Abstract—This paper investigates the problem of fixed-time stabilization for a class of multivariable uncertain nonlinear sys-tems. A new approach is proposed by skillfully revamping the technique of adding a power integrator whereby a state feedback controller and a suitable Lyapunov function for verifying fixed-time convergence can be explicitly constructed to render the closed-loop system fixed-time stable. The novelty of this paper owes to the development of a subtle strategy that provides a new solution to the problem of fixed-time stabilization for multivariable nonlinear systems. Finally, the developed approach is applied to the attitude stabilization of a spacecraft to show the effectiveness of the resultant controller.
Keywords—Uncertain nonlinear systems, adding a power inte-grator technique, fixed-time stabilization.
I. INTRODUCTION
HE stabilization control of nonlinear system has always been crucial in performing additional control tasks, such as output tracking, disturbance attenuation and/or decoupling.
Global asymptotic stabilization of nonlinear systems has gained tremendous progress due to the development of mathematical tools, including backstepping design [1], feedback linearization [2], sliding mode control [3, 4], fuzzy control [5, 6], has tremendous progress by mathematical tools.
As is well-known, finite-time stabilization is more attrac-tive compared with asymptotic stabilization [7] because the systems with finite-time convergence usually exhibit superi-or properties [7-10], which are rather impsuperi-ortant fsuperi-or demand-ing applications. Bedemand-ing aware of these features, the fi-nite-time stabilization problem has been intensively studied, and numerous interesting results have been proposed in the past decades [11-15]. For instance, owing to the benefits including fast response and ease of implementation, terminal sliding mode control design [14] is one of most important techniques for finite-time stabilization of nonlinear system.
By constructing a discontinuous controller while design a suitable nonlinear sliding surface, the phase of terminal sliding mode can be achieved in finite-time, thereby guaran-teeing finite-time stabilization of the closed-loop system [14-16].
It should be mentioned that the information of initial states is critical for the settling-time estimates of finite-time stabilization schemes; however, the availability of initial states will prevent us from applying finite-time schemes [17,
1C.-C. Chen is an Assistant Professor with Department of Systems and Naval Mechatronic Engineering, National Cheng Kung University, Tainan 70101, Taiwan (e-mail: ccchenevan@mail.ncku.edu.tw).
2C.-H. Ding and G.-S. Chen are master students with Department of Systems and Naval Mechatronic Engineering, National Cheng Kung Uni-versity, Tainan 70101, Taiwan.
18]. Fortunately, the notion of fixed-time stability together with its Lyapunov-like criteria has been recently presented in the seminal work [17] in which the potential obstruction of finite-time schemes was resolved effectively. To be more specific, as stated in [17], by fixed-time controller design, it implies global uniform finite-time stability while providing a settling time function to be uniformly bounded by a tunable constant, which independent of initial states [17, 18-24].
Due to the complexity of multivariable nonlinear systems and the lack of systematic strategies for ensuring the fixed-time convergence, the problem on how to design a fixed-time stabilizing controller for multivariable nonlinear systems remains unclear and largely open. In this paper, by introducing extra manipulations in the feedback domination to delicately revamp the technique of adding a power inte-grator [18], a new approach is developed to the synthesis the fixed-time stabilizer together with the Lyapunov function for multivariable uncertain nonlinear systems.
II. PRELIMINARIES
A. Problem Formulation
Consider a class of nonlinear systems described by 𝐱 𝐱𝟐 worth mentioning that a very large class of physical systems can be represented by system (1). Besides, the solutions of system (1) are understood in the sense of Filippov [26] since the control input 𝐮 𝐮 𝑡, 𝐱 is admitted to be discontinu-ous (piecewise continudiscontinu-ous) and 𝐝 𝑡, 𝐱 is assumed to be piecewise continuous and bounded as follows.
Assumption 1. There exists a constant 𝜌 0 such that
|𝑑 𝑡, 𝐱 | 𝜌 for all 𝑡, 𝐱 ∈ ℝ ℝ and 𝑖 1, . . . , 𝑛.
Under Assumption 1, the main objective of this paper is to design a controller 𝐮 𝐮 𝑡, 𝐱 that renders the origin of system (1) fixed-time stable in the sense of the following definition.
