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Universal computation and other capabilities of hybrid and continuous dynamical systems (1995)

by Michael Branicky
Venue:Theoretical Computer Science
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A Unified Framework for Hybrid Control: Model and Optimal Control Theory

by Michael S. Branicky, Vivek S. Borkar, Sanjoy K. Mitter - IEEE TRANSACTIONS ON AUTOMATIC CONTROL , 1998
"... Complex natural and engineered systems typically possess a hierarchical structure, characterized by continuousvariable dynamics at the lowest level and logical decision-making at the highest. Virtually all control systems today---from flight control to the factory floor---perform computer-coded chec ..."
Abstract - Cited by 305 (9 self) - Add to MetaCart
Complex natural and engineered systems typically possess a hierarchical structure, characterized by continuousvariable dynamics at the lowest level and logical decision-making at the highest. Virtually all control systems today---from flight control to the factory floor---perform computer-coded checks and issue logical as well as continuous-variable control commands. The interaction of these different types of dynamics and information leads to a challenging set of "hybrid" control problems. We propose a very general framework that systematizes the notion of a hybrid system, combining differential equations and automata, governed by a hybrid controller that issues continuous-variable commands and makes logical decisions. We first identify the phenomena that arise in real-world hybrid systems. Then, we introduce a mathematical model of hybrid systems as interacting collections of dynamical systems, evolving on continuous-variable state spaces and subject to continuous controls and discrete transitions. The model captures the identified phenomena, subsumes previous models, yet retains enough structure on which to pose and solve meaningful control problems. We develop a theory for synthesizing hybrid controllers for hybrid plants in an optimal control framework. In particular, we demonstrate the existence of optimal (relaxed) and near-optimal (precise) controls and derive "generalized quasi-variational inequalities" that the associated value function satisfies. We summarize algorithms for solving these inequalities based on a generalized Bellman equation, impulse control, and linear programming.
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...ng insight. Indeed, Branicky has shown that there are smooth, Lipschitz continuous ODE’s in 3, which possess the power of universal computation, hence yielding most control questions in 3 undecidable =-=[9]-=- (one such question is constructed in Section IX-B). Continuation’s second route may treat the discrete dynamics as small unmodeled dynamics (and then use robust control), slowly-varying (and gain-sch...

A Survey of Computational Complexity Results in Systems and Control

by Vincent D. Blondel, John N. Tsitsiklis , 2000
"... The purpose of this paper is twofold: (a) to provide a tutorial introduction to some key concepts from the theory of computational complexity, highlighting their relevance to systems and control theory, and (b) to survey the relatively recent research activity lying at the interface between these fi ..."
Abstract - Cited by 187 (18 self) - Add to MetaCart
The purpose of this paper is twofold: (a) to provide a tutorial introduction to some key concepts from the theory of computational complexity, highlighting their relevance to systems and control theory, and (b) to survey the relatively recent research activity lying at the interface between these fields. We begin with a brief introduction to models of computation, the concepts of undecidability, polynomial time algorithms, NP-completeness, and the implications of intractability results. We then survey a number of problems that arise in systems and control theory, some of them classical, some of them related to current research. We discuss them from the point of view of computational complexity and also point out many open problems. In particular, we consider problems related to stability or stabilizability of linear systems with parametric uncertainty, robust control, time-varying linear systems, nonlinear and hybrid systems, and stochastic optimal control.

Reachability Analysis of Dynamical Systems having Piecewise-Constant Derivatives

by Eugene Asarin, Oded Maler, Amir Pnueli - Theoretical Computer Science , 1995
"... In this paper we consider a class of hybrid systems, namely dynamical systems with piecewise-constant derivatives (PCD systems). Such systems consist of a partition of the Euclidean space into a finite set of polyhedral sets (regions). Within each region the dynamics is defined by a constant vector ..."
Abstract - Cited by 129 (20 self) - Add to MetaCart
In this paper we consider a class of hybrid systems, namely dynamical systems with piecewise-constant derivatives (PCD systems). Such systems consist of a partition of the Euclidean space into a finite set of polyhedral sets (regions). Within each region the dynamics is defined by a constant vector field, hence discrete transitions occur only on the boundaries between regions where the trajectories change their direction. With respect to such systems we investigate the reachability question: Given an effective description of the systems and of two polyhedral subsets P and Q of the state-space, is there a trajectory starting at some x 2 P and reaching some point in Q? Our main results are a decision procedure for two-dimensional systems, and an undecidability result for three or more dimensions. 1 Introduction 1.1 Motivation Hybrid systems (HS) are systems that combine intercommunicating discrete and continuous components. Most embedded systems belong to this class since they operate...
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...or A. Remark: There is a variety of other notions of simulation between discrete and continuous dynamical systems, but these semantical issues are subject of an independent ongoing research (see [2], =-=[5]-=-). The simple definition we use here is sufficient for the purpose of proving undecidability. 6.1 Simulation of Finite-State Machines Here we show how every finite-state automaton can be simulated by ...

