Decentralized adaptive tracking control for a class of interconnected nonlinear systems with input quantization

doi:10.1016/j.automatica.2017.03.010

Automatica

Volume 81, July 2017, Pages 359-368

https://doi.org/10.1016/j.automatica.2017.03.010 Get rights and content

Abstract

In this paper, a decentralized output-feedback adaptive control scheme is proposed for a class of interconnected nonlinear systems with input quantization. Both logarithmic quantizers and improved hysteretic quantizers are studied, and a linear time-varying model is introduced to handle the difficulty caused by quantization. The proposed scheme allows the parameters of the quantizers to be freely changed during operation, and can guarantee global stability of the overall closed-loop system regardless of the coarseness of the quantizers and the existence of interactions among subsystems. Moreover, with the aid of a kind of high-gain K-filters, it is shown that all tracking errors converge to a residual set which can be made arbitrarily small by adjusting some design parameters. Simulation results are presented to illustrate the effectiveness of the proposed scheme.

Introduction

Motivated by great interest for applications in complex engineering systems such as electric power systems and chemical reactors, decentralized adaptive control for uncertain interconnected systems has long been an active issue in the control community. Different from centralized controllers, decentralized controllers are designed independently for subsystems and use only local signals for feedback, which brings challenge to the design and analysis in face of uncertain interactions among subsystems. In the early stage of the research, decentralized adaptive control schemes were developed mainly based on the certainty equivalence principle (Ioannou and Kokotovic, 1985, Shi and Singh, 1992, Wen and Hill, 1992). Since the middle 1990s, the research has been accelerated with the development of backstepping design (Krstic, Kanellakopoulos, & Kokotovic, 1995) and considerable achievements have been made; see, for instance, Jiang (2000), Li, Tong, and Li (2015), Wang and Lin (2015), Wen (1994) and Zhou and Wen (2008) and the references therein for more details.

On the other hand, signals in modern control systems are often quantized before being transmitted through communication channels and recent years have witnessed an increasing amount of attention in quantized control. A quantizer can be regarded as a map from a continuous region to a discrete set of numbers, making the control signal or the measurement of the system to be controlled a piecewise constant function of time. Quantization introduces strong nonlinear characteristics, which may degrade system performance or even result in instability. Aiming at understanding the required quantization resolution and mitigating the effect of quantization errors, much attention has been paid to quantized control of systems whose models are completely known or suffer from uncertainties composed of disturbances only (Ceragioli et al., 2011, De Persis, 2005, Fu and Xie, 2005, Liberzon, 2014, Liberzon and Hespanha, 2005).

In practice, it is often required to consider systems with general uncertainties such as unknown parameters and uncertain nonlinearities. In Corradini and Orlando (2008), De Persis (2009) and Liu et al., 2012a, Liu et al., 2012b, such uncertainties were studied for quantized control systems via robust control approaches, under the assumption that the bounds of the uncertainties are known. As we know, adaptive control is also useful to handle uncertainties and specially suitable for those without bound knowledge. Considering input quantization, adaptive control schemes were developed for uncertain linear and nonlinear systems in Hayakawa et al., 2009a, Hayakawa et al., 2009b, respectively, but the resulting stability conditions depend on control signals and are hard to be checked in advance. In Zhou, Wen, and Yang (2014), an adaptive backstepping control scheme was proposed for a class of strict-feedback systems and global stability was guaranteed by choosing the parameters of the quantizer and the controller to satisfy a derived inequality, which relaxes the stability conditions in Hayakawa et al., 2009a, Hayakawa et al., 2009b. A drawback of Zhou et al. (2014) lies in that the system nonlinearities are required to be globally Lipschitz. This restriction was later removed in Xing, Wen, Su, Cai, and Wang (2015). However, the same as Hayakawa et al., 2009a, Hayakawa et al., 2009b, and Zhou et al. (2014), the scheme in Xing et al. (2015) requires the measurement of full states. Recently, adaptive quantized control via output-feedback was studied in Xing, Wen, Zhu, Su, and Liu (2016) for a class of nonlinear systems. Nevertheless, to the best of our knowledge, existing adaptive quantized control schemes generally assume that the parameters of quantizers are constant (Xing et al., 2015, Xing et al., 2016, Zhou et al., 2014) or changeable but satisfy some inequalities involving control signals (Hayakawa et al., 2009a, Hayakawa et al., 2009b). In other words, these schemes do not consider the more general case that the parameters of quantizers may be freely changed during operation, which is an important issue from both theoretical and practical viewpoints. For example in tracking control, when the tracking error is small, the quantizer may be adjusted to be coarser by changing its parameters to decrease the communication burden; on the other hand, when the tracking error is large, the quantizer may be adjusted to be finer to improve the tracking performance. Moreover, existing adaptive quantized control schemes are mainly focused on controlling single-input single-output systems without interactions with other systems, and decentralized adaptive quantized control for interconnected systems needs to be further investigated.

