Design of a robust active fuzzy parallel distributed compensation anti-vibration controller for a hand-glove system

The equations of motion of the active hand-glove model

In Fig. 1, since the actuator is assumed to be placed on the palm under the glove, the actuator force F_a is applied between the soft tissue of the hand, mass m₂ and glove material with mass m₃. The hand pushing the tool handle is represented by F_p.

The dynamic equations of motion of the system can be written as follows:

(1a) $$m_1(t){\ddot{z}}_1+(c_1+c_2){\dot{z}}_1-c_2{\dot{z}}_2+(k_1+k_2)z_1-k_2{z}_2=0$$

(1b) $$m_2{\ddot{z}}_2-c_2{\dot{z}}_1+(c_2+c_3){\dot{z}}_2-c_3{\dot{z}}_3-k_2{z}_1+(k_2+k3)z_2-k_3{z}_3=F_a-F_p$$

(1c) $$m_3{\ddot{z}}_3-c_3{\dot{z}}_2+c_3{\dot{z}}_3-k_3{z}_2+k_3{z}_3=-F_a+F_w$$where $z_i,{\dot{z}}_i$ and ${\ddot{z}}_i$ represent the displacement, velocity and acceleration of mass m_i; F_a is the actuator force, F_p is the push force and F_w is the input vibration or disturbance to the system that needs to be controlled.

In this model, the mass of the hand is assumed to be non-constant, m₁ (t), i.e. the model can be used to represent the hand system for different users. The Eqs. (1), (1b) and (1c) can be arranged in the following form:

(2a) $${\ddot{z}}_1=-{\displaystyle \frac{c_1+c_2}{m_1(t)}{\dot{z}}_1+{\displaystyle \frac{c_2}{m_1(t)}{\dot{z}}_2-{\displaystyle \frac{k_1+k_2}{m_1(t)}{z}_1+{\displaystyle \frac{k_2}{m_1(t)}{z}_2}}}}$$

(2b) $${\ddot{z}}_2={\displaystyle \frac{c_2}{m_2}{\dot{z}}_1-{\displaystyle \frac{c_2+c_3}{m_2}{\dot{z}}_2+{\displaystyle \frac{c_3}{m_2}{\dot{z}}_3+{\displaystyle \frac{k_2}{m_2}{z}_1-{\displaystyle \frac{k_2+k_3}{m_2}{z}_2+{\displaystyle \frac{k_3}{m_2}{z}_3+{\displaystyle \frac{F_a}{m_2}-{\displaystyle \frac{F_p}{m_2}}}}}}}}}$$

(2c) $${\ddot{z}}_3={\displaystyle \frac{c_3}{m_3}{\dot{z}}_2-{\displaystyle \frac{c_3}{m_3}{\dot{z}}_3+{\displaystyle \frac{k_3}{m_3}{z}_2-{\displaystyle \frac{k_3}{m_3}{z}_3-{\displaystyle \frac{F_a}{m_3}+{\displaystyle \frac{F_w}{m_3}}}}}}}$$

Considering x₁ = z₁, x₂ = z₂, x₃ = z₃, $x_4={\dot{z}}_1$, $x_5={\dot{z}}_2$ and $x_6={\dot{z}}_3$ for each of the state variables in Eqs. (2a), (2b) and (2c), it can be rewritten in the state-space equation given by Eq. (3).

(3) $$\begin{array}{cc}\hfill & \dot{\mathbf{x}}(t)=\mathbf{A}(t)\mathbf{x}(t)+\mathbf{B}(t)\mathbf{u}(t)+\mathbf{E}(t)\mathbf{w}(t)\hfill \\ \hfill & \mathbf{y}(t)=\mathbf{C}\mathbf{x}(t)\hfill \end{array}$$where matrices A, B, C and E are the system matrix, the input matrix, output matrix and the disturbance matrix, respectively. The hand vibration, ${\dot{x}}_4$ or ${\ddot{z}}_1$ is considered as the performance output reference. Equation (3) can be further written in the following form:

(4) $$\begin{array}{cc}\hfill & \left[\begin{array}{c}{\dot{x}}_1\\ {\dot{x}}_2\\ {\dot{x}}_3\\ {\dot{x}}_4\\ {\dot{x}}_5\\ {\dot{x}}_6\end{array}\right]=\left[\begin{array}{cccccc}0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\\ -{\displaystyle \frac{k_1+k_2}{m_1(t)}}& {\displaystyle \frac{k_2}{m_1(t)}}& 0& -{\displaystyle \frac{c_1+c_2}{m_1(t)}}& {\displaystyle \frac{c_2}{m_1(t)}}& 0\\ {\displaystyle \frac{k_2}{m_2}}& -{\displaystyle \frac{k_2+k_3}{m_2}}& {\displaystyle \frac{k_3}{m_2}}& {\displaystyle \frac{c_2}{m_2}}& -{\displaystyle \frac{c_2+c_3}{m_2}}& {\displaystyle \frac{c_3}{m_2}}\\ 0& {\displaystyle \frac{k_3}{m_3}}& -{\displaystyle \frac{k_3}{m_3}}& 0& {\displaystyle \frac{c_3}{m_3}}& -{\displaystyle \frac{c_3}{m_3}}\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\\ x_4\\ x_5\\ x_6\end{array}\right]\hfill \\ \hfill & +\left[\begin{array}{cc}0& 0\\ 0& 0\\ 0& 0\\ 0& 0\\ {\displaystyle \frac{1}{m_2}}& -{\displaystyle \frac{1}{m_2}}\\ -{\displaystyle \frac{1}{m_3}}& 0\end{array}\right]\left[\begin{array}{c}{F}_a\\ F_p\\ \end{array}\right]+\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ {\displaystyle \frac{1}{m_3}}\end{array}\right]F_w\hfill \end{array}$$

(5) $$\begin{array}{c}y(t)=\left[\begin{array}{cccccc}0& 0& 0& 1& 0& 0\end{array}\right]{\left[\begin{array}{cccccc}{x}_1& x_2& x_3& x_4& x_5& x_6\end{array}\right]}^T\end{array}$$

Another approach that can be used to model the active hand-glove system is the fuzzy T-S model and is described next.

Fuzzy Takagi-Sugeno model

Fuzzy logic has the ability to deal with nonlinearities and uncertainties in the system where standard analytical models are usually ineffective. It can be used for both modeling and design of the controller. Most nonlinear systems can be approximated by a T-S fuzzy model. The T-S fuzzy model of a dynamic system with control input and disturbance input can be represented by the following rules:

Model Rule i:

IF s₁(t) is M_i1 and ⋯ and s_p(t) is M_ip,

(6) $$\mathbf{T}\mathbf{H}\mathbf{E}\mathbf{N}\{\begin{array}{c}\dot{\mathbf{x}}(t)={\mathbf{A}}_i\mathbf{x}(t)+{\mathbf{B}}_i\mathbf{u}(t)+{\mathbf{E}}_i\mathbf{w}(t),\\ \mathbf{y}(t)={\mathbf{C}}_i\mathbf{x}(t),\end{array}i=1,2,\dots ,r,$$where r shows the number of rules, M_ij denotes the fuzzy set, and x(t) ∈ R^{n × n}, u(t) ∈ R^{n × 1} and w(t) ∈ R^{n × 1} are state vectors, controlled input vector and disturbance input vector, respectively. Note that s_i(t) is a hypothetical variable which can be a function of state variables, disturbance inputs, system parameters and/or time.

By considering each set of (A_i, B_i, E_i, C_i) as a subsystem, the overall output of the fuzzy system can be represented as

(7a) $$\begin{array}{cc}\hfill & \dot{\mathbf{x}}(t)={\displaystyle \frac{\sum _{i=1}^r{\alpha }_i(\mathbf{s}(t))({\mathbf{A}}_i\mathbf{x}(t)+{\mathbf{B}}_i\mathbf{u}(t)+{\mathbf{E}}_i\mathbf{w}(t))}{\sum _{i=1}^r{\alpha }_i(\mathbf{s}(t))}}\hfill \\ \hfill & =\sum _{i=1}^r{h}_i(\mathbf{s}(t))({\mathbf{A}}_i\mathbf{x}(t)+{\mathbf{B}}_i\mathbf{u}(t)+{\mathbf{E}}_i\mathbf{w}(t)),\hfill \end{array}$$

(7b) $$\begin{array}{cc}\hfill & \mathbf{y}(t)={\displaystyle \frac{\sum _{i=1}^r{\alpha }_i(\mathbf{s}(t))({\mathbf{C}}_i\mathbf{x}(t))}{\sum _{i=1}^r{\alpha }_i(\mathbf{s}(t))}}\hfill \\ \hfill & =\sum _{i=1}^r{h}_i(\mathbf{s}(t))({\mathbf{C}}_i\mathbf{x}(t)),\hfill \end{array}$$in which for all t

