STABLE DESIGN OF H OUTPUT FEEDBACK CONTROL SYSTEMS WITH TIME DELAY

Transcription

1 3 th World Conferene on Earthquake Engineering Vanouver, B.C., Canada August -6, 4 Paper No. 44 SABLE DESIGN OF H OUPU FEEDBACK CONROL SYSEMS WIH IME DELAY Chi-Chang LIN, Chang-Ching CHANG, Huang-Lin CHEN SUMMARY In this paper, a H diret output feedbak ontrol algorithm through minimizing the entropy, a performane index measuring the tradeoff between H optimality and H optimality, is developed to redue the earthquake response of strutures. he ontrol fores are obtained from the multipliation of diret output measurements by a pre-alulated time-invariant feedbak gain matrix. o ahieve optimal ontrol performane and assure ontrol system stability, the strategy to selet both ontrol parameters and is extensively investigated onsidering the ontrol fore exeution time delay. It is found that the seletion of smaller or larger will result in better ontrol performane, but larger ontrol fores requirement. However, a lower bound of and an upper bound of exist. he seletion beyond these values will ause the ontrol system instability. For a SDOF damped struture, analytial expressions of diret output feedbak gains are derived. It an be proved that the onventional LQR ontrol is a speial ase of the developed H ontrol. Diret veloity feedbak ontrol is effetive in reduing strutural responses with muh fewer sensors and ontrollers than the degrees of freedom of the struture. In real ative ontrol, ontrol fore exeution time delay annot be avoided. Small delay time not only an render the ontrol ineffetive, but also may ause the system instability. In this paper, expliit formulas of the maximum allowable delay time and ritial ontrol parameters are derived for the design of a stable ontrol system. Some solutions are also proposed to lengthen the maximum allowable delay time. INRODUCION Sine 97s, remarkable progress has been made in the field of ative ontrol of ivil engineering strutures subjeted to environmental loadings suh as winds and earthquakes []. Among those researhes and real appliations, various ontrol algorithms have been investigated in designing ontrollers, for instane LQ [, 3], LQR optimal ontrol [4, 5] and H ontrol [6, 7]. he H ontrol Dean, College of Engineering & Professor, Department of Civil Engineering, National Chung-Hsing University, aihung, aiwan 47, R.O.C., lin3@dragon.nhu.edu.tw, el: ext 5, Fax: Graduate Student, Department of Civil Engineering, National Chung-Hsing University, aihung, aiwan 47, R.O.C.

2 theory onsiders the worst ase of external disturbanes to design the optimal ontroller to ahieve the desired performane. he results from both numerial simulations and experimental tests indiate that H ontrol is quite effetive [6, 7]. In addition, it had been applied to an atual building in okyo using a pair of mass dampers to redue the bending-torsion motions of the building due to earthquakes [8]. However, there are still many problems that affet ative ontrol tehnique towards large-sale pratial appliation. ime delay and limited number of sensors and ontrollers are two of these problems. Stritly speaking, a real struture has an infinite number of degrees of freedom (DOFs). It is impossible to aquire full-state measurements and feedbak to alulate the required ontrol fore. herefore, a diret output feedbak ontrol algorithm beomes neessary from pratial point of view. Moreover, in real ative ontrol systems, time is onsumed in data aquisition, data proessing, on-line alulation, and ontrol fore exeution. here is always a delay between the time at whih the ontrol fore is assumed to be applied and atually applied, and whih may ause degradation in ontrol effiieny or even render the system unstable. herefore, a robust ontrol with the onsideration of time delay effet is neessary. In the literature, the time delay effet on the H ative ontrol system has not been investigated by researhers as muh as those on the LQR optimal ontrol [9-]. Mahmoud et al. [] designed the H ontroller for a lass of dynamial systems onsidering state and input delays. hey expressed the ontrol design proedures in the form of linear matrix inequalities. No time-delayed H output feedbak ontrol system was studied. In this paper, a H diret output feedbak ontrol algorithm through minimizing the entropy, a performane index measuring the trade-off between H optimality and H optimality, is developed to redue the strutural responses due to seismi loads. o ahieve optimal ontrol performane, the strategy to selet both ontrol parameters and is extensively studied. he exat solution of output feedbak gain for a single-degree-of-freedom (SDOF) damped struture is derived. It an be proved analytially that LQR ontrol is a speial ase of H ontrol. Diret veloity feedbak ontrol is effetive in reduing strutural responses with very small number of sensors and ontrollers ompared with the degrees of freedom of the struture. Moreover, expliit formulas are obtained to alulate the maximum allowable delay time to avoid system instability. A formula is also derived to determine the ritial ontrol weighting fator of to assure the system stability. Finally, we propose some solutions to inrease the maximum allowable delay time. H DIREC OUPU FEEDBACK CONROL ALGORIHM he equation of motion of an n -DOF disrete-parameter struture under dynami loading and ative ontrol fore an be written as Mx&& () t + Cx& () t + Kx() t = Bu() t + Ew() t () where M, C, K are the n n mass, damping and stiffness matries, respetively. x () t is the n - dimensional displaement vetor, w () t is the r -dimensional external exitation vetor and u () t is the q -dimensional ontrol fore vetor. he n q matrix B and n r matrix E define the loations of ontrol fores and exitations, respetively. Represented in state-spae form, equation () an be rewritten as X & () t = AX() t + Bu() t + Ew() t ()

