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On Generating Safe Controllers for Discrete-Time Linear Systems

EE 290N Project UC Berkeley December 10, 2004. On Generating Safe Controllers for Discrete-Time Linear Systems. By Adam Cataldo. unsafe state. disable this transition. Talk Outline. Research Question Background Transition Systems Discrete-Time Systems

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On Generating Safe Controllers for Discrete-Time Linear Systems

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  1. EE 290N Project UC Berkeley December 10, 2004 On Generating Safe Controllers for Discrete-Time Linear Systems By Adam Cataldo unsafe state disable this transition

  2. Talk Outline • Research Question • Background • Transition Systems • Discrete-Time Systems • Relation Between Models of Computation • Future Directions/Conclusions

  3. The Question • For what discrete-time linear systems can I compute a controller which will guarantee a safety constraint? • Safety constraint specified as a linear temporal logic constraint over the state space • I must have a method to compute the desired controller or know that no such controller exits

  4. Transition Systems:A Concurrent Model of Computation • The set of tags is T = {0, 1, 2, …}

  5. Behavior • Initialized runs: • Language (Behavior):

  6. Fixed-Point Computation of the Language • Computing the set of all initialized runs: • F is monotonic and • Knowing the set of all initialized runs gives us the language

  7. Composing Transition Systems

  8. Simulation • If there are simulation relations from P2 to P1 and P1 to P2, then P1 and P2 are bisimilar and L(P1) = L(P2)

  9. Linear Temporal Logic • Given a set of predicates P over the set of values, we are interested in enforcing certain time-dependent safety properties • Example: w always satisfies predicate p • We can use linear temporal logic express these properties • When we have finite number of states, we can compute a “controller” whose composition with our system enforces these constraints

  10. A Discrete-Time, Real-ValuedConcurrent Model of Computation • This is actually a special class of discrete-time, real-valued systems (LTI)

  11. Feedback Composition • Feedback composition holds if (I – BH) and (I – FD) are invertible

  12. Feedback Composition • Equivalent system: • We can start with initial values to compute fixed-point behavior

  13. Another Feedback Composition • The following feedback system also makes a valid composition: • Our problem is to design f to make x satisfy a safety property

  14. Discrete-Time Systemsas Transition Systems • We will be interested in the case where V is finite

  15. A Nice Result (Tabuada, Pappas) V is a finite partition of W

  16. A Nice Result (Tabuada, Pappas) • There exists a bisimilar transition system to P with a finite number of states • We can compute c by first computing a controller for the finite-state system

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