Mechanical Systems and Signal Processing 209 (2024) 111084
0888-3270/© 2024 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
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Mechanical Systems and Signal Processing
journal homepage:www.elsevier.com/locate/ymssp
Structural system modelling from base excitation measurements using swarm intelligence
Corinna Cerini
∗, Vladimir Yotov, Guglielmo S. Aglietti
Te P ¯unaha ¯Atea - Space Institute, The University of Auckland, Auckland, 1010, New Zealand
A R T I C L E I N F O
Communicated by S. De Rosa Keywords:
Structural dynamics Mechanical vibrations System identification Particle swarm optimization
A B S T R A C T
Structural system identification is a fundamental task in mechanical engineering since it enables the characterization of the dynamic behaviour of structures from experimental data. The identification of the mass, stiffness, and damping matrices of a structure is essential for the design, analysis, and control of mechanical systems. This paper proposes a novel method for identifying the spatial properties of a structure using base excitation measurements and swarm intelligence. Maintaining relevant physical properties is ensured by placing algebraic constraints on factors of the mass and stiffness matrices. This formulation circumvents the complexities of additional constraints in metaheuristic techniques. Consistency of the eigenvalues with respect to the test data is assured through an integrated refinement process during the stiffness matrix estimation. A healing process enables the rescaling of the mass matrix, guaranteeing an accurate representation of the structure’s total mass. The search space definition only requires prior knowledge of the total mass of the structure and an estimate of anticipated maximum natural frequencies within the targeted frequency range. The efficacy of this method is tested on a numerical example.
1. Introduction 1.1. Background
In the space industry, one of the most important sources of information for the design and analysis of a Spacecraft (S/C) structure is its mathematical model. A dynamic test of the structure, which mimics the actual environmental conditions the system might experience, is necessary to validate the model and ensure confidence in subsequent analyses and the design of the hardware.
Consequently, the mathematical model must undergo a validation, which often involves an iterative process of model updating, intended to enhance the model’s accuracy and minimize the discrepancies between numerical simulations and experimental results.
Due to the high complexity of typical structures, this process can be costly and may not always yield satisfactory agreement between the two data sets. Recently, a comprehensive approach known as Virtual Testing, which models the complete testing facility, including the spacecraft, shaker, and control system, has shown promising results in pre-test simulations and post-test correlations [1–4].
An accurate identification of structure’s mass, stiffness, and damping matrices is vital for designing, analysing, and controlling mechanical systems. Various system identification techniques have been developed over the years to derive a mathematical model of a dynamic system directly from input and output observations [5]. This problem is classified as an inverse problem [6], and more
∗ Corresponding author.
E-mail address: [email protected](C. Cerini).
https://doi.org/10.1016/j.ymssp.2023.111084
Received 5 October 2023; Received in revised form 13 December 2023; Accepted 26 December 2023
specifically, it falls into the subcategory ofmodellingproblems: given a set of inputs and corresponding outputs from a system, find a mathematical description of the system [7]. Traditionally, these matrices have been identified using Experimental Modal Analysis, which requires measuring the structure’s response to different excitations, such as impulse, sine, or random. However, the entire procedure, from setup to data analysis, can be costly, time-consuming, and may not be feasible or practical in certain scenarios.
Optimization techniques can be used to address problems of this nature. The selection of a method for a particular application depends on the nature of the problem and the desired outcome.
1.2. Literature survey
The field of system identification encompasses a diverse array of methodologies, aimed at the development of dynamic system models based on observed data. The study in [8] exemplifies this diversity by comparing five distinct time-domain system identification techniques. These methods provide valuable insights into the structural vibration of a single-link flexible manipulator with curved geometry. Building upon these foundational methodologies, the system identification domain also encompasses optimization techniques. While effective, traditional optimization methods, such as gradient-based optimization algorithms, may encounter limitations addressing complicated engineering problems, e.g., multimodal, non-convex, and large-scale. Additionally, they require a good initial guess to start the process. Modern deterministic algorithms such as the Globally Convergent Method of Moving Asymptotes [9,10] have been shown to work very well on highly non-convex, non-linear, large-scale optimization problems with hundreds of thousands of design variables. Nature-Inspired Algorithms (NIAs) offer an alternative approach that has been adopted within this work. They are a subset of the expansive landscape of Artificial Intelligence (AI), contributing valuable tools for solving complex optimization and decision-making problems. In structural engineering, AI has emerged as an efficient alternative to classical modelling techniques, as elaborated in [11]. This comprehensive review highlights how AI, encompassing machine learning, pattern recognition, and deep learning, addresses engineering problems associated with uncertainties, accelerates the decision-making process, reduces error rates, and enhances computational efficiency. AI-based solutions have the potential to determine engineering design parameters when testing is not feasible, resulting in significant savings in terms of human time and effort spent on experiments. Among these new generations of NIAs, the Evolutionary Algorithms (EAs) and the Swarm-Based Algorithms (SBAs) have emerged to be the most promising, and they have been successfully applied to a wide range of optimization problems, including system identification. NIAs include a broad class of computational algorithms inspired by natural processes and phenomena. EAs draw inspiration from natural selection and evolution, using mechanisms such as mutation, crossover, and selection to evolve a population of candidate solutions towards an optimal solution. Prominent examples of EAs [12] includes the Genetic Algorithm (GA) [13,14], the Differential Evolution [15], Evolutionary Strategy [16], Genetic Programming [17], and Evolutionary Programming [18]. On the other hand, SBAs originate from the concept of Swarm Intelligence (SI), a research field focusing on the collective behaviour of self-organized systems comprising simple agents.
SI algorithms derive their inspiration from the behaviour of social animals like birds, ants, and bees to enhance a system’s performance. By interacting with their environment and fellow agents, these agents make decisions that contribute to optimizing a global objective. Consequently, the defining characteristic of SI is the decentralized nature of the system, in which decision-making is distributed among the agents. In recent years, SI has gained significant interest in various domains such as engineering, computer science, and finance, owing to its efficacy in resolving complex problems. Examples of SI algorithms include the Artificial Bee Colony [19] and Ant Colony Optimization [20], inspired respectively by the foraging behaviour of honeybees and ants. The Firefly Algorithm [21] is based on fireflies’ bioluminescent behaviour, wherein their flashing light patterns serve to attract mates or prey, and a firefly’s brightness signifies the quality of a solution within the search space. Drawing from the principles of electromagnetism, the Electromagnetism-like Mechanism algorithm [22] simulates the behaviour of charged particles, which attract or repel each other, as dictated by Coulomb’s law. The most representative SI algorithm is the Particle Swarm Optimizer (PSO) [23,24], which has been effectively employed in diverse structural engineering problems, including structural design optimization [25], composite structure optimization, model updating [26], structural health monitoring [27], and system identification and parameter estimation [28–31].
