A new fuzzy sliding mode controller for vibration control systems using integrated-structure smart dampers

In order to control ISSmDCSs, the I-SmD shown in figure 1 needs to be established. As mentioned in Introduction, it is quite challenging to develop an accurate mathematical model using the parametric models for SmDs and ISSs in the presence of UAD [11, 18–22]. Therefore, non-parametric models using ANFIS have been widely utilized for modeling of uncertain control systems [7, 17, 27–33]. In this work, the I-SmDs typed ANFIS-I-SmDs is built using the ANFIS. The ANFIS-I-SmD expresses the relationship between dynamic responses and applied voltage or current value of the SmD (or the ISSs). Building an ANFIS-I-SmD is performed via two steps. The first step is to measure dynamic responses of the damper to build an IDS and the second one is to identify the damper via the IDS and ANFIS training algorithm. Figure 3 shows the IDS for building I-MRD, where, ${d}_{{\rm{r}}{\rm{e}}}$ is the relative piston-cylinder displacement of the MRD; $a$ is the acceleration of the piston; ${f}_{{\rm{M}}{\rm{R}}}\equiv {f}_{{\rm{s}}{\rm{d}}}$ is damping force; and $I$ is current supporting the MRD. In this example, for the I-MRD, $({d}_{{\rm{r}}{\rm{e}}},a,{f}_{{\rm{s}}{\rm{d}}})$ is input, while $I$ is the output signal which needs to be estimated to generate the desired damping force, ${f}_{{\rm{s}}{\rm{d}}}.$ The ANFIS is used to identify and hence the result of this work can be indicated by I = ANFIS-I-SmD (d_re, a, f_sd).

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Generally, the IDS of input–output samples $({\bar{{\bf{x}}}}_{i},{y}_{i}),i=\mathrm{1...}P,$ ${\bar{{\bf{x}}}}_{i}={[{x}_{i1},\mathrm{...},{x}_{i{n}_{0}}]}^{{\rm{T}}}\in {\Re }^{{n}_{0}}$ and ${y}_{i}\in {\Re }^{1},$ expresses an unknown mapping f : X → Y from the input data space X to the output data space Y. The result of the training ANFIS based on the ISD is an approximation of f : X → Y by the ANFIS. To train the ANFIS, firstly, clustering the IDS needs to be done. This is acknowledged as an important step to analyze the IDS and build a CDS. Based on this CDS, fuzzy sets are created to take part in the fuzzy inference system as a framework to which establishing fuzzy laws and defuzzification are all organized, associated and operated. In the previous study [27], to identify dynamic characteristics of the SmDs, clustering in a data space ${X}\times {Y}$ via an algorithm for establishing ANFIS named ENFS was presented. Thus, the separating data samples in data spaces X and Y was carried out distinctly with a mutual result reference, step by step. This way, however, results in difficulties related to deploying a fuzzy clustering strategy along with the high calculating cost. To overcome this, in this work an algorithm named im-ENFS is proposed. First, a jointed input–output data space (JDS) deriving from the IDS is presented, and a kernel-based fuzzy clustering process using a new cluster-centroid update law is then carried out in the JDS. Thus, a close interaction between X and Y is established and always upholden during the clustering process to increase the accuracy rate of the ANFIS. Based on the created CDS and the im-ENFS, the ANFIS is generated. Subsequently, the ANFIS works as the ANFIS-D-SmD or ANFIS-I-SmD.

Consider the ISSmDCS shown in figure 1 which is subjected to uncertainty coming from the model error. The dynamic response of the ISSmDCS can be expressed as follows:

$$\begin{eqnarray}&&\left\{\begin{array}{r}{x}^{(n)}=f(x,\dot{x},\mathrm{...},{x}^{(n-1)},t)+g(x,\dot{x},\mathrm{...},{x}^{(n-1)},t)\\ \,\times \,u(t) & \\ y={\bf{x}}(t),\end{array}\right.\end{eqnarray} \tag{ 1 }$$

where, f(·), g(·) are unknown functions; $u(t)$ is control signal; $y(t)$ is the output; ${\bf{x}}(t)={[{x}_{1},{x}_{2},\mathrm{...},{x}_{n}]}^{{\rm{T}}}\ ={[x,\dot{x},\mathrm{...},{x}^{(n-1)}]}^{{\rm{T}}}\in {R}^{n}$ is the state variable vector. Figure 4 shows the operating principle of the proposed controller. Based on the plant dynamic response depicted by ${\bf{x}}(t)$ and the desired control force $u(t)$ estimated by the estimator FSMC, the ANFIS-I-SmD specifies the corresponding current $I(t)$ (or voltage U) so that the SmD can generate the damping force ${f}_{S}(t)\to u(t).$ The aim of the FSMC is to specify the control law $u(t)$ in presence of uncertainty such that ${\bf{x}}(t)$ tracks stably the desired reference state ${{\bf{x}}}_{{\rm{d}}}(t)$.

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Let ${\bf{e}}(t)$ is the error vector defined as follows.

$$\begin{eqnarray}&&{\bf{e}}(t)={\bf{x}}(t)-{{\bf{x}}}_{{\rm{d}}}(t)={[e,\dot{e},\mathrm{...},{e}^{(n-1)}]}^{{\rm{T}}}\in {R}^{n}.\end{eqnarray} \tag{ 2 }$$

A sliding surface $S({\bf{x}})$ is then defined using the ${\bf{e}}(t)$ as follows.

$$\begin{eqnarray}&&S({\bf{x}})={{\bf{a}}}^{{\rm{T}}}{\bf{e}}(t)\end{eqnarray} \tag{ 3 }$$

where, ${\bf{a}}={[{a}_{1},\mathrm{...},{a}_{n-1}]}^{{\rm{T}}}$ is the vector of Hurwitizian polynomial, to which, poles locate in the left half of the complex co-ordinate plane. By using $S({\bf{x}}),$ a Lyapunov candidate function is defined as ${V}_{0}({\bf{x}})=S{({\bf{x}})}^{2}/2.$ With reference to ${V}_{0}(0)=0$ and ${V}_{0}({\bf{x}})\geqslant 0\,\forall {\bf{x}}(t)| t\gt 0,$ it can infer that if the control law $u(t)$ is used so that $\dot{S}({\bf{x}})={\rm{d}}S({\bf{x}})/{\rm{d}}t=-\rho \mathrm{sgn}(S({\bf{x}})),$ where $\rho$ is the positive coefficient, then $S(x)\to 0$ is a stable Lyaponov process. Based on this relation together with the equations of (3), (2) and (1), the feedback control signal such that ${\bf{e}}(t)\to 0$ or ${\bf{x}}\to {{\bf{x}}}_{{\rm{d}}}$when $t\to \infty$ can be inferred as below

$$\begin{eqnarray}&&u(t)=g{({\bf{x}},t)}^{-1}\left(-\displaystyle \sum _{i=1}^{n-1}{a}_{i}{{\rm{e}}}^{(i)}-f({\bf{x}},t)+{x}_{{\rm{d}}}^{(n)}-h({\bf{x}},t)\right),\end{eqnarray} \tag{ 4 }$$

where,

$$\begin{eqnarray}&&h({\bf{x}},t)=\rho \,\mathrm{sgn}(S({\bf{x}})).\end{eqnarray} \tag{ 5 }$$

