Suha S. Husain* | Taghreed MohammadRidha
© 2022 IIETA. This article is published by IIETA and is licensed under the CC BY 4.0 license (http://creativecommons.org/licenses/by/4.0/).
OPEN ACCESS
In this work the robustness properties of Integral Sliding Mode Controller (ISMC) are studied for the problem structural vibration. Selecting the sliding manifold which takes into account the attenuation of the unmatched perturbation is studied as well. This study has two scenarios the first is comparing two types of sliding mode controllers which designed to control 3-story scaled structure supported by Magneto Rheological Damper (MRD). ISMC is compared to classical Sliding Mode Control (SMC) performance, the two controllers checked under effect of Mexico city earthquake. The second scenario compares the proposed controller with other controllers from literature under effect of time scaled El Centro 1940 earthquake. The results show that ISMC performance is better than SMC in scenario one and better than other controllers in scenario two. All the simulation results are obtained by MATLAB.
sliding mode control, integral sliding mode control, unmatched disturbance, magnetorheological damper, MRD, seismic effect, regulation buildings, earthquakes, structural building
In the long run, buildings have suffered significant damage due to earthquakes. There are a lot of statistics on human and material damage caused by earthquakes, on average, about 10,000 people lose their lives due to earthquakes each year [1]. So many researchers in this field have sought to reduce the vibrations of structures to ensure the safety of humans and structures. Several techniques have emerged to reduce vibrations of structures, one of these techniques is passive control, which dissipates energy caused from seismic effect [2], this type of control vibration does not have feedback signal, therefore it is not sufficient [3]. New type of controller is active control, Active control dissipates energy and have feedback signal to correct the structure displacement [4]. Active devices have been used in many studies [5-9]. Active control needs a high power source [10-12], so this challenge led to a semi-active control which does not need a high power source because it works on the battery power [13-15].
Several control techniques have been applied with the semi-active device such as MRD, where MRD contains a liquid that quickly changes from liquid to semi solid in parts of seconds whether a magnetic field or an electric field is applied to it [16]. The control techniques are applied with MRD are divided into three sections, classic control, intelligent control and robust control. Jagadisha et al. [17] designed Proportional Integral Derivative controller (PID) to drive MRD to control three-story scaled structure. Zizouni et al. [18] proposed Linear Quadratic Regulator (LQR) with MRD to reduce the seismic effect for three-story scaled structure.
In terms of intelligent control, Zizouni et al. [19] designed neural network controller to minimize earthquake effect on a scaled structure with MRD. The design is tested under two types of simulated earthquakes, Tōhoku earthquake and Boumerdès earthquake. The maximum redaction for the top floor displacement under the two types effect is 67.14% and 57.64% respectively as compared to the uncontrolled displacement.
Finally, we mention some of the studies that designed robust control to reduce the seismic effect on structures. Humaidi et al. [20] and Fali et al. [21] proposed Adaptive Sliding Mode Controller (ASMC) with MRD to control scaled structure, they compared ASMC performance with SMC, from the obtained results they concluded that ASMC is better than SMC in reducing the structure displacement.
From the observation of previous studies SMC has proven its effectiveness in overcoming the seismic effect but there are some points that have not been studied before that are covered here. Firstly, ISMC is designed for the first time for reducing structural vibrations. This controller does not have reaching phase that is found in classical SMC [22-25]. Secondly, the unmatched disturbance is taken into consideration for the first time in the design for this system.
In this study ISMC is designed to control MRD for the first time to reduce seismic structural vibrations. ISMC is compared with other controllers from literature to show its efficiency [26, 27].
This paper is organized as follows: the mathematical model of a building with 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.
The mathematical model of a building is shown in Eq. (1) [21, 27]:
$\boldsymbol{M} \ddot{\boldsymbol{x}}(t)+\boldsymbol{C} \dot{\boldsymbol{x}}(t)+\boldsymbol{K} \boldsymbol{x}(t)=\boldsymbol{M} \boldsymbol{\Lambda} \ddot{x}_q(t)-\boldsymbol{\Gamma} F_c(t)$ (1)
where, $\boldsymbol{x}, \dot{\boldsymbol{x}}$ and $\ddot{\boldsymbol{x}}$ are displacement, velocity and acceleration vectors of the structure respectively. $\boldsymbol{x}=\left[x_1, x_2, x_3, \ldots, x_n\right]^T$, n is the number of floors and in this work $n=3. \boldsymbol{C}, \boldsymbol{K}$ and $\boldsymbol{M} \in R^{n * n}$ are damping, stiffness and mass matrices. $\ddot{x}_g$ is the unknown earthquake acceleration. $\boldsymbol{\Lambda} \in R^{n * 1}$ is unity vector, Fc is the force produced by the dampers, $\boldsymbol{\Gamma} \in R^{n * 1}$ represent the location of each damper. In this work one damper will be considered, hence:
$\Gamma=[0,0,0,0,0,1]^T$ (2)
State space representation for (1) is as follows:
$\dot{\boldsymbol{x}}=\boldsymbol{A \boldsymbol { x }}+\boldsymbol{B} u+\boldsymbol{D} \ddot{x}_q$ (3)
where, $\boldsymbol{B}$ and $\boldsymbol{D}$ are $\epsilon R^{2 n * 1}, \boldsymbol{A} \in R^{2 n * 2 n}, u=F_c$. These matrices are as follows:
$A=\left[\begin{array}{cc}0 & I \\ -M^{-1} K & -M^{-1} C\end{array}\right], B=\left[\begin{array}{c}0 \\ -M^{-1} \Gamma\end{array}\right], D=-\left[\begin{array}{l}0 \\ \Lambda\end{array}\right].$
For the system in Eq. (3) the vector D contains two parts the first on is D1 which is represent the vector of match disturbance, while the second part represented by D2 which is the vector of unmatched disturbance as the following:
$D=D_1+D_2$
Rewrite the system formulation as follows:
$\dot{\boldsymbol{x}}=\boldsymbol{A} \boldsymbol{x}+\boldsymbol{B} u+\boldsymbol{B} D_0 \ddot{x}_g+f_u$ (4)
where, $D_1=\boldsymbol{B} D_0$ and $f_u=D_2 \ddot{x}_g$.
$f_u=D_2 \ddot{x}_g$ (5)
The nonlinear model of MRD which described by the modified Bouc–Wen model, this model was proposed by Zizouni et al. [18, 26]. The applied force suggested by this model is governed by the following equations:
$F_c=c_1 \dot{y}+k_0(x-y)+k_1\left(x-x_0\right)+\alpha z$ (6)
$\dot{y}=\frac{1}{c_0+c_1}\left(c_o \dot{x}+k_o(x-y)+\alpha z\right)$ (7)
$\dot{z}=-\Upsilon|\dot{x}-\dot{y}| z|z|^{r-1}-\beta(\dot{x}-\dot{y})|z|^r+a(\dot{x}-\dot{y})$ (8)
where, $x$ and $\dot{x}$, are displacement and velocity of the damper respectively, $F_c, z, k_o$ and $k_1$ are generated force, hysteretic component, accumulator stiffness respectively at low and high velocity. $\gamma, \beta, r$ and $a$ are parameters giving the shape and scale of the hysteresis loop. $c_o$ and $c_1$ are the viscous damping at low and high velocity respectively, which depend on control voltage as see in Eqns. (9), (10) (11) and (12) respectively:
$\alpha=\alpha_a+\alpha_b \mu$ (9)
$c_1=c_{1 a}+c_{1 b} \mu$ (10)
$C_o=C_{o a}+C_{o b} \mu$ (11)
$\dot{\mu}=-\eta\left(\mu-v_c\right)$ (12)
where, $\eta$ is time response factor, $\mu$ is a phenomenological variable enveloping the system and $v_C$ is the command voltage applied to the control circuit of the damper.
