Observer based robust H_∞ fuzzy tracking control: application to an activated sludge process

The activated sludge process model

The activated sludge schematized in Fig. 1 is used as a biological purification in waste-water treatment, consisting essentially of flocculating microorganisms, mixed with dissolved oxygen and waste-water. Thus, the microorganisms come into contact with the organic pollutants presents in the wastewater, as well as with dissolved oxygen, and are kept in suspension. Based on the natural metabolism, These microorganisms convert the organic matter into new cells, carbon dioxide and water. The process of the purification is done into tow tanks called aerator and settler. The energy required by the process is provided by the dissolved oxygen. Subsequently, carbon dioxide is released in return. The mathematical model that represents the process is given using the mass balance around the aerator and the settler as follows (Nejjari et al., 1999):

(1) $$\begin{array}{ccc}{\displaystyle \frac{dX}{dt}}\hfill & =\hfill & \mu (.)X-D(1+q_r)X+q_{r}D{X}_r\hfill \\ {\displaystyle \frac{dS}{dt}}\hfill & =\hfill & -{\displaystyle \frac{1}{Y}\mu (.)X-(1+q_r)DS+D{S}_{in}}\hfill \\ {\displaystyle \frac{d{C}_o}{dt}}\hfill & =\hfill & -{\displaystyle \frac{K_0}{Y}\mu (.)X-D(1+q_r)C_o+D{C}_{oin}+K_{La}(C_s-C_o)}\hfill \\ {\displaystyle \frac{d{X}_r}{dt}}\hfill & =\hfill & D(1+q_r)X-D(\beta +q_r)X_r\hfill \end{array}$$where

• X(t), S(t), C_o(t) and X_r(t) are respectively the biomass, the substrate, the dissolved oxygen and the recycled biomass concentrations.

• μ(.) corresponds to the biomass specific growth rate. It is assumed to follow the following model:

$$\mu (S,C_o)={\mu }_{max}{\displaystyle \frac{S}{K_s+S}{\displaystyle \frac{C_o}{K_c+C_o}}}$$

μ_max is the maximum specific growth rate, K_c is the saturation constant and K_s is the affinity constant.

• D and K_La represent respectively the dilution rate and the aeration flow rate.

• S_in and C_oin are the influent substrate and the dissolved oxygen concentrations.

• Y is a constant yield coefficient, K₀ is a constant and C_s is the maximum concentration of the dissolved oxygen concentration.

• q_r and β represent respectively the ratio of recycled flow and the ratio of waste flow to influent flow.

For this model the states, the inputs and the output vectors are given respectively by:

(2) $$\begin{array}{cc}x(t)={\left[\begin{array}{cccc}X(t)& X_r(t)& S(t)& C_o(t)\end{array}\right]}^T\hfill & \hfill \\ u(t)={\left[\begin{array}{cc}D(t)& K_{La}(t)\end{array}\right]}^T\hfill & \hfill \\ y(t)=C_o(t)\hfill & \hfill \end{array}$$

Problem formulation and preliminaries

The TS fuzzy approach consists in transcribing the dynamic of a nonlinear process into a finite weighted sum of linear models. There exist three approaches in the literature to obtain the TS fuzzy model (Tanaka & Wang, 2003b): the black box identification, the linearization technique and non-linearity sector method. We are interested in the third method which gives an accurate TS fuzzy model description of nonlinear model without information loss.

Let consider the following nonlinear disturbed system:

(3) $$\begin{array}{cc}\dot{x}(t)\hfill & =f(x(t),u(t),d(t))\hfill \\ y(t)\hfill & =Cx(t)\hfill \end{array}$$where x(t)∈ Rⁿ is the state vector, u(t)∈ R^m is the input vector, d(t)∈ R^l is the disturbance, y(t)∈ R^q is the output vector and C a matrix ∈ R^{q × n} Consider a TS fuzzy model. The system (3) can be approximated or represented (according to the number n_r of sub-models) by the Takagi–Sugeno structure:

(4) $$\dot{x}(t)=\sum _{i=1}^{n_r}{h}_i(z)(A_{i}x(t)+B_{i}u(t)+Gd(t))$$where A_i∈ R^{n× n}, B_i∈ R^{n× m} and G∈ R^n,l. z∈ R^p denotes the so-called decision variables (premise variables) that can be available when it depends on measurable variable such as u(t) or y(t) i.e., z = z(u(t),y(t)) or unavailable when it depends on non-measured system state x(t) i.e., z = z(x(t)). The weighting functions h_i (z) called the membership functions satisfy the convex sum property expressed in the following equations:

$$\sum _{i=1}^{n_r}{h}_i(z)=10\le h_i(z)\le 1$$

The weighing functions h_i(z) are generally nonlinear and depend on the premise variables z. Let us consider the following partition $x(t)=\left[\begin{array}{c}{x}_1(t)\\ x_2(t)\end{array}\right]$ of the system (4) with the corresponding matrices $A_i=\left[\begin{array}{cc}{A}_i^{11}& A_i^{12}\\ A_i^{21}& A_i^{22}\end{array}\right]$, $B_i=\left[\begin{array}{c}{B}_i^1\\ B_i^2\end{array}\right]$, $G=\left[\begin{array}{c}{G}^1\\ G^2\end{array}\right]$ and where x₂(t) corresponds to the part of the state vector to be controlled and x₁(t) corresponds to the remaining state variables.

The dynamical model can be rewritten as follows:

(5a) $${\dot{x}}_1(t)=\sum _{i=1}^{n_r}{h}_i(z)(A_i^{11}{x}_1(t)+A_i^{12}{x}_2(t)+B_i^{1}u(t)+G^{1}d(t))$$

(5b) $${\dot{x}}_2(t)=\sum _{i=1}^{n_r}{h}_i(z)(A_i^{21}{x}_1(t)+A_i^{22}{x}_2(t)+B_i^{2}u(t)+G^{2}d(t))$$

Consider a linear reference model given by the following equation:

(6) $${\dot{x}}_2^r(t)=A_r{x}_2^r(t)+r(t)$$where $x_2^r(t)$ is the reference state which should be tracked by the system (5b). A_r is a stable matrix and r(t) is a bounded input reference.

Our goal is to synthesize a control law based on the state estimation capable to reduce the error between the reference trajectory $x_2^r(t)$ and the state x₂(t). The Parallel Distributed Compensation concept can be used to design a fuzzy controller where the main idea consists to design a local controller for each sub-model based on local control rule, which shares with the fuzzy model the same fuzzy sets.

The controller we choose in this paper is expressed by an observer-based law with reference model as follows:

(7) $$u(t)=\sum _{i=1}^{n_r}{h}_i(\hat{z})K_i({\hat{x}}_2(t)-x_2^r(t))$$where ${\hat{x}}_2(t)$ is the estimation of x₂(t) and the K_i’s represent the local feedback gains that should be determined.

Stability conditions

Before starting the stability analysis, some useful lemmas are recalled.

Lemma 1 (Guerra et al., 2006) For any matrices X,Y of appropriate dimensions and for any positive scalar η the following inequality holds:

(8) $$X^{T}Y+Y^{T}X\le \eta X^{T}X+{\eta }^{-1}{Y}^{T}Y$$

Lemma 2 (Guerra et al., 2006) Considering π (as in inequality (9)) < 0 a matrix X and a scalar λ, the following inequality holds:

(9) $$X^T\mathrm{\Pi }X\le -\lambda (X^T+X)-{\lambda }^2{\mathrm{\Pi }}^{-1}$$

Using (7) into (5), the closed loop system can then be rewritten as follows:

(10) $$\begin{array}{c}{\dot{x}}_1(t)=\sum _{i,j=1}^{n_r}{h}_i(\hat{z})h_j(\hat{z})(A_i^{11}{x}_1(t)+A_i^{12}{x}_2(t)+B_i^1{K}_j({\hat{x}}_2(t)-x_2^r(t))+{\omega }_1(t)\hfill \\ {\dot{x}}_2(t)=\sum _{i,j=1}^{n_r}{h}_i(\hat{z})h_j(\hat{z})(A_i^{21}{x}_1(t)+A_i^{22}{x}_2(t)+B_i^2{K}_j({\hat{x}}_2(t)-x_2^r(t))+{\omega }_2(t)\hfill \end{array}$$where

$${\omega }_1(t)=\sum _{i=1}^{n_r}(h_i(z)-h_i(\hat{z}))(A_i^{11}{x}_1(t)+A_i^{12}{x}_2(t)+B_i^{1}u(t)+G^{1}d(t))$$

$${\omega }_2(t)=\sum _{i=1}^{n_r}(h_i(z)-h_i(\hat{z}))(A_i^{21}{x}_1(t)+A_i^{22}{x}_2(t)+B_i^{2}u(t)+G^{2}d(t))$$

