G. J. Pappas, G. Lafferriere, and S. Sastry. Hierarchically consistent control systems. Technical Report UCB/ERL M98/16, University of California at Berkeley, Berkeley, CA, April 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....systems. The interest in low complexity representations of mechanical control systems can be related to numerous previous e#orts, including work on hybrid models for motion control systems [1] motion description languages [2] oscillatory motion primitives [3] consistent control abstractions [4], hierarchical steering algorithms [5] and maneuver automata [6] In Section 3, we introduce the notion of kinematic reduction as a model reduction technique adapted to mechanical control systems. This novel concept extends and unifies our previous results in [7, 8] A kinematic model for a ....

G. J. Pappas, G. La#erriere, and S. S. Sastry, "Hierarchically consistent control systems," IEEE Transactions on Automatic Control, vol. 45, no. 6, pp. 1144--60, 2000.

....at the lower level while incorporating modeling detail. Purely discrete abstractions of continuous systems have been considered in [2, 4] Hierarchical systems for discrete event systems have been formally considered in [10] Continuous abstractions of continuous systems is avery recent activity[7]. More precisely, for linear control systems the abstraction problem is formulated as follows. Problem 1.1 [Linear Abstractions( 7] Given a control system # = ## ## # # # (1) ######## ######### ######### ## ##### ###### ##### ################# ### ############################# ### ....

....[2, 4] Hierarchical systems for discrete event systems have been formally considered in [10] Continuous abstractions of continuous systems is avery recent activity[7] More precisely, for linear control systems the abstraction problem is formulated as follows. Problem 1. 1 [Linear Abstractions([7]) Given a control system # = ## ## # # # (1) ######## ######### ######### ## ##### ###### ##### ################# ### ############################# ### ################# and an onto map # = ##, de ne a control system (2) which can produce as trajectories all functions of the ....

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G.J. Pappas, G. Laerriere, and S. Sastry. Hierarchically consistent control systems. IEEE Transactions on Automatic Control, 45(6):1141-1160, June 2000.

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G. J. Pappas, G. Lafferriere, and S. Sastry. Hierarchically consistent control systems. Technical Report UCB/ERL M98/16, University of California at Berkeley, Berkeley, CA, April 1998.

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George J. Pappas, Gerardo Lafferriere, and Shankar Sastry. Hierarchically consistent control systems. IEEE Transactions on Automatic Control, 45(7), July 2000.

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George J. Pappas, Gerardo Lafferriere, and Shankar Sastry. Hierarchically consistent control systems. In Proceedings of the 37th IEEE Conference in Decision and Control, pages 4336--4341. Tampa, FL, December 1998.

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G.J. Pappas, G. Lafferriere, and S. Sastry. Hierarchically consistent control systems. IEEE Transactions on Automatic Control, June 2000. To appear.

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G.J. Pappas, G. Lafferriere, and S. Sastry. Hierarchically consistent control systems. In Proceedings of the 37th IEEE Conference in Decision and Control. Tampa, FL, December 1998.

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George J. Pappas, Gerardo La#erriere, and Shankar Sastry. Hierarchically consistent control systems. IEEE Transactions on Automatic Control, 45(6):1144--1160, June 2000.

....system while incorporating modeling detail. This approach critically depends on whether we are able to construct hierarchies of abstractions as well as characterize conditions under which various properties of interest propagate from the original to the abstracted system and vice versa. In [19], hierarchical abstractions This paper was not presented at any IFAC meeting. This research is partially supported by DARPA under grant F33615 00 C 1707, and by Fundacao para a Ciencia e Tecnologia under grant PRAXIS XXI BD 18149 98. Corresponding author Paulo Tabuada, Phone 215 746 2857, Fax ....

....Clearly, it is di#cult to determine whether a control system is an abstraction of another at the level of trajectories. One is then interested in a characterization of abstractions which is equivalent to Definition 3.1 but algebraic. Such a description is given in the next result adapted from [19]: Theorem 3.2 Let #M and #N be two Hamiltonian control systems defined on manifolds M and N , respectively and # : M N a Poisson map. Control system #N is a # abstraction of #M if and only if #N is # related to #M , that is for every x M : T x # (DM (x) Making use of the above result ....

George J. Pappas, Gerardo La#erriere, and Shankar Sastry. Hierarchically consistent control systems. IEEE Transactions on Automatic Control, 45(6):1144--1160, June 2000.

