1. CS 367: Model-Based Reasoning Lecture 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: a 1 precedes a 2 iff b 1 precedes b 2 Build trim automata H a such that L m (H a ) contains only those strings that contain the above ordering constraints Is H a 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 the execution 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 S 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 E uc 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 E uc  E, and an admissible language L a = L a  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 language L r  L(G)

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

13. Non Blocking Supervisors Controllable: Non blocking: L m (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., S 1 and S 2 We can always build R = R 1  R 2, but the point is to use R 1 and R 2 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 2f (1T,

16. Modular Control Example: Dining Philosophers Modular supervisor S mod12 = 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 A B efef

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

22. Generic Variables: Signalseffort, eelec.mechanical flow, fvoltageforce 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. EffortFlowPowerEnergy MechanicsForce, FVelocity, VFxV  F. V. ElectricityVoltage, VCurrent, IVxI  VI HydraulicPressure, P VolumePxQ  PQ (Acoustic)flow rate (Q) Thermo- Temperature, EntropyQ  Q dynamics Tflow rate (thermal flow rate) Pseudo Examples: