H Observer-Based Control of Uncertain Neutral Systems with Mixed Delays

In this paper, the robust In this paper, the robust H control problem of output dynamic observerbased control for a class of uncertain neutral systems with discrete and distributed time delays is considered. Linear matrix inequality (LMI) optimization approach is used to design the new H output dynamic controls. The minimal H -norm bound and the maximal perturbed bound are given. Based on the result of this paper, the constraint of matrix equality is not necessary for designing the H observer-based controls. Finally, a numerical example is given to illustrate the efficiency of the proposed approach.


Introduction
The existence of the time-delay phenomena in a dynamic system may cause instability or bad performances in dynamic systems [1][2][3].In some practical systems, the system models can be described by functional differential equation of neutral type, which the models depend on the state delay but also depend on the state derivatives.Physical examples for neutral system have distributed networks, population ecology, process including steam, heat exchanges, lossless transmission line.Hence the stability and stabilization problems for neutral time-delay system have received some attenuation.By increasing in the equation number of summands and simultaneously decreasing the differences between neighbouring argument values, one naturally arrives at equations with distributed (or continuous) and mixed (both distributed and discrete) delay arguments [2].In view of Fridman [2], the distributed delays, play an important role about the stability of system.
In the many real-world sysyems, state feedback control will fail to guarantee the stabilizability when some of system states are not measurable.In the observer-based control, output dynamic feedback control is provided and the system state can be estimated from the process.The observer-based controls are often be utilized to stabilize unstable systems or improve the system proformance.Hence, the observer-based control for systems has been an interesting topic in control theory.Lyapunov stability theory is used to design the nonlinear state observers for linear time varying systems [4].In [5], the LMI approach was introduced to design the observerbased controls for uncertain systems.
On the other hand, the f H control concept was proposed to reduce the effect of the disturbance input on the regulated output to within a prescribed level.The state feedback f H controls for uncertain neutral time- delay systems had been considered in [6][7].The output f

H
filtering design for neutral system without uncertainties had been proposed in [8].To the best of the knowledge of author, the robust f H observer-based control for neutral systems with discrete and distributed time delays has never been considered in the past.In this paper, we will adopt this useful methodology to the design of the robust f H observer-based controls for a class of uncertain neutral time-delay systems.The three classes of f H observer-based controls with known and unknown (uncertain) time-delay values will be considered.Moreover, the minimal f H -norm bound and the maximal perturbed bound for the observer-based controls are provided.

Problem formulations
Consider the following uncertain neutral system with discrete and distributed time delays: , , that appear in system (1) will represent the impossibility for exact mathematical model of a dynamic system due to the system complexity.The uncertainty has been widely used in many practical systems which can be either exactly modeled or overbounded by the condition dQ .The Q is the bounding parameter on the uncertain perturbation t F , the matrix t F contains the uncertain parameters, and constant matrices , specify how the uncertain parameter t F affect the nominal matrix of system (1).We wish to design the following modified observer-based control with the known time-delay values h , W and K for system (1): , for some 0 !J .In this condition, the system (1) is said to be stabilizable with disturbance attenuation J and degree Q by observer-based control (2), and the control law ( 2) is said to be an f H observer- based control for system (1).The parameter J is said to be the f H -norm bound for the f H observer-based control, and the parameter Q is said to be the perturbed bound for the f H observer-based control.
Lemma 1. [9] For a given assume that nxn X is a symmetric matrix, then there exists a matrix 3 Robust f H control By (2), system (1) and ( 2) can be rewritten as The minimal disturbance attenuation J , the allowable maximal bound Q , and the f H observer-based control (2) could be solved simultanously from the following result.Theorem 1.Consider the system (1) with the observerbased control (2).Suppose and the following optimization problem: , and ) , and ) . Then the system (1) is robustly stabilizable with disturbance attenuation N J and degree 1 U X by the f H observer-based control (2).The matrices , and
Proof: Due to the limitation of page, it is omitted.

Example
Consider the neutral system (1) with the parameters:  In this situation, the states of the system (1) with (4) could not be measurable or have no any practical sense.Hence, the state feedback control schemes in [7] cannot be applied to design the f H control of system.Obviously, the design scheme derived in this paper is more efficient and flexible.
In this paper, the problem of f H control design for a class of uncertain systems has been considered.LMI optimization approach has been developed to construct the output f H dynamic observer-based feedback control.A numerical example has been given to demonstrate the merits of the proposed results.Finally, we have noticed that the useful results on the robust observer-based control problem for linear systems with perturbations [11].Based on the LMI approach, the robust observerbased control has been derived.However, the conditions including the constraint of matrix equality are not the classic LMI feasible form.Their results can not use the efficient software MATLAB to solve the LMI problem with equality constraint.Our obtained results can be performed directly by LMI toolbox of MATLAB.