Discrete abstractions of continuous control systems - Google Sites

Discrete abstractions of continuous control systems Giordano Pola Center of Excellence for Research DEWS Department of Electrical and Information Engineering University of L’Aquila

G. Pola ( DEWS - UNIVAQ )

Discrete abstractions

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Correct-by-Design Controller Synthesis

A three step approach: Computation of a symbolic abstraction of the physical system Discrete controller synthesis for the symbolic abstraction Hybrid controller synthesis via discrete controller refinement



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Equivalence notions Several notions of equivalence have been introduced in computer science The work R.J. van Glabbeek The linear time-branching time spectrum CONCUR’90, 1990 enumerated a dozen of equivalence notions

A central notion in the construction of discrete abstractions is the one of bisimulation equivalence G. Pola ( DEWS - UNIVAQ )


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Equivalence notions Example Consider the following finite system:

x2 u1

b

u3 x3 c

x1 a u2

x4

u1

b



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Equivalence notions Example Consider the following finite system: Sa = (X , X0 , U,

X = {x1 , x2 , x3 , x4 }

x2 u1

b

- , Y , H)

u3 x3 c

x1 a u2

x4

u1

b



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X = {x1 , x2 , x3 , x4 }

x2 u1

b

- , Y , H)

X0 = {x1 } u3 x3 c

x1 a u2

x4

u1

b



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X = {x1 , x2 , x3 , x4 }

x2 u1

b

- , Y , H)

X0 = {x1 } u3

U = {u1 , u2 , u3 } x3 c

x1 a u2

x4

u1

b



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X = {x1 , x2 , x3 , x4 }

x2 u1

b

X0 = {x1 } u3

U = {u1 , u2 , u3 } x3 c

x1 a u2

x4

- , Y , H)

1 - = {x1 ux2 , u3 u2 x2 x3 , x1 x4 , x4

u1

- x3 }

u1

b



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X = {x1 , x2 , x3 , x4 }

x2 u1

b

X0 = {x1 } u3

U = {u1 , u2 , u3 } x3 c

x1 a u2

x4

- , Y , H)

u1

1 - = {x1 ux2 , u3 u2 x2 x3 , x1 x4 , x4

u1

- x3 }

Y = {a, b, c}

b



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X = {x1 , x2 , x3 , x4 }

x2 u1

b

X0 = {x1 } u3

U = {u1 , u2 , u3 } x3 c

x1 a u2

x4 b


- , Y , H)

u1

1 - = {x1 ux2 , u3 u2 x2 x3 , x1 x4 , x4

u1

- x3 }

Y = {a, b, c} H(x1 ) = a, H(x2 ) = b, H(x3 ) = b, H(x4 ) = c


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X = {x1 , x2 , x3 , x4 }

x2 u1

b

X0 = {x1 } u3

U = {u1 , u2 , u3 } 1 - = {x1 ux2 , u3 u2 x2 x3 , x1 x4 , x4

x3 c

x1 a u2

x4

- , Y , H)

u1

- x3 }

Y = {a, b, c}

u1

H(x1 ) = a, H(x2 ) = b, H(x3 ) = b, H(x4 ) = c

b

System Sa is "equivalent" to system Sb described by: x5 a G. Pola ( DEWS - UNIVAQ )

u1

x6 b


u3

x7 c 4 / 59

Equivalence notions Definition (Bisimulation Relation) Consider systems Sa and Sb with Ya = Yb . A relation R ⊆ Xa × Xb is a bisimulation relation between Sa and Sb if the following three conditions are satisfied:

a) for every xa0 ∈ Xa0 , there exists xb0 ∈ Xb0 with (xa0 , xb0 ) ∈ R; b) for every xb0 ∈ Xb0 , there exists xa0 ∈ Xa0 with (xa0 , xb0 ) ∈ R; for every (xa , xb ) ∈ R we have Ha (xa ) = Hb (xb ); for every (xa , xb ) ∈ R we have that:

b)

ua

- xa0 in a (xa0 , xb0 ) ∈ R. ub xb - xb0 in b (xa0 , xb0 ) ∈ R.

a) xa


Sa implies the existence of xb Sb implies the existence of xa


ub

- x 0 in Sb satisfying b

b

ua

- xa0 in Sa satisfying

a

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b)

ua


a) xa


ub


b

ua


a

We say that Sa and Sb are bisimilar, denoted by Sa ∼ =S Sb , if there exists a bisimulation relation between Sa and Sb .



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b)

ua


a) xa


ub


b

ua


a

We say that Sa and Sb are bisimilar, denoted by Sa ∼ =S Sb , if there exists a bisimulation relation between Sa and Sb . Bisimulation equivalence is an equivalence relation on the class of systems G. Pola ( DEWS - UNIVAQ )


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Equivalence notions Example

x1 a

u1

x2

x3 c

u3

b u2

u1

x5 a

u1

x6 b

u3

x7 c

x4 b



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x1 a

x2

u1

x3 c

u3

b u2

u1

x5 a

u1

x6 b

u3

x7 c

x4 b Systems Sa and Sb are bisimilar with bisimulation relation: R


=


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x1 a

x2

u1

x3 c

u3

b u2

u1

x5 a

u1

x6 b

u3

x7 c



=

{(x1 , x5 )


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x1 a

x2

u1

x3 c

u3

b u2

u1

x5 a

u1

x6 b

u3

x7 c



=

{(x1 , x5 ), (x2 , x6 )


