Optimum fractional order calculus–based sliding mode control for seismic control of structures equipped with active tuned mass damper

Abstract

An optimum fractional order calculus–based sliding mode control (OFOSMC) is proposed to suppress earthquake-induced vibrations on structures equipped with active tuned mass damper (ATMD). The proposed method features a novel integration of fractional-order sliding mode control with Linear Quadratic Regulator (LQR)–based optimization. Unlike conventional integer-order sliding mode control (SMC), the approach employs a fractional-order sliding surface, introducing an additional degree of freedom that enhances control flexibility and robustness. Furthermore, the LQR-based optimization scheme is utilized to systematically obtain optimal control parameters by minimizing a cost function that balances structural response reduction and control effort, thereby eliminating the need for heuristic tuning commonly used in existing fractional-order SMC methods. For the first time, the proposed OFOSMC strategy is applied to the control of building structures equipped with an ATMD system, offering a novel and robust method for mitigating seismic responses of structures. For this end, the design of the proposed OFOSMC strategy is implemented for a benchmark 11-story shear frame equipped with an ATMD system under an artificial earthquake record. Finally, the performance of the designed OFOSMC strategy is assessed through analyzing its ability in the reduction of the building structure’s seismic responses subjected to both near-fault and far-fault earthquake excitations. A comparative study is conducted against several existing control strategies. Comparative results demonstrate that the OFOSMC strategy exhibits superior effectiveness in mitigating the structural responses during real earthquake excitations.

Keywords

structural control active tuned mass damper fractional-order calculus sliding mode control optimal control strategy

1. Introduction

The earthquake phenomenon, as one of the most highly destructive natural disasters, causes substantial damage to human lives and property due to various direct and indirect effects, including intense ground shaking, structural collapse, landslides, tsunamis, fires, and disruptions to essential services (Jarrahi et al., 2022; Khatibinia et al., 2019). Earthquake intensity affects damage to structures by reducing their resistance and stability. Therefore, ensuring adequate resistance and stability through proper design and construction is crucial to prevent both local and global damage. This involves various engineering strategies such as strengthening connections, improving building codes, and implementing innovative seismic-resistant designs, materials, and control devices. Seismic control devices can mitigate vibrations and protect buildings and bridges from earthquake damage by absorbing or dissipating seismic energy and reducing structural response during seismic events.

Tuned Mass Damper (TMD) is a common strategy for controlling structural vibrations particularly induced by wind and earthquakes (Shi et al., 2018). The system is composed of a mass, spring, and damper, which is tuned to resonate with the natural frequency of the structure, thereby dissipating vibrational energy and reducing oscillations. While the effectiveness of TMD generally increases with large mass ratios, its practical use in buildings and bridges is hindered by the inconvenience of accommodating and supporting a significant mass. Further, its effective damping is only provided within a narrow frequency band (Elias and Matsagar, 2017; Ozturk et al., 2022). To overcome the limitations of the TMD system, various types of TMD have been proposed such as multiple TMD (Lu et al., 2017b), nonlinear viscous damping (Chung et al., 2009), semi-active TMD (Rafieipour et al., 2014), hybrid TMD (Salvi et al., 2015), eddy-current TMD (Lu et al., 2017a; Zhou et al., 2018), electromagnetic TMD (Lin and Lin, 2023), bidirectional pendulum TMD (Matta, 2019), particle TMD (Lu et al., 2017c, 2017d, 2018), and TMD inerter (TMDI), (Giaralis and Marian, 2016; Patsialis et al., 2023). However, despite these passive and semi-active improvements, active TMD (ATMD) systems have been proposed as a highly effective control strategy and offer distinct advantages over passive control methods (Ricciardelli et al., 2003; Zhan et al., 2013). ATMD systems can adapt to varying frequencies and handle uncertainties, unlike passive control systems. In an ATMD system, a control force is generated by an actuator and is applied to the ATMD. The reaction force is transmitted to the main structure. The control force is provided by a control algorithm (or control law), which is the core of the ATMD system and is crucial for the successful operation of ATMD (Eliasi et al., 2022).

Numerous control methods have been developed and utilized for the design of active control systems in structural engineering. Several popular control methods include linear quadratic regulator (LQR) (Alavinasab et al., 2006; Chang and Soong, 1980; Li et al., 2004), linear quadratic Gaussian (LQG) (Chen et al., 2011; Samali et al., 2004), proportional-integral-derivative (PID) controller (Etedali and Tavakoli, 2017; Guclu, 2006; Guclu and Yazici, 2009), robust saturation control (Lim, 2008), bang–bang control (Collins et al., 2006), H₂ and H_∞ (Huo et al., 2007; Palazzo and Petti, 1999; Park et al., 2009), fuzzy logic controller (FLC) (Pourzeynali et al., 2007; Shariatmadar et al., 2014; Wang and Lin, 2007), sliding mode control (SMC) (Khatibinia et al., 2020; Monajemi-Nezhad and Rofooei, 2007; Yakut and Alli, 2011), sliding sector control (SSC) (Saadatfar et al., 2023, 2024b, 2025). Further, the intelligent and metaheuristic techniques have been used to enhance seismic control of structures. Yang et al. (2006) explored the application of neural networks for the system identification and vibration control in structures equipped with an ATMD system. Eliasi et al. (2022) presented the optimum design of a SMC strategy for simultaneously mitigating the control force and the structural seismic response in of an 11-story shear building equipped with ATMD. Lavasani and Shangapour (2022) investigated the capability of the interval type-2 fuzzy logic controller (IT2FLC) for improving the seismic performance of structures under near-field and far-field earthquake excitations. Hosseini Lavasani et al. (2022) presented a hybrid control method including the IT2FLC and fractional-order PID strategies. This hybrid control method was used to control the TMD system with a magnetorheological (MR) damper-equipped a 15-story shear building and tune the voltage of the MR damper. Saadatfar et al. (2024a) developed the optimal design of the SSC strategy for improving the seismic performance of an 11-story shear building structure equipped with an ATMD system. Recent studies have increasingly applied artificial intelligence (AI)–based controllers to ATMD systems, demonstrating that methods such as neural networks, fuzzy logic, and reinforcement learning can significantly enhance adaptability and vibration mitigation performance under nonlinear behavior and external uncertainties (Chen and Chien, 2020; Ghanemi et al., 2024; Yao et al., 2024). These AI-driven approaches learn complex dynamic patterns and thus offer improved efficiency and robustness compared with conventional control strategies. However, conventional ATMD controllers suffer from robustness limitations in the presence of nonlinearities and uncertainties. This motivates the use of fractional-order sliding mode control, as will be discussed in the following sections.

