Comparative Study Between Active AMD and ABS Devices by Using μ-Synthesis Robust Control

Abstract

The field of civil engineering has witnessed significant development since the emergence of innovative control strategies that enhanced the construction of structures, imparting valuable resistance against dynamic loads like wind or earthquakes. Despite numerous articles highlighting the potential of various control approaches to reduce vibration, their effectiveness in mitigating the dynamic effects on structures under real-world conditions appears limited once implemented. A variety of factors, including practical constraints, the choice of the control system device, the shape of the structure, and the amount of control energy deployed, contribute to this lack of efficiency. Within this context, the literature primarily addressed the discrepancy between the mathematical model and the actual structure model, commonly referred to as parameter uncertainties, in the controller design process. In other words, logical continuity in this field involves the application of a more adapted control approach, which enhances performance by incorporating more practical aspects in the controller synthesis procedure. These aspects include the dynamics of the control device, high-frequency neglected modes, and the inherent limitations or constraints of the control equipment. Thus, this study treats two main active control systems, ABS and AMD. While applying an approach known as μ-synthesis, the robust control was retained because of its ability to include all these considerations when they act simultaneously. We used this control to make sure that a three-degree-of-freedom structure responds as little as possible to seismic requests, which are shown by an uncertain model. We then conducted a comparative study between these two systems, focusing on displacement reduction and control force, while exploring a classic AMD control system at the top of the structure and an ABS control system at the bottom. This approach proved to be a powerful way to deal with the uncertainties affecting the structure and achieve the stability design objectives, given the satisfying simulation results.

1. Introduction

The field of earthquake protection for civil buildings has received significant attention due to the remarkable advancements in the active control of structures. The global occurrence of numerous earthquakes, which continue to result in human casualties and substantial property losses, amplifies this interest. In recent years, the integration of active control systems capable of handling such natural hazards has significantly advanced the construction of civil engineering structures. A number of works have shown the effectiveness of the active control of structures in dissipating the energy of dynamic seismic loads. Indeed, a large number of papers in the literature have investigated the active vibration control of structures [1,2,3,4,5,6]. However, despite the wide range of controllers tested, some practical considerations continue to pose an important engineering challenge that has received little attention to date. In a different version, the computed controllers showed promising results in reducing vibrations during simulations, but their real-world implementation did not fully live up to the same level of effectiveness. This disparity underscores the challenges and intricacies involved in transitioning from theoretical models and virtual environments to practical applications. Additional adjustments and optimizations of the control algorithms may be required to achieve the desired level of vibration reduction in the physical system.

In the field of active structural control for seismic protection, various control strategies are employed. Among these, linear quadratic regulator (LQR) control is widely used due to its ability to optimize energy consumption while reducing structural displacements [7,8]. Indeed, LQR control can be suitable for structures where the model is reliable, and energy consumption is a strict constraint. However, the LQR was not specifically designed for uncertain systems, so it is often combined with robust techniques to manage the inherent uncertainties of seismic loads and structural variations. H∞ control, on the other hand, offers a solution well-suited for applications where structural uncertainties are significant, such as in seismic control, wind turbines, and other structures exposed to extreme and unpredictable environmental loads [9]. Additionally, the H∞ approach explicitly incorporates structured uncertainties into the control design, providing a robust solution to seismic forces. Therefore, the choice between the two depends on priorities: robustness to uncertainties or the optimization of energy consumption and performance. In our work, the focus was on structural uncertainties and the rejection of random disturbances such as seismic events. The μ-synthesis control method we employed is not strictly an H∞ control, though it uses H∞ principles in its design. It can be considered a robust extension of H∞ control, specifically more suited to applications where uncertainty is both significant and structured. It provides robust performance by minimizing the worst-case impact of disturbances and uncertainties, making it ideal for seismic protection in highly variable environments.

Moreover, as with all previously discussed points, the choice of the control system device significantly influences the control performance. This choice appears to be just as important as the control method itself, as the actuator device has a tangible impact on the control design. Effectively, the actuators serve as active systems, translating control into a force that directly acts on the structure to counteract seismic effects; it is crucial to select the most suitable actuator for each structure. Two main active control devices in the active vibration control (AVC) field are the active bracing system (ABS) and the active mass damper (AMD). Base isolators are also specialized devices installed between a building’s foundation and superstructure to reduce the amount of energy transferred to the structure by decoupling it from ground movements. Together, ABS, AMD, and seismic isolation provide seismic protection, each with unique advantages depending on the specific requirements and characteristics of the structure.

The ABS device has an extra braking system with tendons connected to a servo-hydraulic actuator that is placed between the ground and the first floor of a building or between floors that follow each other. The AMD device, on the other hand, has an extra load on the last floor of the building [10].

An actuator connects the AMD to the structure. Thus, Samali et al. [11] compared the results of a regular TMD with the use of an AMD to control the active vibrations of a 40-story building shook by strong winds. Abdel-Rahman [12] proposed a guideline for the construction of active mass dampers (AMDs) to regulate a tall structure exposed to stationary random wind forces. Wang and Lin [13] used two controllers: fuzzy sliding mode control and variable structure control, for seismic-protected edifices equipped with active mass damper control systems. Guclu and Yazici [14] evaluated the efficacy of fuzzy logic and PD controllers in managing a 15-story frame fitted with AMDs on the first and fifteenth floors. Zhang et al. [15] conducted an experimental investigation on the fuzzy control of seismic structures utilizing an active mass damper (AMD). Tu et al. [16] conducted computational and experimental evaluations of the AMD control system utilizing the model reference adaptive control method.

ABS can decrease building vibrations by including active features between the ground and the first level. So, Preumont et al. [17] looked into how to use positive position feedback (PPF), direct velocity feedback (DVF), and integral force feedback (IFF) to control the active bracing of a seven-story building. However, Achour-Olivier and Arfa [18] investigated the Lyapunov method for vibration control in a SDOF building. Lu [19] proposed a discrete-time modal control strategy, which is a very useful method to control the seismic response of building structures equipped with ABS. Blachowski and Pnevmatikos [20] demonstrated a neural network-based vibration control method that effectively mitigated the shaking of an earthquake-damaged 3D multi-story building. They compared it to the traditional linear quadratic regulator (LQR) in terms of how it responded to displacement and how much force it used to control the building.