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x1 a

x2

u1

x3 c

u3

b u2

u1

x5 a

u1

x6 b

u3

x7 c



=

{(x1 , x5 ), (x2 , x6 ), (x3 , x7 )


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x1 a

x2

u1

x3 c

u3

b u2

u1

x5 a

u1

x6 b

u3

x7 c



=

{(x1 , x5 ), (x2 , x6 ), (x3 , x7 ), (x4 , x6 )}


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Equivalence notions

Bisimulation is a popular tool for complexity reduction of finite systems Consider a system S and a copy of itself The maximal bisimulation relation is a relation R ∗ ⊆ X × X so that R ⊆ R ∗ for any bisimulation relation R



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Equivalence notions

Bisimulation is a popular tool for complexity reduction of finite systems Consider a system S and a copy of itself The maximal bisimulation relation is a relation R ∗ ⊆ X × X so that R ⊆ R ∗ for any bisimulation relation R R ∗ = ∪i Ri where {Ri }i is the collection of all bisimulation relations



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Equivalence notions

Bisimulation is a popular tool for complexity reduction of finite systems Consider a system S and a copy of itself The maximal bisimulation relation is a relation R ∗ ⊆ X × X so that R ⊆ R ∗ for any bisimulation relation R R ∗ = ∪i Ri where {Ri }i is the collection of all bisimulation relations R ∗ is an equivalence relation on the set X of states of S



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Equivalence notions

Bisimulation is a popular tool for complexity reduction of finite systems Consider a system S and a copy of itself The maximal bisimulation relation is a relation R ∗ ⊆ X × X so that R ⊆ R ∗ for any bisimulation relation R R ∗ = ∪i Ri where {Ri }i is the collection of all bisimulation relations R ∗ is an equivalence relation on the set X of states of S Not any bisimulation relation R is an equivalence relation on the set X of states of S



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Equivalence notions Given a system S = (X , X0 , U, - , Y , H), an equivalence and bisimulation relation R between S and itself, induces the quotient S/R on S defined by S/R = (X/R , X0,/R , U/R ,


- , Y/R , H/R )

/R


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- , Y/R , H/R ),

/R

where: X/R is the collection of equivalence classes Ci induced by R on X , so that x, y ∈ Ci iff (x, y ) ∈ R



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- , Y/R , H/R ),

/R

where: X/R is the collection of equivalence classes Ci induced by R on X , so that x, y ∈ Ci iff (x, y ) ∈ R X0,/R = {Ci ∈ X/R : Ci ∩ X0 6= ∅}



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- , Y/R , H/R ),

/R

where: X/R is the collection of equivalence classes Ci induced by R on X , so that x, y ∈ Ci iff (x, y ) ∈ R X0,/R = {Ci ∈ X/R : Ci ∩ X0 6= ∅} U/R = U



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- , Y/R , H/R ),

/R

where: X/R is the collection of equivalence classes Ci induced by R on X , so that x, y ∈ Ci iff (x, y ) ∈ R X0,/R = {Ci ∈ X/R : Ci ∩ X0 6= ∅} U/R = U Ci

u

- Cj if there exists x

/R


u

- y with x ∈ Ci and y ∈ Cj


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- , Y/R , H/R ),

/R


u


/R

u


Y/R = Y



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- , Y/R , H/R ),

/R


u


/R

u


Y/R = Y H/R (Ci ) = H(x) for any x ∈ Ci



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- , Y/R , H/R ),

/R


u


/R

u


Y/R = Y H/R (Ci ) = H(x) for any x ∈ Ci Some remarks: S∼ =S S/R



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- , Y/R , H/R ),

/R


u


/R

u


Y/R = Y H/R (Ci ) = H(x) for any x ∈ Ci Some remarks: S∼ =S S/R ∼ =S S/R ∗



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- , Y/R , H/R ),

/R


u


/R

u


Y/R = Y H/R (Ci ) = H(x) for any x ∈ Ci Some remarks: S∼ =S S/R ∼ =S S/R ∗ S/R ∗ is the minimal bisimilar system of S G. Pola ( DEWS - UNIVAQ )


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Equivalence notions Example Consider the following finite system S: x1 a

u1

x2

u3

b u2

x3 c

u1 x4 b

R∗ =



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u1

x2

u3

b u2

x3 c

u1 x4 b

R ∗ = {(x1 , x1 ),



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u1

x2

u3

b u2

x3 c

u1 x4 b

R ∗ = {(x1 , x1 ), (x2 , x2 ), (x3 , x3 ), (x4 , x4 ),



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u1

x2

u3

b u2

x3 c

u1 x4 b

R ∗ = {(x1 , x1 ), (x2 , x2 ), (x3 , x3 ), (x4 , x4 ), (x2 , x4 ), (x4 , x2 )}



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u1

x2

u3

b u2

x3 c

u1 x4 b

R ∗ = {(x1 , x1 ), (x2 , x2 ), (x3 , x3 ), (x4 , x4 ), (x2 , x4 ), (x4 , x2 )} R ∗ is an equivalence relation on X



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u1

x2

u3

b u2

x3 c

u1 x4 b

R ∗ = {(x1 , x1 ), (x2 , x2 ), (x3 , x3 ), (x4 , x4 ), (x2 , x4 ), (x4 , x2 )} R ∗ is an equivalence relation on X Equivalence classes induced by R ∗ are C1 = {x1 }, C2 = {x2 , x4 } and C3 = {x3 }



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u1

x2

u3

b u2

x3 c

u1 x4 b

R ∗ = {(x1 , x1 ), (x2 , x2 ), (x3 , x3 ), (x4 , x4 ), (x2 , x4 ), (x4 , x2 )} R ∗ is an equivalence relation on X Equivalence classes induced by R ∗ are C1 = {x1 }, C2 = {x2 , x4 } and C3 = {x3 } The minimal system bisimilar to system S is the system S/R ∗ described by C1 a G. Pola ( DEWS - UNIVAQ )

u1

C2 b


u3

C3 c 9 / 59

Discrete abstractions Efficient bisimulation algorithms have been studied for complexity reduction of large–scale finite systems Challenge: From finite systems to infinite systems!