SMC based on the variable structure control is known as a powerful and robust control scheme and handles uncertainties and disturbances (Gambhire et al., 2021; Yu et al., 2021). The design process of SMC involves two key steps: (1) designing a sliding surface that defines the desired dynamics and specifications; (2) developing a control law to drive the system towards and maintain it on this surface, ensuring stable operation. The sliding surface is a crucial design element that dictates the system’s desired behavior and can guarantee the global stability of a dynamical system (Banks et al., 2000). However, the classical SMC suffers from significant practical challenges such as the singularity problem and chattering phenomenon. In recent years, the integration of fractional-order control (FOC) with the classical SMC has received extensive attention in order to eliminate its drawbacks (Duc et al., 2018). In fact, the FOC strategy can enhance system performance compared to integer-order control by leveraging additional parameters from fractional-order operators, leading to more flexible and superior controller designs in terms of steady-state error, disturbance immunity, and response time (Wang et al., 2022). Fractional-order SMC strategies generally offer superior performance compared to the classical SMC by providing faster tracking, higher control accuracy, and improved robustness against uncertainties and disturbances. Several studies demonstrate the benefits of the fractional-order SMC strategies with in comparison with those of classical SMC. Sun et al. (2018) developed a discrete-time fractional-order SMC scheme which ensured the desired tracking performance of a linear motor control system. Aghababa (2017) proposed a fractional-order switching sliding surface for a type of fractional-order chaotic systems to guarantee the existence of the sliding motion in finite time. Fei and Lu (2018) introduced an adaptive fractional order SMC with neural estimator which could improve the tracking performance for a class of systems with nonlinear disturbances. Nguyen et al. (2020) presented a fractional-order derivative-based SMC for semi-active vehicle suspensions and confirmed its ability to achieve robust and high-performance control with finite-time convergence. Eray and Tokat (2020) investigated the effectiveness of a fractional-order SMC with a time-varying sliding surface for the control of nonlinear models subject to parameter variations and external disturbances. Zheng et al. (2022) developed an enhanced fractional order sliding mode control (FOSMC) method for improving the control performance of the fractional order uncertain systems with multiple mismatched disturbances. Recently, Bingol (2025) has proposed a hybrid of fractional-order SMC and neural network–based estimation in order to enhance robustness and adaptability in stabilizing complex systems. Additionally, the successful application of fractional calculus has been demonstrated in improving the performance of traditional SMCs (Chen et al., 2023; Xu et al., 2023).

To the best of the authors’ knowledge, the studies mentioned above indicate that the effectiveness of the SMC strategies with fractional-order calculus concept has not been investigated in the design of ATMD systems. Hence, this study proposes an optimum fractional-order sliding mode control (OFOSMC) for the seismic control of structures equipped with an ATMD system. The proposed OFOSMC strategy is designed by integrating the fractional-order calculus concept into the SMC. In the proposed strategy, the Linear Quadratic Regulator (LQR) method utilizes a cost function to design a control law that optimizes system performance. By minimizing this cost function, the LQR ensures the system’s states and control inputs converge to desired values, leading to improved overall system behavior. Further, the OFOSMC strategy enhances control system performance by improving accuracy and energy efficiency, and increasing adaptability to unexpected events like earthquakes. To investigate the effectiveness of the proposed controller, a benchmark 11-story building frame is considered and controlled by an ATMD system installed on the top floor. The design of the proposed OFOSMC is first conducted for the reduction of the structural responses under an artificial earthquake ground motion. Finally, a comparative simulation study is performed to demonstrate the superiority of the designed OFOSMC in mitigating structural responses compared to other control techniques under four real ground excitations.

2. Motion equations of controlled building structure

2.1. Building structure equipped with a TMD

An N-story shear structure subjected to an earthquake excitation, ${\ddot{x}}_{g} (t),$ is assumed to be controlled using a TMD system (positioned on the upper story of the structure). The motion equations of the building structure can be expressed as (Etedali et al., 2018):

M_{T} \ddot{x} (t) + C_{T} \ddot{x} (t) + K_{T} x (t) = - M_{T} r {\ddot{x}}_{g} (t)

(1)

where

M_{T}, C_{T},

and

K_{T}

represent the

(N + 1) \times (N + 1)

mass, damping and stiffness matrices of the structure equipped with a TMD, respectively. r is an (N+1) vector of ones. The (N+1) x (t) vector is the relative displacement of each story and the TMD system and is expressed as

x (t) = {[x_{1} (t), x_{2} (t), . . ., x_{N} (t), x_{d} (t)]}^{T}

(2)

Assume that lumped mass is considered for the stories of the structure and for the TMD system. Hence, the mass matrix, $M_{T},$ is defined as follows:

M_{T} = d i a g (m_{1}, m_{2}, \dots, m_{N}, m_{d})

(3)

where m_i is the mass of the ith story, and m_d is the mass of the TMD system.

The stiffness matrix, $K_{T},$ is expressed as

K_{T} = [\begin{array}{l} \begin{array}{l} k_{1} + k_{2} & - k_{2} \\ - k_{2} & k_{2} + k_{3} & - k_{3} \end{array} & 0 \\ ⋱ \\ s y m & \begin{array}{l} - k_{N} & k_{N} + k_{d} & - k_{d} \\ - k_{d} & k_{d} \end{array} \end{array}]

(4)

where

k_{i}

is the stiffness of the ith story, and

k_{d}

is the TMD stiffness.

The damping matrix, $C_{T},$ is expressed as

C_{T} = [\begin{array}{l} \begin{array}{l} c_{1} + c_{2} & - c_{2} \\ - c_{2} & c_{2} + c_{3} & - c_{3} \end{array} & 0 \\ ⋱ \\ s y m & \begin{array}{l} - c_{N} & c_{N} + c_{d} & - c_{d} \\ - c_{d} & c_{d} \end{array} \end{array}]

(5)

where

c_{i}

is the damping of the ith story, and

c_{d}

is the stiffness of TMD.

The damping matrix of the structure, $C_{S},$ is determined based on the Rayleigh damping method as

C_{S} = \frac{2 ξ ω_{1} ω_{2}}{ω_{1} + ω_{2}} M_{S} + \frac{2 ξ}{ω_{1} + ω_{2}} K_{S}

(6)

where ω₁ and ω₂ are the first and second frequencies of the structure, respectively. ξ is expressed as the ratio of critical damping of the structure. M_S and K_S are the mass and stiffness matrix of the structure, respectively. The matrices are defined as follows:

M_{S} = d i a g (m_{1}, m_{2}, \dots, m_{N})

(7)

K_{S} = [\begin{array}{l} k_{1} + k_{2} & - k_{2} & \dots & 0 & 0 \\ - k_{2} & k_{2} + k_{3} & \dots & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & \dots & - k_{N} & k_{N} \end{array}]

(8)

2.2. Building structure equipped with an ATMD

The motion equations of the N-story shear frame equipped with an ATMD system installed on the top floor (shown in Figure 1) are represented as

M_{T} \ddot{x} (t) + C_{T} \ddot{x} (t) + K_{T} x (t) = - M_{T} r {\ddot{x}}_{g} (t) + Λ u (t)

(9)

where

u (t) \in R

denotes the control force. Also,

Λ_{N \times 1} = {0, 0, \dots, 0, - 1, 1}^{T}

is the location vector of the control force.

Figure 1.

N-story shear frame equipped with an ATMD system.

For the controlled building structure equipped with the ATMD system, the control force is provided by an actuator in real time and is applied to the ATMD system. The equations of the ATMD-equipped structure defined in equation (6) can be transformed into the state-space form and written as

z (t) = {\begin{cases} x (t) \\ \dot{x} (t) \end{cases}}_{2 (N + 1) \times 1}

(10)

\dot{z} (t) = A z (t) + B u (t) + E {\ddot{x}}_{g} (t)

(11)

y = C z (t) + D u (t)

(12)

The state matrix A, input matrix B, and matrix E are defined as

\begin{array}{l} A = {[\begin{array}{l} 0_{(N + 1) \times 1} & I_{(N + 1) \times (N + 1)} \\ - {M_{T}}^{- 1} K_{T} & - {M_{T}}^{- 1} C_{T} \end{array}]}_{2 (N + 1) \times 2 (N + 1)} \\ B = {[\begin{array}{l} 0_{(N + 1) \times 1} \\ {M_{T}}^{- 1} Λ \end{array}]}_{2 (N + 1) \times 1}; E = {[\begin{array}{l} 0_{(N + 1) \times 1} \\ - r \end{array}]}_{2 (N + 1) \times 1} \end{array}

(13)

In addition, y is the output vector containing the floor displacements of the structure. Since the system is strictly proper, the matrix D in state-space presentation is zero and C is given by

C = {I_{(N + 1) \times (N + 1)} 0_{(N + 1) \times (N + 1)}}

(14)

3. A brief introduction on fractional calculus

Fractional calculus is introduced as a generalization of classical calculus, which deals with the integration and differentiation of non-integer orders (Oldham and Spanier, 1974; Podlubny, 1998). The fractional integral, ${}_{a}{D_{\tilde{t}}^{α},}$ in a single expression is defined as

{}_{a}D_{\tilde{t}}^{α} = {\begin{cases} \frac{d^{a}}{{d \tilde{t}}^{a}} α > 0 \\ 1 α = 0 \\ \int_{α}^{\tilde{t}} {(d τ)}^{- α} α < 0 \end{cases}

(15)

where a and

\tilde{t}

are the limits of the operator.