Moreover, SI methods have demonstrated their potential to enhance the performance of control systems. For instance, a recent study applied SI to tune a modified linear quadratic Gaussian controller, resulting in significant improvements during earthquake events [32]. Although only a few algorithms are mentioned here, a more comprehensive overview of available SI methods can be found in [33].
1.3. Current limitations
System identification using optimization techniques often necessitates employing restrictive simplifications. Defining the search space constitutes a typical example. While literature often recommends setting the upper and lower bounds of the search space based on target values to highlight algorithm efficiency, this method falls short in the absence of initial system knowledge. Also, this leads to a restricted search space where the solution (i.e., the target values) inherently exists within the predefined bounds. In [34], for example, the author uses an EA to solve the system identification problem of a 3- and 10-Degrees of Freedom (DoFs) mass, spring and damper systems. The upper and lower bounds of the variables are set at twice and 0.1 times the target values. This approach might not be applicable to real-world scenarios. Therefore, this approach holds only close to the target solution. Another simplification in system identification is the assumption of matrix sparsity. A common practice is to consider mass, spring, and damper systems, where the mass matrix is diagonal [35,36], thereby neglecting any coupling terms. In some cases, the mass distribution is assumed to be known, such as in [37,38]. In such systems, the stiffness matrix is tridiagonal [38,39]. Additionally, these methods attempt
to estimate the physical properties, such as the stiffness of springs. However, when constructing the stiffness matrix, a detailed understanding of its entries derived from these physical properties is essential. For instance, the (1–1) entry might be represented as𝑘1+𝑘2+𝑘3, and so on, [40]. Achieving such specificity can be challenging. Moreover, these methods cannot reproduce matrices originating from Finite Element (FE) method discretization of real-world systems, wherein trivial elementwise bounds on the matrix entries are unrealistic. Many techniques also assume known forces. The reliance on force measurements is a recurring (underlying) assumption [36]. While this might offer some valuable insights, it also necessitates additional measuring devices, adding layers of complexity and potential inaccuracies. It would be helpful to reduce or eliminate the need for gathering such data. The management of constraints in metaheuristic algorithms is another challenge, requiring extra care [13,41].
Several challenges associated with system identification have been addressed in [42–44]. In [42], an algorithm is introduced which relies solely on acceleration measurements, with the focus being on a 5-DoFs structure having a diagonal mass and full stiffness matrix. The same mass matrix configuration is further explored in [43], with an evolution in the damping approach, transitioning from arbitrary matrix entries to a simplified Rayleigh damping model with only two variables. [44] extends the investigation to incorporate a full mass matrix, emphasizing the significance of the mass coupling term and introducing the concept of equivalent forces. Across these works, the stiffness remains consistently full, and the damping treatment evolves. Moreover, since its initial introduction in [42], the healing process, has seen progressive refinements across the subsequent studies.
1.4. Scope and contribution of this study
This study introduces a novel approach to structural system identification, addressing the traditional challenges and limitations outlined in Section1.3. This methodology overcomes the problem of defining a search space for the mass, stiffness and damping matrices. By converting the Generalized Eigenvalue Problem (GEP) into an ordinary eigenproblem, the search space for the stiffness matrix can be defined independently of prior system knowledge, as detailed in Section 3.2.4. This approach uses information only on the last eigenvalue within the relevant frequency range. Additionally, transformations simplify the search space definition for the mass matrix (Section3.2.3), and the damping search space is based on plausible value ranges from existing literature (Section3.2.5). The search space is defined in Section3.2.6. This method also addresses the sparsity of matrices. In fact, it allows for any meaningful fully populated mass and stiffness matrices, incorporating coupling terms to provide a more detailed and realistic system representation, as explained in Sections3.2.1,3.2.3and3.2.4. This is particularly evident in the mass matrix, where coupling terms link the matrix elements associated with the DoFs of the structure and the interface where input is applied, or in the stiffness matrix, where the tridiagonal representation is relieved. This comprehensive approach brings this system identification process closer to real-world applications, offering a more realistic and detailed representation of the system. Furthermore, the common assumption on known forces is effectively alleviated. This proposed approach is distinguished by the use of acceleration data alone, both as input and output, in the optimization process. This focus on acceleration data simplifies the identification process, substantially reducing the need for elaborate measuring setups and minimizing the potential inaccuracies typically associated with force measurements.
This aspect, not only streamlines the process, but also broadens its applicability, especially in scenarios where force measurements is challenging or impractical. Importantly, in cases where force measurements are available, the proposed method offers the flexibility to adapt. The cost function can be slightly modified to integrate the presence of these known forces. Regarding the handling of the constraints, this approach inherently ensures the positive definiteness of the mass and stiffness matrices, thereby eliminating the need for separately imposed constraints to achieve this condition. In fact, the algorithm, instead of acting directly on these matrices, works with their Cholesky factors. Explanation on how is done can be found in Sections3.2.2to3.2.4. Furthermore, a rescaling of the mass matrix, here calledhealing process, ensures that the constraint on the total mass is met.
In summary, this paper proposes a novel method for identifying a structural system’s mass, stiffness, and damping matrices using only base excitation measurements. Building upon the algorithm introduced in [42], the refined Hybrid PSO with Local Search (hPSO-LS) outlined here optimizes the cost function based on the difference between measured and predicted structure accelerations. The study refines thehealing processfor mass matrix rescaling and introduces a mathematical model formulation that enhances constraint handling. hence, key features of the proposed method include the use of only acceleration data, the ability to define search spaces without prior knowledge of the structure, the handling of arbitrary mass and stiffness matrices, the inclusion of structure-to-interface mass coupling terms, and a simplified approach to constraint management.
