Universal adaptive control for uncertain nonlinear systems via output feedback

doi:10.1016/j.ins.2019.05.087

Information Sciences

Volume 500, October 2019, Pages 140-155

https://doi.org/10.1016/j.ins.2019.05.087 Get rights and content

Abstract

In this paper, the universal adaptive control problem for a class of uncertain nonlinear systems is solved by the output feedback control approach. Firstly, a homogenous high-gain observer is proposed to estimate the system states based on the homogenous theory. Then, by using adding one power integrator method, a homogeneous controller is designed. It can be shown that all signals of the whole system are bounded and at the same time the system states globally asymptotically converge to the origin. In the end, we extend the proposed method to a class of upper-triangular nonlinear systems. Two examples are provided to illustrate the effectiveness of the proposed method.

Introduction

With the development of science and technology, more and more control objects, control devices have become more complex, and various requirements have been put forward for control accuracy. It is obvious that the linear system model will not be applicable. Therefore, the control problem of nonlinear systems has received much attention and has achieved a series of research results, for example [4], [5], [6], [10], [11], [12], [13], [14], [16], [19] and the references therein. In practical nonlinear systems, not all states are measurable and it is a very meaningful work to study its output feedback control problem. It is well known that the separation principle does not apply to nonlinear systems in general. Thus, some growth conditions of the unmeasurable states are indispensable for global output feedback stabilization as shown in [13]. In this note, we focus on the universal adaptive control problem of nonlinear systems described by $\begin{matrix} {\dot{x}}_{i} & = x_{i + 1} + f_{i} (t, x, ν), i = 1, \dots, n - 1 \\ {\dot{x}}_{n} & = ν + f_{n} (t, x, ν) \\ y & = x_{1} \end{matrix}$ where $x = {(x_{1}, \dots, x_{n})}^{T} \in R^{n},$ $ν \in R$ and $y \in R$ are the system state, control input and measured output, respectively. $f_{i} (\cdot), i = 1, \dots, n$ are uncertain continuous nonlinear functions.

The work [15] proposed a new feedback domination approach to achieve global output feedback stabilization for system (1) whose nonlinear functions f_i( · )’s satisfy the linear growth condition. Then, this condition was extended to the homogenous growth condition in [2], [14] and [19]. However, the controller design process needs the precise knowledge of the growth rate of all the above works. When the growth rate is unknown, Lei and Lin [7] proposed a universal control scheme for system (1). Subsequently, the works [17], [20] extended the above result to nonlinear time-delay systems. Ai et al. [1] discussed the universal adaptive regulation problem for nonlinear systems with not only time-delay but also unknown output function. Recently, Li and Liu [8] considered the finite-time stabilization problem for lower-triangular nonlinear systems via time-varying output feedback control. The work [11] investigated the robust adaptive control problem for a class of non-triangular nonlinear systems with unmodeled dynamics and stochastic disturbances. Compared with the existing results, the main contributions of this paper are summarized as follows: (i) A homogenous high-gain observer and a new adaptive output feedback controller are proposed under weak conditions; (ii) Two novel dynamic gains are introduced into the observer and controller to handle the unknown homogenous growth rate. (iii) Based on the homogeneity theory and the Lyapunov stability theory, it is shown that all the signals in the closed-loop system are bounded and the system states globally asymptotically converge to the origin.

Section snippets

Homogeneous systems and useful lemmas

Definition 2.1

[3] For real numbers $r_{i} > 0, i = 1, \dots, n,$ fixed coordinates $(x_{1}, \dots, x_{n}) \in R^{n},$ and ∀ε > 0, one has

•
the family dilation Δ_ε(x) is defined as $Δ_{ε} (x) = (ε^{r_{1}} x_{1}, \dots, ε^{r_{n}} x_{n})$ with r_i being the weight of x_i. For the sake of simplicity, the dilation weight is defined as $Δ = (r_{1}, \dots, r_{n})$ .
•
a function $V (x) : R^{n} \to R$ is said to be homogeneous of degree τ, if $V (Δ_{ε} (x)) = ε^{τ} V (r_{1}, \dots, r_{n})$ for a real number $τ \in R$ .
•
a homogeneous p-norm is defined as ${∥ x ∥}_{Δ, p} = {(\sum_{i = 1}^{n} {| x_{i} |}^{p / r_{i}})}^{1 / p}, \forall x \in R^{n},$ for a constant p ≥ 1. For simplicity, we choose $p = 2$ and write ‖x‖_Δ for

