**4.1. Design of IBBM model for the proposed force-sensorless damping control method**

**Remark 3** (*inverse model selection*). Based on the advantages of the direct modeling method for the MR fluid damper using the black-box model, the inverse black-box model has been derived as the damping force controller. The BBM optimized by using neural network technique has been used to set initial parameters for the IBBM controller. The proposed inverse model estimates current levels should be applied to the MR fluid damper to create desired damping forces.

**Remark 4** (*force-sensorless control solution and model optimization*). The proposed control method is designed for damping systems without using any force sensor. Hence, the optimized BBM is integrated in the control system as a virtual force sensor to estimate the generated damping force. This estimated force with the desired force are fed back to the IBBM controller to compute a control signal (current signal) to supply the MR fluid damper and, subsequently, to perform a closed-loop force-sensorless controller. In order to improve the control performance using the inverse model, the self-learning mechanism based on

neural network technique is also integrated into the IBBM of which the parameters are continuously adjusted during the damping control process with respect to the control error minimization.

**Figure 22.** Structure of proposed force-sensorless damping control system based on BBM and IBBM.

Consequently, the force-sensorless control system is suggested as in Fig. 22. As seen in this figure, the proposed IBBM model contains two parts. The first part is a neural-fuzzy inference (NFI\* ), which was derived from NFI system of the optimized BBM, to estimate the damping force (*u*) caused by the damper rod displacement/velocity. The second part is an inverse scheduling gain fuzzy inference (ISGFI) which was developed from SGFI system of the optimized BBM. The ISGFI selects the current (*IMR\_est*) level needed to supply for the MR fluid damper for obtaining the damping force level (*k*). This damping force level can be computed from the damping force (*fMR\_est*) estimated by using the BBM and the estimated damping force *(u)* caused by the damper rod displacement/velocity (see Fig. 22) as:

$$k = \begin{cases} f\_{\text{MR}\_{-\text{est}}} / u & \text{if } u \neq 0 \\ 0 & \text{if } u = 0 \end{cases} \tag{24}$$

For improving the IBBM control accuracy, an error function (*E\** ) was derived from the difference between the damping force (*fMR\_est*) measured by the BBM sensor and the desired force (*fMR\_ref*). Therefore, the error function is defined as following equation:

$$E^\* = 0.5(f\_{\text{MR\\_est}} - f\_{\text{MR\\_ref}})^2 \tag{25}$$

#### *4.1.1. Neural-Fuzzy inference (NFI\* )*

414 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges

minimization.

inference (NFI\*

neural network technique is also integrated into the IBBM of which the parameters are continuously adjusted during the damping control process with respect to the control error

**Figure 22.** Structure of proposed force-sensorless damping control system based on BBM and IBBM.

Consequently, the force-sensorless control system is suggested as in Fig. 22. As seen in this figure, the proposed IBBM model contains two parts. The first part is a neural-fuzzy

damping force (*u*) caused by the damper rod displacement/velocity. The second part is an inverse scheduling gain fuzzy inference (ISGFI) which was developed from SGFI system of the optimized BBM. The ISGFI selects the current (*IMR\_est*) level needed to supply for the MR fluid damper for obtaining the damping force level (*k*). This damping force level can be computed from the damping force (*fMR\_est*) estimated by using the BBM and the estimated

> \_ / 0 0 0 *MR est f u if u <sup>k</sup>*

(24)

\_ \_ 0.5( *<sup>M</sup>* ) *R est MR ref Ef f* = − (25)

) was derived from the

 <sup>≠</sup> <sup>=</sup> =

difference between the damping force (*fMR\_est*) measured by the BBM sensor and the desired

\* 2

*if u*

damping force *(u)* caused by the damper rod displacement/velocity (see Fig. 22) as:

For improving the IBBM control accuracy, an error function (*E\**

force (*fMR\_ref*). Therefore, the error function is defined as following equation:

), which was derived from NFI system of the optimized BBM, to estimate the

The NFI in the optimized BBM was used to construct the inverse model. Hence, the structure of the NFI*\** in the IBBM is the same as the descriptions in section 3.2.1a except the initial MFs of the fuzzy inputs/output (Fig. 13). In the IBBM, these initial MFs of the NFI*\** are set as the optimized MFs of the NFI in the BBM which were shown in Fig. 15 of section 3.2.2.

