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Fixed-Time Stabilization via Second Order Sliding Mode control

IFAC Proceedings Volumes 2012年第9期45卷 254-258页

作者： Andrey Polyakov Laboratory of Adaptive and Robust Control Systems Institute of Control Sciences Russian Academy of Sciences Profsouyznaya 65 Moscow Russia

The paper addresses the finite-time stabilization problem with possible predefining of the convergence time independently on the initial conditions. The corresponding extension of finite-time stability concept is called by fixed-time stability. The second order sliding mode control algorithm provided fixed-time stabilization of the multi-input linear control system is presented. Theoretical results are supported by numerical simulations.

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On settling time function and stability of vector relay systems

On settling time function and stability of vector relay syst...

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International Workshop on Variable Structure systems, VSS

作者： Andrey Polyakov Laboratory of Adaptive and Robust Control Systems Institute of Control Sciences Russian Academy of Sciences Moscow Russia

The paper addresses the problem of discovering the necessary and sufficient stability conditions of vector relay systems and studying the settling time function of this system. It shows that this function is piecewise linear in some cases and the linearity domains of this function are some convex cones. This fact helps to prove that the existence of the piecewise linear Lyapunov function for such vector relay system is a criterion of its stability. Some theoretical results are supported by numerical example.

关键词： Vectors Relays Numerical stability Stability criteria Asymptotic stability Lyapunov methods

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Fixed-time stabilization of linear systems via sliding mode control

Fixed-time stabilization of linear systems via sliding mode ...

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International Workshop on Variable Structure systems, VSS

作者： Andrey Polyakov Laboratory of Adaptive and Robust Control Systems Institute of Control Sciences Russian Academy of Sciences Moscow Russia

The paper presents some new sliding mode control algorithms for uncertain linear plants. The first class of controllers provide global asymptotic stability of the closed-loop system and allow to adjust a guaranteed convergence time of system trajectories into selected neighborhood of the origin independently on domain of initial system states. The design procedure essentially uses block reduction principles and finite-time attractivity properties for polynomial switching surface. The second class of controllers is meant to guarantee global finite-time stability of the closed-loop system with globally bounded settling-time function. The control design algorithms are presented for both single-input and multi-input systems. Theoretical results are supported by numerical simulations.

关键词： Stability analysis Numerical stability Asymptotic stability Polynomials Circuit stability Algorithm design and analysis Closed loop systems

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A robust dynamic controller for a class of nonlinear systems with sample-data outputs

A robust dynamic controller for a class of nonlinear systems...

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International Conference on Electrical Engineering, Computing Science and Automatic control (CCE)

作者： Manuel Mera Alex Poznyak Vadim Azhmyakov Andrey Polyakov Department of Control and Automation CINVESTAV Mexico Mexico Laboratory of Adaptive and Robust Control Systems Institute of Control Sciences RAS Moscow Russia

In this paper we study a class of continues nonlinear control systems associated with the specific sampled outputs. The dynamical model under consideration is described by ordinary differential equations in the presence of additive bounded uncertainties. We propose a linear-type control design procedure that guarantee robustness property of the closed-loop realization. The proposed design method incorporates a concept of the dynamic feedback controller and a newly elaborated extended version of the classic invariant ellipsoid method. The applicability of the proposed control design scheme is illustrated by a computational example. A brief discussion on the numerical issues is also included.

关键词： Ellipsoids Optimization robustness Linear matrix inequalities control systems Asymptotic stability Symmetric matrices

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Mirror Descent Algorithm for homogeneous finite controlled Markov chains with unknown mean losses

Mirror Descent Algorithm for homogeneous finite controlled M...