(8) $$\begin{array}{c}\mathbf{s}(t)=[s_1(t)s_2(t)\dots s_p(t)],\hfill \end{array}$$

(9) $$\begin{array}{c}{\alpha }_i(\mathbf{s}(t))=\prod _{j=1}^v{M}_{ij}(s_j(t))\hfill \end{array}$$and

(10) $$\begin{array}{c}{h}_i(\mathbf{s}(t))={\displaystyle \frac{{\alpha }_i(\mathbf{s}(t))}{\sum _{i=1}^r{\alpha }_i(\mathbf{s}(t))}.}\hfill \end{array}$$

Note that the term M_ij(s_j(t)) shows the membership grade of s_j(t) in M_i. Also, since α_i(s(t)) ≥ 0 and ${\sum }_{i=1}^r{\alpha }_i(s(t))\ge 0$ for i = 1,2,…,r, then we have h_i(s(t))) ≥ 0 and ${\sum }_{i=1}^r{h}_i(s(t))=1$ for all t.

Next, different types of active vibration control techniques applied to the system to suppress the vibration to the desired level are described.

Proportional-integral-derivative (PID) controller

The PID controller to be used in the AVC system is is a standard PID control algorithm given by Eq. (11):

(11) $$\begin{array}{cc}\hfill & u(t)=K_{p}e(t)+K_i\int e(t)dt+K_d\dot{e}(t)\hfill \end{array}$$where u(t) is the control signal, e(t) is the error signal and K_p, K_i and K_d are the proportional, integral and derivative gains, respectively. Figure 3 shows the feedback control system employing the PID controller. Equation (11) can be written in the following transfer function form:

(12) $$\begin{array}{cc}\hfill & U(s)=K_p+K_i/s+K_{d}s\hfill \end{array}$$

The PID gains K_p, K_i and K_d need to be tuned, by trial and error tuning method, since we get the desired response.

Active force controller

Active force control (AFC) proposed in Zain, Mailah & Priyandoko (2008) is an effective method designed to carry out the robust control of dynamic systems in the presence of disturbance and uncertainties. It is proved that in AFC, the stability and robustness of the system remain unchanged by compensation action of the control strategy when a number of disturbances are applied to the system (Abdelmaksoud, Mailah & Abdallah, 2020; Gohari & Tahmasebi, 2017).

The schematic of the AFC controller for a suspension system, which has a similar representation of the hand-glove system, is shown in Mailah et al. (2009).

One practical example of applying AFC to a suspension system to control the unwanted vibration is given in Mohamad, Mailah & Muhaimin (2006) and it showed the effectiveness of the applied AFC to cancel the vibration with different disturbances levels.

Fuzzy parallel distributed compensation (PDC) controller

Fuzzy logic is an effective way to decompose a nonlinear system control into a group of local linear controls based on a set of design-specific model rules. PDC is one of the many varieties of fuzzy logic that can be implemented in control systems. For a given fuzzy T-S framework, the state feedback based on PDC is usually applied. The key idea of the PDC technique is to divide the nonlinear system to some linear subsystems and then design some controllers for each of the linear subsystems and subsequently obtain the overall controller by the fuzzy blending of the local controllers (Sadeghi, Safarinejadian & Farughian, 2014; Sadeghi, Rezaei & Mardaneh, 2015; Nguyen, Dambrine & Lauber, 2016) i.e. designing a compensator for each rule of the fuzzy model.

The block diagram of the system with fuzzy PDC controller is illustrated in Fig. 4. As can be seen, the overall PDC controller is applied to the original nonlinear system.

The PDC technique proposes a method to design a fuzzy controller associated with the fuzzy T-S model Eqs. (7a) and (7b). In PDC, r controllers, usually state feedback, associated with each rule i are designed as

Control Rule i:

IF s₁(t) is M_i1 and ⋯ and s_p(t) is M_ip,

(13) $$\mathbf{T}\mathbf{H}\mathbf{E}\mathbf{N}u(t)=-F_{i}x(t),i=1,2,\dots ,r.$$

The overall control output of the fuzzy PDC controller is then obtained as

(14) $$\begin{array}{c}\begin{array}{cc}\hfill & u(t)=-{\displaystyle \frac{\sum _{i=1}^r{\alpha }_i(s(t))(F_{i}x(t))}{\sum _{i=1}^r{\alpha }_i(s(t))}}\hfill \\ \hfill & =-\sum _{i=1}^r{h}_i(s(t))(F_{i}x(t)),\hfill \end{array}\hfill \end{array}$$

