Global continuous output-feedback stabilization for a class of high-order nonlinear systems with multiple time delays

doi:10.1016/j.jfranklin.2014.05.003

Journal of the Franklin Institute

Volume 351, Issue 8, August 2014, Pages 4334-4356

https://doi.org/10.1016/j.jfranklin.2014.05.003 Get rights and content

Abstract

This paper addresses the global output-feedback stabilization for a class of high-order nonlinear systems with multiple time delays. A distinct property of the systems to be investigated is that powers on the upper bound restrictions of nonlinearities are allowed to take values on a continuous interval, another remarkable one rests with the existence of multiple time delays in growth conditions. By introducing a combined method of sign function, homogeneous domination and adding a power integrator, an output-feedback controller based on Lyapunov-Krasviskii theorem is designed recursively to guarantee the equilibrium of the closed-loop system globally uniformly asymptotically stable.

Introduction

The study on the high-order nonlinear systems has attracted a lot of attention during the past decade, and achieved a series of results with the aid of the adding a power integrator approach, see References [1], [2], [3], [4], [5], [6], [7], [8], [9] and the references therein.

It is worth pointing out that [2] has obtained breakthrough results on output-feedback control design of high-order nonlinear systems by using the homogeneous domination method, the novelty of which is that no precise information about the nonlinearities is needed. Furthermore, [3] generalized the homogeneous domination method through relaxing the conditions imposed on nonlinearities, based on such a method, [5], [10] investigated stochastic nonlinear systems and upper-triangular nonlinear systems, respectively. However, a common assumption of [2], [3] is that powers on the upper bound restrictions of nonlinearities are required to take values on some isolated points, therefore, an interesting question is naturally proposed:

Is it possible to allow powers to take values continuously on an interval? Moreover, under the weaker conditions, if system state includes multiple time delays, how to design an output-feedback stabilization controller?

In this paper, we shall investigate the aforementioned question and give an affirmative answer to a class of high-order nonlinear systems with multiple time delays. The contributions of this article are characterized by four aspects. (i) This paper may be the first investigation on output-feedback stabilization problem for high-order nonlinear systems with multiple time delays, where system nonlinearities depend on not only the time-delay output but also the unmeasurable time-delay states. (ii) The growth restriction on system nonlinearities is relaxed, the sign functions are introduced in control design, several crucial lemmas and propositions are reestablished. (iii) Due to the existence of time-delay terms and the introduction of the sign functions, it is hard to find a new positive definite and radially unbounded Lyapunov–Krasviskii (L–K) functional, whose time derivative along the solutions of the resulting closed-loop system is negative definite. (iv) By successfully solving several troublesome obstacles (see Remark 1, Remark 2, Remark 3, Remark 4, Remark 5 for the details) in the design and the analysis procedure, a global continuous output-feedback controller is constructed to guarantee that the resulting closed-loop system is globally asymptotically stable.

Notations: $R^{+}$ stands for the set of all the nonnegative real numbers. For a vector $x (t) \in R^{n}$ , we let ${\bar{x}}_{i} (t) = {[x_{1} (t), \dots, x_{i} (t)]}^{⊤} \in R^{i}$ , $i = 1, \dots, n - 1$ . $∥ x (t) ∥$ denotes the Euclidean norm of a vector $x (t) \in R^{n}$ , that is, $∥ x (t) ∥ = \sqrt{\sum_{i = 1}^{n} x_{i}^{2} (t)}$ , and $∥ x_{t} ∥_{C} = \sup_{- τ \leq θ \leq 0} ∥ x (t + θ) ∥$ , $\forall t \geq 0$ , τ is a known positive constant. In Section 4 and Appendix B, we frequently use a generic constant c which represents any finite positive constant and may be implicitly changed in various places, and use $c ({\bar{g}}_{i})$ to indicate its dependence on $g_{1}, \dots, g_{i}$ with $i = 1, \dots, n$ . A continuous function $h : R^{+} \to R^{+}$ satisfying $h (0) = 0$ is called a $K_{\infty}$ function if it is strictly increasing and $\lim_{s \to + \infty} h (s) =+ \infty$ . A sign function $sgn (x)$ is defined as follows: $sgn (x) = 1$ , if $x > 0$ ; $sgn (x) = 0$ , if x=0; and $sgn (x) = - 1$ , if $x < 0$ . Besides, the arguments of functions (or functionals) are sometimes omitted or simplified, for instance, we sometimes denote a function $f (x (t))$ by simply f(x) or $f (\cdot)$ or f.

