MULTIOBJECTIVE OPTIMAL STRUCTURAL CONTROL OF THE NOTRE DAME BUILDING MODEL BENCHMARK *

Transcription

1 Thi d d i h F M k Earthqake Engineering and Strctral Dynamics, in press. MULTIOBJECTIVE OPTIMAL STRUCTURAL CONTROL OF THE NOTRE DAME BUILDING MODEL BENCHMARK * E. A. JOHNSON, P. G. VOULGARIS, AND L. A. BERGMAN Department of Aeronatical and Astronatical Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 680, U.S.A. SUMMARY Redced-order, mltiobjective optimal controllers are developed for the Notre Dame strctral control bilding model benchmark. Standard H /LQG optimal control excels at noise and distrbance rejection, bt may have difficlty with actator satration and plant ncertainty. The benchmark problem is adapted to a mltiobjective optimal control framework, sing l and H constraints to improve controller performance, especially attempting to redce peak responses, avoid satration, and improve robstness to nmodelled dynamics. The tradeoffs between H performance, otpt peak magnitdes, and robst stability are examined. Several optimal controllers and their performance on the benchmark are given. KEY WORDS: active strctral control; earthqake response; mltiobjective optimal control; H control; l control INTRODUCTION Spencer et al., detailed a bilding model benchmark problem for stdying active strctral control strategies. The problem defines a base plant that models the strctral behavior of a threestory, single-bay, scale model of a bilding 3, as well as the dynamics of the sensors (an LVDT, several accelerometers, and a force transdcer) and the actator (an active mass driver or an active tendon member). Sch a benchmark problem is invalable for stdying varios control strategies with realistic limitations sch as actator satration, limited actator stroke, digital controller implementation, redced-order controllers, and mltiple measres of controller performance and robstness. Standard H /LQG optimal control design methods typically excel at noise and distrbance rejection, bt may have difficlty with actator satration and plant ncertainty. The addition of l constraints may help accommodate reqirements sch as limiting control otpt or other peak * This paper is dedicated to Franz Ziegler, Professor of Rational Mechanics and Head of the Institt für Allgemeine Mechanik at the Vienna University of Technology, on the occasion of his 60 th birthday. Formerly, Gradate Research Assistant; crrently, Visiting Research Assistant Professor, Department of Civil Engineering and Geological Sciences, University of Notre Dame, Notre Dame, IN 46556; johnsone@nd.ed Associate Professor; petros@control.csl.ic.ed Professor; lbergman@ic.ed Mltiobjective Optimal Strctral Control of the Notre Dame Bilding Model Benchmark

2 measres of system performance. Frthermore, H robst control design methods are advantageos since the model of a plant always contains some measre of ncertainty for any real-world application, whether de to modelling errors or long-term changes in system dynamics. With a small sacrifice in the system H norm from that fond sing an H (LQG) optimal controller, giving slightly larger root mean sqare response to white noise excitation, the l and H norms can often be significantly redced, leading to better performance and more robst stability. The se of all three of these norms to devise an optimal controller can combine the strengths of each. The general formlation of the mixed H /H optimal control design problem has been explored in a nmber papers (e.g., Khargonekar and Rotea 4, Kaminer et al. 5, Bernstein and Haddad 6 ), bilding on H and H control methods in Doyle et al. 7-9 The l optimal control problem has also been detailed in varios papers and texts (e.g., Dahleh and Pearson 0 ; Dahleh and Diaz-Bobillo ; Khammash ). A nmber of soltion methods have been developed to solve certain mixed-norm control design problems (e.g., Whorton et al. 3, Volgaris 4 ). For example, the homotopy algorithm of Whorton et al. 3 ses a linear combination of the objective H and H norms, varying the ratio of the linear coefficients and searching incrementally in the H and H directions to define a family of mixed-norm designs. To facilitate nmerical soltions to general mixed-norm optimal control problems, Jacqes, Canfield, Ridgely, and Spillman 5-9 developed a set of nmerical algorithms and accompanying MATLAB code to search for fixed-order controllers that minimize an H norm while constraining one or more H, l (or L ), and H norms below some critical vales. These algorithms, which can be sed for continos- or discrete-time systems, have been adapted to the crrent problem with alteration of the associated software. BENCHMARK PROBLEM REQUIREMENTS The base plant, is described by the block diagram in Fig. and by the state eqations ẋ = Ax + B + Eẋ g z = C z x+ D z + F z ẋ g y = C y x+ D y + F y ẋ g + v z [ x x x 3 x m ] T = [ ẋ ẋ ẋ 3 ẋ m ] T y = v + x m ẋ a ẋ a ẋ 3a ẋ ma ẋ g [ ẋ a ẋ a ẋ 3a ẋ ma ] T T () where x is the state vector (in R 8 for the active mass driver (AMD) system and in R 0 for the active tendon system (TEN)), y is the measrement vector, z is a vector of responses to be reg- Mltiobjective Optimal Strctral Control of the Notre Dame Bilding Model Benchmark

