Variable-structure systems and sliding-mode control

Studies in Systems, Decision and Control 271

Martin Steinberger

Martin Horn

Leonid Fridman   Editors

Variable-Structure

Systems and

Sliding-Mode

Control

From Theory to Practice

Studies in Systems, Decision and Control

Volume 271

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Martin Steinberger • Martin Horn •

Leonid Fridman

Editors

Variable-Structure Systems

and Sliding-Mode Control

From Theory to Practice

123

Editors

Martin Steinberger

Institute of Automation and Control

Graz University of Technology

Graz, Austria

Martin Horn

Institute of Automation and Control

Graz University of Technology

Graz, Austria

Leonid Fridman

Control Engineering and Robotics

Department

National Autonomous University of Mexico

Mexico City, Distrito Federal, Mexico

ISSN 2198-4182 ISSN 2198-4190 (electronic)

Studies in Systems, Decision and Control

ISBN 978-3-030-36620-9 ISBN 978-3-030-36621-6 (eBook)

https://doi.org/10.1007/978-3-030-36621-6

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Dedicated to

Stefanie, Georg and Klemens

Astrid, Ilona and Lorenz

and Millie

Preface

Variable-structure systems (VSS) and sliding-mode control (SMC) and observation

are known to be one of the most efficient robust control and observer design

techniques. Especially the ability to completely reject matched perturbations, i.e.,

perturbations that act in the input channels, is a stand-alone feature of these

approaches compared to other robust methods.

In the last years, research was pushed further leading to new algorithms and

important extensions of existing methods and more and more applications exploit

the nice properties of sliding-mode techniques. This was also evident at the 15th

International Workshop on Variable Structure Systems and Sliding Mode Control

(VSS 2018) which was held at Graz University of Technology, Austria. The con￾ference featured five sessions focusing on sliding-mode theory and four sessions on

applications based on submissions from 30 countries.

The present book covers theoretical and practical aspects related to VSS and

SMC. It is divided into four parts comprising 14 self-contained chapters that allow

separate reading in any preferred order.

The first part introduces New HOSM Algorithms. New homogeneous controllers

obtained by means of a recursive procedure for any relative degree of the system are

proposed. The controllers are accompanied by new filtering sliding-mode-based

differentiators. In addition, discontinuous integral control for systems with arbitrary

relative degree capable of tracking smooth but unknown reference signals under the

presence of Lipschitz continuous perturbations is presented.

Part II addresses Properties of Continuous Sliding-Mode Algorithms such as the

computation and estimation of the reaching time of the super-twisting algorithm and

two chapters dealing with the issue of constrained sliding-mode control, which is

inevitable for real-world implementations. Also, an analysis of the orbital stability of

self-excited periodic motions in a Lure system as well as a comparison of the

chattering using continuous and discontinuous sliding-mode controllers is treated.

Part III covers the Usage of VSS Controllers for Solving Other Control

Problems. Sliding-mode stabilization of SISO bilinear systems with delays is

considered where Volterra operator theory is exploited to perform stability and

robustness analysis. After that, a comparison of classical results and recent methods

vii

using integral and HOSM is carried out in the next chapter in order to investigate

their ability to compensate for unmatched disturbances.

The last part of the book is dedicated to Applications of VSS. Three chapters

related to power electronics show the capability of sliding-mode techniques in this

field. First, a grid-connected shunt active LCL control via continuous SMC and

HOSM control techniques is presented. After that, the robust distributed secondary

control of islanded inverter-based microgrids as well as local and wide-area

sliding-mode observers in power systems is investigated.

The last two chapters deal with the application of sliding-mode-based methods

for vehicle platooning, i.e., the task to form tight vehicle formations on one lane on

a highway, and an application to a single-loop integrated guidance and control

intercept strategy that makes use of HOSM control.

Enjoy reading!

Graz, Austria Martin Steinberger

Graz, Austria Martin Horn

Mexico City, Mexico

July 2019

Leonid Fridman

viii Preface

Acknowledgements

We gratefully acknowledge the financial support of (i) the Christian Doppler

Research Association, the Austrian Federal Ministry for Digital and Economic Affairs

and the National Foundation for Research, Technology and Development;

(ii) CONACYT (Consejo Nacional de Ciencia y Tecnologia) grant 282013;

PAPIIT-UNAM (Programa de Apoyo a Proyectos de Investigación e Innovación

Tecnológica) grant IN 115419; and (iii) the European Union’s Horizon 2020 research

and innovation programme under the Marie Skłodowska-Curie grant agreement

No 734832.

