3. Formulation of HAC-IFV

γ~\_ <sup>f</sup> ¼

γ~\_ <sup>g</sup> ¼

<sup>Γ</sup>\_ <sup>¼</sup>

8 >>>><

10 Adaptive Robust Control Systems

>>>>:

8 >>>>><

>>>>>:

8 >>>>>><

>>>>>>:

�μ1M3σsξ<sup>f</sup> if γ~<sup>f</sup>

�μ1M3σsξ<sup>f</sup> þ μ<sup>1</sup>

�μ2M3σsξgu if γ~<sup>g</sup>

�μ2M3σsξgu þ μ<sup>2</sup>

�μ3M3σsξz~xPS2S<sup>T</sup>

�μ3M3σsξz~xPS2S<sup>T</sup>

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

if k k¼ <sup>Γ</sup> ΘΓ and <sup>M</sup>3σsξz~xPS2S<sup>T</sup>

� � �

� � � � �

γ~f � � � � � � 2 � ℵ<sup>f</sup> � �

γ~g � � � � � � 2 � ℵ<sup>g</sup> � �

<sup>2</sup> <sup>P</sup>~x<sup>T</sup> <sup>þ</sup> <sup>μ</sup><sup>3</sup>

� < ℵ<sup>f</sup> or γ~<sup>f</sup> � � �

> δ<sup>1</sup> γ~<sup>f</sup> � � � � � �

> > � � � � �

� <sup>&</sup>lt; <sup>ℵ</sup><sup>g</sup> or <sup>γ</sup>~<sup>g</sup>

δ<sup>2</sup> γ~<sup>g</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> � �

� � �

� <sup>¼</sup> <sup>ℵ</sup><sup>g</sup> and <sup>M</sup>3σsξguγ~<sup>g</sup> <sup>≥</sup> <sup>0</sup> � �

> � � � � �

M3σsξz~xPS2S<sup>T</sup>

� ¼ ℵ<sup>f</sup> and M3σsξ<sup>f</sup> γ~<sup>f</sup> < 0

� <sup>¼</sup> <sup>ℵ</sup><sup>g</sup> and <sup>M</sup>3σsξguγ~<sup>g</sup> <sup>&</sup>lt; <sup>0</sup>

<sup>2</sup> <sup>P</sup>~x<sup>T</sup><sup>Γ</sup> <sup>≥</sup> <sup>0</sup> � �

<sup>2</sup> P~x<sup>T</sup>Γ

(35)

(36)

(37)

M3σsξ<sup>f</sup> γ~<sup>f</sup>

M3σsξguγ~<sup>g</sup>

k k<sup>Γ</sup> <sup>2</sup> � ΘΓ � �

<sup>2</sup> Px~<sup>T</sup>Γ < 0

<sup>2</sup> if γ~<sup>f</sup>

<sup>2</sup> if γ~<sup>g</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>δ</sup>3k k<sup>Γ</sup> <sup>2</sup>

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

$$
\dot{\mathbf{x}} = \mathbf{f\_0(x)} + \mathbf{g\_0(x)}u(t) + \mathbf{D} \tag{38}
$$

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

$$\mathbf{f}(\mathbf{x}) = \mathbf{f}\_{\mathbf{0}}(\mathbf{x}) + \mathbf{\delta}\mathbf{f}(\mathbf{x}); 0 < |\mathbf{\delta}\mathbf{f}(\mathbf{x})| < \|\mathbf{\delta}\mathbf{f}\|\_{\text{ov}}, \mathbf{g}(\mathbf{x}) = \mathbf{g}\_{\mathbf{0}}(\mathbf{x}) + \mathbf{\delta}\mathbf{g}(\mathbf{x}); 0 < |\mathbf{\delta}\mathbf{g}(\mathbf{x})| < \|\mathbf{\delta}\mathbf{g}\|\_{\text{ov}}.$$

