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Journal of Control Science and Engineering
Volume 2012 (2012), Article ID 823290, 8 pages
http://dx.doi.org/10.1155/2012/823290
Research Article

Stabilization of Neutral Systems with Saturating Actuators

1Department of Mathematic and Informatics, Faculty Multidisciplinary, University Mohamed Premier, P.O. Box 300, Nador, Selouane 62700, Morocco
2LESSI, Department of Physics, Faculty of Science Dhar El Mehraz, P.O. Box 1796, Fes-Atlas 30000, Morocco
3Departamento Ingenieria de Sistemas y Automatica, Universidad de Valladolid, Valladolid 47005, Spain

Received 13 September 2011; Accepted 22 November 2011

Academic Editor: L. Z. Yu

Copyright © 2012 F. El Haoussi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A method is proposed for stabilization, using static state feedback, of systems subject to time-varying delays in both the states and their derivatives (i.e., neutral systems), in the presence of saturating actuators. Delay-dependent conditions are given to determine stabilizing state-feedback controllers with large domain of attraction, expressed as linear matrix inequalities, readily implementable using available numerical tools and with tuning parameters that make possible to select the most adequate solution. These conditions are derived by using a Lyapunov-Krasovskii functional on the vertices of the polytopic description of the actuator saturations. Numerical examples demonstrate the effectiveness of the proposed technique.

1. Introduction

Systems with time delays constitute basic mathematical models of many real phenomena in circuits theory, economics, mechanics, and so forth, so they have been extensively studied in the literature: see, for example, [14] and references therein for several theoretical studies on this subject. An aspect that has not been frequently taken into account is the fact that, in many of these systems, the actuators have strict limitations. Thus, stabilization of time-delay systems with actuator saturation is an important issue, already addressed by several authors [59]. From those, we can emphasize [9], where a delay-independent condition was obtained, but for systems with no uncertainty and with known time-invariant delays, situation which is not frequent in practice. Generally speaking, the delay-independent scheme for control design does not use any information on the magnitude of the delay, whereas the delay-dependent approach employs such information. Moreover, when the delay is not big, delay-independent criteria tend to be conservative, so a delay-dependent approach is considered in this paper. In this context, we can cite [6], where a delay-dependent stabilization problem has been introduced for time-delay systems with actuators constraints and 𝐻 control. The methodology followed in this paper follows the frequent approach of using Lyapunov-Krasovskii functionals, providing a set of LMIs that can be easily solved using dedicated solvers [10].

This paper concentrates on the specific class of neutral systems, that is, delayed systems in which both the state and its derivative are affected by time delays. Specific examples of these neutral systems appear in population ecology, transmission lines, and other practical systems [11]. These neutral systems are difficult to handle, so although the control design problem has been studied [12], it is not yet completely solved [13]. In particular, neutral systems are particularly sensitive to delays and can be easily destabilized [14]. Stabilization of neutral systems has already been studied in the literature [13, 1517]. For example, in [18], the stabilizing controller is given for local stability but without giving the domain of initial condition. In any case, stabilization is not yet fully explored for the general class studied in this paper of constrained systems: as it has been mentioned, this situation appears frequently in practical applications, including neutral systems [19].

The proposed methodology is developed in this paper as follows: results are provided that guarantee the local stability of the closed loop system when the initial states are taken within a given region of attraction, by using a method based on the Lyapunov-Krasovskii (L-K) approach. When the proposed stability conditions are derived, a specific method is followed to avoid cross products of the state and delayed state. Moreover, to reduce conservatism, some free matrices 𝑃𝑖 are used (only one of them restricted to be positive definite). This introduction of free matrices can be viewed as an extension of the model description introduced in [6], or similar to the slack variables used in [1, 3], albeit in a different context. Finally, a LMI optimization approach is proposed to design the state feedback gain that maximizes the size of the estimated domain of attraction. It is shown for these systems that the proposed results are less conservative than those in the literature.

