A Counterexample in Stochastic Optimum Control

Witsenhausen, H. S.

doi:10.1137/0306011

Cited by 771 publications

(566 citation statements)

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“…In fact, the optimal solution to this coupled problem can be formulated as a decentralized stochastic problem with information constraints and imperfect observations, and the solution to such a problem is known not to be modular. Witsenhausen's counterexample shows, in fact, that separation of estimation and controller design fails to hold even in simple settings [1]. In addition, recently, we have seen a surge of interesting results [2], [3], [4], [5], [6], [7] addressing "old" communications questions such as channel capacity and quantization in the context of stabilization and control of linear systems (see [8] and [4] for a nice summary).…”

Section: Introductionmentioning

confidence: 99%

“…In fact, the optimal solution to this coupled problem can be formulated as a decentralized stochastic problem with information constraints and imperfect observations, and the solution to such a problem is known not to be modular. Witsenhausen's counterexample shows, in fact, that separation of estimation and controller design fails to hold even in simple settings [1] [7] addressing "old" communications questions such as channel capacity and quantization in the context of stabilization and control of linear systems (see [8] and [4] for a nice summary). Our work differs from these sets of work in that we propose a practical modularization motivated by [9] to integrate practical communication questions with coordination and control of nonlinear vehicles.…”

Section: Index Termsmentioning

confidence: 99%

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Integration of communication and control using discrete time Kuramoto models for multivehicle coordination over broadcast networks

Klein

Lee

Morgansen

et al. 2008

IEEE J. Select. Areas Commun.

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This paper considers the integration of communication and control with respect to the task of coordinated heading control for a group of N vehicles with the energy efficiency of communications in mind. The heading control employed on each vehicle is a discretization of the well-known Kuramoto model of nonlinearly coupled oscillators over a sequence of logical graphs. Stability for both all-to-all and random one-to-all broadcasts is shown to be dependent on the coupling strength, K, and the time discretization, ∆T . For desired system performance characteristics, ∆T imposes a tight deadline by which the state information (M bits) must be propagated through the communication network. Routing optimization with respect to minimizing energy consumption is formulated considering the ∆T deadline. Due to tight time deadline a one-to-all single-hop broadcasting scheme is shown to be more energy efficient for practical choices of M/∆T . The proposed modularization is illustrated via a set of simulations where the overall communication energy to reach alignment is optimized. Index TermsCoordinated Control, Discrete Time Kuramoto, Network Routing, Energy Optimal Communication I. INTRODUCTION A fundamental challenge in designing networked control systems is that the tasks of communication and control cannot, in general, be considered decoupled from each other without loss of optimality. In fact, the optimal solution to this coupled problem can be formulated as a decentralized stochastic problem with information constraints and imperfect observations, and the solution to such a problem is known not to be modular. Witsenhausen's counterexample shows, in fact, that separation of estimation and controller design fails to hold even in simple settings [1] [7] addressing "old" communications questions such as channel capacity and quantization in the context of stabilization and control of linear systems (see [8] and [4] for a nice summary). Our work differs from these sets of work in that we propose a practical modularization motivated by [9] to integrate practical communication questions with coordination and control of nonlinear vehicles. An important feature of our work is that we relax the power constraint on the radios while minimizing energy consumption. Availability of power guarantees a suitably large channel capacity to communicate control variables. We believe that the obtained degree of modularity, despite introducing sub-optimality, can result in solutions that give insight into the problem. Following this philosophy, the work in this paper addresses the joint tasks of communication and coordinated control of an N -vehicle system where the component designs are simultaneously modularized and coupled via a common time discretization variable. Specifically, we assume that at regular finite intervals, control messages are to be generated and transmitted reliably over a network (potentially involving power/rate adaptations as well as multiple transmission over multiple hops, etc) in which the spatial location of th...

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Section: Introductionmentioning

confidence: 99%

“…In fact, the optimal solution to this coupled problem can be formulated as a decentralized stochastic problem with information constraints and imperfect observations, and the solution to such a problem is known not to be modular. Witsenhausen's counterexample shows, in fact, that separation of estimation and controller design fails to hold even in simple settings [1] [7] addressing "old" communications questions such as channel capacity and quantization in the context of stabilization and control of linear systems (see [8] and [4] for a nice summary). Our work differs from these sets of work in that we propose a practical modularization motivated by [9] to integrate practical communication questions with coordination and control of nonlinear vehicles.…”

Section: Index Termsmentioning

confidence: 99%

Integration of communication and control using discrete time Kuramoto models for multivehicle coordination over broadcast networks

Klein

Lee

Morgansen

et al. 2008

IEEE J. Select. Areas Commun.

View full text Add to dashboard Cite

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“…This is mainly due to the fact that the event-trigger and the controller have different information available, which prevents the direct application of dynamic programming. It is well known that such optimization problems with so-called distributed information patterns generally lack of systematic solution algorithms [23]. However, in case of the absence of packet dropouts and time-delays, it has been shown in [15] for finite horizon problems that the optimal solution exhibits structural properties that allow an efficient design by separating the minimization into a set of subproblems.…”

Section: Structural Propertiesmentioning

confidence: 99%

Suboptimal event‐triggered control for networked control systems

Molin

Hirche

2013

Z Angew Math Mech

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Key words Networked control systems, control over packet-switched communications, event-triggered control, stochastic optimal control, long-run average cost optimization, stochastic stability, drift criteria.The advent of networked control systems urges the digital control design to incorporate communication constraints efficiently. In order to accommodate this requirement, this article studies the joint design of controller and event-trigger for linear stochastic systems in the presence of a resource-limited communication channel which exhibits packet dropouts and time-delay. The event-trigger situated at the sensor decides at every sampling instance, whether to send information over the communication channel to the controller. The design approach is formulated as a stochastic average-cost optimization problem, where the communication constraints are reflected as an additional cost penalty of the average transmission rate. Different conditions on the communication model are given where the joint optimal design can be split into a separate control and event-trigger design. Based on these results, two suboptimal design approaches are developed. By using drift criteria, stability guarantees of the closed-loop system for both approaches are derived in terms of bounded moment stability. Numerical simulations illustrate the efficacy of the event-triggered approach compared with optimal time-triggered controllers.

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“…It turns out that such seemingly innocuous differences can significantly alter the resulting constructions of optimal controllers. (A classic illustration is the Witsenhausen example (Ho 1980;Witsenhausen 1968).) For example, under standard assumptions, optimal controllers are of the same dimension as the state dynamics.…”

Section: Effects Of Information Patternsmentioning

confidence: 99%