In equation (4), the functions $g({\bf{x}},t),$ $f({\bf{x}},t)$ and $h({\bf{x}},t)$ need to be determined to specify the control law. For $g({\bf{x}},t)$ and $f({\bf{x}},t),$ approximation functions $\hat{g}({\bf{x}},t)$ and $\hat{f}({\bf{x}},t)$ via the FL presented in [50] are adopted. For estimating $h({\bf{x}},t),$ the calculating time, or time delay should be paid attention. Therefore, more attention is needed to estimate $h({\bf{x}},t).$ In [41, 42, 50], in order to avoid the chattering phenomenon, the fuzzy gains were used. In [45], the PI controller was established to keep the state in a limited boundary layer. However, in these previous works the increment of the calculating cost related to the fuzzy structures was occurred as a common difficulty to solve. To resolve this problem, in this work an adaptive gain ${\rho }_{{\rm{a}}{\rm{d}}},$ and a desired boundary layer of $S({\bf{x}})$ are used as follows (refer to theorem 1 in section 4.2 for the details):

$$\begin{eqnarray}&&h({\bf{x}},t)={\rho }_{{\rm{a}}{\rm{d}}}\,{\rm{sat}}(S({\bf{x}})/{\rm{\Phi }}),\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{\rho }_{{\rm{a}}{\rm{d}}}={k}_{1}(t)(1-\exp (-{k}_{2}| S({\bf{x}})| )),\end{eqnarray} \tag{ 7 }$$

where, ${k}_{2}$ is the adaptive positive coefficient chosen by designer and ${k}_{1}={k}_{1}(t)$ is the adaptive coefficient. The update law of ${k}_{1}(t)$ is to be performed based on the convergence status of $S({\bf{x}})\to 0,$ which is addressed in theorem 1. It is noted that in order to avoid the chattering phenomenon, the saturation function ${\rm{sat}}(\cdot )$ defined in [46] with the desired boundary layer Φ of $S({\bf{x}})$ is used instead of the signum function $\mathrm{sgn}(\cdot )$ in (5);

$$\begin{eqnarray}{\rm{sat}}({S}/{\rm{\Phi }})=\left\{\begin{array}{lll}\mathrm{sgn}({S}/{\rm{\Phi }}) & {\rm{i}}{\rm{f}} & | {S}/{\rm{\Phi }}| \gt 1\\ {S}/{\rm{\Phi }} & {\rm{i}}{\rm{f}} & | {S}/{\rm{\Phi }}| \leqslant 1.\end{array}\right.\end{eqnarray} \tag{ 8 }$$

Remark 1. Parameter ${\rho }_{{\rm{a}}{\rm{d}}}$ $(0\leqslant {\rho }_{{\rm{a}}{\rm{d}}}\leqslant {k}_{1}(t))$ always varies exponentially compared with $| S({\bf{x}})| .$ As a result, if $| S({\bf{x}})| \to 0$ then ${\rho }_{{\rm{a}}{\rm{d}}}\to 0$ rapidly to prevent from the chattering phenomenon; inversely, if $| S({\bf{x}})|$ increases then ${\rho }_{{\rm{a}}{\rm{d}}}$ increases exponentially to keep the stable status of the system.

In order to establish a close interaction between X and Y, and increase the accuracy rate of the ANFIS during the clustering process, in this work a new clustering process is performed using a new data space called JDS $({\Re }^{{n}_{0}+1})$ derived from the IDS $({\Re }^{{n}_{0}}).$ The JDS is built as follows. Firstly, the output signals of the IDS are normalized:

$$\begin{eqnarray}&&{\tilde{y}}_{i}={y}_{i}/\mathop{\max }\limits_{P}| {y}_{j}| .\end{eqnarray} \tag{ 9 }$$

Normalized values ${\tilde{y}}_{i},i=\mathrm{1...}P,$ are then added to the IDS in a new column. The JDS has P rows according to that of the IDS and $\bar{n}={n}_{0}+1$ columns, in which the (n₀ + 1)th one is the added column. Thus, each input data sample belongs to ${\Re }^{{n}_{0}+1}$ while the corresponding output data sample belongs to ${\Re }^{1}.$ Based on a defined objective function and a clustering algorithm, the clustering process is then carried out in the JDS. As a result, a CDS, for an ANFIS is established in next subsection.

The result of the fuzzy clustering process is the CDS with C cluster centroids ${\bar{{\bf{x}}}}_{1}^{0},\mathrm{...},{\bar{{\bf{x}}}}_{C}^{0}$ in the JDS such that the following objective function is minimized [51]:

$$\begin{eqnarray}&&{J}_{{\rm{K}}{\rm{F}}{\rm{C}}{\rm{M}}}(U,{\bar{{\bf{x}}}}^{0})=\displaystyle \sum _{i=1}^{{\rm{C}}}\displaystyle \sum _{j=1}^{P}\,{\mu }_{ij}^{{m}_{0}}{\parallel \varphi ({\bar{{\bf{x}}}}_{j})-\varphi ({\bar{{\bf{x}}}}_{i}^{0})\parallel }^{2}\end{eqnarray} \tag{ 10 }$$

subjected to

$$\begin{eqnarray}\begin{array}{cc}{\mu }_{ij}\in [\mathrm{0,1}]\forall i,j; & \displaystyle \sum _{i=1}^{C}\,{\mu }_{ij}=1\,\forall j\end{array},\end{eqnarray} \tag{ 11 }$$

where, ${\bar{{\bf{x}}}}_{j}=[{x}_{j1},\mathrm{...},{x}_{j\bar{n}}]\in {\Re }^{{n}_{0}+1},j=\mathrm{1...}P,$ is the jth data sample, and ${\bar{{\bf{x}}}}_{i}^{0}=[{x}_{i1}^{0},\mathrm{...},{x}_{i\,\bar{n}}^{0}]\in {\Re }^{{n}_{0}+1},$ $i=\mathrm{1...}C,$ is the ith cluster center in the JDS; ${\mu }_{ij}\in [\mathrm{0,1}]$ denotes membership degree of the jth data sample belonging to the ith cluster; ${m}_{0}\gt 1$ is the fuzzy factor; ${\parallel \varphi ({\bar{{\bf{x}}}}_{j})-\varphi ({\bar{{\bf{x}}}}_{i}^{0})\parallel }^{2}$ denotes the squared distance between ${\bar{{\bf{x}}}}_{j}$ and ${\bar{{\bf{x}}}}_{i}^{0}$ calculated in the kernel space. By choosing Gaussian kernel function, ${\parallel \varphi ({\bar{{\bf{x}}}}_{j})-\varphi ({\bar{{\bf{x}}}}_{i}^{0})\parallel }^{2}$ and ${J}_{{\rm{K}}{\rm{F}}{\rm{C}}{\rm{M}}}(U,{\bar{{\bf{x}}}}^{0}),$ the following equations are obtained.