Section snippets

Problem formulation

It is well known that time-delay phenomena are usually encountered in mechanical, electrical, biological and chemical systems, and its emergence is often a significant reason of instability or serious deterioration in the system performance. Therefore, the design and analysis of controllers for time-delay systems have received much attention, one can consult References [11], [12], [13], [14], [15], [16] and the references therein. However, until now, there are few results on high-order

Key transformations

Introduce the following scaling transformations: $x_{i} (t) = Γ^{- d_{i}} η_{i} (t), i = 1, \dots, n, u (t) = Γ^{- d_{n + 1}} v (t),$ where $Γ \geq 1$ is a positive constant to be determined later, and $d_{1}, \dots, d_{n}$ are defined as $d_{1} = 0$ , $d_{i} = (d_{i - 1} + 1) / p_{i - 1}$ , $i = 2, 3, \dots, n + 1$ .

With the scaling transformations (3) in hand, system (1) can be rewritten as ${\begin{matrix} {\dot{x}}_{i} (t) = Γ x_{i + 1}^{p_{i}} (t) + {\tilde{f}}_{i} (x (t), x_{1} (t - τ_{1}), \dots, x_{n} (t - τ_{n})), \\ {\dot{x}}_{n} (t) = Γ u^{p_{n}} (t) + {\tilde{f}}_{n} (x (t), x_{1} (t - τ_{1}), \dots, x_{n} (t - τ_{n})), \\ y (t) = x_{1} (t), \end{matrix}$ where for each $i = 1, \dots, n$ , ${\tilde{f}}_{i} (\cdot) ≜ Γ^{- d_{i}} f_{i} (\cdot)$ satisfies $| {\tilde{f}}_{i} (\cdot) | = | Γ^{- d_{i}} f_{i} (\cdot) | \leq C Γ^{1 - ς_{i}} \sum_{j = 1}^{i} (| x_{j} (t) |^{(r_{i} + ω) / r_{j}} + | x_{j} (t - τ_{j}) |^{(}$

State observer and output-feedback control design

In this section, we shall design a reduced-order, observer-based output-feedback controller. Motivated by [3], we construct the following reduced-order observer: ${\dot{\hat{ξ}}}_{i} = - Γ L_{i - 1} {\hat{x}}_{i}^{p_{i - 1}}, i = 2, 3, \dots, n,$ where $L_{i} > 0, i = 1, \dots, n - 1$ , are observer gains to be specified in later design, and ${\hat{x}}_{1} ≜ x_{1}, {\hat{x}}_{i} = sgn ({\hat{ξ}}_{i} + L_{i - 1} {\hat{x}}_{i - 1}) | {\hat{ξ}}_{i} + L_{i - 1} {\hat{x}}_{i - 1} |^{r_{i} / r_{i - 1}}, i = 2, 3, \dots, n .$ Thus, applying the certainty equivalence principle to Eq. (14), we get an output-feedback controller $u^{p_{n}} ({\hat{z}}_{n}) = - sgn ({\hat{z}}_{n}) g_{n}^{r_{n + 1} p_{n} / σ} | {\hat{z}}_{n} |^{r_{n + 1} p_{n} / σ},$ where ${\hat{z}}_{n} = sgn ({\hat{x}}_{n}) | {\hat{x}}_{n} |^{σ / r_{n}} + \sum i$

Main results

Before stating the main results of this paper, we would rather give a fundamental proposition which reveals the dynamic characteristic of the resulting closed-loop system via a L–K functional. In proving it, we present the recursive relationship between the gain parameters $L_{1}, \dots, L_{n - 1}, Γ$ and the virtual controller coefficients $g_{1}, \dots, g_{n}$ .

Proposition 4

There exist $K_{\infty}$ functions $π_{1} (\cdot)$ , $π_{2} (\cdot)$ and $π_{3} (\cdot)$ , such that the continuously differentiable functional $V : R^{+} \times R^{n} \times R^{n - 1} \to R$ defined by $V (t, X_{t} (θ)) = V_{n} (t, x_{t}) + U_{n} (x (t), {\hat{ξ}}_{2} (t), \hat{ξ}$

