Role of Control-Structure Interaction in Protective System Design

Transcription

1 Published in the ASCE Journal of Engineering Mechanics, ol. 121, No. 2, Feb pp Role of Control-Structure Interaction in Protective Syste Design S. J. Dyke 1, Student Meber ASCE, B. F. Spencer, Jr. 2, Meber ASCE, P. Quast 3 and M. K. Sain 4 Abstract Most of the current research in the field of structural control for itigation of responses due to environental loads neglects the effects of control-structure interaction in the analysis and design. The iportance of including control-structure interaction when odeling a control syste is discussed herein. A specific odel for hydraulic actuators typical of those used in any protective systes is developed, and experiental verification of this odel is given. Exaples are provided which eploy seisically excited structures configured with active bracing, active tendon, and active ass driver systes. These exaples show that accounting for control-structure interaction and actuator dynaics can significantly iprove the perforance and robustness of a protective syste. Introduction The concept of structural control for civil engineering applications originated in the early 7 s (Yao, 1972). In the two decades since, uch progress has been ade toward exploiting the potential benefits offered by control for protection of structures against environental loads such as strong earthquakes and high winds (e.g., Soong, 199; Housner and Masri, 199, 1993; ATC-17, 1993). Nearly all of the current literature on control of civil engineering structures does not directly account for control-structure interaction and actuator/sensor dynaics in the analysis and design of protective systes. Unodeled control-structure interaction (CSI) effects can severely liit both the perforance and robustness of protective systes. This is true for both active and sei-active systes. To study effectively the control-structure interaction proble, one ust have good odels for the dynaics of the associated actuators. This paper presents a general fraework within which one can study the effect of control-structure interaction. Specific odels are developed for hydraulic actuators typical of those used in any active structural control situations. A natural velocity feedback link is shown to exist, which tightly couples the dynaic characteristics of a hydraulic actuator to the dynaics of the structure to which it is attached. Neglecting this feedback interaction can produce poor, or perhaps catastrophic, perforance of the controlled syste due to the unodeled or isodeled dynaics of the actuator-structure interaction. In addition, the tie lag in generation of control forces is accoodated through appropriate odeling of the actuator and the associated CSI. Experiental verification of the ain concepts is presented. The iplications on protective syste design are illustrated through exaples of seisically excited structures. Active bracing, active tendon and active ass driver (AMD) systes are considered. 1. Doctoral Candidate and Graduate Research Assistant, Departent of Civil Engineering and Geological Sciences, University of Notre Dae, Notre Dae, IN Associate Professor, Departent of Civil Engineering and Geological Sciences, University of Notre Dae, Notre Dae, IN Doctoral Candidate and Graduate Research Assistant, Departent of Electrical Engineering, University of Notre Dae, Notre Dae, IN Freiann Professor, Departent of Electrical Engineering, University of Notre Dae, Notre Dae, IN

2 w Structure y f Feedforward Link Control Actuator Feedback Link u Controller Figure 1. Block diagra of active/sei-active structural control syste. w u f y G a G yf Actuator Structure H i Feedback Interaction Figure 2. Model of interaction between the actuator and the structure. Proble Forulation Figure 1 provides a scheatic diagra of a general active/sei-active structural control proble. The controller receives easureents fro the sensors and fors a coand input vector u to the control actuator. The control actuator then applies a force vector f to the structure. However, when echanical actuators (e.g., hydraulic actuators) are used to control structures, there is generally a dynaic coupling between the actuator and the structure, as represented by the dotted arrow in Fig. 1. This coupling indicates that it is not possible to separate dynaical representations of the structure and the actuator, and odel the as independent systes connected in series. To understand the iplications of the dynaical coupling, exaine ore closely the actuator and structural portions of the control syste (i.e., the region in Fig. 1 enclosed in the dashed box). Consider the case in which the syste has one actuator, with a single coand input u generating a single output force f. Fig. 2 provides a block diagra description of this case, in which the interaction can be odelled fro the output. Here G a is the transfer function of the actuator, and G yf is the transfer function fro the force applied by the actuator to the structural responses. Because the dashed line in Fig. 1, representing the feedback interaction between the structure and the actuator, often has associated dynaics, these dynaics are 2

3 represented by the transfer function H i in Fig. 2. Thus, the overall transfer function fro the control input u to the structural response y is given by G yu G yf G a (1) 1 + G yf G a H i If these dynaic systes are represented by nuerator and denoinator polynoials in s, the representation of the syste becoes G yu n yu d yu n yf n a d i (2) d yf d a d i + n yf n a n i The transfer function fro the coand input u to the force f applied to the structure is given by G fu G a (3) 1 + G yf G a H i Fro Eq. (3), it is clear that the dynaics of the transission fro u to f are not siply the actuator dynaics G a, but contain dynaics due to the structure and the actuator. More insight can be gained by rewriting Eq. (3) in ters of the associated nuerator and denoinator polynoials, i.e., G fu n fu d fu d yf n a d i (4) d yf d a d + n i yf n a n i Coparing Eqs. (2) and (4), notice that unless pole/zero cancellation occurs, the transfer functions G yu and G fu have the sae poles, and that the poles of the structure (i.e., the poles of G yf ) are zeros of G fu. Cancellation is ost unlikely, and therefore this possibility is disregarded in the reainder of the paper. Because the poles of the structure are zeros of G fu, actuators attached to lightly daped structures will have a greatly liited ability to apply forces at the structure s natural frequencies; and if the structure is undaped, the actuator will not be able to apply a force at its natural frequencies. Also note that poles of the structure do not appear as poles in G yu, because they are cancelled by the zeros in G fu. These results occur regardless of how fast the dynaics of the actuator are (including the case when G a is a constant gain). Before closing the section, the effect of neglecting the interaction between the control actuator and the structure is considered. Using the definition of G fu in Eq. (4), an alternative block diagra to that in Fig. 2 can be deterined as shown in Fig. 3. Because the dynaics of an actuator, as given in G a, ay often be fast relative to the structure, one ight argue that the block G fu ay be reasonably represented as a constant both in phase and agnitude. However, as shown previously, neither the phase nor the agnitude of G fu will be constant in general. Moreover, the phase and agnitude characteristics of G fu will vary depending on the structure. Neglecting phase differences between the coand input u and the resulting u f y G fu G yf Figure 3. Equivalent block diagra odel of the actuator/structure. 3