The controller design process within a robust control system follows a structured method to create a robust controller capable of maintaining performance even under extreme conditions or variations, making it particularly useful in applications such as seismic control or other high-stakes engineering systems. These steps include the following:

-

Identifying the system uncertainties by modeling the uncertainties present in the system;

-

Setting performance and stability specifications, which involves defining performance objectives, such as minimizing vibration amplitude and establishing stability margins that the controller must achieve despite uncertainties;

-

Using μ-analysis for robustness assessment with the modeled uncertainties;

-

Applying μ-analysis to measure the system’s robustness;

-

Synthesizing a controller using iterative synthesis, which involves minimizing the worst-case value across all the frequencies.

The structured singular value μ indicates how much uncertainty the system can withstand before performance degrades or instability occurs. This helps determine the level of robustness the controller needs to achieve.

The primary aim of our robust control system for structural applications is to minimize displacement under external disturbances, such as seismic events. We achieved this by setting performance objectives that limit the allowable displacement and optimizing the controller to maintain movement within these limits, thereby ensuring stability and safety. We also imposed constraints on the control force to prevent actuator saturation and reduce energy consumption. Through careful tuning of the controller gains and the use of a cost function that penalizes excessive control force, we achieved a balance between robustness and performance. Our method successfully integrated these goals, achieving effective damping while minimizing both structural displacement and control force, resulting in a robust response to the disturbances.

Some researchers have proposed a simple control force estimation approach to reduce control costs. The authors in [21] developed a response spectrum-based design strategy to incorporate active control, base isolation, and damping technologies. Viscous dampers dissipate energy through fluid motion, while hysteretic dampers absorb energy through material deformation. Similarly, [22] presents a spectral technique that relates the structural response of base-isolated buildings to seismic inputs, simplifying the estimation of LQR control force for passive base-isolated structures.

This paper examines the application of μ-synthesis, a robust control law, to actively regulate the response of three-story buildings to actual earthquakes, utilizing AMD and ABS as the primary active systems. Consequently, this study seeks to address this issue by comparing the two primary active control methods, ABS and AMD. We utilize the renowned efficiency of robust synthesis control, effectively implemented in several engineering applications over the past decade [23,24,25], to mitigate seismic impacts on structures.

Indeed, the selected control is renowned for its ability to incorporate various practical considerations simultaneously in the form of achievable performance objectives. The method’s ability to handle and measure uncertainties from various sources, as well as translate performance goals into clearly stated weighting functions in the control law criteria, makes it highly appealing [1,23]. So, we performed a comparison study on how to control the seismic vibration response of a building model that includes both parametric and dynamic uncertainties. Our main goals were to find the best way to reduce the building’s displacement and control the force that would be needed.

In this study, we aimed to reduce the response of a three-degree-of-freedom uncertain model of a building subjected to the earthquake of El Centro (1979) via the application of µ-synthesis control. We examined two active parts and compared them, with a focus on reducing the displacement and the necessary control force. The two parts were an AMD control system at the top of the structure and an ABS control system at the bottom. We derived the controllers via an extensive assessment procedure that included several uncertainties affecting the structural model. When figuring out how well the controller worked, it was important to look at both the parametric uncertainties (like mass, stiffness, and damping) in the structural model and the non-parametric uncertainties (like the actuator’s dynamics and constraints).

This approach, through simulation results, demonstrates a powerful method for managing all uncertainties affecting the structure system, particularly when the structure is simultaneously subject to various types of perturbations. These include parametric uncertainties that model variations in stiffness, mass, and damping, as well as non-parametric uncertainties that model the dynamic and limits of the control system, including sensor noise measurements.

2. Materials and Methods

2.1. Fundamentals Concepts Associated with μ-Synthesis Control

The parameter μ is the fundamental component upon which the theory of μ-synthesis is constructed. μ-synthesis is a control law based on the calculation of the structured singular value, referred to as μ and frequently abbreviated as SSV μ. The “D-K iterations” approach identifies certain functions (or matrices in the multivariable context) relevant to this computation, thereby facilitating the synthesis of a controller. Additionally, it serves as a tool to establish robustness and stability requirements through a process known as analysis.

Initially, we converted the effects of the uncertainties on the system into weighting functions to define their limits in terms of amplitudes and frequency range. Consequently, we utilized a linear fractional transformation (LFT) for the modeling of uncertainty. This representation facilitated the establishment of a connection that delineated the relationship between the system and the uncertainties.

The calculation of the H∞ bounds weighting functions served as the method for converting design objectives into performance specifications and constraints. Consequently, it was evident that a sequence of fundamental and complex stages was required to precisely translate the performance design objectives. This guaranteed the creation of a controller that complied with the tenets of μ-Synthesis theory.

2.1.1. Uncertainties Representation

The resilient controller design may have struggled to delineate the uncertainty regions. Based on the model parameter’s dependability, we could approximate these ranges for linearly modeled structures based on the dependability of the model parameter. The first step in implementing the synthesis was to delineate the uncertainty. Identifying their origins facilitated their categorization into two types: parametric (real) uncertainties and dynamic (frequency-dependent) uncertainties. The initial category of uncertainties stemmed from fluctuations in the physical system parameters, including temporal constants and natural frequencies, as well as the structural attributes such as mass, stiffness, and damping coefficients. The second category of uncertainty arose from the simplifications in the complex system models, including model order reduction and the omission of high-frequency dynamics. The uncertainties, referred to as frequency-dependent or complex uncertainty, were intrinsic to the model’s dynamic nature. Irrespective of the sources of uncertainty, we expressed them in one of two distinct forms: additive or multiplicative, as illustrated in Figure 1, following four combinations. We formulated Equations (1)–(4) to encompass all the identified uncertainty.

$$k=k_0+r_a\delta , \left|\delta \right|\le 1,$$

(1)

$$k=k_0\left(1+r_m\delta \right), \left|\delta \right|\le 1,$$

(2)

where |δ| = norm of the actual parameter deviation.

$$k_{min}\le k_0 \le k_{max},$$

$$k_0:={\displaystyle \frac{k_{max}+k_{min}}{2}}, r_a:={\displaystyle \frac{k_{max}-k_{min,}}{2}}, r_m:={\displaystyle \frac{k_{min}+k_{max}}{k_{min}-k_{max}}}$$

k_min and k_max are extreme values of uncertainty.

Equation (1) expresses the parametric uncertainties in additive form, while Equation (2) expresses parametric uncertainties in multiplicative form.

Equation (3) expresses dynamic uncertainties in additive form, while Equation (4) expresses them in multiplicative form.

$$M\left(s\right)=M_0\left(s\right)+W\left(s\right)\Delta \left(s\right)$$

(3)

$$M\left(s\right)=M_0\left(s\right)\left(1+W\left(s\right)\Delta \left(s\right)\right)$$

(4)

where

• $M\left(s\right)$: represents the global model of the system, which includes uncertainties,
• $M_0\left(s\right)$: represents the nominal matrix model devoid of uncertainty,
• $W\left(s\right)$: represents the weighting function that translates the effect of uncertainties,
• $\Delta \left(s\right)$: represents a set of all the stable transfer functions such that $\Delta ϵH_{\infty ,} ||\Delta {||}_{\infty }\le 1$.

