2. Formulation of HAC-PP

control algorithm is one of the most important components which brings comfortable requirements to the consumer. The development of control algorithms in recent years is abundantly being undertaken from the aspect of classical control to salient characteristics of intelligent control. The classical control methods are frequently combined with modern control technique to resolve parameter uncertainties and disturbances those are existed in most of control devices. A controller which is formulated using more than two different control schemes is called "a hybrid controller" or "composite controller" [1, 2]. Among many candidates of the hybrid controller, the type of hybrid adaptive controller is the most popular since its structure is relatively simple and its control performance is very robust against the uncertainties or/and external disturbances. A hybrid adaptive control with fuzzy model and wavelet neural networks was presented in [1, 3] in which the sliding mode control was used to connect the parameters of the fuzzy model and the neural networks. This method is the typical model to develop the adaptive control in the last few years. Besides of uncertain nonlinear system, the problem of unknown input nonlinearity such as dead-zone or backlash-like hysteresis was also studied through the hybrid adaptive control [4]. It has been also shown that the neural works can be designed for a good performance of the hybrid adaptive control to deal with the uncertain system [5]. A hybrid adaptive controller possessing the robustness against input and parameter uncertainties was studied using the sliding mode controller associated with the fuzzy model [6, 7]. When a hybrid adaptive controller is formulated, in general the adaptation laws are simultaneously calculated. Furthermore, the back-stepping method was integrated with the fuzzy mode

As mentioned earlier, both the fuzzy model and the neural networks model are frequently used for the formulation of high performance of a hybrid adaptive controller [9]. Recently, a modified type of the fuzzy model called interval type 2 was combined with the back-stepping method to design of a hybrid adaptive control [10, 11]. It is remarked that the fixed fuzzy model always provides a safe choice in design of a hybrid adaptive control. However, this choice may cause a large error in finding the final values. To resolve this problem, an adaptive interval type 2 fuzzy neural network was developed on the basis of the online technique which can strengthen the flexibility of design parameters against the uncertainties [12]. Besides the above, there are many approaches to formulate new hybrid adaptive controllers such as output feedback control approach to take account for unknown hysteresis [13]. From the aspect of experimental implementation of hybrid adaptive controllers, several dynamic systems featuring magneto-rheological (MR) mount and MR damper are adopted for vibration control [2, 14–18]. Most of hybrid adaptive controllers used in these experimental realizations have been formulated by combining the models of interval type 2 fuzzy and interval type 2 fuzzy neural networks, and the control techniques of H-infinity control and sliding mode control. The advantage of using the interval type 2 fuzzy model is its flexibility in which optimized fuzzy values can be achieved unlike the classical fuzzy rule with the fixed value [19]. In order to improve the fuzzy model, clustering method [20] and data-driven for fuzzy

As a subsequent work to develop a new hybrid adaptive controller, in this work two different new hybrid adaptive controllers are developed and their control performances are evaluated by investigation on vibration control of a semi-active seat suspension system installed with MR damper. The first hybrid adaptive controller is designed by combing online interval type 2

to achieve high performance of the hybrid adaptive controller [8].

rules [21] were also introduced.

4 Adaptive Robust Control Systems

As mentioned in Introduction, the online interval type 2 fuzzy neural networks (OIT2FNN in short) model is used to formulate two adaptive controllers. The rule base of OIT2FNN can be expressed as follows [22].

$$R\_f^\dagger: \text{If}\\h\_1 \text{ is } \mathbf{H}\_{f\mathbf{1}}^\dagger \text{ and}... \text{and}\\h\_n \text{ is } \mathbf{H}\_{f\mathbf{n}}^\dagger \text{ Then } g \text{ is } a\_0^j + \sum\_{i=1}^n a\_i^j h\_i \tag{1}$$

where, H<sup>j</sup> fi ð Þ i ¼ 1;…; n; j ¼ 1;…; m are fuzzy sets, m is the number of rules, and a j <sup>i</sup> are interval sets. The calculation process of OIT2FNN is clearly explained in [22]. The defuzzified output is then determined by

$$\mathbf{g}\_f = \frac{\mathbf{g}\_l + \mathbf{g}\_r}{2} = \frac{\mathbf{\Theta}\_l^T \mathbf{\xi}\_l^f + \mathbf{\Theta}\_r^T \mathbf{\xi}\_r^f}{2} \tag{2}$$

In the above, θ<sup>T</sup> <sup>l</sup> <sup>¼</sup> <sup>w</sup><sup>l</sup> <sup>1</sup> wl <sup>2</sup> w<sup>l</sup> 3…w<sup>l</sup> n � � and θ<sup>T</sup> <sup>r</sup> <sup>¼</sup> wr <sup>1</sup> w<sup>r</sup> <sup>2</sup> w<sup>r</sup> 3…w<sup>r</sup> n � � are the weighting vectors, which symbolize the relation of the rule layer and type-reduction, and the weighted firing strength vectors given by

$$\underline{\mathsf{S}}\_{l}^{f} = \left[ \frac{\underline{\underline{f}}\_{1}}{\sum\_{i=1}^{n} \underline{\underline{f}}\_{i}} \, \frac{\underline{\underline{f}}\_{2}}{\sum\_{i=1}^{n} \underline{\underline{f}}\_{i}} \, \frac{\underline{\underline{f}}\_{3}}{\sum\_{i=1}^{n} \underline{\underline{f}}\_{i}} \dots \, \frac{\underline{\underline{f}}\_{n}}{\sum\_{i=1}^{n} \underline{\underline{f}}\_{i}} \right]^{\mathsf{T}}, \\ \underline{\mathsf{S}}\_{r}^{f} = \left[ \frac{\overline{\underline{f}}\_{1}}{\sum\_{i=1}^{n} \overline{\underline{f}}\_{i}} \, \frac{\overline{\underline{f}}\_{2}}{\sum\_{i=1}^{n} \overline{\underline{f}}\_{i}} \, \frac{\overline{\underline{f}}\_{3}}{\sum\_{i=1}^{n} \overline{\underline{f}}\_{i}} \, \dots \, \frac{\overline{\underline{f}}\_{n}}{\sum\_{i=1}^{n} \overline{f}\_{i}} \right]^{\mathsf{T}}$$

