Seismic motion control of structures using an adaptive optimal model-free controller

Abstract

One of the major challenges in the real-time control of structures is the uncertainty in identifying the mechanical properties of the system, including mass, stiffness, and damping. However, the presence of uncertainty in time-varying systems (under severe environmental factors, the dynamic characteristics of the system change due to damage) is more noticeable. In this article, to deal with the problem of parametric uncertainties in seismic motion control of structures, an adaptive optimal controller is applied. The design procedure of the controller is straightforward, and there is no need to have the values of the structural parameters. In the proposed scheme, only the states of the system are measured, and it is not necessary to use the ground acceleration data which are difficult to be measured in real time. It is shown that the performance of the designed controller converges to the well-known linear quadratic regulator. Simulation results reveal that considerable reductions of the dynamic responses during the earthquakes are achieved using the utilized adaptive optimal controller, and also it is robust against any variation in the structural parameters due to the damages.

Keywords

Optimal controller linear quadratic regulator smart structure

1. Introduction

In recent decades, the vibration control strategy has been an efficient and a safe approach to protect civil structures against strong environmental excitations (Amini and Karami, 2011). Important challenges in this research area, which have been studied a lot in the literature, include the unpredictability of inputs (for instance, earthquake and wind), difficulty and complexity of the real-time measurement of input excitations (Yu et al., 2015), the limitation of sensor numbers to identify the system (Abazarsa et al., 2013; Amini and Karami, 2012; Mesquita et al., 2016; Nagarajaiah and Erazo, 2016; Yang and Nagarajaiah, 2013; Yao et al., 2018), the uncertainty in system information (Amini et al., 2017; Giaralis and Taflanidis, 2018; Huo et al., 2016; Nguyen et al., 2018), the computation time required in controllers (Gutierrez Soto and Adeli, 2017), and the time delay in generating forces by control devices (Chae et al., 2013; Chen et al., 2017; Morales-Beltran and Paul, 2015; Shao and Chen, 2013; Shen et al., 2013; Spencer and Nagarajaiah, 2003; Khoshbin et al., 2018). Moving to create a smart structure is developing at a rapid pace due to the evolution of smart devices, actuators, control strategies, novel designs, and calculation tools. To make structures smart, it is needed to use both the vibration control and the structural health monitoring strategies simultaneously (Amini et al., 2015; Bitaraf et al., 2010; He et al., 2014, 2017; Karami and Amini, 2012; Karami and Akbarabadi, 2016; Karami et al., 2016, 2019, 2020; Yang et al., 2013). Structural specifications of a smart structure system have to be measured in real time during extreme excitations. Then, proper control forces should be applied to the structure by the control unit to decrease the dynamic responses and compensate for the probable damages of the structure. A wide range of control approaches is studied to control the behavior of the civil structures (Chae et al., 2012; Lei et al., 2020; Lezgy-Nazargah et al., 2020; Mamat et al., 2020; Yanik et al., 2014). These control methods can be organized in the classic and modern control areas. Utilizing the optimization algorithms in novel control systems has attracted many interests (Miyamoto et al., 2018b). Among this, the linear quadratic regulator (LQR) is the most important achievement of the modern control in the area of the linear analysis of control systems which has been widely used to control the structural vibration (Basu and Nagarajaiah, 2008, 2010; Heidari et al., 2018; Miyamoto et al., 2016; Miyamoto et al., 2018a; Shi et al., 2014). For example Kumar and Narayanan (2008) proposed a control system based on the LQR to decrease the acceleration and the displacement of a base-isolated structure. Amini et al. (2013) suggested an LQR-based state feedback controller to optimize a cost function comprising acceleration, drift, and velocity.