The resulting supplied control voltage of MRD is as follows [18, 26, 27]:
$v_c=v_{\max } H\left[\left(f_s-F_c\right) \cdot F_c\right]$ (13)
where, $v_{\max }$ is maximum applied voltage, fs the output force of controller (control force) and Fc force generated by MRD. H(.) is a Heaviside step function.
SMC is a robust control, which reject matched perturbation, So it is widely used due to its preferred robust performance [28-31]. SMC has two phases, reaching phase and sliding phase. During reaching phase the system is affected by the perturbation, while during sliding phase the system is not affected by it [15, 25]. Therefore, it is desired to reduce or even remove the reaching phase. ISMC eliminates the reaching phase in it is design [32], in addition ISMC can attenuate the unmatched disturbance by suitable selection of sliding manifold [23, 33].
For the system in Eq. (4) with $\boldsymbol{A} \in R^{6 * 6}$, and $\boldsymbol{D}, \boldsymbol{B}, \boldsymbol{D}_1, \boldsymbol{D}_2, \boldsymbol{M} \in R^{6 * 1}$..
Assume the upper bound of matched and unmatched disturbance is known.
Classical SMC and ISMC are designed next to compare their performances.
3.1 Classical SMC design
In this sub section design of SMC for the system in Eq. (4) will be discussed. Firstly the design of sliding surface is defined as follows:
$\sigma=G \boldsymbol{x}$ (14)
where, G vector is designed as $G=\left[\begin{array}{llllll}g_1 & g_2 & g_3 & g_4 & g_5 & g_6\end{array}\right]$, and GB is nonsingular square matrix. In order to analyze the sliding motion associated with the sliding manifold, The derivative of sliding manifold is given in Eq. (15):
$\dot{\sigma}=G \dot{\boldsymbol{x}}=G\left(A \boldsymbol{x}+\boldsymbol{B} u+\boldsymbol{B} D_0 \ddot{x}_g+f_u\right)$ (15)
To ensure the sliding manifold attractiveness the following condition must be satisfied [23, 34];
$\dot{\sigma}=G \dot{\boldsymbol{x}}=G\left(A \boldsymbol{x}+\boldsymbol{B} u+\boldsymbol{B} D_0 \ddot{x}_g+f_u\right)$ (16)
Select control low as
$u=(G B)^{-1}\left(u_n+u_d\right)$ (17)
where, $u_n$ and $u_d$ are the nominal controller and discontinuous controller respectively which are designed as follows;
$u_n=-G A x, u_d=-K_0 \operatorname{sign}(\sigma)$ (18)
K0 is positive switching gain, will be designed next, after substuating Eq. (18) in Eq. (17) than in Eq. (15), the result is Eq. (19);
$\sigma \dot{\sigma} \leq-\|\sigma\|\left[K_0-G \boldsymbol{B} D_0 \ddot{x}_g-G f_u\right]$ (19)
Switching gain must satisfy the following condition in order to ensure the sliding manifold attractiveness:
$K_0>G \boldsymbol{B} D_0 \ddot{x}_g+G f_u+\eta$ (20)
where, $\eta$ is very small positive constant. During sliding u=ueq as the follows:
$u_{e q}=(G B)^{-1}\left[-G A x-G B D_0 \ddot{x}_g-G f_u\right]$ (21)
The system dynamics during sliding is obtained by substituting Eq. (21) in Eq. (4) the result is as follows:
$\dot{\boldsymbol{x}}=\left[I-B(G B)^{-1} G\right] A \boldsymbol{x}+\left[I-B(G B)^{-1} G\right] B D_0 \ddot{x}_g+\left[I-B(G B)^{-1} G\right] f_u$ (22)
Define $\Gamma=\left[I-B(G B)^{-1} G\right]$, then Eq. (22) will become:
$\dot{\boldsymbol{x}}=\Gamma A \boldsymbol{x}+\Gamma B D_0 \ddot{x}_g+\Gamma f_u$ (23)
Note that when the sliding manifold is reached $\Gamma B=0$ for $t \geq t_s$, (where $t_s$ is represent reaching time to the sliding manifoled) then Eq. (23) becomes:
$\dot{\boldsymbol{x}}=\Gamma A \boldsymbol{x}+\Gamma f_u$ (24)
From the above analysis note that the matched disturbance is completely rejected while the unmutched disturbance depends on selection of sliding manifold.