Note that although in the PDC approach the controllers are local (Eq. (14)), the overall controller scheme should be designed globally to ensure global stability and suitable control performance. By substituting Eq. (14) in (7a) and (7b) the closed loop system can be expressed as

(15) $$\begin{array}{cc}\hfill & \dot{x}(t)=\sum _{i=1}^r\sum _{j=1}^r{h}_i(s(t))h_j(s(t))((A_i-B_i{F}_j)x(t)+E_{i}w(t)).\hfill \end{array}$$

Now, by considering a positive definite matrix P and the quadratic Lyapunov function V(x) = x^T(t)Px(t), the closed-loop system (15) is asymptotically stable if the following condition is satisfied (Tanaka & Wang, 2004)

(16) $$\begin{array}{ccc}\hfill & {\{A_i-B_i{F}_j\}}^T\{A_i-B_i{F}_j\}-P<0\hfill & \mathrm{\forall }i,j=1,2,\dots ,r,\hfill \end{array}$$

for h_i(s(t)) h_j(s(t)) ≠ 0, ∀ t.

To design a PDC controller for the hand-glove system, Eqs. (4) and (5), it is required to obtain the T-S fuzzy model of the dynamic system (4). Since our goal is to design an active controller that guarantees suitable performance for different users, i.e. different hand masses m₁, it is supposed that $m_1\in [m_1^{min},m_1^{max}]$. By defining the premise variable ψ_h(t) as

(17) $$\begin{array}{c}{\psi }_h(t):={\displaystyle \frac{1}{m_1(t)}}\hfill \end{array}$$and using the definition in Eq. (17) and the possible range of m₁(t), the minimum and maximum range of ψ_h(t) can be obtained as

(18a) $${\psi }_h^{min}=min\left\{{\displaystyle \frac{1}{m_1(t)}}\right\}={\displaystyle \frac{1}{m_1^{max}},}$$

(18b) $${\psi }_h^{max}=max\left\{{\displaystyle \frac{1}{m_1(t)}}\right\}={\displaystyle \frac{1}{m_1^{min}}.}$$

Then ψ_h(t) can be written in terms of ${\psi }_h^{min}$ and ${\psi }_h^{max}$ as follows:

(19) $$\begin{array}{c}{\psi }_h(t)=M_1({\psi }_h(t)){\displaystyle \frac{1}{m_1^{min}}+M_2({\psi }_h(t)){\displaystyle \frac{1}{m_1^{max}}}}\hfill \end{array}$$where

(20) $$\begin{array}{c}{M}_1({\psi }_h(t))+M_2({\psi }_h(t))=1.\hfill \end{array}$$

By using Eqs. (19) and (20), the membership functions can be obtained as follows:

(21a) $$M_1({\psi }_h(t))={\displaystyle \frac{{\psi }_h(t)-{\displaystyle \frac{1}{m_1^{max}}}}{{\displaystyle \frac{1}{m_1^{min}}-{\displaystyle \frac{1}{m_1^{max}}}}},}$$

(21b) $$M_2({\psi }_h(t))={\displaystyle \frac{{\displaystyle \frac{1}{m_1^{min}}-{\psi }_h(t)}}{{\displaystyle \frac{1}{m_1^{min}}-{\displaystyle \frac{1}{m_1^{max}}}}}}$$which are demonstrated in Fig. 5. Note that the membership functions M₁ and M₂ are called “Small” and “Large”, respectively.

Then, the hand-glove system model Eq. (4) can be represented as the following fuzzy T-S model for variable hand masses:

Model Rule 1:

IF ψ_h(t) is “Small”,

(22a) $$\mathbf{T}\mathbf{H}\mathbf{E}\mathbf{N}\{\begin{array}{c}\dot{x}(t)=A_{1}x(t)+B_{1}u(t)+E_{1}w(t)\\ y(t)=C_{1}x(t)\end{array}$$

Model Rule 2:

IF ψ_h(t) is “Large”,

(22b) $$\mathbf{T}\mathbf{H}\mathbf{E}\mathbf{N}\{\begin{array}{c}\dot{x}(t)=A_{2}x(t)+B_{2}u(t)+E_{2}w(t)\\ y(t)=C_{2}x(t)\end{array}$$where the fuzzy system’s submatrices associated with (22a) and (22b) are obtained as

$$A_1=\left[\begin{array}{cccccc}0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\\ -{\displaystyle \frac{k_1+k_2}{m_1^{min}}}& {\displaystyle \frac{k_2}{m_1^{min}}}& 0& -{\displaystyle \frac{c_1+c_2}{m_1^{min}}}& {\displaystyle \frac{c_2}{m_1^{min}}}& 0\\ {\displaystyle \frac{k_2}{m_2}}& -{\displaystyle \frac{k_2+k_3}{m_2}}& {\displaystyle \frac{k_3}{m_2}}& {\displaystyle \frac{c_2}{m_2}}& -{\displaystyle \frac{c_2+c_3}{m_2}}& {\displaystyle \frac{c_3}{m_2}}\\ 0& {\displaystyle \frac{k_3}{m_3}}& -{\displaystyle \frac{k_3}{m_3}}& 0& {\displaystyle \frac{c_3}{m_3}}& -{\displaystyle \frac{c_3}{m_3}}\end{array}\right]$$

$$A_2=\left[\begin{array}{cccccc}0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\\ -{\displaystyle \frac{k_1+k_2}{m_1^{max}}}& {\displaystyle \frac{k_2}{m_1^{max}}}& 0& -{\displaystyle \frac{c_1+c_2}{m_1^{max}}}& {\displaystyle \frac{c_2}{m_1^{max}}}& 0\\ {\displaystyle \frac{k_2}{m_2}}& -{\displaystyle \frac{k_2+k_3}{m_2}}& {\displaystyle \frac{k_3}{m_2}}& {\displaystyle \frac{c_2}{m_2}}& -{\displaystyle \frac{c_2+c_3}{m_2}}& {\displaystyle \frac{c_3}{m_2}}\\ 0& {\displaystyle \frac{k_3}{m_3}}& -{\displaystyle \frac{k_3}{m_3}}& 0& {\displaystyle \frac{c_3}{m_3}}& -{\displaystyle \frac{c_3}{m_3}}\end{array}\right]$$

$$B_1=B_2={\left[\begin{array}{cccccc}0& 0& 0& 0& {\displaystyle \frac{1}{m_2}}& -{\displaystyle \frac{1}{m_3}}\\ 0& 0& 0& 0& -{\displaystyle \frac{1}{m_2}}& 0\end{array}\right]}^T,$$

$$E_1=E_2={\left[\begin{array}{cccccc}0& 0& 0& 0& 0& {\displaystyle \frac{1}{m_3}}\end{array}\right]}^T,$$

(23) $$C_1=C_2=\left[\begin{array}{cccccc}0& 0& 0& 1& 0& 0\end{array}\right]$$

The designed fuzzy controller shares the same input variables and fuzzy sets with the T-S fuzzy model in the IF parts. So, for the fuzzy models given in Eqs. (22a) and (22b), the following controller rules are designed via PDC:

Control Rule 1:

IF ψ_h(t) is “Small”,

(24a) $$\mathbf{T}\mathbf{H}\mathbf{E}\mathbf{N}u(t)=-F_{1}x(t)$$

Control Rule 2:

IF ψ_h(t) is “Large”,

(24b) $$\mathbf{T}\mathbf{H}\mathbf{E}\mathbf{N}u(t)=-F_{2}x(t)$$where F₁ and F₂ are the feedback gains.

Now, by considering that the disturbance input, i.e. vibration from the hand-held tool, has a limited energy it can be assumed that w ∈ L₂ [0,∞) and $\parallel w{\parallel }_2^2\le w_{max}$. Then, the following control performances are aimed to be obtained:

1. The closed-loop system remains stable.

2. The transmitted vibration to the hand will be reduced significantly, i.e. the effect of disturbance input in the system output will be minimized or mathematically

   (25) $$\begin{array}{c}\parallel y{\parallel }_2^2<\gamma \parallel w{\parallel }_2^2\hfill \end{array}$$

   for all w ≠ 0 where γ is a predefined scalar.

3. The designed control input is enforced to a limited value, i.e.

   (26) $$\begin{array}{c}\parallel u{\parallel }_2\le u^{max}.\hfill \end{array}$$

4. The designed controller performs well for different users/hand masses.

Theorem: The feedback gains, F_i, of the state feedback controller (24a) and (24b) that stabilize the fuzzy T-S system (22a) and (22b) while minimizing the γ in (25) and satisfying the control effort constraint (26) can be obtained by solving the following optimization linear matrix inequality (LMI)-based problem as follows:

$$\underset{\mathrm{\Gamma },M_1,\dots ,M_r}{min}{\gamma }^2$$

$$\text{subject to}\left[\begin{array}{ccc}\left(\begin{array}{c}-{\displaystyle \frac{1}{2}\{\mathrm{\Gamma }{A}_i^T-M_j^T{B}_i^T+A_i\mathrm{\Gamma }-B_i{M}_j}\\ +\mathrm{\Gamma }{A}_j^T-M_i^T{B}_j^T+A_j\mathrm{\Gamma }-B_j{M}_i\}\\ \end{array}\right)& -{\displaystyle \frac{1}{2}(E_i+E_j)}& -{\displaystyle \frac{1}{2}\mathrm{\Gamma }{(C_i+C_j)}^T}\\ & & \\ -{\displaystyle \frac{1}{2}{(E_i+E_j)}^T}& {\gamma }^{2}I& 0\\ & & \\ -{\displaystyle \frac{1}{2}(C_i+C_j)\mathrm{\Gamma }}& 0& I\end{array}\right]\ge 0,$$

(27a) $$\mathrm{\forall }i,j=1,\dots ,r,i\le j,$$

(27b) $$\left[\begin{array}{cc}\mathrm{\Gamma }& M_i^T\\ M_i& u_{max}^{2}I\end{array}\right]\ge 0,\mathrm{\forall }i=1,\dots ,r,$$

(27c) $$\left[\begin{array}{cc}1& x{(0)}^T\\ x(0)& \mathrm{\Gamma }\end{array}\right]\ge 0$$

(27d) $$\mathrm{\Gamma }>0,$$

n which Γ = P⁻¹, where P is a positive definite matrix and M_i = F_i Γ.

Proof:

Part 1: Consider the quadratic Lyapunov function V((x(t))) = x^T(t) P x(t), P > 0, and γ > 0 and u_max > 0 exist. To prove the disturbance rejection LMI (27a), suppose that for the system (7a) and (7b) the following inequality

(28) $$\begin{array}{cc}\hfill & \dot{V}(x(t))+y^T(t)y(t)-{\gamma }^2{w}^T(t)w(t)\le 0\hfill \end{array}$$is true. By integrating (28), one can write

(29) $$\begin{array}{cc}\hfill & {\int }_0^T(\dot{V}(x(t))+y^T(t)y(t)-{\gamma }^2{w}^T(t)w(t))dt\le 0.\hfill \end{array}$$

Assuming that x(0) = 0, then we have

(30) $$\begin{array}{cc}\hfill & V(x(t))+{\int }_0^T(y^T(t)y(t)-{\gamma }^2{w}^T(t)w(t))dt\le 0.\hfill \end{array}$$

Since V((x(t))) > 0, it can be obtained from (25) that

(31) $$\begin{array}{cc}\hfill & {\displaystyle \frac{\parallel y(t){\parallel }_2}{\parallel w(t){\parallel }_2}\le \gamma .}\hfill \end{array}$$

Thus the L₂ norm (31) constraint is true for the fuzzy system (7a) and (7b) if the inequality (28) holds.

Now the LMI condition (27a) can be derived from the inequality (28) as follows

$${\dot{x}}^T(t)Px(t)+x^T(t)P\dot{x}(t)$$

$$+\sum _{i=1}^r\sum _{j=1}^r{h}_i(s(t))h_j(s(t))x^T(t)C_i^T{C}_{j}x(t)-{\gamma }^2{w}^T(t)w(t)$$

$$=\sum _{i=1}^r\sum _{j=1}^r{h}_i(s(t))h_j(s(t))x^T(t)(A_i-B_i{F}_j{)}^{T}Px(t)$$

$$+\sum _{i=1}^r\sum _{j=1}^r{h}_i(s(t))h_j(s(t))x^T(t)P(A_i-B_i{F}_j)x(t)$$

$$+\sum _{i=1}^r\sum _{j=1}^r{h}_i(s(t))h_j(s(t))x^T(t)C_i^T{C}_{j}x(t)-{\gamma }^2{w}^T(t)w(t)$$

$$+\sum _{i=1}^r{h}_i(s(t))x^T(t)E_i^{T}Px(t)+\sum _{i=1}^r{h}_i(s(t))x^T(t)P{E}_{i}w(t)$$

$$=\sum _{i=1}^r\sum _{j=1}^r{h}_i(s(t))h_j(s(t))\left[\begin{array}{cc}{x}^T(t)& w^T(t)\end{array}\right]$$