4 force f, i.e., neglecting the CSI, will result in an apparent tie delay associated in the literature with generation of the control forces. The next section presents a siple odel which shows that a feedback interaction path is always present in hydraulic actuators. Experiental verification of the odel is also provided. Hydraulic Actuator Modeling In the case of hydraulically actuated systes, a feedback path exists between the velocity of the actuator and the coand input to the actuator. Fro DeSilva (1989), the equations describing the fluid flow rate in an actuator can be linearized about the origin to obtain f alve: q k q c k c --, (5) A Hydraulic Actuator: q Aẋ ḟ, (6) 2βA where q is the flow rate, c is the valve input, f is the force generated by the actuator, A is the cross-sectional area of the actuator, β is the bulk odulus of the fluid, is the characteristic hydraulic fluid volue for the actuator, x is the actuator displaceent, and k q, k c are syste constants. Equating Eqs. (5) and (6) and rearranging yields ḟ β Ak c k f A 2 ẋ q c, (7) which shows that the dynaics of the force applied by the actuator are dependent on the velocity response of the actuator, i.e., the feedback interaction path is intrinsic to the dynaical response of a hydraulic actuator. Figure 4 is a block diagra representation of the hydraulic actuator odel given in Eq. (7) attached to a structure. Here, denotes the transfer function fro the force generated by the actuator to the displaceent of the point on the structure where the actuator is attached, and f L is the external load on the structure. Notice the presence of the natural velocity feedback in the open-loop syste. Through this natural velocity feedback, the dynaics of the structure directly affect the characteristics of the control actuator. alve Input c k q k c 2β A G s xf ( s) f f L x A ẋ s Natural elocity Feedback Figure 4. Block diagra of open-loop servovalve/actuator odel. 4

5 The portion of the syste in Fig. 4 identified as has the following transfer function: A s+ k 2β c A k c , τ h s + 1 (8) where τ h 2βk c is the tie constant of the actuator. Thus, the transfer function fro the valve input c to the force f is given by G fc k q , (9) 1 + sa and the transfer function fro the valve input c to the actuator displaceent x is given by G xc k q (1) 1 + sa Representing these transfer functions in ters of their respective nuerator and denoinator polynoials gives G fc n fc d fc k q n h d xf , (11) d h d xf + san h n xf G xc n xc d xc k q n h n xf (12) d h d xf + san h n xf As discussed in the previous section, Eqs. (11) and (12) show that these two transfer functions have the sae poles and that the poles of the structure (i.e., the poles of G yf ) will be zeros of the transfer function fro the input to the applied force. Because the open-loop syste identified in Fig. 4 is typically unstable, position, velocity and/or force feedback ay be used to stabilize the syste. Here, a unity-gain position feedback loop, i.e., position control, is considered. Figure 5 is the block diagra for the hydraulic actuator with the position feedback included. This configuration is typical of those found in active structural control systes (see, for exaple, Chung, et al., 1988, 1989). The transfer function fro the coand u to the actuator force f and to the displaceent x for the syste in Fig. 5, including the unity-gain position feedback loop, are given respectively by G fu γg fc γg fc γk q ( γk q + sa), (13) G xu γg fc γg fc γk q ( γk q + sa), (14) where γ is the proportional feedback gain stabilizing the actuator. Rewriting the Eqs. (13) and (14) in ters of their nuerator and denoinator polynoials yields 5

6 G fc u γ c k q f x A ẋ s Natural elocity Feedback Figure 5. Block diagra of closed-loop servovalve/actuator odel. G fu n fu d fu γk q n h d xf d h d xf + ( γk q + sa)n h n xf, (15) G xu n xu d xu γk q n h n xf d h d xf + ( γk q + sa)n h n xf. (16) Again, fro Eqs. (15) and (16), the poles of the structure,, becoe zeros of the transfer function fro the coand u to the actuator force f, and are then cancelled in the transfer function fro coand to actuator position. As an alternative to using position feedback to stabilize the hydraulic actuator, a velocity and/or force feedback loop can be added. Considering the general case in which a cobination of all three easureents is used, the resulting transfer functions are G fu γk q , (17) 1 + sa + δγk q + ( α+ ηs)γk q G xu γk q , (18) 1 + sa + δγk q + ( α+ ηs)γk q where α is the position feedback gain, η is the velocity feedback gain, and δ is the force feedback gain. These transfer functions can be written in ters of their respective nuerator and denoinator polynoials as 6

