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Monolithic time-integration algorithms for Hamiltonian systems suitable for

real-time hybrid simulations

Giuseppe Abbiati 1,a, Oreste S. Bursi1,b, Alessio Bonelli1,c, Md Shahin Reza1,d

1University of Trento, Trento, Italy

agiuseppe.abbiati@unitn.it, boreste.bursi@ing.unitn.it, calessio.bonelli@ing.unitn.it, drezam@ing.unitn.it

1. Abstract Although the structural engineering community is traditionally more familiar with time integration algorithms conceived for second order systems, state space models based on extended state vectors let more flexible first order methods to be very attractive for possible implementations of nonlinear Numerical Substructures (NSs). In this perspective, Simulink implementations of two time integration algorithms belonging to the monolithic class and tailored to first order systems are provided: i) the linearly implicit Rosenbrock-based L-Stable Real-Time compatible (LSRT) algorithm with two stages; ii) the Modified Generalized-α (MG-α) method. For a proper selection of parameters, both the algorithms are characterized by second order accuracy and linear stability. In particular, the MG-α allows for user controlled algorithmic damping. In order to validate the implementations, RTDS experiments conducted at the Laboratory for Materials and Structural Testing (LPMS) of the University of Trento, Italy, on a full-scale industrial piping system and on a 2-DoFs split mass system are presented.

Keywords: Monolithic time integration; real-time hybrid simulations; user controller algorithmic damping; Internal Modal Control.

2. The LSRT2 algorithm In recent years, hybrid simulation techniques like Real-Time (RT) and Pseudo-Dynamic (PsD) testing with Dynamic Substructuring (DS) have become increasingly popular to study in depth the performance of structures subject to dynamic loads. With regard to relevant time-stepping methods, they can be broadly classified into two groups: monolithic and partitioned. According to the RTDS philosophy, in the monolithic approach, only the NS is integrated while the remainder part, i.e. the Physical Substructure (PS), is considered as a black box. Let us consider the state space representation of a linear system as:

n n n M y K y f (1)

where,

= , =

= ,

N N N

nn n e c

n n n

I 0 0 IM K

0 M K C

u 0y f

u f f

(2)

in which KN, CN and MN are stiffness, damping and mass matrices of the NS whereas, and are external load vector and the restoring force vector measured on the PS, respectively. With reference to monolithic time integration algorithms for Hamiltonian systems, we selected the Rosenbrock algorithm which results to be more competitive than popular Runge-Kutta methods in terms of stability, accuracy and ease of implementation (Bursi et al., 2008). This method is unconditionally stable for uncoupled problems and entails a moderate computational cost for real-time performance. For ease of notation, the following generalization is introduced:

1,n n n n nt y g y M f K y (3)

Hence, the LSRT2 algorithm can be summarized in algorithmic form as follows:

Stage-1

1

1 ,n nt t t k I J g y

2 21 1n n y y k

The displacement command2n y is sent to actuators

and the restoring force is fed back to2 2

( , )n ny t g.

Stage-2

2 2

1

2 21 1,n nt t t

k I J g y J k

1 1 1 2 2n n b b y y k k

The displacement command 1ny is sent to actuators

and the restoring force is fed back to 1 1( , )n ny t g .In order to preserve A-Stability, the Jacobian J is evaluated on the Global emulated System (GS):

1 1G G G G

0 IgJ

M K M Cy (4)

The following parameters are recommended for the LSRT2 method in order to achieve L-stability, second order accuracy and reduce algorithmic damping in the low frequency range:

2 21 21

1 2

1 2 2 , 1 2 , ,

0, 1b b

(5)

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The LSRT2 algorithm is L-Stable and Real Time (RT) compatible. No initialization tasks are required. Fig. 1 depicts a Simulink implementation of the proposed tool.

Fig. 1- Simulink implementation of the LSRT2algorithm: (a) force feed-back corresponding to yn; (b) stage-1 block; (c) displacement

command corresponding to .

3. The Modified Generalized-alpha algorithm The balance equation of the MG-α method proposed by Abbiati et al. (2012) and based on the Generalized-α (G-α) method of Jansen et al. (2004), is forced at the end of the time step by means of additional state variables or velocity-like quantities. These quantities satisfy the recurrence relation formulated by Brüls and Golinval (2006):

1 11 1m n m n f n f n v v y y (6)

The parameters αm, αf and γ involved can be expressed in terms of , i.e. the spectral radius for an infinite

time-step length:

31 1 1, ,

2 1 1 2m f m f

(7)

If is chosen to be zero, the method annihilates the

highest frequency components of the response. Conversely, if is chosen to be one, all frequency

components of the response are preserved in the linear problem. The MG-α method can be summarized in algorithmic form as follows:

11 1 1n n n

y D f K y

(8)

1 1 1n n n f mt y y y

(9)

1 1 1

1

n f m n f m n

m m n

v y y

v

(10)

2 1 1

1 1

n n n m m

n f m

t

t

y y v

y

(11)

The displacement command 2ny is sent to actuators.