Notation.The following notations will be used throughout the paper: denotes the set of real numbers, 𝑛 denotes the 𝑛 dimensional Euclidean space, and 𝑚×𝑛 denotes the set of all 𝑚×𝑛 real matrices. The notation 𝑋𝑌 (resp., 𝑋>𝑌), where 𝑋 and 𝑌 are symmetric matrices, means that 𝑋𝑌 is positive semidefinite (resp., positive definite). The symbol * stands for symmetric block in matrix inequalities. 𝜆(𝑃) and 𝜆(𝑃) denote, respectively, the maximal and minimal eigenvalues of a matrix 𝑃. refers to either the Euclidean vector norm, or the induced matrix 2-norm. The symbol 𝐶1([𝑑,0],𝑛) denotes the Banach space of continuous vector functions mapping the interval [𝑑,0] into 𝑛. 𝐼 denotes the identity matrix of appropriate dimensions. For a matrix 𝐾, the 𝑖th row of 𝐾 is denoted by 𝑘𝑖. For any vector 𝑢𝑚, the saturation function is defined by sat(𝑢)=[sat(𝑢1)sat(𝑢2)sat(𝑢𝑚)]𝑇, where sat(𝑢𝑖)=sign(𝑢𝑖)min{|𝑢𝑖|,𝑢𝑖}, with given bounds 𝑢𝑖>0. The convex hull of a set is the minimal convex set containing it. Thus, for a set of points 𝑥1,𝑥2,,𝑥𝑛𝑛, its convex hull is 𝐶𝑜{𝑥1,𝑥2,,𝑥𝑛}={𝑛𝑖=1𝛼𝑖𝑥𝑖,𝑛𝑖=1𝛼𝑖=1,𝛼𝑖0}.

2. Problem Formulation and Definitions

Consider the following state-space linear system, with time-varying delays in the state and its derivative:̇𝑥(𝑡)𝐶̇𝑥(𝑡𝜏(𝑡))=𝐴0𝑥(𝑡)+𝐴1𝑥(𝑡(𝑡))+𝐵sat(𝑢(𝑡)),(1) where 𝑥(𝑡)𝑛 is the state, 𝑢(𝑡)𝑚 is the control input, 𝐶,𝐴0,𝐴1, and 𝐵 are known real constant matrices.

It must be noticed that throughout the paper, following [1517], the delays 𝜏(𝑡) and (𝑡) are assumed to be unknown but bounded functions of time, continuously differentiable, with their respective rates of change bounded as follows:0(𝑡)𝑚̇,0𝜏(𝑡)<,(𝑡)𝑑1,.𝜏(𝑡)𝑑2,(2) where 𝑚>0, 𝑑1<1, and 𝑑2<1 are given positive constants (these bounds are strictly smaller than one to ensure causality: see [20]). The initial condition of system (1) is given by𝑥𝑡0+𝜃=𝜙(𝜃),𝜃,0,(3) where =max𝑡0{𝜏(𝑡),(𝑡)} and 𝜙() is a vector of differentiable functions of initial values (i.e., 𝜙𝐶1[,0]).

Now, suppose that the solution 𝑥(𝑡)=0 is asymptotically stable, for all delays satisfying (2), then the domain of attraction of the origin isΨ=𝜙𝐶1,0lim𝑡𝑥(𝑡)=0.(4) The exact determination of Ψ is generally difficult. Consequently, it is useful to search for an estimate Ξ𝛿Ψ of the domain of attraction, whereΞ𝛿=𝜙𝐶1,0max,0𝜙𝛿(5) and the stability radius 𝛿>0 is a scalar to be determined.

Throughout this paper, we assume the following.

A1. All the eigenvalues of matrix 𝐶 are inside the unit circle.

Controllers in this paper are linear state feedback of the form𝑢(𝑡)=𝐾𝑥(𝑡).(6) For a given gain matrix 𝐾, we define the polyhedron of states that do not cause saturation as follows:𝐷𝐾,𝑢=𝑥𝑛;||𝑘𝑖𝑥||𝑢𝑖,𝑖=1,,𝑚.(7) A similar approach as the one proposed in [5] is used to represent the saturated system by a polytopic model. Denote by Θ the set of all diagonal matrices in 𝑚×𝑛 with diagonal elements that are 1 or 0. Thus, there are 2𝑚 elements 𝐷𝑖 in Θ, and the matrix 𝐷𝑖=𝐼𝐷𝑖 is also an element of Θ.