Differential Dynamic Logic for Hybrid Systems

by André Platzer , 2007
"... Hybrid systems are models for complex physical systems and are defined as dynamical systems with interacting discrete transitions and continuous evolutions along differential equations. With the goal of developing a theoretical and practical foundation for deductive verification of hybrid systems, ..."
Abstract - Cited by 78 (46 self) - Add to MetaCart
Hybrid systems are models for complex physical systems and are defined as dynamical systems with interacting discrete transitions and continuous evolutions along differential equations. With the goal of developing a theoretical and practical foundation for deductive verification of hybrid systems, we introduce a dynamic logic for hybrid programs, which is a program notation for hybrid systems. As a verification technique that is suitable for automation, we introduce a free variable proof calculus with a novel combination of real-valued free variables and Skolemisation for lifting quantifier elimination for real arithmetic to dynamic logic. The calculus is compositional, i.e., it reduces properties of hybrid programs to properties of their parts. Our main result proves that this calculus axiomatises the transition behaviour of hybrid systems completely relative to differential equations. In a case study with cooperating traffic agents of the European Train Control System, we further show that our calculus is well-suited for verifying realistic hybrid systems with parametric system dynamics.

Complexity of Stability and Controllability of Elementary Hybrid Systems

by Vincent D. Blondel, John N. Tsitsiklis , 1997
"... this paper, weconsider simple classes of nonlinear systems and provethatbasic questions related to their stabilityandcontrollabilityare either undecidable or computationally intractable (NP-hard). As a special case, weconsider a class of hybrid systems in which the state space is partitioned into tw ..."
Abstract - Cited by 57 (9 self) - Add to MetaCart
this paper, weconsider simple classes of nonlinear systems and provethatbasic questions related to their stabilityandcontrollabilityare either undecidable or computationally intractable (NP-hard). As a special case, weconsider a class of hybrid systems in which the state space is partitioned into two halfspaces, and the dynamics in eachhalfspace correspond to a differentlinear system

Computational aspects of feedback in neural circuits

by Wolfgang Maass, Prashant Joshi, Eduardo D. Sontag - PLOS Computational Biology , 2007
"... It has previously been shown that generic cortical microcircuit models can perform complex real-time computations on continuous input streams, provided that these computations can be carried out with a rapidly fading memory. We investigate the computational capability of such circuits in the more re ..."
Abstract - Cited by 37 (7 self) - Add to MetaCart
It has previously been shown that generic cortical microcircuit models can perform complex real-time computations on continuous input streams, provided that these computations can be carried out with a rapidly fading memory. We investigate the computational capability of such circuits in the more realistic case where not only readout neurons, but in addition a few neurons within the circuit, have been trained for specific tasks. This is essentially equivalent to the case where the output of trained readout neurons is fed back into the circuit. We show that this new model overcomes the limitation of a rapidly fading memory. In fact, we prove that in the idealized case without noise it can carry out any conceivable digital or analog computation on time-varying inputs. But even with noise, the resulting computational model can perform a large class of biologically relevant real-time computations that require a nonfading memory. We demonstrate these computational implications of feedback both theoretically, and through computer simulations of detailed cortical microcircuit models that are subject to noise and have complex inherent dynamics. We show that the application of simple learning procedures (such as linear regression or perceptron learning) to a few neurons enables such circuits to represent time over behaviorally relevant long time spans, to integrate evidence from incoming spike trains over longer periods of time, and to process new information contained in such spike trains in diverse ways according to the current internal state of the circuit. In particular we show that such generic cortical microcircuits with feedback provide a new model for working memory that is consistent with a large set of biological constraints.
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...m the fact that every Turing machine (hence any conceivable digital computation, most of which require a persistent memory) can be simulated by systems of equations of the form (4) (this was shown in =-=[29]-=- for the case with continuous time, and in [30, 31] for recurrent neural networks with discrete time; see [32] for a review). But possibly more relevant for applications to biological systems is the f...

