On asymmetric periodic solutions in relay feedback systems

doi:10.1016/j.jfranklin.2020.10.024

Journal of the Franklin Institute

Volume 358, Issue 1, January 2021, Pages 363-383

https://doi.org/10.1016/j.jfranklin.2020.10.024 Get rights and content

Abstract

Asymmetric self-excited periodic motions or periodic solutions which are produced by relay feedback systems that have symmetric characteristics are studied in the paper. Two different mechanisms of producing an asymmetric oscillation by a system with symmetric properties are noted and analyzed by the locus of a perturbed relay system (LPRS) method. Bifurcation between the ability to excite symmetric and asymmetric oscillation with variation of system parameters is analyzed. An algorithm of finding asymmetric solutions is proposed.

Introduction

Design of a control system usually involves an assumption of the existence of a unique globally stable equilibrium point. However, even for a system with a unique equilibrium point and bounded solutions, the absence of attractors revealed through excitation of the oscillations from the vicinity of the equilibrium point cannot guarantee global stability. The reason is the existence of the so-called hidden attractors (limit cycles and chaotic attractors) that reveal themselves only in certain circumstances. If an attractor is revealed as a motion that can be excited from an initial point in the vicinity of an equilibrium point, we shall call this attractor “self-revealing”; otherwise it is hidden.¹ In contrast to the self-revealing attractor, the basin of attraction for a hidden attractor is not connected with equilibria. Therefore, special analytical and numerical methods must be developed to discover these hidden attractors through finding suitable initial points for solving the differential equations of the system or by using other means.

In relay feedback systems that usually do not have stable equilibrium points but have limit cycles, the problem of finding hidden attractors is of high importance too. Undesirable operating modes due to the existence of hidden attractors may occur in power inverters and converters. It is necessary to have methods that might identify their existence. Most of the developed approaches to analysis of relay feedback systems are aimed at finding or designing symmetric or asymmetric unimodal limit cycles² [2], [3], [4], [5], [6], [7], [8]. More general methods of analysis of nonlinear systems are used for this purpose too (see [9], [10]). These methods proved their efficiency in the analysis of symmetric periodic solutions. Some of them can also be used for analysis of asymmetric solutions caused by an asymmetric relay or an external biasing signal. However, the problem of finding and analyzing asymmetric oscillations in a relay feedback system that has fully symmetric properties was not addressed by these or other methods. Analysis of stability of the periodic solutions is done by the methods earlier proposed in [4], [5], and for dead-zone relays and relay systems with delays in [11], [12].

In the present paper, two classes of relay feedback systems that are capable of producing asymmetric oscillations in a system having fully symmetric properties are considered. This formulation of the problem is different from analysis of asymmetric oscillations caused by external signals or using asymmetric oscillations for process dynamics identification [13], [14]. A method of analysis of these oscillations is proposed. It is based on the Poincare map, the locus of a perturbed relay system (LPRS) method [15] and an extension of the latter that are being introduced in the present paper. It is also proposed that analysis of bifurcation of the modes of the oscillations be done through finding the orbital stability boundary conditions. Analysis of stability of the periodic solutions is done by the methods earlier proposed in [4], [5] and extended to systems with dead-zone relays and delays in [11], [12]. Further, by using the LPRS method the mechanism of generation of asymmetric oscillations in one of the considered classes is explained and precisely analyzed. Despite the concepts of the LPRS method were developed for a system having an external input signal, it is found in the present paper that it can be successfully used for analysis of asymmetric oscillations in a relay system having fully symmetric properties and no external signals. Finally, an algorithm of finding asymmetric limit cycles in a symmetric relay feedback system is proposed.