1.5. Organization of the paper
In addition to Section1, which sets the stage for this research, the structure of this manuscript is outlined as follows. Section2 lays out the foundational concepts from the mathematical and the optimizer point of view. Section3shows the core of this work, discussing the algebraic foundation and the settings of the optimization algorithm. In Section4, a test case is defined and the results are shown and discussed. Finally, the conclusions in Section5, where the key findings are summarized and potential avenues for future research are provided.
2. Technical background 2.1. Structural dynamics
A common challenge in applying system identification to real-world structures is the problem complexity and thus large number of DoFs needed to characterize the relevant dynamics. Constraints in sensor availability and other practical considerations hinder access to comprehensive measurement data. As highlighted in Section1, this research aims to estimate the spatial properties of a structure from base excitation analysis. To approach this, it is relevant to consider the concept of a reduced model, presented in Section2.1.1. While complete information acquisition from tests is infeasible, even numerical simulations often rely on some reduced models. Following this, in Section2.1.2, the base-shake analysis technique, is detailed.
2.1.1. Domain decomposition
Building upon the earlier discussion, the concept of domain decompositiondeserves introduction. This computational strategy involves analysing complex problems by separately examining their components and interface solutions [45]. It serves as the basis for several techniques for handling structures within a limited number of DoFs. Pioneering concepts in Dynamic Substructuring were showcased in [46–48], primarily focusing on reduction techniques. A notable advancement within this field is Component Mode Synthesis (CMS), where substructures transition from discretized models to system components represented by generalized responses [49]. This method comprises three main steps. First, dividing the structure into components, then defining sets of components modes and finally integrating the component mode models to construct a reduced-order system model [50]. Among all the CMS techniques, the Craig-Bampton (CB) method [51] remains the most popular within the space industry, despite numerous improvements that have been suggested [52,53].
2.1.2. Mathematical formulation
In this section, the expression for the final mathematical model is derived, aiming to comprehensively represents the structural system under investigation. The theoretical development is based on the formulation in [54]. The DoFs can be partitioned in two sets:
•The𝑆-set, representing the free DoFs, is associated with the displacement𝐮𝑆
•The0-set, where the input is applied, is associated with the displacement𝐮0, which comprises both translational and rotational support motions:
𝐮0= [𝐮0𝑥,𝐮0𝑦,𝐮0𝑧, 𝜃0𝑥, 𝜃0𝑦, 𝜃0𝑧] (1) The equation of motion in the partitioned form is:
[𝐌𝑆𝑆 𝐌𝑆0 𝐌0𝑆 𝐌00
] [𝐮̈𝑆 𝐮̈0 ]
+
[𝐊𝑆𝑆 𝐊𝑆0 𝐊0𝑆 𝐊00
] [𝐮𝑆 𝐮0 ]
= [𝐟𝑆
𝐟0 ]
(2) Under the assumption that no external forces act on the structure, i.e.,𝐟𝑆 = 0, it is possible to compute the response of the unrestrained DoFs and the reaction forces between the structure and its supports, respectively, as:
𝐌𝑆𝑆𝐮̈𝑆+𝐊𝑆𝑆𝐮𝑆= −𝐌𝑆0𝐮̈0−𝐊𝑆0𝐮0 (3a)
𝐟0=𝐊0𝑆𝐮𝑆+𝐊00𝐮0+𝐌0𝑆𝐮̈𝑆+𝐌00𝐮̈0 (3b) Since the structural vibrations occur as motions relative to the rigid-body movements, it is beneficial to divide displacements and accelerations into their absolute and relative components. An extensive derivation of the kinematic equation from the description of the location of a generic point on a deformed structure is given in [55,56]. The kinematic relation results in:
[𝐮𝑆 𝐮0 ]
= [𝐈 𝐆𝑆0
𝟎 𝐈 ] [𝐯𝑆
𝐮0 ]
(4) 𝐆𝑆0is the static condensation matrix at foundation level describing the static deformations on𝐮𝑆for unit displacements on𝐮0.𝐯𝑆 represents the dynamic part of the response, i.e. the perturbations relative to the quasi-static response describing the vibration of the structure on its support, hence defined as relative displacement. By substituting Eq.(4)into Eq.(3a), and using𝐆𝑆0= −𝐊−1
𝑆𝑆𝐊𝑆0 from static reduction, the equation governing the motion of the𝑆-set takes the form:
𝐌𝑆𝑆𝐯̈𝑆+𝐊𝑆𝑆𝐯𝑆= −(𝐌𝑆𝑆𝐆𝑆0+𝐌𝑆0)̈𝐮0 (5)
The right side of Eq.(5)represents the equivalent force computed in terms of the acceleration of the support and the static modes that its motion generates.
2.2. Particle swarm optimization algorithm
This section aims to describe the PSO algorithm. It starts with an exploration of its foundational principles as outlined in the original version, progressing to discussions on its evolutions through adaptation and hybridization techniques to further refine and expand its capabilities. Moreover, this section highlights some on the inherent complexities tied to these algorithms, when dealing with system identification, such as the definition of a search space and addressing constraints.
Fig. 1.Schematic illustration of the particles’ dynamics in a multidimensional space, with the three components of Eq.(7a)represented by coloured arrows:
blue for thememory term, orange for thecognitive componentand green for thesocial component. The black arrow indicates the resultant updated velocity, leading to the new position as defined in Eq.(7b).
2.2.1. The original version
PSO is a stochastic, population-based search and optimization technique first introduced by Kennedy and Eberhart in [23,24].
Drawing inspiration from the social and cooperative behaviours exhibited by various species, PSO simulates the movements of the individuals, the so-calledparticles, within a flock of birds or a school of fish. The ultimate goal is to converge to the global optima of a multidimensional, potentially nonlinear function. In PSO, particles represent potential solutions, with the𝑛th particle at iteration tbeing associated with a vector of its current position and velocity,𝐩𝑛𝑡 and𝐯𝑛𝑡 respectively:
𝐩𝑛𝑡 =[
𝑃𝑡𝑛,1, 𝑃𝑡𝑛,2,…, 𝑃𝑡𝑛,𝐷]
(6a) 𝐯𝑛𝑡 =[
𝑉𝑡𝑛,1, 𝑉𝑡𝑛,2,…, 𝑉𝑡𝑛,𝐷 ]
(6b) where D is the number of parameters to be optimized.