Simulation example

Consider the following example: ${\dot{η}}_{1} (t) = η_{2}^{5 / 3} (t) + \frac{1}{20} η_{1}^{4 / 3} (t - 1) \cdot \cos (η_{2} (t)), {\dot{η}}_{2} (t) = v^{3} (t), y (t) = η_{1} (t) .$ Obviously, Assumption 1 holds with $p_{1} = \frac{5}{3}$ , $p_{2} = 3$ , $C = \frac{1}{20}$ , $r_{1} = 1$ , $ω = \frac{1}{3}$ , $r_{2} = \frac{4}{5}$ , $σ = \frac{7}{5}$ . If we introduce the coordinate transformations $x_{1} (t) = η_{1} (t)$ , $x_{2} (t) = Γ^{- 3 / 5} η_{2} (t)$ , $v^{3} (t) = Γ^{- 8 / 5} u^{3} (t)$ , then the original system can be rewritten as $x_{1} (t) = Γ x_{2}^{5 / 3} (t) + \frac{1}{20} x_{1}^{4 / 3} (t - 1) \cdot \cos (Γ^{3 / 5} x_{2} (t)), x_{2} (t) = Γ u^{3} (t), y = x_{1} (t) .$ In terms of the control design in 3 State-feedback control design, 4 State observer and output-feedback control

Concluding remarks

Under the weaker assumptions, with the help of the constructions of an appropriate reduced-order observer and a delicate L–K functional, a class of high-order nonlinear systems with multiple time delays are globally stabilized via an output-feedback controller by means of the extended adding a power integrator method. In this direction, there are still remaining problems to be investigated. For example, an interesting problem is how to design a finite-time output-feedback stabilization

References (20)

Z.Y. Sun et al.
Adaptive state-feedback stabilization for a class of high-order nonlinear uncertain systems
Automatica
(2007)
X.J. Xie et al.
Further results on output feedback stabilization for stochastic high-order nonlinear systems with time-varying delay
Automatica
(2012)
L. Liu et al.
State feedback stabilization for stochastic feedforward nonlinear systems with time-varying delay
Automatica
(2013)
C.J. Qian et al.
A continuous feedback approach to global strong stabilization of nonlinear systems
IEEE Trans. Autom. Control
(2001)
C.J. Qian et al.
Recursive observer design, homogeneous approximation, and nonsmooth output feedback stabilization of nonlinear systems
IEEE Trans. Autom. Control
(2006)
J. Polendo et al.
A generalized homogeneous domination approach for global stabilization of inherently nonlinear systems via output feedback
Int. J. Robust Nonlinear Control
(2007)
W.Q. Li et al.
Output-feedback stabilization of stochastic high-order nonlinear systems under weaker conditions
SIAM J. Control Optim.
(2011)
X.J. Xie et al.
Output tracking of high-order stochastic nonlinear systems with application to benchmark mechanical system
IEEE Trans. Autom. Control
(2010)
X.J. Xie et al.
State-feedback control of high-order stochastic nonlinear systems with SiISS inverse dynamics
IEEE Trans. Autom. Control
(2011)
X.H. Yan et al.
Global practical tracking for high-order uncertain nonlinear systems with unknown control directions
SIAM J. Control Optim.
(2010)

There are more references available in the full text version of this article.

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^☆: Supported by National Natural Science Foundation of China (61004013, 61203013, 61273125, 61374004), the Shandong Provincial Natural Science Foundation of China (ZR2010FQ003, ZR2012FM018), the National Specialized Research Fund for the Doctoral Program of Higher Education under Grant 20113705120003 and the Program for the Scientific Research Innovation Team in Colleges and Universities of Shandong Province.

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Global continuous output-feedback stabilization for a class of high-order nonlinear systems with multiple time delays☆

Abstract

Introduction

Section snippets

Problem formulation

Key transformations

State observer and output-feedback control design

Main results

Simulation example

Concluding remarks

Automatica

Automatica

Automatica

A continuous feedback approach to global strong stabilization of nonlinear systems

IEEE Trans. Autom. Control

Recursive observer design, homogeneous approximation, and nonsmooth output feedback stabilization of nonlinear systems

IEEE Trans. Autom. Control

A generalized homogeneous domination approach for global stabilization of inherently nonlinear systems via output feedback

Int. J. Robust Nonlinear Control

Output-feedback stabilization of stochastic high-order nonlinear systems under weaker conditions

SIAM J. Control Optim.

Output tracking of high-order stochastic nonlinear systems with application to benchmark mechanical system

IEEE Trans. Autom. Control

State-feedback control of high-order stochastic nonlinear systems with SiISS inverse dynamics

IEEE Trans. Autom. Control

Global practical tracking for high-order uncertain nonlinear systems with unknown control directions

SIAM J. Control Optim.