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Discrete abstractions Efficient bisimulation algorithms have been studied for complexity reduction of large–scale finite systems Challenge: From finite systems to infinite systems! Discrete abstractions of dynamical and hybrid systems: In 1992, Alur and Dill show that Timed Automata admit bisimilar symbolic models In 1993, Alur et al. and Nicollin et al. show that Multirate Automata admit bisimilar symbolic models In 1994, Puri and Varaiya, and in 1998 Henziger et al. show that Rectangular Automata admit finite quotient induced by language equivalence In 2000, Lafferriere et al. show that o-minimal hybrid systems admit bisimilar symbolic models



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Discrete abstractions Discrete abstractions of control systems: X.D. Koutsoukos, P.J. Antsaklis, J.A. Stiver, and M.D. Lemmon, Supervisory control of hybrid systems, Proceedings of the IEEE, 88, 2000 T. Moor, J. Raisch, and S.D. O0 Young, Discrete supervisory control of hybrid systems based on l-complete approximations, Journal of Discrete Event and Dynamic Systems, 12, 2002 D. Forstner, M. Jung, and J. Lunze, A discrete-event model of asynchronous quantised systems, Automatica, 38, 2002 P.E. Caines and Y.J. Wei, Hierarchical hybrid control systems: A lattice-theoretic formulation, Special Issue on Hybrid Systems, IEEE Transactions on Automatic Control, 43, 1998 P. Tabuada and G.J. Pappas, Linear time logic control of discrete-time linear systems, IEEE Transactions on Automatic Control, 51, 2006 P. Tabuada, Symbolic models for control systems, Acta Informatica, 43, 2007 ...



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Discrete abstractions An approximating variation of bisimulation equivalence, termed approximate bisimulation, has been introduced in Girard and Pappas, Approximation metrics for discrete and continuous systems, IEEE Transactions on Automatic Control, 52 (2007), pp. 782–798 Tabuada, An approximate simulation approach to symbolic control approaches to control, IEEE Transactions on Automatic Control, 53(6) (2008), pp. 1406–1418 Control systems admitting approximately bisimilar symbolic models at present: Incrementally–ISS stable nonlinear control systems [Pola, Girard, Tabuada, et al., Automatica, 2008] [Pola, Tabuada, SIAM Journal on Control and Optimization, 2009] Incrementally–GUAS stable nonlinear switched systems [Girard, Pola, Tabuada, IEEE Transactions on Automatic Control, 2010] Incrementally–ISS stable time–delay systems [Pola, Pepe, Di Benedetto, Tabuada, Systems & Control Letters, 2010] [Pola, Pepe, Di Benedetto, This Conference: FrC20.6] G. Pola ( DEWS - UNIVAQ )


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Outline

Approximate bisimulation Nonlinear control systems Incremental stability Symbolic models Control design of a pendulum Conclusion



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Outline




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Approximate equivalence notions

Definition (Metric system) A system S is said to be a metric system if the set of outputs Y is equipped with a metric d : Y × Y → R+ 0. Using a metric, we can generalize bisimulation to approximate bisimulation.



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Approximate equivalence notions Definition (Approximate bisimulation) Consider two metric systems Sa and Sb with Ya = Yb , and let ε ∈ R+ 0.



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Approximate equivalence notions Definition (Approximate bisimulation) Consider two metric systems Sa and Sb with Ya = Yb , and let ε ∈ R+ 0 . A relation R ⊆ Xa × Xb is an ε-approximate bisimulation relation between Sa and Sb if the following three conditions are satisfied:



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a) for every xa0 ∈ Xa0 , there exists xb0 ∈ Xb0 with (xa0 , xb0 ) ∈ R; b) for every xb0 ∈ Xb0 , there exists xa0 ∈ Xa0 with (xa0 , xb0 ) ∈ R;



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a) for every xa0 ∈ Xa0 , there exists xb0 ∈ Xb0 with (xa0 , xb0 ) ∈ R; b) for every xb0 ∈ Xb0 , there exists xa0 ∈ Xa0 with (xa0 , xb0 ) ∈ R; for every (xa , xb ) ∈ R we have d(Ha (xa ), Hb (xb )) ≤ ε;



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a) for every xa0 ∈ Xa0 , there exists xb0 ∈ Xb0 with (xa0 , xb0 ) ∈ R; b) for every xb0 ∈ Xb0 , there exists xa0 ∈ Xa0 with (xa0 , xb0 ) ∈ R; for every (xa , xb ) ∈ R we have d(Ha (xa ), Hb (xb )) ≤ ε; for every (xa , xb ) ∈ R we have that:

b)

ua


a) xa




ub


b

ua


a

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b)

ua


a) xa


ub


b

ua


a

We say that Sa and Sb are ε–approximately bisimilar, denoted by Sa ∼ =εS Sb , if there exists a ε–approximate bisimulation relation between Sa and Sb .



