Dynamic Force Control with Hydraulic Actuators Using Added Compliance and Displacement Compensation

Transcription

1 CUNEES0815 NEES at CU Boulder The George E Brown, Jr. Network for Earthquake Engineering Simulation Dynamic Force Control with Hydraulic Actuators Using Added Compliance and Displacement Compensation By Mettupalayam V. Sivaselvan University of Colorado, Boulder Andrei M. Reinhorn, Xiaoyun Shao, and Scot Weinreber University at Buffalo October 2008 Center for Fast Hybrid Testing Department of Civil Environmental and Architectural Engineering University of Colorado UCB 428 Boulder, Colorado

2 EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS Earthquake Engng Struct. Dyn. 2007; 00:1 10 [Version: 2002/11/11 v1.00] Dynamic force control with hydraulic actuators using added compliance and displacement compensation Mettupalayam V. Sivaselvan 1, Andrei M. Reinhorn 2, Xiaoyun Shao 2 and Scot Weinreber 2 1 Department of Civil, Environmental and Architectural Engineering, University of Colorado at Boulder, Boulder, CO Department of Civil, Structural and Environmental Engineering, University at Buffalo, Buffalo, NY SUMMARY A new approach to dynamic force control of mechanical systems, applicable in particular to frame structures, over frequency ranges spanning their resonant frequencies is presented. This approach is implemented using added compliance and displacement compensation. Hydraulic actuators are inherently velocity sources, that is, an electrical signal regulates their velocity response. Such systems are therefore by nature highimpedance (mechanically stiff) systems. In contrast for force control, a force source is required. Such a system logically would have to be a lowimpedance (mechanically compliant) system. This is achieved by intentionally introducing a flexible mechanism between the Correspondence to: 428 UCB, University of Colorado at Boulder, Boulder, CO siva@colorado.edu, Phone: (303) , FAX: (303) Contract/grant sponsor: George E. Brown Network for Earthquake Engineering Simulation, National Science Foundation; contract/grant number: #CMS and #CMS Received Copyright c 2007 John Wiley & Sons, Ltd. Revised

3 2 M. V. SIVASELVAN ET. AL. actuator, and the structure to be excited. In addition, in order to obtain force control over frequencies spanning the structure s resonant frequency, a displacement compensation feedback loop is needed. The actuator itself operates in closedloop displacement control. The theoretical motivation as well as the laboratory implementation of the above approach is discussed along with experimental results. Having achieved a means of dynamic force control, it can be applied to various experimental seismic simulation techniques such as the Effective Force Method and the Realtime Dynamic Hybrid Testing Method. Copyright c 2007 John Wiley & Sons, Ltd. key words: Dynamics Force Control, Hydraulic Actuators, Natural Velocity Feedback, Smith Predictor, Advanced Seismic Testing 1. INTRODUCTION Advanced seismic testing techniques such as the effective force method [3] and forms of realtime dynamic hybrid testing [15] require the implementation of dynamic force control in hydraulic actuators. Dynamic force control with hydraulic actuators is however a challenging problem. By its physical nature, a hydraulic actuator is a velocity source, i.e., a given controlled flow rate into the actuator results in a certain velocity. Moreover, hydraulic actuators are typically designed for good position control, i.e., to move heavy loads quickly and accurately. They are therefore by construction, high impedance (mechanically stiff) systems [12]. In contrast a force source is required for force control. Such a system logically would have to be a lowimpedance (mechanically compliant) system. Force control with hydraulic actuators is associated with many problems. Actuators designed for position control have stiff oil columns, making force control very sensitive to control

