Modified Runge-Kutta Integration Algorithm for Improved Stability and Accuracy in Real Time Hybrid Simulation

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1 Journal of Earthquake Engineering ISSN: (Print) X (Online) Journal homepage: Modified Runge-Kutta Integration Algorithm for Improved Stability and Accuracy in Real Time Hybrid Simulation Ge Ou, Arun Prakash & Shirley Dyke To cite this article: Ge Ou, Arun Prakash & Shirley Dyke (25) Modified Runge-Kutta Integration Algorithm for Improved Stability and Accuracy in Real Time Hybrid Simulation, Journal of Earthquake Engineering, 9:7, 2-39, DOI:.8/ To link to this article: Published online: 29 Jun 25. Submit your article to this journal Article views: 222 View related articles View Crossmark data Full Terms & Conditions of access and use can be found at

2 Journal of Earthquake Engineering, 9:2 39, 25 Copyright Taylor & Francis Group, LLC ISSN: print / X online DOI:.8/ Modified Runge-Kutta Integration Algorithm for Improved Stability and Accuracy in Real Time Hybrid Simulation GE OU, ARUN PRAKASH, and SHIRLEY DYKE 2 School of Civil Engineering, Purdue University, West Lafayette, Indiana, USA 2 School of Mechanical Engineering, School of Civil Engineering, Purdue University, West Lafayette, Indiana, USA Stability in Real Time Hybrid Simulation (RTHS) has been shown to be largely affected by system dynamics and associated phase lags. This lag typically originates in the physical components and considerable research has been conducted to compensate for it. Within the computational component of RTHS, different time integration algorithms are employed to achieve a more stable and accurate solution, mostly focusing on dissipating the high frequency content in the model. However, in RTHS, an inherent computational delay exists in the force measurement due to the sequential nature of communication between the numerical and experimental sub- structures. In this article, it is demonstrated that this computational delay affects performance and stability of closed loop RTHS even when no other delays or phase lags are present. This finding is validated through theoretical derivation and simulation results. A modified Runge-Kutta (MRK) integration algorithm is proposed to reduce the effect of computational delay. The MRK integration involves a three-stage computation: () the pseudo response is calculated using the delayed force measurement; (2) feedback force from the physical component for the next step is predicted using the pseudo response; and (3) the corrected structural response is then computed using the predicted feedback force. Both analytical and simulation results confirm that the MRK integration scheme is stable and accurate for a wide range time steps and is robust with respect to modeling error and nonlinearity in the experimental substructure. A moment-resisting frame is used as the experimental substructure in different cases of RTHS to validate the MRK integration method. This approach can also be adapted to other existing numerical integration schemes by applying the proposed three-stage computation process approach. Keywords Real-Time; Hybrid Simulation; Runge-Kutta Algorithm; Integration Method; Displacement Prediction; Stability. Introduction In contrast to conventional simulations of structures where one uses a numerical model of the entire structure, in hybrid simulation (HS), the structure is decomposed into two parts namely the numerical part (which is usually well understood) and the physical part (whose behavior is to be studied), as shown in Fig. The two parts of HS are coupled together with a transfer system to enforce continuity of the solution at the boundary/interface between them. This coupling is usually achieved by sending displacement commands generated in the numerical to the physical part and transferring measured forces from the physical to Received 29 July 24; accepted 3 March 25. Address correspondence to Ge Ou, School of Civil Engineering, Purdue University, West Lafayette, IN 4796, USA. gou@purdue.edu 2

3 Modified Runge-Kutta Integration Algorithm 3 Earthquake Structure Response Complete Simulation/Complete Physical Test Earthquake NUM Structure Response i th step restoring force for (i+) th step earthquake EXP Structure i th step response i = i + iteration HS/RTHS Implementation FIGURE Schematic drawing for traditional simulation/shake table test and HS/RTHS. the numerical part. It is considered to be a cost/space/time efficient approach compared to traditional shake table testing [Takanashi et al., 975; Nakashima, 2]. For over two decades, hybrid simulation had been performed on an extended timescale, neglecting the effects of rate dependent behavior [Mahin and Shing, 985; Takanashi and Nakashima, 987; Shing et al.,996]. Several promising load-rate-dependent auxiliary devices have been developed recently. These developments, combined with recent innovations in embedded systems and real-time execution (named Real Time Hybrid Simulation or RTHS) have enabled the earthquake engineering community to embrace this new technology [Christenson et al., 28]. Therefore, executing a hybrid simulation in real time is necessary and possible. In RTHS, many challenges exist since all calculations and measurements are executed in a very small time interval. One major issue in RTHS is the delay/lag in the system, including lags due to actuator dynamics, communication delay, and computational delay. These delays and lags are equivalent to negative damping in the system and can further destabilize the entire RTHS system [Horiuchi et al., 999]. Several studies have been conducted to understand the stability and performance of the RTHS closed loop system [Maghareh et al., 23, 24; Christenson et al., 23]. In the early stages of RTHS development, only the system delay (D/A and A/D conversion) was considered for delay compensation where the servo-mechanism required a larger time interval to achieve the desired command and dominated the unit delay due to the integration time step. Nakashima et al. [992] proposed a staggered central difference integration method to compensate for this unit delay (2 ms in their case) while the actuator outer loop control required 2 ms to realize the command. Following advances in hardware, outer loop actuator control later could be performed at a much higher rate and delay compensation methods were further extended to extrapolation and interpolation schemes considering actuator dynamics [Nakashima and Masaoka, 999]. In most implementations to date, it is still assumed, that the most significant source of lags (sometimes modeled as delays) in RTHS can be attributed to the transfer system. Significant effort has been devoted to estimating the transfer system lag and further compensated it during testing [Darby et al., 22; Wallace et al., 25; Ahmadizadeh et al., 28; Nguyen and Dorka, 28]. As the community develops new control algorithms [Carrion and Spencer, 26;Gao et al.,23; Phillips and Spencer, 23; Ou et al., 25], it is possible to achieve almost zero phase delay (time lag of ms or less) in tracking after proper design. However, elimination of the system lags still does not guarantee the stability of the entire RTHS loop [Maghareh et al., 23].

