On the Feasibility of Dynamic Substructuring for Hybrid Testing of Vibrating Structures

Hu, An; Paoletti, Paolo

doi:10.1115/1.4062256

Abstract

Dynamically substructured systems (DSS) are a typical technique to achieve real-time numerical simulations combined with physically tested components. However, a rigorous feasibility analysis before the implementation is missing. This paper is aimed to fill this gap by establishing rigorous conditions for when DSS is suitable for dynamic testing. The proposed method is based on novel symbolic recursive formulations for the transfer functions describing a generic lumped parameter vibrating structure, enabling the analysis of structural and other properties without requiring the computation of explicit symbolic expressions for the transfer functions involved, representing a significant breakthrough as it allows to perform feasibility analysis in analytical form, rather than solely relying on numerical approaches. The series of analytical conclusions presented in this paper, and future ones unlocked by the proposed approach, will significantly enrich the research in the community of DSS and structural vibrations. In particular, the proposed approach allows performing analysis of causality, controllability, and observability using much reduced knowledge of the structure, thus significantly simplifying such analysis. Analytical conclusions on stability can also be made with the help of novel recursive form, removing the need of repeatedly calculating the roots of characteristic equations, a task that can be performed only via numerical approaches and for which analytical results are not available. The proposed methodology can be applied to a whole class of vibration problems and is not linked to any specific structure, going beyond the specific examples available in the literature.

1 Introduction

Experimental testing of large, nonlinear or complex structures can be very expensive and time consuming. To alleviate these issues, several techniques have been proposed in the literature to replace part of the structure by a numerical simulator and then design a control system so that the combination of numerical and physical subsystems exhibits the same dynamical response of the original structure.

Dynamically substructured systems (DSS) are part of this family of techniques, alongside similar approaches also known as hybrid testing, hybrid simulation, or real-time hybrid simulation [1–3]. In this paper, we focus on the DSS framework shown in Fig. 1, where the original structure is decomposed into two main parts, the physical part and the numerical part. The numerical part simulates a subsystem of the original structure. On the other hand, the physical part is composed of a physical structure and of an actuator which is used to simulate the presence of the subsystem that has been replaced by the numerical part. The challenge is then to control the actuator at the interface so that the closed loop behavior of the DSS emulates the dynamic behavior of the complete original structure [4,5].

Fig. 1

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DSS schematic: the original structure is split into a numerical part and a physical part. The synchronization signal S_s(t) is measured on the physical part and transmitted as input to the numerically simulated subsystem. The goal of a DSS controller is to control the actuator at the interface with the goal of minimizing the error e(t) between the numerical response S_n and the physical signal S_p so that the overall behavior of the DSS is the same of the original structure.

Two main approaches have been proposed in the literature to tackle this challenge: force control and position control. In the force control setting, the displacement of the physical part at the interface (indicated by synchronized signalS_s in Fig. 1) is measured and transmitted as an input to the numerical part. The simulator used in the numerical part then calculates the force at the interface (numerical signalS_n) which is compared with the force generated by the actuator situated at the corresponding interface in the physical part (physical signalS_p). A feedback controller is then designed to minimize the synchronization errore between the simulated force and the actuator force, so that the interface becomes transparent and the DSS behaves as the original structure. In the displacement control setting, the role of force and displacement is swapped. The interested reader is referred to the review paper by Klerk et al. [5] for an extensive review of DSS approaches.

The presence of delays, disturbances, and model uncertainties prevents exact cancelation of the error between numerical and physical signals, therefore a feedback controller is needed to control the actuator in order to minimize the synchronization error. Typical linear feedback controller design techniques used for such purpose include linear substructuring control (LSC) and minimal control with error feedback [6]. Robustness of these techniques has been thoroughly studied, see for example the results by Gawthrop et al. [7] and Tu et al. [8]. Traditional control strategies—such as H_∞ [9], control sliding mode [10], and real-time model updating [11]—have also been applied to nonlinear or uncertain DSS problems. Similarly, delay compensation techniques have been proposed to minimize the detrimental effects of delays introduced by the control logic [12–14], as well as the efforts trying to predict and compensate delays with generalized methodologies [15,16]. Nonlinear substructuring control was also proposed as an extension to traditional LSC to deal with nonlinear dynamic problems [17]. Issues related to application of hybrid testing to systems exhibiting chaotic behaviors, such as a nonlinear pendulum model, have also been analyzed in Ref. [18].

These control strategies have been successfully used to test dynamical systems across different domains. For example, in the railway industry, Stoten and colleagues tested a pantograph using a DSS approach where only the pantograph was physical, whereas the electric line was simulated [19]. Similarly, Facchinetti and Bruni used DSS to study the pantograph-catenary interaction [20], whereas Hong et al. used DSS to analyze the characterization of rail tracks [21]. Allen and Mayes exploited DSS to aid the design of NASA rocket launcher [22] and Mayes and Arviso developed the applications of DSS to transmission structures, which is helpful for experiments on rotational structures such as wind turbines [23]. In addition, the DSS approach can be applied to vehicle development as well; for example, van der Seijs and Rixen proposed a DSS framework for a variety of experiments on vehicle structures [24]. Similarly, models of motorcycles have been tested using DSS techniques [4,25]. Dynamic substructuring is also widely used in civil and structural engineering to test, for example, nonlinear components [26–28], seismic responses [29–31], piping systems [32], and soil–structure interaction [33]. A review of dynamic substructuring techniques can be found, for example, in the review paper by Klerk et al. [5].

Although DSS and the relative common techniques have been extensively studied for different applications, one of the common issues which should not be neglected and could seriously jeopardize future studies is the excessive focus on specific models. Indeed, most of the available literature has been focused on specific benchmark systems or on specific examples. Systems with only two or three degrees-of-freedom were typically selected for analyses, thus posing the fundamental questions of, for example, whether the conclusions can be expanded to systems with different degrees-of-freedom and whether and how the system parameters affect the conclusions. Without clearly addressing these issues, most of the conclusions reported in the literature can only be regarded as the summaries for those specific models. The path to generalization to other models and application is therefore unclear, a factor that can severely limit the applications and future studies of DSS. Another gap in the current literature is the lack of a comprehensive feasibility analysis, with only limited exceptions such as the recent paper by Terkovics et al. [34] where the effect of the interface location on DSS performance was studied, as well as the paper by Gawthrop et al. [35] where the causality analysis for DSS was studied. However, both studies still focused on specific models only, meaning a comprehensive analysis of generic structural properties affecting feasibility of DSS design is lacking in the available literature. Even before starting the control design process for a given DSS decomposition, one needs to know if a controller can be designed in the first place, and this feasibility analysis is missing for generic structures in the literature.

In this paper, an approach to fill this gap is proposed. Two main contributions are made in this paper. The first one is the derivation of a novel recursive formulation to derive transfer functions for lumped parameter systems composed of a chain of mass-spring-damper systems, which allows a greatly simplified analysis of structural properties of lumped parameter DSS, irrespective of the number of degrees-of-freedom and specific values of system parameters. The benefits of the proposed formulation include, but are not limited to, analytical conclusions on stability analysis for similar structures with different degrees-of-freedom [36], which are impossible to be drawn using numerical methodologies. This is a significant advancement compared to the traditional approach solely relying on numerical analyses in DSS and vibration communities, with more analytical conclusions being possible to be found in future to further enrich the achievements. Although a mass-spring-damper chain is selected for demonstration and the results presented might only be suitable for this class of systems, this class already describes a wide range of vibration models [25,34] considered by the vibration community. This is the first step towards a more generalized analysis framework of DSS properties for generic vibrating structures. Moreover, given the structural similarities of equations of motion for lumped parameter systems, the proposed formulation can be expanded to different types of structures, which can significantly broaden the applications of DSS. The methodology proposed in this paper also enables several analytical results on feasibility of DSS decomposition for generic vibrating structures, the second major contribution of this paper. For example, it allows assessing whether a given DSS decomposition and control strategy can be designed for a generic lumped parameter vibration system with n degrees-of-freedom. This, in turns, allows the proposal of general guidelines regarding the type of DSS decomposition and actuator/sensor arrangements to be used in any particular real world application. The focus here is on structural properties and the scope of this paper does not include implementation details—such as the delays imposed by the electronics used to implement the controller and the potential effects of the actuator dynamics—that may hinder the main message of this paper. Extensions to cover these aspects can be easily obtained by, for example, including the terms related to delays and actuator transfer functions in the proposed methodology, as briefly mentioned throughout the paper.

The problem analyzed in this manuscript is formally stated in Sec. 2 for a widely used class of benchmark systems, together with a brief review of the main concepts of structural stability, controllability, causality, and observability that will be used for the analysis presented in this paper. A causality study to obtain feasible DSS configurations is reported in Sec. 4 and provides constraints on the choice of using either directly measured or indirectly estimated synchronized signals for control purposes. The identified causal configurations are then further analyzed in Sec. 5 to assess their structural properties of observability and controllability, and hence assess if a synchronizing controller can be designed. The proposed methodology is applicable to any DSS design for vibrating structures, and hence is not linked to any physical or experimental realization of such class of structures. However, for the sake of completeness, numerical examples are briefly discussed in Sec. 6 to show how the outcomes of the analyses in Secs. 4 and 5 may inform control design for real applications. These include a complex frame structure which does not fall within the class of benchmark systems used to motivate most of the analysis, but for which structural analogies with the class of system used through the paper can be used to successfully infer DSS feasibility. Finally, some concluding remarks and suggestions for future research directions are reported in Sec. 7.

2 Problem Statement and Methodology

This paper initially focuses on a generic class of lumped parameter vibration models, which can be schematically represented as chain of n spring-damper-mass systems, as shown in Fig. 2. As mentioned earlier, although the results presented in this paper strictly apply only to this class of systems, the final results may still be exploited by considering structural similarities with the equations of motions of more generic vibratory systems. An example of such extension is presented in Sec. 6.2. In our framework, a disturbance d(t) is applied as displacement of the support at one end of the structure, whereas an external force F(t) may be applied to the other end (such force will represent, for example, the action of the actuator used in DSS substructuring in the following sections of the paper).

