207067 CHOOSING SENSOR CONFIGURATION FOR A FLEXIBLE STRUCTURE USING FULL CONTROL SYNTHESIS Rick Lind t Volkan Nalbantoglu 2 Gary Balas a NASA Dryden University of Minnesota University of Minnesota Abstract The issue of choosing sensor locations has been stud- ied by considering grammians for observability coupled with minimizing a cost function. Skelton and DeLorenzo Optimal locations and types for feedback sensors which choose a cost function as an LQG performance metric meet design constraints and control requirements are dif- formulated as the root mean square contribution of each ficult to determine. This paper introduces an approach sensor output. Sensors associated with small cost func- to choosing a sensor configuration based on Full Control tions may be removed due to their low effectiveness [15]. synthesis. A globally optimal Full Control compensator is computed for each member of a set of sensor configura- Similar approaches are developed using modal proper- tions which are feasible for the plant. The sensor configu- ties. Kim and Jenkins choose a performance metric ration associated with the Full Control system achieving on modal controllability weighted by the modal the best closed-loop performance is chosen for feedback cost of Skelton [8]. This approach emphasizes both the measurements to an output feedback controller. A flexi- degree of controllability and modal participation in the ble structure is used as an example to demonstrate this performance criteria. Lim defines a performance metric procedure. Experimental results show sensor configura- using a weighted modal projection [9]. This approach is tions chosen to optimize the Full Control performance based on a relationship between grammian singular val- are effective for output feedback controllers. ues and modal observability. Actuator and sensor pairs are chosen with principal directions parallel to the modes with large singular values. This paper considers an approach to choosing a sensor Introduction configuration based on Full Control synthesis. The Full Control system allows the controller to independently Choosing an effective set of sensor measurements is es- affect every state and error signal. Computing the op- sential for designing controllers to achieve stringent per- timal Full Control controller is equivalent to computing formance and robustness goals. Often control require- the optimal controller for a given set of sensors. Synthe- ments are not anticipated in the design stage for physi- sis of globally optimal controllers to minimize an 7/_ or cal systems and sensor configurations are chosen in an ad p upper bound is formulated in the Linear Matrix In- hoc manner. Flexible structures are especially challeng- equality (LMI) framework for Full Information feedback ing systems for choosing sensor locations and types due and extended here to the dual problem of Full Control to the large number of mode shapes. Also, tradeoffs must synthesis [12, 13]. be met by balancing the number of sensors needed to observe the large number of closely spaced modes while The issue of sensor configuration is closely associated simultaneously considering the added weight and cost of with the issue of control design. The optimal closed- these additional sensors. loop system requires an optimal configuration of sensors and optimal gains in the compensator. Optimality in only one of these areas will restrict the achievable perfor- mance and robustness of the closed-loop system. Utiliz- 1NRC PostDoctoral Research Fellow, Structural Dynamics, ing the Full Control system is advantageous for synthesis MS 4840D/RC, Edwards, CA 93523-0273, rick.lindq_tfrc.nasa.gov, Member AIAA and analysis of sensor configuration since a Full Control 2Graduate Student, Aerospace Engineering and Mechanics, compensator can be computed which is globally optimal. Minneapolis, MN 55455, nalbantaaem.umn.edu The procedure will not be affected by local minima as- 3Associate Professor, Aerospace Engineering and Mechanics, sociated with control synthesis. Minneapolis, MN 55455, balasQaem.umn.edu, Member AIAA °AIAA Guidance, Navigation and Control Conference, New Or- leans LA, AIAA-97-3745, August 1997. Thetechniquepresented in this paper considers a cho- The structured singular value, p, can be used to de- sen set of sensor locations. Globally optimal Full Con- termine robustness of the closed-loop system to struc- trol compensators are computed at each of these loca- tured modeling uncertainty and the achievable perfor- tions to determine the maximum performance and ro- mance level in the presence of real and complex uncer- bustness