(32) $$\left[\begin{array}{cc}{(A_i-B_i{F}_j)}^{T}P+P(A_i-B_i{F}_j)+C_i^T{C}_j& P{E}_i\\ & \\ E_i^{T}P& -{\gamma }^{2}I\end{array}\right]\left[\begin{array}{c}x(t)\\ \\ w(t)\end{array}\right]\le 0.$$

From the inequality (32), it can be obtained

(33) $$\begin{array}{cc}\hfill & \left[\begin{array}{cc}\left(\begin{array}{c}-\sum _{i=1}^r\sum _{j=1}^r{h}_i(s(t))h_j(s(t))\{(A_i-B_i{F}_j{)}^{T}P\\ +P(A_i-B_i{F}_j)+C_i^T{C}_j\}\end{array}\right)& -P\sum _{i=1}^r{h}_i(s(t))E_i\\ & \\ -\sum _{i=1}^r{h}_i(s(t))E_i^{T}P& {\gamma }^{2}I\end{array}\right]\ge 0,\hfill \end{array}$$which can be decomposed as

$$\left[\begin{array}{cc}\left(\begin{array}{c}-\sum _{i=1}^r\sum _{j=1}^r{h}_i(s(t))h_j(s(t))\{\\ {(A_i-B_i{F}_j)}^{T}P+P(A_i-B_i{F}_j)\}\end{array}\right)& -P\sum _{i=1}^r{h}_i(s(t))E_i\\ & \\ -\sum _{i=1}^r{h}_i(s(t))E_i^{T}P& {\gamma }^{2}I\end{array}\right]$$

$$-\left[\begin{array}{ccc}\sum _{i=1}^r\sum _{j=1}^r{h}_i(s(t))h_j(s(t))C_i^T{C}_j& & 0\\ & & \\ 0& & 0\end{array}\right]$$

$$=\left[\begin{array}{cc}\left(\begin{array}{c}-\sum _{i=1}^r\sum _{j=1}^r{h}_i(s(t))h_j(s(t))\{\\ {(A_i-B_i{F}_j)}^{T}P+P(A_i-B_i{F}_j)\}\end{array}\right)& -P\sum _{i=1}^r{h}_i(s(t))E_i\\ & \\ -\sum _{i=1}^r{h}_i(s(t))E_i^{T}P& {\gamma }^{2}I\end{array}\right]$$

(34) $$-\left[\begin{array}{c}-\sum _{i=1}^r{h}_i(s(t))C_i^T\\ \\ 0\end{array}\right]\left[\begin{array}{ccc}\sum _{i=1}^r{h}_i(s(t))C_i& & 0\end{array}\right]\ge 0.$$

The inequality condition (34) is equivalent to

(35) $$\begin{array}{cc}\hfill & \left[\begin{array}{ccc}\left(\begin{array}{c}-\sum _{i=1}^r\sum _{j=1}^r{h}_i(s(t))h_j(s(t))\\ \{(A_i-B_i{F}_j{)}^{T}P\\ +P(A_i-B_i{F}_j)\}\end{array}\right)& -P\sum _{i=1}^r{h}_i(s(t))E_i& \sum _{i=1}^r{h}_i(s(t))C_i^T\\ & & \\ -\sum _{i=1}^r{h}_i(s(t))E_i^{T}P& {\gamma }^{2}I& 0\\ & & \\ \sum _{i=1}^r{h}_i(s(t))C_i& 0& I\end{array}\right]\ge 0,\hfill \end{array}$$which can be rewritten as

(36) $$\begin{array}{cc}\hfill & \sum _{i=1}^r\sum _{j=1}^r{h}_i(s(t))h_j(s(t))\hfill \\ \hfill & \left[\begin{array}{ccc}\left(\begin{array}{c}-{\displaystyle \frac{1}{2}\{(A_i-B_i{F}_j{)}^{T}P+P(A_i-B_i{F}_j)}\\ +{(A_j-B_j{F}_i)}^{T}P+P(A_j-B_j{F}_i)\}\end{array}\right)& -{\displaystyle \frac{1}{2}P(E_i+E_j)}& {\displaystyle \frac{1}{2}{(C_i+C_i)}^T}\\ & & \\ -{\displaystyle \frac{1}{2}{(E_i+E_j)}^{T}P}& {\gamma }^{2}I& 0\\ & & \\ {\displaystyle \frac{1}{2}(C_i+C_i)}& 0& I\end{array}\right]\ge 0.\hfill \end{array}$$