Fig. 2

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Schematic representation of the benchmark system considered initially in this paper

2.1 Original Dynamics.

The equations of motion describing the behavior of the structure in Fig. 2 (sometime referred to as emulated system in the DSS literature [4]) can be obtained by imposing force balance at each mass and read

{\begin{matrix} m_{n} {\ddot{y}}_{n} = - k_{n} (y_{n} - y_{n - 1}) - c_{n} ({\dot{y}}_{n} - {\dot{y}}_{n - 1}) - F \\ ⋮ \\ m_{i} {\ddot{y}}_{i} = - k_{i} (y_{i} - y_{i - 1}) - c_{i} ({\dot{y}}_{i} - {\dot{y}}_{i - 1}) + k_{i + 1} (y_{i + 1} - y_{i}) + c_{i + 1} ({\dot{y}}_{i + 1} - {\dot{y}}_{i}) \\ ⋮ \\ m_{1} {\ddot{y}}_{1} = - k_{1} (y_{1} - d) - c_{1} ({\dot{y}}_{1} - \dot{d}) + k_{2} (y_{2} - y_{1}) + c_{2} ({\dot{y}}_{2} - {\dot{y}}_{1}) \end{matrix}

(1)

where m_i is the ith mass, c_i is the ith damping coefficient, k_i is the ith spring stiffness, y_i is the displacement of the ith mass, and the dot indicates time derivative. The displacement of the support is indicated as y₀ = d and is considered unknown. An equivalent expression in the frequency domain can then be written as

Y_{i} (s) = \frac{N u m_{i} (s)}{D e n_{i} (s)} Y_{i - 1} (s) - \frac{N u m 1_{i} (s)}{D e n 1_{i} (s)} F (s)

(2)

where Num_i(s) and Den_i(s) represent, respectively, the numerator and the denominator of the transfer function between the position of the ith and (i − 1)-th masses. Similarly, Num1_i(s) and Den1_i(s) refer to the transfer function between the position of the ith mass and the applied force F.

The linearity of the system dynamics and implies that the superposition principle holds, hence the influence of the disturbance d and the force F can be assessed independently. For example, the transfer function between ith mass displacement y_i and the disturbance d can be written as

\begin{aligned} \frac{Y_{i} (s)}{d (s)} = \frac{Y_{1} (s)}{d (s)} \frac{Y_{2} (s)}{Y_{1} (s)} \dots \frac{Y_{i} (s)}{Y_{i - 1} (s)} = \frac{N u m_{i}^{d}}{D e n_{1}} = {\begin{matrix} \frac{D e n_{i + 1} \prod_{1}^{i} (c_{i} s + k_{i})}{D e n_{1}}, & i \leq n - 1 \\ \frac{\prod_{1}^{i} (c_{i} s + k_{i})}{D e n_{1}}, & i = n \end{matrix} \end{aligned}

(3)

where

\begin{aligned} D e n_{n} & = m_{n} s^{2} + c_{n} s + k_{n} \\ D e n_{n - 1} & = [m_{n - 1} s^{2} + (c_{n - 1} + c_{n}) s + k_{n - 1} + k_{n}] (m_{n} s^{2} + c_{n} s + k_{n}) - (c_{n} s + k_{n})^{2} \\ D e n_{i} & = [m_{i} s^{2} + (c_{i} + c_{i + 1}) s + k_{i} + k_{i + 1}] D e n_{i + 1} - (c_{i + 1} s + k_{i + 1})^{2} D e n_{i + 2} \end{aligned}

(4)

A similar procedure can also be applied to derive the transfer function between the ith displacements y_i and the external force F, thus obtaining

\begin{aligned} \frac{Y_{i} (s)}{F (s)} & = \frac{Y_{n} (s)}{F (s)} \frac{Y_{n - 1} (s)}{Y_{n} (s)} \dots \frac{Y_{i} (s)}{Y_{i + 1} (s)} = - \frac{N u m_{i}^{F}}{D e n 1_{n}} = {\begin{matrix} - \frac{\prod_{i = 2}^{n} (c_{i} s + k_{i})}{D e n 1_{n}}, & i = 1 \\ - \frac{D e n 1_{i - 1} \prod_{i + 1}^{n} (c_{i + 1} s + k_{i + 1})}{D e n 1_{n}}, & 1 < i \leq n - 1 \\ - \frac{D e n 1_{i - 1}}{D e n 1_{n}}, & i = n \end{matrix} \end{aligned}

(5)

where

D e n 1_{i} = {\begin{matrix} m_{1} s^{2} + (c_{1} + c_{2}) s + k_{1} + k_{2}, & i = 1 \\ [m_{1} s^{2} + (c_{1} + c_{2}) s + k_{1} + k_{2}] [m_{2} s^{2} + (c_{2} + c_{3}) s + k_{2} + k_{3}] - (c_{2} s + k_{2})^{2}, & i = 2 \\ [m_{i} s^{2} + (c_{i} + c_{i + 1}) s + k_{i} + k_{i + 1}] D e n 1_{i - 1} - (c_{i} s + k_{i})^{2} D e n 1_{i - 2}, & 2 < i \leq n - 1 \\ (m_{n} s^{2} + c_{n} s + k_{n}) D e n 1_{n - 1} - (c_{n} s + k_{n})^{2} D e n 1_{n - 2}, & i = n \end{matrix}

(6)

Note that Den1_n is equal to Den₁, as they both are the characteristic equations of the system. The degrees of the numerators and the denominators are summarized in Table 1 for reference.

Table 1

Degrees of numerators and denominators for the transfer functions described in (3) and (5)

Index i	$∠ N u m_{i}$	$∠ D e n_{i}$	$∠ N u m 1_{i}$	$∠ D e n 1_{i}$
n	n	2	2n − 2	2n
n − 1	n + 1	4	2n − 3	2n − 2
n − 2	n + 2	6	2n − 4	2n − 4
⋮	⋮	⋮	⋮	⋮
i	2n − i	2n − 2i + 2	n + i − 2	2i
⋮	⋮	⋮	⋮	⋮
1	2n − 1	2n	n − 1	2

The recursive expressions derived in Eqs. (3) and (5), and the associated Table 1, will play a key role in the subsequent analysis, as they allow the assessment of causality and structural properties without requiring detailed knowledge of the structure parameters. Here, a dynamical system is called causal if its response depends only on past and current inputs [37]. This assessment, in turn, will allow the derivation of general conditions for DSS feasibility for hybrid testing of generic vibrating structures, thus simplifying the analysis and going well beyond the specific examples found in the current literature. It is also worth noting that actuator dynamics can be easily included in the analysis by considering the transfer function between the force F and the actuator input (e.g., voltage or current) and adding the relative degrees of such transfer function in Table 1. Similar considerations apply to potential control–structure interaction. However, such both actuator dynamics and control–structure interaction heavily depend on details of specific hardware implementation, and therefore are beyond the scope of the generic analysis presented in this paper.

2.2 Structural Stability Analysis.

Structural stability plays an important role in assessing if a given system or DSS decomposition strategy will exhibit a stable and bounded dynamic response for bounded inputs. As recently shown by the authors in Ref. [36], if a polynomial with even maximum degree has only strictly positive coefficients, then it can only admit roots with negative real parts and purely imaginary roots. In this case, Eq. can be rearranged as

D e n_{i} = \prod_{q = i}^{n} (m_{q} s^{2} + c_{q} s + k_{q}) + b_{i} (s), i \leq n - 1

(7)

where the polynomial b_i(s) includes the common factor s² and all of its coefficients are strictly positive. For example, given that all the parameters m_i, c_i, and k_i in are strictly positive, the proof shown in Ref. [36] implies that the original system only admit roots with negative real parts and hence is asymptotically stable. Although this is expected given the non-zero energy dissipation via damping, equation will play a key role in assessing the structural stability of physical and numerical parts of a given DSS decomposition, as shown in Sec. 4.

2.3 Summary of the Proposed Methodology for Dynamically Substructured Systems Feasibility Analysis.

The first property that needs to be checked before designing any controller is causality. As defined in Sec. 2.1, this mathematically translates to imposing that the degree of the numerator of the input–output transfer function is less or equal to the degree of the denominator and that the region of convergence (ROC) of the Laplace transform of the impulse response is a right half plane. However, all the physical structures considered in this paper are causal by definition, therefore the ROCs of the original system and of the decomposed structure always satisfy this constraint. Therefore, the causality analysis will only focus on the relative degrees of the numerators and denominators involved in the DSS controller design.

If any of the signals used in a DSS decomposition is not causal with respect to some of the inputs, then such decomposition cannot be physically implemented and should be discarded. However, an alternative decomposition of the same original structure may involve only causal signals. Therefore, the first step of the methodology proposed in this paper is aimed at obtaining analytical conditions for which a given DSS decomposition involves only causal signals and can be physically implemented. An example of such analysis has been performed by Li for a specific dynamical system [38], but to the extent of our knowledge no general analysis for generic vibration problems with an arbitrary number of degrees-of-freedom is present in the literature.

Once all the signals required for DSS decomposition have been proven to be causal, the next step is to check if the dynamical system to be controlled has the structural properties of controllability and observability. A system is called controllable if an external input can move the internal states of the system from any initial state to any other final state in a finite time interval. Similarly, a system is observable if its initial state can be determined based on the sequence of inputs and output signals [39]. Such structural properties are sufficient conditions to ensure that a stabilizing controller can be designed.^²

A traditional result of control theory states that a linear dynamical system is controllable and observable if there are no pole-zero cancelations between the numerator and the denominator of the transfer function describing the system behavior [40]. This analytical test will then be applied in Sec. 5.2 to obtain conditions under which there exists a stabilizing controller capable of synchronizing a given DSS decomposition.

3 Dynamically Substructured Systems Decomposition: Types and Structural Stability

Within the DSS framework, the original system shown in Fig. 2 is split into two parts, a physical part implemented in hardware and a numerical part which is numerically simulated. However, different choices can be made on what subsystem should be simulated, where the interface between the two subsystems should be placed, and whether force or position control should be used. Different choices give raise to different feasibility conditions, which are discussed in the rest of the paper.

For the sake of reference, in this paper the decomposition shown in Fig. 3(a) will be called type 1 decomposition, whereas the scheme shown in Fig. 3(b) will be referred to as type 2 decomposition. The difference between these two options lies in the location of the interface between numerical and physical subsystems; in type 1 decomposition the interface is placed right after the lth mass, whereas in type 2 decomposition the interface is placed after the spring-damper connected to the lth mass.