These particles navigate through the parameter space of the problem, seeking improved search regions by learning from historical information gathered throughout their journey. Key pieces of this information are theirpersonal bestposition (orlocal attractor),𝐥𝑛𝑡, and theglobal best(orglobal attractor),𝐠𝑡, which is the best position visited so far by the swarm. The particles’ dynamics within the search space are governed by updating equations of their velocity and position. Consequently, the velocity of a particle in the next iteration is computed as a function of its previous velocity (memoryterm), its best personal position (cognitivecomponent), and the swarm’s best position (socialcomponent). The swarm’s behaviour is influenced by the combination of these three components, each weighted by scalars𝑤,𝑐1, and𝑐2as follows:
(7a) (7b) A schematic representation of the movements of the particles in the search space is shown inFig. 1, where the three different components of the velocity updating of Eq.(7a)are highlighted.𝑤, is the so-calledinertia weight [57,58], introduced to avoid the phenomenon ofexplosion[59], where particles’ velocities move toward infinity. Alternative techniques, shown in previous works, contain the explosion by the so-calledvelocity clamping, with a user defined maximum velocity𝑣max [60], or by defining some constriction coefficients[61]. The acceleration coefficients,𝑐1(personal acceleration coefficient) and𝑐2(social acceleration coefficient), together with the random vectors𝐫𝑛𝑡 and𝐬𝑛𝑡, which are drawn from a uniform distribution on [0,1], control the stochastic influence of the cognitive and social components on the overall velocity of a particle [23]. Choosing the appropriate values for the parameters in PSO can be challenging. To address this issue, adaptation techniques have been introduced in the original algorithm. A brief introduction on the topic is presented in Section2.2.2, as well as the strategies adopted in this paper.
A typical stopping criterion for the PSO is set as a maximum number of iterations𝑡max. Although a predetermined maximum number of iterations is not usually known in advance, guidelines are available [62]. Alternatively, one can conduct trial and error tuning of the minimum𝑡max needed to achieve the optimum. Another common criterion is the value of the cost function, if the desired value is known. The criteria adopted in this paper are discussed in Section3.3.3.
2.2.2. Adaptation techniques
The term adaptation refers to the ability of the process to gain knowledge about the problem during the iterations. It can be seen as a learning process, in which some information about the status of the optimization is used to vary the parameters𝑐1,𝑐2and 𝑤. Regarding the latter, the strategy used in this paper has been presented in [29] and it is known as Improved PSO. This strategy
is based on the change rate of focusing distance of the particle introduced in [63]. The focusing distance can be seen as a measure of the spread or dispersion of the particles during the optimization process. The change rate of this metric is defined as:
k=𝛥max−𝛥mean
𝛥max (8)
where𝛥max and𝛥mean are, respectively, the maximum and the average focus distance between all particles and the historical optimum in the population. The formulas are described as follows:
𝛥max= max
𝑛=1,2,…,𝑁‖‖𝐠𝑛−𝐩𝑛𝑡‖‖ (9)
𝛥mean=
∑𝑁
𝑛=1‖‖𝐠𝑛−𝐩𝑛𝑡‖‖
𝑁 (10)
Theself-adaptivefunction that describes the variation of the inertia weight is:
𝑤=
⎧⎪
⎪⎪
⎨⎪
⎪⎪
⎩ (
𝛽1+|b| 2
)
k>1 𝛽1𝛽2+|b|
2 0.05≤|k|≤1 (
𝛽2+|b| 2
) 1
|lnk| |k|<0.05
(11)
where𝛽1and𝛽2can be adjusted to suit the specific problem at hand and b is a random number in [0,1]. Variations to that adaptation scheme are shown in [64]. Note that the updating strategy of the inertia weight considered in this paper, as well as the choice of the described parameters, are the ones presented in [29]. The acceleration coefficients are instead updated during the iteration process using the information on the Evolutionary State Estimation (ESE), as presented in [65], in the Adaptive PSO. There are four identified states in which the particle might be in:
1. Exploration: During the exploration state, the particles move randomly in search of new solutions. The objective of this state is to cover the entire search space and find new solutions that may be better than the current ones.
2. Exploitation: During the exploitation state, the particles move towards the solutions with the best objective function values found. This state aims to find the best solutions in the exploration state.
3. Convergence: When the exploitation state reaches its optimum, the particles may converge towards a solution, leading to stagnation in the optimization process.
4. Jumping out: When the optimization process reaches the convergence state, the optimization may become stuck in a local optimum. Therefore the jumping-out strategy is introduced, where the particles are randomly moved to different regions of the search space. The objective of this state is to allow for further exploration of the search space.
For further information on ESE, the reader can refer to [66,67], where the first concept of that was introduced. The details of the implemented strategy are presented in [65]. Additionally, [42] shows this adaptation technique applied to the hPSO-LS.
2.2.3. Hybridization techniques
Hybridization in PSO may refer to the inclusions of some operators that characterize other optimization techniques, such as the well-known mutation or crossover operators of the GA. Results have shown that these additions to PSO can enhance optimization performance. Hence, several variants of the PSO have been proposed, incorporating the capabilities of other evolutionary algorithms.
In [68] a review on that topic is given, along with insight on the dependency of the design of the operators to the nature of the problem. Local search techniques offer another avenue of hybridization. They can be integrated to fine-tune solutions in the vicinity of potential optima. The terms of hybridization pertinent to the proposed algorithm are clarified in Section3.3.4.
2.2.4. Search space and penalty techniques
The PSO was originally developed to deal with unconstrained optimization problems. Hence, a central problem for EAs, GAs, PSO, etc., lies in handling constraints inherent to real-world optimization problems. The general form of a Constrained Optimization Problem (COP) is as follows:
min𝑝∈(𝐩)
s.t. 𝑗(𝐩)≤0 ∀𝑗= 1,2,…, 𝐽
𝑖(𝐩) = 0 ∀𝑖= 1,2,…, 𝐼
(12)
where𝐼is the number of equality constraints and𝐽 is the number of inequalities constraints. A point that satisfies all equalities and inequalities constraints is called a feasible point. The search space is a multidimensional space defined by the upper and lower bounds of the admissible region. In general, solution space contains a feasible area and an infeasible area. The most common technique to solve COP is introducing a penalty function [14,69,70]. This approach transforms the constrained problem into an unconstrained one by penalizing infeasible solutions. Due to the difficulty in designing an effective penalty function that can guide the algorithm towards promising areas of the solution space, researchers have proposed different methods, such as constraint- preserving methods [70,71] and repairing methods [13,41]. The latter transforms an infeasible solution into a feasible solution
by replacing the particles. In [72], the author presents a healing process based on the mutation operator from GA, to heal thesick particles. The state of the art is covered in [73,74]. Since there are no standard heuristics for the design of repair algorithms, this issue needs to be addressed.