The accessibility of structural parameters such as mass, stiffness, and damping is required for an LQR-based controller. That is while these parameters are not accessible in many available structures, which are retrofitted to enhance the safety level and only they exist in new structures if the materials and construction are fully matched with the design. It is noteworthy that the information of such structures is not reliable if they had experienced former earthquakes. On the other hand, online measurement of ground motions, as the system input, is difficult and also the earthquake by itself has an uncertain and random nature. It should be mentioned that, in this article, the earthquake is considered as a bounded process. Therefore, the aforementioned uncertainties are caused to decline the effectiveness of control algorithms. To tackle these challenges, control methods considering uncertainty have been proposed. For example Amini and Vahdani (2007) proposed a random optimal control using the linear quadratic Gaussian and a modified sliding mode control for structures with uncertainty; the flexibility of choosing different kinds of control devices is the advantage of their proposed method. A self-tuning and an adaptive proportional-derivative fuzzy controller was developed by Zamani et al. (2017) to control piezoelectric friction dampers which in both the control systems, multiobjective cuckoo search algorithm was used to adjust fuzzy rules and member functions. A model predictive control system for online identification of a structure equipped with nonlinear base isolations using wavelet neural networks and back-propagation learning algorithm was presented by Khodabandehlou et al. (2018) which had satisfactory performance due to the general structure of neural networks. Zamani et al. (2018) used multiobjective modified clipped optimal and adaptive fractional-order fuzzy proportional-integral-derivative strategies as intelligent control for a base-isolated structure equipped with an magneto-rheological (MR) damper. Tu et al. (2014) proposed an active control system containing active mass damper using model reference adaptive control (MRAC) based on minimal controller synthesis in which there was no need to know the exact values of the system parameters. Zhang et al. (2018) introduced an adaptive optimal model reference control using integrated LQR and MRAC based on Lyapunov stability theorem, and its performance was evaluated in a base-isolated structure equipped with an MR damper. Ghaderi and Amini (2017) presented a design procedure based on the adaptive backstepping control to determine the desired control force for a structure under earthquake, and it was assumed that the values of the structural parameters including mass, stiffness, and damping matrices of the controlled structure are unknown, but system states are measurable at each time step. Cheng et al. (2008) proposed a dual-loop adaptive controller by considering the time delay of providing the current for the MR damper; in which, the inner loop was designed to diminish the response time of the MR damper and the outer loop was designed to suppress the vibration of the structure under the earthquake. Amini and Ghaderi (2013) presented a robust control scheme taking into account the unavailability of the ground motion acceleration and structural parameters. In this manner, the unknown parameters were first approximated by adopting appropriate adaptive laws, and then, the backstepping control strategy was exerted to regulate the vibration of multiple-degrees-of-freedom structures using the displacement and the velocity feedback vectors.

In most studies, it has been assumed that structural parameters are time invariant during the system vibration. However, there is a high possibility of structural damage in strong earthquakes. This feature may cause a time-varying structural characteristics system, and consequently, the performance of the smart structure is reduced. Because the mechanical parameters of a structure are not exactly known, the conventional control methods such as the LQR cannot be used to optimize the performance of the control system. Hence, in this article, assuming the lack of structural parameters and the information of input excitations, a novel adaptive optimal controller is used to reduce the dynamic responses of the structure and compensate the possible damages. This controller is designed based on the adaptive dynamic programming (ADP) methodology. The design procedure of the controller is completely independent of structural parameters. The proposed model-free scheme has a state feedback structure and leads to a closed-loop system, where its performance converges to the well-known LQR controller. It is worth noting that in the LQR technique, the optimal state feedback gain is obtained by solving a Riccati equation which needs to know the exact values of the structural parameters of the system. Nevertheless, in the proposed ADP-based technique, an iterative algorithm is used to find this state feedback gain without any knowledge of the structural parameters and just based on the measured state variables. In the implementation of the algorithm, the controller is updated in real time during the earthquake. In this method, an algebraic Riccati equation is solved using an iterative pattern by sampling the system states (displacement and velocity). Each iteration can be conducted by a distinct sampling or only a sample can be used for all iterations. The approximate solution of the Riccati equation is converged to its optimal value after a few iterations in a very short time. Therefore, the high convergence rate at a fraction of a second is an able advantage of this manner. This property provides a situation that sampling can be performed in given intervals, and the controller is updated during ground excitations. Hereupon, beyond the assumption of unknown system parameters, unlike other methods such as the backstepping controller, the system parameters can be assumed as time varying. Also, the other benefits of the suggested control system include nonstochastic nature (unlike the metaheuristic optimization algorithms) and no challenge to select a reference model. The design controller using the mentioned algorithm is called adaptive optimal model-free (AOMF) controller. In this study, 20 ton MR dampers are used (Spencer et al., 1997) to provide the required control force to suppress the dynamic responses of the structure. It should be mentioned that there are many model-free control techniques in the literature (see, e.g., (Åström and Hägglund, 1995; Fliess and Join, 2013) and references therein). However, the proposed AOMF controller has the advantage of achieving the optimal performance of the LQR technique.

The performance of the AOMF controller is evaluated by considering a system with time-varying structural parameters through numerical examples during earthquakes. In closing, the main contributions of this article are:

Optimal performance of the smart structure without any information about the system properties.

Considering changes in structural parameters due to damages during the earthquake.

Presenting online implementation of the applied controller.

Systematically limiting the amplitude of the control forces and the state variables to some predefined values by tuning two weighting functions.

The rest of this article is organized as follows. In the next section, the mathematical model of the considered structure is presented. In addition, the control objectives are formulated in this section. The third section presents a brief overview of the utilized ADP technique. In the fourth section, the obtained numerical simulation results are presented and analyzed in detail and the performance of the utilized controller is compared with that of the LQR-based controller. Finally, the last section is dedicated to the conclusions of the article.

