n-vector

Abstract: The following system is considered: \dot{x}= Ax + Bu y = Cx where x is an n vector describing the state of the system, u is an m vector of inputs to the system, and y is an l vector ( l \leq n ) of output variables. It is shown that if rank C = l , and if (A,B) are controllable, then a linear feedback of the output variables u = K*y, where K*is a constant matrix, can always be found, so that l eigenvalues of the closed-loop system matrix A + BK*C are arbitrarily close (but not necessarily equal) to l preassigned values. (The preassigned values must be chosen so that any complex numbers appearing do so in complex conjugate pairs.) This generalizes an earlier result of Wonham [1]. An algorithm is described which enables K*to be simply found, and examples of the algorithm applied to some simple systems are included.

Abstract: A technique is developed for explicitly eliminating the constraint torques from a canonical system of n vector equations for the attitude dynamics of a satellite consisting of n arbitrarily interconnected rigid bodies. This elimination reduces the number of scalar second order differential equations from 3n to r, the number of degrees of rotational freedom of the satellite. At the same time, the number of dependent variables in these equations is reduced from the full set of 3n angular velocity components to just 3 such components for one body, together with r — 3 relative angular rates. This elimination and reduction saves computer time when the equations are integrated, and also avoids a possible build-up of numerical errors violating the constraints. The final equations resemble those obtained from a Lagrangian approach, but are simpler to derive and to modify to account for additional effects.

Abstract: We make a general study of symmetry and stability of the fixed points of the quartic Hamiltonian of an $n$-component field (or order parameter) for $n\ensuremath{ e}4$. Simple proofs of known results are given. Among new results, we show that when it exists the stable fixed point is unique; we give some precision on its symmetry and on its attractor basin.

Abstract: model representations can be used to describe more than one particular object exemplar, e.g., a single “R” chosen from a particular font. Finally, we’ve discussed two approaches to 3-D model representation offering yet another trade-off: the potentially large number of views needed to reduce a 3-D model to a 2-D model versus the problem of inferring a compact 3-D model’s features from 2-D image features. Armed with this insight, let’s go back and see how the recognition problems in Figure 1 were solved. 5.1 A Viewer-Centered Approach Using Pixels In the approach of Murase and Nayar [39], each image is represented as a point in a high dimensional space. Imagine laying all the rows of the image side by side until the entire image is a 1 × n vector, where n is the number of pixels in the image. Murase and Nayar take the viewer-centered approach to object modeling that models a 3-D object as a set of views. Each of these views is an image of the object taken from a different viewpoint, as shown in Figure 7, and each image becomes a vector, as described above. You may ask why we need a set of different views of the object, and why one view won’t suffice? The reason is that as you move the camera around the object, its appearance may change dramatically. If we’re going to store an object as a collection of views, we need to make sure that we have one view for every possible appearance of the object. Imagine now that we have a database of objects, with each object modeled as a collection of vectors (images). If we’re presented with an arbitrary view of an unknown object and asked to identify the object (and perhaps the viewpoint at which the image was acquired), our task is to find the “closest” image in our database. One simple approach would be to compute the vector distance between the input image vector and all the vectors in the database, choosing the closest database vector as the object’s identity. However, this would be a very computationally expensive procedure, since each vector has n components, where n is the size of the image in pixels. Murase and Nayar’s approach is an extension of a very clever approach proposed by Turk and Pentland [53]. Consider an n-dimensional space, and let each image in the database be a point in that space. It turns out that for the resulting cloud of points, there is a more convenient coordinate system with which to represent the points. Although this coordinate system has the same number of dimensions as the original one, it has the property that the positions of the points can be sufficiently approximated by a small number of the coordinates (e.g., ≤ 20 instead of 16K, for the images in Figure 7). This new coordinate system is defined by the eigenvectors of the covariance matrix derived from the cloud of points. Furthermore, these eigenvectors can be prioritized according to their corresponding eigenvalues. The vectors that have high eigenvalues represent more definitive axes or coordinates in our new coordinate system. Each view of a model object can now be represented in this new coordinate system using only a small number of coordinates. This is an inexpensive process called projection. As we move around the object, its different views will trace out a curve in this new coordinate system, Since each object looks different, each object will have its own characteristic curve in