Most of the signal processing that we will study in this course involves local operations on a signal, namely transforming the signal by applying linear combinations of values in the neighborhood of each sample point. You are familiar with such operations from Calculus, namely, taking derivatives and you are also familiar with this from optics namely blurring a signal. We will be looking at sampled signals only. Let's start with a few basic examples. Local difference Suppose we have a 1D image and we take the local difference of intensities, DI(x) = 1 2 (I(x + 1) − I(x − 1)) which give a discrete approximation to a partial derivative. (We compute this for each x in the image.) What is the effect of such a transformation? One key idea is that such a derivative would be useful for marking positions where the intensity changes. Such a change is called an edge. It is important to detect edges in images because they often mark locations at which object properties change. These can include changes in illumination along a surface due to a shadow boundary, or a material (pigment) change, or a change in depth as when one object ends and another begins. The computational problem of finding intensity edges in images is called edge detection. We could look for positions at which DI(x) has a large negative or positive value. Large positive values indicate an edge that goes from low to high intensity, and large negative values indicate an edge that goes from high to low intensity. Example Suppose the image consists of a single (slightly sloped) edge:

http://ieeexplore.ieee.org/iel5/5/31307/01456462.pdf

Linear systems

Recent developments in computer and communication technologies have led to a new type of large-scale resource-constrained wireless embedded control systems. It is desirable in these systems to limit the sensor and control computation and/or communication to instances when the system needs attention. However, classical sampled-data control is based on performing sensing and actuation periodically rather than when the system needs attention. This paper provides an introduction to event- and self-triggered control systems where sensing and actuation is performed when needed. Event-triggered control is reactive and generates sensor sampling and control actuation when, for instance, the plant state deviates more than a certain threshold from a desired value. Self-triggered control, on the other hand, is proactive and computes the next sampling or actuation instance ahead of time. The basics of these control strategies are introduced together with a discussion on the differences between state feedback and output feedback for event-triggered control. It is also shown how event- and self-triggered control can be implemented using existing wireless communication technology. Some applications to wireless control in process industry are discussed as well.

/pdf/an-introduction-to-event-triggered-and-self-triggered-2ij6tsuvdo.pdf

An introduction to event-triggered and self-triggered control

Event-triggered control (ETC) is a control strategy that is especially suited for applications where communication resources are scarce. By updating and communicating sensor and actuator data only when needed for stability or performance purposes, ETC is capable of reducing the amount of communications, while still retaining a satisfactory closed-loop performance. In this paper, an ETC strategy is proposed by striking a balance between conventional periodic sampled-data control and ETC, leading to so-called periodic event-triggered control (PETC). In PETC, the event-triggering condition is verified periodically and at every sampling time it is decided whether or not to compute and to transmit new measurements and new control signals. The periodic character of the triggering conditions leads to various implementation benefits, including a minimum inter-event time of (at least) the sampling interval of the event-triggering condition. The PETC strategies developed in this paper apply to both static state-feedback and dynamical output-based controllers, as well as to both centralized and decentralized (periodic) event-triggering conditions. To analyze the stability and the L2-gain properties of the resulting PETC systems, three different approaches will be presented based on 1) impulsive systems, 2) piecewise linear systems, and 3) perturbed linear systems. Moreover, the advantages and disadvantages of each of the three approaches will be discussed and the developed theory will be illustrated using a numerical example.

/pdf/periodic-event-triggered-control-for-nonlinear-systems-4semh9nxhe.pdf

Periodic event-triggered control for nonlinear systems

https://heemels.tue.nl/content/papers/HeeDon_AUT13a.pdf

Model-based periodic event-triggered control for linear systems

We consider sensor data scheduling for remote state estimation. Due to constrained communication energy and bandwidth, a sensor needs to decide whether it should send the measurement to a remote estimator for further processing. We propose an event-based sensor data scheduler for linear systems and derive the corresponding minimum squared error estimator. By selecting an appropriate event-triggering threshold, we illustrate how to achieve a desired balance between the sensor-to-estimator communication rate and the estimation quality. Simulation examples are provided to demonstrate the theory.

