## Generating polynomial invariants for hybrid systems (2005)

Venue: | In HSCC |

Citations: | 10 - 1 self |

### BibTeX

@INPROCEEDINGS{Rodríguez-carbonell05generatingpolynomial,

author = {Enric Rodríguez-carbonell and Ashish Tiwari},

title = {Generating polynomial invariants for hybrid systems},

booktitle = {In HSCC},

year = {2005},

pages = {590--605},

publisher = {Springer}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. We present a powerful computational method for automatically generating polynomial invariants of hybrid systems with linear continuous dynamics. When restricted to linear continuous dynamical systems, our method generates a set of polynomial equations (algebraic set) that is the best such over-approximation of the reach set. This shows that the set of algebraic invariants of a linear system is computable. The extension to hybrid systems is achieved using the abstract interpretation framework over the lattice defined by algebraic sets. Algebraic sets are represented using canonical Gröbner bases and the lattice operations are effectively computed via appropriate Gröbner basis manipulations. 1

### Citations

1977 | A theory of timed automata
- Alur, Dill
- 1994
(Show Context)
Citation Context ...ension of our method to hybrid systems with more general continuous dynamics would require the use of heuristics, such as [21,18]. Region graphs suffice to compute exact reach sets for timed automata =-=[2]-=-. Polygonal sets have been used as representations for computing reachable states for linear hybrid automata [1]. For more complex continuous dynamics, various representations have been used for compu... |

1890 |
Abstract interpretation: A unified lattice model for static analysis of programs by construction or approximation of fixpoints
- Cousot, Cousot
- 1977
(Show Context)
Citation Context ... u1v1 − 1, ..., ukvk − 1, w 2 1 + z 2 1 − 1, ..., w 2 l + z 2 l − 1〉 ∩ R[X] In this section we extend the technique for generating algebraic invariants to hybrid systems using abstract interpretation =-=[6]-=-. At each location, we restrict ourselves to linear continuous dynamics. A hybrid system HS = (L, X, T , (Init)ℓ∈L, (A)ℓ∈L, (b)ℓ∈L) consists of a finite set L of locations; a finite set of continuous ... |

881 | Symbolic Boolean Manipulation with Ordered Binary-Decision Diagrams
- Bryant
- 1992
(Show Context)
Citation Context ... an ideal and has a finite basis representation. Furthermore, there is a canonical fully-reduced basis, called a Gröbner basis, which can be effectively computed (cf. ordered binary decision diagrams =-=[4]-=-). The set union, set intersection, and set inclusion operators are efficiently computable on these canonical bases. The same is also true of the quantifier-elimination (projection) operator. Using th... |

361 | Hybrid automata: an algorithmic approach to the specification and verification of hybrid systems
- Alur, Courcoubetis, et al.
- 1993
(Show Context)
Citation Context ..., such as [21,18]. Region graphs suffice to compute exact reach sets for timed automata [2]. Polygonal sets have been used as representations for computing reachable states for linear hybrid automata =-=[1]-=-. For more complex continuous dynamics, various representations have been used for computing over-approximations of the reach sets, such as, union of convex polytopes [5], union of hyper-rectangles [8... |

322 |
Macaulay 2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2
- Grayson, Stillman
(Show Context)
Citation Context ...we did not compute the closure IV always; nonetheless, the obtained invariants sufficed for proving the properties of interest. We implemented our techniques in the algebraic geometry tool Macaulay 2 =-=[9]-=- using a PC running Linux with a 2.5 GHz. processor and 512 MB of memory. (on, 2) ✬ ✩ (off ) ✬ ✩ (on, 1) ✬ ✩ ˙t = 1 ˙y = 1 ˙z = 1 ✫ ˙t = 1 t = a − b → t := 0✲ ˙y = 1 ˙z = 0 ✪ ✫ t = b → t := 0 ˙t = 1 ✛... |

207 |
Ideals, Varieties and Algorithms
- Cox, Little, et al.
- 1992
(Show Context)
Citation Context ...pture all algebraic invariants of CS. Furthermore, to compute the algebraic invariants, the expensive step that involves doing quantifier elimination over the reals can be replaced by a Gröbner basis =-=[7]-=- computation step, which is simpler and often more efficient in practice. Since we use Gröbner bases to eliminate variables, we need to employ an elimination term ordering in which the auxiliary varia... |

120 | Ellipsoidal techniques for reachability analysis
- Kurzhanski, Varaiya
- 2000
(Show Context)
Citation Context ... continuous dynamics, various representations have been used for computing over-approximations of the reach sets, such as, union of convex polytopes [5], union of hyper-rectangles [8], and ellipsoids =-=[14]-=-. Similar in the spirit of the result presented here, Kurzhanski andsVaraiya [14] show that the best ellipsoidal over-approximation of the reach set for certain linear systems can be computed. We also... |

105 | Algorithmic analysis of nonlinear hybrid systems
- Henzinger, Ho, et al.
- 1998
(Show Context)
Citation Context ...˙t = 1 t = a − b → t := 0✲ ˙y = 1 ˙z = 0 ✪ ✫ t = b → t := 0 ˙t = 1 ✛ ˙y = 1 ✲ t = a → t := 0 ˙z = 1 ✪ ✫ ✪ Fig. 2. Hybrid system for a thermostat Thermostat. Figure 2 shows a hybrid system, taken from =-=[13]-=-, modeling a thermostat. The system has three locations: in (on, 1) and (on, 2) the thermostat is on, while in (off ) the thermostat is off. There are three clocks: t tracks the time elapsed at the cu... |

