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Computational Approaches to Analysis and Control of Hybrid Systems - - PowerPoint PPT Presentation

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Computational Approaches to Analysis and Control of Hybrid Systems Antoine Girard Laboratoire Jean Kuntzmann Universit e Joseph Fourier, Grenoble Habilitation ` a Diriger des Recherches Universit e de Grenoble 19 Novembre 2013 A.

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Computational Approaches to Analysis and Control of Hybrid Systems

Antoine Girard

Laboratoire Jean Kuntzmann Universit´ e Joseph Fourier, Grenoble Habilitation ` a Diriger des Recherches Universit´ e de Grenoble 19 Novembre 2013

  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 1 / 43

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Research activity

Positions: 01-04: PhD in applied mathematics, Laboratoire de Mod´ elisation et Calcul, Grenoble. 04-06: Postdoctoral researcher

University of Pennsylvania, Dept. of ESE, Philadelphia (10/04-12/05); Verimag, Grenoble (01/06-08/06).

06-...: Maˆ ıtre de conf´ erences, Universit´ e Joseph Fourier, Research appointment at Laboratoire Jean Kuntzmann. Publications, projects and research supervision: 20 journal and 49 conference papers; 6 funded local or national research projects; 6 PhD students and 5 postdoctoral researchers.

  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 2 / 43

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Hybrid systems

Dynamical systems with continuous and discrete behaviors. In the physical sciences:

Models are traditionally continuous (ODE, PDE...); Numerous sources of hybrid behaviors: mechanical impacts, electrical diodes, biological switches...

In computer science:

Models of reactive systems are discrete (Automata, DEDS...); Models become hybrid when reactive systems are subject to timing constraints (Timed Automata) or are interacting with the physical world (Hybrid Automata).

Theory of hybrid systems has rapidly grown since the 90s at the interface

  • f computer science and control theory.
  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 3 / 43

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Hybrid systems

Computation Control

Automata Logics Model checking Abstraction Differential equations Stability Robustness Lyapunov functions

&

  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 4 / 43

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Outline of the talk

1 Approximation metrics for discrete and continuous systems

Approximate bisimulation Symbolic approach to controller synthesis

2 Reachability analysis of hybrid systems

Approximation of reachable sets of linear systems Efficient set representation and algorithm

  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 5 / 43

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Systems approximation

In control theory, approximation is often captured through metrics e.g. between transfer functions (H2, H∞, Hankel norms):

Mostly for linear systems; Extensions to hybrid behaviors not straightforward.

In computer science, approximation is characterized by behavioral relationships (language inclusion or equivalence, bi-simulation):

Extended to continuous and hybrid systems... For these systems with real-valued variables, metrics seem to be more suitable.

Our contribution: a new approximation framework, based on metrics and capturing these behavioral relationships as special cases.

[TAC 2007, Axelby Award 2009].

  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 6 / 43

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Transition systems

Abstract description of discrete, continuous or hybrid dynamical systems.

Definition

A transition system T = (X, U, S, X 0, Y , O) is given by a set of states X; a set of inputs U; a transition relation S : X × U → 2X; a set of initial states X 0 ⊆ X; a set of outputs Y ; an ouput map O : X → Y .

  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 7 / 43

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Trajectories

A state trajectory of the transition system T is a sequence: s = x0, u0, x1, u1 . . . where x0 ∈ X 0, xk+1 ∈ S(xk, uk), ∀k. The associated output trajectory is

  • = y0, u0, y1, u1, y2, u2 . . . where yk = O(xk), ∀k.

The set L(T) of observed trajectories is the language of T. If Y is equipped with a metric d, we can define the distance between two output trajectories o1 and o2 as: d(o1, o2) =

  • sup

k

d(y1

k , y2 k )

if u1

k = u2 k, ∀k

+∞

  • therwise
  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 8 / 43

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Language metric

Let Ti = (Xi, U, Si, X 0

i , Y , Oi), i ∈ {1, 2}, be transition systems with

a common set of inputs U and outputs Y equipped with a metric d.

Definition

The language metric between T1 and T2 is the Hausdorff distance between L(T1) and L(T2): dL(T1, T2) = max

  • sup
  • 1∈L(T1)

inf

  • 2∈L(T2) d(o1, o2),

sup

  • 2∈L(T2)

inf

  • 1∈L(T1) d(o1, o2)
  • .

Language metric = matching game on trajectories; Non anticipative matching strategies lead to bisimulation metric.