Fig. 3

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Potential choices for the location of the substructuring interface: (a) type 1 decomposition where the interface is right after mass l and (b) type 2 decomposition where the interface is located after the spring-mass damper system connected to mass l

Once the location of the interface has been finalized, each part can be tested either numerically or physically. However, in type 1 decomposition, causality implies that S₁ can only be force and S₂ can only be displacement, whereas the opposite holds for type 2 decomposition. Therefore, only four potential DSS decomposition approaches will be studied in this paper.

First of all, let us note that all the subsystems in Fig. 3, with the only exception of subsystem B in type 2 decomposition, share the same structure of the original system. Therefore, their dynamics can be described by the transfer functions similar to and they are all structurally stable, as expected. On the other hand, subsystem B in type 2 decomposition is lacking a fixed support and therefore is not asymptotically stable. Indeed, its generalized equation of motion can be derived by imposing force balance at each mass, thus obtaining

\begin{aligned} \frac{Y_{i} (s)}{F (s)} & = \frac{Y_{l 2} (s)}{F (s)} \frac{Y_{l + 1} (s)}{Y_{l 2} (s)} \dots \frac{Y_{i} (s)}{Y_{i - 1} (s)} = \frac{N u m_{i}^{F}}{D e n_{l + 1}^{'}} \\ = {\begin{matrix} \frac{D e n_{i + 1}^{'}}{D e n_{l + 1}^{'}}, & i = l + 1 \\ \frac{D e n_{i + 1}^{'} \prod_{l + 2}^{i} (c_{i} s + k_{i})}{D e n_{l + 1}^{'}}, & l + 2 < i \leq n - 1 \\ \frac{\prod_{l + 2}^{i} (c_{i} s + k_{i})}{D e n_{l + 1}^{'}}, & i = n \end{matrix} \end{aligned}

(8)

where

D e n_{i}^{'} = {\begin{matrix} m_{n} s^{2} + c_{n} s + k_{n}, & i = n \\ [m_{n - 1} s^{2} + (c_{n - 1} + c_{n}) s + k_{n - 1} + k_{n}] (m_{n} s^{2} + c_{n} s + k_{n}) - (c_{n} s + k_{n})^{2}, & i = n - 1 \\ ⋮ \\ [m_{i} s^{2} + (c_{i} + c_{i + 1}) s + k_{i} + k_{i + 1}] D e n_{i + 1}^{'} - (c_{i + 1} s + k_{i + 1})^{2} D e n_{i + 2}^{'}, & l + 2 \leq i \leq n - 2 \\ (m_{l + 1} s^{2} + c_{l + 2} s + k_{l + 2}) D e n_{l + 2}^{'} - (c_{l + 2} s + k_{l + 2})^{2} D e n_{l + 3}^{'}, & i = l + 1 \end{matrix}

(9)

and F is the internal force applied at the interface (F = S₂ due to input constraint). Similar to the original system, Den′_l+1 can be rearranged as

D e n_{l + 1}^{'} = m_{l + 1} s^{2} \prod_{q = l + 2}^{n} (m_{q} s^{2} + c_{q} s + k_{q}) + b_{l + 1}^{'}

(10)

where b′_l+1 has the common factor s² and its coefficients are strictly positive. Therefore, according to the approaches mentioned in Sec. 2.2, Den′_l+1 can only have roots with positive real parts and two repeated roots on the origin, confirming that subsystem B in type 2 decomposition is only marginally stable. It is worth noting that in the scenario considered here, there is no fixed reference point in the physical part which can prevent drifts when the force F is applied to the (l + 1)th mass. Therefore, simulations are prone to drifts if the initial conditions are not known and fully consistent with the external forcing. Figure 4 shows an example of such issue, where a four-mass structure with parameters as in Table 2 is tested with a sinusoidal external physical force F_p(t) = 3sin (6πt).

Fig. 4

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Displacement of fourth mass with type 2 decomposition. The drift is due to its marginal stability.

Table 2

Numerical values of parameters used for results of Fig. 4

Index	Mass, m_i	Stiffness, k_i	Damping, c_i
1	380 kg	N/A	N/A
2	350 kg	700 N/m	250 N s/m
3	320 kg	600 N/m	240 N s/m
4	290 kg	500 N/m	230 N s/m

4 Dynamically Substructured Systems Decomposition: Causality Analysis

In this section, a causality analysis for all combination of substructuring types and force/position control is performed. Each feasible case is discussed in a separate subsection to improve clarity, whereas the discussion of infeasible cases is summarized in the final subsection for simplicity.

4.1 Force Control, Physical Subsystem A, Type 1 Decomposition.

The general scheme for type 1 decomposition is shown in Fig. 3(a) and the control diagram is the same as in Fig. 1. In this section, we consider the case

S_{1} = S_{p} = F_{p}, S_{2} = S_{s} = y_{l}, S_{n} = F_{n}

(11)

i.e., the subsystem A is a physical subsystem and the physical displacement of the interface is passed to the numerical subsystem as an input. The goal of the DSS control is therefore to minimize the error between the numerical force F_n at the interface and the physical force F_p that the actuator exerts on the physical subsystem.

The equation of motion describing mass displacements in the physical part can be written as

y_{i} = \frac{N u m_{i}^{d}}{D e n_{d}} d - \frac{N u m_{i}^{F_{p}}}{D e n_{F_{p}}} F_{p}, 1 \leq i \leq l

(12)

Note that Eqs. – and Table 1 imply

\begin{aligned} ⌊ \frac{N u m_{i}^{d}}{D e n_{d}} ⌋ = \frac{2 l - i}{2 l} \\ ⌊ \frac{N u m_{i}^{F_{p}}}{D e n_{F_{p}}} ⌋ = \frac{l + i - 2}{2 l} \end{aligned}

(13)

and therefore all the transfer functions related to quantities to be measured at the interface are causal.^³

Similarly, the equations of motion for the numerical subsystem read

\begin{aligned} y_{j} & = \frac{N u m_{j}^{y_{l}}}{D e n_{y_{l}}} y_{l} \\ = \frac{N u m_{j}^{y_{l}}}{D e n_{y_{l}}} (\frac{N u m_{l}^{d}}{D e n_{d}} d - \frac{N u m_{l}^{F_{p}}}{D e n_{F_{p}}} F_{p}), l + 1 \leq j \leq n \end{aligned}

(14)

Thanks to Table 1, the degree of numerators and denominators of these transfer functions can then be written as

\begin{matrix} ⌊ \frac{N u m_{j}^{y_{l}}}{D e n_{y_{l}}} ⌋ = \frac{2 n - j - l}{2 (n - l)} \\ ⌊ \frac{N u m_{j}^{y_{l}}}{D e n_{y_{l}}} \frac{N u m_{i}^{d}}{D e n_{d}} ⌋ = \frac{2 n - j}{2 n} \\ ⌊ \frac{N u m_{j}^{y_{l}}}{D e n_{y_{l}}} \frac{N u m_{i}^{F_{p}}}{D e n_{F_{p}}} ⌋ = \frac{2 n - j + l - 2}{2 n} \end{matrix}

(15)

and therefore all the transfer functions involved in calculating mass displacements in the numerical subsystem are causal only if either the interface displacement y_l is passed as an input (first substitution in Eq. ) or the combination of physical force F_p and disturbance d are passed as inputs (second substitution in Eq. ).

On the other hand, the numerical force F_n at the interface also needs to be simulated to compute the synchronization error fed to the DSS controller. Such numerical force can be calculated as

\begin{aligned} F_{n} & = (c_{l + 1} s + k_{l + 1}) (y_{l} - y_{l + 1}) \\ = (c_{l + 1} s + k_{l + 1}) (1 - \frac{N u m_{l + 1}^{y_{l}}}{D e n_{y_{l}}}) y_{l} \\ = (c_{l + 1} s + k_{l + 1}) (1 - \frac{N u m_{l + 1}^{y_{l}}}{D e n_{y_{l}}}) (\frac{N u m_{l}^{d}}{D e n_{d}} d - \frac{N u m_{l}^{F_{p}}}{D e n_{F_{p}}} F_{p}) \end{aligned}

(16)

Once again, Table 1 implies that the relative degrees of such transfer functions are

\begin{matrix} ⌊ (c_{l + 1} s + k_{l + 1}) (1 - \frac{N u m_{l + 1}^{y_{l}}}{D e n_{y_{l}}}) ⌋ = \frac{2 n - 2 l + 1}{2 (n - l)} \\ ⌊ (c_{l + 1} s + k_{l + 1}) (1 - \frac{N u m_{l + 1}^{y_{l}}}{D e n_{y_{l}}}) \frac{N u m_{l}^{d}}{D e n_{d}} ⌋ = \frac{2 n - l + 1}{2 n} \\ ⌊ (c_{l + 1} s + k_{l + 1}) (1 - \frac{N u m_{l + 1}^{y_{l}}}{D e n_{y_{l}}}) \frac{N u m_{l}^{F_{p}}}{D e n_{F_{p}}} ⌋ = \frac{2 n - 1}{2 n} \end{matrix}

(17)

Equation implies that the numerical force F_n cannot be estimated in a causal way if information on y_l only is passed to the numerical subsystem. On the other hand, F_n can be obtained in a causal way if information about F_p and d is provided, as indicated by the last substitution in Eq. .

4.2 Position Control, Numerical Subsystem A, Type 1 Decomposition.

The case analyzed in this subsection refers to the same schematics shown in Sec. 4.1, but with the role of the physical part and the numerical part swapped, i.e.,

S_{1} = S_{s} = F_{p}, S_{2} = S_{p} = y_{l}^{p}, S_{n} = y_{l}^{n}

(18)

The equations of motion for the physical subsystem can be written as

y_{j} = \frac{N u m_{j}^{y_{l}}}{D e n_{y_{l}}} y_{l}^{p}, i + 1 \leq j \leq n

(19)

where

y_{l}^{p}

is the physical displacement of the lth mass,

N u m_{j}^{y_{l}}

and

D e n_{y_{l}}

represent the numerator and the denominator of the transfer function relating input

y_{l}^{p}

to output y_j. According to Eq. , all of the transfer functions are causal.

On the other hand, the equation of motion for the numerical part reads

y_{i}^{n} = \frac{N u m_{i}^{d}}{D e n_{d}} d - \frac{N u m_{i}^{F_{p}}}{D e n_{F_{p}}} F_{p}, 1 \leq i \leq l

(20)

where

y_{i}^{n}

is the numerical displacement of ith mass in subsystem A.

