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## Providing a basin of attraction to a target region by computation of Lyapunov-like functions (2006)

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Venue: | In IEEE Int. Conf. on Computational Cybernetics |

Citations: | 11 - 5 self |

### Citations

765 |
A decision method for elementary algebra and geometry.
- Tarski
- 1948
(Show Context)
Citation Context ...ability of ordinary differential equations is the existence of a Lyapunov function [6]. In cases where the differential equation is polynomial, due to decidability of the theory of real-closed fields =-=[24]-=-, one can always check, whether for a given polynomial with parametric coefficients, there are instantiations of these parameters resulting in a Lyapunov function. However, all the existing decision p... |

287 |
Stability of Motion,
- Hahn
- 1967
(Show Context)
Citation Context ...o hyper-rectangles. Tests on an implementation are promising. I. INTRODUCTION A sufficient condition for verifying stability of ordinary differential equations is the existence of a Lyapunov function =-=[6]-=-. In cases where the differential equation is polynomial, due to decidability of the theory of real-closed fields [24], one can always check, whether for a given polynomial with parametric coefficient... |

130 | Redlog: Computer algebra meets computer logic.
- Dolzmann, Sturm
- 1997
(Show Context)
Citation Context ...s whether there are instantiations of these parameters resulting in a Lyapunov function. However, all the existing decision procedures (e.g., implemented in the software packages QEPCAD [4] or REDLOG =-=[11]-=-), while being able to solve impressively difficult examples, are not efficient enough to be able to solve this problem in practice. Recently, a method based on sum of squares decomposition [26, 27] h... |

89 | The construction of Lyapunov functions using the sum of squares decomposition.
- Papachristodoulou, Prajna
- 2002
(Show Context)
Citation Context ...for rewriting the strict to non-strict inequalities depend on the degree of the chosen parametric polynomial of each example. Example 1: An example of the simplified model of a chemical oscillator in =-=[10]-=-. The original system is: � ˙x1 = a − x1 + x 2 1x2 ˙x2 = b − x 2 1x2 Let (a, b) = (0.5, 0.5), then the equilibrium is (1, 0.5). Let V (x1, x2) = ax 2 1+bx1+cx1x2+dx2+ex 2 2+f, then ˙ V (u, v) = 2ax 3 ... |

89 | Safety verification of hybrid systems using barrier certificates.
- PRAJNA, JADBABAIE
- 2004
(Show Context)
Citation Context ...d-relax constraint solving algorithm. It seems that similar constraints have to be solved in many other areas (e.g., proving the termination of term-write systems, computation of barrier certificates =-=[13]-=-, invariant generation [21], [18], [9], [19], [14], and analysis of FEM [23]), and it is interesting work to apply our algorithms in these areas. We will further increase the efficiency of our method,... |

85 | Quantifier Elimination and Cylindrical Algebraic Decomposition. - Caviness, Johnson - 1998 |

64 |
Decidable theories
- Rabin
- 1977
(Show Context)
Citation Context ...ging the basin of 8sattraction. Note that—in the case where both f and V are polynomials—Constraint 1 is a formula in the predicate logical theory of the real numbers with addition and multiplication =-=[21]-=-. Hence, in theory, as for classical Lyapunov functions, one could compute set Lyapunov functions using decision procedures for the theory of real-closed fields [34]. However, the current methods are ... |

59 | Constructing invariants for hybrid systems
- Sankaranarayanan, Sipma, et al.
(Show Context)
Citation Context ...algorithm. It seems that similar constraints have to be solved in many other areas (e.g., proving the termination of term-write systems, computation of barrier certificates [13], invariant generation =-=[21]-=-, [18], [9], [19], [14], and analysis of FEM [23]), and it is interesting work to apply our algorithms in these areas. We will further increase the efficiency of our method, for example, by improving ... |

58 | Subdivision direction selection in interval methods for global optimization - Csendes, Ratz - 1997 |

35 | Continuous and interval constraints. In
- Behamou, Granvilliers
- 2006
(Show Context)
Citation Context ...stead of using the relaxation technique described in this paper, it deduces information from the input constraints using a generalization of interval arithmetic called interval constraint propagation =-=[2]-=-. We simply added our relaxation technique to the branching loop of RSolver. As a result, we have an algorithm that can in some cases infer slightly more information from the input than Algorithm 1 du... |

34 | Automatic Generation of Polynomial Loop Invariants: Algebraic Foundations
- Rodriguez, Kapur
- 2004
(Show Context)
Citation Context ...thm. It seems that similar constraints have to be solved in many other areas (e.g., proving the termination of term-write systems, computation of barrier certificates [13], invariant generation [21], =-=[18]-=-, [9], [19], [14], and analysis of FEM [23]), and it is interesting work to apply our algorithms in these areas. We will further increase the efficiency of our method, for example, by improving the us... |

