# Cooperative Control Method of Active and Semiactive Control: New Framework for Vibration Control

Authors: Kazuhiko Hiramoto
Journal: Mathematical Problems in Engineering
 (2014)

---

## Abstract

A new control design framework for vibration control, the cooperative control of active
and semiactive control, is proposed in the paper. In the cooperative control, a structural system
having both of an actuator and a semiactive control device, for example, MR damper and so forth, is
defined as the control object. In the proposed control approach, the higher control performance
is aimed by the cooperative control between the active control with the actuator and
the semiactive control with the semiactive control device. A design method to determine the
active control input and the command signal to drive the semiactive control device based on
the one-step prediction of the control output is proposed. A simulation example of a control
system design for a benchmark building is presented to show the effectiveness of the proposed
control framework.


---

## Body

## 1. Introduction

Enormous methods for vibration control system design have been proposed so far for mechanical and/or structural systems and those methods are applied to variety of practical applications, for example, civil structures, vehicle suspensions and robot manipulators, and so forth. As is well known, methodologies of vibration control have been classified as the passive [1, 2], active [3, 4], and semiactive control [5]. Especially, the active and semiactive control methodologies have been well studied for the last several decades because of the higher control performance compared to that of the passive control. Those methods have been practically applied to the above systems thanks not only to the development of the control theory and the computer technology for digital implementation from the software point of view, but also, from the hardware viewpoint, that is, the development of high performance actuators for active control and semiactive control devices that utilize so-called smart materials, for example, ER or MR dampers [6, 7], variable stiffness devices [8–10], and so forth, for semiactive control.In the vibration control system design, a model of a control object to be controlled is assumed as a some degree of freedom mechanical system that is represented by a connection of masses, springs, and dampers. In the active or semiactive control, the control object is defined as a connected model with the above model of mechanical systems, sensors to measure the current state of the control object, and the actuator for the active control or the semiactive control devices for the semiactive control. Generally, the semiactive control system is always asymptotically stable because of the energy dissipating nature of semiactive control devices and the mechanical system to be controlled itself while the active control system can be unstable because of the sensor/actuator failure and the modeling error of the control object.The active or semiactive control law has been obtained based on the above model of the control object, that is, the mechanical system, sensors, and the actuator (active control) or the semiactive control device (semiactive control). In fact a lot of methods to obtain the active and semiactive control systems have been proposed so far [3, 5] including the author’s works [11–13].However, until now, a mechanical system having both of an actuator and a semiactive control device has not been considered as the control object for the vibration control. If there exists a good control theory for the control object with the actuator and the semiactive device, control characteristics that cannot be achieved by the conventional semiactive and active control systems can be expected logically because of the higher degree of the design freedom.Although the control system with the above control object having both of the actuator and the semiactive control device is potentially capable of realizing the higher control performance, there are no methodologies for controlling the actuator and the semiactive control device coordinately and complementarily to realize more sophisticated control characteristics.In this study, a new vibration control system design framework for the control object having the actuator and the semiactive control device is considered. The control system design problem is formulated as a cooperative control problem between the actuator and the semiactive control device so that the control performance on vibration suppression and the energy to drive the actuator and/or semiactive control devices are optimized. A switching-based cooperative active and semiactive control law with a one-step prediction of the control output is proposed. A simulation example for a vibration control design of a benchmark building is presented.The rest of the paper is organized as follows. In Section2, the cooperative control problem of the active and semiactive control is formulated. A design method of the cooperative control composed of a robust active control and a switching semiactive control is presented in Section 3. A simulation example is given in Section 4. The summary of the paper and the discussion about the future research direction of the proposed framework are shown in Section 5.Notations are as follows.t: time, R
n (R
m
×
n): the set of n-dimensional (m
×
n-dimensional) real vector (matrix), 0
m (0
m
×
n): an m-dimensional (m
×
n-dimensional) zero vector (matrix), 1
m: 1
m
=
[
1
⋯
1
]
T
︸
m, I
n: an n-dimensional identity matrix, M
T: transposition of a matrix M, trace
(
N
): trace of a square matrix N, diag
⁡
(
a
1
,
…
,
a
n
): a diagonal matrix whose diagonal elements are a
1
,
…
,
a
n,  and  RMS
(
a
(
t
)
): RMS value of a scalar signal a
(
t
), 0
≤
t
≤
T; that is, RMS
(
a
(
t
)
)
=
(
1
/
T
)
∫
0
T
‍
a
2
(
t
)
d
t, Peak
(
a
(
t
)
)
=
max
⁡
t
|
a
(
t
)
|.


