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Control of Dynamic Discrete-Event Systems. Lenko Grigorov Master’s Thesis, QU supervisor: Dr. Karen Rudie. Discrete-Event Systems (background). Discrete-Event Systems are systems where events (changes of state) occur: spontaneously logically ordered relative to each other
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Control of DynamicDiscrete-Event Systems Lenko Grigorov Master’s Thesis, QU supervisor: Dr. Karen Rudie
Discrete-Event Systems(background) • Discrete-Event Systems are systems where events (changes of state) occur: • spontaneously • logically ordered relative to each other • not tied to a continuous global time • Common representation of DESs: • Finite-state machines • G=(,Q,,q0,Qf) (Cassandras and Lafortune, Introduction to Discrete Event Systems, 1999) Lenko Grigorov, QU
Motivation • Control of a specific class of DESs • dynamic (change with time) • relatively large • with continuous lifecycle • with requirements with different levels of stringency • Such systems are common in real life • Classical DES control methods are not suitable Lenko Grigorov, QU
Outline • Definition of Dynamic Discrete-Event Systems • Redundancy for Modular Architecture • Optimal DDES control • Experiment Lenko Grigorov, QU
DES Modular Architecture(background) • Separate small DES modules • Combined using a synchronous composition • an event can happen in a module it can happen in the system • if modules have common events, these events happen simultaneously (Cassandras and Lafortune, Introduction to Discrete Event Systems, 1999) Lenko Grigorov, QU
Dynamic DES Model • Time • discrete • increases by one after every event • Sets of modules • Mi = {M1i, M2i, …, Mni}, i{0, 1, 2, …} • ||Mi = M1i || M2i || … || Mni • DDES G={(||Mi, i) | i{0, 1, 2, …}) • at time i, Gi = ||Mi • No restrictions on the sets Mi Lenko Grigorov, QU
Redundancy forModular Architecture • Mi Mi+1 • thus some part of ||Mi may be reused to compute ||Mi+1 • Given operation • commutative • associative • How to compute A = A1 A2 … An so that recomputing A after a structure change is least expensive? • redundant storage of intermediate results Lenko Grigorov, QU
Redundancy Structures • Stack structure – simple • use when: the result of the operation does not increase exponentially, older modules are stable • disadvantages: large size when used with synchronous composition • Tree structure – robust to random changes • use when: the oldest elements have highest chance to change • Hybrid structure – small footprint • use when: there is small storage space • disadvantages: may demand more computations, while savings in space are insignificant Lenko Grigorov, QU
Complexity ofRedundancy Structures Lenko Grigorov, QU
Standard Online Control(background) • Construct a limited-depth tree of the possible future behavior of the system • For each node, determine if the string leading to it is acceptable • Propagate the information back to the root • Disable events leading to “unsafe” states • where we cannot prevent the execution of an unacceptable string • Repeat this after each execution of an event (Chung, Lafortune, and Lin, “Limited lookahead policies in supervisory control of discrete event systems”, 1992) Lenko Grigorov, QU
Valuation of Event Strings • Two functions are defined by the user • The value function gives the “value” of event strings, according to some criteria • v(s) R, s L(G) • greater v(s) string is more desirable • v(s) = - string is unacceptable • The goal function indicates which strings accomplish a task • g(s) {0,1}, s L(G) • no need to investigate the look-ahead tree further • similar to final (marked) states, but works on strings Lenko Grigorov, QU
Optimal DDES Control Algorithm • Online control using the value and goal functions • exploration of a branch in the tree is carried until • a goal is generated, • an unacceptable string is generated, or • the depth limit is reached • the value function is used to obtain the benefit of the different paths (event strings) • the controller selects the path which may yield the greatest benefit Lenko Grigorov, QU
Advantages (1) • Attempts to guide the system to the maximal benefit for the user • quality depends on the way the system evolves • The use of the value function renders the control process more robust to failures • it does the best possible with the available resources Lenko Grigorov, QU
Advantages (2) • The algorithm adapts automatically to different types of dynamics in the system • structural changes • the constituent modules change • changes of goals • changes in the requirements for the system behavior • changes in the evaluation of events • events have varying costs • depending on the event string • depending on time Lenko Grigorov, QU
Advantages (3) • Requirements on the system behavior can have many levels of stringency • not only acceptable/unacceptable • The method does not need access to the complete system model • can work with large systems • The algorithm can be implemented as modular software Lenko Grigorov, QU
Issues inOptimal DDES Control • The control may not be optimal • if the tree depth is too limited • the controller cannot observe far enough along the event strings to compute relevant costs and payoffs • if the tree depth is too big • the controller bases its decisions on the current system, while the system may change in the future • The complexity of the algorithm is affected by the particular value and goal functions used • may be significantly greater than the complexity of standard online control • O(kNv(s)g(s)) Lenko Grigorov, QU
Experiment • Different number of trains enter or leave a system of railroads • A set of requirements on the system behavior • no trains can be at the same section of a track, etc. • Comparison between optimal DDES control and simple online control (Experiment based on: Chung, Lafortune, and Lin,“Supervisory control using variablelookaheadpolicies”, 1994) Lenko Grigorov, QU
Experiment Results • The overall system behavior is much closer to the requirements • strings have higher value • more trains arrive at train stations per unit time • more balanced use of the resources • Disadvantages • significant increase in the time to make a decision • can work only with a much smaller tree-depth Lenko Grigorov, QU
Conclusion and Contributions • The use of redundancy structures can reduce the number of computations needed to rebuild the model of a system after it changes • different types of redundancy structures available for different applications • can be used in other areas • not limited to the synchronous composition of modules Lenko Grigorov, QU
Conclusion and Contributions • The proposed control method can be used successfully to supervise dynamic discrete-event systems • achieves near-optimal control • adapts automatically to dynamics in the system • allows a very flexible definition of requirements • is more robust to system failures • is easily implementable as modular software Lenko Grigorov, QU
Selected References • C. G. Cassandras and S. Lafortune. Introduction to Discrete Event Systems. Kluwer Academic Publishers, Norwell, Massachusetts, USA, 1999, • Sheng-Luen Chung, Stéphane Lafortune, and Feng Lin. Limited lookahead policies in supervisory control of discrete event systems. IEEE Transactions on Automatic Control, 37(12):1921–1935, 1992, • Sheng-Luen Chung, Stéphane Lafortune, and Feng Lin. Supervisory controlusing variable lookahead policies. Discrete Event Dynamic Systems: Theory andApplications, 4:237–268, 1994. Lenko Grigorov, QU