/pdf/event-based-sensor-data-scheduling-trade-off-between-1p4cfey1du.pdf

Event-Based Sensor Data Scheduling: Trade-Off Between Communication Rate and Estimation Quality

This paper considers a distributed estimation problem in which a sensor sporadically transmits information to a remote-observer. An event-triggered approach is used to trigger the transmission of information from the sensor to the remote-observer. The event-trigger is chosen to minimize the mean square estimation error at the remote-observer subject to a constraint on how frequently the information can be transmitted. This problem was studied by O.C. Imer et al. [1] and M. Rabi et al. [2] where the observed process was a scalar linear system over a finite time interval. This paper extends those earlier results by relaxing the prior assumption that the initial condition is zero-mean with no measurement noise. It extends those earlier results to vector linear systems through a computationally efficient way of computing sub-optimal event-triggering thresholds.

/pdf/event-triggered-state-estimation-in-vector-linear-processes-5ce0lfnqcl.pdf

Event-triggered state estimation in vector linear processes

This paper examines output feedback control of wireless networked control systems where there are separate links between the sensor-to-controller and controller-to-actuator. The proposed triggering events only rely on local information so that the transmissions from the sensor and controller subsystems are not necessarily synchronized. This represents an advance over recent work in event-triggered output feedback control where transmission from the controller subsystem was tightly coupled to the receipt of event-triggered sensor data. The paper presents an upper bound on the optimal cost attained by the closed-loop system. Simulation results demonstrate that transmissions between sensors and controller subsystems are not tightly synchronized. These results are also consistent with derived upper bounds on overall system cost.

https://www3.nd.edu/~lemmon/projects/NSF-05-1518/Publications/wncs11_submission_3.pdf

Weakly coupled event triggered output feedback control in wireless networked control systems

Event-triggered systems sample and transmit data when a state dependent criterion is violated. Many event-triggered systems are efficiently attentive, i.e., longer inter-sampling intervals are achieved as the system gets closer to its equilibrium. This property allows event-triggered systems to use fewer communication resources since control systems are usually operated around their equilibria. Efficient attentiveness, however, is not a necessary property of all event-triggered systems. To ensure efficient attentiveness, this technical note proposes an approach to design event triggers and quantization maps for nonlinear systems with transmission delays. We present sufficient conditions to guarantee input-to-state stability (ISS) of the resulting systems without exhibiting Zeno behavior.

Efficiently Attentive Event-Triggered Systems With Limited Bandwidth

Event-triggered control systems are systems in which the control signal is recomputed when the plant's output signal leaves a triggering-set. There has been recent interest in event-triggered control systems as a means of reducing the communication load in control systems. This paper re-examines a problem [1] whose solution characterizes triggering-sets that minimize a quadratic control cost over a finite horizon subject to a hard constraint on the number of times the feedback control is computed. Computational complexity confined prior solutions of this problem to scalar linear systems. This paper presents an approximate solution that is suitable for multi-dimensional linear systems. This approximate solution uses families of quadratic forms to bound the value functions generated in solving the problem. This approach has a computational complexity that is polynomial in state-space dimension and horizon length. This paper's results may therefore provide a basis for developing practical methods for the event-triggered output control of multi-dimensional discrete-time linear systems.

/pdf/event-triggered-output-feedback-control-of-finite-horizon-1wnqqvuqem.pdf

Event-triggered output feedback control of finite horizon discrete-time multi-dimensional linear processes

Event-triggered approaches to control and estimation have the sensor transmit processed information when a measure of information ‘novelty’ exceeds a threshold. Prior work has empirically demonstrated that event-triggered systems may have significantly longer average sampling intervals than comparably performing periodically triggered systems. There are, however, few results that analytically characterize the tradeoff that event-triggering introduces between average sampling period and system performance. This paper examines that tradeoff for a sub-optimal solution to the constrained state-estimation problem considered by Xu and Hespanha. The sub-optimal solution is comparable to that used by Cogill, and extends the earlier work to unstable systems. In particular, the paper derives a sub-optimal solution that guarantees the specified least average sampling period. The paper also derives upper and lower bounds on the event-triggered estimator performance. Simulation results are used to demonstrate the utility of these bounds.

/pdf/performance-and-average-sampling-period-of-sub-optimal-2j0ewabtw0.pdf

Performance and average sampling period of sub-optimal triggering event in event triggered state estimation

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