On Generating Safe Controllers for DiscreteTime Linear Systems

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On Generating Safe Controllers for Discrete-Time
Linear Systems

• By Adam Cataldo

EE 290N Project UC Berkeley December 10, 2004
unsafe state
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Talk Outline

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

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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

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Transition SystemsA Concurrent Model of
Computation

• The set of tags is T 0, 1, 2,

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Behavior

• Initialized runs

• Language (Behavior)

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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

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Composing Transition Systems
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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)

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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

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A Discrete-Time, Real-ValuedConcurrent Model of
Computation

• This is actually a special class of
  discrete-time, real-valued systems (LTI)

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Feedback Composition

• Feedback composition holds if (I BH) and (I
  FD) are invertible

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Feedback Composition

• Equivalent system

• We can start with initial values to compute
  fixed-point behavior

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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

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Discrete-Time Systemsas Transition Systems

• We will be interested in the case where V is
  finite

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