A Mechanism Design Approach for the Stabilization of Networked dynamical systems

1. 48th IEEE Conference on Decision and Control Shanghai, ChinaDecember 16-18, 2009 A Mechanism Design Approachfor the StabilizationofNetworkeddynamicalsystems L. Galbusera, N. Gatti, C. Romani Dipartimento di Elettronica e Informazione – Politecnico di Milanoe-mail: galbusera, ngatti, romani@elet.polimi.it

2. Networkedcontrol 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).

3. Networkedcontrol system (NCS) • 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 t* AUCTION FOR ACCESSING THE MEDIUM AT TIME t* PLAYERS REPORT THEIR(NOT-NECESSARILY TRUE)CURRENT STATES S1 S2 SN

4. Networkedcontrol system (NCS) t t* AUCTION FOR ACCESSING THE MEDIUM AT TIME t* Controlobjectives Stabilityof the NCS Efficientallocationof the communication medium Avoidingstrategicbehaviorsof the players S1 SN S2 PLAYERS ARESELF-INTERESTED THEY REPORT THEIR(NOT-NECESSARILY TRUE)CURRENT STATES

5. Preliminaries: stability in NCS Dynamical model of subsystem i: Control law: T Time: t S1 j-th time interval of lenght T S2

6. Preliminaries: stability in NCS Stability condition: Lower bound to control subintervals T Furtherassumption: Period T isdiscretized in M regular timeintervalsforexecuting the auctions. t 1 2 3 4 5 … M AUCTIONS

7. Groundings on mechanism design Reference: [Fudemberg & Tirole, Game theory, The MIT Press, 1991] MECHANISM (otherplayers) REPORTED EVALUATION of player iover the set ofalternatives PAYMENT of player i PLAYER i TRUE EVALUATIONof player iover the set ofalternatives if player ipays MONETARY RESOURCESof player i A player can participateto the auctiononlyif ALTERNATIVES(= possibleoutcomesof the game)

8. Groundings on mechanism design Reference: [Fudemberg & Tirole, Game theory, The MIT Press, 1991] PLAYER 1 WINNING ALTERNATIVE PLAYER 2 MECHANISM PLAYER N Maximizationof the social welfare

9. Groundings on mechanism design Reference: [Fudemberg & Tirole, Game theory, The MIT Press, 1991] PLAYER 1 DEFINITION OF PAYMENTS WINNING ALTERNATIVE PLAYER 2 MECHANISM PLAYER N Maximizationof the social welfare

10. Groundings on mechanism design • 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:

11. Groundings on mechanism design • 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 ruleCavallo’s pivot = Clarke’s pivot + redistribution mechanism> Truthfulness is preserved> Budget balancing is enhanced The second and thirdclassified in the bidincreasetheirresources

12. A mechanismfor NCS • Two-layer structure: • First layer efficient allocation of the medium • (with no stability guarantees); • Second layer for ensuring stability. PLAYER i • 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.

13. A mechanismfor NCS • 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 a priori information online information t Set of alternatives:

14. A mechanismfor NCS Evaluation function (common to both layers): if the subplant is choosen if the subplant is not choosen Depends on the state evolution of the closed-loop subsystem along the next time subinterval • 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 (truthfulness)

15. A mechanismfor NCS Social-efficiency based selection criterion: In view of truthfulness, the subplant i* with the highest evaluation value maximizes the social efficiency and is thus selected. Payment mechanism (related to standard resources) Cavallo’s redistributions Limited communication requirements:each player only sends its own evaluation

16. A mechanismfor NCS • 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

17. A mechanismfor NCS • 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) currentresources payment Cavallo’s redistribution Stability-preserving resources (csi) Eachtime the subplant is chosen, the resources are reduced by one unit until they reach zero.

18. A mechanismfor NCS • 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).

19. Simulationexample • 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. Simulationexample (B) Alternative solution (A) Proposed mechanism • Solution (A): • more marked alternation among subsystems in the scheduling; • penalization of the subsystem that “lies” (S1), in view of the resource-exhaustion phenomenon.

21. Simulationexample (B) Alternative solution (A) Proposed mechanism • Solution (A): • Better overall state performance.

22. Conclusions • 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.

23. Simulationexample (A) Proposed mechanism Ordinary tokens Stability tokens Stability tokens zeroed before the end of the period. Switching to the second layer does not occur in this example.