CS 367: Model-Based Reasoning Lecture 11 (02/19/2002)

1. CS 367: Model-Based ReasoningLecture 11 (02/19/2002) Gautam Biswas

2. Today’s Lecture • Today’s Lecture: • Finish up Supervisory Control • Onto Modeling of Continuous Systems: The Bond Graph Approach

3. Supervisory Controller: Examples • Admissible strings: a1 precedes a2 iff b1 precedes b2 • Build trim automata Ha such that Lm(Ha) contains only those strings that contain the above ordering constraints • Is Ha blocking? • In general, how do we build supervisors? If all events controllable and observable:

4. Realizing Supervisors • How to build an automaton that realizes S? • Build an automaton that marks K, i.e., Note that R has the same event set as G, therefore, Control action S(s) is encoded into transition structure of R

5. Standard Realization of S • Start with G in state x, R in state y, following theexecution of • G generates that is currently enabled, i.e., this event set is present in R’s active event set at y • R executes the event as a passive observer of G and the system now moves into states x’ and y’ • Set of enabled events of G given by active event set of R at y’

6. Induced Supervisor • Reverse Question: Given C, can the product CG imply that C is controlling G • Depends on the controllability of L(C) • The supervisor for G induced by C is

7. Reduced State Realization L(S/G) = K may not be the most economical way to represent S in terms of an automata (memory requirements) Relax requirements L(R) = K, and Come up with Collapse 2,5,6,7, and 8 into one state

8. Controllable sub languages and super languages of an uncontrollable language • K is not controllable wrt M and Euc • Two languages derived from K: • The supremal controllable sub language K: (Inside K) • The infimal prefix-closed and controllable super language of K: (Outside K)

9. Example: Supremally Controllable Language

10. Infimal Prefix-closed controllable language

11. Supervisory Control Problems • BSCP: Basic Supervisory Control Problem • Given G with event set E, and Euc E, and an admissible language La = La  L(G) find supervisor such that Look up standard realization presented couple of lectures ago (sec. 3.4.2) • DuSCP: Dual Version of SCP:minimum required languageLr L(G)

12. Supervisory Controller Problems • SCPT: Supervisory Controller with Tolerance • Ldes: desired language, try and achieve as much of it as possible • Ltot: tolerated language, do not exceed tolerated langauge • Solution:

13. Non Blocking Supervisors • Controllable: • Non blocking: Lm(G) closure: typically holds by construction of K • Supervisory Controller with Blocking Typically use two measures: Blocking Measure: Satisficing Measure: BM(S) and SM(S) conflicting, i.e., reducing one may increase the other

14. Modular Control • Supervisor S combines the actions of two or more supervisors, e.g., S1 and S2 We can always build R = R1 R2, but the point is to use R1 and R2 and take the active event sets of both at their respective states after execution of s

15. Modular Control Example: Dining Philosophers • Philosopher i picks up for j is controllable • Philosopher putting down fork is uncontrollable • Remember there is only one marked state • Design two supervisors: one for each fork (1T, 2f

16. Modular Control Example: Dining Philosophers • Modular supervisor Smod12 = R1R2G

17. Did not cover • Unobservability • Decentralized Control

18. Modeling of Continuous Dynamic Systems The Bond Graph

19. Bond Graph Methodology • From Systems Dynamics • formal and systematic method for modeling physical systems • forces one to make explicit: issues about system functionality and behavior assumptions • unlike other modeling schemes… • directly grounded in physical reality… • 1-1 correspondence with components and mechanisms of the physical system modeled… • (as opposed to formal languages, such as logic)

20. Bond Graphs… Modeling Language (Ref: physical systems dynamics – Rosenberg and Karnopp, 1983) NOTE: The Modeling Language is domain independent… Bond Connection to enable Energy Transfer among components (directed bond from A to B). each bond: two associated variables effort, e flow, f e f B A

21. Bond Graphs • modeling language (based on small number of primitives) • dissipative elements: R • energy storage elements: C, I • source elements: Se, Sf • Junctions: 0, 1 • physical systemmechanisms • R C, I Se, Sf 0,1 • forces you to make assumptions • explicit uniform network – like representation: domain indep.

22. Generic Variables: Signals effort, e elec. mechanical flow, f voltageforce currentvelocity NOTE: power = effort × flow. energy = (power) dt. state/behavior of system: energy transfer between components… rate of energy transfer = power flow Energy Varibles momentum, p= e dt : flux, momentum displacement, q =  f dt : charge, displacement

23. Examples: Effort Flow Power Energy Mechanics Force, F Velocity, V FxV F. V. Electricity Voltage, V Current, I VxI VI Hydraulic Pressure, P Volume PxQ PQ (Acoustic) flow rate (Q) Thermo- Temperature, Entropy Q Q dynamics T flow rate (thermal flow rate) Pseudo