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Modeling and Analysis of Hybrid Systems Introduction Prof. Dr. - - PowerPoint PPT Presentation

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Modeling and Analysis of Hybrid Systems Introduction Prof. Dr. Erika brahm Informatik 2 - Theory of Hybrid Systems RWTH Aachen University SS 2013 brahm - Hybrid Systems 1 / 28 Organizational Lecture: Tuesday 13:15-14:15 in 5056

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Modeling and Analysis of Hybrid Systems

Introduction

  • Prof. Dr. Erika Ábrahám

Informatik 2 - Theory of Hybrid Systems RWTH Aachen University

SS 2013

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Organizational

Lecture: Tuesday 13:15-14:15 in 5056 Friday 13:15-14:30 in 5056 Exercise: Tuesday 14:15-15:00 in 5056 Exam dates will be chosen by Doodle vote: 1st: 26.07.2013 09:45-12:15 01.08.2013 13:45-16:15 02.08.2013 09:45-12:15 2nd: 18.09.2013 15:45-18:15 19.09.2013 10:45-13:15

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Learning materials and contact persons

Learning materials available in L2P: Slides Lecture notes Video recordings Some research publications Exercise sheets, solutions Lecture: Erika Ábrahám room: 2U07 (Hauptbau, basement), phone: 0241/80-21242 email: abraham@informatik.rwth-aachen.de Exercise: Xin Chen room: 2U08 (Hauptbau, basement), phone: 0241/80-21243 email: xin.chen@informatik.rwth-aachen.de Further information (topic, evaluations etc.):http: //www-i2.informatik.rwth-aachen.de/i2/hybrid_lecture/

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Contents

1 Hybrid systems 2 Modeling 3 Specification 4 Analysis

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Contents

1 Hybrid systems 2 Modeling 3 Specification 4 Analysis

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“Hybrid”

Wikipedia: “A hybrid is the combination of two or more different things, aimed at achieving a particular objective or goal.”

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A hybrid rose

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A hybrid car

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Hybrid in computer science

discrete continuous

t f(t)

+

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The discrete part

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Combined with the continuous part

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Example: Thermostat

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Example: Thermostat

Temperature x is controlled by switching a heater on and off x is regulated by a thermostat:

17◦≤ x ≤ 18◦ “heater on” 22◦≤ x ≤ 23◦ “heater off”

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Example: Thermostat

Temperature x is controlled by switching a heater on and off x is regulated by a thermostat:

17◦≤ x ≤ 18◦ “heater on” 22◦≤ x ≤ 23◦ “heater off”

Continuous: temperature Discrete: switching

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Example: Thermostat

Temperature x is controlled by switching a heater on and off x is regulated by a thermostat:

17◦≤ x ≤ 18◦ “heater on” 22◦≤ x ≤ 23◦ “heater off”

Continuous: temperature Discrete: switching t x

20 18 17 22 23

t

  • n
  • ff

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Contents

1 Hybrid systems 2 Modeling 3 Specification 4 Analysis

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Modeling

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Modeling

To be able to apply formal (mathematical) methods to a real system, we need a formal model of it. A model never exactly corresponds to the modeled real system. Abstract away unnecessary details.

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Modeling

To be able to apply formal (mathematical) methods to a real system, we need a formal model of it. A model never exactly corresponds to the modeled real system. Abstract away unnecessary details. What you probably already know: Kripke structures (state transition systems)

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Modeling

To be able to apply formal (mathematical) methods to a real system, we need a formal model of it. A model never exactly corresponds to the modeled real system. Abstract away unnecessary details. What you probably already know: Kripke structures (state transition systems) What you probably also know: Transition systems

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Modeling

To be able to apply formal (mathematical) methods to a real system, we need a formal model of it. A model never exactly corresponds to the modeled real system. Abstract away unnecessary details. What you probably already know: Kripke structures (state transition systems) What you probably also know: Transition systems What you perhaps know: Timed automata

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Example: Timed automaton

q1 x = 0 x ≥ 2, reset(x)

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Example: Timed automaton

q1 x = 0 x ≥ 2, reset(x) t x 2 3

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Example: Timed automaton

q2 x ≤ 3 x = 0 x ≥ 2, reset(x)

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Example: Timed automaton

q2 x ≤ 3 x = 0 x ≥ 2, reset(x) t x 2 3

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Modeling general hybrid systems: Hybrid automata

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Modeling general hybrid systems: Hybrid automata

Let’s take again the thermostat as an example.

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Modeling general hybrid systems: Hybrid automata

Let’s take again the thermostat as an example.

  • n

˙ x = −x + 50 x ≤ 23

  • ff

˙ x = −x x ≥ 17 22 ≤ x ≤ 23 17 ≤ x ≤ 18 x := 20

t x

20 18 17 22 23

t

  • n
  • ff

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Contents

1 Hybrid systems 2 Modeling 3 Specification 4 Analysis

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Logic

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Logic

We want to specify how a hybrid system is expected to behave.