## 2. Problem Formulation

The control object in the paper is defined as the followingn-dof linear mechanical system with n
u actuators and n
r semiactive control devices given as the following:
(1)
M
q
¨
(
t
)
+
D
(
r
(
t
)
)
q
˙
(
t
)
+
K
(
r
(
t
)
)
q
(
t
)
=
E
(
r
(
t
)
)
w
(
t
)
+
F
(
r
(
t
)
)
u
(
t
)
,
where q
(
t
)
∈
R
n, w
(
t
)
∈
R
n
w, and u
(
t
)
∈
R
n
u are the displacement, disturbance or reference signal, and active control input vectors, respectively. The vector r
(
t
)
=
[
r
1
(
t
)
⋯
r
n
r
(
t
)
]
T
∈
R
n
r is the n
r-dimensional time varying parameter vector of semiactive control devices installed on the mechanical system. As some examples of such semiactive control devices, we can assume electrorheological (ER) [6], magnetorheological (MR) dampers [7] and variable stiffness devices [8, 9]. The vector r
(
t
) is the signal representing the state of the physical properties of the semiactive control devices, for example, the damping coefficient of the ER and MR dampers and the stiffness of the variable stiffness devices. Without loss of generality, the vector r
(
t
) be in a set C
r defined as the following:
(2)
C
r
=
{
r
(
t
)
∈
R
n
r
∣
0
n
r
≤
r
(
t
)
≤
1
n
r
}
.
For example, if the actual maximum and minimum values of n
r variable damping semiactive control devices are given as vectors d
max
⁡
∈
R
n
r and d
min
⁡
∈
R
n
r, respectively, the actual damping coefficient vector d
r
(
t
) is expressed with r
(
t
)
∈
C
r as the following:
(3)
d
r
(
t
)
=
(
1
n
r
-
r
(
t
)
)
d
min
⁡
+
r
(
t
)
d
max
⁡
=
d
min
⁡
+
r
(
t
)
(
d
max
⁡
-
d
min
⁡
)
,
r
(
t
)
∈
C
r
.
Some elements in the damping and stiffness matrices can be changed in the range of their minimum to maximum values as the function on the signal r
(
t
)
∈
C
r. Also, the vibration of the mechanical system in (1) is controlled by the actuator that produces the active control input u
(
t
). In other words, the dynamic behavior of the mechanical system in (1) can be controlled by the n
r signals of the semiactive control devices and the n
u actuator input signals simultaneously. In the present study, the control system design problem for such mechanical systems is formulated as the following cooperative control between active and semiactive control.Cooperative Control Problem of Active and Semiactive Control. For the mechanical system with the actuator and the semiactive control device in (1), find the active control input u
(
t
) and the signal r
(
t
)
∈
C
r to determine the state of the semiactive control devices so that the control system satisfies given control specifications on vibration control.A cooperative and complementary control law of the active control inputu
(
t
) and the signal of the semiactive control device r
(
t
)
∈
C
r is desired in the formulated control design problem. Mathematically, control specifications are given as the form of an optimization problem with some constraints, for example, the minimization of the norm of the closed-loop system with a robustness constraint or satisfying some indices on the closed-loop response, for example, the decay rate of the impulse response or the constraint on the active control input and so forth.As described in Section1, the control framework considered in the paper is potentially capable of achieving more advanced control performance than that achievable in the conventional passive, semiactive, and active control because of its higher degree of design freedom. However, unfortunately, we do not have any systematic control design methodologies in the present design problem unlike the optimal dynamic absorber design in passive control [1, 2] and the controller design based on the linear control theory in active control [3] and so forth. Because the (potentially) higher control performance under the present framework can be expected and there are no established methods to obtain the cooperative control law, the author claims that the study of the cooperative control problem is theoretically interesting.Besides, because the present cooperative control scheme can directly incorporate the rapidly growing outcomes of the research and development on smart materials [14] used for semiactive control devices, in this sense, the development of the control design method for the formulated cooperative control problem is also expected to be practically useful under the rapid upgrading of specifications on vibration control [15].In the next section, a switching control of the command signal of the semiactive control device with a robust active feedback control law will be presented as an example of the cooperative and complementary control method for the formulated corporative control design problem.


## 3. Design Method: Switching Semiactive Control with Robust Active Control

For the formulated cooperative control problem in the previous section, a design method of the control system is proposed in this section. The control law is composed of a robust active feedback control and the switching semiactive control based on the one-step prediction of the control output. The detail of the control method will be presented in the following subsections.


### 3.1. Robust Active Feedback Control

To obtain the control law, let us define the generalized plantS
G given as
(4)
S
G
:
{
x
˙
(
t
)
=
A
(
r
(
t
)
)
x
(
t
)
+
B
1
(
r
(
t
)
)
w
(
t
)
+
B
2
u
(
t
)
,
z
(
t
)
=
C
1
x
(
t
)
+
D
12
u
(
t
)
,
y
(
t
)
=
x
(
t
)
,
where x
(
t
)
∈
R
n
x and z
(
t
)
=
R
n
z are the state and control output vectors, respectively. Note that all the state variables are assumed to be measured for the active feedback control in this paper. The generalized plant S
G consists of the model of the mechanical systems in (1) with time varying signal r
(
t
)
∈
C
r and some weighting functions for representing the control specifications. Accordingly, coefficient matrices of the generalized plant S
G become functions on the time varying vector r
(
t
)
∈
C
p. That is, the generalized plant S
G is represented as an LPV (linear parameter varying) model with time varying parameter vector r
(
t
).In obtaining the active control law generating the control inputu
(
t
), the active control law must stabilize the mechanical system with the arbitrary value of the time varying damping and/or stiffness characteristics; that is, r
(
t
)
∈
C
r in (2). In the present paper, the active control law is defined as a time invariant state-feedback control given as
(5)
C
A
:
u
(
t
)
=
-
K
A
x
(
t
)
,
K
A
∈
R
n
u
×
n
x
.
Then the closed-loop system with the generalized plant S
G in (4) and the control law C
A in (5) is defined as follows:
(6)
S
C
:
{
x
˙
(
t
)
=
A
c
(
r
(
t
)
)
x
(
t
)
+
B
1
(
r
(
t
)
)
w
(
t
)
,
z
(
t
)
=
C
c
x
(
t
)
,
A
c
(
r
(
t
)
)
=
A
(
r
(
t
)
)
-
B
2
K
A
,
C
c
=
C
1
-
D
12
K
A
.
The control law C
A in (5) is designed so that the closed-loop system S
C in (6) is asymptotically stable for all r
(
t
)
∈
C
r with an arbitrary rate; that is, there are no restrictions on r
˙
(
t
). As a state-feedback control, a robust H
2 state-feedback-control law is used. That is, the closed-loop system S
C in (6) is robustly stable and its H
2 norm is less than η
>
0 for all r
(
t
)
∈
C
p with an arbitrary rate r
˙
(
t
). Assume that all the r
(
t
)-dependent coefficient matrices of S
G, that is, A and B
1, are linear functions on the element of r
(
t
). Then the robust H
2 state-feedback-control law C
A in (5) exists if there exist solution matrices X
=
X
T
>
0, Q
=
Q
T
>
0, and K
A
∈
R
n
u
×
2
n satisfying
(7)
[
{
A
(
r
(
t
)
)
-
B
2
K
A
}
X
+
X
{
A
(
r
(
t
)
)
-
B
2
K
A
}
T
B
1
(
r
(
t
)
)
B
1
T
(
r
(
t
)
)
-
I
]
<
0
,
(8)
[
Q
(
C
1
-
D
12
F
)
X
X
(
C
1
-
D
12
F
)
T
X
]
>
0
,
(9)
trace
(
Q
)
<
η
2
,
for r
(
t
)
=
0 and r
(
t
)
=
1 [16]. With the change of variable W
=
K
A
X, (7) and (8) become LMIs (linear matrix inequalities) on X and W given as follows [17]:
(10)
[
A
(
r
(
t
)
)
X
-
B
2
W
+
X
A
T
(
r
(
t
)
)
-
W
T
B
2
T
B
1
(
r
(
t
)
)
B
1
T
(
r
(
t
)
)
-
I
]
<
0
,
(11)
[
Q
C
1
X
-
D
12
W
X
C
1
T
-
W
T
D
12
T
X
]
>
0
,
(12)
trace
(
Q
)
<
η
2
,
r
(
t
)
=
0,1
.
Once solution matrices X and W are found, we can obtain the state-feedback gain matrix K
A
=
W
X
-
1 that robustly stabilizes the generalized plant S
G in (4).


