Quantised Systems Research Articles

Two-level quantised control systems: sliding-mode approach

ABSTRACTSliding mode quantisers (SMQs), where signals are converted to switching signals represented by the sequence of binary values 's and 's, are increasingly being used in various applications including control of power converters, networked control systems (NCSs), one-bit control processing, etc. This paper derives the stability conditions of nonlinear control system embedded with SMQ-based Δ-M. The equivalence conditions of the proposed SMQ, which assure that its binary outputs are equivalent to its inputs, are established. The results of theoretical analysis are validated through simulation considering an example of the underwater vehicle control system, where the transmission bit-rate is limited by . The results of the simulation demonstrates that the quantised feedback signals effectively control the system and achieves both the desired specifications and the bandwidth reduction.

A new way of visualising quantum fields

Quantum field theory (QFT) is the basis of some of the most fundamental theories in modern physics, but it is not an easy subject to learn. In the present article we intend to pave the way from quantum mechanics to QFT for students at early graduate or advanced undergraduate level. More specifically, we propose a new way of visualising the wave function Ψ of a linear chain of interacting quantum harmonic oscillators, which can be seen as a model for a simple one-dimensional bosonic quantum field. The main idea is to draw randomly chosen classical states of the chain superimposed upon each other and use a grey scale to represent the value of Ψ at the corresponding coordinates of the quantised system. Our goal is to establish a better intuitive understanding of the mathematical objects underlying quantum field theories and solid state physics.

Reversible adiabatic and isentropic changes in quantum systems

In traditional thermodynamics it is assumed that isentropic, reversible, adiabatic processes can be summoned up on demand and straightforwardly accomplished. By contrast, taking entropy as the maximised uncertainty of a final equilibrium state of a quantised system, it is not obvious that an associated process can always be found that is both rigorously isentropic and reversibly adiabatic. In fact, we find that linear relations between generalized forces Xj (such as pressures Pj) and energies Ej are necessary and sufficient conditions for a reversible quasi-static and adiabatic change to be truly isentropic. However, such relationships only hold for a few especially simple systems, such as the perfect gas and the idealised paramagnet. They do not generally hold to all orders for more complicated systems. By considering the associated entropy increases up to second order in small changes of the conjugate displacements (such as the volume Vj) we argue that the consequences are nevertheless in practice negligible.

Adaptive integral sliding mode control with bounded ℒ 2 gain performance of uncertain quantised control systems

Adaptive integral sliding mode control (ISMC) is investigated for a class of non-linear uncertain systems considering the mismatch relation between quantisation sensitivity parameters. First, by analysing the operation principle of the encoder/decoder, a mismatched relation between the quantisation parameters is established by using a time-varying ratio model, where the boundaries of the time-varying ratio can be unknown, regarding the existing cases as special ones. Second, using linear matrix inequality techniques, the prescribed bounded ℒ2 gain performance condition is obtained for the sliding dynamics. Sequently, based on the formed boundary unknown time-varying ratio relation, adaptive ISMC laws are designed to ensure the reachability of the sliding manifold, then the establishment of the stability of the closed-loop system. Finally, an example is provided to illustrate the effectiveness of the proposed methods.

Stability of an improved dynamic quantised system with time‐varying delay and packet losses

The stability problem of a linear dynamic quantised system with time-varying delay and packet losses is studied in this article. An optimal dynamic quantiser which is able to minimise the maximum output error between the quantised system and unquantised systems is designed, and the minimum upper bound of the quantised error is also given. Moreover, the system is lifted into a switched system for stability analysis and a sufficient condition for asymptotic stability is developed in terms of matrix inequalities. Finally, an illustrative example demonstrates the effectiveness of the proposed method.

Output-feedback quantised control of decentralised systems

The problem of designing decentralised H ∞ feedback control for a class of linear interconnected continuous-time systems with quantised signals in the subsystem control channels is fully examined. The system has unknown-but-bounded couplings and interval delays. A decentralised static output-feedback controller is designed at the subsystem level to render the closed-loop system is delay-dependent asymptotically stable with guaranteed γ-level. When the local output measurements are quantised, we develop a local output-dependent procedure for updating the quantiser parameters to attain similar asymptotic stability and guaranteed performance of the closed-loop quantised system. Several special cases of interest are derived. Simulation of two representative examples are presented to show the effectiveness of the developed approach.