  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 9 / 43

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Approximate bisimulation

Definition

Let ε ≥ 0, a relation R ⊆ X1 × X2 is an ε-approximate bisimulation relation if for all (x1, x2) ∈ R :

1 d(O1(x1), O2(x2)) ≤ ε; 2 ∀u ∈ U, ∀x′

1 ∈ S1(x1, u), ∃x′ 2 ∈ S2(x2, u), such that (x′ 1, x′ 2) ∈ R;

3 ∀u ∈ U, ∀x′

2 ∈ S2(x2, u), ∃x′ 1 ∈ S1(x1, u), such that (x′ 1, x′ 2) ∈ R.

Definition

T1 and T2 are ε-approximately bisimilar (T1 ∼ε T2) if :

1 For all x1 ∈ X 0

1 , there exists x2 ∈ X 0 2 , such that (x1, x2) ∈ R;

2 For all x2 ∈ X 0

2 , there exists x1 ∈ X 0 1 , such that (x1, x2) ∈ R.

  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 10 / 43

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Bisimulation metric

Definition

The bisimulation metric between T1 and T2 is given by dB(T1, T2) = inf {ε ≥ 0 | T1 ∼ε T2} . Characterization of bisimulation metric using Lyapunov-like functions called bisimulation functions.

Theorem

The following inequality holds (equality for deterministic systems): dL(T1, T2) ≤ dB(T1, T2).

  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 11 / 43

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A simple example

1 1 1 2 2 4 4

T1 T2

1

T3

3

dL(T1, T2) = 0, dB(T1, T2) = 2. dL(T1, T3) = 1, dB(T1, T3) = 1.

  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 12 / 43

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Applications of approximate bisimulation

Model reduction of continuous and hybrid systems;

[Automatica 2007, DEDS 2007].

Reported applications in biology, mechanics, multi-agent robotics... Hierarchical hybrid control of linear and differentially flat systems;

[HSCC 2007, Automatica 2009, ECC 2013].

Trajectory based verification;

[HSCC 2006, FORMATS 2006, TECS 2012].

Symbolic approaches to controller synthesis.

  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 13 / 43

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Symbolic approaches to controller synthesis

Controller synthesis using a discrete system (symbolic model) which is approximately bisimilar to the dynamics of the physical system:

Physical System:

≈

˙ x(t) = f (x(t), u(t)) Symbolic Model:

  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 14 / 43

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Symbolic approaches to controller synthesis

Controller synthesis using a discrete system (symbolic model) which is approximately bisimilar to the dynamics of the physical system:

Discrete Controller: Symbolic Model: Physical System: ˙ x(t) = f (x(t), u(t))

  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 14 / 43

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Symbolic approaches to controller synthesis

Controller synthesis using a discrete system (symbolic model) which is approximately bisimilar to the dynamics of the physical system:

Discrete Controller: Symbolic Model: Hybrid Controller: Physical System:

≈

˙ x(t) = f (x(t), u(t)) Refinement q(t+) = g(q(t), x(t)) u(t) = k(q(t), x(t))

  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 14 / 43

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Switched systems

We consider a switched system Σ: ˙ x(t) = fp(t)(x(t)), x(t) ∈ Rn, p(t) ∈ P = {1, . . . , m}. Let τ > 0, the sampled dynamics of Σ is described by transition system Tτ(Σ) = (Rn, P, S, Rn, Rn, id) where the transition relation is given by x′ = S(x, p) ⇐ ⇒ x′ = x(τ), where ˙ x(t) = fp(x(t)), x(0) = x.

  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 15 / 43

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Computation of the symbolic model

We start by approximating the set of states Rn by: [Rn]η =

  • q ∈ Rn
  • qi = ki

2η √n, ki ∈ Z, i = 1, ..., n

  • ,

where η > 0 is a state sampling parameter. Approximation of the transition relation by quantization:

Sa(q, p) S(q, p)

  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 16 / 43

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Computation of the symbolic model

In general, the computed symbolic model Tτ,η(Σ) and Tτ(Σ) are not approximately bisimilar. It holds when Σ satisfies an incremental stability property: intuitively, asymptotic forgetfulness of past history.

t x(t, x1, p) x(t, x2, p)

Incremental global uniform asymptotical stability (δ-GUAS) of switched systems can be proved using a common δ-GAS Lyapunov function.

  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 17 / 43

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Approximation theorem

Theorem

Let V : Rn × Rn → R+

0 be a common δ-GUAS Lyapunov function for Σ.