level achievable. The sensor locations chosen tainty. The uncertainty description is structured with for implementation on the physical system correspond to two types of blocks. The blocks are repeated scalar or the sensor locations achieving the best Full Control per- full block matrices. Let integers m,n,p define the number formance. There is no guarantee that the optimal Full of real scalar, complex scalar, and complex full blocks. Control sensor locations are equivalent to the optimal Define integers R1,... ,/_ such that the i_h repeated sensor locations for ageneral output feedback controller; scalar block of real parametric uncertainty is of dimen- however, experiments indicate this technique can choose sion R_ x/?_. Define similar integers C1,..., Cn to de- effective configurations for a physical system. note the dimension of the complex repeated scalar blocks. The structured uncertainty description A is assumed to This approach easily allows a sensor configuration to be be norm bounded and belonging to the following set. determined by considering variations in both type and ° ° location of sensors. The plant model used to design con- .... 6_IR= _xlc_ .. •6. Ic_ Ax... trollers for these configurations may be generated from R,*7E c } experimental data transfer functions or from a compu- tational finite element model using a package such as Real parametric uncertainty is allowed to enter the prob- NASTRAN. Additionally, choosing actuator configuration lem as scalar or repeated scalar blocks. Complex uncer- using globally optimal Full Information synthesis is a tainty enters the problem as scalar, repeated scalar or natural extension to this technique [10]. full blocks. Sensor configurations are chosen for a flexible structure The function p is defined as using the method described in this paper. Several sets of sensor locations are considered for feedback measure- 1 ments to achieve vibration attenuation at different po- _(P) = m_m{O(A) : det(I - Pa) = 0} sitions on the structure. Globally optimal Full Control compensators are computed to determine the best sensor with #(P) = 0 if no A exists such that det(I - PA) = 0. configuration of the sets. Output feedback controllers are generated and implemented on the experimental struc- Upper and lower bounds for # have bee derived which ture using feedback from these sensor configurations. utilize two sets of scaling matrices which are structured similar to the uncertainty block structures. Robust Control Synthesis V = {diag (D L ..., Din_, D °l ,..., D_, d_Io,,. .., dp°I_) : 0 < D = D*,D_ 6 C_×m,D7 6 cC'xC',d_ 6 C} Consider a state-space description of a linear time- invariant plant P(s). The second set of scalings in G affect only the real para- metric uncertainty blocks. z = Cx En E12 d = {diag(G1,... ,Gin,O,...,0) :Gi e C m×R' } y C2 E_I F-_2 u An upper bound for p is computed as an optimization [4]. where A E R'_Xn, B1 E R'_Xna,B2 E RnXn',Cl E The real/complex/_ upper bound reduces to the well Rn'xn,c2 E R nyxn ,and the E matrices of appropriate known complex p upper bound when there are no real similar dimensions. parametric uncertainty blocks. Define Ep as the set of all real, rational, proper con- < inf (Ww' +G2)+) trollers, K(s), which stabilize the closed-loop system. Ge_ Analyzing performance using the induced _oo norm D_T_ leads to the following minimization problem for Fi(P, K) which is the linear fractional transformation (LFT) for The structured singular value provides a measure of ro- the lower loop of P closed with the controller K. bustness in the presence of the defined structured un- certainty. The Z) and _ are restricted to be constant inf sup_[Fl (P(y.o),K(y_))] = inf HFi (P,K)I]oo matrices to scale with time-varying uncertainty in this KE_ wER K EIC paper. The objective of control design is to maximize This is an 7_oo optimal controller synthesis problem robust performance which corresponds to minimizing p which has been solved using state-space equations [3, 5]. in this framework. Feasibility : Output Feedback System Theorem 1 Given the state-space plant P(s) and asso- ciated constant matrix Mp along with the set l) of scaling matrices, then the following are equivalent. 