Integral Sliding Mode Controlled ATMD for Buildings under Seismic Effect

One of the main goals of civil engineering is to design safe and comfortable buildings to their users. To achieve this goal, buildings should maintain stability by resisting external disturbances such as wind and earthquakes. The increase in population led to the necessity of constructing multi-story buildings. Conventional seismic structural design attempts to construct buildings that do not collapse under strong earthquake excitations, but damage of non-structural elements may not be avoided. This can render the building non-functional after the earthquake, which may be problematic in some structures, like hospitals, which need to remain functional after earthquake.

Two basic technologies are used to protect buildings from damaging earthquake effects. These are base isolation devices (passive control) and seismic dampers (active, hybrid and semi-active control) [1]. The idea behind base isolation is to detach (isolate) the building from the ground in such a way that earthquake motions are not transmitted up through the building, or at least greatly reduced. Isolation techniques are good choice but not sufficient in some cases and they are passive devices (uncontrolled) [2].

On the other hand, there are active control devices that work in the presence of external forces generated by actuators such as Active Tuned Mass Damper (ATMD). These are transmitted through the structure and give a response that reduces the seismic effects [3]. Active control devices are areal time control [4]. The combination of the two previous types produces the so called hybrid control system. Its active portion is only used when there is high building excitation, otherwise, it behaves passively. The force of these devices is adjustable based on the control of fluid viscosity using electrical or magnetic fields supplied by low-power batteries such as a magneto rheological fluid damper (MRD) [5].

Many researchers have been involved in this field, some of them adopted classical control methods like [6] where the authors suggested MRD to control a three story scaled structure with Linear Quadratic Regulator (LQR) controller, El Ouni et al. [7] studied the effect of adding shear walls with active control. Others relied on robust control. In previous two studies the authors proposed classical controllers which are not sufficient to control the systems with perturbations.

Moreover, Wasilewski et al. [8] designed an active control system for 20-story large scale structure and compared three types of controllers: Linear-quadratic-Gaussian (LQG) regulator, H_∞ and adaptive optimal control. These types of controllers were checked under different earthquake simulations (Kobe earthquake, El Centro earthquake, Sine signal and Poly harmonic signal). The acceleration drift was minimized to 50% using adaptive control as compared to the other controllers.

Fali et al. [9] and Humaidi et al. [10] proposed adaptive sliding mode control (ASMC) to control a three story scaled structure then compared it with classical SMC. They used MRD as an actuator. They concluded that the adaptive gain of ASMC varies between the upper and lower bounds of sliding gain depending on the perturbations bounds. Khatibinia et al. [11] designed Optimal Sliding Mode Control (OSMC) to control a 11-story building with ATMD installed on the top floor. They compared OSMC with PID controller, LQR and fuzzy logic controller (FLC) under simulated earthquakes. They concluded that OSMC can further reduce the displacement by 36.7%. Concha et al. [12] proposed an automatic tuning algorithm to design SMC to control an ATMD to reduce vibration effect on structures. They compared SMC with LQR and concluded that SMC performance dominated the LQR results.

As presented above, SMC is proved to be one of the reliable robust controllers for such systems under severe external disturbances. However, from the previous studies it is noticed that SMC, ASMC and OSMC need the perturbations upper bounds in the design procedure. Moreover, these robust controllers usually need kind of approximation to avoid chattering phenomenon caused by discontinuous term in controller design.

In this work, Integral Sliding Mode Control Based on Barrier Function (ISMCbf) is designed for the first time for this application. It is chosen due to its robustness and design simplicity where the upper bound of external disturbances is not required in the design procedure [13]. Moreover, ISMCbf does not have a discontinues term hence, it is a chattering-free SMC type as will be shown later. ISMCbf is designed to control ATMD that is placed on the top floor of a 5-story scaled building exposed to El Centro earthquake. The results are compared to SMC performance [12]. Simulation results show that ISMCbf is efficient in reducing structural displacement and control effort as well. Moreover, the proposed control algorithm is more robust and simpler in design as compared to other SMC types.

This paper organized as follows: the mathematical model of a building with ATMD section 2. Sliding mode control design explained in section 3. Thereby, in section 4 the results were presented and discussed. Finally, this study was concluded with the conclusion section 5.

Sliding mode control is a robust controller that has two phases, reaching phase and sliding phase [14, 15]. It can reject the effect of matched disturbances and uncertainties during the sliding phase. The sliding manifold is designed to be attractive and invariant. Therefore, SMC is a good choice for systems where the controller and the disturbance are in the same channel [16].

ISMC has no reaching phase and sliding occurs from the first instance. Hence, the system is not affected by the undesired external inputs as it is imposed to remain on the sliding manifold. Moreover, ISMC can guarantee that unmatched disturbances are not amplified via a good selection of sliding surface [17, 18]. For both classical SMC and ISMC, the bounds of the matched disturbances and uncertainties are needed. A new ISMCbf [19] is designed here which does not require the prior knowledge of the bounds of disturbances and uncertainties. Only one parameter is to be chosen, as will be shown in subsection 3.3.

Classical SMC, ISMC and ISMCbf are designed next to compare their performances.

