Time-delay systems : design of delay-free and low-order observers

This note provides a comprehensive treatment on the design of functional observers for linear systems having a time-varying delay in the state variables. The designed observers possess attractive features of being low-order and delay-free and hence they are cost effective and easy to implement. Existence conditions are derived and a design procedure for finding low-order observers is given.

Partial state and unknown input estimation for time-delay systems

This article considers the problem of estimating a partial set of the state vector and/or unknown input vector of linear systems driven by unknown inputs and time-varying delay in the state variables. Three types of reduced-order observers, namely, observers with delays, observers without internal delays and delay-free observers are proposed in this article. Existence conditions and design procedures are presented for the determination of parameters for each case of observers. Numerical examples are presented to illustrate the design procedures.

A delay-dependent functional observer for linear time-invariant systems with input delay

A delay-dependent functional observer is designed for linear time-invariant (LTI) systems with time-varying input delay. Compared to delay-free observers, delay-dependent functional observers are less conservative and cover more systems. The designed functional observer is with minimum possible order (minimal). Necessary and sufficient conditions of the existence of the observer and asymptomatic stability of it are illustrated. The proposed observer is extended to multiple input delayed systems with time-varying delays. An algorithm is developed for designing of the minimal order observer based on the methodology of this paper. Two numerical examples and simulations are used to support our proposed methodology.

Reduced order functional observers with application to partial state estimation of linear systems with input delay

Reduced order multi-functional observer design for multi-input multi-utput (MIMO) linear time-invariant (LTI) systems with constant delayed inputs is studied. This research is useful in the input estimation of LTI systems with actuator delay, as well as system monitoring and fault detection of these systems. Two approaches for designing an asymptotically stable functional observer for the system are proposed: delay-dependent and delay-free. The delay-dependent observer is infinite-dimensional, while the delay-free structure is finite-dimensional. Moreover, since the delay-free observer does not require any information on the time delay, it is more practical in real applications. However, the delay-dependent observer contains less restrictive assumptions and covers more variety of systems. The proposed observer design schemes are novel, simple to implement, and have improved numerical features compared to some of the other available approaches to design (unknown-input) functional observers. In addition, the proposed observers usually possess lower order than ordinary Luenberger observers, and the design schemes do not need the observability or detectability requirements of the system. The necessary and sufficient conditions of the existence of an asymptoticobserver in each scenario are explored. The extensions of the proposed observers to systems with multiple delayed-inputs are also discussed. Several numerical examples and simulation results are employed to support our theories.

Full-rate full-diversity finite feedback space-time schemes with minimum feedback and transmission duration

A Finite Feedback Scheme (FFS) for a quasi-static MIMO block fading channel with finite N-ary delay-free noise-free feedback consists of N Space-Time Block Codes (STBCs) at the transmitter, one corresponding to each possible value of feedback, and a function at the receiver that generates N-ary feedback. A number of FFSs are available in the literature that provably attain full-diversity. However, there is no known full-diversity criterion that universally applies to all FFSs. In this paper a universal necessary condition for any FFS to achieve full-diversity is given, and based on this criterion the notion of Feedback-Transmission duration optimal (FT-optimal) FFSs is introduced, which are schemes that use minimum amount of feedback N for the given transmission duration T, and minimum T for the given N to achieve full-diversity. When there is no feedback (N = 1) an FT-optimal scheme consists of a single STBC, and the proposed condition reduces to the well known necessary and sufficient condition for an STBC to achieve full-diversity. Also, a sufficient criterion for full-diversity is given for FFSs in which the component STBC yielding the largest minimum Euclidean distance is chosen, using which full-rate (N-t complex symbols per channel use) full-diversity FT-optimal schemes are constructed for all N-t > 1. These are the first full-rate full-diversity FFSs reported in the literature for T < N-t. Simulation results show that the new schemes have the best error performance among all known FFSs.

Thalamic stimulation for tremor. Subtle changes in episodic memory are related to stimulation per se and not to a microthalamotomy effect

The aim of this study was to investigate the impact of unilateral deep brain stimulation (DBS) of the ventrointermediate (Vim) thalamic nucleus on neuropsychological functioning comparing stimulation-on with stimulation-off conditions. Nine patients [five patients with Parkinson's Disease (PD), two patients with essential tremor (ET) and 2 patients with multiple sclerosis (MS)] underwent comprehensive neuropsychological testing for cognitive functions, including general mental impairment, aphasia, agnosia, executive and constructional abilities, learning, memory, cognitive processing speed and attention as well as depression. The neuropsychological assessments were performed at least 6 months postoperatively (mean 9 months). Testing in the stimulation-on and stimulation-off condition was obtained within a period of 3 to 4 weeks. Unilateral DBS resulted in improvement of tremor in all patients. There were no significant differences between the stimulation-on and the stimulation-off condition with the exception of a decrement of word-recall in the short delay free-recall subtest of the Rey Auditory-Verbal Learning Test (RAVLT). Subgroup analysis indicated that the impairment in word-recall was related to left-sided thalamic stimulation. Our study confirms that chronic unilateral DBS is a safe method with regard to cognitive function. The subtle changes in episodic memory are related to stimulation per se and not to a microthalamotomy effect.

