Limits of highly oscillatory controls and the approximation of general paths by admissible trajectories

Sussmann, Héctor J.; Liu, W.

doi:10.1109/cdc.1991.261338

Cited by 160 publications

(116 citation statements)

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“…The proof in [5] does not give a process to build these sequences of approximating sequence of oscillatory control, and although the case of a simple bracket (approximating [b 1 , b 2 ] by switching between b 1 and b 2 ) is elementary and well known, the case above of brackets of order 3 is more complex. The more recent work by Liu [9] and Sussmann and Liu [25] gives an explicit construction of the approximating sequence. The process of building this sequence is amazingly intricate compared to the simplicity of the existence proof in [5].…”

Section: Outline Of the Methodmentioning

confidence: 99%

“…, according to the process given in [25,9] so that when ε tends to zero, the system (1.1) controlled with these controls "tends to" the extended system (1.3) controlled with the controls v i (x).…”

Section: Outline Of the Methodmentioning

confidence: 99%

“…The meaning of "tends to" in point (b) is very imprecise. In [5], and in [25,9], only uniform convergence of the trajectories on finitetime intervals are considered. This is not adequate for asymptotic stabilization.…”

Section: Outline Of the Methodmentioning

confidence: 99%

“…This method gives homogeneous feedbacks, which ensure global stability if the control vector fields are homogeneous and local stability otherwise. The fact that it relates controllability with the construction of a stabilizing control law in a more direct way than previous designs also makes it conceptually appealing, all the more so as it may be viewed as converting the open-loop control techniques reported by Liu and Sussmann in [25] and Liu [9] into closed-loop techniques.…”

Section: Related Work and Contributionmentioning

confidence: 99%

“…When applied to particular systems for which explicit solutions have long been available, the present method often yields solutions which are significantly more complicated. This comes partly from the complexity of the approximation algorithm proposed in [25,9], which we use. This is also a consequence of the modifications that we have made to adapt this algorithm to our feedback control objective.…”

Section: Related Work and Contributionmentioning

confidence: 99%

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Design of Homogeneous Time-Varying Stabilizing Control Laws for Driftless Controllable Systems Via Oscillatory Approximation of Lie Brackets in Closed Loop

Morin¹

1999

SIAM J. Control Optim.

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Abstract.A constructive method for time-varying stabilization of smooth driftless controllable systems is developed. It provides time-varying homogeneous feedback laws that are continuous and smooth away from the origin. These feedbacks make the closed-loop system globally exponentially asymptotically stable if the control system is homogeneous with respect to a family of dilations and, using local homogeneous approximation of control systems, locally exponentially asymptotically stable otherwise.The method uses some known algorithms that construct oscillatory control inputs to approximate motion in the direction of iterated Lie brackets that we adapt to the closed-loop context.

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Section: Outline Of the Methodmentioning

confidence: 99%

Section: Outline Of the Methodmentioning

confidence: 99%

Section: Outline Of the Methodmentioning

confidence: 99%

Section: Related Work and Contributionmentioning

confidence: 99%

Section: Related Work and Contributionmentioning

confidence: 99%

See 3 more Smart Citations

Design of Homogeneous Time-Varying Stabilizing Control Laws for Driftless Controllable Systems Via Oscillatory Approximation of Lie Brackets in Closed Loop

Morin¹

1999

SIAM J. Control Optim.

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Polynomial motion of non-holonomic mechanical systems of chained form

Wang

1999

Math. Meth. Appl. Sci.

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In this study a simple general motion planning approach for non‐holonomic mechanical systems of chained form is proposed, in which the intricate motion planning problem (steering the system from an initial state to a final state) is converted to a simple curve‐fitting problem (satisfying a set of end‐point conditions). By means of this approach, other geometric constraints, such as passing a channel and avoiding collision with obstacles, and minimizing some cost functions, such as the minimum path length, can be easily handled, while the control inputs can be derived directly from the smooth path planned. For verifying the effectiveness of the proposed approach, a number of simulations are conducted for various task require ments and environments, with respect to a four‐wheel mobile chart (one‐chain system) and a three‐input firetruck (two‐chain system). Copyright © 1999 John Wiley & Sons, Ltd.

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Stabilization by unbounded‐variation noises

Nishimura

2016

Intl J Robust & Nonlinear

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In this paper, we claim the availability of deterministic noises for stabilization of the origins of dynamical systems, provided that the noises have unbounded variations. To achieve the result, we first consider the system representations of rough systems based on rough path analysis; then, we provide the notion of asymptotic stability for rough systems to analyze the stability for the systems. In the procedure, we also confirm that the system representations include stochastic differential equations; we also found that asymptotic stability for rough systems is the same property as uniform almost sure asymptotic stability provided by Bardi and Cesaroni. After the discussion, we confirm that there is a case that deterministic noises are capable of making the origin become asymptotically stable for rough systems while stochastic noises do not achieve the same stabilization results. ‡ In this paper, the term 'noises' is used when deterministic noises and stochastic noises are stated in a bundle. The deterministic noises are non-probabilistic signals having high-frequency oscillations, and the stochastic noises are probabilistic vibration signals such as Wiener processes.STABILIZATION BY UNBOUNDED-VARIATION NOISES 4127 subsets of Euclidean spaces. The stochastic stability property 'almost the same' as AS is uniform almost sure asymptotic stability (UASAS) [13,14]; all the sublevel sets of the related Lyapunov functions are invariant sets with probability one. However, achieving the property is generally difficult because the necessary and sufficient conditions are restrictive. Furthermore, there is a negative result that the non-AS origin never becomes the UASAS origin by the addition of any diffusion term [15]; that is, as long as employing the stabilization by noise with the use of Wiener processes, we should allow the possibility of all sample paths traveling to points very far away from the origin. Nevertheless, this paper claims that the stabilization by noise has still possibility to provide the non-AS origins with AS, provided that the resulting systems are neither ordinary nor stochastic differential equations. The assertion will be confirmed by the addition of deterministic noises having unbounded variations; this plan needs to employ a system representation by using rough path analysis [16][17][18][19] because the theory enables considering the dynamics with unbounded-variation functions such as Wiener processes and a particular kind of deterministic processes. The key point of the analysis is to classify external inputs by calculating the orders of the variations that is finite. Based on the information of the orders, rough paths and their dynamics-generally said to be rough differential equations, and rough systems in this paper-are obtained.Recently in [20], the stability notions for rough systems are defined with some important discussions; however, the claims have not been proven completely. Therefore, in this paper, we provide the proof of a concrete system representation of rough systems, show that ...

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Limits of highly oscillatory controls and the approximation of general paths by admissible trajectories

Cited by 160 publications

References 5 publications

Design of Homogeneous Time-Varying Stabilizing Control Laws for Driftless Controllable Systems Via Oscillatory Approximation of Lie Brackets in Closed Loop

Design of Homogeneous Time-Varying Stabilizing Control Laws for Driftless Controllable Systems Via Oscillatory Approximation of Lie Brackets in Closed Loop

Polynomial motion of non-holonomic mechanical systems of chained form

Stabilization by unbounded‐variation noises

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