Stabilization of Neutral Systems with Saturating Actuators

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, [1–4] 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 [5–9]. 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, 15–17]. 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 𝑃_𝑖 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: ℜ denotes the set of real numbers, ℜ^𝑛 denotes the 𝑛 dimensional Euclidean space, and ℜ^𝑚×𝑛 denotes the set of all 𝑚×𝑛 real matrices. The notation 𝑋≥𝑌 (resp., 𝑋>𝑌), where 𝑋 and 𝑌 are symmetric matrices, means that 𝑋−𝑌 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],ℜ^𝑛) denotes the Banach space of continuous vector functions mapping the interval [−𝑑,0] into ℜ^𝑛. 𝐼 denotes the identity matrix of appropriate dimensions. For a matrix 𝐾, the 𝑖th row of 𝐾 is denoted by 𝑘_𝑖. For any vector 𝑢∈ℜ^𝑚, the saturation function is defined by sat(𝑢)=[sat(𝑢₁)sat(𝑢₂)⋯sat(𝑢_𝑚)]^𝑇, where sat(𝑢_𝑖)=sign(𝑢_𝑖)min{|𝑢_𝑖|,𝑢_𝑖}, with given bounds 𝑢_𝑖>0. The convex hull of a set is the minimal convex set containing it. Thus, for a set of points 𝑥₁,𝑥₂,…,𝑥_𝑛∈ℜ^𝑛, its convex hull is 𝐶𝑜{𝑥₁,𝑥₂,…,𝑥_𝑛}={∑^𝑛𝑖=1𝛼_𝑖𝑥_𝑖,∑^𝑛𝑖=1𝛼_𝑖=1,𝛼_𝑖≥0}.

2. Problem Formulation and Definitions

Consider the following state-space linear system, with time-varying delays in the state and its derivative:𝑥̇(𝑡)−𝐶𝑥̇(𝑡−𝜏(𝑡))=+𝐴𝐵₀𝑥s(a𝑡t)+(𝑢𝐴(𝑡₁)𝑥),(𝑡−ℎ(𝑡))(1) where 𝑥(𝑡)∈ℜ^𝑛 is the state, 𝑢(𝑡)∈ℜ^𝑚 is the control input, 𝐶,𝐴₀,𝐴₁, and 𝐵 are known real constant matrices.

It must be noticed that throughout the paper, following [15–17], the delays 𝜏(𝑡) and ℎ(𝑡) are assumed to be unknown but bounded functions of time, continuously differentiable, with their respective rates of change bounded as follows:0≤ℎℎ̇((𝑡𝑡))≤≤ℎ𝑑_𝑚1,0≤𝜏(𝑡)<∞,,𝜏.(𝑡)≤𝑑₂,(2) where ℎ_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 by𝑥𝑡₀+𝜃=𝜙(𝜃),𝜃∈−ℎ,0,(3) where ℎ=max_𝑡≥0{𝜏(𝑡),ℎ(𝑡)} and 𝜙(⋅) is a vector of differentiable functions of initial values (i.e., 𝜙∈𝐶¹[−ℎ,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=𝜙∈𝐶¹−ℎ,0∶lim_𝑡→∞𝑥(𝑡)=0.(4) The exact determination of Ψ is generally difficult. Consequently, it is useful to search for an estimate Ξ_𝛿⊂Ψ of the domain of attraction, whereΞ_𝛿=⎧⎪⎨⎪⎩𝜙∈𝐶¹−ℎ,0∶max_{−ℎ,0}‖𝜙‖≤𝛿⎫⎪⎬⎪⎭(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:𝐷𝐾,𝑢=𝑥∈ℜ^𝑛;||𝑘_𝑖𝑥||≤𝑢_𝑖,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 ℜ^𝑚×𝑛 with diagonal elements that are 1 or 0. Thus, there are 2^𝑚 elements 𝐷_𝑖 in Θ, and the matrix 𝐷^−𝑖=𝐼−𝐷_𝑖 is also an element of Θ.

Lemma 1 (see [5]). Given 𝐾 and 𝐻 in ℜ^𝑚×𝑛, then sat𝐾𝑥,𝑢∈𝐶𝑜𝐷_𝑖𝐾𝑥+𝐷^−𝑖𝐻𝑥,𝑖=1,…,2^𝑚(8) for all 𝑥∈ℜ^𝑛 that satisfy |ℎ_𝑖𝑥|≤𝑢_𝑖, 𝑖=1,…,𝑚.

