Funnel control for electrical circuits

We study the zero dynamics and funnel control for linear passive electrical circuits. We show that asymptotic stability of the zero dynamics can be characterized by criteria on the circuit topology. Thereafter we consider the output regulation problem for electrical circuits by funnel control. We show that for circuits with asymptotically stable zero dynamics, the funnel controller achieves tracking of a class of reference signals within a pre-specified funnel. This result can be relaxed to the case of non-autonomous zero dynamics by requiring the reference trajectory to evolve within a certain subspace. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Edge-wise funnel synchronization

Recently, it was suggested in [Shim & Trenn 2015] to use the idea of funnel control in the context of synchronization of multi-agent systems. In that approach each agent is able to measure the difference of its own state and the average state of its neighbours and this synchronization error is used in a typical funnel gain feedback law, see e.g. [Ilchmann & Ryan 2008]. Instead of considering one error signal for each node of the coupling graph (corresponding to an agent) it is also possible to consider one error signal for each edge of the graph. In contrast to the node-wise approach this edgewise funnel synchronization approach results (at least in simulations) in a predictable consensus trajectory. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Well-Posed State/Signal Systems in Continuous Time

We introduce a new class of linear systems, the L ^p -well-posed state/signal systems in continuous time, we establish the foundations of their theory and we develop some tools for their study. The principal feature of a state/signal system is that the external signals of the system are not a priori divided into inputs and outputs. We relate state/signal systems to the better-known class of well-posed input/state/output systems, showing that state/signal systems are more flexible than input/state/output systems but still have enough structure to provide a meaningful theory. We also give some examples which point to possibilities for further study.     

Inversion of linear dynamical systems in the presence of additive noise

We consider the problem of inversion of a stationary dynamical system (i.e., the generation of an estimate of the unknown output signal on the basis of the known output signal) in the presence of an unknown perturbation acting “not in the same channel” as the input to be estimated. We present necessary and sufficient conditions for the inversion of such systems and suggest inversion algorithms.     

Analogue Implementation of the Funnel Controller

In many tracking control problems, pre-specified bounds for the evolution of the tracking error should be met. The ‘funnel controller’ addresses this requirement and guarantees transient performance for a fairly large class of systems. In addition, only structural assumptions on the underlying system are made; the exact knowledge of the system parameters is not required. This is in contrast to most classical controllers where only asymptotic behaviour can be guaranteed and the system parameters must be known or estimated. Until now, the funnel controller was only studied theoretically. We will present the results of an analogue implementation of the funnel controller. The results show that the funnel controller works well in reality, i.e. it guarantees the pre-specified error bounds. The implementation is an analogue circuit composed of standard devices and is therefore suitable for a broad range of applications. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Connections Between Classical and Generalised Trajectories of a State/Signal System

In our earlier article “Well-posed state/signal systems in continuous time”, we originally defined the notion of a trajectory of a state/signal system by means of a generating subspace. However, it was left as an open problem whether the generating subspace is uniquely determined by a given family of all generalised trajectories of a well-posed state/signal system. In this article we give a positive answer to this question and show how this insight simplifies some formulations in the theory of well-posed state/signal systems. The main contribution of the article is an explicit convolution scheme for constructing classical trajectories approximating an arbitrary generalised trajectory. We apply this scheme by studying relationships between classical and generalised trajectories of continuous-time state/signal systems under very weak assumptions. Among others, we show that there exists a space of classical trajectories that is invariant under differentiation and dense in the space of generalised trajectories. Some of our results generalise known results for strongly continuous semigroups and input/state/output systems, but we make no use of decompositions of the signal space into an input space and an output space, and in particular, none of our results depend on well-posedness.     

Stability of bilinear time-delay systems

In this paper, the stability of the differential bilinear time-delaysystems is first studied. We consider time-varying bilineartime-delay systems with output feedback. The input or controlu(t)is not only a signal but also an input with output feedback.The analysis is given by using norm-transformation methods.     

