The Practical Stabilization for a Class of Networked Systems with Actuator Saturation and Input Additive Disturbances

Abstract

The practical stabilization problem is investigated for a class of linear systems with actuator saturation and input additive disturbances. Firstly, the case of the input additive disturbance being a bounded constant and a variety of different situations of system matrices are studied for the three-dimensional linear system with actuator saturation, respectively. By applying the Riccati equation approach and designing the linear state feedback control law, sufficient conditions are established to guarantee the semiglobal practical stabilization or oscillation for the addressed system. Secondly, for the case of the input additive disturbances being time-varying functions, a more general class of systems with actuator saturation is investigated. By employing the Riccati equation approach, a low-and-high-gain linear state feedback control law is designed to guarantee the global or semiglobal practical stabilization for the closed-loop systems.

1. Introduction

Actuator saturation (control saturation), as a common and typical nonlinear constraint for control systems, is often encountered in various industrial systems, especially in many physical-controlled systems with magnitude limitation in the input. In general, linear systems can be completely controlled by using the linear state feedback, and the semiglobal (or local) stabilization can be achieved [1–3]. Linear systems with actuator saturation are also a special class of nonlinear systems [4]. The occurrence of actuator saturation inevitably affects the control systems performance and may even result in the instability of the controlled systems. Consequently, the actuator saturation has attracted significant attention, and a variety of approaches have been developed in the literature with respect to various types of systems [5–13]. Specifically, the problems of the global (or semiglobal) asymptotic stabilization, the attraction domain estimation, and the practical stabilization have been extensively investigated.

When the dimension of the integrator is greater than or equal to 3, the linear systems with input saturation cannot be stabilized by using the linear state feedback control law, so the global asymptotic stabilization of the system cannot be attained. However, if all eigenvalues of the open-loop system have negative real parts, the global asymptotic stabilization for the addressed system can be guaranteed by designing a globally stable boundary control law. Accordingly, the exponentially semiglobal stabilization problems have been widely studied in [1–3] by designing a linear state feedback control law. By employing the linear matrix inequality (LMI) technique, much work has been done in finding the condition of invariant set and the estimation of attraction domain, see for example, [14–18]. Very recently, networked control systems (NCSs) have attracted considerable attention due to their successful applications in a wide range of areas with the advantage of decreasing the installation cost and implementation difficulties [19–27]. It should be pointed out that, in most related literature concerning the actuator saturation problems, the networked systems with actuator saturation and input additive disturbances have not been thoroughly studied yet.

On the other hand, the study of the linear systems with actuator saturation also covers the practical stabilization problem, that is, a controller is designed such that the trajectories of closed-loop system can enter into an arbitrarily small prescribed neighborhood of the origin in finite time and remain thereafter. The practical stabilization problems have been gaining an increasing research interest, and many important results have been reported, see, for example, [28–30]. To mention a few, the problems of global practical stabilization for planar linear systems have been studied [28, 29] in the presence of actuator saturation and input additive disturbances. By tuning the value of the parameter and designing a parameterized linear state feedback law, sufficient conditions have been established such that all trajectories of the closed-loop systems approach to an arbitrarily small neighborhood of the origin in a finite time and remain thereafter. Moreover, the global stabilization problem has also been studied in [30] for a class of second-order switched systems with input saturation. It is worth mentioning that, because of the mathematical complexity and computational difficulty, the corresponding results for general multidimensional systems with actuator saturation and input additive perturbations have been not reported. By designing the low-and-high-gain and the scheduled low-and-high-gain control laws, the semiglobal asymptotic stabilization problems have been investigated in [31, 32] for general multidimensional linear systems. The feedback control law has been designed to deal with the matched uncertainties and input additive disturbances. However, strict assumptions on the input additive uncertainties and disturbances have been imposed on the systems. It is, therefore, our aim to address the multidimensional systems with general constraints on the input additive disturbances.

Motivated by the above discussion, we aim to investigate the practical stabilization problem for three-dimensional (multidimensional) system with actuator saturation and input additive disturbances. By employing the Riccati equation approach, the linear state feedback control law is designed to guarantee the practical stabilization for the system with actuator saturation and time-invariant input additive disturbances. Moreover, the practical stabilization of a general multidimensional system with actuator saturation and time-varying input additive disturbances is studied, and the low-and-high-gain linear state feedback control law is synthesized by using the Riccati equation approach. The main contributions of this paper can be highlighted as follows: (1) the practical stabilization problem of three-dimensional linear systems is investigated for the first time, which covers actuator saturation as well as input additive disturbances; (2) the low-and-high-gain linear state feedback control law is designed for multidimensional system with actuator saturation and time-varying input additive disturbances.

Notations.The notations in this paper are quite standard except where otherwise stated. The superscript “𝑇” stands for matrix transposition; ℝ^𝑛 (ℝ^𝑛×𝑚) denote, respectively, the 𝑛-dimensional Euclidean space, the set of all 𝑛×𝑚 matrices; the notation 𝑃>0(𝑃≥0) means that matrix 𝑃 is real symmetric and positive definite (positive semi-definite); 0 represents a zero matrix with appropriate dimension, respectively; ‖⋅‖ denotes the Euclidean norm of a vector and its induced norm of a matrix. In symmetric block matrices or long matrix expressions, we use a star “∗” to represent a term that is induced by symmetry. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.

2. The Case of Bounded Constant Input Disturbance

Consider the following three-dimensional system with actuator saturation and input additive disturbance:𝑥̇=𝐴𝑥+𝐵𝜎(𝑢+𝑑), (2.1) where 𝑥∈ℝ³ is the state, 𝐴 and 𝐵 are real matrices of appropriate dimensions, 𝑢∈ℝ is the control input, and 𝑑∈ℝ is the disturbance.

