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## Approximately bisimilar symbolic models for nonlinear control systems (2007)

Venue: | In 46th IEEE Conference on Decision and Control |

Citations: | 52 - 19 self |

### Citations

3945 |
Communication and Concurrency
- Milner
- 1989
(Show Context)
Citation Context ...community with the purpose of mitigating the complexity of software verification. A central notion when dealing with complexity reduction, is the one of bisimulation equivalence, introduced by Milner =-=[16]-=- and Park [17] in the 80s’. The key idea is to find and compute an equivalence relation on the state space of the system, that respects the system dynamics. This equivalence relation induces a new sys... |

3248 | Model Checking
- Clarke, Grumberg, et al.
- 1999
(Show Context)
Citation Context ...lent to T(Σ). The notion of equivalence that we consider is the one of bisimulation equivalence [16, 17]. Bisimulation relations are standard mechanisms to relate the properties of transition systems =-=[8]-=-. Intuitively, a bisimulation relation between a pair of transition systems T1 and T2 is a relation between the corresponding state sets explaining how a sequence of transitions r1 of T1 can be transf... |

2650 | Timed Automata
- Alur
- 1999
(Show Context)
Citation Context ...olic models allows one to leverage the rich literature on supervisory control [19] and algorithmic approaches to game theory [27, 5], for controller design. After the pioneering work of Alur and Dill =-=[1]-=- that showed existence of symbolic models for timed automata, researchers tried to identify more general classes of continuous systems admitting finite bisimulations. For example, [2] showed that o–mi... |

2000 |
Nonlinear Systems
- Khalil
- 1996
(Show Context)
Citation Context ...llowing condition is satisfied: (2.1) ‖x(t, x,u) − x(t, y,u)‖ ≤ β(‖x − y‖,t). Definition above can be thought of as an incremental version of the classical notion of global asymptotic stability (GAS) =-=[13]-=-. Furthermore when f satisfies f(0, 0) = 0, δ–GAS implies GAS of Σ with U = {0}, by just comparing a trajectory of Σ with any x ∈ Rn and u(t) = 0, t ∈ R + 0 , with the null trajectory x(t) = 0, t ∈ R ... |

709 |
Supervisory control of a class of discrete event processes,
- Ramadge, Wonham
- 1987
(Show Context)
Citation Context ...ntinuous systems as well as, hardware and software interacting with the physical environment. Furthermore, the use of symbolic models allows one to leverage the rich literature on supervisory control =-=[19]-=- and algorithmic approaches to game theory [27, 5], for controller design. After the pioneering work of Alur and Dill [1] that showed existence of symbolic models for timed automata, researchers tried... |

354 | Qualitative Reasoning: modeling and simulation with incomplete knowledge - �Kuipers - 1994 |

327 | Discrete abstractions of hybrid systems.
- Alur, Henzinger, et al.
- 2000
(Show Context)
Citation Context ...of Alur and Dill [1] that showed existence of symbolic models for timed automata, researchers tried to identify more general classes of continuous systems admitting finite bisimulations. For example, =-=[2]-=- showed that o–minimal hybrid systems admit symbolic models, while discrete–time control systems have been considered in [24, 22]. In particular [24] showed that discrete–time controllable linear syst... |

169 | Infinite Games on Finitely Coloured Graphs with Applications to Automata on Infinite Trees.
- Zielonka
- 1998
(Show Context)
Citation Context ...are interacting with the physical environment. Furthermore, the use of symbolic models allows one to leverage the rich literature on supervisory control [19] and algorithmic approaches to game theory =-=[27, 5]-=-, for controller design. After the pioneering work of Alur and Dill [1] that showed existence of symbolic models for timed automata, researchers tried to identify more general classes of continuous sy... |

123 | SOSTOOLS: Sum of squares optimization toolbox for MATLAB
- Prajna, Papachristodoulou, et al.
- 2004
(Show Context)
Citation Context ...f the control system. The computation of δ–ISS Lyapunov functions is in general a hard task. However, one can resort to numerical tools available in the literature, as for example the one proposed in =-=[PPP02]-=-. Furthermore, a way for mitigating the exponential grow in the sizes of Q2 and L2 is to adapt techniques from on-the-fly verification of transition systems [TA99] to the construction of the proposed ... |