....system #M = UM , FM ) if there exists a curve c : I making the following diagrams commutative: I M # I TM T c # (2.5) where we have identified I with T I. 3 The Category of Control Systems We start by reviewing the notion of # related control systems introduced in [11] and which motivates the construction of the category of control systems to be later presented. Definition 3.1 (# related Control Systems) Let #M and #N be two control systems defined on manifolds M and N , respectively. Given a map # : M N we say that #N is # related to #M i# for every x M ....

....the category of control systems to be later presented. Definition 3.1 (# related Control Systems) Let #M and #N be two control systems defined on manifolds M and N , respectively. Given a map # : M N we say that #N is # related to #M i# for every x M : T x #(SM (x) # SN # #(x) 3. 6) In [11] it is shown that this notion, local in nature, is equivalent to a more intuitive and global relation between #M and #N . Proposition 3.1 ( 11] Let #M and #N be two control systems defined on manifolds M and N , respectively and let # : M N be a map. Control system #N is # related to #M i# ....

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George J. Pappas, Gerardo La#erriere, and Shankar Sastry. Hierarchically consistent control systems. IEEE Transactions on Automatic Control, 45(6):1144--1160, June 2000.

....a controller synthesis tool for discrete event systems [5] As mentioned in [23] notions similar to bisimulation have escaped the world of purely continuous systems. Recently, a notion of abstraction, that is essentially the notion of simulation [16] was introduced for continuoustime systems in [19]. In [19] a formal construction was provided for extracting abstractions of linear systems, Preprint submitted to Automatica 19 February 2003 and have, furthermore, characterized linear quotient maps that preserve control theoretic properties such as controllability [19] and stabilizability ....

....synthesis tool for discrete event systems [5] As mentioned in [23] notions similar to bisimulation have escaped the world of purely continuous systems. Recently, a notion of abstraction, that is essentially the notion of simulation [16] was introduced for continuoustime systems in [19] In [19], a formal construction was provided for extracting abstractions of linear systems, Preprint submitted to Automatica 19 February 2003 and have, furthermore, characterized linear quotient maps that preserve control theoretic properties such as controllability [19] and stabilizability [18] The ....

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G. J. Pappas, G. La#erriere, and S. Sastry. Hierarchically consistent control systems. IEEE Transactions on Automatic Control, 45(6):1144--1160, June 2000.

....in Con from a control system X (UM, X) to Y (Uv,Yv) is given by a pair (qb, qb2) of smooth maps with qb: M x Ux N x UN and qb2: M N, such that M x U qS. N x Uv M x U N x Uv X M YN 711 711 T52 52 TM TN M N both commute. Thus related control systems are said to be (qb, qb2) related [16]. Note that since r is a surjective map, qb2 is uniquely determined given qb. The identity morphism idx: X X for an object X in Con is given by idx = idMxrrM,id) Given f: X Y and g: Y Z, the composite gf : X Z is given by gf = gf, g2f2) The path category P is defined as the ....

George J. Pappas, Gerardo Lafferriere, and Shankar Sastry. Hierarchically consistent control systems. IEEE Transactions on Automatic Control, 45(6):1144-1160, June 2000.

....are established notions of abstraction for discrete systems that preserve properties expressed in various temporal logics [25] For purely continuous systems, the notions of simulation, and bisimulation had no counterparts. Recently, similar notions were introduced in the collection of papers [32, 31, 33, 30, 42, 43]. This research resulted in automatic constructions of abstractions for linear control systems [31] while characterizing abstracting maps that preserve properties of interest such as controllability. Such notions were furthermore generalized for nonlinear control ane systems [33] and fully ....

....logics [25] For purely continuous systems, the notions of simulation, and bisimulation had no counterparts. Recently, similar notions were introduced in the collection of papers [32, 31, 33, 30, 42, 43] This research resulted in automatic constructions of abstractions for linear control systems [31], while characterizing abstracting maps that preserve properties of interest such as controllability. Such notions were furthermore generalized for nonlinear control ane systems [33] and fully nonlinear systems [43] Notions of bisimulation for purely continuous control systems were introduced in ....

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George J. Pappas, Gerardo Laerriere, and Shankar Sastry. Hierarchically consistent control systems. IEEE Transactions on Automatic Control, 45(6):1144-1160, June 2000.

