Survey paper: Set invariance in control

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

Model Predictive Control (MPC), the dominant advanced control approach in industry over the past twenty-five years, is presented comprehensively in this unique book. With a simple, unified approach, and with attention to real-time implementation, it covers predictive control theory including the stability, feasibility, and robustness of MPC controllers. The theory of explicit MPC, where the nonlinear optimal feedback controller can be calculated efficiently, is presented in the context of linear systems with linear constraints, switched linear systems, and, more generally, linear hybrid systems. Drawing upon years of practical experience and using numerous examples and illustrative applications, the authors discuss the techniques required to design predictive control laws, including algorithms for polyhedral manipulations, mathematical and multiparametric programming and how to validate the theoretical properties and to implement predictive control policies. The most important algorithms feature in an accompanying free online MATLAB toolbox, which allows easy access to sample solutions. Predictive Control for Linear and Hybrid Systems is an ideal reference for graduate, postgraduate and advanced control practitioners interested in theory and/or implementation aspects of predictive control.

Predictive Control for Linear and Hybrid Systems

The study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems. The objective of this paper is to outline some of these problems, to review progress made in solving them in a number of diverse communities, and to review some problems that remain open. An important contribution of our work is to bring together material from several areas of research and to present results in a unified manner. We begin our review by relating the stability problem for switched linear systems and a class of linear differential inclusions. Closely related to the concept of stability are the notions of exponential growth rates and converse Lyapunov theorems, both of which are discussed in detail. In particular, results on common quadratic Lyapunov functions and piecewise linear Lyapunov functions are presented, as they represent constructive methods for proving stability and also represent problems in which significant progress has been made. We also comment on the inherent difficulty in determining stability of switched systems in general, which is exemplified by NP-hardness and undecidability results. We then proceed by considering the stability of switched systems in which there are constraints on the switching rules, through both dwell-time requirements and state-dependent switching laws. Also in this case the theory of Lyapunov functions and the existence of converse theorems are reviewed. We briefly comment on the classical Lur'e problem and on the theory of stability radii, both of which contain many of the features of switched systems and are rich sources of practical results on the topic. Finally we present a list of questions and open problems which provide motivation for continued research in this area.

/pdf/stability-criteria-for-switched-and-hybrid-systems-cmu7vlfa6a.pdf

Stability Criteria for Switched and Hybrid Systems

We study model predictive control (MPC) schemes for discrete-time linear time-invariant systems with constraints on inputs and states, that can be formulated using a linear program (LP). In particular, we focus our attention on performance criteria based on a mixed 1 -norm, namely, 1-norm with respect to time and -norm with respect to space. First we provide a method to compute the terminal weight so that closed-loop stability is achieved. We then show that the optimal control profile is a piecewise affine and continuous function of the initial state and briefly describe the algorithm to compute it. The piecewise affine
form allows to eliminate online LP, as the computation associated with MPC becomes a simple function evaluation. Besides practical advantages, the availability of the explicit structure of the MPC controller provides an insight into the type of control action in different
regions of the state space, and highlights possible conditions of degeneracies of the LP, such as multiple optima.

Model predictive control based on linear programming - the explicit solution

This paper describes an integrated Bayesian approach to maneuver-based trajectory prediction and criticality assessment that is not limited to specific driving situations. First, a distribution of high-level driving maneuvers is inferred for each vehicle in the traffic scene via Bayesian inference. For this purpose, the domain is modeled in a Bayesian network with both causal and diagnostic evidences and an additional trash maneuver class, which allows the detection of irrational driving behavior and the seamless application from highly structured to nonstructured environments. Subsequently, maneuver-based probabilistic trajectory prediction models are employed to predict each vehicle's configuration forward in time. Random elements in the designed models consider the uncertainty within the future driving maneuver execution of human drivers. Finally, the criticality time metric time-to-critical-collision-probability (TTCCP) is introduced and estimated via Monte Carlo simulations. The TTCCP is a generalization of the time-to-collision (TTC) in arbitrary uncertain multiobject driving environments and valid for longer prediction horizons. All uncertain predictions of all maneuvers of every vehicle are taken into account. Additionally, the criticality assessment considers arbitrarily shaped static environments, and it is shown how parametric free space (PFS) maps can advantageously be utilized for this purpose.

An Integrated Approach to Maneuver-Based Trajectory Prediction and Criticality Assessment in Arbitrary Road Environments

The authors propose a Bayesian trajectory prediction and criticality assessment system that allows to reason about imminent collisions of a vehicle several seconds in advance. The authors first infer a distribution of high-level, abstract driving maneuvers such as lane changes, turns, road followings, etc. of all vehicles within the driving scene by modeling the domain in a Bayesian network with both causal and diagnostic evidences. This is followed by maneuver-based, long-term trajectory predictions, which themselves contain random components due to the immanent uncertainty of how drivers execute specific maneuvers. Taking all uncertain predictions of all maneuvers of every vehicle into account, the probability of the ego vehicle colliding at least once within a time span is evaluated via Monte-Carlo simulations and given as a function of the prediction horizon. This serves as the basis for calculating a novel criticality measure, the Time-To-Critical-Collision-Probability (TTCCP) - a generalization of the common Time-To-Collision (TTC) in arbitrary, uncertain, multi-object driving environments and valid for longer prediction horizons. The system is applicable from highly-structured to completely non-structured environments and additionally allows the prediction of vehicles not behaving according to a specific maneuver class.

Bayesian, Maneuver-based, Long-term Trajectory Prediction and Criticality Assessment for Driver Assistance Systems

Soft variable-structure controls: a survey

A unified theory of quadratic and piecewise-linear Lyapunov functions for continuous and discrete-time linear systems is presented. The key to this work is the description of these Lyapunov functions by vector norms. The main results are sufficient and necessary conditions for a vector norm to be a Lyapunov function as well as a method (based on these conditions) of constructing such Lyapunov functions. >

Vector norms as Lyapunov functions for linear systems

This paper proposes a biologically inspired and technically implemented sound localization system to robustly estimate the position of a sound source in the frontal azimuthal half-plane. For localization, binaural cues are extracted using cochleagrams generated by a cochlear model that serve as input to the system. The basic idea of the model is to separately measure interaural time differences and interaural level differences for a number of frequencies and process these measurements as a whole. This leads to two-dimensional frequency versus time-delay representations of binaural cues, so-called activity maps. A probabilistic evaluation is presented to estimate the position of a sound source over time based on these activity maps. Learned reference maps for different azimuthal positions are integrated into the computation to gain time-dependent discrete conditional probabilities. At every timestep these probabilities are combined over frequencies and binaural cues to estimate the sound source position. In addition, they are propagated over time to improve position estimation. This leads to a system that is able to localize audible signals, for example human speech signals, even in reverberating environments

Jürgen Adamy

Papers

An Integrated Approach to Maneuver-Based Trajectory Prediction and Criticality Assessment in Arbitrary Road Environments

Bayesian, Maneuver-based, Long-term Trajectory Prediction and Criticality Assessment for Driver Assistance Systems

Soft variable-structure controls: a survey

Vector norms as Lyapunov functions for linear systems

A Probabilistic Model for Binaural Sound Localization