Symbolic Control of Linear Systems Based on Symbolic Subsystems

Tabuada, Paulo

doi:10.1109/tac.2006.876946

Cited by 22 publications

(14 citation statements)

References 56 publications

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“…This more flexible notion of bisimulation allows the identification of more classes of systems, admitting symbolic models. Indeed, the work in [Tab06] showed that for every asymptotically stabilizable linear control system it is possible to construct a symbolic model, which is based on an approximate notion of simulation (one-sided version of approximate bisimulation). Extensions of the results in [Tab06], from approximate simulation to approximate bisimulation can be found in [Gir07,PGT07].…”

Section: Introductionmentioning

confidence: 99%

“…Indeed, the work in [Tab06] showed that for every asymptotically stabilizable linear control system it is possible to construct a symbolic model, which is based on an approximate notion of simulation (one-sided version of approximate bisimulation). Extensions of the results in [Tab06], from approximate simulation to approximate bisimulation can be found in [Gir07,PGT07]. In particular [PGT07] showed that, for the class of (incrementally globally) asymptotically stable nonlinear control systems, symbolic models exist which are This work has been partially supported by the National Science Foundation CAREER award 0717188.…”

Section: Introductionmentioning

confidence: 99%

“…[AHLP00] and the references therein) or control systems (e.g. [TP06,Tab06,Tab07b,Gir07,PGT07]) not affected by exogenous disturbances. However, in many realistic situations, physical processes are characterized by a certain degree of uncertainty which is often modeled by additional exogenous disturbance inputs.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Symbolic Models for Nonlinear Control Systems: Alternating Approximate Bisimulations

Pola¹,

Tabuada²

2009

SIAM J. Control Optim.

141

189

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Abstract. Symbolic models are abstract descriptions of continuous systems in which symbols represent aggregates of continuous states. In the last few years there has been a growing interest in the use of symbolic models as a tool for mitigating complexity in control design. In fact, symbolic models enable the use of well known algorithms in the context of supervisory control and algorithmic game theory, for controller synthesis. Since the 1990's many researchers faced the problem of identifying classes of dynamical and control systems that admit symbolic models. In this paper we make a further progress along this research line by focusing on control systems affected by disturbances. Our main contribution is to show that incrementally globally asymptotically stable nonlinear control systems with disturbances admit symbolic models. When specializing these results to linear systems, we show that these symbolic models can be easily constructed.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Symbolic Models for Nonlinear Control Systems: Alternating Approximate Bisimulations

Pola¹,

Tabuada²

2009

SIAM J. Control Optim.

141

189

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“…The basic idea is that incremental stability of can be equivalently described in terms of gain stability of a new system (referred to as a "supersystem" in [40]) consisting of two copies of driven in parallel by two different inputs and initial conditions. Indeed, let be a deterministic finite state machine defined by (15) and (16) (29) and output function defined by (30) Thus, is described by the following state transition (31) and output (32) equations:…”

Section: Incremental Stability Of Dfm Modelsmentioning

confidence: 99%

A Framework for Robust Stability of Systems Over Finite Alphabets

Tarraf

Megretski

Dahleh

2008

IEEE Trans. Automat. Contr.

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Abstract-Systems over finite alphabets are discrete-time systems whose input and output signals take their values in finite sets. Three notions of input/output stability (gain stability, incremental stability and external stability) that are particularly applicable to this class of systems are proposed and motivated through examples. New formulations for generalized small gain and incremental small gain theorems are presented, thus showing that gain stability and incremental stability are useful robustness measures. The paper then focuses on deterministic finite state machine (DFM) models. For this class, the problems of verifying gain stability, incremental stability, and corresponding gain bounds are shown to reduce to searching for an appropriate storage function. These problems are also shown to be related to the problem of verifying the nonexistence of negative cost cycles in an appropriately constructed network. Using this insight and based on a solution approach for discrete shortest path problems, a strongly polynomial algorithm is proposed. Finally, incremental stability and external stability are shown to be equivalent notions for this class of systems.Index Terms-Dissipative system, finite state machine, incremental stability, input/output stability, shortest path problem, small gain theorem, storage functions, system over finite alphabet.

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“…Contrary to these approaches, the abstractions we build are valid for an infinite time-horizon. In [12], a similar technique is proposed to build finite abstractions for stabilizable linear systems, however only a one-sided approximation result is provided making these abstractions suitable for control synthesis but not for verification.…”

Section: Introductionmentioning

confidence: 99%

Approximately Bisimilar Finite Abstractions of Stable Linear Systems

Girard

Hybrid Systems: Computation and Control

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Abstract. The use of bisimilar finite abstractions of continuous and hybrid systems, greatly simplifies complex computational tasks such as verification or control synthesis. Unfortunately, because of the strong requirements of bisimulation relations, such abstractions exist only for quite restrictive classes of systems. Recently, the notion of approximate bisimulation relations has been introduced, allowing the definition of less rigid relationships between systems. This relaxed notion should certainly allow us to build approximately bisimilar finite abstractions for more general classes of continuous and hybrid systems. In this paper, we show that for the class of stable discrete-time linear systems with constrained inputs, there exists an approximately bisimilar finite state system of any desired precision. We describe an effective procedure for the construction of this abstraction, based on compositional reasoning and samples of the set of initial states and inputs. Finally, we briefly show how our finite abstractions can be used for verification or control synthesis.

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Symbolic Control of Linear Systems Based on Symbolic Subsystems

Cited by 22 publications

References 56 publications

Symbolic Models for Nonlinear Control Systems: Alternating Approximate Bisimulations

Symbolic Models for Nonlinear Control Systems: Alternating Approximate Bisimulations

A Framework for Robust Stability of Systems Over Finite Alphabets

Approximately Bisimilar Finite Abstractions of Stable Linear Systems

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