Therefore, if we consider any compact set 𝑆_𝑐⊂ℜ^𝑛, for any 𝑥∈𝑆_𝑐 and 𝐻 in ℜ^𝑚×𝑛 such that |ℎ_𝑖𝑥|≤𝑢_𝑖, then the closed loop system of (1) and (6) may be written as follows:𝑥̇(𝑡)−𝐶𝑥̇(𝑡−𝜏(𝑡))=^2𝑚_𝑗=1𝜆_𝑗𝐴_𝑗𝑥(𝑡)+𝐴₁𝑥(𝑡−ℎ(𝑡)),(9) where 𝐴_𝑗=𝐵(𝐷_𝑗𝐾+𝐷^−𝑗𝐻)+𝐴₀,∑^2𝑚_𝑗=1𝜆_𝑗=1, and 𝜆_𝑗≥0.

The following L-K functional candidate will be used throughout the paper:𝑉(𝑡)=+𝑥^𝑇(𝑡)𝑃₁𝑥(𝑡)+^{𝑡𝑡−ℎ(𝑡)}𝑥^𝑇(𝑠)𝑄𝑥(𝑠)𝑑𝑠_0−ℎ𝑚^{𝑡𝑡+𝜃}+𝑥̇^𝑇(𝑠)𝑅𝑥̇(𝑠)𝑑𝑠𝑑𝜃_{𝑡𝑡−𝜏(𝑡)}𝑥̇^𝑇(𝑠)𝑊𝑥̇(𝑠)𝑑𝑠,(10) where 𝑃₁=𝑃^𝑇1>0,𝑄=𝑄^𝑇>0,𝑅=𝑅^𝑇>0 and 𝑊=𝑊^𝑇>0.

Finally, for a positive scalar 𝛽 and a positive definite symmetric matrix 𝑃₁, the ellipsoid 𝐷_𝑒 is defined as follows:𝐷_𝑒≡𝑥(𝑡)∈ℜ^𝑛;𝑥^𝑇(𝑡)𝑃₁𝑥(𝑡)≤𝛽⁻¹.(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 Ξ_𝛿, 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=𝑃^𝑇1>0,𝑄=𝑄^𝑇>0,𝑅=𝑅^𝑇>0,𝑊=𝑊^𝑇>0, and appropriately dimensioned matrices 𝑃_𝑖,𝑖=2,…,6 such that the following condition holds: Γ_𝑗=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝Γ^11(𝑗)Γ_21(𝑗)Γ^𝑇21(𝑗)Γ^𝑇31−𝑃^𝑇4𝑃^𝑇2𝐶Γ₃₁Γ₂₂Γ^𝑇32−𝑃^𝑇5𝑃^𝑇3𝐶−𝑃₄Γ₃₂Γ₃₃−𝑃^𝑇60−𝑃₅−𝑃₆−1ℎ_𝑚𝐶𝑅0_𝑇𝑃₂𝐶^𝑇𝑃₃00𝑗=1𝑑,₂…−,12_𝑚𝑊⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠<0,,(12) where Γ||ℎ_𝑖𝑥||≤𝑢_𝑖,∀𝑥∈𝐷_𝑒,(13)_11(𝑗)Γ=𝑃^𝑇2𝐴_𝑗+𝐴^𝑇𝑗𝑃₂+𝑃₄+𝑃^𝑇4+𝑄,_21(𝑗)Γ₂₂=𝑃₁+𝑃^𝑇3𝐴_𝑗+𝑃^𝑇5−𝑃₂,Γ=₃₁ℎ_𝑚𝑅+𝑊−𝑃₃−𝑃^𝑇3,Γ=₃₂𝐴^𝑇1𝑃₂−𝑃₄+𝑃^𝑇6,Γ₃₃=𝐴^𝑇1𝑃₃−𝑃₅,=𝑑₁−1𝑄−𝑃₆−𝑃^𝑇6.(14)