2. Mathematical model and control objectives

2.1. Mathematical model of the structure

The dynamic equation of motion of an N-degrees-of-freedom (DOFs) controlled structure underground excitation is as follows

M \ddot{x} (t) + C \dot{x} (t) + K x (t) = - M l {\ddot{x}}_{g} (t) + Γ B_{u} u (t)

(1)

where

M \in ℝ^{N \times N}

is the mass matrix,

K \in ℝ^{N \times N}

is the stiffness matrix, and

C \in ℝ^{N \times N}

is the damping matrix. Moreover,

x (t) \in ℝ^{N}

\dot{x} (t) \in ℝ^{N}

, and

\ddot{x} (t) \in ℝ^{N}

are the dynamic responses of the structure including displacement, velocity, and acceleration vectors, respectively. It is assumed that the structure is affected by the ground acceleration

{\ddot{x}}_{g} (t)

. In the above model, the vector

l \in ℝ^{N}

is the influence of

{\ddot{x}}_{g} (t)

on the structure mass. In addition,

u (t) \in ℝ^{m}

is the control force which is applied to the structure.

B_{u} \in ℝ^{N \times m}

is the location of DOFs which control forces are exerted, and

Γ \in ℝ^{N \times N}

is the interactions of the control forces between DOFs. Because the control device is installed between two floors of the structure, the lower floor acts as a support for the control device. As a result, a generated force by the control device is applied to the lower floor as a reaction in the opposite direction. This interaction is considered by using the matrix Γ. Defining z₁(t) = x(t) and z₂(t) =

\dot{x} (t)

and using (1), a state space representation of the considered structure equation is found as follows

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

(2)

where

z (t) = {[\begin{matrix} z_{1} (t) \\ z_{2} (t) \end{matrix}]}_{2 N \times 1}, A = {[\begin{matrix} 0 & I \\ - M^{- 1} K & - M^{- 1} C \end{matrix}]}_{2 N \times 2 N}, B = {[\begin{matrix} 0 \\ M^{- 1} Γ B_{u} \end{matrix}]}_{2 N \times m} L = {[\begin{matrix} 0 \\ l \end{matrix}]}_{2 N \times 1}

Here, 0 and I are the zero and the identity matrices with appropriate dimensions, respectively. It should be noted that the places of the control forces are determined by the elements of B_u, and in general, there is no need to have a control force in each story.

2.2. Problem statement and control objectives

The problem is to find the control law u(t) in such a way that the smart structure not only remains close enough to its normal equilibrium (z^∗ = 0) but also eventually converges to this equilibrium. In addition, it is of our interest to find the best possible solution for this problem. From a control engineering point of view, u(t) should be found to achieve an asymptotically stable structure, while a predefined cost function is simultaneously minimized. In this article, the following standard quadratic cost function is used

J = \int_{t}^{\infty} (z^{T} (t) Q z (t) + u^{T} (t) R u (t)) d t

(3)

where

Q \in ℝ^{2 n \times 2 n}

and

R \in ℝ^{m \times m}

are two user-defined weighting matrices which are, respectively, positive semidefinite and positive definite. It should be noted that the first term of the cost function (3), that is

\int_{t}^{\infty} z^{T} (t) Q z (t) d t

, is related to the quality of the closed-loop response, whereas the second term

\int_{t}^{\infty} u^{T} (t) R u (t) d t

is related to the energy consumed by the controller. These two terms are in conflict with each other, and the matrices R and Q are used to make an acceptable trade-off between the closed-loop performance and the control effort. For example for achieving a faster response, it is possible to select larger values for the elements of Q. Simple and reasonable choices for the matrices R and Q are given by Bryson’s rule (Franklin et al., 1994). Based on his rule, diagonal matrices R = [r_i] i = 1: m and Q = [q_j] j = 1: 2N are selected with

r_{i} = \frac{1}{u_{m, i}}, q_{j} = \frac{1}{z_{m, j}}

(4)

where u_m,i and z_m,i are the maximum acceptable values of the ith input and jth state, respectively. It should be noted that u_m,i and z_m,i are indeed constraints on their corresponding input and state.

In closing, the considered control problem is as follows. Optimal control problem with unknown dynamics (OCPUD) for N-DOFs building: For system (2), find u(t) such that quadratic cost function (3) is minimized, while the parameters of the structure and as a result, the matrices A and B in (2) are unknown to the controller.

3. Controller design

Consider the following linear dynamical system

\dot{z} (t) = A z (t) + B u (t)

(5)

where z(t) and u(t) are the state vector and the control input, respectively. A and B are assumed to be unknown matrices. It is also assumed that the pair (A, B) is stabilizable.

The control problem is to find the control input u(t), t > 0 such that the following quadratic cost function is minimized

J = \int_{t}^{\infty} (z^{T} (t) Q z (t) + u^{T} (t) R u (t)) d t

(6)

where Q and R are, respectively, positive semidefinite and positive definite symmetric matrices with appropriate dimensions. Using the LQR technique (Franklin et al., 1994), the optimal solution of this problem is obtained as u(t) = −Gz(t), where the state feedback gain G is calculated as G = R⁻¹B^TP. The symmetric positive definite matrix P is the unique solution of the following Riccati equation (Franklin et al., 1994)

A^{T} P + P A - P B R^{- 1} B^{T} P + Q = 0

(7)

From (7), one can see that the above standard solution needs to have complete knowledge of the system dynamics, that is A and B. However, as it has been mentioned earlier, the state matrix A and the input matrix B are both unknown in the defined OCPUD for N-DOFs structure. Let us present a theorem which is used to establish an iterative model-free approach to solve this OCPUD problem.

