### Citations

477 |
Quantifier elimination for real closed fields by cylindric algebraic decomposition.
- Collins
- 1975
(Show Context)
Citation Context ...nt formula always exists. In fact, it can even be computed algorithmically. There are tools (some are freely available) that perform quantifier elimination in the theory of reals, such as qepcad [9], =-=[10]-=-, [3], [11], [6], and the computer algebra systems redlog [14] and Mathematica. However, quantifier elimination is a computationally hard problem: the time complexity of real quantifier elimination is... |

465 | HyTech: A Model Checker for Hybrid Systems. In CAV.
- Henzinger, Ho, et al.
- 1997
(Show Context)
Citation Context ...lant Block diagram of a PI Controller and a plant system. 1) reach-set methods that compute the set of all reachable states of the system, either exactly [20], or approximately [7], [35], [19], [12], =-=[17]-=-, [15] 2) abstraction-based methods that first abstract the system and then analyze the abstraction [33], [1], [8] 3) certificate-based methods that directly search for certificates of correctness (su... |

432 |
Nonlinear Systems Analysis.
- Vidyasagar
- 1993
(Show Context)
Citation Context ...estion we answer in this paper is the following: what is the class of plant models that can be stabilized by a given PI controller. This problem is related to the so-called absolute stability problem =-=[37]-=-, [22], but there are important differences that we will discuss later. We focus on systems of the form shown in Figure 1. We assume the PI controller is being used to drive the plant to a given setpo... |

373 |
Linear matrix inequalities in systems and control theory.
- Boyd, LE, et al.
- 1994
(Show Context)
Citation Context ...n [33], [1], [8] 3) certificate-based methods that directly search for certificates of correctness (such as inductive invariants and Lyapunov functions) of systems [30], [25], [27], [23], [16], [31], =-=[4]-=-, [24] While all these techniques have had some success, the certificate-based methods are turning out to be particularly effective in proving deep properties of complex systems. Certificate-based met... |

225 |
Partial cylindrical algebraic decomposition for quantifier elimination
- Collins, Hong
- 1991
(Show Context)
Citation Context ...always exists. In fact, it can even be computed algorithmically. There are tools (some are freely available) that perform quantifier elimination in the theory of reals, such as qepcad [9], [10], [3], =-=[11]-=-, [6], and the computer algebra systems redlog [14] and Mathematica. However, quantifier elimination is a computationally hard problem: the time complexity of real quantifier elimination is (both lowe... |

211 | PHAVer: algorithmic verification of hybrid systems past HyTech.
- Frehse
- 2005
(Show Context)
Citation Context ...lock diagram of a PI Controller and a plant system. 1) reach-set methods that compute the set of all reachable states of the system, either exactly [20], or approximately [7], [35], [19], [12], [17], =-=[15]-=- 2) abstraction-based methods that first abstract the system and then analyze the abstraction [33], [1], [8] 3) certificate-based methods that directly search for certificates of correctness (such as ... |

162 | On ellipsoidal techniques for reachability analysis.”
- Kurzhanski, Varaiya
- 2000
(Show Context)
Citation Context ...Controller Plant Block diagram of a PI Controller and a plant system. 1) reach-set methods that compute the set of all reachable states of the system, either exactly [20], or approximately [7], [35], =-=[19]-=-, [12], [17], [15] 2) abstraction-based methods that first abstract the system and then analyze the abstraction [33], [1], [8] 3) certificate-based methods that directly search for certificates of cor... |

129 | REDLOG: Computer algebra meets computer logic
- Dolzmann, Sturm
- 1997
(Show Context)
Citation Context ...orithmically. There are tools (some are freely available) that perform quantifier elimination in the theory of reals, such as qepcad [9], [10], [3], [11], [6], and the computer algebra systems redlog =-=[14]-=- and Mathematica. However, quantifier elimination is a computationally hard problem: the time complexity of real quantifier elimination is (both lower and upper) bounded by a double exponential functi... |

122 |
Real quantifier elimination is doubly exponential.
- Davenport, Heintz
- 1988
(Show Context)
Citation Context ...ation is a computationally hard problem: the time complexity of real quantifier elimination is (both lower and upper) bounded by a double exponential function in the length of the input formula [38], =-=[13]-=-. As a result, when faced with problems that contains large number of variables (about 10) with large (about 3) degrees, quantifier elimination tools can fail to produce a result. In our experiments, ... |

98 |
The complexity of linear problems in fields.
- Weispfenning
- 1988
(Show Context)
Citation Context ...elimination is a computationally hard problem: the time complexity of real quantifier elimination is (both lower and upper) bounded by a double exponential function in the length of the input formula =-=[38]-=-, [13]. As a result, when faced with problems that contains large number of variables (about 10) with large (about 3) degrees, quantifier elimination tools can fail to produce a result. In our experim... |