As a problem formulation, consider a single-input and single-output (SISO) nonlinear system governed by the following equation:

$$
\dot{\mathbf{x}} = \mathbf{f}(\mathbf{x}) + \mathbf{g}(\mathbf{x})u(t) + \mathbf{d}(\mathbf{t})\tag{3}
$$

where f(x)∈ Rn and g(x)∈ Rn are two unknown non-linear function vectors, u(t)∈ R<sup>1</sup> is control function, d(t) ∈R<sup>n</sup> is an external disturbance vector, |d(t)| ≤ δd where δd∈R<sup>n</sup> is upper bound of d(t), x ¼ ½ �¼ x1; x2;…; xn x1; x\_ <sup>1</sup>;…; x1 ð Þ <sup>n</sup>�<sup>1</sup> � �<sup>T</sup> ∈ Rn is the state vector of the system. The first sliding surface ss is defined as follows:

$$\mathbf{x}\_s = k\_1 \mathbf{x}\_1 + k\_2 \mathbf{x}\_2 + k\_3 \mathbf{x}\_3 + \dots + k\_n \mathbf{x}\_n = \sum\_{i=1}^n k\_i \mathbf{x}\_i \tag{4}$$

where, K = [kn, kn � 11, kn � 2, …, k1] is defined as the coefficients such that all of the roots of the polynomial <sup>σ</sup><sup>n</sup> <sup>+</sup> kn � <sup>1</sup>σ<sup>n</sup> � <sup>1</sup> <sup>+</sup> kn � <sup>2</sup>σ<sup>n</sup> � <sup>2</sup> <sup>+</sup> …<sup>+</sup> <sup>k</sup><sup>1</sup> are in the open left-half complex plane. The sliding surface (4) is rewritten using the state variables as follows:

$$\mathbf{x}\_n = -k\_1 \mathbf{x}\_1 - k\_2 \mathbf{x}\_2 - k\_3 \mathbf{x}\_3 - \dots - k\_{n-1} \mathbf{x}\_{n-1} + \mathbf{s}\_s \tag{5}$$

A new vector ~x is defined by ~x ¼ ½ � x<sup>1</sup> x<sup>2</sup> x3…xn�<sup>1</sup> T , and thus the system (3) is rewritten as follows:

$$\dot{\tilde{\mathbf{x}}} = \mathbf{S}\_1 \tilde{\mathbf{x}} + \mathbf{S}\_2^T \mathbf{s}\_s \tag{6}$$

From the above conditions (10), the function S(ϕ) can be determined as follows:

S ϕ

Then using Eq. (8), the tracking error is obtained by

<sup>ϕ</sup> <sup>¼</sup> <sup>1</sup> 2

<sup>M</sup><sup>1</sup> <sup>¼</sup> λ λ €ð Þ� <sup>þ</sup> <sup>e</sup> <sup>λ</sup>\_ <sup>þ</sup> <sup>e</sup>\_ � �<sup>2</sup> 2ð Þ λ þ e

where,

where γ<sup>f</sup> = f(x) � f

rewritten as:

ln <sup>1</sup> <sup>þ</sup> <sup>S</sup> <sup>1</sup> � <sup>S</sup> <sup>¼</sup> <sup>1</sup> 2

Hence, the derivatives of Eq. (13) are obtained as:

where cs > 0. The derivative of Eq. (16) is obtained as:

The lumped uncertainty of system is defined as:

(x), <sup>γ</sup><sup>g</sup> <sup>=</sup> <sup>g</sup>(x) � <sup>g</sup><sup>∗</sup>

σ\_<sup>s</sup> ¼ M<sup>1</sup> þ M<sup>2</sup> þ M3f

∗

Based on Eq. (2), the relationship between Eq. (19) and OIT2FNN is expressed as follows:

∗

�λð Þt < λð Þt S ϕ

for t! ∞}. On the other hand, the inverse function of (11) is expressed as:

ln <sup>1</sup> <sup>þ</sup> <sup>e</sup>ð Þ <sup>=</sup><sup>λ</sup> 1 � <sup>e</sup>ð Þ =<sup>λ</sup>

> <sup>ϕ</sup>\_ <sup>¼</sup> <sup>1</sup> 2

<sup>2</sup> , M<sup>2</sup> ¼ � λ λ €ð Þ� � <sup>e</sup> <sup>λ</sup>\_ � <sup>e</sup>\_ � �<sup>2</sup>

In order to realize ϕ! 0, the second sliding surface is defined as follows:

� � <sup>¼</sup> <sup>e</sup><sup>ϕ</sup> � <sup>e</sup>�<sup>ϕ</sup>

Hence, the tracking error can be summarized as Ξ = {e∈ R : |e(t)| < λ ∀ t ≥ 0 and e(t) < λ<sup>∞</sup>

ln <sup>λ</sup> <sup>þ</sup> <sup>e</sup> λ � e

" #

� <sup>λ</sup>\_ � <sup>e</sup>\_ λ � e

¼ 1 2

¼ 1 2

> <sup>λ</sup>\_ <sup>þ</sup> <sup>e</sup>\_ λ þ e

<sup>ϕ</sup>€ <sup>¼</sup> <sup>M</sup><sup>1</sup> <sup>þ</sup> <sup>M</sup><sup>2</sup> <sup>þ</sup> <sup>M</sup>3<sup>e</sup>

2ð Þ λ � e

<sup>σ</sup>\_<sup>s</sup> <sup>¼</sup> <sup>ϕ</sup>€ <sup>þ</sup> csϕ\_ <sup>¼</sup> <sup>M</sup><sup>1</sup> <sup>þ</sup> <sup>M</sup><sup>2</sup> <sup>þ</sup> <sup>M</sup>3ðf xð Þþ g xð Þu tð Þþ d tð Þ� <sup>x</sup>€dÞ þ csϕ\_ (17)

<sup>e</sup><sup>ϕ</sup> <sup>þ</sup> <sup>e</sup>�<sup>ϕ</sup> (11)

Robust Adaptive Controls of a Vehicle Seat Suspension System

http://dx.doi.org/10.5772/intechopen.71422

½ � ln ð Þ� λ þ e ln ð Þ λ � e (13)

€ (15)

<sup>2</sup> þ

!

λ � e 2ð Þ λ � e 2

2ð Þ λ þ e

<sup>σ</sup><sup>s</sup> <sup>¼</sup> <sup>ϕ</sup>\_ <sup>þ</sup> cs<sup>ϕ</sup> (16)

(14)

7

:

� � <sup>&</sup>lt; <sup>λ</sup>ð Þ<sup>t</sup> <sup>⇔</sup> � <sup>λ</sup>ð Þ<sup>t</sup> <sup>&</sup>lt; e tð Þ <sup>&</sup>lt; <sup>λ</sup>ð Þ<sup>t</sup> (12)

<sup>2</sup> , M<sup>3</sup> <sup>¼</sup> <sup>λ</sup> <sup>þ</sup> <sup>e</sup>

w ¼ M3γ~<sup>f</sup> ξ<sup>f</sup> þ M3γ~gξgu þ M3d tð Þ (18)

(x). Using Eqs. (17) and (18), the derivative Eq. (17) is

ð Þþ <sup>x</sup> <sup>M</sup>3g<sup>∗</sup>ð Þ<sup>x</sup> u tð Þ� <sup>M</sup>3x€<sup>d</sup> <sup>þ</sup> csϕ\_ <sup>þ</sup> <sup>w</sup> (19)

where,

$$\mathbf{S}\_1 = \begin{bmatrix} 0 & 1 & 0 & \dots & 0 \\ 0 & 0 & 1 & \dots & 0 \\ \cdot & \cdot & \cdot & \dots & \cdot \\ -k\_1 & -k\_2 & -k\_3 & \dots & -k\_{n-1} \end{bmatrix}, \mathbf{S}\_2 = \begin{bmatrix} 0 \\ 0 \\ \cdot \\ 1 \end{bmatrix}$$