4 DYNAMIC FORCE CONTROL 3 parameters often leading to instability. Moreover friction, stickslip, breakaway forces on seals, backlash etc. cause noise in the force measurement, making force a difficult quantity to control. Several strategies have been introduced to work around this problem. For instance, a dual compensation scheme [11] uses a primary displacement feedback loop with force as a secondary tracking feedback. This scheme also supports features such as acceleration compensation to overcome some of the effects that distort the force measurements. In robotics, the impedance control strategy has been employed wherein the forcedisplacement relationship is controlled at the actuator interface [7, 19]. Pratt et. al. [14] have used the idea of series elastic actuators where a flexible mechanism is intentionally introduced between the actuator and the point of application of force, along with force feedback. They applied this to nonresonant systems. Furthermore, in force control the dynamics of the structure on which force is applied, is coupled in a feedback system with the dynamics of the actuator, resulting in a natural velocity feedback. When the structure is resonant, this results in a set of complex conjugate zeros of the open loop transfer function. Shield et. al. [3, 17] in their work on the effective force method, compensate for this effect by using velocity feedforward. It was also recognized by Conrad and Jensen [1], that closedloop control with force feedback is ineffective without velocity feedforward, or full state feedback. In this paper, a new approach to dynamic force control is presented, in which a compliance in the form of a spring is intentionally introduced between the actuator and the structure, and a displacement feedforward compensation is used. The method does not use direct force feedback. It also allows for an added physical design parameter in the control system, namely the stiffness of the added compliance. In the following, a standard linearized dynamic model of a hydraulic actuator is first presented. The natural velocity feedback problem and the solution of Shield et.

5 4 M. V. SIVASELVAN ET. AL. al. [3] are then discussed side by side to emphasize the differences and commonalities with the approach presented in this paper. The motivation behind the proposed solution using added compliance and displacement compensation is then discussed. The analysis of the proposed solution and some experimental results are then presented. 2. LINEAR MODEL OF A SERVOHYDRAULIC ACTUATOR A hydraulic actuator driving a single degree of freedom structure is shown in Figure 1. The analysis is this paper is based on linear models of the actuator and of the structure. For this, the dynamics of the actuator are linearized about the equilibrium point at the midstroke of the actuator. The linearized equations are standard (see for example [9, 2, 5, 18]) and are given by ẋ p = v p v p = A p M P ω2 stx p 2ζ st ω st v p P = 2 κ A p L ( A pv p γ 1 P γ 2 x v ) (1) ẋ v = 1 τ v K v τ v u Here, x p and v p are respectively the displacement of the SDOF structure, P is the differential pressure between the actuator chambers, x v is the valve spool displacement, M is the combined mass of the actuator piston and the SDOF system, L is half the stroke of the actuator, A p is the area of the actuator piston, τ v is the servovalve time constant, K v is the servovalve gain, κ is the bulk modulus of the oil, γ 1 is a dissipative constant that depends on the chamber and valve leakage flows, γ 2 is a gain coefficient, ω st and ζ st are the natural frequency and damping ratio of the SDOF structure and u is the control input to the servovalve. A block diagram

6 DYNAMIC FORCE CONTROL 5 Supply P S Return P R ~ x v x p kst P 1 P 2 M c st Figure 1. Model of a hydraulic actuator driving a SDOF structure Valve Command, u K τ s 1 v Servovalve Flow A p C s γ 12 1 Actuator Applied Force, f Natural Velocity Feedback A p 1 s M s s 2 2 2ζ st ω st ω st Figure 2. Block diagram representation of the linear model of equation (1). h C 12 = ApL 2κ, K = Kvγ2 i model of this linear system is shown in Figure 2. The quantity ω oil = 2κAp LM (2) is referred to as the oil column frequency. This is the imaginary part of a complex conjugate eigenvalue pair of the linearization, in the absence of a structure stiffness.