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b)

ua


a) xa


ub


b

ua


a

We say that Sa and Sb are ε–approximately bisimilar, denoted by Sa ∼ =εS Sb , if there exists a ε–approximate bisimulation relation between Sa and Sb . What happens when ε = 0? G. Pola ( DEWS - UNIVAQ )


16 / 59



b)

ua


a) xa


ub


b

ua


a

We say that Sa and Sb are ε–approximately bisimilar, denoted by Sa ∼ =εS Sb , if there exists a ε–approximate bisimulation relation between Sa and Sb . What happens when ε = 0? d(Ha (xa ), Hb (xb )) ≤ ε implies Ha (xa ) = Hb (xb ) G. Pola ( DEWS - UNIVAQ )


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Some remarks: (Reflexivity) Sa ∼ =εS Sa



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Some remarks: (Reflexivity) Sa ∼ =εS Sa (Symmetry) Sa ∼ =εS Sb implies Sb ∼ =εS Sa



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Some remarks: (Reflexivity) Sa ∼ =εS Sa (Symmetry) Sa ∼ =εS Sb implies Sb ∼ =εS Sa ∼εS Sb and Sb ∼ (Transitivity?) Sa = =εS Sc does not imply Sa ∼ =εS Sc



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Some remarks: (Reflexivity) Sa ∼ =εS Sa (Symmetry) Sa ∼ =εS Sb implies Sb ∼ =εS Sa ∼εS Sb and Sb ∼ (Transitivity?) Sa = =εS Sc does not imply Sa ∼ =εS Sc Hence, ε-approximate bisimulation is not an equivalence relation on the class of systems!



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Some remarks: (Reflexivity) Sa ∼ =εS Sa (Symmetry) Sa ∼ =εS Sb implies Sb ∼ =εS Sa ∼εS Sb and Sb ∼ (Transitivity?) Sa = =εS Sc does not imply Sa ∼ =εS Sc Hence, ε-approximate bisimulation is not an equivalence relation on the class of systems! ε ∼εab +εbc Sc ∼εbc Sc imply Sa = It is easy to see that Sa ∼ =Sab Sb and Sb = S S



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Outline




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Nonlinear control systems A nonlinear control system is a quintuple Σ = (X , X0 , U, U, f ),



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Nonlinear control systems A nonlinear control system is a quintuple Σ = (X , X0 , U, U, f ), where: X ⊆ Rn is the state space X0 ⊆ X is the set of initial states U ⊆ Rm is the input space U is the collection of input functions u :]a, b[→ U f : Rn × U → Rn is a continuous map



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Nonlinear control systems A nonlinear control system is a quintuple Σ = (X , X0 , U, U, f ), where: X ⊆ Rn is the state space X0 ⊆ X is the set of initial states U ⊆ Rm is the input space U is the collection of input functions u :]a, b[→ U f : Rn × U → Rn is a continuous map A curve ξ :]a, b[→ Rn is a trajectory of Σ if there exists u ∈ U satisfying ˙ = f (ξ(t), u(t)), ξ(t) for almost all t ∈ ]a, b[.



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Nonlinear control systems A nonlinear control system is a quintuple Σ = (X , X0 , U, U, f ), where: X ⊆ Rn is the state space X0 ⊆ X is the set of initial states U ⊆ Rm is the input space U is the collection of input functions u :]a, b[→ U f : Rn × U → Rn is a continuous map A curve ξ :]a, b[→ Rn is a trajectory of Σ if there exists u ∈ U satisfying ˙ = f (ξ(t), u(t)), ξ(t) for almost all t ∈ ]a, b[. The symbol ξxu (τ ) denotes the point reached at time τ under the input u from the initial condition x G. Pola ( DEWS - UNIVAQ )


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Nonlinear control systems Nonlinear control systems as systems Given a nonlinear control system Σ = (X, X0 , U, U, f) consider the following system S(Σ) = (X , X0 , U,

- , Y , H)

where: X = X is the set of states



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- , Y , H)

where: X = X is the set of states X0 = X0 is the set of initial states



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- , Y , H)

where: X = X is the set of states X0 = X0 is the set of initial states U = U is the set of inputs



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- , Y , H)

where: X = X is the set of states X0 = X0 is the set of initial states U = U is the set of inputs x

u

- y if there exists τ ∈ R+ so that ξxu (τ ) = y



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- , Y , H)


u


Y = X is the output set



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- , Y , H)


u


Y = X is the output set H = 1X is the output function



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- , Y , H)


u


Y = X is the output set H = 1X is the output function System S(Σ) is: metric



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- , Y , H)


u


Y = X is the output set H = 1X is the output function System S(Σ) is: metric neither countable nor symbolic



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Outline




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Discrete abstractions of nonlinear control systems Class K, K∞ and KL functions + A continuous function γ : R+ 0 → R0 belongs to class K if it is strictly increasing and γ(0) = 0



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Discrete abstractions of nonlinear control systems Class K, K∞ and KL functions + A continuous function γ : R+ 0 → R0 belongs to class K if it is strictly increasing and γ(0) = 0 function γ belongs to class K∞ if γ ∈ K and γ(r ) → ∞ as r → ∞



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Discrete abstractions of nonlinear control systems Class K, K∞ and KL functions + A continuous function γ : R+ 0 → R0 belongs to class K if it is strictly increasing and γ(0) = 0 function γ belongs to class K∞ if γ ∈ K and γ(r ) → ∞ as r → ∞ + + A continuous function β : R+ 0 × R0 → R0 belongs to class KL if

for each s, β(r , s) belongs to class K∞ with respect to r



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for each s, β(r , s) belongs to class K∞ with respect to r for each r , β(r , s) is decreasing wrt s and β(r , s) → 0 as s → ∞



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Examples 1

Does function γ(r ) = tan−1 (r ) belong to class K?