96 | QEPCAD B: a program for computing with semi-algebraic sets using CADs
- Brown
(Show Context)
Citation Context ... exists. In fact, it can even be computed algorithmically. There are tools (some are freely available) that perform quantifier elimination in the theory of reals, such as qepcad [9], [10], [3], [11], =-=[6]-=-, and the computer algebra systems redlog [14] and Mathematica. However, quantifier elimination is a computationally hard problem: the time complexity of real quantifier elimination is (both lower and... |

92 |
Cylindrical algebraic decomposition I: The basic algorithm
- Arnon, Collins, et al.
- 1984
(Show Context)
Citation Context ...mula always exists. In fact, it can even be computed algorithmically. There are tools (some are freely available) that perform quantifier elimination in the theory of reals, such as qepcad [9], [10], =-=[3]-=-, [11], [6], and the computer algebra systems redlog [14] and Mathematica. However, quantifier elimination is a computationally hard problem: the time complexity of real quantifier elimination is (bot... |

90 | Series of Abstractions for Hybrid Automata.
- Tiwari, Khanna
- 2002
(Show Context)
Citation Context ...ll reachable states of the system, either exactly [20], or approximately [7], [35], [19], [12], [17], [15] 2) abstraction-based methods that first abstract the system and then analyze the abstraction =-=[33]-=-, [1], [8] 3) certificate-based methods that directly search for certificates of correctness (such as inductive invariants and Lyapunov functions) of systems [30], [25], [27], [23], [16], [31], [4], [... |

89 | Reachability analysis via face lifting.
- Dang, Maler
- 1998
(Show Context)
Citation Context ...ller Plant Block diagram of a PI Controller and a plant system. 1) reach-set methods that compute the set of all reachable states of the system, either exactly [20], or approximately [7], [35], [19], =-=[12]-=-, [17], [15] 2) abstraction-based methods that first abstract the system and then analyze the abstraction [33], [1], [8] 3) certificate-based methods that directly search for certificates of correctne... |

89 | Symbolic reachability computation for families of linear vector fields
- Lafferriere, Pappas, et al.
(Show Context)
Citation Context ...shish.tiwari@sri.com Fig. 1. x PI Controller Plant Block diagram of a PI Controller and a plant system. 1) reach-set methods that compute the set of all reachable states of the system, either exactly =-=[20]-=-, or approximately [7], [35], [19], [12], [17], [15] 2) abstraction-based methods that first abstract the system and then analyze the abstraction [33], [1], [8] 3) certificate-based methods that direc... |

89 | Safety verification of hybrid systems using barrier certificates
- Prajna, Jadbabaie
(Show Context)
Citation Context ...nd then analyze the abstraction [33], [1], [8] 3) certificate-based methods that directly search for certificates of correctness (such as inductive invariants and Lyapunov functions) of systems [30], =-=[25]-=-, [27], [23], [16], [31], [4], [24] While all these techniques have had some success, the certificate-based methods are turning out to be particularly effective in proving deep properties of complex s... |

86 | Applying linear quantifier elimination.
- Loos, Weispfenning
- 1993
(Show Context)
Citation Context ...a recent paper [28]. Our approach can be briefly described as follows: (1) First, we use the virtual substitution method of redlog to eliminate quantified variables. Virtual substitution method [38], =-=[21]-=-, [39] can eliminate variables whose degrees are bounded by 2, and hence it can fail to eliminate some variables. However, virtual substitution is an efficient method that scales to large formulas. (2... |

79 |
Frequency Domain Criteria for Absolute Stability
- Narendra, Taylor
- 1973
(Show Context)
Citation Context ... we answer in this paper is the following: what is the class of plant models that can be stabilized by a given PI controller. This problem is related to the so-called absolute stability problem [37], =-=[22]-=-, but there are important differences that we will discuss later. We focus on systems of the form shown in Figure 1. We assume the PI controller is being used to drive the plant to a given setpoint. W... |

76 | Differential dynamic logic for hybrid systems.,
- Platzer
- 2008
(Show Context)
Citation Context ...], [1], [8] 3) certificate-based methods that directly search for certificates of correctness (such as inductive invariants and Lyapunov functions) of systems [30], [25], [27], [23], [16], [31], [4], =-=[24]-=- While all these techniques have had some success, the certificate-based methods are turning out to be particularly effective in proving deep properties of complex systems. Certificate-based methods w... |

70 | Computational techniques for the verification of hybrid systems.
- Tomlin, Mitchell, et al.
- 2003
(Show Context)
Citation Context ... x PI Controller Plant Block diagram of a PI Controller and a plant system. 1) reach-set methods that compute the set of all reachable states of the system, either exactly [20], or approximately [7], =-=[35]-=-, [19], [12], [17], [15] 2) abstraction-based methods that first abstract the system and then analyze the abstraction [33], [1], [8] 3) certificate-based methods that directly search for certificates ... |