Theorem 1

Assume G₀ is a stabilizing feedback gain, that is system (5) is asymptotically stable under the control law u(t) = G₀z(t). Assume further that P_i is the unique symmetric positive definite solution of the following equation

{(A - B G_{i})}^{T} P_{i} + P_{i} (A - B G_{i}) = - (G_{i}^{T} R G_{i} + Q)

(8)

Let us define G_i for i = 1, 2, … as follows

G_{i} = R^{- 1} B^{T} P_{i - 1}

(9)

Then, G_i is also a stabilizing feedback gain for system (5) and $lim_{i \to \infty} G_{i} = R^{- 1} B^{T} P$ (Vrabie et al., 2009). ■

It should be noted that the above iterative method, that is (8) and (9), needs to know all the system dynamics. To relax this problem, let us first rewrite cost function (6) as follows (Vrabie et al., 2009)

J = z^{T} (t) P_{i} z (t)

Using this equation, the following equality is obtained for any T > 0

z^{T} (t) P_{i} z (t) - z^{T} (t + T) P_{i} z (t + T) = \int_{t}^{t + T} (z^{T} (τ) Q z (τ) + u^{T} (τ) R u (τ)) d τ

(10)

Remark 1

It is also noteworthy that finding P_i from (10) instead of (8) leads to an iterative solution proposed in Vrabie et al. (2009). Nevertheless, this solution is not completely model free, and it needs to know the input matrix B.

Now, (5) can be written as follows

\dot{z} (t) = A_{i} z (t) + B (G_{i} z (t) + u (t))

(11)

where A_i = A − BG_i. Defining

Q_{i} = Q + G_{i}^{T} R G_{i}

and using (8) and (11), the following equality is obtained

z^{T} (t) P_{i} z (t) - z^{T} (t + T) P_{i} z (t + T) = \int_{t}^{t + T} (z^{T} (τ) Q_{i} z (τ) - 2 {(u (τ) + G_{i} z (τ))}^{T} R G_{i + 1} z (τ)) d τ

(12)

It should be mentioned that the system dynamics, that is A and B, are not appeared in (12), and therefore, it is possible to obtain both the state feedback gain G_i+1 and the positive matrix P_i just using this equation. To this end, we should extract at least n (n + 1)/2 + mn samples from (12) because the symmetric matrix P_i and the state feedback gain G_i+1 have n(n + 1)/2 and mn independent elements, respectively. Then, we should apply the least square (LS) technique to the obtained system of equations and find $P_{i} = [p_{r, j}^{i}]$ (r, j = 1: n) and $G_{i + 1} = [g_{r, j}^{i + 1}]$ (r = 1: m, j = 1: n). To convert (12) to a system of linear equations, in the following some definitions are presented

\begin{array}{l} {\bar{P}}_{i} = {[p_{11}^{i}, 2 p_{12}^{i}, \dots, 2 p_{1 n}^{i}, p_{22}^{i}, 2 p_{23}^{i}, \dots, 2 p_{(n - 1) n}^{i}, p_{n n}^{i}]}^{T}, \\ \bar{z} (t) = {[z_{1}^{2} (t), z_{1} (t) z_{2} (t), \dots, z_{1} (t) z_{n} (t), z_{2}^{2} (t), z_{2} (t) z_{3} (t), \dots, z_{(n - 1)} (t) z_{n} (t), z_{n}^{2} (t)]}^{T} \end{array}

(13)

In addition, vec(W) is defined to be the rj vector formed by stacking the columns of $W \in ℝ^{r \times j}$ on top of one another. For example

vec (G_{i + 1}) = {[\begin{matrix} g_{1,1}^{i + 1}, \dots, g_{1, n}^{i + 1}, \dots, g_{m, 1}^{i + 1}, \dots, g_{m, n}^{i + 1} \end{matrix}]}^{T}

(14)

Using the Kronecker product, it is possible to have the following equalities (Jiang and Jiang, 2012)

z^{T} (t) Q_{i} z (t) = (z^{T} (t) \otimes z^{T} (t)) vec (Q_{i}), {(u (t) + G_{i} z (t))}^{T} R G_{i + 1} z (t) = ((z^{T} (t) \otimes z^{T} (t)) (I_{n} \otimes G_{i}^{T} R) + (z^{T} (t) \otimes u^{T} (t)) (I_{n} \otimes R)) vec (G_{i + 1})

(15)