The tracking error is defined as e = x1� xd with the desired states of xd. Then the error performance function is defined as follows [23]:

$$
\lambda(t) = (\lambda(0) - \lambda\_{\circ})e^{-lt} + \lambda\_{\circ} \tag{7}
$$

where, l > 0, 0 < |e(0)| < λ(0), λ<sup>∞</sup> > 0, λ<sup>∞</sup> < λ(0) then λ<sup>t</sup> > 0 and λ(t) tend to λ<sup>∞</sup> exponentially. In order to guarantee fast convergence of tracking error, and obtain a certain convergence accuracy, the tracking error is set as follows:

$$e(t) = \lambda(t)S(\phi)\tag{8}$$

In the above, the prescribed error performance function S(ϕ) found as follows:

$$S(\phi) = \frac{\varepsilon(t)}{\lambda(t)}\tag{9}$$

The function S(ϕ) must satisfy the following conditions.

(i) S(ϕ) is smooth continuous and monotone increasing function

$$1 \text{ (ii)} - 1 < \mathcal{S} \text{(\(\phi\))}<1\tag{10}$$

(iii) lim<sup>ϕ</sup> ! <sup>+</sup> <sup>∞</sup>S(ϕ) = 1 and limϕ! � <sup>∞</sup>S(ϕ) = � 1

From the above conditions (10), the function S(ϕ) can be determined as follows:

$$S(\phi) = \frac{e^{\phi} - e^{-\phi}}{e^{\phi} + e^{-\phi}} \tag{11}$$

Then using Eq. (8), the tracking error is obtained by

$$-\lambda(t) < \lambda(t)S(\phi) < \lambda(t) \Leftrightarrow -\lambda(t) < e(t) < \lambda(t) \tag{12}$$

Hence, the tracking error can be summarized as Ξ = {e∈ R : |e(t)| < λ ∀ t ≥ 0 and e(t) < λ<sup>∞</sup> for t! ∞}. On the other hand, the inverse function of (11) is expressed as:

$$\phi = \frac{1}{2}\ln\frac{1+S}{1-S} = \frac{1}{2}\ln\frac{1+\left(\epsilon/\lambda\right)}{1-\left(\epsilon/\lambda\right)} = \frac{1}{2}\ln\frac{\lambda+e}{\lambda-e} = \frac{1}{2}\left[\ln\left(\lambda+e\right)-\ln\left(\lambda-e\right)\right] \tag{13}$$

Hence, the derivatives of Eq. (13) are obtained as:

$$\dot{\phi} = \frac{1}{2} \left[ \frac{\dot{\lambda} + \dot{e}}{\lambda + e} - \frac{\dot{\lambda} - \dot{e}}{\lambda - e} \right] \tag{14}$$

$$
\ddot{\vec{\phi}} = M\_1 + M\_2 + M\_3 \ddot{\vec{e}} \tag{15}
$$

where,

of d(t), x ¼ ½ �¼ x1; x2;…; xn x1; x\_ <sup>1</sup>;…; x1

sliding surface ss is defined as follows:

6 Adaptive Robust Control Systems

ð Þ <sup>n</sup>�<sup>1</sup> � �<sup>T</sup>

sliding surface (4) is rewritten using the state variables as follows:

~\_

010 … 0 001 … 0 ::: … : �k<sup>1</sup> �k<sup>2</sup> �k<sup>3</sup> … �kn�<sup>1</sup>

λðÞ¼ t ð Þ λð Þ� 0 λ<sup>∞</sup> e

<sup>x</sup> <sup>¼</sup> <sup>S</sup>1~<sup>x</sup> <sup>þ</sup> <sup>S</sup><sup>T</sup>

The tracking error is defined as e = x1� xd with the desired states of xd. Then the error perfor-

where, l > 0, 0 < |e(0)| < λ(0), λ<sup>∞</sup> > 0, λ<sup>∞</sup> < λ(0) then λ<sup>t</sup> > 0 and λ(t) tend to λ<sup>∞</sup> exponentially. In order to guarantee fast convergence of tracking error, and obtain a certain convergence accu-

e tðÞ¼ λð Þt S ϕ

In the above, the prescribed error performance function S(ϕ) found as follows:

S ϕ � � <sup>¼</sup> e tð Þ

ð Þ� ii 1 < S ϕ

A new vector ~x is defined by ~x ¼ ½ � x<sup>1</sup> x<sup>2</sup> x3…xn�<sup>1</sup>

S<sup>1</sup> ¼

mance function is defined as follows [23]:

racy, the tracking error is set as follows:

The function S(ϕ) must satisfy the following conditions.

(iii) lim<sup>ϕ</sup> ! <sup>+</sup> <sup>∞</sup>S(ϕ) = 1 and limϕ! � <sup>∞</sup>S(ϕ) = � 1

(i) S(ϕ) is smooth continuous and monotone increasing function

follows:

where,

ss <sup>¼</sup> <sup>k</sup>1x<sup>1</sup> <sup>þ</sup> <sup>k</sup>2x<sup>2</sup> <sup>þ</sup> <sup>k</sup>3x<sup>3</sup> <sup>þ</sup> … <sup>þ</sup> knxn <sup>¼</sup> <sup>X</sup><sup>n</sup>

where, K = [kn, kn � 11, kn � 2, …, k1] is defined as the coefficients such that all of the roots of the polynomial <sup>σ</sup><sup>n</sup> <sup>+</sup> kn � <sup>1</sup>σ<sup>n</sup> � <sup>1</sup> <sup>+</sup> kn � <sup>2</sup>σ<sup>n</sup> � <sup>2</sup> <sup>+</sup> …<sup>+</sup> <sup>k</sup><sup>1</sup> are in the open left-half complex plane. The

T

∈ Rn is the state vector of the system. The first

, and thus the system (3) is rewritten as

<sup>2</sup> ss (6)

�lt <sup>þ</sup> <sup>λ</sup><sup>∞</sup> (7)

� � (8)

<sup>λ</sup>ð Þ<sup>t</sup> (9)

� � < 1 (10)

kixi (4)

i¼1

xn ¼ �k1x<sup>1</sup> � k2x<sup>2</sup> � k3x<sup>3</sup> � … � kn�<sup>1</sup>xn�<sup>1</sup> þ ss (5)