Consider time sampling parameter τ > 0 and a desired precision ε > 0, then there exists a state sampling parameter η > 0 such that Tτ(Σ) ∼ε Tτ,η(Σ). The ε-approximate bisimulation relation is given by R = {(x, q) ∈ Rn × [Rn]η| V (x, q) ≤ α(ε)} . Explicit relation between τ, ε, and η. Main idea of the proof: show that accumulation of errors due to successive quantizations is contained by incremental stability.

  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 18 / 43

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Examples of incrementally stable systems

Power converters. Thermal dynamics in buildings. Road traffic.

  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 19 / 43

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Controllers for safety specifications

Definition

Let T = (X, U, S, X 0, Y , O), and Ys ⊆ Y be a set of safe outputs. C : X → 2U is a safety controller for specification Ys if, for all x ∈ X such that C(x) = ∅, O(x) ∈ Ys (safety); For all u ∈ C(x), for all x′ ∈ S(x, u), C(x′) = ∅ (deadend freedom).

Definition

Controller C1 is more permissive than controller C2 (C2 C1) if, for all x ∈ X, C2(x) ⊆ C1(x). C∗ is the maximal safety controller for specification Ys if, C∗ is safe and for all safety controllers C, C C∗.

  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 20 / 43

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Safety controller synthesis via symbolic models

The maximal safety controller exists and is unique. Maximal safety controllers are easy to compute for symbolic models... We need a controller refinement procedure.

Definition

The ε-contraction of Ys is Cε(Ys) =

  • y′ ∈ Ys| ∀y ∈ Y , d(y, y′) ≤ ε =

⇒ y ∈ Ys

  • .

Ys ε Cε(Ys)

  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 21 / 43

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Safety controller synthesis via symbolic models

Theorem (”Correct by design”)

Let T1 ∼ε T2 with R ⊆ X1 × X2 the ε-approximate bisimulation relation. Let C∗

2,ε be the maximal safe controller for T2 for the specification Cε(Ys).

Let C1 be the controller for T1 given by ∀x1 ∈ X1, C1(x1) =

  • x2∈R(x1)

C∗

2,ε(x2)

Then, C1 is safe for specification Ys.

Theorem (”Maximal in the limit”)

Let C∗

1 and C∗ 1,2ε be the maximal safe controllers for T1 for specifications

Ys and C2ε(Ys), respectively. Then, C∗

1,2ε C1 C∗ 1.

  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 22 / 43

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Example: DC-DC converter

Power converter with switching control:

il s1 vs rl xl s2 xc rc vc r0 v0

System dynamics: ˙ x(t) = Ap(t)x(t) + b, x(t) ∈ R2, p(t) ∈ P = {1, 2}. Common δ-GUAS Lyapunov function of the form: V (x1, x2) = (x1 − x2)TM(x1 − x2).

  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 23 / 43

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Example: DC-DC converter

Abstraction parameters: τ = 1, η = 10−3 = ⇒ ε = 0.05. Ys = [1.1, 1.6] × [5.4, 5.9] = ⇒ Cε(Ys) = [1.15, 1.55] × [5.45, 5.85].

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 5.3 5.4 5.5 5.6 5.7 5.8 5.9

C∗

a,ε

  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 24 / 43

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Example: DC-DC converter

The synthesized controller is non-deterministic. Several implementations of the controller are possible.

  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 25 / 43

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Results on symbolic control

Symbolic models for incrementally stable systems:

Several classes of systems:

Linear, nonlinear and switched systems;

[HSCC 2007, Automatica 2008, TAC 2010].

Time-delay systems, stochastic systems...

Improvement of scalability:

Multi-scale symbolic models;

[HSCC 2011, CDC 2011, HSCC 2013], Tool CoSyMA.

Grid-free symbolic models.

[CDC 2013]...

“Correct by design, optimal in the limit” controller synthesis:

Maximal safety controllers, time-optimal reachability controllers;

[Automatica 2012].

Complexity reduction via determinization.

[NAHS 2013].

  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 26 / 43

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Outline of the talk

1 Approximation metrics for discrete and continuous systems

Approximate bisimulation Symbolic approach to controller synthesis

2 Reachability analysis of hybrid systems

Approximation of reachable sets of linear systems Efficient set representation and algorithm

  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 27 / 43

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Hybrid automaton

Popular mathematical description of hybrid systems.

˙ x ∈ F1(x) x ∈ I1 ˙ x ∈ F2(x) x ∈ I2

x ∈ G12 x+ ∈ R12(x−) x ∈ G21 x+ ∈ R21(x−) x(t0) x(t−

1 )

x(t+

1 )

x(t−

2 )

x(t+

2 )

I1 G12 I2 G21

  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 28 / 43

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Hybrid automaton

Several sources of non-determinism:

Differential inclusions, enabling guards, set-valued resets... Result of disturbance modeling, system abstraction, partial specification...

