On stability and application of extremum seeking control without steady-state oscillation

doi:10.1016/j.automatica.2016.01.009

Automatica

Volume 68, June 2016, Pages 18-26

https://doi.org/10.1016/j.automatica.2016.01.009 Get rights and content

Abstract

To enhance the dynamic and static performance of the extremum seeking control (ESC) scheme, a novel fast ESC scheme without steady-state oscillation is proposed, in which the structure of the classic ESC scheme is adjusted to make the sinusoidal excitation signal amplitude locally exponentially converge to zero. The improved ESC scheme can speed up the convergence to shorten the seeking time dramatically, and enlarge the search area to avoid falling to local extrema effectively. It can also reduce the sinusoidal excitation signal amplitude to eliminate the adverse effects of the steady-state oscillation eventually. The rigorous stability analysis and proof of the improved ESC scheme are provided in detail, and the simulation results are presented to illustrate its effectiveness and superiority. Finally, the application of the improved ESC scheme to antilock braking systems (ABS) has been discussed through comparing with classical perturbation-based ESC scheme and sliding-mode-based ESC scheme to illustrate the practicability of the improved scheme.

Introduction

Extremum seeking is a kind of adaptive control which can drive and maintain the input and output of the controlled object to their respective extrema. Extremum seeking control will work without any explicit knowledge about the input–output characteristics as long as the extrema exist, which is its greatest advantage. Therefore, extremum seeking is a model independent control scheme. Though extremum seeking scheme has been developed for several decades, the first rigorous stability analysis of the classic extremum seeking scheme was published by Wang and Krstić, 2000a, Wang and Krstić, 2000b. It appears that this paper renewed research interest in the theory of extremum seeking, and consequently the last decade has witnessed the significant development and numerous applications (Moase, Manzie, Nešić, & Mareels, 2010). Among them, Tan, Nešić, and Mareels (2008) studied how the different excitation signal in extremum seeking scheme affects the system performance. Lavretsky, Hovakimyan, and Calise (2003) pointed out that the amplitude of the excitation signal plays an important role in the system performance.

Extremum seeking scheme is essentially a method that achieves and maintains the function extremum by obtaining gradient information of the unknown function. Therefore, the extremum seeking scheme could easily converge to one of the local extrema if they exist. Committed to finding solutions to the problem, Tan, Nešić, Mareels, and Astolfi (2009) proposed a global extremum seeking scheme. A monotonically decreasing time function was designed to adjust the amplitude of the excitation signal, so that the searching arguments are expanded to get a certain ability to overcome the possible convergence to a local extremum. However, the performance of the scheme proposed by Tan et al. (2009) is not prominent because the excitation signal amplitude only varies with time. The same as the ordinary extremum seeking scheme, the improved scheme still has a large steady-state oscillation, which is undesirable, even not allowed for most practical systems.

In order to achieve the purpose of real-time optimality, the scheme is required to have a fast convergence rate. Krstić (1999) proposed a fast adaptive extremum seeking scheme to solve this problem. By introducing a dynamic compensator, the gain and phase margins of the feedback loop were improved to enhance the stability of the system and increase the response speed of the system. Compared with ordinary extremum seeking scheme it has better dynamic performance (Zuo, Hu, & Shi, 2006). However, the dynamic compensator design is dependent on the prior knowledge of the Wiener–Hammerstein model of the plant (Ariyur & Krstić, 2003), which brings inconvenience to its application.

As an application of extremum seeking control, the antilock braking systems with extremum seeking control have been investigated extensively (Dinçmen et al., 2014, Drakunov et al., 1995, Tunay, 2001, Yu and Özgüner, 2002, Zhang and Ordóñez, 2007). They all have reached the control purpose that the braking time is short and the tires are not locked during braking. However, most of them do bring additional oscillations in steady-state due to the perturbation signal and the sliding mode, respectively. Zhang and Ordóñez (2007) proposed an extremum seeking scheme based on numerical optimization through combining the numerical optimization algorithm with state regulation. The steady-state oscillation is successfully avoided when using the asymptotic state regulator numerical optimization based extremum seeking control. However, its real-time implementation will be affected when using a sophisticated optimization algorithm. Besides, the complex parameter adjustment of the scheme is not convenient for its application, too.

Motivated by the above study, we proposed a novel extremum seeking scheme in which the excitation signal amplitude can change adaptively with the extremum estimation error (Wang, Chen, & Zhao, 2014). The same as the scheme proposed by Tan et al. (2009), our scheme can avoid falling into local extrema effectively. The difference from the scheme proposed by Tan et al. (2009) is that our scheme can eliminate the steady-state oscillation due to the fact that the excitation signal amplitude will fast converge to zero with the decrease of the extremum estimation error. Furthermore it also has a strong adaptability to the extremum perturbation. In this paper, we improve the scheme further and give the rigorous stability proof of this improved scheme using Singular Perturbation Theory, Averaging Method, The Center Manifold Theorem and Lyapunov Method. We also give a simulation example to validate the characteristics of non-oscillating steady state of the improved scheme. Besides, the application of the improved scheme to antilock braking systems sufficiently illustrates the practicability of the scheme.

This paper is organized as follows. The problem formulation is given in Section 2. The main results are stated in Section 3. In Section 4 we discuss the improved scheme in detail. Section 5 consists of simulation example and application to ABS. Finally, a brief summary of the full text is given in Section 6. Some lemmas which are used in proof and auxiliary results are presented in the Appendix.

