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

This paper is a tutorial on the positive realization problem, that is the problem of finding a positive state-space representation of a given transfer function and characterizing existence and minimality of such representation. This problem goes back to the 1950s and was first related to the identifiability problem for hidden Markov models, then to the determination of internal structures for compartmental systems and later embedded in the more general framework of positive systems theory. Within this framework, developing some ideas sprang in the 1960s, during the 1980s, the positive realization problem was reformulated in terms of a geometric condition which was recently exploited as a tool for finding the solution to the existence problem and providing partial answers to the minimality problem. In this paper, the reader is carried through the key ideas which have proved to be useful in order to tackle this problem. In order to illustrate the main results, contributions and open problems, several motivating examples and a comprehensive bibliography on positive systems organized by topics are provided.

A tutorial on the positive realization problem

This paper investigates the stabilizability of linear systems with closed-loop positivity. A necessary and sufficient condition for the existence of desired state-feedback controllers guaranteeing the resultant closed-loop system to be asymptotically stable and positive is obtained. Both continuous and discrete-time cases are considered, and all of the conditions are expressed as linear matrix inequalities which can be easily verified by using standard numerical software. Numerical examples are provided to illustrate the proposed conditions.

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Control for stability and positivity: equivalent conditions and computation

This paper is devoted to the study of a stochastic linear-quadratic (LQ) optimal control problem where the control variable is constrained in a cone, and all the coefficients of the problem are random processes. Employing Tanaka's formula, optimal control and optimal cost are explicitly obtained via solutions to two extended stochastic Riccati equations (ESREs). The ESREs, introduced for the first time in this paper, are highly nonlinear backward stochastic differential equations (BSDEs), whose solvability is proved based on a truncation function technique and Kobylanski's results. The general results obtained are then applied to a mean-variance portfolio selection problem for a financial market with random appreciation and volatility rates, and with short-selling prohibited. Feasibility of the problem is characterized, and efficient portfolios and efficient frontier are presented in closed forms.

Constrained Stochastic LQ Control with Random Coefficients, and Application to Portfolio Selection

Technological innovation pushes towards the consideration of dynamical systems of a mixed continuous and discrete nature, which are called “hybrid systems” Hybrid systems arise, for instance, from the combination of an analog continuous-time process and a digital time-asynchronous controller Many consumer products (cars, micro-wave units, washing machines and so on) are controlled by digital embedded software, rendering the overall process a system with mixed dynamics Also many physical systems display hybrid behavior: the description of multi body dynamics depends crucially on the presence or absence of a contact, models of friction phenomena distinguish between slip and stick phases and electrical circuits contain switching elements like diodes that can be blocking (open circuit) or conducting (short circuit) From these examples it is obvious that a too general study of hybrid systems will lack decisive power: it will not result in detailed information on individual elements in the studied class Therefore, one has to consider a subclass of hybrid systems carrying a clear additional structure allowing analysis of its behavior (eg well-posedness, simulation methods, stability) and facilitating systematic controller synthesis However, the chosen subclass must also contain many interesting examples from an application point of view The class of (linear) complementarity systems satisfies both requirements and is the subject of the thesis Complementarity systems are described by differential equations, inequalities and logic expressions and form dynamical extensions of the linear complementarity problem (LCP) of mathematical programming The study of the complementarity class is motivated by a broad range of physically interesting systems that can be reformulated in terms of the complementarity formalism Examples include mechanical systems subject to unilateral constraints, Coulomb friction or one-sided springs; electrical networks with diodes; control systems with saturation or deadzones; piecewise linear and variable structure systems; relay systems; hydraulic processes with one-way valves; and sets of equations resulting from optimal control problems with state or control constraints Moreover, in Chapter 6 it is shown that the class of “projected dynamical systems” also fits into the complementarity framework To obtain a well-founded theory, it is essential to define a physically relevant solution concept and answer the classical questions of existence and uniqueness of solutions Because of the “jump-phenomena” in the system variables and the multimodal behavior, formulating a solution concept for linear complementarity systems (LCS) is non-trivial The solution trajectories are defined by combining a hybrid point of view and a distributional framework After the formal introduction of the solution concept, connections are established with the existing literature on mechanical systems and electrical circuits It is shown that the proposed solution concept is not an artificial one, but that it is in accordance with well-known rules specified for subclasses of complementarity systems

/pdf/linear-complementarity-systems-a-study-in-hybrid-dynamics-49npiz06dm.pdf

Linear complementarity systems : a study in hybrid dynamics

In this paper, the Linear Quadratic Regulator Problem with a positivity constraint on the admissible control set is addressed. Necessary and sufficient conditions for optimality are presented in terms of inner products, projections on closed convex sets, Pontryagin's maximum principle and dynamic programming. The main results are concerned with smoothness of the optimal control and the value function. The maximum principle will be extended to the infinite horizon case. Based on these analytical methods, we propose a numerical algorithm for the computation of the optimal controls for the finite and infinite horizon problem. The numerical methods will be justified by convergence properties between the finite and infinite horizon case on one side and discretized optimal controls and the true optimal control on the other.

Linear quadratic regulator problem with positive controls

In this paper, the Linear Quadratic Regulator Problem with a positively constraint on the admissible control set is addressed. Necessary and sufficient conditions for optimality are presented in terms of inner products, projections on closed convex sets. Pontryagin's maximum principle and dynamic programming. Sufficient and sometimes necessary conditions for the existence of positive stabilizing controls are incorporated. Convergence properties between the finite and infinite horizon case are presented. Besides these analytical methods, we describe briefly a method for the approximation of the optimal controls for the finite and infinite horizon problem.

/pdf/linear-quadratic-regulator-problem-with-positive-controls-2x3bpolzj0.pdf

van Sjl Stef Eijndhoven

Papers

Linear quadratic regulator problem with positive controls

Linear quadratic regulator problem with positive controls

Linear quadratic regulator problem with positive controls