The saturation function 𝜎(⋅) is defined as𝜎(𝑢)=sign(𝑢)min{1,|𝑢|}, (2.2) where the notation of “sign” denotes the signum function.

Before proceeding further, we make the following assumptions.

Assumption 2.1. The matrix pair (𝐴,𝐵) is controllable, and the eigenvalues of 𝐴 have nonpositive real parts.

Assumption 2.2. The input disturbance is bounded, that is, |𝑑|≤𝐷, where 𝐷 is an arbitrarily large positive scalar.

Based on Assumption 2.1, let 𝐵=[001]^𝑇, matrix 𝐴 is in one of the following forms:(𝑎),(𝑏),(𝑐),(𝑑),(𝑒),𝑓,𝑔,(ℎ),𝐴=010001000𝐴=01000100−𝑎𝐴=0100010−𝑎0𝐴=010001−𝑎00𝐴=010001−𝑎−𝑏0𝐴=010001−𝑎0−𝑏𝐴=0100010−𝑎−𝑏𝐴=010001−𝑎−𝑏−𝑐 (2.3) where 𝑎, 𝑏, and 𝑐 are positive scalars.

These forms of the matrix pair (𝐴,𝐵) correspond to eight different cases for system (2.1). In the following, the problem of practical stabilization for system (2.1) will be investigated by designing the linear state feedback control law. To facilitate our development, we introduce the following definition.

Definition 2.3. Define an ellipsoid Ω(𝑃,𝜌) as follows: Ω𝑃,𝜌∶=𝑥∈ℝ³,𝑥^𝑇𝑃𝑥≤𝜌, (2.4) where 𝑃∈ℝ^3×3 is a symmetric positive definite matrix and 𝜌 is a positive scalar.

For system (2.1), the control law under consideration is of the following structure:1𝑢=−𝜀𝐵^𝑇𝑃𝑥, (2.5) where 𝜀∈(0,1], and 𝑃 is a symmetric positive definite matrix to be determined according to the form of (𝐴,𝐵) defined in (2.3).

Lemma 2.4 (see [10]). Letting 𝑢,𝑣∈ℝ with |𝑣|≤1, 𝐸₁=0 and 𝐸₂=1, one has 𝜎(𝑢)∈co{𝑢,𝑣}=co𝐸_𝑖𝑢+𝐸⁻_𝑖𝑣,𝑖=1,2, (2.6) where “co” denotes the convex hull and 𝐸⁻_𝑖=1−𝐸_𝑖.

Theorem 2.5. Consider the system (2.1) with (𝐴,𝐵) defined in (2.3) and the control law (2.5).(i)If 𝐴 satisfies case (𝑔) with 𝑏²>2𝑎 or case (ℎ) with 𝑏𝑐>𝑎 in (2.3), then for any given arbitrarily small set 𝜒₀⊂ℝ³ containing origin in its interior and for any positive scalar 𝐷, there always exists 𝜀^∗∈(0,1] such that, for any 𝜀∈(0,𝜀^∗], all trajectories of the closed-loop system will enter into the set 𝜒₀ in finite time and remain thereafter.(ii)If 𝐴 satisfies cases (𝑎) or (𝑏) or (𝑐) in (2.3), when the initial state is in some bounded set, then for any given arbitrarily small set 𝜒₀⊂ℝ³ containing origin in its interior and any positive number 𝐷, there always exists 𝜀^∗∈(0,1], such that, for any 𝜀∈(0,𝜀^∗], all trajectories of the closed-loop system will enter into the set 𝜒₀ in finite time and remain thereafter.(iii)If 𝐴 satisfies cases (𝑑) or (𝑒) or (𝑓) in (2.3), then all trajectories of the closed-loop system are oscillatory.