Lemma 1 (see [5]). Given 𝐾 and 𝐻 in 𝑚×𝑛, then sat𝐾𝑥,𝑢𝐷𝐶𝑜𝑖𝐾𝑥+𝐷𝑖𝐻𝑥,𝑖=1,,2𝑚(8) for all 𝑥𝑛 that satisfy |𝑖𝑥|𝑢𝑖, 𝑖=1,,𝑚.

Therefore, if we consider any compact set 𝑆𝑐𝑛, for any 𝑥𝑆𝑐 and 𝐻 in 𝑚×𝑛 such that |𝑖𝑥|𝑢𝑖, then the closed loop system of (1) and (6) may be written as follows:̇𝑥(𝑡)𝐶̇𝑥(𝑡𝜏(𝑡))=2𝑚𝑗=1𝜆𝑗𝐴𝑗𝑥(𝑡)+𝐴1𝑥(𝑡(𝑡)),(9) where 𝐴𝑗=𝐵(𝐷𝑗𝐾+𝐷𝑗𝐻)+𝐴0,2𝑚𝑗=1𝜆𝑗=1, and 𝜆𝑗0.

The following L-K functional candidate will be used throughout the paper:𝑉(𝑡)=𝑥𝑇(𝑡)𝑃1𝑥(𝑡)+𝑡𝑡(𝑡)𝑥𝑇(+𝑠)𝑄𝑥(𝑠)𝑑𝑠0𝑚𝑡𝑡+𝜃̇𝑥𝑇+(𝑠)𝑅̇𝑥(𝑠)𝑑𝑠𝑑𝜃𝑡𝑡𝜏(𝑡)̇𝑥𝑇(𝑠)𝑊̇𝑥(𝑠)𝑑𝑠,(10) where 𝑃1=𝑃𝑇1>0,𝑄=𝑄𝑇>0,𝑅=𝑅𝑇>0 and 𝑊=𝑊𝑇>0.

Finally, for a positive scalar 𝛽 and a positive definite symmetric matrix 𝑃1, the ellipsoid 𝐷𝑒 is defined as follows:𝐷𝑒𝑥(𝑡)𝑛;𝑥𝑇(𝑡)𝑃1𝑥(𝑡)𝛽1.(11)

3. Main Results

This section presents sufficient conditions that guarantee the convergence to the origin of all the trajectories of system (1), starting from the domain Ξ𝛿, that is included in the ellipsoid (11). First the main results are derived, which are later extended to neutral systems, and a practical algorithm is presented to design controllers that enlarge the size of the domain of initial conditions.

3.1. Neutral Systems with Time-Varying Delays

Theorem 2. The system described by (9) is asymptotically stable if there exist 𝑃1=𝑃𝑇1>0,𝑄=𝑄𝑇>0,𝑅=𝑅𝑇>0,𝑊=𝑊𝑇>0, and appropriately dimensioned matrices 𝑃𝑖,𝑖=2,,6 such that the following condition holds: Γ𝑗=Γ11(𝑗)Γ𝑇21(𝑗)Γ𝑇31𝑃𝑇4𝑃𝑇2𝐶Γ21(𝑗)Γ22Γ𝑇32𝑃𝑇5𝑃𝑇3𝐶Γ31Γ32Γ33𝑃𝑇60𝑃4𝑃5𝑃61𝑚𝐶𝑅0𝑇𝑃2𝐶𝑇𝑃3𝑑002𝑊1<0,𝑗=1,,2𝑚,(12) where ||𝑖𝑥||𝑢𝑖,𝑥𝐷𝑒Γ,(13)11(𝑗)=𝑃𝑇2𝐴𝑗+𝐴𝑇𝑗𝑃2+𝑃4+𝑃𝑇4Γ+𝑄,21(𝑗)=𝑃1+𝑃𝑇3𝐴𝑗+𝑃𝑇5𝑃2,Γ22=𝑚𝑅+𝑊𝑃3𝑃𝑇3,Γ31=𝐴𝑇1𝑃2𝑃4+𝑃𝑇6,Γ32=𝐴𝑇1𝑃3𝑃5,Γ33=𝑑11𝑄𝑃6𝑃𝑇6.(14)

