Integral Sliding Mode Control for Seismic Effect Regulation on Buildings Using ATMD and MRD

The reduction of vibrations in structures resulting from seismic effects is the primary objective of engineers in this field. Seismic effects have impacts on people and structures, and many of these structures are very effective in serving people, they should remain functional even after earthquakes. For this purpose, several techniques have been proposed to minimize the seismic impact.

All these devices and seismic control methods have benefits and flaws. One type of these devices is the passive control, which depends on energy dissipation and has no external energy source. Passive control principle is to isolate the building from the ground by means of energy dissipation devices such as Tuned Mass Damper (TMD) [1]. It doesn't have a feedback correction signal of a controller, so it is not sufficient to control external perturbations [2]. Therefore, active control dampers were designed which operate on an external power source directed by a control algorithm. The control signal is generated by a control algorithm that used feedback measurements of the state variables of the structure [3].

Active control dampers like Active Tuned Mass Damper (ATMD), are efficient but has the drawback of requiring a high power source [4]. As a result, semi-active control has emerged, which require low power supply to work, like Magneto Rheological Damper (MRD) [5]. Semi active control is a combination of passive control and active control. 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 [6].

The objective of many studies in this field was to design a suitable robust control algorithm for active and also semi active dampers. Some researchers designed classical controllers, others designed intelligent control. Kavyashree and Rao [7] proposed proportional–integral–derivative (PID) controller with MRD to control a scaled structure to reduce earthquake effect. The controller was checked under three types of earthquakes. While Zizouni et al. [8] they designed Linear Quadratic Regulator (LQR) to control a scaled structure with MRD. In the two previous studies researchers used classical control for the purpose of reducing seismic effect. As is known, classical control is insufficient to control the systems that is exposed to external disturbances. Yan et al. [9] proposed Fuzzy Neural Network algorithm (FNN) and compared it with LQR to control a symmetric building with ATMD on the top floor, they concluded that FNN performance was better than LQR. However, FNN needs training in design and requires prior knowledge of the perturbations upper bound. Concha et al. [10] designed Sliding Mode Control (SMC) to control a scaled structure with ATMD and compared its performance with the traditional LQR, they checked the two designs under effect of time scaled El Centro 1940 earthquake, they concluded that SMC was better than LQR performance. This study used ATMD as actuator which needs high power supply. Khatibinia et al. [11] proposed Optimal Sliding Mode Control (OSMC) to control 11- story building with ATMD in the top floor, the proposed controller compared to PID, LQR and fuzzy logic controller, the concluded OSMC has the better performance than other controllers. Humaidi et al. [12] and Fali et al. [13] the authors designed Adaptive Sliding Mode Control (ASMC) with MRD to control a scaled structure. They compared ASMC with SMC response to show the efficiency of the former. In the previous studies SMC, ASMC and OSMC required the prior knowledge of disturbance upper bounds.

In summary highlighted from the robust controllers in previous studies that SMC, ASMC and OSMC all these controllers need the upper bound of the disturbance in its design. Moreover, these robust controllers have discontinuous term that causes chattering phenomenon which means these controllers need some kind of approximation to attenuate it.

From the previous drawbacks, the motivated that lead to overcomes this problem by Husain and Ridha [14] to design Integral Sliding Mode Based on barrier function (ISMCbf). They compared three types of controllers: SMC, Integral Sliding Mode Controller (ISMC) and ISMCbf to control the behavior of ATMD placed on the top floor of a scaled structure. The results showed that SMC, ISMC and ISMCbf succeeded in reducing the maximum displacement, but ISMCbf reduced the applied force by 52 percent i.e. it reduced the required energy. The most important advantage in designing this new proposed controller (ISMCbf) is that it does not require the upper bounds of the external disturbance nor the uncertainties in it design [15]. As mention earlier ATMD needs high power supply and high cost. Moreover, it’s difficult maintain, and there are constraints in its performance. Therefore, in this study ISMCbf is designed for the first time to control MRD to reduce seismic structural vibrations and also minimizing the required control energy. ISMCbf has shown to be simple in design and does not need prior knowledge of the perturbations upper bound. Moreover, this controller is not discontinuous, therefore it is not needing any kind of approximation to avoid chattering. This new control strategy is compared to the results of ISMCbf with ATMD under two different simulated earthquakes. Both dampers were placed on the top floor of a three-story scaled structure. To show the efficiency of ISMCbf control algorithm, the performance of ISMCbf with MRD is compared to other control algorithms from the literature [16-18] which used the same MRD for the same scaled structure. ISMCbf algorithm has proven its efficiency in reducing both the displacement and also reducing the control force compared to other control algorithms.

This paper is organized as follows: The mathematical model of a building with ATMD and MRD is presented in section 2. Integral sliding mode control design is explained in section 3. In section 4, the results are presented and discussed. Finally, the conclusion is presented in section 5.