α \in ℝ

represents the fractional-order of the operator.

For general fractional-order operator, Grunwald–Letnikov (GL), Riemann–Letnikov (RL), and Caputo definitions are most commonly used (Podlubny, 1998). Hence, these definitions are expressed in the following section.

Definition 1

The αth order GL fractional derivative of function $f (\tilde{t})$ is expressed as follows:

{}_{a}D_{\tilde{t}}^{α} f (\tilde{t}) = \lim_{h \to 0} h^{- α} \sum_{j = 0}^{[(\tilde{t} - a) / h]} {(- 1)}^{j} (\begin{array}{l} α \\ j \end{array}) f (\tilde{t} - j h)

(16)

where

[(\tilde{t} - a) / h]

is the maximum integer figure which is less than

(\tilde{t} - a) / h .

Definition 2

The αth order RL fractional derivative of function $f (\tilde{t})$ is defined as

{}_{a}D_{\tilde{t}}^{α} = \frac{d^{m}}{d {\tilde{t}}^{m}} [\frac{1}{Γ (m - α)} \int_{0}^{\tilde{t}} \frac{f (τ)}{{(\tilde{t} - τ)}^{α - m + 1}} d τ]

(17)

where

m - 1 < α < m \in Z^{+} . Γ (\cdot)

is the gamma function and is expressed as

Γ (γ) = \int_{0}^{\infty} e^{- \tilde{t}} {\tilde{t}}^{(γ - 1)} d \tilde{t}

(18)

Definition 3

The αth order Caputo fractional derivative of function $f (\tilde{t})$ is expressed as

{}_{a}D_{\tilde{t}}^{α} = \frac{1}{Γ (m - α)} \int_{0}^{\tilde{t}} \frac{f (τ)}{{(\tilde{t} - τ)}^{α - m + 1}} d τ

(19)

Contrary to the integral and derivatives of integer order, the numerical simulation of fractional integrals and derivatives is not simple. The Laplace transform method as a suitable and conformable tool is widely developed for solving fractional-order differential equations (Oldham and Spanier, 1974). The Laplace transform of equation (15) can be represented as (Oldham and Spanier, 1974):

L {_{a} D_{\tilde{t}}^{α}} = \int_{0}^{\infty} e^{- s \tilde{t}} D^{α} f (\tilde{t}) d \tilde{t} = s^{α} F (s) - \sum_{k = 0}^{m - 1} s^{α - k - 1} D^{k} f (0)

(20)

where

L {\cdot}

is the Laplace transform of

f (\tilde{t}) .

Lemma 1

Fractional-order type TypeMittag–Leffler Stability (Jiang et al., 2020): Let $\tilde{x} = 0$ be an equilibrium point of the fractional-order system $D_{\tilde{t}}^{α} \tilde{x} (\tilde{t}) = f (\tilde{t}, \tilde{x} (\tilde{t}))$ . Assume that there exists a Lyapunov function $V (\tilde{t}, \tilde{x} (\tilde{t}))$ and and class-κ function $β_{1}$ satisfying:

{}_{a}D_{\tilde{t}}^{α} V (\tilde{t}, \tilde{x} (\tilde{t})) \leq 0, β_{1} (‖ \tilde{x} (\tilde{t}) ‖) \leq V (\tilde{t}, \tilde{x} (\tilde{t}))

(21)

where

0 \leq α_{1} \leq α \leq α_{2} < 1, \tilde{t} \in [0, \infty)

. Then, the equilibrium point of the system

{}_{a}D_{\tilde{t}}^{α} \tilde{x} (t) = f (\tilde{t}, \tilde{x} (\tilde{t}))

is globally stable.

Lemma 2

Let $\tilde{x} (\tilde{t})$ be a continuous and derivable function. Then, for any time instant $\tilde{t} \geq 0$ , the following inequality holds (Jiang et al., 2020):

\frac{1}{2} {}_{a}D_{\tilde{t}}^{α} {\tilde{x}}^{2} (\tilde{t}) \leq \tilde{x} (\tilde{t}) {}_{a}D_{t}^{α} \tilde{x} (\tilde{t}), 0 < α < 1

(22)

In this study, the fractional-order element $s^{α}$ is determined using the Oustaloup’s method which is considered as an approximation method (Oustaloup et al., 2000). The Oustaloup’s method is presented as

s^{α} = k_{0} \prod_{k = 1}^{N} \frac{s + ω_{k}^{z}}{s + ω_{k}^{p}}

(23)

where k₀,

ω_{k}^{z},

and

ω_{k}^{p}

are gain, zero, and pole, respectively. The more detail of the Oustaloup’s method can be found in Oustaloup et al. (2000).

4. Design of control strategy

In this section, the fractional-order calculus-based sliding mode surface and the design of the proposed OFOSMC strategy are expressed.

4.1. Fractional-order calculus-based sliding mode surface

For the system defined in equation (11), the fractional order–based sliding mode surface can be defined as follows:

S (t) = P_{1} D^{1 - α} z (t) + P_{2} z (t) = P \hat{z}

(24)

where

S \in R^{2 (N + 1)}, 0 < α_{1} \leq α \leq α_{2} < 1, \hat{z} = {[z_{1} z]}^{T}, z_{1} = D^{1 - α} z (t),

and

P = [P_{1} P_{2}],

with P₁ and P₂ undetermined coefficient matrices of the sliding surface.

By taking the (1–α)th derivative on both sides of equation (11), it gives:

D^{2 - α} z (t) = A D^{1 - α} z (t) + B u (t)

(25)

Substituting $z_{1} = D^{1 - α} z (t),$ into (25) yields:

z_{1} (t) = A z_{1} (t) + B u (t)

(26)

For calculating the matrix P, the following state transformation is assumed:

y = Γ \hat{z} \to \hat{z} = Γ^{- 1} y

(27)

where

Γ

is the transition matrix and can be expressed as

Γ = [\begin{array}{l} Γ_{1} & 0 \\ 0 & Γ_{2} \end{array}], Γ_{1} = Γ_{2} = [\begin{array}{c} I_{2 N - 1} & - B_{1} {B_{2}}^{- 1} \\ 0 & 1 \end{array}], B = [\begin{array}{l} B_{1} \\ B_{2} \end{array}]

(28)

and the dimension of the vectors B₁ and B₂ is

(2 N - 1) \times 1

and

1 \times 1 .