Proof of Theorem 2. Calculating the time derivative of the proposed Lyapunov function (10) along the trajectory of the system (9) gives ̇𝑉(𝑡)=2𝑥𝑇(𝑡)𝑃1̇𝑥(𝑡)+𝑥𝑇̇𝑥(𝑡)𝑄𝑥(𝑡)1(𝑡)𝑇(𝑡(𝑡))𝑄𝑥(𝑡(𝑡))+𝑚̇𝑥𝑇(𝑡)𝑅̇𝑥(𝑡)𝑡𝑡𝑚̇𝑥𝑇(𝑠)𝑅̇𝑥(𝑠)𝑑𝑠+̇𝑥𝑇.𝜏(𝑡)𝑊̇𝑥(𝑡)1(𝑡)̇𝑥𝑇(𝑡𝜏(𝑡))𝑊̇𝑥(𝑡𝜏(𝑡)).(15) From (2), it is clear that the following is true: 𝑡𝑡𝑚̇𝑥𝑇(𝑠)𝑅̇𝑥(𝑠)𝑑𝑠𝑡𝑡(𝑡)̇𝑥𝑇(𝑠)𝑅̇𝑥(𝑠)𝑑𝑠.(16) Also, from the Leibniz-Newton formula 0=𝑥(𝑡)𝑥(𝑡(𝑡))𝑡𝑡(𝑡)̇𝑥(𝑠)𝑑𝑠 and using (9), we can write the first term of (15) as follows: 2𝑥𝑇(𝑡)𝑃1̇𝑥(𝑡)=2̃𝑥𝑇(𝑡)𝑃𝑇00̇𝑥(𝑡)=2̃𝑥𝑇(𝑡)𝑃𝑇×̇𝑥(𝑡)̇𝑥(𝑡)+𝐶̇𝑥(𝑡𝜏(𝑡))+2𝑚𝑗=1𝜆𝑗𝐴𝑗𝑥(𝑡)+𝐴1𝑥(𝑡(𝑡))𝑥(𝑡)𝑥(𝑡(𝑡))𝑡𝑡(𝑡),̇𝑥(𝑠)𝑑𝑠(17) where ̃𝑥(𝑡)=(𝑥𝑇(𝑡)̇𝑥𝑇(𝑡)𝑥𝑇(𝑡(𝑡)))𝑇 and 𝑃=𝑃1𝑃002𝑃30𝑃4𝑃5𝑃6.
Since Σ2𝑚𝑗=1𝜆𝑗=1, substituting (17) into (15) gives ̇𝑉(𝑡)=2𝑚𝑗=1𝜆𝑗̃𝑥𝑇(𝑡)Ξ𝑗̃𝑥(𝑡)+2̃𝑥𝑇(𝑡)𝑃𝑇00𝐼𝑡𝑡(𝑡)̇𝑥(𝑠)𝑑𝑠+𝑥𝑇̇𝑥(𝑡)𝑄𝑥(𝑡)1(𝑡)𝑇.𝜏(𝑡(𝑡))𝑄𝑥(𝑡(𝑡))1(𝑡)̇𝑥𝑇(𝑡𝜏(𝑡))𝑊̇𝑥(𝑡𝜏(𝑡))+𝑚̇𝑥𝑇(𝑡)𝑅̇𝑥(𝑡)+̇𝑥𝑇(𝑡)𝑊̇𝑥(𝑡)+2̃𝑥𝑇(𝑡)𝑃𝑇0𝐶0̇𝑥(𝑡𝜏(𝑡))𝑡𝑡(𝑡)̇𝑥𝑇,(𝑠)𝑅̇𝑥(𝑠)𝑑𝑠(18) where Ξ𝑗=𝑃𝑇0𝐼0𝐴𝑗𝐼𝐴1𝐼0𝐼+0𝐴𝑇𝑗𝐼𝐼𝐼00𝐴𝑇1𝐼𝑃.
Using Jensen’s inequality [21], the last term in (18) can be bounded as follows: 𝑡𝑡(𝑡)̇𝑥𝑇(1𝑠)𝑅̇𝑥(𝑠)𝑑𝑠𝑚𝑡𝑡(𝑡)̇𝑥𝑇(𝑠)𝑑𝑠𝑅𝑡𝑡(𝑡)̇𝑥(𝑠)𝑑𝑠.(19) Therefore, we get ̇𝑉(𝑡)2𝑚𝑗=1𝜆𝑗̃𝑥𝑇(𝑡)Ξ𝑗̃𝑥(𝑡)+2̃𝑥𝑇(𝑡)𝑃𝑇00𝐼𝑡𝑡(𝑡)̇𝑥(𝑠)𝑑𝑠+𝑥𝑇(𝑡)𝑄𝑥(𝑡)1𝑑1𝑥𝑇(1(𝑡))𝑄𝑥(1(𝑡))1𝑑2̇𝑥𝑇(1𝜏(𝑡))𝑊̇𝑥(1𝜏(𝑡))+𝑚̇𝑥𝑇(𝑡)𝑅̇𝑥(𝑡)+̇𝑥𝑇(𝑡)𝑊̇𝑥(𝑡)+2̃𝑥𝑇(𝑡)𝑃𝑇0𝐶01̇𝑥(𝑡𝜏(𝑡))𝑚𝑡𝑡(𝑡)̇𝑥𝑇(𝑠)𝑑𝑠𝑅𝑡𝑡(𝑡).̇𝑥(𝑠)𝑑𝑠(20) By simple manipulation, the inequality (20) can be rewritten as: ̇𝑉(𝑡)2𝑚𝑗=1𝜆𝑗Ω𝑇(𝑡,𝑠)Γ𝑗Ω(𝑡,𝑠)<0,withΩ(𝑡,𝑠)=̃𝑥(𝑡)𝑡𝑡(𝑡),̇𝑥(𝑠)𝑑𝑠̇𝑥(𝑡𝜏(𝑡))(21) and Γ𝑗 defined in (12). Therefore, if the condition (13) holds, then ̇𝑉(𝑡) is negative definite, which ensures the asymptotic stability of the polytopic system (9) [22].
This result gives a general solution for testing stability. We present now the following result that permits to calculate a stabilizing controller.