Now, let us divide the time interval [t, t + T] into n^′ ≥ (n(n + 1)/2) + mn subintervals [t, t + ΔT], …, [t + (n^′ − 1)ΔT, t + T], where ΔT = T/n^′. By defining

\begin{array}{l} δ_{z z} = {[\bar{z} (t + Δ T) - \bar{z} (t), \bar{z} (t + 2 Δ T) - \bar{z} (t + Δ T), \dots, \bar{z} (t + (n^{'} - 1) Δ T) - \bar{z} (t + T)]}^{T} \in ℝ^{(n (n + 1) / 2) \times n^{'}}, \\ I_{zz} = {[\int_{t}^{t + Δ T} z (τ) \otimes z (τ) d τ, \int_{t + Δ T}^{t + 2 Δ T} z (τ) \otimes z (τ) d τ, \dots, \int_{(t + (n^{'} - 1) Δ T)}^{+ T} z (τ) \otimes z (τ) d τ]}^{T} \in ℝ^{n^{2} \times n^{'}}, \\ I_{z u} = {[\int_{t}^{t + Δ T} z (τ) \otimes u (τ) d τ, \int_{t + Δ T}^{t + 2 Δ T} z (τ) \otimes u (τ) d τ, \dots, \int_{(t + (n^{'} - 1) Δ T)}^{t + T} z (τ) \otimes u (τ) d τ]}^{T} \in ℝ^{n m \times n^{'}} \end{array}

(16)

it is possible to obtain the following system of linear equations with n(n + 1)/2 + mn unknown variables and n^′ equations

Θ_{i} [\begin{matrix} {\bar{P}}_{i} \\ vec (G_{i + 1}) \end{matrix}] = Ξ_{i}

(17)

where

Θ_{i} \in ℝ^{n^{'} \times (n (n + 1) / 2 + m n)}

and

Ξ_{i} \in ℝ^{n^{'}}

are as follows

\begin{array}{l} Θ_{i} = [δ_{z z}, - 2 I_{z z} (I_{n} \otimes G_{i}^{T}) - 2 I_{z u} (I_{n} \otimes R)] \\ Ξ_{i} = - I_{z z} vec (Q_{i}) \end{array}

(18)

The standard solution of (17) using the LS technique is as follows (Jiang and Jiang, 2012)

[\begin{matrix} {\bar{P}}_{i} \\ vec (G_{i + 1}) \end{matrix}] = {(Θ_{i}^{T} Θ_{i})}^{- 1} Θ_{i}^{T} Ξ_{i}

(19)

Now, based on (16)–(19), it is possible to present an adaptive algorithm to find the solution of the optimal control problem (5) and (6). In this algorithm, which is called AOMF, we start with a stabilizing control law u(t) = −G₀z(t) and apply it to system (5) during the time interval [t, t + T]. Then, using (16)–(19), the state feedback gain G₁ is obtained, and the control law is updated as u(t) = −G₁z(t) which is applied to the system during the interval [t + T, t + 2T]. This iterative process is repeated till an update of the control policy will no longer improve the system performance. In other words, we stop the adaptation when the current state feedback gain G_i and its previous value G_i−1 are close to each other. Indeed, if ∥ G_i − G_i−1 ∥ is smaller than a predefined positive scalar ϵ, the iterative process is stopped. To sum up, the following algorithm is presented.

Algorithm 1

AOMF

Select the updating time T > 0 and a stopping criterion index ϵ > 0.

Start with u⁰(t) = −G₀z(t) which stabilizes system (5) and set i = 0.

Divide the time interval [t + iT, t + (i + 1)T] into n^′ ≥ (n(n + 1)/2) + mn subintervals [t + iT, t + iT + ΔT], …, [t + iT + (n^′ − 1)ΔT, t + (i + 1)T], where ΔT = T/n^′.

Collect the known parts of (17), that is Θ_i and Ξ_i based on (16) and (18), where the system is under the control law u(t) = −G_iz(t).

Calculate ${\bar{P}}_{i}$ and vec (G_i+1) using (19).

If ∥ G_i − G_i−1 ∥ > ϵ for i ≥ 1, set i = i + 1 and go back to the third step; else stop.

4. Numerical examples

In this article, the five- and nine-story structures of the SAC¹ benchmark building (Ohtori et al., 2004) are used to assess the performance of the utilized method. The specifications of the used structures are listed in Tables 1 and 2. The damping rate of both the structures is considered as 0.05.

Table 1.

Structural characteristics of the 5-story building.

Story no	Mass	Stiffness
Story no	$(\times 10^{3} kg)$	$(\times 10^{6} N / m)$
1	215.2	147
2	209.2	113
3	207	99
4	204.8	89
5	266.1	84

Table 2.

Structural characteristics of the 9-story building.

Story no	Mass	Stiffness
Story no	$(\times 10^{3} kg)$	$(\times 10^{6} N / m)$
1	505	159.7
2	494.5	193.3
3	494.5	179.6
4	494.5	162.9
5	494.5	152.8
6	494.5	138.8
7	494.5	105.4
8	494.5	79.9
9	535	73.8

Four different (in intensity and duration) earthquake records are used to evaluate the efficiency of the applied controller. The selected accelerograms are: (1) El-Centro-FN (Imperial Valley 10/15/79, Brawley Airport, USGS#5060, 225), (2) Newhall-FN (Northridge AFT 03/20/94, LA-Wonderland Ave, USC#17, 095), (3) Sylmar-FN (San Fernando 02/09/71, Fairmont DAM, CDMG#121, 056), and (4) Kobe-FP (Kobe 01/16/95, OSA, 090) with different scaling factors 1.3, 0.6, 0.45, and 0.7, respectively.