119 | O-minimal hybrid systems.
- Lafferriere, Pappas, et al.
- 2000
(Show Context)
Citation Context ... q‖, τ) + η/2 ≤ β(ε, τ) + η/2 ≤ ε. This result represents a substantial improvement over previously known classes of control systems admitting symbolic models, which included o–minimal hybrid systems =-=[14]-=-, controllable linear systems in discrete– time [22] and stable linear systems in discrete–time [9]. However Theorem 4.2 relates Tτ(Σ) to the symbolic model in (4.4), whose construction is in general ... |

117 | Games for synthesis of controllers with partial observation.
- Arnold, Vincent, et al.
- 2003
(Show Context)
Citation Context ...are interacting with the physical environment. Furthermore, the use of symbolic models allows one to leverage the rich literature on supervisory control [19] and algorithmic approaches to game theory =-=[27, 5]-=-, for controller design. After the pioneering work of Alur and Dill [1] that showed existence of symbolic models for timed automata, researchers tried to identify more general classes of continuous sy... |

113 | A Lyapunov approach to incremental stability properties.
- Angeli
- 2002
(Show Context)
Citation Context ...ented in this paper will assume certain stability assumptions on the control systems. We briefly recall these notions and the results that will be used in the subsequent developments. Definition 2.2. =-=[3]-=- A control system Σ is incrementally globally asymptotically stable (δ–GAS) if it is forward complete and there exist a KL function β such that for any t ∈ R + 0 , any x, y ∈ Rn and any u ∈ U the foll... |

105 | Approximation metrics for discrete and continuous systems,”
- Girard, Pappas
- 2007
(Show Context)
Citation Context ...n this paper and the above lines of research. In this paper we follow the line of research based on simulation/bisimulation by making use of the recently introduced notion of approximate bisimulation =-=[GP07]-=-, that captures equivalence of systems in an approximate setting. By relaxing the usual notion of bisimulation to approximate bisimulation, a larger class of control systems can be expected to admit s... |

97 | Reachability of uncertain linear systems using zonotopes.
- Girard
- 2005
(Show Context)
Citation Context ...tion of the symbolic models can be performed even for non-constant inputs. This can be done by using results on polytopic approximation of reachable sets for linear control systems (see e.g. [Var98], =-=[Gir05]-=-) with compact input space. It is known from [Var98] that for any desired precision ν ∈ R + , the reachable set R(τ, q) of (4.2) can be approximated by a polytope P (τ, q), so that dh(P (τ, q), R(τ, q... |

96 | Input to state stability: basic concepts and results.
- Sontag
- 2007
(Show Context)
Citation Context ...tions of Theorems 4.2 and 5.1 simplify. In fact given a linear control system: ˙x = Ax + Bu, x ∈ R n , u ∈ U ⊆ R m , notions of δ–GAS and δ–ISS reduce to asymptotic stability of matrix A; moreover by =-=[20]-=-, functions β and γ appearing in inequalities (2.1) and (2.2) can be chosen as: β(r, s) = ‖e As ( ∫ ∞ ‖r; γ(r) = ‖B‖ ‖e As ) (7.1) ‖ds r. Notice that functions in (7.1) are well defined, because of as... |

77 | Forward completeness, unboundedness observability, and their Lyapunov characterizations
- Angeli, Sontag
- 1999
(Show Context)
Citation Context ...ntrol system Σ is said to be forward complete if every trajectory is defined on an interval of the form ]a, ∞[. Sufficient and necessary conditions for a system to be forward complete can be found in =-=[4]-=-. Simpler, but only sufficient, conditions for forward completeness are also available in the literature. These include linear growth or compact support of the vector field (see e.g. [15]). 2.3. Stabi... |