7 G fu n fu d fu γk q n h d xf d h d xf + san h n xf + δγk q n h d xf + ( α+ ηs)γk q, (19) G xu n xu d xu γk q n h n xf d h d xf + san h n xf + δγk q n h d xf + ( α+ ηs)γk q. (2) Siilarly to the case of position control, the poles of G fu are those of the overall transfer function G xu, and the poles of the structure (i.e., the poles of ) are the zeros of the transfer function fro the actuator coand u to the applied actuator force f. Experiental erification To deonstrate the validity of this odel of a hydraulic actuator, the above results were copared to experiental data obtained at the Earthquake Engineering/Structural Dynaics and Control Laboratory at the University of Notre Dae. A scale odel of the prototype building discussed in Chung, et al. (1988, 1989) was the test structure, shown in Fig. 6. The total ass of the floors of the odel is 1,1 kg (5 lb), distributed evenly between the three levels, and the structure is c (62 in.) tall. The tie scale was decreased by a factor of five, aking the natural frequency of the structure five ties that of the prototype. Cross-braces can be attached to the top two floors, causing the structure to respond priarily as a SDOF structure (see Fig. 6). A hydraulic control actuator with a ± 5.8 c ( ± 2 in.) stroke was placed at the first floor of the building and attached to the seisic siulator table via a rigid frae. For this syste, the actuator displaceent is equivalent to the displaceent of the first floor. A position sensor was therefore used to easure the displaceent of the first floor and to provide feedback for the control actuator. The force 3 2 x 3 () t reovable cross-braces x 2 () t 1 x 1 () t f(t) ẋ g () t Figure 6. Three degree-of-freedo structure with active bracing. 7

8 4 3 2 G fu Gain (db) G xu Frequency (Hz) Figure 7. Magnitude of the transfer functions of the structure, the actuator, and the cobination for the SDOF odel Gain (db) G fu G xu Frequency (Hz) Figure 8. Magnitude of the transfer functions of the structure, the actuator, and the cobination for the MDOF odel. transitted to the building by the control actuator was easured with a piezoelectric force ring anufactured by PCB Piezotronics, Inc. Experiental transfer functions were found for the test structure and actuator using a Tektronix 263 spectru analyzer. Figure 7 shows the agnitude of the building and actuator transfer functions for the SDOF case (i.e., with the cross-braces attached). The fundaental frequency of the structure in the SDOF configuration is 7 Hz. Exaining the transfer function fro the input coand to the applied force, G fu, it is clear that significant odeling error would be incurred if one took this transfer function to be constant (i.e., neglected the actuator dynaics and the CSI). Notice that the zeros of G fu coincide with the poles of the structure. Also, in the transfer function fro the coand input u to the actuator displaceent x, G xu, 8

9 Gain (/) Calculated 2 Experiental Frequency (Hz) Figure 9. Experiental and calculated transfer functions G fc for SDOF odel. the poles and zeros cancel and a new coplex pole pair appears at a higher frequency. This behavior is exactly what is predicted by the odel presented in this section. Figure 8 shows the agnitude of the building and actuator transfer functions when the braces are reoved and the odel is in the three degree-of-freedo configuration. The actuator transfer function G fu in this case is significantly different than it is with the SDOF structure. Notice, as in the SDOF case, that the poles of the structure are cancelled by the zeros of the actuator in the transfer function fro the control input u to the structural displaceent x, G xu, and new poles appear at poles of the actuator transfer function. To verify the actuator odel, the transfer function was deterined fro the experiental data in the SDOF case. The proportional feedback gain stabilizing the actuator, γ, was set at 2.5. This user-defined constant can be changed through adjustent of the potentioeter of the servo-valve aplifier. By using Eq. (9) and the experiental data for and G fc, A k q was deterined as a function of frequency. Although k q is a nonlinear paraeter dependent on the operating point and the response aplitude of the hydraulic actuator (DeSilva, 1989), it can reasonably be assued constant over the operating and frequency range of interest. The value of A k q which best fits the experiental data below 4 Hz was deterined to be.15. Knowing A k q, ratios of the various paraeters in Eq. (8) were deterined to fit G fc. The values of the ratios which define k q are k q A k c 25 and 2βk c.15, which deterine the actuator transfer function as 25 k q (21).15s + 1 Substituting Eq. (21) into Eq. (9) and using the experientally obtained transfer function for, the transfer function G fc can be obtained. A coparison between these results and those obtained directly fro the experiental data is given in Fig. 9. Notice the excellent agreeent between the experiental and calculated results. Because of the excellent agreeent between the odel and the experient, the odel of the hydraulic actuator in Eq. (7) is concluded to be valid. 9