4 4 G. Ou, A. Prakash, and S. Dyke Due to the nature of RTHS the numerical and experimental substructures are linked in sequential order and the response of the experimental substructure is not available instantaneously. For example, even for computing response at the first time step, the restoring force is zero since no input has yet been sent to the experimental substructure. However, the true restoring force should be R (x, ẋ ) instead. Consequently, a unit delay, as shown in Fig., exists in the experimental force measurement. This delay is normally considered as a component of the computational delay [Mosqueda et al., 25]. Once the transfer system lag is compensated properly to enforce the boundary conditions accurately with almost zero delay, the effect of this computational delay dominates the stability and accuracy of RTHS. This effect is most pronounced for stiff, lightly-damped structures that may have relatively high natural frequencies associated with the first few dominant modes (potentially higher than the common RTHS execution rates of 4 khz) where this delay can lead to instability of the RTHS closed loop. Even for stable RTHS, the presence of even small computational delays can have detrimental effect on the accuracy of RTHS. Further, as some researchers are focusing on the fidelity of RTHS results, there is a need to use larger, more sophisticated numerical models within RTHS. Such high-fidelity models often take more time to run than the conventional RTHS execution rate of 24 Hz, creating a need for exploring RTHS at lower execution rates and consequently higher computational delays. Many integration schemes have been used in HS and RTHS, including both explicit and implicit methods. Due to the need for fast computation, most of the integration schemes developed for RTHS are explicit. Some explicit algorithms are unconditionally stable and are not affected by the highest natural frequency of the structure [Zhang et al., 25; Wu et al., 25, 26; Chen et al., 29]. However, when the computational delay is considered, stability condition for the integration scheme may be affected leading to constrains on the integration step size. Other methods, such as predictor-corrector methods, are also some of the numerical integration techniques used to solve ordinary differential equations. There are some predictor-corrector-based numerical integration algorithms that help reduce delays in RTHS, however, they are more focused on the delay in the transfer system rather than the inherent delay in RTHS [Wu et al., 22]. Some implicit integration algorithms that have been investigated by researchers include an equivalent force control for solving nonlinear equations of motion [Wu et al., 27] and HHT-α method with fixed number of substep iteration [Shing et al., 22] which provides stable experimental results for RTHS. However, the aforementioned HHT-α method requires numerical-experimental information exchange at each substep, where the substep displacement commands is calculated based on the measured restoring force from the previous substep [Chen and Ricles, 2]. In this article, computational delay is evaluated analytically using different integration algorithms, including the Newmark-β algorithm, CR algorithm, a discrete state-space method, and the conventional Runge-Kutta Method. The stability characteristics of RTHS closed loop using these integration schemes are studied here and compared to a no delay/pure simulation case. Results indicate that computational delay in RTHS affects the stability and accuracy of the test which also depend on the partitioning between experimental substructure and numerical substructure. To improve the accuracy of RTHS, the Runge-Kutta algorithm is also studied here, which is a fourth-order accurate integration algorithm and it takes the measured force for both i and i + time steps to reduce the prediction error in the experimental force. A modified Runge-Kutta (MRK) integration scheme is proposed in this article consisting of three computational stages: () a pseudo experimental response is calculated by solving the equation of motion using the force measured at both time steps i and i and (2) a pseudo feedback force is predicted at the time step i + using pseudo response from stage ; and (3) the corrected system response is calculated by solving the equation of motion

5 Modified Runge-Kutta Integration Algorithm 5 again using the predicted feedback force for time step i + combined with the measured experimental force at time step i. This modified integration procedure can be applied to other existing integration schemes by repeating stages 3. This MRK method can be considered as a model-based predictor-corrector integration method modified specifically to compensate for the inherent unit delay in RTHS. The performance of the MRK algorithm, proposed here, is theoretically examined and is shown to have improved the stability and accuracy of RTHS compared to regular integration schemes. For illustrating the capabilities of the proposed algorithm, an example is considered using a lightly damped SDOF structure with different cases of partitioning of stiffness in experimental substructure considered. A moment-resisting frame is used as the experimental substructure for experimental verification of the MRK algorithm, with a fixed stiffness partition ratio in the experimental substructure to illustrate the improved accuracy obtained using MRK. 2. Definition of Terms Some terminology used in this article is defined in this section [Nakata et al., 24]. Additional descriptions can be found in the Hybrid Simulation Wiki on NEEShub. nees.org/topics/rthswiki Actuator Control Algorithm: The algorithm used for controlling the transfer system to impose accurate desired signal (displacement/acceleration) between numerical substructure and experimental substructure. Reference/Whole Structure: This is the entire structure to be studied using HS/RTHS. Experimental Substructure: Structural component attached to the transfer system which is tested physically during HS/RTHS. Numerical Substructure: The portion that remains after subtracting the experimental substructure from the reference structure. Partitioning: Portion of mass, damping, stiffness in the experimental substructure, denoted P M, P C, P K. Stiffness partitioning of P K =.5 indicates 5% of whole structural stiffness is present in the experimental substructure. Delay: Delay in the system usually causes a time-shift in the input signal, but does not affect the signal characteristic. Delay in RTHS may refer to processing delay, communication delay or computational delay. Lag: Lag is the small amount of time that a physical system (e.g., actuator) lags behind by, when commanded to reach a specific displacement compared to when it actually achieves it [Callender et al., 936; Hartree et al., 937]. Tracking Performance: How the real measured signal (displacement/acceleration) from the transfer system interface performs compared to the desired signal from integration in the numerical substructure. Transfer System: Any physical interface that links the experimental substructure and the numerical substructure such as hydraulic actuator, electronic motor, shake table. 3. Effect of Computational Delay on Stability In this section, computation delay is described and discussed in detail. Several integration algorithms commonly used in RTHS are studied such as Newmark β, CR method, and a discrete state space method. The effect of computational delay is analyzed by comparing the stability characteristics of the RTHS closed loop using different integration algorithms. The following assumptions are made for the analysis:

6 6 G. Ou, A. Prakash, and S. Dyke. mass partitioning usually introduces an additional degree of freedom and experimental mass is relatively small in most RTHS cases, thus experimental mass is ignored and all the mass is assumed to be contained only in the numerical portion. Nevertheless, the proposed predictor-corrector approach is applicable to systems with mass partitioning as well; 2. transfer system (hydraulic actuator) that links the numerical and experimental substructures is considered to be perfectly controlled, i.e., the desired displacement is assumed to be accurately imposed on the experimental substructure; 3. reference system is considered to be linear time invariant (LTI), including both numerical and experimental substructure; and 4. since multiple DOFs (MDOFs) systems can be decomposed into several decoupled single DOF systems, theoretical stability characteristics are only derived for a single DOF system. The equation of motion for the reference system can be written as: Mẍ + Cẋ + Kx = MƔẍ g () where M, C, K are the mass, damping, stiffness matrices of the reference structure, and ẍ g denotes earthquake excitation. We can write the equations of motion of an RTHS system in the form: Mẍ + C N ẋ + K N x + F E (x, ẋ) = MƔẍ g (2) C E ẋ + K E x F E (x, ẋ) = (3) where the superscript ( ) N and ( ) E denote the portions in numerical and experimental substructures, and F E denotes the measured force in the experimental substructure. RTHS typically follows the steps below: Initialize F E =, loop over step, 2, 3:. Calculate x i+ using the measured force F E i from the experimental side and ground acceleration ẍ g,i+. 2. Impose x i+ on the experimental substructure and measure the feedback force F E i+. 3. Set i = i +, go to step. In step above, ideally the feedback force from the experimental substructure, should be obtained at i +, i.e., Fi+ E. However, FE i+ is not yet available at step and only FE i is available. In step 2, the desired displacement x i+ is assumed to be accurately imposed using the transfer system (hydraulic actuator). In this step, researchers usually assume a delay/lag resulting from actuator dynamics and compensate for it with extrapolation. However, even when perfect tracking can be achieved in the transfer system, i.e., the desired displacement x i+ can be enforced without any delay, the fact that Fi E was utilized to compute x i+, leads to the inherent unit time delay of RTHS. The discrete equation of motion for RTHS with perfect tracking can then be written as: Mẍ i+ + C N ẋ i+ + K N x i+ = MƔẍ g,i+ F E (x i, ẋ i ), (4) where F E (x i, ẋ i ) is obtained from Eq. (3). Since x i = x i+ and, ẋ i =ẋ i+, thus, Eq. (4) is not automatically equivalent to Eq. () and the existence of the unit delay is clearly evident.

7 Modified Runge-Kutta Integration Algorithm 7 To understand the differences introduced by the computational delay, different algorithms used to solve the numerical model are studied in the following sections. Mathematical presentations for both pure simulation and RTHS are derived individually for: Newmark-β (NB) method, Chen-Ricles (CR) method, and classic Runge-Kutta (RK) method. 3.. Newmark-β Method Newmark-β algorithm is discussed in many textbooks on structural dynamics, and a brief derivation is presented here. Consider the SDOF case in Eq. (), the calculated displacement, velocity and acceleration are: ẋ i+ =ẋ i + t [ ( γ ) ẍ i + γ ẍ i+ ] (5) x i+ = x i + tẋ i + ( 2β) /2 t 2 ẍ i + β t 2 ẍ i+ (6) ẍ i+ = Kx i C ẋ i M ẍ i ˆM (7) where, t is the integration time interval, C = (K t + C), M = [ tc ( γ ) + K (/2 β) t 2, ˆM = M + tγ C + Kβ t 2. Assuming no excitation force ẍ g,i =, displacement, velocity and acceleration between two adjacent time steps can be related in a recursive form: X i = A NB X i. (8) where, A NB is the amplification matrix, X i = [x i, ẋ i, ẍ i ] T is the system state. The amplification matrix can be expressed as: A NB = [ β t 2 K/ ˆM t β t 2 C / ˆM ( 2β) t 2 /2 β t 2 M ] / ˆM Kγ t/ ˆM γ tc / ˆM ( γ ) t γ tm / ˆM K/ ˆM C / ˆM M / ˆM Considering RTHS with unit delay, solving the finite difference equation using Newmark-β integration as in Eq. (4), Eqs. (5) and (6) are preserved but the acceleration calculation is slightly different : ẍ i = Kx i C R x. i M Rẍi ˆM R () where, C R = ( K N t + C ), M R = [ tc N ( γ ) + K N (/2 β) t 2, ˆM R = M + tγ C N + K N β t 2. Based on Eqs. (5), (6), and (), a similar amplification matrix for RTHS can be written in the recursive form: (9) where the amplification matrix A NB,R is defined as: X i = A NB,R X i ()

8 8 G. Ou, A. Prakash, and S. Dyke A NB,R = [ β t 2 K/ ˆM R t β t 2 CR / ˆM R ( 2β) t 2 /2 β t 2 MR / ] ˆM R Kγ t/ ˆM R γ tc R / ˆM ( γ ) t γ tm R / ˆM R K/ ˆM R C R / ˆM R M R / ˆM R Newmark-β method has been proved to be unconditionally stable for 2β γ /2, and is equivalent to the average acceleration method (denoted as AA in the figures) for β = /4, γ = /2. Evaluation of RTHS integration stability is based on the comparison of spectral radii obtained from Eq. (9) and Eq. (2) for β = /4, γ = /2. Figures 2 and 3 show these spectral radii for different amounts of damping (ζ ) in the system and different stiffness partition ratios P K. Where = ω dt, where dt is the time interval and ω is structural natural frequency in rad/s. (2) Stability Limit AA without delay AA with unit delay, P K =. AA with unit delay, P K =.3 AA with unit delay, P K =.5 AA with unit delay, P K =.7 AA with unit delay, P K = FIGURE 2 Spectral radii comparison between ordinary numerical integration and RTHS, Newmark β AA. Damping ζ = Stability Limit AA without delay AA with unit delay, P =. K AA with unit delay, P =.3 K AA with unit delay, P =.5 K AA with unit delay, P =.7 K AA with unit delay, P =.9 K FIGURE 3 Spectral radii comparison between ordinary numerical integration and RTHS, Newmark β AA. Damping ζ =..