Proof of Theorem 2. Calculating the time derivative of the proposed Lyapunov function (10) along the trajectory of the system (9) gives 𝑉̇(𝑡)=−2𝑥^𝑇1(−𝑡)ℎ𝑃̇₁(𝑡𝑥)̇(𝑡𝑥)_𝑇+𝑥^𝑇(𝑡)𝑄𝑥(𝑡)+ℎ_𝑚(𝑡−ℎ(𝑡))𝑄𝑥(𝑡−ℎ(𝑡))−𝑥̇^𝑇(𝑡)𝑅𝑥̇(𝑡)_{𝑡𝑡−ℎ𝑚}−1−𝑥𝜏.̇^𝑇(𝑡()𝑠)𝑅𝑥̇_𝑇𝑥̇(𝑠)𝑑𝑠+𝑥̇^𝑇(𝑡)𝑊𝑥̇(𝑡)(𝑡−𝜏(𝑡))𝑊𝑥̇(𝑡−𝜏(𝑡)).(15) From (2), it is clear that the following is true: −^{𝑡𝑡−ℎ}_𝑚𝑥̇^𝑇(𝑠)𝑅𝑥̇(𝑠)𝑑𝑠≤−^{𝑡𝑡−ℎ(𝑡)}𝑥̇^𝑇(𝑠)𝑅𝑥̇(𝑠)𝑑𝑠.(16) Also, from the Leibniz-Newton formula 0=𝑥(𝑡)−𝑥(𝑡−ℎ(𝑡))−∫^{𝑡𝑡−ℎ(𝑡)}𝑥̇(𝑠)𝑑𝑠 and using (9), we can write the first term of (15) as follows: 2=𝑥^𝑇2𝑥(𝑡̃_𝑇)𝑃₁𝑥̇(𝑡)(𝑡)𝑃^𝑇=2𝑥̃_𝑇⎛⎜⎜⎜⎜⎝𝑥̇00(𝑡)⎞⎟⎟⎟⎟⎠(𝑡)𝑃^𝑇×⎛⎜⎜⎜⎜⎜⎜⎝−𝑥̇(𝑡)+𝐶𝑥𝑥(̇𝑡()𝑡−−𝑥𝜏((𝑡𝑡)−)+ℎ(𝑥𝑡̇^2𝑚)(_𝑗=1)𝑡𝜆)−_𝑗𝐴_𝑗𝑥(𝑡)_{𝑡𝑡−ℎ(𝑡)}+𝐴₁𝑥(𝑡−ℎ(𝑡))𝑥̇(𝑠)𝑑𝑠⎞⎟⎟⎟⎟⎟⎟⎠,(17) where 𝑥̃(𝑡)=(^{𝑥𝑇(𝑡)̇𝑥𝑇(𝑡)𝑥𝑇(𝑡−ℎ(𝑡))})^𝑇 and 𝑃=^𝑃1𝑃00₂^𝑃₃^0𝑃_4𝑃5𝑃6.
Since Σ^2𝑚_𝑗=1𝜆_𝑗=1, substituting (17) into (15) gives 𝑉̇(𝑡)=^2𝑚_𝑗=1𝜆_𝑗⎧⎪⎪⎨⎪⎪⎩𝑥̃_𝑇(𝑡)Ξ_𝑗𝑥̃(𝑡)+2𝑥̃^𝑇(𝑡)𝑃^𝑇⎛⎜⎜⎜⎜⎝−00𝐼⎞⎟⎟⎟⎟⎠_{𝑡𝑡−ℎ(𝑡)}+𝑥_𝑇𝑥̇(𝑠)𝑑𝑠−1−(𝑡)ℎ𝑄̇(𝑡𝑥)(𝑡𝑥)_𝑇−1−𝜏.(𝑡)𝑥̇_𝑇(𝑡−ℎ(𝑡))𝑄𝑥(𝑡−ℎ(𝑡))+ℎ_𝑚(𝑡−𝜏(𝑡))𝑊𝑥̇(𝑡−𝜏(𝑡))+2𝑥̃_𝑇𝑥̇^𝑇(𝑡)𝑅𝑥̇(𝑡)+𝑥̇^𝑇(𝑡)𝑊𝑥̇(𝑡)(𝑡)𝑃^𝑇−⎛⎜⎜⎜⎜⎝𝐶00⎞⎟⎟⎟⎟⎠𝑥̇(𝑡−𝜏(𝑡))_{𝑡𝑡−ℎ(𝑡)}𝑥̇^𝑇(𝑠)𝑅𝑥̇(𝑠)𝑑𝑠⎫⎪⎪⎬⎪⎪⎭,(18) where Ξ_𝑗=𝑃^𝑇^0𝐼0𝐴_{𝑗−𝐼𝐴1𝐼0−𝐼}+⁰𝐴^𝑇𝑗_{𝐼𝐼−𝐼00𝐴𝑇1−𝐼}𝑃.
