State-of-the-Art Advances in Structural Control of Buildings: Methods and Algorithms for Active Control of Seismic Response (Preprint)

State-of-the-Art Advances in Structural Control of Buildings: Methods and Algorithms for Active Control of Seismic Response Author: Ruben Iacob Munteanu Ferdinand I Military Technical Academy, Bucharest, Romania (Doctoral research conducted at National University of Science and Technology Politehnica Bucharest) [email protected] October 2023 Abstract This state-of-the-art review presents an overview of structural control strategies for mitigating the seismic response of buildings, including passive, semi-active, active, and hybrid systems, with reference to real-world implementations. Emphasis is placed on active control, which allows real-time adaptation of control forces based on structural response, enhancing resilience during seismic events. The review discusses numerical modeling of multi-degree-of-freedom structures, with equations of motion formulated in state-space form to support implementation of control algorithms. Preprint note: This manuscript is derived from the doctoral thesis of Ruben Iacob Munteanu (defended at National University of Science and Technology Politehnica Bucharest). It is released as an open-access preprint to support dissemination and citation of the research. A version of this document may also appear on arXiv and/or Zenodo for archival and DOI purposes. License: CC BY 4.0 — This work is distributed under the Creative Commons Attribution 4.0 International License. https://creativecommons.org/licenses/by/4.0/ 1.1 Systems for Structural Control (Passive, Semi-Active, Active, Hybrid) Structural control is a topic of great interest for both the community of civil engineers and automation engineers. The intense concern manifested over the years is fueled by the need to achieve safer constructions that remain operational after the occurrence of extreme events. Although the development of constructions has been a principal element in society throughout history, the accelerated vertical expansion of cities in recent decades generates continuous challenges in this field. Current approaches in the design of structural systems assume that constructions are designed to ensure uninterrupted functioning only in the event of extreme phenomena with a short return period. In the case of more destructive events, with a longer return period, the primary objective is the safety of the occupants, even at the expense of functional and structural integrity. In other words, extensive controlled but damage of structural and non-structural elements is accepted, leading to the temporary decommissioning of the building. More precisely, the energy generated by tectonic movement is transmitted through the foundation to the structural system of the building. This energy is absorbed by the structural elements in the form of kinetic energy, damping energy, and elastic and inelastic (hysteretic) deformation energy. To illustrate this energy exchange, [1] presents the subject in analogy with the process of rainwater collection by a building’s drainage system. Figure 1.1: Analogy regarding the distribution of seismic energy (adapted after [1]) In Figure 1.1, the analogy illustrating the distribution of seismic energy during an earthquake can be visualized. The collected rainwater represents the input seismic energy transmitted to the structural system. Initially, this energy is transformed into kinetic energy, generated by the mass of the structure mobilizing inertial forces. As the structure begins to vibrate, the structural elements deform and absorb energy. At the moment the structure stops, at maximum displacement, all kinetic energy is transformed into deformation energy. This process is illustrated in Figure 1.1 by a pump connecting the lower and upper parts of the containers representing kinetic energy and elastic deformation energy. Thus, the vibration of the structure can be visualized as a continuous transformation of kinetic energy into deformation energy and vice versa. Furthermore, it can also be observed that as this exchange occurs, some of the energy is lost, transforming into damping energy. This triple energy exchange characterizes systems with elastic response. In this case, the transfer between kinetic energy and deformation energy occurs until all incoming seismic energy equals the damping energy. On the other hand, if the deformation energy exceeds a certain limit, it permanently transforms into hysteretic energy. This limit is determined by the elastic deformations capable of the structural elements, beyond which non-linear behavior begins to manifest through damage to the constructive elements of the building. In this case, at the end of the seismic movement, the incoming seismic energy will equal the sum of the energy dissipated through damping and the hysteretic energy dissipated through inelastic deformations. In this context, the implementation of structural control aims to minimize the total amount of hysteretic energy dissipated through inelastic deformations, utilizing systems for structural control. Depending on how they operate, systems for structural control are divided into: •Passive control systems; •Active control systems; •Semi-active control systems; •Hybrid control systems; 1.1.1 Passive Control Systems Passive systems encompass equipment attached to the structural framework that serves to dissipate/isolate part of the energy induced by seismic movement. They operate without their own energy source, and their behavior characteristics remain constant during the dynamic response of the structure. Thus, this type of system does not have the capacity to adapt its characteristics based on the global seismic response of the structural framework, but can be designed to have an optimal effect only under specific conditions, depending on the vibration characteristics of the building. As proposed by [2], passive control systems can be divided into two major categories: seismic isolation systems and dissipative systems. Seismic Isolation Systems Seismic isolation involves the introduction of a system with high vertical stiffness at the base, having significantly greater horizontal flexibility than that of the