3.2 ISMC design
To design ISMC, the sliding manifold is designed first as follows: The basic idea of ISMC design is to enforce a sliding mode from the first instant, as well as therefore ISMC eliminate reaching phase [22]. The first step in the design procedure is to design the sliding surface:
$\sigma=G x+Z$ (25)
where, $\sigma$ is the sliding manifold, Z is the integral term, $G \in R^{1 * 6}$ is to be designed. The derivative of sliding manifold and the integral term which will be used to prove attractiveness of sliding manifold is:
$\dot{\sigma}=G \dot{x}+\dot{Z}$ (26)
$\dot{Z}=-G A x-(G B) u_n$ (27)
Finally, ISMC control action produced in Eq. (28);
$u=u_n+u_d$ (28)
where, un and ud defined as
$u_n=B^{-1}[-\boldsymbol{A x}-K \boldsymbol{x}], u_d=-K_0 \operatorname{sign}(\sigma)$ (29)
where, K is gain vector which can calculated by Ackermann’s formula. The choice of nominal controller un depend on some desired performance during sliding.
To ensure attractiveness of sliding surface $\sigma$ the condition Eq. (19) must be satisfied [23, 34];
$\sigma \dot{\sigma} \leq-\|\sigma\|\left[K_0-D_0 \ddot{x}_g-G f_u\right]$ (30)
Switching gain must satisfy the following condition in order to ensure the sliding manifold attractiveness:
$K_0>D_0 \ddot{x}_g+G f_u+\eta$ (31)
During sliding, the equivalent control of ud is yielded:
$\left[u_{e q}\right]_d=(G B)^{-1}\left[-G B D_0 \ddot{x}_g-G f_u\right]$ (32)
Then the system dynamics during sliding is obtained by substituting Eq. (32) in Eq. (4):
$\dot{\boldsymbol{x}}=(A-B K) \boldsymbol{x}+\Gamma f_u$ (33)
For matched disturbance need only (GB) nonsigular, but for unmatched disturbance need specific selection of sliding surface. From Eq. (33) note that:
$G=\left(B^T B\right)^{-1} B^T$ (34)
This choice of G not only avoid amplification of unmatched disturbance but also has the simplifying property as follows [23]:
$G B=\left(B^T B\right)^{-1} B^T B=I_m$ (35)
To ensures that the square matrix GB is nonsingular. With the choice of G in Eq. (34) the projection operator $\Gamma$ becomes:
$\Gamma=\left[I-B\left(B^T B\right)^{-1} B^T\right]$ (36)
Note that $\Gamma$ symmetric and idempotent, $\Gamma=\Gamma^2$. The properties of symmetry and idempotency imply that $\|\Gamma\|=1$ which means that the effect of fu is not amplified since $\left\|\Gamma f_u\right\| \leq\left\|f_u\right\|$.
In fact the choice of G in Eq. (34) represent the optimal choice of sliding manifold [23, 33]. This choice of G is compared to a classical choice that is used in the literature.
In this section a three-story scaled structure is given as a case study with MRD as actuator whose parameters are given in Table 1 and Table 2 respectively. All the results are obtained by MATLAB/ SIMULINK. All the initial conditions are set to zero, which means that system starts from rest. The results are categorized into three scenarios. Firstly, ISMC performance is compared to SMC. In the second scenario ISMC with MRD is compared to other control algorithms from literature [26, 27].