By substituting equation (27) into equations (11) and (26), one has

{\begin{cases} \dot{y} = Γ \dot{\hat{z}} = [\begin{array}{l} \hat{A} y_{1} + \hat{B} u \\ \hat{A} y_{2} + \hat{B} u \end{array}] \\ S (t) = P Γ^{- 1} y = \hat{P} y = 0 \end{cases}

(29)

where

\hat{A} = Γ A Γ^{- 1}

and

B = {[\begin{array}{l} 0 & B_{2} \end{array}]}^{T}

The decomposition of equation (29) leads to

\begin{array}{l} \hat{A} = [\begin{array}{l} {\hat{A}}_{11} & {\hat{A}}_{12} \\ {\hat{A}}_{21} & {\hat{A}}_{22} \end{array}], y = [\begin{array}{l} y_{1} \\ y_{2} \end{array}] = {[\begin{array}{l} \begin{array}{l} y_{11} & y_{12} \end{array} & y_{21} & y_{22} \end{array}]}^{T} \\ \hat{P} = [{\begin{array}{l} {\hat{P}}_{1} & \hat{P} \end{array}}_{2}] = [\begin{array}{l} \begin{array}{l} {\hat{P}}_{11} & {\hat{P}}_{12} \end{array} & {\hat{P}}_{21} & {\hat{P}}_{22} \end{array}] \end{array}

(30)

From equations (29) and (30), it yields:

\begin{array}{l} S (t) = {\hat{P}}_{1} y_{1} + {\hat{P}}_{2} y_{2} = 0 \\ {\dot{y}}_{11} = {\hat{A}}_{11} y_{11} + {\hat{A}}_{12} y_{12} \\ {\dot{y}}_{12} = {\hat{A}}_{11} y_{21} + {\hat{A}}_{12} y_{22} \end{array}

(31)

For simplicity, letting

{\hat{P}}_{12} = {\hat{P}}_{22} = I,

one gives:

\begin{array}{l} y_{12} = - {\hat{P}}_{11} y_{11}, y_{22} = - {\hat{P}}_{21} y_{21} \\ {\dot{y}}_{11} = ({\hat{A}}_{11} - {\hat{A}}_{12} {\hat{P}}_{12}) y_{11} \\ {\dot{y}}_{12} = ({\hat{A}}_{11} - {\hat{A}}_{12} {\hat{P}}_{21}) y_{21} \end{array}

(32)

The LQR method as a systematic method can be employed for obtaining the matrix P. This method consists of minimizing the integral of the quadratic function of the state vector as follows:

J = \int_{t_{0}}^{\infty} {\hat{z}}^{T} Y \hat{z} d t; Y = [\begin{array}{l} Q 0 \\ 0 R \end{array}]

(33)

and using equation (27) into equation (33), it obtains:

J = \int_{t_{0}}^{\infty} ([\begin{array}{l} y_{11}^{T} & y_{12}^{T} \end{array}] \hat{Q} [\begin{array}{l} y_{11} \\ y_{12} \end{array}] + [\begin{array}{l} y_{21}^{T} & y_{22}^{T} \end{array}] \hat{R} [\begin{array}{l} y_{21} \\ y_{22} \end{array}])

(34)

where

\hat{Q} = {Γ_{1}^{- 1}}^{T} Q Γ_{1}^{- 1}

and

\hat{R} = {Γ_{2}^{- 1}}^{T} R Γ_{2}^{- 1} .

Q and R are positive definite weight matrices.

According to the LQR method, minimizing equation (33) is the function of the constraint of motion on the slip surface (equation (31)), so:

y_{12} = - \frac{1}{2} {\hat{Q}}_{22}^{- 1} ({\hat{A}}_{12}^{T} \tilde{P} + 2 {\hat{Q}}_{21}) y_{11}

(35)

where the martix

\tilde{P}

2 N \times 2 N

and is obtained using the solution of the matrix Riccati equation as

{\tilde{A}}^{T} \tilde{P} + \tilde{P} \tilde{A} - \frac{1}{2} \tilde{P} {\hat{A}}_{12} {\hat{Q}}_{22}^{- 1} {\hat{A}}_{12}^{T} \tilde{P} = - 2 ({\hat{Q}}_{11} - {\hat{Q}}_{12} {\hat{Q}}_{22}^{- 1} {\hat{Q}}_{12}^{T})

(36)

in here

\tilde{A} = {\hat{A}}_{11} - {\hat{A}}_{12} {\hat{Q}}_{22}^{- 1} {\hat{Q}}_{21}

(37)

By comparing equations (32) and (35), it yields:

{\hat{P}}_{11} = - \frac{1}{2} {\hat{Q}}_{22}^{- 1} ({\hat{A}}_{12}^{T} \tilde{P} + 2 {\hat{Q}}_{21})

(38)

Therefore, the matrices P₁ and P₂ are obtained as follows:

\begin{array}{l} P_{1} = {\hat{P}}_{1} Γ_{1} = [\begin{array}{l} {\hat{P}}_{11} & I \end{array}] Γ_{1} \\ P_{2} = {\hat{P}}_{2} Γ_{2} = [\begin{array}{l} {\hat{P}}_{21} & I \end{array}] Γ_{2} = [\begin{array}{l} β {\hat{P}}_{11} & β I \end{array}] Γ_{2} \end{array}

(39)

4.2. Design of the OFOSMC strategy

For the suppression of the seismic vibration defined in equation (11), the fractional-order derivative of the sliding surface, S (t), is considered as follows:

D^{α} S (t) = P_{1} \dot{z} (t) + P_{2} D^{α} z (t)

(40)

Substituting the system model in equation (11) into equation (40) and setting

D^{α} S (t) = 0

yields:

D^{α} S (t) = P_{1} (A z (t) + B u (t) + E {\ddot{x}}_{g} (t)) + P_{2} D^{α} z (t) = 0

(41)

Then, the equivalent controller is solved as

u_{e q} (t) = {(P_{1} B)}^{- 1} (- P_{1} A z (t) - P_{2} D^{α} z (t))

(42)

For the suppression of the seismic vibration, the switching control law can be obtained as

u_{s w} (t) = {(P_{1} B)}^{- 1} (- η sgn (S (t)))

(43)

where

sgn (\cdot)

is the sign function.

η > 0

is constant.

Thus, the comprehensive controller of the proposed OFOSMC strategy is designed as

u (t) = u_{e q} (t) + u_{s w} (t)

(44)

From equations (42) to (44), the control law of the proposed OFOSMC strategy is

u (t) = {(P_{1} B)}^{- 1} (- P_{1} A z (t) - P_{2} D^{α} z (t) - η sgn (S (t)))

(45)

The proposed OFOSMC strategy for the seismic control of the N-story shear frame equipped with the ATMD is shown in Figure 2.

Figure 2.

The proposed OFOSMC strategy for the seismic control of the N-story shear frame equipped with an ATMD system.

4.3. Stability analysis of closed-loop system

Let us consider the Lyapunov function:

V (t) = \frac{1}{2} S^{2} (t) .

(46)

According to Lemma 1, it yields:

D^{α} V (t) \leq S (t) D^{α} S (t),

(47)

Using equation (Equation 40) in equation (Equation 47) results in:

D^{α} V (t) \leq S (t) {P_{1} \dot{z} (t) + P_{2} D^{α} z (t)} = S (t) {P_{1} (A z (t) + B u (t)) + P_{2} D^{α} z (t)}

(48)

Substituting the control law (45) into equation (48) leads to

\begin{array}{l} D^{α} V (t) \leq S (t) {P_{1} \dot{z} (t) + P_{2} D^{α} z (t)} \\ = S (t) {P_{1} A z (t) + P_{1} B {(P_{1} B)}^{- 1} (- P_{1} A z (t) - P_{2} D^{α} z (t) - η sgn (S (t))) + P_{2} D^{α} z (t)} \\ = - η | S (t) | < 0 \end{array}

(49)

According to Lemma 2, the building structure under OFOSMC is stable.