Proof. Firstly, let us prove (i).
(i₁) Consider the following system 001𝑥̇=()0100010−𝑎−𝑏𝑥+sat𝑢+𝑑, (2.7) that is, 𝐴 satisfies the case (𝑔) with 𝑏²>2𝑎 in (2.3).
Let 𝑃 be the solution to the following algebra Riccati equation (ARE):𝐴^𝑇𝑃+𝑃𝐴−𝑃𝐵𝐵^𝑇𝑎𝑃=−²000𝑏²−2𝑎0001, (2.8) we have 𝑎𝑃=²+𝑎𝑏𝑎𝑏+𝑎𝑎∗𝑏²+𝑏𝑏∗∗1. (2.9)
Define Ω⁺∶=𝑥∈ℝ³,1𝜀𝐵^𝑇,Ω𝑃𝑥≥1+𝐷₋∶=𝑥∈ℝ³,1𝜀𝐵^𝑇,Ω𝑃𝑥≤−1−𝐷₀∶=𝑥∈ℝ³,1𝜀𝐵^𝑇𝑃𝑥<1+𝐷. (2.10) For any 𝑥∈Ω⁺, we have 𝐵^𝑇𝑃𝑥≥𝜀(1+𝐷), that is, 𝑎𝑥₁+𝑏𝑥₂+𝑥₃≥𝜀(1+𝐷). From (2.7), we obtain 𝑥̇₁=𝑥₂,𝑥̇₂=𝑥₃,𝑥̇₃=−𝑎𝑥₂−𝑏𝑥₃−1. (2.11) Furthermore, there is 𝑎𝑥̇₁+𝑏𝑥̇₂+𝑥̇₃=−1. It means that 𝑎𝑥₁+𝑏𝑥₂+𝑥₃ will decrease in a constant speed and will be less than 𝜀(1+𝐷) in finite time. Hence, any trajectory departing from 𝑥∈Ω⁺ will enter into the interior of Ω⁰ in finite time. Similarly, any trajectory departing from 𝑥∈Ω⁻ will enter into Ω⁰ in finite time.
We choose the Lyapunov function 𝑉(𝑥)=𝑥^𝑇𝑃𝑥 with 𝑃>0 being the solution to ARE (2.8). For any 𝑥∈Ω⁰, we have |(1/𝜀(1+𝐷))𝐵^𝑇𝑃𝑥|<1. It follows from Lemma 2.4 that 1sat𝜀𝐵^𝑇𝑃𝑥−𝑑∈co𝐸_𝑖1𝜀𝐵^𝑇𝑃𝑥−𝑑+𝐸⁻_𝑖1𝜀()𝐵1+𝐷^𝑇𝑃𝑥,𝑖=1,2. (2.12) Noticing that 𝜀>0 is a sufficiently small scalar, we have 𝑉̇(𝑥)=𝑥^𝑇𝑃𝐴𝑥−𝐵sat1𝜀𝑥^𝑇+1𝑃𝑥−𝑑𝐴𝑥−𝐵sat𝜀𝑥^𝑇𝑃𝑥−𝑑^𝑇𝑃𝑥≤max_𝑖∈{1,2}𝑥^𝑇𝑃𝐴𝑥−𝐵𝐸_𝑖1𝜀𝐵^𝑇𝑃𝑥−𝑑+𝐵𝐸⁻_𝑖1𝜀()𝐵1+𝐷^𝑇+𝑃𝑥𝐴𝑥−𝐵𝐸_𝑖1𝜀𝐵^𝑇𝑃𝑥−𝑑+𝐵𝐸⁻_𝑖1𝐵𝜀(1+𝐷)^𝑇𝑃𝑥^𝑇𝑃𝑥≤𝑥^𝑇𝐴^𝑇𝑃+𝑃𝐴−2𝐷𝜀(1+𝐷)𝑃𝐵𝐵^𝑇𝑃𝑥+2𝑥^𝑇𝑃𝐵𝑑≤𝑥^𝑇𝐴^𝑇𝑃+𝑃𝐴−𝑃𝐵𝐵^𝑇𝑃𝑥+2𝜀𝐷(1+𝐷)=−𝑥^𝑇𝑎²000𝑏²−2𝑎0().001𝑥+2𝜀𝐷1+𝐷 (2.13) Define Ω⁷_𝑑∶=𝑥∈ℝ³,𝑥^𝑇𝑎²000𝑏²−2𝑎0()001𝑥≤2𝜀𝐷1+𝐷,𝑏²>2𝑎, (2.14) then there exists a scalar 𝜌>0 such that Ω(𝑃,𝜌) is the smallest ellipsoid satisfying Ω(𝑃,𝜌)∩Ω⁰⊃Ω⁷_𝑑∩Ω⁰. Hence, we have 𝑉̇(𝑥)<0 for all 𝑥∈Ω(𝑃,𝜌)∩Ω⁰, and Ω(𝑃,𝜌)∩Ω⁰ is an invariant set. Hence, any trajectory of the closed-loop system departing from Ω(𝑃,𝜌)∩Ω⁰ will enter into Ω(𝑃,𝜌)∩Ω⁰ in finite time and remain thereafter. If 𝜀→0, then Ω⁷_𝑑 tends to origin. Therefore, Ω(𝑃,𝜌)∩Ω⁰ approaches to the origin.
(i₂) Assuming that matrix 𝐴 satisfies the case (ℎ) and 𝑏𝑐>𝑎, then the system (2.1) can be written as follows: 001𝑥̇=()010001−𝑎−𝑏−𝑐𝑥+sat𝑢+𝑑. (2.15) It follows from 𝑏𝑐>𝑎 that 𝐴 is stable. Then there exists 𝑃>0 satisfying 𝐴^𝑇𝑃+𝑃𝐴=−𝐼₃. (2.16) When 𝑥∈Ω⁺, noting that 𝐵^𝑇𝑃𝑥≥𝜀(1+𝐷), we have sat(𝑢+𝑑)=−1 and𝑉̇(𝑥)=𝑥^𝑇𝑃𝐴+𝐴^𝑇𝑃𝑥−2𝑥^𝑇𝑃𝐵≤−𝑥^𝑇𝑥−2𝜀(1+D)<0. (2.17) Similarly, for any 𝑥∈Ω⁻, it is easy to obtain 𝑉̇(𝑥)<0.
When 𝑥∈Ω⁰, considering |(1/𝜀(1+𝐷))𝐵^𝑇𝑃𝑥|<1 and noticing that 𝜀>0 is a sufficiently small scalar, we have 𝑉̇(𝑥)=𝑥^𝑇𝑃𝐴+𝐴^𝑇𝑃𝑥−2𝜀𝑥^𝑇𝑃𝐵𝐵^𝑇𝑃𝑥+2𝜀𝐷(1+𝐷)<−𝑥^𝑇𝑥+2𝜀𝐷(1+𝐷). (2.18) Set Ω⁸_𝑑∶=𝑥∈ℝ³,‖𝑥‖²≤2𝜀𝐷(1+𝐷). (2.19) Then there exists a scalar 𝜌>0 such that Ω(𝑃,𝜌) is the smallest ellipsoid which satisfies Ω(𝑃,𝜌)∩Ω⁰⊃Ω⁸_𝑑∩Ω⁰. Now, we can conclude that 𝑉̇(𝑥)<0 holds for all 𝑥∉Ω(𝑃,𝜌)∩Ω⁰. Therefore, all trajectories will enter into Ω(𝑃,𝜌)∩Ω⁰ in finite time. When 𝜀→0, Ω(𝑃,𝜌)∩Ω⁰ approaches to the origin.