Stability analysis of optimally quantised LFT-feedback systems

This article is concerned with the stability analysis of quantised feedback control systems which contain the optimal dynamic quantisers recently proposed by the authors. First, it is shown that a sort of separation property of quantiser-controller design holds as far as the feedback system stability concerns. This allows us to design feedback controllers independently of the choice of quantisers. Then, based on this property, a necessary and sufficient condition is derived for the optimally quantised feedback system to be stable, which is described in terms of the poles/zeros of a linear feedback system to be quantised. Finally, we provide a class of suboptimal dynamic quantisers for which the resulting quantised control systems are guaranteed to be stable.

Behaviour Prediction and Simulation of Partition-Based Biological Models – Some Open Issues of Lattice Paradigms

The state space partition problem for hybrid systems, closely related to prediction of systems' behaviour, consists of finding the conditions under which a given quantised system is deterministic . Some biological processes present a hybrid nature, involving the interaction between the continuous evolution of internal cell variables, such as concentrations, and abrupt changes of discrete cells characteristics. In these classes of models, the basic modelling paradigm, implying a choice of the state space structure, is a lattice, with elements generally represented by single cells. This paper comparatively presents two approaches from the literature, both with a hybrid nature but involving two distinct lattice types. The description of their behaviour prediction mechanisms are overviewed and simulation of an abstract quasi-random invasion process, inspired from the second model of a brain tumor growth, is finally discussed, emphasising the difficulties in solving the general state space partition problem. Also a tumor-growth limiting model is proposed, as a dynamic nutrient inhibitor. Some aspects driving to a general lattice-based model of a growing process are finally discussed.

Comparison of Interval Models and Quantised Systems in Fault Detection with Application to the DAMADICS Actuator Benchmark Problem

Model-based fault detection relies on checking the discrepancy between the measurements obtained from a monitored process and its model. In general, the model representing the process behaviour has substantial uncertainties. In this contribution, two fault detection methods that explicitly take the modelling uncertainties into account are presented and compared. The first approach uses the interval model while the second tackles the process to be monitored as a quantised system. The main ideas of both approaches are discussed and the results of the fault detection using each approach will be analysed and compared with respect to the DAMADICS actuator benchmark.

A discrete-event model of asynchronous quantised systems

The paper deals with the discrete-event modelling of continuous-variable dynamical systems whose input and state signals can only be measured through quantisers. A change of the quantised input or state value is called an input or a state event. The paper concerns the general case in which these events may occur asynchronously to one another. A nondeterministic automaton is used as the discrete-event model of such quantised systems. The paper presents an abstraction algorithm for determining the state transition relation of the automaton from the state-space description of the continuous-time continuous-variable system and the signal space partitions introduced by the quantisers. The abstraction is based on the reachability analysis of quantised systems. A hierarchy of models with respect to the set of spurious behaviours is obtained by varying an abstraction parameter. The discrete-event models are proved to be complete models of the quantised system with the best possible transition relation.

Diagnosis of transient faults in quantised systems

This paper concerns the diagnosis and identification of faults that occur in systems where signals can only be measured through a quantiser. A qualitative model is used that represents the discrete-event behaviour of the quantised system. Three different diagnostic algorithms are presented for determining the fault probabilities, the first concerning faults currently affecting the system and the second determining the probabilities that faults occurred at any time in the past. The third algorithm is based on the assumption that the faults occurred only during a time interval in the past. Due to this assumption, the algorithm is applicable for continuously running processes and for the online identification of transient faults. The approach is illustrated by an example of a manufacturing cell.

Diagnosis of Quantised Systems

Quantised systems are continuous-variable systems whose input and output signals can only be measured through quantisers. The paper surveys methods for modelling and diagnosing such systems. In the modelling step, a unique discrete-event description of the quantised system is set up, which has the form of a stochastic automaton. The paper shows how the behavioural relation of the automaton has to be selected in order to obtain a complete model of the quantised system. The diagnosis is based on a general method for diagnosing stochastic automata, which is applied to the automaton that represents the quantised system. The paper ends with a list of practical applications and with bibliographical notes on the subject.