7 6 M. V. SIVASELVAN ET. AL. 3. NATURAL VELOCITY FEEDBACK AND THE ASSOCIATED CONTROL PROBLEM It can be seen from the third part of equations (1), that the velocity of the mass affects the rate of change of the differential pressure. This feedback can also be seen in the block diagram of Figure 2. This has been termed natural velocity feedback [3]. If the dissipation related to leakage flows, γ 1 is assumed to be zero, then the resulting transfer function H uf from the control input u to the applied force f is given by K s 2 2ζ st ω st s ωst 2 H uf = s (τ v s 1) s 2 2ζ st ω st s ωst 2 ωoil 2 (3) It can be seen that this transfer function has a complex conjugate pair of zeros corresponding to the natural frequency and damping ratio of the SDOF structure. This implies that the force applied on the structure at this frequency becomes small. Feedback control using for example a PID controller does not improve the performance because these zeros persist in the closedloop transfer function also. Therefore, additional control strategies are necessary Strategy of Velocity Feedback Compensation (Shield et. al. [3, 17]) It can be seen that there is a negative feedback of velocity at the summing junction in Figure 1. If we can add a positive feedback at this junction of the same amount, then the effect of the natural velocity feedback can be nullified. But since this is a physical junction that in inaccessible, the strategy of Shield et. al. [3, 17] is to add this positive feedback to the valve command. However, this signal has to now be preconditioned by the pseudoinverse of the servovalve transfer function. This is done using a leadlag compensator. In addition force feedback is also used. Figure 1 shows the resulting control strategy [3].

8 DYNAMIC FORCE CONTROL 7 Force Feedback Desired Force Valve Command K Velocity Compensation Servovalve Flow Compensator Natural Velocity Feedback A p A p C s 12 Actuator 1 s M s ζ ω ω st st s st Achieved Force Figure 3. Block diagram showing velocity feedforward correction loop 4. MOTIVATION FOR SOLUTION BASED ON ADDED COMPLIANCE AND DISPLACEMENT COMPENSATION It is known from experience that hydraulic actuators are more conveniently tuned in closedloop position control, than in force control. It is therefore suggested to indirectly control force by controlling position. To do this, a compliance, a spring of stiffness k LC, is introduced between the actuator and the structure. In this section, for simplicity of illustration, the effect of the reaction force from the spring on the actuator is ignored, i.e., perfect disturbance rejection is assumed. Perfect tracking is also assumed over all frequencies of interest. The full linear dynamics of the actuator is however considered in the analysis in section 5. First, the scenario the scenario of applying a force f on a rigid structure is considered as shown in Figure 4. It is easily seen that to apply a force f, the actuator piston needs to move an amount f/k LC. Thus the actuator can be operated in closedloop position control, and be commanded to the position f/k LC. If the structure were not rigid but flexible, then the applied force would cause it to displace by an amount x st. Thus the actuator needs to be commanded to the position f/k LC x st. This leads to the need for displacement compensation. The structure displacement x st may be obtained by from a model of the structure, or by measurement. These

9 8 M. V. SIVASELVAN ET. AL. Measured Force, f Desired Force, f Position 1 / k LC Command Added Actuator in Closedloop Position Control Compliance, k LC Rigid Figure 4. Applying a desired force on a rigid structure by controlling the position of the actuator possibilities are shown in Figure 5. It will be seen later that a mix of the two approaches leads to the Smith Predictor approach. In addition, since the assumptions of perfect tracking and perfect disturbance rejection in the above discussion are not realistic, additional compensation is needed for the dynamics of the actuator. This is presented in section 5 below. However, first the relationship of this approach to that of Shield et. al. is shown Comparison of the Proposed Approach to Velocity Compensation The relationship of the proposed approach to the velocity feedback compensation strategy of Shield et. al. [3, 17] can be shown by rearranging terms in the block diagram in Figure 3. Factoring A p s suitably in Figure 3, the block diagram in Figure 6 is obtained. Comparing the block diagrams in Figures 6 and 5(c), it is seen that the in the absence of added compliance, the oil column behaves as a spring providing the compliance required for force control. Relative deformation occurs across this spring and force is applied through it. However, the compliance of the oil column spring is fixed for a given actuator. In the approach proposed here, this compliance becomes an additional physical design parameter for the control system.