To avoid chattering phenomena which caused by discontinuous part of controller (ud) [35], the following approximation will be used:
$\operatorname{sign}(\sigma) \approx \frac{\sigma}{|\sigma|+\varepsilon}$ (37)
where, $\varepsilon$ is a very small positive constant. The three-story scaled structure parameter and MRD parameters are given in Table 1 and Table 2 respectively [26].
Table 1. System parameters
|
Parameter name |
Parameter value |
|
Mass matrix (M) Kg |
$\left[\begin{array}{ccc}98.3 & 0 & 0 \\ 0 & 98.8 & 0 \\ 0 & 0 & 98.3\end{array}\right]$ |
|
Damping matrix (C) N.s/m |
$\left[\begin{array}{ccc}175 & -50 & 0 \\ -50 & 100 & -50 \\ 0 & -50 & 50\end{array}\right]$ |
|
Stiffness matrix (K) N/m |
$10^5 \times\left[\begin{array}{ccc}12 & -6.84 & 0 \\ -6.84 & 13.7 & -6.84 \\ 0 & -6.84 & 6.84\end{array}\right]$ |
Table 2. MRD parameters
|
Parameter name |
Parameter value |
|
$c_{0 a}, c_{0 b}$ $N . s / c m$ |
$21 \mathrm{~N} . \mathrm{s} / \mathrm{cm} 3.5$ |
|
$k_0, a$ |
46.9 N/cm, 301 |
|
$c_{1 a}, c_{1 b}$ |
$283 \mathrm{~N} . \mathrm{s} / \mathrm{cm}$ $2.95 \mathrm{~N} . \mathrm{s} / \mathrm{cm}$ |
|
r |
2 |
|
$\alpha_a, \quad \alpha_b$ |
$140 \frac{\mathrm{N}}{\mathrm{cm}}, \quad 695 \mathrm{~N} / \mathrm{cm}$ |
|
$\gamma, \quad \beta$ |
$363 \mathrm{~cm}^{-2}, \quad 363 \mathrm{~cm}^{-2}$ |
|
$\eta, \quad x_0$ |
$190 \mathrm{~s}^{-1}, \quad 14.3 \mathrm{~cm}$ |
|
$v_{\max }$ |
2.25v |
First Scenario: The uncontrolled displacement of three-story scaled structure under effect of time scaled Mexico City earthquake which shown in Figure 1 are shown in Figure 2.
Figure 1. Time scaled Mexico City earthquake
(a)
(b)
(c)
Figure 2. Uncontrolled system displacement under effect of Mexico City earthquake
(a)
(b)
(c)
Figure 3. Displacement of three-story scaled structure controlled by SMS and ISMC
The controlled displacement with SMS and ISMC is shown in Figure 3.
The control force which is applied on the structure is shown in Figure 4.
Figure 4. The control force with SMC and ISMC
The statistical results of the tow controllers results are given in Table 3.
Table 3. Maximum structural responses when structure is subjected to Mexico City earthquake
|
|
Uncontrolled |
SMC |
ISMS |
|
x1(m) |
0.002 |
0.0003 |
0.0003 |
|
x2(m) |
0.003 |
0.00048 |
0.00048 |
|
x3(m) |
0.0034 |
0.00058 |
0.00058 |
|
Control force(N) |
/ |
425 |
314 |
From the previous result it is clear that the two controllers succeeded in reducing the structural displacement at the same rate, but ISMC can reduce the displacement with less control effort by about 33% compared to the maximum force by SMC as shown in Table 3.
Second scenario: To show the efficiency of ISMC control algorithm with the MRD, it will be compared to other controllers from the literature [23, 24] which used the same MRD and same structure which shown in Table 1 and Table 2 respectively under effect of Time scaled El Centro 1940 earthquake which shown in Figure 5.