$$\mathbf{f}\_{\mathbf{0}}(\mathbf{x}) = \begin{bmatrix} \mathbf{x}\_{2}, \dots, \mathbf{x}\_{n}, f\_{0} \end{bmatrix}^{T}, \mathbf{g}\_{\mathbf{0}}(\mathbf{x}) = \begin{bmatrix} 0, \dots, 0, \mathbf{g}\_{0} \end{bmatrix}^{T}, \mathbf{\delta}\mathbf{f} = \begin{bmatrix} 0, 0, \dots, \delta f\_{0} \end{bmatrix}^{T}, \mathbf{\delta}\mathbf{g} = \begin{bmatrix} 0, 0, \dots, \delta \mathbf{g}\_{0} \end{bmatrix}^{T}.$$

In the above, δf and δg are two positive vectors. It is noted that D = δf + δgu(t) + d(t) denotes the uncertain disturbance and D = [0, 0,…, D0] T . In order to formulate the controller, the following 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 E ¼ ½e0;e1;e2;…;en� ¼ e;e\_;e €;…;eð Þ <sup>n</sup>�<sup>1</sup> . The sliding surface ss can be written as s(x, t) = K<sup>T</sup> E, 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\_ <sup>d</sup>�K<sup>T</sup>x\_. Using this derivative function of the sliding surface and Eq. (38), the initial control law u is determined by:

$$\mu = \frac{1}{g\_0(\mathbf{x})} \left( -f\_0(\mathbf{x}) + \dot{\mathbf{x}}\_d + \mathbf{K^T E} + D\_0 \right) \tag{39}$$

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

$$
\mu = \frac{1}{g\_0(\mathbf{x})} \left( -f\_0(\mathbf{x}) + \dot{\mathbf{x}}\_d + \mathbf{K^T E} \right) \tag{40}
$$

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

$$\mu = \frac{1}{g\_{00}(\mathbf{x})} \left( -f\_{00}(\mathbf{x}) + \dot{\mathbf{x}}\_d + \mathbf{K^T E} \right) \tag{41}$$

where, f00(x) and g00(x) are the fuzzified functions of f(x) and g(x), respectively. The derivative of E is expressed through Eqs. (40) and (41) as follows:

$$\begin{aligned} \dot{\mathbf{E}} &= \dot{\mathbf{x}}\_{\mathbf{d}} - \dot{\mathbf{x}} = \left( g\_{00}(\mathbf{x}) - g\_{0}(\mathbf{x}) \right) \boldsymbol{\mu} + \left( f\_{00}(\mathbf{x}) - f\_{0}(\mathbf{x}) \right) - \mathbf{K}^{\mathsf{T}} \mathbf{E} \\ &= \mathbf{S\_{1}} \mathbf{E} + \mathbf{S\_{2}} \left[ \left( g\_{00}(\mathbf{x}) - g(\mathbf{x}) \right) \boldsymbol{\mu} + \left( f\_{00}(\mathbf{x}) - f(\mathbf{x}) \right) \right] \end{aligned} \tag{42}$$

Define the minimum approximation error due to fuzzy approximation as follows.

$$w = \left(f\_{\;\;\;00}^\*(\mathbf{x}) - f(\mathbf{x})\right) + \left(\mathbf{g}\_{\;\;\;00}^\*(\mathbf{x}) - \mathbf{g}(\mathbf{x})\right)\mathbf{u} \tag{43}$$

<sup>E</sup>\_ <sup>¼</sup> S1E <sup>þ</sup> S2 <sup>γ</sup>~<sup>f</sup> <sup>ξ</sup><sup>f</sup> <sup>þ</sup> <sup>γ</sup>~gξgu<sup>1</sup> <sup>þ</sup> gou<sup>2</sup> <sup>þ</sup> <sup>w</sup>

where, <sup>γ</sup>~<sup>f</sup> <sup>¼</sup> <sup>γ</sup><sup>f</sup> � <sup>γ</sup>b<sup>f</sup> , <sup>γ</sup>~<sup>g</sup> <sup>¼</sup> <sup>γ</sup><sup>g</sup> � <sup>γ</sup>bg. Consider the Lyapunov function candidate of the system as