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Logic

We want to specify how a hybrid system is expected to behave. We are especially interested in safety and liveness.

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Logic

We want to specify how a hybrid system is expected to behave. We are especially interested in safety and liveness. E.g., we can use the logic TCTL for specification.

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Logic

We want to specify how a hybrid system is expected to behave. We are especially interested in safety and liveness. E.g., we can use the logic TCTL for specification. In TCTL we can express properties like: “The temperature is always below 20◦C.”

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Logic

We want to specify how a hybrid system is expected to behave. We are especially interested in safety and liveness. E.g., we can use the logic TCTL for specification. In TCTL we can express properties like: “The temperature is always below 20◦C.” Or “If the temperature is above 20◦C it will get below 20◦C within 5 seconds.”

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Logic

We want to specify how a hybrid system is expected to behave. We are especially interested in safety and liveness. E.g., we can use the logic TCTL for specification. In TCTL we can express properties like: “The temperature is always below 20◦C.” Or “If the temperature is above 20◦C it will get below 20◦C within 5 seconds.” Or “It is always the case that the temperature will somewhen in the future get above 20◦C.”

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Contents

1 Hybrid systems 2 Modeling 3 Specification 4 Analysis

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The analysis of hybrid systems

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The analysis of hybrid systems

Assume we modeled a hybrid system as a hybrid automaton. Assume we specified a property of the system. Can we prove that the system satisfies the property?

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The analysis of hybrid systems

Assume we modeled a hybrid system as a hybrid automaton. Assume we specified a property of the system. Can we prove that the system satisfies the property? Well, it depends...

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The analysis of hybrid systems

Assume we modeled a hybrid system as a hybrid automaton. Assume we specified a property of the system. Can we prove that the system satisfies the property? Well, it depends... ...on the fact if the logic is decidable for the underlying modeling language.

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The analysis of hybrid systems

Assume we modeled a hybrid system as a hybrid automaton. Assume we specified a property of the system. Can we prove that the system satisfies the property? Well, it depends... ...on the fact if the logic is decidable for the underlying modeling language. We will see for which classes of hybrid automata the reachability question is decidable.

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The analysis of hybrid systems

Assume we modeled a hybrid system as a hybrid automaton. Assume we specified a property of the system. Can we prove that the system satisfies the property? Well, it depends... ...on the fact if the logic is decidable for the underlying modeling language. We will see for which classes of hybrid automata the reachability question is decidable. We will deal with

(unbounded) reachability for timed automata. (unbounded) reachability for initialized rectangular automata. bounded reachability for linear hybrid automata. reachability approximation for general hybrid automata.

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Method for timed automata: Finite abstraction

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Method for timed automata: Finite abstraction

Constructive proof of decidability via finite abstraction:

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Method for timed automata: Finite abstraction

Constructive proof of decidability via finite abstraction: x y

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Method for timed automata: Finite abstraction

Constructive proof of decidability via finite abstraction: x y 0 0 1 2 1 2 3 4

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Method for initialized rectangular automata: Transformation

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Method for initialized rectangular automata: Transformation

Leading back the proof of decidability to a known problem:

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Method for initialized rectangular automata: Transformation

Leading back the proof of decidability to a known problem: Timed automaton ↑ Initialized stopwatch automaton ↑ Initialized singular automaton ↑ Initialized rectangular automaton

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Method for linear hybrid automata: Fixedpoint computation

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Method for linear hybrid automata: Fixedpoint computation

Forward reachability computation:

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Method for linear hybrid automata: Fixedpoint computation

Forward reachability computation: Bad states Initial states

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Method for linear hybrid automata: Fixedpoint computation

Forward reachability computation: Bad states Initial states

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Method for linear hybrid automata: Fixedpoint computation

Forward reachability computation: Bad states Initial states

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Method for linear hybrid automata: Fixedpoint computation

Forward reachability computation: Bad states Initial states

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Method for linear hybrid automata: Fixedpoint computation

Forward reachability computation: Bad states Initial states Note: the method is incomplete

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Method for linear hybrid automata: Fixedpoint computation

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Method for linear hybrid automata: Fixedpoint computation

Backward reachability computation:

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Method for linear hybrid automata: Fixedpoint computation

Backward reachability computation: Initial states Bad states

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Method for linear hybrid automata: Fixedpoint computation

Backward reachability computation: Initial states Bad states

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Method for linear hybrid automata: Fixedpoint computation

Backward reachability computation: Initial states Bad states

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Method for linear hybrid automata: Fixedpoint computation

Backward reachability computation: Initial states Bad states

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Method for linear hybrid automata: Fixedpoint computation

Backward reachability computation: Initial states Bad states Note: also the backward method is incomplete

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Method for hybrid automata: Approximation

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Method for hybrid automata: Approximation

P P

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Polyhedra (left) and oriented rectangular hulls (right) in reachability computation

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Zonotopes in reachability computation

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