No continuity w.r.t. initial conditions, parameters... Simulation not always suitable for exploring effectively the behaviors

  • f a hybrid automaton.

Reachability analysis:

Compute the set of all trajectories for all initial conditions and parameters under all manifestations of non-determinism... Interesting by itself; useful for verification, controller synthesis, computation of symbolic models...

  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 29 / 43

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Reachability analysis

Reachability algorithm, informally:

˙ x ∈ F1(x) x ∈ I1 ˙ x ∈ F2(x) x ∈ I2

x ∈ G12 x+ ∈ R12(x−) x ∈ G21 x+ ∈ R21(x−)

I1 G12 I2 G21

X0

  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 30 / 43

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Reachability analysis

Reachability algorithm, informally:

˙ x ∈ F1(x) x ∈ I1 ˙ x ∈ F2(x) x ∈ I2

x ∈ G12 x+ ∈ R12(x−) x ∈ G21 x+ ∈ R21(x−)

I1 I2 G21

ReachC(X0)

  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 30 / 43

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Reachability analysis

Reachability algorithm, informally:

˙ x ∈ F1(x) x ∈ I1 ˙ x ∈ F2(x) x ∈ I2

x ∈ G12 x+ ∈ R12(x−) x ∈ G21 x+ ∈ R21(x−)

I1 I2 G21

X1−

  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 30 / 43

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Reachability analysis

Reachability algorithm, informally:

˙ x ∈ F1(x) x ∈ I1 ˙ x ∈ F2(x) x ∈ I2

x ∈ G12 x+ ∈ R12(x−) x ∈ G21 x+ ∈ R21(x−)

I1 I2 G21

X1− X1+ = ReachD(X1−)

  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 30 / 43

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Reachability analysis

Reachability algorithm, informally:

˙ x ∈ F1(x) x ∈ I1 ˙ x ∈ F2(x) x ∈ I2

x ∈ G12 x+ ∈ R12(x−) x ∈ G21 x+ ∈ R21(x−)

I1 I2 G21

ReachC(X1+)

  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 30 / 43

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Reachability analysis of linear systems

Consider a linear system of the form: ˙ x(t) = Ax(t) + v(t), x(0) ∈ X 0 v(t) ∈ V , where X 0 and V are compact convex subsets of Rn. Over-approximation of the reachable set R[0,T](X 0):

Let τ = T/N, we have R[0,T](X 0) =

N−1

  • i=0

R[iτ,(i+1)τ](X 0). Each R[iτ,(i+1)τ](X 0) over-approximated by a compact convex set Ωi.

  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 31 / 43

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Approximation scheme

Initialization of the sequence: Rτ(X 0) ≈ eτAX 0 ⊕ τV ; R[0,τ](X 0) ≈ Conv(X 0, eτAX 0 ⊕ τV ); R[0,τ](X 0) ⊆ Conv(X 0, eτAX 0 ⊕ τV ⊕ ατB) =: Ω0 where ατ = O(τ 2). Recurrence relation: R[(i+1)τ,(i+2)τ](X 0) = Rτ(R[iτ,(i+1)τ](X 0)); ≈ eτA R[iτ,(i+1)τ](X 0)

  • ⊕ τV ;

⊆ eτA R[iτ,(i+1)τ](X 0)

  • ⊕ τV ⊕ βτB;

⊆ eτAΩi ⊕ τV ⊕ βτB =: Ωi+1 where βτ = O(τ 2).

  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 32 / 43

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Approximation scheme

Theorem

For all i = 0, . . . , N − 1, R[iτ,(i+1)τ](X 0) ⊆ Ωi and dH(Ωi, R[iτ,(i+1)τ](X 0)) ≤ τeTA A 4 DX 0 + τA2RX0 + eτARV

  • where dH is the Hausdorff distance.

In practice, for computing an over-approximation of the reachable set, we need to:

1 Choose a data structure to represent a class of compact convex sets; 2 Compute (efficiently) for this class of compact convex sets, linear

transformations, Minkowski sum and convex hull.

  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 33 / 43

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Set representation: polytopes

Polytopes form a fairly large (dense) class of compact convex sets. Canonical representations:

H-polytope: P = {x ∈ Rn| ck · x ≤ dk, k = 1, . . . , m}. V-polytope: P = Conv(x1, . . . , xp).

Polytopes are closed under linear transformations, Minkowski sum and convex hull. In high dimension, lack of efficient algorithms to compute Minkowski sum and combinatorial increase of the size of representations.

