## 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].

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 signal**S*_{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 signal**S*_{n}) which is compared with the force generated by the actuator situated at the corresponding interface in the physical part (*physical signal**S*_{p}). A feedback controller is then designed to minimize the *synchronization error**e* 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).

### 2.1 Original Dynamics.

*emulated system*in the DSS literature [4]) can be obtained by imposing force balance at each mass and read

*m*

_{i}is the

*i*th mass,

*c*

_{i}is the

*i*th damping coefficient,

*k*

_{i}is the

*i*th spring stiffness,

*y*

_{i}is the displacement of the

*i*th mass, and the dot indicates time derivative. The displacement of the support is indicated as

*y*

_{0}=

*d*and is considered unknown. An equivalent expression in the frequency domain can then be written as

*Num*

_{i}(

*s*) and

*Den*

_{i}(

*s*) represent, respectively, the numerator and the denominator of the transfer function between the position of the

*i*th and (

*i*− 1)-th masses. Similarly,

*Num*1

_{i}(

*s*) and

*Den*1

_{i}(

*s*) refer to the transfer function between the position of the

*i*th mass and the applied force

*F*.

*d*and the force

*F*can be assessed independently. For example, the transfer function between

*i*th mass displacement

*y*

_{i}and the disturbance

*d*can be written as

*i*th displacements

*y*

_{i}and the external force

*F*, thus obtaining

*Den*1

_{n}is equal to

*Den*

_{1}, 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.

Index i | $\u2220Numi$ | $\u2220Deni$ | $\u2220Num1i$ | $\u2220Den1i$ |
---|---|---|---|---|

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 |

Index i | $\u2220Numi$ | $\u2220Deni$ | $\u2220Num1i$ | $\u2220Den1i$ |
---|---|---|---|---|

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.

*b*

_{i}(

*s*) includes the common factor

*s*

^{2}and all of its coefficients are strictly positive. For example, given that all the parameters

*m*

_{i},

*c*

_{i}, and

*k*

_{i}in (4) 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 (7) 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 *l*th mass, whereas in type 2 decomposition the interface is placed after the spring-damper connected to the *l*th mass.

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 *S*_{1} can only be force and *S*_{2} 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.

*F*is the internal force applied at the interface (

*F*=

*S*

_{2}due to input constraint). Similar to the original system,

*Den*′

_{l+1}can be rearranged as

*b*′

_{l+1}has the common factor

*s*

^{2}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 (6

*πt*).

## 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.

*F*

_{n}at the interface and the physical force

*F*

_{p}that the actuator exerts on the physical subsystem.

^{3}

*only if*either the interface displacement

*y*

_{l}is passed as an input (first substitution in Eq. (14)) or the combination of physical force

*F*

_{p}and disturbance

*d*are passed as inputs (second substitution in Eq. (14)).

*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

*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. (16).

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

*l*th mass, $Numjyl$ and $Denyl$ represent the numerator and the denominator of the transfer function relating input $ylp$ to output

*y*

_{j}. According to Eq. (15), all of the transfer functions are causal.

*i*th mass in subsystem A.

*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

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

*F*

_{n}can be expressed as

*F*

_{n}cannot be causally estimated based on the physical displacement

*y*

_{l+1}(second line of Eq. (27)). However, this causality issue can be avoided if direct measurements of the physical force

*F*

_{p}are available, as suggested by Eq. (28).

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.

*y*

_{i}are causal.

*d*and

*y*

_{i+1}based on Eq. (28) and needs to be measured directly.

### 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.

### 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.

*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.

*y*

_{l}plays the role of the disturbance

*d*in the original system, therefore

*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.

*Den*

_{d}in Eq. (37) considers only the masses up to

*l*, therefore conditions analogous to the first and the last expressions in Eq. (33) are needed.

## 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.

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 |

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 *H*_{2} 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.

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.

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 *H*_{2} 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.

### 6.2 Complex Frame Structure.

*k*

_{34},

*k*

_{13},

*k*

_{12}, and

*k*

_{11}being set equal to

*k*

_{4},

*k*

_{3},

*k*

_{2}, and

*k*

_{1}, and similarly for the damping coefficients. In addition,

*k*

_{24}= 800 N/m,

*k*

_{23}= 500 N/m, and

*c*

_{24}= 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

*m*

_{1}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

*H*

_{2}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.

## 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

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.

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

*i*th mass- $\u230aH(s)\u230b$ =
maximum degree of transfer function

*H*(*s*)- $\u2220P(s)$ =
degree of polynomial

*P*(*s*)- $Den2Fp$ =
generic denominator in subsystem B with

*F*_{p}as input (type 2 decomposition)- $Denyl$ =
generic denominator in subsystem B with

*y*_{l}as input (type 1 decomposition)- $Denyl+1$ =
generic denominator in subsystem A with

*y*_{l+1}as input (type 2 decomposition)- $Numid$ =
numerator of transfer function between

and*d**y*_{i}- $Numid$ =
*i*th generic numerator in subsystem A with*F*_{p}as input (type 2 decomposition)- $Numjyl$ =
*j*th generic numerator in subsystem B with*y*_{l}as input (type 1 decomposition)- $Numiyl+1$ =
*i*th 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.

*Den*

_{n}and

*Den*

_{n−1}in Eq. (4) 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 (4) can be rewritten as

*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.

Index, j | $\u2220b1(j)$ | $\u2220b2(j)$ |
---|---|---|

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 |

Index, j | $\u2220b1(j)$ | $\u2220b2(j)$ |
---|---|---|

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 |

*Den*

_{n−j+1}and

*Den*

_{n−i−1}had common roots, then Eq. (A4) could be rearranged as

*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. (A4) can be rewritten as

*j*=

*i*+ 3 due to (A9)–(A12). Therefore, the following proof will focus on Eq. (A16) for

*j*>

*i*+ 3. To this end, note that given the non-zero polynomials

*X*,

*Y*,

*a*

_{1},

*a*

_{2},

*T*, the relation

*b*

_{1(j)}and

*b*

_{2(j)}as

*Num*

_{i}and

*Den*

_{i}(1 ≤

*i*≤

*n*) if the hypotheses listed in Eq. (33) hold. The original system is then fully controllable and observable.

## Appendix B. *H*_{2} Controllers Used for Numerical Results Shown in the Main Paper

## References

*H*

_{∞}/

*μ*Control Design for Dynamically Substructured Systems