4.4. Chattering attenuation

Chattering as an undesirable phenomenon is a finite-amplitude oscillate with finite frequency that occurs near the desired equilibrium surface. Due to the discontinuous sign function in the control law (equation (45)), the OFOSMC strategy suffers from the chattering phenomena. The phenomena lead to damage in actuators and system itself. In this study, a smooth hyperbolic tangent function, tanh(·/φ), was utilized to replace the discontinuous sign function and attenuate chattering in the OFOSMC strategy. The hyperbolic tangent function is defined as follows:

\tanh (S / φ) = \frac{e^{S / φ} - e^{S / φ}}{e^{S / φ} + e^{S / φ}}

(50)

where φ is assumed as a boundary layer. The hyperbolic tangent function for the different values of φ and the sign function were depicted in Figure 3. As can be seen from Figure 3, the hyperbolic tangent function can smoothly mitigate the chattering effect in the control law. In this study, the value of 0.05 was used for the boundary layer.

Figure 3.

Sign and hyperbolic tangent functions (Eliasi et al., 2022).

4.5. Practical implementation considerations

The proposed OFOSMC, like any full-state feedback controller, relies on accurate and timely measurement of the state vector. In practical implementation, however, imperfections such as sensor failures, measurement noise, and time delays are unavoidable. These issues can degrade control performance or even cause instability if not properly addressed. The following remarks discuss these practical limitations and suggest possible remedies.

4.5.1. Remark 1 (Sensor failures)

It is important to emphasize that the proposed OFOSMC, like any state-feedback controller, is vulnerable to sensor failures. If the measured state vector, $\tilde{z} (t),$ deviates from the true state, $z (t),$ due to sensor malfunction, the computed sliding surface, $\tilde{S} (t),$ will be erroneous, and the reachability condition, $D^{α} V \leq - η | S |,$ may be violated. Therefore, practical implementation requires sensor redundancy, fault detection and isolation (FDI), or an observer-based state reconstruction scheme to ensure reliability under real-world conditions.

4.5.2. Remark 2 (Measurement noise)

In the presence of measurement noise, ideal sliding mode convergence to $S (t) = 0$ is not achievable because noise corrupts the computation of the sliding surface. However, by replacing the sign function with a saturation function, $sat (S / φ),$ the controller guarantees practical convergence to a boundary layer $| S | \leq φ,$ where φ can be chosen as a trade-off between chattering suppression and steady-state error. Furthermore, the fractional-order operator $D^{α - 1}$ in the sliding surface provides inherent low-pass filtering, which attenuates high-frequency sensor noise. For $α$ values closer to 0, this filtering effect is stronger, improving noise robustness at the cost of a slower response. Thus, with appropriate tuning of $α$ and $φ,$ the proposed OFOSMC guarantees practical convergence despite bounded measurement noise.

4.5.3. Remark 3 (Time delays)

The proposed OFOSMC, like all state-feedback sliding mode controllers, is sensitive to state measurement delays. When the controller operates on delayed state information, $z (t - τ_{s}),$ the computed sliding surface, $S (t) = S (t - τ_{s}),$ may differ significantly from the true sliding surface, $S (t),$ especially near the sliding manifold or under rapidly varying disturbances. If the delay exceeds a critical bound, the reaching condition, $S D^{α} S \leq - η | S |,$ is violated, potentially leading to chattering, limit cycles, or complete instability. Therefore, the proposed controller may fail under excessive response delays.

4.6. Rationale for the proposed controller

To justify the selection of the proposed OFOSMC over adaptive data-driven methods such as ANN-based fuzzy logic control (ANN-FLC), several key advantages are highlighted:

(1) Deterministic safety: OFOSMC provides a rigorous Lyapunov-based convergence guarantee for any bounded disturbance, including future earthquakes unseen during training, whereas ANN-FLC may fail unpredictably under novel ground motions.

(2) No training data required: Collecting sufficient representative response data from full-scale structures is often impractical, whereas OFOSMC operates without any offline training.

(3) Low online computational cost: OFOSMC can be implemented on low-cost fixed-point microcontrollers (e.g., STM32 and Arduino Due) with sampling rates exceeding 1 kHz, while ANN-FLC typically demands floating-point hardware and more memory.

(4) Parameter transparency: The few parameters of OFOSMC, $(α, β, λ, η, φ),$ have clear physical interpretations, which simplifies tuning and deployment by practicing structural engineers.

(5) Proven robustness: OFOSMC guarantees sliding motion for any bounded disturbance within the design PGA range, regardless of frequency content or duration.

Consequently, despite the adaptivity of ANN-FLC, the deterministic safety, computational simplicity, and parameter clarity of OFOSMC make it a more suitable choice for seismic protection of building structures.

5. Simulation studies

In the design of the proposed OFOSMC strategy, selecting an appropriate earthquake record and tuning the control parameters are crucial steps. These factors significantly influence the effectiveness and robustness of the control strategy in mitigating seismic vibrations in structures. The selection of an appropriate earthquake record has been considered as an important challenge in the design process of active control systems (Saadatfar et al., 2023, 2025). In the first stage of the present study, the proposed OFOSMC strategy was designed to control a structure under an artificial earthquake record. Further, the parameters of the OFOSMC strategy were tuned in its design process. In the second stage, the performance of the designed OFOSMC strategy was assessed in the reduction of structural responses subjected to four near- and far-fault earthquake records. In this study, a benchmark 11-story realistic building (Pourzeynali et al., 2007) was selected and controlled using an ATMD system. It was assumed that the diaphragm of each story was rigid. Hence, the realistic building was modeled as a 2-D shear frame with one degree of freedom in each story. The mass and stiffness of the stories are listed in Table 1.

Table 1.

Properties of the benchmark structure.

Floor	1	2	3	4	5	6	7	8	9	10	11
Mass (ton)	215	201	201	200	201	201	201	203	203	203	176
Stiffness (MN/m)	468	476	468	450	450	450	450	437	437	437	312

The natural frequencies of the first two modes of the structure are ω_s,1 = 6.5727 and ω_s,2 = 19.355 rad/s, respectively. The damping ratio of 5% (i.e. ξ = 0.05) was assumed for the first two modes. Therefore, the Rayleigh damping approach expressed in equation (6) was employed for the damping matrix of the structure. For the seismic control of the structure, the TMD and ATMD systems were located on the top story (i.e. roof). The properties of the TMD system consist of the frequency ratio (f_d), damping ratio $(ξ_{d}),$ and the mass ratio $(α_{d}) .$ The properties are defined as follows:

f_{d} = \frac{ω_{d}}{ω_{s, 1}} = \frac{2 π}{ω_{s, 1}} \sqrt{\frac{k_{d}}{m_{d}}}

(51)

ξ_{d} = \frac{c_{d}}{2 \sqrt{m_{d} k_{d}}}

(52)

μ_{d} = \frac{m_{d}}{m_{m}}

(53)

where m_m is the total mass of the main structure.

The optimal values of $μ_{d}, ξ_{d},$ and f_d were respectively assumed to be 3%, 7%, and 1.2., which were obtained using a genetic algorithm in Ref. (Pourzeynali et al., 2007). In addition, the values of 3%, 7%, and 1 were respectively assumed for the parameters $μ_{d}, ξ_{d},$ and f_d, of the ATMD system (Pourzeynali et al., 2007). The adopted optimal tuning parameters are input-dependent (especially for the passive TMD part); therefore, different excitations may theoretically lead to different optimal values. In this study, the same parameters are used for all inputs to enable fair comparison among controllers.