9 Modified Runge-Kutta Integration Algorithm 9 The average acceleration method is unconditionally stable for conventional time integration. However, for the RTHS case, when there is no damping in the structure (ζ = ), due to the computational delay, all partitioning ratios (P K > ) are unstable, as the maximum spectral radius is greater than. This finding supplements the conclusion in Horiuchi et al.[999] which mainly focused on experimental delay/lag. Delay/lag in RTHS system is equivalent to adding negative damping to the structure. For the undamped case, any negative damping makes the RTHS loop unstable. For lightly damped structures, the range of stability for decreases as well CR Method Another direct integration algorithm developed by Chen and Ricles [29]. The variation of dispalcement, velocity over the time step in CR defined as: ẋ i+ =ẋ i + α tẍ i (3) x i+ = x i + t ẋ i + α 2 t 2 ẍ i. (4) Define integration parameters α and α 2 to obtain unconditional stability: α = α = α 2 = ζω n t + ω 2 n t2 (5) where ω n = ( K/M, ζ = C/ 2 ) MK. Write Eqs. (3) and (4) into recursive form, assuming to external force ( x g,i = ) : [ t t 2 ] α A CR = α K/M (C + K t) /M ( C t α + K t 2 α ) /M (6) Similarly, solving for the amplification matrix in the RTHS case: [ t t 2 ] α R A CR,R = α R K/M N ( C + K N t ) /M N ( ) C N t α R + K N t 2 α R /M (7) where α R is calculated based on numerical substructure natural frequency w nr = K N /M N and substructure damping ratio ζ R = C N /(2 M N K N ). For stability analysis, comparisons of the spectral radii for the conventional CR method and its RTHS implementation is shown in Fig. 4 for a lightly damped SDOF system. Similar to the Newmark-β algorithm, the CR integration algorithm shows unconditional stability for conventional time integration. However, when we consider computational delay, RTHS is more likely to go unstable for larger stiffness partitioning ratios and/or larger Discrete State Space Method In control engineering, a state space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential

10 2 G. Ou, A. Prakash, and S. Dyke Stability Limit CR without delay CR with unit delay, P K =. CR with unit delay, P K =.3 CR with unit delay, P K =.5 CR with unit delay, P K =.7 CR with unit delay, P K = FIGURE 4 Spectral radii comparison between ordinary numerical integration and RTHS CR algorithm. Damping ζ =.. equations. For continuous linear time invariant (LTI) systems, the standard continuous state-space representation is given below: ẋ = Ax + Bu (8) y = Cx + Du, (9) where x is the vector (n ) of state, u is the vector (p ) of input, and y is the vector (q ) of output. A is the system matrix (n n) (same as amplification matrix), B is the input matrix (n p), C is the output matrix (q n), and D is the feed-forward matrix (q p). Using the state space representation, Eq. () can be expressed as: Ẋ = A S X + B S U (2) Y = C S X + D S U, (2) where U =ẍ g, and the matrices A S, B S, C S, D S X, and Y are derived from Eq. () as: [ ] [ ] [ ] x A S = B K/M C/M S = X = ẋ [ ] [ C S = D S = K/M C/M ] Y = [ ẋ x ẍ ] (22) (23) Choosing a time step t, the amplification matrix can be written in discrete form as: X (i + ) = e AS t X (i) + A ( S e A S t I ) B S U (i) (24)

11 Modified Runge-Kutta Integration Algorithm 2 Using Tustin s method again to discretize the continuous state space representation above, which assumes: e A S t = ( I + )( 2 A S t I S t) 2 A (25) By combining Eqs. (24) and (25), discretized amplification matrix A DSS can be written: [ A DSS = 2 t ][ 2 tk/m 2 tc/m + 2 t ] 2 tk/m 2 tc/m + (26) Similarly, write governing Eq. (2) for RTHS using continuous state space representation: Ẋ = A R X + B R U R (27) Y = C R X + D R U R, (28) where U R =ẍ g + F E, recall F E = K E x + C E ẋ [ A R = K N /M C N /M ] [ ] B R = X = [ ] [ C R = D R = K N /M C N /M ] [ ] x ẋ Y = [ ẋ x ẍ ] (29) (3) Consider the free response case, where U R = F E. The amplification matrix A DSSR is: where A DSSR = A DSS + B DSS [ KE /M C E /M ] (3) [ A DSS = 2 t ][ 2 tkn /M 2 t ] 2 tcn /M + 2 tkn /M 2 tcn /M + (32) B DSS = ( A R A DSS I ) B R (33) System stability is analyzed again using the spectral radii method. Spectral radii for the RTHS system using discrete state space method are compared to the reference structure, as shown in Fig. 5, and similar results as using Newmark β and CR algorithm were found Classic Runge-Kutta Integration Algorithm The Runge-Kutta (RK) integration method is considered as a single-step method that evolves the solution from X i to X i, without requiring information from the previous time step. The classic fourth-order RK method uses intermediate stages with information and interpolation of excitation at i and i time steps and preserves the fourth-order accuracy. Consider a dynamic system:

12 22 G. Ou, A. Prakash, and S. Dyke Stability Limit DSS without delay DSS with unit delay, P K =. DSS with unit delay, P K =.3 DSS with unit delay, P K =.5 DSS with unit delay, P K =.7 DSS with unit delay, P K = FIGURE 5 Spectral radii comparison between ordinary numerical integration and RTHS, discrete state space using Tustin s method, ζ =.. ẏ = h (y, t) (34) Approximating the derivatives at the beginning, midpoint, and end of an time interval t i to t i+, four increments can be written as: k = h (y i, t i ) (35) k 2 = h (y i + t/2k, t i + t/2) (36) k 3 = h (y i + t/2k 2, t i + t/2) (37) k 4 = h (y i + tk 3, t i + t) (38) The approximated y i+ is determined using the present value y i and the weighted average of the four increments k, k 2, k 3, and k 4 is written as follows: y i+ = y i + t (b k + b 2 k 2 + b 3 k 3 + b 4 k 4 ) (39) where for classic Runge-Kutta, b = b 4 = /6, b 2 = b 3 = /3. Reconsider the state space form of reference structure in Eq. (2): Ẋ = A S X + B S U (4) where U =ẍ g, Y = C S X + D S U, (4)