62 | Reachability analysis of hybrid systems via predicate abstraction
- Alur, Dang, et al.
- 2002
(Show Context)
Citation Context ...chable states of the system, either exactly [20], or approximately [7], [35], [19], [12], [17], [15] 2) abstraction-based methods that first abstract the system and then analyze the abstraction [33], =-=[1]-=-, [8] 3) certificate-based methods that directly search for certificates of correctness (such as inductive invariants and Lyapunov functions) of systems [30], [25], [27], [23], [16], [31], [4], [24] W... |

59 | Constructing invariants for hybrid systems
- Sankaranarayanan, Sipma, et al.
(Show Context)
Citation Context ...n analyze the abstraction [33], [1], [8] 3) certificate-based methods that directly search for certificates of correctness (such as inductive invariants and Lyapunov functions) of systems [30], [25], =-=[27]-=-, [23], [16], [31], [4], [24] While all these techniques have had some success, the certificate-based methods are turning out to be particularly effective in proving deep properties of complex systems... |

44 |
Approximate reachability for linear systems
- Tiwari
(Show Context)
Citation Context ...stem and then analyze the abstraction [33], [1], [8] 3) certificate-based methods that directly search for certificates of correctness (such as inductive invariants and Lyapunov functions) of systems =-=[30]-=-, [25], [27], [23], [16], [31], [4], [24] While all these techniques have had some success, the certificate-based methods are turning out to be particularly effective in proving deep properties of com... |

40 | Constraint-based approach for analysis of hybrid systems
- Gulwani, Tiwari
- 2008
(Show Context)
Citation Context ...e abstraction [33], [1], [8] 3) certificate-based methods that directly search for certificates of correctness (such as inductive invariants and Lyapunov functions) of systems [30], [25], [27], [23], =-=[16]-=-, [31], [4], [24] While all these techniques have had some success, the certificate-based methods are turning out to be particularly effective in proving deep properties of complex systems. Certificat... |

39 | Nonlinear control system design by quantifier elimination
- Jirstrand
- 1997
(Show Context)
Citation Context ...em does not give any useful information when a < 1/25000. We now briefly mention other related work in the literature. Quantifier elimination has been applied to solve nonlinear control system design =-=[18]-=- before. However, most of this early work focused only on continuous dynamical systems (and not switched systems), and moreover, only on simple properties. Properties like safety and stability were no... |

39 | Nonlinear systems: Approximating reach sets
- Tiwari, Khanna
- 2004
(Show Context)
Citation Context ...fety, there is also work on solving other (region-of-attraction) problems using sum-of-squares programming [36] There is plenty of work on certificate-based verification of hybrid systems [30], [27], =-=[34]-=-, [16], but none of it has usedreal quantifier elimination and instead relied on approximate methods to eliminate quantifiers. In a recent paper [29], we used qepcad to solve ∃∀ formulas arising from... |

39 |
Quantifier elimination for real algebra—the quadratic case and beyond. Applicable Algebra in Engineering Communication and Computing
- Weispfenning
- 1997
(Show Context)
Citation Context ...nt paper [28]. Our approach can be briefly described as follows: (1) First, we use the virtual substitution method of redlog to eliminate quantified variables. Virtual substitution method [38], [21], =-=[39]-=- can eliminate variables whose degrees are bounded by 2, and hence it can fail to eliminate some variables. However, virtual substitution is an efficient method that scales to large formulas. (2) The ... |

37 | Verification of hybrid systems based on counterexample-guided abstraction refinement
- CLARKE, FEHNKER, et al.
(Show Context)
Citation Context ...e states of the system, either exactly [20], or approximately [7], [35], [19], [12], [17], [15] 2) abstraction-based methods that first abstract the system and then analyze the abstraction [33], [1], =-=[8]-=- 3) certificate-based methods that directly search for certificates of correctness (such as inductive invariants and Lyapunov functions) of systems [30], [25], [27], [23], [16], [31], [4], [24] While ... |

33 | Computing polyhedral approximations to flow pipes for dynamic systems,” in
- Chutinan, Krogh
- 1998
(Show Context)
Citation Context ...g. 1. x PI Controller Plant Block diagram of a PI Controller and a plant system. 1) reach-set methods that compute the set of all reachable states of the system, either exactly [20], or approximately =-=[7]-=-, [35], [19], [12], [17], [15] 2) abstraction-based methods that first abstract the system and then analyze the abstraction [33], [1], [8] 3) certificate-based methods that directly search for certifi... |