Secondly, let us prove (ii).
(ii₁) Consider the following system:001𝑥̇=()010001000𝑥+sat𝑢+𝑑, (2.20) that is, 𝐴 satisfies the case (a).
In this case, we choose the matrix 𝑃 to be the solution to the following ARE: 𝐴^𝑇2𝑃+𝑃𝐴−𝜀(1+𝐷)𝑃𝐵𝐵^𝑇1𝑃=−𝜀()()1+𝐷100020001−8𝜀1+𝐷4, (2.21) where 0<𝜀<1/(8(1+𝐷)). Then, we obtain √𝑃=2𝜀()11+𝐷()√2𝜀1+𝐷22∗√221−𝜀(1+𝐷)𝜀()1√1+𝐷∗∗24. (2.22) For any 𝑥∈Ω⁺, 𝐵^𝑇𝑃𝑥≥𝜀(1+𝐷) and sat(𝑢+𝑑)=−1. The system (2.20) reduces to 001𝑥̇=010001000𝑥−. (2.23) Hence, the solution to (2.23) can be described as: 𝑥₁1=−6𝑡³+12𝑥₀₃𝑡²+𝑥₀₂𝑡+𝑥₀₁,𝑥₂1=−2𝑡²+𝑥₀₃𝑡+𝑥₀₂,𝑥₃=−𝑡+𝑥₀₃, (2.24) where 𝑥(0)=[𝑥₀₁𝑥₀₂𝑥₀₃]^𝑇 is the initial state.
By noting 𝐵^𝑇𝑃𝑥≥𝜀(1+𝐷), we have (√2/2)𝑥₁+𝑥₂+(√2/4)𝑥₃≥𝜀(1+𝐷). Furthermore, we obtain𝐵^𝑇√𝑃𝑥̇=22𝑥̇₁+𝑥̇₂+√24𝑥̇₃√=−24𝑡²−√1−22𝑥₀₃√𝑡−24+𝑥₀₃+√22𝑥₀₂. (2.25) If 𝐵^𝑇𝑃𝑥̇<0, then any trajectory departing from Ω⁺ will enter into Ω⁰ in finite time. Let √24𝑡²+√1−22𝑥₀₃√𝑡+24−𝑥₀₃−√22𝑥₀₂>0. (2.26) For 𝑡>0, the discriminant of quadratic inequality is less than zero, then it is accessible to the conditions of the initial point set.
Define𝑋⁺₁∶=𝑥=𝑥₁𝑥₂𝑥₃^𝑇∈Ω⁺∶𝑥₁1∈ℝ,2𝑥²₃+𝑥₂+12<0, (2.27) then any trajectory departed from 𝑋⁺₁ will enter into Ω⁰ in finite time and remain thereafter.
Similarly, when 𝑥∈Ω⁻, a set of initial points can also be found such that any trajectory departing from the set will enter into Ω⁰ in finite time and remain thereafter. After some algebraic manipulations, the initial set of points can be expressed as follows: 𝑋⁻₁∶=𝑥=𝑥₁𝑥₂𝑥₃^𝑇∈Ω⁻∶𝑥₁1∈ℝ,2𝑥²₃−𝑥₂+12<0. (2.28) For any 𝑥∈Ω⁰, |(1/𝜀(1+𝐷))𝐵^𝑇𝑃𝑥|<1, we have 𝑉̇(𝑥)≤𝑥^𝑇𝑃𝐴+𝐴^𝑇𝑃−2𝜀(1+𝐷)𝑃𝐵𝐵^𝑇𝑃()−1𝑥+2𝜀𝐷1+𝐷𝜀()𝑥1+𝐷^𝑇()100010001−8𝜀1+𝐷4𝑥+2𝜀𝐷(1+𝐷). (2.29) Let Ω¹_𝑑∶=𝑥∈ℝ³,𝑥^𝑇()100010001−8𝜀1+𝐷4𝑥≤2𝜀²𝐷(1+𝐷)²1,𝜀<8(1+𝐷). (2.30) Then there exists a scalar 𝜌>0 such that Ω(𝑃,𝜌) is the smallest ellipsoid which satisfies Ω(𝑃,𝜌)∩Ω⁰⊃Ω¹_𝑑∩Ω⁰. Now, we can conclude that 𝑉̇(𝑥)<0 holds for all 𝑥∈Ω(𝑃,𝜌)∩Ω⁰. So Ω(𝑃,𝜌)∩Ω⁰ is an invariant set. For the initial state 𝑥∉Ω(𝑃,𝜌)∩Ω⁰, any trajectory of the system departing from 𝑥∈𝑋⁺₁∪𝑋⁻₁∪(Ω⁰⧵Ω(𝑃,𝜌)) will enter into Ω(𝑃,𝜌)∩Ω⁰ in finite time and remain thereafter. When 𝜀→0, Ω(𝑃,𝜌)∩Ω⁰ approaches to the origin.
(ii₂) Consider the following system: 001𝑥̇=()01000100−𝑎𝑥+sat𝑢+𝑑, (2.31) that is, 𝐴 satisfies the case (b).