3 7 7 7 5, <sup>S</sup><sup>2</sup> <sup>¼</sup>

$$M\_1 = \frac{\ddot{\lambda}(\lambda + e) - \left(\dot{\lambda} + \dot{e}\right)^2}{2\left(\lambda + e\right)^2}, \\ M\_2 = -\frac{\ddot{\lambda}(\lambda - e) - \left(\dot{\lambda} - \dot{e}\right)^2}{2\left(\lambda - e\right)^2}, \\ M\_3 = \left(\frac{\lambda + e}{2\left(\lambda + e\right)^2} + \frac{\lambda - e}{2\left(\lambda - e\right)^2}\right).$$

In order to realize ϕ! 0, the second sliding surface is defined as follows:

$$
\sigma\_s = \dot{\phi} + \mathfrak{c}\_s \phi \tag{16}
$$

where cs > 0. The derivative of Eq. (16) is obtained as:

$$\dot{\sigma}\_s = \ddot{\phi} + c\_s \dot{\phi} = M\_1 + M\_2 + M\_3(f(\mathbf{x}) + g(\mathbf{x})u(t) + d(t) - \ddot{\mathbf{x}}\_d) + c\_s \dot{\phi} \tag{17}$$

The lumped uncertainty of system is defined as:

$$
\Delta w = M\_3 \tilde{\boldsymbol{\gamma}}\_f \boldsymbol{\xi}\_f + M\_3 \tilde{\boldsymbol{\gamma}}\_g \boldsymbol{\xi}\_3 \boldsymbol{\mu} + M\_3 d(t) \tag{18}
$$

where γ<sup>f</sup> = f(x) � f ∗ (x), <sup>γ</sup><sup>g</sup> <sup>=</sup> <sup>g</sup>(x) � <sup>g</sup><sup>∗</sup> (x). Using Eqs. (17) and (18), the derivative Eq. (17) is rewritten as:

$$\dot{\sigma}\_s = M\_1 + M\_2 + M\_3 f^\*(\mathbf{x}) + M\_3 g^\*(\mathbf{x}) \mu(t) - M\_3 \ddot{\mathbf{x}}\_d + \mathbf{c}\_s \dot{\boldsymbol{\phi}} + \mathbf{w} \tag{19}$$

Based on Eq. (2), the relationship between Eq. (19) and OIT2FNN is expressed as follows:

$$\dot{\sigma}\_s = M\_1 + M\_2 + M\_3 \theta\_f^\* \, ^\* \xi\_f + M\_3 \theta\_g^\* \, ^\* \xi\_g \mu - M\_3 \ddot{\mathbf{x}}\_d + \mathbf{c}\_s \dot{\phi} + \mathbf{w} \tag{20}$$

γ~\_

<sup>f</sup> ¼ �μ1M3σsξ<sup>f</sup> ; <sup>γ</sup>~\_

Lv <sup>¼</sup> <sup>1</sup> 2 σ2 <sup>s</sup> þ 1 2

<sup>v</sup> ¼ σsσ\_<sup>s</sup> þ

1 μ1 <sup>γ</sup>~<sup>f</sup> <sup>γ</sup>~\_

� �

<sup>þ</sup> <sup>M</sup>3Γξzσs~xPS2S<sup>T</sup>

<sup>x</sup>~<sup>T</sup>Qx~<sup>T</sup> � <sup>1</sup>

2

σs ffiffiffi <sup>β</sup> <sup>p</sup> � ffiffiffi β p w !<sup>2</sup>

Lvð Þ� 0 Lvð Þþ T β

<sup>2</sup> <sup>~</sup>xð Þ<sup>0</sup> Px~<sup>T</sup>ð Þþ <sup>0</sup> <sup>1</sup>

Lvð Þþ 0 β

1 2 \_ <sup>~</sup>xPx~<sup>T</sup><sup>þ</sup> 1 2 <sup>~</sup>xP \_ <sup>~</sup>x<sup>T</sup><sup>þ</sup>

Substituting Eq. (27) into Eq. (30), Eq. (30) is rewritten as follows:

<sup>2</sup> Px~<sup>T</sup> <sup>þ</sup>

The derivative of Eq. (29) is then obtained by

L\_

<sup>v</sup> ¼ M3σsγ~<sup>f</sup> ξ<sup>f</sup> þ

L\_

following is achieved.

<sup>v</sup> ¼ � <sup>1</sup> 2

2 4

L\_

where, Lvð Þ¼ <sup>0</sup> <sup>1</sup>

Ξ<sup>1</sup> ¼ γ~<sup>f</sup> γ~<sup>f</sup> � � � � � � ≤ℵ<sup>f</sup>

� � � n o

redefined as follows:

<sup>2</sup> σ<sup>2</sup>

<sup>s</sup>ð Þþ <sup>0</sup> <sup>1</sup>

From Eqs. (32) and (34), the stability is guaranteed.

, Ξ<sup>2</sup> ¼ γ~<sup>g</sup> γ~<sup>g</sup> � � � � � � ≤ℵ<sup>g</sup>

� � � n o

always positive, so Eq. (33) is determined as:

have:

<sup>g</sup> ¼ �μ2M3σsξgu; <sup>Γ</sup>\_ ¼ �μ3M3σsξz~xPS2S<sup>T</sup>

In order to make a proof, in this work the following Lyapunov function candidate is proposed.

1 2μ<sup>1</sup> γ~2 <sup>f</sup> þ 1 2μ<sup>2</sup> γ~2 <sup>g</sup> þ 1 2μ<sup>3</sup>

> 1 μ1 <sup>γ</sup>~<sup>f</sup> <sup>γ</sup>~\_ <sup>f</sup> þ 1 μ2 <sup>γ</sup>~gγ~\_ <sup>g</sup> þ 1 μ3

� �

It is noted that Eq. (24) is used in finding Eq. (31). Substituting Eq. (28) into Eq. (31), the

<sup>þ</sup> <sup>β</sup>w<sup>2</sup>

Eq. (32) cannot use for conclusion of stability. Hence, it will be integrated from t = 0 to t = T, we

ð T

0 w2 dt ≥ 1 2 ð T

<sup>f</sup>ð Þþ <sup>0</sup> <sup>1</sup> 2μ<sup>2</sup> γ~2

0

� � � � � �

� � �

From the boundedness of the parameters γ~<sup>f</sup> and γ~g, the closed sets are defined as

In here, ℵf, ℵg, δ1, δ<sup>2</sup> are the choosing parameters. Hence, the adjusted adaptation laws are

, Ξδ<sup>1</sup> ¼ γ~<sup>f</sup> γ~<sup>f</sup>

2μ<sup>1</sup> γ~2

ð T

0 w2 dt ≥ 1 2 ð T

ΓΓ\_� þ <sup>σ</sup>sw � <sup>σ</sup><sup>2</sup>

1 μ2 <sup>γ</sup>~gγ~\_ g

� � � (31)

s <sup>β</sup> � <sup>1</sup> 2

ρPS2S<sup>T</sup>

0

<sup>g</sup>ð Þþ <sup>0</sup> <sup>1</sup> 2μ<sup>3</sup> Γ2

≤ℵ<sup>f</sup> þδ<sup>1</sup>

n o

ρPS2S<sup>T</sup>

Robust Adaptive Controls of a Vehicle Seat Suspension System

http://dx.doi.org/10.5772/intechopen.71422

<sup>2</sup> <sup>P</sup> <sup>≤</sup> � <sup>1</sup> 2

<sup>2</sup> <sup>P</sup> � <sup>1</sup> 2 x~<sup>T</sup>Qx~<sup>T</sup>

~xQx~Tdt (33)