building. Thus, the isolator system aims to limit the seismic energy reaching the structural framework. The main objective is to change the natural vibration characteristics of the building by introducing a new predominant mode of vibration with a much longer period, to keep the structure away from the frequency content of the seismic excitation. Seismic isolation systems consist of two main subsystems [1]: • Isolation system, composed of flexible elements that ensure the "decoupling" of the structure from the foundation ground and that modify the natural vibration characteristics of the building. These can include: natural rubber elastomeric isolators (Natural Rubber Bearing - NRB), high damping elastomeric isolators made of synthetic rubber (High 2 from 44 Damping Rubber Bearings - HDRB), lead-core rubber isolators (Lead Rubber Bearings - LRB), sliding bearings (Sliding Bearings - SB) Figure 1.2: Seismic isolators: a) 1seismic isolator concept - LRB b) 2seismic isolator in operation • Damping system, which primarily serves to dissipate part of the seismic energy absorbed by the structural system, to prevent excessive rigid body displacements. Additionally, it is also used to return the building to its initial position when the deformation of the isolators is equal to zero. Figure 1.3: Seismic isolation system composed of isolators and dampers 3 As highlighted in Figure 1.4, the isolation system modifies how the building responds to seismic excitations. The high flexibility of the isolators, coupled with the high stiffness of the structural framework, leads to the building moving as a whole (rigid body displacement). In this way, it avoids the occurrence of significant relative level displacements that can lead to excessive damage of structural and non-structural elements. The implementation of this type of system is effective only for structures with high and medium natural frequencies, for which vibration is transmitted through the foundation ground. 1source: https://www.boomarine.com/products/lead-rubber-bearing, (accessed on 28.01.2023) 2source:https://civildigital.com/base-isolation-system-outline-on-principles-types-advantages-applications (accessed on 28.01.2023) 3source: https://www.fujiengineering.com/yimg/r37.jpg (accessed on 28.01.2023) 3 from 44 Figure 1.4: Structural response: a) without seismic isolation system b) with seismic isolation system For buildings sensitive to wind action or other direct actions, the use of this type of system is not feasible [3]. Dissipative Systems Dissipative systems consist of equipment installed in the superstructure of buildings, operating without an energy source or external automatic control, and are designed to absorb part of the seismic energy taken by the structural system. These systems are most commonly used to reduce structural response, both in new buildings and for improving the seismic performance of existing buildings. To contribute effectively to the structural response, dissipative equipment must be judiciously placed within the structure. Moreover, they must be calibrated so that the dissipative effect is mobilized under specific circumstances. Depending on how their action is engaged, dissipative systems are divided into [1]: displacement-activated systems,velocityactivated systems, and motion-activated systems. • Displacement-activated systems dissipate the seismic energy taken by the structural system based on the relative displacements occurring between the connection points of the dissipative equipment. They come into play proportionally with the development of internal forces in the structural elements and serve to consume part of the elastic deformation energy induced by the earthquake. Some of the most common such devices include: metal hysteretic dampers - represented by highly ductile metal parts that dissipate energy through cyclic inelastic deformations; friction dampers - made of an assembly composed of metal pieces joined with high-strength bolts that dissipate energy when the friction force is exceeded, allowing them to move relative to each other. Also included in this category are self-centering dampers, which, in addition to their dissipative component, have the ability to return to the initial state after the load has ceased, thus minimizing any residual deformations that may occur at the level of the structural elements (Figure 1.6 b)). • Velocity-activated systems dissipate the seismic energy taken by the structural system based on the relative velocities recorded between the connection points of the dissipative equipment, with maximum damping forces being generated out of phase with the moment when the maximum efforts are recorded at the level of the structural elements. Thus, they 4photo source: https://www.technoarete.org/common_abstract/pdf/IJERMCE/v5/i1/ Ext_08356.pdf (accessed on 09.02.2023) 5photo source: https://www.damptech.com/ (accessed on 09.02.2023) 4 from 44 Figure 1.5: Displacement-activated systems: a) 4hysteretic metal damper b) 5friction damper Figure 1.6: a) Hysteretic behavior: a) hysteretic metal damper [4] b) self-centering damper [5] serve to supplement the energy dissipated by the structural system. Velocity-activated systems generally consist of viscous dampers or viscoelastic dampers. Viscous dampers Figure 1.7 are mechanical systems that dissipate energy by deforming a fluid with high viscosity. These devices generally consist of a steel cylinder inside which hydraulic or silicone fluid is subjected to pressure generated by the movement of a piston whose head has passage orifices. They may also have a reaction chamber designed to suppress the general elastic reaction to the pressure applied to the fluid [6]. These systems are not sensitive to temperature; however, they have a shortcoming in their ability to engage early in the structural response due to the fact that the magnitude of the damping force is proportional to the velocity at which the piston moves. Viscoelastic dampers Figure 1.8 are made using materials with viscoelastic properties that dissipate energy through shear deformations. During deformation, the viscoelastic material behaves both as a viscous liquid and as an elastic solid, having the ability to return to its original shape after unloading. The performance