Section snippets

Problem formulation

Consider a general single input and single output (SISO) nonlinear model as follows: $\dot{x} = f (x, u), y = h (x),$ where $f : R^{n} \times R \to R^{n}$ and $h : R^{n} \to R$ are continuously differentiable, $x$ is the state, $u$ is the input and $y$ is the measurable output. Suppose there exists a family control laws of the following form: $u = α (x, θ),$ where $θ \in R$ is a scalar parameter. The closed-loop system $\dot{x} = f (x, α (x, θ)),$ then has an equilibrium point parameterized by $θ$ . For simplicity, $θ$ is assumed to be scalar and (1), (2) is assumed to be

Main results

In this section we will investigate stability of a novel ESCWSSO. This novel scheme is an improvement of the perturbation-based extremum seeking scheme. The analysis of the proposed scheme lends itself to an understanding of this improvement. The proposed ESCWSSO is shown in Fig. 1.

The scheme shown in Fig. 1 introduces two new design parameters $ω_{l}, r$ instead of the excitation signal amplitude in the classic extremum seeking scheme. $ω_{l}$ is the cutoff frequency of the low-pass filter and $r$ is a

Discussions

In this section we will discuss main characteristics of the proposed scheme including non-oscillating steady state, the ability of passing local extremum and the selection of the design parameters.

From (23), (25) we can get the following formula in the vicinity of extreme point. ${\tilde{m}}_{r}^{a} = \frac{\ddot{v} (0)}{2} {({\tilde{θ}}_{r}^{a})}^{2} + \frac{\ddot{v} (0)}{2} {(a_{r}^{a})}^{2}$ $\frac{d a_{r}^{a}}{d κ} = - δ {ω^{'}}_{L} a_{r}^{a} .$ (35) shows that the excitation signal amplitude locally exponentially converges to zero. We can always make $a_{r}^{a} > 0$ by choosing the initial value $a_{r}^{a} (0) > 0$ . It can be seen

Examples

This section consists of two parts. In the first part, we give the simulation results of the ESCWSSO. The application of the scheme to ABS is given in the second part to illustrate the advantage of it through comparing with the ABS with classic perturbation-based extremum seeking scheme and sliding-mode-based extremum seeking scheme.

Conclusion and future work

This paper proposed a novel extremum seeking scheme without steady-state oscillation. The scheme greatly improved the dynamic and static performance of the extremum seeking scheme. This scheme not only can converge fast, but also can effectively avoid falling into local extreme points. The most important is that the scheme eliminates the steady-state oscillation which hinders the online applications of the classical schemes. The rigorous stability analysis of the scheme was given and the

Libin Wang received his B.S. degree in Control Science and Engineering from Hebei University of Technology in 2012, and the M.S. degree in Control Science and Engineering from Harbin Institute of Technology (HIT) in 2014. He is now a Ph.D. student at the Control and Simulation Center, HIT, China. His current research interests include extremum seeking, decoupling control, adaptive control, and sliding mode control.

References (20)

Y. Tan et al.
On the choice of dither in extremum seeking systems: A case study
Automatica
(2008)
Y. Tan et al.
On non-local stability properties of extremum seeking control
Automatica
(2006)
Y. Tan et al.
On global extremum seeking in the presence of local extrema
Automatica
(2009)
K.B. Ariyur et al.
Real-time optimization by extremum-seeking control
(2003)
J. Carr
Applications of centre manifold theory
(1981)
Deschênes, J. S. (2012). Demodulation considerations in extremum seeking control loops. In Proceedings of the American...
E. Dinçmen et al.
Extremum-seeking control of ABS braking in road vehicles with lateral force improvement
IEEE Transactions on Control Systems Technology
(2014)
S. Drakunov et al.
ABS control using optimum search via sliding modes
IEEE Transactions on Control Systems Technology
(1995)
H.K. Khalil
Nonlinear systems
(2002)
Krstić, M. (1999). Toward faster adaptation in extremum seeking control. In Proceeding of the 38th IEEE conference on...

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

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Songlin Chen received the M.S. degree in Control Science and Control Engineering from Harbin University of Science and Technology in 2002, and the Ph.D. degree in Guidance, Navigation and Control from Harbin Institute of Technology (HIT) in 2006. He is currently an associate professor at Control and Simulation Center, HIT. His current research interests include high-performance servo systems, nonlinear and adaptive control, active disturbance rejection control, and robot control.

Kemao Ma received his Bachelor degree in engineering from the Department of Information and Control Engineering, Xi’an Jiaotong University, China, in 1992, and his Ph.D. degree from Harbin Institute of Technology, China, in 1998. Since 2006, he has been a professor at the Control and Simulation Center, School of Astronautics, Harbin Institute of Technology, China. His current research interests are nonlinear control, non-smooth analysis and control, robust control and their applications to guidance and control of flight vehicles.

^☆: This work was partially supported by the Natural Science Foundation of China under grant numbers 61174001, 61321062. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Raul Ordóñez under the direction of Editor Miroslav Krstic.

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On stability and application of extremum seeking control without steady-state oscillation☆

Abstract

Introduction

Section snippets

Problem formulation

Main results

Discussions

Examples

Conclusion and future work

Automatica

Automatica

Automatica

Real-time optimization by extremum-seeking control

Applications of centre manifold theory

Extremum-seeking control of ABS braking in road vehicles with lateral force improvement

IEEE Transactions on Control Systems Technology

ABS control using optimum search via sliding modes

IEEE Transactions on Control Systems Technology

Nonlinear systems