1 2α<sup>1</sup> γ~2 <sup>f</sup> þ 1 2α<sup>2</sup> γ~2

The derivative of Eq. (50), and then substituting Eq. (25) into the derivative, the result is

<sup>þ</sup> <sup>E</sup>TPS2<sup>w</sup> <sup>þ</sup>

<sup>þ</sup> ETPS2<sup>w</sup>

þ 1 4ρ wm<sup>2</sup> <sup>≤</sup> � <sup>1</sup> 2

0

0

2ρ

4ρ ð T

0

<sup>2</sup>α<sup>2</sup> <sup>γ</sup>~<sup>2</sup>

is achieved. From the boundedness of the parameters, γ~<sup>f</sup> and γ~<sup>g</sup> are guaranteed by closed sets

0

wm<sup>2</sup> dt ≥ 1 2 ð T

wm<sup>2</sup> dt ≥ 1 2 ð T

1 α1

<sup>E</sup>TPE <sup>þ</sup>

<sup>V</sup> <sup>¼</sup> <sup>1</sup> 2

follows:

obtained as follows:

<sup>V</sup>\_ ¼ � <sup>1</sup> 2

<sup>V</sup>\_ <sup>≤</sup> � <sup>1</sup> 2

¼ � <sup>1</sup> 2

gm <sup>p</sup> .

where, wm <sup>¼</sup> <sup>w</sup>ffiffiffiffi

where, <sup>V</sup>ð Þ¼ <sup>0</sup> <sup>1</sup>

þ 1 α2

ETQE � gm

From Eq. (51), adaptation laws are established as follows:

Applying Eqs. (52) and (53), Eq. (51) can be written as follows:

Γξ<sup>z</sup>

ffiffiffiffiffiffiffi gm Γξ<sup>z</sup> r

ETQE � gm

ETQE �

<sup>2</sup> ETð Þ<sup>0</sup> PEð Þþ <sup>0</sup> <sup>1</sup>

Γξ<sup>z</sup>

<sup>α</sup>2ETPS2ξgu<sup>1</sup> � <sup>γ</sup>~\_

� �γ~<sup>g</sup>

ETPS2 � �<sup>2</sup>

γ~\_

γ~\_

ETPS2 � �<sup>2</sup>

Now, the integration of (54) from t = 0 to t = T yields the following equation.

<sup>V</sup>ð Þ� <sup>0</sup> V Tð Þþ <sup>1</sup>

<sup>V</sup>ð Þþ <sup>0</sup> <sup>1</sup> 4ρ ð T

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

<sup>2</sup>α<sup>1</sup> <sup>γ</sup>~<sup>2</sup>

The value of V(T) ≥ 0, and thus Eq. (55) is rewritten as follows:

� �<sup>2</sup>

<sup>E</sup>TPS2 � wm

g

h i (49)

Robust Adaptive Controls of a Vehicle Seat Suspension System

<sup>α</sup>1ETPS2ξ<sup>f</sup> � <sup>γ</sup>~\_

<sup>f</sup> ¼ �α1ETPS2ξ<sup>f</sup> (52)

<sup>g</sup> ¼ �α2ETPS2ξgu<sup>1</sup> (53)

<sup>E</sup>TQE <sup>þ</sup>

1 4ρ wm<sup>2</sup>

ETQEdt (55)

ETQEdt (56)

<sup>g</sup>ð Þ0 . Hence the H-infinity tracking performance

� �γ~<sup>f</sup>

<sup>g</sup> (50)

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

f

(51)

13

(54)

Substituting functions of f00(x), g00(x) and (43) into Eq. (42) yields the following equation.

$$\dot{\mathbf{E}} = \mathbf{S\_1E} + \mathbf{S\_2} \left[ \left( \theta\_f^\* - \theta\_f \right) \xi\_f + \left( \theta\_g^\* - \theta\_\mathcal{g} \right) \xi\_\mathcal{g} \mu + w \right] \tag{44}$$

