For the implementations of controllers on digital processors, certain limitations, e.g. in the instruction set and register length, need to be taken into account, especially for safety-critical applications. This work aims to provide a computer-certified inductive definition for the control functions that are implemented on such processors accompanied with the fixed-point data type in a proof assistant. Using these inductive definitions we formally ensure correct realization of the controllers on a digital processor. Our results guarantee overflowfree computations of the implemented control algorithm. The method presented in this paper currently supports functions that are defined as polynomials within an arbitrary fixed-point structure. We demonstrate the verification process in the case study on an example with different scenarios of fixed-point type implementations.
Resonance tracking is an approach to measuring concentrations, forces or viscosities. Such vibration-based measurements appear to be particularly well suited for applications at the micro- or even nano-scale. In order to monitor more than one parameter or parameter ratio simultaneously, a new kind of resonance tracking is developed with methods from adaptive control. It combines parameter estimation methods and state observers to adopt the resonant excitation to vibrating systems with time-varying parameters. At the same time, these parameters are measured. This approach is exemplified at two single-input-single-output (SISO) systems: a linear spring-mass-damper oscillator and a weakly nonlinear oscillator of Duffing-type.
Reliably determining system trajectories is essential in many analysis and control design approaches. To this end, an initial value problem has to be usually solved via numerical algorithms which rely on a certain software realization. Because software realizations can be error-prone, proof assistants may be used to verify the underlying mathematical concepts and corresponding algorithms. In this work we present a computercertified formalization of the solution of the initial value problem of ordinary differential equations. The concepts are performed in the framework of constructive analysis and the proofs are written in the Minlog proof system. We show the extraction of a program, which solves an ODE numerically and provide some possible optimization regarding the efficiency. Finally, we provide numerical experiments to demonstrate how programs of a certain high level of abstraction can be obtained efficiently. The presented concepts may also be viewed as a part of preliminary work for the development of formalized nonlinear control theory, hence offering the possibility of computer-assisted controller design and program extraction for the controller implementation.
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