In this case, we choose the matrix 𝑃 to be the solution to the following ARE:𝐴^𝑇2𝑃+𝑃𝐴−𝜀𝑃𝐵𝐵^𝑇2𝑃=−𝜀1𝑎16⁴01002𝑎²0001. (2.32) Then, we obtain 𝑎𝑃=³𝑎2𝜀²12𝜀4𝑎²∗2𝜀1𝑎−4𝑎²𝑎∗∗1, (2.33) where, if 𝑎>6, then 𝜀<6/𝑎, and, if 0<𝑎≤6, then 𝜀∈(0,1].
Define𝑋⁺₂∶=𝑥=𝑥₁𝑥₂𝑥₃^𝑇∈Ω⁺∶𝑥₁∈ℝ,𝑥₂≤4𝑎²,𝑥₃1≥−𝑎,𝑋⁻₂∶=𝑥=𝑥₁𝑥₂𝑥₃^𝑇∈Ω⁻∶𝑥₁∈ℝ,𝑥₂≥4𝑎²,𝑥₃1≤−𝑎,Ω²_𝑑∶=𝑥∈ℝ³,𝑥^𝑇1𝑎16⁴01002𝑎²0001𝑥≤𝜀²𝐷()61+𝐷,𝜀<𝑎,𝑎>6,𝜀∈(0,1],0<𝑎≤6. (2.34) Then there exists a scalar 𝜌>0 such that Ω(𝑃,𝜌) is the smallest ellipsoid which satisfies Ω(𝑃,𝜌)∩Ω⁰⊃Ω²_𝑑∩Ω⁰. Therefore, 𝑉̇(𝑥)<0 holds for all 𝑥∈Ω(𝑃,𝜌)∩Ω⁰. Hence Ω(𝑃,𝜌)∩Ω⁰ is an invariant set. For the initial state 𝑥 outside Ω(𝑃,𝜌)∩Ω⁰, any trajectory departing from 𝑥∈𝑋⁺₂∪𝑋⁻₂∪(Ω⁰⧵Ω(𝑃,𝜌)) will enter into Ω(𝑃,𝜌)∩Ω⁰ in finite time and remain thereafter. Moreover, when 𝜀→0, Ω(𝑃,𝜌)∩Ω⁰ approaches to the origin.
(ii₃) Consider the following system:001𝑥̇=()0100010−𝑎0𝑥+sat𝑢+𝑑, (2.35) that is, 𝐴 satisfies the case (𝑐).
In this case, we choose the matrix 𝑃 to be the solution to the following ARE:𝐴^𝑇2𝑃+𝑃𝐴−𝜀𝑃𝐵𝐵^𝑇1𝑃=−𝜀2𝑎²0001−6𝜀²𝑎2𝜀³0001. (2.36) Then, we obtain 𝑎𝑃=²+𝑎𝜀²2𝑎𝜀𝑎∗1𝜀²12𝜀∗∗1, (2.37) where, if 𝑎>1/6, then 𝜀<(1/6)𝑎, and, if 0<𝑎≤1/6, then 𝜀∈(0,1].
Set𝑋⁺₃∶=𝑥=𝑥₁𝑥₂𝑥₃^𝑇∈Ω⁺∶𝑥₁∈ℝ,√𝑎𝑥₂+𝑥₃1<2𝜀−√𝑎,𝑋⁻₃∶=𝑥=𝑥₁𝑥₂𝑥₃^𝑇∈Ω⁻∶𝑥₁∈ℝ,√𝑎𝑥₂+𝑥₃1<2𝜀+√𝑎,Ω³_𝑑∶=𝑥∈ℝ³,𝑥^𝑇2𝑎²0001−6𝜀²𝑎2𝜀³0001𝑥≤2𝜀²𝐷(1+𝐷),𝜀<161𝑎,𝑎>6(]1,𝜀∈0,1,0<𝑎≤6. (2.38) Then, there exists a scalar 𝜌>0 such that Ω(𝑃,𝜌) is the smallest ellipsoid which satisfies Ω(𝑃,𝜌)∩Ω⁰⊃Ω²_𝑑∩Ω⁰. Therefore, 𝑉̇(𝑥)<0 is true for all 𝑥∈Ω(𝑃,𝜌)∩Ω⁰. Hence, Ω(𝑃,𝜌)∩Ω⁰ is an invariant set. For the initial state 𝑥∉Ω(𝑃,𝜌)∩Ω⁰, any trajectory of the system departing from 𝑥∈𝑋⁺₃∪𝑋⁻₃∪(Ω⁰⧵Ω(𝑃,𝜌)) will enter into Ω(𝑃,𝜌)∩Ω⁰ in finite time and remain thereafter. Moreover, when 𝜀→0, Ω(𝑃,𝜌)∩Ω⁰ approaches to the origin.
Finally, let us prove (iii).
Assuming that matrix 𝐴 satisfies any case among (𝑑)–(𝑓), when the actuator is saturated, all the system trajectories departed from Ω⁺ and Ω⁻ are oscillatory, that is, all the trajectories cannot back into any arbitrarily small set containing the origin in finite time. When the actuator is unsaturated, that is, 𝑥∈Ω⁰. The positive definite symmetric matrix 𝑃 satisfying certain ARE can be obtained. When 𝜀→0, there exists a small ellipsoid Ω^𝑖_𝑑 which contains origin as its interior and approaches to the origin such that 𝑉̇(𝑥)<0 for all 𝑥∈Ω⁰⧵Ω^𝑖_𝑑. Furthermore, there exists 𝜌>0 such that Ω(𝑃,𝜌) is the smallest ellipsoid which satisfies Ω(𝑃,𝜌)∩Ω⁰⊃Ω^𝑖_𝑑∩Ω⁰, that is, 𝑥∉Ω(𝑃,𝜌)∩Ω⁰, the trajectory only starting from Ω⁰⧵Ω(𝑃,𝜌) can enter into Ω(𝑃,𝜌)∩Ω⁰ in finite time.
Based on (i)–(iii), the proof of this theorem is now complete.