In this paper, a decentralized output-feedback adaptive backstepping control scheme is proposed for a class of interconnected nonlinear systems with input quantization. Both logarithmic quantizers and hysteretic quantizers are studied. The proposed scheme has the following features:

•
Unlike existing adaptive quantized control schemes, in this paper the parameters of quantizers are allowed to be freely changed during operation according to system performance and communication burden. To handle the difficulty caused by quantization, we introduce a linear time-varying model to describe quantizers and estimate the bounds of the resulting time-varying terms. With the aid of such efforts, our controllers need no information about the parameters of quantizers.
•
An improved hysteretic quantizer is introduced, which can enhance the ability to reduce chattering in comparison with the hysteretic quantizer in Zhou et al. (2014).
•
Instead of the traditional K-filters employed in the existing output-feedback adaptive quantized control scheme (Xing et al., 2016), we construct a kind of high-gain K-filters to estimate the unmeasured states, which is shown to be effective to improve the tracking performance.
•
The proposed scheme is totally decentralized and the effect of interactions among subsystems is successfully compensated for by introducing a smooth function. It is proved that the overall closed-loop system is globally stable regardless of the coarseness of quantizers.

The rest of this paper is organized as follows. In Section 2, the control problem is introduced. In Section 3, the adaptive controllers design is presented, followed by the stability analysis in Section 4. Section 5 gives the simulation results to illustrate the effectiveness of the proposed scheme. Finally, we conclude in Section 6.

Section snippets

Problem formulation

Consider an interconnected nonlinear system consisting of $N$ single-input single-output subsystems in output-feedback form, given by ${\dot{x}}_{i} = A_{i} x_{i} + φ_{i} (y_{i}) θ_{i} + b_{i} η_{i} (y_{i}) Q_{i} (u_{i}) + f_{i} (y_{1}, \dots, y_{N}, t),$ $y_{i} = x_{i, 1}, i = 1, \dots, N,$ $A_{i} = [\begin{matrix} 0 \\ ⋮ & I_{n_{i} - 1} \\ 0 & \dots 0 \end{matrix}] \in R^{n_{i} \times n_{i}}, b_{i} = {[0 \dots 0 {\bar{b}}_{i}^{T}]}^{T} \in R^{n_{i}},$ where $x_{i} = {[x_{i, 1}, \dots, x_{i, n_{i}}]}^{T} \in R^{n_{i}}$ , $Q_{i} (u_{i}) \in R$ and $y_{i} \in R$ are the states, input and output of the $i$ th subsystem, respectively; $I_{n_{i} - 1}$ is the $(n_{i} - 1) \times (n_{i} - 1)$ identity matrix; $θ_{i} \in R^{ι_{i}}$ and ${\bar{b}}_{i} = {[b_{i, m_{i}}, \dots, b_{i, 0}]}^{T} \in R^{m_{i} + 1}$ with $b_{i, m_{i}} \neq 0$ are unknown constants; $φ_{i} (y_{i}) \in R^{n_{i} \times ι_{i}}$ and $η_{i} (y_{i}) \in R$

State estimation

In traditional output-feedback adaptive backstepping design, K-filters are widely adopted to estimate the unmeasured states (Krstic et al., 1995). In this paper, we design a set of high-gain K-filters for each subsystem as follows: ${\dot{ξ}}_{i} = Λ_{i} ξ_{i} + K_{i} y_{i},$ ${\dot{Ξ}}_{i} = Λ_{i} Ξ_{i} + φ_{i} (y_{i}),$ ${\dot{ς}}_{i} = Λ_{i} ς_{i} + E_{n_{i}, n_{i}} η_{i} (y_{i}) Q_{i} (u_{i}),$ where $E_{n_{i}, j}$ denotes the $j$ th coordinate vector in $R^{n_{i}}$ , $Λ_{i} = A_{i} - K_{i} E_{n_{i}, 1}^{T}$ , and $K_{i} = {[g_{i} k_{i, 1}, g_{i}^{2} k_{i, 2}, \dots, g_{i}^{n_{i}} k_{i, n_{i}}]}^{T}$ with $g_{i} \geq 1$ being a design parameter and $k_{i, 1}, \dots, k_{i, n_{i}}$ being chosen such that $s^{n_{i}} + k_{i, 1} s^{n_{i} - 1} + \dots + k_{i, n_{i}}$ is

Stability analysis

We are now ready to establish our main theorem.