Theorem 3. The system (1)–(3) is asymptotically stabilized by feedback law 𝑢(𝑡)=𝐾𝑥(𝑡), with Ξ𝛿 inside the domain of attraction if there exist 𝑄=𝑄𝑇>0,𝑅=𝑅𝑇>0,𝑊=𝑊𝑇>0,𝑋1=𝑋𝑇1>0,𝑋2,𝑋3𝑛×𝑛, 𝑈,𝐺𝑚×𝑛, 𝜀1,𝜀2, and positive scalars 𝛽 and 𝛿, satisfying the following conditions: (𝑗)=1121(𝑗)22𝜀1𝑄𝐴𝑇11𝜀2𝑄𝐴𝑇1𝑑11𝑄𝜀1𝑅𝐴𝑇1𝜀2𝑅𝐴𝑇110𝑚0𝑅𝑊𝐶𝑇𝑑0021𝑊𝑚𝑋2𝑚𝑋3000𝑚𝑋𝑅2𝑋30000𝑋𝑊1000000𝑄<0,𝑗=1,,2𝑚𝑔,(22)𝛽𝑇𝑖𝑢2𝑖𝑋1𝛿0,𝑖=1,,𝑚,(23)2max𝜆𝑋11+2𝑚1𝑑1𝜆𝑄1;22𝑚𝜆𝑄1+11𝑑2𝜆𝑊1+𝑚𝜆𝑅1𝛽1,(24) where 11=𝑋2+𝑋𝑇2+𝜀1𝑋1𝐴𝑇1+𝐴1𝑋1,21(𝑗)=𝑋𝑇3𝑋2+𝐴0+𝜀2𝐴1𝑋1𝐷+𝐵𝑗𝑈+𝐷𝑗𝐺,22=𝑋𝑇3𝑋3,(25) and 𝜆 denotes the maximum eigenvalue of the corresponding matrix. The corresponding gain matrix that stabilizes the system is given by 𝐾=𝑈𝑋11.(26)