7too control synthesis involves iterating over a set of fea- sibility conditions. These conditions determine whether a controller exists that achieves a desired closed-loop 7too 1. There exists D E D and stabilizing K E Igp such norm value. A standard bisection search can be used to that find the lowest achievable norm value to within a given IID½E(e,K)D-½11oo < 1 accuracy. The optimal controller is computed using el- g. There exists D E D and stabilizing K E ICp ements of the plant and the solutions to the feasibility along with real X = X T > 0 such that with conditions. g = diag(X, D), The controller feasibility and synthesis may be formu- "_(Z]Fo(Ta,E(Mp, K))Z-½) < 1 lated as state-space equations or in the LMI frame- work [6, 7]. The feasibility conditions in the state-space 3. There exists D E 2)and stabilizing K E ICp framework are two Riccati equations. The comparable along with real X = X T > 0 such that with feasibility conditions in the LMI framework are gener- Z = diag(X, D), ated by applying the Bounded Real Lemma and consid- ering orthogonal subspaces to matrix elements. "_(Z½Ft(F,(T_,Mp),K)Z-½) < 1 A separate LMI formulation is developed for computing optimal full information controllers [12, 13]. This ap- Now perform a change of variables. Denote {R, U, V, T} proach uses algebraic arguments to demonstrate a con- as elements of the constant matrix term involving the stant matrix condition which is equivalent to the state- star product F,(Ta, Mp). Introduce Q to replace K(I + space 7too control problem. LMI feasibility conditions TK) -1 in the closed-loop LFT for notational conve- are generated using a variant of Parrott's theorem [14]. nience. This paper will adopt a standard for denoting plant ma- trices for ease of notation and convenience in theorems. El (Fs (Ta, Mp), K) = F_ V T 'K = R + UQV Denote P as the continuous-time state-space plant ma- The final theorem presents the pair of LMI optimiza- trix with the elements P(s) = {A, B, C, D}. Denote MR tions that represent the 7/_ controller feasibility condi- as the constant matrix whose entries are comprised of tion for a general output feedback system. The variant the state-space elements of P. of Parrott's theorem is applied to the constant matrix condition involving the maximum singular value. P=D+C(sI-A)-IB ._ Mp= 6' D The LMI feasibility conditions utilize a matrix Ta which Theorem 2 Given the state-space plant P(s) and asso- is formulated for a real scalar a > 0. ciated constant matrix MR with the star product elements [, Ft (F, (Ta, Mp) ,K) = R + UQV along with the set l) of scaling matrices, then the following are equivalent. To= v_1 a1 This matrix is used to compute the following star prod- 1. There exists stabilizing K G IC_, and D E l) such uct LFT with ei = (I + c,A) -1 defined for notational that convenience. IID½Ft(P,K)D-½11_ < 1 _. There exists stabilizing K E tgp and D E 1) F'(Ta'MP) = [ (I + aA)ftv/_CAE_/_tBaC.4B ]+ along with rexd X = X T > 0 such that with g = diag(X, D), Computing the star product with T_ has several impor- tant properties. The most immediately noticed property "_(Z½(R + UQV)Z-½) < 1 is the relationship between the star.product and the bi- linear transformation. The matrix P = F, (T_, P) is the 3. There exists stabilizing K E ICp and D E l) discrete-time formulation of the continuous-time plant along with real X = X T > 0 such that with P. The star product also has a commutation property Z = diag(X, D), such that F, (T_,F_(P,K)) = Ft(P,F,(T_,K)). (Rz-'RT- z-') <o The following theorem demonstrates a constant matrix condition, formulated using the star product, which is (R zn - Z)VI) <0 equivalent to an 7/o0 condition [13]. Feasibility : Optimal Full Control This formulation is easily extended to account for real parametric uncertainty [2]. The maximum singular value condition in Theorem 2 is replaced with the correspond- Consider the state-space Full Control plant Pie. ing condition from the real/complex # upper bound. Consider this condition for the matrix system R + UQV PIc = C1 Dn 0 I including the additional scaling matrices _. A[B IO C2 D21 0 0 u(R + UQV) Define Mpl, as the constant matrix associated with the o(W(R + + < state-space elements of Pie. Formulate the R, U, V ele- ments of the star product term F, (Ta, Mpl. ) using the < o(_ +UQV) term A = (I + aA) -1 for notational convenience. where _= (DRD-1 +3G) ff+G_)-½ O=DU R= [ (Iv+_Ca1AA)A EnV+_-_a-C4B1xAB1 ] (" = VD -1 (I + Ga) -½ 0 ] The variant of Parrott's theorem can be applied to this new singular value condition in the variables R, U and V. aCa A I The new matrix 0 retains the desired full rank condition since both D and U are invertible. Thus, a single LMI V= [v/_C2A E12 + aC2ABI] represents the feasibility condition. Consider this LMI. (_±(_._- _)v;) The matrix U is square and invertible for the Full Con- = _ (V±D ((D-'R*D- )G) (DRD-' +3(7) trol system. This full rank condition is anticipated by - (I + G2) ) DV._) the complete controllability of this system. A linearly = _(v±(n'_)n +s(n'_ -_n) - b) v_) independent