Definition 1 ([17]). Consider the following nonlinear system 𝐱 𝐠 𝑡, 𝐱
Fixed-Time Stabilization for a Class of Uncertain Nonlinear Systems
Chih-Chiang Chen1, Chi-Hsuan Ding2, and Guan-Shiun Chen2
T
ICIUS 2019, Beijing, China Paper ID 26
where 𝐱 ∈ ℝ ,𝑡 ∈ ℝ , and 𝐠 ∶ ℝ ℝ → ℝ is discon-tinuous (piecewise condiscon-tinuous). The initial time is 𝑡 0 and the initial state is 𝐱 0 𝐱 . The solutions of system (2) are understood in the sense of Filippov [26]. Then, the origin of system (2) is said to be fixed-time stable if it is globally uniformly finite-time stable (see, e.g., [27]) and the set-tling-time function 𝑇 𝐱 is globally uniformly bounded by a positive constant; i.e., there exists a positive stant 𝑇 0 such that 𝑇 𝐱 𝑇 for all 𝐱 ∈ ℝ . Remark 1. Compared to global uniform finite-time stability, the key feature of fixed-time stability is the uniformity of its settling time. To see this point more clearly, the following two examples are considered. First, the origin of the system 𝑥 3 2⁄ 𝑥 ⁄ is globally uniformly finite-time stable with the settling-time function 𝑇 𝑥 0 𝑥 / 0 because its solutions take the form of
𝑥 𝑡 𝑠𝑖𝑔𝑛 𝑥 0 𝑥 0 𝑡 , 0 𝑡 𝑥 0
0, 𝑡 𝑥 0
. However, with an additional drift term, the solutions of the system 𝑥 3 2⁄ 𝑥 ⁄ 3 2⁄ 𝑥 ⁄ can be found directly Therefore, the origin of system (3) is fixed-time stable with the settling-time function 𝑇 𝑥 0 satisfying 𝑇 𝑥 0 𝜋 2⁄ uniformly in 𝑥 0 .
B. Technical Lemmas
Lemma 1. Let 𝑚 1 is a ratio of two odd integers. For
III. FIXED-TIME STABILIZING CONTROLLER DESIGN
We first summarize our approach to the construction of a fixed stabilizing controller for system (1) as follows.
A. Theorem 1.
Under Assumption 1, the origin of system (1) is fixed-time stable with the settling-time estimate
𝑇 𝐱 𝑇 2𝑛 posi-tive even integers and posiposi-tive odd integers, respecposi-tively, and 𝔏 x ∈ ℝ and 𝔏 x ∈ ℝ are square matrices
Part I— Design the controller
A two-step design approach is developed to construct the controller.
Step 1: Choose V 𝐱 𝐱 𝐱 𝑎𝑠 𝑡ℎ𝑒 scalar function, which is obviously positive definite, proper and continuous-ly differentiable. Select the virtual control 𝐱∗ 𝐱
In addition, it can be shown by Lemma 3, and we have
ICIUS 2019, Beijing, China Paper ID 26
V 𝐱 2 V 𝐱 2𝑛 2 V 𝐱 (10)
for all 𝑡, 𝐱 ∈ ℝ ℝ ∖ 𝒩 ∪ 𝒩 . From [29], we know that (10) is satisfied for almost all 𝑡 ∈ ℝ (almost every-where in 𝑡 ∈ ℝ ). Thus, (18) implies that the origin of sys-tem (1) with the controller (4), which is Lebesgue measura-ble and locally essentially bounded, is uniformly stameasura-ble and 𝐱 𝑡 exists for all 𝑡 ∈ 0, ∞ [29]. Additionally, since positive definite, continuously differentiable and proper (i.e., radially unbounded) implies that the origin of the closed-loop system is globally uniformly asymptotically stable [30].
Part II—Analysis of fixed-time stability
In what follows, the fixed time stability of the closed-loop system will be proven by using a contradiction argument.
First, when V 𝐱 0 1, we assume that there exists decreasing, it follows that
0 2 fixed-time stable with the settling-time estimate
𝑇 𝐱 𝑇 𝑇 𝑇
We take the attitude control model of a spacecraft shown in [25, 31], which has the same from as (1) with n = m = 3. and G. All details are shown in [25, 31] and therefore omit-ted for the sake of space.
The simulation results shown in Figs. 1–2 are conducted for the initial state 𝐱 0 0.5,0.3, 0.48, 1.9, 1.2,2 . Clearly, Fig. 1 shows that the finite-time stabilization task can be successfully performed by the corresponding control signals shown in Fig. 2. It can be found that the settling-time (convergence time) of state trajectories is much less than 𝑇 35 sec. (i.e., the settling-time estimate). This in turn reveals that the fixed-time stabilization can achieved by the controller designed by Theorem 1. Simulations with differ-ent initial states can be also performed to obtain the same conclusion, i.e., the success of the fixed-time stabilization.
Notably, this example exhibits the benefits and effectiveness of the proposed approach.
