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On Generating Safe Controllers for Discrete-Time Linear Systems

This project explores generating safe controllers for discrete-time linear systems, focusing on safety constraints specified using linear temporal logic. It investigates transition systems, feedback composition, and linear temporal logic enforcement of safety properties. The research aims to identify systems for which a safe controller can be computed. Future directions and conclusions are discussed.

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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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