33 | Semidefinite programming relaxations and algebraic optimization in control,”
- Parrilo, Lall
- 2003
(Show Context)
Citation Context ... we choose B = [−0.8, 0.8]×[−0.8, 0.8], δ = 0.1 and ε = 0.000001, we get a relaxed Lyapunov function V (x, y) = x4 + 1.21822257384x3 + 0.60911128692x2 + 0.0250081223629y2 . Example 5: An example from =-=[11]-=- whose Lyapunov function has been constructed using the sum of squares decomposition: { ˙x = −x + (1 + x)y ˙y = −(1 + x)x Let V (x, y) = ax2 + bxy + cy2 + dy3 + ex4 + fx2y 2 + gy4 , then ˙ V (x, y) = ... |

32 | Efficient solving of quantified inequality constraints over the real numbers.
- Ratschan
- 2002
(Show Context)
Citation Context ..., the current methods are by far not efficient enough to solve this problem in practice. Another approach would be to use an interval arithmetic based branch-and-bound or branch-andprune scheme [15], =-=[17]-=-. However, one can do even better, as will be shown in the next section. III. ALGORITHM In this section, we present a method for finding a polynomial relaxed Lyapunov function V , provided that f is a... |

32 | Safe bounds in linear and mixedinteger programming
- Neumaier, Shcherbina
- 2004
(Show Context)
Citation Context ...terval constraint propagation. We solve the resulting linear programs using the GNU linear programming kit (GLPK). A user worried by resulting rounding errors could easily add verification techniques =-=[23, 17]-=- or use a linear programming implementation based on rational number arithmetic, instead. We use the following heuristic for branching: Choose the widest box, and bisect it into two boxes along the va... |

27 | Continuous first-order constraint satisfaction with equality and d isequality constraints
- Ratschan
- 2002
(Show Context)
Citation Context ...owever, the current methods are by far not efficient enough to solve this problem in practice. Another approach would be to use an interval arithmetic based branch-and-bound or branch-andprune scheme =-=[15]-=-, [17]. However, one can do even better, as will be shown in the next section. III. ALGORITHM In this section, we present a method for finding a polynomial relaxed Lyapunov function V , provided that ... |

25 | Computation of Lyapunov Functions for Smooth Nonlinear Systems using Convex Optimization.
- Johansen
- 2000
(Show Context)
Citation Context ...ygonal system characterization [24, 25]. Methods that use approximate discretizations are: • A method based on approximation by radial basis functions [14], and • a method based on linear programming =-=[18]-=-. Methods for computing the region of attraction can be roughly divided into two classes: one class maximizes the size of a region of attraction for a certain given Lyapunov function [42]; another cla... |

24 |
Rigorous lower and upper bounds in linear programming
- Jansson
- 2004
(Show Context)
Citation Context ...terval constraint propagation. We solve the resulting linear programs using the GNU linear programming kit (GLPK). A user worried by resulting rounding errors could easily add verification techniques =-=[23, 17]-=- or use a linear programming implementation based on rational number arithmetic, instead. We use the following heuristic for branching: Choose the widest box, and bisect it into two boxes along the va... |

21 | Computing polynomial program invariants
- Mller-Olm, Seidl
- 2004
(Show Context)
Citation Context ...t seems that similar constraints have to be solved in many other areas (e.g., proving the termination of term-write systems, computation of barrier certificates [13], invariant generation [21], [18], =-=[9]-=-, [19], [14], and analysis of FEM [23]), and it is interesting work to apply our algorithms in these areas. We will further increase the efficiency of our method, for example, by improving the used br... |

20 | Computer generated Lyapunov functions for a class of nonlinear systems
- Ohta, Imanishi, et al.
- 1993
(Show Context)
Citation Context ...ffine functions [15] based on user-provided bounds on second-order derivatives. • Methods that compute piecewise linear Lyapunov function based on some user-provided polygonal system characterization =-=[24, 25]-=-. Methods that use approximate discretizations are: • A method based on approximation by radial basis functions [14], and • a method based on linear programming [18]. Methods for computing the region ... |

19 | Model checking of hybrid systems: from reachability towards stability. - Podelski, Wagner - 2006 |

18 | Quantified constraints under perturbations,
- Ratschan
- 2002
(Show Context)
Citation Context ...es with a solution. Proof. Due to Lemma 2 of [26], an interval arithmetic based branch-andbound process using such a branching strategy succeeds in finding a solution 18sfor which the degree of truth =-=[22]-=- is positive. This precisely corresponds to robustness of the solution [22]. Since due to Theorem 3 our algorithm needs less iterations, it terminates under the above assumptions. � Since actual imple... |