of this type of equipment 6photo source: https://structurae.net/en/products-services/ dampers-for-butterfly-towers-of-bucharest (accessed on 11.02.2023) 5 from 44 Figure 1.7: Viscous damper: a) viscous damper concept b) 6viscous dampers implemented at Orhidea Towers, Bucharest is influenced by temperature and loading frequency. The first buildings equipped with such systems were the World Trade Center towers in New York, which served to reduce acceleration levels to ensure occupant comfort during wind actions [6]. Figure 1.8: Viscoelastic damper: a) viscoelastic damper concept b) viscoelastic damper proposed by [7] • Motion-activated systems do not directly dissipate seismic energy but rather absorb some of it from the building through control equipment. In this case, the kinetic energy at the level of the structural system becomes input energy for the passive system, which in turn absorbs it in the form of kinetic energy, damping energy, and elastic deformation. In general, these systems consist of a mass (1%-2% of the building’s mass) that is attached to the structural system so that its dynamic characteristics are tuned to those of the building’s fundamental vibration mode. Thus, during motion, the system is in resonance with the building, generating inertial forces in the opposite direction to the direction of its displacement. In practice, two such systems are encountered: tuned mass dampers (TMD) and tuned liquid dampers (TLD). Compared to tuned mass dampers, tuned liquid dampers are much simpler, generating lower maintenance costs, are easier to install, and can be easily used to control motion in both directions. Additionally, the much lower friction forces ensure efficiency even for low-frequency and low-amplitude vibrations. However, considering that in the case of 7Photo source: http://www.tesolution.com/uploads/6/9/3/0/69304001/dsc07463_ orig.jpg (accessed on 12.02.2023) 6 from 44 Figure 1.9: Concept of motion-activated systems Figure 1.10: a) 7tuned mass damper b) tuned liquid damper [8] TMD the mass is generally made of steel, a TLD requires a much larger volume of liquid to generate the same inertial force. 1.1.2 Active Control Systems Despite their high reliability and relatively low implementation costs, passive systems have limited effectiveness in reducing structural response. For example, base isolation systems are not effective in the case of direct actions on buildings, while dissipative systems can only dissipate energy from a single mode of vibration. These systems do not have the capacity to adapt to the type of dynamic loading acting on the building or to changes in dynamic characteristics that may occur when the structural system behaves inelastically. Once installed, a passive system will behave in a predefined manner, regardless of the global response of the building. Considering these limitations, the implementation of active control systems naturally emerges as a solution that offers the possibility of monitoring the behavior of the construc7 from 44 tion, with the capability to make decisions and act in real time to ensure a favorable response of the structure. An active control system encompasses three subsystems: •Monitoring system - its role is to measure seismic excitation and the structural response manifested in specific areas in the form of accelerations, velocities, or displacements, using sensors or other measuring equipment. •Control system - aggregates and filters information from sensors, using it within a control algorithm to calculate an appropriate response method from the actuation system. •Actuation system - consists of execution elements that receive and implement the command calculated by the control system. Figure 1.11: Building - active control system interaction [9] As shown in Figure 1.11, the active control system can be seen as equipment within the building, which monitors structural response and applies control forces based on a numerical algorithm. The main active control systems encountered in the specialized literature are: Active tendon systems (ATS),Active bracing systems (ABS),Active mass dampers (AMD) •Active tendon systems - are generally made from a set of tendons connected to the building structure, whose tensioning is controlled through an electro-hydraulic actuation mechanism governed by a control algorithm. The advantage of this system is that it can be easily implemented in constructions that have structural elements in the form of tendons, such as cable-stayed bridges (Figure 1.12) or steel towers with anchorage. For example, the Normandy Bridge in France, with a span of 856 m, has, in addition to classical supporting tendons, active elements of the ATS type [10]. For civil constructions, active tendon systems can be used to reduce relative displacement levels (Figure 1.13). These systems can operate in both continuous and pulsed modes. •Active Bracing Systems - are composed of hydraulic/electric actuation elements mounted within the bracing of a structural framework. Through these elements, using a control algorithm, judicious control forces are applied to reduce the structural response (Figure 1.14). Although initial experimental research conducted by [11] suggested that these systems have great potential for controlling structural systems, it has become evident that, with the development of semi-active control systems, they represent a more reliable solution. 8 from 44 of an additional system aims to reduce excessive lateral displacements of the isolators, which can lead to their damage. One application of this type of system involves the Obayashi Corporation’s Technical Research Institute, for which the Obayashi company implemented, in addition to the base isolation system, an actuator that ensures active control in parallel (Figure 1.22 c). Figure 1.22: a) Hybrid base isolation system - experimental implementation from [45] b) magnetorheological damper implemented by [45] c) actuator for active control of the base isolation system - implemented by Obayashi [46] •Hybrid Systems with Dampers and Actuators - are control systems that include both viscous dampers and automatic actuation elements (actuators), which can operate independently or in parallel to control the structural response. They can be arranged within the same structural element (for example, within a bracing system K, Figure 1.23) or in different areas of the structural framework. Hybrid systems with dampers and actuators have been extensively studied over the years by [47], [48], [49], and [50], with experimental results highlighting that they have higher efficiency than passive systems for reducing structural response, but also that to achieve the same performance, they require smaller control forces compared to active control systems. 