Nonlinear signal-based control with an error feedback action for nonlinear substructuring control

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Nonlinear signal-based control with an error feedback action for nonlinear substructuring control Ryuta Enokida n, Koichi Kajiwara E-Defense, National Research Institute for Earth Science and Disaster Resilience, 1501-21 Nishikameya, Mitsuda, Shijimi-cho Miki, Hyogo, 673-0515 Japan

a r t i c l e i n f o

abstract

Article history: Received 2 March 2016 Received in revised form 29 July 2016 Accepted 19 September 2016 Handling Editor: D.J Wagg

A nonlinear signal-based control (NSBC) method utilises the ‘nonlinear signal’ that is obtained from the outputs of a controlled system and its linear model under the same input signal. Although this method has been examined in numerical simulations of nonlinear systems, its application in physical experiments has not been studied. In this paper, we study an application of NSBC in physical experiments and incorporate an error feedback action into the method to minimise the error and enhance the feasibility in practice. Focusing on NSBC in substructure testing methods, we propose nonlinear substructuring control (NLSC), that is a more general form of linear substructuring control (LSC) developed for dynamical substructured systems. In this study, we experimentally and numerically verified the proposed NLSC via substructuring tests on a rubber bearing used in base-isolated structures. In the examinations, NLSC succeeded in gaining accurate results despite significant nonlinear hysteresis and unknown parameters in the substructures. The nonlinear signal feedback action in NLSC was found to be notably effective in minimising the error caused by nonlinearity or unknown properties in the controlled system. In addition, the error feedback action in NLSC was found to be essential for maintaining stability. A stability analysis based on the Nyquist criterion, which is used particularly for linear systems, was also found to be efficient for predicting the instability conditions of substructuring tests with NLSC and useful for the error feedback controller design. & 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction The nonlinear signal-based control (NSBC) method [1] proposed recently achieves accurate control of nonlinear systems, relying on the ‘nonlinear signal’; this signal is obtained as an error of the outputs of a controlled system and its linear model subjected to the same input signal. The NSBC controllers are currently composed of two transfer functions {Kr, Ks} associated with the reference and nonlinear signals {r, s}, respectively. These controllers are basically designed by an inverse transfer function of a linear model of the controlled system. However, this controller design is not flexible for the enhancement of the robustness for a pure time delay, which inevitably exists in the transfer system and causes the instability of the controlled system. Therefore, in order to enhance the flexibility of the design and robustness of the system, we incorporate an error feedback controller, Ke, into NSBC. In this study, for the first application in the physical experiments, the NSBC method n

Corresponding author. E-mail addresses: [email protected], [email protected] (R. Enokida).

http://dx.doi.org/10.1016/j.jsv.2016.09.023 0022-460X/& 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Please cite this article as: R. Enokida, & K. Kajiwara, Nonlinear signal-based control with an error feedback action for nonlinear substructuring control, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.09.023i

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with the error feedback action is employed in a dynamical substructuring method. Dynamical substructure testing, whereby a dynamical experiment is conducted upon a critical part (referred to as the physical substructure) of an emulated system with real-time simulations of the other parts (referred to as the numerical substructure) of the system, has become a key experimental method in a wide range of engineering fields. The purpose of substructure tests is to study the properties of the critical part, such as its nonlinear characteristics, instead of conducting experiments on the entire emulated system. Many engineering fields require the effects of nonlinearities to be minimised for ease of control of the system, but, in structural and earthquake engineering, nonlinearity is positively utilised to increase the energy dissipation or change the natural frequency of the structures under earthquakes. Therefore, the dynamical substructuring methodology is required to maintain robustness for systems with nonlinearities or unknown parameters. Hybrid system (HS) testing is probably the most intuitive execution of this technique, since this method directly uses feedback of the numerical substructure output as an input to the physical substructure via an actuation system [2–9]. The actuation system is regarded as the system to be controlled. This basic HS realises perfect control of the systems only in an ideal situation where the actuator dynamics are perfectly modelled, and there is no pure time delay. However, in practice, there is always a pure time delay and modelling error for the actuator dynamics in substructured systems. In this case, the basic HS suffers from a significant lack of robustness, in particular, for a low damped system [10–12]. To enhance the robustness, a series of advanced HS methods have been proposed to compensate for pure time delays [2–9] and dynamics changes in the actuation system [10,13–15]. Controllers developed within the dynamically substructured system (DSS) framework are proposed from the perspective of automatic control system design [16] to generate an input signal to the transfer system (a combination of actuators, innerloop controllers and signal-conditioning hardware) that drives the physical substructure output so that it closely matches that of the corresponding output of the numerical substructure. In this method, knowledge of the parameters of the substructures, as well as the transfer system, is required in the formulation of the DSS using a linear substructuring controller (LSC). The advantage of the DSS-LSC strategy is the resultant separation of the emulated system dynamics from those of the closed-loop error dynamics. This enables the representation of even very lightly damped emulated systems using a DSS configuration with large stability margins that are designed into the system. However, the applicability of LSC is limited to relatively well-known systems because it is designed on the basis of a nominal knowledge of the dynamics of the substructures. As a result, adaptive minimal control synthesis (MCS) [17] is typically required in the DSS scheme when system parameters are unknown or poorly known [16,18,19]. In this study, in order to enhance the robustness of DSS for poorly known and highly nonlinear systems, we propose a nonlinear substructuring control (NLSC) method as a generalised form of LSC by incorporating the NSBC method with the error feedback action into DSS. The use of discrete-time computational elements in the transfer system inevitably results in a pure time delay, which tends to destabilise experimental substructure test systems. Therefore, the stability of DSS with NLSC is also discussed in this study, with the critical pure time delay given by the Nyquist criterion, i.e., the value of delay that causes instability, which can therefore be utilised as an index of relative stability [20]. Associated experiments were conducted for the substructure test of a rubber bearing demonstrating a base-isolated structure under a seismic excitation in the Advanced Control and Test Laboratory (ACTLab), at the University of Bristol. In Section 2, NSBC with an error feedback action is discussed together with the details of NLSC and its stability analysis for a pure time delay. In Section 3, NLSC is examined in numerical simulations of substructuring tests for rubber bearings. In Section 4, the real implementation of substructured experiments for the rubber bearing is reported.