17 | Generating polynomial invariants for hybrid systems
- Rodŕıguez-Carbonell, Tiwari
- 2005
(Show Context)
Citation Context ...ms that similar constraints have to be solved in many other areas (e.g., proving the termination of term-write systems, computation of barrier certificates [13], invariant generation [21], [18], [9], =-=[19]-=-, [14], and analysis of FEM [23]), and it is interesting work to apply our algorithms in these areas. We will further increase the efficiency of our method, for example, by improving the used branchin... |

16 | Construction of global Lyapunov functions using radial basis functions. - Giesl - 2007 |

13 |
Practical stability of nonlinear systems
- Laskhmikanthan, Leela, et al.
- 1990
(Show Context)
Citation Context ...in of attraction, we compute a Lyapunov-like function that ensures a basin of attraction to a target region containing the equilibrium. This can be formalized using some notion of practical stability =-=[7]-=-: Definition 1: Given an n-dimensional differential equation ˙x = f(x), and sets U and TR such that TR ⊂ U ⊆ R n , the differential equation is stable with respect to U and the target region TR if eve... |

12 |
Primal-dual tests for safety and reachability
- Prajna, Rantzer
- 2005
(Show Context)
Citation Context ...t similar constraints have to be solved in many other areas (e.g., proving the termination of term-write systems, computation of barrier certificates [13], invariant generation [21], [18], [9], [19], =-=[14]-=-, and analysis of FEM [23]), and it is interesting work to apply our algorithms in these areas. We will further increase the efficiency of our method, for example, by improving the used branching heur... |

11 |
Set invariance in control–a survey
- Blanchini
- 1999
(Show Context)
Citation Context ...ith a cycle TR Lyapunov function of the linearized system to show asymptotic stability [8, 15]. On the other hand, one can use techniques to prove that an invariant set of the system has been reached =-=[3]-=-. The question remains, how for a given set Lyapunov Function V (x) to find an s-sub-level set to which one can apply Corollary 1 to arrive at a basin of attraction. One approach follows an analogy of... |

11 | A constructive converse Lyapunov theorem on asymptotic stability for nonlinear autonomous ordinary differential equations. Dynamical Systems:
- Hafstein
- 2005
(Show Context)
Citation Context ...s some manual intervention to distinguish critical points from optima. Recently, new methods have been introduced for computing Lyapunov functions in the form of continuous piecewise affine functions =-=[10]-=- or using radial basis functions [9]. Methods for computing the region of attraction can be roughly divided into two classes: one class maximizes the size of a region of attraction for a certain given... |

9 |
QEPCAD B: a system for computing with semi-algebraic sets via cylindrical algebraic decomposition
- Brown
(Show Context)
Citation Context ...cients, decides whether there are instantiations of these parameters resulting in a Lyapunov function. However, all the existing decision procedures (e.g., implemented in the software packages QEPCAD =-=[4]-=- or REDLOG [11]), while being able to solve impressively difficult examples, are not efficient enough to be able to solve this problem in practice. Recently, a method based on sum of squares decomposi... |

8 | Construction of Lyapunov functions using Grobner bases,
- Forsman
- 1991
(Show Context)
Citation Context ...ed on sum of squares decomposition using relaxation to linear matrix inequalities [10], [11]. The other method is to use Gröbner bases to choose the parameters in Lyapunov functions in an optimal way =-=[4]-=-. This requires the computation of a Gröbner basis for an ideal with a large number variables, and requires some manual intervention to distinguish critical points from optima. Many methods proposed f... |

8 | Search heuristics for box decomposition methods - Ratschan |

7 | Inclusion functions and global optimization II - Moore, Ratschek - 1988 |

6 |
The behaviour of optimal Lyapunov functions
- Shields, Storey
- 1975
(Show Context)
Citation Context ...et a relaxed Lyapunov function V (x, y) = x2 − 0.70528691608xy +1.34394364271y2 −0.235095638693y3 + 0.17632172902x4 + 0.35264345804x2y2 + 0.17632172902y4 . Example 6: A three-dimensional example from =-=[22]-=-: ⎧ ⎪⎨ ˙x1 = −x2 ˙x2 = −x3 ⎪⎩ ˙x3 = −x1 − 2x2 − x3 + x 3 1 Let V (x1, x2, x3) = ax2 1 + bx2 2 + cx2 3 + dx1x2 + ex1x3 + fx2x3, then ˙ V (x1, x2, x3) = (−d + e)x2 1 − 2fx2 2 + (−2c − f)x2 3 + (−2a − 2e... |

5 |
On the estimation of asympototic stability regions: state of the art and new proposals
- Genesio, Tartaglia, et al.
- 1985
(Show Context)
Citation Context ...e a Lyapunov function of a linearization of the non-linear problem around a given equilibrium point, and compute a basin of attraction for this Lyapunov function wrt. the original, non-linear problem =-=[5]-=-. However, due the information loss introduced by the linearization process, this basin of attraction will usually be very small. In this paper we will provide an algorithm for computing a Lyapunov-li... |