Defining the search space in the context of structural system identification can be challenging, particularly when there is limited or no prior knowledge about the structure. Some bounds can be established based on physical constraints for mass–spring–damper systems. For example, mass and stiffness coefficients must be positive, and damping coefficients should lie within a reasonable range. In literature, the search definition is commonly based on known values of the actual variables, as it often serves to assess and demonstrate the efficiency of a newly proposed algorithm [35,36,38]. In scenarios where there is no knowledge about the system, a common technique is iterative refinement. This involves starting with a broad search space and progressively narrowing it down based on the optimization results [75]. However, it is essential to have some guidance on how to define a search space that is sufficiently large to ensure that the solution is encompassed within it. The stiffness matrix poses a unique challenge in this regard, as the definition of its search space is heavily reliant on the discretization of the structure, i.e., the number of DoFs, and consequently, the number of sensors used during the test campaign.
This paper presents a novel, rigorous approach to concurrently address both the challenges of constraint handling and search space definition. The strategy adopted and its detailed implementation are discussed in Sections3.2.2–3.2.5.
3. Proposed algorithm
3.1. Overview
The hPSO-LS is a two-phase algorithm designed for estimating the spatial properties of a generic structure subject to constraints.
The first phase, theGlobal Phase, employs the PSO algorithm to efficiently explore the design space and provide a rough estimate of the near-optimal solution. In the second phase, called theLocal Phase, the interior-point algorithm is applied using thefmincon function already implemented in MATLAB. This step refines the solution obtained from theGlobal Phase, ensuring that it satisfies the imposed constraints. One of the main advantages of this approach is that it combines the global search capabilities of PSO with the local optimization capabilities offmincon, resulting in a more robust and accurate solution. This method is an extension of the approach first introduced in [43,72], which focused on estimating the spatial properties of a structure under single-axial excitation.
A significant advancement in the current work is the consideration of the mass matrix coupling terms, which are often neglected in the literature. In [44], a strategy for incorporating these coupling terms was proposed by employing the concept of equivalent forces. However, the problem of defining the search space for these equivalent forces remained unresolved. The current work aims to address this open issue and further develop the methodology to enhance the estimation of the structure’s spatial properties under multi-axial excitation. In Section3.2.3, a novel method for addressing the coupling problem is introduced. It builds upon the foundation laid in [44]. This method results in a slightly increased number of unknowns but allows for a more comprehensive handling of the mass matrix.
In identifying a structure’s spatial properties through base excitation, the overall test structure needs to be mounted on a shaker table and subjected to acceleration in different spatial degrees of freedom. The measurement of the multi-axis base acceleration (input) and the structural responses (output) induced by the base excitation is necessary to form the basis for the identification of structural dynamics. In this paper, a simulation of the experimental test is conducted. Hence, the accelerations in input and output are obtained and filled in the cost function, as described in Section3.2.1.
A flowchart illustrating the systematic progression of the proposed hPSO-LS is presented inFig. 2. This graphical representation offers an organized breakdown of the entire procedure. The algorithm starts with theSwarm Initialization, employing the Latine Hypercube sampling (LHS) method, as elaborated in Section3.3.1. The initial global best is identified from the pool of particles generated during initialization, and the termination criteria are assessed. If the termination conditions are met, the solution proceeds to theLocal Phasefor further refinement. Conversely, if the conditions are not satisfied, the iterative process initiates. First, the PSO coefficients are updated, as outlined in Section3.3.2. Subsequently, particles’ position and velocity undergo updates using Eq.(7).
TheSwarm evaluationstep encompasses the algebraic framework central to this work, as introduced in Section3.2. In this phase, the matrices are estimated from the particles’ position, following the process presented in Section3.2.3, and the resulting accelerations are used to feed the cost function, as described in Section3.2.1. The global best solution is continually identified throughout the iterative process, and the termination conditions, detailed in Section3.3.3, are monitored.
3.2. Algebraic foundations
In this section, the mathematical foundations of this work is shown. It starts with the derivation of the cost function, used in the GlobalandLocal Phases. Then it breaks down into the mathematical formulation of all components of the cost function, concluding with the definition of the search space.
Fig. 2.Flowchart depicting the steps of the proposed hPSO-LS algorithm for structural system identification.
3.2.1. Mathematical derivation of the cost function
The underlying mathematics was previously detailed in Section2.1.2. The cost function originates from Eq.(5)but is recast into the frequency domain for streamlined computations, as:
(−𝜔2𝐌𝑆𝑆+𝐊𝑆𝑆) 𝐚̃𝑆=(
𝐌𝑆𝑆𝐆𝑆0+𝐌𝑆0)
𝐮̃0𝜔2 (13)
Here,𝜔represents the angular frequency. In Eq.(13),𝐚̃𝑆is the relative acceleration, while𝐮̃0is the support acceleration. The modal superposition method is employed for the simulations, given its adeptness in managing systems with multiple degrees of freedom. To account for energy dissipation, modal damping is used in the analyses. First, Eq.(13)is converted into modal coordinates and solved for the modal acceleration𝐪̃𝑆. Then, the mode superposition is applied to convert the response back to the physical coordinates:
𝐚̃𝑆=𝚽𝐪̃𝑆 (14)
where𝚽is the matrix of eigenvectors of the GEP associated with the𝑆-set.
