ESE601: Hybrid Systems

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ESE601 Hybrid Systems

• Some tools for verification

Spring 2006
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Software tools for verification

• See the Hybrid Systems wiki at GRASP
• http//wiki.grasp.upenn.edu/graspdoc/hst/index.ph
  p?nMain.HomePage
• Today we are going to discuss
• MATISSE for reachability of constrained linear
  systems.
• SOSTOOLS for computation of barrier certificate
  for nonlinear systems
• UPPAAL for verification of timed automata

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MATISSE

• MATISSE is a MATLAB toolbox.
• Developed by Antoine Girard and George Pappas at
  UPenn.
• Main purpose is to compute abstraction/reduction
  of constrained linear systems, based on
  approximate bisimulation. will be discussed
  later
• Contains a functionality to compute the reachable
  set of a constrained linear system.

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Constrained Linear Systems

• Constraints and reachable set are expressed as
  zonotopes.
• Constrained linear systems are systems of the
  form
• The set I and U are zonotopes.

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What is a zonotope?

• Zonotope Minkowski sum of a finite number of
  segments.
• c is the center of the zonotope, g1,,gp are
  the generators. The ratio p/n is the order of the
  zonotope.

Two dimensional zonotope with 3 generators
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Some Properties of Zonotopes

• The encoding of a zonotope has a polynomial
  complexity with the dimension.
• The set of zonotopes is closed under linear
  transformation
• The set of zonotopes is closed under the
  Minkowski sum

• Exactly what we need for our reachability
  algorithm

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Usage

• A constrained linear system (CLS) is defined as a
  5-tuple, (A,B,C,U,I).
• Example

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Reachable set

• Reachable set is computed using the function
  reach_set. The function returns two arrays of
  zonotopes.
• S is a CLS, dt is the time step, N is the number
  of intervals. The end time of the reachability
  algorithm is thus N x dt.
• Then, a 2-dimensional cross-section of the
  reachability set can be plotted using

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Plotting the reachable set

• Plotting the reachability set in 2D
• P is a 2xm matrix that defines the projection
  from output space to .
• The color of the plot is defined by the last
  option. In this case b means blue, r means
  red, etc

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Example
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SOSTOOLS for barrier certificate

• SOSTOOLS is a MATLAB toolbox for formulating and
  solving sums of squares (SOS) optimization
  programs.
• It is developed by a group from Caltech. Website
  
• http//www.cds.caltech.edu/sostools
• The problems are solved using Sedumi or SDPT3,
  both well-known semidefinite programming solver,
  with SOSTOOLS handling internally all the
  necessary reformulations and data conversion.

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Sum of squares
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Gram matrix representation
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Sum of squares program (SOSP)
The feasible set of solutions is convex.
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Basic steps
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Usage

• Polynomials can be declared symbolically
• A SOSP is initialized using the function
  sosprogram.
• Declaring scalar decision variables

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Usage

• Declaring polynomial variables.
• Declaring SOS polynomial variables is done in
  terms of the constructing vector of monomials.

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Usage

• Adding equality constraints.
• will add the following equality constraint to
  the program
• Adding inequality constraints.
• will add the following inequality constraint to
  the program

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Usage

• The solver is called using sossolve function.
• The output contains
• The solution is then obtained by the function
  sosgetsol.

SOSP variable
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Safety verification
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Safety verification
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Safety verification
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UPPAAL

• UPPAAL is a software tool for modeling and
  verification of timed automata.
• UPPAAL is developed by a group of researchers
  from Uppsala (Sweden) and Aalborg (Denmark).
• It has a graphical interface.
• A complex system can be modeled as a network of
  hybrid automata, sharing some global variables
  (including clocks), and synchronizing with a
  handshake.

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UPPAAL

• Two automata can execute a transition labeled
  lab if the guards are satisfied at both
  automata, and in one automaton the transition is
  label as lab? and the other lab!.
• Verification is done by verifying temporal logic
  formulas. When a formula is invalid, a
  counterexample is provided.

lab?
synchronize
lab!