~xQx~Tdt ≥ 0 (34)

, Ξδ<sup>2</sup> ¼ γ~<sup>g</sup> γ~<sup>g</sup>

� � � � � �

� � �

≤ℵ<sup>g</sup> þδ<sup>2</sup>

.

n o

<sup>~</sup>xPx~<sup>T</sup> <sup>þ</sup>

þ M3σsγ~gξgu þ

1 μ3 <sup>2</sup> Px~<sup>T</sup> (28)

9

Γ<sup>2</sup> (29)

ΓΓ\_ (30)

<sup>x</sup>~<sup>T</sup>Qx~<sup>T</sup> <sup>þ</sup> <sup>β</sup>w<sup>2</sup> (32)

ð Þ0 . The value Lv(T) is

where

θ∗ <sup>f</sup> ¼ arg min<sup>θ</sup><sup>f</sup> <sup>∈</sup> Δθ<sup>f</sup> sup<sup>x</sup><sup>∈</sup> <sup>Δ</sup><sup>x</sup> f xð Þ� f <sup>∗</sup> j j ð Þ<sup>x</sup> � �, <sup>θ</sup><sup>∗</sup> <sup>g</sup> <sup>¼</sup> arg min<sup>θ</sup><sup>g</sup> <sup>∈</sup> Δθ<sup>g</sup> sup<sup>x</sup><sup>∈</sup> <sup>Δ</sup><sup>x</sup> g xð Þ� <sup>g</sup><sup>∗</sup> j j ð Þ<sup>x</sup> � �, Δθ<sup>f</sup> <sup>=</sup> {θf∈ Rn , <sup>k</sup>θf<sup>k</sup> <sup>≤</sup> <sup>Θ</sup>f}, Δθ<sup>g</sup> = {θg∈R<sup>n</sup> , <sup>k</sup>θg<sup>k</sup> <sup>≤</sup> <sup>Θ</sup>g}, <sup>Δ</sup><sup>x</sup> = {<sup>x</sup> <sup>∈</sup>R<sup>n</sup> , kxk ≤ Θx}

Now, an equivalent control is determined from Eq. (20) based on the assumption σ\_<sup>s</sup> ≈ 0:

$$\mu\_1 = \frac{1}{M\_3 \hat{\theta}\_{\mathcal{G}} \xi\_{\mathcal{G}}} \left( -M\_1 - M\_2 - M\_3 \hat{\theta}\_f \xi\_f + M\_3 \ddot{\mathbf{x}}\_d - \mathfrak{c}\_s \dot{\phi} \right) \tag{21}$$

The equivalent control u<sup>1</sup> cannot control the system because it cannot compensate the error from the fuzzy approximation. To guarantee the robustness and stability in control, a robust control part u<sup>2</sup> should be introduced as follows:

$$\mu\_2 = \frac{1}{M\_3 \hat{\theta}\_{\mathcal{S}} \xi\_{\mathcal{S}}} \left( - \sum\_{i=1}^{n-1} P\_{(n-1)i} \mathbf{x}\_i - \frac{\sigma\_s}{\beta} + \frac{1}{2} M\_3 \Gamma\_{\mathbf{S}}^{\mathbf{x}} \mathbf{\tilde{s}} \mathbf{P} \mathbf{S}\_2 \mathbf{S}\_2^T \mathbf{P} \mathbf{\tilde{x}}^T \right) \tag{22}$$

Then, the total control u of the system is determined as follows:

$$
\mu = \mu\_1 + \mu\_2 \tag{23}
$$

The control u<sup>2</sup> is the combination of two sliding surfaces ss and σs. The value Γ is the adaptive parameter where its boundary is given by ΔΓ = {Γ ∈R, kΓk ≤ ΘΓ, σsΓξfz ≤ ρ}, and ΘΓ is constant boundary. The matrix P = P<sup>T</sup> ≥ 0 in which its result is a solution of Riccati-like equation given by

$$\mathbf{P}\mathbf{S}\_1 + \mathbf{S}\_1^\mathsf{T}\mathbf{P} + \mathbf{Q} - \sigma\_s \Gamma \xi\_z \mathbf{P} \mathbf{S}\_2 \mathbf{S}\_2^\mathsf{T}\mathbf{P} + \rho \mathbf{P} \mathbf{S}\_2 \mathbf{S}\_2^\mathsf{T}\mathbf{P} = 0\tag{24}$$

where, ρ ≥ σsΓξz, ρ is the prescribed attenuation level, Q = QT ≥ 0, ξ<sup>z</sup> is consequent membership value of the OIT2FNN. When the value ρ = σsΓξz, the Riccati-like equation is rewritten as:

$$\mathbf{P} \mathbf{S}\_1 + \mathbf{S}\_1^T \mathbf{P} + \mathbf{Q} = 0 \tag{25}$$

Now, Eq. (20) can be analyzed as follows:

$$\dot{\sigma}\_s = M\_1 + M\_2 + M\_3 \ddot{\gamma}\_f \xi\_f + M\_3 \ddot{\gamma}\_g \xi\_g \mu - M\_3 \ddot{\chi}\_d + \mathfrak{c}\_s \dot{\phi} + \mathfrak{w} + \left[ M\_3 \widehat{\Theta}\_f \xi\_f + M\_3 \widehat{\Theta}\_g \xi\_g \mu \right] \tag{26}$$

where γ~<sup>f</sup> ¼ θ<sup>f</sup> <sup>∗</sup> � <sup>θ</sup>b<sup>f</sup> , <sup>γ</sup>~<sup>g</sup> <sup>¼</sup> <sup>θ</sup><sup>g</sup> <sup>∗</sup> � <sup>θ</sup>bg. Using Eqs. (23) and (26), Eq. (26) is rewritten by

$$\dot{\sigma}\_s = \left[ -\sum\_{i=1}^{n-1} P\_{(n-1)i} \mathbf{x}\_i - \frac{\sigma\_s}{\beta} + \frac{1}{2} M\_3 \Gamma \xi\_2 \tilde{\mathbf{x}} \mathbf{PS}\_2 \mathbf{S}\_2^T \mathbf{P} \tilde{\mathbf{x}}^T \right] + \left[ M\_3 \tilde{\boldsymbol{\gamma}}\_f \xi\_f + M\_3 \tilde{\boldsymbol{\gamma}}\_g \xi\_g \mu + w \right] \tag{27}$$

Now, the stability of the proposed adaptive control system can be solidly proved with Eqs. (21)–(23) and adaptation laws as follows:

Robust Adaptive Controls of a Vehicle Seat Suspension System http://dx.doi.org/10.5772/intechopen.71422 9

$$\dot{\tilde{\boldsymbol{\gamma}}}\_{f} = -\mu\_1 M\_3 \sigma\_s \xi\_{\tilde{\boldsymbol{\gamma}}}; \ \dot{\tilde{\boldsymbol{\gamma}}}\_{\mathcal{S}} = -\mu\_2 M\_3 \sigma\_s \xi\_{\mathcal{S}} \mu; \ \dot{\tilde{\boldsymbol{\Gamma}}} = -\mu\_3 M\_3 \sigma\_s \xi\_z \tilde{\mathbf{x}} \mathbf{PS}\_2 \mathbf{S}\_2^T \mathbf{P} \tilde{\mathbf{x}}^T \tag{28}$$

In order to make a proof, in this work the following Lyapunov function candidate is proposed.

$$L\_v = \frac{1}{2}\sigma^2\_{\ s} + \frac{1}{2}\tilde{\mathbf{x}}\mathbf{P}\tilde{\mathbf{x}}^T + \frac{1}{2\mu\_1}\tilde{\boldsymbol{\gamma}}^2{\boldsymbol{\gamma}}\_f + \frac{1}{2\mu\_2}\tilde{\boldsymbol{\gamma}}^2{\boldsymbol{\gamma}}\_{\mathcal{S}} + \frac{1}{2\mu\_3}\boldsymbol{\Gamma}^2 \tag{29}$$

The derivative of Eq. (29) is then obtained by

σ\_<sup>s</sup> ¼ M<sup>1</sup> þ M<sup>2</sup> þ M3θ<sup>f</sup>

<sup>u</sup><sup>1</sup> <sup>¼</sup> <sup>1</sup>

control part u<sup>2</sup> should be introduced as follows:

<sup>u</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup>

M3θbgξ<sup>g</sup>

PS1 <sup>þ</sup> ST

Now, Eq. (20) can be analyzed as follows:

<sup>∗</sup> � <sup>θ</sup>b<sup>f</sup> , <sup>γ</sup>~<sup>g</sup> <sup>¼</sup> <sup>θ</sup><sup>g</sup>

Eqs. (21)–(23) and adaptation laws as follows:

<sup>P</sup>ð Þ <sup>n</sup>�<sup>1</sup> ixi � <sup>σ</sup><sup>s</sup>

β þ 1 2

i¼1

<sup>σ</sup>\_<sup>s</sup> ¼ �X<sup>n</sup>�<sup>1</sup>

where γ~<sup>f</sup> ¼ θ<sup>f</sup>

� Xn�1 i¼1

Then, the total control u of the system is determined as follows:

M3θbgξ<sup>g</sup>

<sup>∗</sup> j j ð Þ<sup>x</sup> � �, <sup>θ</sup><sup>∗</sup>

<sup>f</sup> ¼ arg min<sup>θ</sup><sup>f</sup> <sup>∈</sup> Δθ<sup>f</sup> sup<sup>x</sup><sup>∈</sup> <sup>Δ</sup><sup>x</sup> f xð Þ� f

8 Adaptive Robust Control Systems

, <sup>k</sup>θf<sup>k</sup> <sup>≤</sup> <sup>Θ</sup>f}, Δθ<sup>g</sup> = {θg∈R<sup>n</sup>

where

{θf∈ Rn

θ∗

∗

, <sup>k</sup>θg<sup>k</sup> <sup>≤</sup> <sup>Θ</sup>g}, <sup>Δ</sup><sup>x</sup> = {<sup>x</sup> <sup>∈</sup>R<sup>n</sup>

Now, an equivalent control is determined from Eq. (20) based on the assumption σ\_<sup>s</sup> ≈ 0:

The equivalent control u<sup>1</sup> cannot control the system because it cannot compensate the error from the fuzzy approximation. To guarantee the robustness and stability in control, a robust

<sup>P</sup>ð Þ <sup>n</sup>�<sup>1</sup> ixi � <sup>σ</sup><sup>s</sup>

The control u<sup>2</sup> is the combination of two sliding surfaces ss and σs. The value Γ is the adaptive parameter where its boundary is given by ΔΓ = {Γ ∈R, kΓk ≤ ΘΓ, σsΓξfz ≤ ρ}, and ΘΓ is constant boundary. The matrix P = P<sup>T</sup> ≥ 0 in which its result is a solution of Riccati-like equation given by

where, ρ ≥ σsΓξz, ρ is the prescribed attenuation level, Q = QT ≥ 0, ξ<sup>z</sup> is consequent membership value of the OIT2FNN. When the value ρ = σsΓξz, the Riccati-like equation is rewritten as:

<sup>σ</sup>\_<sup>s</sup> <sup>¼</sup> <sup>M</sup><sup>1</sup> <sup>þ</sup> <sup>M</sup><sup>2</sup> <sup>þ</sup> <sup>M</sup>3γ~<sup>f</sup> <sup>ξ</sup><sup>f</sup> <sup>þ</sup> <sup>M</sup>3γ~gξgu � <sup>M</sup>3x€<sup>d</sup> <sup>þ</sup> csϕ\_ <sup>þ</sup> <sup>w</sup> <sup>þ</sup> <sup>M</sup>3θb<sup>f</sup> <sup>ξ</sup><sup>f</sup> <sup>þ</sup> <sup>M</sup>3θbgξgu

M3Γξz~xPS2S<sup>T</sup>

Now, the stability of the proposed adaptive control system can be solidly proved with

h i "

<sup>1</sup> <sup>P</sup> <sup>þ</sup> <sup>Q</sup> � <sup>σ</sup>sΓξzPS2S<sup>T</sup>

PS1 <sup>þ</sup> <sup>S</sup><sup>T</sup>

ξ<sup>f</sup> þ M3θ<sup>g</sup>

∗

�M<sup>1</sup> � <sup>M</sup><sup>2</sup> � <sup>M</sup>3θb<sup>f</sup> <sup>ξ</sup><sup>f</sup> <sup>þ</sup> <sup>M</sup>3x€<sup>d</sup> � csϕ\_ � �

β þ 1 2

<sup>ξ</sup>gu � <sup>M</sup>3x€<sup>d</sup> <sup>þ</sup> csϕ\_ <sup>þ</sup> <sup>w</sup> (20)

(21)

(22)

(26)

(27)

<sup>g</sup> <sup>¼</sup> arg min<sup>θ</sup><sup>g</sup> <sup>∈</sup> Δθ<sup>g</sup> sup<sup>x</sup><sup>∈</sup> <sup>Δ</sup><sup>x</sup> g xð Þ� <sup>g</sup><sup>∗</sup> j j ð Þ<sup>x</sup> � �, Δθ<sup>f</sup> <sup>=</sup>