### 3.2. Switching Semiactive Control

The closed-loop systemS
C in (6) is always asymptotically stable for all r
(
t
)
∈
C
r with arbitrary rate r
˙
(
t
). It means that we can employ any semiactive control laws if r
(
t
)
∈
C
r; the time varying parameter vector representing the state of the semiactive device remains in the prescribed range. In this paper, a semiactive control method to control r
(
t
) based on the one-step ahead prediction of the control output [18] is proposed.For the closed-loop systemS
C in (6), define t
0 and Δ
t
>
0 as the current time and the small prediction length, respectively. From (6), the predicted control output of the closed-loop system S
C is given as the following:
(13)
z
(
t
0
+
Δ
t
)
=
C
c
x
(
t
0
+
Δ
t
)
.
The state vector of the closed-loop system S
C at t
=
t
0
+
Δ
t is approximated to
(14)
x
(
t
0
+
Δ
t
)
≃
x
(
t
0
)
+
Δ
t
x
˙
(
t
0
)
.
With the model of the actively controlled system S
C in (6), the time derivative of the state vector of the active control system is obtained as the following:
(15)
x
˙
(
t
0
)
=
A
c
(
r
(
t
0
)
)
x
(
t
0
)
+
B
1
(
r
(
t
0
)
)
w
(
t
0
)
.
In calculating x
(
t
0
+
Δ
t
) with (14), we can employ some numerically stable integration methods, for example, Runge-Kutta method and Newmark’s β method and so forth.To obtain the predicted control outputz
(
t
0
+
Δ
t
) in (13), the time varying signal r
(
t
0
)
∈
C
r that represents the state of the semiactive control device at t
=
t
0 also needs to be determined. In general semiactive control devices, the physical characteristics, for example, the variable damping coefficient of the MR damper, are varied but delayed to the variation of their command signals to change the physical characteristics of semiactive devices. In this paper, such dynamic delay is modeled by a first order lag system given as the following:
(16)
r
˙
i
(
t
)
=
-
1
T
i
r
i
(
t
)
+
1
T
i
r
i
c
(
t
)
,
where T
i, r
i
(
t
), and r
i
c
(
t
)
∈
R
n
r, i
=
1
,
…
,
n
r, are the time constant of the ith semiactive control device, the ith element of the time varying signal r
(
t
), and its command signal, respectively. The time varying parameter r
(
t
) can be varied by changing the command signal r
c
(
t
)
=
[
r
1
c
(
t
)
⋯
r
n
r
c
(
t
)
]
T
∈
C
r.In the present study, candidates of the command signal of theith semiactive device at t
=
t
0 are assumed to be as the following:
(17)
r
i
c
(
t
0
)
=
{
1
,
0
,
i
=
1
,
…
,
n
r
.
Then, predicted control output signals of the control system at t
=
t
0
+
Δ
t, denoted by z
(
t
0
+
Δ
t
)
[
j
]
=
C
c
(
r
(
t
0
+
Δ
t
)
[
j
]
), j
=
1
,
…
,
2
n
r, can be obtained with (13)–(16). From the obtained 2
n
r predicted control output signals, the optimal command signal vector of the semiactive control devices (denoted by r
(
t
0
)
[
j
opt
]) is selected in a real time manner so that the performance indices J
p
(
r
c
(
t
0
)
[
j
]
), j
=
1
,
…
,
2
n
r, defined as the following is minimized:
(18)
J
p
(
r
c
(
t
0
)
[
j
]
)
=
(
z
(
t
0
+
Δ
t
)
[
j
]
)
T
(
z
(
t
0
+
Δ
t
)
[
j
]
)
,
j
=
1
,
…
,
2
n
r
.The command signal ofn
r semiactive control devices at t
=
t
0 is selected from all the 2
n
r combinations of (candidates of) command signals based on the performance index J
p. The detail is given as follows.Step 1.
For all the candidates of the command signal vectors of the semiactive control devices, that is,r
c
(
t
0
)
[
j
], j
=
1
,
…
,
2
n
r, whose ith element is given as (17), the predicted control output signals z
(
t
0
+
Δ
t
)
[
j
], j
=
1
,
…
,
2
n
r, are computed by (13)–(16).Step 2.
Obtain performance indicesJ
p
(
r
c
(
t
0
)
[
j
]
), j
=
1
,
…
,
2
n
r, in (18) for corresponding r
c
(
t
0
)
[
j
], j
=
1
,
…
,
2
n
r.Step 3.
The optimal command signal vector of the semiactive control devicesr
opt
c
(
t
0
) is selected as the following:
(19)
r
opt
c
(
t
0
)
=
argmin
r
c
(
t
0
)
[
j
]
,
j
=
1
,
…
,
2
n
r
⁡
J
p
(
r
c
(
t
0
)
[
j
]
)
.
The control algorithm to select the command signal of the semiactive control devices is carried out in a real time manner from the initial time t
i (the time when the control system is turned on), with the continuing measurement of the disturbance signal w
(
t
), t
≥
t
i, and the state vector x
(
t
) while running the active control law C
A in (5) that robustly stabilizes the generalized plant S
G in (4) for all r
(
t
)
∈
C
r with any rate r
˙
(
t
). In other words, the robust active control law C
A and the predictive semiactive control law cooperatively operate for minimizing the control output z
(
t
).As design parameters that are adjustable in the cooperated control system design, some elements of the coefficient matricesC
1 and D
12 in (4) can be defined. By tuning those design parameters, we obtain the cooperative control system of the active and semiactive control laws satisfying the specifications on vibration control.Remark 1.
In the proposed cooperative control method, a nonstationary disturbance signalw
(
t
) such as an earthquake wave or a wind gust is assumed. Therefore, only one-step prediction of the control output z
(
t
0
+
Δ
t
) is employed in the semiactive control law. On the other hand, in the case that the disturbance signal w
(
t
) is known in advance (e.g., the disturbance signal w
(
t
) is defined as a step reference signal), more than one-step prediction of the control output z
(
t
) is possible such as z
(
t
0
+
Δ
t
)
,
…
,
z
(
t
0
+
M
p
Δ
t
),  M
p
>
1. With the multistep prediction data, a cooperative control method between the active and semiactive control may be obtained under the model predictive control framework [19]. Such a control problem is also interesting and will be addressed in the future study.


