Consistent Discretization of Locally Homogeneous Finite-time Stable Control Systems

Polyakov, Andrey; Efimov, Denis; Brogliato, Bernard; Reichhartinger, Markus

doi:10.23919/ecc.2019.8795633

Cited by 3 publications

(3 citation statements)

References 32 publications

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“…In both of the numerical examples, the performance of discretized implementation is at par with the theoretical results, i.e., the convergence is super-linear and the time of convergence is upper bounded by the theoretically established upper bound. Yet, how to prove that the convergence properties are preserved after discretization is an open problem, and is an active field of research (see [36,37,38]). In [36], the authors study a particular class of homogeneous systems and show that there exists a consistent discretization scheme that preserves the finite-time convergence.…”

Section: Discussionmentioning

confidence: 99%

Fixed-Time Stable Gradient Flows: Applications to Continuous-Time Optimization

Garg,

Panagou

2018

Preprint

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In this paper, continuous-time optimization methods are studied and a novel gradient-flow scheme is designed that yields convergence to the optimal point of the convex objective function in a fixed time from any given initial point. It is shown that the solutions of the modified gradient flow exist and are unique under certain regularity conditions on the objective function, and Lyapunov-based analysis is used to show fixed-time convergence. The unconstrained optimization problem is considered under two different assumptions, namely, strong-convexity and gradient-dominance, and fixed-time convergence is proven for both cases. Next, a modified Newton's method is presented that exhibits fixed-time convergence under some mild conditions on the objective function. Then, a method for solving convex optimization problems with linear equality constraints is proposed that yields convergence to the optimal point in fixed time. The constrained optimization problems are formulated as min-max problems and a novel method of computing the optimal solution is proposed using the Lagrangian dual. Finally, the general min-max problem is considered and a modified scheme for the saddle-point dynamics is proposed so that the optimal solution can be obtained in fixed time. Numerical illustrations are included to corroborate the efficacy of the proposed methods.

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Section: Discussionmentioning

confidence: 99%

Fixed-Time Stable Gradient Flows: Applications to Continuous-Time Optimization

Garg,

Panagou

2018

Preprint

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“…Actually, this behavior holds for all the numerical examples considered in Section 5. To show that the convergence properties are still preserved after applying a suitable discretization scheme to a continuous-time dynamical system, is an active area of research (see [33][34][35]). In [33], the authors study a particular class of homogeneous systems and show that there exists a consistent discretization scheme that preserves the finitetime convergence property.…”

Section: Discussionmentioning

confidence: 99%

Fixed-Time Stable Proximal Dynamical System for Solving MVIPs

Garg,

Baranwal,

Gupta

et al. 2019

Preprint

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In this paper, the fixed-time stability of a novel proximal dynamical system is investigated for solving mixed variational inequality problems. Under the assumptions of strong monotonicity and Lipschitz continuity, it is shown that the solution of the proposed proximal dynamical system exists in the classical sense, is uniquely determined and converges to the unique solution of the associated mixed variational inequality problem in a fixed time. As a special case, the proposed proximal dynamical system reduces to a novel fixed-time stable projected dynamical system. Furthermore, the fixed-time stability of the modified projected dynamical system continues to hold, even if the assumptions of strong monotonicity are relaxed to that of strong pseudomonotonicity. Connections to convex optimization problems are discussed, and commonly studied dynamical systems in the continuous-time optimization literature are shown as special cases of the proposed proximal dynamical system considered in this paper. Finally, several numerical examples are presented that corroborate the fixed-time convergent behavior of the proposed proximal dynamical system.

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“…Being efficient for numerical simulations, the mentioned schemes do not allow a consistent discretization (sampled-time implementation) of finitetime controllers in the general case. To the best of authors' knowledge, such implementations are developed only for first order ( [1], [20]) and second order systems ( [21], [6], [46]). This paper presents a consistent discretization for a homogeneous controller designed in [44], [55] for multidimensional linear plants.…”

Section: Introductionmentioning

confidence: 99%

Consistent Discretization of Finite-Time and Fixed-Time Stable Systems

Polyakov¹,

Efimov²,

Brogliato³

2019

SIAM J. Control Optim.

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Algorithms of implicit discretization for generalized homogeneous systems having a possible discontinuity only at the origin are developed. They are based on the transformation of the original system to an equivalent one which admits an implicit or a semi-implicit discretization schemes preserving the stability properties of the continuous-time system. Namely, the discretized model remains finite-time stable (in the case of negative homogeneity degree), and practically fixed-time stable (in the case of positive homogeneity degree). The theoretical results are supported with numerical examples.

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Consistent Discretization of Locally Homogeneous Finite-time Stable Control Systems

Cited by 3 publications

References 32 publications

Fixed-Time Stable Gradient Flows: Applications to Continuous-Time Optimization

Fixed-Time Stable Gradient Flows: Applications to Continuous-Time Optimization

Fixed-Time Stable Proximal Dynamical System for Solving MVIPs

Consistent Discretization of Finite-Time and Fixed-Time Stable Systems

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