3 where x() t I X () t = A = B = E = x& () t, M K M C, M B, M E are n state vetor, n n system matrix, n q ontroller loation matrix and n r external exitations loation matrix, respetively. Define a p ontrol output vetor Z () t and a s output measurement vetor y () t as Z() t = C X() t + Du () t (3) y() t = C X () t (4) where C, D and C are p n, p q and s n matries. In the absene of time delay, the diret output feedbak ontrol fore is alulated by u() t = G y () t (5) where G is a q s time-invariant feedbak gain matrix. Aording to H ontrol algorithm, the H norm of transfer funtion matrix Zw ( jω) of ontrol output with respet to external exitation, takes the form ( jω) ( jω) = Z Zw sup ( jω) < (6) w where j =, and sup is defined as the supremum over all w () t ; is a positive attenuation onstant whih denotes a measurement of ontrol performane. * denotes the H norm. From equations ()-(6), it is derived that the transfer funtion matrix ( jω) is expressed by Zw ( jω) = ( C + DGC )[ jω I ( A+ BGC )] E (7) Zw And, the optimal H ontroller is designed suh that the H norm of Zw ( jω) satisfies the following onstraint Zw ( jω) = sup σ[ ( jω)] < Zw (8) where supσ is the largest singular value of Zw ( jω). It has been proved [3] that an optimal H ontrol system is asymptotially stable if there exists a matrix P that satisfies the following Riati equation ( A + BGC ) ( ) P + P A + BGC + PEE P + ( C + ) ( + ) = DGC C DGC (9) he ontrollers that satisfy equation (8) are not unique and may have an unbounded losed-loop H norm. hus, one way to design the optimal H output feedbak gain is to solve equation (9) with onstraint of the H norm. For example, the so-alled ombined H / H ontrol problems whih the bound of H norm is imposed and the upper bound of H norm is minimized. Another example is to minimize the Entropy of transfer funtion ( jω) Zw * En( Zw, ) ln det[ ( jω) ( jω)] dω π I Zw () Zw where * Zw ( jω ) is the onjugate transform of Zw ( jω). It has been shown that the Entropy of a omplex funtion is an upper bound of its H norm. herefore, the minimization of Entropy is equivalent to limit the magnitude of H norm. In addition, the Entropy of a funtion is a useful measurement of how its

4 singular values are lose to the upper bound,. By minimizing the Entropy, we push all singular values of ( jω) away form. From the results by Stoorvogel [4], the Entropy is also expressed by Zw En( Zw, ) = tr{ E PE } () where tr{-} denotes trae of a square matrix. hen, the optimization problem is onverted to minimize the Entropy of equation () subjet to the onstraint of equation (9). Inorporated with the onstraint, the Lagrangian an be introdued as L( GPλ,, ) tr{ E PE + λ [( A + BGC ) P + P( A + BGC ) + PEE P + ( C + ) ( + )]} DGC C DGC () where λ is a n n Lagrangian multiplier matrix. For simpliity and without loss of generality, let DC = and D D= I, the neessary and suffiient onditions for minimization of L( GPλ,, ) are ( A + BGC) P + P( A + BGC) + PEE P + C + = C CG GC (3a) ( A + BGC + EE P) λ ( ) + λ A + BGC + EE P + EE = (3b) B PλC + GC λc = (3) hus, the proedures to obtain the H diret output feedbak gain matrix G are: (i) deide a ontrol output vetor of equation (3), (ii) selet a disturbane attenuation onstant, (iii) solve equations (3a- 3) to obtain P, λ and G by any iterative sheme. Aording to Yaesh and Shaked [3], the Riatti equation (9) an be rewritten as A P+ PA P( BB EE ) p P+ C + = C ν PBB Pν (4) where ν = I λc ( ) CλC C For full-state measurement, C = I, equations (3a)-(3) redue to A P+ PA+ P( EE BB ) P+ C = C (5) whih is the same expression derived by Lin and Wei [5] for the H state feedbak ontrol. hus, diret output feedbak ontrol is a general form of the state feedbak ontrol. Furthermore, the Entropy is a performane index measuring the trade-off between H optimality and H optimality. When approahes infinity ( ), the Entropy of equation () takes the form lim En( Zw, ) = σi ( ( jω)) dω = ( jω) π Zw (6) Zw i and equations (3a)-(3) redue to ( A + BGC ) P + P( A + BGC ) + C C + C G GC = (7a) ( A+ BGC ) λ+ λ( A+ BGC ) + EE = (7b) B PλC + GC λc = (7)

5 Above equations have the same forms as those of the optimal LQR diret output feedbak ontrol [4]. It shows that the LQR diret output feedbak ontrol is a speial ase of the developed H diret output feedbak ontrol. Control parameters Consider a n -story linear shear building equipped with an ative tendon ontrol system as shown in Figure. he ontrol fore u () t is expressed as u() t = 4 k U ()os t θ (8) where U () t is atuator stroke. k and θ are the stiffness and inlination angle of tendons, respetively. (t) x n x ( t n ) x ( t) Ative endon θ u(t) Atuator W(t) Figure. n -story shear building with ative tendon ontrol system Define the ontrol output vetor satisfying D D= I and DC = as () t Γ () t Z = X + U() t (9) I Where Z () t denotes the ombination of strutural displaement response and atuator stroke and Γ is a salar matrix with element or. In equation (9), the ontrol weighting fator,, determines the relative importane between response redution and ontrol fore requirement. he larger value of is, the greater redution of responses. = represents unontrolled ase. For a SDOF struture with mass m, natural frequeny ω and damping ratio ξ as example, Γ = [ ]. he analytial expression of optimal state feedbak gain matrix G is obtained by solving equation (5) as 4k osθ G = [ g g] = [ Pd Pv] () m

6 And, the ontrolled frequeny, ω, and damping ratio, ξ, are expressed as P ξω + ( ξω + 4ξ ω + PP d ) P ω = ω + P Pd, ξ = ω + P P where d () 4 4k osθ ω + ω + P ξω + 4ξ ω + PP d P =, P = P, P d =, Pv = m P P It is seen from equation () that, both ontrol parameters and determine the ontrol performane. = leads to G = and ξ = ξ indiating the unontrolled ase, as expeted. As dereases or inreases, the ontrolled damping ratio ξ inreases. For the extreme ase of =, the ontrolled damping ratio is expressed as ξ = /( P/ P) whih is a onstant value for a hosen. At the same time, if =, ξ redues to / (i.e. 7.7%) as obtained by Chung et al. [4] and Lin et al. [] for the LQR ontrol. his proves that the LQR state feedbak ontrol is a speial ase of the H state feedbak ontrol as =. In addition, under the onstraints of P d and P v to be real numbers, and ξ less than %, the seleting ranges of and are given as 4 + ω P 4 + 4( ξ) ξω P when ξ (a) when ξ (b) Equation (b) shows that in pratie, a minimum value of, lim, exists and takes the form lim = 4 P + 4( ξ ) ξ ω Above equation shows that lim dereases as dereases or ω and ξ inrease. For the extreme ase of =, lim equals to zero, as expeted. However, for a general struture, as =, lim equals to a onstant as m lim ( = ) = = (3b) P 4k osθ whih depends on the strutural system parameters. he stiffer tendon system, the smaller lim. Moreover, equations (3a) and (3b) reveals that if a larger than lim ( = ) is used, the ontrol system is always stable no matter what is seleted. Similarly, for only veloity measurement, the analytial expression of diret veloity feedbak gain G is obtained by solving equations (3a)-(3) as 4k osθ G = [ Pv ] (4) m And, the ontrolled frequeny and damping ratio are expressed as ω = ω, P PP P ω (3a) ξ = ξ + ( ξ + ξ + ) (5)

