usef property of OVC design, since in this way it works as a self-tuning mechanism, which makes the lab test easier. With what has bem discussed above, whether the c chosen for the desip loop is achievable or not is unimportant, since it depends on the noise covariance W used in the design. If W is too small thcn a can easily be achieved, but this gives only a low gain (estimator gain) controller and no big improvement can be expected. However, ifw is too large then a cannot be achieved. As a result, by choosing a small ca and a small updatng power n (step 4 in the OVC algorithm), we can get a complete sequence of controllers, and by checking them one by one we can get an appropriate contrller. To summarize, the integration of MCA model reduction and between modeling and OVC controller design algorithm iterates contrller design. Fis, a design model is obtained from the evaluadon model by MCA model reduction with an output cost function weighted by some Q. Then, the OVC controller design algorithm is applied to the design model by choosing small c and nm A sequence of controllers will be produced at this step. Check the stability and performance of thse controllers with the evaluation model and find an appropriate one. Using the corrsponding Q for the selected controller as the new weight of the output cost funcion, do MCA again to get a new design model. If the modes kept in the new design model are the same as the previous one (this is equivalent to say that Q is converged), the procedure is completed and the sequence of controlers produced in the last Q-itertion loop can be teed in the lab. In the next section, we use this design strategy to design the controllers for the Mini-Mat System. 5. Controlr Design for Mini-Mast 5.1 System Description (11 The Mini-Mast (see Figure 2) is a generic deployablereracte space tuss with a triangular section. This truss is located at the NASA Langley Research Center (LaRC) and is epresntative of future deployable usses to be used in space. Two istrmentation platfrins have been insutalled at bay 1 and bay 18. The atuators md sensors used for the control loop are all mounted on these two platfofns. The configuration of these sensors and actuas is shown in Figure 2. Three torque wheel actuators (TWA) serving as the input actuators are located at tip platform (bay 18) in x, y and z directions; they are noed as TWAx,TWAy,TWAz, rspectively. Three types of sensors are used in this experiment: accelerometer, rate gyro and displacement sensr. Tle accelerometer and me gyro outputs ae fed back for closed loop control. However, displacement sensor outputs are not fed back but are for post-proessing. There are six accelerometers: four of them are located at tip platm and the oter two are located at mid-platform (bay 1). All accelerometers ae mounted in either x or y direction for linear acceleration measurments. These accelerometers are labeled as AISXI, A18YI, A18X2, A18Y2, AIOX, AIOY, respetvely. One rte gyro labeled R18Z is mounted at tip plate measuring the rotation rate in the z direction. The controled outputs in this experiment are the torsion and corner displacements at bays 1, 14 and 18. Three Kaman proximity probes are instaled at each bay. They are mounted parallel to the flat face on the comer joints of the structure and positioned to measure deflections normal to the face of the probe. The torsion angle at each bay is then calculated from the corrsponding three Kaman sensor outputs by assuming that the structure cross section is rigid. A smmary of the contrlled outputs and their limits is contained in Table I. For disturbance signals to the structure, three 5 lb. shakers (labeled as SHA, SHB and SHC) are atched at bay 9 of the Mini-Mast, providing force disturbances to the system. The evaluation model for the system consists of the structure finite element model and sensor actuator dynamics. There are ttally 123 states. In this design the system noise w is the noise from TWA's and shakers with the following covariance W = diag (9.19eI, 9.19e1, 9.19el, 9.89e4, 9.89e4, 9.89e4) and Xt measurement noise v of rate gyro and accelerometers has the following covariance V= diag (.13e-2, 2.22e-1, 2.22e-1, 2.22e-l, 2.22e-1, 2.22e-1, 2.22e-l) 5.2 Contrller design The design strategy used in this project is shown in Figure 2123