Let <sup>γ</sup><sup>f</sup> <sup>¼</sup> <sup>θ</sup><sup>∗</sup> <sup>f</sup> � θ<sup>f</sup> � �, <sup>γ</sup><sup>g</sup> <sup>¼</sup> <sup>θ</sup><sup>∗</sup> <sup>g</sup> � θ<sup>g</sup> � �. From Eq. (44), the equivalence control <sup>u</sup><sup>1</sup> established without the minimum approximation error w is defined as follows:

$$
\mu\_1 = \frac{1}{\widehat{\mathcal{V}}\_{\mathcal{S}} \xi\_{\mathcal{S}}} \left( -\widehat{\mathcal{V}}\_f \xi\_f \right) \tag{45}
$$

where, <sup>γ</sup>b<sup>f</sup> and <sup>γ</sup>b<sup>g</sup> are the estimates of <sup>γ</sup><sup>f</sup> and <sup>γ</sup>g, respectively. The control <sup>u</sup><sup>1</sup> cannot use for control the system because of the error from the fuzzy approximation. To deal with this problem, a new robust compensator based on the inversely fuzzified value is suggested as follows:

$$\mu\_2 = -\frac{1}{\Gamma \xi\_z} \mathbf{E}^{\mathsf{T}} \mathbf{P} \mathbf{S}\_2 \tag{46}$$

where, Γ is a constant, and P = PT ≥ 0 is the solution of the following Riccati-like equation.

$$\mathbf{P}\mathbf{S}\_1 + \mathbf{S}\_1^T\mathbf{P} + \mathbf{Q} - \frac{1}{\Gamma\xi\_z}\mathbf{P}\mathbf{S}\_2\mathbf{S}\_2^T\mathbf{P} + \rho\mathbf{P}\mathbf{S}\_2\mathbf{S}\_2^T\mathbf{P} = 0\tag{47}$$

where, ρ ≥ <sup>1</sup> Γξ<sup>z</sup> , ρ is the prescribed attenuation level, Q = Q<sup>T</sup> ≥ 0, ξ<sup>z</sup> is consequent membership value of the OIT2FNN. When the value <sup>ρ</sup> <sup>¼</sup> <sup>1</sup> Γξ<sup>z</sup> , the Riccati-like equation is obtain as given in Eq. (25). It is noteworthy that Eq. (25) is objective to guarantee the stability of the system. If this condition is obtained, the fuzzy approximation error is removed, and then the control u<sup>1</sup> is the main controller to retain the stability of the system. From Eqs. (45) and (46), the final fuzzy control of the system is determined as follows:

$$
\mu = \mu\_1 + \mu\_2 = \frac{1}{\widehat{\mathcal{V}}\_{\mathcal{S}} \mathbb{E}\_{\mathcal{S}}} \left( -\widehat{\mathcal{V}}\_f \mathbb{E}\_f \right) - \frac{1}{\Gamma \xi\_z} \mathbf{E}^{\mathsf{T}} \mathbf{P} \mathbf{S}\_2 \tag{48}
$$

Now, substituting Eq. (48) into (44) yields he following.

Robust Adaptive Controls of a Vehicle Seat Suspension System http://dx.doi.org/10.5772/intechopen.71422 13

$$\dot{\mathbf{E}} = \mathbf{S\_1E} + \mathbf{S\_2} \left[ \tilde{\boldsymbol{\gamma}}\_f \boldsymbol{\xi}\_f + \tilde{\boldsymbol{\gamma}}\_g \boldsymbol{\xi}\_g \boldsymbol{\mu}\_1 + \mathbf{g}\_o \boldsymbol{\mu}\_2 + \mathbf{w} \right] \tag{49}$$

where, <sup>γ</sup>~<sup>f</sup> <sup>¼</sup> <sup>γ</sup><sup>f</sup> � <sup>γ</sup>b<sup>f</sup> , <sup>γ</sup>~<sup>g</sup> <sup>¼</sup> <sup>γ</sup><sup>g</sup> � <sup>γ</sup>bg. Consider the Lyapunov function candidate of the system as follows:

$$V = \frac{1}{2} \mathbf{E}^{\mathsf{T}} \mathbf{P} \mathbf{E} + \frac{1}{2\alpha\_1} \tilde{\boldsymbol{\gamma}}\_f^2 + \frac{1}{2\alpha\_2} \tilde{\boldsymbol{\gamma}}\_g^2 \tag{50}$$