Remark 2.6. It should be pointed out that, because of the mathematical complexity and computational difficulty, almost all papers concerning the actuator saturation and input additive disturbances have considered the two-dimensional systems. In this paper, we make the first attempt to investigate the practical stabilization problem for three-dimensional system with actuator saturation and input additive disturbances. To the best of our knowledge, the research topic addressed in this paper is new and meaningful. The above attempts distinguish our research results from the existing ones.

Remark 2.7. The sufficient conditions are established in Theorem 2.5 for three-dimensional system with actuator saturation and input additive disturbance. By applying the Riccati equation approach, the linear state feedback control law is designed such that the semiglobal practical stabilization can be guaranteed. It should be pointed out that the simultaneous consideration of actuator saturation and input additive disturbance leads to essential difficulties in the technical development, and the corresponding derivations are nontrivial. More specifically, some constructive strategies are introduced to facilitate the derivation of the main results.

3. The Case of Time-Varying Uncertain Input Disturbance

In this section, we investigate a more general class of systems with actuator saturation and time-varying disturbance input. A low-and-high-gain is designed to guarantee the global or semiglobal practical stabilization for the closed-loop systems.

Consider the following linear system with actuator saturation and time-varying disturbance input𝑥̇=𝐴𝑥+𝐵𝜎𝑢+𝑔(𝑥,𝑡), (3.1) where 𝑥∈ℝ^𝑛 is the state vector, 𝑢∈ℝ^𝑚 is the control input, 𝑔(𝑥,𝑡)∶ℝ^𝑛×ℝ→ℝ^𝑚 represents the time-varying uncertainty, and 𝐴 and 𝐵 are known real matrices with appropriate dimensions. 𝜎(⋅)∶ℝ^𝑚→ℝ^𝑚 is the standard saturated function, that is, 𝜎(𝑠)=𝜎₁𝑠₁𝜎₂𝑠₂⋯𝜎_𝑚𝑠_𝑚^𝑇,𝜎_𝑖𝑠_𝑖=sign𝑠_𝑖min1,𝑠_𝑖,𝑠∈ℝ^𝑚. (3.2)

Before proceeding further, we make the following assumptions.

Assumption 3.1. The matrix pair (𝐴,𝐵) is asymptotically null controllable with bounded control, that is, all the eigenvalues of 𝐴 are in the left half plane (LHP) and (𝐴,𝐵) is stabilizable.

Assumption 3.2. The uncertain element 𝑔(𝑥,𝑡) is piecewise continuous in 𝑡 and locally Lipschitz in 𝑥 satisfying 𝑔(𝑥,𝑡)≤𝑔₀(‖𝑥‖)+𝐷₀,∀(𝑥,𝑡)∈ℝ^𝑛×ℝ⁺, (3.3) where 𝐷₀ is a known positive scalar, and the known function 𝑔₀(⋅)∶ℝ⁺→ℝ⁺ is local Lipschitz and satisfies 𝑔₀(‖𝑥‖)≤𝐿‖𝑥‖ with 𝐿>0 being a constant.

Problem 1. For the system (3.1), the purpose of this section is to design a state feedback control law 𝑢=𝐹𝑥 such that the following requirements are satisfied.
(1)If 𝐷₀=0, the closed-loop system is globally asymptotically stable at the point of 𝑥=0.(2)If 𝐷₀>0, the trajectories of the system enter into a given set which contains the origin as an interior point and remain thereafter.

Remark 3.3. As discussed in [31], corresponding to the specific values for 𝐷₀ and 𝑔₀, Problem 1 refers to different names. Specifically, when 𝑔₀=0, 𝐷₀=0, it is called the globally asymptotical stabilization by state feedback problem. When 𝑔₀=0, 𝐷₀>0, it is called the global disturbance rejection by state feedback problem. When 𝑔₀≠0, 𝐷₀=0, it is called the robust globally asymptotical stabilization by state feedback problem. When 𝑔₀≠0, 𝐷₀>0, it is called the robust global disturbance rejection by state feedback problem.

For Problem 1, the design of low-and-high gain state feedback control law is composed of three steps. In Steps 1 and 2, the low-gain and high-gain state feedback control law is designed, respectively. Accordingly, the low-and-high-gain control law is designed in Step 3.

Step 1 (low-gain state feedback design). Let Q∶(0,1]→ℝ^𝑛×𝑛 be a continuously differentiable and strictly increasing function such that 𝑄(𝜀) is symmetric positive definite for each 𝜀∈(0,1] and lim_𝜀→0𝑄(𝜀)=0. Consider the following ARE: 𝐴^𝑇𝑃(𝜀)+𝑃(𝜀)𝐴−𝑃(𝜀)𝐵𝐵^𝑇𝑃(𝜀)=−𝑄(𝜀). (3.4) Then, the following lemma is obtained.

Lemma 3.4 (see [32]). Let Assumption 2.1 be satisfied, then, for all 𝜀∈(0,1], there exists a unique symmetric positive definite matrix 𝑃(𝜀)>0 which solves the ARE (3.4) and lim_𝜀→0𝑃(𝜀)=0.

Now, we construct the low-gain state feedback law as follows:𝑢_𝐿=−𝐵^𝑇𝑃(𝜀)𝑥, (3.5) where 𝑃(𝜀) is a symmetric positive definite solution to ARE (3.4).

Based on Lemma 3.4, we can conclude that the above low-gain state feedback control law can be arbitrarily small when 𝜀 is arbitrarily small.