Furthermore in the IBBM, the decisive factors of the NFI*\** input MFs, *aj*, *bj*, and the weights of NFI*\** output, *wj*, are automatically online-trained by using neural network technique during the damping force control process. Therefore, these affecting factors are updated as follows

$$\begin{aligned} \left. a\_{ji} \right|\_{t+1} &= a\_{ji} \Big|\_{t} - \eta\_a^\* \frac{\partial E^\*}{\partial a\_{ji}} \Big|\_{t} \\ \left. b\_{ji} \right|\_{t+1} &= b\_{ji} \Big|\_{t} - \eta\_b^\* \frac{\partial E^\*}{\partial b\_{ji}} \Big|\_{t} \\ \left. \left. \pi v\_k \right|\_{t+1} &= \left. w\_k \right|\_{t} - \eta\_w^\* \frac{\partial E^\*}{\partial w\_k} \Big|\_{t} \right| \end{aligned} \tag{26}$$

where and are the learning rate which determine the speed of learning;

*E\** is the error function defined by (25).

The factor \* *k E w* ∂ ∂ in equation (25) can be calculated as

$$\frac{\partial E}{\partial w\_k} = \frac{\partial E}{\partial f\_{\text{MR\\_est}}} \frac{\partial f\_{\text{MR\\_est}}}{\partial I\_{\text{MR\\_est}}} \frac{\partial I\_{\text{MR\\_est}}}{\partial k} \frac{\partial k}{\partial u} \frac{\partial u}{\partial w\_k} \tag{27}$$

where:

$$\frac{\partial E^{\*}}{\partial f\_{MR\\_est}} = e^{\*}\left(t\right) = f\_{MR\\_est}\left(t\right) - f\_{MR\\_ref}\left(t\right) \tag{28}$$

$$\frac{\partial f\_{\text{MR\\_est}}}{\partial I\_{\text{MR\\_est}}} = \frac{\delta f\_{\text{MR\\_est}}}{\delta I\_{\text{MR\\_est}}} \bigg|\_{t} \tag{29}$$

$$\left. \frac{\partial I\_{MR\\_est}}{\partial k} = \frac{\delta I\_{MR\\_est}}{\delta k} \right|\_{t} \tag{30}$$

$$\frac{\partial k\left(t\right)}{\partial u\left(t\right)} = -\frac{f\_{\text{MR\\_est}}}{u^2} \tag{31}$$

$$\frac{\partial \boldsymbol{u}}{\partial \boldsymbol{w}\_{k}} = \frac{\mu \left(\boldsymbol{w}\_{k}\right)}{\left(\sum\_{l=1}^{M} \mu \left(\boldsymbol{w}\_{l}\right)\right)}\tag{32}$$

The next factors \* *ji E a* ∂ <sup>∂</sup> in (26) can be computed by:

$$\frac{\partial E^\*}{\partial a\_{ji}} = \frac{\partial E^\*}{\partial u} \frac{\partial u}{\partial \mu\_j(\mathbf{x}\_i)} \frac{\partial \mu\_j(\mathbf{x}\_i)}{\partial a\_{ji}} \tag{33}$$

where:

\* *E u* ∂ ∂ is calculated by using equations from (28) to (31).

$$\frac{\partial u}{\partial \mu\_j(\mathbf{x}\_i)} = \frac{\partial u}{\partial \mu(w\_k)} = \frac{\sum\_{l=1}^{M} \mu(w\_l)(w\_k - w\_l)}{\left(\sum\_{l=1}^{M} \mu(w\_l)\right)^2} \tag{34}$$

$$\frac{\partial \mu\_j(\mathbf{x}\_i)}{\partial a\_{ji}} = \text{sign}(\mathbf{x}\_i - a\_{ji}) \frac{\mathbf{2}}{b\_{ji}} \tag{35}$$

The final factor \* *ji E b* ∂ ∂ in (26) can be found by:

$$\frac{\partial E^\*}{\partial \boldsymbol{\theta}\_{ji}} = \frac{\partial E^\*}{\partial \boldsymbol{u}} \frac{\partial \boldsymbol{u}}{\partial \boldsymbol{\mu}\_j(\boldsymbol{x}\_i)} \frac{\partial \boldsymbol{\mu}\_j(\boldsymbol{x}\_i)}{\partial \boldsymbol{\theta}\_{ji}} \tag{36}$$

where:

$$\frac{\partial E}{\partial u} \text{ and } \frac{\partial u}{\partial \mu\_j(\mathbf{x}\_i)} \text{ are calculated by using equations from (28) to (31), and (34), respectively.}$$

$$\frac{\partial \mu\_j \left( x\_i \right)}{\partial b\_{ji}} = \frac{2 \left| x\_i - a\_{ji} \right|}{\left| b\_{ji} \right|^2} \tag{37}$$

By using the above self-learning algorithm (from (26) to (37)), the NFI*\** can work more precisely in estimating the damping force (*u*) with respect to the applied displacement/velocity.

### *4.1.2. Inverse scheduling gain fuzzy inference (ISGFI)*

416 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges

<sup>∂</sup> in (26) can be computed by:

( ) ( )

 μ

*x*

∂μ

*u u x w*

*ji k M*

∂ is calculated by using equations from (28) to (31).