10 Illustrative Exaples This section provides exaples that deonstrate the iportance of accounting for the interaction between the control actuator and the structure in protective syste design. An active bracing, active tendon and active ass driver syste are considered. Several controllers are designed for each of the exaple systes. Three control odels are eployed in the design of the various controllers. The first odel, designated Model 1, uses the full equations of otion. In the second odel the fluid in the hydraulic actuator was assued to be incopressible (i.e., ḟ in equation Eq. (7)). In this pseudo static odel, designated Model 2, the hydraulic stiffness and daping ters are still included. In ost studies of the control of civil engineering structures, G fu is considered to be constant in agnitude and phase. This assuption is eployed for the third odel considered (Model 3). In each exaple, the value of K, the constant agnitude of G fu used for control design, was found by deterining the DC value of G fu for the coplete odel which considers actuator dynaics and CSI (Model 1). Controllers were designed based on each of these odels. Several perforance objectives were also considered. The objective for the type A controller was to iniize the relative displaceents of each floor by equally weighting the relative displaceent easureents in the perforance function. For the type B controller, the perforance objective was to iniize the absolute accelerations of each floor by equally weighting the respective absolute acceleration easureents. The identifications and descriptions for the various controllers are given in Table 1. Because the easured output vector y is not the full state vector, the controllers are observer based and are designed using H 2 /Linear Quadratic Gaussian (LQG) design ethods (Spencer, et. al., 1991, 1994; Suhardjo, et. al. 1992). Both low authority (Case 1) and high authority (Case 2) controllers were considered. To allow for direct coparison of each type of controller, the respective weightings on the displaceents and accelerations are deterined such that, when the ground acceleration is taken to be a given broadband excitation, the RMS control force for each case has the sae agnitude. The RMS responses are deterined through solution of the associated Lyapunov equation (Soong and Grigoriu, 1991). For all control studies, the odel for which the responses are calculated is Model 1, incorporating both actuator dynaics and CSI. Table 1: Controller Design Descriptions. Model used for Controller Design Displaceent Weighting (Type A) Acceleration Weighting (Type B) Including CSI (Model 1) 1A 1B Neglecting copressibility (Model 2) 2A 2B Constant G fu (Model 3) 3A 3B Exaple 1: Active Bracing Consider the three-story, single-bay building subjected to a one-diensional earthquake excitation ẋ g with active bracing as depicted in Fig. 6. The equations of otion are ẋ 1 c 11 c 12 c 13 ẋ 1 k 11 k 12 k 13 ẋ 2 + c 12 c 22 c ẋ k 12 k 22 k 23 ẋ 3 c 13 c 23 c ẋ 33 3 k 13 k 23 k 33 x 1 x 2 x f ẋ g, 1 (22) 1

11 where x i and i are the displaceent relative to the ground and the ass of the ith floor of the building, c ij and k ij are the daping and stiffness coefficients, respectively, and f is the control force applied by the hydraulic actuator. Strictly speaking, 1 includes the weight of the actuator rod and piston. However, this additional ass is usually negligible in coparison with the first floor ass. Equation (22) can be written in atrix for as M s ẋ + C s ẋ + K s x B s f M s G s ẋ g. (23) Incorporating unity gain displaceent feedback into the hydraulic actuator odel given in Eq. (7) yields ḟ 2β Ak q γ ( u x 1 ) k c f A 2 ẋ 1, (24) where u is the control coand. Defining the state vector of the syste as equation is z 1 [x ẋ f], the state ż 1 I 1 M s Ks 1 M s Cs 1 M s Bs 2βAk q γ βA2 2βk c z 1 + u + 2βAk q γ G s ẋ g (25) Az 1 + Bu + Eẋ g. Assuing that the actuator displaceent (i.e., the relative displaceent of the first floor) x 1, the absolute accelerations of each floor, ẋ 1a, ẋ 2a and ẋ 3a, and the applied control force, f, are easured, the easureent equation is x 1 v 1 y ẋ 1a ẋ 2a ẋ 3a M s Ks M s Cs M s Bs 1 z 1 + v 2 v 3 v 4. (26) f v 5 Cz 1 + v The vector v contains the noises in each easureent. The odel given in Eqs. (25) and (26) (Model 1) includes actuator dynaics and CSI. Model 2 for this exaple is found by setting ḟ in Eq. (24). The resulting algebraic relation for the force is then given by f Aγk q Aγk q A u x k c k ẋ c k 1 c. (27) Using the state vector z 2 [x ẋ ], the state equation reduces to 11

12 I ż M s Ks M s Cs + Ak q γ A k c 1 k z c 2 + Aγk q 1 u M k s Bs c Using the sae easured outputs as above, the easureent equation can be written y 1 1 M s Ks 1 M s Cs Ak q γ k A c k c + G ẋ g s Ak q γ A Aγk 1 k c 1 k q M c z k s Bs 2 + c u + v Ak q γ k c. (28). (29) Model 3 considers G fu to be constant in agnitude and phase. The constant gain, K is deterined fro the coplete odel given in Eqs. (25) and (26). Using the state vector z 3 [x ẋ ] and the sae easureents as above, the state-space representation reduces to I ż M s Ks M s Cs z K u + M s Bs G s ẋ g, (3) y M s Ks M s Cs z M. (31) s Bs K u + v 1 The structural paraeters for the three degree-of-freedo odel reported in Chung, et. al. (1989) were eployed in this exaple with an active bracing syste. The paraeters associated with the control actuator were chosen to correspond to the odel presented previously. Here, the ground acceleration was odeled as a broadband disturbance with a constant two-sided spectral density of agnitude c 2 /s 3 ( S in 2 /s ). The structural responses to this disturbance are shown in Table 2 for the type A controllers (i.e., weighting the displaceents of the structure) and in Table 3 for type B controllers (i.e., weighting the accelerations of the structure). Here, the uncontrolled configuration has the active bracing syste copletely reoved fro the structure. The zeroed controller corresponds to the case in which the active bracing syste is attached, but the input coand signal is set equal to zero. This configuration is included because it has been used as a basis for coparison in previously reported control studies. Exaining the zeroed control case, one observes that the stiffness of the structure increases (as copared to the uncontrolled case), thus causing the first floor displaceent to decrease. Notice that if the relative displaceents are weighted (Table 2), the controller which includes actuator dynaics and CSI produces noticeably better results than either of the other two controllers. For the high authority controller (Case 2), the closed-loop syste created with Controller 3A becoes unstable before the chosen force level is reached. When absolute accelerations are weighted, the responses using Controller 1B (i.e., including actuator dynaics and CSI, and designed based on Model 1) and Controller 2B (i.e., designed based on Model 2 which considers the fluid to be incopressible) are very close for both the low and high authority cases 12