Let us define𝐀target as the target relative acceleration matrix and𝐀estimatedas the estimated relative acceleration. The latter is derived from Eq.(13). Note that sensors measure absolute accelerations, therefore it is possible to derive the relative accelerations of the target system from Eq.(4)and collect them in the matrix form,𝐀target. The cost function, based on the Frobenius norm of
, which is the difference between target and estimated acceleration matrices, is given by:
=‖‖𝐹=
√√
√√
√
𝑁𝑝
∑
𝑖=1 𝑁𝑓
∑
𝑗=1
|𝑒𝑖𝑗|2 (15)
with𝑁𝑝and𝑁𝑓respectively denoting the number of physical DoFs and the count of excitation frequencies. The Frobenius norm is chosen because it offers a straightforward way to quantify the discrepancy between matrices.
In order to achieve a minimized discrepancy between the target and estimated accelerations, a precise estimation of the system matrices is paramount. These matrices, encapsulated within the optimization vector𝐱, are central to the introduced analytical approach. The methodology to reconstruct the mass, stiffness, and damping matrices from𝐱will be expounded upon in the following subsections. The optimization strategy then seeks to find the optimal configuration of𝐱∈, such that: →R𝑁𝑃×𝑁𝐹, defined in Eq.(15), is minimized. The solution of the minimization problem is denoted as𝐱∗:
𝐱∗∈ (𝐱∗)≤ (𝐱) ∀𝐱∈ (16)
Note that Eq.(13)includes the presence of a mass coupling term between the𝑆-set (structural DoFs) and the0-set, where the input is applied. An earlier approach presented in [44], employed the concept of equivalent forces to account for these mass coupling terms.
This method condensed the quantification of coupling into a vector of length𝑁𝑃, reducing the number of variables to be estimated by abstracting the complexity of coupling into this vector representation. However, the challenge of defining a meaningful search
space for the force persisted, particularly without prior knowledge of the structure. To address this issue, this paper introduces an expanded mass matrix, that considers the presence of the additional node0, which contributes an additional𝑁𝐷by𝑁𝐷 matrix, with𝑁𝐷being the number of excitation directions. As such, the mass matrix that the optimizer aims at estimating is not only the partition related to the𝑆-set, but:
𝐌=
[𝐌𝑆𝑆 𝐌𝑆0 𝐌𝑇𝑆0 𝐌00 ]
(17) It is worth noting that the node0can be assigned without loss of generality and serves a dual role. Firstly, it acts as the application point for inputs. Secondly, it forms the interface that accounts for a portion of the mass not captured by the reduced mathematical model.
3.2.2. Physical constraints
In the pursuit of modelling real-world systems, it is imperative that the analytical constructs reflects the inherent physical properties of such systems. While mathematical formulations provide a framework, not all solutions they produce may correspond to a physically plausible scenario. In fact, the physically possible solutions are a subset of the obtainable values of the parameters in. However, restricting the search space to the feasible region is nearly always impossible, given the nature of the constrictions.
Therefore, several specific constraints must be acknowledged when estimating the mass, stiffness, and damping matrices of a mechanical system:
•Positive definiteness of the mass and stiffness matrices.
•Ensuring the correct total mass of the structure.
•Boundedness of the mass, stiffness and damping matrices.
In particular, the positive definiteness is addressed using the Cholesky decomposition, and the complex issue of the boundedness of the stiffness matrix is tackled by transforming the generalized eigenproblem into an ordinary one. For the damping matrix, modal damping is leveraged to define it, followed by a reverse transformation into physical coordinates. The rationale behind this choice is that in many applications, the plausible values of modal damping are well-known. This makes it more straightforward to define a search space, as opposed to guessing the bounds on the entries of the matrices or the values that delineate the proportional damping.
Furthermore, modal damping is strictly a superset of proportional damping, which ca be restrictive in many practical situations.
3.2.3. Mass matrix
Rather than directly estimating the mass matrix𝐌, this procedure determines its Cholesky factor,𝐋𝑀. By doing so, the constraint of positive definiteness of𝐌is inherently satisfied. This guarantees the physical plausibility of the derived mass matrix, without the need to impose any extra constraints. The procedure for estimating𝐋𝑀 does not directly yield the matrix itself; instead, it is derived through two distinct steps, as shown inFig. 3.
1. Mass distribution. The primary aim of this initial step is to ensure that the mass matrix is, in a statistical sense, over a large number of optimizer guesses, uniformly distributed. This is crucial to prevent a concentration of mass in particular regions, which could distort the physical representation of the system.
2. Enforcement of total mass. In this subsequent phase, the entries of the Cholesky factor undergo a rescaling process. The purpose is to guarantee that the mass matrix aligns with a predetermined total mass value. This step goes under the name of healing process.
First, define the subset𝑚 of the vector of unknowns,𝐱, related to the entries of𝐋𝑀. The initial step is specifically designed to enhance the optimization process. While in a previous work [44] norm inequalities were utilized to define the boundaries of the entries, the results, although mathematically sound, can sometimes lead to an uneven distribution, with a substantial mass concentration in the DoFs with higher IDs. This observation prompted a revision to the approach. Instead of norm inequalities, an auxiliary search space for𝐱𝑚 on [−1,1] is defined. A variable transformation𝑘(𝐱𝑚)maps𝐱𝑚 to𝐲, such that𝐲resides in the same interval, but now the mass diagonal entries have the same expectation, instead of being dependent on the row number k. The function under consideration is the following:
𝑘(𝐱𝑚) =sgn(𝐱𝑚)[ 1 −(
1 −||𝐱𝑚||)𝛼(𝑘)]
(18) The parameter𝛼(𝑘)is crucial as it stabilizes the norm of each row of the Cholesky factor. The relationship between𝛼(𝑘)and the expectation of the𝑘th row norm, assuming optimizer sampling induces uniform distributions for the variables, is:
∫
1
−1
2(𝐱𝑚)𝑑𝑥= 𝑐
𝑘 (19)
Evaluating the above integral yields to two possible solutions for the parameter𝛼:
𝛼1,2=3𝑐±√ 𝑐2+ 16𝑐𝑘
2(2𝑘−𝑐) (20)
Here, the parameter𝑐offers a degree of flexibility in its choice, subject to the constraint𝑐 <2𝑘, where𝑘being the row-number (or number of non-zero entries on that row). Thus, for a fixed𝑐across the problem domain, and as𝑘spans from 1 up to the largest
Fig. 3.Flowchart illustrating the transformation process from initial Cholesky factor,𝐋𝑀(𝐱𝑚), to the final matrix,𝐋𝑀(̂𝐲), via an intermediary transformation, 𝐋𝑀(𝐲).