Graz, Austria Martin Steinberger

Graz, Austria Martin Horn

Mexico City, Mexico Leonid Fridman

ix

Contents

Part I New HOSM Algorithms

1 New Homogeneous Controllers and Differentiators............. 3

Avi Hanan, Adam Jbara and Arie Levant

2 Discontinuous Integral Control for Systems with Arbitrary

Relative Degree ........................................ 29

Jaime A. Moreno, Emmanuel Cruz-Zavala and Ángel Mercado-Uribe

Part II Properties of Continuous Sliding-Mode Algorithms

3 Computing and Estimating the Reaching Time

of the Super-Twisting Algorithm .......................... 73

Richard Seeber

4 Saturated Feedback Control Using Different Higher-Order

Sliding-Mode Algorithms ................................ 125

Mohammad Ali Golkani, Stefan Koch, Markus Reichhartinger,

Martin Horn and Leonid Fridman

5 Constrained Sliding-Mode Control: A Survey ................ 149

Massimo Zambelli and Antonella Ferrara

6 Analysis of Orbital Stability of Self-excited Periodic Motions

in Lure System ........................................ 177

Igor Boiko

7 Chattering Comparison Between Continuous and Discontinuous

Sliding-Mode Controllers ................................ 197

Ulises Pérez-Ventura and Leonid Fridman

xi

Part III Usage of VSS Controllers for Solving Other Control

Problems

8 Sliding-Mode Stabilization of SISO Bilinear Systems

with Delays ........................................... 215

Tonametl Sanchez, Andrey Polyakov, Jean-Pierre Richard

and Denis Efimov

9 Compensation of Unmatched Disturbances via Sliding-Mode

Control .............................................. 237

Kai Wulff, Tobias Posielek and Johann Reger

Part IV Applications of VSS

10 Grid-Connected Shunt Active LCL Control via Continuous SMC

and HOSMC Techniques ................................ 275

Mohamad A. E. Alali, Yuri. B. Shtessel and Jean-Pierre Barbot

11 On the Robust Distributed Secondary Control of Islanded

Inverter-Based Microgrids ............................... 309

Alessandro Pilloni, Milad Gholami, Alessandro Pisano and Elio Usai

12 Local and Wide-Area Sliding-Mode Observers

in Power Systems ...................................... 359

Gianmario Rinaldi, Prathyush P. Menon, Christopher Edwards

and Antonella Ferrara

13 Sliding-Mode-Based Platooning: Theory and Applications ....... 393

Astrid Rupp, Martin Steinberger and Martin Horn

14 Single-Loop Integrated Guidance and Control Using High-Order

Sliding-Mode Control ................................... 433

Michael A. Cross and Yuri B. Shtessel

xii Contents

Nomenclature

N The natural numbers

R The field of real numbers

C The field of complex numbers

Re[z] The real part of the complex number z

Im[z] The imaginary part of the complex number z

R þ The set of strictly positive real numbers

Rnm The set of real matrices with n rows and m columns

jaj The absolute values of the real (or complex) number a

signðÞ The signum function

AT The transpose of the matrix A

detðAÞ The determinant of the square matrix A

A1 The inverse of the square matrix A

A þ The (left) pseudo-inverse of the matrix A

rankðAÞ The rank of the matrix A

kðAÞ The spectrum of the square matrix A; i.e., the set of eigenvalues

kmaxðAÞ The largest eigenvalue of the square matrix A

kminðAÞ The smallest eigenvalue of the square matrix A

RðAÞ The range space of the matrix A (viewed as a linear operator)

NðAÞ The null space of the matrix A (viewed as a linear operator)

In The n n identity matrix

A [0 The square matrix A is symmetric positive definite

A [B The square matrix A  B is symmetric positive definite

k k Euclidean norm for vectors and the spectral norm for matrices

xðnÞ n-th derivative of the variable x with respect to time

 Equivalent to

Cartesian product

? Orthogonal complement

:¼ Equal to by definition

xiii

Part I

New HOSM Algorithms

Chapter 1

New Homogeneous Controllers

and Differentiators

Avi Hanan, Adam Jbara and Arie Levant

Abstract Unlimited number of new homogeneous output regulators are produced

by means of a recursive procedure in the form of control templates. The templates

are valid for any relative degree of the system, any homogeneity degree and include a

number of parameters to be found recursively by simulation. In particular, infinitely

many new sliding-mode (SM) controllers are produced. The controllers are accom￾panied by new filtering SM-based differentiators which are exact in the absence of

noises, robust, and asymptotically optimal in the presence of bounded noises, and

filter out unbounded noises of small average values.

1.1 Introduction

Sliding-mode (SM) control (SMC) [24, 27, 70, 71, 78] remains one of the main

approaches to control uncertain systems. The idea is to suppress the uncertainty by

exactly keeping a proper output σ (the sliding variable) at zero.

Due to the uncertainty of the system, one does not know the exact control which

would accomplish the task. The solution is to keep σ ≡ 0 by applying a sufficient

control effort each time a deviation of σ from zero is detected. It results in the

infinite-frequency switching of control. The corresponding motion is called SM.

The high-frequency control switching generates undesirable system vibrations

called chattering effect [8, 12, 16, 33, 78]. Another restriction of the classical SMC

[27, 78] is the requirement that the control explicitly appear already in σ˙.

In the following, we restrict ourselves to scalar sliding variables σ ∈ R. Introduce

the notation

A. Hanan · A. Jbara · A. Levant (B)

Applied Mathematics Department, Tel-Aviv University, 6997801 Tel Aviv, Israel

e-mail: [email protected]

A. Hanan

e-mail: [email protected]

A. Jbara

e-mail: [email protected]

© Springer Nature Switzerland AG 2020

M. Steinberger et al. (eds.), Variable-Structure Systems and Sliding-Mode Control,

Studies in Systems, Decision and Control 271,

https://doi.org/10.1007/978-3-030-36621-6_1

3

4 A. Hanan et al.

σk = (σ, σ˙,..., σ(k)

)

T , k ∈ N.

The appearance of the control in σ˙ means that the relative degree r [37] of the sliding

variable σ is 1, and the SMC keeps σr−1 = (σ) at zero.

Both described restrictions are connected. Indeed, the discontinuity of σ˙ due to the

control switching exaggerates the chattering of σ [48] and diminishes the accuracy

of the SM [43]. In the case when r > 1, the discontinuous control appears in σ(r)

and one usually introduces the auxiliary variable

Σ = βT σr−1, β = (β0,..., βr−2, 1)

T ∈ Rr

of the relative degree 1. Unfortunately, keeping Σ ≡ 0 by discontinuous control

results in the same vibration magnitude level [75] of all the components of σr−1

and prevents the establishment of σ ≡ 0 in finite time (FT). It still diminishes the

vibration energy (i.e., the chattering [48]) of lower derivatives.

High-order SMs (HOSMs) were introduced to deal with the restrictions of the

relative degree and the chattering, and to improve the accuracy of keeping σ = 0.

The corresponding rth-order SM (r-SM) controllers establish the equality σr−1 ≡ 0

in FT by means of discontinuous control. It potentially improves the accuracy of

keeping σ = 0 [43] and can significantly diminish the chattering by moving the

highest chattering level to σ(r−1) [9, 48]. The HOSM approach has become popular

in the last two decades [3, 10, 18, 20, 22, 25, 28, 32, 35, 41, 42, 60, 62, 65–69, 72,

74, 81] to mention just a few publications.

Establishing the r-SM σr−1 ≡ 0 does not automatically improve the SM accuracy

[43, 75]. In particular, such improvement is attained by the homogeneity approach

[45, 46] and its modifications, but other methods are also possible [1, 77].

This paper deals with the homogeneity-based stabilization of σ. For that end

one intentionally increases system uncertainty reducing the output dynamics to an

autonomous differential inclusion (DI) of the form σ(r) ∈ H(σr−1) + G(σr−1)u. The

homogeneity theory [7, 13–15, 38, 46, 63] is further applied to make σ vanish in

FT or asymptotically.

We present a simple, effective method of general homogeneous feedback stabiliza￾tion of such DIs. The method does not require validation by complicated Lyapunov￾stability analysis. The generated controllers contain a number of parameters to usually

be found by experiment, and therefore we call them control templates. We demon￾strate the technique by designing a completely new family of controllers valid for any

relative degree r, and any possible homogeneity degree including positive, negative,

and zero values.

The approach naturally requires estimating the derivatives σr−1 in real time. More￾over, the exact nth-order differentiation problem can itself be reformulated as FT

stabilizing the output σ = z − f (t) of the auxiliary control system z(n+1) = u. The

differentiation problem lies in the core of the observation methods [6, 31]. SM obser￾vation and differentiation always were a significant part of SMC research [24, 34,

71, 76, 79, 82].