13 (Table 3). These results are considerably better than those corresponding to Controller 3B (i.e., neglecting the actuator dynaics and CSI). Table 2: Coparison of RMS responses for active bracing syste with weighting on the relative displaceents. Configuration σ x1 σ x2 σ x3 σ ẋ a1 σ ẋ a2 σ ẋ a3 σ f (c) (c) (c) (c/s 2 ) (c/s 2 ) (c/s 2 ) (N) (a) noinal configurations Uncontrolled 3.478e e e Zeroed control 3.14e e e (b) Case 1: low authority controller Controller 1A 5.77e e e Controller 2A 4.14e-2 1.4e e Controller 3A 2.982e e e (c) Case 2: high authority controller Controller 1A 6.528e e e Controller 2A 5.169e e e Controller 3A Table 3: Coparison of RMS responses for active bracing syste with weighting on the absolute accelerations. Configuration σ x1 σ x2 σ x3 σ ẋ a1 σ ẋ a2 σ ẋ a3 σ f (c) (c) (c) (c/s 2 ) (c/s 2 ) (c/s 2 ) (N) (a) noinal configurations Uncontrolled 3.478e e e Zeroed control 3.14e e e (b) Case 1: low authority controller Controller 1B 2.56e e e Controller 2B 2.55e e e Controller 3B 1.442e e e (c) Case 2: high authority controller Controller 1B 3.416e e e Controller 2B 3.44e e e Controller 3B 2.568e e e

14 One should point out that the systes eploying controllers designed based on Model 2 or Model 3 can quickly becoe unstable if one tries to reduce the RMS responses through increasing the weighting on the relative displaceents or the absolute accelerations in the perforance function. However, by accounting for the actuator dynaics/csi a significantly ore authoritative control design (i.e., higher perforance) can be achieved without such an instability occurring. The results given in Tables 2 and 3 indicate that, for a given level of RMS control action, the ability of the controller to reduce the relative displaceents of the structure is greatest when the relative displaceents of the structure are weighted in the control perforance function. Siilarly, these tables indicate that the absolute accelerations of the structure are ost efficiently reduced when the absolute accelerations of the structure are weighted. Exaple 2: Active Tendon Syste Consider the single story structure subjected to a one-diensional earthquake excitation ẋ g with an active tendon syste as shown in Fig. 1. In this syste a tendon/pulley syste is used to transit the force generated by the hydraulic actuator to the first floor of the structure. A stiff frae is included to connect the actuator to the four pretensioned tendons. The linearized equations of otion are ẋ c+ c 2 cos θ ẋ k + 2 k cos θ + + x + c ( )ȧ + k ( )a ẋ g ȧ + c ȧ + k a+ c ẋ + k x ẋ g + f (32) (33) where is the ass of the building, o is the cobined ass of the stiff frae and the actuator rod/piston, c and k are the daping and stiffness coefficients of the structure, respectively, c o and k o are the total daping and stiffness of the four tendons, x is the displaceent of the building relative to the ground, a is the displaceent of the actuator, and f is the force applied by the hydraulic actuator to the rigid frae. In Fig. 1, the force designated f o is the force in the tendons. Under unity gain feedback of the actuator position, the dynaics of the hydraulic syste can be written as ḟ 2β Ak q γ ( u a) k c f A 2 ȧ. (34) Defining the state vector of the syste as z 1 [a ȧ x ẋ f], the state equation is x() t c o, k o f o θ ck, a(t) o f(t) ẋ g () t Figure 1. Single story building with active tendon syste. 14

15 ż 1 1 k c k c k c k cos 2 θ + k c cos 2 θ + c β 2β Ak q γ A 2 2β k c z 1 + u + 2β Ak q γ ẋ g. (35) The easureents are chosen to include the displaceent of the actuator and the absolute acceleration of the building, i.e., y [ a ẋ a ]. The easureent equation is thus y 1 k c k cos 2 θ + k c cos 2 θ + c z 1 + v (36) where the vector v contains the noises in each easureent. The odel in Eqs. (35) and (36) includes actuator dynaics and CSI and is designated Model 1. Model 2 is found by setting ḟ in Eq. (34). Using this approach will result in a controller that is equivalent to that used in Reinhorn, et. al. (1989). The resulting algebraic relation for the force is then given by f Ak q γ ( u a) k c A ȧ k c. (37) Using the state vector z 2 [a ȧ x ẋ] and the sae easured outputs as above, the state-space representation of Model 2 is ż 2 1 k Ak q γ c A 2 k c k c k c 1 k c k cos 2 θ + k c cos 2 θ + c z 2 + Ak q γ k c u + ẋ g, (38) y 1 k c k cos 2 θ + k c cos 2 θ + c z 2 + v (39) 15