Due to Eq. , all the transfer functions involved in the substructuring procedures are causal in this scenario. It is worth noting that the decomposition described in this section is feasible only if the physical force F_p is directly measured. In fact, if one tries to estimate it from measurements of displacements at the interface, namely

\begin{aligned} F_{p} & = (c_{l + 1} s + k_{l + 1}) (y_{l}^{p} - y_{l + 1}) \\ = (c_{l + 1} s + k_{l + 1}) (1 - \frac{N u m_{l}^{y_{l}}}{D e n_{y_{l}}}) y_{l}^{p} \end{aligned}

(21)

then such an estimation is not causal.

4.3 Force Control, Numerical Subsystem A, Type 2 Decomposition.

In this scenario, the original structure is decomposed as shown in Fig. 3(b) with

S_{1} = S_{s} = y_{l + 1}, S_{2} = S_{p} = F_{p}, S_{n} = F_{n}

(22)

The equation of motion for the physical part can then be written as

y_{j} = \frac{N u m_{j}^{F_{p}}}{D e n_{2 F_{p}}} F_{p}, l + 1 \leq j \leq n

(23)

and therefore, according to Table 1,

⌊ \frac{N u m_{j}^{F_{p}}}{D e n_{2 F_{p}}} ⌋ = \frac{2 n - j - l - 1}{2 (n - l)}

(24)

and the physical part is causal, as expected.

Similarly, the displacements of the masses in the numerical part can be described as

y_{i} = \frac{N u m_{i}^{d}}{D e n_{d}} d - \frac{N u m_{i}^{y_{l + 1}}}{D e n_{y_{l + 1}}} y_{l + 1}, 1 \leq i \leq l

(25)

which implies

⌊ \frac{N u m_{i}^{y_{l + 1}}}{D e n_{y_{l + 1}}} ⌋ = \frac{l + i - 1}{2 l}

(26)

and due to Eqs. and , all the displacements in the numerical part can be simulated using causal transfer functions.

On the other hand, the numerical force F_n can be expressed as

\begin{aligned} F_{n} & = (c_{l + 1} s + k_{l + 1}) (y_{l} - y_{l + 1}) \\ = (c_{l + 1} s + k_{l + 1}) [\frac{N u m_{l}^{d}}{D e n_{d}} d - (\frac{N u m_{l}^{y_{l + 1}}}{D e n_{y_{l + 1}}} + 1) y_{l + 1}] \\ = (c_{l + 1} s + k_{l + 1}) [\frac{N u m_{l}^{d}}{D e n_{d}} d - (\frac{N u m_{l}^{y_{l + 1}}}{D e n_{y_{l + 1}}} + 1) \frac{N u m_{l + 1}^{F_{p}}}{D e n_{2 F_{p}}} F_{p}] \end{aligned}

(27)

which implies

\begin{matrix} ⌊ (c_{l + 1} s + k_{l + 1}) \frac{N u m_{l}^{d}}{D e n_{d}} ⌋ = \frac{l + 1}{2 l} \\ ⌊ (c_{l + 1} s + k_{l + 1}) (\frac{N u m_{l}^{y_{l + 1}}}{D e n_{y_{l + 1}}} + 1) ⌋ = \frac{2 l + 1}{2 l} \\ ⌊ (c_{l + 1} s + k_{l + 1}) (\frac{N u m_{l}^{y_{l + 1}}}{D e n_{y_{l + 1}}} + 1) \frac{N u m_{l + 1}^{F_{p}}}{D e n_{2 F_{p}}} ⌋ = \frac{2 n - 1}{2 n} \end{matrix}

(28)

Therefore, the numerical force F_n cannot be causally estimated based on the physical displacement y_l+1 (second line of Eq. ). However, this causality issue can be avoided if direct measurements of the physical force F_p are available, as suggested by Eq. .

Given that subsystem B is marginally stable and its displacements may drift as shown in Sec. 3, the DSS decomposition considered in this section is dangerous to be implemented and should be avoided if possible.

4.4 Position Control, Physical Subsystem A, Type 2 Decomposition.

The setup is similar to the one considered in Sec. 4.3, but with the role of the physical part and the numerical part swapped, i.e.,

S_{1} = S_{p} = y_{l + 1}^{p}, S_{2} = S_{s} = F_{p}, S_{n} = y_{l + 1}^{n}

(29)

In this case, the equation of motion for the physical part A reads

y_{i} = \frac{N u m_{i}^{d}}{D e n_{d}} d - \frac{N u m_{i}^{y_{l + 1}}}{D e n_{y_{l + 1}}} y_{l + 1}, 1 \leq i \leq l

(30)

and according to Eqs. and , all the transfer functions related to y_i are causal.

On the other hand, the physical force can be expressed as

\begin{aligned} F_{p} & = (c_{l + 1} s + k_{l + 1}) (y_{l} - y_{l + 1}) \\ = (c_{l + 1} s + k_{l + 1}) [\frac{N u m_{l}^{d}}{D e n_{d}} d - (\frac{N u m_{l}^{y_{l + 1}}}{D e n_{y_{l + 1}}} + 1) y_{l + 1}] \end{aligned}

(31)

Therefore, the physical force cannot be estimated in a causal way from d and y_i+1 based on Eq. and needs to be measured directly.

Similarly, the equations of motion describing the numerical part B read

y_{j} = \frac{N u m_{j}^{F_{p}}}{D e n_{2 F_{p}}} F_{p}, i + 1 \leq j \leq n

(32)

Therefore, no causality problems arise as expected according to Eqs. and .

4.5 Infeasible Cases.

It is worth noting that for each type of decomposition, there are four possible control strategies, namely position or force control with subsystem A or subsystem B being physically tested. However, out of the eight potential strategies, only the four discussed in Secs. 4.1–4.4 give raise to feasible realizations. Other cases are infeasible because, for example, the required signal cannot be applied at the interface. This is the case for force control in type 1 decomposition with subsystem A numerically simulated; the physical force F_p cannot be measured or reconstructed in a causal way given that only displacement related signals are available. Similar considerations apply for: (i) position control in type 1 decomposition with subsystem B being physically tested, (ii) force control in type 2 decomposition with subsystem A being physically tested, and (iii) position control in type 2 decomposition with subsystem A being implemented numerically. A summary of feasible and infeasible cases is presented in Table 3.

Table 3

Summary of causality analysis

Control type	Type 1	Type 2
Force control
(A physical)	✓	×
(A numerical)	×	✓ (Drift)
Position control
(A physical)	×	✓
(A numerical)	✓	×

4.6 Summary of Causality Analysis.

The summary of the results obtained in this section is reported in Table 3, showing that only three out of four feasible decomposition strategies admit a strictly causal implementation with no potential or experimental drifts. In addition, in type 1 decomposition, F_p can either be measured or estimated for the force control while it can only be measured for the position control. In type 2 decomposition, F_n can only be estimated through F_p for the force control. Moreover, this strategy should be avoided if possible, due to the occurrence of potential drifts in the physical part posing experimental safety issues unless when subsystem B only has single degree-of-freedom and a carefully tuned controller is used [41–43]. Finally, F_p can only be measured for the position control in type 2 decomposition. Note that the analysis presented so far does not rely on detailed knowledge of the structure under test (e.g., no numerical values for the parameters are required to be known), unlike other approaches described in the literature for specific examples. This is one of the major advantages of using the recursive formulations derived in Sec. 2 and it simplifies the whole analysis significantly.

5 Dynamically Substructured Systems Decomposition: Structural Properties

In this section, the structural properties of controllability and observability of the various causal DSS decomposition strategies are derived to assess what strategies are controllable and observable, and hence admit the existence of a synchronizing controller. As mentioned in Sec. 4, a frequency domain approach will be taken. Therefore, the analysis of structural properties reduces to obtaining conditions under which no pole-zero cancelations occur in the transfer functions used for control purposes.

5.1 Original Structure Structural Properties.

The structural properties of the original structure (before decomposition) are discussed at first in this subsection, as they form the basis for the analysis of the decomposed structure. As shown in Appendix A, if the following conditions hold

\begin{aligned} {\begin{matrix} c_{n} s + k_{n} ⊥ m_{n} s^{2} + c_{n} s + k_{n}, & i = n \\ c_{i} s + k_{i} ⊥ m_{i} s^{2} + (c_{i} + c_{i + 1}) s + (k_{i} + k_{i + 1}), & i \leq n - 1 \\ c_{i + 1} s + k_{i + 1} ⊥ m_{i} s^{2} + (c_{i} + c_{i + 1}) s + (k_{i} + k_{i + 1}), & i \leq n - 1 \\ c_{j} s + k_{j} ⊥ D e n_{i}, & j = i \dots n; i \leq n \end{matrix} \end{aligned}

(33)

then the original system is fully controllable and observable. In Eq. , the notation A(s)⊥B(s) indicates that the polynomials A(s) and B(s) do not share any common root.

From a physical point of view, most of the conditions listed in Eq. (33) are satisfied if none of the masses, spring, or damping coefficients are zero in the original structure. Moreover, if the system is underdamped, then all conditions are automatically satisfied. It is worth noting that, in any case, checking the hypotheses listed in Eq. (33) is simpler and computationally less onerous than the traditional approaches based on controllability/observability Gramians or on rank conditions on controllability matrices [39]. Indeed, testing Eq. (33) does not require neither computing all the terms in the transfer functions associated with DSS design nor a state-space representation of DSS subsystems.

5.2 Structural Properties of Causal Dynamically Substructured Systems Decompositions.

Results shown in Sec. 5.1 can be used to analyze the structural properties of the various DSS decomposition strategies. Only strategies deemed feasible according to the analysis provided in Sec. 4.6 are considered here. The main focus in this section is ensuring that all the signals used by the DSS controller, and in particular the synchronization error e(t), are observable and controllable, so that the controller can effectively synchronize the two subsystems.

Let us then start with the case of force control with physical subsystem A in type 1 decomposition. In the numerical subsystem B, y_l plays the role of the disturbance d in the original system, therefore

\begin{aligned} y_{l + 1} & = \frac{N u m_{l + 1}^{y_{l}}}{D e n_{y_{l}}} y_{l} \\ F_{n} & = (c_{l + 1} s + k_{l + 1}) (y_{l} - y_{l + 1}) \\ = (c_{l + 1} s + k_{l + 1}) (1 - \frac{N u m_{l + 1}^{y_{l}}}{D e n_{y_{l}}}) y_{l} \end{aligned}

(34)

and the synchronization error can be expressed as

\begin{aligned} e & = F_{p} - F_{n} \\ = G_{u} (s) u - (c_{l + 1} s + k_{l + 1}) (1 - \frac{N u m_{l + 1}^{y_{l}}}{D e n_{y_{l}}}) y_{l} \end{aligned}

(35)

where G_u(s) is the transfer function of the actuator (usually assumed to be a first-order system). Note that all the transfer functions considered here are a subset of the transfer functions considered for the original structure and in Appendix A, therefore no pole-zero cancelations occur and the system is completely observable and controllable.