The block ∆ corresponds to different uncertainty set formulations of stable functions; three cases of ∆ are possible:

• Unstructured uncertainty

  $$\Delta =\left\{\left[\begin{array}{cc}{\Delta }_{11}\left(s\right)& {\Delta }_{12}\left(s\right)\\ {\Delta }_{21}\left(s\right)& {\Delta }_{22}\left(s\right)\end{array}\right],{\Delta }_{ij}\subset H_{\infty }\right\}$$

  (5)
• Real uncertainty

  $$\Delta =\left\{\left[\begin{array}{cc}{\delta }_1& 0\\ 0& {\delta }_2\end{array}\right],{\delta }_i \in ℝ\right\}$$

  (6)
• Mixed uncertainty

  $$\Delta =\left\{\begin{array}{c} \left[\begin{array}{ccc}{\delta }_1{I}_{r1}& & \\ & \ddots & \\ & & \begin{array}{cc}{\delta }_s{I}_{rs}& \\ & \begin{array}{cc}{\Delta }_1& \\ & \begin{array}{cc}\ddots & \\ & {\Delta }_F\begin{array}{c} \\ \end{array}\end{array}\end{array}\end{array}\end{array}\right] {\delta }_i \in R,\end{array}\right\} {\Delta }_{ij}\subset H_{\infty }$$

  (7)

where

• ${\delta }_j$ represents the frequency responses of parametric uncertainty;
• ${\Delta }_i$ represents the frequency responses of dynamic uncertainty.

2.1.2. The Linear Fractional Transformation

The linear fractional transformation (LFT) is one of the most important tools for representing an uncertain system because it clearly separates the uncertainties in the ∆(s) block from the model as a whole. A mapping is used instead of a block diagram to show the analysis process. It makes a connection between a transfer matrix M(s), which shows how the system responds to feedback, and the new uncertainties block ∆(s).

Essentially, this represents prior knowledge of the process by illustrating how uncertainties ∆(s) influence the system transfer matrix M(s). Two forms of LFT are distinguished:

• Upper LFT defined by $\mathcal{Y}={\mathcal{F}}_{\mathcal{U}}\left(M,\Delta \right)\mathcal{U}$, as represented in Figure 2:

  $${\mathcal{F}}_u:=M_{11}+M_{12} \Delta {\left( I-M_{22}\Delta \right)}^{-1}{M}_{21}$$

  (8)

  $$\left\{\begin{array}{c}\left[\begin{array}{c}\mathcal{z}\\ \mathcal{y}\end{array}\right]=\left[\begin{array}{cc}{M}_{11}& M_{12}\\ M_{21}& M_{22}\end{array}\right] \left[\begin{array}{c}\mathcal{w}\\ \mathcal{U}\end{array}\right]\\ \mathcal{w}=\Delta \mathcal{z} \end{array}\right.$$

  (9)
• Lower LFT defined by $\mathcal{Y}={\mathcal{F}}_{\mathcal{L} }\left(M,\Delta \right)\mathcal{U}$, as represented in Figure 2:

  $${\mathcal{F}}_L:=M_{22}+M_{21} \Delta {\left( I-M_{11}\Delta \right)}^{-1}{M}_{12}$$

  (10)

  $$\left\{\begin{array}{c}\left[\begin{array}{c}\mathcal{y}\\ \mathcal{z}\end{array}\right]=\left[\begin{array}{cc}{M}_{11}& M_{12}\\ M_{21}& M_{22}\end{array}\right] \left[ \begin{array}{c}\mathcal{U}\\ \mathcal{w}\end{array}\right]\\ \mathcal{w}=\Delta \mathcal{z} \end{array}\right.$$

  (11)

where $M\left(s\right)=\left[\begin{array}{cc}{M}_{11}& M_{12}\\ M_{21}& M_{22}\end{array}\right]$.

2.2. Criteria for Robustness in Stability and Performance

The μ-analysis is a procedure that extends the Nyquist stability criterion to robust stability conditions for multi-input multi-output (MIMO) systems. To do this, one has to find the structured singular value μ of a system’s uncertain transfer matrix, which is written as μ(M(s)), for any structured complex block Δ of uncertainties across frequencies, usually at point s = jω.

In fact, the structured singular value (SSV μ) and the different ways of showing uncertainty that were talked about in the last section are very important for expressing robustness criteria. They provide a special means of measuring the size of any complex matrix M (n-by-n), thereby establishing a link between uncertainties and the physical system.

Hence, to define the μ value of the transfer matrix of a system M, consider a set of structured complex matrices ∆, such that

$$\Delta =\left\{\begin{array}{c} \left[\begin{array}{ccc}{\delta }_1{I}_{r1}& & \\ & \ddots & \\ & & \begin{array}{cc}{\delta }_s{I}_{rs}& \\ & \begin{array}{cc}{\Delta }_1& \\ & \begin{array}{cc}\ddots & \\ & {\Delta }_F\begin{array}{c} \\ \end{array}\end{array}\end{array}\end{array}\end{array}\right] \begin{array}{ccc}{\delta }_i& \in & ℂ\\ {\Delta }_j& \in & {ℂ}^{m_{j}x{m}_j}\end{array}\end{array}\right\}\subset {ℂ}^{nxn}$$

(12)

where ${\displaystyle \sum }_{i=1}^s{r}_i+{\displaystyle \sum }_{j=1}^F{m}_j=n$.

Then, μ is defined as the following:

$${\mu }_{\Delta }\left(M\right)={\displaystyle \frac{1}{\underset{\Delta \in \Delta }{\min }\left\{\overline{\sigma }\left(\Delta \right) : \det \left(I-M\Delta \right)=0\right\}}}$$

(13)

Equation (13) represents a minimization problem that is challenging to resolve both analytically and numerically; hence, upper and lower bounds are calculated in lieu of the μ-value itself. The robustness criteria for stability, resulting from the μ-analysis, is articulated through the new definitions based on the SSV μ as follows:

• Robustness condition on stability

M(s) is internally stable for any structured block Δ, $||\Delta {||}_{\infty }\le 1 \mathrm{and}$ $\det \left(I-M\left(j\omega \right)\Delta \left(j\omega \right)\right)\ne 0,$ if and only if

$$\mu \left(M\left(j\omega \right)\right):={\displaystyle \frac{1}{\begin{array}{c}min\left\{\overline{\sigma }\left(\Delta \left(j\omega \right)\right):\det \left(I-M\left(j\omega \right)\Delta \left(j\omega \right)\right)=0\right\}\\ \Delta \in \Delta \end{array}}}<1,\forall \omega$$

(14)

where ω is the frequency and $\overline{\sigma }$ is the maximum singular value of a matrix.

