Control of Dynamic Discrete-Event Systems Lenko Grigorov Master’s Thesis, QU supervisor: Dr. Karen Rudie

Dec. 2003Lenko Grigorov, QU2 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, ,q 0,Q f ) (Cassandras and Lafortune, Introduction to Discrete Event Systems, 1999)

Dec. 2003Lenko Grigorov, QU3 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

Dec. 2003Lenko Grigorov, QU4 Outline 1. Definition of Dynamic Discrete-Event Systems 2. Redundancy for Modular Architecture 3. Optimal DDES control 4. Experiment

Dec. 2003Lenko Grigorov, QU5 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)

Dec. 2003Lenko Grigorov, QU6 Dynamic DES Model Time discrete increases by one after every event Sets of modules M i = {M 1i, M 2i, …, M ni }, i  {0, 1, 2, …} ||M i = M 1i || M 2i || … || M ni DDES G={(||M i, i) | i  {0, 1, 2, …}) at time i, G i = ||M i No restrictions on the sets M i

Dec. 2003Lenko Grigorov, QU7 Redundancy for Modular Architecture M i  M i+1   thus some part of ||M i may be reused to compute ||M i+1 Given operation  commutative associative How to compute A = A 1  A 2  …  A n so that recomputing A after a structure change is least expensive? redundant storage of intermediate results

Dec. 2003Lenko Grigorov, QU8 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

Dec. 2003Lenko Grigorov, QU9 Complexity of Redundancy Structures

Dec. 2003Lenko Grigorov, QU10 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)

Dec. 2003Lenko Grigorov, QU11 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

Dec. 2003Lenko Grigorov, QU12 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

Dec. 2003Lenko Grigorov, QU13 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

Dec. 2003Lenko Grigorov, QU14 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

Dec. 2003Lenko Grigorov, QU15 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

Dec. 2003Lenko Grigorov, QU16 Issues in Optimal 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(k N v(s)g(s))

Dec. 2003Lenko Grigorov, QU17 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 variable lookahead policies”, 1994)

Dec. 2003Lenko Grigorov, QU18 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

Dec. 2003Lenko Grigorov, QU19 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

Dec. 2003Lenko Grigorov, QU20 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

Dec. 2003Lenko Grigorov, QU21 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 control using variable lookahead policies. Discrete Event Dynamic Systems: Theory and Applications, 4:237–268, 1994.