13 Modified Runge-Kutta Integration Algorithm 23 [ A S = K/M C/M ] [ ] B S = X = [ ] x ẋ (42) The state space form of the Runge-Kutta method can be expressed as: Ẋ (i+), = k = A S X i + B S U i (43) Ẋ (i+),2 = k 2 = A S (X i + t/2k ) + B S (U i + U i+ ) /2 (44) Ẋ (i+),3 = k 3 = A S (X i + t/2k 2 ) + B S (U i + U i+ ) /2 (45) leading to the update equation for step (i + ) as: Ẋ (i+),4 = k 4 = A S (X i + tk 3 ) + B S U i+ (46) X i+ = X i + t/6 (k + 2k 2 + 2k 3 + k 4 ). (47) Assuming free vibration, the ground acceleration is set to zero (U i = ) and the amplification matrix for the RK method takes the form: where A RK = I + t/6 ( A S, + A S,2 + A S,3 + A S,4 ) (48) A S, = A S (49) A S,2 = A S + A S t/2a S, (5) A S,3 = A S + A S t/2a S,2 (5) A S,4 = A S + A S ta S,3. (52) For RTHS with unit delay, solving Eq. (4) under free vibration leads to the following equations for the RK method: Ẋ (i+), = k,r = A R, X i + B S U i (53) Ẋ (i+),2 = k 2,R = A R, (X i + t/2k ) + B S (U i + U i+ ) /2 (54) Ẋ (i+),3 = k 3,R = A R, (X i + t/2k 2 ) + B S (U i + U i+ ) /2 (55) Ẋ (i+),4 = k 4,R = A R, (X i + tk 3 ) + B S U i+ (56) where, A R, is defined as [ A R, = K N /M C N /M ]. (57)

14 24 G. Ou, A. Prakash, and S. Dyke Note that in RTHS, the force from experimental substructure has a unit time delay and U i+ is not immediately available, thus the external force U i+ = F E i, U i = F E i are restoring forces both with unit step delay. The amplification matrix for the RK method in RTHS is obtained as: A RKR = [ ] [ ] I + I A R B R (58) where A R = I + t/6 ( A R, + A R,2 + A R,3 + A R,4 ) B R = t/6 ( B R, + B R,2 + B R,3 + B R,4 ) [ ] B R, = K E /M C E /M (59) (6) (6) with A R, defined in Eq. (57) and A R,2 = A R, + t/2a 2 R, B R,/2, B R,2 = t/2a R, B R, + B R, /2 (62) A R,3 = A R, + t/2a R,2 A R, B R, /2, B R,3 = t/2a R, B R,2 + B R, /2 (63) A R,4 = A R, + ta R,3 A R, B R,, B R,4 = ta R, B R,2. (64) Spectral radii for the Runge-Kutta method under ordinary numerical integration and RTHS are compared in Fig. 6 and similar results as other aforementioned methods were found. This indicates that the unit step delay affects all conventional integration methods presented in this paper Stability Limit RK without delay RK with unit delay, P K =. RK with unit delay, P K =.3 RK with unit delay, P K =.5 RK with unit delay, P K =.7 RK with unit delay, P K = FIGURE 6 Spectral radii comparison between ordinary numerical integration and RTHS, Runge-Kutta algorithm. Damping ζ =..

15 Modified Runge-Kutta Integration Algorithm 25 In this section, effect of computational delay in RTHS has been studied using stability analysis of the closed RTHS loop using different integration algorithms. From the analysis, it can be concluded that the inherent unit delay in RTHS affects closed loop stability. Even assuming perfect tracking in the transfer system and ignoring actuator dynamics or controlstructure interaction effects, RTHS will be unstable when large partitioning of stiffness is present in the experimental substructure or when the analyzed structure has large natural frequency. 4. Modified Runge-Kutta Algorithm To reduce the effect of computational delay, a modified Runge-Kutta (MRK) integration algorithm is proposed. The MRK integration algorithm is built based on the classic Runge- Kutta method and it is aimed at minimizing the stability issue brought on by computational delay described in Sec. 2. Mathematical formulation of MRK is derived in this section, theoretical analysis is performed by comparing stability characteristics of classic Runge Kutta algorithm in reference structure simulation and stability characteristics of MRK in RTHS. First the formulation is derived for a single degree of freedom (SDOF) case and then it is extended to multiple degree of freedom (MDOF) case. In the conventional Runge-Kutta integration scheme, it is noted that the measured experimental force Fi E and FE i used as external force in the computation of the numerical substructure does not match excitation earthquake record ẍ g,i and ẍ g,i+. To minimize the effect of this unit step delay in the measured force F E, a pseudo measured force is predicted using the calculated response. In the beginning, this next time step response is calculated under delayed experimental force, the earthquake excitation records utilized in this stage are ẍ g,i and ẍ g,i, the calculated response labeled pseudo response is written as x i+.the pseudo measured force is predicted using the identified initial stiffness of the specimen and damping of the experimental substructure: F E i+ = KE x i+ + C E x i+ (65) The measured experimental force at the current step, F E i and the predicted force F E i+ are used as the external forces in the computation. Note that these also match the earthquake excitation records ẍ g,i and ẍ g,i+. Then a second Runge-Kutta iteration is performed to calculate the corrected next step response x i+. A detailed time diagram for using MRK in RTHS is shown in Fig. 7. The three stages of the MRK algorithm can be outlined as follows. Stage. Calculate the psuedo response of the RTHS same as regular RK integration in RTHS X i+ = X i + t/6 ( k,r + 2k 2,R + 2k 3,R + k 4,R ) (66) where k i,r has been derived in the classic RK algorithm. Stage 2. Predict the pseudo experimental substructure force using Eq. (65). Stage 3. The corrected response is calculated using the predicted pseudo force Ũ i+ = F E i+ and measured force Ũ i = F E i : Ẋ (i+), = k = A R, X i + B S Ũ i (67)