The robust stability requirement is not the sole feature that the controller must guarantee; additional conditions, termed robust performance, should also be evaluated to reflect the degree of degradation linked to specific levels of disturbances and uncertainties. The basic concept is clear: addressing the robust performance issue as a component of the robust stability challenge.

Consequently, if all the external disturbances affecting a system induce trucking and regulation errors in a weighted matrix, represented by the transfer matrix T, then in a multivariable control system, optimal performance is achieved when the robust stability requirement indicated by the H∞ norm in Equation (15) is satisfied:

$$||T{||}_{\infty }:= \underset{\omega \in R}{\max }\overline{\sigma }\left(T\left(j\omega \right)\right)\le 1$$

(15)

• Robust condition on performance

If T is an uncertain matrix represented as T = F_U (M, ∆), the upper linear fractional transformation (LFT) illustrates the extraction of uncertainty from the system. Consequently, as per Equation (15), a robust performance criterion is established.

$$||F_U\left(M,\Delta \right){||}_{\infty }:= \underset{\omega \in R}{\max }\overline{\sigma }\left(F_U\left(M,\Delta \right) \right)\le 1$$

(16)

2.3. μ-Synthesis and D-K Iteration Algorithm

The µ-synthesis technique aims to develop a controller K for a feedback uncertain system (G, ∆), as illustrated in Figure 2, by integrating H∞ control design with diagonal scaling methods from the SSV μ framework. The ∆ block illustrates both parametric and non-parametric uncertainty. These uncertainties are termed “structured uncertainties” and are constrained by the H∞ norm of uncertain gains. These gains illustrate the impact of various alterations on one or more inputs or outputs of a system.

The controller’s design process will specifically target the minimization of the structured singular value µ of a complex cost matrix. The structured singular value µ serves as an effective instrument for evaluating the system’s resilience via the µ analysis technique, addressing both stability and performance, as demonstrated in the preceding section.

Figure 2 represents the μ-synthesis control configuration with the following input–output structure:

• Inputs: w (disturbances), u (control inputs);
• Outputs: z (performance variables), y (measured outputs).

$\Delta$ represents the uncertainty block, $M=F_U\left(G,\Delta \right)$ represents the generalized uncertain plant, K is the controller and, $F_L\left(M,K\right)$ represents the interconnections of M, and K forms the closed-loop system.

The control objective is often formulated in terms of minimizing the H∞ norm of the transfer function from w to z, ensuring that the system can handle worst-case disturbances.

$${\mu }_{\Delta }\left(M\right)\le \underset{D\in \mathcal{D}}{\min }\overline{\sigma }\left(DM{D}^{-1}\right)$$

(17)

The µ-synthesis approach poses the problem of finding a matrix D for a complex matrix $M={\mathcal{F}}_{\mathcal{l}}\left(G,K \right),$ as shown in Figure 2, with

$$\mathcal{D}=\left\{\begin{array}{c} \left[\begin{array}{ccc}{d}_1{I}_{r1}& & \\ & \ddots & \\ & & \begin{array}{cc} d_s I_{rs}& \\ & \begin{array}{cc}{\mathcal{D}}_1& \\ & \begin{array}{cc}\ddots & \\ & {\mathcal{D}}_F\begin{array}{c} \\ \end{array}\end{array}\end{array}\end{array}\end{array}\right] \begin{array}{c}{D}_i\in {ℂ}^{{\Delta }_i\times {\Delta }_i}\\ \begin{array}{c}{D}_i=D_i^{*}>0\\ d_j\in R\\ d_j>0\end{array}\end{array} \end{array}\right\} ,$$

(18)

where M is the uncertain system and K is the controller to be designed.

The µ-synthesis problem consists of designing a nominally stabilizing controller K and solving the following minimization problem, which represents the µ upper bound over frequencies.

$$\underset{K}{\min }\underset{\omega }{\sup }\underset{D\left(j\omega \right)\in \mathcal{D}}{\min }\overline{\sigma }D\omega {\mathcal{F}}_{\mathcal{l}}\left(G,K\right)\left(j\omega \right)D^{-1}$$

(19)

This problem is equivalent to a minimizing scaled H∞ norm, which can be expressed as

$$\underset{K}{\min }\underset{D,D^{-1}ϵ\mathcal{R}{H}_{\infty },D\left(j\omega \right)ϵ\mathcal{D}}{\min }||D{\mathcal{F}}_{\mathcal{l}}\left(M,K\right)D^{-1}{||}_{\infty }\le 1$$

(20)

Equation (20) yields the cost function, which is a non-convex minimization problem involving D and K. The synthesis is executed using an algorithm known as D-K iteration to find a local minimum. Once the controller K is fixed and constitutes a convex problem, the µ-analysis procedure can be derived. However, if D is fixed, the problem becomes convex, and we are in front of an H∞ optimal controller design. If the minimized µ value is less than one, the resulting controller K is a robust stabilizing controller.

For the numerical computation of a μ-controller, an efficient algorithm called D-K iteration, summarized in seven steps, is executed, as follows:

Step 1: H∞ synthesis is used to find a controller that minimizes the nominal system’s closed-loop gain;

Step 2: The system conducts a robustness analysis to gauge the closed-loop system’s robust H∞ performance. This quantity is expressed as a scaled H∞ norm involving the D and G scaling, called the D step;

Step 3: Step 2 finds a new controller, known as the K step, to minimize the H∞-norm;

Step 4: Repeat steps 2 and 3 until the robust performance stops improving.

In summary, this approach can be characterized by its goal of synthesizing a controller that minimizes the structured singular value μ, which is associated with an uncertain system model. More precisely, it addresses the challenge of minimizing the closed-loop gain (μ < 1) for a weighted combination of desired and controlled system signals, all while maximizing the tolerance for allowable uncertainties within the system.

2.4. Mathematical Representation of the Structural Model

We model the structural system using established geometry, material properties, and boundary conditions. This representation of the structure constitutes the nominal model.