Step 2 (high-gain state feedback design). We design the high-gain state feedback control law as follows: 𝑢_𝐻=−𝜌𝐵^𝑇𝑃(𝜀)𝑥, (3.6) where 𝜌≥0 is called high-gain parameter, 𝜌=𝜌₀(1+𝐿+𝐷₀)²/𝜆_min(𝑄(𝜀)) with 𝜌₀≥0 being an adjustable constant, 𝐿 and 𝐷₀ are defined in Assumption 3.2.

Step 3 (low-and-high-gain state feedback design). Taking the low-gain and high-gain state feedback control law into account, the low-and-high-gain state feedback control law is constructed as follows: 𝑢_𝐿𝐻=−1+𝜌𝐵^𝑇𝑃(𝜀)𝑥. (3.7) Concluded from [32], the state feedback control law (3.7) is one of the optimal control for the linear system (3.1) without any saturation constraint. The following theorem presents a criterion about the solution to Problem 1.

Theorem 3.5. Let Assumption 3.1 be true. There exist positive scalars 𝜀 and 𝜌^∗₀ such that, for all 𝜀∈(0,𝜀^∗] and 𝜌₀≥𝜌^∗₀, the low-and-high-gain state feedback control law (3.7) is a solution to Problem1.

Proof. To begin with, the cases of saturation and unsaturation are discussed respectively. For convenience, set 𝑃(𝜀)=𝑃 and 𝑄(𝜀)=𝑄.
Firstly, we choose the Lyapunov function 𝑉(𝑥)=𝑥^𝑇𝑃𝑥. By calculating the derivative of 𝑉(𝑘) along the trajectory of closed-loop system, we have 𝑉̇(𝑥)=𝑥^𝑇𝐴^𝑇𝑃+𝑃𝐴𝑥+2𝑥^𝑇𝑃𝐵𝜎−1+𝜌𝐵^𝑇𝑃𝑥+𝑔(𝑥,𝑡)=−𝑥^𝑇𝑄𝑥+𝑥^𝑇𝑃𝐵𝐵^𝑇𝑃𝑥+2𝑥^𝑇𝑃𝐵𝜎−1+𝜌𝐵^𝑇𝑃𝑥+𝑔(𝑥,𝑡)≤−𝑥^𝑇𝑄𝑥+2𝑥^𝑇𝑃𝐵𝐵^𝑇𝑃𝑥+2𝑥^𝑇𝑃𝐵𝜎−1+𝜌𝐵^𝑇𝑃𝑥+𝑔(𝑥,𝑡). (3.8) (1) The actuator is with saturation. Here, it is proved only when (𝐵^𝑇𝑃𝑥)_𝑖<0, where (⋅)_𝑖 denotes the 𝑖th element of vector (⋅).
When 1+𝜌≥(−(1+𝜌)𝐵^𝑇𝑃𝑥)_𝑖≥1+𝐿‖𝑥‖+𝐷₀, we arrive at−(1+𝜌)𝐵^𝑇𝑃𝑥+𝑔(𝑥,𝑡)_𝑖≥1,∀(𝑥,𝑡)∈ℝ^𝑛×ℝ⁺,𝜎_𝑖−1+𝜌𝐵^𝑇𝑃𝑥+𝑔(𝑥,𝑡)=1,𝑖=1,2,…,𝑚,∀(𝑥,𝑡)∈ℝ^𝑛×ℝ⁺,−1≤𝐵^𝑇𝑃𝑥_𝑖‖𝑥‖≤−1+𝐿+𝐷₀11+𝜌≤−1+𝜌<0. (3.9)(3.10) Thus, we get 𝑥^𝑇𝑃𝐵_𝑖𝐵^𝑇𝑃𝑥_𝑖+1≤0. (3.11) Hence, 𝑥^𝑇𝑃𝐵𝐵^𝑇𝑃𝑥+2𝑥^𝑇𝑃𝐵𝐻≤2^𝑚_𝑖=1𝑥^𝑇𝑃𝐵_𝑖𝐵^𝑇𝑃𝑥_𝑖+1<0, (3.12) where 𝐻=[1⋯1]^𝑇.
It follows from (3.8) that𝑉̇(𝑥)=−𝑥^𝑇𝑄𝑥+𝑥^𝑇𝑃𝐵𝐵^𝑇𝑃𝑥+2^𝑚_𝑖=1𝑥^𝑇𝑃𝐵_𝑖≤−𝑥^𝑇𝑄𝑥+2^𝑚_𝑖=1𝑥^𝑇𝑃𝐵_𝑖𝐵^𝑇𝑃𝑥_𝑖+1<−𝑥^𝑇𝑄𝑥<0. (3.13) If (𝐵^𝑇𝑃𝑥)_𝑖<−1, for each (𝑥,𝑡)∈ℝ^𝑛×ℝ⁺, it can be easily seen that (−(1+𝜌)𝐵^𝑇𝑃𝑥+𝑔(𝑥,𝑡))_𝑖≥1, that is, 𝜎_𝑖(−(1+𝜌)𝐵^𝑇𝑃𝑥+𝑔(𝑥,𝑡))=1(𝑖=1,2,…,𝑚). Then, we have 𝑉̇(𝑥)=𝑥^𝑇𝐴^𝑇𝑃+𝑃𝐴𝑥+2^𝑚_𝑖=1𝑥^𝑇𝑃𝐵_𝑖≤𝑥^𝑇𝐴^𝑇𝑃+𝑃𝐴𝑥−2𝑛<0. (3.14) According to Assumption 3.1, we have 𝐴^𝑇𝑃+𝑃𝐴<0 for any 𝑃>0.
When −(1+𝜌)≤(−(1+𝜌)𝐵^𝑇𝑃𝑥)_𝑖≤−1−𝐿‖𝑥‖−𝐷₀, we obtain−1+𝜌𝐵^𝑇𝑃𝑥+𝑔(𝑥,𝑡)_𝑖≤−1,∀(𝑥,𝑡)∈ℝ^𝑛×ℝ⁺,𝜎_𝑖−1+𝜌𝐵^𝑇𝑃𝑥+𝑔(𝑥,𝑡)=1,𝑖=1,2,…,𝑚,∀(𝑥,𝑡)∈ℝ^𝑛×ℝ⁺,1≥B^𝑇𝑃𝑥_𝑖≥1+𝐿‖𝑥‖+𝐷₀≥11+𝜌1+𝜌>0. (3.15) That is (𝐵^𝑇𝑃𝑥)_𝑖>0. Along the same line of the above proof, we can discuss the cases 0<(𝐵^𝑇𝑃𝑥)_𝑖≤1 and (𝐵^𝑇𝑃𝑥)_𝑖>1, respectively. Hence, we can conclude that 𝑉̇(𝑥)<0 for (𝐵^𝑇𝑃𝑥)_𝑖>0.
(2) The actuator is not saturated. When |(−(1+𝜌)𝐵^𝑇𝑃𝑥)_𝑖+(𝑔(𝑥,𝑡))_𝑖|<1, according to (3.8), we have 𝑉̇(𝑥)=−𝑥^𝑇𝑄𝑥−2𝜌𝑥^𝑇𝑃𝐵𝐵^𝑇𝑃𝑥+2𝑥^𝑇𝑃𝐵𝑔(𝑥,𝑡)≤−𝑥^𝑇𝑄𝑥+2^𝑚_𝑖=1𝑔(𝑥,𝑡)_𝑖−𝜌𝐵^𝑇𝑃𝑥_𝑖𝐵^𝑇𝑃𝑥_𝑖. (3.16) Next, we discuss the following two cases.
If |(𝜌𝐵^𝑇𝑃𝑥)_𝑖|≥|𝑔_𝑖|, by noting (𝐵^𝑇𝑃𝑥)_𝑖(−𝜌(𝐵^𝑇𝑃𝑥)_𝑖+(𝑔(𝑥,𝑡))_𝑖)<0, we obtain 𝑉̇(𝑥)=−𝑥^𝑇𝑄𝑥−2𝑥^𝑇𝑃𝐵−1+𝜌𝐵^𝑇𝑃𝑥+𝑔(𝑥,𝑡)+𝐵^𝑇𝑃𝑥≤−𝑥^𝑇𝑄𝑥+2^𝑚_𝑖=1−𝜌𝐵^𝑇𝑃𝑥_𝑖+𝑔(𝑥,𝑡)_𝑖≤−𝑥^𝑇𝑄𝑥<0. (3.17) If |(𝜌𝐵^𝑇𝑃𝑥)_𝑖|≤|𝑔_𝑖|, then 𝑉̇(𝑥)=−𝑥^𝑇𝑄𝑥+2𝑥^𝑇𝑃𝐵𝑔(𝑥,𝑡)≤−𝑥^𝑇𝑄𝑥+2^𝑚_𝑖=1𝑔_𝑖²𝜌≤−𝑥^𝑇𝐿‖𝑥‖𝑄𝑥+2+𝐷₀𝜌≤−𝜆_min(𝑄)−4𝐿²𝜌‖𝑥‖²+4𝐷²₀𝜌. (3.18) Suppose 𝑊₀∶=𝑥∈ℝ^𝑛∶‖𝑥‖²≤4𝐷²₀𝜌₀1+𝐿+𝐷₀²−4𝐿². (3.19) Taking 𝜌^∗=4, then all trajectories will enter into 𝑊₀ in finite time and remain thereafter for all 𝜌₀≥𝜌^∗. Here, 𝑊₀ is an arbitrarily small set including the origin point, and 𝜌₀ is a sufficiently large scalar.
Specifically, it follows from 𝐷₀=0 that 𝑉̇(𝑥)<0, that is, 𝑥=0 is globally asymptotically stable. To this end, the proof of this theorem is complete.