Using Jensen’s inequality [21], the last term in (18) can be bounded as follows: −^{𝑡𝑡−ℎ(𝑡)}𝑥̇^𝑇(𝑠)𝑅𝑥̇(𝑠)𝑑𝑠≤−1ℎ_𝑚^{𝑡𝑡−ℎ(𝑡)}𝑥̇^𝑇(𝑠)𝑑𝑠𝑅^{𝑡𝑡−ℎ(𝑡)}𝑥̇(𝑠)𝑑𝑠.(19) Therefore, we get 𝑉̇(𝑡)≤^2𝑚_𝑗=1𝜆_𝑗⎧⎪⎪⎨⎪⎪⎩𝑥̃_𝑇(𝑡)Ξ_𝑗𝑥̃(𝑡)+2𝑥̃^𝑇(𝑡)𝑃^𝑇⎛⎜⎜⎜⎜⎝−00𝐼⎞⎟⎟⎟⎟⎠_{𝑡𝑡−ℎ(𝑡)}+𝑥_𝑇𝑥̇(𝑠)𝑑𝑠−1−(𝑡)𝑑𝑄₁𝑥(𝑡)−1−𝑑₂𝑥^𝑇(1−ℎ(𝑡))𝑄𝑥(1−ℎ(𝑡))+ℎ_𝑚𝑥̇^𝑇(1−𝜏(𝑡))𝑊𝑥̇(1−𝜏(𝑡))+𝑥̇_𝑇𝑥̇^𝑇(𝑡)𝑅𝑥̇(𝑡)(𝑡)𝑊𝑥̇(𝑡)+2𝑥̃^𝑇(𝑡)𝑃^𝑇−1⎛⎜⎜⎜⎜⎝𝐶00⎞⎟⎟⎟⎟⎠𝑥̇(𝑡−𝜏(𝑡))ℎ_𝑚^{𝑡𝑡−ℎ(𝑡)}𝑥̇^𝑇(𝑠)𝑑𝑠𝑅^{𝑡𝑡−ℎ(𝑡)}𝑥̇(𝑠)𝑑𝑠⎫⎪⎪⎬⎪⎪⎭.(20) By simple manipulation, the inequality (20) can be rewritten as: 𝑉̇(𝑡)≤^2𝑚_𝑗=1with𝜆Ω_𝑗Ω(𝑡,^𝑇𝑠()𝑡,=𝑠)⎛⎜⎜⎜⎜⎜⎝Γ_𝑗Ω(𝑥𝑡,̃(𝑠𝑡))_{𝑡𝑡−ℎ(𝑡)}<0,𝑥̇(𝑡−𝑥𝜏̇(𝑠(𝑡))𝑑)𝑠⎞⎟⎟⎟⎟⎟⎠,(21) and Γ_𝑗 defined in (12). Therefore, if the condition (13) holds, then 𝑉̇(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 Ξ_𝛿 inside the domain of attraction if there exist 𝑄=𝑄^𝑇>0,𝑅=𝑅^𝑇>0,𝑊=𝑊^𝑇>0,𝑋₁=𝑋^𝑇1>0,𝑋₂,𝑋₃∈ℜ^𝑛×𝑛, 𝑈,𝐺∈ℜ^𝑚×𝑛, 𝜀₁,𝜀₂∈ℜ, and positive scalars 𝛽 and 𝛿, satisfying the following conditions: _(𝑗)=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝^{11∗∗∗∗∗∗∗21(𝑗)22∗∗∗∗∗∗−𝜀1𝑄𝐴𝑇11−𝜀2𝑄𝐴𝑇1𝑑1−1𝑄∗∗∗∗∗−𝜀1𝑅𝐴𝑇1−𝜀2𝑅𝐴𝑇110−}_ℎ𝑚^{0𝑅∗∗∗∗}_{𝑊𝐶𝑇𝑑002−1ℎ𝑊∗∗∗𝑚𝑋2ℎ𝑚𝑋3000−ℎ𝑚𝑋𝑅∗∗2𝑋30000−𝑋𝑊∗1000000−𝑄}⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠<0,𝑗=1,…,2_𝑚⎛⎜⎜⎝𝑔𝛽∗_𝑇𝑖,(22)𝑢^2𝑖𝑋₁𝛿⎞⎟⎟⎠≥0,𝑖=1,…,𝑚,(23)₂max𝜆𝑋¹⁻¹+2ℎ^𝑚1−𝑑₁+1𝜆𝑄⁻¹;2ℎ^2𝑚𝜆𝑄⁻¹1−𝑑₂𝜆𝑊⁻¹+ℎ^𝑚𝜆𝑅⁻¹≤𝛽⁻¹,(24) where ₁₁₂_1(𝑗)=𝑋₂+𝑋^𝑇2+𝜀₁𝑋₁𝐴^𝑇1+𝐴₁𝑋₁,₂₂=𝑋^𝑇3−𝑋₂+𝐴₀+𝜀₂𝐴₁𝑋₁+𝐵𝐷_𝑗𝑈+𝐷^−𝑗𝐺,=−𝑋^𝑇3−𝑋₃,(25) and 𝜆 denotes the maximum eigenvalue of the corresponding matrix. The corresponding gain matrix that stabilizes the system is given by 𝐾=𝑈𝑋¹⁻¹.(26)

Proof of Theorem 3. From the requirement that 𝑃₁=𝑃^𝑇1>0, if condition (12) is satisfied, then −𝑃₃−𝑃^𝑇3 must be negative definite. Thus, it follows that 𝑃 is nonsingular, where 𝑃⁻¹=𝑋=⎛⎜⎜⎝𝑃¹𝑃₂0𝑃₃⎞⎟⎟⎠⁻¹=⎛⎜⎜⎝𝑋¹𝑋₂0𝑋₃⎞⎟⎟⎠.(27) Then, multiplying (12) on the left by diag{𝑋^𝑇,𝐼,𝐼,𝐼}, on the right by diag{𝑋,𝐼,𝐼,𝐼}, introducing the following changes of variables: 𝑋₁=𝑃¹⁻¹,𝑄=𝑄⁻¹,𝑅=𝑅⁻¹,𝑊=𝑊⎛⎜⎜⎝𝑁⁻¹₁,𝑈=𝐾𝑋₁,𝐺=𝐻𝑋₁,𝑁₂⎞⎟⎟⎠=⎛⎜⎜⎝𝑋₁𝑃^𝑇4+𝑋^𝑇2𝑃^𝑇5𝑋^𝑇3𝑃^𝑇5⎞⎟⎟⎠𝑋₁,(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 to⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝑋₂+𝑋^𝑇2+𝑁₁+𝑁^𝑇1∗∗∗∗∗∗∗_21(𝑗)−𝑋₁₋₁−𝑋₃−𝑋^𝑇3∗∗∗∗∗∗−𝑋₁₋₁𝑁^𝑇1𝐴^𝑇1−𝑋¹⁻¹𝑁^𝑇2𝑑₁−1𝑄⁻¹∗∗∗∗∗𝑁^𝑇1−𝑋¹⁻¹𝑁^𝑇20−1ℎ_𝑚0𝑅⁻¹∗∗∗∗ℎ𝑊𝐶^𝑇00𝑑₂−1𝑊∗∗∗_𝑚𝑋𝑋₂ℎ_𝑚𝑋₃000−ℎ_𝑚𝑅∗∗₂𝑋𝑋₃0000−𝑊∗₁00000𝑗=10,…−,2𝑄_𝑚⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠<0,,(29)with Π_21(𝑗)=𝑋^𝑇3−𝑋₂+𝑁₂+𝐴₀𝑋₁+𝐵(𝐷_𝑗𝑈+𝐷^−𝑗𝐺).
This condition (29) cannot be solved directly, due to the presence of the cross products in 𝑋¹⁻¹𝑁^𝑇1 and 𝑋¹⁻¹𝑁^𝑇2. 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 |ℎ_𝑖𝑥|≤𝑢_𝑖,∀𝑥∈𝐷_𝑒, 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 𝜋₁‖Δ𝜙‖²≤𝑉(𝜙)≤𝜋₂max_{−ℎ,0}‖𝜙‖²,(31) with 𝜋₁=𝜆(𝑋¹⁻¹) and 𝜋₂=max𝜆𝑋¹⁻¹+2ℎ^𝑚1−𝑑₁+1𝜆𝑄⁻¹;2ℎ^2𝑚𝜆𝑄⁻¹1−𝑑₂𝜆𝑊⁻¹+ℎ^𝑚𝜆𝑅⁻¹.(32) From 𝑉̇(𝑡)<0, it follows that 𝑉(𝑡)<𝑉(𝜙), and, therefore, 𝑥^𝑇(𝑡)𝑋¹⁻¹𝑥(𝑡)≤𝑉(𝑡)<𝑉(𝜙)≤max_{−𝜃∈ℎ,0}‖𝜙(𝜃)‖²𝜋₂≤𝛽⁻¹.(33) Then, the inequality (25) guarantees that the trajectories of 𝑥(𝑡) remain within 𝐷_𝑒 for all initial functions 𝜙∈Ξ_𝛿; moreover, 𝑉̇(𝑡)<0 along the trajectories of (9), which implies that lim_𝑡→∞𝑥(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 𝑋¹⁻¹,𝑄⁻¹,𝑅⁻¹, and 𝑊⁻¹ and constructs a feasibility problem, for given ℎ_𝑚, as follows. Find𝑄,𝑅,𝑊,𝑋₁,𝑋₂,𝑋₃,𝑈,𝐺,𝛽,𝜀₁,𝜀₂,𝛿,𝜎₁,𝜎₂,𝜎₃,𝜎₄subjectto𝑋𝛽₁>>0,0,𝛿𝑄>>0,0,𝜎₁𝑅>0,𝑊>0,𝜎>0,𝜎₂>0,𝜎₃>0,₄⎛⎜⎜⎝𝜎>0,(22),(23),₁𝐼𝐼𝑋𝐼₁⎞⎟⎟⎠≥0,⎛⎜⎜⎝𝜎𝐼₂𝐼𝐼⎛⎜⎜⎝𝜎𝑄⎞⎟⎟⎠≥0,₃𝐼𝐼𝐼𝑅⎞⎟⎟⎠≥0,⎛⎜⎜⎝𝜎⁴𝐼𝐼𝐼𝛿𝑊⎞⎟⎟⎠≥0,₂max𝜎₁+2ℎ^𝑚1−𝑑₁𝜎₂;2ℎ^2𝑚𝜎₂+ℎ_𝑚𝜎₃+11−𝑑₂𝜎₄≤𝛽⁻¹.