2. Nonlinear substructuring control In the physical experiments, transfer systems are crucially important to excite controlled systems, based on the control signal generated at computation. However, the transfer system has certain dynamics and a pure time delay, mainly caused by the discrete-time computational elements. In this study, the transfer system and its model are expressed by: −τs ⎧ ⎪ G TS ( s ) = Gts ( s )e ⎨ ⎪ ¯ −τs ¯ ⎩ G TS ( s ) = Gts( s )e

(1)

where s is the Laplace operator, {GTS, Gts} represents the transfer system dynamics including a pure time delay τ, and the undelayed component of the transfer system dynamics, respectively, and G¯ TS , G¯ ts represents the linear models. Although dynamics in the transfer system is influenced by a controlled system due to the interaction, the error feedback action, in general, can compensate the error caused by the change of dynamics. Thus, for the simplification, our controller design is based on the separation of the transfer system and controlled system. The controller designs of the NSBC and NLSC methods are discussed in Sections 2.1 and 2.2, together with the transfer system including the pure time delay term.

{

}

2.1. Nonlinear signal-based control NSBC relies on a nonlinear signal obtained from the output signals of a nonlinear system and its linear model under the same input signal [1], as shown in Fig. 1. In NLSC, the nonlinear system is equivalently expressed, as follows: Please cite this article as: R. Enokida, & K. Kajiwara, Nonlinear signal-based control with an error feedback action for nonlinear substructuring control, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.09.023i

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Fig. 1. Nonlinear signal-based control method used for the physical experiments.

G( s ) = G¯ ( s ) + ΔG( s )

(2)

where G is the equivalent nonlinear system, G¯ is the linear model and ΔG is the modelling error. Now, the outputs of the linear model and the nonlinear system under an input signal are described as follows:

⎧ y¯ ( s ) = G¯ ( s )y¯ ( s ) = G¯ ( s )G¯ ( s )u( s ) ⎪ TS ts ⎨ ¯ ¯ ⎪ ⎩ y( s ) = G( s ) + ΔG( s ) yts ( s ) = G( s ) + ΔG( s ) G TS ( s )u( s )

(

)

(

)

(3)

where { y, y¯ } is a set of outputs from the nonlinear system and its linear model, respectively, { yts, y¯ts } is a set of outputs in the transfer system and its linear model, respectively, and u is the input signal. Here, the nonlinear signal is expressed by:

σ( s ) = y( s ) − y¯ ( s )

(4)

where s is the nonlinear signal. Now, the error signal is obtained as:

e( s ) = r ( s ) − y( s ) = r ( s ) − G¯ u( s )e−τs u( s ) − σ ( s )

(5)

where G¯ u( s ) = G¯ ( s )G¯ ts( s ), r is the reference signal and e is the error signal. In Eq. (5), the input signal is clearly related to the three signals {e, r, s}. Thus, the input signal in NLSC is proposed to be:

u( s ) = Kr ( s )r ( s ) + Ke( s )e( s ) + Kσ ( s )σ ( s )

(6)

where {Kr, Ke, Ks} is the controller transfer function set related to the reference, error and nonlinear signals, respectively. Substituting Eq. (6) into Eq. (5), the error signal is rewritten as:

e( s ) =

1 − G¯ u( s )e−τs Kr ( s ) 1 + G¯ u( s )e−τs Kσ ( s ) r ( s) − σ ( s) −τs ¯ 1 + Gu( s )e Ke( s ) 1 + G¯ u( s )e−τs Ke( s )

(7)

According to Eq. (7), the following transfer functions are found to be suitable for the NSBC controllers:

Kr ( s ) = G¯ u( s )−1Fr ( s ),

Kσ ( s ) = − G¯ u( s )−1Fσ ( s )

(8)

where Fr and Fs are filters to obtain proper controller transfer functions. When the pure time delay is zero and G¯ u( s )−1 is proper (i.e., Fr ¼Fs ¼1.0 in Eq. (8)), zero-error is achieved in Eq. (7). When G¯ ( s )−1 is a non-proper transfer function, Fr and Fs u

have to be filters to obtain proper controller transfer functions. Note that these controller transfer functions are not always in the relation of Kr ¼  Ks, especially in the case where the reference signal and nonlinear signal have different dimensions (e.g., displacement, velocity, or acceleration). Even when Gu(s)  1 can be directly adopted as the NSBC controllers, a pure time delay causes the generation of error, as follows:

e( s ) =

1 − e−τs ⎡⎣ r ( s ) − σ ( s )⎤⎦. ¯ 1 + Gu( s )e−τs Ke( s )

(9)

In this case, the error controller transfer function, Ke contributes to minimising the error. The design of Ke must be based on stability criteria (e.g., Nyquist stability criterion), because the pure time delay normally leads the destabilisation of the controlled systems. Please cite this article as: R. Enokida, & K. Kajiwara, Nonlinear signal-based control with an error feedback action for nonlinear substructuring control, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.09.023i

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Fig. 2. Nonlinear substructuring control.