Figure 1 depicts the schematics of the utilized structures. Here, it is assumed that in the 5-story building, the locations of the installation of control devices are within the stories (Figure 1(a)). Also, it is supposed that the 9-story building is adjacent to another 5-story building which is enough rigid to use as a support to install the control actuators (Figure 1(b)). Therefore, the interaction matrix of the control forces in the stories, that is Γ in equation (1), for the 9-story building is equal to the identity matrix with proper dimensions and for the 5-story building is as follows

5 - story : Γ = [\begin{matrix} 1 & - 1 & 0 & 0 & 0 \\ 0 & 1 & - 1 & 0 & 0 \\ 0 & 0 & 1 & - 1 & 0 \\ 0 & 0 & 0 & 1 & - 1 \\ 0 & 0 & 0 & 0 & 1 \end{matrix}]

(20)

Figure 1.

Schematic of (a) 5- and (b) 9-story buildings.

All stories of the 5-story building are equipped with 20-ton MR dampers. The maximum control force in each story is considered as 6 × 10⁵ N. Thus, three 20-ton MR dampers should be installed in each floor. For the 9-story building, the maximum control force is presumed 8 × 10⁵ N. Therefore, in the first five stories, four 20-ton MR dampers are installed.

For both the structures, an AOMF controller is designed by means of Algorithm 1. A conventional LQR-based controller is also designed to appraise more and compare the obtained results. The design procedure is in such a manner that the state variables are limited to a predefined value. In this study, the aim is to decrease 40% or more of the maximum displacement of the floors. To avoid the saturation phenomenon of the actuators, the control forces must be limited too. Utilizing these values and based on the Brayson’s rule, discussed in the second section, the following weighting matrices are selected and applied in both the controllers

\begin{array}{l} 5 - story : R = 10^{- 8} I_{5}, Q = 10^{5} I_{10} \\ 9 - story : R = 3 \times 10^{- 8} I_{5}, Q = 10^{5} I_{18} \end{array}

(21)

As mentioned, the considered method estimates the optimal gain using an iterative algorithm and converges to its exact value after a few iterations. Therefore, high convergence speed is one of the main advantages of the utilized AOMF controller. The convergence of the gain matrix G to its optimal value G^∗ is shown by Figure 2. In this diagram, the convergence for sampling in multiple intervals and the difference between calculated gain and the optimal value are plotted. To determine G in each step, only one sample can be used and all the iterations of the step be conducted with that sample. Also, each iteration can be done with a distinct sample.

Figure 2.

Convergence of G to its optimal value for some steps in different intervals.

To evaluate the performance of the designed controllers, the following five performance indices are defined and calculated for each earthquake (Ghaderi and Amini, 2017; Hashemi et al., 2016).

1. Peak floor displacement

J_{1} = \frac{\max_{t, i} | x_{i} (t) |}{\max_{t, i} | {\hat{x}}_{i} (t) |}

(22)

2. Peak interstory drift

J_{2} = \frac{\max_{t, i} | d_{i} (t) |}{\max_{t, i} | {\hat{d}}_{i} (t) |}

(23)

3. Peak floor acceleration

J_{3} = \frac{\max_{t, i} | {\ddot{x}}_{i} (t) |}{\max_{t, i} | {\hat{\ddot{x}}}_{i} (t) |}

(24)

4. Peak control force

J_{4} = \frac{\max_{t} | \sum_{i} F_{i} (t) |}{\max_{t} | V_{0} (t) |}

(25)

5. Peak base shear

J_{5} = \frac{\max_{t} | V_{0} (t) |}{\max_{t} | {\hat{V}}_{0} (t) |}

(26)

In equations (22)–(26), x_i(t), d_i(t), and ${\ddot{x}}_{i} (t)$ are the displacement, drift, and acceleration of the ith story, respectively. F_i(t) and V₀(t) are the control force and base shear of the controlled structure, respectively. “ $\hat{}$ ” declares the corresponding responses in the uncontrolled structure.

In the following, to consider both the linear time invariant and the linear time variant systems, simulations of the control algorithms are carried out in two cases, including: (1) the healthy condition (structural properties are remained constant during the earthquakes) and (2) the damaged condition (reduction in story stiffness due to damage occurrence during the earthquakes).

Case 1

The healthy condition

The values of the performance indices using both the LQR and AOMF controllers for the five- and nine-story buildings under the earthquakes are shown in Figures 3 and 4, respectively. The comparison of the values of both the controllers demonstrates that the proposed control system converges to the LQR. The values of J₁ illustrate that the reduction of the peak floor displacement for both the structures is about 40% or more which meet the considered value for the design procedure. The obtained values for the J₂ and J₃ indices elucidate that the controllers are caused to reduce the maxima of the interstory drift and the floors acceleration, respectively. Because the nonstructural damages are related to these values, the considered controller can limit the risk of this type of damage. The obtained results for J₄ clarify that the maximum control forces which have been applied to the structures are less than the base shear subjected to all the earthquakes. As a result, the considered maximum control forces are appropriate values. The similarity of the corresponding indices values for the controllers exemplifies that solving the Riccati equation of the LQR using the applied numerical method is accomplished with a reliable approximation.