Similarly, for the case of position control with numerical subsystem A in type 1 decomposition, the equations relative to the controller signals can be written as

y_{l}^{n} = \frac{N u m_{l}^{d}}{D e n_{d}} d - \frac{N u m_{l}^{F_{p}}}{D e n_{F_{p}}} F_{p}

(36)

e = y_{l}^{p} - y_{l}^{n} = G_{u} (s) u - \frac{N u m_{l}^{d}}{D e n_{d}} d + \frac{N u m_{l}^{F_{p}}}{D e n_{F_{p}}} F_{p}

(37)

The same steps discussed in Appendix A can then be followed, yielding that the system is fully controllable and observable if

\begin{aligned} {\begin{matrix} c_{l} s + k_{l} ⊥ m_{l} s^{2} + c_{l} s + k_{l} \\ c_{j} s + k_{j} ⊥ D e n_{d}, j = 1, \dots, l \end{matrix} \end{aligned}

(38)

are satisfied in addition to Eq. . These additional constraints are introduced because the denominator Den_d in Eq. considers only the masses up to l, therefore conditions analogous to the first and the last expressions in Eq. are needed.

Finally, for the case of position control with physical subsystem A in type 2 decomposition, the relevant equations read

\begin{aligned} y_{l + 1}^{n} & = \frac{N u m_{l + 1}^{F_{p}}}{D e n_{2 F_{p}}} F_{p} \\ e & = y_{l + 1}^{p} - y_{l + 1}^{n} = G_{u} (s) u - \frac{N u m_{l + 1}^{F_{p}}}{D e n_{2 F_{p}}} F_{p} \end{aligned}

(39)

Once again, such equations are a subset of the ones considered in Appendix A, therefore also this decomposition strategy is fully controllable and observable. In summary, all the causal DSS decompositions identified in Sec. 4 are fully controllable and observable, and therefore a synchronizing controller can be successfully designed for them.

6 Numerical Examples

In this section, numerical examples are discussed to highlight how the analysis performed in this paper can be used to guide the design of synchronizing controllers in DSS problems. The analysis presented in this paper allows some conclusions to be drawn even before the implementation of DSS to avoid infeasible cases as well as to predict potential difficulties during implementation. A benchmark system, falling within the class represented in Fig. 2, is considered at first in Sec. 6.1 to show performance of the different control strategies deemed feasible via the proposed methodology. DSS control of a more complex frame structure is then analyzed to show that the conclusions drawn from the analysis described in previous sections can still be used to inform feasibility by looking at, for example, the physical constraints at the interface. For the sake of clarity, detailed expressions for the controllers used in this section are reported in Appendix B.

6.1 Benchmark Structure.

In this section, three different cases are simulated to show the control performance that can be obtained when using the DSS decompositions deemed feasible according to the analysis of Secs. 4 and 5. Detailed tuning of the controller parameters and experimental validation are beyond the scope of this paper. To perform such a numerical study, a system composed of n = 6 masses is considered and the corresponding DSS decomposition is obtained by assigning three masses to the physical system and three masses to the numerical system. The numerical values of the parameters used for simulation are reported in Table 4.

Table 4

Numerical values of parameters used for the simulation examples

Index	Mass, m_i	Stiffness, k_i	Damping, c_i
1	500 kg	1200 N/m	300 N s/m
2	470 kg	1100 N/m	290 N s/m
3	440 kg	1000 N/m	280 N s/m
4	410 kg	900 N/m	270 N s/m
5	380 kg	800 N/m	260 N s/m
6	350 kg	700 N/m	250 N s/m

The disturbance d(t) is a chirp signal of amplitude 1 mm, and having a frequency increasing from 0 Hz to 0.5 Hz over 45 s and then maintained constant for further 15 s. For each case, the synchronization controller is designed as an H₂ controller having e(t) as input and using the energy of e(t) as performance index to be minimized.

The results obtained for force control in type 1 decomposition, where the physical part sits in subsystem A, are reported in Fig. 5, with the physical force F_p being measured. In this case, perfect synchronization is achieved between the numerical force F_n and the physical force F_p and the response is also close to the one exhibited by the original system.

Fig. 5

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Results obtained for DSS with force control in the type 1 decomposition with physical subsystem A

Similarly, the results for position control in type 1 decomposition, where the physical part now sits in subsystem B, are reported in Fig. 6. Also in this case, almost perfect synchronization is achieved, with a negligible error between physical displacement y_p and numerical displacement y_n, as well as a good match with the original response.

Fig. 6

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Results obtained for DSS with position control in the type 1 decomposition with numerical subsystem A

Finally, results for position control in type 2 decomposition with numerical subsystem B are reported in Fig. 7. In this case, synchronization is more challenging, with some noticeable discrepancies at the local minima and maxima for the displacement. Potential reasons include a non-optimal choice of poles for the closed loop system and, most importantly, the marginal stability of subsystem B causing drifts that are not compensated well by the controller. This is in accordance with the analysis in Sec. 4.4 where type 2 decomposition was shown to be harder to control due to the potential drifts in the numerical part. Exact tuning of the H₂ controller to increase performance or design of more robust controllers is beyond the scope of this paper. Figure 7 highlights the poorer performance obtainable with position control in type 2 decomposition, in agreement with the advice made in the summary of causality analysis to not use this DSS decomposition strategy for real applications.

Fig. 7

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Results obtained for DSS with position control in the type 2 decomposition with physical subsystem A

6.2 Complex Frame Structure.

In order to demonstrate that the analysis presented in this paper applies to generic complex structures, beyond the benchmark system illustrated in Fig. 3 used to motivate the analysis, a frame structure with stronger coupling across different structural elements is considered in this section. The structure, shown in Fig. 8(a), is composed of several masses connected by elastic elements and may be used to model transverse oscillations in multi-storey buildings [44]. The equivalent mathematical representation of such a structure is shown in Fig. 8(b) and its equations of motion read

m_{4} {\ddot{y}}_{4} = - k_{34} (y_{4} - y_{3}) - c_{34} ({\dot{y}}_{4} - {\dot{y}}_{3}) - k_{24} (y_{4} - y_{2}) - c_{24} ({\dot{y}}_{4} - {\dot{y}}_{2})

(40)

\begin{aligned} m_{3} {\ddot{y}}_{3} & = k_{34} (y_{4} - y_{3}) + c_{34} ({\dot{y}}_{4} - {\dot{y}}_{3}) - k_{23} (y_{3} - y_{2}) - k_{13} (y_{3} \\ - y_{1}) - c_{13} ({\dot{y}}_{3} - {\dot{y}}_{1}) \end{aligned}

(41)

\begin{aligned} m_{2} {\ddot{y}}_{2} & = k_{24} (y_{4} - y_{2}) + c_{24} ({\dot{y}}_{4} - {\dot{y}}_{2}) + k_{23} (y_{3} - y_{2}) - k_{12} (y_{2} - y_{1}) \\ - c_{12} ({\dot{y}}_{2} - {\dot{y}}_{1}) \end{aligned}

(42)

\begin{aligned} m_{1} {\ddot{y}}_{1} & = k_{13} (y_{3} - y_{1}) + c_{13} ({\dot{y}}_{3} - {\dot{y}}_{1}) + k_{12} (y_{2} - y_{1}) + c_{12} ({\dot{y}}_{2} - {\dot{y}}_{1}) \\ - k_{11} (y_{1} - d) - c_{11} ({\dot{y}}_{1} - \dot{d}) \end{aligned}

(43)

Numerical values of the parameters are the same as in Table 4, with k₃₄, k₁₃, k₁₂, and k₁₁ being set equal to k₄, k₃, k₂, and k₁, and similarly for the damping coefficients. In addition, k₂₄ = 800 N/m, k₂₃ = 500 N/m, and c₂₄ = 260 Ns/m. The disturbance d(t) is a 1 mm chirp signal whose frequency increases from 0 Hz to 0.7 Hz over 50 s and then is maintained constant for further 30 s. Type 1 decomposition can be implemented by placing the interface above m₁ given the similarities of the conditions at the interface: the absence of spring and damper at the interface implies that only the external force can be applied, therefore, results from Secs. 4 and 5 suggest that force and position control are feasible. Indeed, it was possible to design H₂ controllers to eliminate the error between forces or displacements at the interface, in accordance with the causality and structural properties analysis. Results about force and position control are shown, respectively, in Figs. 9 and 10. As expected, signals at the interface are well synchronized and the DSS responses are very close to those exhibited by the original structure. These numerical results demonstrate that the analysis discussed in this paper is valid for generic vibrating structures.

Fig. 8

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Complex frame structure: (a) real structure and (b) equivalent mathematical representation

Fig. 9

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Results obtained for the complex structure with force control in the type 1 decomposition

Fig. 10

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Results obtained for the complex structure with position control in the type 1 decomposition

7 Conclusions

A comprehensive analysis of DSS decomposition strategies for vibration problems was presented in this paper. A control-theoretic approach was explored to derive rigorous conditions for which a given vibration structure can be tested within the DSS framework. The proposed method is based on a novel recursive formulation of the transfer functions involved in DSS design. The analysis presented in this paper is independent from the number of degrees-of-freedom present in the original structure, so it goes beyond all the specific cases described in the literature so far. Moreover, it allows feasibility analysis of DSS control for whole classes of vibrating structures, without detailed knowledge of the numerical values of their parameters. Indeed, structural stability for different subsystems can be analyzed without calculating the roots of characteristic equations and causality can be studied without knowing any of the system parameters. Moreover, the proposed methodology also provides guidelines for choosing the best DSS decomposition strategy for any given structure. Simple conditions to ensure controllability and observability of DSS design have been derived in Eqs. (33) and (9). These conditions are automatically satisfied if the vibrating structure is underdamped and, in any case, can be tested with less computational effort compared to standard controllability and observability tests. The potential applicability of the results to more generic structures than the benchmark system used for the initial analysis is demonstrated on a numerical example regarding DSS control of a complex frame structure. This suggests that the proposed framework can potentially be extended to more complex structures, and investigating this problem by combining modal analysis with our framework represents a promising avenue for future work. Overall, the results presented in this paper provide rigorous guidelines for guiding the choice of DSS decomposition strategies for hybrid testing of vibrating structures.