set of control vectors are available to affect the states and error outputs of the plant. Correspond- where /)=D 2ED ingly, the perpendicular subspace, U±, utilized in the = DGD E LMI conditions for 7too controller feasibility is null. This feasibility condition to determine existence of a con- The feasibility condition for existence of an 7/o0 con- troller that satisfies a closed-loop p condition may be less troller for Full Control feedback is reduced to a single conservative than the previous condition since it directly LMI. The LMI involving U± in Theorem 2 is vacuous accounts for real parametric uncertainty. Theorem 4 and automatically satisfied. The remaining LMI involv- combines Theorem 3 with the real/complex p bound. ing variables V and V± constitutes the only condition for Full Control feasibility as demonstrated in Theorem 3. Theorem 4 Given the Full Control plant Pie and scal- ing sets D, _ define the following : Theorem 3 Given the t%11 Control plant Pfc and scal- ing set D, define the following : I. The augmented scaling matrices ZD I. The augmented scaling matrices Z 0 D :O<X= 6 D6D 0 D :0<X= 6 ED _. The augmented scaling matrices ZG {[00] } 2. Real scalar a > O, so that (I - aA) is invertible Z¢= 0 G :G6g 3. R and V as defined above 3. R, V, V± and a as defined in Theorem 3. 4. V± such that vTV± = 0 and [ VV± ] isinvertible Then, there exists a stabilizing K 6 ICpIosuch that Then, there exists a stabilizing K E ICv, o and a constant #(FI(PIc,K)) < 1 D E D such that if the following convex set is nonempty. [D½FI(PIc,K) D-_[oo<I {Zo eZv,Za E Z_ : if and only if the following convex set is nonempty. -_[v± (R*Zoa + 3(R'Za - ZGR) - zo) v__]< o} {Z _ Z: _,_, [V± (R*ZR - Z) Vl] < 0} # {0} Choosin_ Sensor Configuration The plant model and associated set of sensor configura- tions used in this algorithm may be generated using com- putational packages, such as RASTR£11,or experimentally Synthesis of optimal Full Control compensators can be derived transfer functions. Finite element models give used to determine efficient sensor configurations. The the freedom to easily compute a large number of sensor Full Control system allows the controller to indepen- configurations but may be subject to modeling errors. dently affect every state and error signal. The previous Experimental data gives a more accurate representation section demonstrates globally optimal controllers can be of the true system but it may be difficult to physically computed for the Full Control system. Computing opti- reposition a large number of sensors and identify models mal Full Control compensators is equivalent to comput- for each location. ing the optimal controller for a given set of sensors. Location, type, and number of sensors used in each con- An optimal sensor configuration can be chosen by min- figuration, along with the total number of configurations imizing the achievable Full Control performance level to be considered in the set, may be chosen using several with respect to a set of possible sensor configurations. criteria. The set may reflect physical requirements lim- Plant models are generated for each sensor configuration iting the sensors to select configurations due to size of under consideration and a Full Control compensator is the sensor, direction of sensing, wiring connections and computed for each plant using Theorem 4. The optimal weight restrictions. The set may also be chosen based on sensor configuration chosen with this method may not a priori knowledge of the system and properties such as be globally optimal over every possible sensor location symmetries, mode shapes, and previous control design in the system; rather, it is optimal with respect to the experience. considered set of locations. This algorithm is not guaranteed to generate a globally The following algorithm demonstrates this procedure. optimal sensor configuration for output feedback con- trollers. The computed optimal sensor location may not Algorithm 1 even be optimal among the discrete set of sensor configu- rations when considering an output feedback controller. The computed sensor configuration is only optimal with • Define state-space elements of transfer function respect to a Full Control compensator; however, there are from disturbances to errors. many plants which can effectively utilize this method. If the dominant modes affecting the achievable perfor- mance are controllable, or nearly so, then the physical P = C1 Dll actuator sets used for output feedback controllers will be able to achieve, or nearly so, the performance of the • Define set of n sensor configurations. Full Control