2.2. Nonlinear substructuring control method 2.2.1. General configuration The NSBC method can be incorporated into substructuring tests when a signal corresponding to the nonlinear signal is given. Thus, NLSC shown in Fig. 2 is developed as an application of NSBC. A set of two substructures {Gp, Gn} in NLSC corresponds to the nonlinear system in NSBC, and the output of the numerical substructure, xn, in NLSC corresponds to the reference signal, r, in NSBC. Although the reference signal is explicitly given in NSBC, xn in NLSC is simultaneously calculated during the real-time substructured experiments. In general, transfer systems are normally well studied in advance of the experiments, and this configuration is based on G¯ TS ¼GTS. Moreover, even when the modelling of the transfer system does not perfectly match its real dynamics, NLSC has the function of minimising the error caused by the inaccurate modelling; this function derives from LSC, which was particularly developed for reducing the error [16]. In NLSC, two substructures with nonlinearities, or unknown parameters, are equivalently expressed by:

⎧ ¯ ⎪ Gp( s ) = Gp( s ) + ΔGp( s ) ⎨ ⎪ ⎩ Gn( s ) = G¯ n( s ) + ΔGn( s )

(10)

where {Gp, G¯ p , ΔGp} is the physical substructure, its linear model and modelling error, respectively, and {Gn, G¯ n , ΔGn} is the numerical substructure, its linear model and modelling error, respectively. In Fig. 2, output forces related to the physical substructure and its linear model can be expressed by:

⎧ f ( s ) = G ( s )G ( s )u( s ) p TS ⎪ p ⎨ ¯ ( s ) = G¯ ( s )G¯ ( s )u( s ) ⎪ f p TS ⎩ p

(11)

where f¯p is the force on the physical linear model and fp is the force on the physical substructure. Now, outputs from the numerical substructure and its linear model can be expressed by:

⎧ ⎪ x n( s ) = Gn( s ) −fp ( s ) − fd ( s ) ⎨ ⎪ x¯ ( s ) = G¯ ( s ) −f¯ ( s ) − f ( s ) n p d ⎩ n

( (

) )

(12)

where xn is the displacement of the numerical substructure, x¯ n is the displacement of the numerical linear model and fd is the force deriving from the disturbance and directly applying to the numerical substructure. The nonlinear signal in NLSC is expressed as:

σ( s ) = x n( s ) − x¯ n( s )

(13)

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Under displacement control, displacement of the physical substructure is equal to that of the transfer system. In addition, the outputs of the transfer system and its model become nearly identical, when the dynamics of the transfer system are well known. Based on these conditions, the outputs of the substructures can be rewritten as:

⎧ ¯ n( s ) + σ ( s ) = G¯ n( s )fd ( s ) − G¯ n( s )G¯ p( s )G¯ ts( s )e−τs u( s ) + σ ( s ) ⎪ x n( s ) = x ⎨ −τs ⎪ ¯ ⎩ x p( s ) = x¯ p( s ) = Gts( s )e u( s )

(14)

where xp is the displacement of the physical substructure and x¯ p is the displacement of the physical linear model. Now, the error between the outputs of the substructures becomes:

e( s ) = x n( s ) − x p( s ) = G¯d( s )d( s ) − G¯ u( s )e−τs u( s ) + σ where G¯ d( s ) = G¯ n( s )Hd( s ) , G¯ u( s ) = G¯ n( s )G¯ p( s )G¯ ts( s ) + G¯ ts( s ) , Hd( s ) =

(15) fd ( s ) d( s )

and d is the external disturbance. In Eq. (15), the

input signal is related to the three signals {e, d, s}, thus, the input signal in NLSC is proposed as follows:

u( s ) = Kd( s )d( s ) + Ke( s )e( s ) + Kσ ( s )σ ( s )

(16)

Substituting Eq. (16) into Eq. (15), the error dynamics can be expressed by:

e( s ) =

G¯d( s ) − G¯ u( s )e−τs Kd( s ) 1 − G¯ u( s )e−τs Kσ ( s ) d( s ) + σ ( s) −τs ¯ 1 + Gu( s )e Ke( s ) 1 + G¯ u( s )e−τs Ke( s )

(17)

Based on Eq. (17), the controller transfer functions, Kd and Ks, need to be determined so that the error becomes zero when τ ¼ 0. This is achieved by the following equation, particularly when Fd ¼Fs ¼1:

Kd( s ) =

G¯d( s ) 1 Fd( s ), Kσ ( s ) = Fσ ( s ) G¯ u( s ) G¯ u( s )

(18)

where Fd is the filter required to realise a proper transfer function, Kd. Similar to NSBC, the controller transfer function, Ke, needs to be designed by following the stability criterion for a pure time delay.

2.2.2. Stability analysis for pure time delay A pure time delay in transfer systems contributes to the destabilisation of the controlled systems. This effect is particularly significant in dynamical substructuring tests. To enhance the robustness of NLSC against the pure time delay, an error feedback action is fully utilised. The design of the controller transfer function, Ke, is discussed here. In order to maintain the stability of the system, Ke must be designed so that the transfer functions of s/d and e/d satisfy the stability conditions. However, this requires stability analysis of nonlinear systems that include a posteriori information. Therefore, in order to make the stability analysis simple, we assume that s is an additional bounded disturbance. When the controller transfer functions, Kd and Ks, are designed to be Eq. (18) with Fd ¼Fs ¼1, the error dynamics in Eq. (17) can be rewritten as:

( 1 − e−τs)G¯d( s) e(s ) = d +σ ( s ) 1 + G¯ u( s )e−τs Ke( s )

(19)

−1

where d+σ ( s ) = d( s ) + G¯ d( s ) σ ( s ). Now, the closed-loop characteristic equation (CLCE) of Eq. (19) is expressed by:

1 + e−τs G¯ u( s )Ke( s ) = 0

(20)

When Ke is a simple proportional gain, its stability significantly relies on the transfer function, G¯ u( s ) even under the condition of τ ¼ 0. Now, to enhance the stability, we design the transfer function Ke to be as follows:

Ke( s ) = G¯ u( s )−1Fe( s )

(21)

where Fe is the error feedback filter. Hence, the CLCE becomes:

1 + e−τs Fe( s ) = 1 + H ( s ) = 0

(22)

−τs

where H ( s ) = e Fe( s ). Here, we design the error feedback filter to be:

Fe( s ) =

s+b β s2

where β is the error feedback gain and b is a controller coefficient. In order to obtain the critical pure time delay, following requirement of the Nyquist stability criterion [11], is required to be solved:

(23)

τc, the

Please cite this article as: R. Enokida, & K. Kajiwara, Nonlinear signal-based control with an error feedback action for nonlinear substructuring control, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.09.023i

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Fig. 3. Dynamical substructuring test of a base-isolated structure with a rubber bearing.