1.2 Methodologies/Algorithms for Implementing Active Control The previous section briefly presented the main structural control systems that can be implemented to reduce the seismic response of buildings. Passive systems, while having lower efficiency, are the most accessible and easiest to implement approaches, while semi-active and hybrid systems offer the advantage of a middle ground between effectiveness and resource requirements. Thus, active systems represent the ideal option for controlling the behavior of structures. In terms of effectiveness, they have the ability to adapt the implemented control forces 15 from 44 Figure 1.23: Concept of hybrid system with damper and actuator (proposed in [50]) to the requirements imposed by the structural response, thereby being capable of counteracting destructive effects for a wide range of actions and excitations. Although the high energy requirements for implementing control forces and the level of confidence in their reliability have led to a relatively low number of active control implementations in the last decade, recent technological advancements, especially regarding electric motors and energy storage equipment, are bringing these systems back to the forefront for research and large-scale implementation in the near future. As previously mentioned, an active control system consists of three main subsystems that focus on: monitoring the structural response, calculating the actuation command using a control algorithm, and implementing control forces through execution elements (Figure 1.24). Given the purpose of this doctoral thesis, namely the study of the effectiveness of implementing active control for structural moment resisting - frames, it is necessary to develop integrated numerical models that can capture both the structural response and the effects of the control system’s action. In this regard, within the numerical models commonly used by structural engineers, it is necessary to implement a control algorithm that uses the structural response to calculate, at each time step, the corresponding magnitude of the applied control forces. Thus, in the continuation of the content of this subsection, the main methodologies/algorithms used for implementing active control of buildings will be reviewed. Looking at the overall evolution of research regarding the algorithms for active control of buildings, the late 1990s and early 2000s represent reference periods in which the subject enjoyed widespread popularity. However, numerous recent experimental studies and practical implementations suggest that the topic has once again become one of the main concerns of researchers in the field [34], [38], [51], [52], and [53]. Below are briefly presented the main methodologies addressed in the specialized literature: 1. Active Structural Control through Judicious Pole Allocation 2. Independent Control of Modal Space 3. Active Structural Control through Successive Impulse 4. Sliding Mode Active Structural Control 5. Active Structural Control Using Fuzzy Logic 6. Market - Based Structural Control 16 from 44 Figure 1.24: Conceptual diagram of an active control system 7. Optimal Structural Control 8. Instantaneous Optimal Structural Control 9. Active Structural Control Using Artificial Neural Networks Since the concepts underlying active control of structures are inherently interdisciplinary, it is necessary to first reformulate the equations commonly used in building dynamics into an equivalent form that aligns with the conventions of the control system field before reviewing the corresponding control methodologies. Thus, the following presents the procedure for transitioning classical equation of motion to state-space format. The motion of damped systems subjected to seismic excitations can be modeled by an inhomogeneous system of nsecond-order differential equations with constant coefficients, as presented in Eq: 1.1. M·¨ u(t)+C·˙ u(t)+K(t)·u(t) = −M·h·¨ ag(t)(1.1) The components of Eq: 1.1 can be divided into two categories: 1. Components that determine and describe the structural response: •u(t)- the instantaneous displacement vector (n×1) response in the direction of the degrees of freedom (DOF); •˙ u(t)- the instantaneous velocity vector response in the direction of the DOF; •¨ u(t)- the instantaneous acceleration vector response in the direction of the DOF; •¨ ag(t)- the ground acceleration during the earthquake; •h– is a vector that serves to scale the ground acceleration in the direction of the DOF; 2. Components that describe the properties of the structural system: 17 from 44 •K(t)– is the stiffness matrix of the structure; •M– is the mass matrix of the structure; •C– is the damping matrix of the structure; Multiplying Eq: 1.1 by M−1and considering that the stiffness matrix is constant throughout the simulation, K(t) = K, results in: ¨ u(t)+M−1·C·˙ u(t)+M−1·K·u(t) = −h·¨ ag(t)(1.2) Noting: u(t) = u1(t)(1.3) ˙ u(t) = u2(t) = ˙ u1(t)(1.4) By identification, it results: ˙ u1(t) = u2(t)(1.5) ˙ u2(t) = −M−1·C·˙ u(t)−M−1·K·u(t)−h·¨ ag(t)(1.6) Integrating in matrix form Eq: 1.5 and Eq: 1.6: ˙ u1(t) ˙ u2(t)=˙ u(t) ¨ u(t)=O I −M−1K−M−1C·u(t) ˙ u(t)+0 -h·¨ ag(t)(1.7) Noting: ˙ z(t) = ˙ u(t) ¨ u(t),z(t) = u(t) ˙ u(t), A=O I −M−1K−M−1C, F(t) =0 −h·¨ ag(t) = H¨ ag(t); (1.8) Rewriting Eq: 1.7, the equation of motion in state format results: ˙ z(t) = Az(t) + F(t)(1.9) This formulation is relatively new in the field of structural engineering, even though the mathematical model is frequently used in electrical engineering and automation. The main advantage of this expression is that the system of second-order differential equations is transformed into a system of first-order differential equations, which leads to obtaining an explicit time integration relationship, through which the structural response can be determined numerically. In Eq: 1.9,˙ z(t)and z(t)are called state variables and describe the dynamic model’s response at a given moment in time. Introducing the effect of control forces into Eq: 1.9: ˙ z(t) = Az(t) + F(t)+Bfc(t)(1.10) Where: B=0 M−1D(1.11) And the matrix Dserves to scale the control forces in the desired degrees of freedom. 