Section snippets

System model

Consider the following relay feedback system having hysteresis in the relay switching: $\begin{matrix} \dot{x} & = A x + B u, \\ y & = C x, \\ u (t) & = {\begin{matrix} + c, & if σ = f - y \geq b or {σ > - b and u (t - 0) = c}, \\ - c, & if σ = f - y \leq - b or {σ < b and u (t - 0) = - c}, \end{matrix} \end{matrix}$ where $x \in R^{n}$ is a state vector, $y \in R^{1}$ system output, $σ \in R^{1}$ error, $A \in R^{n \times n},$ $B \in R^{n \times 1},$ $C \in R^{1 \times n}$ are matrices, all quantities are real, $f$ is an input signal, $u$ is the control and $u (t - 0)$ is the control value at the time immediately preceding the current time, $y$ is the output, $b$ and $c$ are the hysteresis value and the amplitude of the relay.

Mathematical description of asymmetric oscillations

Here we introduce an analytical-numerical algorithm for localizing unimodal asymmetric periodic trajectories and determining the values of their parameters, namely, the frequency, the amplitudes and relative pulse duration, as well as possible initial states for producing an oscillation through simulations. Consider the solution of a linear plant given by Eq. (1) having a constant control $u$ : $x (t) = e^{A t} x (0) + A^{- 1} (e^{A t} - I) B u .$

Now if we assume that asymmetric limit cycle exists and time $t = 0$ corresponds

The system considered by Atherton [20]

Consider the following relay control system proposed by Atherton in [20]: $\begin{matrix} A = (\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 10 d & 2 d - 10 & d - 2 \end{matrix}), B = (\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}), C = {(\begin{matrix} 1 \\ 0 \\ 0 \end{matrix})}^{T}, \end{matrix}$ and $d$ is a parameter.

The linear part of the system (25) is defined by the transfer function $W_{Ath} (s) = \frac{1}{(s - d) (s^{2} + 2 s + 10)} .$ The system has two symmetric unstable saddle-focus equilibrium points: $S_{\pm} = \pm (\frac{1}{10 d}, 0, 0)$ with one-dimensional unstable manifold.

It is known [20] that depending on the parameter $d$ there can exist either symmetric or asymmetric periodic motions. According to [22], two

Conclusion

Analysis of asymmetric limit cycles in relay feedback systems with fully symmetric properties is undertaken in the paper. Among the examples known from the literature, two types of systems are identified: with non-zero constant term (bias) in the output oscillation and with zero constant term. It is shown that different mechanisms of asymmetry generation in the oscillations are involved. One of these mechanisms is explained through the LPRS method and supported by analysis of eigenvalues of the

Acknowledgements

This work was supported by the Russian Science Foundation 19-41-02002 and the Khalifa University, Abu Dhabi Project CIRA-2018-104.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

References (26)

I. Boiko
Oscillations and transfer properties of relay feedback systems with time-delay linear plants
Automatica
(2009)
I. Boiko
Autotune identification via the locus of a perturbed relay system approach
IEEE Trans. Control Syst. Technol.
(2008)
M. Ruderman
Computationally efficient formulation of relay operator for Preisach hysteresis modeling
IEEE Trans. Magn.
(2015)
N.V. Kuznetsov
Theory of hidden oscillations and stability of control systems
J. Comput. Syst. Sci. Int.
(2020)
Y. Tsypkin
Relay Control Systems
(1984)
D. Anosov
On the stability of the equilibrium positions of relay systems
Autom. Remote Control [in Russian]
(1959)
K. Astrom et al.
Fast switches in relay feedback systems
Automatica
(1999)
S. Varigonda et al.
Dynamics of relay relaxation oscillators, automatic control
IEEE Trans. Autom. Control
(2001)
J. Goncalves et al.
Global stability of relay feedback systems
IEEE Trans. Autom. Control
(2001)
L. Fridman
Singularly perturbed analysis of chattering in relay control systems
IEEE Trans. Autom. Control
(2002)

I. Boiko

Discontinuous Control Systems: Frequency-Domain Analysis and Design

(2008)

B. van der Pol

On relaxation-oscillations

Philos. Mag. J. Sci.