Proof of Theorem 3. From the requirement that 𝑃1=𝑃𝑇1>0, if condition (12) is satisfied, then 𝑃3𝑃𝑇3 must be negative definite. Thus, it follows that 𝑃 is nonsingular, where 𝑃1𝑃=𝑋=10𝑃2𝑃31=𝑋10𝑋2𝑋3.(27) Then, multiplying (12) on the left by diag{𝑋𝑇,𝐼,𝐼,𝐼}, on the right by diag{𝑋,𝐼,𝐼,𝐼}, introducing the following changes of variables: 𝑋1=𝑃11,𝑄=𝑄1,𝑅=𝑅1,𝑊=𝑊1,𝑈=𝐾𝑋1,𝐺=𝐻𝑋1,𝑁1𝑁2=𝑋1𝑃𝑇4+𝑋𝑇2𝑃𝑇5𝑋𝑇3𝑃𝑇5𝑋1,(28) and then using the Schur complement [10], some conditions are obtained that are bilinear due to cross products of 𝑃6 with 𝑃1,𝑃2, and 𝑃3. To avoid such terms, first we select 𝑃6=0, which leads to𝑋2+𝑋𝑇2+𝑁1+𝑁𝑇121(𝑗)𝑋3𝑋𝑇3𝑋11𝑁𝑇1𝐴𝑇1𝑋11𝑁𝑇2𝑑11𝑄1𝑋11𝑁𝑇1𝑋11𝑁𝑇210𝑚𝑅10𝑊𝐶𝑇𝑑0021𝑊𝑚𝑋2𝑚𝑋3000𝑚𝑋𝑅2𝑋30000𝑋𝑊1000000𝑄<0,𝑗=1,,2𝑚,(29)with Π21(𝑗)=𝑋𝑇3𝑋2+𝑁2+𝐴0𝑋1+𝐵(𝐷𝑗𝑈+𝐷𝑗𝐺).
This condition (29) cannot be solved directly, due to the presence of the cross products in 𝑋11𝑁𝑇1 and 𝑋11𝑁𝑇2. To overcome this, we select 𝑁1=𝜀1𝐴1𝑋1,𝑁2=𝜀2𝐴1𝑋1,(30) where 𝜀1 and 𝜀2 are decision variables. Substituting (30) into (29), the condition in (22) is obtained (which can be solved using the procedure presented in Remark 3).
Moreover, the satisfaction of LMIs (23) guarantee that |𝑖𝑥|𝑢𝑖,𝑥𝐷𝑒, 𝑖=1,,𝑚. This can be proven in the same manner as in [5, 9].
Furthermore, following [2], the Lyapunov functional defined in (10) satisfies 𝜋1Δ𝜙2𝑉(𝜙)𝜋2max,0𝜙2,(31) with 𝜋1=𝜆(𝑋11) and 𝜋2=max𝜆𝑋11+2𝑚1𝑑1𝜆𝑄1;22𝑚𝜆𝑄1+11𝑑2𝜆𝑊1+𝑚𝜆𝑅1.(32) From ̇𝑉(𝑡)<0, it follows that 𝑉(𝑡)<𝑉(𝜙), and, therefore, 𝑥𝑇(𝑡)𝑋11𝑥(𝑡)𝑉(𝑡)<𝑉(𝜙)max𝜃,0𝜙(𝜃)2𝜋2𝛽1.(33) Then, the inequality (25) guarantees that the trajectories of 𝑥(𝑡) remain within 𝐷𝑒 for all initial functions 𝜙Ξ𝛿; moreover, ̇𝑉(𝑡)<0 along the trajectories of (9), which implies that lim𝑡𝑥(𝑡)=0, completing the proof.

Remark 1. The result of Theorem 3 is derived by using Theorem 2, when 𝑃6 is fixed to be zero, in order to simplify the numerical solution: although this makes the solution only slightly more conservative, it reduces significantly the computational cost.

3.2. Practical Algorithms

In this section, we give as remarks some practical procedures to design controllers using the results derived in this paper.