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作者： Nazin, Alexander V. Miller, Boris M. Laboratory for Adaptive and Robust Control Systems Institute of Control Sciences RAS Profsoyuznaya str. 65 117997 Moscow Russia School of Mathematical Sciences Monash University Clayton VIC 3800 Australia Institute for Information Transmission Problems RAS 19 B. Karetny Moscow Russia

ISBN: (纸本)9783902661937

We consider the adaptive stochastic problem for a system described by a controlled Markov chain (CMC) with a finite number of states. The novelty of the approach consists in adaptation technique for optimization of the system with unknown distribution of the cost function. This approach is applicable to the Internet congestion control with active users, motivating our study. In fact, we consider the homogenous finite CMC, introduce the assumptions of processes independence, stationarity of positive random losses and the bounds for their second moments, and the model regularity. The uncertainty is that the mean loss matrix is unknown. The related convex stochastic optimization problem as well as ideas of Mirror Descent Algorithm (MDA) give us the ability to design the control algorithm and to prove the upper bound for the difference between the expectation of time-averaged losses and their theoretical minimum. Asymptotically, the upper bound equals (ln T/T σ2N ln(NK) )1/3 and weakly depends on the state num ber K (standing inside the logarithm);here N and T stand for the number of control actions and the time horizon, respectively, parameter σ2 bounds the conditional second moments of the losses© 2011 IFAC.

关键词： Mirrors

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The Mirror Descent control Algorithm for Weakly Regular Homogeneous Finite Markov Chains with Unknown Mean Losses

The Mirror Descent Control Algorithm for Weakly Regular Homo...

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IEEE Conference on Decision and control

作者： Alexander V. Nazin Boris Miller Laboratory for Adaptive and Robust Control Systems Institute of Control Sciences RAS 65 Profsoyuznaya str. 117997 Moscow Russia School of Mathematical Sciences Monash University Clayton Victoria Vic 3800 Australia

ISBN: (纸本)9781612848006

We address the adaptive stochastic control problem for a discrete time system described by controlled Markov chain with finite number of states. The mirror descent randomized control algorithm on the class of controlled homogeneous finite Markov chains with unknown mean losses has been proposed and studied. Here we develop the approach represented in Nazin and Miller (2011). The main assumptions are the following: processes are independent and stationary, nonnegative random losses are almost surely bounded by a given constant, and the connectivity assumption for the controlled Markov chain holds. The uncertainty is that the mean loss matrix is unknown. The novelty of the approach is in extension of the class of controlled homogeneous finite Markov chains to the chains with connectivity assumption. The main result consists in demonstration of the asymptotical upper bound (that is asymptotic by time) and in determining the explicit constant which is weakly depending on the logarithm of the number of states.

关键词： Markov chain Computer algorithms Tree graphs discrete time systems Finite descent asymptotic Mirrors Unknown Standing Orders

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Mirror Descent Algorithm for Homogeneous Finite controlled Markov Chains with Unknown Mean Losses

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IFAC Proceedings Volumes 2011年第1期44卷 12421-12426页

作者： Alexander V. Nazin Boris M. Miller Laboratory for Adaptive and Robust Control Systems Institute of Control Sciences RAS Profsoyuznaya str. 65 117997 Moscow Russia (Tel: +7-495-334-7641) School of Mathematical Sciences Monash University Clayton Victoria Vic 3800 Australia (Tel: +61 3 9905 5870) Institute for Information Transmission Problems RAS 19 B. Karetny Moscow Russia

Abstract We consider the adaptive stochastic problem for a system described by a controlled Markov chain (CMC) with a finite number of states. The novelty of the approach consists in adaptation technique for optimization of the system with unknown distribution of the cost function. This approach is applicable to the Internet congestion control with active users, motivating our study. In fact, we consider the homogenous finite CMC, introduce the assumptions of processes independence, stationarity of positive random losses and the bounds for their second moments, and the model regularity. The uncertainty is that the mean loss matrix is unknown. The related convex stochastic optimization problem as well as ideas of Mirror Descent Algorithm (MDA) give us the ability to design the control algorithm and to prove the upper bound for the difference between the expectation of time-averaged losses and their theoretical minimum. Asymptotically, the upper bound equals ( In T T σ 2 N In ( N K ) ) 1 / 3 and weakly depends on the state number K (standing inside the logarithm); here N and T stand for the number of control actions and the time horizon, respectively, parameter σ 2 bounds the conditional second moments of the losses.