⎧ H jω = 1 ⎪ ( c) ⎨ ⎪ ⎩ ∠H ( jωc ) = π

(24)

where ωc is the corresponding critical frequency. Then, the critical pure time delay and corresponding frequency are obtained as:

⎧ ⎛ω ⎞ 1 tan−1⎜ c ⎟ ⎪ τc = ⎝ b⎠ ω c ⎪ ⎨ 2 ⎪ 1 + 1 + 4( b / β ) ⎪ ωc = β ⎩ 2

(25)

The error feedback gain, β and controller coefficient, b can be designed by roots’ loci method applied to the CLCE of Eq. (22) with τ ¼0. In the roots’ loci plot for the CLCE, the dominant roots of the break point (i.e., s ¼  2b) correspond to the critically damped closed-system whose settling time is approximately 2/b. Since the settling time of the transfer system is approximately 4/a in many cases, the suitable design is found to be b¼4/a, obtained from the relation of 2/b¼8/a. In addition, β is supposed to be the gain relating to s ¼  2b, although a higher gain results in a smaller error. Eq. (25) is identical to that for the stability analysis of LSC [11], indicating its eligibility to both linear and nonlinear systems. Thus, the examination of the stability analysis for substructuring tests on nonlinear systems is a matter of interest in this study. This is numerically and experimentally examined in the following sections.

3. Numerical simulations of NLSC for rubber bearings Base-isolated structures with rubber bearings are a good example of a substructuring test, since the rubber bearings are clearly a critical part of the structure. The emulated model of the base-isolated structure is normally expressed by a singledegree-of-freedom system with a mass, dashpot and spring. A substructuring test for a base-isolated structure can be divided into numerical and physical substructures, as shown in Fig. 3. The numerical substructure comprises the mass of the superstructure and the physical substructure comprises a rubber bearing. Therefore, the numerical linear model basically becomes G¯ n ¼Gn, whereas the physical substructure becomes G¯ p ≠Gp.

Fig. 4. Numerical models: (a) tri-linear hysteresis, and (b) inaccurate modelling.

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In substructuring tests for rubber bearings, two models are considered in Fig. 4. These cases correspond to substructuring tests of a rubber bearing with modelling error and nonlinear materials inside it. A configuration of NLSC for a rubber bearing is presented in Section 3.1. Numerical simulations for two examples are presented in Sections 3.2 and 3.3.

3.1. Dynamical substructuring system for a nonlinear rubber bearing The equation of motion for the nonlinear emulated model is expressed by:

mx¨ ( t ) + fc ( ẋ, t ) + fk ( x, t ) = − md¨( t )

(26)

where t is time and {x, m, fc, fk} is the relative displacement, mass, force owing to damping and restoring force of the emulated model, respectively. The emulated model can be divided into numerical and physical substructures as follows:

⎧ ⎪ mnx¨ n( t ) = fn ( t ) = − fp ( t ) − fd ( t ) ⎨ ⎪ mpx¨ p( t ) + f ẋp , t + f x p , t = f ( t ) pc pk p ⎩

(

(

)

)

(

)

(27)

where fd ( t ) = md¨( t ), m = mn + mp . In addition, mn is the mass of the numerical substructure and {mp, fpc, fpk} is the mass, force owing to damping and restoring force of the physical substructure, respectively. Linear models of the substructures are based on the linearization of the stiffness and damping terms, as follows:

⎧ ¯ nx¯¨ n( t ) = f¯n ( t ) = − f¯p ( t ) − fd ( t ) ⎪m ⎨ ¯ ¨ ¯ ¨ ̇ ⎪m ⎩ ¯ px¯ p( t ) + c¯ px¯ p( t ) + k px¯ p( t ) = fp ( t )

(

)

(28)

¯ n is the mass of the numerical linear model, and {m ¯ p ,c¯p ,k¯ p } is the mass, damping coefficient and stiffness coefficient of where m ¯ n = mn . the physical linear model, respectively. Here, the masses of the numerical substructure and its linear model can be m The nonlinear signal in NLSC can be obtained from the outputs of the numerical substructure and its linear model, as follows:

1 σ¨ ( t ) = x¨ n( t ) − x¯¨ n( t ) = − f ( t ) − f¯p ( t ) ¯n p m

(

)

(29)

Based on Eq. (29) and the second equation in Eq. (28), the force in the physical substructure can be expressed by:

(

)

¯ nσ¨ ( t ) = m ¯ px¯¨ p( t ) + c¯ px¯ ̇p( t ) + k¯ px¯ p( t ) − m ¯ nσ¨ ( t ) fp ( t ) =f¯p ( t ) − m

(30)

Substituting Eq. (30) into the first equation of Eq. (27), it can be rewritten as follows:

¯ px¯¨ p( t ) + c¯ px¯ ̇p( t ) + k¯ px¯ p( t ) − m ¯ nσ ( t )⎤⎦ − md¨( t ) mnx¨ n( t ) = − ⎡⎣ m

(31)

The dynamics of the transfer system, including inner-loop (proportional and integral) controllers, is well described by the first-order transfer function and a pure time delay term [11,16], as follows:

G TS ( s ) =

a −τs e s+a

(32)

where a is the transfer system coefficient. Note that the displacement of the physical substructure in this case is equal to that of the transfer system. Based on Eqs. (31) and (32), the outputs of the two substructures in the Laplace domain are expressed by:

⎧ ⎛ ¯ 2 ¯ ¯ ⎞ ⎛ ¯ ¯ ⎞ ⎪ x ( s ) = ⎜ − mn + mp ⎟d( s ) − ⎜ mps + cps + k p ⎟⎜⎛ a ⎟⎞e−τs u( s ) + σ ( s ) n ⎜ ⎟⎝ s + a ⎠ 2 ⎪ ¯n ⎠ m ¯ ns ⎝ m ⎝ ⎠ ⎨ ⎪ ⎛ a ⎞ −τs ⎟e ⎪ x p( s ) = ⎜ u( s ) ⎝ s + a⎠ ⎩

(33)

Then, the error signal can be obtained from Eq. (33), as follows:

e( s ) = G¯d( s )d( s ) − G¯ u( s )e−τs u( s ) + σ ( s )

(34)

⎛ m¯ ps2 + c¯ps + k¯ p ⎞ a ¯n + m ¯p m where G¯ u( s ) = ⎜ ⎟ s + a , G¯ d( s ) = − m¯ . 2 ¯ n m s ⎝ ⎠ n In the substructuring tests on a rubber bearing with nonlinearity or unknown parameters, the input signal generated by NLSC is determined with the control law in Eq. (35) and the corresponding transfer functions in Eq. (36).