16 26 G. Ou, A. Prakash, and S. Dyke FIGURE 7 Timing diagram of modified Runge-Kutta integration in RTHS. Ẋ (i+),2 = k 2 = A R, ( X i + t/2 k ) + B S (Ũi + Ũ i+ ) /2 (68) Ẋ (i+),3 = k 3 = A R, ( X i + t/2 k 2 ) + B S (Ũi + Ũ i+ ) /2 (69) Ẋ (i+),4 = k 4 = A R, ( X i + t k 3 ) + B S Ũ i+ (7) ) X i+ = X i + t/6 ( k + 2 k k 3 + k 4 Writing the MRK integration for RTHS in a recursive way: [ ] I A MRKR = A 2 R B R A R B R (7) (72) where the derivation of A R and B R has been presented previously with the classic Runge- Kutta algorithm., The spectral radii for the MRK method are plotted in Figs. 8 and 9. It can be concluded that, compared to results in Sec. 2, MRK largely improves the stability performance of RTHS in the presence of computational delay. This algorithm can be considered as a model-based predictor-corrector method, which compensates specifically the inherent unit delay in RTHS. However, the effect of model accuracy (K E, C E ) and specimen nonlinear dynamics needs further analysis.

17 Modified Runge-Kutta Integration Algorithm Stability Limit RK without delay MRK with unit delay, P K =. MRK with unit delay, P K =.3 MRK with unit delay, P K =.5 MRK with unit delay, P K =.7 MRK with unit delay, P K = FIGURE 8 Spectral radii comparison between ordinary numerical integration and RTHS, modified Runge Kutta algorithm. Damping ζ = Stability Limit RK without delay MRK with unit delay, P K =. MRK with unit delay, P K =.3 MRK with unit delay, P K =.5 MRK with unit delay, P K =.7 MRK with unit delay, P K = FIGURE 9 Spectral radii comparison between ordinary numerical integration and RTHS, modified Runge Kutta algorithm. Damping ζ = Robustness Analysis of MRK in RTHS The MRK algorithm utilizes a psuedo measured force from Eq. (65) using a high fidelity model of the experimental substructure, i.e. no modeling error in K E, C E. Therefore, the effect of modeling error in the experimental substructure and the effect of specimen nonlinearity on the performance of this method should be examined. Assuming that the true stiffness of the experimental substructure is Kt E and modeling error only exists in the specimen stiffness (with estimated stiffness Kest E ), the equivalent stiffness can be written as K = Kest N + KE est and the amplification matrix for MRK with modeling error in RTHS can be derived as before. As two illustrative examples, Figs. and represent the most critical cases of partitioning ratio 9% (P K =.9) and %, respectively, with % damping ratio in both systems. The investigated modeling errors are ±(%, 2%, 3%, 4%) of the true specimen stiffness Kt E. Results with stiffness over-estimation and under-estimation distribute evenly about the reference line (MRK with no error). It is observed that the under-estimation of specimen stiffness ( Kest E < ) KE t potentially may destabilize the RTHS

18 28 G. Ou, A. Prakash, and S. Dyke Stability Limit RK without delay MRK with unit delay, P K =.9, no error MRK with unit delay, P K =.9, K est real E <K E MRK with unit delay, P K =.9, K est real E >K E FIGURE Spectral radii comparison for MRK with modeling error. Damping ζ = Stability Limit RK without delay MRK with unit delay, P K =.9, no error MRK with unit delay, P K =.9, K E MRK with unit delay, P K =.9, K E est <KE real est >KE real FIGURE Spectral radii comparison for MRK with modeling error. Damping ζ =.. closed loop when there is no damping in the system. However, the results are still significantly better compared to other conventional integration schemes (c.f. Fig. 2). For lightly damped system, the stability of the RTHS is preserved and the performance is similar to MRK with no error. Inevitably, solution accuracy is also largely affected by modeling error. However, this may not be due to MRK itself, but because of the existence of modeling error in general. The effect of specimen nonlinearity is also studied. Up to 8% stiffness degradation (equivalent system with K = Kest N + KE est with KE t =.2Kest E ) is analyzed for 9% partitioning ratio (P K =.9). Figure 2 shows the spectral radii for stiffness degrading (stiffness over estimation), for a system with no damping. It is evident that the RTHS closed loop stability is still preserved even with 8% stiffness degradation. 5. Numerical Examples Several numerical examples are presented to illustrate the performance of MRK integration scheme in RTHS. The reference structure used in these example was a lightly damped

19 Modified Runge-Kutta Integration Algorithm Stability Limit RK without delay MRK with delay, P K =.9,no nonlinearity MRK with delay, P K =.9, degrading FIGURE 2 Spectral radii comparison for MRK with stiffness degrading. Damping ζ =. FIGURE 3 SDOF RTHS numerical example and its interpretation using SIMULINK block. SDOF system with stiffness partitioned into numerical and experimental substructures. The RTHS examples are first completely simulated in MATLAB SIMULINK as shown in Fig. 3. In the Numerical Sub block, different integration algorithms are evaluated. MRK performance is compared with traditional integration methods discussed, including: central difference method and average acceleration method using Newmark-β, CR method, discrete state space method and classic Runge-Kutta method. In the experimental sub block, since integration stability and accuracy may be affected by partitioning in RTHS, different partitioning cases are studied and simulated. In this manuscript, the main focus is the delay introduced by computational loop, and so dynamics and lags in the transfer system are intentionally not modeled in the simulation. Consequently, only a unit delay block is added to the force generated in the experimental substructure to simulate the computational delay in a real test scenario.

