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.
SourceAvailable from: Danielle C. Tarraf[Show abstract] [Hide abstract]
ABSTRACT: We consider discrete-time 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 approxi-mate 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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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
Conference Paper: On a control-oriented notion of finite state approximation[Show abstract] [Hide abstract]
ABSTRACT: We consider the problem of approximating discrete-time plants with finite-valued sensors and actuators by deterministic finite memory systems for the purpose of certified-by-design controller synthesis. We propose a control-oriented notion of input/output approximation for these systems, building on ideas from classical robust control theory, and we explain the relevance of the proposed notion of approximation to the control synthesis problem.Control (MSC); 09/2011