( )

Please cite this article as: R. Enokida, & K. Kajiwara, Nonlinear signal-based control with an error feedback action for nonlinear substructuring control, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.09.023i

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u( s ) = Kd¨( s )d¨( s ) + Ke( s )e( s ) + Kσ¨ ( s )σ¨( s )

(35)

⎧ ⎪ K ¨( s ) = ⎪ d ⎪ ⎪ ⎪ ⎨ Kσ¨ ( s ) = ⎪ ⎪ ⎪ ⎪ Ke( s ) = ⎪ ⎩

(36)

(

)

¯p + m ¯ n ( s + a )/a m G¯d( s ) ⎛ 1 ⎞ ⎜ ⎟=− G¯ u( s ) ⎝ s 2 ⎠ ¯p + m ¯ n s 2 + c¯ ps + k¯ p m

(

)

¯ n( s + a )/ a m 1 ⎛ 1⎞ ⎜ ⎟= G¯ u( s ) ⎝ s 2 ⎠ ¯ ¯ n s 2 + c¯ ps + k¯ p mp + m

(

)

¯ n /a( s + a)( s + b) m

1 β Ke( s ) = G¯ u( s ) ¯p + m ¯ n s 2 + c¯ ps + k¯ p m

(

)

Note that the use of σ¨ and d¨ , instead of s and d, enables one to easily gain proper controller transfer functions of Kd¨ and Kσ¨ . As for LSC, the input signal is determined by Eq. (37) and the corresponding transfer functions in Eq. (38). u( s ) = Kd¨( s )d¨( s ) + Ke( s )e( s ) (37)

⎧ mp + mn ( s + a )/a ⎪ K ¨( s ) = − d ⎪ mp + mn s 2 + cps + k p ⎪ ⎨ ⎪ mn /a( s + a)( s + b) β ⎪ Ke( s ) = 2 ⎪ m p + mn s + cps + k p ⎩

(

)

(

(

)

)

(38)

As Eq. (38) shows, the controllers of LSC rely on the accurate properties of the physical substructure. However, the controllers of NLSC in Eq. (36) are independent from the properties of the physical substructure and rely on the parameters of its linear model, which users can flexibly assign; i.e., NLSC does not require accurate knowledge of the dynamic properties of the physical substructure to be tested.

3.2. Example 1: Nonlinear hysteresis The purpose of the substructuring test is to examine the properties of the physical substructure or its nonlinearity. Thus, here, NLSC is numerically examined via a substructuring test with a nonlinear hysteresis. Based on a test rig built for the substructured test on a rubber bearing, [11], we set the basic properties of the physical substructure and transfer system as: mp ¼115 kg, cp ¼354.6 Ns/m, kp ¼158.4 kN/m, a ¼75.0 s  1 and τ ¼6.0ms. The numerical substructure is set to be mn ¼20mp , resulting in natural frequency, ω ¼1.29  2π rad/s, and damping, ζ ¼0.0091, in the emulated model within the linear stage. As nonlinearity, a tri-linear hysteresis in Fig. 4(a), commonly used in structural engineering, is embedded into the physical substructure. The parameters of the tri-linear hysteresis are chosen such that the stiffness decreases to 1/2 of the initial value at the deformation δ1(¼100 mm) and to 1/10 of the initial value at the deformation δ2(¼ 200 mm). Japan Meteorological Agency Kobe (JMA Kobe [21]) and Takatori ground motions in Fig. 5(a), which were recorded during the 1995 Hyogo-ken Nanbu Earthquake, are adopted as input motions. According to the acceleration response spectra in Fig. 5(b), Takatori peaks at 1.2 s, while JMA Kobe peaks at 0.4 s. These ground motions enable one to examine NLSC under excitations having different frequency components. Emulated responses obtained from this emulated system subjected to the ground motions are shown in Fig. 6, together with the nonlinear hysteresis obtained. As Fig. 6 shows, the maximum displacement of the emulated response reaches 350 mm under JMA Kobe motion and 384 mm under Takatori motion. Substituting the above-mentioned parameters {mp, cp, γkkp, a¼75, b¼18.75} into Eq. (36) as the linear model parameters, the NLSC controllers are obtained as:

Fig. 5. Ground motions: (a) acceleration time history, and (b) acceleration response spectra.

Please cite this article as: R. Enokida, & K. Kajiwara, Nonlinear signal-based control with an error feedback action for nonlinear substructuring control, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.09.023i

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Fig. 6. Example 1: Response and hysteresis of the emulated system: (a) JMA Kobe, and (b) Takatori.

Fig. 7. Example 1: Numerical results of LSC for a model with hysteresis: (a) JMA Kobe, β¼ 75, (b) JMA Kobe, β ¼150, (c) Takatori, β ¼75, and (d) Takatori, β ¼150.

Please cite this article as: R. Enokida, & K. Kajiwara, Nonlinear signal-based control with an error feedback action for nonlinear substructuring control, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.09.023i

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Fig. 8. Example 1: Numerical results of NLSC for a model with hysteresis: (a) JMA Kobe, β ¼0, (b) JMA Kobe, β ¼75, (c) JMA Kobe, β¼ 150, (d) Takatori, β ¼ 0, (e) Takatori, β ¼75, and (f) Takatori, β ¼150.

