Elsevier

Automatica

Volume 149, March 2023, 110830
Automatica

Brief paper
On the transient behavior and gain design for a class of second order homogeneous systems

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Abstract

In this note, we consider a class of homogeneous second order systems which contains two relevant sub-classes: the first one in a controller form and the second one in an observer form. For both of them, we provide necessary and sufficient conditions on the gain parameters to guarantee oscillatory or non-oscillatory behavior of the solutions. This is in the same spirit of tuning of second order linear systems, where the gains can be designed to obtain an overdamped, critically damped or underdamped behavior. The analysis of all the cases is performed in a unified framework based on a search for invariant sets which are analogous to the invariant eigenspaces of linear systems.

Introduction

Homogeneity has proven to be useful for the analysis and design of nonlinear control systems (Polyakov, 2020). For example, since linear systems are a special case of homogeneous systems, the latter ones are useful to approximate a wider set of nonlinear systems by keeping relevant nonlinear characteristics from the original one (Andrieu et al., 2008, Hahn, 1967, Hermes, 1986, Zavala-Río and Fantoni, 2014, Zubov, 1964). Other interesting feature of homogeneous systems are: local properties turn out to hold globally; in case of asymptotic stability of the origin, the homogeneity degree determines the type of convergence: exponential, finite-time, and nearly-fixed-time (Bacciotti and Rosier, 2005, Hahn, 1967, Hong et al., 1999, Kawski, 1990, Nakamura et al., 2002).

On the other hand, the aims of the process of tuning the gains of a control system are, in general, twofold: to guarantee some kind of stability and to guarantee a desired or an acceptable performance. Regarding nonlinear homogeneous control systems, most of the works found in the literature are devoted to: the study of general properties, e.g. the existence of homogeneous controllers (Grüne, 2000, Hermes, 1991, Kawski, 1988, Sepulchre and Aeyels, 1996), the existence of homogeneous Lyapunov functions (Nakamura et al., 2002, Rosier, 1992), and robustness (Bernuau, Polyakov, Efimov, & Perruquetti, 2013); synthesis of controllers (Cruz-Zavala and Moreno, 2017, Levant, 2005, Orlov, 2004, Polyakov et al., 2016); synthesis of observers and differentiators (Andrieu et al., 2008, Levant, 2003, Moreno, 2022, Qian and Lin, 2006); or construction of Lyapunov functions (Efimov et al., 2018, Sanchez and Moreno, 2019). However, most of the results that drive the problem of gain design are only to guarantee some stability properties. Thus, in general, there is a lack of results on the gain selection that provide guaranteed behavior of the trajectories.

The aim of this paper is to go in the performance direction. We show, by means of two relevant structures of second order nonlinear homogeneous systems, that it is possible to select the parameters of the system in order to obtain a guaranteed behavior of the trajectories, indeed, in a very similar way as it is done for linear systems. In particular, we provide necessary and sufficient conditions on the gain parameters to guarantee oscillatory or non-oscillatory behavior of the solutions. The analysis of all the cases is performed in a unified framework based on a search for invariant sets which are analogous to the invariant eigenspaces of linear systems. This simple analysis allows us to obtain the design conditions and a sort of characteristic function (which coincides with the characteristic polynomial in the linear case). It is important to mention that such an achievement has not been possible by means of the approach of generalized eigenvalues for homogeneous systems (see, e.g., Nakamura, Yamashita, & Nishitani, 2006), nor by the approach of the projected dynamics on the homogeneous sphere (see, e.g., ch. III Hahn, 1967).

Paper organization: In Section 2 we provide the preliminary definitions and results. In Section 3 we study the Controller form, and the Observer form is studied in Section 4. Finally, in Section 5, we provide some concluding remarks.

Notation: Real numbers are denoted as R. R>0 denotes the set {xR:x>0}, analogously for the sign . For xRn, |x| denotes the Euclidean norm. For xR, pR>0, we use the notation xp|x|psign(x) and x0sign(x).

Section snippets

Fundamentals

In this section, we recall the definition of weighted homogeneity, we describe the class of systems treated in the paper, and we state a result, which is the basic tool for the developments in Sections 3 Controller form, 4 Observer form.