Observer design

Motivated by Li et al. [9], Zhai and Karimi [18], we design the following observer with two novel dynamic gains as $\begin{matrix} {\dot{\hat{x}}}_{i} & = {\hat{x}}_{i + 1} + L^{i} a_{i} {(x_{1} - {\hat{x}}_{1})}^{r_{i + 1}}, i = 1, \dots, n - 1 \\ {\dot{\hat{x}}}_{n} & = ν + L^{n} a_{n} {(x_{1} - {\hat{x}}_{1})}^{r_{n + 1}} \end{matrix}$ $\begin{matrix} \dot{L} & = \frac{L {(x_{1} - {\hat{x}}_{1})}^{μ + τ}}{M^{μ}}, L (0) = 1 \\ \dot{M} & = λ \frac{L {(x_{1} - {\hat{x}}_{1})}^{μ + τ}}{M^{μ - 1}}, M (0) = 1 \end{matrix}$ where $\hat{x} = {({\hat{x}}_{1}, \dots, {\hat{x}}_{n})}^{T},$ $a_{1} > 0, \dots, a_{n} > 0$ are chosen by Lemma 2.5. τ ≥ 0, $μ = 2 r_{n},$ and r_i’s are given in (3). L and M are dynamic gains. λ is a constant which will be determined later.

To simplify the analysis and design, we introduce the coordinate changes as follows $\begin{matrix} z_{i} & = \frac{x_{i}}{L^{i - 1} M^{r_{i}}}, {\hat{z}}_{i} = \end{matrix}$

Controller design

Initial Step: Choose the following Lyapunov function $V_{1} ({\hat{z}}_{1}) = \int_{{\hat{z}}_{1}^{*}}^{{\hat{z}}_{1}} {(s^{\frac{r_{n + 1}}{r_{1}}} - {\hat{z}}_{1}^{* \frac{r_{n + 1}}{r_{1}}})}^{\frac{μ - r_{1}}{r_{n + 1}}} d s$ where ${\hat{z}}_{1}^{*} = 0$ .

Letting $ψ_{i} = - (i - 1) \frac{\dot{L}}{L} {\hat{z}}_{i} - \frac{\dot{M}}{M} r_{i} {\hat{z}}_{i}, i = 1, \dots, n,$ one has ${\dot{V}}_{1} = M^{τ} L ({\hat{z}}_{1}^{\frac{μ - r_{1}}{r_{1}}} {\hat{z}}_{2} + a_{1} {\hat{z}}_{1}^{\frac{μ - r_{1}}{r_{1}}} e_{1}^{r_{2}}) + \frac{\partial V_{1}}{\partial {\hat{z}}_{1}} ψ_{1} .$

By Lemma 2.4, one has $a_{1} {\hat{z}}_{1}^{\frac{μ - r_{1}}{r_{1}}} e_{1}^{r_{2}} \leq {\tilde{d}}_{1} | {\hat{z}}_{1} |^{\frac{μ + τ}{r_{1}}} + \frac{{\bar{d}}_{1}}{2 n} {| e_{1} |}^{μ + τ}$ where ${\tilde{d}}_{1} > 0$ .

Let $ξ_{1} = {\hat{z}}_{1}^{r_{n + 1} / r_{1}}$ and design the first virtual controller ${\hat{z}}_{2}^{*} = - β_{1} ξ_{1}^{\frac{r_{2}}{r_{n + 1}}}, β_{1} = {\tilde{d}}_{1} + n .$

Then, one has ${\dot{V}}_{1} \leq - M^{τ} L (n | ξ_{1} |^{\frac{μ + τ}{r_{n + 1}}} - \frac{{\bar{d}}_{1}}{2 n} {| e_{1} |}^{μ + τ} - ξ_{1}^{\frac{μ - r_{1}}{r_{n + 1}}} ({\hat{z}}_{2} - {\hat{z}}_{2}^{*})) + \frac{\partial V_{1}}{\partial {\hat{z}}_{1}} ψ_{1} .$

Inductive Step: Assume at step $j - 1,$ there is a

Boundedness of the whole system

In this subsection, for any initial value of $(x (0), \hat{x} (0)) \in R^{n} \times R^{n}$ and $L (0) = 1, M (0) = 1,$ it will be shown that: (i) The solution of $(x (t), \hat{x} (t), L (t), M (t))$ exists on $[0, + \infty)$ and is unique and bounded; (ii) $\lim_{t \to + \infty} (x (t), \hat{x} (t)) = 0,$ $\lim_{t \to + \infty} L (t) = \bar{L},$ and $\lim_{t \to + \infty} M (t) = \bar{M}$ .

We recall that the solution of $(x (t), \hat{x} (t), L (t), M (t))$ exists and is unique on [0, t_f). If it can be proved that $(e (t), \hat{z} (t), L (t), M (t))$ is bounded on [0, t_f), then conclusions (i) and (ii) follow at once. We use the counter-evidence method to

Extension and discussion

It should be pointed out that the proposed scheme can be extended to deal with upper-triangular nonlinear systems whose nonlinearities satisfy the following assumption.

Assumption 6.1

There exists an unknown positive constant c, such that $| f_{i} (\cdot) | \leq c (\sum_{j = i + 2}^{n} | x_{j} |^{\frac{r_{i} + τ}{r_{j}}} + {| ν |}^{\frac{r_{i} + τ}{r_{n + 1}}}), i = 1, \dots, n - 1$ where τ ≥ 0 and r_i’s are defined in (3).

Conclusion

In this article, we proposed a novel universal adaptive control scheme for nonlinear systems under lower-triangular and upper-triangular homogenous growth condition with unknown growth rates. There are still some unresolved issues that need to be further investigated, such as dynamic output feedback control for high-order nonlinear systems with unknown coefficients.

Declaration of Conflict of Interest

We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in the manuscript entitled Universal adaptive control for uncertain nonlinear systems via output feedback.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grants 61873061, 61473082.

References (20)

W. Ai et al.
Universal adaptive regulation for a class of nonlinear systems with unknown time delays and output function via output feedback
J. Franklin Ins.
(2013)
V. Andrieu et al.
High gain observers with updated gain and homogeneous correction terms
Automatica
(2009)
B. Jiang et al.
Reduced-order adaptive sliding mode control for nonlinear switching semi-Markovian jump delayed systems
Inf. Sci.
(2019)
H. Lei et al.
Universal output feedback control of nonlinear systems with unknown growth rate
Automatica
(2006)
Y. Li et al.
Robust adaptive output feedback control to a class of non-triangular stochastic nonlinear systems
Automatica
(2018)
F. Mazenc et al.
Global stabilization by output feedback: examples and counter examples
Syst. Control Lett.
(1994)
J. Zhai et al.
Global adaptive output feedback control for a class of nonlinear time-delay systems
ISA Trans.
(2014)
A. Bacciotti et al.
Liapunov Functions and Stability in Control Theory
(2001)
F. Esfandiari et al.
Output feedback stabilization of fully linearizable systems
Int. J. Control
(1992)
Z. Jiang et al.
A small-gain control method for nonlinear cascaded systems with dynamic uncertainties
IEEE Trans. Autom. Control
(1997)

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

Cited by (56)