$$\begin{eqnarray}&&{\parallel \varphi ({\bar{{\bf{x}}}}_{j})-\varphi ({\bar{{\bf{x}}}}_{i}^{0})\parallel }^{2}=2(1-\exp (-{\parallel {\bar{{\bf{x}}}}_{j}-{\bar{{\bf{x}}}}_{i}^{0}\parallel }^{2}/{\sigma }^{2})),\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&{J}_{{\rm{K}}{\rm{F}}{\rm{C}}{\rm{M}}}(U,{\bar{{\bf{x}}}}^{0})=2\displaystyle \sum _{i=1}^{{\rm{C}}}\displaystyle \sum _{j=1}^{P}\,{\mu }_{ij}^{{m}_{0}}(1-\exp (-{\parallel {\bar{{\bf{x}}}}_{j}-{\bar{{\bf{x}}}}_{i}^{0}\parallel }^{2}/{\sigma }^{2})).\end{eqnarray} \tag{ 13 }$$

From (13), the optimal centers are the solution of the following equation:

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial {\bar{{\bf{x}}}}_{i}^{0}}{J}_{{\rm{K}}{\rm{F}}{\rm{C}}{\rm{M}}}(U,{\bar{{\bf{x}}}}_{i}^{0})\\ &&\,=\displaystyle \frac{-4}{{\sigma }^{2}}\displaystyle \sum _{j=1}^{P}\,{\mu }_{ij}^{{m}_{0}}({\bar{{\bf{x}}}}_{j}-{\bar{{\bf{x}}}}_{i}^{0})\exp (-{\parallel {\bar{{\bf{x}}}}_{j}-{\bar{{\bf{x}}}}_{i}^{0}\parallel }^{2}/{\sigma }^{2})=0.\end{eqnarray} \tag{ 14 }$$

As a result, the iterative function used to update ${\bar{{\bf{x}}}}_{i}^{0}$ can be expressed as follows:

$$\begin{eqnarray}&&{\bar{{\bf{x}}}}_{i}^{0}=\displaystyle \frac{\displaystyle {\sum }_{j=1}^{P}{{\mu }_{ij}}^{{m}_{0}}\,{\bar{{\bf{x}}}}_{j}\,K({\bar{{\bf{x}}}}_{j},{\bar{{\bf{x}}}}_{i}^{0})\,}{\displaystyle {\sum }_{j=1}^{P}{{\mu }_{ij}}^{{m}_{0}}K({\bar{{\bf{x}}}}_{j},{\bar{{\bf{x}}}}_{i}^{0})\,},i=\mathrm{1...}C.\end{eqnarray} \tag{ 15 }$$

Based on (11), (13) and the method of Lagrange multipliers, the optimal values of the memberships are identified. Thus, the following iterative function is obtained for updating:

$$\begin{eqnarray}{\mu }_{ij} & = & {\left[{\left(\displaystyle \sum _{h=1}^{C}\displaystyle \frac{1-K({\bar{{\bf{x}}}}_{j},{\bar{{\bf{x}}}}_{i}^{0})}{1-K({\bar{{\bf{x}}}}_{j},{\bar{{\bf{x}}}}_{h}^{0})}\right)}^{1/({m}_{0}-1)}\right]}^{-1},\\ & & i=\mathrm{1...}C;j=\mathrm{1...}P,\end{eqnarray} \tag{ 16 }$$

where, ${\mu }_{ij}$ expresses the membership of the jth sample belonging to the ith data cluster. The stop condition of the clustering phase is $ts\leqslant [ts],$ where index $ts$ is calculated by

$$\begin{eqnarray}&&ts=({{J}_{{\rm{K}}{\rm{F}}{\rm{C}}{\rm{M}}}}^{(r)}-{{J}_{{\rm{K}}{\rm{F}}{\rm{C}}{\rm{M}}}}^{(r-1)})/{{J}_{{\rm{K}}{\rm{F}}{\rm{C}}{\rm{M}}}}^{(r-1)}.\end{eqnarray} \tag{ 17 }$$

In the above, $[ts]$ is the required value of $ts;$ r denotes the order of loops.

Remark 2. As above mentioned, the JDS is only used in the clustering process to determine the cluster centroids belonging to ${\Re }^{{n}_{0}+1}$ as below.

$$\begin{eqnarray}&&{\bar{{\bf{x}}}}_{i}^{0}=[{x}_{i1}^{0},\mathrm{...},{x}_{i{n}_{0}}^{0},{x}_{i({n}_{0}+1)}^{0}],i=\mathrm{1...}C.\end{eqnarray} \tag{ 18 }$$

When the clustering process has accomplished, the (n₀ + 1)^th column (corresponding to ${\tilde{y}}_{i},i=\mathrm{1...}P$ (9)) of these cluster centroids is deleted and they resume to the ${\Re }^{{n}_{0}}$:

$$\begin{eqnarray}&&{\bar{{\bf{x}}}}_{i}^{0}=[{x}_{i1}^{0},\mathrm{...},{x}_{i{n}_{0}}^{0}]\in {\Re }^{{n}_{0}},i=\mathrm{1...}C.\end{eqnarray} \tag{ 19 }$$

All of the next steps will be carried out based on the cluster centroids (19) and the IDS in ${\Re }^{{n}_{0}}$.