compensator. A similar procedure can be formulated based on minimiz- ing the achievable performance level of output feedback controllers; however, there are advantages to using Full C_ _)_I Control synthesis. The Full Control compensators are guaranteed to be globally optimal while output feedback for i=l:n{ controllers, computed with methods such as D-K itera- tion, are only locally optimal. A poor local minimum may capture the output feedback controller synthesis for Formulate P}_= C1 Dll 0 I a given sensor configuration. The resulting optimal sen- cAl B1 oI 0o] sor configuration would be incorrectly computed due to deficiencies in the control synthesis procedure. Compute 7i = inf p (Ft(P}e,K)) K EJC Additionally, the Full Control synthesis is an LMI which } is easily solved with no user interaction using convex op- timization algorithms. D-K iteration requires the user to monitor the process and select weighting functions. Suc- cessive D-K iterations might produce better, or worse, corresponding to 7j = mi.'nTi. performing controllers depending on these weight selec- $ tions and initial conditions. Flexible Structure : Model Accelerometers may be placed along the edges ofany platein any Bay. These accelerometersare ICSensor 3145-002 witha bandwidth of300 Hz. These sensorsare Sensor configurations are chosen for vibration attenu- alignedalongthe horizontalcomponent ofan actuator ation of an experimental flexible structure. The flexi- and do not measure any verticalcomponent of move- ble structure is constructed at the Dynamics and Con- ment. trois Laboratory in the Department of Aerospace Engi- neering and Mechanics at the University of Minnesota. Figure 2 shows the control elements in Bay 3. The colo- This structure models a space truss for potential satel- cated actuators and displacement sensors are seen along lite and space platform applications. The structure is the diagonal rods. The accelerometers are placed atop designed to place 12 lightly damped modes between 0 the horizontal edges of the triangular frame plate. and 100 rad/sec. The flexible structure is represented in Figure 1and Figure 2. \ Figure 2: Control Elements in Bay 3 An analytical model is generated for the structure. A NASTRANmodel is available; however, this paper uses ex- / perimental data to formulate the model. Experimental transfer functions are computed from the actuators to the accelerometers on Bays 3 and 4by commanding sinu- Figure 1: University of Minnesota Flexible Structure soids of varying frequency to the actuators. System iden- tification algorithms based on curve fitting techniques The structure consists of a rigidly held fixed top plate and model reduction via balanced realization computes and four hanging bays numbered 1through 4 with Bay 1 a 38°* order model. Transfer functions plots of the open- being at the top of the structure and Bay 4 at the bot- loop peak gain for Bay 3 and Bay 4 accelerometers is tom. Each bay contains an aluminum plate and thin hol- given in Figure 3. low rods connecting corners of neighboring plates. The top four plates are spaced .62 m apart while the bottom plate is .47 m below the third bay. Plates in Bay 2 and t0' ,._,,op._.o_-souo _, | Bay 4 are triangular frames while the others are solid. The actuators for control are contained in Bay 3. Three I I 1 # I actuators are colocated along diagonal rods connecting J10e the top and bottom plates of this bay. These force actu- ,' , :', ators are voice-coil type actuators produced by Northern Magnetics as ML3-1310-020LB with a limit of +2 pounds 10-1 of force. The working linear stroke is +12 inch with an effective bandwidth of 200 Hz. Lineardisplacementsensorsarecolocatedalongthe di- lff .... " rectionoftheforceactuatorsinBay 3.These sensorsare I0' Trans-Tek0242 typesensorswitha working rangeof:i:_ inch.They havezerohysteresisand arelineartowithin Figure 3: Open-Loop Peak Gains f_om Bay 3 Actuators to Bay 3and Bay 4Accelerometers +.5% up tothebandwidth of100 Hz. Flexible Structure : Objective Flexible Structure : Full Control Synthesis It is desired to formulate controllers to attenuate vibra- Full Control compensators are computed for systems de- tion and lower the peak gains of the open-loop system. rived as subsets of the flexible structure model in Fig- Accelerometers ate used as the performance signals with ure 4. This block diagram is designed with 6 accelerom- a peak gain of approximately 10 for the 65 rad/sec mode. eters and 3 displacement sensors for performance errors Bay 3 accelerometers ate located in the same bay as the and feedback measurements. These 9 sensors ate divided actuators so it is anticipated more performance can be into 3 groups of 3 with the following notation. achieved for these sensors. Bay 3 attenuation is desired to be approximately 4.6 while the performance request d ad ad displacement sensors in Bay 3 for Bay 4 is a factor of 3.5 for attenuation. Performance weightings, w,,ba_y3r! and L"w"bwayr14' are included with the ac- 31,32,33 acceleration sensors in Bay 3 41,4a, 43 acceleration sensors in Bay 4 celerometer error signals to specify the desired attenua- tion levels. The subscript on the sensor designations indicates the W b_¢3 4.6 wbaT/4 3.5 ,I = i"6 "',,.