Let us define by $e_r(t)=x_2(t)-x_2^r(t)$ the tracking error and $e_{o2}(t)=x_2(t)-{\hat{x}}_2(t)$ the state estimation error and consider the augmented state: $x_a(t)=\left[\begin{array}{c}{x}_1(t)\\ e_r(t)\end{array}\right]$. Using (10), the dynamical model of the augmented system is given by:

(11) $${\dot{x}}_a(t)=\sum _{i=1}^{n_r}\sum _{j=1}^{n_r}{h}_i(\hat{z})h_j(\hat{z})({\mathcal{A}}_{ij}{x}_a(t)+{\mathcal{D}}_{ij}\xi (t))$$which can be rewritten as:

(12) $${\dot{x}}_a(t)=\sum _{i=1}^{n_r}{h}_i^2(\hat{z})({\mathcal{A}}_{ii}{x}_a(t)+{\mathcal{D}}_{ii}\xi (t))+2\underset{i<j}{\sum _{i,j=1}^{n_r}}{h}_i(\hat{z})h_j(\hat{z})({\displaystyle \frac{{\mathbb{A}}_{ij}{x}_a(t)+{\mathbb{D}}_{ij}\xi (t)}{2})}$$where

(13) $$\begin{array}{ccc}{\mathcal{A}}_{ij}\hfill & =\left[\begin{array}{cc}{A}_i^{11}& A_i^{12}+B_i^1{K}_j\\ A_i^{21}& A_i^{22}+B_i^2{K}_j\end{array}\right],{\mathbb{A}}_{ij}={\mathcal{A}}_{ij}+{\mathcal{A}}_{ji}\hfill & \hfill \\ {\mathcal{D}}_{ij}\hfill & =\left[\begin{array}{ccccc}-B_i^1{K}_j& A_i^{12}& 0& I& 0\\ -B_i^2{K}_j& (A_i^{22}-A_{ref})& -I& 0& I\end{array}\right],{\mathbb{D}}_{ij}={\mathcal{D}}_{ij}+{\mathcal{D}}_{ji}\hfill & \hfill \\ \xi (t)\hfill & ={\left[\begin{array}{ccccc}{e}_{o2}{(t)}^T& x_2^r{(t)}^T& r{(t)}^T& {\omega }_1{(t)}^T& {\omega }_2{(t)}^T\end{array}\right]}^T\hfill & \hfill \end{array}$$

The term ξ(t) is acting like a disturbance affecting the augmented state x_a(t). Thus to attenuate its effect, we propose the use of the H_∞ technique applied to System (12). The weighted H_∞ performance to minimize can be presented as follows:

(14) $${\int }_0^{t_f}{x}_a^T(t)Q{x}_a(t)dt\le {\gamma }^2{\int }_0^{t_f}{\xi }^T(t)\xi (t)dt$$where Q is a positive definite matrix and γ is a prescribed positive scalar that defines the attenuation level of the disturbance ξ(t).

To realize Condition (14), one has to use a Lyapunov function for System (12) given by:

(15) $$V(x_a)=x_a(t{)}^{T}P{x}_a(t)(P>0)$$

To achieve the performance (14) and ensure the stability of augmented System (12), the following condition must be realized:

(16) $$\dot{V}(x_a)+x_a^T(t)Q{x}_a(t)-{\gamma }^2\xi (t{)}^T\xi (t)\le 0$$

The following result can be announced:

Lemma 3 If there exist positive definite matrices P, Q₁ and Q₂, and positive scalars α₁,α₂, γ₁ and γ₂, the augmented system in (12) is asymptotically stable, such that the following conditions are satisfied:

(17) $$\begin{array}{ccc}{\mathcal{A}}_{ii}^{T}P+P{\mathcal{A}}_{ii}+Q_1+{\alpha }_1^{-1}P{\mathcal{D}}_{ii}{\mathcal{D}}_{ii}^{T}P& <0& \mathrm{\forall }i=1,\dots ,n_r\\ {\alpha }_1-{\gamma }_1& <0& \end{array}$$

(18) $$\begin{array}{ccc}{\displaystyle \frac{1}{2}({\mathbb{A}}_{ij}^{T}P+P{\mathbb{A}}_{ij}+Q_2+{\alpha }_2^{-1}P{\mathbb{D}}_{ij}{\mathbb{D}}_{ij}^{T}P)}& <0& \mathrm{\forall }i<j=1,\dots ,n_r\\ {\displaystyle \frac{1}{2}({\alpha }_2-{\gamma }_2)}& <0& \end{array}$$

The H_∞ performmance criteria (14) is guaranteed where the scalar γ and the matrix Q are given by $\gamma =\sqrt{{\gamma }_1+{\gamma }_2}$ and Q = Q₁ + Q₂

Proof:

Using (12), the derivative of the Lyapunov function (16) is:

(19) $$\begin{array}{cc}\dot{V}(x_a)\hfill & =\sum _{i=1}^{n_r}{h}_i^2(\hat{z})[x_a^T(t)({\mathcal{A}}_{ii}^{T}P+P{\mathcal{A}}_{ii})x_a(t)+{\xi }^T(t){\mathcal{D}}_{ii}^{T}P{x}_a(t)+x_a^T(t)P{\mathcal{D}}_{ii}\xi (t)]+\hfill \\ \hfill & 2\underset{i<j}{\sum _{i,j=1}^{n_r}}{h}_i(\hat{z})h_j(\hat{z}){\displaystyle \frac{1}{2}[x_a^T(t)({\mathbb{A}}_{ij}^{T}P+P{\mathbb{A}}_{ij})x_a(t)+{\xi }^T(t){\mathbb{D}}_{ij}^{T}P{x}_a(t)+x_a^T(t)P{\mathbb{D}}_{ij}\xi (t)]}\hfill \end{array}$$Denote by J the expression:

(20) $$J=\dot{V}(x_a(t))+x_a^T(t)Q{x}_a(t)-{\gamma }^2\xi (t{)}^T\xi (t)$$by setting Q = Q₁ + Q₂ and γ² = γ₁ + γ₂ and using (19), J can be written as the sum of two terms:

(21) $$\begin{array}{cc}J\hfill & =\sum _{i=1}^{n_r}{h}_i^2(\hat{z})[x_a^T(t)({\mathcal{A}}_{ii}^{T}P+P{\mathcal{A}}_{ii})x_a(t)+x_a^T(t)Q_1{x}_a(t)-{\gamma }_1{\xi }^T(t)\xi (t))\hfill \\ \hfill & +{\xi }^T(t){\mathcal{D}}_{ii}^{T}P{x}_a(t)+x_a^T(t)P{\mathcal{D}}_{ii}\xi (t)]\hfill \\ \hfill & +2\underset{i<j}{\sum _{i,j=1}^{n_r}}{h}_i(\hat{z})h_j(\hat{z}){\displaystyle \frac{1}{2}[x_a^T(t)({\mathbb{A}}_{ij}^{T}P+P{\mathbb{A}}_{ij})x_a(t)+x_a^T(t)Q_2{x}_a(t)-{\gamma }_2{\xi }^T(t)\xi (t)}\hfill \\ \hfill & +{\xi }^T(t){\mathbb{D}}_{ij}^{T}P{x}_a(t)+x_a^T(t)P{\mathbb{D}}_{ij}\xi (t)]\hfill \end{array}$$

Using the Lemma 1 on the crossed terms yields:

(22) $$\begin{array}{cc}{\xi }^T(t){\mathcal{D}}_{ii}^{T}P{x}_a(t)+x_a^T(t)P{\mathcal{D}}_{ii}\xi (t)\hfill & \le {\alpha }_1^{-1}{x}_a^T(t)P{\mathcal{D}}_{ii}{\mathcal{D}}_{ii}^{T}P{x}_a(t)+{\alpha }_1{\xi }^T(t)\xi (t)\hfill \\ {\xi }^T(t){\mathbb{D}}_{ij}^{T}P{x}_a(t)+x_a^T(t)P{\mathbb{D}}_{ij}\xi (t)\hfill & \le {\alpha }_2^{-1}{x}_a^T(t)P{\mathbb{D}}_{ij}{\mathbb{D}}_{ij}^{T}P{x}_a(t)+{\alpha }_2{\xi }^T(t)\xi (t)\hfill \end{array}$$(22) into (21) leads to the following inequality:

(23) $$\begin{array}{cc}J\le \hfill & \sum _{i=1}^{n_r}{h}_i^2(\hat{z})[x_a^T(t)({\mathcal{A}}_{ii}^{T}P+P{\mathcal{A}}_{ii}+Q_1+{\alpha }_1^{-1}P{\mathcal{D}}_{ii}{\mathcal{D}}_{ii}^{T}P)x_a(t)\hfill \\ \hfill & +{\xi }^T(t)({\alpha }_1-{\gamma }_1)\xi (t))]\hfill \\ \hfill & +2\underset{i<j}{\sum _{i,j=1}^{n_r}}{h}_i(\hat{z})h_j(\hat{z}){\displaystyle \frac{1}{2}[x_a^T(t)({\mathbb{A}}_{ij}^{T}P+P{\mathbb{A}}_{ij}+Q_2+{\alpha }_2^{-1}P{\mathbb{D}}_{ij}{\mathbb{D}}_{ij}^{T}P)x_a(t)}\hfill \\ \hfill & +{\xi }^T(t)({\alpha }_2-{\gamma }_2)\xi (t)]\hfill \end{array}$$

This implies that (16) is satisfied if the following sufficient conditions hold:

(24) $$\begin{array}{ccc}{\mathcal{A}}_{ii}^{T}P+P{\mathcal{A}}_{ii}+Q_1+{\alpha }_1^{-1}P{\mathcal{D}}_{ii}{\mathcal{D}}_{ii}^{T}P& <0& \mathrm{\forall }i=1,\dots ,n_r\\ {\alpha }_1-{\gamma }_1& <0& \end{array}$$

(25) $$\begin{array}{ccc}{\displaystyle \frac{1}{2}({\mathbb{A}}_{ij}^{T}P+P{\mathbb{A}}_{ij}+Q_2+{\alpha }_3^{-1}P{\mathbb{D}}_{ij}{\mathbb{D}}_{ij}^{T}P)}& <0& \mathrm{\forall }i<j=1,\dots ,n_r\\ {\displaystyle \frac{1}{2}({\alpha }_2-{\gamma }_2)}& <0& \end{array}$$

This ends the lemma proof.

The main result

To determine the controller gains K_i, we present new conditions in terms of LMIs. These conditions are developed through the use of separation Lemma 2, the introduction of some slack variables and other calculations leading to the following results.

Theorem 1 There exists an observer based controller (7) for the system (12) guaranteeing the H_∞ performance criteria (14) if there exists positive matrices $X_1=X_1^T,X_2=X_2^T$, matrices Y_i, i = 1,2,…,n_r, positive matrices ${\tilde{Q}}_1^1,{\tilde{Q}}_1^2,{\tilde{Q}}_2^1$ and ${\tilde{Q}}_2^2$ and prescribed positive scalars α₁, α₂, α₃, α₄, γ₁ and γ₂ such that α₁ < γ₁ and α₂ < γ₂ and that the following conditions hold: for i = 1,…,n_r

(26) $$\left[\begin{array}{cccccccc}{M}_1& M_2& -B_i^1{Y}_i& A_i^{12}& 0& I& 0& 0\\ \ast & M_3& -B_i^2{Y}_i& A_i^{22}-A_r& -I& 0& I& 0\\ \ast & \ast & -2{\alpha }_3{X}_2& 0& 0& 0& 0& -{\alpha }_{3}I\\ \ast & \ast & \ast & -{\alpha }_{1}I& 0& 0& 0& 0\\ \ast & \ast & \ast & \ast & -{\alpha }_{1}I& 0& 0& 0\\ \ast & \ast & \ast & \ast & \ast & -{\alpha }_{1}I& 0& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & -{\alpha }_{1}I& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & \ast & -{\alpha }_{1}I\end{array}\right]<0$$for $i<j=1,\dots ,n_r$