3 ẋ g F z E C z z D z v B I/s x F y A C y y D y Figre. Block diagram of benchmark plant. lated, is the actator inpt, ẋ g is the seismically-indced grond acceleration, and v is the sensor noise (independent, zero-mean, Gassian white noise processes). The components of the otpt measrement y and reglated response z, the former corrpted by sensor noise, inclde the displacement x and velocity of the i th i ẋ i floor relative to the grond, absolte accelerations of the i th floor ẋ ia and the grond ẋ g, relative actator displacement x m and velocity ẋ m, and either the active mass absolte acceleration ẋ ma or the active tendon force f. In the active tendon system, the tendon force f replaces the active mass acceleration ẋ ma in both y and z in (). Interstory drifts are defined as d () t = x () t, d () t = x () t x () t, and d 3 () t = x 3 () t x () t. The coefficient matrices in () represent the inpt-otpt behavior of the bilding models in the Notre Dame Strctral Dynamics and Control / Earthqake Engineering Laboratory (SDC/ EEL) p to 00Hz (the AMD system) and the National Center for Earthqake Engineering Research (NCEER) p to 50Hz (the TEN system). Both sets of coefficient matrices were distribted in a MATLAB MAT-file as a part of the benchmark problem definition. 0 The goal, then, is to design a stable, discrete-time controller with sampling period T = ms, sbject to several implementation constraints: (a) the sensor noise components are Gassian rectanglar plse processes of ms dration and root mean sqare magnitde 0.0 V. (b) the A/D and D/A converters at the controller inpt and otpt have -bit precision and a span of ±3V; (c) the controller may have no more than states; and (d) the reslting actator response mst be within the bonds of the 6 hard constraints, C i, i 6, as defined below. Mltiobjective Optimal Strctral Control of the Notre Dame Bilding Model Benchmark 3

4 The performance of the controller is then jdged by 0 performance measres, J i, i 0, as smmarized below. RMS Responses to Kanai-Tajimi Excitation Spectrm The first five performance measres are root mean sqare responses to a family of Kanai- Tajimi excitation spectra. The two-sided spectral density S ( ω) is given by ẋ gẋ g 4ζ S ( ω) S g ω g ω + ω 4 g s = ẋ gẋ g S 0 ζ g ( ω ω g ) + 4ζ g ω g ω 0 = πω g ( 4ζ g + ) () where the freqency and damping ratio of the bedrock-grond connection are in given ω g ranges ( ζ g [ 0.3, 0.75], as per Table I). (The intensity of the excitation is sch that, the ω g root mean sqare grond acceleration, is given in Table I and is independent of and.) The performance measres, weighted by the nondimensionalization parameters ζ g in Table I, are root mean sqare strctral response (maximm interstory drifts and maximm floor acceleration) and actator response (actator displacement, velocity, and acceleration or force), defined by j i ω g ζ g σ ẋ g J σ d i = max J ω g, ζ g, i j σ ẋ ia = max J σ x m 3 = max ω g, ζ g, i j ω g, ζ g j 3 (3) J σ ẋ m 4 = max J 5 ω g, ζ g j 4 = max ω g, ζ g σ or σ ẋ ma f j 5 (4) with the constraints C σ max ω σ g, ζ g σ = C ẋ ma = max or C 3 ω g, ζ g σ ẋ ma σ f σ f = σ xm max ω g, ζ g σ xm (5) where σ =V, σ =g, σ, and. RMS responses are compted via ẋ ma f =4kN g = 9.8m/s a SIMULINK block similar to that in Spencer et al., p to 300 (AMD) or 750 (TEN) onds. Table I. Performance measre normalization parameters and benchmark constants. s 0 ω g σ ẋ g σ xm x m j j j 3 j 4 j 5 j 6 j 7 j 8 j 9 j 0 actator earthqake cm AMD 0.03 V 3.37 cm 5.05 g 3.37 cm rads cm 5.05 g El Centro [0,0] g 3 cm 9 cm.3 cm.79 g.3 cm g cm.66 cm.58 g.66 cm g Hachinohe TEN (mv) [8,50] g cm 3 cm.34 cm g.34 cm kn 6.45 cm.57 g 6.45 cm 99.9 cm rads cm El Centro cm 89 kn 3.78 cm g 3.78 cm Hachinohe Mltiobjective Optimal Strctral Control of the Notre Dame Bilding Model Benchmark 4

5 Peak Responses to NS Records of the 940 El Centro and 968 Hachinohe Earthqakes The remaining five performance measres are peak responses to the scaled grond acceleration records of two actal earthqakes. (The records and appropriate scaling were defined in the benchmark. 0 ) The performance measres are J 6 d i () t ẋ ia () t x = max J 7 = max J m () t 8 = max t, i El Centro Hachinohe j 6 t, i El Centro Hachinohe j 7 t El Centro Hachinohe j 8 (6) J 9 ẋ m () t ẋ ma ( t) or ft () = max J 0 = max t El Centro Hachinohe j 9 t El Centro Hachinohe j 0 (7) with constraints C 4 t () ẋ ma () t ft () x = max C 5 = max or C m () t 6 = max t El Centro Hachinohe t El Centro Hachinohe ẋ ma f t El Centro Hachinohe x m (8) where =3V, ẋ ma =6g, and f =kn. Performance of a Sample LQG Controller for the Active Mass Driver System One sample controller, based on an LQG design that weights the three floor accelerations eqally and feeds back the accelerations of the three floors and the active mass, was compted by Spencer et al. ; the reslting vales of the performance measres are given in Table II. Table II. Performance of sample LQG controller on the AMD system. J J J 3 J 4 J 5 J 6 J 7 J 8 J 9 J 0 C C C 3 C 4 C 5 C NOTATION AND MATHEMATICAL DEFINITIONS Varios signal and system norms are sed in formlating the mltiobjective optimization problem to solve for mixed-norm optimal controllers. A brief mathematical description of these norms and their se in the crrent problem are as follows. Let x() t be a real-valed vector process in R m, with its discrete-time correspondent, () k, x with sampling time T. The p-norm of this signal may be defined as Mltiobjective Optimal Strctral Control of the Notre Dame Bilding Model Benchmark 5