....control system M = UM ; FM ) if there exists a curve c : I UM making the following diagrams commutative: I M I TM T c (2.5) where we have identi ed I with T I. 3 The Category of Control Systems We start by reviewing the notion of related control systems introduced in [11] and which motivates the construction of the category of control systems to be later presented. De nition 3.1 ( related Control Systems) Let M and N be two control systems de ned on manifolds M and N , respectively. Given a map : M N we say that N is related to M i for every x 2 M ....

....of the category of control systems to be later presented. De nition 3.1 ( related Control Systems) Let M and N be two control systems de ned on manifolds M and N , respectively. Given a map : M N we say that N is related to M i for every x 2 M : T x (SM (x) SN (x) 3. 6) In [11] it is shown that this notion, local in nature, is equivalent to a more intuitive and global relation between M and N . Proposition 3.1 ( 11] Let M and N be two control systems de ned on manifolds M and N , respectively and let : M N be a map. Control system N is related to M i ....

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George J. Pappas, Gerardo Laerriere, and Shankar Sastry. Hierarchically consistent control systems. IEEE Transactions on Automatic Control, 45(6):1144-1160, June 2000.

....to that work, in this paper we seek not to unify, but to characterize the notion by easily checkable (algebraic) conditions. A characterization of bisimulation for nonlinear systems is important for several reasons that go beyond its application in hybrid systems. In the series of pa pers [17, 18, 21], a methodology has been introduced to compute abstractions of linear and nonlinear control systems. These abstractions are clearly important for verification problems, but also for hierarchical syn thesis. For example, in [16] hierarchical stabilization of linear systems is discussed in the ....

....on the original def initions in [tt] We then focus on control affne systems and relations between them defined by subimmersions, and provide algebraic characterizations for these no tions. These characterizations turn out to be related with the notion of ) related control systems introduced in [17]. We then show that by factoring out certain in variant distributions one obtains bisimilar systems and that all bisimilar systems are obtained in this way. The distinguishing power of the two introduced notions is also discussed by showing that, locally, they are equivalent up to a feedback ....

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George J. Pappas, Gerardo Lafferriere, and Shankar Sastry. Hierarchically consistent control systems. IEEE Transactions on Automatic Control, 45(6):114&1160, June 2000.

....of bisimilarity concepts for dynamical, control, and hybrid systems. Other attempts to formulate the notion of bisimulation in categorical language, include the coalgebraic approach of [1, 13] In this paper we propose a new equivalence relation for dynamical and control systems (see also [12]) that we call bisimulation and further show that this equivalence relation is captured by the abstract bisimulation relation of JNW [2] This extends the latter abstract framework to the continuous domain. In this paper, our main focus, besides introducing a new equivalence relation for dynamical ....

....space. A morphism in Con from a control system X = UM , XM ) to Y = UN , YN ) is given by a pair (# 1 , # 2 ) of smooth maps with # 1 : M UN and # 2 : M N , such that UN M UN TM T# 2 YN # 2 both commute. Thus related control systems are said to be (# 1 , # 2 ) related [12]. Note that since # 1 is a surjective map, # 2 is uniquely determined given # 1 . The identity morphism id X : X X is given by id X = id MU , id M ) for any object X in Con and given f : X and g : Y Z, the composite gf : X Z is given by gf = g 1 f 1 , g 2 f 2 ) The path category ....

George J. Pappas, Gerardo La#erriere, and Shankar Sastry. Hierarchically consistent control systems. IEEE Transactions on Automatic Control, 45(6):1144--1160, June 2000.

....map or aggregation map that will induce a new Hamiltontan control system S on the lower dimensional Potsson manifold (N, v) having as trajectories b(cM) where c M are Sr trajectories. The concept of abstraction map for continuous, not necessarily Hamiltontan, control sys tems is defined in [10] as: Definition 4.1 (Abstracting Maps) Let SM and S be two control systems on manifolds M and N, 2A section of r U: U M is a smooth map cr: M U such that ru ocr =identity on M. 3From now on, Hr = UM,HM) or simply Hr denotes an Hamiltontan control system on Potsson manifold (M, M) ....

....4.3 Let SHM and SHay be Hamiltonian control systems on Poisson manifolds (M, M) and (N, N) respectively, and q) M N a smooth Poisson map. Then SHand SH are q) related if and only if SH is a q) abstraction of SH Before proving this result we recall the following The orem from [10]: Theorem 4.4 ( 10] Let SM and SN be control sys tems on manifolds M and N, respectively, and q) M N a smooth map. Then SN is a q) abstraction of SM iff Tx(Z)M(X) C Z)N o (x) 4.2) We now return to the proof of Proposition 4.3. Proof: It is enough to show that if b is a Poisson map then ....