Table 1 Maximum errors of LSC and NLSC in example 1. Input motion

Error (mm)

xn  xp xe  xp xe  xn xn  xp xe  xp xe  xn

JMA Kobe

Takatori

a

LSC

NLSC

β¼ 75

β ¼150

β ¼0

β¼ 75

β ¼150

8.8 12.7 6.5 9.2 10.6 3.7

4.5 6.7 3.5 4.7 5.5 2.0

NaNa

0.5 1.1 0.8 0.4 0.7 0.7

0.2 0.6 0.6 0.2 0.6 0.6

NaNa

Not a number owing to instability.

⎧ ⎪ Kd¨( s ) = ⎪ ⎪ ⎪ ⎨ Kσ¨ ( s ) = ⎪ ⎪ ⎪ ⎪ Ke( s ) = ⎩

−

2415( s + 75)/75 2415s 2 + 354.6s + γk158400 2300( s + 75)/75

2415s 2 + 354.6s + γk158400 2300( s + 18.75)( s + 75)/75 2415s 2 + 354.6s + γk158400

β (39)

where γk is the modelling error of stiffness, kp. Note that Kd¨ and Kσ¨ are the proper transfer functions related to the ground motion acceleration, d¨ , and the nonlinear signal, σ¨ . Kd¨ and Ke in Eq. (39) are taken as the LSC controller for the comparison with the NLSC controller. To focus on the nonlinearity owing to the hysteresis in Example 1, the modelling error parameter in the LSC and NLSC controllers is fixed to be γk ¼ 1.0. According to the roots’ loci of Eq. (22) with the parameters {b¼18.75, τ ¼0.0}, the suitable controller gain is found to be β ¼ 75. In order to study the effect of β, numerical simulations are conducted with β ¼75 and 150 in LSC and NLSC, respectively; NLSC with β ¼0 is additionally considered for the study of the nonlinear signal feedback action. Numerical results are shown in Figs. 7 and 8, and the maximum errors are summarised in Table 1. As shown in Table 1, LSC with β ¼75 results in an error of 12.7 mm under the JMA Kobe motion and 10.6 mm under the Takatori motion, which are 3.6 percent and 2.8 percent of the maximum values of each emulated response. When the controller gain is increased to β ¼150, the error becomes nearly half of the error in the case of β ¼75, as shown in Fig. 7. Please cite this article as: R. Enokida, & K. Kajiwara, Nonlinear signal-based control with an error feedback action for nonlinear substructuring control, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.09.023i

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Fig. 9. Example 2: Response of the emulated system.

The result of NLSC for β ¼0 (i.e. Ke ¼0) becomes unstable for both ground motions in Fig. 8(a) and (d), while the results for β ¼ 75 and 150 (i.e. Ke≠0) become stable and show a small error in Fig. 8(b), (c), (e) and (f). According to Table 1, NLSC with the error feedback action achieves accurate control with the maximum error of 1.1 mm for the JMA Kobe motion and 0.7 mm for the Takatori Motion. These errors are 0.3 percent and 0.2 percent of the maximum values of each emulated response and even smaller than 1/10 of the errors obtained by LSC. Now, it is clear that the nonlinear signal feedback in NLSC is effective in minimising the error and the error feedback is essential for maintaining stability regardless of the different frequency components in the input motions. Under the condition of τ ¼6 ms and b¼18.75, the maximum gain for the stability becom es βmax ¼248 according to Eq. (25). In fact, additional simulations conducted with the two ground motions and NLSC, setting β as a variable parameter maintained stability up to β ¼ βmax. Now, the analysis is found to be accurate for predicting instability of substructured tests with nonlinear hysteresis. 3.3. Example 2: Inaccurate modelling Substructuring tests, in general, have to be conducted with an assumed model that, to one degree or another, contains a modelling error, because an accurate model is rarely obtained in advance of the experiments. Here, NLSC is examined for large modelling errors, with the same system and conditions considered in example 1, apart from the nonlinear hysteresis.

Fig. 10. Example 2: Numerical results of LSC based on inaccurate modelling: (a) γk ¼1/5, β ¼75, (b) γk ¼ 1/5, β ¼ 150, (c) γk ¼ 5, β ¼ 75, and (d) γk ¼ 5, β ¼ 150.

Please cite this article as: R. Enokida, & K. Kajiwara, Nonlinear signal-based control with an error feedback action for nonlinear substructuring control, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.09.023i

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Fig. 11. Example 2: Numerical results of NLSC based on inaccurate modelling: (a) γk ¼1/5, β ¼ 0, (b) γk ¼1/5, β¼ 75, (c) γk ¼ 1/5, β¼ 150, (d) γk ¼ 5, β ¼0, (e) γk ¼5, β¼ 75, and (f) γk ¼5, β¼ 150.

Table 2 Maximum errors of LSC and NLSC in example 2. Modelling error

γk ¼ 1/5

γk ¼ 5

a

Error (mm)

xn  xp xe  xp xe  xn xn  xp xe  xp xe  xn

LSC

NLSC

β ¼ 75

β¼ 150

β¼0

β ¼75

β ¼150

1.4 14.9 14.7 5.8 35.4 38.8

0.7 6.8 6.7 3.0 23.2 24.0

NaNa

0.1 0.8 0.8 0.3 3.2 3.2

0.04 0.5 0.4 0.2 1.5 1.5

5.4 26.1 27.4

Not a number owing to instability.