The output value of the ANFIS is calculated via output clusters typed hyper-planes, ${w}_{k}(\cdot ),$ which is the kth hyper-plane constituted by an index vector $\bar{{\bf{a}}}.$ Firstly, by using the obtained cluster centroids $({\bar{{\bf{x}}}}_{1}^{0},\mathrm{...},{\bar{{\bf{x}}}}_{C}^{0})$ in (19), the membership of each data sample in the IDS belonging to each cluster can be determined by ${\pi }_{k}({\bar{{\bf{x}}}}_{i}),i=\mathrm{1...}P,k=\mathrm{1...}C.$ These memberships are then used to specify the hard distribution status of the data samples in each cluster deputized by cluster centroids $({\bar{{\bf{x}}}}_{1}^{0},\mathrm{...},{\bar{{\bf{x}}}}_{C}^{0}).$ Namely, the ith sample belongs to the kth cluster if the following equation is satisfied.

$$\begin{eqnarray}&&{\pi }_{k}({\bar{{\bf{x}}}}_{i})=\mathop{\max }\limits_{h=\mathrm{1...}C}({\pi }_{h}({\bar{{\bf{x}}}}_{i})),i=\mathrm{1...}P,k=\mathrm{1...}C.\end{eqnarray} \tag{ 20 }$$

Let ${t}_{k}$ be the number of data samples being hard distributed to the kth cluster. By using the well-known least mean squares method for ${t}_{k},$ the index vector $\bar{{\bf{a}}}={[{a}_{0},{a}_{1},\mathrm{...},{a}_{{n}_{0}}]}^{{\rm{T}}}={[{a}_{0},{\bf{a}}]}^{{\rm{T}}}$ of the hype-plane ${w}_{k}$ will be specified (see [27] for more detail). Thus, the value of hype-plane ${w}_{k}$ associated with data sample ${\bar{{\bf{x}}}}_{i}$ is calculated as below.

$$\begin{eqnarray}&&{w}_{k}({\bar{{\bf{x}}}}_{i})={a}_{0}+{{\bf{a}}}^{{\rm{T}}}{\bar{{\bf{x}}}}_{i}.\end{eqnarray} \tag{ 21 }$$

The result of the training process provides five layers as shown in figure 5; (i) the data (D) layer: this layer has n₀ nodes. (ii) The input cluster layer: the result of the clustering process is C clusters with cluster centroids ${\bar{{\bf{x}}}}_{i}^{0},\mathrm{...},{\bar{{\bf{x}}}}_{C}^{0},$ to which C fuzzy sets, ${A}^{1},$..., ${A}^{C}\,,$ are established. The output of this layer is the membership values of ${\bar{{\bf{x}}}}_{i},$ calculated for each direction $({x}_{i1},\mathrm{...},{x}_{i{n}_{0}}).$ Based on (16), the membership value of ${x}_{il}$ belonging to ${A}^{k}\,,$ ${A}_{l}^{k}(k=\mathrm{1...}C,l=\mathrm{1...}{n}_{0},)$ is calculated as follows:

$$\begin{eqnarray}{A}_{l}^{k}({\bar{{\bf{x}}}}_{i})\equiv {\bar{\mu }}_{ki}({x}_{il}) & = & {\left[{\left(\displaystyle \sum _{h=1}^{C}\displaystyle \frac{1-K({x}_{il},{x}_{kl}^{0})}{1-K({x}_{il},{x}_{hl}^{0})}\right)}^{1/(m-1)}\right]}^{-1}\\ & & (k=\mathrm{1...}C;i=\mathrm{1...}P,l=\mathrm{1...}{n}_{0}.)\end{eqnarray} \tag{ 22 }$$

(iii) The product layer, ∏: each node in this layer calculates the product of membership values of its inputs corresponding to all directions for each fuzzy rule:

$$\begin{eqnarray}&&{\pi }_{k}({\bar{{\bf{x}}}}_{i})=\displaystyle \prod _{l=1}^{{n}_{0}}{\bar{\mu }}_{ki}({x}_{il}),\,k=\mathrm{1...}C,i=\mathrm{1...}P.\end{eqnarray} \tag{ 23 }$$

(iv) The hype-plane (Hp) layer: this is used to specify hype-planes mentioned in section 3.3. (v) The specifying (S) layer: this one is used to specify the output of the ANFIS via defuzzification functions (DF). DF can come from the well-known methods such as the center-average method or the 'the winner takes all'. If 'the winner takes all' is used, the output signal is calculated as follows:

$$\begin{eqnarray}&&{\hat{y}}_{i}={\rm{DF}}({\bar{{\bf{x}}}}_{i})={w}_{k}({\bar{{\bf{x}}}}_{i}),i=\mathrm{1...}P,\end{eqnarray} \tag{ 24 }$$

where,

$$\begin{eqnarray}&&{\pi }_{k}({\bar{{\bf{x}}}}_{i})=\mathop{\text{arg}\,\max }\limits_{h=\mathrm{1...}C}({\pi }_{h}({\bar{{\bf{x}}}}_{i})).\end{eqnarray} \tag{ 25 }$$

Now, the building alogoritm for ANFIS can be completed by separating it into three phases: establish a training data set JDS, building a CDS and building ANFIS.

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Establish the JDS: this was mentioned in section 3.1.

Initialize: the initial index of loops, r = 1; the number of clusters $C\ll P-1;$ the initial cluster centroids are chosen randomly ${\bar{{\bf{x}}}}_{i}^{0}(r)=({x}_{i1}^{0},\mathrm{...},{x}_{i\bar{n}}^{0})\in {\Re }^{{n}_{0}+1},1\leqslant i\leqslant C;$ ${{J}_{{\rm{K}}{\rm{F}}{\rm{C}}{\rm{M}}}}^{(r)}={{\rm{\Omega }}}_{A},$ in which ${{\rm{\Omega }}}_{A}$ is any real number larger than the desired value, ${{\rm{\Omega }}}_{A}\gt [ts]$.

The im-ENFS:

• 1.  
  Calculate ${\mu }_{ij}$ based on (16);
• 2.  
  Update cluster centroids ${\bar{{\bf{x}}}}_{i}^{0}(r)=({x}_{i1}^{0},\mathrm{...},{x}_{i\bar{n}}^{0})\in {\Re }^{{n}_{0}+1},1\leqslant i\leqslant C$ using (15);
• 3.  
  Calculate the stop coefficient $ts$ via (17);

  • -  
    Either $(ts\leqslant [ts])$ or ($ts\gt [ts]$ and $r=[r]):$ go to Step 4;
  • -  
    If $ts\gt [ts]$ and $r\lt [r]:$ setup r =: r + 1 and return to Step 1;
• 4.  
  Build ANFIS as in figure 5 based on (22)–(25) with ${\bar{{\bf{x}}}}_{i}^{0}\in {\Re }^{{n}_{0}},1\leqslant i\leqslant C$ (Remark 2);
• 5.  
  Calculate root-mean-square error (RMSE) and evaluate:

  $$\begin{eqnarray}&&{\rm{R}}{\rm{M}}{\rm{S}}{\rm{E}}=\sqrt{\displaystyle \frac{\displaystyle {\sum }_{i=1}^{P}{({\hat{y}}_{i}({\bar{{\bf{x}}}}_{i})-{y}_{i})}^{2}}{P}};\end{eqnarray} \tag{ 26 }$$

  • -  
    If $RMSE\leqslant [E]:$ Stop and use the ANFIS as an optimal fuzzy system;
  • -  
    If $RMSE\gt [E]$ and $C=P-1:$ Stop (the ANFIS cannot converge to $[E]$);
  • -  
    If $RMSE\gt [E]$ and $C\lt P-1,$ then (1) C =: C + 1; (2) establish a new cluster centroid ${\bar{{\bf{x}}}}_{C}^{0}$ at the kth data sample specified by $| {\hat{y}}_{k}-{y}_{k}| =\mathop{\max }\limits_{h=\mathrm{1...}P}| {\hat{y}}_{h}-{y}_{h}| ;$ and (3) return to Step 1.