## 3.1. Robust Active Feedback Control

To obtain the control law, let us define the generalized plantS
G given as
(4)
S
G
:
{
x
˙
(
t
)
=
A
(
r
(
t
)
)
x
(
t
)
+
B
1
(
r
(
t
)
)
w
(
t
)
+
B
2
u
(
t
)
,
z
(
t
)
=
C
1
x
(
t
)
+
D
12
u
(
t
)
,
y
(
t
)
=
x
(
t
)
,
where x
(
t
)
∈
R
n
x and z
(
t
)
=
R
n
z are the state and control output vectors, respectively. Note that all the state variables are assumed to be measured for the active feedback control in this paper. The generalized plant S
G consists of the model of the mechanical systems in (1) with time varying signal r
(
t
)
∈
C
r and some weighting functions for representing the control specifications. Accordingly, coefficient matrices of the generalized plant S
G become functions on the time varying vector r
(
t
)
∈
C
p. That is, the generalized plant S
G is represented as an LPV (linear parameter varying) model with time varying parameter vector r
(
t
).In obtaining the active control law generating the control inputu
(
t
), the active control law must stabilize the mechanical system with the arbitrary value of the time varying damping and/or stiffness characteristics; that is, r
(
t
)
∈
C
r in (2). In the present paper, the active control law is defined as a time invariant state-feedback control given as
(5)
C
A
:
u
(
t
)
=
-
K
A
x
(
t
)
,
K
A
∈
R
n
u
×
n
x
.
Then the closed-loop system with the generalized plant S
G in (4) and the control law C
A in (5) is defined as follows:
(6)
S
C
:
{
x
˙
(
t
)
=
A
c
(
r
(
t
)
)
x
(
t
)
+
B
1
(
r
(
t
)
)
w
(
t
)
,
z
(
t
)
=
C
c
x
(
t
)
,
A
c
(
r
(
t
)
)
=
A
(
r
(
t
)
)
-
B
2
K
A
,
C
c
=
C
1
-
D
12
K
A
.
The control law C
A in (5) is designed so that the closed-loop system S
C in (6) is asymptotically stable for all r
(
t
)
∈
C
r with an arbitrary rate; that is, there are no restrictions on r
˙
(
t
). As a state-feedback control, a robust H
2 state-feedback-control law is used. That is, the closed-loop system S
C in (6) is robustly stable and its H
2 norm is less than η
>
0 for all r
(
t
)
∈
C
p with an arbitrary rate r
˙
(
t
). Assume that all the r
(
t
)-dependent coefficient matrices of S
G, that is, A and B
1, are linear functions on the element of r
(
t
). Then the robust H
2 state-feedback-control law C
A in (5) exists if there exist solution matrices X
=
X
T
>
0, Q
=
Q
T
>
0, and K
A
∈
R
n
u
×
2
n satisfying
(7)
[
{
A
(
r
(
t
)
)
-
B
2
K
A
}
X
+
X
{
A
(
r
(
t
)
)
-
B
2
K
A
}
T
B
1
(
r
(
t
)
)
B
1
T
(
r
(
t
)
)
-
I
]
<
0
,
(8)
[
Q
(
C
1
-
D
12
F
)
X
X
(
C
1
-
D
12
F
)
T
X
]
>
0
,
(9)
trace
(
Q
)
<
η
2
,
for r
(
t
)
=
0 and r
(
t
)
=
1 [16]. With the change of variable W
=
K
A
X, (7) and (8) become LMIs (linear matrix inequalities) on X and W given as follows [17]:
(10)
[
A
(
r
(
t
)
)
X
-
B
2
W
+
X
A
T
(
r
(
t
)
)
-
W
T
B
2
T
B
1
(
r
(
t
)
)
B
1
T
(
r
(
t
)
)
-
I
]
<
0
,
(11)
[
Q
C
1
X
-
D
12
W
X
C
1
T
-
W
T
D
12
T
X
]
>
0
,
(12)
trace
(
Q
)
<
η
2
,
r
(
t
)
=
0,1
.
Once solution matrices X and W are found, we can obtain the state-feedback gain matrix K
A
=
W
X
-
1 that robustly stabilizes the generalized plant S
G in (4).


