Approximate reduction of dynamical systems

Proceedings of the IEEE Conference on Decision and Control 08/2007; DOI: 10.1109/CDC.2006.377156
Source: arXiv


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.

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Available from: Agung Julius, Oct 24, 2012
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    • "The past decade has witnessed increasing interest in using finite state approximate models of more complex dynamics, such as switching dynamics, for the purpose of certifiedby-design control synthesis [1], [2], [3], [4], [5], [6]. In particular, a notion of finite state ρ/µ approximation was proposed in [7], explicitly identifying three properties that finite state approximations should satisfy to ensure they are compatible with the objective of certified-by-design control synthesis for systems over finite alphabets. "
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    ABSTRACT: We revisit the problem of constructing finite state ρ/µ approximations for the purpose of certified-by-design 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
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    • "Finite state machines have been extensively studied as potential abstractions or approximate models of more complex hybrid dynamics. Indeed the past two decades have witnessed activity in several distinct and complementary directions, including the work on qualitative models [9]–[12], [14], simulation and bisimulation abstractions [1], [8], [13], [17]– [19], symbolic models [3], [6], [7] and ⇢/µ approximations [20]–[22]. Finite state machines also have a long history as models of discrete event systems [5]. "
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    ABSTRACT: We consider finite state machines whose states evolve in a vector space defined over a finite field, and whose dynamics are linear time-invariant. For this class of systems, we show how the linear structure may be exploited to reduce the complexity of solving a class of finite horizon optimal control problems.
    Allerton Conference on Communication, Control and Computing, Allerton, Il; 10/2013
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    • "in hybrid systems to study model reduction [15]-[23]. Some works have been focused on ordinary model reduction methods that have potential applications in modeling and analysis of hybrid systems [15]-[21] motivated by reachability analysis and safety verification problem. Some researches addresses the problem of model reduction of switched and hybrid systems directly [22], [23]. "

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