Remark 2. Theorem 3 provides a condition allowing us to compute both a control law and a domain of attraction in which the closed loop neutral system is asymptotically stable. It would be interesting to develop a methodology to estimate the largest possible domain of initial conditions that ensure stability of the system. Unfortunately, this is very difficult, due to the nonlinearity in the system. An interesting solution consists in imposing conditions on the maximal eigenvalues of 𝑋11,𝑄1,𝑅1, and 𝑊1 and constructs a feasibility problem, for given 𝑚, as follows. Find𝑄,𝑅,𝑊,𝑋1,𝑋2,𝑋3,𝑈,𝐺,𝛽,𝜀1,𝜀2,𝛿,𝜎1,𝜎2,𝜎3,𝜎4subjectto𝑋1>0,𝑄>0,𝑅>0,𝑊>0,𝛽>0,𝛿>0,𝜎1>0,𝜎2>0,𝜎3𝜎>0,4𝜎>0,(22),(23),1𝐼𝐼𝐼𝑋1𝜎0,2𝐼𝐼𝐼𝑄𝜎0,3𝐼𝐼𝐼𝑅𝜎0,4𝐼𝐼𝐼𝑊𝛿0,2𝜎max1+2𝑚1𝑑1𝜎2;22𝑚𝜎2+𝑚𝜎3+11𝑑2𝜎4𝛽1.(34) If the above problem has a solution for a given 𝑚, then there exists a controller 𝑢(𝑡)=𝑈𝑋11𝑥(𝑡) that guarantees stability of the saturated neutral system (1)–(3).

Remark 3. When the scalar parameters 𝜀1 and 𝜀2 are fixed, the condition (22) of Theorem 3 becomes LMI. However, choosing arbitrary 𝜀1 and 𝜀2 does not lead to the best result. In the following, a tuning procedure for the parameters 𝜀1 and 𝜀2 is proposed to enlarge the bound 𝑚 on the time varying delay. If we select as optimization parameters 𝜀1 and 𝜀2 and choose a cost function 𝑡min, with Σ(𝑗)𝑡min𝐼, where Σ(𝑗) is defined in (22), then if there exists a combination of parameters 𝜀1 and 𝜀2 that gives a negative 𝑡min, these parameters give a feasible solution of the conditions in Theorem 3 (finding this combination can be carried out by solving the corresponding feasibility problem). Finally, applying a numerical optimization algorithm, it is possible to obtain a locally convergent solution to the problem (using, e.g., fminsearch in the Optimization Toolbox [23]). If the resulting minimum value of the cost function is negative, then a combination of tuning parameters that solves the problem is found.
Thus, this procedure to look for a feasible solution of the conditions in Theorem 3 can be summarized as follows.
Algorithm (maximization of 𝑚>0). Step 1. Fix initial values 𝜀1=𝜀10, 𝜀2=𝜀20, and 𝑚=𝑚0, where 𝑚0 must be small enough to have a feasible solution, and set a step variation 𝑚step.Step 2. Solve the following problem: min𝜀1,𝜀2𝑡minsuchthat(𝑗)𝑡min𝐼,(35) and obtain new values of 𝜀1 and 𝜀2.Step 3. If 𝑡min>0, the previous values of 𝜀1 and 𝜀2 give the largest domain of attraction; otherwise (𝑡min0), to improve the solution, set 𝑚=𝑚+𝑚step and repeat from Step 2.

4. Numerical Examples

This section provides some numerical examples to illustrate that the proposed method is less conservative than previous results in the literature.

Example 1. Consider a neutral system described by (1)–(3), with the following parameters: 𝐴0=0.510.50.5,𝐴1=,11,0.60.400.5𝐵=,𝐶=0.1000.2𝑢=5.(36) It can be seen that this system is unstable, so, for stabilization, a controller was designed based on Theorem 3. In particular, using the algorithm proposed in Section 3.2 to enlarge the bound on the state delay, it was found that, when 𝑑1=0 and 𝑑2=0, the system is stabilizable for all state delays (𝑡)1.566, when the state feedback gain is 𝐾=(2.59390.0653). For this case, the stability radius is 𝛿=0.2472, obtained when 𝜀1=0.1350 and 𝜀2=0.9241.
To check the stability and the corresponding time responses, the closed-loop system was simulated, starting from different initial values inside the domain of attraction given by (5). To check the closed-loop stability, we show in Figure 1 some trajectories of the saturated closed-loop system, together with the ellipsoid 𝐷𝑒.