7 In above equations, ξω + 4ξω + PP P = ω ( ), P 4 v = ω P Same as the state feedbak ontrol, as dereases or inreases, the ontrolled damping ratio ξ inreases for diret veloity feedbak ontrol. However, due to only veloity feedbak, the ontrolled frequeny, ω, remains unhanged. From equation (5), it shows that the ontrolled damping ratio an reah %. For the SDOF ative tendon ontrol struture, if the desired ontrolled damping ratio, ξ obj, is given, the required ontrol weighting fator an be obtained from equation (5) as ξ, obj ξ, obj = ω ξ ξ (6) P where ω 4ξω ξ = 4 ( ξobj ξ) + ( ξobj ξ). If ξ obj = %, the maximum value of,% an P P be obtained from equation (6) after determining.however, under the onstraints of P and P v to be real numbers and ξ less than %, the seleting ranges of and are ξ r P ω ( + ) min[ ω, ωξ ξ ] when ξ /P (7a) r min[ ω, ωξr ξ ] when r > ξ /P (7b) ω where P 4ξω ξr = ( ξ 4 ) + ( ξ ). P P Based on above derivations and disussions, to ahieve optimal ontrol performane and assure ontrol system stability, the appropriate ontrol strategy is to adjust ontrol weighting fator after the seletion of based on equations () and (7). hat makes the seletion strategy of ontrol parameters more effiient and flexible. Numerial simulation A SDOF struture (S) whih system parameters listed in able is used for a parametri study to demonstrate the effetiveness of the proposed ontrol algorithm. he analytial and numerial solutions of G and ξ of diret veloity feedbak (DVF) ontrol for different and are shown in Figure. It is seen that the absolute values of G and ξ inrease dramatially as dereases or inreases. he ontrolled frequeny ω is always equal to the original natural frequeny ω. Figure 3 illustrates the transfer funtion of absolute aeleration with respet to support aeleration for =. and =.,.5, and.. It is found that the larger, the smaller peak amplitude of the transfer funtion. he time histories of aeleration response and ontrol fore with ( =., =.) and without ontrol under the strong motion part of the free field ground aeleration (EW omponent) reorded at the ampus of National Chung-Hsing University (NCHU) during the 999 aiwan Chi-Chi earthquake (Figure 4) are illustrated in Figure 5. Beause the system damping ratio inreases (from.4% to 4.58%), the aeleration response is signifiantly redued. It is also seen that the required maximum ontrol fore is only 6% of the strutural weight.

8 For a three DOF ontrol struture (S3), whih system properties listed in able, the patterns of ontrol fore and sensor loation are symbolized by F and V. For example, FV indiates that one ontroller is plaed at the first floor and the first floor veloity is measured. he more number of ontrollers and sensors, the more redution in strutural responses. However, it is found that one pair of olloated ontroller and sensor at the first floor (FV) is effetive in signifiantly inreasing all modal dampings and reduing the total strutural responses, as illustrated in able and Figures 6-8 for the ase of =. and =.. he required maximum ontrol fore is % of strutural total weight. able. System parameters of S struture Mass, 3 m (Kg).935 Stiffness, (N m) k Damping, (N s m).588 Natural frequeny, ω (Hz) 3.47 Natural period, (se).88 endon stiffness, (N m) k endon inlination, θ ( o ) 36 4 =. exat solution numerial solution..5 =. exat solution numerial solution G -4 G ζ (%) 6 4 ζ (%) Figure. Variation of veloity feedbak gain G and ontrolled damping ratio ξ for S struture