(1926)

N. Krylov et al.

Introduction to Non-Linear Mechanics (in Russian)

(1937)

Cited by (8)

Andronov-Vyshnegradsky problem on Watt governor and Kalman conjecture on global stability
2023, IFAC-PapersOnLine
The development of modern control theory, theory of differential inclusions, theory of global stability and hidden oscillations offers a new perspective on a number of classical problems in the field of discontinuous control systems analysis. It allows us to make solutions of these problems more efficient as well as to advance in previously unstudied areas. In this paper, we study an extended model of Watt governor, namely, a model that takes self-regulation of the engine into account, and estimate inner and outer bounds of the global stability boundary.
Efficient disturbance rejection of gimbaled inertial stabilization platform via relay controller
2022, Journal of the Franklin Institute
Citation Excerpt :
The primary mission of ISP is to maintain the line-of-sight jitterless under a certain level of disturbance torque, as we discussed in Section 2.1. If a piecewise linear system like ISP has a stable limit cycle, then the limit cycle is known to suppress the error signal [10,12]. This adaptive feature is explained by the incremental gain of a relay, or the equivalent gain [5].
In this article a relay controller is introduced in the simple motor driven gimbaled inertial stabilization platform (ISP) to improve disturbance rejection capacity. This improvement is possible through the manipulation of the adaptive feature of a limit cycle inherent in relay control systems. We present the sufficient condition for the existence of a limit cycle and its stability analysis in the relay control system, along with how this limit cycle can suppress the exogenous torque disturbance efficiently even in the presence of randomly distributed exogenous inputs. As a disturbance rejection measure, line-of-sight stabilization accuracy was obtained with an ISP driven by a typical linear controller and one by a relay controller, respectively. The results of numerical examples show the effectiveness and efficiency of the relay controller over a typical linear controller.
Fixed-time SOSM controller design subject to an asymmetric output constraint
2021, Journal of the Franklin Institute
Citation Excerpt :
The constraint problem may also arise from the perspective of safety such as the maximum output voltage of circuits and the maximum rotation angle of the manipulator joints [19–22]. It can be observed from the existing literatures that there are some results about asymmetric constraints, such as [23,24]. Nevertheless, these methods can only be used to solve asymmetric input constraint rather than asymmetric output constraint.
This paper addresses the problem of fixed-time controller design for the second-order sliding mode (SOSM) dynamics with asymmetric output constraints. Based on the construction of a barrier Lyapunov function (BLF) and the technique of adding a power integrator, a fixed-time SOSM controller that can be used to handle the asymmetric output constraint issue is developed. A strict Lyapunov stability analysis shows that the developed SOSM controller guarantees that the sliding variable can converge to the origin within a fixed time that is irrelevant to the system initial conditions. Meanwhile, the system output will never invade the boundary of the preset asymmetric constrained area. Finally, two examples are given to verify the effectiveness and the feasibility of the proposed scheme.
Sliding-Mode Control of Phase Shift for Two-Rotor Vibration Setup
2023, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Synthesis of Simple Relay Controllers in Self-Oscillating Control Systems
2022, Automation and Remote Control
Global Stability Boundaries and Hidden Oscillations in Dynamical Models with Dry Friction
2022, Advanced Structured Materials

View all citing articles on Scopus

View full text

On asymmetric periodic solutions in relay feedback systems

Abstract

Introduction

Section snippets

System model

Mathematical description of asymmetric oscillations

The system considered by Atherton [20]

Conclusion

Acknowledgements

Declaration of Competing Interest

Automatica

IEEE Trans. Control Syst. Technol.

IEEE Trans. Magn.

Theory of hidden oscillations and stability of control systems

J. Comput. Syst. Sci. Int.

Relay Control Systems

On the stability of the equilibrium positions of relay systems

Autom. Remote Control [in Russian]

Fast switches in relay feedback systems

Automatica

Dynamics of relay relaxation oscillators, automatic control

IEEE Trans. Autom. Control

Global stability of relay feedback systems

IEEE Trans. Autom. Control

Singularly perturbed analysis of chattering in relay control systems

IEEE Trans. Autom. Control

Discontinuous Control Systems: Frequency-Domain Analysis and Design

On relaxation-oscillations

Philos. Mag. J. Sci.

Introduction to Non-Linear Mechanics (in Russian)