On vanishing gains in robust adaptation of switched systems: A new leakage-based result for a class of Euler–Lagrange dynamics

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Abstract

In the presence of unmodelled dynamics and uncertainties with no a priori constant bounds, conventional robust adaptation strategies for switched systems cannot allow the control gains of inactive subsystems to remain constant during inactive intervals: vanishing gains are typically required in order to prove bounded stability. As a consequence, these strategies, known in literature as leakage-based adaptive methods, might introduce poor transients at each switching instant. Leakage-based adaptive control becomes even more problematic in the switched nonlinear case, where non-conservative state-dependent upper bounds for uncertainties and unmodelled dynamics are required. This work shows that, for a class of switched Euler–Lagrange systems, such difficulties can be mitigated: a novel leakage-based stable mechanism is introduced which allows the gains of inactive subsystems to remain constant. At the same time, unmodelled dynamics and uncertainties with no a priori bounds can be handled by a quadratic state-dependent upper bound structure that reduces conservativeness as compared to state-of-the-art structures. The proposed design is validated analytically and its performance is studied in simulation with a pick-and-place robotic manipulator example.

Introduction

Switched systems represent an important class of hybrid systems consisting of subsystems with continuous dynamics and a switching law to regulate the switching among subsystems. The switching can be state-dependent or time-driven, being dwell-time (DT) or average dwell-time (ADT) the most studied classes of time-driven switching [1], [2]. Over the last decade, several works have been reported for control of linear [3], [4], [5], [6] and nonlinear [7], [8], [9], [10], [11], [12], [13] switched systems (see also references therein). Here, we focus specifically on adaptive control of uncertain switched systems, i.e. control of switched systems with possibly large parametric uncertainties. Recent advances in the field include [14], [15], [16], [17] for switched linear systems and [18], [19], [20], [21], [22], [23], [24] for classes of switched nonlinear systems.

In the presence of unmodelled dynamics and uncertainties with no a priori constant bounds, it is well known that leakage-based adaptive control is the only robust adaptive mechanism able to prove bounded stability [25, Chap. 8], since projection, switching σ-modification, dead-zone and dynamic normalization all require knowledge of the bounds of the unmodelled dynamics/uncertainties. Efforts have been made recently to design leakage-based adaptive methods for uncertain switched systems. However, it was recently demonstrated that switched leakage-based strategies face serious drawbacks as compared to their non-switched counterpart [26], [27], [28]. Most notably, [26] showed that the control gains of the inactive subsystems should decrease exponentially as a consequence of leakage, otherwise bounded stability cannot be proven. This will create poor transients whenever a subsystem that remained inactive for sufficiently long time is activated again. One would desire a situation in which the inactive gains are kept fixed during inactive intervals. Unfortunately, this was shown to be possible only in restrictive cases, such as the class of globally Lipschitz nonlinear dynamics in [28].

Leakage-based adaptive control becomes even more challenging for switched nonlinear systems, where the presence of unmodelled dynamics and uncertainties with no a priori constant bounds requires suitable (possibly non-conservative) state-dependent upper bound structures. It is worth mentioning that conservative upper bound structures typically require high inputs, e.g. achieved by monotonically increasing control gains [19], [20], [28]. This work focuses on how conservative structures arise for the class of switched Euler–Lagrange (EL) systems, relevant in many application domains and recurring motif in adaptive switched literature. For example, the switched linear uncertain systems considered in [14], [15], [17], [26] (aircraft, electromechanical systems etc.) are linearized switched dynamics that should be more appropriately described as switched EL dynamics. Even the state–space linear-in-the-parameter (LIP) dynamics in [18], [22], [23], [24] can cover only a small class of EL dynamics, since the state–space EL form is in general nonlinear-in-the-parameter (NLIP) due to the inversion of the mass matrix. Even the NLIP structures in [19], [20] might be conservative for EL systems: while being extremely useful to attain strong stability results, the EL examples in [19], [20] reveal that such structures, relying on the parameter separation method pioneered in [29], require detailed structural knowledge of the system dynamics and result in a state-dependent quartic polynomial upper bound to the uncertainties. But it is known that, under mild assumptions [30], uncertainties in EL dynamics can be upper bounded by a less conservative state-dependent quadratic polynomial.

In light of the above discussions, leakage-based adaptive switched control presents unsolved challenges. This work proposes a new adaptation method in this direction with the following contributions:

  • A novel leakage-based adaptive mechanism is proposed which avoids the undesirable phenomenon of vanishing control gains. This is achieved by introducing auxiliary gains specifically for leakage purpose, which allow the control gains of inactive subsystems to be kept at the same value they had at switch-out instant.

  • Such leakage-based strategy is embedded in an adaptation framework for switched EL systems where uncertainties are upper bounded by a less conservative state-dependent quadratic polynomial structure, requiring less structural knowledge than LIP or parameter separation-based structures proposed in literature.