Zonotopic interval estimation for nonlinear systems with event-triggered protocols
2024, Journal of the Franklin Institute
This paper investigates the zonotope-based interval estimation problem for event-triggered nonlinear systems with incrementally quadratic constraint and unknown-but-bounded uncertainties from process disturbance and measurement noise. Considering the restricted communication bandwidth, an event-triggered communication protocol is deployed in the sensor-to-observer channel, based on which an event-triggered observer is constructed for nonlinear systems having incremental quadratic constraints. A co-design method of nonlinear observer and event-triggered protocol is obtained to ensure the stability and $ℓ_{\infty}$ index of the estimation error dynamic. Based on the designed event-triggered observer, a zonotope-based interval estimation algorithm is designed for the purpose of giving the intervals enclosing admissible values of system state. The mean value extension coalesce into natural interval extension is utilized to deal with the nonlinear transformation of system, and the enclosing zonotope of ellipsoid is introduced to tackle the extra uncertainties induced by event-triggered protocol. Lastly, the results of a simulation on a hyper-chaotic Chen system are provided to confirm the validity of the presented methodology.
Nonsingular recursive-structure sliding mode control for high-order nonlinear systems and an application in a wheeled mobile robot
2022, ISA Transactions
This work investigates the problem of fast tracking control for a class of high-order nonlinear systems subject to the matched disturbances. More particularly, a novel practical fixed-time disturbance observer is first presented by using a smooth hyperbolic tangent function. Then, a new nonsingular recursive-structure sliding mode surface is proposed based on the terminal sliding mode surface. With the reconstructed information deriving from the designed disturbance observer, a nonsingular recursive-structure sliding mode based finite-time tracking control approach incorporating with a new adaptive law is proposed to ensure the tracking errors converge to a small region of the origin in finite time. The finite-time stability of the closed-loop tracking control system driven by the proposed control scheme is analyzed and proved utilizing Lyapunov theory. And also, the proposed generalized control approach is applied to a mobile robotic experimental platform to achieve accurate trajectory tracking on the uneven ground. Finally, the numerical simulation and comparative experiment results demonstrate the effectiveness and superiority of the proposed approach.
Adaptive output feedback controller design for high-order stochastic nonlinear systems with uncertain output function
2022, ISA Transactions
This paper concentrates on the adaptive output feedback controller design for a class of high-order stochastic nonlinear systems(SNSs) with uncertain output function. Firstly for the nominal system, a homogeneous observer and output feedback controller is put forward through adding a power integrator method. Secondly, a dynamic gain technique is adopted into the observer and controller to guarantee the convergence of original system states and boundedness of the dynamic gain in probability. Besides, the proposed controller design scheme can be also broadened to upper-triangular SNSs. Finally, two numerical examples are provided to indicate the effectiveness of the proposed method.
Adaptive output feedback control for a class of switched stochastic nonlinear systems via an event-triggered strategy
2022, Applied Mathematics and Computation
This article addresses the adaptive event-triggered output-feedback control issue for a class of switched stochastic nonlinear systems (SSNSs) under arbitrary switchings. To cope with the unknown growth rate, two dynamic gains are introduced. Different from most existing event-triggered strategies, the developed dynamic event-triggered strategy can be dynamically adjusted, which is delicately designed by exploiting two dynamic gains and a decaying function. Then, a new event-triggered output-feedback controller (ETOFC) is established, which can guarantee that all states of the closed-loop system converge to the origin almost surely (a.s.). Finally, two examples are given to verify the effectiveness of the presented control scheme.
Adaptive output feedback tracking for time-delay nonlinear systems with unknown control coefficient and application to chemical reactors
2021, Information Sciences
Citation Excerpt :
Further, the problem of adaptive state regulation (or adaptive stabilization) has been investigated under the weakened growth constraints on the nonlinearities by redesigning dynamic gain instead of constant high gain such as [20–23]. In recent works [24,25], homogenous dynamic–high-gain observers and universal adaptive controllers were presented to achieve state asymptotical regulation for nonlinear systems with homogenous growth conditions. In addition, [26,30,27–29] concerned unknown control coefficient or measurement output uncertainty, where [27,28] dealt with the case of unknown output function that can be converted somewhat to the unknown control coefficient and vice versa.
In this paper, the problem of global adaptive output feedback tracking is considered for a class of highly nonlinear time-delay systems with unknown control coefficient and relaxed lower-triangular growth constraints. Based on the non-separation principle, a non-identifier-based adaptive output feedback controller is proposed by designing a novel update law of the dynamic gain in observer and controller. The novel gain can simultaneously compensate the uncertain parameters, the output-polynomial growth rate and the unknown time-varying delay. The proposed controller is well universal attributed to using the non-identification adaptive mechanism, and all design parameters are easy to get by simple calculation. With the help of Lyapunov–Krasovskii functionals and Barbalat’s lemma, it is shown that the solutions of the resulting closed-loop system are globally uniformly ultimately bounded by improving the backstepping design and dynamic high-gain scaling approach; the adaptive tracking is achieved under any small pre-given tracking error. Finally, the effectiveness of the proposed controller is illustrated by two examples.
Adaptive neural control for uncertain switched nonlinear systems with a switched filter-contained hysteretic quantizer
2021, Information Sciences
Citation Excerpt :
Uncertain switched nonlinear systems adopting adaptive tracking control via a backstepping method has attracted extensive attention and study in recent decades [1–6].
This paper demonstrates an adaptive neural control strategy for the uncertain nonlinear switched systems with dwell time. The unknown nonlinear functions, hysteresis nonlinear and external disturbances are taken into account in this system. A filter-contained hysteretic quantizer is introduced to filter out the high frequency components, leading to a durable solution for the conflict between hysteresis actuator and quantization, that is the hysteresis loop deforms when the input signal goes beyond the boundary. In addition, the adaptive neural network control strategy is developed using backstepping techniques to eliminate external disturbances and unknown nonlinear function. This strategy can guarantee that the tracking error converges to an adjustable field near zero.

View all citing articles on Scopus

View full text

Universal adaptive control for uncertain nonlinear systems via output feedback

Abstract

Introduction

Section snippets

Homogeneous systems and useful lemmas

Observer design

Controller design

Boundedness of the whole system

Extension and discussion

Conclusion

Declaration of Conflict of Interest

Acknowledgments

J. Franklin Ins.

Automatica

Inf. Sci.

Automatica

Automatica

Syst. Control Lett.

ISA Trans.

Liapunov Functions and Stability in Control Theory

Output feedback stabilization of fully linearizable systems

Int. J. Control

A small-gain control method for nonlinear cascaded systems with dynamic uncertainties

IEEE Trans. Autom. Control