μ

∂ in (26) can be found by:

The next factors

The final factor

 and ( ) *j i u* μ *x* ∂

displacement/velocity.

where:

\* *E u* ∂ ∂

\*

*ji E b* ∂

where:

\* *E u* ∂

\*

*ji E a* ∂

( )

*k*

1

μ

 

( ) ( ) \* \* *j i ji j i ji EE u x au a x*

1

μ

*l*

( ) <sup>2</sup> ( ) *j i*

*sign x a a b*

( ) ( ) \* \* *j i ji j i ji EE u x bu b x*

∂ are calculated by using equations from (28) to (31), and (34), respectively.

2

μ

( )

∂μ

By using the above self-learning algorithm (from (26) to (37)), the NFI*\**

2 *j i i ji*

*x x a*

*ji ji*

precisely in estimating the damping force (*u*) with respect to the applied

*b b*

<sup>−</sup> ∂ ∂ = = ∂ ∂

=

*M*

μ

*l*

=

*M*

<sup>∂</sup> <sup>=</sup> <sup>∂</sup>

μ

*u w*

*k*

*w*

( )

*w*

*l*

μ

( )( )

*ww w*

*lkl*

2

= − <sup>∂</sup> (35)

<sup>∂</sup> ∂∂ ∂ <sup>=</sup> ∂∂ ∂ <sup>∂</sup> (36)

<sup>−</sup> <sup>=</sup> <sup>∂</sup> (37)

( )

*w*

*l*

1

μ

 

*l*

=

*i ji ji ji*

μ

<sup>∂</sup> ∂∂ ∂ <sup>=</sup> ∂∂ ∂ <sup>∂</sup> (33)

(32)

(34)

can work more

The ISGFI system, designed with a single input (*k*) and a single output (*IMR\_est*), works as an intelligent switch to select the current (*IMR\_est*) level needed to supply for the MR fluid damper with respect to the damping force level (*k*). This fuzzy system was obtained from the SGFI mechanism of the BBM design (section 3.2.1) in which the SGFI input/output became the ISGFI output/input. Consequently, the input/output MFs and rule table of the ISGFI system were set as in Fig. 23 and Table 6, respectively.

**Figure 23.** MFs of the ISGFI inputs and output.


**Table 6.** Rules table for the inverse scheduling gain inference of the inverse black box model

#### **4.2. Verifications**

#### *4.2.1. Checking the control ability of a damping system using the designed IBBM*

As the descriptions for the proposed force-sensorless control method, the designed IBBM controller was firstly examined by simulations before its application to the real-time damping control. Therefore, a simulating scheme for the IBBM controller validation was built in Simulink/MATLAB as in Fig. 24. As seen in this figure, the validating damping control system contains three main blocks. The two blocks labeled as 'BBM1' and 'BBM2' are similar and represent for the optimized BBM model which was obtained in section 3.2. These blocks then function as the actual MR fluid dampers. The remained block is the IBBM controller which was designed from section 4.1.

**Figure 24.** Simulation program for testing the IBBM controller.

The goal of the validation process is using the IBBM controller to control the second damper model, BBM2, to create the damping force to follow the reference force given from the first damper model, BBM1. Therefore, a displacement/velocity signal was generated and input into both the two damper models and the force controller. A current command signal was applied to the first damper model, BBM1. The output of this model, called the first simulated damping force, was used as a reference force signal for the damping system based on the second damper model, BBM2, and the IBBM controller. Hence, corresponding to a force command sent from the BBM1, the IBBM generated a simulated current command to control the damper model BBM2. This simulated current was then fed into the BBM2 together with the applied displacement/velocity to produce the second simulated damping force. As a result, the validation process carried out the comparison between the reference current command and simulated current command obtained from the IBBM, and the comparison between the first and second simulated damping forces.

The simulation results are displayed in Fig. 25. The results show that almost not only the simulated current command coincided with the reference current command, but also the simulated damping force of the BBM2 coincided with the simulated damping force obtained from the BBM1. As the result, it points out that the damping system using the self-learning IBBM controller with the optimized BBM virtual sensor can control accurately the MR fluid damper for a good tracking force performance. In the next section, experiments on the test rig TR02 have been carried out in order to verify the real-time control ability of the proposed force-sensorless damping control method.

controller which was designed from section 4.1.