13 | A.: Verification and synthesis using real quantifier elimination
- Sturm, Tiwari
- 2011
(Show Context)
Citation Context ...mber of variables (about 10) with large (about 3) degrees, quantifier elimination tools can fail to produce a result. In our experiments, we used a combination approach as described in a recent paper =-=[28]-=-. Our approach can be briefly described as follows: (1) First, we use the virtual substitution method of redlog to eliminate quantified variables. Virtual substitution method [38], [21], [39] can elim... |

13 | A.: Switching logic synthesis for reachability
- Taly, Tiwari
(Show Context)
Citation Context ...-based verification of hybrid systems [30], [27], [34], [16], but none of it has usedreal quantifier elimination and instead relied on approximate methods to eliminate quantifiers. In a recent paper =-=[29]-=-, we used qepcad to solve ∃∀ formulas arising from certificatebased synthesis. Recently, Anai [2] used a combination of numerical methods (sum-of-squares) and symbolic quantifier elimination methods t... |

11 | Stability region analysis using simulations and sum-of-squares programming
- Topcu, Packard, et al.
- 2007
(Show Context)
Citation Context ...icates [25] moves this overall approach from stability to safety. Apart from stability and safety, there is also work on solving other (region-of-attraction) problems using sum-of-squares programming =-=[36]-=- There is plenty of work on certificate-based verification of hybrid systems [30], [27], [34], [16], but none of it has usedreal quantifier elimination and instead relied on approximate methods to el... |

4 |
SLFQ (Simplifying Large Formulas with QEPCAD B). http://www.cs.usna.edu/~qepcad/SLFQ/Home.html
- Brown
(Show Context)
Citation Context ...tion plots consider the case when the PI controller parameters are not fixed, but they vary with time. Specifically, Kp is still fixed to 500, but Ki is set to 100 ∗ t, where t is the time. Since t ∈ =-=[0, 5]-=- in our simulation, we always satisfy the condition that Ki ≤ Kp. We use the same three plant models. Simulations with the varying parameter PI controller are shown in Figure 3. Note that the error er... |

3 |
A symbolic-numeric approach to nonlinear gain analaysis of a class of dynamical systems
- Ichihara, Anai
(Show Context)
Citation Context ...ier elimination and instead relied on approximate methods to eliminate quantifiers. In a recent paper [29], we used qepcad to solve ∃∀ formulas arising from certificatebased synthesis. Recently, Anai =-=[2]-=- used a combination of numerical methods (sum-of-squares) and symbolic quantifier elimination methods to solve problems arising in control, and such an integration is left for future work here. VI. CO... |

3 |
SOS methods for semi-algebraic games and optimization
- Parrilo
- 2005
(Show Context)
Citation Context ...yze the abstraction [33], [1], [8] 3) certificate-based methods that directly search for certificates of correctness (such as inductive invariants and Lyapunov functions) of systems [30], [25], [27], =-=[23]-=-, [16], [31], [4], [24] While all these techniques have had some success, the certificate-based methods are turning out to be particularly effective in proving deep properties of complex systems. Cert... |

3 | Generating box invariants
- Tiwari
- 2008
(Show Context)
Citation Context ...raction [33], [1], [8] 3) certificate-based methods that directly search for certificates of correctness (such as inductive invariants and Lyapunov functions) of systems [30], [25], [27], [23], [16], =-=[31]-=-, [4], [24] While all these techniques have had some success, the certificate-based methods are turning out to be particularly effective in proving deep properties of complex systems. Certificate-base... |

2 |
Certificate-based verification: Tools and benchmarks
- Tiwari
- 2011
(Show Context)
Citation Context ...630 2.53GHz single-core processor (x86 64 arch) with 4G RAM running Ubuntu Linux 2.6.32-26.) The scripts containing quantifier elimination problems generated in the experiments are publicly available =-=[32]-=-. V. RELATED WORK Our results are analogous to the results on absolute stability and the Lur’e problem [37], [22]. In fact, in retrospect, now that we know the result of Theorem 1 and Theorem 2, we ca... |

1 |
The SAC-1 polynomial system
- Collins
- 1968
(Show Context)
Citation Context ...ivalent formula always exists. In fact, it can even be computed algorithmically. There are tools (some are freely available) that perform quantifier elimination in the theory of reals, such as qepcad =-=[9]-=-, [10], [3], [11], [6], and the computer algebra systems redlog [14] and Mathematica. However, quantifier elimination is a computationally hard problem: the time complexity of real quantifier eliminat... |

1 |
http://www.cds. caltech.edu/sostools
- Prajna, Papachristodoulou, et al.
- 2002
(Show Context)
Citation Context ...ratic Lyapunov function) and the solver can find a concrete quadratic function that proves stability of the given system. Most commonly used solvers include ones for sums-of-squares (SOS) programming =-=[26]-=- and real quantifier elimination. Numeric approaches, such as SOS programming, have two limitations: first, they may give incorrect answers, and second, they have their limitations when solving Boolea... |