4 |
The Role of State-space Partitioning in Automated Verification of Affine Hybrid System Stability
- Burchardt, Oehlerking, et al.
(Show Context)
Citation Context ...s. We will further increase the efficiency of our method, for example, by improving the used branching heuristics, and we will generalize our results to the stability analysis of hybrid systems [12], =-=[2]-=-. [12] A. Podelski and S. Wagner. Model checking of hybrid systems: From reachability towards stability. In J. Hespanha and A. Tiwari, editors, Hybrid Systems: Computation and Control, volume 3927 of ... |

4 |
Semidefinite programming relaxations and algebraic optimization in control
- Parrilo, Lall
- 2003
(Show Context)
Citation Context ...choose B = [−0.8, 0.8]×[−0.8, 0.8], δ = 0.1 and ε = 0.000001, we get a relaxed Lyapunov function V (x, y) = x 4 + 1.21822257384x 3 + 0.60911128692x 2 + 0.0250081223629y 2 . Example 5: An example from =-=[11]-=- whose Lyapunov function has been constructed using the sum of squares decomposition: � ˙x = −x + (1 + x)y ˙y = −(1 + x)x Let V (x, y) = ax2 + bxy + cy2 + dy3 + ex4 + fx2y 2 + gy4 , then ˙ V (x, y) = ... |

4 | Linear interval inequalities
- Rohn, Kreslová
(Show Context)
Citation Context ...ts of the polynomial. The algorithm for solving this constraint employs a branchand-relax scheme. It relaxes the constraint to a system of linear interval inequalities that can then be solved exactly =-=[20]-=-, and iteratively reduces the relaxation error by recursively decomposing the state space into hyper-rectangles. We implemented our algorithm and tested our implementation on eight examples. The struc... |

4 | Estimating the region of attraction of ordinary differential equations by quantified constraint solving,
- Burchardt, Ratschan
- 2007
(Show Context)
Citation Context ...n cases where an equilibrium point exists, one can use as a target region in our method a small basin of attraction computed from a Lyapunov function of the linearization of the differential equation =-=[8, 15, 5]-=-. 4sIn order to ensure this stability notion, we use the following adaption of the notion of a Lyapunov function: Definition 2 For a given differential equation ˙x = f(x) with sets B and TR such that ... |

4 | A posteriori direction selection rules for interval optimization methods - Csendes, Klatte, et al. - 2000 |

2 | Stability analysis of nonlinear systems using interval analysis
- Delanoue, Jaulin, et al.
- 2009
(Show Context)
Citation Context ...v function of a linearization of the non-linear problem around a given equilibrium point, and compute a basin of attraction for this Lyapunov function with respect to the original, non-linear problem =-=[13, 10]-=-. However, due to the information loss introduced by the linearization process, this basin of attraction will usually be very small. In this paper we will provide an algorithm for computing a Lyapunov... |

2 | Stability analysis by using piecewise linear Lyapunov functions
- Ohta, Onishi
- 1999
(Show Context)
Citation Context ...ffine functions [15] based on user-provided bounds on second-order derivatives. • Methods that compute piecewise linear Lyapunov function based on some user-provided polygonal system characterization =-=[24, 25]-=-. Methods that use approximate discretizations are: • A method based on approximation by radial basis functions [14], and • a method based on linear programming [18]. Methods for computing the region ... |

2 | Discrete maximum principle for higher-order nite elements in 1D
- ·Y, SOL·IN
(Show Context)
Citation Context ...ts have to be solved in many other areas (e.g., proving the termination of term-write systems, computation of barrier certificates [29], invariant generation [41, 38, 22, 39, 30], and analysis of FEM =-=[43]-=-), and it is interesting work to apply our algorithms in these areas. The presentation in this paper is restricted to polynomial ordinary differential equations and polynomial Lyapunov-like functions.... |

1 |
On a discrete maximum principle for onedimensional hp-FEM
- ˇSolín, Vejchodsk´y
(Show Context)
Citation Context ... to be solved in many other areas (e.g., proving the termination of term-write systems, computation of barrier certificates [13], invariant generation [21], [18], [9], [19], [14], and analysis of FEM =-=[23]-=-), and it is interesting work to apply our algorithms in these areas. We will further increase the efficiency of our method, for example, by improving the used branching heuristics, and we will genera... |

1 |
Delanoue, Luc Jaulin. Stability analysis of a nonlinear system using interval analysis
- Nicolas
(Show Context)
Citation Context ...e a Lyapunov function of a linearization of the non-linear problem around a given equilibrium point, and compute a basin of attraction for this Lyapunov function wrt. the original, non-linear problem =-=[8, 15]-=-. However, due to the information loss introduced by the linearization process, this basin of attraction will usually be very small. In this paper we will provide an algorithm for computing a Lyapunov... |