For the inaccurate model, two cases are considered: the stiffness in the physical substructure is assumed to be 1/5 and 5 times the real value. These two cases correspond to the lower and upper bounds of inaccurate modelling, respectively. In order to focus on the modelling error, only the JMA Kobe motion is selected as an input motion of example 2. In addition, the numerical simulations in this example are to be compared with the experimental study shown in Section 4. Thus, for comparison, the amplitude of the input motion (i.e. JMA Kobe) is scaled down to 10 percent; this amplitude is determined by the maximum stroke of the actuator in the test rig. The response of the emulated system is shown in Fig. 9, resulting in the maximum value of 35.6 mm. Numerical simulations are conducted with the LSC and NLSC controllers shown in Eq. (39) for the two cases of γk ¼1/5 and 5. In the simulations, the controller gain, β is again set to be β ¼75 and 150, and the case of β ¼ 0 is additionally considered for the NLSC controller. The numerical results are shown in Figs. 10 and 11, with maximum errors for each case summarised in Table 2. In the numerical results of LSC for γk ¼1/5, shown in Fig. 10(a), (b) and Table 2, the errors become 14.9 mm and 6.8 mm, which correspond to 41.9 percent and 19.1 percent of the maximum emulated response (i.e., 35.6 mm), respectively. As shown in Fig. 10(c) and (d), the results of LSC for γk ¼5 show an even larger error than the results for γk ¼1/5. Now, LSC is found to be clearly problematic to these inaccurate models. The simulation of NLSC for {γk ¼1/5, β ¼0} shows an unstable result in Fig. 11(a). However, the simulations for β ¼75 and 150 become stable in Fig. 11(b) and (c), resulting in errors of 0.8 mm and 0.5 mm in Table 2, respectively; these errors are ten times smaller than the results of LSC, resulting in 2.2 percent and 1.4 percent of the maximum emulated response. Although Please cite this article as: R. Enokida, & K. Kajiwara, Nonlinear signal-based control with an error feedback action for nonlinear substructuring control, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.09.023i

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Fig. 12. Test rig including the computational signal flow of the substructured experiment.

the simulation of NLSC for {γk ¼5, β ¼0} shows a stable result in Fig. 11(d), its error reaches 77.0 percent of the maximum emulated response. When the controller gain is increased to β ¼75 and 150, as shown in Fig. 11(e) and (f), the error feedback controller clearly contributes smaller errors; even the largest error (3.2 mm) remains within 9.0 percent of the maximum emulated response. According to the above results, the feedback actions of nonlinear and error signals in NLSC are fundamentally important for minimising the error as well as maintaining stability. In terms of error minimisation, NLSC is clearly more effective than LSC. In additional simulations conducted with NLSC, setting β as a variable, stability has been maintained up until βmax ¼248, which is obtained as the maximum value for stability in Eq. (25). Here, the stability analysis is found to accurately predict the instability condition of the systems with larger modelling error. It is noticeable that, in Table 2, the smaller the error of xn xp, the outputs of xn and xp become closer to the emulated response, xe. This indicates that reliability of substructured tests can be evaluated based upon the value of xn  xp; this is very helpful for the implementation of substructured experiments, because emulated responses in many cases are unknown.

4. Experiment Substructuring experiments were conducted with the test rig shown in Fig. 12 in the ACTLab at the University of Bristol. The specification of the test rig is described in Section 4.1, and substructured experiments using the rig are shown in Section 4.2.

Fig. 13. Experimental results of LSC with inaccurate modelling: (a) γk ¼ 1/5, β ¼75, (b) γk ¼ 1/5, β ¼150, (c) γk ¼ 5, β ¼75, and (d) γk ¼5, β ¼ 150.

Please cite this article as: R. Enokida, & K. Kajiwara, Nonlinear signal-based control with an error feedback action for nonlinear substructuring control, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.09.023i

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Fig. 14. Experimental results of NLSC based on inaccurate modelling: (a) γk ¼1/5, β ¼ 0, (b) γk ¼ 1/5, β ¼75, (c) γk ¼1/5, β¼ 150, (d) γk ¼ 5, β ¼ 0, (e) γk ¼ 5, β ¼ 75, and (f) γk ¼5, β ¼ 150.

Table 3 Maximum errors of LSC and NLSC in substructured experiments. Modelling error

γk ¼ 1/5 γk ¼ 5 a

Error (mm)

xn  xp xn  xp

LSC

NLSC

β ¼ 75

β¼ 150

β¼0

β ¼75

β ¼150

1.3 6.5

0.7 3.4

NaNa

0.3 0.4

0.3 0.2

Not a number owing to instability.

4.1. Test rig The test rig in Fig. 12 [11] was constructed at the University of Bristol for the substructured experiments on a rubber bearing. This rig consisted of a servohydraulic actuator, a steel plate and a natural rubber bearing. The rubber bearing had a diameter of 200.0 mm and height of 125.0 mm, and the mass of the steel plate was directly measured to be mp ¼115 kg. The steel plate rigidly connected the actuator and rubber bearing, so that the measured data via an LVDT displacement transducer and load cell attached to the actuator were equal to the displacement and force applied in the rubber bearing. The transfer system in this test rig comprised the actuator and a proprietary inner-loop discrete-time controller. DSS was implemented as an outer-loop configuration using dSPACE 1104 hardware, operating with a sampling interval of 1.0 ms. Here, the properties of the test rig obtained via system identification [11] are again summarised: cp ¼354.6 Ns/m, kp ¼ 158.4 kN/m, a¼75.0 s  1 and τ ¼6.0 ms. 4.2. Substructured experiments In substructuring experiments, the mass of the numerical substructure was set to mn ¼20mp, following the numerical studies in Section 3. The JMA Kobe NS ground motion was adopted for the experiments, but its amplitude was scaled down to 10.0 percent of the original value because of the 7120 mm stroke limitation in the experimental apparatus. Substructure experiments were conducted with the LSC and NLSC controllers shown in Eq. (39) for the two cases of the modelling error parameters (γk ¼1/5 and 5). In the experiments, the controller gain was set to β ¼75 and 150 in both NLSC and LSC controllers, and the case of β ¼0 was particularly examined for NLSC. Experimental results are shown in Figs. 13 and 14, and the maximum errors are summarised in Table 3. Note that the emulated response is unknown and the errors related to xe are not listed in Table 3. The experimental results of LSC in Table 3 showed a similar tendency to those observed in the numerical simulations of Please cite this article as: R. Enokida, & K. Kajiwara, Nonlinear signal-based control with an error feedback action for nonlinear substructuring control, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.09.023i

