A Mechanism Design Approach for the Stabilization of Networked dynamical systems

A Mechanism Design Approachfor the Stabilization of

Networked dynamical systems

L. Galbusera, N. Gatti, C. Romani

Dipartimento di Elettronica e Informazione – Politecnico di Milanoe-mail: galbusera, ngatti, [email protected]

48th IEEE Conference on Decision and Control48th IEEE Conference on Decision and Control

Shanghai, ChinaDecember 16-18, 2009

2Networked control system (NCS)Networked control system (NCS)

Elements:

N linear continuous-time subplants with unstable uncontrolled dynamics.

A bus communication medium.

N controllersdesigned to stabilize each subplant.

Standing assumption: at each time instant, only one subplant is connected to its

controller

Control objective:

Synthesis of an effective dynamic scheduling policy (not preassigned).


Previous literature on dynamic scheduling policies:

the scheduling is usually assigned in a centralized manner by comparing systems’ states and parameters(e.g., the CLS-ε policy in [Hristu-Varsakelis, CDC 2001]).

Real-world applications:

the subplants can be modeled as strategic players in a game for having access to the communication medium.

t

AUCTION FOR ACCESSING THE

MEDIUM AT TIME t*

S1 S2 SN

t*

PLAYERS REPORT THEIR(NOT-NECESSARILY

TRUE)CURRENT STATES


t

AUCTION FOR ACCESSING THE

MEDIUM AT TIME t*

S1

S2

SN

t*

Control objectivesControl objectives

1.Stability of the NCS

2.Efficient allocation of the communication medium

3.Avoiding strategic behaviors of the players

PLAYERS ARESELF-INTERESTED

THEY REPORT THEIR(NOT-NECESSARILY

TRUE)CURRENT STATES

5Preliminaries: stability in NCSPreliminaries: stability in NCS

t

Dynamical model of subsystem i:

Control law:

Time: T

S1

S2

j-th time interval of lenght T

6Preliminaries: stability in NCSPreliminaries: stability in NCS

Stability condition:

Further assumption:

Period T is discretized in M regular time intervals for executing the auctions.

Lower bound to control subintervals

tT

1 2 3 4 5 … M

AUCTIONS

7

if player i pays

Groundings on mechanism designGroundings on mechanism design

ALTERNATIVES(= possible outcomes of the game)

PLAYER i

MECHANISM

TRUE EVALUATIONof player i over the set of alternatives

REPORTED EVALUATION of player i over the set of alternatives

(other players)

PAYMENT of player i

MONETARY RESOURCES

of player i

A player can participate to the auction only if

Reference: [Fudemberg & Tirole, Game theory, The MIT Press, 1991]

8Groundings on mechanism designGroundings on mechanism design

WINNING ALTERNATIVE

PLAYER 1

MECHANISM

PLAYER N

PLAYER 2

Maximization of the social welfare



WINNING ALTERNATIVE

PLAYER 1

MECHANISM

PLAYER N

PLAYER 2

Maximization of the social welfare

DEFINITION OF PAYMENTS



Key features:

Player i ’s utility:

Truthful mechanism: a mechanism in which each player cannot increase its utility by misreporting its true evaluation, i.e., a mechanism in which

VCG mechanisms (Vickrey, Clarke and Grove):a class of mechanisms which is guaranteed to be truthful by means of a suitable definition of the payment function:


Key features:

Clarke’s pivot rule for specifying the payment:

the winner’s payment equals the second-highest bid

VCG mechanisms are weakly budget-balanced, i.e.,

Therefore, the iterated application of the mechanism (non-strictly) decreases the players’ resources.

A solution: Cavallo’s pivot rule

Cavallo’s pivot = Clarke’s pivot + redistribution mechanism

> Truthfulness is preserved> Budget balancing is enhanced The second and third classified in

the bid increase their resources

12A mechanism for NCSA mechanism for NCS

Two-layer structure:

First layer efficient allocation of the medium

(with no stability guarantees);

Second layer for ensuring stability.

The allocation procedure is governed by two sets of monetary sources:

Standard resources (ci)used at the first layer, in order to allocate the medium;

Stability-preserving resources (csi)used at the second layer, in order to preserve stability.

PLAYER i


What does the mechanism need to know in order to work?

The quantities

The period T

The standard resources andstability-preserving resources of the players

The true value of the state of eachsubsystem (=player) at the beginning of each period

Set of alternatives:

t

a priori information

online information


Evaluation function (common to both layers):

if the subplant is choosenif the subplant is not choosen

Remarks:

Subplant i has a positive evaluation only if it is chosen to be controlled.

The monetary resources do not directly affect the value of the evaluation function, but only enable the participation of the subplants to the mechanism.

VCG mechanism (truthfulnes

s)

Depends on the state evolution of the closed-loop subsystem along the next time subinterval


Social-efficiency based selection criterion:

Payment mechanism (related to standard resources)

In view of truthfulness, the subplant i* with the highest evaluation value maximizes the social efficiency and is thus selected.

Cavallo’s redistributions

Limited communication requirements:each player only sends its own evaluation


Initialization of monetary resources

At the beginning of each period of length T, ci and csi are initialized as follows:

Standard resources (ci)depend on the state at the beginning of the same period

Stability-preserving resources (csi) equal the minimum number of subintervals subsystem i needs to be controlled in order to preserve stability


Update rules for monetary resources

Both standard resources and stability-preserving resources are updated at each execution of the mechanism during the period ( ).

Standard resources (ci)

Stability-preserving resources (csi)

Each time the subplant is chosen, the resources are reduced by one unit until they reach zero.

current resources payment Cavallo’s redistribution


Mechanism design switching rule

IDEAallocation based on standard resources (efficiency-based) until the stabilization requirement becomes critical.

At each time step both resources are updated;

Standard resources are used for the bid until the number of remaining time step before the end of the period are just enough to complete the stabilization of all subsystems (i.e., zeroing the stability-preserving resources).

19Simulation exampleSimulation example

Three first-order unstable linear subplants, each of them associated with a controller that stabilizes the respective subplant.Open- and closed-loop eigenvalues:

A comparison between different allocation methods over a time period T:

(A) The proposed mechanism-based allocation method;

(B) A modified allocation method obtained by removing induced payments and standard monetary resources.

In order to emphasize the difference in the resulting control action, we assume that subplant S1 reports the following altered evaluation function value:

uncontrolled plants

controlled plants

20

Solution (A): more marked alternation among subsystems in the

scheduling;

penalization of the subsystem that “lies” (S1), in view of the resource-exhaustion phenomenon.

Simulation exampleSimulation example

(A) Proposed mechanism (B) Alternative solution


Solution (A): Better overall state performance.

(A) Proposed mechanism (B) Alternative solution

22ConclusionsConclusions

Main features:

Application of mechanism design to the stabilization issue of networked control systems;

synthesis of a dynamic scheduling policy in a game-theoretical setting;

our scheme avoids strategic behaviors of the players and efficiently allocates the communication;

the mechanism needs limited information to properly work.


Stability tokens zeroed before the end of the period.

Switching to the second layer does not occur in this example.

Stability tokens Ordinary tokens

(A) Proposed mechanism

A Mechanism Design Approach for the Stabilization of Networked dynamical systems

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