As given in equation (4), the functions $g({\bf{x}},t)$ and $f({\bf{x}},t)$ are approximated via MISO fuzzy systems; n input variables and m fuzzy laws [50]. The ith fuzzy law is written as below.

$$\begin{eqnarray}&&{R}^{(i)}\,:{\rm{IF}}\,{x}_{1}\,{\rm{is}}\,{A}_{1}^{i},{\rm{and}}\mathrm{...},\,{\rm{and}}\,{x}_{n}\,{\rm{is}}\,{A}_{n}^{i}\\ &&\,{\rm{then}}\,y\,{\rm{is}}\,{B}^{i}(i=\mathrm{1...}m),\end{eqnarray} \tag{ 27 }$$

where, ${A}_{j}^{i},j=\mathrm{1...}n,$ is the fuzzy set in the input space related to the physical parameter ${x}_{j}$ and the ith fuzzy law, while ${B}^{i}$ is the corresponding fuzzy set in the output space. Using the center-average defuzzification, the output is calculated by

$$\begin{eqnarray}&&y({\bf{x}})=\left(\displaystyle \sum _{i=1}^{m}{y}^{i}{\mu }_{{A}^{i}}({\bf{x}})\right)/\displaystyle \sum _{i=1}^{m}{\mu }_{{A}^{i}}({\bf{x}}).\end{eqnarray} \tag{ 28 }$$

In (28), ${\mu }_{{A}^{i}}({\bf{x}})$ is the value of the membership function in the input fuzzy space of ${\bf{x}}(t).$ If the product law of ${A}^{i}={A}_{1}^{i}\times \mathrm{...}\times {A}_{n}^{i}$ is used to ${\mu }_{{A}^{i}}({\bf{x}})=\displaystyle {\prod }_{j=1}^{n}{\mu }_{{A}_{j}^{i}}({x}_{j}),$ equation (28) becomes as follows.

$$\begin{eqnarray}&&y({\bf{x}})=\left(\displaystyle \sum _{i=1}^{m}{y}^{i}\displaystyle \prod _{j=1}^{n}{\mu }_{\underset{j}{\overset{i}{A}}}({x}_{j})\right)/\displaystyle \sum _{i=1}^{m}\displaystyle \prod _{j=1}^{n}{\mu }_{{A}_{j}^{i}}({x}_{j}).\end{eqnarray} \tag{ 29 }$$

Value ${y}^{i},i=\mathrm{1...}m,$ can be determined by the well-known methods via the fuzzy set in the output space, which is the singleton fuzzification. Then, equation (29) is shortly re-expressed as follows.

$$\begin{eqnarray}&&y({\bf{x}})={\varphi }^{T}\lambda ({\bf{x}}).\end{eqnarray} \tag{ 30 }$$

In the above,

$$\begin{eqnarray}\begin{array}{ccc}\varphi ={[{y}^{1},\mathrm{...},{y}^{m}]}^{{\rm{T}}} & ; & \lambda ({\bf{x}})={[{\lambda }^{1}({\bf{x}}),\mathrm{...},{\lambda }^{m}({\bf{x}})]}^{{\rm{T}}}\end{array},\end{eqnarray} \tag{ 31 }$$

$$\begin{eqnarray}&&{\lambda }^{i}({\bf{x}})=\phantom{\rule{}{1.0ex}}(\displaystyle \prod _{j=1}^{n}{\mu }_{\underset{j}{\overset{i}{A}}}({x}_{j})\phantom{\rule{}{1.0ex}})/\displaystyle \sum _{i=1}^{m}\displaystyle \prod _{j=1}^{n}{\mu }_{{A}_{j}^{i}}({x}_{j})\end{eqnarray} \tag{ 32 }$$

Consider the approximation of the functions $g({\bf{x}},t),$ $f({\bf{x}},t)$ in (4) by the fuzzy system (27) given in (30). The obtained functions, respectively, are expressed as follows:

$$\begin{eqnarray}&&{\hat{g}}_{1}({\bf{x}},{\varphi }_{g})={{\varphi }_{g}}^{{\rm{T}}}\lambda ({\bf{x}}),\hat{f}({\bf{x}},{\varphi }_{f})={{\varphi }_{f}}^{{\rm{T}}}\lambda ({\bf{x}}),\end{eqnarray} \tag{ 33 }$$

where, $\lambda ({\bf{x}}),$ ${\varphi }_{f}$ and ${\varphi }_{g}$ are specified by (31). Let ${\varphi }_{f}^{* }$ and ${\varphi }_{g}^{* }$ be the optimal vectors of ${\varphi }_{f}$ and ${\varphi }_{g}$ as follows.

$$\begin{eqnarray}&&{\varphi }_{f}^{* }=\mathop{\text{arg}\,\min }\limits_{{\varphi }_{f}\subset {\Im }_{f}}(\sup | \hat{f}({\bf{x}},{\varphi }_{f}^{* })-f({\bf{x}},t)| ),\end{eqnarray} \tag{ 34 }$$

$$\begin{eqnarray}&&{\varphi }_{g}^{* }=\mathop{\text{arg}\,\min }\limits_{{\varphi }_{g}\subset {\Im }_{g}}(\sup | {\hat{g}}_{1}({\bf{x}},{\varphi }_{g}^{* })-{g}_{1}({\bf{x}},t)| ),\end{eqnarray} \tag{ 35 }$$

where, ${\Im }_{f}=\{{\varphi }_{f}| \parallel {\varphi }_{f}\parallel \leqslant {M}_{f}\},$ ${\Im }_{g}=\{{\varphi }_{g}| \parallel {\varphi }_{g}\parallel \leqslant {M}_{g}\};$ ${M}_{f}$ and ${M}_{g}$ are design parameters. Let ${\rm{\Omega }}({\bf{x}},{\varphi }_{f},{\varphi }_{g})$ be a function defined by

$$\begin{eqnarray}{\rm{\Omega }}({\bf{x}},{\varphi }_{f},{\varphi }_{g}) & = & f({\bf{x}},t)-\hat{f}({\bf{x}},{\varphi }_{f}^{* })\\ & & +({g}_{1}({\bf{x}},t)-{\hat{g}}_{1}({\bf{x}},{\varphi }_{g}^{* }))u(t),\end{eqnarray} \tag{ 36 }$$

where, $u(t)$ is the control law. Assume that ${\rm{\Omega }}$ is a bounded function; $| {\rm{\Omega }}({\bf{x}},{\varphi }_{f},{\varphi }_{g})| \leqslant {{\rm{\Omega }}}_{0}$.