18 from 44 1.2.1 Active Structural Control through Judicious Pole Placement The controlled motion of a structural model with a finite number of degrees of freedom can be described by the system in Eq: 1.12. It is observed that, in addition to Eq: 1.9, an additional equation has been introduced. This serves to establish the relationship between the state variable and the measured output data (the measured structural response). (˙ z(t) = Az(t) + F(t)+Bfc(t) y(t) = Cz(t)(1.12) Where Cis the output matrix. Considering that the control force fc(t)is dependent on the state vector, according to the relationship fc(t) = G·z(t), where Gis the reaction matrix: (˙ z(t) = Az(t) + F(t)+BGz(t) y(t) = Cz(t)(1.13) Equivalent to: (˙ z(t) = (A+BG)·z(t) + F(t) y(t) = Cz(t)(1.14) Noting that (A+BG)=Wand substituting in Eq: 1.14 results in: (˙ z(t) = Wz(t) + F(t) y(t) = Cz(t)(1.15) An equivalent representation of the system from relation Eq: 1.14 is the input-output format in the frequency domain, using a transfer matrix. This is obtained by using the Laplace transform. Figure 1.25: Transition from the time domain to frequency domain Thus, Eq: 1.14 becomes: (s·Z(s)−z(0) = W·Z(s)+F(s) Y(s) = C·Z(s)(1.16) 19 from 44 Considering that, before the incidence of seismic motion, the structural model is considered to be at rest, we can evaluate the initial state vector as being equal to zero z(0) = 0. (s·Z(s)−W·Z(s)=F(s) Y(s)=C·Z(s)(1.17) Eq: uivalently: (Z(s)·(sI −W) = F(s) Y(s)=C·Z(s)(1.18) Substituting Z(s)into the second equation results in: Y(s)=C·(sI −W)−1·F(s)(1.19) Noting H(s)as the transfer matrix of the model and considering that the matrix Cis equal to the identity matrix: H(s) = Y(s) F(s)= (sI −W)−1=(sI −W)∗ |sI −W|(1.20) By definition, the poles of the transfer matrix are those values of sfor which H(s)tends to infinity. At the same time, the values of sfor which |(sI −W)|= 0 represent the eigenvalues λi of the matrix W. These describe the structural response, being related to the circular frequency ωiand the critical damping ratios ζiassociated with the vibration modes, in complex conjugate pairs. Thus, considering that W= (A+BG), structural control through judicious pole allocation implies controlling the behavior of the system/model by favorably adjusting the vibration characteristics associated with the significant modes affecting the structural response. Essentially, the eigenvalues of the matrix A, which characterize the eigenmodes of the uncontrolled building, are conveniently modified by the judicious choice of the reaction matrix G[54]. λi=ζiωi±iωiq1−ζi2(1.21) Numerous works that focused on this topic describe different methods for determining the command/reaction matrix. Among the early research, those by W. M. Wonham in 1967 [55] and E. J. Davison in 1970 [56] proposed that, for a system of the form Eq: 1.22, the input size should be fc(t) = G·z(t). Defining z(t)as a vector of dimension ndescribing the system state, fc(t) as a vector of dimension mrepresenting the input size, and y(t)as the output size of dimension l, Wonham showed that, for the case where l=m(the state size is completely measurable), the pair (A,B)is controllable if and only if, for an arbitrarily chosen set of values, there exists a constant matrix Gsuch that these represent the eigenvalues of the matrix (A+BG). Davison generalized this result for situations where the rank of the matrix Bis l≤m(fc(t) = G·y(t)). (˙ z(t) = Az(t) + Bfc(t) y(t) = Cz(t)(1.22) In 1972, O. A. Solheim [57] proposed a new method for determining the reaction matrix G (fc(t) = G·y(t)), combining the main approaches up to that time, namely: • Calculate the Gmatrix such that it minimizes a quality index (cost function) expressed through a quadratic functional of states and inputs. J(t) = 1 2Zt 0 [z(t)T·Q·z(t)+fc(t)T·R·fc(t)]dt (1.23) 20 from 44 • Determine the matrix Gsuch that the system ˙ z(t)=Az(t) + Bfc(t)has predetermined eigenvalues. He emphasized that, in the case of the first approach, the constant matrices Qand Rgenerate a unique set of eigenvalues for (A+BG), which does not necessarily guarantee the desired degree of stability. On the other hand, the second method ensures system stability by judiciously choosing the poles of the transfer function. However, in most cases, since the matrix Gis not unique, a "favorable" choice of it is necessary. Therefore, Solheim proposes a new method that encompasses both approaches and involves determining the reaction matrix such that, on the one hand, it ensures certain eigenvalues, and on the other hand, minimizes the quadratic functional. Specifically, this materializes through finding the tuning matrices Qand R, corresponding to a predetermined set of eigenvalues. Other important contributions to the development and understanding of these system control techniques were made by the works published by W.L. Brogan in 1974 [58], H. Kimura in 1975 [59], B. C. Moore in 1976 [60], O. A. Sebakhy, and N.N. Sorial in 1979 [61]. The application of these techniques in the field of structural engineering has its beginnings in "Modal control of multistory structures" published by Martin R. C. and Soong T. T. in 1976 [62]. Other representative works include those developed by P. C. Wang et al. in 1983 [63], H.H.E. Leipholz and M. Abdel-Rohman in 1986 [64], and T.T. Soong in 1992 [65]. Among the most recent references are N. G. Pnevmatikos and C. J. Gantes in 2006 [66], who proposed the implementation of a control strategy based on the distribution of the eigenvalues of the system according to the frequency content of seismic excitation, N. G. Pnevmatikos in 2017 [67], who presented a new algorithm for determining the reaction matrix, as well as the works [68], [69]. 