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

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

Cataldo 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

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

Cataldo 5 Behavior Initialized runs: Language (Behavior):

Cataldo 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

Cataldo 7 Composing Transition Systems

Cataldo 8 Simulation If there are simulation relations from P 2 to P 1 and P 1 to P 2, then P 1 and P 2 are bisimilar and L(P 1 ) = L(P 2 )

Cataldo 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

A Discrete-Time, Real-Valued Concurrent Model of Computation This is actually a special class of discrete- time, real-valued systems (LTI)

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

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

Cataldo 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

Cataldo 14 Discrete-Time Systems as Transition Systems We will be interested in the case where V is finite

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

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