6 The H norm of a system reflects the root mean sqare response to Gassian white noise excitation, and ths it is central to control design for the benchmark problem; minimizing the closed loop H norm G will minimize the root mean sqare otpt y. The H norm is a worst-case magnification of excitation energy into the system response (i.e., G = max y ); it l also can be sed to measre the worst-case sensitivity of the stability of the closed-loop system to nmodelled dynamics, sch as high-freqency modes that are often neglected or have low signalx p m i= / p = x i () t p dt p = T i x x () k p k= m i= / p (9) The special cases for the -, -, and -norms are x x m = x i () t dt = T i x x () k i= k= m i= = x * ()x t () t dt x = T * k () k x ()x k= x = sp max x i () t = sp max i x x () k t i k i (0) where () * denotes complex-conjgate transpose. (The sampling time is inclded in the discretetime signal norms so that, assming T is sfficiently small, the reslting norms are eqivalent to that of the continos-time signal.) A continos-time signal is in discrete-time signal in l p, if its p -norm is finite. (a Lebesqe space), or a Let G() s and () z denote the continos- and discrete-time transfer fnctions of a system, G respectively, with corresponding implse response fnctions, Ĝ() t and () k. Frthermore, let Ĝ and y be excitation and response (inpt and otpt of the system), respectively. Then several key norms may be written as ( σ [] denotes the largest singlar vale) L p n L : G = max ĝ ij () t dt l : = max ij G ĝ () k i j= i k= H : G = trace [ G π * ( jω)g( jω) ] dω H : G = trace [ * ( e πt G jω ( e )G jω )] dω H : G = sp σ [ G( jω) ] H : = sp σ [ ( e G G jω )] ω ω π π n j= () Mltiobjective Optimal Strctral Control of the Notre Dame Bilding Model Benchmark 6

7 to-noise ratios (making them difficlt to model accrately). Ths, to minimize the H norm arond an nknown block of nmodelled dynamics for robst control design will minimize the sensitivity of the closed-loop stability to the ncertainties in the model. Finally, the L norm (or the l norm for discrete-time systems), is the worst-case magnification of inpt magnitde; minimizing the L norm can often decrease peak responses. MULTIOBJECTIVE OPTIMAL CONTROL The Mixed-Norm Toolbox 8,9, also known as MXTOOLS, is designed to solve mixed-norm, fixedorder, optimal control problems by choosing a controller of a selected order to minimize an H norm sbject to one or more H, L or l, and H norm constraints. Assming a base H problem, one L or l constraint, and one H constraint, the problem is described by the the interrelated systems in Fig. and the corresponding continos-time state-space descriptions ẋ = A x + B w w + B ẋ = A x + B r r + B ẋ = A x + B d d + B z = C z x + D zw w + D z m = C m x + D mr r + D m e = C e x + D ed d + D e () y = C x + D yw w + D y y = C x + D yr r + D y y = C x + D yd d + D y (or similar discrete-time eqations), where state vectors x, x, and x may have some or all states in common and the measrements y, y, and y are eqal in the absence of distrbances w, r, and d, i.e., C x = C x = C x. The inpt w is a zero-mean Gassian white noise of nit intensity (i.e., E [ w()w t T () s ] = Iδ( t s) or E w()w k T [ () l ] =, where T is the sampling period), the inpt r is assmed to be an nknown magnitde-bonded signal with r, and the inpt d is assmed to be an nknown energy-bonded signal with d. No relationships are assmed between w, r, and d or between z, m, and e. The H /L /H (or H /l /H ) optimization problem is then to find an admissible, stabilizing controller K, with state space description ( A c, B c, C c, D c ), of a fixed order, that achieves inf sbject to T mr ν and T (3) ed γ K admissible T zw T --Iδ kl w Base H Problem L or l Constraint H Constraint T zw T z T yw T y z r y T mr T m T yr T y m d T ed T e y T yd T y y e Figre. MXTOOLS plant with a base H system, an L or l constraint, and an H constraint. Mltiobjective Optimal Strctral Control of the Notre Dame Bilding Model Benchmark 7

8 The reqirements for soltion with the fixed order at least that of the base H sbproblem are the base H sbproblem mst have ( A, B ) stabilizable and ( A, C ) detectable, the base H sbproblem mst be well posed, i.e., the overall problem mst be non-singlar, and ( I+ D c D y ) is invertible, for continos-time systems, D zw + D z D c D yw = 0 so that T zw is finite. For a controller of lesser order, an additional reqirement is the existence of a stabilizing controller of the given order that satisfies the constraints. MXTOOLS parameterizes the controller in modal form to redce the nmber of design variables (generally disallows repeated controller eigenvales; one cold modify MXTOOLS to se a fll state-space, or any other, parameterization if repeated eigenvales are thoght to be reqired). The objective fnction, the constraints, and their gradients, are constrcted 7 and passed to any constrained optimization algorithm. The defalt solver ses a seqential qadratic programming (SQP) method similar in fnction to MATLAB s Optimization Toolbox constr fnction, bt handles larger problems more effectively (constr often has difficlty maintaining a positive definite approximation to the Lagrangian Hessian for problems with more than 40 or 50 design variables). The formlation above assmes one L or l constraint and one H constraint. Similar formlations for an arbitrary nmber of constraints may be performed. The crrent implementation of MXTOOLS is limited to one MIMO L or l constraint, thogh this is of no difficlty since several L or l constraints can be combined into one with appropriate scaling. A few caveats in the se of MXTOOLS mst be noted. First, no information is available as to how distant a compted fixed-order soltion is from the tre optimal, order-free soltion. 9 For the benchmark problem, this is not of great concern since the controller order is restricted. Second, the search for redced-order controllers may be more ssceptible to the problem of local minima, bt may be significantly easier to solve de to the redced search space dimension. 9 Third, the l norm tends to be qite conservative for most magnitde-bonded, bt otherwise nknown, inpts, and therefore its vales oght to be regarded in a relative, qalitative sense, rather than a measre of the actal performance on magnitde-bonded otpt constraints. 7 The MXTOOLS software was a sefl tool in solving the mltiobjective problems in this stdy, thogh a fair amont of tweaking and a few bg fixes were necessary in the process. Mltiobjective Optimal Strctral Control of the Notre Dame Bilding Model Benchmark 8