The derivative of Eq. (50), and then substituting Eq. (25) into the derivative, the result is obtained as follows:

$$\begin{split} \dot{V} &= -\frac{1}{2} \mathbf{E}^{\mathsf{T}} \mathbf{Q} \mathbf{E} - \frac{\mathcal{g}\_{m}}{\Gamma \underline{\boldsymbol{\zeta}}\_{z}} \left( \mathbf{E}^{\mathsf{T}} \mathbf{P} \mathbf{S}\_{2} \right)^{2} + \mathbf{E}^{\mathsf{T}} \mathbf{P} \mathbf{S}\_{2} w + \frac{1}{\alpha\_{1}} \left( \alpha\_{1} \mathbf{E}^{\mathsf{T}} \mathbf{P} \mathbf{S}\_{2} \underline{\boldsymbol{\xi}}\_{f} - \dot{\boldsymbol{\mathcal{V}}}\_{f} \right) \ddot{\boldsymbol{\mathcal{V}}}\_{f} \\ &+ \frac{1}{\alpha\_{2}} \left( \alpha\_{2} \mathbf{E}^{\mathsf{T}} \mathbf{P} \mathbf{S}\_{2} \underline{\boldsymbol{\xi}}\_{g} \boldsymbol{u}\_{1} - \dot{\boldsymbol{\mathcal{V}}}\_{\mathcal{g}} \right) \ddot{\boldsymbol{\mathcal{V}}}\_{\mathcal{g}} \end{split} \tag{51}$$

From Eq. (51), adaptation laws are established as follows:

$$\dot{\boldsymbol{\tilde{\gamma}}}\_f = -\boldsymbol{\alpha}\_1 \mathbf{E}^{\mathsf{T}} \mathbf{P} \mathbf{S}\_2 \boldsymbol{\xi}\_f \tag{52}$$

$$\dot{\mathcal{V}}\_{\mathcal{S}} = -\alpha\_2 \mathbf{E}^{\mathsf{T}} \mathbf{P} \mathbf{S}\_2 \boldsymbol{\xi}\_{\mathcal{S}} \boldsymbol{\mu}\_1 \tag{53}$$

Applying Eqs. (52) and (53), Eq. (51) can be written as follows:

$$\begin{split} \dot{V} &\leq -\frac{1}{2} \mathbf{E}^{\mathsf{T}} \mathbf{Q} \mathbf{E} - \frac{g\_{m}}{\Gamma \xi\_{z}} \left( \mathbf{E}^{\mathsf{T}} \mathbf{P} \mathbf{S}\_{2} \right)^{2} + \mathbf{E}^{\mathsf{T}} \mathbf{P} \mathbf{S}\_{2} w \\ &= -\frac{1}{2} \mathbf{E}^{\mathsf{T}} \mathbf{Q} \mathbf{E} - \left( \sqrt{\frac{g\_{m}}{\Gamma \xi\_{z}}} \mathbf{E}^{\mathsf{T}} \mathbf{P} \mathbf{S}\_{2} - \frac{w\_{m}}{2\rho} \right)^{2} + \frac{1}{4\rho} w\_{m}{}^{2} \leq -\frac{1}{2} \mathbf{E}^{\mathsf{T}} \mathbf{Q} \mathbf{E} + \frac{1}{4\rho} w\_{m}{}^{2} \end{split} \tag{54}$$

where, wm <sup>¼</sup> <sup>w</sup>ffiffiffiffi gm <sup>p</sup> .