5.1. Design process of the proposed OFOSMC

As described in Section (4), the design of the OFOSMC strategy for the seismic control of structures essentially depends on the tuning parameters (i.e. ${α, β, η} .)$ The tuning parameters can be selected by using an optimization process or the trial-and-error process. Using a trial-and-error procedure, the different combinations of the parameters were first selected in this study. The OFOSMC strategy was then designed based on these combinations and applied to the structure equipped with an ATMD. Finally, the combination that produced the lowest peak displacement of the top story among the designed OFOSMC strategies was identified as the optimal combination. Thus, the optimal values of $α, β,$ and $η$ were selected to be 0.75, 0.025, and 0.5, respectively. Furthermore, the most important challenge of the FOSMC design is to appropriately select the positive-definite symmetric matrix Q defined in equation (33). The matrix Q can be defined based on the stiffness and damping values of the stories of the structure and the TMD system (Eliasi et al., 2022; Khatibinia et al., 2020). In this study, the matrix Q was considered as follows:

Q = [\begin{array}{l} J_{1} & 0 \\ 0 & J_{2} \end{array}]

(54)

where

J_{1} = 2 \times 10^{2} \times diag [1000 k_{d}, 1000 k_{11}, \dots, 4000 k_{3}, 1000 k_{2}, 10 k_{1}]

(55)

J_{2} = 2 \times 10^{4} \times diag [100 c_{d}, 100 c_{11}, . . ., 400 c_{3}, 100 c_{2}, c_{1}]

(56)

In this study, the OFOSMC strategy was designed for the seismic control of the structure under an artificial earthquake ground motion which was simulated through a white-noise signal with constant spectral density,

{\bar{S}}_{0} .

In this method, the power-spectrum density (PSD) function of the Kanai–Tajimi model (Tajimi, 1960) was used in this study as

S_{K . T} (ω) = {\bar{S}}_{0} [\frac{{ω_{g}}^{4} + 4 {ζ_{g}}^{2} {ω_{g}}^{2} ω^{2}}{{(ω^{2} - {ω_{g}}^{2})}^{2} + 4 {ζ_{g}}^{2} {ω_{g}}^{2} ω^{2}}]; {\bar{S}}_{0} = \frac{0.03 ζ_{g}}{π ω_{g} (4 {ζ_{g}}^{2} + 1)}

(57)

where

ζ_{g}

and

ω_{g}

are the damping and frequency of ground, respectively.

In the present study, the ground damping and frequency were assumed to be 0.3 and 37.3 rad/s, respectively. The values were selected based on literature (Eliasi et al., 2022; Etedali et al., 2018; Khatibinia et al., 2020). Based on the values of the parameters, the PGA of the artificial earthquake record is equal to 1.5 g. It should be noted that the characteristics and graphical representations of this artificial record have been previously presented by Saadatfar et al. (Saadatfar et al., 2023, 2024a). To perform the analysis of the structures under the artificial earthquake record, MATLAB and Simulink tools were used. Figure 4 shows the time history of the displacements of the top floor for the cases of the uncontrolled structure (Unctrl’d) and the controlled structure using the TMD and ATMD systems. Results show that the OFOSMC strategy–featured ATMD can efficiently mitigate the maximum displacement of the top story in compared with that of the uncontrolled structure and the TMD-equipped structure. In fact, the comparison of the uncontrolled and controlled structure indicates that the OFOSMC strategy–featured ATMD considerably reduces the peak displacement of the top story (0.114 m) by 61.6%. In addition, a reduction of 44.4% and 51% was obtained by the OSSC (0.165 m) and ASSC (0.145 m) strategies (Saadatfar et al., 2024a, 2024b), respectively.

Figure 4.

Comparison of the displacement time history of the top story during artificial earthquake.

The time history of the acceleration of the top floor was also depicted in Figure 5. As can be seen from Figure 5, the maximum acceleration of the top floor is 25.33, 25.25, and 15.51 m/s² for the cases of the uncontrolled structure and the controlled structure equipped with the TMD and ATMD systems, respectively. Hence, the maximum acceleration of the top floor for the structure equipped with the ATMD in comparison with the uncontrolled one was reduced by 38.8%. Using the OSSC and ASSC controllers, the maximum acceleration of the top floor was reported by 21.47 and 23.15 m/s² (Saadatfar et al., 2024a, 2024b), which represented a reduction of 15.2% and 8.6% relative to the uncontrolled structure. Therefore, the results demonstrate that the OFOSMC strategy outperformed the OSSC and ASSC controllers in the reduction of the maximum displacement and acceleration of the top story.

Figure 5.

Comparison of the acceleration time history of the top story during artificial earthquake.

The time history of the control force obtained by the OFOSMC strategy was depicted in Figure 6. It can be seen from Figure 6 that the maximum control force is 8230 kN. Further, the maximum control force obtained using the OSSC and ASSC controllers was reported by 8000 kN (Saadatfar et al., 2023, 2024a), so the difference between the maximum control force of the FOSMC and that of the OSSC and ASSC controllers is less than about 3%.

Figure 6.

Time history of the control force during the artificial excitation.

Figure 7 shows the performance assessment of the OFOSMC strategy in terms of the time history of the input, damping, and strain energies for the uncontrolled structure and the structure equipped with the TMD and ATMD systems. For the ATMD-equipped structure, the maximum value of the input, damping, and strain energies is achieved by 1287.32, 2115.56, and 399 kJ, respectively. In addition, the ATMD system can mitigate the maximum value of the input, damping, and strain energies by 46%, 70%, and 82%, respectively, relative to the uncontrolled structure.

Figure 7.

Comparison of the time history of different energies of the structure during artificial earthquake.

5.2. Comparative simulation studies

In this section, the performance of the designed OFOSMC strategy is investigated under a number of real earthquake excitations. The real earthquake excitations include El Centro (California, USA–1940), Hachinohe (Tokachi–Oki, Japan–1968), Northridge (California, USA–1994), and Kobe (Hyōgo, Japan–1995). The peak ground acceleration (PGA) of the El Centro, Hachinohe, Northridge and Kobe earthquakes are 0.342, 0.225, 0.827, and 0.818 g, respectively (Etedali et al., 2018). Further, the performance of the designed FOSMC was compared with that of the LQR (Pourzeynali et al., 2007), FLC (Pourzeynali et al., 2007), PID (Etedali et al., 2018), OSMC (Eliasi et al., 2022), OSSC (Saadatfar et al., 2024a), and ASSC (Saadatfar et al., 2024b) controllers. To ensure a fair and practically relevant comparison, the other controllers (LQR, FLC, PID, OSMC, OSSC, and ASSC) were implemented using their standard tuning approaches as documented in the literature. Unlike the proposed OFOSMC, these benchmark controllers do not possess a built-in optimization mechanism; therefore, applying any external optimization method to them would alter their fundamental nature and would not represent their conventional implementation. A key novelty of the proposed OFOSMC lies precisely in the integration of optimal tuning (via LQR) into the fractional-order sliding mode control framework. Consequently, comparing the optimally tuned proposed controller against these conventionally tuned benchmark controllers fairly demonstrates the added value of this novel feature and accurately reflects real-world engineering practice.

Tables 2 –5 report the peak displacement of all stories for the various controllers under El Centro, Hachinohe, Kobe and Northridge earthquakes, respectively. It can be seen from Tables 2 –5 that the reduction in the peak displacement of all stories can be achieved using the OFOSMC strategy. For a comparative performance of all controllers, the mean value of the peak displacement of all stories is obtained and shown in Tables 2 –5. Results demonstrate that in comparison with the other controllers, the OFOSMC strategy exhibits the effective performance in the reduction of the peak displacement of all stories. However, the performance of the ASSC strategy under the Northridge earthquake is trivially better (by about 8%) than that of the OFOSMC strategy.