9 FORMULATION OF THE SUBPROBLEMS The benchmark problem can be agmented by varios static and dynamic weights to transform it into a form sitable for a mixed-norm optimal controller search. One aspect of the benchmark problem rather indirectly addressed in the control design here is the 00 µs comptational delay at the plant inpt (time to compte the control action and perform the A/D and D/A conversions); it is inclded here only in the H sbproblem as a part of the plant ncertainty. If the delay were larger or considered more important, the base plant cold be explicitly agmented from the start (at the plant inpt) with a Padé approximation of the delay. 3 Some of the formlation given below is in continos-time terms, bt all sch transfer fnctions were converted to discrete-time for the control optimization problem. Base H Problem The base H problem can be formlated by adding to the base plant varios weighting fnctions for normalization, as shown in Fig. 3, where w is a 7 vector of zero-mean, Gassian white noises with E [ w()w t T ( t + τ) ] = Iδ() τ. One element, or channel, of w is filtered throgh the ond-order system W KT () s to model the Kanai-Tajimi spectrm W KT ( s, ω g, ζ g ) = πs 0 ζ g ω g s + ω g s + ζ g ω g s + ω g (4) sing the grond parameters ω g and ζ g that case the worst ncontrolled response ( ζ worst g = 0.3 ; ω worst g = 37.8 rads for the active mass driver system and ω worst g = 4.5 rads for the active tendon system). The remaining elements of w are weighted to model the sensor noise. At the otpt side, W z is a weight on the control effort and W is a normalization factor. M zz is a matrix sed to rearrange the otpts of the base plant into the elements to be reglated; Z V converts from Volts to the dimensional nits and then J -5 nondimensionalizes. W z is a diagonal matrix to weight the normalized criteria differently as reqired for improved controller performance (the w D Base Plant E W KT M ẋ g U W X v v z M zz J -5 M UX W z [ d d d 3 ] T [ ẋ a ẋ a ẋ 3a ] z [ weights] T = [ x m ẋ m ẋ ma ] T y W W z Figre 3. Block diagram of the base H problem. (Note: DEMUX is a demltiplexer that separates two sets of signals in a mlti-signal line; MUX, or mltiplexer, combines several channels into one vector signal.) Mltiobjective Optimal Strctral Control of the Notre Dame Bilding Model Benchmark 9

10 identity matrix was fond acceptable in the final designs herein). The reslting otpt vector z contains weighted forms of the three interstory drifts, the three floor accelerations, and the actator relative displacement, relative velocity, absolte acceleration (actator force for the active tendon system), and control effort. J -5 Z V AMD = = [ j j j ] T diag [ j j j ] T [ j 3 j 4 j 5 ] T V [ ] cm T diag Vs [ ] cm T --- [ 0.] T V g W v = 0.0 T I W = σ M zz = Z V (5) Z V TEN = V [ 3.48 ] cm T diag Vs [ ] cm T 0.5 V g -- [ ] T ḡ kn H Constraints The benchmark reqires that the root mean sqare controller otpt, actator displacement, and actator acceleration/force be less than critical vales, to remain within the operational range of the actator. This can be formlated as an H constraint on the system shown in Fig. 4, where the new weights are given by W q = σ or σ ẋ ma f 0 q 0 σ xm a or f ẋ m 0000 = M zz = 000 x m Z V (6) and where W z is a diagonal matrix to be determined for best performance. More properly, three H constraints, one for each element of z will be sed herein (thogh one may sbstitte the single constraint in the interests of comptational feasibility). w D E M U X W KT W v ẋ g v Base Plant z M M zz q W q UX W z ẋ ma z = [ weights] x m y W Figre 4. Block diagram of the H constraints. Mltiobjective Optimal Strctral Control of the Notre Dame Bilding Model Benchmark 0

11 l Constraints The se of l constraints permits the inclsion of the hard peak constraints of the actator (to avoid satration) as well as to facilitate bonding the performance of the peak response portion of the benchmark as measred via J 6 throgh J 0. The block diagram of the l constraint sbproblem is shown in Fig. 5, where the weights, chosen to reflect actator peak constraints and the desired performance of the peak responses, are given by W m M mz = max El Centro Hachinohe = M zz M zz ẋ g [ j 6 j 6 j 6 ] T [ j 7 j 7 j 7 ] T J 6-0 = min diag max( j 8, x m ) El Centro ẋ g Hachinohe j 9 max( j 0, ẋ ma or f ) (7) The otpt vector m, the system responses whose peak magnitdes shold be constrained or minimized, is the same as z in the base H problem bt with different weights. The inpt r is the distrbance, which shold be a bonded-magnitde signal for this sbproblem (i.e., in l ); the problem was stdied with two inpt weights W r : nity and a Kanai-Tajimi filter to more closely model earthqake inpt spectra (in which case r corresponds to the acceleration of bedrock in the Kanai-Tajimi earthqake model). The matrix serves as a final scaling matrix and to select W m elements to inclde in the otpt vector m. It was observed in evalation of several controllers that for of the ten elements of m were critical (weighted measres of the nd and 3 rd floor accelerations, m 5 and m 6 ; actator velocity m 8 ; and actator acceleration/force m 0 ); de to the expense of compting the l norm and its gradient, the matrix was sed to select only those for otpts in m dring the optimization. W m r W r ẋ g v Base Plant z M mz J 6-0 M UX [ d d d 3 ] T m [ ẋ a ẋ a ẋ 3a ] W m m [ weights] T = [ x m ẋ m ẋ ma ] T y W m Figre 5. Block diagram of the l constraints. H Constraints One typical prpose for involving H in many problems is to minimize the error in tracking given command inpts, generally by minimizing the weighted sensitivity of the system. The Mltiobjective Optimal Strctral Control of the Notre Dame Bilding Model Benchmark