<sup>E</sup>\_ <sup>¼</sup> <sup>x</sup>\_<sup>d</sup> � <sup>x</sup>\_ <sup>¼</sup> g00ð Þ� <sup>x</sup> g0ð Þ<sup>x</sup> � �<sup>u</sup> <sup>þ</sup> <sup>f</sup> <sup>00</sup>ð Þ� <sup>x</sup> <sup>f</sup> <sup>0</sup>ð Þ<sup>x</sup> � � � <sup>K</sup>TE

(42)

(44)

(45)

<sup>00</sup>ð Þ� <sup>x</sup> g xð Þ � �<sup>u</sup> (43)

ETPS<sup>2</sup> (46)

, the Riccati-like equation is obtain as given in

<sup>2</sup> P ¼ 0 (47)

ETPS2 (48)

<sup>¼</sup> S1E <sup>þ</sup> S2 g00ð Þ� <sup>x</sup> g xð Þ � �<sup>u</sup> <sup>þ</sup> <sup>f</sup> <sup>00</sup>ð Þ� <sup>x</sup> f xð Þ � � � �

Define the minimum approximation error due to fuzzy approximation as follows.

<sup>00</sup>ð Þ� <sup>x</sup> f xð Þ � � <sup>þ</sup> <sup>g</sup><sup>∗</sup>

Substituting functions of f00(x), g00(x) and (43) into Eq. (42) yields the following equation.

<sup>ξ</sup><sup>f</sup> <sup>þ</sup> <sup>θ</sup><sup>∗</sup>

�γb<sup>f</sup> <sup>ξ</sup><sup>f</sup> � �

where, <sup>γ</sup>b<sup>f</sup> and <sup>γ</sup>b<sup>g</sup> are the estimates of <sup>γ</sup><sup>f</sup> and <sup>γ</sup>g, respectively. The control <sup>u</sup><sup>1</sup> cannot use for control the system because of the error from the fuzzy approximation. To deal with this problem, a new robust compensator based on the inversely fuzzified value is suggested as

h i

<sup>g</sup> � θ<sup>g</sup> � �

ξgu þ w

. From Eq. (44), the equivalence control u<sup>1</sup> established

<sup>f</sup> � θ<sup>f</sup> � �

<sup>u</sup><sup>1</sup> <sup>¼</sup> <sup>1</sup> <sup>γ</sup>b<sup>g</sup>ξ<sup>g</sup>

<sup>u</sup><sup>2</sup> ¼ � <sup>1</sup> Γξ<sup>z</sup>

where, Γ is a constant, and P = PT ≥ 0 is the solution of the following Riccati-like equation.

Γξ<sup>z</sup>

Γξ<sup>z</sup>

<sup>γ</sup>b<sup>g</sup>ξ<sup>g</sup>

Eq. (25). It is noteworthy that Eq. (25) is objective to guarantee the stability of the system. If this condition is obtained, the fuzzy approximation error is removed, and then the control u<sup>1</sup> is the main controller to retain the stability of the system. From Eqs. (45) and (46), the final fuzzy

> �γb<sup>f</sup> <sup>ξ</sup><sup>f</sup> � �

PS2ST

<sup>2</sup> <sup>P</sup> <sup>þ</sup> <sup>ρ</sup>PS2ST

� 1 Γξ<sup>z</sup>

, ρ is the prescribed attenuation level, Q = Q<sup>T</sup> ≥ 0, ξ<sup>z</sup> is consequent membership

<sup>1</sup> <sup>P</sup> <sup>þ</sup> <sup>Q</sup> � <sup>1</sup>

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

w ¼ f ∗

<sup>E</sup>\_ <sup>¼</sup> S1E <sup>þ</sup> <sup>S</sup><sup>2</sup> <sup>θ</sup><sup>∗</sup>

<sup>g</sup> � θ<sup>g</sup> � �

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

value of the OIT2FNN. When the value <sup>ρ</sup> <sup>¼</sup> <sup>1</sup>

control of the system is determined as follows:

Now, substituting Eq. (48) into (44) yields he following.

without the minimum approximation error w is defined as follows:

, <sup>γ</sup><sup>g</sup> <sup>¼</sup> <sup>θ</sup><sup>∗</sup>

Let <sup>γ</sup><sup>f</sup> <sup>¼</sup> <sup>θ</sup><sup>∗</sup>

follows:

where, ρ ≥ <sup>1</sup>

Γξ<sup>z</sup>

<sup>f</sup> � θ<sup>f</sup> � �

12 Adaptive Robust Control Systems

Now, the integration of (54) from t = 0 to t = T yields the following equation.

$$V(0) - V(T) + \frac{1}{4\rho} \int\_0^T w\_m^2 dt \ge \frac{1}{2} \int\_0^T \mathbf{E}^T \mathbf{Q} \mathbf{E} dt\tag{55}$$

The value of V(T) ≥ 0, and thus Eq. (55) is rewritten as follows:

$$V(0) + \frac{1}{4\rho} \int\_0^T w\_m^{-2} dt \ge \frac{1}{2} \int\_0^T \mathbf{E}^T \mathbf{Q} \mathbf{E} dt\tag{56}$$

where, <sup>V</sup>ð Þ¼ <sup>0</sup> <sup>1</sup> <sup>2</sup> ETð Þ<sup>0</sup> PEð Þþ <sup>0</sup> <sup>1</sup> <sup>2</sup>α<sup>1</sup> <sup>γ</sup>~<sup>2</sup> <sup>f</sup> ð Þþ <sup>0</sup> <sup>1</sup> <sup>2</sup>α<sup>2</sup> <sup>γ</sup>~<sup>2</sup> <sup>g</sup>ð Þ0 . Hence the H-infinity tracking performance is achieved. From the boundedness of the parameters, γ~<sup>f</sup> and γ~<sup>g</sup> are guaranteed by closed sets

in Eqs. (7)–(13). The parameters of both the desired and the applied prescribed performance are listed in Table 1. The damping force of the MR damper is designed 1000 N (5%) at 2 A. The fuzzy model is established based on the online model with the centroid vector as shown in [25]. It is noted that two main variables for the fuzzy models are displacement and acceleration. The fuzzy models include 6 clusters, and then the outputs of fuzzy rules become also 6. The sigma value for Gaussian function of the fuzzy model is chosen as 0.4 [22, 25], and this value is not changed through the simulation. The values of the sliding surface [k1, k2] are chose by [1, 20] for both random step wave road and regular bump road. The constant value Γ of the

Riccati-like equation is chosen by 10 for both roads. The constant cs is 500 and 5000 for regular bump road and the random step wave road, respectively. In addition, the matrix Q of the Riccati-like equation is chosen as Q = [�2 0; 0 � 2]. The constants μ1, μ2, μ<sup>3</sup> of adaptation laws are chosen as 10 for two road profiles. The values of ℵ<sup>f</sup> ; ℵg; ΘΓ of the expanded adaptation laws are chosen by 0.1 and the values of δ1, δ2, δ<sup>3</sup> are chosen by 0.1. In this simulation, the initial states for the dynamic states are used as 0½ � :035 2:5 , 0½ � :035 2:5 for random regular bump, and random step wave bump, respectively. The initial states for the observer are ½ � 0:035 0 for two excitations. It is noted that the observer is applied to evaluate the results of

Robust Adaptive Controls of a Vehicle Seat Suspension System

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

17

Figures 5–8 present control responses of the HAC-PP. It is clearly observed from Figures 5 and 6 that the initial excitation has been significantly reduced by activation the proposed adaptive controller in terms of both displacement and acceleration. In addition, it is seen that the proposed control well tracks the objective trajectory which directly indicates high performance of the prescribed performance of the sliding surface. Figure 7 presents the error of performance of the proposed adaptive controller which is always less than the boundary of the prescribed

(a1) (a2)

(b1) (b2)

Figure 6. Control results with the HAC-PP at the driver (x1): (a1, a2) random step wave road, (b1, b2) regular bump road.

the proposed controller.


Table 1. Parameters of desired prescribed performance and applied prescribed performance.

Figure 5. Control results with the HAC-PP at the seat (xs): (a1, a2) random step wave road, (b1, b2) regular bump road.