16 Model 3 is found by assuing the transfer function G fu is constant in agnitude and has zero phase. Using the state vector z 3 [a ȧ x ẋ] and the sae easureents as above, the state-space representation for Model 3 is deterined as ż 3 1 k c k c k c k cos 2 θ + k c cos 2 θ + c z 3 + K u + ẋ g (4) y 1 k o c o k cos 2 θ + k c cos 2 θ + c z 3 + v (41) In this exaple, the structural paraeters were chosen to correspond to the SDOF test structure described in Chung, et. al. (1988). Their values were: 2,924 kg (16.69 lb-s 2 /in), c 15.8 N s/c (9.25 lb-s/in), k 13,895 N/ (7934 lb/in), θ 36 degrees, k o 14,879 N/ [8496 lb/in (i.e., )], and c. The ass of the frae,, was chosen to be.417 lb-s 2 o o /in. The ground acceleration was odeled as a broadband excitation with a constant two-sided spectral density of agnitude S c 2 /s 3 ( in 2 /s ). In all cases, the response calculations were based on the odel in Eqs. (35) and (36). Here, the uncontrolled configuration refers to the case in which the tendons are present, but are fixed to the ground. The zeroed configuration refers to the case in which the actuator is attached to the tendons, but the coand signal of the control actuator is set equal to zero. The response statistics are provided in Table 4 for the various type A controllers (displaceent weighting) and in Table 5 for the various type B controllers (acceleration weighting). Notice that in the case of displaceent weighting (Table 4), a stable controller for Model 2 cannot be designed at the force levels chosen. Also, application of both the low and high authority controllers designed using Model 3 (Controller 3A) has a detriental effect on the displaceent response of the syste copared to the uncontrolled syste. All of the control designs which iniize the absolute acceleration produce a significant reduction in the acceleration as well as the displaceent (see Table 5), except the controller which is based on Model 3 (Controller 3B). The low authority controller designed based on Model 3 (Controller 3B) has little effect of the acceleration response of the structure copared to the uncontrolled case, and the high authority controller actually increases the acceleration response. At this force level, the control design based on Model 2 (Controller 2B) perfors coparably to Controller 1B, but this controller becoes unstable if the weighting on the acceleration is increased above this value. As in the previous exaple, the displaceents are ost efficiently reduced when the displaceents are weighted, and the accelerations are ost efficiently reduced when the accelerations are weighted. To better understand the effects of CSI and actuator dynaics in this exaple, the transfer function fro the actuator coand to the actuator displaceent is provided in Fig. 11 for the active tendon syste. Notice that a coplex pole pair is present at approxiately 9 Hz due to the stabilizing position feedback. Also, the poles of the uncontrolled structure (i.e., with the actuator reoved and the tendons attached to the ground) have becoe zeros of the actuator transfer function, G au. The transfer functions fro the actuator coand to the actuator force, G fu, and to the tendon force, G fo u, are shown in Fig. 12. Copar- 16

17 Table 4: Coparison of RMS values of controlled responses for the active tendon syste with weighting on the relative displaceent. Configuration σ x σ a σ ẋ a σ f (c) (c) (c/s 2 ) (N) Uncontrolled 4.295e Zeroed Control 2.748e e Case 1: Low Authority Controller Controller 1A 3.752e e Controller 2A 3.692e-2 4.9e Controller 3A 3.71e e Case 2: High Authority Controller Controller 1A 2.744e e Controller 2A Controller 3A 1.111e e Table 5: Coparison of RMS values of controlled responses for the active tendon syste with weighting on the absolute acceleration. Configuration σ x σ a σ ẋ a σ f (c) (c) (c/s 2 ) (N) Uncontrolled 4.295e Zeroed Control 2.748e e Case 1: Low Authority Controller Controller 1B 3.748e e Controller 2B 3.722e e Controller 3B 3.43e e Case 2: High Authority Controller Controller 1B 1.91e e Controller 2B 1.77e e Controller 3B 1.58e e

18 Magnitude (db) Phase (deg) Frequency (Hz) Frequency (Hz) Figure 11. Transfer Function fro Actuator Coand to Actuator Displaceent. Magnitude (db) Phase (deg) Tendon Force Actuator Force Frequency (Hz) Frequency (Hz) Figure 12. Transfer Functions fro Actuator Coand to Tendon Force and Actuator Force. ing these two transfer functions, it is evident that the two forces are not the sae. Due to the presence of a coplex zero pair in G fu, the two transfer functions are significantly different in agnitude at high frequencies, and above approxiately 23 Hz they are 18 degrees out of phase. 18

19 Exaple 3: Active Mass Driver Consider the single story structure subjected to a one-diensional earthquake excitation ẋ g active ass driver as shown in Fig. 13. The equations of otion are with an Mẋ 1 + ( c 1 + c 2 )ẋ 1 c 2 ẋ 2 + ( k 1 + k 2 )x 1 k 2 x 2 f Mẋ g, (42) ẋ 2 + c 2 ( ẋ 2 ẋ 1 ) + k 2 ( x 2 x 1 ) f ẋ g, (43) where M is the ass of the building, is the ass of the AMD (including the ass of the actuator rod/piston), x 1 and x 2 are the displaceent relative to the ground of the building and the oving ass, respectively, c 1 and k 1 are the daping and stiffness coefficients of the building, respectively, c 2 and k 2 are the daping and stiffness coefficients of the active ass driver syste, respectively, and f is the control force applied by the hydraulic actuator. Fro above, the equation governing the dynaics of the hydraulic syste under unity-gain feedback of the actuator displaceent (i.e., x 2 x 1 ) can be written as ḟ 2β Ak q γ { u ( x 2 x 1 )} k c f A 2 ( ẋ 2 ẋ 1 ). (44), the equations of otion can be writ- Defining the state vector of the syste as ten in atrix for as z 1 [x 1 x 2 ẋ 1 ẋ 2 f] ż ( k 1 + k 2 ) k 2 ( c c 2 ) c M M M M M k 2 c 2 c k βAk q γ βAk q γ βA βA βk c z 1 + 2βAk q γ u + ẋ g. (45) k 2 c 2 f(t) x 2 () t M k 1, c 1 x 1 () t ẋ g Figure 13. Single story building with active ass driver. 19