  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 34 / 43

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Set representation: zonotopes

Smallest class of polytopes containing intervals and closed under linear transformations and Minkowski sum. Canonical representation: Z =

  • x ∈ Rn| x = c +

i=p

  • i=1

xigi, −1 ≤ xi ≤ 1

  • =

(c, g1, . . . , gp).

g1 g2 g3 c

  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 35 / 43

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Reachability analysis using zonotopes

Efficient computation of linear transformations and Minkowski sum: AZ = (Ac, Ag1, . . . , Agp); Z ⊕ Z ′ = (c + c′, g1, . . . , gp, g′

1, . . . , g′ p′).

Easy implementation of Ωi+1 := eτAΩi ⊕ τV ⊕ βτB. Zonotopes not closed under convex hull: compute a zonotope Ω0 such that Conv(X 0, eτAX 0 ⊕ τV ⊕ ατB) ⊆ Ω0.

  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 36 / 43

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Efficient implementation

Direct implementation of the recurrence relation: Ωi+1 = eτAΩi ⊕ Wτ where Wτ = τV ⊕ βτB. Space and time complexity in O(N2). Alternative implementation: Z0 = Ω0, Zi+1 = eτAZi, W0 = Wτ, Wi+1 = eτAWi, S0 = {0}, Si+1 = Si ⊕ Wi, Ωi+1 = Zi+1 ⊕ Si+1. Space and time complexity in O(N).

  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 37 / 43

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Further results on reachability analysis

Reachability using zonotopes: hundreds of variables can be handled.

[HSCC 2005, HSCC 2006].

Alternative set representation: support functions.

[NAHS 2010].

ρΩ(ℓ) = max

x∈Ω ℓ · x

Extension to hybrid systems:

Intersection between reachable sets and guards;

[HSCC 2008, CAV 2009].

Further improvements in the verification platform SpaceEx.

[CAV 2011].

  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 38 / 43

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Discussion

Reachability analysis of nonlinear systems via hybridization:

[Acta Inf. 2007].

Over-approximate the nonlinear dynamics by piecewise affine system with disturbances; Reachability analysis on the approximation.

Reachability analysis of polynomial systems:

[Automatica 2012, ATVA 2012].

Polynomial optimization for computing polytopic invariant sets and

  • ver-approximations of the reachable sets;

LP relaxations based on the Bernstein form of polynomials.

−2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 2.5 −1.5 −1 −0.5 0.5 1 1.5 2 2.5 3 3.5

  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 39 / 43

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Multi-agent systems

Sufficient conditions for consensus in continuous-time:

[SICON 2013].

Generalizing several existing conditions; Estimation of the convergence rate towards consensus.

Opinion dynamics with decaying confidence:

[TAC 2011].

Consensus reached only locally; Decentralized algorithm for community detection in networks.

  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 40 / 43

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SLIDE 47

Conclusion

A multi-disciplinary approach:

Confrontation and marriage of concepts and tools of computer science and control theory; Background in applied mathematics was instrumental for my ability to build bridges between disciplines.

Theoretical research has led to the development of computational tools: CoSyMA, SpaceEx... Research done through collaborative work:

Ph.D. Students: C. Le Guernic, S. Martin, M.A. Ben Sassi, P.O. Lamare, P.J. Meyer, Y. Tang. Postdocs: G. Zheng, C. Morarescu, J. Camara, S. Mouelhi, E. Le Corronc. Projects: Val-AMS, VeDeCy, CompACS (ANR), CoHyBa (Rhˆ

  • ne-Alpes),

CARESSE, SymbAD (UJF-MSTIC).

  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 41 / 43

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SLIDE 48

Perspectives

Approximation metrics for discrete and continuous systems: Scalability improvement of symbolic control approaches

Alternatives to gridding of the state-space

  • Partial gridding (slow manifolds in multiple time-scales systems);
  • Mode sequences as symbolic states (infinite dimensional systems);
  • Partitions generated by reachability analysis.

Explore other applications of approximate bisimulation

Robustness and continuity analysis of computer programs:

  • Characterization of continuity and sensitivity of programs via

bisimulation functions.

  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 42 / 43

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SLIDE 49

Perspectives

Reachability analysis of hybrid systems: Develop specific algorithms for specific problems

Trade generality for efficiency and accuracy Motivated by applications

  • Computation aware control systems design (CompACS project);
  • Design contracts (implementation specification), event-based control,

scheduling...

For verification and for (parameter and controller) synthesis Provide application oriented computational tools

  • A. Girard (LJK-UJF)

Analysis and Control of Hybrid Systems 43 / 43