number of rows, the mentioned relation leads to𝑐 <2. The rationale behind this limitation stems from the requirement to maintain a positive𝛼 across all rows. This is crucial as it ensures that the transformation function(𝑥)retains the sign of𝑥. Out of the two solutions derived from Eq.(20), only one emerges as positive. It is worth mentioning that these parameters are predetermined and set at the commencement of the optimization process. As a result, in the context of computational overhead, this approach introduces only the evaluation of Eq.(18). At this stage, the only constraint not satisfied by design is the one on the total mass. This is achieved in a second step, by rescaling the entries of𝐋𝑀. Considering only the translational DoFs, three scaling constants must be determined to appropriately rescale the Cholesky factor, ensuring the desired total mass for each direction. Let𝐥𝑀 ,𝑘denote the 𝑘th row of𝐋𝑀, where𝑘= 1,…, 𝑁𝑃+𝑁𝐷. The rows of𝐋𝑀 are grouped according to the specific direction they represent, i.e.𝑥,𝑦, or𝑧. Let𝑖denote the set of indices corresponding to the rows of𝐋𝑀associated with direction𝑖. Then,
trace(𝐌) = ∑
𝑖∈𝑥,𝑦,𝑧
trace(𝐌𝑖) =
𝑁𝑃∑+𝑁𝐷
𝑘=1
𝐥𝑀 ,𝑘𝐥𝑇𝑀 ,𝑘 (21)
Using the shorthandTr𝑖= trace(𝐌𝑖), Tr𝑖=∑
𝑘∈𝑖
𝐌𝑘,𝑘, 𝑖=𝑥, 𝑦, 𝑧 (22)
Given that𝑀is the total mass of the structure, the𝑖-set rows of𝐋𝑀 are scaled by factors𝑠𝑖=√
𝑀∕ Tr𝑖. Thishealing processplays a crucial role in the optimization, ensuring that the total mass complies with the constraint and is physically meaningful. Finally, the mass matrix can be computed as follows:
𝐌=𝐋𝑀(̂𝐲)𝐋𝑇𝑀(̂𝐲) (23)
Breaking𝐋𝑀 down block-wise:
𝐋𝑀=
[ 𝐋𝑀𝑆𝑆 0 𝐌𝑇𝑆0𝐋−𝑇𝑀
𝑆𝑆 𝐋𝑀̄00
]
(24) where𝐋𝑀̄00 is the Cholesky factor of the Schur complement𝐌̄00:
𝐌̄00=𝐌00−𝐌𝑇𝑆0𝐌−1𝑆𝑆𝐌𝑆0 (25)
3.2.4. Stiffness matrix
In this section, the process to estimate the stiffness by using the Cholesky decomposition is detailed. The primary aim is to ensure positive definiteness and symmetry. As touched upon in Section3.1, one of the key challenges lies in adequately defining the search space for the stiffness matrix. In this paper, the problem is addressed through the introduction of a matrix𝐇, which arises from the reduction of the GEP
𝐊𝚽=𝐌𝚽𝚲 (26)
to an ordinary one and has the property𝚲𝐻 =𝚲𝐾,𝑀, where𝚲𝐾,𝑀 contains the eigenvalues of the GEP and𝚲𝐻 the ones of𝐇.
Indeed, premultiplying Eq.(26)by𝐋𝑇𝑀𝐌−1and writing𝐈=𝐋−𝑇𝑀𝐋𝑇𝑀yields 𝐋𝑇𝑀(
𝐋𝑀𝐋𝑇𝑀)−1
𝐊( 𝐋−𝑇𝑀𝐋𝑇𝑀)
𝚽=𝐋𝑇𝑀𝐌−1𝐌𝚽𝚲 (27)
which leads to
𝐇𝚿=𝚿𝚲 (28a)
𝐇=𝐋−𝑇𝑀𝐊𝐋−𝑇𝑀 (28b)
𝚿=𝐋𝑇𝑀𝚽 (28c)
Fig. 4.Flowchart illustrating the transformation process from initial Cholesky factor,𝐋𝐻(𝐱ℎ), to the final matrix,𝐋𝐻(̂𝐱).
Observe that𝐇is Gramian, thus nonnegative definite. Then its Cholesky factor is
𝐇= (𝐋−1𝑀𝐋𝐾)(𝐋−1𝑀𝐋𝐾)𝑇=𝐋𝐻𝐋𝑇𝐻 (29)
and is simply related to that of𝐊by
𝐋𝐾=𝐋𝑀𝐋𝐻 (30)
To fully exploit the information from𝐇for the estimation of the stiffness matrix, an intermediary step is introduced. An inverse problem is addressed using the optimization function,fmincon. Given the knowledge of the system’s natural frequencies, either sourced from response observations or operational modal analysis, it is possible to construct a target eigenvalues matrix,𝚲∗. This realistic assumption aids in guiding the optimization process: from an initial state𝐋𝐻(𝐱ℎ)to an optimized state𝐋𝐻(̂𝐱), whereℎis the subset of𝐱related to the entries of𝐋𝐻. The end goal is to ensure that the ordinary eigenproblem, and consequently the GEP, reflect the eigenvalues in𝚲∗. To achieve this, any potential solution identified by the optimizer – be it during the global or local optimization phase – is considered as an initial estimate for a subsequent optimization phase. The solution’s fitness is assessed based on the discrepancy between𝚲∗and the eigenvalues of the estimated matrix,𝚲est. A cost function is therefore defined as follows:
eig=√∑𝑁𝑀
𝑖=1(𝜆est,𝑖−𝜆∗,𝑖)2 (31)
Through this approach, the original parameter set𝐱ℎundergoes a refinement, yielding a modified𝐱̂ that inherently preserves the eigenvalues from𝚲∗(seeFig. 4). By employing𝐱ℎas the starting point, instead of adopting a straightforward diagonal matrix bearing the desired eigenvalues, allows to integrate and build upon the improvements already made in response-based optimization stages. This sequential approach, as opposed to transforming the challenge into a multi-objective optimization problem, retains the simplicity and clarity of the optimization process. While a multi-objective framework holds potential advantages, it might inadvertently introduce superfluous complexities. At this stage Eq.(30)is evaluated and then stiffness matrix is computed as follows:
𝐊=𝐋𝐾(̂𝐱)𝐋𝑇𝐾(̂𝐱) (32)
3.2.5. Damping matrix
In structural dynamics, the representation of damping can vary depending on the approach. For instance, the work presented in [72] seeks to directly estimate the individual entries of the damping matrix. Conversely, the studies detailed in [43,44] employ proportional damping, which reduces the complexity by introducing just two unknowns and eliminates the need for additional constraints on the damping matrix, provided the ones on𝐌and𝐊are satisfied. However that approach poses the formerly discussed limitations of proportional damping. Here, modal damping ratios,𝜻are used to describe the damping properties. The modal damping matrix,𝐂modalis reconstructed using the modal damping coefficients𝑐𝑖= 2𝜁𝑖𝜇𝑖𝜔0𝑖, where𝑖= 1,…, 𝑁𝑀, with𝑁𝑀being the number of modal DoFs,𝜔0𝑖the𝑖th natural frequency, and𝜇𝑖the modal mass of the𝑖th mode. The damping matrix, if needed, can be then converted into physical coordinates:
𝐂=𝚽−𝑇𝐂modal𝚽−1 (33)
This methodology pathway provides a systematic approach to bridge the gap between intuitive modal damping ratios and their equivalent physical representation, allowing for a physically meaningful definition of the search space. In fact, the range is defined based on existing literature, which is very likely to capture the solution in most cases and allows for trivial modifications based on engineering judgment for highly atypical structures.