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Fig. 15. Experimental results of NLSC for {γk ¼5, β ¼ 150}: (a) hysteresis and (b) comparison of xp and xap.

example 2. As shown in Fig. 13, the maximum errors obtained for γk ¼ 5 were larger than those for γk ¼1/5 and a larger gain β clearly resulted in smaller errors. The experimental results of NLSC for β ¼0 (i.e. Ke ¼0) became unstable, as shown in Fig. 14(a) and (d). However, the controller gains of β ¼75 and 150 (i.e. Ke≠0) produced stable results, as shown in Fig. 14(b), (c), (e) and (f); this clearly indicates the importance of the error feedback action for stability. The errors in the experiment of NLSC with β≠0 were clearly smaller than the results of LSC, and the efficiency of NLSC was experimentally verified. Additional experiments conducted for NLSC with {γk ¼1/5, β ¼175} and {γk ¼5, β ¼175} produced unstable results. This instability happened with a smaller controller gain than βmax, as normally happens in real control practices with high gains. Thus, the controller gain in the NLSC controller should be determined with enough relative stability margin, following the stability analysis. 4.3. Examination of substructured experiments The experiment based on NLSC with {γk ¼5, β ¼150} shows the smallest error in Table 3, thus, the reliability of the experiment is presumably the highest among the experiments conducted. However, in the implementation of the substructured experiments, the correct response to be compared is unknown. Now, we examine the reliability of the experimental result, based on an approximation of the emulated model from the experimental result. The approximated model is built to be me ¼2415 kg, 354.6 Ns/m, and 136.6 kN/m; the damping coefficient is the value obtained at system identification, and the stiffness is obtained by curve fitting in a hysteresis of the rubber bearing in the experiment of NLSC with {γk ¼5, β ¼150}, as shown in Fig. 15(a). This stiffness is 86.2 percent of the obtained value at system identification, indicating a slight existence of nonlinearity in the rubber bearing. Now the stiffness assumed for the controllers is found to be 11.6 times larger than the stiffness obtained by the substructured experiment, as shown in Fig. 15(a). A numerical simulation based on the approximated model is conducted, using the JMA Kobe ground motion with an amplitude of 10 percent. Fig. 15(b) shows that the amplitudes of experimental and numerical results are reasonably close. The difference may primarily derive from the nonlinearity, which slightly existed in the rubber bearing. In order to quantitatively evaluate the similarity between the experimental result and the response of the approximated model, we used the following equation:

⎛ ∑ S( f ) − S( f ) ⎜ EX AP ⎜1 + 2 ⎜ ∑ S( f ) AP ⎝

(

2 −1

) ⎞⎟⎟ ⎟ ⎠

× 100% (40)

where S(f)EX and S(f)AP are the Fourier amplitude spectra of the output, xp in the experiment and the output, xap of the approximated model, respectively. Based on Eq. (40), NLSC with {γk ¼ 5, β ¼150} results in the highest similarity of 72.5 percent and all other cases of NLSC become less than 70.0 percent; as for LSC, the case with {γk ¼5, β ¼150}, which shows the smallest error in Table 3 regarding the experiments with LSC, becomes 64.0 percent. Now, the experiment of NLSC with {γk ¼5, β ¼150} is found to clearly have the highest reliability.

5. Conclusions This study proposed NLSC as an application of NSBC with an error feedback action into DSS. NLSC was numerically and experimentally examined via a substructuring test on a rubber bearing. The conclusions obtained in this study are summarised below. Please cite this article as: R. Enokida, & K. Kajiwara, Nonlinear signal-based control with an error feedback action for nonlinear substructuring control, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.09.023i

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 As an application of NSBC into physical experiments, NLSC was developed for a general form of LSC. NLSC is composed of

 







three controllers {Kd, Ke, Ks} related to the reference, error and nonlinear signals, respectively, while LSC does not have the controller Ks. In NLSC, the controller Ks minimises the error caused by the nonlinearity or unknown parameters in the controlled system, while the controller Ke contributes to maintaining stability and further reduces the error caused by other factors. The stability of NLSC for pure time delays was analysed with the Nyquist criterion. When the error feedback controller Ke was designed to be Eq. (23), the critical pure time delay in NLSC was found to be identical to that in LSC [11]. Thus, the stability analysis was numerically and experimentally examined via substructuring tests on nonlinear systems. In example 1, a tri-linear hysteresis, commonly used for structural buildings, was embedded in the physical substructure. LSC and NLSC controllers were designed on the basis of the initial stiffness of the tri-linear model. Based on the conditions, substructured tests with a pure time delay of 6 ms and two ground motions (JMA Kobe and Takatori motions) are numerically simulated. The results obtained for NLSC clearly showed much smaller errors than those of LSC. In example 2, the stiffness was assumed to be 1/5 and 5 times its real value in order to introduce deliberately large modelling errors in both the NLSC and LSC controllers. In the numerical simulations for the substructured test with a pure time delay of 6 ms and JMA Kobe motion, NLSC succeeded in generating accurate results even with inaccurate modelling, while LSC was found to be inadequate for such poor modelling. A series of substructuring experiments on a rubber bearing was conducted with exactly the same conditions as those in example 2. As the numerical simulation suggested, NLSC succeeded in gaining accurate results even under the testing conditions, while LSC failed to do so. Via the experiments, the efficiency and practicality of NLSC were clearly demonstrated. Through the numerical and experimental examinations, the nonlinear signal feedback and error feedback actions employed in NLSC were found to be fundamentally important for minimisation of the error and maintaining stability in substructured systems with nonlinearities or unknown parameters. The stability analysis was found to be useful for the design of the error feedback controller, particularly for the determination of the controller gain.

Acknowledgements This work has been supported by Professor David Stoten at the University of Bristol. The authors gratefully acknowledge the support of the Japanese Society for the Promotion of Science (Postdoctoral Fellowships for Research Abroad, No.: 20140023).

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Please cite this article as: R. Enokida, & K. Kajiwara, Nonlinear signal-based control with an error feedback action for nonlinear substructuring control, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.09.023i