Theorem 1. The ISSmDCS (1) is controlled by the following control law.

$$\begin{eqnarray}&&u(t)=\hat{g}{({\bf{x}},t)}^{-1}\left(-\displaystyle \sum _{i=1}^{n-1}{a}_{i}{{\rm{e}}}^{(i)}-\hat{f}({\bf{x}},t)+{{x}_{{\rm{d}}}}^{(n)}-h({\bf{x}},t)\right),\end{eqnarray} \tag{ 37 }$$

where, $h({\bf{x}},t)$ is estimated as in (6); ${\hat{g}}_{1}({\bf{x}},{\varphi }_{g})$ and $\hat{f}({\bf{x}},{\varphi }_{f})$ are specified in (33). Then, ${\bf{e}}(t)\to 0$ $({\bf{x}}\to {{\bf{x}}}_{{\rm{d}}})$ when $t\to \infty .$ Thus, the proposed control system satisfies the stable Lyaponov process if the following update laws are adopted:

$$\begin{eqnarray}&&{\dot{\varphi }}_{f}=S({\bf{x}})\lambda ({\bf{x}});{\dot{\varphi }}_{g}=S({\bf{x}})\lambda ({\bf{x}}){u}_{s}(t),\end{eqnarray} \tag{ 38 }$$

$$\begin{eqnarray}&&{k}_{1}(t)\\ =\left\{\begin{array}{ll}\xi \displaystyle \frac{{{\rm{\Omega }}}_{0}{\rm{\Phi }}}{(| S({\bf{x}})| +\varepsilon )(1+\varepsilon -{\rm{\exp }}(-{k}_{2}| S({\bf{x}})| ))} & {\rm{i}}{\rm{f}}\,| S({\bf{x}})| \leqslant {\rm{\Phi }}\\ \xi \displaystyle \frac{{{\rm{\Omega }}}_{0}}{1-{\rm{\exp }}(-{k}_{2}| S({\bf{x}})| )+\varepsilon } & {\rm{i}}{\rm{f}}\,| S({\bf{x}})| \gt {\rm{\Phi }},\end{array}\right.\end{eqnarray} \tag{ 39 }$$

where, ${{\rm{\Omega }}}_{0}$ is the bound value of ${\rm{\Omega }}({\bf{x}},{\varphi }_{f},{\varphi }_{g})$ defined in (36); $\xi \gt \mathrm{1,}0\lt \varepsilon \ll 1$ are adaptive parameters chosen by designer.

Proof. From equations (3) and (1), the following is given.

$$\begin{eqnarray}&&\dot{S}({\bf{x}})=\displaystyle \sum _{i=1}^{n-1}\,{a}_{i}{{\rm{e}}}^{(i)}+f({\bf{x}},t)+g({\bf{x}},t)u(t)-{{x}_{{\rm{d}}}}^{(n)}.\end{eqnarray} \tag{ 40 }$$

Using the control law (37) with reference to (40), (36) and (33), the followings are obtained.

$$\begin{eqnarray}\dot{S}({\bf{x}}) & = & (f({\bf{x}},t)-\hat{f}({\bf{x}},{\varphi }_{f}))\\ & & +\,({g}_{1}({\bf{x}},t)-{\hat{g}}_{1}({\bf{x}},{\varphi }_{g}))u(t)-h({\bf{x}},t),\end{eqnarray} \tag{ 41 }$$

$$\begin{eqnarray}\dot{S}({\bf{x}}) & = & (\hat{f}({\bf{x}},{\varphi }_{f}^{* })-\hat{f}({\bf{x}},{\varphi }_{f}))+({\hat{g}}_{1}({\bf{x}},{\varphi }_{g}^{* })-{\hat{g}}_{1}\\ & & \times ({\bf{x}},{\varphi }_{g})){u}_{s}(t)-h({\bf{x}},t)+{\rm{\Omega }}({\bf{x}},{\varphi }_{f},{\varphi }_{g}),\end{eqnarray} \tag{ 42 }$$

$$\begin{eqnarray}\dot{S}({\bf{x}}) & = & {{\psi }_{f}}^{{\rm{T}}}\lambda ({\bf{x}})+{{\psi }_{g}}^{{\rm{T}}}\lambda ({\bf{x}}){u}_{s}(t)\\ & & -\,h({\bf{x}},t)+{\rm{\Omega }}({\bf{x}},{\varphi }_{f},{\varphi }_{g}),\end{eqnarray} \tag{ 43 }$$

where, ${\psi }_{f}={\varphi }_{f}^{* }-{\varphi }_{f};$ ${\psi }_{g}={\varphi }_{g}^{* }-{\varphi }_{g}.$ Using the following Lyaponov function

$$\begin{eqnarray}&&{V}_{1}=\displaystyle \frac{1}{2}S{({\bf{x}})}^{2}+\displaystyle \frac{1}{2}{{\psi }_{f}}^{{\rm{T}}}{\psi }_{f}+\displaystyle \frac{1}{2}{{\psi }_{g}}^{{\rm{T}}}{\psi }_{g},\end{eqnarray} \tag{ 44 }$$

with reference to (6), (7) and (43), the following equations are obtained.