Controller form

In this section we consider (1) with k11=0, k12=1, p12=1. To simplify the notation we rename some parameters as follows: k21=k1, k22=k2, p21=a and p22=b. Thus, (1) is rewritten as ẋ1=x2,ẋ2=k1x1ak2x2b.For this case, (2) can be reduced to aR0,b=2a1+a,guaranteeing that (8) is r-homogeneous of degree μ=ab with r=[b,a] for a>0, and of degree μ=1 with r=[2,1] for a=0. Observe that:

  • if a=1, then b=1 and μ=0 ((8) is linear);

  • if a(0,1), then 1>b>a and μ<0;

  • if a>1, then min{a,2}>b>1 and μ>0

Observer form

We consider in this section the subclass of (1) with k12=1, p12=1, and k22=0. To simplify the notation we rename the following parameters: k11=k1, k21=k2, p11=a and p21=b. Hence, (1) is rewritten as ẋ1=k1x1a+x2,ẋ2=k2x1b.From (2) we have that the exponents a and b satisfy the relation bR0,a=1+b2,which guarantees that (17) is an r-homogeneous system of degree μ=a1 with weights r=[1,a]. Observe that:

  • if b=1, then a=1 and μ=0 ((17) is linear);

  • if b(0,1), then max12,b<a<1 and μ<0;

  • if b>1

Conclusion

From our point of view, the contribution in this paper can be discussed in two different directions. The first one is about the procedural nature of the results: they provide useful criteria to choose the gains of two classes of homogeneous systems in order to guarantee the type of transient response of the system. This is done in a way which is analogous to that for standard second order linear systems. The second one is about the theoretical properties of homogeneous systems: the results in

Tonametl Sanchez received his Ph.D. in Electrical Engineering from the National and Autonomous University of Mexico (UNAM) in 2016 under the supervision of Prof. Jaime A. Moreno. He was a postdoctoral researcher at Institute of Engineering of UNAM, Mexico, from 2016 to 2017, and at Inria Lille – Nord Europe, France, from 2017 to 2019. Since 2019 he is associate researcher at IPICYT, Mexico. His research interests include Lyapunov function design methods, homogeneous control systems,

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    Tonametl Sanchez received his Ph.D. in Electrical Engineering from the National and Autonomous University of Mexico (UNAM) in 2016 under the supervision of Prof. Jaime A. Moreno. He was a postdoctoral researcher at Institute of Engineering of UNAM, Mexico, from 2016 to 2017, and at Inria Lille – Nord Europe, France, from 2017 to 2019. Since 2019 he is associate researcher at IPICYT, Mexico. His research interests include Lyapunov function design methods, homogeneous control systems, discrete-time nonlinear control systems, and sliding mode control.

    Arturo Zavala-Río received his B.S. degree in Electronic Systems Engineering and M.S. degree in Control Engineering from the Instituto Tecnologico y de Estudios Superiores de Monterrey, Mexico, in 1989 and 1992, respectively, and his DEA and Ph.D. degrees in Automatic Control from the Institut National Politechnique de Grenoble, France, in 1994 and 1997, respectively. He held professor-researcher positions at Universidad Autonoma de Queretaro (1999–2000), Mexico, and Universidad Autonoma de San Luis Potosi (2001), Mexico. He has been visiting researcher at Mechanical Engineering Laboratory (1998), Japan, and Universite de Technologie de Compiegne (2001–2002; 2013–2014), France. Since 2002, he is a full-time researcher at the Instituto Potosino de Investigacion Cientifica y Tecnologica, Mexico. He has been member of the National Researcher System, Conacyt, Mexico, since 1999. His research topics focus on the modeling, analysis, and control of nonlinear systems, with particular emphasis on the control of Euler–Lagrange systems.

    Griselda I. Zamora-Gómez received the Master and Ph.D. degrees from Instituto Potosino de Investigación Científica y Tecnológica A.C. (IPICyT), San Luis Potosi, Mexico, in 2015 and 2020 respectively, both of them in Control and Dynamical Systems. She is currently a Professor in the Engineering Department at Cuauhtémoc University, San Luis Potosi, Mexico. Her research interests involve dynamical and control system analysis, control design and programming for robotic systems.

    The authors acknowledge the support from CONACYT, Mexico CVU 371652. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Denis Efimov under the direction of Editor Daniel Liberzon.

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