Error bounds in the gap metric for dissipative balanced approximations

We derive an error bound in the gap metric for positive real balanced truncation and positive real singular perturbation approximation. We prove these results by working in the context of dissipative driving-variable systems, as in behavioral and state/signal systems theory. In such a framework no prior distinction is made between inputs and outputs. Dissipativity preserving balanced truncation of dissipative driving-variable systems is addressed and a gap metric error bound is obtained. Bounded real and positive real input–state–output systems are manifestations of a dissipative driving-variable system through particular decompositions of the signal space. Under such decompositions the existing bounded real and positive real balanced truncation schemes can be seen as special cases of dissipative balanced truncation and the new positive real error bounds follow.     

Optimal Mathematical Models for Nonlinear Dynamical Systems

We propose a new method for the optimal causal representation of nonlinear systems. The proposed approach is based on the best constrained approximation of mappings in probability spaces by operators constructed from matrices of special form so that the approximant preserves the causality property. It is supposed that the observable input is contaminated with noise. The approximant minimises the mean-square difference between a desired output signal and the output signal of the approximating model. The method provides a numerically realisable mathematical model of the system. An analysis is given of the error associated with this representation.     

A philosophy for the modelling of realistic nonlinear systems

A nonlinear dynamical system is modelled as a nonlinear mapping from a set of input signals into a corresponding set of output signals. Each signal is specified by a set of real number parameters, but such sets may be uncountably infinite. For numerical simulation of the system each signal must be represented by a finite parameter set and the mapping must be defined by a finite arithmetical process. Nevertheless the numerical simulation should be a good approximation to the mathematical model. We discuss the representation of realistic dynamical systems and establish a stable approximation theorem for numerical simulation of such systems.

     

Inversion of linear delay dynamical systems

We consider the problem of inversion of a dynamical system, that is, the problem of real-time reconstruction of the unknown input of the system on the basis of measurements of its output. We consider linear systems of functional-differential equations in the case of commensurable delays. We obtain necessary conditions for the invertibility of this class of systems and suggest an inversion algorithm that permits one to obtain an estimate of the unknown input signal with given accuracy.     

Asymptotic observers for <Emphasis Type="Italic">n</Emphasis>-dimensional bilinear systems

We consider the construction of exponential observers for n-dimensional bilinear dynamical systems with arbitrary n. An algorithm is proposed to construct observers with any prespecified degree of stability for a system with a multidimensional output. Observers are constructed for systems with a scalar output and a singular bilinearity matrix. __________ Translated from Nelineinaya Dinamika i Upravlenie, No. 3, pp. 19-26, 2003.     

Spurious correlation as an approximation of the mutual information between redundant outputs and an unknown input

Stochastic resonance (SR) is a counterintuitive phenomenon, observed in a wide variety of nonlinear systems, for which the addition of noise of opportune magnitude can improve signal detection. Tuning the noise for maximizing the SR effect is important both for artificial and biological systems. In the case of artificial systems, full exploitation of the SR effect opens the possibility of measuring otherwise unmeasurable signals. In biology, identification of possible SR maximization mechanisms is of great interest for explaining the low-energy high-sensitivity perception capabilities often observed in animals. SR maximization approaches presented in literature use knowledge on the input signal (or stimulus, in the case of living beings), and maximize the mutual information between the input and the output signal. The input signal, however, is unknown in many practical settings. To cope with this problem, this paper introduces an approximation of the input–output mutual information based on the spurious correlation among a set of redundant units. A proof of the approximation, as well as numerical examples of its application are given.     

Transfer Function and the State Space Model Identification of a Funnel Shaped Piezoelectric Structure for the Controller Design Purposes

For the controller design purposes in order to suppress vibration magnitudes of a funnel shaped piezoelectric shell structure the mathematical model of the structure was identified in the form of the transfer function and the state space representation. The excitation of the structure with different signal types as well as the measurement of the responses were achieved using the piezoelectric actuator and sensor patches attached to the funnel. From the frequency responses obtained as a ratio of the Fast Fourier Transforms between the appropriate sensor and the actuator signals, discrete-time transfer functions were determined by best curve fitting of the model-based frequency response. For the state space model identification the subspace based identification approach was used. Obtained models were used for the optimal LQ controller design. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stochastic resonance with tuning system parameters: the application of bistable systems in signal processing

A thorough evaluation of stochastic resonance with tuning system parameters in bistable systems is presented as a nonlinear signal processor. It is shown that the output signal-to-noise ratio obtained by adjusting systems parameters can exceed that by tuning noise intensity, especially when the input noise intensity is already beyond the resonance region. It is demonstrated that the theory and the method presented here can markedly improve the output signal-to-noise ratio, and minimize phase lag as well as the distortion of the system output signal with multi-frequency.     