(34) If the above problem has a solution for a given ℎ_𝑚, then there exists a controller 𝑢(𝑡)=𝑈𝑋¹⁻¹𝑥(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 ℎ_𝑚 on the time varying delay. If we select as optimization parameters 𝜀₁ and 𝜀₂ and choose a cost function 𝑡_min, with Σ_(𝑗)≤𝑡_min𝐼, where Σ_(𝑗) is defined in (22), then if there exists a combination of parameters 𝜀₁ and 𝜀₂ that gives a negative 𝑡_min, 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 ℎ_𝑚>0). Step 1. Fix initial values 𝜀₁=𝜀₁₀, 𝜀₂=𝜀₂₀, and ℎ_𝑚=ℎ_𝑚0, where ℎ_𝑚0 must be small enough to have a feasible solution, and set a step variation ℎ_𝑚step.Step 2. Solve the following problem: min_𝜀1,𝜀2𝑡_minsuchthat_(𝑗)≤𝑡_min𝐼,(35) and obtain new values of 𝜀₁ and 𝜀₂.Step 3. If 𝑡_min>0, the previous values of 𝜀₁ and 𝜀₂ give the largest domain of attraction; otherwise (𝑡_min≤0), to improve the solution, set ℎ_𝑚=ℎ_𝑚+ℎ_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=𝐵⎛⎜⎜⎝=00..55⎛⎜⎜⎝11−⎞⎟⎟⎠−,01.5⎞⎟⎟⎠,𝐶=𝐴⎛⎜⎜⎝−₁00=.1⎛⎜⎜⎝0−0.600.2−⎞⎟⎟⎠00.,4.5⎞⎟⎟⎠,𝑢=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 ℎ(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 𝐷_𝑒.

Figure 1 

Trajectories and ellipsoid 𝐷𝑒 for a maximum delay of ℎ𝑚=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 ℎ_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 ℎ(𝑡)=ℎ_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 𝐷_𝑒, and the inner ellipsoid is the stability circle of radius 𝛿=3.0092.

Table 1 

Comparison of bounds on ℎ_𝑚, maximum radius 𝛿, and stabilizing control gains 𝐾 using different approaches.

Figure 2 

Trajectories and estimated domain of attraction for a maximum delay of ℎ𝑚=0.35.

Example 3. Consider the example studied in [5, 6, 9], where the plant can be described by (1)–(3), where ℎ(𝑡)𝐴=₀ℎ=_𝑚⎛⎜⎜⎝0𝐵1.3=1−⎛⎜⎜⎝.2511⎞⎟⎟⎠0⎞⎟⎟⎠,,𝐴𝐶₁==⎛⎜⎜⎝000,−01⎞⎟⎟⎠,=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 Ξ_𝛿 with 𝛿≤42.3308, 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 𝛿≤58.395. In [5], stabilization by a saturated memoryless state feedback law was obtained for all initial conditions in Ξ_𝛿 with a 𝛿≤67.0618. 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 [5–9].

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.