1.2.2 Independent Control of Modal Space This control method involves, through the introduction of the modal matrix Φ Φ Φand the modal coordinates q(t), decoupling the system vibrations into independent modes and applying the control strategy distinctly for each. Thus, starting from the equation of motion of a controlled structural model, described by a system of linear second-order differential equations with constant, non-homogeneous coefficients that are dependent on each other Eq: 1.24. M·¨ u(t)+C·˙ u(t)+K(t)·u(t) = −M·h·¨ ag(t)+D·fc(t)(1.24) Noting: ¨ u(t) = Φ Φ Φ·¨ q(t) ˙ u(t) = Φ Φ Φ·˙ q(t)(1.25) u(t) = Φ Φ Φ·q(t) The modal matrix composed of the eigenvectors has the form: Φ Φ Φ = [ϕ ϕ ϕ1ϕ ϕ ϕ2.... ϕ ϕ ϕn−1ϕ ϕ ϕn](1.26) Considering the eigenvectors to be orthogonal with respect to the stiffness, inertia, and damping matrices, rand srepresenting the number corresponding to the vibration mode: ϕ ϕ ϕT r·M·ϕ ϕ ϕs=(0,if r=s m∗ i,if r=s(1.27) 21 from 44 ϕ ϕ ϕT r·K·ϕ ϕ ϕs=(0,if r=s k∗ i,if r=s(1.28) ϕ ϕ ϕT r·C·ϕ ϕ ϕs=(0,if r=s c∗ i,if r=s(1.29) Replacing Eq: 1.26 in Eq: 1.24 and multiplying from the left by Φ Φ Φ−1results in: Φ Φ ΦT·M·Φ Φ Φ·¨ q(t)+Φ Φ ΦT·C·Φ Φ Φ·˙ q(t)+Φ Φ ΦT·K·Φ Φ Φ·q(t)=−Φ Φ ΦT·M·h·¨ ag(t)+Φ Φ ΦT·D·fc(t)(1.30) Considering that (Φ Φ ΦT·M·Φ Φ Φ), (Φ Φ ΦT·C·Φ Φ Φ), and (Φ Φ ΦT·K·Φ Φ Φ) are diagonal matrices with components m∗ i,c∗ i, and k∗ i, one can arrive at the relationship: ¨qi(t)+2·ζiωi˙qi(t)+ω2 iqi(t) = −Γi·¨ ag(t)+vi(t)(1.31) where: Γ Γ Γ = Φ Φ ΦT·M·h ϕ ϕ ϕT i·K·ϕ ϕ ϕi (1.32) v(t) = Φ Φ ΦT·D ϕ ϕ ϕT i·K·ϕ ϕ ϕi ·fc(t) = [v1v2.... vn−1vn](1.33) and Γ Γ Γis the matrix that contains the mass activated independently on each vibration mode, v(t)being the vector of control forces in the modal space. It is observed that the system remains coupled through the control component, as the value of v(t)idepends on multiple modal coordinates. However, if v(t)iis defined according to [9], based on ˙qi(t)and qi(t), the equations become independent and can be treated separately for each vibration mode. The optimal modal control forces v(t)can be obtained by minimizing the functional: J= n X i=1 Ji(1.34) where Jiis a cost function that can take the form [9]: Ji=Zt 0 [Q1i·qi(t)2+Q2i·˙qi(t)2+Ri·vi(t)2]dt (1.35) The control force can be obtained using Eq: 1.36: fc(t) = ( Φ Φ ΦT·D ϕ ϕ ϕT i·K·ϕ ϕ ϕi )−1·v(t)(1.36) It should be noted that this method is effective only when there are a limited number of vibration modes that influence the structural response. This methodology was initially proposed by L. Meirovitch et al. in 1977 in [70] and [71] for applications in the aerospace field and for certain distributed gyroscopic systems. In [72], the method is specialized for the dynamics of structures subjected to seismic excitations. Additionally, analyzing Eq: 1.33 from the perspective of the number of actuating devices associated with the controlled modes, [9] concludes that the most appropriate choice is to set the number of controllers equal to the number of considered modes. Recently, alternative approaches have also been proposed by [73], [74], and [75]. 22 from 44 From an experimental perspective, A. Baz et al. [76] analyzed the response of a singledegree-of-freedom system, proposing a modified version of the method that incorporates energy criteria for calculating the control forces. 1.2.3 Active Structural Control through Successive Impulses During dynamic loading, the response of a structure transitions between the elastic behavior zone and the inelastic behavior zone. Structural control through the application of successive impulses proposes an approach based on the efficiency of using energy to apply control forces, namely activating the control mechanism only when the state variables reach a certain value. The fundamental idea of this technique is to apply a train of forces (impulses), which aim to keep the response of the structural system within certain parameters (Figure 1.26). For buildings, these can be determined based on displacements/angular drifts or absolute level accelerations. In the case of violent seismic excitations, the main objective becomes maintaining structural elements within the linear behavior domain or limiting, as much as possible, the structural damage. Figure 1.26: Structural control through the application of successive impulses In 1981, F.E. Udwadia et al.[77], utilizing modal decomposition and superimposing the effect of control forces in the equation of motion, proposed a method for determining the impulse vector, such that the functional J(t)in relation Eq: 1.37 is minimized. A criterion based on velocities and displacements was used to calculate the response ceiling threshold. J(t) = 1 2Zt0+Tp t0 [q(t)T·A1·q(t) + ˙ q(t)T·A2·˙ q(t)]dt (1.37) Where q(t)and ˙ q(t)are modal coordinates, t0is the time step at which the structural response meets the ceiling criteria, and Tpis the time interval during which control forces (impulses) are applied. S.F. Masri et al. [78] published a similar approach in the same year. They proposed a cost function that can encompass potential deformation energy or the kinetic energy of the system. Additionally, they performed a simulation regarding the efficiency of the arrangement of actuating devices. Using a model with three degrees of freedom, they analyzed the response of the controlled system, both sequentially and simultaneously, in the direction of each dynamic 23 from 44 coordinate. The results showed that for the analyzed model, positioning the actuating device at the top level represents the most energy-efficient option. A.M. Reinhorn et al. [79] studied the application of this control methodology in structures considered to exhibit inelastic behavior and that can be modeled as single-degree-of-freedom systems. They considered implementing a strategy based on associating the Newmark-beta numerical integration method with the Newton-Raphson method. In 1988, R.K. Miller et al. [80] demonstrated the effectiveness of the methodology in an experimental study using a reduced model of a six-story structure, with a floor height of 0.3 m and plan dimensions of 0.61 m x 0.91 m. The actuating devices, using compressed air, were placed at the top level along with the structural response measurement sensor and the external force application mechanism. The control strategy proposed by them is based on applying impulses in the direction of the degrees of freedom where the velocity reaches a maximum value. In this regard, control impulses are applied when the relative displacement changes sign and are calculated based on Eq: 1.38. pi(t) = (−ci·sgn(vi)· | vi|ni,if t0i≤t≤(t0i+Tdi) 0,if (t0i+Tdi)< t < t0i (1.38) Where ciis a scaling coefficient, viis the absolute or relative velocity at point i,niis a power considered for the velocity at point i,t0iis the time at which point ihas a displacement of 0, and Tdi is the duration of the action. It can be observed that the method presented above does not depend on knowledge of the structure’s characteristics (M, C, K) nor on the influence of incursions into the plastic domain. Other approaches combining optimal instantaneous control methods and the use of artificial neural networks have been proposed by C.P. Pantelides et al. [81] and S-L Hung et al. [82]. 