9 g.... H X X 4 3 =. Unontrolled =. =.5 =. Aeleration(g) ω (Hz) Figure 3. Absolute aeleration transfer funtion for different g ime (se) Figure 4. NCHU ampus reord (EW) of 999 aiwan Chi-Chi earthquake Aeleration (g) =. =. Unontrolled DVF 5 5 ime(se) Control Fore (N) - - =. =. 5 5 ime(se) Figure 5. ime histories of aeleration response and ontrol fore of S struture able. System parameters of S3 struture with and without ontrol Mass matrix, M (Kg) Stiffness matrix, K (N m) Damping matrix, C (N s m)

10 able. System parameters of S3 struture with and without ontrol (ontinued) unontrolled FV =., =. Natural frequeny (Hz) ω, ω, = ω 3, ω,.5 ω, 6.83 = ω 3,.38 Damping ratio (%) ξ,.6 ξ, =.39 ξ 3,.36 ξ, 6.6 ξ,.86 = ξ 3, 4.5 ξ (%) FV =. st mode nd mode 3rd mode.. Figure 6. Variation of ontrolled damping ratio of S3 struture for different. H X3 Ẋ. g FV =. Unontrolled =.5 =. =. Displaement (m) =., =. Unontrolled FV 5 5 Frequeny (Hz) Figure 7. op floor displaement transfer funtion of S3 struture for different ime(se) Figure 8. ime histories of top floor displaement response

11 CONROL FORCE EXECUION IME DELAY In real ative ontrol systems, time is onsumed in data aquisition, data proessing, on-line alulation, and ontrol fore exeution. here is always a delay between the time at whih the ontrol fore is assumed to be applied and really applied. hat means the ontrol fore at time instant t is expressed as u() t = 4kos θ G y ( t td) (8) where t d is the delay time. Substituting equation (8) into equations () and (3), the system poles or eigenvalues are obtained by solving the following sets of homogeneous algebrai equations t d λ λ I ( A+ e B G C ) = (9) where I is the identity matrix and λ represents omplex eigenvalues of the time-delayed ontrol system. he orresponding ontrolled frequeny and damping ratio are given as ω = λ ξ = Re( λ) ω (3) Where and Re ( ) denote absolute value and real part of a omplex number, respetively. For S struture with diret veloity feedbak, when delay time t d inreases, the ontrolled damping ratio, ξ, varies slightly for small t d and then drops to zero very fast as shown in Figure 9 for the ase of =. and =.,.5, and.. he larger, the faster degradation of ontrolled damping. he maximum allowable delay time ( t d,max ) whih auses system instability ( ξ = ) will also derease as inreases. As =., the omparison of t d,max for state feedbak (SFB) and diret veloity feedbak (DVF) is given in able 3. It is found that the smaller, the longer maximum delay time for both SFB and DVF ontrols. However, with the same ontrol weighting fator, diret veloity feedbak allows longer delay time than state feedbak. ξ (%) 6 4 t d,max =48. mse =. =. =.5 =. t d,max =6.6 mse t d,max =86.7 mse...3 t d / Figure 9. Variation of ξ for different td / (S, DVF)

12 Similarly, for MDOF ontrol systems with time delay, all modal ontrolled damping ratios derease as the delay time or inreases. he higher modal damping ratio redues faster than those of lower modes. For the S3 struture with Fstate and FV ontrols, the results are given in able 4 and able 5. It shows that, when a large is used, the highest damping ratio, ξ 3,, drops to zero first and results in instability of the whole ontrol system for both ontrol types. However, it is also found that, when is smaller than a ertain value, to be determined later, the seond modal damping redues to zero rather than the third and thus, the maximum delay time inreases. able.3 Maximum delay time of S with SFB and DVF ontrols ( =.) t d,max (mse) =. =.5 =. SFB DVF SFB DVF SFB DVF able.4 Controlled results of S3 with FState ontrol for different ( =.) ξ,(%) ξ (%), ξ3,(%).36 t d,max (mse) =.5 =. =. =.5 = able.5 Controlled results of S3 with FV ontrol for different ( =.) ξ (%), ξ,(%) ξ3, (%).8 t d,max (mse) =. =.3 =.5 =. = Maximum delay time For the S ontrol struture, the analytial expression of t d,max for diret veloity feedbak ontrol is given as,max os ω ξ d ω 4ωξ PP v 4 ω [ω 4 ωξ PP v ] PP v t = (3)