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George J. Pappas, Gerardo Lafferriere, and Shankar Sastry. Hierarchically consistent control systems. IEEE Transactions on Automatic Control, 45(6):1144-1160, June 2000.

....decidability. This approach has been successfully applied to timed automata [2] multirate automata [1] and initialized rectangular automata [19,12] It should be mentioned that the notion of bisimulation is closely related to the various consistency notions for discrete and continuous systems [7,8,18]. Since the discrete dynamics are already finite, it is clear that decidability results for hybrid systems depend crucially on the success of obtaining finite bisimulations for continuous dynamics. The cases considered so far in the literature dealt with simple dynamics: x = 1 for timed automata ....

G.J. Pappas, G. La#erriere, and S. Sastry, Hierarchically consistent control systems, Proceedings of the 37th IEEE Conference in Decision and Control, Tampa, FL, December 1998, Submitted.

....simulation, and bisimulation are established notions of abstraction for discrete systems that preserve properties expressed in various temporal logics. For purely continuous systems, the notions of simulation, and bisimulation had no counterparts. Recently, similar notions were intro duced in [7] and this research resulted in automatic constructions of abstractions for linear and nonlinear analytic control systems [8] while characterizing ab stracting maps that preserve properties of interest such as controllability. Based on these results, in [10] we took the first steps towards ....

....X x Yx x X. Furthermore every time a continuous flow crosses the boundary between adjacent covering sets, the required discrete transitions are captured by the set J. Finally in the last step the continuous generator of Hy is obtained from the continuous generator of Hx by the methods described in [7, 8]. The above algorithm does compute a simulation of Hx as asserted in the next theorem whose proof we were forced to omit due to lack of space: Theorem 3.5 Let Hx be an hybrid control system over X and R Ax x A an admissible relation. Then hybrid control system H obtained through Algorithm 3.6 ....

George J. Pappas, Gerardo Lafferriere, and Shankar Sastry. Hierarchically consistent control systems. IEEE Transactions on Automatic Control, 45(6):1144-1160, June 2000.

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G.J. Pappas, G. La#erriere, and S. Sastry, Hierarchically Consistent Control Systems, IEEE Transactions on Automatic Control, 45(6):1144-1160, June 2000.

....This would ensure that the higher level model is a consistent abstraction of the lower level. Different properties may require placing different conditions on the quotient maps. In previous work, we have focused on extracting continuous abstractions from continuous systems. In particular, in [10] a hierarchical framework for continuous control systems was formally proposed, and easily checkable characterizations were obtained for constructing reachability preserving abstractions of linear control systems. In [11] we extended our hierarchical approach to a significant class of nonlinear ....

....In this paper, we review our framework and illustrate its complexity reduction properties on a detailed search and rescue case study. 2 Abstraction Methodology In this section we will review the hybrid abstraction framework of [13] which builds on top of the continuous abstraction framework of [10, 11]. We assume that the reader is familiar with differential geometric concepts at the level of [1] 2.1 Problem Formulation We start with a continuous control system , affine in control and defined on a analytic manifold 1 1 # ( ....

[Article contains additional citation context not shown here]

George J. Pappas, Gerardo Lafferriere, and Shankar Sastry. Hierarchically consistent control systems. IEEE Transactions on Automatic Control, 45(6):1144--1160, June 2000.

No context found.

G. Pappas, G. La#erriere, and S. Sastry. Hierarchically consistent control systems. In Proceedings of the 37th IEEE Conference on Decision and Control, pages 4336--4341, Tampa, FL, 1998.

No context found.

G. J. Pappas, G. La#erriere, and S. S. Sastry. Hierarchically consistent control systems. IEEE Transactions on Automatic Control, 45(6):1144--60, 2000.

No context found.

G. J. Pappas, G. Lafferriere, and S. S. Sastry, "Hierarchically consistentcontrol systems," IEEE Transactions on Automatic Control,vol. 45, no. 6, pp. 1144--60, 2000.

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G. Pappas, G. Lafferiere, and S. Sastry. "Hierarchically Consistent Control Systems," IEEE Transactions on Automatic Control , 45: 1145--1160, 2000.

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