10 DYNAMIC FORCE CONTROL 9 Measured Force, f Desired Force, f 1 / k LC Model Position Command Actuator in Closedloop Position Control Added Compliance, k LC Flexible (a) Using a model to obtain the structure displacement Measured Force, f Desired Force, f 1 / k LC Position Command Actuator in Closedloop Position Control Added Compliance, k LC Flexible Displacement, x st (b) Using measured structure displacement Desired Force 1/k LC Displacement Command Actuator in Position Control Actuator Displacement k LC Added Compilance Achieved Force Displacement Compensation Compensator Displacement 1 ( 2 2ζ 2 st ω st s ω st ) M s (c) Block diagram representation of (b) Figure 5. Applying a desired force on a flexible structure 5. ANALYSIS OF THE PROPOSED CONTROL SOLUTION The analysis of the proposed strategy for dynamic force control with added compliance and displacement compensation is based on a linearized model, which is first presented.

11 10 M. V. SIVASELVAN ET. AL. Force Feedback Desired Force Valve Command K A s p Servovalve Actuator Actuator Displacement κ 2 Ap L Oil spring Achieved Force Displacement Compensation Compensator Displacement 1 ( 2 2ζ 2 st ω st s ω st ) M s Figure 6. Refactoring of block diagram in Figure Linear Modeling Modifying the model in equation (1) suitably, the linearized model of the actuator and the structure with the added compliance is obtained as ẋ p = v p v p = A p m p P k LC (x p x st ) P = 2 κ A p L ( A pv p γ 1 P γ 2 x v ) ẋ v = 1 τ v K v τ v u (4) ẋ st = v st v st = ωstx 2 p 2ζ st ω st v p k LC (x st x p ) Here, m p is the mass of the piston, x xt and v st are the displacement and velocity of the structure (which are now different from those of the actuator piston because of the added compliance), k LC is the stiffness of the added compliance and the other symbols are as defined before. The block diagram representation of this system along with the position controller C 1 and the displacement feedforward compensator C 2 are shown in Figure 7. In the figure, A 1 and A 2 are actuator transfer functions respectively from the valve command to the actuator

12 DYNAMIC FORCE CONTROL 11 A 2 Desired Force, f 1/k LC C 2 Position Command C 1 A 1 x p k LC Applied Force x st S Figure 7. Block diagram of the linear model of the actuator and structure with added compliance and the inner and outer loop controllers Desired Force, f 1/k LC C 2 Position Command ( ) ( ) A 1 k S x 1 LC p C 1 k LC 1 k A S LC 2 Applied Force x st S Figure 8. Block diagram of Figure 7 rearranged displacement, and from the force on the piston to the actuator displacement. These are given by A 1 = A 2 = 4Kκ m p Ls (τ v s 1)(s 2 2ζ a ω oil s ω 2 oil ) s 2ζ a ω oil m p s (s 2 2ζ a ω oil s ω 2 oil ) (5) where ω oil = 2Apκ m is the oil column frequency, ζ mpκ pl a = γ 1 a 3 pl is the actuator damping ratio, and S is the transfer function of the SDOF structure, S = 1 m st (s 2 2ζ st ω st s ω 2 st ) (6) The block diagram in Figure 7 can be rearranged as shown in Figure 8. The block diagram consists of an inner loop with controller C 1 whose role is to track the position command, and an outer loop which provides displacement feedforward compensation. The role of the C 2 is to compensate for the dynamics of the inner loop. The inner loop dynamics and the controller C 1

13 12 M. V. SIVASELVAN ET. AL. Position Command C 1 1 ( 1 LC ) ( ) A k S 1 k A S LC 2 x p Figure 9. Inner Loop are presented in section 5.2. The outer loop and the compensator C 2 are discussed in section Inner Loop The inner loop is shown in Figure 9. It can be seen that a more compliant spring, i.e. a lower k LC relative to the structure stiffness and the oil column stiffness, results in reducing the influence of the structure dynamics S, and of the effect of the reaction force A 2 on the dynamics of the actuator A 1. Physically, this can be interpreted as the compliant spring isolating the dynamics of the actuator from that of the structure for displacement tracking. The role of the controller C 1 is to track the position command. For this purpose, a proprietary control system, typically implementing a PID control can be used. The control system can be tuned with the structure connected to the actuator through the spring. Experience shows that the controller C 1 can be tuned in most cases so that the inner loop dynamics has a nearly flat frequency response magnitude with a linearly increasing phase lag over the bandwidth of interest. The inner loop dynamics can therefore be modeled reasonably as a pure time delay. This approach is used in modeling the inner loop dynamics.