Footnotes

2

Note that observability and controllability are not necessary conditions. In fact, a stabilizing controller can still be designed in the presence of unobservable/uncontrollable states as long as these latter are stable. However, such weaker conditions of stabilizability and detectability can only be studied on a case by case basis, therefore they are not useful to develop the generic framework considered in this paper.

3

Note that for simplicity, all the degrees considered in this analysis are the maximum degrees and no pole-zero cancelations are taken into account. However, the same result holds even in the presence of pole-zero cancelations, as the relative degree of the transfer functions is preserved also in these cases.

Conflict of Interest

There are no conflicts of interest.

Data Availability Statement

The authors attest that all data for this study are included in the paper. The authors attest that all data for this study are included in the paper.

Nomenclature

Symbols

d =: disturbance acting on the system
l =: location of DSS interface
n =: number of masses
F =: force of DSS actuator
y_i =: displacement of ith mass
$⌊ H (s) ⌋$ =: maximum degree of transfer function H(s)
$∠ P (s)$ =: degree of polynomial P(s)
$D e n_{2 F_{p}}$ =: generic denominator in subsystem B with F_p as input (type 2 decomposition)
$D e n_{y_{l}}$ =: generic denominator in subsystem B with y_l as input (type 1 decomposition)
$D e n_{y_{l + 1}}$ =: generic denominator in subsystem A with y_l+1 as input (type 2 decomposition)
$N u m_{i}^{d}$ =: numerator of transfer function between d and y_i
$N u m_{i}^{d}$ =: ith generic numerator in subsystem A with F_p as input (type 2 decomposition)
$N u m_{j}^{y_{l}}$ =: jth generic numerator in subsystem B with y_l as input (type 1 decomposition)
$N u m_{i}^{y_{l + 1}}$ =: ith generic numerator in subsystem A with y_l+1 as input (type 2 decomposition)

Appendix A. Controllability and Observability of Original Dynamics

The complete proof of controllability and observability for the original vibrating structure discussed in Sec. 5.1 is reported in this Appendix.

In particular, an induction procedure is used to show that under the hypotheses listed in Eq. , there are no common roots between any denominator and numerator of the transfer functions considered in DSS design. As first step, let us note that the constraints listed in Eq. imply that the first two denominators Den_n and Den_n−1 in Eq. do not have any common root. Let us then assume that two adjacent denominators Den_n−i+1 and Den_n−i do not share any common roots. Equation can be rewritten as

D e n_{n - i - 1} = K_{1 (n - i - 1)} D e n_{n - i} - K_{2 (n - i - 1)} D e n_{n - i + 1}

(A1)

where

K_{1 (n - i - 1)} = m_{n - i - 1} s^{2} + (c_{n - i} + c_{n - i}) s + k_{n - i - 1} + k_{n - i}

(A2)

K_{2 (n - i - 1)} = (c_{n - i} s + k_{n - i})^{2}

(A3)

Then Eq. can be plugged into the following denominators to obtain

\begin{aligned} D e n_{n - j + 1} & = K_{1 (n - j + 1)} D e n_{n - j + 2} - K_{2 (n - j + 1)} D e n_{n - j + 3} \\ = (\prod_{q = i + 2}^{j} K_{1 (n - q + 1)} - b_{1 (j)}) D e n_{n - i} \\ - (K_{2 (n - i - 1)} \prod_{q = i + 3}^{j} K_{1 (n - q + 1)} - b_{2 (j)}) D e n_{n - i + 1}, i + 3 \leq j \leq n \end{aligned}

(A4)

where

K_{1 (n - j + 1)} = m_{n - j + 1} s^{2} + (c_{n - j + 1} + c_{n - j + 2}) s + k_{n - j + 1} + k_{n - j + 2}

(A5)

K_{2 (n - j + 1)} = (c_{n - j + 2} s + k_{n - j + 2})^{2}

(A6)

b_{1 (i + 2)} = 0

(A7)

b_{1 (i + 3)} = K_{2 (n - i - 2)}

(A8)

\begin{aligned} b_{1 (j)} & = K_{1 (n - j + 1)} b_{1 (j - 1)} + K_{2 (n - j + 1)} \prod_{q = i + 2}^{j - 2} K_{1 (n - q + 1)} - K_{2 (n - j + 1)} b_{1 (j - 2)}, \\ i + 4 \leq j \leq n \end{aligned}

(A9)

b_{2 (i + 2)} = b_{2 (i + 3)} = 0

(A10)

b_{2 (i + 4)} = K_{2 (n - i - 1)} K_{2 (n - i - 3)}

(A11)

\begin{aligned} b_{2 (j)} & = K_{1 (n - j + 1)} b_{2 (j - 1)} + K_{2 (n - j + 1)} K_{2 (n - i - 1)} \prod_{q = i + 3}^{j - 2} K_{1 (n - q + 1)} \\ - K_{2 (n - j + 1)} b_{2 (j - 2)}, i + 5 \leq j \leq n \end{aligned}

(A12)

In the above equations, K_1(n−j+1) and K_2(n−j+1) are used to describe the explicit terms, while b_1(j) and b_2(j) are used to implicitly describe the rest of the division of Den_n−j+1 by, respectively, K_1(n−j+1) and K_2(n−j+1). The degrees of the polynomials b_1(j) and b_2(j) are reported in Table 5 for reference.

Table 5

Degrees of polynomials b_1(j) and b_2(j)

Index, j	$∠ b_{1 (j)}$	$∠ b_{2 (j)}$
i+2	0	0
i+3	2	0
i+4	4	4
i+5	6	6
⋮	⋮	⋮
j	2j − 2i − 4 (j ≥ i + 2)	2j − 2i − 4 (j ≥ i + 4)
⋮	⋮	⋮
n	2n − 2i − 4	2n − 2i − 4

Let us then proceed with a proof by contradiction. To this end, note that if Den_n−j+1 and Den_n−i−1 had common roots, then Eq. could be rearranged as

D e n_{n - j + 1} = A \cdot D e n_{n - i - 1}

(A13)

When A is a constant or a polynomial, all roots of Den_n−i−1 are included in the roots of Den_n−j+1. On the other hand, if A is a ratio of polynomials and its denominator shares some roots with Den_n−i−1, then some roots of Den_n−i−1 are not included in the roots of Den_n−j+1. Note that Eq. can be rewritten as

\begin{aligned} D e n_{n - j + 1} & = K_{1 (n - i - 1)} (\prod_{q = i + 3}^{j} K_{1 (n - q + 1)} - \frac{b_{1 (j)}}{K_{1 (n - i - 1)}}) D e n_{n - i} \\ - K_{2 (n - i - 1)} (\prod_{q = i + 3}^{j} K_{1 (n - q + 1)} - \frac{b_{2 (j)}}{K_{2 (n - i - 1)}}) D e n_{n - i + 1} \end{aligned}

(A14)

which, once combined with Eqs. and , implies

A = \prod_{q = i + 3}^{j} K_{1 (n - q + 1)} - \frac{b_{1 (j)}}{K_{1 (n - i - 1)}} = \prod_{q = i + 3}^{j} K_{1 (n - q + 1)} - \frac{b_{2 (j)}}{K_{2 (n - i - 1)}}

(A15)

and therefore the following condition holds:

\frac{b_{1 (j)}}{K_{1 (n - i - 1)}} = \frac{b_{2 (j)}}{K_{2 (n - i - 1)}}

(A16)

It is obvious that Eq. is trivially not satisfied for j = i + 3 due to –. Therefore, the following proof will focus on Eq. for j > i + 3. To this end, note that given the non-zero polynomials X, Y, a₁, a₂, T, the relation

\frac{X}{Y} = \frac{T X + a_{1}}{T Y + a_{2}}

(A17)

holds if and only if

\frac{a_{1}}{a_{2}} = \frac{X}{Y}

(A18)

Let us then rewrite Eq. as

\frac{b_{1 (j)}}{b_{2 (j)}} = \frac{K_{1 (n - i - 1)}}{K_{2 (n - i - 1)}}

(A19)

and use Eqs. and to express b_1(j) and b_2(j) as

b_{1 (j)} = T X + a_{1}, b_{2 (j)} = T Y + a_{2}

(A20)

where

T = K_{2 (n - j + 1)} \prod_{q = i + 3}^{j - 2} K_{1 (n - q + 1)}

(A21)

X = K_{1 (n - i - 1)}

(A22)

Y = K_{2 (n - i - 1)}

(A23)

a_{1} = K_{1 (n - j + 1)} b_{1 (j - 1)} - K_{2 (n - j + 1)} b_{1 (j - 2)}

(A24)

a_{2} = K_{1 (n - j + 1)} b_{2 (j - 1)} - K_{2 (n - j + 1)} b_{2 (j - 2)}

(A25)

Then, holds if and only if

\frac{K_{1 (n - j + 1)} b_{1 (j - 1)} - K_{2 (n - j + 1)} b_{1 (j - 2)}}{K_{1 (n - j + 1)} b_{2 (j - 1)} - K_{2 (n - j + 1)} b_{2 (j - 2)}} = \frac{K_{1 (n - i - 1)}}{K_{2 (n - i - 1)}}

(A26)

or, by rearranging the terms in a more convenient form,

\frac{K_{1 (n - j + 1)}}{K_{2 (n - j + 1)}} = \frac{K_{2 (n - i - 1)} b_{1 (j - 1)} - K_{1 (n - i - 1)} b_{2 (j - 1)}}{K_{2 (n - i - 1)} b_{1 (j - 2)} - K_{1 (n - i - 1)} b_{2 (j - 2)}}

(A27)

However, according to and and Table 5, the left-hand side is a ratio of second-order polynomials, whereas the right-hand side is a ratio of polynomials of different degrees. Therefore, cannot hold and the proof is concluded, indicating that in the original structure no common roots exist between Num_i and Den_i (1 ≤ i ≤ n) if the hypotheses listed in Eq. hold. The original system is then fully controllable and observable.