61 | G.J.: Linear time logic control of discrete-time linear systems.
- Tabuada, Pappas
- 2006
(Show Context)
Citation Context ... classes of continuous systems admitting finite bisimulations. For example, [2] showed that o–minimal hybrid systems admit symbolic models, while discrete–time control systems have been considered in =-=[24, 22]-=-. In particular [24] showed that discrete–time controllable linear systems admit symbolic models. The common denominator of the work [1, 2, 24, 22] is the employed notion of bisimulation that is essen... |

50 | Reach set computation using optimal control.
- Varaiya
- 1998
(Show Context)
Citation Context ...truction of the symbolic model of (4.4) can be performed for this class of systems. This can be done by using results on polytopic approximation of reachable sets for linear control systems (see e.g. =-=[26]-=-, [12]) with compact input space. In fact by following [26], given any desired precision ν ∈ R + , the reachable set R(τ, q) of (4.2) can be approximated by a polytope P(τ, q), so that: dh(P(τ, q), R(... |

46 |
On the reachability of quantized control systems,”
- Bicchi, Marigo, et al.
- 2002
(Show Context)
Citation Context ...lic model can be obtained by quantizing the space of inputs. As a by–product, our results also shed some light into the construction of finite abstractions in the context of quantized control systems =-=[6, 7]-=-. Indeed, by performing a quantization on the input space, we can guarantee that the resulting symbolic model admits a lattice structure in the state space. Similar ideas to the ones of this paper, ha... |

45 | Symbolic models for nonlinear control systems: Alternating approximate bisimulations,”
- Pola, Tabuada
- 2009
(Show Context)
Citation Context ...obtained by quantizing the space of inputs. As an illustrative example, we apply the proposed techniques to a control design problem for a pendulum. A preliminary version of these results appeared in =-=[PGT07]-=-. 2.1. Notations. The symbols N, Z, R, R + and R + 0 2. Control systems and stability notions denote the natural, integers, real, positive and nonnegative real numbers, respectively. Given a vector x ... |

39 | On-the-fly controller synthesis for discrete and timed systems. - Tripakis, Altisen - 1999 |

31 | Approximate bisimulations for nonlinear dynamical systems
- Pappas
- 2005
(Show Context)
Citation Context ...], [11] and [23] to reduce continuous control systems to continuous control systems. A new twist in this research line has been recently given by the so–called approximate bisimulation, introduced in =-=[10]-=-, that captures equivalence of systems in an approximate setting. By relaxing the usual notion of bisimulation to approximate bisimulation, a larger class of control systems can be expected to admit s... |

28 |
der Schaft, “Equivalence of dynamical systems by bisimulation
- van
- 2004
(Show Context)
Citation Context ...common denominator of the work [1, 2, 24, 22] is the employed notion of bisimulation that is essentially equivalent to the classical one of Milner [16] and Park [17]. The same notion has been used in =-=[25]-=-, [11] and [23] to reduce continuous control systems to continuous control systems. A new twist in this research line has been recently given by the so–called approximate bisimulation, introduced in [... |

25 | Bisimilar control affine systems
- Tabuada, Pappas
- 2004
(Show Context)
Citation Context ...tor of the work [1, 2, 24, 22] is the employed notion of bisimulation that is essentially equivalent to the classical one of Milner [16] and Park [17]. The same notion has been used in [25], [11] and =-=[23]-=- to reduce continuous control systems to continuous control systems. A new twist in this research line has been recently given by the so–called approximate bisimulation, introduced in [10], that captu... |

21 | Approximate Simulation Relations and Finite Abstractions of Quantized Control Systems,
- Tabuada
- 2007
(Show Context)
Citation Context ...ms in an approximate setting. By relaxing the usual notion of bisimulation to approximate bisimulation, a larger class of control systems can be expected to admit symbolic models. In fact the work in =-=[21]-=- shows that for every asymptotically stabilizable control system it is possible to construct a symbolic model, which is based on an approximate notion of simulation (one–sided version of This work has... |