20 3 G. Ou, A. Prakash, and S. Dyke TABLE RMS error for all integration method in SDOF RTHS, 52 Hz,. damping ratio, =.5 Partitioning AA-NB CR DSS RK MRK % % % UNS UNS UNS UNS.32 7% UNS UNS UNS UNS.45 9% UNS UNS UNS UNS.57 TABLE 2 RMS error for all integration method in SDOF RTHS, 24 Hz,. damping ratio, =.25 Partitioning AA-NB CR DSS RK MRK % % % % UNS UNS UNS UNS. 9% UNS UNS UNS UNS.5 A schematic diagram for the SDOF RTHS simulation is shown in Fig. 3. The reference structure has a mass m = 343 lb, stiffness k = 6 lb/in, and damping ratio %. Sampling frequencies are selected to be 52 Hz and 24 Hz. The accuracy of each integration algorithm is quantified using an RMS error indicator: E RMS = ( ) i n 2 DRTHS D Ref /Max ( ) D Ref = RMS (De ) /Max ( ) D Ref n Based on simulation results, performance for different integration algorithms in different RTHS cases is listed in Tables and 2, where UNS in the table indicates an unstable simulation. For a lightly damped structure (ζ =.), when a smaller partitioning ratio is chosen (%, 3%), the stability is preserved, however, the accuracy is affected using traditional algorithms. Note that sampling frequency affects the accuracy as well. The MRK algorithm, in all scenarios, matches well with the reference solution. The accuracy is affected by partitioning, damping ratio and sampling rate as well. The worst performance of MRK among all scenarios was obtained corresponding to the lightly damped reference structure with 9% stiffness in experimental substructure, sampled at 52 Hz which is still only.57%. This finding agrees with results in Figs. 8 and 9. Detailed responses obtained for partitioning ratios of % and 9% are presented in Figs (73) 6. Experimental Validation Actual experimental RTHS is also conducted to demonstrate the effectiveness of using modified Runge-Kutta integration algorithm over conventional integration. The experimental substructure chosen is a steel frame located in the Intelligent Infrastructure System

21 Modified Runge-Kutta Integration Algorithm 3 displacement(m) CR RTHS NB RTHS DSS RTHS RK RTHS MRK RTHS 6 x 3 MRK RTHS time(sec) FIGURE 4 SDOF RTHS numerical example: P K = %, sampling 52 Hz, damping ratio=., =.5. displacement(m).5 x CR RTHS NB RTHS DSS RTHS RK RTHS MRK RTHS 6 x 3 MRK RTHS time(sec) FIGURE 5 SDOF RTHS numerical example: P K = 9%, sampling 52 Hz, damping ratio=., =.5. displacement(m) CR RTHS NB RTHS DSS RTHS RK RTHS MRK RTHS 6 x 3 MRK RTHS time(sec) FIGURE 6 SDOF RTHS numerical example: P K = %, sampling 24 Hz, damping ratio=., =.25.

22 32 G. Ou, A. Prakash, and S. Dyke displacement(m) CR RTHS NB RTHS DSS RTHS RK RTHS MRK RTHS 6 x 3 MRK RTHS time(sec) FIGURE 7 SDOF RTHS numerical example: P K = 9% sampling 24 Hz, damping ratio=., =.25. FIGURE 8 Experimental substructure setup for RTHS validation. Laboratory located at Purdue University is shown in Fig. 8. The stiffness of the experimental substructure K E is identified through a predefined displacement test, K E is identified as 8.25 kips/in (444.7 kn/m) as shown in FIG. 9. The actuator attached has a maximum force capacity of 2 kips (8.82k kn), which limited the maximum displacement response of the RTHS under.25 inch (6.35 mm). The structure s dynamic properties are identified using a hammer test, with results shown in FIG. 2. The structure is highly damped with damping ratio ζ E of 5.54%. Note that experimental mass M E of the structure is 8.3 lb (36.4 kg), which is negligible compared to M N noted in Table 3. Four different RTHS cases are designed with natural frequencies of.5 Hz, Hz,.5 Hz, and 2 Hz. Each is excited with both El-Centro and Kobe earthquakes, and the earthquake intensities in both cases were chosen in order to keep the response displacement under.2 inch. In each case, 2/3 of the structural stiffness is assumed to be contributed by experimental substructure, and the damping ratio of reference structure in the three cases was set to 2%. Test parameter details are shown in Table 3, and in the following expressions: K N =.5K E, P K = 67% (74)

23 Modified Runge-Kutta Integration Algorithm force(lb) 5 force(lb) 5 measured 5 curve fitted result, stiffness kips/in displacemen(in) time(sec) FIGURE 9 Experimental stiffness identification for RTHS. 5 System Identification open loop transfer function Magnitude (db) Phase (deg) Frequency (Hz) Transfer Functions Phase identified model experimental data Frequency (Hz) FIGURE 2 Experimental dynamic property for RTHS. TABLE 3 Partition mass damping and stiffness for experimental validation EQ Intensity f = ω/2π M N C N K N M E C E K E Units Hz 3 Lb kips/in s kips/in 3 Lb kips/in s kips/in 3% % % %