Table 2.

Comparison of the peak displacement (m) of stories for the different control systems in El Centro earthquake.

Story	Unctrl’d	Passive	LQR	FLC	PID	OSMC	ASSC	OSSC	OFOSMC
1	0.019	0.013	0.009	0.009	0.014	0.008	0.007	0.008	0.008
2	0.039	0.025	0.018	0.016	0.028	0.017	0.015	0.015	0.014
3	0.057	0.037	0.027	0.023	0.041	0.026	0.022	0.022	0.020
4	0.074	0.048	0.035	0.028	0.053	0.034	0.028	0.029	0.026
5	0.090	0.058	0.043	0.034	0.064	0.041	0.036	0.035	0.030
6	0.100	0.067	0.050	0.039	0.074	0.049	0.039	0.041	0.034
7	0.120	0.074	0.058	0.043	0.081	0.055	0.044	0.047	0.037
8	0.130	0.083	0.060	0.047	0.087	0.060	0.049	0.051	0.039
9	0.140	0.094	0.067	0.049	0.091	0.063	0.052	0.054	0.041
10	0.140	0.094	0.070	0.050	0.094	0.065	0.054	0.056	0.043
11	0.147	0.099	0.072	0.051	0.096	0.067	0.055	0.057	0.046
Mean	0.096	0.063	0.046	0.035	0.066	0.044	0.036	0.038	0.031

Table 3.

Comparison of the peak displacement (m) of stories for the different control systems in Hachinohe earthquake.

Story	Unctrl’d	Passive	LQR	FLC	PID	OSMC	ASSC	OSSC	OFOSMC
1	0.014	0.012	0.011	0.008	0.008	0.010	0.008	0.009	0.006
2	0.028	0.024	0.021	0.017	0.015	0.020	0.016	0.017	0.012
3	0.040	0.035	0.032	0.024	0.022	0.029	0.024	0.025	0.019
4	0.053	0.046	0.041	0.030	0.027	0.037	0.031	0.032	0.024
5	0.064	0.055	0.050	0.036	0.031	0.044	0.036	0.038	0.030
6	0.074	0.064	0.058	0.040	0.036	0.050	0.040	0.043	0.034
7	0.085	0.073	0.065	0.046	0.042	0.054	0.044	0.047	0.038
8	0.094	0.081	0.071	0.050	0.047	0.057	0.048	0.051	0.040
9	0.100	0.089	0.076	0.053	0.052	0.061	0.052	0.055	0.042
10	0.110	0.095	0.079	0.055	0.055	0.063	0.055	0.057	0.043
11	0.110	0.099	0.083	0.057	0.058	0.065	0.056	0.058	0.045
Mean	0.070	0.061	0.053	0.038	0.036	0.045	0.037	0.039	0.030

Table 4.

Comparison of the peak displacement (m) of stories for the different control systems in Northridge earthquake.

Story	Unctrl’d	Passive	LQR	FLC	PID	OSMC	ASSC	OSSC	OFOSMC
1	0.046	0.040	0.033	0.031	0.032	0.031	0.026	0.030	0.033
2	0.088	0.080	0.063	0.058	0.062	0.058	0.049	0.057	0.063
3	0.123	0.109	0.109	0.080	0.087	0.081	0.070	0.079	0.090
4	0.150	0.140	0.110	0.099	0.107	0.100	0.089	0.100	0.114
5	0.180	0.160	0.130	0.119	0.126	0.117	0.106	0.118	0.132
6	0.194	0.178	0.149	0.130	0.143	0.136	0.122	0.134	0.145
7	0.204	0.190	0.169	0.139	0.156	0.156	0.139	0.150	0.153
8	0.210	0.200	0.181	0.143	0.165	0.174	0.154	0.169	0.157
9	0.220	0.220	0.189	0.156	0.168	0.195	0.163	0.189	0.157
10	0.230	0.230	0.209	0.168	0.188	0.212	0.167	0.205	0.155
11	0.230	0.230	0.219	0.170	0.207	0.227	0.167	0.220	0.155
Mean	0.170	0.162	0.142	0.118	0.131	0.135	0.114	0.132	0.123

Table 5.

Comparison of the peak displacement (m) of stories for the different control systems in Kobe earthquake.

Story	Unctrl’d	Passive	LQR	FLC	PID	OSMC	ASSC	OSSC	OFOSMC
1	0.060	0.049	0.050	0.046	0.050	0.034	0.030	0.034	0.025
2	0.120	0.098	0.101	0.092	0.100	0.069	0.060	0.068	0.047
3	0.180	0.149	0.144	0.131	0.144	0.103	0.088	0.102	0.067
4	0.240	0.199	0.192	0.180	0.191	0.138	0.115	0.136	0.086
5	0.290	0.238	0.241	0.229	0.231	0.171	0.140	0.168	0.102
6	0.340	0.289	0.269	0.258	0.272	0.201	0.164	0.197	0.115
7	0.390	0.332	0.308	0.293	0.306	0.228	0.187	0.222	0.125
8	0.430	0.361	0.344	0.335	0.338	0.252	0.205	0.243	0.134
9	0.460	0.391	0.363	0.354	0.359	0.270	0.218	0.256	0.143
10	0.480	0.408	0.374	0.360	0.375	0.283	0.225	0.269	0.151
11	0.500	0.420	0.390	0.370	0.384	0.293	0.228	0.273	0.165
Mean	0.317	0.267	0.252	0.241	0.250	0.186	0.151	0.179	0.105

Figure 8 compares the performance of all controllers in terms of the average value of the reduction percentages obtained for the peak displacement of all stories subjected to the earthquake excitations. Figure 8 shows that for the various controllers, the highest average values of the reduction percentage are achieved using the OFOSMC-featured ATMD, with values of 66%, 56%, and 65.4% under El Centro, Hachinohe, and Kobe earthquakes, respectively. Under the Northridge earthquake, the ASSC-controlled ATMD leads to the highest average value of the reduction percentage by 35.4, while the OFOSMC controller achieves an average reduction percentage of 27.6%. It can be observed from Figure 8 that the total average of the reduction percentage for the passive, LQR, FLC, PID, OSMC, ASSC, and OSSC controllers are obtained by 17.6%, 28.5%, 40.6%, 31%, 38.6%, 48.7%, and 42.9%, respectively. In contrast, the total reduction average for the OFOSMC-controlled ATMD is yielded by 53.8%.

Figure 8.

Average of the reduction percentages for the maximum displacement for all stories.

To show the performance of the OFOSMC strategy, Figures 9 –12 present the time history of the top story displacement and acceleration of the ATMD-equipped structure and compare this displacement for the uncontrolled and TMD-controlled structures. As observed from Figures 9 -12, the OFOSMC controller considerably mitigated the peak displacement and acceleration of the controlled structure subjected to the real earthquake excitations.

Figure 9.

Comparison of (a) displacement and (b) acceleration time history of the top story during El Centro earthquake.

Figure 10.

Comparison of (a) displacement and (b) acceleration time history of the top story during Hachinohe earthquake.

Figure 11.

Comparison of (a) displacement and (b) acceleration time history of the top story during Northridge earthquake.

Figure 12.

Comparison of (a) displacement and (b) acceleration time history of the top story during Kobe earthquake.