, kxk ≤ Θx}

M3Γξz~xPS2S<sup>T</sup>

<sup>2</sup> <sup>P</sup> <sup>þ</sup> <sup>ρ</sup>PS2S<sup>T</sup>

<sup>∗</sup> � <sup>θ</sup>bg. Using Eqs. (23) and (26), Eq. (26) is rewritten by

u ¼ u<sup>1</sup> þ u<sup>2</sup> (23)

<sup>1</sup> P þ Q ¼ 0 (25)

<sup>2</sup> Px~<sup>T</sup>� þ <sup>M</sup>3γ~<sup>f</sup> <sup>ξ</sup><sup>f</sup> <sup>þ</sup> <sup>M</sup>3γ~gξgu <sup>þ</sup> <sup>w</sup>

h i

<sup>2</sup> Px~<sup>T</sup><sup>Þ</sup>

<sup>2</sup> P ¼ 0 (24)

$$\dot{L}\_v = \sigma\_s \dot{\sigma}\_s + \frac{1}{2} \ddot{\mathbf{x}} \mathbf{P} \dot{\mathbf{x}}^T + \frac{1}{2} \ddot{\mathbf{x}} \mathbf{P} \dot{\mathbf{x}}^T + \frac{1}{\mu\_1} \ddot{\boldsymbol{\gamma}}\_f \dot{\boldsymbol{\gamma}}\_f + \frac{1}{\mu\_2} \ddot{\boldsymbol{\gamma}}\_g \dot{\boldsymbol{\gamma}}\_g + \frac{1}{\mu\_3} \Gamma \dot{\Gamma} \tag{30}$$

Substituting Eq. (27) into Eq. (30), Eq. (30) is rewritten as follows:

$$\begin{aligned} \dot{L}\_{v} &= \left[M\_{3}\sigma\_{s}\ddot{\boldsymbol{\gamma}}\_{f}\boldsymbol{\xi}\_{f} + \frac{1}{\mu\_{1}}\ddot{\boldsymbol{\gamma}}\_{f}\dot{\boldsymbol{\dot{\gamma}}}\right] + \left[M\_{3}\sigma\_{s}\ddot{\boldsymbol{\gamma}}\_{g}\boldsymbol{\xi}\_{g}\boldsymbol{u} + \frac{1}{\mu\_{2}}\ddot{\boldsymbol{\gamma}}\_{g}\dot{\boldsymbol{\dot{\gamma}}}\_{\boldsymbol{\beta}}\right] \\ &+ \left[M\_{3}\Gamma\boldsymbol{\xi}\_{z}\sigma\_{s}\ddot{\mathbf{x}}\mathbf{PS}\_{2}\mathbf{S}\_{2}^{T}\mathbf{P}\ddot{\mathbf{x}}^{T} + \frac{1}{\mu\_{3}}\Gamma\dot{\boldsymbol{\Gamma}}\right] + \left[\sigma\_{s}w - \frac{\sigma\_{s}^{2}}{\beta} - \frac{1}{2}\rho\mathbf{PS}\_{2}\mathbf{S}\_{2}^{T}\mathbf{P} - \frac{1}{2}\ddot{\mathbf{x}}^{T}\mathbf{Q}\ddot{\mathbf{x}}^{T}\right] \end{aligned} \tag{31}$$

It is noted that Eq. (24) is used in finding Eq. (31). Substituting Eq. (28) into Eq. (31), the following is achieved.

$$\dot{L}\_v = \left[ -\frac{1}{2} \ddot{\mathbf{x}}^T \mathbf{Q} \ddot{\mathbf{x}}^T - \frac{1}{2} \left( \frac{\sigma\_s}{\sqrt{\beta}} - \sqrt{\beta} w \right)^2 + \beta w^2 \right] - \frac{1}{2} \rho \mathbf{P} \mathbf{S}\_2 \mathbf{S}\_2^T \mathbf{P} \le -\frac{1}{2} \ddot{\mathbf{x}}^T \mathbf{Q} \ddot{\mathbf{x}}^T + \beta w^2 \tag{32}$$

Eq. (32) cannot use for conclusion of stability. Hence, it will be integrated from t = 0 to t = T, we have:

$$L\_v(0) - L\_v(T) + \beta \int\_0^T w^2 dt \ge \frac{1}{2} \int\_0^T \tilde{\mathbf{x}} \mathbf{Q} \tilde{\mathbf{x}}^T dt \tag{33}$$

where, Lvð Þ¼ <sup>0</sup> <sup>1</sup> <sup>2</sup> σ<sup>2</sup> <sup>s</sup>ð Þþ <sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>~</sup>xð Þ<sup>0</sup> Px~<sup>T</sup>ð Þþ <sup>0</sup> <sup>1</sup> 2μ<sup>1</sup> γ~2 <sup>f</sup>ð Þþ <sup>0</sup> <sup>1</sup> 2μ<sup>2</sup> γ~2 <sup>g</sup>ð Þþ <sup>0</sup> <sup>1</sup> 2μ<sup>3</sup> Γ2 ð Þ0 . The value Lv(T) is always positive, so Eq. (33) is determined as:

$$L\_v(0) + \beta \int\_0^T w^2 dt \ge \frac{1}{2} \int\_0^T \|\mathbf{Q}\|^T dt \ge 0 \tag{34}$$

From Eqs. (32) and (34), the stability is guaranteed.

From the boundedness of the parameters γ~<sup>f</sup> and γ~g, the closed sets are defined as Ξ<sup>1</sup> ¼ γ~<sup>f</sup> γ~<sup>f</sup> � � � � � � ≤ℵ<sup>f</sup> � � � n o , Ξ<sup>2</sup> ¼ γ~<sup>g</sup> γ~<sup>g</sup> � � � � � � ≤ℵ<sup>g</sup> � � � n o , Ξδ<sup>1</sup> ¼ γ~<sup>f</sup> γ~<sup>f</sup> � � � � � � ≤ℵ<sup>f</sup> þδ<sup>1</sup> � � � n o , Ξδ<sup>2</sup> ¼ γ~<sup>g</sup> γ~<sup>g</sup> � � � � � � ≤ℵ<sup>g</sup> þδ<sup>2</sup> � � � n o .