823290.fig.001
Figure 1: Trajectories and ellipsoid 𝐷𝑒 for a maximum delay of 𝑚=1.566.

Example 2. Consider the system of Example 1, in which we set 𝐶=0. This example gives the system studied in [5, 6, 19], where upper bounds 𝑚 and maximum radius 𝛿 were calculated for which a state feedback control 𝐾 stabilizes (36). Their results are listed in Table 1 along with the results obtained by Theorem 3 (with 𝜀1=0.4569 and 𝜀2=0.8166, selected following the algorithm provided in Section 3.2).
Note that, in [19] the search of the values of 𝜀1 and 𝜀2 is achieved by using an iterative algorithm, while, in this paper, we adopt an optimization procedure which leads to less conservative results.
In order to compare with the results of [5, 6, 19], we take (𝑡)=𝑚=0.35. Theorem 3 yields the stability radius of 𝛿=3.0092 (when 𝜀1=0.0015, 𝜀2=0.9984, and 𝐾=(1.71500.7143)), whereas the results of [5, 6, 19] give, respectively, 𝛿=0.9680, 𝛿=2.9089, and 𝛿=2.852. It is clear that our method gives the largest stability radius. This confirms that the stabilization conditions in this note are less conservative than those of [5, 6, 19].
Figure 2 shows some trajectories of the closed-loop system and the domain of attraction with this controller. The outer ellipsoid is 𝐷𝑒, and the inner ellipsoid is the stability circle of radius 𝛿=3.0092.

tab1
Table 1: Comparison of bounds on 𝑚, maximum radius 𝛿, and stabilizing control gains 𝐾 using different approaches.
823290.fig.002
Figure 2: Trajectories and estimated domain of attraction for a maximum delay of 𝑚=0.35.

Example 3. Consider the example studied in [5, 6, 9], where the plant can be described by (1)–(3), where 𝐴0=11.50.32,𝐴1=,10100𝐵=10,𝐶=0,(𝑡)=𝑚=1,𝑑1=𝑑2=0,𝑢=15,(37) (i.e., the system is nonneutral with constant delays). In [9] stabilization via state feedback was achieved for all initial conditions in Ξ𝛿 with 𝛿42.3308, when the origin of the saturated system is requested to be asymptotically stable and the unsaturated system be 𝛼-stable with 𝛼=1. If we only require that the saturated system be asymptotically stable (i.e., 𝛼=0), it is found that 𝛿58.395. In [5], stabilization by a saturated memoryless state feedback law was obtained for all initial conditions in Ξ𝛿 with a 𝛿67.0618. Following [6], this domain can be still enlarged, with stability radius 𝛿=79.43, in [7], is 79.54, and in [8], is 83.55.
The application of Theorem 3 in the present paper gives a larger stability region: when 𝜀1=0.3307, 𝜀2=1.3307, 𝛽=1, the stability radius is 𝛿=96.1645 when the state feedback gain is 𝐾=10.19700.9550.(38) It is clear that this estimation is less conservative than previous results in [59].

5. Conclusions

This paper has presented a new approach for delay-dependent stabilization of neutral systems with saturating actuators, under time-varying delays. This was accomplished by combining the Lyapunov-Krasovskii technique and the transformation of a system with actuator saturation into a convex polytope of linear systems. An estimation of the domain of attraction is proposed that can be numerically solved using linear matrix inequalities.

The derived conditions depend on the tuning parameters 𝜀1 and 𝜀2 that can be used to enlarge the domain of attraction. A simple iterative procedure based on numerical optimization has been proposed to obtain adequate values for these parameters. This procedure has been illustrated using an example. Finally, additional examples have shown that the particularization of the results for standard delayed systems gives less conservative results than previous results proposed in the literature.

Acknowledgment

This work has been funded by AECI Projects A/024215/09, AECI A/030426/10, and MiCInn DPI2010-21589-c05.

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