3.2.6. Search space and design variables
The nonzero entries of Cholesky factors of𝐋𝑀,𝐋𝐻and the modal damping ratios𝜻, are collected together in the vector𝐱={ 𝑥𝑑}
, with𝑑∈ 1,2,…, 𝐷and𝐷being the number of variables (and unknowns) of the system.is bounded by the maximum and minimum values that the parameters can assume. It comprises two vectors of size𝐷,𝐱maxand𝐱min, respectively, such that
={
𝐱∈R𝐷∣𝐱min,𝑑≤𝐱𝑑≤𝐱max,𝑑,1≤𝑑≤𝐷}
(34)
Table 1
Number of unknowns needed to reconstruct each matrix in the proposed technique.
Mass matrix Stiffness matrix Damping matrix
Unknowns 𝑁𝑃(𝑁𝑃+1)
2 +𝑁𝐷(𝑁𝐷+1)
2
𝑁𝑃(𝑁𝑃+1)
2 𝑁𝑀
Drawing upon the discussion in Section3.2.3, the definition of the search space for the mass matrix is quite straightforward. The search space for the elements of𝐋𝑀 can be articulated as:
−1≤𝐱𝑚
𝑜 ≤1 (35a)
0≤𝐱𝑚𝑑 ≤1 (35b)
where𝑚𝑜and𝑚𝑑indicate the off-diagonal and diagonal entries, respectively.
Regarding the stiffness matrix, specifying a reasonable search space for𝐊is highly problematic for an actual structure. It strongly depends on the discretization of the model, e.g., the number of physical DoFs intended to be used to identify the spatial properties.
This problem is circumvented by the method in Section3.2.4. The entries of𝐋𝐻 used in𝐱𝑑 admit meaningful constraints, using norm inequalities:
‖‖𝐋𝐻‖‖max≤‖‖𝐋𝐻‖‖2=𝜎max(𝐋𝐻) =√
𝜆max(𝐇) (36)
where𝜎and𝜆denote singular values and eigenvalues, respectively. Invoking𝚲𝐻=𝚲𝐾𝑆𝑆,𝑀𝑆𝑆, it follows that the individual entries of𝐋𝐻can be bounded by the maximum generalized eigenvalue that would be found in the identified system, say,𝜆lim=𝜔2lim. Unlike elements of the stiffness matrix,𝜔limis readily available, as it can be immediately asserted from the known test results or specified as the maximum natural frequency of interest. Therefore
−𝜔lim≤𝐱ℎ
𝑜≤𝜔lim (37a)
0≤𝐱ℎ𝑑≤𝜔lim (37b)
whereℎ𝑜andℎ𝑑indicate the off-diagonal and diagonal entries of𝐋ℎ, respectively.
As for the damping coefficients, the search space is defined following the guidelines for aerospace structures, [76]. Typical values are between 0.5% and 5%, therefore:
0.005≤𝐱𝑐≤0.05 (38)
where𝑐denotes the subset of𝐱𝑑used to describe the damping properties.
The number of unknowns of the proposed technique are thereby as inTable 1.
3.3. Optimization aspects
In this section, the problem is examined from an algorithmic perspective. With the mathematical formulation now established, attention turns to the subsequent steps outlined inFig. 2. It is important to note that the updating process for the particle’s position and velocity follows Eq.(7). This section clarifies the strategy for swarm initialization and defines the initial parameters that guide the adaptation techniques throughout iterations. The discussion then shifts to the termination criteria employed during theGlobal Phase, thehealing process, and the terms of hybridization within the PSO. The section concludes with an introduction to theLocal Phase.
3.3.1. Swarm initialization and settings
The LHS [77] is employed to generate the initial swarm. This method ensures that each dimension is sampled uniformly and independently, minimizing the correlation between sample points and representing all distribution regions. This technique is particularly advantageous for high-dimensional problems, as it ensures that the entire search space is explored in the initial stage of the optimization process. The swarm size is a crucial parameter that influences the performance of the PSO algorithm. As shown in [78], the best swarm size is weakly related to problem dimensionality. A larger size does not necessarily yield better results. Based on extensive sensitivity analysis, the swarm size is set to be 5 times the number of design variables, offering a good trade-off between the computational cost and the accuracy of optimization results. Velocity settings have the potential to influence the optimization process. In this context, the following considerations are made:
•Velocity initialization. This determines the direction and speed of the initial search by particles. In this work, the initial velocity of each particle is set to zero, which allows the particles to start the search in a smooth and controlled manner.
•Maximum velocity. This parameter ensures the stability of the PSO algorithm. The maximum velocity of each particle is set to a value proportional to the difference between the upper and lower bounds of the variables, with a proportionality constant of 0.4.
For themass distributionstep in the algorithm, the parameter c has been set to 1. This choice ensures compatibility with any number of DoFs.