=.TIG horizontal component of the sensing direction. The accelerometers are placed along edges of the horizon- Additive uncertainty between the control inputs and sen- tal plates while the displacement sensors are colocated sor measurements is included to account for unmodeled along the diagonal rods connecting corners of neighbor- dynamics and neglected high frequency modes. A dy- ing plates with the same horizontal direction of sensing namic weighting, Wadd, is affected on each displacement as the accelerometers. Sensors 3_, 31 and 41 axe located and accelerometer sensor for feedback. along the same edges of plates on the same side of the structure and consequently have the same horizontal di- s2 - 72s + 4790 W_dd rection of sensing. 482 -- 2058 -t- 28910 The physical actuator positions ate affected by a distur- The uncertainty structure for the system in Figure 4has bance input. A constant weighting, Wdiot = .5, is in- a single uncertainty block. This block represents addi- tive uncertainty on the sensor signals used for feedback cluded to normalize the disturbance signal affecting each measurements to the controller. This uncertainty, A_dd, control channel. Sensor noise is also included in the sys- tem to affect the accelerometer feedback measurements. is a complex operator to allow variations in both magni- tude and phase. Aadd is treated as an unstructured full Constant weightings of W_i,_ = .01 are included to nor- block uncertainty for controller synthesis. realize the noise affecting each sensor. A performance block will also be included in the con- The magnitude of the control signal for each physical actuator is included as a performance error to limit the troller synthesis procedure. This block relates the noise and disturbance inputs to the performance errors. There amount of control actuation. A weighting of W_,t = .2 are 3 noise disturbances affecting the 3 physical actua- is used as the performance penalty for each actuator. tors and 9 noise inputs for the sensors. The performance The open-loop flexible structure model with uncertainty errors are composed of 3 penalties on the amount of ac- blocks and weightings in given in Figure 4. tuation and 6 weightings on the accelerometers. •disturbances Initial controllers are synthesized to determine the type actuator, _-7 of sensors to use for feedback. Full Control compensators are computed for the system with Bay 3 accelerome- ters, 31,32,33, used as the error signals to be minimized and feedback measurements of either the Bay 3displace- ments, 3d, 3_,3 d, or the Bay 3 accelerations, 31,32,33. The optimal achievable 7too norm was lower using the ac- celerometers than the displacement sensors for feedback. _ _ input This result agrees with previous analysis of the structure indicating the accelerometers axe generally more effective than the displacement sensors for vibration attenuating noi_ _ meafseuerdebmaecnkts controllers [11]. The control designs presented in this pa- per will ignore the displacement sensors and only utilize the accelerometers. Figure 4: Flexible Structure Block Diagram Full Control compensators are computed for several con- Robust stability is plotted for each controller in Figure 5. figurations of accelerometers. The performance errors The peak robust stability p values are .445 for K 3 and used for vibration attenuation are chosen to be either .456 for K_. /_ is less than one for each controller to the Bay 3 accelerometers, {31, 32,32}, or the Bay 4 ac- indicate the desired robustness objectives are achieved. celerometers, {41,42, 43). Several combinations of these sensors are used for feedback measurements with either 2 or 3 sensors in each combination. The number of 2 ik/$ _*,e,*_ -SOLID iw/4 bettmak- D_HED combinations is reduced by symmetry arguments which indicate {31,32} should be as effective as {31,33) and {32,33}. The achievable p performance levels for the optimal Full Control compensators are given in Table 1. i I Feedback Bay 3 Bay 4 Accelerometers Performance Performance 