And, eventually, it can be derived from (36) that

(37) $$\begin{array}{cc}\hfill & \left[\begin{array}{ccc}\left(\begin{array}{c}-{\displaystyle \frac{1}{2}\{(A_i-B_i{F}_j{)}^{T}P+P(A_i-B_i{F}_j)}\\ +{(A_j-B_j{F}_i)}^{T}P+P(A_j-B_j{F}_i)\}\end{array}\right)& -{\displaystyle \frac{1}{2}P(E_i+E_j)}& {\displaystyle \frac{1}{2}{(C_i+C_j)}^T}\\ & & \\ -{\displaystyle \frac{1}{2}{(E_i+E_j)}^{T}P}& {\gamma }^{2}I& 0\\ & & \\ {\displaystyle \frac{1}{2}(C_i+C_j)}& 0& I\end{array}\right]\ge 0\hfill \end{array}$$

Now by multiplying the both side of Eq. (37) by block-diagonal {Γ I I}, LMI (27a) is obtained, where Γ = P⁻¹.

Part 2: To prove the bounding LMI condition (27b), from ||u||₂ ≤ u^max one can write

(38) $$\begin{array}{cc}\hfill & u^T(t)u(t)=\sum _{i=1}^r\sum _{j=1}^r{h}_i(s(t))h_j(s(t))x^T(t)F_i^T{F}_{j}x(t)\le u_{max}^2,\hfill \end{array}$$and then

(39) $$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{u_{max}^2}\sum _{i=1}^r\sum _{j=1}^r{h}_i(s(t))h_j(s(t))x^T(t)F_i^T{F}_{j}x(t)\le 1.}\hfill \end{array}$$

Since x^T(t) Γ⁻¹ x(t) < x^T(0) Γ^{− 1} x(0) ≤ 1 for all t > 0, the inequality (39) holds if

(40) $$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{u_{max}^2}\sum _{i=1}^r\sum _{j=1}^r{h}_i(s(t))h_j(s(t))x^T(t)F_i^T{F}_{j}x(t)\le x^T(t){\mathrm{\Gamma }}^{-1}x(t),}\hfill \end{array}$$and consequently (39) holds. Thus, we have

(41) $$\begin{array}{c}\sum _{i=1}^r\sum _{j=1}^r{h}_i(s(t))h_j(s(t))x^T(t)\left({\displaystyle \frac{1}{u_{max}^2}{F}_i^T{F}_j-{\mathrm{\Gamma }}^{-1}}\right)x(t)\le 0.\end{array}$$

The left-hand-side of (41) is equivalent to

(42) $$\begin{array}{cc}& {\displaystyle \frac{1}{2}\sum _{i=1}^r\sum _{j=1}^r{h}_i(s(t))h_j(s(t))x^T(t)\left({\displaystyle \frac{1}{u_{max}^2}{F}_i^T{F}_j+{\displaystyle \frac{1}{u_{max}^2}{F}_j^T{F}_i-2{\mathrm{\Gamma }}^{-1}}}\right)x(t)}\\ & ={\displaystyle \frac{1}{2}\sum _{i=1}^r\sum _{j=1}^r{h}_i(s(t))h_j(s(t))x^T(t)}\\ & \left({\displaystyle \frac{1}{u_{max}^2}(F_i^T{F}_j+F_j^T{F}_i)-{\displaystyle \frac{1}{u_{max}^2}(F_i^T-F_j^T)(F_i-F_j)-2{\mathrm{\Gamma }}^{-1}}}\right)x(t)\\ & \le {\displaystyle \frac{1}{2}\sum _{i=1}^r\sum _{j=1}^r{h}_i(s(t))h_j(s(t))x^T(t)\left({\displaystyle \frac{1}{u_{max}^2}(F_i^T{F}_j+F_j^T{F}_i)-2{\mathrm{\Gamma }}^{-1}}\right)x(t)}\\ & =\sum _{i=1}^r{h}_i(s(t))x^T(t)\left({\displaystyle \frac{1}{u_{max}^2}{F}_i^T{F}_i-{\mathrm{\Gamma }}^{-1}}\right)x(t).\end{array}$$

Then, the inequality (41) holds if

(43) $$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{u_{max}^2}{F}_i^T{F}_i-{\mathrm{\Gamma }}^{-1}\le 0.}\hfill \end{array}$$

By defining M_i = F_i Γ, we have

(44) $$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{u_{max}^2}{M}_i^T{M}_i-\mathrm{\Gamma }\le 0.}\hfill \end{array}$$and using the Schur Complement, the LMI condition (27b) is obtained. The proof is completed.

The overall modeling steps and controller design stages are depicted in Fig. 6.