Chaos time-domain reflectometry for fault location on live wires

We propose a chaos time-domain reflectometry (CTDR) for locating faults on live wires. This method uses a chaotic output of an improved Colpitts oscillator as probe signal, and detects wire faults by correlating a duplicate with the echo of the probe signal. Benefiting from the anti-jamming of the correlation function of the wideband chaos, fault location on live wires can be achieved. We experimentally demonstrate the detection for live wires in a digital communication system, in which a type of digital signal named high density bipolar of order 3 (HDB3) is transmitted. The effects of the chaotic probe signal on the bit error rate (BER) of the transmitted HDB3 at different rates are analyzed. Meanwhile, the influences of the backward HDB3 reflected by wiring faults on the signal-noise-ratio (SNR) of CTDR measurement are examined experimentally. The results show that fault detection on live wires is achieved when the power of the chaotic probe signal is about from -24.8 dB to -13.5 dB lower than that of the transmitted digital signal. In this case, the BER is kept less than 3E-10, and the SNR of CTDR is higher than 3 dB. Besides, the auto-correlation properties of the improved Colpitts oscillator at different states are investigated experimentally to explore the suitable chaotic states for the CTDR.     

Construction of a Class of Finite H ∞ Controllers for Infinite-Dimensional Systems

We consider the suppression of forced oscillations in distributed systems of a hyperbolic type by finite-dimensional controllers using an H objective. The system is split into a finite-dimensional and an infinite-dimensional subsystems. The controller receives a signal from the output of both systems. The class of controllers is described in the form of a system of ordinary differential equations.     

Global Optimization on Funneling Landscapes

Molecular conformation problems arising in computational chemistry require the global minimization of a non-convex potential energy function representing the interactions of, for example, the component atoms in a molecular system. Typically the number of local minima on the potential energy surface grows exponentially with system size, and often becomes enormous even for relatively modestly sized systems. Thus the simple multistart strategy of randomly sampling local minima becomes impractical. However, for many molecular conformation potential energy surfaces the local minima can be organized by a simple adjacency relation into a single or at most a small number of funnels. A distinguished local minimum lies at the bottom of each funnel and a monotonically descending sequence of adjacent local minima connects every local minimum in the funnel with the funnel bottom. Thus the global minimum can be found among the comparatively small number of funnel bottoms, and a multistart strategy based on sampling funnel bottoms becomes viable. In this paper we present such an algorithm of the basin-hopping type and apply it to the Lennard–Jones cluster problem, an intensely studied molecular conformation problem which has become a benchmark for global optimization algorithms. Results of numerical experiments are presented which confirm both the multifunneling character of the Lennard–Jones potential surface as well as the efficiency of the algorithm. The algorithm has found all of the current putative global minima in the literature up to 110 atoms, as well as discovered a new global minimum for the 98-atom cluster of a novel geometrical class.     

Multivariable fuzzy control applied to the physical–chemical treatment facility of a Cellulose factory

Feedback linearization is a well-known technique in nonlinear control in which known system nonlinearities are canceled by the control input leaving a linear control problem. Feedback linearization requires an exact model for the system. Fundamental and advanced developments in neuro-fuzzy synergy for modeling and control are used to apply the feedback linearization control law on second-order plants. In the models that are used, the nonlinear plant is decomposed on six fuzzy systems necessary to apply the control signal to allow the following of a reference value. A practical application is also presented using a waste water plant. This method can be extended to multiple input–multiple output (MIMO) plants based on input–output data pairs collected directly from the plant.