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Examples 1

Does function γ(r ) = tan−1 (r ) belong to class K? + 1 Yes, because γ(0) = 0 and γ 0 (r ) = 1+r 2 > 0 for any r ∈ R0



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Examples 1

2

Does function γ(r ) = tan−1 (r ) belong to class K? + 1 Yes, because γ(0) = 0 and γ 0 (r ) = 1+r 2 > 0 for any r ∈ R0 Does function γ(r ) = tan−1 (r ) belong to class K∞ ?



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Examples 1

2

Does function γ(r ) = tan−1 (r ) belong to class K? + 1 Yes, because γ(0) = 0 and γ 0 (r ) = 1+r 2 > 0 for any r ∈ R0 Does function γ(r ) = tan−1 (r ) belong to class K∞ ? No, because γ(r ) → π/2 as r → ∞



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Examples 1

2

3

Does function γ(r ) = tan−1 (r ) belong to class K? + 1 Yes, because γ(0) = 0 and γ 0 (r ) = 1+r 2 > 0 for any r ∈ R0 Does function γ(r ) = tan−1 (r ) belong to class K∞ ? No, because γ(r ) → π/2 as r → ∞ Does function γ(r ) = 2r belong to class K∞ ?



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Examples 1

2

3

Does function γ(r ) = tan−1 (r ) belong to class K? + 1 Yes, because γ(0) = 0 and γ 0 (r ) = 1+r 2 > 0 for any r ∈ R0 Does function γ(r ) = tan−1 (r ) belong to class K∞ ? No, because γ(r ) → π/2 as r → ∞ Does function γ(r ) = 2r belong to class K∞ ? Yes, because γ(0) = 0, γ 0 (r ) = 2 > 0 and γ(r ) → ∞ as r → ∞



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Examples 1

2

3

4

Does function γ(r ) = tan−1 (r ) belong to class K? + 1 Yes, because γ(0) = 0 and γ 0 (r ) = 1+r 2 > 0 for any r ∈ R0 Does function γ(r ) = tan−1 (r ) belong to class K∞ ? No, because γ(r ) → π/2 as r → ∞ Does function γ(r ) = 2r belong to class K∞ ? Yes, because γ(0) = 0, γ 0 (r ) = 2 > 0 and γ(r ) → ∞ as r → ∞ Does function β(r , s) = e−s r belong to class KL?



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Examples 1

2

3

4

Does function γ(r ) = tan−1 (r ) belong to class K? + 1 Yes, because γ(0) = 0 and γ 0 (r ) = 1+r 2 > 0 for any r ∈ R0 Does function γ(r ) = tan−1 (r ) belong to class K∞ ? No, because γ(r ) → π/2 as r → ∞ Does function γ(r ) = 2r belong to class K∞ ? Yes, because γ(0) = 0, γ 0 (r ) = 2 > 0 and γ(r ) → ∞ as r → ∞ Does function β(r , s) = e−s r belong to class KL? Yes



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Discrete abstractions of nonlinear control systems Input–to–State Stability

Definition [Sontag, IEEE TAC 1989] A nonlinear control system Σ is Input-to-State Stable (ISS) if there exists a KL function β and a K∞ function γ so that for any initial condition x0 ∈ X , for any input u ∈ U and for any time t ∈ R+ 0 the solution ξx0 u (t) exists and kξx0 u (t)k ≤ β(kx0 k, t) + γ(kuk∞ )



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Some remarks: ISS is a notion of stability wrt an equilibrium point



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Some remarks: ISS is a notion of stability wrt an equilibrium point A linear control system is ISS if and only if it is asymptotically stable.



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Some remarks: ISS is a notion of stability wrt an equilibrium point A linear control system is ISS if and only if it is asymptotically stable. Hence, if and only if the eigenvalues λi of the dynamical matrix A are so that Re(λi ) < 0 G. Pola ( DEWS - UNIVAQ )


23 / 59



Some remarks: ISS is a notion of stability wrt an equilibrium point A linear control system is ISS if and only if it is asymptotically stable. Hence, if and only if the eigenvalues λi of the dynamical matrix A are so that Re(λi ) < 0 Lyapunov characterization of ISS well known in the literature G. Pola ( DEWS - UNIVAQ )


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Discrete abstractions of nonlinear control systems Incremental Input–to–State Stability

Definition [Angeli, IEEE TAC 2002] A nonlinear control system Σ is Incrementally Input-to-State Stable (δ–ISS) if there exists a KL function β and a K∞ function γ so that for any pair of initial conditions x01 , x02 ∈ X , for any pair of inputs u1 , u2 ∈ U and for any time t ∈ R+ 0 the solutions ξx 1 u1 (t) and ξx 2 u2 (t) exist and 0

0

kξx 1 u1 (t) − ξx 2 u2 (t)k ≤ β(kx01 − x02 k, t) + γ(ku1 − u2 k∞ ) 0


0


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0


0

Some remarks: δ–ISS is a notion of stability wrt trajectories



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0


0

Some remarks: δ–ISS is a notion of stability wrt trajectories If f (0, 0) = 0 then δ–ISS implies ISS, while the converse is not true in general G. Pola ( DEWS - UNIVAQ )


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0


0

Some remarks: δ–ISS is a notion of stability wrt trajectories If f (0, 0) = 0 then δ–ISS implies ISS, while the converse is not true in general A linear control system is δ–ISS if and only if it is ISS G. Pola ( DEWS - UNIVAQ )


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Discrete abstractions of nonlinear control systems




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Digitizing a continuous signal ...