Fitting model-based psychometric functions to simultaneity and temporal-order judgment data: MATLAB and R routines

Research on temporal-order perception uses temporal-order judgment (TOJ) tasks or synchrony judgment (SJ) tasks in their binary SJ2 or ternary SJ3 variants. In all cases, two stimuli are presented with some temporal delay, and observers judge the order of presentation. Arbitrary psychometric functions are typically fitted to obtain performance measures such as sensitivity or the point of subjective simultaneity, but the parameters of these functions are uninterpretable. We describe routines in MATLAB and R that fit model-based functions whose parameters are interpretable in terms of the processes underlying temporal-order and simultaneity judgments and responses. These functions arise from an independent-channels model assuming arrival latencies with exponential distributions and a trichotomous decision space. Different routines fit data separately for SJ2, SJ3, and TOJ tasks, jointly for any two tasks, or also jointly for the three tasks (for common cases in which two or even the three tasks were used with the same stimuli and participants). Additional routines provide bootstrap p-values and confidence intervals for estimated parameters. A further routine is included that obtains performance measures from the fitted functions. An R package for Windows and source code of the MATLAB and R routines are available as Supplementary Files.

On the discrepant results in synchrony judgment and temporal-order judgment tasks: a quantitative model.

Research on the perception of temporal order uses either temporal-order judgment (TOJ) tasks or synchrony judgment (SJ) tasks, in both of which two stimuli are presented with some temporal delay and observers must judge the order of presentation. Results generally differ across tasks, raising concerns about whether they measure the same processes. We present a model including sensory and decisional parameters that places these tasks in a common framework that allows studying their implications on observed performance. TOJ tasks imply specific decisional components that explain the discrepancy of results obtained with TOJ and SJ tasks. The model is also tested against published data on audiovisual temporal-order judgments, and the fit is satisfactory, although model parameters are more accurately estimated with SJ tasks. Measures of latent point of subjective simultaneity and latent sensitivity are defined that are invariant across tasks by isolating the sensory parameters governing observed performance, whereas decisional parameters vary across tasks and account for observed differences across them. Our analyses concur with other evidence advising against the use of TOJ tasks in research on perception of temporal order.

Linear functional observers with guaranteed ε-convergence for discrete time-delay systems with input/output disturbances

The problem of designing linear functional observers for discrete time-delay systems with unknown-but-bounded disturbances in both the plant and the output is considered for the first time in this paper. A novel approach to design a minimum-order observer is proposed to guarantee that the observer error is ϵ-convergent, which means that the estimate converges robustly within an ϵ-bound of the true state. Conditions for the existence of this observer are first derived. Then, by utilising an extended Lyapunov-Krasovskii functional and the free-weighting matrix technique, a sufficient condition for ϵ-convergence of the observer error system is given. This condition is presented in terms of linear matrix inequalities with two parameters needed to be tuned, so that it can be efficiently solved by incorporating a two-dimensional search method into convex optimisation algorithms to obtain the smallest possible value for ϵ. Three numerical examples, including the well-known single-link flexible joint robotic system, are given to illustrate the feasibility and effectiveness of our results.

Network Error Correction for Unit-Delay, Memory-Free Networks Using Convolutional Codes

A single source network is said to be memory-free if all of the internal nodes (those except the source and the sinks) do not employ memory but merely send linear combinations of the symbols received at their incoming edges on their outgoing edges. In this work, we introduce network-error correction for single source, acyclic, unit-delay, memory-free networks with coherent network coding for multicast. A convolutional code is designed at the source based on the network code in order to correct network- errors that correspond to any of a given set of error patterns, as long as consecutive errors are separated by a certain interval which depends on the convolutional code selected. Bounds on this interval and the field size required for constructing the convolutional code with the required free distance are also obtained. We illustrate the performance of convolutional network error correcting codes (CNECCs) designed for the unit-delay networks using simulations of CNECCs on an example network under a probabilistic error model.

Design of linear functional observers for time-delay systems of the neutral-type

This paper presents the design of reduced-order linear functional observers for a class of linear time-delay systems of the neutral-type. The type of the observer proposed in this paper is without internal delay and its order is the same as the number of linear functions to be estimated. First, conditions for the existence of the reduced-order functional observers that are capable of asymptotically estimating any given function of the state vector are derived. Then, based on the newly derived existence conditions, a procedure is given for the determination of the observer parameters. The results derived in this paper include a range of linear systems and extend some existing results of linear functional observers to linear neutral delay systems. A numerical example is given to illustrate the design procedure.

Design of reduced-order observers for neutral time-delay systems

This paper presents a method for the design of reduced-order observers for a class of linear time-delay systems of the neutral-type. Conditions for the existence of reduced-order observers that are capable of asymptotically estimating any given function of the state vector are derived. A step-by-step design procedure is given for the determination of the observer parameters. A numerical example is given to illustrate the design procedure.

Design of distributed functional observers for interconnected time-delay systems

This paper considers the design of reduced-order distributed functional observers for interconnected linear systems with the presence of time delays in the interconnections. Unlike observers which consider only the ideal non-delayed output information transfer, the proposed observer is capable of dealing with delayed output information from geographically separated subsystems. It is shown that by accepting measurement data from other subsystems, the conditions under which an observer exists can be made less conservative. Existence conditions, systematic and a straightforward procedure for the synthesis of the observers are given along with numerical examples illustrating the effectiveness and simplicity of the design algorithm.

Approach to fault detection of time-delay systems using functional observers

A new methodology is reported for designing functional observers to detect actuator faults of a class of time-delay systems where the matrix pair (A, C) is not observable. First, a generalised state transformation is used to transform the system into new coordinates where the delay term associated with the state vector is injected into the system's output and input. Then, a minimum-order functional observer is designed to construct a residual function that can trigger system faults. The finding is significant as it is now possible to detect faults of time-delay systems where the pair (A, C) is not required to be observable. A numerical example is given to illustrate the effectiveness of the proposed design approach. © The Institution of Engineering and Technology 2014.