Given the configuration of a linear building structure subject to horizontal seismic excitation in Figure 3, we can formulate the equation of motion governing this system as follows:

$$M\ddot{y}\left(t\right)+C\dot{\dot{y}\left(t\right)+Ky\left(t\right)=} Hf\left(t\right)+E{\ddot{u}}_g\left(t\right)$$

(21)

The matrices of mass (see Equation (22)), damping (see Equation (23)), and stiffness (see Equation (24)) represent the mass, stiffness, and damping of the ith storey, respectively, where n is the number of structure stores.

$$M=\left[\begin{array}{ccc}{m}_1& \cdots & 0\\ ⋮& \ddots & ⋮\\ 0& \cdots & m_n\end{array}\right]$$

(22)

$$C=\left[\begin{array}{ccc}{c}_1+c_2& -c_2& 0\\ -c_2& c_2+c_3& -c_3\\ \begin{array}{c}0\\ \begin{array}{c}⋮\\ 0\end{array}\end{array}& \begin{array}{c}-c_3\\ \begin{array}{c}\cdots \\ \cdots \end{array}\end{array}& \begin{array}{c}\ddots \\ \begin{array}{c}-c_{n-1}\\ 0\end{array}\end{array}\end{array} \begin{array}{cc}\cdots & 0\\ \cdots & ⋮\\ \begin{array}{c}-c_{n-1}\\ \begin{array}{c} c_{n-1+}{c}_n\\ -c_n\end{array}\end{array}& \begin{array}{c}0\\ \begin{array}{c}-c_n\\ c_n\end{array}\end{array}\end{array}\right]$$

(23)

$$K=\left[\begin{array}{ccc}{k}_1+k_2& -k_2& 0\\ -k_2& k_2+k_3& -k_3\\ \begin{array}{c}0\\ \begin{array}{c}⋮\\ 0\end{array}\end{array}& \begin{array}{c}-k_3\\ \begin{array}{c}\cdots \\ \cdots \end{array}\end{array}& \begin{array}{c}\ddots \\ \begin{array}{c}-k_{n-1}\\ 0\end{array}\end{array}\end{array} \begin{array}{cc}\cdots & 0\\ \cdots & ⋮\\ \begin{array}{c}-k_{n-1}\\ \begin{array}{c} k_{n-1+}{k}_n\\ -k_n\end{array}\end{array}& \begin{array}{c}0\\ \begin{array}{c}-k_n\\ k_n\end{array}\end{array}\end{array}\right]$$

(24)

where $y\left(t\right), \dot{y}\left(t\right)$ and $\ddot{y}\left(t\right)$ are, respectively, the displacement, velocity, and acceleration vectors; $f\left(t\right)$ is the force provided by the actuator depending on the control law, and $g\left(t\right)$ is the ground acceleration. $H \in {ℝ}^{nxr }$ is a vector that gives the location of the force while E characterizes the impact of the external forces, namely the seismic excitation.

We use the state space representation to re-express Equation (21) in a more suitable form, allowing the new system model to preserve the temporal notions of its physical properties.

Moreover, we can model measurements of the structural system using the system states and exogenous inputs, such as noise, disturbance, and control inputs.

Therefore, we can substitute $x\left(t\right)=\left[y\left(t\right) \dot{y}\left(t\right)\right]$ in Equation (21) to create the following state vector:

$$\left\{ \begin{array}{c}\dot{x}\left(t\right)=Ax\left(t\right)+Bu\left(t\right)+Bw w\left(t\right)\\ y_n=Cx\left(t\right)+Du\left(t\right)+Dw w\left(t\right)+n\end{array}\right.$$

(25)

The system matrices A, B, and B_w are defined in the following way:

$$A=\left[\begin{array}{cc}0& I\\ -M^{-1}K& -M^{-1}C\end{array}\right]$$

(26)

A is the system matrix that encapsulates the relationships between the states and their derivatives. Matrix A converts the second-order linear differential equation (refer to Equation (21)) into a first-order differential equation (refer to Equation (25)). Equation (25) converts the unknown variables from Equation (21) into six unknowns.

$$B=\left[\begin{array}{c}0\\ M^{-1} H\end{array}\right]$$

(27)

$$B_w=-\left[\begin{array}{c}0\\ M^{-1}E\end{array}\right]$$

(28)

The desired output variable, y_n, forms the basis for C, D, and D_w, which are weighting matrices. B and B_w represent the disturbance and control input weighting vectors, respectively.

2.5. Active Vibration Control Devices

To mitigate the structural response, we strategically implemented structural control devices to attenuate the dynamic effects, particularly displacements and internal forces. Because the internal forces and structural deformations were naturally linked, these devices were mostly used to reduce the sideways movements caused by earthquakes and wind loads.

These control devices, distinguished by their unique operational mechanisms, fall into four primary categories: active, semi-active, passive, and hybrid controllers. We can group active structural control systems into three categories: the active tuned mass damper (ATMD), the active bracing system (ABS), and the active tendons system (ATS). In our work, we specifically focused on two main pieces of equipment used in the active vibration control of structures (AVC): the active mass damper (AMD) and the active bracing system (ABS). These active devices, also called actuators, produced the required forces via a power supply.

2.5.1. Control with AMD System

One of the first versions of active structural vibration control approaches was the improvement of TMD systems to actively adjustable mass–damper (ATMD) systems. This system is also known as the Active Mass Driver (AMD). This type of system primarily aims to counteract a structure’s lateral movements by positioning a steel pendulum or mass at its top, as shown in Figure 3. Between the structure and the TMD system, an actuator applied a calculated force in real time [26,27,28,29,30,31,32]. In the general case of the AMD, the matrices [M], [C], and [K] were of dimension (n + 1), and the displacement vector {x} had (n + 1) entries.

In our scenario, adding the AMD to the last floor produced the following matrix:

$$M=\left[ \begin{array}{cc}{M}_{nxn}& 0_{nx1}\\ 0_{1xn}& m_a\end{array}\right]$$

(29)

$$C=\left[\begin{array}{ccc}{C}_{n-1xn-1}& 0_{n-1x1}& 0_{n-1x1}\\ 0_{1xn-1}& C_{nxn}+c_a& -c_a\\ 0_{1xn-1}& -c_a& c_a\end{array}\right]$$

(30)

$$K=\left[\begin{array}{ccc}{K}_{n-1xn-1}& 0_{n-1x1}& 0_{n-1x1}\\ 0_{1xn-1}& K_{nxn+c_a}& -k_a\\ 0_{1xn-1}& -k_a& k_a\end{array}\right]$$

(31)

with m_a, k_a, and c_a as the mass, stiffness, and damping of the added AMD mass, respectively. Vector B represents the location of the device control and the seismic influence.

$$B={\left\{0,0 \dots ,-1,1\right\}}_{1x\left(n+1\right)}^T$$

(32)

2.5.2. Control with ABS

As shown in Figure 4, these systems typically consist of a set of pre-stressed bracings connected to a structure, with electro-hydraulic serving mechanisms controlling their tensions. By utilizing existing structural elements, active bracing control can minimize additions and significant modifications, preserving the structure in its original form. As a result, active control using an active bracing system (ABS) has been one of the most studied control mechanisms [26,33]. The added active damping ABS generates opposing forces between the interconnected degrees of freedom. When the structure is subjected to horizontal seismic excitation, the first floor experiences the most energy deformation. Therefore, it would be prudent to position the active bracing system between the ground and the first floor to maximize seismic protection.