20 The easureents are assued to be: displaceent of the first floor ass relative to the ground, displaceent of the AMD relative to the first floor ass, the absolute accelerations of both asses, and the force applied by the actuator, i.e., y [ x 1 ( x 2 x 1 ) ẋ a1 ẋ a2 f]. Thus, the easureent equation is y ( k 1 + k 2 ) k 2 ( c c 2 ) c M M M M M z 1 + v 1 v 2 v 3, (46) k 2 k c 2 c v 4 v 5 where v i is the noise in the ith easureent. This odel of the AMD syste accounts for both the dynaics of the actuator and CSI and is designated Model 1. Model 2 is fored by setting ḟ in Eq. (44), resulting in the algebraic relation for the force given by f Aγk q Aγk q u ( x k c k 2 x 1 ) c A ( ẋ k 2 ẋ 1 ) c. (47) Using the state vector z 2 [x 1 x 2 ẋ 1 ẋ 2 ] and the sae easured outputs as above, the state-space representation of this odel is ż ( k 1 + k 2 ) + d 1 k 2 + d 1 ( c c 2 ) + d 2 c 2 + d M M M M k 2 + d 1 k d 1 c 2 + d 2 c d d z u + M d ẋ g, (48) y ( k 1 + k 2 ) + d 1 k 2 + d 1 ( c c 2 ) + d 2 c 2 + d M M M M k 2 + d 1 k d 1 c 2 + d 2 c d d 1 d 1 d 2 d 2 z 2 + d M d d 1 u + v (49) where d 1 Aγk q k c, and d 2 A 2 k c. Model 3 is fored by assuing G fu has a constant agnitude and zero phase. Using the state vector z 3 [x 1 x 2 ẋ 1 ẋ 2 ] and the sae easureents as above, the state-space representation for Model 3 is 2

21 ż ( k 1 + k 2 ) k 2 ( c c 2 ) c M M M M k 2 k c 2 c z K u + M ẋ g (5) y ( k 1 + k 2 ) k 2 ( c c 2 ) c M M M M k 2 k c 2 c z M K u + v (51) In this exaple, the structural paraeters were chosen to correspond to the experiental odel in the SDOF configuration described in the experiental verification section. The values were: M 245 kg (1.4 lb-s 2 /in), c N-s/ (1.75 lb-s/in), and k 1 4,991 N/ (285 lb/in). The AMD was chosen such that the ass was 2% that of the structure, the stiffness k N/ (57 lb/in) was chosen to tune the AMD to the natural frequency of the structure, and the daping c 2 was considered to be negligible and set equal to zero. Using the constants for the hydraulic actuator odel presented in experiental verification section of this paper and the above paraeters, the atrices in the above odels were fored. For all of the controllers, the odel on which the responses calculations were based is given in Eqs. (45) and (46). In each case, the ground acceleration was odeled as a broadband excitation with a constant two-sided spectral density of agnitude S c 2 /s 3 ( in 2 /s ). Herein, the controlled response using the AMD is copared to the uncontrolled structure (i.e., with the AMD reoved). The zeroed control configuration is not presented for the AMD syste because the results are siilar to the responses of the uncontrolled configuration. The response statistics for the various type A controllers are provided in Table 6. For the high authority controller (Case 2), notice that while requiring the sae RMS control force, Controller 1A reduced the relative displaceent of the first floor by 51.3%, whereas Controller 3A only produced a 38.5% reduction in relative displaceent. The syste fored using Controller 2A becae unstable before this force level was achieved. Response statistics for the three type B controllers are shown in Table 7. Notice that the RMS absolute accelerations for Controllers 1B and 2B are the very close in both the high and low authority cases, and these controllers produce significantly better results than Controller 3B which considers the actuator transfer function to be constant over all frequencies. Also notice that for this exaple, the relative displaceent response with Controller 1B (acceleration weighting) is nearly the sae as for Controller 1A (displaceent weighting) when the sae aount of actuator force is eployed. For this exaple, it would be slightly ore beneficial to use a control strategy which weights the acceleration of the structure than one which weights the relative displaceent. In addition, the controlled systes fored by using Controllers 2A, 2B, 3A, and 3B becoe unstable at high displaceent/acceleration weighting in the perforance function (i.e., high authority control). If the copressibility of the hydraulic fluid is accounted for, a ore authoritative (i.e., higher perforance) controller can be designed without such an instability occurring. 21

22 Table 6: Coparison of RMS values of controlled responses for the AMD odel with weighting on the relative displaceent. Configuration σ x1 σ x1 x 2 σ ẋ 1 σ f (c) (c) (c/s 2 ) (N) (a) noinal configuration Uncontrolled 6.787e (b) Case 1: low authority controller Controller 1A 4.321e e Controller 2A 5.59e-2 5.1e Controller 3A 5.255e e (c) Case 2: high authority controller Controller 1A 3.353e e Controller 2A Controller 3A 4.229e e Table 7: Coparison of RMS values of controlled responses for the AMD odel with weighting on the absolute acceleration. Configuration σ x1 σ x1 x 2 σ ẋ 1 σ f (c) (c) (c/s 2 ) (N) (a) noinal configuration Uncontrolled 6.787e (b) Case 1: low authority controller Controller 1B 4.181e e Controller 2B 4.181e e Controller 3B 5.936e e (c) Case 2: high authority controller Controller 1B 3.272e-2 2.4e Controller 2B 3.277e e Controller 3B 4.663e e