3.1 Classical ISMC design

Classical ISMC is designed for the dominant mode model in Eq. (17). First define the switching variable [17]:

$\sigma=G \boldsymbol{z}+Y$      (18)

The vector $G$ is given by:

$G=\left[\begin{array}{llll}g_1 & g_2 & g_3 & g_4\end{array}\right]$      (19)

where, $g_1, \ldots g_4$ are design parameters. $Y$ is the integral term whose dynamics is described in Eq. (20) [17];

$\dot{Y}=-G A z-u_n$      (20)

ISMC control law is as follows:

$u_{=}(G B)^{-1}\left(u_n+u_s\right)$      (21)

$u_s=-K_0 \operatorname{sign}(S)$       (22)

$u_n=-K \boldsymbol{z}$      (23)

where, $u_n$ is a nominal controller which provides the nominal system performance, $u_s$ is the discontinuous control which handles disturbances and uncertainties [20]. $K_0$ is a positive switching gain that will be designed depending on the upper bound of the perturbation. In this way the discontinuous control term $u_s$ will compensate the perturbation effect. $K$ is a gain vector which can be designed by using Ackermann's formula. $G$ is designed such that $(G B)$ is invertible.

To ensure the attractiveness of the sliding manifold $\sigma$ the condition in Eq. (24) must be satisfied [21]:

$\sigma \dot{\sigma}<0$      (24)

The dynamics of the sliding manifold $\sigma$ which used in Eq. (24) is:

$\dot{\sigma}=G \dot{\boldsymbol{z}}+\dot{Y}=G\left(A \boldsymbol{z}+B\left(u-f\left(z_3\right)\right)+D \ddot{x}_g\right)+\dot{Y}$       (25)

The condition (24) becomes:

$\sigma \dot{\sigma}=\left(\sigma G\left(A Z+B\left(u-f\left(z_3\right)\right)+D \ddot{x}_g\right)+\dot{Y}\right)<0$      (26)

Assumption 1. The earthquake acceleration $\ddot{x} g$ and the non-linear friction $f\left(z_3\right)$ are assumed to be unknown but bounded by known positive quantities $\delta$ and $\tilde{\omega}$ respectively [12]:

$|\ddot{x} g| \leq \delta$     (27)

$\left|f\left(z_3\right)\right| \leq \tilde{\omega}$      (28)

Using Eqns. (21), (22) and (23), and taking the above upper bounds into consideration in Eq. (26), the result is;

$\sigma \dot{\sigma} \leq-|\sigma|\left(K_0+G B \tilde{\omega}-G D \delta\right)$      (29)

$K_0$ is designed to ensure attractiveness condition (24) i.e.:

$K_0>G D \delta-G B \tilde{\omega}$     (30)

During sliding $\sigma=0, \dot{\sigma}=0$, the equivalent of the discontinuous control is yielded $\left[u_s\right]_{e q}$

$\left[u_s\right]_{e q}=G B \tilde{\omega}-G D \delta$     (31)

Thus, the equivalent system dynamics during the sliding phase becomes:

$\dot{\boldsymbol{z}}=(A-B K) \boldsymbol{z}$     (32)

Finally, ISMC is as follows:

$u=(G B)^{-1}\left(-G A \boldsymbol{z}-K \boldsymbol{z}-K_0 \operatorname{sign}(\sigma)\right)$      (33)

3.2 Integral sliding mode based on barrier function

In this study ISMCbf is used as it does not require the perturbations bounds and it is inherently continuous. The design principle is based on replacing the discontinuous control term $u_c$ by a barrier function. The barrier function provides a simpler ISMC design as it does not require any information regarding the bounds of the unknown disturbance as classical SMC and ISMC do [19, 22]. Moreover, it provides a continuous control without the high frequency chattering that is present in the classical SMC and ISMC as shown in Eq. (33).

Two types of barrier function exist: Positive definite BFs (PBFs) and Positive Semi-definite BFs (PSBFs) [19, 22]. PSBFs is used in this work. ISMCbf design procedure does not differ from ISMC design presented earlier, only the discontinuous term is replaced as follows [22]:

$\sigma=G \boldsymbol{z}+Y$     (34)

$\dot{Y}=-G A \boldsymbol{z}-u_n$      (35)

$u_n=-K \boldsymbol{z}$     (36)

$u_s=-\frac{\sigma}{\epsilon-|\sigma|}$     (37)

$u=(G B)^{-1}\left(u_n+u_s\right)$     (38)

where, $\epsilon$ is very small positive number. Finally, ISMCbf is as follows:

The attractiveness of the sliding manifold is verified by:

$\sigma \dot{\sigma}=\sigma\left(-\frac{\sigma}{\epsilon-|\sigma|}-G B f\left(z_3\right)+G D \ddot{x}_g\right)<0$      (39)

Let $-G B f\left(z_3\right)+G D \ddot{x}_g=\delta_1$

$\sigma \dot{\sigma} \leq-|\sigma|\left(\frac{|\sigma|}{|\epsilon-| \sigma||}-\left|\delta_1\right|\right)$     (40)

So $\sigma \dot{\sigma}<0$ for $|\sigma|$ sufficiently near $\epsilon$ where $\frac{|\sigma|}{|\epsilon-| \sigma||}>\left|\delta_1\right|$.

As shown, the main advantage of ISMCbf is that the upper bound of parameters uncertainty and disturbances are not required [19, 21]. Only one control parameter is to be chosen that is $\epsilon$, which is very small positive constant.

To compare the performance of the proposed control algorithm, classical SMC will also be designed in the following subsection.

3.3 Classical SMC design

The design procedure of SMC which used in the study of Concha et al. [12] will be followed in this work. The sliding manifold is defined as [12]:

$\sigma=G \boldsymbol{Z}$     (41)

SMC control action is as follows:

$u=-K_0 \operatorname{sign}(\sigma)$     (42)

The switching gain $K_0$ is choose to insure the attractiveness of sliding manifold [12]:

$K_0>\tilde{\omega}+h_0$     (43)

where, $h_0$ is defined as follows [12]:

$h_0=\left|-K z+\alpha_1 \ddot{x}_g\right|$     (44)

$\alpha_1=b_0\left(g_4-g_3\right)+g_3$     (45)

For more details see the study of Concha et al. [12].