1.2.4 Sliding Mode Active Structural Control The method of controlling systems in sliding mode (sliding mode control - SMC) has been developed for application in nonlinear systems, providing robustness against parametric uncertainties. The strategy is based on discontinuous control and aims to steer and maintain the trajectories of state variables toward a suitably chosen surface, called the sliding surface. Determination of the control law occurs in two steps. 1. A sliding surface (sliding function) is established that generates an appropriate response, restricting the system’s trajectory along it. 2. The command is determined so that the state trajectories of the system are brought to and maintained on the sliding surface. Thus, a system, starting from non-zero initial conditions, can be in two situations: either driving toward the sliding surface or on the sliding surface, following its dynamics. Among the first reference works addressing this control strategy are those of V. Utkin (1977) [83], E. Slotine [84], and K. Furuta [85]. In the case of structures exhibiting linear and nonlinear behavior subjected to seismic excitations, sliding mode control has been described in detail by J. N. Yang et al. in Technical Report NCEER-94-0017 [86]. Starting from the state-space equation of motion Eq: 1.39, they define the sliding surface sas a linear combination of the state variables, s=P·z(t)=0, where s= [s1s2.... sr−1sr],ris the number of controllers, and Pof size (r×2n)is determined such that motion on the surface is stable. ˙ z(t) = Az(t) + F(t)+Bfc(t)(1.39) To determine the equation of motion on the sliding surface, the effect of the external action F(t)is neglected, but is being considered to calculate the control law that drives the motion 24 from 44 the system of differential equations characterizing motion. A more recent approach regarding the implementation of instantaneous optimal control is proposed by [136]. Also, another important work is that published by R.C. Lin et al. [137], which presents experimental results obtained by analyzing the response of a single-degree-of-freedom structure subjected to seismic action using a shaking table. They concluded that it is feasible to implement instantaneous optimal control within single-degree-of-freedom structures. It was also noted that the values of the structural response are very close to those obtained using classical/global optimal control. 1.2.9 Active Structural Control Using Artificial Neural Networks (ANN) Artificial neural networks are defined in analogy with the functioning of the human brain and represent systems that can perform specific tasks by adapting their internal operating rules through learning. Figure 1.30 presents the conceptual structure of an ANN. Figure 1.30: Neural Network (concept) Each unit in the input layer receives information from sensors, which it transmits to the units in the hidden layers. At this level, the information is processed and transmitted further to the units in the output layer, which connect to the actuation element. Additionally, the connections between units are characterized, depending on their importance, by certain weights W, which represent the essential elements of the learning process. Among the first works that present the implementation of structural control based on artificial neural networks are those of L. Faravelli et al. [138] and Y.-K. Wan et al. [139]. They approached the subject in a similar manner. Initially, the control algorithm uses as input data the information obtained from sensors (the state variable at different time steps, along with the control force from the previous step). Then, after traversing the network, the output layer generates input data for the actuation devices, which apply the control forces. To train the ANN controller, a neural emulator network was introduced, which connects the input generated by the actuation device and the output represented by the measurements from the sensors. Additionally, J. Ghaboussi et al. [140] proposed a control strategy that takes into account the dynamic coupling between the structure and the actuation devices and utilizes a variable number of units in the hidden layer, depending on necessity. A "quickprop" type algorithm was used for learning. Furthermore, K. Bani-Hani et al. [141] developed this neurocontroller for cases where the nonlinear behavior of the system is considered. 31 from 44 T.J. Kim et al. [142] proposed implementing an optimal method for adjusting the weights of the neural network. They used a cost function that incorporates the state vector and the control force vector, eliminating the need for empirical determination of a standard for the structural response. A different approach was proposed by M. M. Rao et al. [143]. They developed a control algorithm that manages the natural modes of vibration with significant influence on the structural response. The control strategy uses two sets of artificial neural networks. The first set is used to obtain generalized accelerations for each considered vibration mode, utilizing the accelerations measured at specific points and the seismic excitation. In other words, the first set aims to determine the modal properties of the structure using the measurements of the structural response. The second set uses the generalized accelerations and the seismic input to calculate the control forces. Additionally, the control methodology is numerically applied to a ten-story structural system. Other works addressing the subject include those by S. Saad et al. [144], T.K. Lin et al. [145], Y.A. He et al. [146], Y. Tang et al. [147], S.K. Chang et al. [148], Z. Khaled et al. [149], and T.Yu et al. [150]. 