## 3.2. Switching Semiactive Control

The closed-loop systemS
C in (6) is always asymptotically stable for all r
(
t
)
∈
C
r with arbitrary rate r
˙
(
t
). It means that we can employ any semiactive control laws if r
(
t
)
∈
C
r; the time varying parameter vector representing the state of the semiactive device remains in the prescribed range. In this paper, a semiactive control method to control r
(
t
) based on the one-step ahead prediction of the control output [18] is proposed.For the closed-loop systemS
C in (6), define t
0 and Δ
t
>
0 as the current time and the small prediction length, respectively. From (6), the predicted control output of the closed-loop system S
C is given as the following:
(13)
z
(
t
0
+
Δ
t
)
=
C
c
x
(
t
0
+
Δ
t
)
.
The state vector of the closed-loop system S
C at t
=
t
0
+
Δ
t is approximated to
(14)
x
(
t
0
+
Δ
t
)
≃
x
(
t
0
)
+
Δ
t
x
˙
(
t
0
)
.
With the model of the actively controlled system S
C in (6), the time derivative of the state vector of the active control system is obtained as the following:
(15)
x
˙
(
t
0
)
=
A
c
(
r
(
t
0
)
)
x
(
t
0
)
+
B
1
(
r
(
t
0
)
)
w
(
t
0
)
.
In calculating x
(
t
0
+
Δ
t
) with (14), we can employ some numerically stable integration methods, for example, Runge-Kutta method and Newmark’s β method and so forth.To obtain the predicted control outputz
(
t
0
+
Δ
t
) in (13), the time varying signal r
(
t
0
)
∈
C
r that represents the state of the semiactive control device at t
=
t
0 also needs to be determined. In general semiactive control devices, the physical characteristics, for example, the variable damping coefficient of the MR damper, are varied but delayed to the variation of their command signals to change the physical characteristics of semiactive devices. In this paper, such dynamic delay is modeled by a first order lag system given as the following:
(16)
r
˙
i
(
t
)
=
-
1
T
i
r
i
(
t
)
+
1
T
i
r
i
c
(
t
)
,
where T
i, r
i
(
t
), and r
i
c
(
t
)
∈
R
n
r, i
=
1
,
…
,
n
r, are the time constant of the ith semiactive control device, the ith element of the time varying signal r
(
t
), and its command signal, respectively. The time varying parameter r
(
t
) can be varied by changing the command signal r
c
(
t
)
=
[
r
1
c
(
t
)
⋯
r
n
r
c
(
t
)
]
T
∈
C
r.In the present study, candidates of the command signal of theith semiactive device at t
=
t
0 are assumed to be as the following:
(17)
r
i
c
(
t
0
)
=
{
1
,
0
,
i
=
1
,
…
,
n
r
.
Then, predicted control output signals of the control system at t
=
t
0
+
Δ
t, denoted by z
(
t
0
+
Δ
t
)
[
j
]
=
C
c
(
r
(
t
0
+
Δ
t
)
[
j
]
), j
=
1
,
…
,
2
n
r, can be obtained with (13)–(16). From the obtained 2
n
r predicted control output signals, the optimal command signal vector of the semiactive control devices (denoted by r
(
t
0
)
[
j
opt
]) is selected in a real time manner so that the performance indices J
p
(
r
c
(
t
0
)
[
j
]
), j
=
1
,
…
,
2
n
r, defined as the following is minimized:
(18)
J
p
(
r
c
(
t
0
)
[
j
]
)
=
(
z
(
t
0
+
Δ
t
)
[
j
]
)
T
(
z
(
t
0
+
Δ
t
)
[
j
]
)
,
j
=
1
,
…
,
2
n
r
.The command signal ofn
r semiactive control devices at t
=
t
0 is selected from all the 2
n
r combinations of (candidates of) command signals based on the performance index J
p. The detail is given as follows.Step 1.
For all the candidates of the command signal vectors of the semiactive control devices, that is,r
c
(
t
0
)
[
j
], j
=
1
,
…
,
2
n
r, whose ith element is given as (17), the predicted control output signals z
(
t
0
+
Δ
t
)
[
j
], j
=
1
,
…
,
2
n
r, are computed by (13)–(16).Step 2.
Obtain performance indicesJ
p
(
r
c
(
t
0
)
[
j
]
), j
=
1
,
…
,
2
n
r, in (18) for corresponding r
c
(
t
0
)
[
j
], j
=
1
,
…
,
2
n
r.Step 3.
The optimal command signal vector of the semiactive control devicesr
opt
c
(
t
0
) is selected as the following:
(19)
r
opt
c
(
t
0
)
=
argmin
r
c
(
t
0
)
[
j
]
,
j
=
1
,
…
,
2
n
r
⁡
J
p
(
r
c
(
t
0
)
[
j
]
)
.
The control algorithm to select the command signal of the semiactive control devices is carried out in a real time manner from the initial time t
i (the time when the control system is turned on), with the continuing measurement of the disturbance signal w
(
t
), t
≥
t
i, and the state vector x
(
t
) while running the active control law C
A in (5) that robustly stabilizes the generalized plant S
G in (4) for all r
(
t
)
∈
C
r with any rate r
˙
(
t
). In other words, the robust active control law C
A and the predictive semiactive control law cooperatively operate for minimizing the control output z
(
t
).As design parameters that are adjustable in the cooperated control system design, some elements of the coefficient matricesC
1 and D
12 in (4) can be defined. By tuning those design parameters, we obtain the cooperative control system of the active and semiactive control laws satisfying the specifications on vibration control.Remark 1.
In the proposed cooperative control method, a nonstationary disturbance signalw
(
t
) such as an earthquake wave or a wind gust is assumed. Therefore, only one-step prediction of the control output z
(
t
0
+
Δ
t
) is employed in the semiactive control law. On the other hand, in the case that the disturbance signal w
(
t
) is known in advance (e.g., the disturbance signal w
(
t
) is defined as a step reference signal), more than one-step prediction of the control output z
(
t
) is possible such as z
(
t
0
+
Δ
t
)
,
…
,
z
(
t
0
+
M
p
Δ
t
),  M
p
>
1. With the multistep prediction data, a cooperative control method between the active and semiactive control may be obtained under the model predictive control framework [19]. Such a control problem is also interesting and will be addressed in the future study.