The mass, stiffness, and damping matrices [M], [K], and [C] are defined by Equations (22)–(24), respectively. We then represented the influence vector B, which shows the location of the ABS active system and the forces it generates, as follows:

$$B={\left\{0 \dots ,0,-1,0 \dots ,0,1, \dots ,0\right\}}_{1xn}^T$$

(33)

2.6. μ-Synthesis Control of a Three-Story Building

2.6.1. Properties and Parameter Uncertainties

Through an application consisting of a three-story shear building [34], we aimed to demonstrate the efficiency of the robust μ-synthesis method presented in our study. We obtained the nominal model of this structure based on known geometry, material properties, and boundary conditions.

Therefore, as described in the previous section, the structure is a rectangular building with 3-DOF. Each floor measures 4.5 m by 3 m, resulting in three floors measuring 9 m in height.

The floor’s mass is identical from bottom to the top, equaling 1000 kg per floor. The structure’s damping coefficients remained constant across all the floors, at 1407 N.m/s, respectively. In terms of stiffness coefficients, these parameters had a significant impact on the structure’s rigidity. To make the structure more flexible, we set each floor’s stiffness in our study 10 times lower than in [24], specifically 98 kN/m instead of 980 kN/m. However, the three natural frequencies of the structure were 1.19 Hz, 3.58 Hz, and 5.97 Hz, respectively. However, we used the design parameters of the AMD as proposed by Sadek et al. 1997 [35]. These parameters encompass the mass ratio (µ), frequency ratio (γ), and the state parameters of the AMD (m_a, c_a, and k_a) as follows:

$$\mu ={\displaystyle \frac{m_a}{M_t}}; f={\displaystyle \frac{{\omega }_a}{{\omega }_s}}={\displaystyle \frac{1}{1+\mu }}\left[1-\beta .\sqrt{{\displaystyle \frac{\mu }{1+\mu }}}\right]; \xi ={\displaystyle \frac{\beta }{1+\mu }}+\sqrt{{\displaystyle \frac{\mu }{1+\mu }}}$$

(34)

$$m_a=\mu .M_t;c_a=2.\gamma .m_a.{\omega }_a; k_a={\omega }_a^2.m_a$$

(35)

where

$\beta$ represents the structure’s damping, which is equal to 0.02;

μ represents the mass ratio, which is equal to 0.05;

M_t represents the total mass of the structure, which is equal to 3000 kg;

f represents the tuning ratio;

ω_a represents the AMD’s natural frequency;

ω_s represents the natural frequency of the structure’s first mode.

Therefore, the design parameters of the AMD used in this study are as follows:

m_a = 150 kg, c_a = 1185.948 N.m/s, and k_a = 2608.33 kN/m.

When we placed the AMD actuator on the last floor, it generated the necessary active force to counter seismic loads, whereas we placed the ABS actuator on the first floor, which exerted a force on the entire structure through the other floors. If we neglected to factor in potential uncertainties in the structure’s parameters, such as mass, stiffness, and damping, the controller’s control force might not have been effective in its actual implementation conditions. Indeed, any uncertainties in these parameters would induce a less accurate control. That is why, to ensure robust seismic protection with the μ-controller, we considered that there were 30% of uncertainties in the stiffness of the three floors of the structure.

2.6.2. Actuators Modelization and Related Dynamic Uncertainties

Typically, we use filters that accurately capture the behavior of the system when modeling the dynamics of AMD or ABS actuators. In fact, the filter choice depends on the specific actuator characteristics and application requirements. To model the dynamics of such systems, we use some common filters. Usually, we employ low-pass filters because they effectively handle low-frequency inputs like building oscillations caused by wind or seismic activity. This work considers the first-order filter G_act, as its roll-off aligns with the frequency range under consideration.

$$G_{act}={\displaystyle \frac{K_{act}}{T_{act }s+1 }}={\displaystyle \frac{10}{0.2 s+1}}$$

(36)

The transfer function G_act models the nominal actuator dynamic, but a significant error exists between this model and the physical device, reflecting 10% and 20% uncertainty on K_act and T_act, respectively. It creates a group of actuator models, including the nominal model, with an amount of uncertainty that changes with frequency. Each model shows a possible behavior of the uncertain actuator.

These uncertainties are regarded as unmodeled dynamics characterized by the input multiplicative uncertainty constrained in magnitude. The weighting function W_unc in Equation (37) adjusts the degree of uncertainty and is derived by a graphical trial-and-error approximation:

$$W_{unc}={\displaystyle \frac{0.38s-0.5475}{s+5.475}}$$

(37)

2.6.3. Control Design Objectives

The main objective of this study was to achieve a compromise between producing a low control effort and significantly reducing the seismic effect on the structure. To achieve this goal through synthesis control, we expressed the design objectives as a single cost function that requires minimization. When it came to actively controlling the structures against seismic activity, we could formulate the design objectives as disturbance rejection problems, with the primary aim of control being to reduce both the impact of the earthquake and the effort required for control.

The diagram in Figure 5 summarizes these objectives: the model inputs were the seismic disturbance and actuator force, while the outputs were the displacements of the three floors. The goal was then to minimize (μ < 1) the impact of the signals on a weighted combination of the control signal and the measured outputs.

A crucial step in μ-synthesis control is the careful selection of frequency-dependent weighting filters to achieve the desired performance objectives. This study formulates the performance based on two main requirements:

• Attenuate the disturbance’s effect on system outputs by a factor of 80 below the system’s roll-off frequency;
• At all the frequencies, constrain the control effort to restrict the actuator’s command signal gain to a factor of 0.001.

3. Description of the Model

We conducted numerical simulations on the three-story seismic-excited building structure model, presented in the last section and defined by Equation (22), to show the effectiveness of the robust control approach in this work.