12 problem at hand, however, is involved with agmenting the inherent damping in the strctre. There is some concern that the nominal plant may not be entirely accrate, partly de to the inaccracies involved with any model derived from experimental data, and partly de to the fact that high freqency dynamics (above 00 Hz for the AMD system, and above 50Hz for the TEN system) were trncated in forming the state-space model given in the benchmark problem. For this reason, a mltiplicative ncertainty at the plant inpt is assmed, as seen in Fig. 6, with the dynamic weight W e ()= s 3s + 90s to accont for the primarily high-freqency nmodeled s + 300s dynamics, the 00 µs comptational delay, and the inpt loop gain robstness reqirement above 35Hz (5Hz was sed for the TEN system). The reslting H constraint wold be to inclde the nmodelled dynamics block in Fig. 6, where e and d and reflect the inpt and otpt of the nmodelled dynamics. If taken by itself as an H control design problem, this H constraint has the trivial soltion of the zero controller becase the system is already stable, and the controller that minimizes the infinity norm of the transfer fnction from d to e is identically zero. This is not an inherent problem with the mltiobjective problem, bt for prposes of comparison, it is convenient to add an additional ncertainty block so that a base H controller can be designed. A robst performance control design problem can be solved sing a fictitios ncertainty block, so sch a block is placed at the plant otpt. The reslting design will not only be robstly stable, bt its otpt performance shold also be robst. The weights for this additional block inclde the static diagonal weight normalize by the ncontrolled infinity norm for each channel, and the weight dynamic, bt here will be set to identity. W es W e and its inverse to, which cold be robst performance K nmodelled dynamics W e e d P y d e W es W e W es d D E d M U X d W es ẋ g v Base Plant W e z e W e e W es M UX e y Figre 6. Block diagram of the H constraints. Mltiobjective Optimal Strctral Control of the Notre Dame Bilding Model Benchmark

13 CONTROLLER DESIGNS: ACTIVE MASS DRIVER SYSTEM Performance with Zero Controller In order to properly jdge sperior control designs, the performance of the system with the zero controller (a 0V command signal always sent to the actator) was first evalated. Since the closed-loop system is linear, the RMS responses to the Kanai-Tajimi spectra can be efficiently compted by solving a Lyapnov eqation (if a mlti-inpt, single-otpt (MISO) system has Laplace transform Ys () = G()W s () s, where w() t is a zero-mean, Gassian white noise vector process with E [ w()w t T () s ] = Iδ( t s), then the expected otpt is E[ y () t ] = y RMS = G ). The peak responses to the known earthqakes can be simlated qickly via lsim in MATLAB. The reslting performance data are given in colmn # of Table IV (inclding the Kanai- Tajimi parameters ω g and ζ g that maximized the RMS vales). For prposes of verification, the worst-case RMS and peak third-floor responses, sed in the j i performance weights, were compted and are shown in Table III; they are fairly consistent with the given above in Table I. j i Table III. Worst-case third-story responses with the zero controller actator σ worst x3 σ worst ẋ3 σ worst ẋ 3a x worst 3 ẋ worst 3 ẋ 3aworst earthqake AMD.39 cm rads ω g = cm rads ω g = g rads ω g = cm cm g El Centro cm cm g Hachinohe TEN cm rads ω g = cm rads ω g = g rads ω g = cm cm g El Centro cm cm g Hachinohe Performance with Sample Controller In order to verify that the evalation software was performing properly, the sample controller spplied with the benchmark proposal was evalated. The reslts, shown in colmn # of Table IV, are consistent with those detailed in the benchmark and given above in Table II (with the exception of C 5, which in Table II may reflect jst one of the two earthqakes in the benchmark), demonstrating that the evalation software operates satisfactorily. Base H (LQG) Design The primary parameter to be determined in a base H design is the relative weight on the control effort, W z. Fll-order, discrete-time H -optimal controllers were compted sing a wide Mltiobjective Optimal Strctral Control of the Notre Dame Bilding Model Benchmark 3

14 0 0 J 5 C Performance and Constraint vales 0 - J J (drifts) J (accelerations) J 3 (actator disp) J 4 (actator veloc) J 5 (actator accel) C (control) C (actator accel) C 3 (actator disp) C J 3 J 4 J C H control weight W z Figre 7. Control weight tradeoffs for the base H controller in the AMD system. range of control effort vales to minimize T zw. The reslting RMS performance and constraint vales, assming the system is sfficiently linear to apply the appropriate Lyapnov eqations, are shown in Fig. 7. It is a worthwhile tradeoff to allow a larger control effort and greater actator dynamics for the sake of redcing the larger performance and constraint vales by letting W z be nity. A choice of W z < gains little in the way of the RMS performance and constraints, bt may limit the ability to increase robstness and otpt magnitdes in sbseqent optimization. Two initial redced order H controllers were compted, one by a Hankel norm model redction of the fll H controller, and the other by first redcing the plant and then designing an H controller. These two designs were then sed as initial conditions in a minimization of the H norm of the closed loop formed with the fll-order plant and the redced-order controller. The performance on the benchmark for the reslting (redced-order) H optimal controller is given in colmn #3 of Table IV. Two additional redced-order controllers are sed as starting points in the H /H optimization below, compted in a manner similar, bt doing H control designs by minimizing T ed. The base H problem contains somewhat conflicting goals, since the response of the strctre and the response of the actator cannot be simltaneosly minimized. Figre 8 shows the tradeoff between the RMS strctral response Str (simply the first six otpts of in the base H T zw z Mltiobjective Optimal Strctral Control of the Notre Dame Bilding Model Benchmark 4