14 DYNAMIC FORCE CONTROL Response without OuterLoop Compensation If an explicit outer loop compensator is not used, i.e. C 2 is set equal to 1, then the transfer function from the desired force to the measured force is given by f achieved f desired = s 2 2ζ st ω st s ωst 2 s 2 2ζ st ω st s ωst 2 klc m st (1 IL) where IL is the inner loop transfer function. If the inner loop dynamics is modeled by a pure timedelay, and a first order Taylor series approximation of the delay is used (i.e., (1 IL) τs), then this transfer function reduces to f achieved f desired = s 2 2ζ st ω st s ωst ( 2 s 2 2ζ st ω st klc m st τ d )s ωst 2 where τ d is the timedelay of the inner loop dynamics. It is seen that the delay, to a first order approximation, has effect of increasing the damping of the poles of the transfer function. For a lightly damped structure, lightly damped zeros still exist in the transfer function. These zeros manifest as a drop in the frequency response magnitude at the resonant frequency of the SDOF structure as shown in Figure 15(a). This necessitates the design of the outer loop compensator C 2. (7a) (7b) 6. OUTER LOOP COMPENSATOR DESIGN Motivated by the fact that the inner loop dynamics can be reasonably modeled as a pure timedelay, we consider the Smith predictor is considered as an approach to design the compensator C 2 of Figure 8. The Smith predictor was developed as a timedelay compensation algorithm in chemical process control [6]. It is however applicable to compensate for other types of dynamics as well. In the following, the basic idea of the Smith predictor is first reviewed, followed by a description of how it can be used to compensate for the inner loop dynamics.

15 14 M. V. SIVASELVAN ET. AL The Smith Predictor The basic idea of the Smith predictor is described by constructing it based on motivation. Figure 10(a) shows a standard feedback control system where a controller C has been designed for the plant P in such a way that the closed loop system has certain desired characteristics. The P C closedloop transfer function is 1PC. However, the control input cannot be applied directly to the plant, but has to be applied through an actuator A. The dynamics of the actuator may be thought of as undesirable dynamics in the feedback path. In order to regain the original closed loop structure, feedback is obtained from a model of the plant, ˆP instead of from the plant itself as shown in Figure 10(b). However due to modeling error, the feedback obtained from the model of the plant ˆP will not be the same as what would have been obtained from the actual plant P in the absence of the undesirable dynamics. Therefore, an additional error feedback is used as shown in Figure 10(c). Here, Â is the transfer function model of the actuator dynamics. This leads to the Smith Predictor architecture. It can be seen that if the models were exact, i.e. Â = A and ˆP = P, then the transfer function from reference to output PC is 1 PC A, and the Smith Predictor has the effect of moving the undesirable dynamics out of the feedback loop. The Smith Predictor is also intimately related to the Internal Model Control idea (see for example, [10]) Smith Predictor for Compensation of Inner Loop Dynamics As discussed in section 4, a desired force f is applied on the SDOF structure by imposing a displacement of f/k LC x st to the end of the added compliance. Thus the feedback structure is as shown in Figure 11, corresponding to the idea depicted in Figure 5(b). This is the desired feedback structure corresponding to Figure 10(a). However in reality, also present in this

16 DYNAMIC FORCE CONTROL 15 reference output C P (a) Standard feedback control system reference C A P output ˆP (b) Using feedback from the model to avoid undesirable dynamics A in the feedback path reference C A ˆP P Â output error (c) The Smith Predictor Figure 10. The concept of the Smith Predictor Desired Force, f 1/k LC 1 f k LC x st k LC Added Compilance Achieved Force x st 1 ( 2 2ζ 2 st ω st s ω st ) M s Figure 11. Desired feedback structure for displacement compensation