## 4. Simulation Example

A 3-dof benchmark building in NCREE, Taiwan [20], is employed as an example of the simulation study. The schematic diagram of the model is shown in Figure 1. The building is subjected to the earthquake disturbance w
(
t
)
=
[
v
(
t
)
v
˙
(
t
)
]
T, where v
(
t
) is the absolute displacement of the earthquake wave. Between the first floor and the base, a semiactive damper whose variable damping coefficient is d
1
v
(
t
) and, moreover, an actuator that produces the active control input u
(
t
) are installed. The objective of the control system design in the present example is to suppress the vibration of the building affected by the earthquake disturbance by using the proposed cooperative control method between the semiactive damper and the actuator.Figure 1
3-dof model with a semiactive damper and an actuator.The state vector and coefficient matrices of the equation of motion in (1) are given as follows:
(20)
q
=
[
q
1
q
2
q
3
]
T
,
M
=
diag
⁡
(
m
1
,
m
2
,
m
3
)
,
D
=
D
0
+
r
(
t
)
D
v
,
D
0
=
[
d
1
+
d
2
+
(
d
1
v
)
min
⁡
-
d
2
0
-
d
2
d
2
+
d
3
-
d
3
0
-
d
3
d
3
]
,
D
v
=
[
(
d
1
v
)
max
⁡
-
(
d
1
v
)
min
⁡
0
0
0
0
0
0
0
0
]
,
r
(
t
)
∈
C
r
,
K
=
[
k
1
+
k
2
-
k
2
0
-
k
2
k
2
+
k
3
-
k
3
0
-
k
3
k
3
]
,
E
=
E
0
+
r
(
t
)
E
v
,
E
0
=
[
d
1
+
(
d
1
v
)
min
⁡
k
1
0
2
×
2
]
,
E
v
=
[
(
d
1
v
)
max
⁡
-
(
d
1
v
)
min
⁡
0
0
2
×
2
]
T
F
=
[
1
0
0
]
T
,
where (
d
1
v
)
max
⁡ and (
d
1
v
)
min
⁡ are the maximum and minimum values of the variable damping coefficients of the semiactive damper, respectively. Values of structural parameters are summarized in Table 1 [20]. Also, the maximum and minimum values of the variable damping coefficients of the semiactive damper are (
d
1
v
)
max
⁡
=
20 (kNs/m) and (
d
1
v
)
min
⁡
=
0 (kNs/m), respectively.Table 1
Structural Parameters of the 3-dof model.
Floor
m
i (kg)
d
i (kNs/m)
k
i (kN/m)
1
6000
9.88
1211
2
6000
17.12
1148
3
6000
10.84
1366With the equation of motion in (1), the state vector and coefficient matrices of the generalized plant S
G in (4) are defined as follows:
(21)
x
=
[
q
q
˙
]
,
A
=
A
0
+
r
(
t
)
A
v
,
B
1
=
B
1
0
+
r
(
t
)
B
1
v
,
A
0
=
[
0
3
×
3
I
3
-
M
-
1
K
-
M
-
1
D
0
]
,
A
v
=
[
0
3
×
6
0
3
×
3
-
M
-
1
D
v
]
,
B
1
0
=
[
0
3
×
2
M
-
1
E
0
]
,
B
1
v
=
[
0
3
×
2
M
-
1
E
v
]
,
B
2
=
[
0
6
×
1
M
-
1
F
]
,
C
1
=
[
G
0
1
×
6
]
,
G
=
diag
⁡
(
g
1
,
g
2
,
…
,
g
6
)
,
g
i
≥
0
,
i
=
1
,
…
,
6
,
D
12
=
[
0
6
×
1
ρ
]
,
ρ
>
0
,
where g
i
≥
0, i
=
1
,
…
,
6, are the weighting factors to be adjusted for optimizing the control performance of the cooperative control scheme.The control performance of the proposed cooperative control system is evaluated with the performance index and the inequality constraint given as follows:(22)
J
=
max
⁡
(
J
c
/
off
⁡
,
J
c
/
on
⁡
)
J
c
/
off
⁡
=
∑
k
=
1
n
E
∑
i
=
1
3
(
RMS
(
k
s
i
c
)
RMS
(
k
s
i
off
⁡
)
+
RMS
(
k
a
i
c
)
RMS
(
k
a
i
off
⁡
)
+
Peak
(
k
s
i
c
)
Peak
(
k
s
i
off
⁡
)
+
Peak
(
k
a
i
c
)
Peak
(
k
a
i
off
⁡
)
)
,
J
c
/
on
⁡
=
∑
k
=
1
n
E
∑
i
=
1
3
(
RMS
(
k
s
i
c
)
RMS
(
k
s
i
on
⁡
)
+
RMS
(
k
a
i
c
)
RMS
(
k
a
i
on
⁡
)
+
Peak
(
k
s
i
c
)
Peak
(
k
s
i
on
⁡
)
+
Peak
(
k
a
i
c
)
Peak
(
k
a
i
on
⁡
)
)
,
(23)
Peak
(
u
k
)
≤
u
¯
,
k
=
1
,
…
,
n
E
,
(24)
Peak
(
f
d
k
)
≤
f
d
¯
,
k
=
1
,
…
,
n
E
,
where k
s
i
* and k
a
i
*, *
=
c
,
off
⁡ and on
⁡ are the relative displacements between ith and (
i
-
1
)th floor of the building including that between the 1st floor and the ground and the absolute acceleration of the ith floor for kth earthquake waves (k
=
1
,
…
,
n
E, n
E: the number of earthquake waves used for obtaining simulated structural responses), respectively. Superscripts c, off
⁡, and on
⁡ show the method of the vibration control; that is, c is the proposed cooperative control, off
⁡ is passive off (the variable damping coefficient d
1
v is always kept at its minimum value (
d
1
v
)
min
⁡), and on
⁡ is passive on (the variable damping coefficient d
1
v is always kept at its maximum value (
d
1
v
)
max
⁡), respectively. Equations (23) and (24) are inequality constraints on the active control input k
u and the axial force of the semiactive damper for the kth earthquake wave. The axial force of the damper f
d for the kth earthquake is obtained by
(25)
k
f
d
(
t
)
=
d
1
v
(
t
)
{
q
˙
1
(
t
)
-
v
˙
(
t
)
}
.Inequality constraints in (23) and (24) are, respectively, posed to avoid the excessive active control force produced by the actuator and the excessive axial load that causes the damage to the damper.The objective functionJ in (22), peak values of the active control input Peak
(
u
k
), and the axial force generated by the semiactive damper Peak
(
f
k
d
) are obtained by simulated structural responses of the cooperative control system. In the present example, design parameters g
i
≥
0, i
=
1
,
…
,