$$\begin{eqnarray}{\dot{V}}_{1} & = & S({\bf{x}})\,({{\psi }_{f}}^{{\rm{T}}}\lambda ({\bf{x}})+{{\psi }_{g}}^{{\rm{T}}}\lambda ({\bf{x}}){u}_{s}(t)-h\\ & & \times ({\bf{x}},t)+{\rm{\Omega }}\,({\bf{x}},{\varphi }_{f},{\varphi }_{g}))+{{\psi }_{f}}^{{\rm{T}}}{\dot{\psi }}_{f}+{{\psi }_{g}}^{{\rm{T}}}{\dot{\psi }}_{g},\end{eqnarray} \tag{ 45 }$$

$$\begin{eqnarray}{\dot{V}}_{1} & = & {{\psi }_{f}}^{{\rm{T}}}(S({\bf{x}})\lambda ({\bf{x}})+{\dot{\psi }}_{f})+{{\psi }_{g}}^{{\rm{T}}}(S({\bf{x}})\lambda ({\bf{x}})\\ & & \times {u}_{s}(t)+{\dot{\psi }}_{g})-{k}_{1}S({\bf{x}}){\rm{sat}}(S({\bf{x}})/{\rm{\Phi }})\\ & & \times (1-\exp (-{k}_{2}| S({\bf{x}})| ))+S({\bf{x}})({\rm{\Omega }}({\bf{x}},{\varphi }_{f},{\varphi }_{g})).\end{eqnarray} \tag{ 46 }$$

In (46), due to ${\dot{\psi }}_{f}=-{\dot{\varphi }}_{f}$ and ${\dot{\psi }}_{g}=-{\dot{\varphi }}_{g},$ the following equation can be derived.

$$\begin{eqnarray}{\dot{V}}_{1} & = & {{\psi }_{f}}^{{\rm{T}}}(S({\bf{x}})\lambda ({\bf{x}})-{\dot{\varphi }}_{f})+{{\psi }_{g}}^{{\rm{T}}}(S({\bf{x}})\lambda ({\bf{x}})\\ & & \times {u}_{s}(t)-{\dot{\varphi }}_{g})-{k}_{1}S({\bf{x}}){\rm{sat}}(S({\bf{x}})/{\rm{\Phi }})(1\\ & & -\exp (-{k}_{2}| S({\bf{x}})| ))+S({\bf{x}})({\rm{\Omega }}({\bf{x}},{\varphi }_{f},{\varphi }_{g})).\end{eqnarray} \tag{ 47 }$$

Then, using update law (38), (47) becomes as follows.

$$\begin{eqnarray}{\dot{V}}_{1} & = & -{k}_{1}S({\bf{x}}){\rm{sat}}(S({\bf{x}})/{\rm{\Phi }})\\ & & \times (1-\exp (-{k}_{2}| S({\bf{x}})| ))+S({\bf{x}}){\rm{\Omega }}(x,{\phi }_{f},{\phi }_{g}).\end{eqnarray} \tag{ 48 }$$

- If $| {S}/{\rm{\Phi }}| \gt 1,$ from (8) and (48), the followings can be inferred.

$$\begin{eqnarray}{\dot{V}}_{1} & = & -{k}_{1}| S({\bf{x}})| (1-\exp (-{k}_{2}| S({\bf{x}})| ))\\ & & +\,S({\bf{x}}){\rm{\Omega }}(x,{\varphi }_{f},{\varphi }_{g})\lt -{k}_{1}| S({\bf{x}})| \\ & & \times (1-\exp (-{k}_{2}| S({\bf{x}})| ))+| S({\bf{x}})| {{\rm{\Omega }}}_{0}.\end{eqnarray} \tag{ 49 }$$

In this case, based on (49), the update law of ${k}_{1}(t)$ is derived as follows.

$$\begin{eqnarray}&&{k}_{1}(t)=\xi \displaystyle \frac{{{\rm{\Omega }}}_{0}}{1-\exp (-{k}_{2}| S({\bf{x}})| )+\varepsilon },\,\xi \gt \mathrm{1,}0\lt \varepsilon \ll 1.\end{eqnarray} \tag{ 50 }$$

- If $| {S}/{\rm{\Phi }}| \leqslant 1,$ from (8) and (48), the followings can be inferred.

$$\begin{eqnarray}{\dot{V}}_{1} & = & -{k}_{1}\displaystyle \frac{S{({\bf{x}})}^{2}}{{\rm{\Phi }}}(1-\exp (-{k}_{2}| S({\bf{x}})| ))\\ & & +\,S({\bf{x}}){\rm{\Omega }}(x,{\varphi }_{f},{\varphi }_{g})\lt -{k}_{1}\displaystyle \frac{S{({\bf{x}})}^{2}}{{\rm{\Phi }}}\\ & & \times (1-\exp (-{k}_{2}| S({\bf{x}})| ))+| S({\bf{x}})| {{\rm{\Omega }}}_{0}\\ & = & | S({\bf{x}})| \left(-{k}_{1}\displaystyle \frac{| S({\bf{x}})| }{{\rm{\Phi }}}\right.\times (1-\exp (-{k}_{2}| S({\bf{x}})| ))+{{\rm{\Omega }}}_{0}\phantom{\rule{}{3.0ex}}).\end{eqnarray} \tag{ 51 }$$

So, the update law of ${k}_{1}(t)$ is obtained as follows.

$$\begin{eqnarray}{k}_{1}(t) & = & \xi \displaystyle \frac{{{\rm{\Omega }}}_{0}{\rm{\Phi }}}{(| S({\bf{x}})| +\varepsilon )(1+\varepsilon -\exp (-{k}_{2}| S({\bf{x}})| ))}\\ & & \xi \gt \mathrm{1,}0\lt \varepsilon \ll 1.\end{eqnarray} \tag{ 52 }$$

Finally, deriving from (50) and (52), the update law proposed in (39) can be inferred. □

Using the measured database and the im-ENFS, ANFISs working as the inverse MRD models are established. The results related to this work are presented in figures 9–13 and table 2. Figure 9 shows the time-series prediction output of the ANFISs which is the damping force at different current levels, from 0 to 2.4 A. Identifying an extraction from this database corresponding to I = 1.6 A is illustrated in figure 10. The results from this figure show that the accuracy rate of the ANFIS is high showing a very low RMSE of 0.1652 (N).

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Figure 11(a) illustrates the damping force versus the relative MRD piston-cylinder velocity. This is an extraction from the inverse MRD model expressing $I(t)={\rm{ANFIS}}({f}_{{\rm{M}}{\rm{R}}},{v}_{{\rm{r}}{\rm{e}}}).$ The time-series prediction error using the im-ENFS and the data set corresponding to I = 0.8 A as is quite small as shown in figure 11(b). The RMSE in this phase is 0.0802 (N). Figure 12 displays an extraction from the inverse MRD model using ANFIS expressing $I(t)={\rm{ANFIS}}({f}_{{\rm{M}}{\rm{R}}},{d}_{{\rm{r}}{\rm{e}}})$ corresponding to six current levels from 0 to 2.4 A. Figure 13 shows the time-series prediction error of the ANFIS built by the im-ENFS and the data set deriving from figure 12 when I = 0.8 A. The RMSE in this phase is 0.0241 (N). In order to more fully evaluate the effectiveness of the proposed im-ENFS, identification results coming from the im-ENFS and the other algorithms are considered. They are the ENFS [27], ANFIS-PF (synthesizing an ANFIS via the proposed data potential field) [52], optimized hyperplane clustering synthesis [53], OD-T2FLS (optimally designed IT-2FLS from the dataset) [54]. The data sets used for figures 10, 11, and 13 (called Data-10, Data-11, and Data-13, respectively) are re-utilized for these ANFIS training algorithms. The compared RMSEs are listed in table 2 which shows the best identification accuracy rate of the proposed im-ENFS algorithm. Specially, the im-ENFS can deal well with the very large data structure such as the 25.000-data-sample set (Data-10). This is the inherent feature for building inverse smart models typed data-driven ones.