Iteration, Inequalities, and Differentiability in Analog Computers

by Manuel Lameiras Campagnolo, Cristopher Moore, Jose Felix Costa , 1999
"... Shannon's General Purpose Analog Computer (GPAC) is an elegant model of analog computation in continuous time. In this paper, we consider whether the set G of GPAC-computable functions is closed under iteration, that is, whether for any function f(x) 2 G there is a function F (x; t) 2 G s ..."
Abstract - Cited by 36 (19 self) - Add to MetaCart
Shannon's General Purpose Analog Computer (GPAC) is an elegant model of analog computation in continuous time. In this paper, we consider whether the set G of GPAC-computable functions is closed under iteration, that is, whether for any function f(x) 2 G there is a function F (x; t) 2 G such that F (x; t) = f t (x) for non-negative integers t. We show that G is not closed under iteration, but a simple extension of it is. In particular, if we relax the definition of the GPAC slightly to include unique solutions to boundary value problems, or equivalently if we allow functions x k (x) that sense inequalities in a dierentiable way, the resulting class, which we call G + k , is closed under iteration. Furthermore, G + k includes all primitive recursive functions, and has the additional closure property that if T (x) is in G+k , then any function of x computable by a Turing machine in T (x) time is also.

An analog characterization of the Grzegorczyk hierarchy

by Manuel Lameiras Campagnolo, Cristopher Moore - Journal of Complexity , 2002
"... We study a restricted version of Shannon's General . . . ..."
Abstract - Cited by 35 (18 self) - Add to MetaCart
We study a restricted version of Shannon's General . . .

Achilles and the Tortoise climbing up the hyper-arithmetical hierarchy

by Olivier Bournez , 1997
"... We pursue the study of the computational power of Piecewise Constant Derivative (PCD) systems started in [5, 6]. PCD systems are dynamical systems defined by a piecewise constant differential equation and can be considered as computational machines working on a continuous space with a continuous tim ..."
Abstract - Cited by 32 (8 self) - Add to MetaCart
We pursue the study of the computational power of Piecewise Constant Derivative (PCD) systems started in [5, 6]. PCD systems are dynamical systems defined by a piecewise constant differential equation and can be considered as computational machines working on a continuous space with a continuous time. We prove that the languages recognized by rational PCD systems in dimension d = 2k + 3 (respectively: d = 2k + 4), k 0, in finite continuous time are precisely the languages of the ! k th (resp. ! k + 1 th ) level of the hyper-arithmetical hierarchy. Hence the reachability problem for rational PCD systems of dimension d = 2k + 3 (resp. d = 2k + 4), k 1, is hyper-arithmetical and is \Sigma ! k-complete (resp. \Sigma ! k +1 -complete).
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...constant. Hybrid systems can be considered either as computational machines working on a continuous space with a discrete time or as machines working on a continuous space with a continuous time: see =-=[2, 3, 4, 9, 10]-=-. Several theoretical computational models of machines working on a continuous space with a discrete time are known: in particular, in [5], Blum, Shub and Smale introduce the real Turing machine (see ...

A Survey of Continuous-Time Computation Theory

by Pekka Orponen - Advances in Algorithms, Languages, and Complexity , 1997
"... Motivated partly by the resurgence of neural computation research, and partly by advances in device technology, there has been a recent increase of interest in analog, continuous-time computation. However, while special-case algorithms and devices are being developed, relatively little work exists o ..."
Abstract - Cited by 31 (6 self) - Add to MetaCart
Motivated partly by the resurgence of neural computation research, and partly by advances in device technology, there has been a recent increase of interest in analog, continuous-time computation. However, while special-case algorithms and devices are being developed, relatively little work exists on the general theory of continuous-time models of computation. In this paper, we survey the existing models and results in this area, and point to some of the open research questions. 1 Introduction After a long period of oblivion, interest in analog computation is again on the rise. The immediate cause for this new wave of activity is surely the success of the neural networks "revolution", which has provided hardware designers with several new numerically based, computationally interesting models that are structurally sufficiently simple to be implemented directly in silicon. (For designs and actual implementations of neural models in VLSI, see e.g. [30, 45]). However, the more fundamental...
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...in the corresponding discrete-time model, asymmetric and symmetric networks are computationally equivalent.) Also, the simulation in [34] uses a similar twophasing trick as Branicky's construction in =-=[8, 9]-=-, and is thus in the same way somewhat unsatisfactory. And finally, the result is only a lower bound on the computational power: the possibility still remains that polynomial-size Hopfield networks mi...

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