82 | S.: A new class of decidable hybrid systems
- Lafferriere, Pappas, et al.
- 1999
(Show Context)
Citation Context ... sin(qt) and cos(qt), there is no finite set (basis) of inequalities that captures all the semi-algebraic relationships between these functions. This also partly explains why the decidability results =-=[15,16]-=- are not easy to generalize. 3.2 Generalization to Arbitrary Eigenvalues Let L be the set of all eigenvalues of the matrix A. We now drop the assumption that a, b ∈ Q for all a + bι ∈ L by generalizin... |

79 | P.: Verification of Linear Hybrid Systems by Means of Convex Approximations
- Halbwachs, Proy, et al.
- 1994
(Show Context)
Citation Context ... y 2 − 1) = 0}. Its solution, which defines the variety V(〈x(x 2 + y 2 − 1) ,y(x 2 + y 2 − 1)〉), is the union of the circle x 2 + y 2 = 1 and the origin. Notice that this set, unlike convex polyhedra =-=[10,5]-=-, is not convex or even connected. Reciprocally, given a set of points S ⊆ K n , the polynomials vanishing on this set form the ideal I(S) = {P ∈ K[X] | P (s) = 0 ∀s ∈ S}, called the ideal of S. Notic... |

72 | O.: Reachability analysis via face lifting
- Dang, Maler
- 1998
(Show Context)
Citation Context ...1]. For more complex continuous dynamics, various representations have been used for computing over-approximations of the reach sets, such as, union of convex polytopes [5], union of hyper-rectangles =-=[8]-=-, and ellipsoids [14]. Similar in the spirit of the result presented here, Kurzhanski andsVaraiya [14] show that the best ellipsoidal over-approximation of the reach set for certain linear systems can... |

64 | Symbolic Reachability Computation for Families of Linear Vector Fields
- Lafferriere, Pappas, et al.
(Show Context)
Citation Context ...s dynamical system CS, the best algebraic over-approximation of the reach set of CS can be computed (Section 3). The proof of this result borrows some key insights from Lafferriere, Pappas and Yovine =-=[16]-=-, who use semi-algebraic sets and show that exact reach sets can be computed for more restricted classes of linear vector fields. (ii) We show that the method for over-approximating reach sets for lin... |

39 |
Approximate reachability for linear systems
- Tiwari
- 2003
(Show Context)
Citation Context ... set can be computed for a subclass of linear continuous dynamical systems [16]. Subsequently, it was shown that invariants could be effectively constructed for more general classes of linear systems =-=[20]-=-. We show here that the most precise equational invariant for arbitrary linear systems can be computed. We focus on a special case in Section 3.1, and generalize to arbitrary linear systems in Section... |

37 | Constructing invariants for hybrid systems
- Sankaranarayanan, Sipma, et al.
(Show Context)
Citation Context ...er bases (Section 4). We also present some experimental results obtained by using our method to generate polynomial invariants for hybrid systems (Section 5). 1.1 Related Work Sankaranarayanan et al. =-=[18]-=- presented an approach for generating polynomial equational invariants for hybrid systems with more general (nonlinear) polynomial dynamics. However, their approach is based on guessing a template for... |

27 | G.: Nonlinear systems: Approximating reach sets
- Tiwari, Khanna
- 2004
(Show Context)
Citation Context ...mplate. In fact, it is complete for linear systems. On the other hand, any extension of our method to hybrid systems with more general continuous dynamics would require the use of heuristics, such as =-=[21,18]-=-. Region graphs suffice to compute exact reach sets for timed automata [2]. Polygonal sets have been used as representations for computing reachable states for linear hybrid automata [1]. For more com... |

26 | A note on abstract interpretation strategies for hybrid automata
- Henzinger, Ho
- 1995
(Show Context)
Citation Context ...ar systems can be computed. We also note here that some of the above works use abstract interpretation ideas, most notably in the form of widening to accelerate reachability (or fixpoint) computation =-=[12,8]-=-. Exact reach sets for a class of linear vector fields were computed as semialgebraic sets over state variables and special variables representing exponential or trigonometric functions [16]. We contr... |

19 | Computing polyhedral approximations to flow pipes for dynamic systems
- Chutinan, Krogh
- 1998
(Show Context)
Citation Context ...es for linear hybrid automata [1]. For more complex continuous dynamics, various representations have been used for computing over-approximations of the reach sets, such as, union of convex polytopes =-=[5]-=-, union of hyper-rectangles [8], and ellipsoids [14]. Similar in the spirit of the result presented here, Kurzhanski andsVaraiya [14] show that the best ellipsoidal over-approximation of the reach set... |

16 | D.: An abstract interpretation approach for automatic generation of polynomial invariants
- Rodríguez-Carbonell, Kapur
- 2004
(Show Context)
Citation Context ...a widening operator to guarantee termination. Specifically, we use the following operations on algebraic sets (represented as ideals) to abstract the corresponding operations on (arbitrary) sets, see =-=[17]-=-: Assignment Transformation → Elimination of Variables. Given an ideal of variety I = 〈P1(X), ..., Pk(X)〉 and a multiple (polynomial) assignment (x1, . . . , xn) := (α1(X), . . . , αn(X)), we introduc... |

1 |
The symbolic approach to hybrid systems
- Henzinger
- 2002
(Show Context)
Citation Context ..., these operations are well understood. Depending on the exact verification procedure, some or all of the set union, set intersection, set complement, subset, and projection operators may be required =-=[11]-=-. In the case of hybrid systems, we additionally require that the representation behaves “nicely” along the continuous evolutions at different locations of the hybrid system. This paper explores the r... |