In this paper, the state feedback and design problem of state observer are presented to stabilize the discrete-time piecewise linear systems. In this article, we have two discrete-time systems that one of them is the disturbed system and the other is the system without disturbance.The linear matrix inequalities approach, piecewise quadratic Lyapunov functions and the Finsler’s lemma are used to design the state observer. In addition to the above mentioned methods, the H¥ control approach used to design the state observer for the system with external disturbance. The H¥control approach undermines the disturbance signal and provides an appropriate estimation of the system states. In this paper, by using the Finsler’s lemma, a set of slack variables are introduced to reduce the design conservatism. The Simulation results show the high performance of the proposed method for stabilize the closed-loop system and achieve to the acceptable estimation of state variables.

In this paper, we establish further improvements  of the Young inequality and its reverse. Then, we assert operator versions corresponding them. Moreover, an application including positive linear mappings is given. For example, if $A,B\in {\mathbb B}({\mathscr H})$ are two invertible positive operators such that $0\begin{align*}& \Phi ^{2} \bigg(A \nabla _{\nu} B+ rMm \left( A^{-1}+A^{-1} \sharp_{\mu} B^{-1} -2 \left(A^{-1} \sharp_{\frac{\mu}{2}} B^{-1} \right)\right)\\& \qquad +\left(\frac{\nu}{\mu} \right) Mm \bigg(A^{-1}\nabla_{\mu} B^{-1} -A^{-1} \sharp_{\mu} B^{-1}\bigg)\bigg) \\& \quad \leq \left( \frac{K(h)}{ K\left( \sqrt{{h^{'}}^{\mu}},2 \right)^{r^{'}}} \right) ^{2} \Phi^{2} (A \sharp_{\nu} B),\end{align*}where $r=\min\{\nu,1-\nu\}$, $K(h)=\frac{(1+h)^{2}}{4h}$,  $h=\frac{M}{m}$, $h^{'}=\frac{M^{'}}{m^{'}}$ and $r^{'}=\min\{2r,1-2r\}$. The results of this paper generalize the results of recent years.

In this paper, we study the finitely many constraints of fuzzy relation inequalities problem and optimize the linear objective function on this region which is defined with fuzzy max-Lukasiewicz operator. In fact Lukasiewicz t-norm is one of the four basic t-norms. A new simplification technique is given to accelerate the resolution of the problem by removing the components having no effect on the solution process. Also, an algorithm and one numerical example are offered to abbreviate and illustrate the steps of the problem resolution ‎process.‎

In this paper, the formation tracking of multi-agent systems is discussed. The model considered for each agent is linear with uncertain parameters. The effect of external disturbances is also considered in the model. To achieve predetermined time-varying formation, the required control input is presented. By applying this input, the closed-loop system will take the desired formation. Establishing the appropriate conditions for the realization of time-varying formation and using the Lyapunov theory and the H index to reduce the disturbance effect, results in some linear matrix inequalities. The designed parameter is then computed by solving those linear matrix inequalities. Finally, a simulation example is presented to illustrate the effectiveness of the designed strategy.

In this paper, we study the finitely many constraints of the fuzzy relation inequality problem and optimize the linear objective function on the region defined by the fuzzy max-product operator. Simplification operations have been given to accelerate the resolution of the problem by removing the components having no effect on the solution process. Also, an algorithm and some numerical and applied examples are presented to abbreviate and illustrate the steps of the problem resolution.

In this paper, the finitely many constraints of a fuzzy relation inequalities problem are studied and the linear objective function on the region defined by a fuzzy max-average operator is optimized. A new simplification technique which accelerates the resolution of the problem by removing the components having no effect on the solution process is given together with an algorithm and a numerical example to illustrate the steps of the problem resolution process.

This paper considers the practical asymptotic stabilization of adesired equilibrium point in discrete-time switched affine systems.The main purpose is to design a state feedback switching rule forthe discrete-time switched affine systems whose parameters can beextracted with less computational complexities. In this regard,using switched Lyapunov functions, a new set of sufficientconditions based on matrix inequalities are developed to solve thepractical stabilization problem. For any size of the switched affinesystem, the derived matrix inequalities contain only one bilinearterm as a multiplication of a positive scalar and a positivedefinite matrix. It is shown that the practical stabilizationproblem can be solved via a few convex optimization problems,including linear Matrix inequalities (LMIs) through gridding of ascalar variable interval between zero and one. The numericalexperiments on an academic example and a DC-DC buck-boost converter,as well as comparative studies with the existing works, prove thesatisfactory operation of the proposed method in achieving betterperformances and more tractable numerical solutions.

This paper considers the practical asymptotic stabilization of adesired equilibrium point in discrete-time switched affine systems.The main purpose is to design a state feedback switching rule forthe discrete-time switched affine systems whose parameters can beextracted with less computational complexities. In this regard,using switched Lyapunov functions, a new set of sufficientconditions based on matrix inequalities are developed to solve thepractical stabilization problem. For any size of the switched affinesystem, the derived matrix inequalities contain only one bilinearterm as a multiplication of a positive scalar and a positivedefinite matrix. It is shown that the practical stabilizationproblem can be solved via a few convex optimization problems,including linear Matrix inequalities (LMIs) through gridding of ascalar variable interval between zero and one. The numericalexperiments on an academic example and a DC-DC buck-boost converter,as well as comparative studies with the existing works, prove thesatisfactory operation of the proposed method in achieving betterperformances and more tractable numerical solutions.