Article
Approximate reduction of dynamical systems
08/2007; DOI: 10.1109/CDC.2006.377156
Source: arXiv

Conference Paper: On Synchronizing Sampling and Quantization for Stabilizing the Double Integrator under Binary Sensing
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ABSTRACT: We revisit the problem of constructing finite state ρ/µ approximations for the purpose of certifiedbydesign control synthesis. We investigate in this context the problem of picking the 'initial partition' to enable successful control design for a benchmark problem, namely that of exponentially stabilizing the double integrator by switching between two available feedback gains under binary sensing constraints. We motivate the problem through two illustrative case studies, provide an analysis of the intuition gleamed from special instances of it, propose a general algorithm for choosing the initial partition taking into account the sampling frequency and available choices of feedback gains, and demonstrate the use of our algorithm in a set of illustrative examples.22nd Mediterranean Conference on Control and Automation, Palermo, Italy; 06/2014  [Show abstract] [Hide abstract]
ABSTRACT: We revisit the problem of designing a fullstate feedback controller for a deterministic finite state machine, so as to maximize a performance parameter R while simultaneously ensuring that the closed loop system satisfies a given performance objective involving a positive scaling parameter τ . Under some additional assumptions, we show that the problem of choosing τ to optimize R in closed loop admits an analytical solution. We demonstrate the use of this approach via a numerical example, showing substantial computational savings over the existing sampling based method. We also provide an intuitive, graphtheoretic interpretation of our result. I. INTRODUCTION The past decades have witnessed increasing research interest in using finite state machines as approximate models of more complex hybrid systems. Such models are easier to work with for analysis and control design. In order for the approximate models to be useful in this context, they need to be constructed so as to allow engineers to quantify the performance of a controller designed for the lower complexity model and then implemented in feedback with the original system. Several distinct and complementary notions of approximation have been developed, including qualitative models [1], [2], [3], [4], [5], simulation and bisimulation abstractions [6], [7], [8], [9], and ρ/µ approximations [10], [11], [12], [13], [14], [15]. Depending on the notion of approximation used, the process of control design can vary, as can the resulting controllers. For instance, controller synthesis for qualitative models can be formulated as a supervisory control problem, addressed using the RamadgeWonham framework [16]. Controller synthesis using simulation and bisimulation abstractions is a two step procedure in which a finite state supervisory controller is first designed and then subsequently refined into a certified hybrid controller [17]. In the ρ/µ approximation framework, control synthesis reduces to designing a full state feedback controller for a deterministic finite state machine approximate model, termed a ρ/µ approximation [13]. In particular, in [12], [11] the authors provide a systematic approach for using ρ/µ approximations to construct a finite state stabilizing controller that maximizes the provable rate of convergence R, for switched second order homogenous systems. Consequently, control synthesis reduces to designing a controller that maximizes the value of R for the ρ/µIEEE Multiconference on Systems and Control, Antibes, France; 10/2014  [Show abstract] [Hide abstract]
ABSTRACT: We consider discretetime plants that interact with their controllers via fixed discrete alphabets in the absence of exogenous inputs. For this class of systems, we revisit a general procedure for constructing a sequence of finite state approximate models starting from finite length sequences of input and output signal pairs. After providing intuition for the proposed construction, we show that it satisfies the desired properties of ρ/µ approximations. We then propose a readily verifiable sufficient condition for ensuring finiteness of the approximation error gain. Finally, we show that the proposed construct satisfies an interesting 'completeness' property, making it attractive as a medium for studying fundamental limitations of finite state approximation and state observation under coarse sensing.IEEE Conference on Decision and Control, Florence, Italy; 12/2013
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