6, diagonal elements of the matrix G in (21), are optimized so that the performance index J in (22) is optimized subject to inequality constraints in (23) and (24). Three recorded earthquake waves, that is, El Centro NS (1940), Hachinohe NS (1968), and JMA Kobe NS (1995) waves, are employed to obtain simulated structural responses. All the earthquake waves are scaled so that their peak ground accelerations (PGA) become 4.0 (m
/
s
2). The maximum peak value of the active control force and the axial force of the damper in the optimal design problem (the minimization problem of the performance index J in (22) subject to inequality constraints in (23) and (24)) are defined as u
¯
=
20 (kN) and f
d
¯
=
10 (kN), respectively. The optimization of design parameters is carried out with the genetic algorithm (GA). Inequality constraints are considered as the penalty function in the process of the optimization.In the optimization with the GA, design parametersg
i, i
=
1
,
…
,
6, in (21) are optimized in the range 0
≤
g
i
≤
1
0
3, i
=
1
,
…
,
6, while ρ
=
1
0
-
3 in the matrix D
12. Values of optimal design parameters obtained with the GA-based optimization are given as follows:
(26)
g
1
=
243.23
,
g
2
=
66.16
,
g
3
=
35.33
,
g
4
=
29.99
,
g
5
=
12.99
,
g
6
=
11.08
.Results of the optimization are summarized in Figures2, 3, 4, 5, 6, 7, 8, 9, and 10 and Table 2 with some alternatives. Specifically, Figures 2, 6, and 10 show the RMS and peak values of s
i, i
=
1,2
,
3: the relative displacement between neighboring floors and a
i, i
=
1,2
,
3: the absolute acceleration of each floor of the proposed cooperative control method and those of alternatives.Table 2
Maximum values of the active control input and the axial force of the damper.
Earthquake wave
Control method
Peak(
u
) (N)
Peak(
f
d
) (N)
El Centro NS (1940)
Cooperative control (proposed)
9867.9
5433.4
Active control with passive on
9115.2
5697.1
Hachinohe NS (1968)
Cooperative control (proposed)
20000
8963.6
Active control with passive on
19988
12595
JMA Kobe NS (1995)
Cooperative control (proposed)
15254
6710.7
Active control with passive on
16091
9392.9Figure 2
RMS and peak values of the relative displacement between neighboring two floors and the absolute acceleration of each floor (El Centro NS, 1940).Figure 3
Time history of the variable damping coefficient of the semiactive damperd
1
v
(
t
) (El Centro NS, 1940).Figure 4
Time history ofs
3
(
t
)
=
q
3
(
t
)
-
q
2
(
t
) and q
¨
3
(
t
) (El Centro NS, 1940).Figure 5
The actuator forceu
(
t
)and the damper axial force f
d
(
t
) (El Centro NS, 1940).Figure 6
RMS and peak values of the relative displacement between neighboring two floors and the absolute acceleration of each floor (Hachinohe NS, 1968).Figure 7
Time history of the variable damping coefficient of the semiactive damperd
1
v
(
t
) (Hachinohe NS, 1968).Figure 8
Time history ofs
3
(
t
)
=
q
3
(
t
)
-
q
2
(
t
) and q
¨
3
(
t
) (Hachinohe NS, 1968).Figure 9
The actuator forceu
(
t
) and the damper axial force f
d
(
t
) (Hachinohe NS, 1968).Figure 10
RMS and peak values of the relative displacement between neighboring two floors and the absolute acceleration of each floor (JMA Kobe NS, 1995).Clearly, the proposed active and semiactive cooperative control achieves the higher performance on vibration suppression compared to two passive control cases, that is, the passive off and passive on cases.Moreover, two conventional active control cases and a semiactive control case are also considered for the comparison purpose. In the first case of the active control denoted by “active control with passive off,” the active control law is obtained for the benchmark building without the semiactive damper. In other words, the active control law is obtained in the passive off case; that is,d
1
v
(
t
)
=
(
d
1
v
)
min
⁡ (=0 (kNs/m)). In the second case of the active control denoted by “active control with passive on,” the active control law is obtained for the building with a passive damper whose damping coefficient is equal to (
d
1
v
)
max
⁡ (=20 (kNs/m)); that is, the active control law is obtained in the passive on case. In both cases the active control laws are obtained with LMIs in (10)–(12) for the single value of r
(
t
); that is, r
(
t
)
=
0 (the active control in the passive off case) and r
(
t
)
=
1 (the active control in the passive on case), respectively.In both cases, design parameters, diagonal elements of the matrixG in (21), are optimized so that the performance index J in (22) is minimized subject to the inequality constraints in (23) and (24) under the same setting (the range of design parameters g
i, i
=
1
,
…
,