03 1131,32,3211oo 1141,42,43Hoo 31,32 1.378 3.031 31,32, 33 0.299 0.719 o IoI 10I 41,42 1.641 0.656 Ir_ (rKVm=) 41,42,43 0.299 0.511 Figure 5: p for Robust Stability 31,41 3.281 1.312 31,42 1.641 0.609 Nominal performance is also calculated for each con- troller as 1.152 for g3a and 1.880 for K 3. The weighted "l_able 1: Achievable Full Control p Performance Levels norms greater than 1indicate neither controller is able to achieve the desired performance objectives. /_ for nomi- nal performance is plotted in Figure 6. Flexible Structure : Bay 3 Attenuation 2 Era/$ _- 80UO _ 4INdba_k -DASHED Analysis of the first performance column of Table 1 in- It dicates the effectiveness of various sensor configurations t t.5 ! for providing feedback to control Bay 3 accelerometers. t t i I i The most effective sensor configuration for Bay 3 vibra- 1 tion attenuation is to use three sensors in the same bay. Feedback configurations using all Bay 3 sensors or us- ing all Bay 4 sensors achieve p = .299 for Full Control O.5 closed-loop performance. These performance levels are A__ , similar since the set of sensors in each bay is able to ob- 0 serve the dynamics of Bay 3. Each set of accelerometers tO1 101 is able to provide sufficient information to the controller to attenuate the Bay 3 vibration responses. Figure 6: /_ for Nominal Performance Restricting the feedback to only two sensors significantly The robust performance p values for each controller is decreases the optimal performance level. The p values in- given in Figure 7. The p upper bounds are computed as crease by approximately a factor of 4 when using {31,32 } 1.296 for K 3 and 2.065 for K_. Each controller gives a as compared to {31,32,32} for feedback. The perfor- peak p greater than 1 indicating robust performance is mance decreases even more if the two sensor are aligned not achieved for either output feedback controller. in the same direction. The p of 3.28 is for {31,41} is twice the p = 1.64 value achieved when using {31,42 }. The robust performance _uplots are of similar shape for each controller with peaks at 104 rad/sec even though Output feedback controllers are generated using D-K it- for K43 is much higher. Both controllers are driven by eration for vibration attenuation of the Bay 3 accelerom- meeting the performance goals as evidenced by Figure 6. eters. Separate controllers are designed for 2 different The open-loop gains in Figure 3 are smaller from the feedback configurations. The first controller, K3, will actuators to the Bay 4 sensors as compared to the Bay 3 use the 3 sensors in Bay 3 to control vibration in Bay 3. sensors. K 3 is unable to increase the controller gains to The second controller, K 3 will feedback the 3 sensors in match the performance of K 3 due to the penalty on the Bay 4 to attenuate vibration in Bay 3. amount of control actuation. Flexible Structure : Bay 4 Attenuation IW/$ bedb_k -SOUD t IW/4 MedQ_lt -DASHED ,,I at| fIrtl The achievable performance levels of optimal Full Con- i I iq i I ii I t I trol compensators to attenuate vibrations measured by Bay 4 accelerometers for various sensor configurations is given in the last column of Table 1. These performance levels clearly indicate some measure of the Bay 4 accelerometers is required for adequate at- tenuation of Bay 4 vibrations. Synthesizing a Full Con- trol compensator using the entire set of Bay 3 accelerom- , . , , , . . . | I0' t_ eters achieves a/_ value of .719 while a p value is .511 is _(_ achieved using the three Bay 4 accelerometers. Figure 7: p for Robust Performance The need for utilizing Bay 4 accelerometers to control Bay 4 vibrations is demonstrated by the open-loop modal The Full Control synthesis results indicated controllers responses in Figure 3. A torsional mode exists at 62 could be computed which achieve similar performance tad/see that is clearly observable by Bay 4 but does not levels using either Bay 3 or Bay 4 feedback. There are appear in the frequency response data of Bay 3. Any several possible explanations to account for the poor per- feedback configuration utilizing only Bay 3 sensors fails formance of K3 in comparison to K 3. The Full Control to provide information to the controller about the tor- results axe based on a globally optimal controller while sional mode dynamics at this frequency. Consequently, D-K iteration may have computed a K 3 far from opti- the controller can not properly cancel these dynamics as mal. Also, the optimal Full Control compensator may is demonstrated by the poor closed-loop performance. be realized as a constant gain controller while K_ was greatly affected