Elsevier

Automatica

Volume 101, March 2019, Pages 27-35
Automatica

Brief paper
Overcoming overshoot performance limitations of linear systems with reset control

https://doi.org/10.1016/j.automatica.2018.11.038Get rights and content

Abstract

It is well-known that for a class of minimum-phase relative degree one linear-time-invariant (LTI) systems, with a unit feedback control structure, overshoot necessarily happens if the plant transfer function has poles at origin or unstable poles. This work aims to overcome this overshoot performance limitation (OPL) by using a novel reset controller, which has a generalized first order reset element (GFORE) structure. By tuning parameters of this reset controller carefully, the non-overshoot performance can be ensured. Furthermore, the implementation of the proposed reset controller with a high-pass filter is provided. Parameter tuning guidelines are also provided and, finally, the proposed design is verified with a simulation example.

Introduction

The concept of reset control was first proposed by Clegg in Clegg (1958) who introduced the so-called Clegg integrator: a linear integrator and a reset mechanism which resets the state of the integrator to zero when its input and output have opposite signs. This reset mechanism provides a describing function of the Clegg integrator similar to the frequency response of a linear integrator but with only 38.1° phase lag. It was suggested by Clegg that this property could overcome some of the fundamental performance limitations of linear control, but it was not until 1990s that this idea was confirmed by an example in Beker, Hollot, and Chait (2001). The Clegg integrator was generalized in Horowitz and Rosenbaum (1975), where the first order reset elements (FORE) was introduced together with a systematic design procedure. The early work on reset control was summarized in Chait and Hollot (2002).

Since the late 1990s, there has been a renewed interest in reset control systems. Various stability analysis and transient performance improvement results are reported in the literature Barreiro and Banos, 2010, Beker et al., 2004, Carrasco and Navarro-Lopez, 2013, Guo et al., 2009, and several experimental applications of reset control such as regulation of exhaust gas recirculation (EGR) valve (Panni, Waschl, Alberer, & Zaccarian, 2014), process control (Vidal & Banos, 2008) and networked control systems (Banos, Perez, & Cervera, 2014) are also presented. The majority of the above-mentioned literature has focused on the stability analysis of the reset control systems as resetting might lead to unstable performance, and the “zero crossing” reset models were used.

In Zaccarian, Nešić, and Teel (2005), a new reset model was proposed using the hybrid systems framework (Goebel & Teel, 2006), and a clock variable was introduced to avoid the Zeno phenomenon. This model allows flows and jumps on more complicated closed sets, which results in more favourable properties of reset systems Fichera et al., 2013, Nešić et al., 2011, Nešić et al., 2008, Prieur et al., 2013, Zaccarian et al., 2011, Zhao and Wang, 2016. For instance, the L2 stability analysis of reset systems was first presented in Nešić et al. (2008), exponential stability analysis results with explicit Lyapunov functions and piecewise quadratic Lyapunov functions were given in Zaccarian et al. (2011). Recently, a novel FORE model was proposed in Nešić et al. (2011), and it was shown that a strictly decreasing Lyapunov function at jumps can be constructed. The performance improvement of reset control systems has been addressed in Prieur et al. (2013), and L2 disturbance attenuation performance of linear systems was reported in Zhao and Wang (2016).

Although reset control is well-known for its ability to overcome the performance limitation of linear controller for linear systems, as demonstrated in a few examples Beker et al., 2001, Hunnekens et al., 2016, Zhao et al., 2013, a systematic design methodology for overcoming certain fundamental limitations is still missing in the literature.