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Estimating the principal eigenvector of a stochastic matrix: Mirror Descent Algorithms via game approach with application to PageRank problem

Estimating the principal eigenvector of a stochastic matrix:...

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IEEE Conference on Decision and control

作者： Alexander Nazin Laboratory of Adaptive and Robust Control Systems Institute of Control Sciences RAS Moscow Russia

The problem of estimating the principal eigenvector related to the largest eigenvalue of a given (left) stochastic matrix A has many applications in ranking search results, multi-agent consensus, networked control and data mining. The well-known power method is a typical tool, but it modifies matrix and, therefore, the related solution. We propose and study both deterministic and randomized game algorithms based on Mirror Descent (MD) method which are intended for bounding the Euclidean norm residual ∥Ax-x∥ 2 on the standard simplex in ℝ N . We prove the explicit uniform upper bounds of type O(√ln(N)/n) with arbitrary horizon n ≥ 1. They improve the similar earlier results with respect to n which have been proved for the squared norm residual, i.e. ∥Ax-x∥ 2 2 . Numerical results for N = 100 illustrate the general decrease of the norm residual ∥Ax̂ t -x̂ t ∥ 2 in time t and corroborate theoretical results.

关键词： Upper bound Games Minimization Linear matrix inequalities Algorithm design and analysis Mirrors Stochastic processes

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adaptive randomized algori for finding eigenvector of stochastic matrix with application to pagerank

Adaptive randomized algori for finding eigenvector of stocha...

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48th IEEE Conference on Decision and control held jointly with 2009 28th Chinese control Conference, CDC/CCC 2009

作者： Nazin, Alexander Polyak, Boris Laboratory for Adaptive and Robust Control Systems Institute of Control Sciences RAS 65 Profsoyuznaya str. 117997 Moscow Russia

ISBN: (纸本)9781424438716

The problem of finding the eigenvector corresponding to the largest eigenvalue of a stochastic matrix has numerous applications in ranking search results, multi-agent consensus, networked control and data mining. The well-known power method is a typical tool for its solution. However randomized methods could be competitors vs standard ones;they require much less calculations for one iteration and are well-tailored for distributed computations. We propose a novel adaptive randomized algorithm and provide an explicit upper bound for its rate of convergence O(√lnN/n), where N is the dimension and n is the number of iterations. The bound looks promising because √lnN is not large even for very high dimensions. The proposed algorithm is based on the mirror-descent method for convex stochastic optimization. ©2009 IEEE.

关键词： Eigenvalues and eigenfunctions

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adaptive Randomized Algorithm for Finding Eigenvector of Stochastic Matrix with Application to Page Rank

Adaptive Randomized Algorithm for Finding Eigenvector of Sto...

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Joint 48th IEEE Conference on Decision and control and 28th Chinese control Conference

作者： Alexander Nazin Boris Polyak Laboratory for Adaptive and Robust Control Systems Institute of Control Sciences RAS

The problem of finding the eigenvector corresponding to the largest eigenvalue of a stochastic matrix has numerous applications in ranking search results,multi-agent consensus,networked control and data *** well-known power method is a typical tool for its *** randomized methods could be competitors vs standard ones;they require much less calculations for one iteration and are well-tailored for distributed *** propose a novel adaptive randomized algorithm and provide an explicit upper bound for its rate of convergence O(（lnN/n）),where N is the dimension and n is the number of *** bound looks promising because（lnN） is not large even for very high *** proposed algorithm is based on the mirrordescent method for convex stochastic optimization.

关键词： adaptive Randomized Algorithm for Finding Eigenvector of Stochastic Matrix with Application to Page Rank Matrix

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