We considered two principal sources of uncertainties in our control design of the structure: the first one, known as parametric uncertainty, accounted for variability in its model parameters. In this case, we focused on the uncertainties in the stiffness parameters k of each floor, including the floors where the control devices were placed. Errors in the physical actuator dynamics, modeled by a first-order transfer function, constituted the second source of uncertainty, known as the dynamic uncertainty.

Because μ-synthesis is a multivariable control method, one of its best features is that the state vector made up of the three stages’ displacements and speeds can be used as input (feedback vector) to figure out the controller that will give you the best results. We first computed the responses of the open-loop system (uncontrolled structure) and compared them with the closed-loop ones (controlled structure).

3.1. Selected Earthquake

A specific factor influenced the control objectives and should be explicitly included in the μ-control design; it is the characteristic of the soil disturbances, notably the earthquake. Therefore, we simulated and subjected the controlled structure to real earthquake loads to assess its performance.

We employed the Kanai–Tajimi seismic filter model [36,37], whose frequency peak corresponds to the maximum energy in a set of near-fault earthquakes of El Centro, Northbrige, and Kishami. We used a second-order filter W_dist model here, with the following transfer function:

$$W_{dist}={\displaystyle \frac{\sqrt{S_0}\ast \left(2\ast z_z\ast s+w_g^2\right)}{s^2+2\ast z_g\ast s+w_g^2 }}$$

(38)

where $w_g=2\pi f{\displaystyle \raisebox{1ex}{$rd$}\!\left/ \!\raisebox{-1ex}{$s$}\right.}, f=1,17 Hz$, $\xi =0.6$, and $S_0={\displaystyle \frac{0.03\ast z_g}{\left(\pi \ast w_g\ast \left(4\ast z_g^2+1\right)\right)}}$.

Equation (38) became:

$$W_{dist}={\displaystyle \frac{0.4216 s+26.21}{s^2+22.38 s+1391}}$$

(39)

3.2. Performance Criteria

When choosing the design parameters, it is important to emphasize that in the realm of active control, the primary performance objective is to achieve a well-balanced compromise between reducing the structure’s response and minimizing the required control effort. The two active components, placed at different structure floors, represented displacements, accelerations, and control forces in the time domain. Similarly to electronics, people often prefer root mean square (RMS) values because they offer a more meaningful measure of a signal’s average power over a specific period. In our study, we employed them to measure the physical quantities, thereby enhancing our understanding of the developed controllers’ performance.

3.3. Simulations and Analysis

Using MATLAB software, we developed two controllers based on synthesis theory to control the described uncertain structure model. The two active systems considered in this study were AMD and ABS.

Concerning uncertainties, to show that the controllers we designed were strong, as we already talked about, we looked at the specific case of parametric and dynamic uncertainties affecting the design of the controllers. When large calculation errors occur in certain parameters, like stiffness, it becomes crucial to protect the structures, as this parameter significantly affects the structure’s rigidity. We assumed 30% of the errors in the stiffness of each floor, modeling these uncertainties as additive errors.

When considering dynamic uncertainties, we specifically took into account the errors in modeling the dynamics of the actuators. The actuators served as active systems, translating the calculated control into a force that directly acted on the structure to counteract the seismic effect. Because they created a mechanical force that caused a non-negligible dynamic, mistakes in the calculation of this dynamic were an important part of the study to make sure that the controller would work well in real-world situations. W_unc models these errors as multiplicative input errors.

The computation of the two controllers was based on the measurements of the feedback state vector, which included the displacements and velocities of the three floors. The first controller, referred to as μ_C-MD for simplicity, synthesized and treated the case of an AMD system placed on the last floor, which was the suited place for this type of system. The second controller, known as μ_C-BS, addressed the ABS on the first floor, which experienced the highest deformation.

Figure 6 represents the seismic signal acceleration of the El Centro earthquake (1979), to which the structure must submit, in the time domain. The examination of this figure informs the maximal acceleration (0.3 g) and the duration (45 s) of the shaking of the earthquake considered in this work.

We computed both the uncontrolled and controlled responses of the excited structure based on the previously described assumptions. The figures presented below illustrate the simulation of the effect of the selected ground acceleration on displacements and accelerations of the studied structure before and after control in the time domain. We also compute and compare the control forces provided for each type of control system.

4. Results

4.1. Analysis of the Controllers Performances with SSV μ

In Section 3, the synthesis approach for structural active control involved identifying a controller that reduced the structured singular value (SSV) of the structure’s uncertain model, enabling the system to manage the most severe disturbances. We then formulated the control using the minimization problem of the transfer function, which connected the controlled signals (displacements, accelerations, etc.) to the disturbances represented by the stochastic seismic signal.

Based on this concept, the structural active control problem could be compared to solving a disturbance rejection problem, where the synthesized controller must minimize the closed-loop gain of the weighted combination of the outputs and control effort by ensuring a value of 1 (Equations (14) and (16)). The SSV μ was then the effective guarantee to reach the required robustness in the performance of the controlled system against structured uncertainties.

Thus, a high SSV μ > 1 indicates increased sensitivity to uncertainties, which may be undesirable. Conversely, a low SSV μ < 1 indicates enhanced robustness to uncertainties. More precisely, as a rule of thumb, if μ is less than 0.8, then the achievable performance can be improved; when it is greater than 1.2, the desired closed loop bandwidth is not achievable for the given amount of system uncertainties.

In conclusion, the value continues to serve as the most reliable indicator of the performance achievable under actual implementation conditions. To evaluate the controllers’ performance achieved with the AMD and ABSs, the one realizing the closest SSV μ to one and respecting the conditions presented above was considered more effective and robust against uncertainties.

The first controller developed, μ_C-MD, reached a SSV μ of 1.0907, which was a value included in the secure interval (0.8 < μ < 1.2), traducing the guarantee that the controller can meet the desired performance objectives (structure responses’ mitigation to seismic load).

The second controller developed, μc-_BS, reached a SSV μ of 1.1861, which was also a value included in the secure interval (0.8 < μ < 1.2).

Based on the SSV realized, the two controllers presented very good indications in their ability to respond to the fixed control requirements; however, one should explore more results to compare their performances and attest which one of the two system used presented the best results.

4.2. Displacements and Accelerations Analysis

• AMD Control System

In this case, recall that μ_C-MD refers to the controller designed with the AMD system, with which Figure 7a–c was obtained. They represent the three floors’ responses from the bottom to the top of the structure. The three floors recorded the displacements, demonstrating the devastating impact of an earthquake without protection, and the significant reduction in the structure through μ_C-MD control. When compared to the first, second, and third floors, the displacement peaks without control were 0.1036 m, 0.1847 m, and 0.2345 m for each floor, respectively. However, after applying μ_C-MD control, these peaks decreased to 0.0540 m, 0.0881 cm, and 0.0975 cm.