15 .8 T Str z w.6.4. H Str /H Act Pareto crve? locally optimal H Str Pareto optimal H Str H Str H Str /H Act /H Act /H Act /H Act (violates Peak constrs.) (violates Peak & RMS constrs.) redced-order H Act redced-order H sample H controller zero controller T Act z w Figre 8. H Str / H Act tradeoffs between RMS strctral and actator responses for the AMD system. plant) and the RMS actator response T Act zw (the three actator responses in z ). (The constraints were ignored in the minimization, bt are noted in the graph where violations occrred.) The benchmark performance of the controller that minimized the RMS actator response is given in colmn #4 of Table IV, while colmn #5 gives the performance of the controller that minimized the RMS strctral response while not violating RMS constraints (it did violate the peak active mass acceleration, bt it is sefl to see what the performance wold be withot the peak constraints). If the relative importance of these two sets of responses were known, one cold appropriately weight the strctral and actator responses in the base H problem; in the mixed-norm designs below, however, they are weighted eqally. Mixed H /H Design Abot 00 different control designs were compted via the optimization software to stdy the tradeoffs between H and H performance. The norms of the reslting closed-loop systems are plotted in Fig. 9; some of the designs obviosly stalled at local minima, bt a nmber of them demonstrate qite clearly that significant robst stability and robst performance can be obtained with near H performance. (Note that the Pareto crve is approximate since the points from compted control designs are pper and right bonds on the crve, while constant lines at the closed-loop -norm with the fll-order H controller and the -norm with the H controller are Mltiobjective Optimal Strctral Control of the Notre Dame Bilding Model Benchmark 5

16 T z w approx. H /H Pareto crve locally optimal H /H Pareto optimal H /H fll-order H and H redced-order H and H sample H controller zero controller T ed Figre 9. Pareto optimal crve for H /H AMD controllers. left and lower bonds, respectively.) The two circled larger s near the minimm H have -norms only 0.86 and 0.05 percent above minimm, bt at one thirteenth and one eighth of the infinity norm of the redced-order H optimal design. Sch design points introdce significant robstness with negligible changes in RMS otpts to white noise excitation. The third circled large has the opposite tradeoff, with a closed-loop infinity norm nearly that of the H optimal controller, bt with a RMS otpt over 30% less. The benchmark performance vales for several of the (near-) Pareto optimal designs are graphed in Fig. 0, where it can be seen that the performance changes little for vast robstness improvements. The performance on the benchmark problem for the first of the above mentioned designs is given in colmn #6 of Table IV, and its inpt loop gain is shown in Fig. ; this design far exceeds the robstness reqirements in the benchmark description with little or no performance degradation. The performance of the controller reslting in the least mean sqare of the 0 6 performance and constraint vales (i.e., arg min J i ( K) + C ) is given in colmn i = i ( K) K i = #7 of Table IV; this controller was one that significantly decreased the peak active mass acceleration. If the robst performance component of the H constraint is removed, the reslting system has the zero controller as the H optimal robst controller. This complicates the characterization of the Pareto optimal tradeoff between the robst stability sensitivity, denoted here with, and () RS Mltiobjective Optimal Strctral Control of the Notre Dame Bilding Model Benchmark 6

17 . Performance and Constraint vales J J J 3 J 4 J 5 C C C 3 J 6 J 7 J 8 J 9 J 0 C 4 C 5 C T ed Figre 0. Linear performance and constraint vales for (near-) Pareto H /H optimal AMD controllers. 30 Singlar Vale Magnitde [db] mixed H /H controller redced-order H controller redced-order H controller sample H controller Freqency [Hz] Figre. Inpt loop gain for a redced-order H /H optimal controller for the AMD system. the H RMS response. Nevertheless, a nmber of mixed H / H RS controllers were fond as shown in Fig. The circled large represents a controller that provides a closed-loop RMS otpt only 0.8% larger than the redced-order H optimal controller bt 50 times as robst. The benchmark performance of this controller is given in colmn #8 of Table IV. Mltiobjective Optimal Strctral Control of the Notre Dame Bilding Model Benchmark 7

18 H /H RS Pareto crve? locally optimal H /H RS Pareto optimal H /H RS fll-order H redced-order H sample H controller T z w T RS ed Figre. H / H RS RMS performance and robst stability sensitivity for the AMD system. Mixed H /l Design The H /l optimization problem is more difficlt than the H /H for several reasons. First, compting the l norm and its gradients with respect to controller parameters is qite expensive, limiting the nmber of trials that cold be tested. The ond is the lack of a good starting point; in the H /H design, it was observed that the optimization algorithm more readily relaxed constraints than restrict them, so sing the H optimal controller as the starting point worked well. Here, the l optimal controller was nknown; several semianalytical methods, inclding those of Khammash, were applied to solve for the l optimal controller, bt the combination of very low damping in some modes and high freqency actator dynamics prevented a soltion. Conseqently, the optimization software was modified to do nconstrained l optimization, which facilitated finding a nmber of controllers with smaller closed-loop l norms (bt with no knowledge of the proximity to the l optimal controller). Two different mixed H /l strategies were attempted to determine if one or the other had an impact on the peak performance and constraint measres defined above. The first sed the l constraint with a constant weight on the inpt. Abot 40 mixed H /l designs were compted and are shown in Fig. 3. The l norm was redced somewhat from the H optimal controller, bt not nearly as significantly as the H norm above. Some of these designs lowered the l norm of the Mltiobjective Optimal Strctral Control of the Notre Dame Bilding Model Benchmark 8