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"Digitizing" a continuous system ...



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Symbolic models Time discretization Given a nonlinear control system Σ = (X, X0 , U, U, f) and a sampling parameter τ consider the following system Sτ (Σ) = (Xτ , X0,τ , Uτ ,

- , Yτ , Hτ )

τ

where: Xτ = X is the set of states



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- , Yτ , Hτ )

τ

where: Xτ = X is the set of states X0,τ = X0 is the set of initial states



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- , Yτ , Hτ )

τ

where: Xτ = X is the set of states X0,τ = X0 is the set of initial states Uτ = {u ∈ U : the time-domain of u is [0, τ ] and u is constant }



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- , Yτ , Hτ )

τ

where: Xτ = X is the set of states X0,τ = X0 is the set of initial states Uτ = {u ∈ U : the time-domain of u is [0, τ ] and u is constant } x

u

- y if ξxu (τ ) = y

τ



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- , Yτ , Hτ )

τ


u


τ

Yτ = X is the output set



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- , Yτ , Hτ )

τ


u


τ

Yτ = X is the output set Hτ = 1X is the output function



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- , Yτ , Hτ )

τ


u


τ

Yτ = X is the output set Hτ = 1X is the output function Some remarks: Sτ (Σ) is metric



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- , Yτ , Hτ )

τ


u


τ

Yτ = X is the output set Hτ = 1X is the output function Some remarks: Sτ (Σ) is metric Sτ (Σ) is neither countable nor symbolic



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- , Yτ , Hτ )

τ


u


τ

Yτ = X is the output set Hτ = 1X is the output function Some remarks: Sτ (Σ) is metric Sτ (Σ) is neither countable nor symbolic What are the relationships between S(Σ) and Sτ (Σ)?



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- , Yτ , Hτ )

τ


u


τ

Yτ = X is the output set Hτ = 1X is the output function Some remarks: Sτ (Σ) is metric Sτ (Σ) is neither countable nor symbolic What are the relationships between S(Σ) and Sτ (Σ)? - ⊆ System Sτ (Σ) is a sub–system of S(Σ), i.e. τ G. Pola ( DEWS - UNIVAQ )


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Symbolic models Space and input quantization

Given a nonlinear control system Σ = (X, X0 , U, U, f), a state space quantization η and input space quantization µ, consider the following system Sτ,η,µ (Σ) = (Xτ,η,µ , X0,τ,η,µ , Uτ,η,µ ,



- , Yτ,η,µ , Hτ,η,µ )

τ,η,µ

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- , Yτ,η,µ , Hτ,η,µ )

τ,η,µ

where: Xτ,η,µ = ηZn ∩ X is the set of states



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- , Yτ,η,µ , Hτ,η,µ )

τ,η,µ




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- , Yτ,η,µ , Hτ,η,µ )

τ,η,µ




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- , Yτ,η,µ , Hτ,η,µ )

τ,η,µ




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- , Yτ,η,µ , Hτ,η,µ )

τ,η,µ

where: Xτ,η,µ = ηZn ∩ X is the set of states X0,τ,η,µ = ηZn ∩ X0 is the set of initial states



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- , Yτ,η,µ , Hτ,η,µ )

τ,η,µ

where: Xτ,η,µ = ηZn ∩ X is the set of states X0,τ,η,µ = ηZn ∩ X0 is the set of initial states Uτ,η,µ = µZm ∩ Uτ is the set of inputs



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- , Yτ,η,µ , Hτ,η,µ )

τ,η,µ

where: Xτ,η,µ = ηZn ∩ X is the set of states X0,τ,η,µ = ηZn ∩ X0 is the set of initial states Uτ,η,µ = µZm ∩ Uτ is the set of inputs x

u

- y if kξxu (τ ) − y k ≤ η

τ,η,µ



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- , Yτ,η,µ , Hτ,η,µ )

τ,η,µ


u


τ,η,µ

Yτ,η,µ = X is the output set



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- , Yτ,η,µ , Hτ,η,µ )

τ,η,µ


u


τ,η,µ

Yτ,η,µ = X is the output set Hτ,η,µ := Xτ,η,µ ,→ Yτ,η,µ is the output function



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- , Yτ,η,µ , Hτ,η,µ )

τ,η,µ


u


τ,η,µ

Yτ,η,µ = X is the output set Hτ,η,µ := Xτ,η,µ ,→ Yτ,η,µ is the output function Some remarks: Sτ,η,µ (Σ) is metric



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- , Yτ,η,µ , Hτ,η,µ )

τ,η,µ


u


τ,η,µ

Yτ,η,µ = X is the output set Hτ,η,µ := Xτ,η,µ ,→ Yτ,η,µ is the output function Some remarks: Sτ,η,µ (Σ) is metric Sτ,η,µ (Σ) is countable and becomes finite when X and U are bounded G. Pola ( DEWS - UNIVAQ )


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- , Yτ,η,µ , Hτ,η,µ )

τ,η,µ


u


τ,η,µ

Yτ,η,µ = X is the output set Hτ,η,µ := Xτ,η,µ ,→ Yτ,η,µ is the output function Some remarks: Sτ,η,µ (Σ) is metric Sτ,η,µ (Σ) is countable and becomes finite when X and U are bounded What are the relationships between Sτ,η,µ (Σ) and Sτ (Σ)? G. Pola ( DEWS - UNIVAQ )


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Symbolic models What are the relationships between Sτ,η,µ (Σ) and Sτ (Σ)?