In here, ℵf, ℵg, δ1, δ<sup>2</sup> are the choosing parameters. Hence, the adjusted adaptation laws are redefined as follows:

γ~\_ <sup>f</sup> ¼ �μ1M3σsξ<sup>f</sup> if γ~<sup>f</sup> � � � � < ℵ<sup>f</sup> or γ~<sup>f</sup> � � � � <sup>¼</sup> <sup>ℵ</sup><sup>f</sup> and <sup>M</sup>3σsξ<sup>f</sup> <sup>γ</sup>~<sup>f</sup> <sup>≥</sup> <sup>0</sup> � � �μ1M3σsξ<sup>f</sup> þ μ<sup>1</sup> γ~f � � � � � � 2 � ℵ<sup>f</sup> � �M3σsξ<sup>f</sup> <sup>γ</sup>~<sup>f</sup> δ<sup>1</sup> γ~<sup>f</sup> � � � � � � <sup>2</sup> if γ~<sup>f</sup> � � � � ¼ ℵ<sup>f</sup> and M3σsξ<sup>f</sup> γ~<sup>f</sup> < 0 8 >>>>< >>>>: (35) γ~\_ <sup>g</sup> ¼ �μ2M3σsξgu if γ~<sup>g</sup> � � � � � � <sup>&</sup>lt; <sup>ℵ</sup><sup>g</sup> or <sup>γ</sup>~<sup>g</sup> � � � � � � <sup>¼</sup> <sup>ℵ</sup><sup>g</sup> and <sup>M</sup>3σsξguγ~<sup>g</sup> <sup>≥</sup> <sup>0</sup> � � �μ2M3σsξgu þ μ<sup>2</sup> γ~g � � � � � � 2 � ℵ<sup>g</sup> � �M3σsξguγ~<sup>g</sup> δ<sup>2</sup> γ~<sup>g</sup> � � � � � � <sup>2</sup> if γ~<sup>g</sup> � � � � � � <sup>¼</sup> <sup>ℵ</sup><sup>g</sup> and <sup>M</sup>3σsξguγ~<sup>g</sup> <sup>&</sup>lt; <sup>0</sup> 8 >>>>>< >>>>>: (36) <sup>Γ</sup>\_ <sup>¼</sup> �μ3M3σsξz~xPS2S<sup>T</sup> <sup>2</sup> Px~<sup>T</sup>if k k<sup>Γ</sup> <sup>&</sup>lt; ΘΓ or k k<sup>Γ</sup> <sup>¼</sup> ΘΓ <sup>þ</sup> <sup>δ</sup><sup>3</sup> and <sup>M</sup>3σsξz~xPS2S<sup>T</sup> <sup>2</sup> <sup>P</sup>~x<sup>T</sup><sup>Γ</sup> <sup>≥</sup> <sup>0</sup> � � �μ3M3σsξz~xPS2S<sup>T</sup> <sup>2</sup> <sup>P</sup>~x<sup>T</sup> <sup>þ</sup> <sup>μ</sup><sup>3</sup> k k<sup>Γ</sup> <sup>2</sup> � ΘΓ � �M3σsξz~xPS2S<sup>T</sup> <sup>2</sup> P~x<sup>T</sup>Γ <sup>δ</sup>3k k<sup>Γ</sup> <sup>2</sup> if k k¼ <sup>Γ</sup> ΘΓ and <sup>M</sup>3σsξz~xPS2S<sup>T</sup> <sup>2</sup> Px~<sup>T</sup>Γ < 0 8 >>>>>>< >>>>>>:

$$\text{(37)}$$

In the above, δ1, δ<sup>2</sup> and δ<sup>3</sup> are choosing parameters related boundaries of f(x), g(x) and Γ. It is noted here that in order to utilize the states of the system, the Luenberger observer [24] has been used in this work. Figure 1 presents a flow chart of the HAC-PP showing the combination

where, the function f0(x) and g0(x) are the functions of f(x) and g(x) which are determined as:

f xð Þ¼ f0ð Þþ x δf xð Þ; 0 < j j δf xð Þ < k k δf <sup>∞</sup>, g xð Þ¼ g0ð Þþ x δg xð Þ; 0 < j j δg xð Þ < k k δg <sup>∞</sup>:

In the above, δf and δg are two positive vectors. It is noted that D = δf + δgu(t) + d(t) denotes

T

lowing assumption is made: There exists a constant gm ∈ ℜ<sup>+</sup> to satisfy |g(x)| > gm. Without loss of generality, it is assumed that the equation g(x) > gm. The error between a desired output xd and the measured output x is e = xd � x. Hence, the error vector is defined by

g0ð Þ<sup>x</sup> �<sup>f</sup> <sup>0</sup>ð Þþ <sup>x</sup> <sup>x</sup>\_ <sup>d</sup> <sup>þ</sup> KTE <sup>þ</sup> <sup>D</sup><sup>0</sup>

where, f00(x) and g00(x) are the fuzzified functions of f(x) and g(x), respectively. The derivative

x\_ ¼ f0ð Þþ x g0ð Þx u tð Þþ D (38)

Robust Adaptive Controls of a Vehicle Seat Suspension System

http://dx.doi.org/10.5772/intechopen.71422

, δg ¼ 0; 0;…; δg<sup>0</sup> <sup>T</sup>

. In order to formulate the controller, the fol-

<sup>d</sup>�K<sup>T</sup>x\_. Using this derivative function of the

(39)

g0ð Þ<sup>x</sup> �<sup>f</sup> <sup>0</sup>ð Þþ <sup>x</sup> <sup>x</sup>\_ <sup>d</sup> <sup>þ</sup> KTE (40)

g00ð Þ<sup>x</sup> �<sup>f</sup> <sup>00</sup>ð Þþ <sup>x</sup> <sup>x</sup>\_ <sup>d</sup> <sup>þ</sup> <sup>K</sup>TE (41)

:

11

E,

, δf ¼ 0; 0;…; δf <sup>0</sup> <sup>T</sup>

€;…;eð Þ <sup>n</sup>�<sup>1</sup> . The sliding surface ss can be written as s(x, t) = K<sup>T</sup>

As a first step to design the controller, consider the system (3) rewritten by

, g0ð Þ¼ x 0;…; 0; g<sup>0</sup>

sliding surface and Eq. (38), the initial control law u is determined by:

<sup>u</sup> <sup>¼</sup> <sup>1</sup>

Assuming the disturbance of D ≈ 0, then Eq. (39) can be rewritten as:

<sup>u</sup> <sup>¼</sup> <sup>1</sup>

<sup>u</sup> <sup>¼</sup> <sup>1</sup>

The relationship of Eq. (40) and OIT2FNN is expressed by

of E is expressed through Eqs. (40) and (41) as follows:

<sup>T</sup>

of each controller and the prescribed performance.

3. Formulation of HAC-IFV

f0ð Þ¼ x x2;…; xn; f <sup>0</sup>

E ¼ ½e0;e1;e2;…;en� ¼ e;e\_;e

<sup>T</sup>

the uncertain disturbance and D = [0, 0,…, D0]

and its derivative is found as s x \_ð Þ¼ ; <sup>t</sup> <sup>K</sup><sup>T</sup><sup>E</sup> <sup>¼</sup>\_ <sup>K</sup><sup>T</sup>x\_

Figure 1. Control flow chart of the HAC-PP.

In the above, δ1, δ<sup>2</sup> and δ<sup>3</sup> are choosing parameters related boundaries of f(x), g(x) and Γ. It is noted here that in order to utilize the states of the system, the Luenberger observer [24] has been used in this work. Figure 1 presents a flow chart of the HAC-PP showing the combination of each controller and the prescribed performance.