This paper focuses on proposing new reset control laws that can overcome the overshoot performance limitation (OPL) without sacrificing the stability. This is motivated from engineering applications in which overshoot is unacceptable. Classic examples in flight control are the landing and taking off maneuvers, in which an overshoot of the pitch angle can have a disastrous effect. In order to overcome OPL, the tracking error cannot change its sign during the transient response. Hence the reset control law will depend on the tracking error e(t) and its derivative ė(t). This leads to a novel reset control algorithm which is a generalized form of FORE. The main result of this paper shows that through a careful selection of the parameters, the proposed reset controller can overcome OPL and achieve semi-global practical asymptotic stability. When the derivative of the tracking error is not directly measurable, a high-pass filter is introduced to approximate ė(t). The tuning guideline of the proposed reset controller, as well as the design trade-off are also provided. The effectiveness of the proposed reset controller is demonstrated through a numerical example.

The rest of this paper is organized as follows. Preliminaries and problem formulation are given in Section 2. A GFORE is proposed in Section 3. Section 4 presents the main results including practical stability and non-overshoot performance, and the implementation of the proposed reset controller. Section 5 presents a systematic parameter tuning procedure. Section 6 provides an illustrative example and Section 7 concludes this paper.

Notation: The set of real (integer) numbers is denoted as R (N). sup{S} denotes the supremum of a set S. |x| denotes the Euclidean norm of x. Given a state variable x with jumps, its derivative with respect to time is denoted as ẋ. At jump instants, we denote the value of state after the jump by x+, and the value of the state before the jump simply by x. B denotes the closed unit ball in a Euclidean space of a given dimension. For a set SRn, denote |x|SinfyS|xy|. sgn() denotes the sign function. dom(x) denotes the domain of x.

Section snippets

Preliminaries and problem formulation

In this work, we are interested in a class of single-input-single-output (SISO) LTI systems with closed-loop structure shown in Fig. 1. The plant and controller are represented by transfer functions G(s) and C(s) respectively. u is the control input and y is the plant output. The reference set-point is denoted as r, which is assumed to be constant. Assuming that C(s) can stabilize the loop, the following performance index is introduced to characterize the transient performance of the

Reset controller design

Several types of reset controllers exist in literature and majority of them contain a FORE. Thus it is instructive to revisit the main FORE models before introducing the proposed reset controller.

  • (i)

    The first one is taken from Horowitz and Rosenbaum (1975) and is described by the following equations ẋr=acxr+bcur,ur0,xr+=0,ur=0,where ac and bc are respectively the pole and input gain of the FORE, and xrR and urR are respectively the state and input.

  • (ii)

    A second FORE model was proposed in Zaccarian

Stability and non-overshoot performance

The plant (4) and GFORE (10) are interconnected with u=xr, e=ry and (8). The arising closed-loop system can be written in the following form τ̇=1ẋ=Ax+Br if xForτρ,τ+=0x+=Arx+Brα if xJandτρ,y=Cx

where x[zTyxr]TRn, C=[010] and F={xRn|(xEr)TM(xEr)2ap(xEr)TEr0}J={xRn|(xEr)TM(xEr)2ap(xEr)TEr0}A=AzBzy0Czapbp0bcac,B=00bc,Ar=In200010Czbpapbp0,Br=001bp,Er=0r0,M=0CzT0Czε2apbp0bp0.

The reset controller design consists of two steps: (1) Design parameters (ac,bc,ε,ρ2

Parameter tuning

The following procedure presents the tuning guidelines for the selection of parameters {ac,bc,ε,α,ρ,σ}. Step 1 and step 2 are taken from Nešić et al. (2011), while the last two steps are novel and present the selection of parameters α,ρ,σ.

Step 1: If the plant (4) includes one or more integrators (respectively, does not include integrators), select (ac,bc) (respectively, ac=0,bc) such that at least one of the following conditions holds.

  • (1)

    There exist γz>0,γv>0 and symmetric positive definite

An illustrative example

Consider the following minimum-phase relative degree one plant G(s)=(s+1)(s2+s+1)s(s2)(s22s+5),which can be represented in the state space form (4) as ż=010001122z+001y,ẏ=6y+u6119z.The plant includes an integrator, an unstable real pole and unstable complex conjugate poles. Therefore, the fundamental limitations reported in Corollary 1 are satisfied.