By combining the databases deriving from the $I(t)={\rm{ANFIS}}\,({f}_{{\rm{M}}{\rm{R}}},{v}_{{\rm{r}}{\rm{e}}})$ illustrated in figure 11 and $I(t)={\rm{ANFIS}}\,({f}_{{\rm{M}}{\rm{R}}},{d}_{{\rm{r}}{\rm{e}}})$ shown in figure 12, a new database with the input to be $({f}_{{\rm{M}}{\rm{R}}},{d}_{{\rm{r}}{\rm{e}}},{v}_{{\rm{r}}{\rm{e}}})$ and the output to be the corresponding current, $I(t),$ is generated. Based on this and the algorithm of the im-ENFS, an inverse MRD model expressing $I(t)={\rm{ANFIS}}\,({f}_{{\rm{M}}{\rm{R}}},{d}_{{\rm{r}}{\rm{e}}},{v}_{{\rm{r}}{\rm{e}}})$ is established which works as the ANFIS-I-MRD of the controller presented in figure 4. Using this ANFIS-I-MRD, the FSMC formulated in this work is experimentally relaized for vibration control of the semi-active vehicle suspension system.

In this test, in order to demonstrate the effectiveness of the proposed FSMC, the passive result which is obtained without any control action and the control result achieved from one existing ANFIS associated with the sliding mode controller named NFSmUoC [6] are presented together. It is noted here that all MRD structures and experimental conditions for three cases are same. The aim of the control system is to support the control force u(t) such that vibration of the chassis mass is damped out as soon as possible. This means the required state to be ${{\bf{x}}}_{{\rm{d}}}={[{x}_{1},{x}_{2}]}_{{\rm{d}}}{}^{{\rm{T}}}={[{z}_{{\rm{s}}},{\dot{z}}_{{\rm{s}}}]}_{{\rm{d}}}{}^{{\rm{T}}}={[0,0]}^{{\rm{T}}}.$ To show quantitative results, the maximum magnitude of the chassis displacement and acceleration ${A}_{{\rm{d}}},{A}_{{\rm{a}}}$ are defined as follows.

$$\begin{eqnarray}&&{A}_{{\rm{d}}}=\mathop{{\rm{\max }}}\limits_{\,i=\mathrm{1...}P}| {z}_{{\rm{s}}}^{i}| ;{A}_{{\rm{a}}}=\mathop{{\rm{\max }}}\limits_{\,i=\mathrm{1...}P}| {\ddot{z}}_{{\rm{s}}}^{i}| ,\end{eqnarray} \tag{ 59 }$$

where, P is the number of samples; ${z}_{{\rm{s}}}$ and ${\ddot{z}}_{{\rm{s}}}$ are the vertical chassis displacement and acceleration.

As a first test, the bump road shown in figure 14 is applied. Obtained survey results related to the passive, NFSmUoC and FSMC suspensions are shown in figures 15–17 and table 3. The maximum vertical displacement of the sprung mass ${m}_{{\rm{s}}}$ shows that the vibration amplitudes of the two controlled suspensions are much smaller than that of the passive one. And it is clearly seen that the amplitude of 0.0020 (m) achieved from the proposed controller FSMC is the smallest. The vertical accelerations of ${m}_{{\rm{s}}}$ depicted in figure 16 also shows the smallest amplitude (0.5332 m s⁻²) for the case of the proposed FSMC, while the largest amplitude (4.8402 m s⁻²) is occurred for the passive case as expected. It is known that the evaluation of control force (or power) is equivalently important to the evaluation of control performances. Figure 17 compares control force applied to the suspension system to get the results shown in figures 15 and 16. It is clear observed that the proposed controller is rquired smaller control force, but provides better vibration control results than the NFSmUoC. The quantitative results for vibration control and required force are identified and lisred in table 3.

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Table 3.  The maximum magnitudes.

	${A}_{{\rm{d}}}$ (m)	${A}_{{\rm{a}}}$ (m s−2)	Control force (N)
Passive	0.0254	4.8402	0.0
NFSmUoC	0.0025	0.9812	1225.51
FSMC	0.0020	0.5332	956.79

As a second test, a sinusoidal road profile is adopted as follows.

$$\begin{eqnarray}y(t) & = & {Y}_{0}\,\sin (\omega \,t)+{\rm{random}}(0;\,0.003)({\rm{m}}),\\ \omega & = & 2\pi V/D,\end{eqnarray} \tag{ 60 }$$

where, $y(t)$ denotes the vertical displacement of the road surface; ${Y}_{0}$ is the amplitude; D is the traveling length of the vehicle along the road with the speed of $V$ corresponding to one sine-wave cycle; ${\rm{random}}(0;\,0.003)$ denotes a random value belonging to $[-0.003,+0.003].$ Related to this road profile, ${\rm{random}}(0;\,0.003)$ describes the road disturbance surface. This implies the unknown shake status of the lower bed in experimental apparatus shown in figure 6. In (60), by choosing ${Y}_{0}=0.04\,{\rm{m}},$ $D=8\,{\rm{m}},$ and $V=30$ km h⁻¹, the road profile is generated as shown in figure 18. Using this excitation, vibration control results are obtained and presented in figures 19–21. The compared results show that the smallest ${A}_{{\rm{d}}}$ and ${A}_{{\rm{a}}}$ of the chassis vibration are obtained from the proposed FSMC. The maximum displacement amplitudes of the passive, NFSmUoC and FSMC, respectively are identified by 0.0253, 0.0058, and 0.0030 m, respectively. The maximum acceleration amplitudes of them, in this order, are evaluated as 2.2297, 0.8067, and 0.4948 m s⁻², respectively. In this case, the control force applied to MRD is almost same for the FSMC and NFSmUoC as shown in figure 21. The results presented in this tests indicate that the FSMC proposed in this work can provide the smallest displacement and acceleration amplitudes of the vehicle suspension system which are directly related to the ride comfort of the vehilce.

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