Figure 5. Time scaled El Centro 1940 earthquake acceleration
(a)
(b)
(c)
Figure 6. Uncontrolled displacement of three-story structure under effect of time scaled El Centro 1940 earthquake
(a)
(b)
(c)
Figure 7. Displacement of three-story scaled structure under effect of El Centro 1940 earthquake controlled by ISMC
The uncontrolled displacement of three-story scaled structure under effect of time scaled El Centro 1940 earthquake are shown in Figure 6.
The displacement of three-story scaled structure under effect of time scaled El Centro 1940 earthquake controlled by ISMC shown in Figure 7.
The statistical results of the proposed controller under effect of El Centro 1940 earthquake are given in Table 4.
Table 4. Comparison of maximum structural responses between ISMC performance and the proposed controllers in [24, 25]
|
|
Control strategy |
x3(m) |
Fc(N) |
|
1 |
Uncontrolled |
0.0055 0.0083 0.0097 |
/ |
|
2 |
Passive off |
0.0021 0.0036 0.0045 |
259.2 |
|
3 |
Passive on |
0.0008 0.002 0.0031 |
992.8 |
|
4 |
Lyapunov controller (A) |
0.0009 0.0021 0.0031 |
1023 |
|
5 |
Lyapunov controller (B) |
0.0013 0.0018 0.0023 |
993.3 |
|
6 |
Quasi-bang-bang controller |
0.0013 0.0016 0.0023 |
1002 |
|
7 |
Decentralized bang-bang controller |
0.0015 0.0025 0.0032 |
923 |
|
8 |
Modulated homogenous friction controller |
0.0019 0.0029 0.0038 |
503 |
|
9 |
Maximum energy dissipation controller |
0.0008 0.0020 0.0031 |
993 |
|
10 |
Clipped-optimal controller |
0.0014 0.0021 0.0026 |
918 |
|
11 |
Modified Quasi-bang-bang controller |
0.0012 0.0019 0.0027 |
848.9 |
|
12 |
ISMC |
0.000787 0.00126 0.00136 |
682.2 |
From the comparison with the results that were obtained by [24, 25], notice that the proposed control technique is better in terms of control force than all control techniques except those proposed control algorithms No.2 and 8 respectively, but by observing the displacement which obtained by ISMC, ISMC result is better than the above two algorithms as well as from all control algorithms which proposed in [24, 25].
In this study ISMC is designed to drive MRD to reduce structure vibration for the first time. The main advantage which distinguishes ISMC from SMC is that ISMC does not have a reaching phase which means the system is in sliding from the first instant. ISMC can also attenuate the effect of unmatched disturbance by suitable choice of the sliding manifold. In addition, ISMC has better performance as compared to other control algorithms from the literature under the same simulation conditions.
As a future work, ISMC will be designed based on the barrier function to ensure a better robust behavior and simpler design requirement.
To show the efficiency of ISMC control algorithm with optimal design of sliding manifold, ISMC with classical and optimal design of sliding manifold are compared under effect of El Centro 1940 earthquake. The controlled displacement of three-story scaled structure which is parameter shown in Table 1, supported by MRD driven by ISMC are shown in Figure 8.
The statistical results of the ISMC with optimal and classical choice of sliding manifold under effect of El Centro 1940 earthquake are given in Table 5.
From the previous result show that performance of ISMC with optimal design of sliding manifold is relatively better than performance of ISMC with classical design of sliding manifold.
(a)
(b)
(c)
Figure 8. The control displacement by ISMC with optimal and classical design G
Table 5. Maximum structural responses with ISMC with classical and optimal design of sliding manifold
|
Classical G |
Optimal G |
|
|
x1(m) |
0.00086 |
0.000787 |
|
x2(m) |
0.00136 |
0.00126 |
|
x3(m) |
0.00144 |
0.00136 |
|
Control force(N) |
704.4 |
682.2 |
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