24 34 G. Ou, A. Prakash, and S. Dyke M N = ( K E + K N) /ω 2 M E (75) C N = 2ζ KM C E (76) K = K N + K E (77) C = C N + C E (78) M = ( K E + K N) /ω 2, (79) where ω is the natural frequency of the reference structure in rad/s. RTHS results are further compared to a simulation of the reference structure, with structural properties M, C, K. Sampling frequency is kept at 24 Hz for RTHS and 248 for reference structure simulation. In addition to inner PID control loop, the actuator is controlled using the robust integration actuator control (RIAC) algorithm as the external control loop [Ou et al., 25]. The RIAC uses the H as feedback core controller, Kalman filter to minimize effect of noise on feedback measurement and a delay compensation block for online tuning. Tracking performance of band limited white noise signal of Hz bandwidth using RIAC algorithm is shown in Fig. 2. The time delay of the actuator after RIAC is shown to be under ms which indicates that the actuator lag is of a similar magnitude compared to the unit computation delay. Three integration methods, Newmark-β (with γ =.5 and β =.25), conventional Runge-Kutta, and the modified Runge-Kutta, are examined with this RTHS setup. The RTHS results from these cases are presented in Figs. 22 and 23. The CR method and the discrete state-space method yields very similar responses compared to the Newmark-β and the conventional RK method, and therefore were omitted from these plots. All these results are generated using the same actuator control algorithm and testing took place on the same day. It is clear from the responses in Figs. 22 and 23 that the proposed MRK method performs far better than any of the conventional integration schemes. Computed values of the RMS error are listed in Tables 4 and 5. In general, the errors are lower for low displacement(in) measured displacement desired displacement time(sec) (a) Tracking time history using RIAC Gain (abs) Phase ( ) Delay (sec) 2 Transfer Function Magnitude Transfer Functions Phase Control System Delay Trial Trial 2 Unit Delay Frequency (Hz) (b) Tracking transfer function using RIAC FIGURE 2 Controller tracking performance for experimental substructure.

25 Modified Runge-Kutta Integration Algorithm time(sec) NB RTHS AA RK RTHS MRK RTHS (a) RTHS results: f n =.5Hz NB RTHS AA RK RTHS. MRK RTHS time(sec) (b) RTHS results: f n =Hz NB RTHS AA.5 RK RTHS. MRK RTHS time(sec) (c) RTHS results: f n =.5Hz time(sec) NB RTHS AA RK RTHS MRK RTHS (d) RTHS results: f n =2Hz displacement(in) displacement(m) MRK measured Reference time (sec) (e) RTHS results (MRK): f n =2.5Hz displacement(in) displacement(m) MRK measured Reference time (sec) (f) RTHS results (MRK): f n =3Hz FIGURE 22 Experimental RTHS: El-Centro Earthquake. natural frequencies of the reference structure and increase for higher natural frequencies. As demonstrated in the numerical examples, RTHS with RK, CR, and NB integration algorithm yields unstable results with P K =.7 at =.25, in the experimental validation. For the case where the reference structure has a natural frequency at 2 Hz, RTHS with conventional integration is only marginally stable and thus, it was decided not to continue

26 36 G. Ou, A. Prakash, and S. Dyke NB RTHS AA RK RTHS MRK RTHS time(sec) (a) RTHS results: f n =.5Hz NB RTHS AA RK RTHS MRK RTHS time(sec) (b) RTHS results: f n =Hz time(sec) NB RTHS AA RK RTHS MRK RTHS (c) RTHS results: f n =.5Hz time(sec) NB RTHS AA RK RTHS MRK RTHS (d) RTHS results: f n =2Hz FIGURE 23 Experimental RTHS: Kobe Earthquake. TABLE 4 Experimental RTHS error comparison: El-Centro Earthquake f = ω/2π Error using NB Error using RK Error using MRK.5 Hz % 5.62% 3.6%. Hz.6 4.5% 3.65%.76%.5 Hz % 2.4% 4.48% 2. Hz.23.86% 2.9% 3.37% TABLE 5 Experimental RTHS error comparison: Kobe Earthquake f = ω/2π Error using NB Error using RK Error using MRK.5 Hz % 6.68% 3.97%. Hz % 5.% 2.42%.5 Hz %.2% 4.% 2. Hz % 28.28% 7.44% increasing the natural frequency of the reference structure. However, for RTHS using MRK, the test pt is extended to reference structures with natural frequencies of 2.5 Hz ( =.53) and3hz( =.84) (shown in Figs. 22e and 22f) without any concerns of instability.

27 Modified Runge-Kutta Integration Algorithm 37 Compared to simulated RTHS results, the errors in the actual RTHS are observed to be slightly higher because of the inexact enforcement of the boundary conditions and the underlying complexity of RTHS. Factors such as two feedback loops (force feedback in RTHS and displacement feedback in actuator control), noise and system uncertainties (such as actuator dynamics and test specimen variabilities) also affect the accuracy of RTHS results. Nevertheless, all the results indicate that the RTHS performance is affected negatively in the presence of a single-step computational delay and that performance is significantly improved by using the MRK algorithm. 7. Conclusion An inherent unit delay exists in the force measurement of the experimental substructure in RTHS due to the sequential order of communication between the numerical and experimental substructures and this may cause instability or performance degradation of the test. To explicitly evaluate the effect of this unit delay, different integration algorithms were employed in this study. Compared to pure simulation, both analytical and simulation results indicate that, the unit delay in the closed loop RTHS affects the stability and is highly dependent upon integration step size, structure natural frequency and structure partitioning ratio. Therefore, a modified Runge-Kutta integration algorithm is proposed to predict the feedback force measurement and minimize the effect of this inherent delay. The MRK integration includes three computation stages: () pseudo response calculation; (2) prediction of the measured force; and (3) corrected response calculation. Results illustrate that the modified Runge-Kutta improves the performance of RTHS. Further, a robustness analysis, considering modeling error in the experimental substructure, demonstrates that only under-estimation of structure stiffness (specimen stiffening) may affect MRK stability for the undamped case. For lightly damped structures with high partitioning ratio, the MRK method is shown to be robust for up to 4% modeling error. Experimental RTHS is also implemented to verify the effectiveness of the modified Runge-Kutta integration algorithm over conventional integration algorithms. A moment-resisting frame with a large stiffness is tested as the experimental substructure in RTHS. Results indicated that the modified Runge-Kutta algorithm improves the accuracy of the RTHS and extends the stability limit of the test. It may be noted that this method can directly be applied to other existing integration algorithms by adapting the three computation stages to that method in a similar manner, as shown here. Acknowledgment Special thanks goes to Dr. Wei Song ( from the University of Alabama for advice on the closed loop RTHS analysis. Funding This material is based upon work supported by the National Science Foundation under Grant No. CMMI 92778, CNS Support for the development of the laboratory was made possible through NSF Grant CNS and the School of Mechanical Engineering at Purdue.