5 1. Our o*ctive is to design a 12th order conrller for the systm. The output ostaints (a) used for the OVC algorithm are those limits shown in Table I. We first use MCA to reduce the evahlation model to get a 12th order design model. As the tating weighting matix, we choose the weight for the particular output to be the inverse of the scuare of tat out limilt i.e., Q = (diag. (o)-2. The modal cost is shwn in FIgur 3. We choose the most impora 6 modes to cxrnprise the design model. They are the fist 5 modes and the 16th mode. (hy are system modes.) In the OVC design step, we chose Q()= I as the ardng weightin matix fbr each Q-fiteinL The hipt - outu varine curve for te losed lop system ae shown in Figur 4. The solid carve is the erfimance of the cwtrolkl obtined fromn the OVC algoitldn ev lu with the design model. We Stained 72 cnm llers By evaluating these cat es with the evaluation model we get the dashed ine. Only the first 15 controlers can stabilize t evauatio model as rued by "+". Now we reed to chooe an ate controler from the firt Q-iratio klop aid dten use the corresponding output weighting marix to do the MCA for the nex loop. No mater what coroller we dcose, the most importu 6 modes are the same according to the MCA with the sew ouu weigng matix. As a result, dtis design is nvrged in on loop. Call controller I to be the first ntrole (lowes gain) on the dashed line in Figure 4, controler 2 to be the 2nd condller, etc. We choose wrwromle 12 as our finl owror. The analytil and e results of impulse response for the cosed loop system with c r 12 ar own in Figure 5. Very good agren is obsved between t m erical and experimenal data A step by step design poed and discussion about thfis ex et is nined in [12,13]. 6. Concdon A new design strategy whch i MCA model eduction and OVC cnnlr design has povided a practical method for Lag Spac Stuctu coitrollr synesis. This sategy has the 1fowing feaures: 1. Closed form model cos formuas for model reduction make it pssie to deal with high order stms. 2. Itratio of modeling and conroller design gives us a better design model 3. The OVC algorithm is a self-tning mechanim which automatically creaes a sequence of contrlers from low gain to high gain. This featr is very helpful for ral time lab testing of Ue MIMO syste. 4. All Ue needed softwar is wull developed A wors disturbance formula is also provided to obtain a practical dape to check system performa, with Ut assurance that if the perfonnac specifications are met with this disuxrban, they will be met for any dis of equal 12 norm. This stategy has been appied to the Mini-Ma system at LaRC with very s essgfl results NASA points out that this is the first sabilizing controuer for Ute Mfini-Mast, using "on-board" sersors. yi Y2 Y3 Y4 Ys Y6 Y7 Ys Y9 Yio Yi1 Y12 Notation D1OA DIOB DIOC D1B D14A D14B D14C D18A D18B D18C THl TH14 TH18 Admowledgement The help of NASA personnel, R Pappa, J. Sulla, S. Perez and i Kim at Langley Rech Center is gtflly acknowledged. This study wa suppored by NASA grnt NAGI-958. [1] Skelton, R E., Si KR. nd Ranaistman, J., "Componen Model Reduction by Componen Cost Analysis," AIM Gai Contol Cor(, Minneapolis Mimn., (2] Kim, J. IL and Sketo, R E., "Model Redution by Weighted Component Cost Analysis," AIM Dynamics Spcialsts Co., Long Beach, CA, Apnri199. [31 Hsieh, C, Skelon, R. E and Damra, F. M, "Minimum Frergy CSorll with bquly I Constrain on Output Variaces," Opmal Cotrol Appicatons and Methds, Vol. 1, No. 4, pp , Oct-Dec [41 Kwakenaak, H. and Sivan, R, Linear Optmal Control Systm, Wiley, New Yo*, [5] flt, G. and Skeltn, R E, "Chois of Weighting Matrie m LQI Proems," AfIM, Navgatn Guidance and Control Co., Portland, Ogon, 199. [61 Zfh, G., Cods,1M. and Skelo R. K, "New Romess Properties of Linear Systems," Submited for publicatio [7] Hsieh, C, Kim, L H. ad Sktton, R. E., "NASA GI Project - Purdue Anual Repot" NASA GC project amal report meeting, Hanptn, VA, Jum 199. [8] Shepard, G. D., "Transient Modal Twuing," 4th lnternational Modal Aalysiy Corf, Feb [9] khns, T. L aid mns, "On the Dsign of Optimal Constrined Dynanic Compestors for Liner Constant System," IEEE Trans. Auomc Control, AC-IS, pp , 197. [11 Went, C. J. and Knapp, C H., "Parameter Optimizaton in Linear Systems with Arbitrrily Constrined Controller Structure," IEEE Tns. Aurwatic Control, AC-25, pp , 198. [11] Pappa, R., Sulla, J. et al, "Mini-Mast CSI Testbed - User's Guide," NASA Langley Research Center, Hampton, VA, March 23,1989. [12] Liu, K. and Skeltn, R E., "q-markov COVER Idendfication and its Appication on Large Flexible Strucre," in i rion [13] Hsieh, C., Kim, J. I., Zhu, G., Skelton, R E.,"Control of NASA's Mini-Mast - An Apprach Integrting Modeling aid Control," NASA Guest-Investigator pogram annual rport [14] Hsieh, C., Kim, I. H., Skelt R. E.,"Closed Loop Lab Tests of NASA's Mini-Mast," Proc. ofacc, Log no. TA- 12-3, San Diego, CA, May 199. Table I - Controlled Output for the Mni-Mast Iten Location Limits Kaman Proxinity Probe Bay IO Comer A 1.27e-3 M (.5") Kaman Proximity Probe Bay IO Comer B 1.27e-3 M (.5") Kaman Proximiity Probe Bay 1 Comer C 1.27e-3 M (.5") Kaman Proxity Probe Bay 14 Comer A 4.45e-3 M (.175") Kaman Proximity Probe Bay 14 Corer B 4.45e-3 M (.175") Kaman Proximity Probe Bay 14 Comer C 4.45e-3 M (.175") Kanan Proximity Probe Bay 18 Comer A 7.62e-3 (.3") Kaman Proximity Probe Bay 18 Comer B 7.62e-3 M (.3") Kaman Proximity Probe Bay 18 Comer C 7.62e-3 M (.3") Torsion Angle Bay e-3 rad (.7") Torsion Angle Bay e-3 rad (. 11) Torsion Angle Bay e-3 rad (.15") 2124

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