(27) $${\displaystyle \frac{1}{2}\left[\begin{array}{cccccccc}{N}_1& N_2& -B_i^1{Y}_j-B_j^1{Y}_i& A_i^{12}+A_j^{12}& 0& 2I& 0& 0\\ \ast & N_3& -B_i^2{Y}_j-B_j^2{Y}_i& A_i^{22}+A_j^{22}-2A_r& -2I& 0& 2I& 0\\ \ast & \ast & -2{\alpha }_4{X}_2& 0& 0& 0& 0& -{\alpha }_{4}I\\ \ast & \ast & \ast & -{\alpha }_{2}I& 0& 0& 0& 0\\ \ast & \ast & \ast & \ast & -{\alpha }_{2}I& 0& 0& 0\\ \ast & \ast & \ast & \ast & \ast & -{\alpha }_{2}I& 0& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & -{\alpha }_{2}I& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & \ast & -{\alpha }_{2}I\end{array}\right]<0}$$where

$$M_1=A_i^{11}{X}_1+X_1{A}_i^{{11}^T}+{\tilde{Q}}_1^1$$

$$M_2=X_{1}A{}_i^{{21}^T}+A_i^{12}{X}_2+B_i^1{Y}_i$$

$$M_3=A_i^{22}{X}_2+X_{2}A{}_i^{{22}^T}+B_i^2{Y}_i+(B_i^2{Y}_i{)}^T+{\tilde{Q}}_1^2$$

$$N_1=(A_i^{11}+A_j^{11})X_1+X_1(A_i^{11}+A_j^{11}{)}^T+{\tilde{Q}}_2^1$$

$$N_2=X_1(A_i^{21}+A_j^{21}{)}^T+(A_i^{12}+A_j^{12})X_2+B_i^1{Y}_j+B_j^1{Y}_i$$

$$N_3=(A_i^{22}+A_j^{22})X_2+X_2(A_i^{22}+A_j^{22}{)}^T+(B_i^2{Y}_j+B_j^2{Y}_i)+{\tilde{Q}}_2^2+(B_i^2{Y}_j+B_j^2{Y}_i{)}^T$$and * stands for the symmetric term of the corresponding off-diagonal term.

Solving LMIs (26) and (27) the controller gains K_i, the attenuation level γ and the matrix Q are given by: $K_i=Y_i{X}_2^{-1},i=1,\dots ,n_r$ and $\gamma =\sqrt{{\gamma }_1+{\gamma }_2}$ Q = Q₁ + Q₂ where $Q_1=\left[\begin{array}{cc}{Q}_1^1& 0\\ 0& Q_1^2\end{array}\right]$ and $Q_2=\left[\begin{array}{cc}{Q}_2^1& 0\\ 0& Q_2^2\end{array}\right]$

Proof:

For the proof of the theorem 1, we will start from the sufficient conditions given in the Lemma 3. Let us consider the first condition (17). Multiplying it post and prior by P ⁻¹ and using Schur lemma, the following inequality is obtained:

(28) $$\left[\begin{array}{cc}{P}^{-1}{\mathcal{A}}_{ii}^T+{\mathcal{A}}_{ii}{P}^{-1}+{\tilde{Q}}_1& {\mathcal{D}}_{ii}\\ {\mathcal{D}}_{ii}^T& -{\alpha }_{1}I\\ & \end{array}\right]<0$$

By choosing matrices P and Q₁ as follows: $P=\left[\begin{array}{cc}{P}_1& 0\\ 0& P_2\end{array}\right]$ and ${\tilde{Q}}_1=P^{-1}{Q}_1{P}^{-1}=\left[\begin{array}{cc}{\tilde{Q}}_1^1& 0\\ 0& {\tilde{Q}}_1^2\end{array}\right]$ and Replacing ${\mathcal{D}}_{ii},{\mathcal{A}}_{ii}$ by their expressions in (13), (28) becomes:

(29) $$\left[\begin{array}{ccccccc}{M}_1& M_2& -B_i^1{K}_i& A_i^{12}& 0& I& 0\\ \ast & M_3& -B_i^2{K}_i& A_i^{22}-A_r& -I& 0& I\\ \ast & \ast & -{\alpha }_{1}I& 0& 0& 0& 0\\ \ast & \ast & \ast & -{\alpha }_{1}I& 0& 0& 0\\ \ast & \ast & \ast & \ast & -{\alpha }_{1}I& 0& 0\\ \ast & \ast & \ast & \ast & \ast & -{\alpha }_{1}I& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & -{\alpha }_{1}I\end{array}\right]<0$$

where $$M_1=A_i^{11}{P}_1^{-1}+P_1^{-1}A{}_i^{{11}^T}+{\tilde{Q}}_1^1$$

$$M_2=P_1^{-1}A{}_i^{{21}^T}+(A_i^{12}+B_i^1{K}_i)P_2^{-1}$$

$$M_3=(A_i^{22}+B_i^2{K}_i)P_2^{-1}+P_2^{-1}(A_i^{22}+B_i^2{K}_i{)}^T+{\tilde{Q}}_1^2$$

Multiplying (29) left and right respectively by $diag(\left[\begin{array}{cccccccc}I& & I& P_2^{-1}& I& I& I& I\end{array}\right])$ and its transpose yields to:

(30) $$\left[\begin{array}{ccccccc}{M}_1& M_2& -B_i^1{K}_i{P}_2^{-1}& A_i^{12}& 0& I& 0\\ \ast & M_3& -B_i^2{K}_i{P}_2^{-1}& A_i^{22}-A_r& -I& 0& I\\ \ast & \ast & -P_2^{-1}{\alpha }_1{P}_2^{-1}& 0& 0& 0& 0\\ \ast & \ast & \ast & -{\alpha }_{1}I& 0& 0& 0\\ \ast & \ast & \ast & \ast & -{\alpha }_{1}I& 0& 0\\ \ast & \ast & \ast & \ast & \ast & -{\alpha }_{1}I& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & -{\alpha }_{1}I\end{array}\right]<0$$

Using Lemma 2 we have:

(31) $$P_2^{-1}(-{\alpha }_1)P_2^{-1}\le -2{\alpha }_3{P}_2^{-1}+{\alpha }_3^2({\alpha }_1{)}^{-1}I$$and Schur complement yields to:

(32) $$\left[\begin{array}{cccccccc}{M}_1& M_2& -B_i^1{K}_i{P}_2^{-1}& A_i^{12}& 0& I& 0& 0\\ \ast & M_3& -B_i^2{K}_i{P}_2^{-1}& A_i^{22}-A_r& -I& 0& I& 0\\ \ast & \ast & -{\alpha }_3{P}_2^{-1}& 0& 0& 0& 0& {\alpha }_{3}I\\ \ast & \ast & \ast & -{\alpha }_{1}I& 0& 0& 0& 0\\ \ast & \ast & \ast & \ast & -{\alpha }_{1}I& 0& 0& 0\\ \ast & \ast & \ast & \ast & \ast & -{\alpha }_{1}I& 0& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & -{\alpha }_{1}I& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & \ast & -{\alpha }_1\end{array}\right]<0$$

Using the following variable change $X_1=P_1^{-1},X_2=P_2^{-1}$ and Y_i = K_iX₂, the conditions (26) of the theorem is fulfilled.

To carry out the second LMI of the theorem 1, we proceed in the same way with the second sufficient condition of Lemma 3. Multiplying (18) post and prior with P ⁻¹ and using the Schur lemma, the following inequality is obtained:

(33) $${\displaystyle \frac{1}{2}\left[\begin{array}{cc}{P}^{-1}{\mathbb{A}}_{ij}^T+{\mathbb{A}}_{ij}{P}^{-1}+{\tilde{Q}}_2& {\mathbb{D}}_{ij}\\ {\mathbb{D}}_{ij}^T& -{\alpha }_{2}I\end{array}\right]<0}$$By choosing the matrix ${\tilde{Q}}_2=P^{-1}{Q}_2{P}^{-1}=\left[\begin{array}{cc}{\tilde{Q}}_2^1& 0\\ 0& {\tilde{Q}}_2^2\end{array}\right]$ and replacing D_ij and a_ij by their expression in (13), (33) becomes:

(34) $${\displaystyle \frac{1}{2}\left[\begin{array}{ccccccc}{N}_1& N_2& -(B_i^1{K}_j+B_j^1{K}_i)& A_i^{12}+A_j^{12}& 0& 2I& 0\\ \ast & N_3& -(B_i^2{K}_j+B_j^2{K}_i)& A_i^{22}+A_j^{22}-2A_r& -2I& 0& 2I\\ \ast & \ast & -{\alpha }_{2}I& 0& 0& 0& 0\\ \ast & \ast & \ast & -{\alpha }_{2}I& 0& 0& 0\\ \ast & \ast & \ast & \ast & -{\alpha }_{2}I& 0& 0\\ \ast & \ast & \ast & \ast & \ast & -{\alpha }_{2}I& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & -{\alpha }_{2}I\end{array}\right]<0}$$

$$N_1=(A_i^{11}+A_j^{11})P_1^{-1}+P_1^{-1}(A_i^{11}+A_j^{11}{)}^T+{\tilde{Q}}_2^1$$

$$N_2=P_1^{-1}(A_i^{21}+A_j^{21}{)}^T+(A_i^{12}+A_j^{12})P_2^{-1}+(B_i^1{K}_j+B_j^1{K}_i)P_2^{-1}$$

$$N_3=(A_i^{22}+A_j^{22}+B_i^2{K}_j+B_j^2{K}_i)P_2^{-1}+P_2^{-1}(A_i^{22}+A_j^{22}+B_i^2{K}_j+B_j^2{K}_i{)}^T+{\tilde{Q}}_2^2$$

Multiplying (34) left and right respectively by: $diag(\left[\begin{array}{ccccccc}I& I& P_2^{-1}& I& I& I& I\end{array}\right])$ and its transpose we get:

(35) $${\displaystyle \frac{1}{2}\left[\begin{array}{ccccccc}{N}_1& N_2& -(B_i^1{K}_j+B_j^1{K}_i)P_2^{-1}& A_i^{12}+A_j^{12}& 0& 2I& 0\\ \ast & N_3& -(B_i^2{K}_j+B_j^2{K}_i)P_2^{-1}& A_i^{22}+A_j^{22}-2A_r& -2I& 0& 2I\\ \ast & \ast & -P_2^{-1}{\alpha }_2{P}_2^{-1}& 0& 0& 0& 0\\ \ast & \ast & \ast & -{\alpha }_{2}I& 0& 0& 0\\ \ast & \ast & \ast & \ast & -{\alpha }_{2}I& 0& 0\\ \ast & \ast & \ast & \ast & \ast & -{\alpha }_{2}I& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & -{\alpha }_{2}I\\ & & & & & & \end{array}\right]<0}$$using Lemma 2 we have:

(36) $$P_2^{-1}(-{\alpha }_2)P_2^{-1}\le -2{\alpha }_4{P}_2^{-1}+{\alpha }_4^2({\alpha }_2{)}^{-1}I$$and Schur complements yiels to:

(37) $${\displaystyle \frac{1}{2}\left[\begin{array}{cccccccc}{N}_1& N_2& -(B_i^1{K}_j+B_j^1{K}_i)P_2^{-1}& A_i^{12}+A_j^{12}& 0& 2I& 0& 0\\ \ast & N_3& -(B_i^2{K}_j+B_j^2{K}_i)P_2^{-1}& A_i^{22}+A_j^{22}-2A_r& -2I& 0& 2I& 0\\ \ast & \ast & -2{\alpha }_4{P}_2^{-1}& 0& 0& 0& 0& {\alpha }_{4}I\\ \ast & \ast & \ast & -{\alpha }_{2}I& 0& 0& 0& 0\\ \ast & \ast & \ast & \ast & -{\alpha }_{2}I& 0& 0& 0\\ \ast & \ast & \ast & \ast & \ast & -{\alpha }_{2}I& 0& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & -{\alpha }_{2}I& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & \ast & -{\alpha }_{2}I\end{array}\right]<0}$$

Using the following variable change $X_1=P_1^{-1},X_2=P_2^{-1}$ and Y_i = K_iX₂, the conditions (27) of the theorem is fulfilled. This achieves the proof of the theorem.