Figure 5 represents the comparison of peak floor displacement for the healthy 9-story structure in the controlled and the uncontrolled situations. It is shown that the suggested controller has effectively suppressed the peak floor displacement, and its performance is almost the same as the LQR-based control system, with the difference that there is no knowledge about the system parameters in the former.

The comparison of the peak interstory drift of the 9-story structure in the healthy condition is shown in Figure 6 for the controlled and the uncontrolled situations. Results are similar to the displacement and demonstrate that the performance of the AOMF controller and the LQR is equivalent in reducing the peak interstory drift.

Figure 7 depicts the time history diagram of the top floor displacement for the structures, in the healthy condition, subjected to the earthquakes and harmonic sine loads. The frequency of the harmonic loads is equal to the first frequency of the considered structures, and the resonance phenomenon is occurred. It is shown that the system is stable subjected to a load with a frequency equal to the resonant frequency. It can be revealed that the utilized controller has considerably decreased the displacement of the top floor in comparison with uncontrolled situation during the earthquake beyond the decrement of the peak values.

To generate the required control forces, a sequence is adopted that shares the force generation of each semiactive device as the same. In this way, the saturation phenomenon is avoided in each actuator, if the provided control force is less than the considered maximum value. The proposed method calculates the required control forces, that is vector u(t), which are generated by MR dampers. The provided force by the MR damper can be calculated through the following equations (Spencer et al., 1997)

\begin{array}{l} u (t) = α z_{h} (t) + c_{0} \dot{x} (t), \\ {\dot{z}}_{h} (t) = - γ | \dot{x} (t) | z_{h} (t) {| z_{h} (t) |}^{n_{h} - 1} - β {| z_{h} (t) |}^{n_{h}} + A_{h} \dot{x} (t) \end{array}

(27)

where z_h(t) is an evolutionary dimensionless hysteric variable;

\dot{x} (t)

is the velocity at the location of the MR damper; α and c₀ are linear functions of control drive voltage of the MR damper and they are given as follows

α = α_{a} + α_{b} v (t), c_{0} = c_{0 a} + c_{0 b} v (t)

(28)

The relation between v(t) and the input voltage of the MR damper is defined by a first-order filter as follows

\dot{v} (t) = - η (v (t) - v_{s} (t))

(29)

where v_s(t) is the input voltage which is applied to the MR damper to generate the calculated force by the control algorithm. To obtain the input voltage v_s(t), considering some approximations, an inverse model for v(t) and v_s(t) can be derived from (27)–(29) as follows

v (t) \approx \frac{u (t) - c_{0 a} \dot{x} (t) - α_{a} z_{h} (t)}{c_{0 b} \dot{x} (t) + α_{b} z_{h} (t)}, v_{s} (t) = v (t) + \frac{\dot{v} (t)}{η}

(30)

With this description, in the proposed procedure, the required forces to control the structures are first calculated via the AOMF or the LQR controllers. Then, the voltage v_s(t) is obtained through (30) at each time step. It is worth mentioning, the variables z_h(t), v(t), and $\dot{x} (t)$ are available in each time step. The values of constant parameters of the 20 ton MR damper are listed in Table 3 (Jung et al., 2003).

Figure 8 shows the time history of the total generated control forces by the located MR damper at the 5th story of the structures, in the healthy condition, under the earthquakes. From Figure 8, it can be realized that the performance of the AOMF controller is very close to the LQR. The saturation of the MR dampers is an adverse phenomenon, and it can be kept away by selecting the weights R and Q, appropriately. Figure 8 reveals that considering the weighting matrices in (21), the saturation is avoided or the desired performance of the control system is preserved, if the MR dampers are saturated. It should be mentioned that the problem of the input saturation can also be addressed using some other model-free techniques such as the algorithms developed in (Modares et al., 2013, 2014).

Figures 3–8 demonstrates that using the considered numerical solution, the Riccati equation of the LQR method can be solved with an acceptable approximation, without the availability of the system parameters.

Figure 3.

Performance indices for the 5-story building in the healthy condition.

Figure 4.

Performance indices for the 9-story building in the healthy condition.

Figure 5.

Peak floor displacement for the 9-story building in the healthy condition subjected to the earthquakes.

Figure 6.

Peak interstory drift for the 9-story building in the healthy condition subjected to the earthquakes.

Figure 7.

Time history response of the top floor displacement in the healthy condition subjected to the earthquakes.

Table 3.

Values of the 20 ton MR damper constant parameters.