1.3 Discussions/Conclusions Regarding the State of Research in Active Structural Control The diversity of approaches presented highlights the ongoing concern of researchers and engineers towards the development and implementation of structural control to enhance the dynamic performance of buildings. Regarding the control methodologies presented, it can be observed that most of them are implemented for simplified numerical models, with a limited number of degrees of freedom, elastic behavior, and where all values in the state vector are considered known/measurable. Furthermore, most of the listed methodologies utilize modeled dynamic characteristics (inertia matrix, stiffness matrix, damping matrix) for determining control forces. In reality, however, structures are complex systems with nonlinear behavior under seismic incidence, for which it is necessary to use elaborate numerical models to accurately estimate the seismic response. At the same time, measuring the structural response can be performed with a limited number of sensors, placed in specific areas of interest, and the numerical modeling of the dynamic characteristics of buildings is of an estimative nature. In this sense, the innovative elements proposed in the doctoral thesis, which contribute to expanding knowledge in the field of active control, involve the development and simulation of complex numerical models (2D and 3D) that consider the inelastic behavior of structural elements and that have implemented control algorithms that operate independently of the variation of building characteristics using a reduced number of states. Additionally, it should be noted that, besides the nine classical approaches to structural control presented, the specialized literature continues to propose other methodologies, such as those presented by [151] and [152], some of which are still in the early research phases for application in active control of buildings. From a practical implementation perspective, until now, active control has been applied in reality to a limited number of buildings. In most cases, these control systems are used to reduce the structural response generated only by wind actions and earthquakes with a low recurrence interval. Recently, [34] conducted a significant study that includes a systematic evaluation of the literature regarding practical applications where structural control has been implemented. They show that out of a total of 208 identified applications, 57.7% of the control systems are located in Asia, 22.1% in America (Latin and North), 17.3% in Europe, 2.4% in Australia, and 0.5% in 32 from 44 Africa. Furthermore, the authors show that 47% of the systems were implemented before the year 2000, of which 7% (7 applications) were active systems, while 53% were implemented after 2000, of which 0.9% (one application) were active systems. Thus, the eight applications of active structural control were identified by them and are presented in Table 1.1. It should be mentioned that these include only control systems with active mass. Additionally, they identified 14 cases where structural control was implemented using a hybrid system (composed of a passive component (TMD) and an active one (AMD)). Table 1.1: Applications of Active Control - Active Mass Systems Building Control Algorithm Country Kyobashi Seiwa Building Optimal control Japan Riverside Sumida Building Optimal control Japan Nanjing Communication Tower Optimal control China Sendagaya INTES Building Optimal control Japan Applause Tower Optimal control Japan Porte Kanazawa Optimal control Japan Herbis Osaka Optimal control Japan Optimal control Kingkey Finance Tower Control through judicious placement of poles China Fuzzy-Neural Networks They also present a discussion that starts from the question "Are the research results in structural control applied in industry?" from which the following points can be noted regarding active control: • Although there are numerous theoretical studies related to modern algorithms for implementing active control, in industry there is a preference for using classical optimal control algorithms. Thus, there is a reluctance to adopt new control algorithms proposed by research initiatives. • Despite the superior performance of active or hybrid systems, the reason they are not used to the same extent as passive systems is the large amount of energy required for operation, as well as the fact that they involve the use of high-capacity actuation devices, which incur significant additional costs. • Although there is no clear evidence in this regard, another reason why their implementation is not popular may be the malfunctioning of existing systems during extreme events they have undergone. Additionally, although there are works in the specialized literature regarding the behavior of active systems, they do not detail the elements related to their actual performance, possible improper operations, or the costs of implementation and maintenance required. Moreover, as mentioned by [153], disseminating data regarding the performance of active control systems in real situations would lead to increased confidence in their large-scale implementation. • To date, there are no technical prescriptions regarding the implementation of active systems, and the practical applications realized so far have been based on very specific approaches. • Last but not least, the low popularity of active control is also caused by the lack of theoretical training among design engineers, as knowledge of both structural design and automatic control is required in this field. 33 from 44 34 from 44 Bibliography [1] C. Christopoulos and A. Filiatrault, Principle of Passive Supplemental Damping and Seismic Isolation. 01 2006. [2] T. E. Saaed, G. Nikolakopoulos, J.-E. Jonasson, and H. Hedlund, “A state-of-the-art review of structural control systems,” Journal of Vibration and Control, vol. 21, no. 5, pp. 919–937, 2015. [3] S. 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