13 It is observed that t d,max inreases as system original damping ratio, ξ, inreases and ontrol weighting fator dereases as seen in Figure. For a damped struture, a ritial (or maximum) value of, max, exists. When < max is seleted, the ontrol system will remain stable even with long delay time. In addition, as observed previously, the stability of the S3 ontrol system, is generally determined by the third (highest) mode. However, when dereases or the third original modal damping ratio, ξ 3,, inreases, the ontrolled damping ratio of the seond or first mode will derease to zero earlier than that of the third mode. Under this irumstane, t d,max will thus be inreased as seen in Figure. his finding eliminates the question that a real struture allows very small delay time beause of inherent large frequenies in higher modes, and thus inreases the onfidene of the appliation of ative ontrol. t d,max (se).6..8 =. ξ = % ξ = % ξ = 5 % t d,max (se) FV ξ 3, = % ξ 3, = % ξ 3, = 5 % Figure. Variation of t d,max for different and ξ (S, DVF) - - Figure. Variation of t d,max for different ξ (S3, FV) and 3, Critial ontrol weighting fator As found in preeding setion, the max delay time may be lengthened by seleting appropriate ontrol weighting fators and/or inreasing the damping ratio of higher modes. For a given SDOF damped struture, the maximum ontrol weighting fator ( max ) exists to satisfy equation (4). hat means the value of the funtion in are osine equals to minus one, i.e., ( ωξ ) ( P P v ) =. For the S struture with diret veloity feedbak, takes the form as max ξ = (3) max ω (3 ) Pω whih orresponds to ξ = ξ. It indiates the ontrol system will always be stable if < max or say ξ < ξ. he variation of max with different ω or ξ for S struture with DVF ontrol is shown in Figure. he larger ω or ξ is, the larger an be used. For the S3 struture with FV ontrol, the max that system stability is dominated by the seond mode, versus the third original damping ratio ξ 3, for the third original frequeny ω 3, =l, 5, and Hz is shown in Figure 3. max inreases with the inrease of ω 3, andξ 3,. he larger ω 3, or ξ 3, is, the larger an be used. his reonfirms the fat that the stability of MDOF ontrol systems turns to be dominated by lower modes if the higher modes have some dampings for ertain value of.

14 ω = Hz ω = 5 Hz ω = Hz - ω 3, = Hz ω 3, = 5 Hz ω 3, = Hz FV max - max Damping Ratio, ξ Figure. Variation of max for different ξ and ω (S, DVF) Damping Ratio, ξ 3, Figure3. Variation of max for different ξ 3, ω (S3, FV) and 3, CONCLUSIONS his paper develops an optimal H diret output feedbak ontrol algorithm with the onsideration of limited number of sensors and ontrollers and ontrol fore exeution time delay. he losed form solutions of optimal output feedbak ontrol gains, ontrolled modal frequeny and damping ratio are obtained. From the modal parameters, frequeny domain and time domain analysis of ontrolled systems under earthquake exitations, it is demonstrated that one pair of olloated veloity sensor and ontroller suh as FV is suffiient and effetive in reduing the dynami responses of MDOF strutures. Analytial expression of ontrol parameter,, is also derived to make the ontrol strategy more effiient and flexible. Moreover, the expliit formula is obtained to alulate the maximum allowable delay time. his quantity is a useful parameter for the design of ontrol devies. Finally, the ritial ontrol weighting fator is also derived to avoid ontrol system instability. he allowable delay time is lengthened by adding strutural dampings through passive dampers and/or seleting a ontrol weighting fator smaller than the ritial one. he ontrol performane an thus be signifiantly improved even with time delay. ACKNOWLEDGEMENS his researh was supported in part by Nation Siene Counil of the Republi of China under the grant No. NSC 9--E-5-3. hese supports are greatly appreiated. REFERENCES. Soong, Spener Jr BF. Ative, semi-ative and hybrid ontrol of strutures, Proeedings of the th World Conferene on Earthquake Engineering, Aukland, New Zealand,.. Kobori, Koshika N, Yamada K, Ikeda Y. Seismi-response-ontrolled struture with ative mass driver system. part : design, Earthquake Engineering and Strutural Dynamis 99; :

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