23 Conclusions The role of control-structure interaction in the design of protective systes has been investigated, and the iportance of accounting for actuator dynaics and control-structure interaction has been deonstrated. For the case of hydraulic control actuation, a natural velocity feedback interaction path has been shown to exist. This feedback, together with the stabilizing displaceent (and/or force, velocity) feedback, causes control-structure interaction to be intrinsic to the device. The dynaic odel presented for the hydraulic actuator was verified experientally, as well as the predicted control-structure interaction behavior. Exaples eploying seisically excited structures have been provided which show that considering actuator dynaics and control-structure interaction in the design of a controller significantly iproves the perforance of the controlled syste. The following conclusions fro this work should be ephasized: When the feedback interaction path is present, the poles of the structure will appear as zeros of the transfer function fro the coand input to the force applied to the structure. This result occurs regardless of how fast the dynaics of the actuator are (including the case when G a is a constant gain). For actuators attached to lightly daped structures in which the feedback interaction path is present, the ability of the actuator to apply forces at the structure s natural frequencies is greatly liited. The actuator cannot apply a force at the natural frequencies of an undaped structure. Hydraulic actuators, both active and sei-active, have an iplicit feedback interaction path that occurs due to the natural velocity feedback of the actuator response. This interaction occurs for actuators configured in both displaceent, velocity and/or force control. Siple odels can be eployed to represent the dynaics of the hydraulic actuator and the associated control-structure interaction. Most researchers in the control of civil engineering structures have neglected the dynaics of the actuator, as well as the control-structure interaction effect. This approach is equivalent to assuing that the transfer function G fu is constant in agnitude with zero phase. In general, neither the phase nor the agnitude of G fu will be constant. Neglecting phase differences between the coand input u and the resulting force f, will result in a tie lag associated with generation of the control forces. Appropriate odeling of the actuator dynaics and control-structure interaction accoodates this tie lag. In a structural control syste for a given level of control action, neglecting the actuator dynaics generally results in larger responses than in the case where the control-structure interaction is considered. Also, neglecting actuator dynaics and control-structure interaction results in less achievable perforance of the controller because the closed loop syste ore easily becoes unstable. Better results are obtained if the copressibility of the hydraulic fluid is taken into account than if the fluid is treated as incopressible. Also, in the latter case, the achievable perforance level of the controller is significantly reduced due to instabilities created in the closed-loop syste. For the exaples of active bracing and active tendon systes considered herein, the structural response quantities are ore efficiently reduced if they are directly weighted in the control perforance function. Thus, to reduce the absolute structural accelerations, one should directly weight these acceleration responses in the control perforance function. For the active ass driver exaple, the relative displaceents and absolute accelerations are ost efficiently reduced by weighting the absolute accelerations of the structure. In general, odeling errors resulting fro neglecting actuator dynaics and control-structure interaction can be expected to decrease both the stability and perforance robustness of the controlled structure. Acknowledgent This research is partially supported by National Science Foundation Grants No. BCS and No. BCS

24 References ATC 17-1, 1993, Proceedings on Seisic Isolation, Passive Energy Dissipation, and Active Control, Applied Technology Council, Redwood City, California. DeSilva, Clarence W., 1989, Control Sensors and Actuators, Prentice Hall, Inc., Englewood Cliffs, New Jersey, pp Chung, L.L., Reinhorn, A.M. and Soong, T.T., 1988, Experients on Active Control of Seisic Structures, Journal of Engineering Mechanics, ASCE, ol. 114, pp Chung, L.L., Lin, R.C., Soong, T.T., and Reinhorn, A.M., 1989, Experients on Active Control for MDOF Seisic Structures, Journal of Engineering Mechanics, ASCE, ol. 115, No. 8, pp Housner, G.W. and Masri, S.F., Eds., 199, Proceedings of the U.S. National Workshop on Structural Control Research, USC Publications No. M913, University of Southern California, October Housner, G.W. & Masri, S.F., Eds Proc. of the Int. Workshop on Structural Control, Univ. of Southern California. Reinhorn, A.M., Soong, T.T., Lin, R.C., Wang, Y.P., Fukao, Y., Abe, H., and Nakai, M., 1989, 1:4 Scale Model Studies of Active Tendon Systes and Active Mass Dapers for Aseisic Protection, Technical Report NCEER Soong, T.T., 199, Active Structural Control: Theory and Practice, Longan Scientific and Technical, Essex, England. Soong, T.T. and Grigoriu, M., 1992, Rando ibration of Mechanical and Structural Systes, Prentice Hall, Englewood Cliffs, New Jersey. Spencer Jr., B.F., Suhardjo, J. and Sain, M.K., 1991, Frequency Doain Control Algoriths for Civil Engineering Applications. Proceedings of the International Workshop on Technology for Hong Kong s Infrastructure Developent, Hong Kong, Deceber 19-2, 1991, pp Spencer Jr., B.F., Suhardjo, J. and Sain, M.K., 1993, Frequency Doain Optial Control Strategies for Aseisic Protection. Journal of Engineering Mechanics, ASCE, ol. 12, No. 1, pp , Suhardjo, J., Spencer Jr., B.F. and Karee, A., 1992, Frequency Doain Optial Control of Wind Excited Buildings. Journal of Engineering Mechanics, ASCE, ol. 118, No. 12, pp Yao, J.T.P., 1972, Concept of Structural Control, Journal of the Structural Division, ASCE, 98(ST7),