21 |
Mathematical Control Theory, volume 6 of Texts in Applied Mathematics
- Sontag
- 1998
(Show Context)
Citation Context ...to denote the point reached at time t ∈]a, b[ under the input u from initial condition x; this point is uniquely determined, since the assumptions on f ensure existence and uniqueness of trajectories =-=[Son98]-=-. A control system Σ is said to be forward complete if every trajectory is defined on an interval of the form ]a, ∞[. Sufficient and necessary conditions for a system to be forward complete can be fou... |

20 | Approximately bisimilar finite abstractions of stable linear systems.
- Girard
- 2007
(Show Context)
Citation Context ...ntization on the input space, we can guarantee that the resulting symbolic model admits a lattice structure in the state space. Similar ideas to the ones of this paper, have been recently explored in =-=[9]-=- for the class of discrete–time linear control systems. A discussion on relationships between our results and the ones in [9], can be found in the last section of this paper. A preliminary version of ... |

20 |
Concurrency and Automata on Infinite Sequences, volume 104 of LNCS
- Park
- 1980
(Show Context)
Citation Context ... the purpose of mitigating the complexity of software verification. A central notion when dealing with complexity reduction, is the one of bisimulation equivalence, introduced by Milner [16] and Park =-=[17]-=- in the 80s’. The key idea is to find and compute an equivalence relation on the state space of the system, that respects the system dynamics. This equivalence relation induces a new system on the quo... |

16 | Modelling, State Observation and Diagnosis of Quantised Systems, - Schroder - 2002 |

11 |
Feedback encoding for efficient symbolic control of dynamical systems,” Automatic Control,
- Bicchi, Marigo, et al.
- 2006
(Show Context)
Citation Context ...lic model can be obtained by quantizing the space of inputs. As a by–product, our results also shed some light into the construction of finite abstractions in the context of quantized control systems =-=[6, 7]-=-. Indeed, by performing a quantization on the input space, we can guarantee that the resulting symbolic model admits a lattice structure in the state space. Similar ideas to the ones of this paper, ha... |

9 | Polytopic Approximations of Reachable Sets Applied to Linear Dynamic Games and to a Class of Nonlinear Systems,”
- Hwang, Stipanovic, et al.
- 2005
(Show Context)
Citation Context ...on of the symbolic model of (4.4) can be performed for this class of systems. This can be done by using results on polytopic approximation of reachable sets for linear control systems (see e.g. [26], =-=[12]-=-) with compact input space. In fact by following [26], given any desired precision ν ∈ R + , the reachable set R(τ, q) of (4.2) can be approximated by a polytope P(τ, q), so that: dh(P(τ, q), R(τ, q))... |

9 | Representation of Hybrid Systems by means of Stochastic Automata - Lunze, Nixdorf |

9 | Equivalence of dynamical systems by bisimulation. - Schaft - 2004 |

6 |
Foundations of optimal control theory. The SIAM series in applied mathematics
- Lee, Markus
- 1967
(Show Context)
Citation Context ...n be found in [4]. Simpler, but only sufficient, conditions for forward completeness are also available in the literature. These include linear growth or compact support of the vector field (see e.g. =-=[15]-=-). 2.3. Stability notions. The results presented in this paper will assume certain stability assumptions on the control systems. We briefly recall these notions and the results that will be used in th... |

6 | Equivalence of switching linear systems by bisimulation - Pola, Schaft, et al. - 2006 |

5 | Symbolic models for control systems - Tabuada - 2007 |

3 |
Simulation and bisimulation of nonlinear control systems with admissible classes of inputs and disturbances
- Grasse
(Show Context)
Citation Context ... denominator of the work [1, 2, 24, 22] is the employed notion of bisimulation that is essentially equivalent to the classical one of Milner [16] and Park [17]. The same notion has been used in [25], =-=[11]-=- and [23] to reduce continuous control systems to continuous control systems. A new twist in this research line has been recently given by the so–called approximate bisimulation, introduced in [10], t... |

3 | Qualitative heterogeneous control of higher order systems - Ramamoorthy, Kuipers - 2003 |