Theorem 1

Consider the overall closed-loop system consisting of the interconnected system (1), the quantizers (2) and/or (3), the high-gain K-filters (6), (7), (8), the adaptive laws (29), (31), (49), (53), and the control laws (52). Suppose thatAssumption 1, Assumption 2 hold. Then, all signals of the overall closed-loop system are globally uniformly bounded, and the tracking errors converge to a residual set that can be made arbitrarily

Simulation results

To illustrate the effectiveness of the proposed scheme, we consider the double inverted pendulums connected by a spring in Fig. 3, whose dynamics are given by Bechlioulis and Rovithakis (2011) $J_{1} {\ddot{y}}_{1} = Q_{1} (u_{1}) + 0.5 M_{1} \bar{g} H sin y_{1} - 0.5 k (X - X_{0}) H cos (y_{1} - y_{0}),$ $J_{2} {\ddot{y}}_{2} = Q_{2} (u_{2}) + 0.5 M_{2} \bar{g} H sin y_{2} + 0.5 k (X - X_{0}) H cos (y_{2} - y_{0}),$ where $y_{1}$ and $y_{2}$ are the pendulum angles, $Q_{1} (u_{1})$ and $Q_{2} (u_{2})$ are quantized control torques, $J_{1}$ and $J_{2}$ are the moment of inertia, $M_{1}$ and $M_{2}$ are the pendulum masses, $H$ is the pendulum length, $\bar{g}$ is the

Conclusion

In this paper, a decentralized adaptive tracking control scheme has been proposed for a class of interconnected nonlinear systems with input quantization. Both logarithmic quantizers and improved hysteretic quantizers have been studied. The parameters of quantizers are allowed to be freely changed during operation according to system performance and communication burden and a linear time-varying model has been introduced to cope with the difficulty caused by quantization. We have shown that all

Chenliang Wang was born in Hunan Province, China, in 1986. He received the B.Eng. and Ph.D. degrees from the School of Automation Science and Electrical Engineering, Beihang University, Beijing, China, in July 2008 and January 2013, respectively, where he is currently a lecturer. From April 2015 to April 2016, he was a visiting scholar at National University of Singapore, Singapore. His research interests include adaptive control, fault-tolerant control, multivariable systems and multi-agent

References (25)

M.L. Corradini et al.
Robust quantzied feedback stabilization of linear systems
Automatica
(2008)
C. De Persis
Robust stabilization of nonlinear systems by quantized and ternary control
Systems & Control Letters
(2009)
T. Hayakawa et al.
Adaptive quantized control for linear uncertain discrete-time systems
Automatica
(2009)
T. Hayakawa et al.
Adaptive quantized control for nonlinear uncertain systems
Systems & Control Letters
(2009)
P.A. Ioannou et al.
Decentralized adaptive control of interconnected systems with reduced-order models
Automatica
(1985)
D. Liberzon
Finite data-rate feedback stabilization of switched and hybrid linear systems
Automatica
(2014)
T. Liu et al.
A sector bound approach to feedback control of nonlinear systems with state quantization
Automatica
(2012)
C. Wang et al.
Decentralized adaptive tracking control for a class of interconnected nonlinear time-varying systems
Automatica
(2015)
C. Wen et al.
Global boundedness of discrete-time adaptive control just using estimator projection
Automatica
(1992)
L. Xing et al.
A new adaptive control scheme for uncertain nonlinear systems with quantized input signal
Journal of the Franklin Institute
(2015)

L. Xing et al.

Output feedback control for uncertain nonlinear systems with input quantization

Automatica

(2016)

C.P. Bechlioulis et al.