**Figure 24.** Simulation program for testing the IBBM controller.

comparison between the first and second simulated damping forces.

force-sensorless damping control method.

control system contains three main blocks. The two blocks labeled as 'BBM1' and 'BBM2' are similar and represent for the optimized BBM model which was obtained in section 3.2. These blocks then function as the actual MR fluid dampers. The remained block is the IBBM

The goal of the validation process is using the IBBM controller to control the second damper model, BBM2, to create the damping force to follow the reference force given from the first damper model, BBM1. Therefore, a displacement/velocity signal was generated and input into both the two damper models and the force controller. A current command signal was applied to the first damper model, BBM1. The output of this model, called the first simulated damping force, was used as a reference force signal for the damping system based on the second damper model, BBM2, and the IBBM controller. Hence, corresponding to a force command sent from the BBM1, the IBBM generated a simulated current command to control the damper model BBM2. This simulated current was then fed into the BBM2 together with the applied displacement/velocity to produce the second simulated damping force. As a result, the validation process carried out the comparison between the reference current command and simulated current command obtained from the IBBM, and the

The simulation results are displayed in Fig. 25. The results show that almost not only the simulated current command coincided with the reference current command, but also the simulated damping force of the BBM2 coincided with the simulated damping force obtained from the BBM1. As the result, it points out that the damping system using the self-learning IBBM controller with the optimized BBM virtual sensor can control accurately the MR fluid damper for a good tracking force performance. In the next section, experiments on the test rig TR02 have been carried out in order to verify the real-time control ability of the proposed

**Figure 25.** Comparison of simulation results between estimated current command with given current command corresponding to the damping force output.

#### *4.2.2. Verification of the force-sensorless control system based on the BBM and IBBM*

Based on the advanced characteristics of the designed BBM and IBBM models which were proven to be effective in the previous sections, the proposed force-sensorless control methodology was applied to the system TR02 for the real-time damping force control. Here, a harmonic excitation was applied to the damping system through the pneumatic cylinder controlled by the proportional valve. Meanwhile, a desired force performance was given in order to validate the damping control ability. Therefore, the control signal (voltage) for the pneumatic valve and the reference damping force were given as equations (38) and (39), respectively.

$$\text{ValveControlSignal} = \mathbb{S}\sin\left(\pi t + \pi / 2\right) \left(V\right) \tag{38}$$

$$\text{ReferencesForce} = A \sin(\pi t) \text{(N)}\tag{39}$$

here: the amplitude *A* of the reference force signal was set at two values: 500N and 1000N to perform two testing cases. The working load condition was set with a 9kg of load.

The control program for the damping system TR02 was built in Simulink with the real-time toolbox of MATLAB as shown in Fig. 26. The program contained two main blocks in which the first one is the motion generating block to generate the system displacement while the second one is the damping control system to ensure that the system could track the desired damping force by only using the IBBM controller and BBM sensor, without using the mechanical force sensor (load cell). In this control system, the optimized BBM estimated the damping force and fed back to the self-learning IBBM controller to create the current control signal for the MR fluid damper. In order to evaluate the control performance, the load cell was used separately with the controller to measure the actual damping force.

Experiments on the TR02 controlled by the system in Fig. 26 had been done under the testing conditions as shown in (38) and (39). Consequently, the validating results for the force-sensorless damping control system based on the BBM and IBBM corresponding to the two testing cases, 500N and 1000N of the desire force amplitude, are shown in figures 27a and 27b, respectively. The figures show that the system using the IBBM controller with the virtual sensor BBM tracked the desired damping force well. From the results, it strongly indicates that the damping force can be completely controllable by this proposed control methodology.

**Figure 26.** Simulink program for damping force control based on IBBM controller and BBM sensor.

methodology.

The control program for the damping system TR02 was built in Simulink with the real-time toolbox of MATLAB as shown in Fig. 26. The program contained two main blocks in which the first one is the motion generating block to generate the system displacement while the second one is the damping control system to ensure that the system could track the desired damping force by only using the IBBM controller and BBM sensor, without using the mechanical force sensor (load cell). In this control system, the optimized BBM estimated the damping force and fed back to the self-learning IBBM controller to create the current control signal for the MR fluid damper. In order to evaluate the control performance, the load cell

Experiments on the TR02 controlled by the system in Fig. 26 had been done under the testing conditions as shown in (38) and (39). Consequently, the validating results for the force-sensorless damping control system based on the BBM and IBBM corresponding to the two testing cases, 500N and 1000N of the desire force amplitude, are shown in figures 27a and 27b, respectively. The figures show that the system using the IBBM controller with the virtual sensor BBM tracked the desired damping force well. From the results, it strongly indicates that the damping force can be completely controllable by this proposed control

**Figure 26.** Simulink program for damping force control based on IBBM controller and BBM sensor.

was used separately with the controller to measure the actual damping force.

(a) Case 1 (Working load: 9kg; Vibrating control signal: (5*V*, 0.5*Hz*, 90°); Reference damping force: (500*N*, 0.5*Hz*, 0°))

(b) Case 2 (Working load: 9kg; Vibrating control signal: (5*V*, 0.5*Hz*, 90°); Reference damping force: (1000*N*, 0.5*Hz*, 0°))

**Figure 27.** Damping force control performance using IBBM controller and BBM sensor.