SMC is a robust control, which deals with matched disturbances and uncertainties. SMC is widely used due to its preferred robust performance [10, 20, 21]. SMC has two phases, reaching phase and sliding phase. In reaching phase, the controller will derive state will trajectory toward the sliding manifold in-finite time. In sliding phase when the state trajectory gets on the sliding manifold, the trajectory will remain on this manifold, and then it moves along the sliding manifold until it reaches the origin asymptotically. During reaching phase the system is affected by external disturbances, while during sliding phase the system is not affected by external disturbances [22, 23]. Therefore, it is desired to reduce the reaching phase. ISMC eliminates the reaching phase in it is design [22]. Both SMC and ISMC need the upper bound of disturbance in the design procedure. A new ISMCbf is recently designed such that it does not require any prior knowledge of the perturbations bounds [23, 24]. This new control algorithm has the following advantages [15, 25]:

1. The strategy does not require any prior knowledge of the perturbations bounds.

2. Simpler in design, where its design needs one parameter to be specified only that defines the steady-state error accuracy.

3. The algorithm is a smooth function, so it does not need any approximation to avoid chattering effect caused by the conventional discontinuous of ISMC.

ISMCbf is designed for the system in Eq. (3) with $A \in$ $R^{6 * 6}$, and $\boldsymbol{D}, \boldsymbol{B} \in R^{6 * 1}$. Firstly, the barrier function will be defined as follows:

Definition [15, 25, 26]: For some given fixed ε>0; the barrier function can be defined as even continuous function $\mathrm{f}: \mathrm{x} \in[-\varepsilon, \varepsilon] \rightarrow \mathrm{g}(\mathrm{x}) \in[\mathrm{b}, \infty]$ strictly increasing on [0, $\mathcal{E}$].

1. $\lim _{|x|} \rightarrow \varepsilon g(x)=+\infty$

2. g(x) has a unique minimum at zero and g(0)= b$\geq$0

there are two different classes of Barrier Functions (BFs) as follows:

1. Positive definite BFs (PBFs):

$g_p(x)=\frac{\varepsilon F}{\varepsilon-|x|}, g_p(0)=F>0$              (14)

2. Positive Semi-definite BFs (PSBFs):

$g_{p s}(x)=\frac{|x|}{\varepsilon-|x|}, g_{p s}(0)=0$         (15)

The $g_{p s}(x)$ is used in this work. The first step in the design procedure is to design the sliding surface:

$\sigma=G x+Z$         (16)

where, $\sigma$ is the sliding manifold, $Z$ is the integral term, $G \in \mathrm{R}^{1 * 6}$ to be designed. The derivative of sliding manifold and the integral term which

$\dot{\sigma}=G \dot{x}+\dot{Z}$                (17)

$\dot{Z}=-G A \, x-u_n$              (18)

Finally, ISMCbf control action produced in Eq. (19);

$u=(G B)^{-1}\left(u_n+u_d\right)$          (19)

where, u_n and u_d are the nominal controller and discontinuous controller respectively which are designed as follows:

$u_n=-K_1 x$               (20)

$u_d=-\frac{\sigma}{\epsilon-|\sigma|}$             (21)

where, $K_1$ is the gain vector designed using Ackermann’s formula, $\epsilon$ is very small positive constant.

After substituting Eq. (19) in Eq. (3) then Eqns. (18) and (3) in Eq. (17) the result as follows:

$\dot{\sigma}=u_d+G D \ddot{x}_g$            (22)

To write the system dynamics during sliding ( $\dot{\sigma}=0$ ), the equivalent control is:

$[\dot{\sigma}]_{e q}=0=\left[u_d\right]_{e q}+G D \ddot{x}_g$                   (23)

$\left[u_d\right]_{e q}=-G D \ddot{x}_g$            (24)

Substituting in Eq. (3), the resulting system dynamics is described as follows:

$\dot{x}=\left(A-B K_1\right) x$             (25)

The above system is stable by choosing $K_1$ matrix to be Hurwitz. To ensure attractiveness of sliding surface $\sigma$ the condition below must be satisfied [27];

$\sigma \dot{\sigma}<0$            (26)

After substituting Eq. (21) in (22) then in (26) the result is;

$\sigma \dot{\sigma}=\sigma\left(-\frac{\sigma}{\epsilon-|\sigma|}+G D \ddot{x}_g\right)<0$            (27)

Let $G D \ddot{x}_g=\delta$ is assumed bounded with unknown upper bound $|\delta|$, taking the bounds of (27):

$\sigma \dot{\sigma} \leq-|\sigma|\left(\frac{|\sigma|}{|\epsilon-| \sigma||}-|\delta|\right)$                   (28)

Therefore $\sigma \dot{\sigma}<0$ for $|\sigma|$ sufficiently near $\epsilon$ where $\frac{|\sigma|}{|\epsilon-| \sigma||}>|\delta|$. Therefore, the set $\Omega=\{x:|\sigma| \leq \epsilon\}$ is positively invariant.

ISMCbf drives the system to start inside the small neighborhood of |σ|<ϵ; and as this set is invariant. Therefore, the controller rejects the disturbance from the first instance. The preferred robustness of ISMC is improved by adding the barrier function where with ISMCbf the upper bound of the system perturbations are not required [15, 25]. Only one control parameter is to be chosen $\epsilon$, which is very small positive constant that defines the barrier function invariant set.