For the uncontrolled and controlled structures, the peak acceleration of the top story under the real earthquake excitations is listed in Table 6 and compared with that of the one controlled by the ASSC (Saadatfar et al., 2023) and OSSC (Saadatfar et al., 2024a) strategies. Base on the average values of the reduction percentage shown in Table 6, the OFOSMC strategy (57.5%) outperforms the ASSC (1.98%) (Saadatfar et al., 2023) and OSSC (29.8%) (Saadatfar et al., 2024a) strategies in the reduction of the peak acceleration of the top story.

Table 6.

Comparison of the peak acceleration of top story for the different control system.

Earthquake	Peak acceleration (m/s²)					Reduction (%)
Earthquake	Unctrl’d	Passive	ASSC	OSSC	OFOSMC	Passive	ASSC	OSSC	OFOSMC
El centro	8.90	6.41	8.15	6.40	4.31	28.0	8.4	25.1	51.4
Hachinohe	7.85	6.93	5.82	4.82	2.62	11.7	25.9	39.3	66.6
Northridge	18.01	17.69	23.68	15.14	8.6	1.8	−31.5	16.2	52.3
Kobe	28.30	24.08	25.87	17.21	11.42	14.9	5.1	38.9	59.7
Mean	15.77	13.78	15.88	10.89	6.74	14.10	1.98	29.88	57.50

Finally, the performance of the OFOSMC, ASSC (Saadatfar et al., 2023), and OSSC (Saadatfar et al., 2024a) controllers is compared in terms of the input and strain energies of the controlled structure under the El Centro, Hachinohe, Northridge and Kobe earthquakes. According to the results shown in Table 7, the OFOSMC-featured ATMD provides the reduction of 10.6%, 51.5%, and 19.7% for the seismic input energy transferred into the controlled structure during the El Centro, Hachinohe, and Kobe earthquake excitations. However, the input energy under the Northridge earthquake is increased by 30.3% compared with that of the uncontrolled structure. The results listed in Table 8 indicate that the values of the strain energy for the structure equipped with the OFOSMC-featured ATMD are 90.5%, 81.0%, 39.7%, and 89.6% higher than those of the uncontrolled structure during the El Centro, Hachinohe, Northridge, and Kobe earthquakes. Based on the average values of the reduction percentage in Table 8, the OFOSMC strategy (75.2%) outperforms the ASSC (72.0%) (Saadatfar et al., 2023) and OSSC (63.7%) (Saadatfar et al., 2024a) strategies in the reduction of the strain energy.

Table 7.

Comparison of the input energy for the different control system.

Earthquake	Input energy (kN)					Reduction (%)
Earthquake	Unctrl’d	Passive	ASSC	OSSC	OFOSMC	Passive	ASSC	OSSC	OFOSMC
El centro	1251.88	758.36	1116.84	1092.51	1119.21	39.4	10.1	12.4	10.6
Hachinohe	1339.68	1095.27	696.91	681.33	649.89	18.2	47.9	49.1	51.5
Northridge	2538.07	2464.25	2801.05	2315.64	3307.81	2.91	−10.4	8.8	−30.3
Kobe	9762.54	8078.09	8662.98	8620.74	7843.93	17.3	11.3	11.7	19.7
Mean	–	–	–	–	–	19.5	14.7	20.5	12.9

Table 8.

Comparison of the strain energy for the different control system.

Earthquake	Input energy (kN)					Reduction (%)
Earthquake	Unctrl’d	Passive	ASSC	OSSC	OFOSMC	Passive	ASSC	OSSC	OFOSMC
El centro	516.47	216.98	73.19	82.83	48.88	58.0	85.8	84.0	90.5
Hachinohe	268.28	202.48	75.54	84.37	50.89	24.5	71.8	68.6	81.0
Northridge	1549.07	1314.65	719.02	998.49	933.46	15.1	53.6	35.5	39.7
Kobe	5461.54	3875.89	1266.64	1811.77	569.94	29.0	76.8	66.8	89.6
Mean	–	–	–	–	–	31.7	72.0	63.7	75.2

Finally, Figure 13 indicates the time histories of the control force applied by the OFOSMC strategy under the real earthquake excitations. Results show that the maximum control force provided using the OFOSMC strategy is 2212, 1928, 7150, and 5567 kN under the El Centro, Hachinohe, Northridge and Kobe earthquakes, respectively. However, the maximum control force obtained using ASSC strategy is 1989, 1686, 8012, and 6517 kN, respectively (Saadatfar et al., 2023). Hence, the maximum force of the OFOSMC controller under the El Centro and Hachinohe earthquakes is almost 11% and 14% more than that of the ASSC controller, respectively. Furthermore, the corresponding value for the OFOSMC controller under Northridge and Kobe earthquakes is almost 11% and 15% less than that of the ASSC controller, respectively.

Figure 13.

Time history of the control force during the real earthquake records.

6. Conclusion

This study proposed the OFOSMC strategy for the seismic control of building structures equipped with ATMD. To this end, the design of the OFOSMC strategy was conducted by incorporating the concept of the fractional-order calculus into the SMC. Then, the LQR was adopted to obtain an optimal control strategy, where the minimization of a cost function was considered in this procedure. In this study, the effectiveness of the proposed OFOSMC strategy was investigated in two stages. In the first stage, the design of the proposed OFOSMC controller was performed for the seismic control of an ATMD-equipped 11-story shear frame structure subjected to an artificial earthquake record. In the second stage, the capability of the designed OFOSMC strategy was assessed under four real earthquake records and was compared with that of other controller techniques including LQR, PID, FLC, OSMC, ASSC, and OSSC. On the basis of the results obtained in this study, the following conclusions can be drawn:

• Results indicated that the OFOSMC strategy–featured ATMD subjected to an artificial earthquake record reduced the peak displacement and acceleration of the top story by almost 62 and 39% in comparison with those of the uncontrolled structure.

• The ATMD system subjected to an artificial earthquake record could mitigate the maximum value of the input, damping, and strain energies by 46%, 70%, and 82%, respectively, relative to the uncontrolled structure.

• By comparison of the structural responses in the first stage, it was concluded that the control performance of the proposed OFOSMC was superior to that obtained with the OSSC and ASSC strategies. However, a comparison of the three control strategies shows that the values of their peak control forces are very similar.

• Results indicated that using the LQR, FLC, PID, OSMC, ASSC, and OSSC controllers the total average values of the reduction percentage of the peak displacement for all stories were obtained by 28.5%, 40.6%, 31%, 38.6%, 48.7%, and 42.9%, respectively. In contrast, the total average value of this reduction percentage using the FOSMC-controlled ATMD was yielded by 53.8%.

• The average value of the reduction percentage in the peak acceleration of the top story was yielded by approximately 58%, 2%, and 30% for the OFOSMC, ASSC, and OSSC strategies, respectively.

• According to the average values of the reduction percentage for the strain energy of the controlled structure, the OFOSMC strategy (75.2%) outperformed the ASSC (72.0%) and OSSC (63.7%) controllers.

• Therefore, comparative simulation results verified the robustness and effectiveness of the proposed OFOSMC strategy for the seismic control of building structures equipped with an ATMD system.

Despite the promising results, the present study involves certain limitations. Practical implementation challenges such as sensor failures, measurement noise, and time delays have been discussed in Section 4.5. However, the control performance was evaluated under idealized modeling assumptions without considering parametric uncertainties, or varying seismic intensity levels. Furthermore, experimental validation on physical ATMD systems remains unexplored. Addressing these limitations-along with robust stability analysis under real-world conditions-constitutes the main direction for future research.

Footnotes

ORCID iD

Mohsen Khatibinia

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

Declaration of conflicting interests

The authors declare no conflict of interest in preparing this article.

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