17 16 M. V. SIVASELVAN ET. AL. feedback loop is the undesirable dynamics of the inner loop as shown in Figure 8. The corresponding Smith Predictor architecture in line with Figure 10(c) is therefore as shown in Figure 12. In this figure, IL is the inner loop transfer function and quantities with hats are the modeled values of the actual physical parameters. The part shown in the dotted box is the controller C 2 defined in Figure 8. In can be seen that this part as a whole has two inputs (the reference and the feedback) and a single output, the actuator command. For digital implementation, the blocks in this part can be therefore composed into two transfer functions, to avoid algebraic loops. These are then transformed to a discrete time transfer functions using the bilinear transform, s = 2 T z 1 z1 [4]. As described above, if the models were exact, then the Smith Predictor has the effect of moving the undesirable inner loop dynamics out of the outer loop. If the model were not exact, it can be verified that the transfer function becomes f achieved f desired = 1 k LC [ ˆm st(1il) m st(1 ˆ 1 IL)]s 2 [ĉ st(1il) c st(1 IL)]s[ˆk ˆ st(1il) k st(1 IL)] ˆ m sts 2 c stsk st As the stiffness of the added compliance decreases relative to the structure stiffness and the oil column stiffness, the sensitivity of the performance of the Smith Predictor to modeling error decreases. This is a further benefit of the added compliance. IL 7. EXPERIMENTAL RESULTS Experiments were performed using a smallscale pilot test setup shown in Figure 13 to study the performance of the proposed force control strategy, before it was applied to largescale actuators. A hydraulic actuator with 1 kn (2.2 kip) force capacity and 100 mm (4 in) stroke was used. The actuator was fitted with a MTS twostage servovalve with a 19 liters/minute (5 gpm) flow capacity. The MTS FlexTest GT system was used for the inner loop controller.

18 DYNAMIC FORCE CONTROL 17 Achieved Force Desired Force, f 1/k LC IL 1 k LC m ( st s ζ stωst s ωst ) Added Compilance 1 kˆ LC 2 ˆ ˆ ˆ st st st LC mˆ s c s k k ˆ IL error Figure 12. Smith Predictor structure for displacement compensation The outer loop controller was implemented using using Simulink and xpc Target [8]. For the SDOF structure, a simple one story shear building model was used. A 305mm x 203mm x 25mm (12in 8in x 1in) steel plate served as the floor, while four 12.7mm (0.5in) diameter aluminum threaded rods served as columns. Braces were installed in the transverse direction on both sides of the structure to limit outofplane motion. Lead blocks are used to provide additional mass. Two different damping scenarios were considered for the structure one with merely the inherent damping in the structure, and another with model dashpots installed as shown in Figure 13(a). Helical springs were used for the added compliance as show in Figure 13(c). Four compressiononly springs were used. They were precompressed so that they could act in both tension and in compression. The properties of the structure and the added compliance are summarized in Table I. The inner loop controller C 1 was tuned for position tracking, and the resulting frequency

19 18 M. V. SIVASELVAN ET. AL. m st k st ω st ζ st k LC Case kg (170lb) 16.67N/m (156lb/in) 3 Hz N/m (111lb/in) Case Table I. Properties Mass Dashpot Braces (a) The SDOF structure Load Cell Stroke Servovalve Hydraulic Supply Displacement Transducer Reaction Frame (b) The hydraulic actuator (c) Spring used for added compliance Figure 13. Experimental Setup

20 DYNAMIC FORCE CONTROL x achieved /x desired Frequency (Hz) (a) Magnitude Phase (degress) Frequency (Hz) (b) Phase Figure 14. Inner loop Frequency Response Function response function of the inner loop is shown in Figure 14. For the frequency range of interest, it is seen that the inner loop dynamics can be modeled as a pure timedelay, τ, of 5.6 ms. The force control performance was studied by measuring the frequency response of the ratio of the applied force to the desired force. This was done using a crest factorminimized multisine input [13] for the desired force. Figure 15(a) shows the results for the structure with low