This work studies the same class of switched dynamics studied by some of the same authors in [31]. In addition to proposing a new leakage-based adaptation law, this work also manages to remove some structural constraints present in [31]. More specifically, as compared to [31], the switching law and leakage terms proposed in this work are independent of system dynamics terms, thus freely tunable. The rest of the paper is organized as follows: Section 2 describes the uncertain switched EL dynamics; Section 3 details the proposed control framework, with stability analysis carried out in Section 4; a simulation study is provided in Section 5, while Section 6 presents concluding remarks.

The following notations are used throughout the paper: λmin(), λmax() and represent minimum eigenvalue, maximum eigenvalue and Euclidean norm of () respectively; I denotes identity matrix with appropriate dimension; R+,N+ denote the set of positive real numbers and set of positive integers, respectively; Ω=[1,2,,N] denotes the set subsystems and N(p) denotes the set of inactive subsystem corresponding to an active subsystem pΩ.

Section snippets

System dynamics and problem formulation

Consider the following class of switched Euler–Lagrange (EL) systems Mσ(q)q̈+Cσ(q,q̇)q̇+Gσ(q)+Fσ(q̇)+dσ=τσ,where qRn is the system state and σ(t):[0)Ω is a piecewise constant function of time, called the switching signal, taking values in Ω=[1,2,,N]; for each subsystem σ, Mσ(q)Rn×n is the mass/inertia matrix; Cσ(q,q̇)Rn×n are Coriolis/centripetal terms; Gσ(q)Rn denotes the gravity vector; Fσ(q̇)Rn represents the vector of damping and friction forces; dσ(t)Rn denotes bounded external

Controller design

Let us consider the tracking problem for a desired trajectory qd(t) satisfying the following assumption.

Assumption 3

The desired trajectories are smooth enough, in particular qd,q̇d,q̈d.

Let e(t)q(t)qd(t) be the tracking error and ξ(t)[e(t),ė(t)]. We define a filtered tracking error variable rσ as rσBTPσξ,σΩwhere Pσ>0 is the solution to the Lyapunov equation AσTPσ+PσAσ=Qσ for some Qσ>0, Aσ0IK1σK2σand B0IT. Here, K1σ and K2σ are two user-defined positive definite gain matrices and their positive

Stability analysis of the proposed switched controller

From the definitions of Λσ and ξ we have Λσξ=K1σe+K2σė. Using this relation, the following error dynamics is obtained from (9) ξ̇=Aσξ+BΨσΔτσEσΔτσ.Before presenting the closed-loop stability result, let us recall the stability concept sought in switched robust adaptive control [26]:

Definition 2 Uniform Ultimate Boundedness (UUB)

The switched system (15) under switching signal σ() is uniformly ultimately bounded if there exists a convex and compact set C such that for every initial condition ξ(t0)=ξ0, there exists a finite time T(ξ0) such

Simulation results

This section studies the effectiveness of the proposed controller using a simplified scenario with pick-and-place robotic manipulator, often modelled via switched EL dynamics with two subsystems with different system parameters (one for the pick phase and one for the place phase) [34]: Mσ(q)q̈+Cσ(q,q̇)q̇+Gσ(q)+Fσ(q̇)+dσ=τσ, Mσ=Mσ11Mσ12Mσ12Mσ22,q=qlqu,Mσ11=(mσl+mσu)lσu2+mσulσl(lσl+2lσucos(qu)),Mσ12=mσulσu(lσu+lσlcos(qu)),Mσ22=mσulσu2,Cσ=mσulσllσusin(qu)q̇umσulσllσusin(qu)(q̇l+q̇u)0mσulσllσusin

Conclusions

A new concept of robust adaptation with leakage mechanism for uncertain switched EL systems was presented in this paper. The issue of vanishing gains of inactive subsystems was completely eliminated by virtue of properly designed auxiliary gains. At the same time, unmodelled dynamics and uncertainties with no a priori bounds could be handled by a quadratic state-dependent upper bound structure that reduces conservativeness as compared to state-of-the-art structures. Bounded stability analysis

CRediT authorship contribution statement

Spandan Roy: Data curation, Formal analysis, Methodology, Software, Validation, Visualization, Writing - original draft, Writing - review & editing. Elias B. Kosmatopoulos: Funding acquisition, Investigation, Project administration, Resources, Supervision, Writing - original draft. Simone Baldi: Conceptualization, Funding acquisition, Investigation, Methodology, Resources, Software, Supervision, Visualization, Writing - original draft, Writing - review & editing.

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.

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    The research leading to these results has been partially funded by the European Commission H2020-SEC-2016-2017-1, Border Security: autonomous control systems, China, under contract #740593 (ROBORDER) and H2020-ICT-2014-1, FIRE+ (Future Internet Research & Experimentation, China), under contract #645220 (RAWFIE), by the Fundamental Research Funds for the Central Universities, China under contract #4007019109 (RECON-STRUCT), and by the special guiding fund for double first-class, China under contract #4007019201.

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