6, and the value of ρ) by using the genetic algorithm. Obviously, values of obtained optimal design parameters in both cases, that is, active control with passive off and active control with passive on, are different from those obtained in the case of the proposed cooperative control.In the semiactive control for comparison, the method used in [18] is applied. In fact, the semiactive control method in the present cooperative control method based on the predicted control output is the same as that in [18] except for the existence of the actuator. The result is denoted by “predictive semiactive control.”We can see that the proposed cooperative control of the active and semiactive control shows the superior control performance compared to the above alternative three cases: active control with passive off, active control with passive off, and predictive semiactive control.The control performance in active control with passive on case seems to be almost the same as that of the proposed cooperative control in the sense of vibration suppression. However, we can see that the inequality constraint on the axial force of the damper given by (24) is not satisfied in the case of the Hachinohe NS (1968) earthquake disturbance from Table 2. In fact, in active control with passive on case, no feasible active control laws satisfying the constraint on the axial force of the damper are found with the GA-based optimization employed in the paper. On the other hand, the axial force of the damper in the proposed cooperative control, that is, the robust active control with the switching semiactive control based on the prediction of the structural response, clearly satisfies the inequality constraint in (24) even in the Hachinohe NS (1968) case.We can see the fact also from time history responses of the actuator forceu
(
t
) and the axial force of the damper for each earthquake disturbance depicted in Figures 5, 9, and 13. From Figure 9, the inequality constraint on the axial force of the damper is violated not only in active control with passive on case (shown in Table 2) but also in predictive semiactive control case in the Hachinohe NS earthquake disturbance.To clarify the advantage of the present cooperative control approach, time history responses of the relative displacements
3
(
t
)
=
q
3
(
t
)
-
q
2
(
t
) and the absolute acceleration q
¨
3
(
t
) for each earthquake waves are shown in Figures 4, 8, and 12 with some alternatives. Time history responses of the variable damping coefficient d
1
v
(
t
) are also shown in Figures 3, 7, and 11. We can see that the best control performance is achieved in the proposed cooperative control by using the capability of the variable damping property of the semiactive damper among all the alternatives.Figure 11
Time history of the variable damping coefficient of the semiactive damperd
1
v
(
t
) (JMA Kobe NS, 1995).Figure 12
Time history ofs
3
(
t
)
=
q
3
(
t
)
-
q
2
(
t
) and q
¨
3
(
t
) (JMA Kobe NS, 1995).Figure 13
The actuator forceu
(
t
) and the damper axial force f
d
(
t
) (JMA Kobe NS, 1995).From the above results, we can conclude that the proposed cooperative control method of the robust active control and the switching semiactive control is effective because of its high control performance that cannot be achieved by conventional passive and active control methods. Moreover, because of the capability of the cooperative control, the proposed control approach can be a new framework for the vibration control that is comparable to conventional and established passive and active control methodologies.


## 5. Summary and Future Research Subjects

The active and semiactive cooperative control scheme for vibration control of mechanical and structural systems has been considered in the present study. The control object was assumed to have both of the actuator and the semiactive control device. As the semiactive control device the variable damper that is now available in practice [6, 7] is employed in the paper. The robust active control law with the switching semiactive control based on the prediction of the controlled output was proposed as the cooperative control method. The design example of the 3-dof benchmark building showed that, with the proposed cooperative control methodology, the superior control performance on vibration suppression compared to the conventional active control method was achieved while satisfying inequality constraints that could not be satisfied by the conventional method.As described above, the framework of the cooperative active and semiactive control potentially has the possibility to achieve the higher control performance compared with that of conventional vibration control methods.Further studies for the development of the design methods of the cooperative active and semiactive control, for example, the cooperative control with other semiactive control devices including the variable stiffness devices [8–10], are the future research subject.



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*Source: 518276-2014-05-28.xml*