by bandwidth constraints. Output feedback controllers are generated using D-K it- eration for vibration attenuation of the Bay 4 accelerom- Peak gains of the experimental closed-loop transfer func- eters. Separate controllers are designed for 2 different tions are plotted in Figure 8. K_ provides better attenu- feedback configurations. The first controller, K4, will ation for the modal response at 64 rad/sec which agrees use the 3 sensors in Bay 3 to control vibration in Bay 4. with Figure 7. Neither controller is able to provide the The second controller, K_, will feedback the 3sensors in desired attenuation above 104 rad/sec. Bay 4 to attenuate vibration in Bay 4. Magnitude plots are given for K4 in Figure 9a and for K4 in Figure 9b. _: -" 10+ - O0"I'rED :: :: . 10I _4_-_o _'_ _1 I : j,' _ ::'_I I •- :./ ..:" , J10I _,0+_ 10_ 101 101 101 10I ___.,..,.,,,,.-/_i,_... .............. lOi "r 'r 10-_ ..,+,Io-" ' w _, 10"101 ........ 1lOre InXl(mS'wc_ lO-a,_o' ,o' to' 1o' i,m (racll_ Figure 8: Experimental Closed-Loop Peak Gains for Bay 3 Accelerometers with X_ (--) and K_ (- - -) Figure 9: Magnitude Gains for K4 (a) and K44 (b) The similar performance levels on the physical struc- Robust stability for the linear plant model with each ture somewhat contradict the output feedback p anal- controller with respect to the uncertainty description in ysis, which anticipated K 3 should provide 40% better Figure 1 is computed using p. All uncertainty operators attenuation, but agree more closely with the Full Con- axe complex and linear, time-invariant. Each controller trol analysis, which anticipated each controller should achieves robust stability with/J values of .789 for K44and provide similar attenuation levels. .707 for K34. Robust stability p is shown in Figure 10. Bff $W -_KM..H_ IJ Ila_$Im_v_m_ -804.1D BIV4ludm_ - DAIHED I_ 4keda_k- D,4SHF.D 0,, j--_. I i I ""...... i/ OA 0.4 t OA tiiI ti i•ri__ 0A tl II _I • iIo .//il;•_t I i • III 1II 1 • 02 II I O.Z 0 , , | . , .... i I0v !0I ioa _¢_ Imm|rll'l,_ Figure 10: p for Robust Stability Figure 12: p for Robust Performance Nominal performance is also calculated for each con- The peak p for robust performance with K44 occurs at troller as .543 for/(44 and 1.199 for K4. The weighted 104 rad/sec. This is the 4th bending mode which is only norm greater than 1indicates K34using the Bay 3sensors weakly observed by Bay 4 and has a high level of additive for feedback is unable to achieve the desired performance uncertainty. K44 does not receive sui_icient information objectives. K_ presents a/_ less than 1 and is able to about this mode in the presence of the noise and addi- achieve nominal performance. Nominal performance p tive uncertainty and thus the closed-loop performance is is plotted in Figure 11. only slightly more attenuated than the open-loop perfor- mance. The controller bandwidths in Figures 9a and 9b show K34 is able to roll off noticeably faster than K44 due I_ kV 1foedl_d_ - 80UO to observance of this mode. am/4 b*dl_m - OASNED 1 Implementing each controller configuration on the ex- perimental flexible structure produces performance lev- eis which agree with the Full Control synthesis results. Using Bay 4 accelerometers as feedback measurements it°" l! allows better vibration attenuation than using Bay 3 0+4 11 feedbacks. Peaks gains of closed-loop transfer functions glI from the experimental flexible structure are presented in 02 ! Figure 13. K4 demonstrates the expected poor perfor- mance in attenuating the 62 rad/sec mode while K4 is , , , , , , i 10q 1o1 able to attenuate each mode to nearly equal peak gains m <mlw_ as expected by the p plots. Figure 11: p for Nominal Performance The robust performance p values for each controller are _o'[ mm,,-iooe- OOT'mV i.++ given in Figure 12. The p upper bounds are computed as .982 for K_ and 1.269 for/(34. The controller using Bay 4 feedback, K4, is able to achieve robust performance while K34 is unable to achieve the desired robustness goals due ,o.F_+ '_ /f_, :_' _.+.._.. to its associated/_ being greater than 1. The/_ plots show controller synthesis of K34is driven by ' the unobserved torsional mode at 62 rad/sec. The con- troller is able to lower the weighted nominal performance measure to less than 1 near this modal frequency; how- ever, the robust stability/_ is raised as atradeoff. Robust ° , , , , i performance p demonstrates K34 is unable to simulta- I0"i0 + 10_ neously achieve the desired performance and robustness goals with the peak/_ occurring at 62 rud/sec. Figure 13: Experimental Closed-Loop Peak Gains forBay 4 Accelerometers with Ks4 (--) and K_ (- - -) 10