19 T z w H /l Pareto crve? locally optimal H /l Pareto optimal H /l redced-order H sample H design H /H designs T mr Figre 3. Mixed H /l optimal controllers for the AMD system (no freqency dependent weight). crve is poorly defined at the low l -norm end de to difficlties with local minima. The large circled is the controller that obtained the smallest closed-loop l norm. The ond set of mixed H /l designs sed a freqency-dependent inpt weight, that of the Kanai-Tajimi spectrm (with the grond freqency and damping that cased the worst response with the zero controller), to more closely model an earthqake record; to denote the freqencydependent weight, these designs will be called H / l KT KT. The reslts of the 55 H / l designs are plotted in Fig. 4. The Pareto crve is a little better defined here, bt withot an optimal l controller to give the lower bond, it is impossible to define the crve with mch certainty. The reslts here are similar in that 5-0% drops in closed-loop l norm can be achieved with only nominal sacrifice in H performance. Unfortnately, little correlation was fond in the end between the closed-loop l norm and the vales of the peak performance J 6 0 and constraint C 4 6 vales, as shown for the non-freqencydependent H /l designs in Fig. 5 over a range of the l norm. The irreglarity of the lines is to be expected, bt the obvios lack of a general proportional trend between the l norm and the performance/constraint vales shows that the l norm of the closed-loop cannot be sed to predict peak responses to the known earthqake records. Mltiobjective Optimal Strctral Control of the Notre Dame Bilding Model Benchmark 9

20 .6.55 T z w H /l KT Pareto crve? locally optimal H /l KT Pareto optimal H /l KT redced-order H sample H design H /H designs T KT mr KT Figre 4. Mixed H / l optimal controllers for the AMD system (Kanai-Tajimi freqency-dependent weight). 0 0 J 0 J 7 Performance and Constraint Vales J 6 (drifts) J 7 (accelerations) J 8 (actator disp) J 9 (actator veloc) J 0 (actator accel) C 4 (control) C 5 (actator accel) C 6 (actator disp) C 5 J 9 J 6 J 8 C 6 C T mr Figre 5. Peak linear performance and constraints are ncorrelated with closed-loop l norm. The lack of correlation was rather srprising at first, bt, in retrospect, it makes some sense. The l norm is the worst-case peak otpt over a very wide class of inpt excitations, those that are magnitde bonded (i.e., those signals in l ), which incldes persistent signals that have DC Mltiobjective Optimal Strctral Control of the Notre Dame Bilding Model Benchmark 0

21 offsets and long-term non-decaying behavior. The earthqake records sed to evalate the peak response of the closed-loop system for the benchmark, however, are finite in dration and more properly belong to the class of energy bonded signals (l ). The worst case otpt magnitde over all possible energy-bonded inpts is governed by the H norm of the system. Ths, the H norm shold be a better measre of peak otpt magnitdes than the l norm for the excitations of interest here. CONTROLLER DESIGNS: ACTIVE TENDON SYSTEM Performance with Zero Controller The performance of the active tendon system with the zero controller is given in colmn #9 of Table IV. This system has significantly tighter constraints, primarily on the actator command voltage C 4 and actator force C 5. Base H (LQG) Design The analysis parallels that of the AMD system. The base H tradeoff between performance and control effort is shown in Fig. 6. The control effort weight has a very narrow admissible range since setting it to larger than.874 reslts in systems that violate the constraint on the W z C Performance and Constraint vales C C 3 nstable controllers J J J (drifts) J (accelerations) J 3 (actator disp) J 4 (actator veloc) J 5 (actator force) C (control) C (actator force) C 3 (actator disp) 0 - J 5 J 3 J H control weight W z Figre 6. Control weight tradeoffs for the base H controller in the TEN system. Mltiobjective Optimal Strctral Control of the Notre Dame Bilding Model Benchmark

22 actator force ( C > ), and for W z < 0.887, the reslting controller is nstable. Ths, the weight chosen here is again nity. Several initial H controllers, first fll-order, then redced order, were compted as described for the AMD system above; the reslting benchmark performance data for the best redced-order H design is given in colmn #0 of Table IV. It mst be noted that the actator velocity here reached its peak in a very different location in ( ω g, ζ g ) space than other RMS performance and constraint vales. A redced-order H optimal controller were also sed as initial conditions for mixed optimization. Mixed H /H Design Abot 00 mixed H /H designs were compted to stdy the performance and robstness tradeoffs of the TEN system. The reslting norms are shown in Fig. 7; as was previosly noted, the optimization algorithm does get stck in local minima for many of these designs, bt some are very good at improving robstness with little loss of the H benefits. In fact, the H optimal controllers have terrible robstness characteristics for the active tendon system. The circled large s are controllers that generate closed-loop plants with -norms. and 0. percent larger than the H optimal controller, bt with infinity norms abot /850 th and /75 th that of the H controller. The inpt loop gains for these two designs are shown in Fig. 8 and the performance.5 T z w approx. H /H Pareto crve locally optimal H /H Pareto optimal H /H fll-order H and H redced-order H and H zero controller T ed Figre 7. Pareto optimal crve for H /H TEN controllers. Mltiobjective Optimal Strctral Control of the Notre Dame Bilding Model Benchmark