Theorem Consider a control system Σ. Suppose that Σ is δ–ISS and choose any desired precision ε ∈ R+ .



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Theorem Consider a control system Σ. Suppose that Σ is δ–ISS and choose any desired precision ε ∈ R+ . For any sampling time τ ∈ R+ , space quantization η ∈ R+ and input quantization µ ∈ R+ satisfying: β(ε, τ ) + γ(µ) + η ≤ ε,



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Theorem Consider a control system Σ. Suppose that Σ is δ–ISS and choose any desired precision ε ∈ R+ . For any sampling time τ ∈ R+ , space quantization η ∈ R+ and input quantization µ ∈ R+ satisfying: β(ε, τ ) + γ(µ) + η ≤ ε, systems Sτ,η,µ (Σ) and Sτ (Σ) are ε–approximately bisimilar.



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Symbolic models Sketch of the proof



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Define R ⊆ Xτ × Xτ,η,µ so that (x, q) ∈ R if and only if kx − qk ≤ ε



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Define R ⊆ Xτ × Xτ,η,µ so that (x, q) ∈ R if and only if kx − qk ≤ ε Consider any x

u

- y in Sτ (Σ) where y = ξxu (τ ).

τ



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Define R ⊆ Xτ × Xτ,η,µ so that (x, q) ∈ R if and only if kx − qk ≤ ε u

Consider any x τ- y in Sτ (Σ) where y = ξxu (τ ). Pick v ∈ Uτ so that: ku − v k ≤ µ



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Consider any x τ- y in Sτ (Σ) where y = ξxu (τ ). Pick v ∈ U so that: ku − v k ≤ µ Consider q

v

- z in Sτ (Σ) where z = ξqv (τ ).

τ



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Consider any x τ- y in Sτ (Σ) where y = ξxu (τ ). Pick v ∈ U so that: ku − v k ≤ µ v

Consider q τ- z in Sτ (Σ) where z = ξqv (τ ). There exists p ∈ Xτ,η,µ so that kp − zk ≤ η



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Consider q τ- z in Sτ (Σ) where z = ξqv (τ ). There exists p ∈ Xτ,η,µ so that kp − zk ≤ η and q

v

- p.

τ,η,µ



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Symbolic models Sketch of the proof Define R ⊆ Xτ × Xτ,η,µ so that (x, q) ∈ R if and only if kx − qk ≤ ε u


Consider q τ- z in Sτ (Σ) where z = ξqv (τ ). There exists p ∈ Xτ,η,µ so that kp − zk ≤ η v

- p. We now show that ky − pk ≤ ε. and q τ,η,µ By δ–ISS one gets: ky − pk


≤

ky − zk + kz − pk


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≤

ky − zk + kz − pk = kξxu (τ ) − ξqv (τ )k + kz − pk



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≤ ≤

ky − zk + kz − pk = kξxu (τ ) − ξqv (τ )k + kz − pk β(kx − qk, τ ) + γ(ku − v k∞ ) + η



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≤ ≤ ≤

ky − zk + kz − pk = kξxu (τ ) − ξqv (τ )k + kz − pk β(kx − qk, τ ) + γ(ku − v k∞ ) + η β(ε, τ ) + γ(µ) + η



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≤ ≤ ≤

ky − zk + kz − pk = kξxu (τ ) − ξqv (τ )k + kz − pk β(kx − qk, τ ) + γ(ku − v k∞ ) + η β(ε, τ ) + γ(µ) + η ≤ ε



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Symbolic models... How much far from necessity?



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Symbolic models... How much far from necessity? Unstable linear systems do not admit approximate bisimilar symbolic models!



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Symbolic models... How much far from necessity? Unstable linear systems do not admit approximate bisimilar symbolic models! 1 Consider Σ described by x˙ = x ∈ R.



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Symbolic models... How much far from necessity? Unstable linear systems do not admit approximate bisimilar symbolic models! 1 Consider Σ described by x˙ = x ∈ R. System Σ is unstable and hence not δ–ISS



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Symbolic models... How much far from necessity? Unstable linear systems do not admit approximate bisimilar symbolic models! 1 Consider Σ described by x˙ = x ∈ R. System Σ is unstable and hence not δ–ISS 2 Consider any ε and any countable metric system S = (X , U,



- , R, H)

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- , R, H)

3 Suppose existence of an ε–approximate bisimulation relation R ⊆ Xτ × X



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- , R, H)

3 Suppose existence of an ε–approximate bisimulation relation R ⊆ Xτ × X 4 By countability of S, there exist z0 ∈ X and x0 , y0 ∈ Xτ = R such that x0 6= y0 ,

and (x0 , z0 ), (y0 , z0 ) ∈ R.



