Original articleFinite time stabilization of a perturbed double integrator with unilateral constraints☆
Introduction
The study of discontinuous systems has received considerable interest amongst control theorists and practitioners. Discontinuous systems are studied in very different research fields such as economics, electrical circuit theory, mechanical engineering, biosciences, systems and control theory. Many different frameworks therefore exist to describe various classes of discontinuous systems, for example, differential inclusions [12], measure differential inclusions [18] and complementarity systems [8] amongst others. Discontinuities appear either due to the nature of the system dynamics or due to the application of discontinuous feedback control. A survey article on discontinuous dynamical systems can be found in [11], which discusses various kinds of discontinuities, their respective solution concepts together with the stability tools available in the literature. The monograph [8] details methods for specifying solutions, the Lyapunov stability framework and control synthesis for non-smooth mechanical systems with friction and collision. The rigorous theoretical developments in the theory of non-smooth mechanics has been accompanied by applications such as biped robotics [6], [13], [15] thereby emphasizing the practical importance of the developed theory. An introduction and a detailed study of non-smooth Lyapunov functions in the context of discontinuous systems can be found in [24].
The main focus of this paper is on mechanical systems with resets in velocity. The well-known second order sliding mode controller [17] is utilized for finite time stabilization of a double integrator with a unilateral constraint surface. The velocity undergoes an instantaneous reset when this inelastic collision occurs. It is assumed in this paper that the restitution or reset map relating the velocities just before and after the time of impact is fully known. However, no such assumption is made on the time of impact. The method of Zhuravlev's non-smooth transformation [30] is utilized to first transform the system into a variable-structure system without jumps (also see references cited in [8, Ch. 1, Sec. 1.4]). Within the domain of engineering applications, such a transformation is very useful in the analysis of vibro-impact systems [3], [30]. The resulting transformed system turns out to be a switched homogeneous system with a negative homogeneity degree [21] where the solutions are well-defined in the sense of Filippov's definition [12], an attribute absent in case of the original jump system (see [28] for solutions concept of systems with jumps and friction). As an immediate consequence, the resulting transformed system turns out to be a valid candidate for stability analysis via smooth and non-smooth Lyapunov functions [8]. Next, a non-smooth Lyapunov function is identified to prove global uniform asymptotic stability. In turn, the quasi-homogeneity principle [21] is shown to be applicable to the transformed system which, while being locally homogeneous with negative homogeneity degree, is shown to be finite time stable.
The main theoretical contributions of this paper are threefold. Firstly, although results exist for asymptotic stabilization of continuous and discrete dynamics [9], finite time stabilization in the presence of velocity jumps is a novel concept. Whereas the existing stability results using Lyapunov methods [8], [4], [6], [7], [11], [14], [16], [25], [29] inevitably involve analysis of jumps of the Lyapunov function and their respective limits in asymptotic time, the proposed method proves the finite time stability of the origin of a perturbed double integrator in the presence of jumps in velocity without having to analyze jumps in the proposed non-smooth Lyapunov function. Secondly, finite time stability is proved using the homogeneity principle for the switched system thereby obviating the need for obtaining a differential inequality of the Lyapunov function. In turn, the quasi-homogeneity principle [21] is extended to the case where jumps in velocity are present. Finally, the ‘twisting’ controller [17] is shown to stabilize the unilaterally constrained perturbed double integrator in finite time. Furthermore, an upper bound on the settling time is also computed. These contributions bridge three streams, namely, non-smooth Lyapunov analysis, homogeneity principle and finite time stability for a class of impact mechanical systems.
The theoretical motivation to propose this new framework to replace the existing Lyapunov methods for the discontinuous systems [17], [21], [26] is that the later do not apply to the case of jumps in the velocity dynamics. In contrast, a class of non-smooth semi-global Lyapunov functions is shown to exist in this paper to prove uniform asymptotic stability of the transformed system in order to take advantage of the existing finite time stability results [21]. In turn, attainment of global uniform finite time stability becomes possible. From a practical viewpoint, the motivation stems from the applicability of the proposed method to the analysis and control of mechanical systems with jumps in velocity such as biped robots [13]. The proposed theory not only covers individual resets but also encompasses the proof of finite time stability under the influence of infinite rebounds of impulses, otherwise known as the so-called ‘zeno mode’. In the context of cyclic gait control of biped robot, the practical significance of this proof is that the stability of all actuated joints under the influence of infinite rebounds at each step is substantiated by mathematically transparent Lyapunov theory.
The rest of the paper is outlined as follows. The problem statement is presented in Section 2. 3 Global finite time stability, 4 Settling time estimate contain the main results, namely, the proof of finite time stability and computation of the upper bound on the settling time respectively. 5 Numerical simulation, 6 Conclusions and future work present numerical simulations and concluding remarks respectively.
Section snippets
Problem statement
Consider the following open loop system [8]: where x1, x2 are the position and the velocity respectively, u is the control input, ω(t) is piece-wise continuous [21, Sec. 2], [12] disturbance, tk is the kth jump time instant where the velocity undergoes a reset or jump, represents the loss of energy and and represent right and left limits respectively of x2 at the jump time tk. The equalities (1a), (1b)
Global finite time stability
This paper employs Zhuravlev–Ivanov's method of non-smooth transformation [30], [8, Sec. 1.4.2] to transform the impact system (1) into a jump-free system. Let the non-smooth coordinate transformation be defined as follows:The variable structure-wise transformed systemis then obtained by employing (5) and using the dynamics (1a), (1b) (see [8, Ch. 1] for the mechanics viewpoint and detailed
Settling time estimate
A finite upper bound on the settling time of the closed-loop system (6), (7) is computed in this section. The concept is graphically depicted in Fig. 1. When the trajectories are initialized on the positive vertical semi-axis at O4, the factor by which it gets closer to the origin after one revolution can be computed. The value of the intercept (point O3) on the positive vertical semi-axis after one revolution should be greater than the radius r1 of the ball containing
Numerical simulation
The numerical simulation result is presented in Fig. 2 which gives a comparison between the system (1), (2) with and the transformed system (8). Appropriate initial conditions x1(t0) = 2, x2(t0) = 1 and are used. The jump in velocity occurs when s changes sign [8]. The simulation is carried out using the event based Runge–Kutta method and it is inherently prone to exhibit departure from the physical behavior for both the discontinuities in the system (1)
Conclusions and future work
Robust finite time stabilization is presented for a unilaterally constrained perturbed double integrator. A non-smooth state transformation is employed to generate a jump-free system which is then shown to be finite time stable. The theoretical contribution of the presented work lies in achieving finite time stabilization of a class of impact mechanical systems without the need to analyze jumps in the Lyapunov function explicitly. A finite upper-bound on the settling time is also estimated.
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This work was supported by EPSRC via research grant EP/G053979/1.
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The second author wishes to thank Consejo Nacional de Ciencia y Tecnología de México for financial support of his sabbatical stay at the University of Kent.