The percentage of these considerable reductions reached about 71% for the first floor, 73% for the second floor, and 75% for the last floor. This constituted an encouraging result, which led us to continue the evaluation of the μ_C-MD controller by computing the structure accelerations.

The accelerations plotted in Figure 8a–c showed the three floors’ responses from the bottom to the top of the structure, before and after control. The figures clearly demonstrated an improvement in the structure’s acceleration responses, with the peak accelerations of the floors before control reaching 3.331 (m/s²), 3.4229 (m/s²), and 4.59 (m/s²) from the bottom to the top, respectively, and reducing to 2.25 (m/s²), 3.12 (m/s²), and 2.1 785 (m/s²) after control.

The acceleration reduction percentages reached nearly 48%, 56%, and 67% for the first, second, and third floors, respectively, confirming the controller’s efficiency and providing satisfying reduction results.

• ABS Control System

In Figure 9 and Figure 10, one can see the three floor displacements and the accelerations of the structure responses before and after control with the controller μ_C-BS, which was made for the ABS control system. In these figures, one can see a significant attenuation of the three floors’ displacements and accelerations.

Indeed, the application of the µ_C-BS controller reduced the peak displacements in Figure 9a–c to 0.0617 m, 0.0787 m, and 0.1063 m. In Figure 10a–c, one can see that after control, the acceleration peaks were also reduced to 3.3860 (m/s²), 5.0413 (m/s²), and 5.7201 (m/s²) for the first, second, and third floors, respectively.

The percentages of the displacement attenuation achieved were about 67%, 71%, and 72%, while the percentages of the acceleration attenuation were about 32%, 54%, and 59% reached by the first, second, and third floors, respectively.

The time domain measurements of the three floor movements and accelerations, following the implementation of the AMD and ABS control systems with the μ_C-BS and μ_C-MD controllers, indicated that both the systems effectively achieved the performance objectives, with the μ_C-MD controller demonstrating marginally superior attenuation results.

4.3. Control Forces Analysis

We checked the two designed controllers based on the control force necessary to achieve the robust performance we previously presented. Figure 11 and Figure 12 were plotted to reflect the control effort in the time domain for the nominal uncertainties realized with µ_C-MD and µ_C-BS, respectively. To reduce the force produced by the control system, the µ-synthesis control allowed for the incorporation of weighing functions to shape the obtained results. Hence, while using the Wu = 10⁻², the control forces produced by the two controllers were computed and compared. Figure 13 represents the two control forces simultaneously to enhance understanding of the results.

We observed a notable distinction from the previously obtained results of displacements and accelerations, which were relatively similar. Indeed, the cost of the control forces reached a maximum of 996.91 N and 1449 N by µC-MD and µC-BS, respectively. Figure 13 shows clearly that the AMD system required the lowest cost of the control force with fewer fluctuations to obtain the previous attenuation results, which clearly confirmed the undeniable advantage of the AMD system versus the ABS.

4.4. RMS Values Evaluation

To more precisely assess the efficacy of the controls, evaluations were conducted using additional indicators such as the maximum peak values or the root mean square (RMS) values of the temporal signals. Therefore, it is important to underline that the time representations obtained previously with the two controllers showed a slight improvement of the results with μ_C-MD compared to those obtained with μ_C-BS. Figure 14 and Figure 15 represent the calculated RMS values of displacements and accelerations of the three floors before and after control. These figures demonstrate the significant attenuation previously noted, confirming that both of the developed controllers effectively reduced the structure’s responses to seismic load, with the AMD controller showing a slight advantage over the ABS controller due to its relatively high percentages.

4.5. Worst Case Analysis

To evaluate the robustness of the synthesis control method in the face of uncertainties, it is essential to analyze the outcomes under two conditions: when uncertainties are at their central values, referred to as the ‘Nominal Case’, and when uncertainties reach their extreme values, termed the ‘Worst Case’.

We calculated the structural RMS quantities before and after controlling the displacements of the floors where the control systems were located, specifically the first and third floors, to thoroughly examine the robustness of the designed controllers. We performed this by varying the system uncertainties, specifically the stiffness parameters, to their lowest values, as shown in Figure 16 and Figure 17, respectively. Thus, these figures represented the RMS quantities of the displacements of the considered floor before control and the two cases after control:

• The nominal situation arose when the stiffness parameters were at their median values, corresponding to 98 kN/m;
• When these parameters fluctuated to their extreme values, ±30% of their nominal values were referred to as the worst scenario, resulting in the poorest performance, which corresponds to 68.6 kN/m.

Both of the figures distinctly illustrate that, even at peak uncertainty, the structure’s responses to seismic loads exhibited consistent levels of attenuation, with just little discrepancies in the attenuation results between the two controller’s μ_C-MD, and μ_C-BS.

Indeed, the controller µ_C-MD achieved the worst-case floors’ displacements attenuation percentages of 64%, 67%, and 70% for the first, second, and third floors, respectively, while the controller µ_C-BS achieved these percentages of 53%, 66%, and 67%. The performance of the two controllers in the bad cases was actually very satisfactory, attesting to the robustness of the control method applied.

5. Conclusions

This research demonstrates that μ-synthesis is effective for creating controllers for both AMD and ABSs. It markedly reduced the structural response while considering the parametric and dynamic uncertainties that influenced the structure–actuator system. As a result, we assumed 30% of the errors in the stiffness of each floor, modeling these uncertainties as additive errors. The results reported in this research allowed us to conclude the following conclusions:

• The performance analyses conducted in this study, regarding the percentage of RMS and peak displacements and accelerations, averaged between 70% and 75%, with significantly improved attenuation observed with the AMD system;
• In the worst-case scenario, we sustained these percentages with an equivalent degree of attenuation, guaranteeing a uniform, marginal disparity between the two control systems;
• The control force results indicated that the AMD exhibited superior structural performance and consumed less energy than the ABS;
• The system that minimized the structure’s reactivity to seismic loads while maintaining minimal control costs was unequivocally the optimal choice. From this perspective, the AMD system, requiring a control force of less than 1000 N, appeared to possess a superior advantage.

Consequently, we must execute experimental testing in the future to corroborate the simulation results on a more precise structure, which will influence the performance of both devices. The ABS and AMD are significantly influenced by financial limitations, energy efficiency objectives, and, crucially, the ease of installation with minimized maintenance requirements. Furthermore, the µ-synthesis control explicitly integrated weighting functions that closely to the intended performance and equilibrium, facilitating further improvement and refinement of the outcomes through more precise modification of the weighting functions.