Parameter	Value
$α_{a}$	26.0 (kN/m)
$α_{b}$	29.1 (kN/mV)
$c_{0 a}$	105.4 (kNs/m)
$c_{0 b}$	131.6 (kNs/mV)
$γ$	141.0 (m⁻²)
$β$	141.0 (m⁻²)
$η$	100 (s⁻¹)
$A_{h}$	2074.5
$n_{h}$	2

Figure 8.

Generated force at the 5th story of the structures in the healthy condition subjected to the earthquakes.

Case 2

The damaged condition

In Case 1, it was assumed that the structural characteristics remain constant during the earthquake and the simulations accomplished with this supposition. In this subsection, the simulations are carried out with the assumption of change in the structural properties, due to damage occurrence, during the earthquake. For this purpose, 40% stiffness reduction in all stories of the five-story building and the 1st to 5th stories of the nine-story building is assumed as damages during the earthquakes. The time occurrence of the damage in each earthquake is considered a few moments before the peak acceleration of that earthquake. With this explanation, for the El-Centro, Newhall, Sylmar, and Kobe earthquakes, the damage times are considered as 1.5, 4, 3.5, and 3 s, respectively.

Figures 9 and 10 represent the values of the performance indices for the structures in the damaged condition. As it can be seen, both the controllers have had a desired performance after the damage occurrence. The considered change in the stiffness results in variation of the damping matrix of the system. In the LQR controller, these values have been obtained based on the known parameters of the structure in the healthy condition; while in the AOMF controller, the structural specifications have been supposed to be unknown before and after the damage, and the optimal gain is calculated in real time during the earthquake. After the damage, the favorable performance has been also maintained by updating the control gain through the applied algorithm. The values of the indices in Figures 9 and 10 illustrate that the AOMF controller is robust against any variations in the system parameters and stables after damage occurrence in the structure.

The diagram of the peak floor displacement in Figures 11 and 12 in the damaged structures depicts that even after damage occurrence, the controllers have significantly decreased the maximum displacement. These figures show that the LQR has a sufficient robustness. Also, the AOMF controller, which its convergence to the LQR is assessed, can be effectively used without any knowledge about time-varying system properties.

Figure 13 shows the peak interstory drift in the 9-story structure. The outcomes of the figure are in line with the displacement results, and the interstory drift is decreased by the controllers.

The displacement of the top floor in the damaged condition during the earthquakes is shown by Figure 14. In this figure, the diagrams related to the harmonic loads are obtained subjected to sine waves in which their frequency is equal to the resonant frequency of the damaged structures. The significant decline of the top floor displacement for the LQR and the AOMF controller in the aforementioned condition has been clearly represented.

The collected control forces located at the 5th level for both the buildings, in the damaged condition, are shown in Figure 15. It can be seen that even in saturation mode, there is no disruption in the control system performance. This is the consequence of determining sufficiently of the weighting matrices to calculate the optimal control gain of both the controllers, with the difference that in the AOMF controller the gain is obtained in an adaptive manner based on the only measurement data from the system states.

Figure 9.

Performance indices for the 5-story building in the damaged condition.

Figure 10.

Performance indices for the 9-story building in the damaged condition.

Figure 11.

Peak floor displacement for the 5-story building in the damaged condition subjected to the earthquakes.

Figure 12.

Peak floor displacement for the 9-story building in the damaged condition subjected to the earthquakes.

Figure 13.

Peak interstory drift for the 9-story building in the damaged condition subjected to the earthquakes.

Figure 14.

Time history response of the top floor displacement in the damaged condition subjected to the earthquakes.

Figure 15.

Generated force at the 5th story of the structures in the damaged condition subjected to the earthquakes.

5. Conclusions

Achieving the optimal performance in the seismic response control of the structure is an important issue, which was studied in this article. Designing a controller based on the LQR approach is an adequate method to meet the optimal performance. On the other hand, to obtain the optimal gain in the LQR, exact values of system parameters are required. In this article, an AOMF controller is applied in which the optimal gain with state feedback structure was calculated using an iterative algorithm. In this method, the optimal control gain was achieved without any knowledge about system parameters which met the LQR. Also, it was illustrated that the considered controller was converged to the optimal value after a few iterations and in a very short period. The implementation of the control system was accomplished in such a way that during the earthquake be updated by measuring the system states in predefined intervals. Therefore, even in the case of variations existence in the structural characteristics, for example due to damage subjected to the strong earthquake, the control gain was appropriately updated with the changes by using new measurements. Utilizing the controller in different conditions was made to happen due to the high rate of convergence. Performing various simulations for the healthy and the damaged structure during the earthquake, the performance of the controller was evaluated. Considerable reduction of the dynamic responses either in time-invariant or time-varying systems during the earthquake demonstrated that the controller is robust against system parameters changes. Consequently, the utilized controller can be used in an existing structure in which their parameters, such as the mass, the stiffness, and the damping, are not available and they need to be retrofitted. Also, for structures in which their parameters have changed due to the former earthquakes, utilizing this controller could be effective.

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Yazdan Batmani

Note

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