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- , R, H)


and (x0 , z0 ), (y0 , z0 ) ∈ R. Set xk = eτ k x0 , yk = eτ k y0 , for any k ∈ N



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- , R, H)


and (x0 , z0 ), (y0 , z0 ) ∈ R. Set xk = eτ k x0 , yk = eτ k y0 , for any k ∈ N 5 Since x0 6= y0 , by selecting λ ∈ R+ such that kx0 − y0 k > λ, we have: kxk − yk k = eτ k kx0 − y0 k > eτ k λ, ∀k ∈ N



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- , R, H)



(1)

0

6 Choose k 0 ∈ N so that eτ k λ − ε > ε.



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- , R, H)



(1)

0

6 Choose k 0 ∈ N so that eτ k λ − ε > ε. There exist zk 0 ∈ X so that,

(xk 0 , zk 0 ), (yk 0 , zk 0 ) ∈ R.



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- , R, H)



(1)

0


(xk 0 , zk 0 ), (yk 0 , zk 0 ) ∈ R. 7 Since (xk 0 , zk 0 ) ∈ R, kxk 0 − H(zk 0 )k ≤ ε



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- , R, H)



(1)

0


(xk 0 , zk 0 ), (yk 0 , zk 0 ) ∈ R. 7 Since (xk 0 , zk 0 ) ∈ R, kxk 0 − H(zk 0 )k ≤ ε By combining inequalities (1) and (2) and by definition of k 0 , we obtain:

(2)

0

kH(zk 0 ) − yk 0 k ≥ kxk 0 − yk 0 k − kxk 0 − H(zk 0 )k > eτ k λ − ε > ε



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- , R, H)



(1)

0


(xk 0 , zk 0 ), (yk 0 , zk 0 ) ∈ R. 7 Since (xk 0 , zk 0 ) ∈ R, kxk 0 − H(zk 0 )k ≤ ε By combining inequalities (1) and (2) and by definition of k 0 , we obtain:

(2)

0

kH(zk 0 ) − yk 0 k ≥ kxk 0 − yk 0 k − kxk 0 − H(zk 0 )k > eτ k λ − ε > ε 8 ... a contradiction holds! G. Pola ( DEWS - UNIVAQ )


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Correct-by-Design Controller Synthesis

A three step approach: Computation of a symbolic abstraction of the physical system Discrete controller synthesis for the symbolic abstraction Hybrid controller synthesis via discrete controller refinement



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Outline




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Discrete abstractions of nonlinear control systems Consider the following nonlinear control system describing the dynamics of a pendulum: x˙ 1 = x2 , Σ: k x2 + u, x˙ 2 = − gl sin x1 − m where: x1 is the point mass angular position x2 is the point mass velocity u is the torque g is the gravity acceleration l is the length of the rod m is the mass k is the coefficient of friction G. Pola ( DEWS - UNIVAQ )


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Control design of a pendulum Symbolic model for ε = 0.25, τ = 2, η = 0.2, µ = 1.5 · 10−4


5

10

15

20

25

4

9

14

19

24

3

8

13

18

23

2

7

12

17

22

1

6

11

16

21


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Control design of a pendulum

Suppose that our objective is to design a controller enforcing a specification P given by the alternation between the two periodic motions P1 and P2 : P1 requires the state to cycle between (−η, 0) and (0, 0) P2 requires the state to cycle between (−η, 0) and (η, 0) Specification P requires the execution of the sequence of periodic motions P1 , P1 , P2 , P1 , P1



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Control design of a pendulum Symbolic controller solving P1 on the symbolic model is: (−η, 0)


- (0, 0) −1.5 - (−η, 0)

1.38


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- (0, 0) −1.5 - (−η, 0)

1.38

Symbolic controller solving P2 on the symbolic model is: (−η, 0)


- (0, η) 1.5- (η, 0) −1.5 - (0, −η) −0.71 - (−η, 0)

1.5


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- (0, 0) −1.5 - (−η, 0)

1.38

Symbolic controller solving P2 on the symbolic model is: (−η, 0)

- (0, η) 1.5- (η, 0) −1.5 - (0, −η) −0.71 - (−η, 0)

1.5

Symbolic controller enforcing P on the symbolic model is: (−η, 0)

- (0, 0) −1.5 - (−η, 0) 1.38 - (0, 0) −1.5 - (−η, 0)

1.38

- (0, η) 1.5- (η, 0) −1.5 - (0, −η) −0.71 - (−η, 0)

1.5

- (0, 0) −1.5 - (−η, 0) 1.38 - (0, 0) −1.5 - (−η, 0)

1.38



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Control design of a pendulum Symbolic model for ε = 0.25, τ = 2, η = 0.2, µ = 1.5 · 10−4


5

10

15

20

25

4

9

14

19

24

3

8

13

18

23

2

7

12

17

22

1

6

11

16

21


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u

x2

x1

Control design of a pendulum 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 0

2

4

6

8

10

12 t

14

16

18

20

22

24

0

2

4

6

8

10

12 t

14

16

18

20

22

24

0

2

4

6

8

10

12 t

14

16

18

20

22

24

0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8

1.5 1 0.5 0 −0.5 −1 −1.5



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Outline




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Conclusion Recap ... Computation of a symbolic abstraction of the physical system (this talk) Discrete controller synthesis for the symbolic abstraction (previous talk) Hybrid controller synthesis via discrete controller refinement (next talk)

Some results on efficient algorithms for correct–by–design embedded control software synthesis also discussed in [This conference, WeB19.2]



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Discrete abstractions of continuous control systems Giordano Pola Center of Excellence for Research DEWS Department of Electrical and Information Engineering University of L’Aquila



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Discrete abstractions of continuous control systems - Google Sites

Discrete abstractions of continuous control systems - Google Sites

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