Robust partial-state feedback prescribed performance control of cascade systems with unknown nonlinearities

IEEE Transactions on Automatic Control

(2011)

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In [10], a decentralized output tracking control scheme for interconnected nonlinear systems was developed based on backstepping technique. In the sequel, the decentralized control was extended to handle interconnected nonlinear systems with nonsmooth nonlinearities, such as quantization [12], hysteresis [13], dead-zone [14], etc. Note that the aforementioned literatures primarily focus on the stability of interconnected nonlinear systems.
This paper addresses the decentralized fault tolerant control problem for interconnected nonlinear systems under a reinforcement learning strategy. The system under consideration includes unknown actuator faults and asymmetric input constraints. By constructing an improved cost function related to the estimation of the actuator faults for each auxiliary subsystem, the original control issue is converted into finding an array of decentralized optimal control policies. Then, we prove that these optimal control policies can ensure the entire system to be stable in the sense of uniform ultimate boundedness. Moreover, a single critic network architecture is developed to acquire the solutions of Hamilton–Jacobi–Bellman equations, which simplifies the architecture of the reinforcement learning algorithm. All signals in the closed-loop auxiliary subsystems are uniformly ultimately bounded based on the Lyapunov theory, and numerical and practical simulation examples are conducted to validate the effectiveness of the designed method.

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Changyun Wen received the B.Eng. degree from Xi’an Jiaotong University, Xi’an, China, in 1983 and the Ph.D. degree from the University of Newcastle, Newcastle, Australia in 1990. From August 1989 to August 1991, he was a research associate and then postdoctoral fellow at University of Adelaide, Adelaide, Australia. Since August 1991, he has been with School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, where he is currently a full professor. His main research activities are in the areas of control systems and applications, intelligent power management system, smart grids, cyber-physical systems, complex systems and networks, model based online learning and system identification, signal and image processing.

He is an associate editor of a number of journals including Automatica, IEEE Transactions on Industrial Electronics and IEEE Control Systems Magazine. He is the executive editor-in-chief of Journal of Control and Decision. He served the IEEE Transactions on Automatic Control as an associate editor from January 2000 to December 2002. He has been actively involved in organizing international conferences playing the roles of General Chair, General Co-Chair, Technical Program Committee Chair, Program Committee Member, General Advisor, Publicity Chair and so on. He received the IES Prestigious Engineering Achievement Award 2005 from the Institution of Engineers, Singapore (IES) in 2005.

He is a fellow of IEEE, was a member of IEEE Fellow Committee from January 2011 to December 2013 and a distinguished lecturer of IEEE Control Systems Society from February 2010 to February 2013.

Yan Lin received the M.S. and Ph.D. degrees from Beihang University, Beijing, China, in 1988 and 1999, respectively. He is currently a professor in the School of Automation Science and Electrical Engineering, Beihang University. His research interests include robust control and adaptive control.

Wei Wang received her B.Eng. degree in electrical engineering and automation from Beihang University (BUAA) in 2005, M.Sc. degree in Radio Frequency Communication Systems with Distinction from University of Southampton (UK) in 2006 and Ph.D. degree from Nanyang Technological University (Singapore) in 2011. From January 2012 to June 2015, she was a lecturer with the Department of Automation at Tsinghua University, China. Currently, she is an associate professor with the School of Automation Science and Electrical Engineering at Beihang University and supported by BUAA Young Talent Recruitment Program. Her research interests include adaptive control of uncertain systems, distributed cooperative control of multi-agent systems, fault tolerant control and security control of cyber-physical systems. She is the receipt of Zhang Si-Ying Outstanding Youth Paper Award in 2013 25th Chinese Control and Decision Conference.

^☆: This work was supported by the National Natural Science Foundation of China under Grants 61673036, 61661136007, 61522301, 61633003 and 61673035. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Shuzhi Sam Ge under the direction of Editor Miroslav Krstic.

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Brief paperDecentralized adaptive tracking control for a class of interconnected nonlinear systems with input quantization☆

Abstract

Introduction

Section snippets

Problem formulation

State estimation

Stability analysis

Simulation results

Conclusion

Automatica

Systems & Control Letters

Automatica

Systems & Control Letters

Automatica

Automatica

Automatica

Automatica

Automatica

Journal of the Franklin Institute

Automatica

Robust partial-state feedback prescribed performance control of cascade systems with unknown nonlinearities

IEEE Transactions on Automatic Control

Brief paper
Decentralized adaptive tracking control for a class of interconnected nonlinear systems with input quantization☆