V. CONCLUDING REMARKS
This paper has solved the problem of fixed-time stabiliza-tion for a second-order uncertain multivariable nonlinear systems. By introducing extra manipulations in the feedback domination to delicately revamp the technique of adding a power integrator, a state feedback fixed-time stabilizing controller together with a Lyapunov function can be orga-nized simultaneously, thereby proving and ensuring the fixed-time convergence of the closed-loop system.
ICIUS 2019, Beijing, China Paper ID 26
Fig. 1. State trajectories of the closed-loop system
Fig. 2. Control signals of the closed-loop system.
ACKNOWLEDGMENT
This work was supported in part by the Ministry of Sci-ence and Technology (MOST), Taiwan, under Grant MOST 107-2221-E-006-231-.
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Fixed-Time Stabilization for a Class of Uncertain Nonlinear Systems
Chih-Chiang Chen, Chi-Hsuan Ding, and Guan-Shiun Chen Department of Systems and Naval Mechatronic Engineering
National Cheng Kung University, Taiwan August, 2019
‘CC’ Chen August, 2019 1 / 20
Outline
1 Motivations
2 Fixed-time stabilization for uncertain nonlinear systems
3 Simulation results
4 Conclusions
Outline
1 Motivations
2 Fixed-time stabilization for uncertain nonlinear systems
3 Simulation results
4 Conclusions
‘CC’ Chen August, 2019 3 / 20
Motivations
Asymptotic convergence
˙x = −x
=⇒ x(t) = e−t
=⇒ x(t) → 0 as t → ∞
Motivations
Finite-time convergence
˙x = −x13
=⇒ x(t) =
sign(x(t0))h|x(t0)|32 −32(t − t0)i3/2 t0 ≤ t ≤ T (x(t0))
0 t > T(x(t0))
=⇒ x(t) = 0 as t ≥ T(x(t0)) := t0+2
3|x(t0)|32
‘CC’ Chen August, 2019 5 / 20
Motivations
Fixed-time convergence
˙x = −x13 − x53
=⇒ x(t) =
( sign(x(t0)) tan32 tan−1|x(t0)|32− (t − t0) t0≤ t ≤ T∗
0 t > T∗
=⇒ x(t) = 0 as t ≥ T∗ := t0+ tan−1|x(t0)|32 Noting that
T∗ ≤ t0+π 2 we have
=⇒ x(t) = 0 as t ≥ t0+π 2
Motivations
Definition of Fixed-time Stability Consider a nonlinear system
˙x = f(x, t), x(t0) = x0, t0 ∈ R+ (1) where f : Rn× R+ → Rn is continuous with f(0, t) = 0 for all t ≥ t0. The origin of the system is said to be fixed time stable if it satisfies:
(i) Globally uniformly asymptotically stable
‘CC’ Chen August, 2019 7 / 20
Motivations
Definition of Fixed-time Stability Consider a nonlinear system
˙x = f(x, t), x(t0) = x0, t0 ∈ R+ (1) where f : Rn× R+ → Rn is continuous with f(0, t) = 0 for all t ≥ t0. The origin of the system is said to be fixed time stable if it satisfies:
(i) Globally uniformly asymptotically stable
(ii) Finite timeconvergent with the settling time T (x0, t0)
Motivations
Definition of Fixed-time Stability Consider a nonlinear system
˙x = f(x, t), x(t0) = x0, t0 ∈ R+ (1) where f : Rn× R+ → Rn is continuous with f(0, t) = 0 for all t ≥ t0. The origin of the system is said to be fixed time stable if it satisfies:
(i) Globally uniformly asymptotically stable
(ii) Finite timeconvergent with the settling time T (x0, t0) (iii) T(x0, t0)< Tmax
‘CC’ Chen August, 2019 7 / 20
Motivations
Existing approaches
SISO → MIMO
Motivations
Existing approaches
SISO → MIMO
Sliding mode technique
Z. Zuo, “Non-singular fixed-time terminal sliding mode control of non-linear systems,” IET control theory and applications, vol. 9, no. 4, pp. 545–552, 2014.
∃ chattering even in the case of having no disturbances
‘CC’ Chen August, 2019 8 / 20
Motivations
Existing approaches
SISO → MIMO
Sliding mode technique
Z. Zuo, “Non-singular fixed-time terminal sliding mode control of non-linear systems,” IET control theory and applications, vol. 9, no. 4, pp. 545–552, 2014.
∃ chattering even in the case of having no disturbances bi-limit homogeneity
B. Tian, Z. Zuo, X. Yan, and H. Wang “A fixed-time output feedback control scheme for double integrator systems,” Automatica, vol. 80, pp. 17–24, 2017.
It is not an MIMO result
Outline
1 Motivations
2 Fixed-time stabilization for uncertain nonlinear systems
3 Simulation results
4 Conclusions
‘CC’ Chen August, 2019 9 / 20