Riccati-like equation is chosen by 10 for both roads. The constant cs is 500 and 5000 for regular bump road and the random step wave road, respectively. In addition, the matrix Q of the Riccati-like equation is chosen as Q = [�2 0; 0 � 2]. The constants μ1, μ2, μ<sup>3</sup> of adaptation laws are chosen as 10 for two road profiles. The values of ℵ<sup>f</sup> ; ℵg; ΘΓ of the expanded adaptation laws are chosen by 0.1 and the values of δ1, δ2, δ<sup>3</sup> are chosen by 0.1. In this simulation, the initial states for the dynamic states are used as 0½ � :035 2:5 , 0½ � :035 2:5 for random regular bump, and random step wave bump, respectively. The initial states for the observer are ½ � 0:035 0 for two excitations. It is noted that the observer is applied to evaluate the results of the proposed controller.

in Eqs. (7)–(13). The parameters of both the desired and the applied prescribed performance are listed in Table 1. The damping force of the MR damper is designed 1000 N (5%) at 2 A. The fuzzy model is established based on the online model with the centroid vector as shown in [25]. It is noted that two main variables for the fuzzy models are displacement and acceleration. The fuzzy models include 6 clusters, and then the outputs of fuzzy rules become also 6. The sigma value for Gaussian function of the fuzzy model is chosen as 0.4 [22, 25], and this value is not changed through the simulation. The values of the sliding surface [k1, k2] are chose by [1, 20] for both random step wave road and regular bump road. The constant value Γ of the

Parameter Desired prescribed performance Applied prescribed performance

Figure 5. Control results with the HAC-PP at the seat (xs): (a1, a2) random step wave road, (b1, b2) regular bump road.

Initial value λ(0) 0.5 0.5 Infinity value λ<sup>∞</sup> 0.001 0.001 Exponential value l 1 0.00047

16 Adaptive Robust Control Systems

Table 1. Parameters of desired prescribed performance and applied prescribed performance.

Figures 5–8 present control responses of the HAC-PP. It is clearly observed from Figures 5 and 6 that the initial excitation has been significantly reduced by activation the proposed adaptive controller in terms of both displacement and acceleration. In addition, it is seen that the proposed control well tracks the objective trajectory which directly indicates high performance of the prescribed performance of the sliding surface. Figure 7 presents the error of performance of the proposed adaptive controller which is always less than the boundary of the prescribed

Figure 6. Control results with the HAC-PP at the driver (x1): (a1, a2) random step wave road, (b1, b2) regular bump road.

Figure 7. Tracking error with the HAC-PP: (a1, a2) random step wave road, (b1, b2) regular bump road.

performance. These results mean that the application of the prescribed performance in design of the hybrid adaptive controller can improve the quality of control with high robustness against severe excitations.

Figures 8–10 present control responses of the HAC-IFV. As similar to the HAC-PP, the initial excitations were remarkably reduced by applying the proposed controller. The displacements at the seat and driver positions are reduced resulting in the improvement of the ride comfort. In order to demonstrate a salient benefit of the proposed controller, its control response is compared obtained from the controller proposed in [17, 25]. It is clearly identified that the convergence time of the displacement of the proposed controller is 2 seconds for both excitations, while that is 15 seconds for the random step wave excitation, 6 seconds for regular bump excitation in [17, 25]. In Figure 8, the sliding surfaces of three controllers are shown. It is observed that the proposed control obtains stable motion much faster than the comparative controls at 0.1 second. It is noted here that the better control responses of the proposed controller comes from the inversely fuzzified values in given Eqs. (46)–(48). In Eq. (48), the independent of the inversely fuzzified value helps the controller to increase its robustness. This new exploration is the outstanding property of the proposed controller in the severe operation

Figure 8. Control results with the HAC-IFV at the seat (xs): (a1, a2) random step wave road, (b1, b2) regular bump road.

Robust Adaptive Controls of a Vehicle Seat Suspension System

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

19

environment subjected to strong and random disturbances.
