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

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Numerical methods for mixed-integer optimal control problems

SUMMARY It has been common practice to find controls satisfying only necessary conditions for optimality, and then to use these controls assuming that they are (locally) optimal. However, sufficient conditions need to be used to ascertain that the control rule is optimal. Second order sufficient conditions (SSC) which have recently been derived by Agrachev, Stefani, and Zezza, and by Maurer and Osmolovskii, are a special form of sufficient conditions which are particularly suited for numerical verification. In this paper we present optimization methods and describe a numerical scheme for finding optimal bang–bang controls and verifying SSC. A straightforward transformation of the bang–bang arc durations allows one to use standard optimal control software to find the optimal arc durations as well as to check SSC. The proposed computational verification technique is illustrated on three example applications. Copyright # 2005 John Wiley & Sons, Ltd.

Optimization methods for the verification of second order sufficient conditions for bang-bang controls

An efficient algorithm, called the time-optimal switching (TOS) algorithm, is proposed for the time-optimal switching control of nonlinear systems with a single control input. The problem is formulated in the arc times space, arc times being the durations of the arcs. A feasible switching control, or as a special case bang-bang control, is found using the STC method previously developed by the authors to get from an initial point to a target point with a given number of switchings. Then, by means of constrained optimization techniques, the cost being considered as the summation of the arc times, a minimum-time switching control solution is obtained. Example applications of the TOS algorithm involving second-order and third-order systems are presented. Comparisons are made with a well-known general optimal control software package to demonstrate the efficiency of the algorithm.

Computational method for time-optimal switching control

A new control method, the switching time computation (STC) method (Kaya, C Y and J L Noakes, Proc 33rd IEEE Conf on Decision and Control, Orlando, FL, 1994, pp 3823-3824), is described in detail for single-input non-linear systems, and the mathematical reasoning behind it is explained A concatenation of constant-input arcs is used to get from an initial point to the target, and an optimization procedure is employed to find the necessary time lengths of the arcs General difficulties of the STC method in finding the necessary arc time lengths are emphasized and exemplified The STC method is incorporated in a time-optimal bang-bang control (TOBC) algorithm The results of the application of the TOBC algorithm to the van der Pol equation, a third-order non-linear system and a non-linear dynamical model of the F-8 aircraft are presented The STC method is shown to be fast by making comparisons with a general optimal control software package

Computations and time-optimal controls

The leapfrog algorithm, so called because of its geometric nature, for solving a class of optimal control problems is proposed. Initially a feasible trajectory is given and subdivided into smaller pieces. In each subdivision, with the assumption that local optimal controls can easily be calculated, a piecewise-optimal trajectory is obtained. Then the junctions of these smaller pieces of optimal control trajectories are updated through a scheme of midpoint maps. Under some broad assumptions the sequence of trajectories is shown to converge to a trajectory that satisfies the maximum principle. The main advantages of the leapfrog algorithm are that (i) it does not need an initial guess for the costates and (ii) the piecewise-optimal trajectory generated in each iteration is feasible. These are illustrated through a numerical implementation of leapfrog on a problem involving the van der Pol system.

Leapfrog for Optimal Control

We study the problem of finding an interpolating curve passing through prescribed points in the Euclidean space. The interpolating curve minimizes the pointwise maximum length, i.e., $L^\infty$-norm, of its acceleration. We reformulate the problem as an optimal control problem and employ simple but effective tools of optimal control theory. We characterize solutions associated with singular and nonsingular controls. Some of the results we obtain are new even for the scalar interpolating function case. We reduce the infinite-dimensional interpolation problem to an ensuing finite-dimensional one and derive closed form expressions for interpolating curves. Consequently we devise efficient numerical techniques and illustrate them with examples.

Finding Interpolating Curves Minimizing $L^\infty$ Acceleration in the Euclidean Space via Optimal Control Theory

In the plane results are given about the structure of closed trajectories which may occur as simple closed curves or general closed curves with self intersections Necessary conditions for the global controllability of nonlinear systems that are in the so called linear analytic form x f x u g x where x IR and juj are given It is proved that if there exists a closed trajectory of the system then either contains a point where f and g are linearly dependent or encloses some zeroes of f u g for all u Then this result is used to prove that if the linear analytic system is controllable and the vector eld g is never zero in W IR then W contains at least one zero of f u g for some u A topological approach is taken Remarks are made about the size of the region where a closed trajectory can lie and about the shape of the closed trajectories Further implications are discussed

Closed trajectories and global controllability in the plane

Using linear approximations of nonlinear systems has long been a practice to design control laws In this paper an analysis is given involving linear approxima tion of the nonlinear control system and small time reachable sets in IR A useful concept the swing out which is a measure of nonlinearity is de ned This is used to examine the relationship between the small time reachable sets of the nonlinear control system and its linear approximation Behaviour of the nonlinear system under a con trol law is examined within this context More facts are given about the swing out for some special cases

J. Lyle Noakes

Papers

Computations and time-optimal controls

Leapfrog for Optimal Control

Finding Interpolating Curves Minimizing $L^\infty$ Acceleration in the Euclidean Space via Optimal Control Theory

Closed trajectories and global controllability in the plane

Linearized control systems and small-time reachable sets