Appendix B. H₂ Controllers Used for Numerical Results Shown in the Main Paper

For the sake of completeness, explicit numerical expressions for the controllers designed in Sec. 6 are reported in this Appendix. Controller for type 1 decomposition with force control in the benchmark system, used to obtain results shown in Fig. 5:

\begin{aligned} C (s) & = \frac{2.9306 \times 10^{6} (s + 10) (s + 3.781) (s + 3.589) (s + 3.328) (s^{2} + 0.002372 s + 2.681 \times 10^{- 6})}{(s + 1.003 \times 10^{5}) (s^{2} + 0.006758 s + 0.1492) (s^{2} + 0.3164 s + 1.124) (s^{2} + 0.8353 s + 2.843)} \\ \times \frac{(s^{2} + 0.6761 s + 1.979) (s^{2} + 2.014 s + 5.917) (s^{2} + 132 s + 8950)}{(s^{2} + 1.399 s + 4.581) (s^{2} + 2.037 s + 6.571) (s^{2} + 2.339 s + 8.129)} \end{aligned}

(B1)

Controller for type 1 decomposition with position control in the benchmark system, used to obtain results shown in Fig. 6:

\begin{aligned} C (s) & = \frac{- 2.6375 \times 10^{10} (s + 65.33) (s + 10) (s + 3.793) (s + 3.571) (s^{2} + 0.1433 s + 0.4533)}{(s + 10^{9}) (s^{2} + 0.4257 s + 0.4533) (s^{2} + 0.3281 s + 1.093) (s^{2} + 0.8342 s + 2.758)} \\ \times \frac{(s^{2} + 1.048 s + 3.141) (s^{2} + 2.158 s + 6.492) (s^{2} + 62.11 s + 5636)}{(s^{2} + 1.446 s + 4.779) (s^{2} + 2.029 s + 6.688) (s^{2} + 2.312 s + 7.938)} \end{aligned}

(B2)

Controller for type 2 decomposition with position control in the benchmark system, used to obtain results shown in Fig. 7:

\begin{aligned} C (s) & = \frac{1.2628 \times 10^{13} (s + 1.173) (s + 10) (s^{2} + 0.145 s + 0.4791) (s^{2} + 1.038 s + 3.103)}{(s + 9.872 \times 10^{7}) (s + 1.279 \times 10^{6}) (s + 2.814) (s^{2} + 0.4184 s + 0.6607) (s^{2} + 0.02365 s + 0.908)} \\ \times \frac{(s^{2} + 3.205 s + 5.738) (s^{2} + 2.172 s + 6.498) (s^{2} + 1.087 s + 12.08)}{(s^{2} + 1.119 s + 3.217) (s^{2} + 2.196 s + 6.525) (s^{2} + 1.621 s + 6.191)} \end{aligned}

(B3)

Controller for type 1 decomposition with force control in the complex frame system, used to obtain results shown in Fig. 9:

C (s) = \frac{1.4101 \times 10^{6} (s + 3.684) (s^{2} + 0.002657 s + 1.534 \times 10^{- 5}) (s^{2} + 1.856 s + 6.01) (s^{2} + 1.231 s + 6.374)}{(s^{2} + 0.1155 s + 0.4773) (s^{2} + 1.517 s + 5.154) (s^{2} + 1.224 s + 6.405) (s^{2} + 2.593 s + 9.28)}

(B4)

Controller for type 1 decomposition with position control in the complex frame system, used to obtain results shown in Fig. 10:

C (s) = \frac{2.2243 \times 10^{12} (s + 1.112 \times 10^{4}) (s + 20) (s^{2} + 0.3835 s + 1.377) (s^{2} + 2.112 s + 6.961) (s^{2} + 1.218 s + 6.357)}{(s + 2 \times 10^{9}) (s^{2} + 0.1242 s + 0.4791) (s^{2} + 1.516 s + 5.15) (s^{2} + 1.224 s + 6.405) (s^{2} + 2.589 s + 9.246)}

(B5)

References

1.

Chen

,

C.

,

Ricles

,

J. M.

,

Marullo

,

T. M.

, and

Mercan

,

O.

,

2009

, “

Real-Time Hybrid Testing Using the Unconditionally Stable Explicit CR Integration Algorithm

,”

Earthquake Eng. Struct. Dyn.

,

38

(

1

), pp.

23

–

44

.

Google Scholar

Crossref

2.

Mosterman

,

P. J.

,

1999

, “

An Overview of Hybrid Simulation Phenomena and Their Support by Simulation Packages

,”

Hybrid Systems: Computation and Control

,

Berlin

,

Jan. 1

, Springer, pp.

165

–

177

.

Google Scholar

Crossref

3.

Maghareh

,

A.

,

Dyke

,

S. J.

,

Prakash

,

A.

, and

Bunting

,

G. B.

,

2014

, “

Establishing a Predictive Performance Indicator for Real-Time Hybrid Simulation

,”

Earthquake Eng. Struct. Dyn.

,

43

(

15

), pp.

2299

–

2318

.

Google Scholar

Crossref

4.

Tu

,

J. Y.

,

2013

, “

Development of Numerical-Substructure-Based and Output-Based Substructuring Controllers

,”

Struct. Control Health Monit.

,

20

(

6

), pp.

918

–

936

.

Google Scholar

Crossref

5.

Klerk

,

D. D.

,

Rixen

,

D. J.

, and

Voormeeren

,

S. N.

,

2008

, “

General Framework for Dynamic Substructuring: History, Review and Classification of Techniques

,”

AIAA J.

,

46

(

5

), pp.

1169

–

1181

.

Google Scholar

Crossref

6.

Lim

,

C. N.

,

Neild

,

S. A.

,

Stoten

,

D. P.

,

Drury

,

D.

, and

Taylor

,

C. A.

,

2007

, “

Adaptive Control Strategy for Dynamic Substructuring Tests

,”

J. Eng. Mech.

,

133

(

8

), pp.

864

–

873

.

7.

Gawthrop

,

P.

,

Wallace

,

M.

,

Neild

,

S.

, and

Wagg

,

D.

,

2007

, “

Robust Real-Time Substructuring Techniques for Under-Damped Systems

,”

Struct. Control Health Monit.

,

14

(

4

), pp.

591

–

608

.

Google Scholar

Crossref

8.

Tu

,

J.-Y.

,

Chen

,

C.-Y.

, and

Hsiao

,

W.-D.

,

2015

, “

Dynamic and Numerical Issues Relating to the Control Robustness of Dynamically Substructured Systems

,”

Struct. Control Health Monit.

,

22

(

3

), pp.

518

–

534

.

Google Scholar

Crossref

9.

Yamaguchi

,

T.

, and

Stoten

,

D. P.

,

2016

, “

Synthesised H_∞/μ Control Design for Dynamically Substructured Systems

,”

J. Phys.: Conf. Ser.

,

744

(

1

), p.

012205

.

10.

Wu

,

B.

, and

Zhou

,

H.

,

2014

, “

Sliding Mode for Equivalent Force Control in Real-Time Substructure Testing

,”

Struct. Control Health Monitor.

,

21

(

10

), pp.

1284

–

1303

.

Google Scholar

Crossref

11.

Wu

,

B.

, and

Wang

,

T.

,

2014

, “

Model Updating With Constrained Unscented Kalman Filter for Hybrid Testing

,”

Smart Struct. Syst.

,

14

(

6

), pp.

1105

–

1129

.

Google Scholar

Crossref

12.

Carrion

,

J. E.

,

2008

, “

Real-Time Hybrid Testing Using Model-Based Delay Compensation

,”

Smart Struct. Syst.

,

4

(

6

), pp.

809

–

828

.

Google Scholar

Crossref

13.

Liu

,

J.

,

Dyke

,

S. J.

,

Liu

,

H.-J.

,

Gao

,

X.-Y.

, and

Phillips

,

B.

,

2013

, “

A Novel Integrated Compensation Method for Actuator Dynamics in Real-Time Hybrid Structural Testing

,”

Struct. Control Health Monit.

,

20

(

7

), pp.

1057

–

1080

.

Google Scholar

Crossref

14.

Shi

,

P.

,

Wu

,

B.

, and

Chang

,

C.-M.

,

2016

, “

Real-Time Hybrid Testing With Equivalent Force Control Method Incorporating Kalman Filter

,”

Struct. Control Health Monit.

,

23

(

4

), pp.

735

–

748

.

Google Scholar

Crossref

15.

Maghareh

,

A.

,

Dyke

,

S.

,

Rabieniaharatbar

,

S.

, and

Prakash

,

A.

,

2017

, “

Predictive Stability Indicator: A Novel Approach to Configuring a Real-Time Hybrid Simulation

,”

Earthquake Eng. Struct. Dyn.

,

46

(

1

), pp.

95

–

116

.

Google Scholar

Crossref

16.

Botelho

,

R. M.

,

Gao

,

X.

,

Avci

,

M.

, and

Christenson

,

R.

,

2022

, “

A Robust Stability and Performance Analysis Method for Multi-actuator Real-Time Hybrid Simulation

,”

Struct. Control Health Monit.

,

29

(

10

), p.

e3017

.

Google Scholar

Crossref

17.

Enokida

,

R.

, and

Kajiwara

,

K.

,

2019

, “

Nonlinear Signal-Based Control for Single-Axis Shake Tables Supporting Nonlinear Structural Systems

,”

Struct. Control Health Monit.

,

26

(

9

), p.

e2376

.

Google Scholar

Crossref

18.

Drazin

,

P. L.

, and

Govindjee

,

S.

,

2017

, “

Hybrid Simulation Theory for a Classical Nonlinear Dynamical System

,”

J. Sound Vib.

,

392

(

18

), pp.

240

–

259

.

Google Scholar

Crossref

19.

Stoten

,

D. P.

,

Yamaguchi

,

T.

, and

Yamashita

,

Y.

,

2016

, “

Dynamically Substructured System Testing for Railway Vehicle Pantographs

,”

J. Phys.: Conf. Ser.

,

744

(

1

), p.

012204

.

20.

Facchinetti

,

A.

, and

Bruni

,

S.

,

2012

, “

Hardware-in-the-Loop Hybrid Simulation of Pantograph–Catenary Interaction

,”

J. Sound Vib.

,

331

(

12

), pp.

2783

–

2797

.

Google Scholar

Crossref

21.

Hong

,

W.

,

Kang

,

S.

,

Lee

,

J.-S.

,

Byun

,

Y.

, and

Choi

,

C.