23 Singlar Vale Magnitde [db] mixed H /H controller mixed H /H controller redced-order H controller redced-order H controller Freqency [Hz] Figre 8. Inpt loop gain for redced-order H /H optimal controller for the TEN system. evalation of the ond is given in colmn # of Table IV. The optimal H controller reslts in a fair sensitivity to high-freqency modelling errors that cold, at worst, case instability if the the trncated high-freqency modes dominate nder some loading conditions. The mixed H /H designs, however, while retaining most of the RMS performance of the H design, also captre some of the robstness of the H optimal controller. Mixed H /l Design De to the comptational intensity and its inability to sppress peak responses of the AMD system, no mixed H /l optimization was done on the active tendon system. OBSERVATIONS AND CONCLUDING REMARKS The mixed H /H optimal control design was able to provide controllers with significantly greater robstness characteristics at little cost to RMS and peak performance measres defined in the benchmark. This was especially noticeable in the active tendon system, where the infinity-norms of the H optimal controllers were qite large. The robstness was a relatively minor point in the benchmark definition, bt it is a critical safety isse for actively-controlled bildings and mst be inclded in an evalation of controller design. Mltiobjective Optimal Strctral Control of the Notre Dame Bilding Model Benchmark 3

24 The performance of the controllers mentioned herein is generally better than the sample controller provided with the benchmark, thogh the definition of better is, of corse, rather application specific and will vary with the reqirements of the strctre. The 50 best AMD reslts here averaged abot 0% less than the sample controller (where all 0 performance measres and the 6 weighted constraints are averaged together), with the best being 8% less. The compter resorces reqired for the mltiobjective optimizations performed for this stdy vary significantly depending on what objective and constraint fnctions are sed. The l norms were rotinely significantly more comptationally intensive than the others, p to the better part of a day of CPU time on an HP9000/780 workstation. Other mixed-norm designs, sch as the RMS strctral/actator response tradeoffs, generally took only a few mintes on the same platform. A fll Simlink simlation (p to 300 onds) of one Kanai-Tajimi spectrm for one controller took abot 05 mintes, so the evalation was at least as expensive as the design process. It was observed in the process of doing H /H mixed designs that the optimization algorithm more readily relaxed constraints than restricted them, being less likely to get stck at local minima. For example, in an H /H problem, starting with an H optimal controller and relaxing the constraint often reslts in soltions that are closer to the Pareto optimal crve than starting with a controller that is closer to H optimal and searching for a more constrained H vale. It may be noted that the H portion of the problem tends toward the conservative side since the overall ncertainty is strctred de to the two separate ncertainty blocks. Walker and Ridgely 6 have done some work in mixed H /µ optimization, so the above development cold replace the H constraint with a µ constraint to more closely model the tre system ncertainty. Investigation into l optimization in the mixed-norm context for this benchmark proved inadeqate for redcing peak responses and accommodating peak hardware constraints. Since an earthqake is generally an energy-bonded excitation (it dies ot after a short time), the l norm might not be the best indced norm to se since that assmes the inpt to be in too broad a class of signals (l ). (In fact, it may be arged that an earthqake excitation is in l, an even narrower class of signals whose l norm is bonded.) The H norm, already inclded for minimizing RMS response, has a bigger impact on peak otpts to the real earthqake records than did controllers with small corresponding l norms. This stdy demonstrates the seflness of mixed-norm optimization to allow the control designer to dramatically improve robstness while retaining the performance of H /LQG optimal controllers. Understanding these tradeoffs is essential to the control design process. Mltiobjective Optimal Strctral Control of the Notre Dame Bilding Model Benchmark 4

25 Table IV. Performance and constraints of the AMD and TEN systems with varios controllers. The worst case (ω g,ζ g ) locations were fond for each RMS vale, sing 300 (AMD) or 750 (TEN) onds of response to a Kanai-Tajimi excitation, and are given below the corresponding J and C vales (ω g is in rads/) Sys: AMD AMD AMD AMD AMD AMD AMD AMD TEN TEN TEN Act Str RS Name: Zero SampleH Base H H H H /H #H /H # H / H Zero Base H H /H Note: J J J 3 J 4 J 5 J ω g= ω g= ω g= ω g= ω g= min. accel. responses ω g= ω g= ω g= ω g= ω g= min. strct. & actator responses ω g= ω g= ω g= ω g= ω g= min. actator responses ω g= ω g= ω g= ω g= ω g= min. strctral responses ω g= ω g= ω g= ω g= ω g= stability/ perform. robstness ω g= ω g= ω g= ω g= ω g= least mean sq. perf. & cnstr. vals ω g= ω g= ω g= ω g= ω g=37.93 stability robstness ω g= ω g= ω g= ω g= ω g= ω g= ω g= ω g= ω g= ω g= min. strct. & actator responses ω g= ω g= ω g= ω g= ζ g = ω g=4.808 stability/ perform. robstness ω g= ω g= ω g= ω g= ζ g = ω g= J 7 J 8 J 9 J 0 C C C 3 C 4 C 5 C ω g= ω g= ω g= ω g= ω g= ω g= ω g= ω g= ω g= ω g= ω g= ω g= ω g= ω g= ω g= ω g= ω g= ω g= ω g= ω g= ω g= ω g= ω g= ω g= ω g= ω g= ω g= ω g= ω g= ω g= ω g= The controllers mentioned herein are available pon reqest. The athors wold like to acknowledge and thank Prof. D.B. Ridgely and his colleages for making the MXTOOLS toolbox available for or se. Mltiobjective Optimal Strctral Control of the Notre Dame Bilding Model Benchmark 5