Remark 3.6. In this section, the practical stabilization problem is investigated for a general multidimensional system. From a practical point of view, it is more significant to consider the high-dimensional systems. It is worth mentioning that the system under consideration is comprehensive that includes the actuator saturation and the time-varying input disturbances. By using the Riccati equation approach, the low-and-high-gain state feedback control law is designed such that the global or semiglobal practical stabilization for the multidimensional system can be guaranteed. On the other hand, we are now researching into a method for the system with uncertainties, time-delay, and/or input disturbance in more general cases. The corresponding results will appear in the near future.

4. Conclusions

In this paper, we have made an attempt to investigate the practical stabilization problem for a class of system with actuator saturation and input additive disturbances. For the case of the input additive disturbance being bounded constant, the three-dimensional system has been studied where the system matrices satisfy a class of the controllability canonical form. Eight different forms of the system matrices have been discussed. Subsequently, by applying the Riccati equation approach and designing the linear state feedback control law, the sufficient conditions of the semiglobal practical stabilization or oscillation for the addressed systems have been established. For the case when the input additive disturbances are time-varying functions, by using of the Riccati equation approach as well as combining the low-gain linear state feedback and high-gain linear state feedback, a low-high-gain linear state feedback control law has been designed such that the global or semiglobal practical stabilization for a general multidimensional system with actuator saturation can be guaranteed. One of the future research topics would be the extension of the main results obtained in this paper to networked control systems [27, 33–36].

Acknowledgment

This work was supported by the National Natural Science Foundation of China under Grant 10771047.