The restoring force is fed back to fn+2. To preserve A-stability, whereas coupling forces depend on the state

vector, the Doperator is based on GS matrices:

f mG G G

t

I 0 0 ID

0 M C K (12)

Since the predictor-corrector structure of the MG-α let actuators track predicted states y , a first displacement

prediction 1y is deemed necessary at the initialization

stage.

0

10 0 0

0 0

1 0 0

0

0

1

m m

f m

t

t

y y

y M f K y

v y

y y v

v

(13)

The MG- is L-Stable for < 1 and RT compatible. Fig. 2 shows the Simulink implementation of the proposed scheme.

Fig. 2 - Simulink implementation of the MG-αmethod: (a) feedback force corresponding to ; (b) time integrator block; (c)

displacement command corresponding to .

Although the linearly implicit implementation requires an extended initialization, the MG-α algorithm is extremely lightweight and robust.

4. Case Study #1: tests on a piping system The abovementioned integration schemes were implemented to perform several RT tests with DS on a typical full-scale industrial piping system at the LPMS of the University of Trento within a European project INDUSE (Karamanos et al., 2013) in order to evaluate its seismic performance under operating conditions. To this end, pipes were filled with water with pressure at 3.2 MPa. The piping system contained several critical components, such as elbows, Tee-joint and bolted flange joints as illustrated in Fig. 3. In order to comply with performances of the transfer system, the seismic record adopted for RT tests was normalized to 0.02g of Peak Ground Acceleration (PGA). The experimental set-up of the PS is presented in Fig. 4.

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(a) (b) Fig. 3 (a) A 3D model of the piping system and its support

structure; (b) PS, NS and coupling nodes.

Fig. 4 Experimental set-up of the PS. As can be observed from Fig. 4, the PS of the piping network was a typical distributed parameters system. Since the experimental dynamic substructuring technique was conceived for lumped mass systems, the SEREP reduction method (Friswell et al., 1995) was applied to the FE model of the PS to obtain stiffness and mass matrices condensed at the two coupling DoFs depicted in Fig. 3b. Both LSRT2 and MG- integration schemes were proved effective for RT testing; they entailed similar experimental responses, which showed a favorable agreement with relevant numerical simulation results, as depicted in Fig. 5 and Fig. 6, respectively.

Fig. 5 LSRT2-based results: displacement of Coupling DoF #2 (Red- Experimental; Blue- Numerical).

 

Fig. 6 MG--based results: displacement of Coupling DoF #2 (Red- Experimental; Blue- Numerical).

5. Case Study #2: the Test Rig TT1 system A versatile system was conceived and installed for examining control techniques and assessing reliability of results in RT simulations for linear/nonlinear M-DoF systems at the LPMS of the University of Trento, Italy.

Fig. 7 – 4-DoFs test rig TT1.

The test rig TT1 system depicted in Fig. 7 consists of four actuators, one dSpace DS1103 control board and other high performance devices some of which are shown in Fig. 8.

(b)

(a) (c) Fig. 8 – Test rig TT1 equipment: a) dSpace control

board; (b) laser distance sensor; (c) load cell.

In view of performance analysis of actuator control systems, hybrid simulations of the 2-DoFs split mass system shown in Fig. 9 were carried out with the LSRT2 algorithm (Bursi et al 2012).

Fig. 9 – Computation schematic of the 2-DoFs system.

Because of rotational nonlinearities, translational and rotational frequency values of 2Hz and 1.5Hz were associated with the linearized system at zero rotation.

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The external excitation was provided by two sinusoidal waves with frequencies of 1Hz and 1.8 Hz, applied to the center of translational and rotational masses, respectively. To ensure both compatibility and equilibrium between the NS and the PS at the interface, two electro-magnetic actuators were distinctly operated in displacement control by means of the Internal Model Control (IMC) system sketched in Fig. 10.

Fig. 10 – IMC block diagram. 

In greater detail, the IMC implementation relied on a

two poles model ( )PG s of the actuator transfer

function ( )PG s -see the relevant characterization in

Fig. 11- and a low-frequency pass filter GF (s) aimed at

keeping proper the transfer function 1F PG G between

the controlled variable and the set point (Levine, 1999).

Fig. 11 – Displacement to dispalcement identified -GP2P, GP3P- and measured -EETF- transfer functions of the plant.

The IMC resulted to be superior to the widely used PID controller owing to both its internal stability and trade-off between robustness and set point tracking performance as well as ease of implementation. Moreover, it lends itself to the use of Model Predictive Control, for dealing with multiple actuator constraints (Li et al., 2010). Finally, in both the two case studies, the optimal delay compensation method proposed by Wu et al. (2012) based on the over prediction of displacement commands was applied.

6. Conclusions This paper presented Simulink implementations of the LSRT2 and the MG-α time integration algorithms for conducting RT tests. The effectiveness of proposed schemes was corroborated by two Case Studies: i) the full-scale industrial piping network; ii) the Test Rig TT1 2-DoFs split-mass system. In both the two cases, the optimal delay compensation method proposed by

Wu et al. (2012) based on the over prediction of displacement commands was applied. Moreover, in order to improve performances of the transfer system, the IMC model based control strategy was applied to the Test Rig TT1 Case Study.

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