21 20 M. V. SIVASELVAN ET. AL. damping (ζ st = 0.01). For the case with C 2 = 1, the analytically obtained FRF considering the actuator as a pure timedelay of 5.6 ms agrees well with the experimentally measured FRF. This implies that that it is in fact reasonable to model the inner loop dynamics as a pure delay. It is also seen that using a Smith predictor for C 2 improves the force control performance. However, the frequency response function still exhibits some drop (about 20 %) at the resonant frequency of the structure and a bump at the resonant frequency of the structure (about 10 %) with the added compliance. This is because the damping, being very small is not known accurately and hence is modeled imprecisely. Figure 15(b) shows the results for the structure with the added dashpots, and hence higher damping (ζ st = 0.17). The frequency response with C 2 = 1 still shows a drop a the resonant frequency of the structure, but the drop is smaller (about 20 %) because the zeros of the transfer function of equation (7) are more highly damped. Since damping is modeled more accurately in the Smith Predictor, the frequency response with compensation is almost ideally at one. 8. SUMMARY AND CONCLUDING REMARKS From both the analytical and the experimental studies, the strategy of adding compliance and providing displacement feedforward compensation appears adequate for dynamic force control using hydraulic actuators over frequencies spanning resonances. The strategy does not use force feedback, the measurement of which generally is noisier and is corrupted by stickslip, breakaway forces on seals, backlash etc. in the hydraulic actuator. The strategy results in two controllers an inner loop controller, a typical PID controller, whose role is to track a position command, and outer loop controller whose role is to compensated for the inner loop dynamics. The inner loop dynamics can be reasonable modeled as a pure timedelay. In this work, the

22 DYNAMIC FORCE CONTROL C 2 = 1 Analytical with C 2 = 1 C 2 = Smith Predictor 1.4 f achieved /f desired Frequency (Hz) (a) Case 1: ζ st = Open Outer Loop C 2 = 1 C 2 = Smith Predictor 1.4 f achieved /f desired Frequency (Hz) (b) Case 2: ζ st = 0.17 Figure 15. Force transfer function outer loop controller has been designed using the Smith predictor approach. This requires approximate models of the SDOF structure as well as of the inner loop dynamics. It is seen that the performance of the system is less sensitive to the accuracy of these models when the added compliance is made more flexible. The tuning of the inner loop controller also becomes less sensitive to the dynamics of the SDOF structure with increase in this flexibility. The added

23 22 M. V. SIVASELVAN ET. AL. compliance thus provides an additional physical parameter in the dynamic force control system design. The flexibility cannot be arbitrarily reduced, for this comes at the expense of increased stroke requirement on the actuator. This method of force control has been successfully used in a unified approach to realtime dynamic hybrid simulations [16] ACKNOWLEDGEMENT The authors gratefully acknowledge the financial support from the National Science Foundation through the George E. Brown Network for Earthquake Engineering Simulation (NEES) development program, grants #CMS and #CMS REFERENCES 1. F. Conrad and C. J. D. Jensen. Design of hydraulic force control systems with state estimate feedback. In IFAC 10th Triennial World Congress, Munich, Germany, J. P. Conte and T. L. Trombetti. Linear dynamic modeling of a uniaxial servohydraulic shaking table system. Earthquake Engineering and Structural Dynamics, 29(9):1375, J. Dimig, C. Shield, C. French, F. Bailey, and A. Clark. Effective force testing: A method of seismic simulation for structural testing. Journal of Structural EngineeringAsce, 125(9): , G. F. Franklin, J. D. Powell, and M. L. Workman. Digital control of dynamic systems. AddisonWesley, Menlo Park, Calif., 3rd edition, J. Kuehn, D. Epp, and W. N. Patten. Highfidelity control of a seismic shake table. Earthquake Engineering and Structural Dynamics, 28(11 Nov): , J. E. Marshall. Control of timedelay systems. P. Peregrinus, Stevenage [Eng.]; New York, M. Mason. Compliant motion. In Michael Brady, editor, Robot motion: planning and control, MIT Press series in artificial intelligence, pages xv, 585 p. MIT Press, Cambridge, Mass., Mathworks. Simulink and xpc target, H. E. Merritt. Hydraulic control systems. Wiley, New York,, M. Morari and E. Zafiriou. Robust process control. Prentice Hall, Englewood Cliffs, N.J., 1989.

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