,

2016

, “

Characterization of Railroad Track Substructures Using Dynamic and Static Cone Penetrometer

,”

Proceedings of the 5th International Conference on Geotechnical and Geophysical Site Characterisation, ISC 2016

,

Gold Coast, Australia

,

Sept. 5–9

, Vol. 1, Australian Geomechanics Society, pp.

707

–

710

.

22.

Allen

,

M. S.

, and

Mayes

,

R. L.

,

2018

, “

Recent advances to estimation of fixed-interface modal models using dynamic substructuring

,”

Proceedings of the 36th International Modal Analysis Conference

,

Orlando, FL

,

May 1

, Vol. 4, Springer, pp.

157

–

170

.

Google Scholar

Crossref

23.

Mayes

,

R. L.

, and

Arviso

,

M.

,

2010

, “

Design Studies for the Transmission Simulator Method of Experimental Dynamic Substructuring

,”

Proceedings of the 24th ISMA

,

Leuven, Belgium

,

Sept. 20–22

, pp.

1929

–

1938

.

24.

van der Seijs

,

M.

, and

Rixen

,

D.

,

2016

, “Experimental Dynamic Substructuring: Analysis and Design Strategies for Vehicle Development,” PhD thesis, Delft University of Technology, Delft.

25.

Stoten

,

D. P.

, and

Hyde

,

R. A.

,

2006

, “

Adaptive Control of Dynamically Substructured Systems: The Single-Input Single-Output Case

,”

Proc. Inst. Mech. Eng. Part I: J. Syst. Control Eng.

,

220

(

2

), pp.

63

–

79

.

26.

Zapateiro

,

M.

,

Karimi

,

H. R.

,

Luo

,

N.

, and

Spencer Jr

,

B. F.

,

2010

, “

Real-Time Hybrid Testing of Semiactive Control Strategies for Vibration Reduction in a Structure With MR Damper

,”

Struct. Control Health Monit.

,

17

(

4

), pp.

427

–

451

.

27.

Chu

,

S.-Y.

,

Lu

,

L.-Y.

, and

Yeh

,

S.-W.

,

2018

, “

Real-Time Hybrid Testing of a Structure With a Piezoelectric Friction Controllable Mass Damper by Using a Shake Table

,”

Struct. Control Health Monit.

,

25

(

3

), p.

e2124

.

Google Scholar

Crossref

28.

Zhu

,

F.

,

Wang

,

J.-T.

,

Jin

,

F.

,

Lu

,

L.-Q.

,

Gui

,

Y.

, and

Zhou

,

M.-X.

,

2017

, “

Real-Time Hybrid Simulation of the Size Effect of Tuned Liquid Dampers

,”

Struct. Control Health Monit.

,

24

(

9

), p.

e1962

.

Google Scholar

Crossref

29.

Mercan

,

O.

,

Ricles

,

J.

,

Sause

,

R.

, and

Marullo

,

T.

,

2008

, “

Real-Time Large-Scale Hybrid Testing for Seismic Performance Evaluation of Smart Structures

,”

Smart Struct. Syst.

,

4

(

5

), pp.

667

–

684

.

Google Scholar

Crossref

30.

Londoño

,

J. M.

,

Serino

,

G.

,

Wagg

,

D. J.

,

Neild

,

S. A.

, and

Crewe

,

A. J.

,

2012

, “

On the Assessment of Passive Devices for Structural Control Via Real-Time Dynamic Substructuring

,”

Struct. Control Health Monit.

,

19

(

8

), pp.

701

–

722

.

Google Scholar

Crossref

31.

Fu

,

B.

,

Jiang

,

H.

, and

Wu

,

T.

,

2019

, “

Experimental Study of Seismic Response Reduction Effects of Particle Damper Using Substructure Shake Table Testing Method

,”

Struct.Control Health Monit.

,

26

(

2

), p.

e2295

.

Google Scholar

Crossref

32.

Bursi

,

O. S.

,

Abbiati

,

G.

, and

Reza

,

M. S.

,

2014

, “

A Novel Hybrid Testing Approach for Piping Systems of Industrial Plants

,”

Smart Struct. Syst.

,

14

(

6

), pp.

1005

–

1030

.

Google Scholar

Crossref

33.

Guo

,

J.

,

Tang

,

Z.

,

Chen

,

S.

, and

Li

,

Z.

,

2016

, “

Control Strategy for the Substructuring Testing Systems to Simulate Soil–Structure Interaction

,”

Smart Struct. Syst.

,

18

(

6

), pp.

1169

–

1188

.

Google Scholar

Crossref

34.

Terkovics

,

N.

,

Neild

,

S.

,

Lowenberg

,

M.

,

Szalai

,

R.

, and

Krauskopf

,

B.

,

2016

, “

Substructurability: the Effect of Interface Location on a Real-Time Dynamic Substructuring Test

,”

Proc. R. Soc. A

,

472

(

2192

), p.

20160433

.

Google Scholar

Crossref

35.

Gawthrop

,

P.

,

Neild

,

S.

,

Gonzalez-Buelga

,

A.

, and

Wagg

,

D.

,

2009

, “

Causality in Real-Time Dynamic Substructure Testing

,”

Mechatronics

,

19

(

7

), pp.

1105

–

1115

.

Google Scholar

Crossref

36.

Hu

,

A.

, and

Paoletti

,

P.

,

2020

, “

A Novel Control Architecture for Marginally Stable Dynamically Substructured Systems

,”

Mech. Syst. Signal Process.

,

143

(

22

), p.

106834

.

Google Scholar

Crossref

37.

Oppenheim

,

A.

,

Willsky

,

A.

,

Nawab

,

H.

, and

Hamid

,

S.

,

1998

,

Signals and Systems

,

Pearson Education

,

London

.

38.

Li

,

G.

,

2014

, “

Dynamically Substructured System Frameworks With Strict Separation of Numerical and Physical Components

,”

Struct. Control Health Monit.

,

21

(

10

), pp.

1316

–

1333

.

Google Scholar

Crossref

39.

Kalman

,

R.

,

1959

, “

On the General Theory of Control Systems

,”

IRE Trans. Autom. Control

,

4

(

3

), pp.

110

–

110

.

Google Scholar

Crossref

40.

Antsaklis

,

P.

, and

Michel

,

A.

,

2007

,

A Linear Systems Primer

,

Birkhäuser

,

Boston, MA

.

41.

Neild

,

S.

,

Stoten

,

D.

,

Drury

,

D.

, and

Wagg

,

D.

,

2005

, “

Control Issues Relating to Real-Time Substructuring Experiments Using a Shaking Table

,”

Earthquake Eng. Struct. Dyn.

,

34

(

9

), pp.

1171

–

1192

.

Google Scholar

Crossref

42.

Tu

,

J.-Y.

,

Hsiao

,

W.-D.

, and

Chen

,

C.-Y.

,

2014

, “

Modelling and Control Issues of Dynamically Substructured Systems: Adaptive Forward Prediction Taken as an Example

,”

Proc. R. Soc. A

,

470

(

2168

), p.

20130773

.

Google Scholar

Crossref

43.

Barton

,

D. A.

,

2017

, “

Control-Based Continuation: Bifurcation and Stability Analysis for Physical Experiments

,”

Mech. Syst. Signal Process.

,

84

(

4

), pp.

54

–

64

.

Google Scholar

Crossref

44.

Del Carpio Ramos

,

M.

,

Mosqueda

,

G.

, and

Hashemi

,

M. J.

,

2016

, “

Large-Scale Hybrid Simulation of a Steel Moment Frame Building Structure Through Collapse

,”

J. Struct. Eng.

,

142

(

1

), p.

04015086

.

Google Scholar

Crossref

On the Feasibility of Dynamic Substructuring for Hybrid Testing of Vibrating Structures

Abstract

1 Introduction

2 Problem Statement and Methodology

2.1 Original Dynamics.

2.2 Structural Stability Analysis.

2.3 Summary of the Proposed Methodology for Dynamically Substructured Systems Feasibility Analysis.

3 Dynamically Substructured Systems Decomposition: Types and Structural Stability

4 Dynamically Substructured Systems Decomposition: Causality Analysis

4.1 Force Control, Physical Subsystem A, Type 1 Decomposition.

4.2 Position Control, Numerical Subsystem A, Type 1 Decomposition.

4.3 Force Control, Numerical Subsystem A, Type 2 Decomposition.

4.4 Position Control, Physical Subsystem A, Type 2 Decomposition.

4.5 Infeasible Cases.

4.6 Summary of Causality Analysis.

5 Dynamically Substructured Systems Decomposition: Structural Properties

5.1 Original Structure Structural Properties.

5.2 Structural Properties of Causal Dynamically Substructured Systems Decompositions.

6 Numerical Examples

6.1 Benchmark Structure.

6.2 Complex Frame Structure.

7 Conclusions

Footnotes

Conflict of Interest

Data Availability Statement

Nomenclature

Symbols

Appendix A. Controllability and Observability of Original Dynamics

Appendix B. H₂ Controllers Used for Numerical Results Shown in the Main Paper

References

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On the Feasibility of Dynamic Substructuring for Hybrid Testing of Vibrating Structures

Abstract

1 Introduction

2 Problem Statement and Methodology

2.1 Original Dynamics.

2.2 Structural Stability Analysis.

2.3 Summary of the Proposed Methodology for Dynamically Substructured Systems Feasibility Analysis.

3 Dynamically Substructured Systems Decomposition: Types and Structural Stability

4 Dynamically Substructured Systems Decomposition: Causality Analysis

4.1 Force Control, Physical Subsystem A, Type 1 Decomposition.

4.2 Position Control, Numerical Subsystem A, Type 1 Decomposition.

4.3 Force Control, Numerical Subsystem A, Type 2 Decomposition.

4.4 Position Control, Physical Subsystem A, Type 2 Decomposition.

4.5 Infeasible Cases.

4.6 Summary of Causality Analysis.

5 Dynamically Substructured Systems Decomposition: Structural Properties

5.1 Original Structure Structural Properties.

5.2 Structural Properties of Causal Dynamically Substructured Systems Decompositions.

6 Numerical Examples

6.1 Benchmark Structure.

6.2 Complex Frame Structure.

7 Conclusions

Footnotes

Conflict of Interest

Data Availability Statement

Nomenclature

Symbols

Appendix A. Controllability and Observability of Original Dynamics

Appendix B. H2 Controllers Used for Numerical Results Shown in the Main Paper

References

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Appendix B. H₂ Controllers Used for Numerical Results Shown in the Main Paper