Showing papers by "Paulo Tabuada published in 2007"

Event-Triggered Real-Time Scheduling of Stabilizing Control Tasks

Abstract: In this note, we revisit the problem of scheduling stabilizing control tasks on embedded processors. We start from the paradigm that a real-time scheduler could be regarded as a feedback controller that decides which task is executed at any given instant. This controller has for objective guaranteeing that (control unrelated) software tasks meet their deadlines and that stabilizing control tasks asymptotically stabilize the plant. We investigate a simple event-triggered scheduler based on this feedback paradigm and show how it leads to guaranteed performance thus relaxing the more traditional periodic execution requirements.

Approximately bisimilar symbolic models for nonlinear control systems

Abstract: Control systems are usually modeled by differential equations describing how physical phenomena can be influenced by certain control parameters or inputs. Although these models are very powerful when dealing with physical phenomena, they are less suitable to describe software and hardware interfacing the physical world. For this reason there is a growing interest in describing control systems through symbolic models that are abstract descriptions of the continuous dynamics, where each "symbol" corresponds to an "aggregate" of states in the continuous model. Since these symbolic models are of the same nature of the models used in computer science to describe software and hardware, they provide a unified language to study problems of control in which software and hardware interact with the physical world. Furthermore the use of symbolic models enables one to leverage techniques from supervisory control and algorithms from game theory for controller synthesis purposes. In this paper we show that every incrementally globally asymptotically stable nonlinear control system is approximately equivalent (bisimilar) to a symbolic model. The approximation error is a design parameter in the construction of the symbolic model and can be rendered as small as desired. Furthermore if the state space of the control system is bounded the obtained symbolic model is finite. For digital control systems, and under the stronger assumption of incremental input-to-state stability, symbolic models can be constructed through a suitable quantization of the inputs.

Symbolic Models for Nonlinear Control Systems: Alternating Approximate Bisimulations

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.

Approximate simulation relations and finite abstractions of quantized control systems

Abstract: In this paper we revisit the construction of quantized models of control systems. Based on an approximate notion of simulation relation and under a stabilizability assumption we show how we can force a lattice structure on the reachable space of a quantized control system for any finite input quantization. When we are only interested in a compact subset of the state space, as is the case in concrete applications, our results immediately provide a finite model for the quantized control system.

Symbolic models for control systems

Abstract: In this paper we provide a bridge between the infinite state models used in control theory to describe the evolution of continuous physical processes and the finite state models used in computer science to describe software. We identify classes of control systems for which it is possible to construct equivalent (bisimilar) finite state models. These constructions are based on finite, but otherwise arbitrary, partitions of the set of inputs or outputs of a control system.

On simulations and bisimulations of general flow systems

Abstract: We introduce a notion of bisimulation equivalence between general flow systems, which include discrete, continuous and hybrid systems, and compare it with similar notions in the literature. The interest in the proposed notion is based on our main result, that the temporal logic GFL* - an extension to general flows of the well-known computation tree logic CTL* - is semantically preserved by this equivalence.

Symbolic models for nonlinear control systems using approximate bisimulation

Abstract: Control systems are usually modeled by differential equations describing how physical phenomena can be influenced by certain control parameters or inputs. Although these models are very powerful when dealing with physical phenomena, they are less suitable to describe software and hardware interfacing the physical world. This has spurred a recent interest in describing control systems through symbolic models that are abstract descriptions of the continuous dynamics, where each "symbol" corresponds to an "aggregate" of continuous states in the continuous model. Since these symbolic models are of the same nature of the models used in computer science to describe software and hardware, they provided a unified language to study problems of control in which software and hardware interact with the physical world. In this paper we show that every incrementally globally asymptotically stable nonlinear control system is approximately equivalent (bisimilar) to symbolic model with a precision that can be chosen a-priori. We also show that for digital controlled systems, in which inputs are piecewise-constant, and under the stronger assumption of incremental input-to-state stability, the symbolic models can be obtained, based on a suitable quantization of the inputs.

Symbolic models for linear control systems with disturbances

Abstract: A recent trend in the control systems community is the study of appropriate symbolic abstractions capturing the behavior of continuous and hybrid systems. This approach provides a common mathematical language to describe physical systems as well as software and hardware, and is therefore particularly appealing when dealing with the design of embedded systems. In this paper we address the construction of symbolic models for the class of linear control systems with politopically bounded states and disturbances. We show that under an asymptotic stabilizability assumption, it is always possible to construct a symbolic model that approximates the control system with a precision that is chosen a priori, as a design parameter. While in previous approaches in the existing literature, the construction of symbolic models relied on a (arbitrary) choice of a finite number of control signals, the symbolic model that we propose, captures any (control and disturbance) input. Therefore, the proposed model provides a finer description of the continuous model than the existing ones and this feature translates into a more efficient controller synthesis process. Furthermore, the computation of the symbolic model can be performed by resorting to linear matrix inequalities.

Controller synthesis for bisimulation equivalence

Abstract: The objective of this paper is to solve the controller synthesis problem for bisimulation equivalence in a wide variety of scenarios including discrete-event systems, nonlinear control systems, behavioral systems, hybrid systems and many others. This will be accomplished by showing that the arguments underlying proofs of existence and methods for the construction of controllers are extraneous to the particular class of systems being considered and thus can be presented in greater generality.

Approximate reduction of dynamical systems

Abstract: The reduction of dynamical systems has a rich history, with many important applications related to stability, control and verification. Reduction of nonlinear systems is typically performed in an exact manner - as is the case with mechanical systems with symmetry--which, unfortunately, limits the type of systems to which it can be applied. The goal of this paper is to consider a more general form of reduction, termed approximate reduction, in order to extend the class of systems that can be reduced. Using notions related to incremental stability, we give conditions on when a dynamical system can be projected to a lower dimensional space while providing hard bounds on the induced errors, i.e., when it is behaviorally similar to a dynamical system on a lower dimensional space. These concepts are illustrated on a series of examples.