Suppose that the set-point reference r=1. Then, we select the parameters of reset controller (18) by the tuning procedure in Section 5.

Conclusion

This paper investigates the problem of overcoming overshoot performance limitations of linear feedback control using reset control. We focus on a class of minimum-phase relative degree one LTI plants under unit feedback control structure. A novel GFORE is proposed with the goal of overcoming the OPLs of linear systems presented in Corollary 1. Systematic reset control design procedures and an implementation of the proposed reset controller with a high-pass filter are given. Finally, an

Guanglei Zhao is an Associate Professor with the Institute of Electrical Engineering, Yanshan University, Qinhuangdao, China. He received the Ph.D. degree in Electrical Engineering from Shanghai Jiaotong University, China, in 2015. From 2012 to 2013, he was a visiting scholar in the University of Melbourne. His research interests include networked control systems, hybrid systems and robust control.

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    Guanglei Zhao is an Associate Professor with the Institute of Electrical Engineering, Yanshan University, Qinhuangdao, China. He received the Ph.D. degree in Electrical Engineering from Shanghai Jiaotong University, China, in 2015. From 2012 to 2013, he was a visiting scholar in the University of Melbourne. His research interests include networked control systems, hybrid systems and robust control.

    Dragan Nešić is a Professor with the Department of Electrical and Electronic Engineering, the University of Melbourne, Australia. He received the B.E. degree in Mechanical Engineering from the University of Belgrade, Belgrade, Serbia, in 1990, and the Ph.D. degree in Systems Engineering from Australian National University, Canberra, Australia, in 1997. His research interests include networked control systems, nonlinear control systems, input-to-state stability, extremum seeking control, hybrid control systems. He was awarded a Humboldt Research Fellowship in 2003 by the Alexander von Humboldt Foundation, an Australian Professorial Fellowship in 2004–2009, and a Future Fellowship in 2010–2014 by the Australian Research Council. He was a Distinguished Lecturer of Control Systems Society, IEEE. He is a fellow of the Institution of Engineers Australia. He served as an Associate Editor for Automatica, IEEE Transactions on Automatic Control, and IEEE Transactions on Control of Networked Systems. Dr. Nešić is Fellow of IEEE.

    Ying Tan is an Associate Professor with the Department of Electrical and Electronic Engineering, the University of Melbourne, Australia. She received her B.E. degree from Tianjin University, China, in 1995, and her Ph.D. degree from the National University of Singapore in 2002. She joined McMaster University in 2002 as a postdoctoral fellow in the Department of Chemical Engineering. Since 2004, she has been with the University of Melbourne. She was awarded an Australian Postdoctoral Fellow (2006–2008) and a Future Fellow (2009–2013) by the Australian Research Council. Her research interests are in intelligent systems, nonlinear control systems, real time optimization, sampled-data distributed parameter systems and formation control.

    Changchun Hua is a Professor with the Institute of Electrical Engineering, Yanshan University, Qinhuangdao, China. He received Ph.D. degree in Electrical Engineering from Yanshan University, Qinhuangdao in 2005. He was a research fellow in the National University of Singapore from 2006 to 2007. From 2007 to 2009, he worked in Carleton University, Canada. From 2009 to 2011, he worked in University of Duisburg—Essen, Germany. His research interests are in nonlinear control systems, control systems design over network, teleoperation systems and intelligent control.

    This work was supported by National Natural Science Foundation of China (61603329, 61673335, 61751309), China Post-doc Science Foundation (2016M601283, 2017T100167), Natural Science Foundation of Hebei Province, China (F2017203145) and Development Project of Qinhuangdao , China (201703A003). The material in this paper was partially presented at the 37th Chinese Control Conference, July 27–29, 2018, Wuhan, China. This paper was recommended for publication in revised form by Associate Editor Luca Zaccarian under the direction of Editor Daniel Liberzon.

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