Reconfigurable Control of Quantised Systems

This paper deals with the reconfiguration of the control after major faults have occurred which require a change of the control structure. The faulty plant is a quantised system, for which a stochastic automaton is derived as purely discrete-event representation. Using a qualitative method, a new controller is designed as an approximate solution to an optimisation problem. This paper focuses on fast numerical methods suitable for on-line use. The Three Tank Reconfiguration Benchmark Problem is solved to illustrate the method.

Diagnosis of Transient Faults in Quantised Systems

Abstract This paper concerns the diagnosis and identification of faults that occur in system where signals can only be measured qualitatively. By using a stochastic automaton to represent the quantised system, the fault probabilities are determined. The diagnostic problem considered here does not only concern faults currently affecting the system but also the faults that occurred at any time in the past. Such faults can be detected only by evaluating past sequences of measurements and consideration of sequences of faults. An algorithm that solves this diagnostic problem is presented. The approach is illustrated by an example of a manufacturing cell.

Resonant Tunnelling Studies of Chaos in Quantised Systems

Resonant Tunnelling Studies of Chaos in Quantised Systems

Solitons in Laser Physics

We survey some of the applications of soliton theory in laser physics. We briefly treat the theory of optical self-focussing and optical filamentation in neutral dielectrics and plasmas where the governing equation is the non-linear Schrödinger equation or one of its generalisations. We establish the connection of this theory with 1-dimensional Langmuir turbulence in plasmas. We treat the theory of optical self-induced transparency (SIT) at greater length and develop the Maxwell-Bloch (MB), reduced Maxwell-Bloch (RMB), SIT and sine-Gordon (s-G) equations to describe it. An optical three-wave interaction is related to the s-G equation; and reference is made to recent work on solitons in stimulated Raman scattering.The RMB equations are solved by a Zakharov-Shabat-AKNS inverse scattering scheme. The inhomogeneously broadened RMB equations have the unusual feature that ln a is not a constant of the motion. However, the sharp line RMB equations have two infinite sets of conserved densities, and the system constitutes one more example of a completely integrable infinite dimensional Hamiltonian system. In terms of scattering data the Hamiltonian of the RMB equations separates into soliton, breather and `background', that is `radiation', contributions. The sine-Gordon equation and its separable Hamiltonian are found as a special case of the RMB equations and its Hamiltonian. Averaged Lagrangian techniques are independently used to relate the c-number MB, RMB and SIT equations and to analyse the slowly varying phase and envelope approximations by which the SIT equations are derived from the MB or RMB equations. The connection of this Lagrangian theory with the Hamiltonian theory is not established.The Hamiltonian formalism in terms of the scattering data is used to quantise the RMB and s-G equations. The RMB, like the s-G, has the discrete energy level spectrum associated with a quantised breather; but only in the s-G limit is the quantised system easy to interpret. The quantised s-G is used to model a `coarse grained' operator theory of strictly resonant sharp line optical pulse propagation. The validity of such a description is examined.The physics of the c-number RMB equations is also discussed and particularly the relation of the c-number breather solutions to the 2π-pulse solutions of the SIT equations. The c-number RMB breather solutions provide a more general theory of SIT valid (within the 2-level atom model) at all electromagnetic field intensities and restricted only by the low density condition which permits the neglect of back scattering.Finally we look at four problems in resonant non-linear optics from a physical point of view. These are degenerate SIT and singular perturbation theory for it; the collision of oppositely directed resonant optical pulses; the theory of super-radiance; and the theory of optical self-focussing in resonant SIT.

Quantisation of soliton solution of a generalised discrete non-linear equation

Soliton solution of a discrete non-linear equation of which the KdV, Volterra and Toda equation are particular cases, have been quantised by the semi-classical method of Dashen et al. the primary feature of the system is the emergence of the same stability angle in all three cases. The discrete energy levels of the quantised system were found to depend on the characteristic constants of the individual non-linear equations.

Distortion of speech by quantising

The proper function of a communication system is to produce, at its output, any one of a number of acceptable alternatives for the exact form of the input message. Certain changes during transmission can therefore be accepted. This principle has been applied to the transmission of speech by a quantised system.