======= What are nD systems? ======= nD systems are simply put a generalization of the 1D systems. More precisely, the information propagates not only in 1 direction (usually the time for 1D systems) but in 2 or more directions. These directions can be either time/space or space/space variables. \\ If you are familiar to view systems as transfer functions, here are simple examples to help you understand the differences between a 2D and a 1D system: $$(1D):\quad H(z)=\frac{P(z)}{Q(z)}=\frac{1}{z+1}\qquad(2D):\quad H(z_1,z_2)=\frac{P(z_1,z_2)}{Q(z_1,z_2)}=\frac{1}{z_1+2z_2}$$ If you are familiar with the state-space approach: $$(1D):\quad x(i+1)=Ax(i)\qquad(2D):\quad x(i+1,j+1)=A_1x(i,j+1)+A_2x(i+1,j)$$ ======= Why study nD systems? ======= Well, firstly because it is **mathematically challenging** and don't we all love challenges!? Note that nD systems are part of the class of infinite dimensional systems. This in turns implies several difficulties whilst dealing with nD systems the first one is that there are an infinite number of 'poles' to the system (from the example given below, all the points such that $z_1=-2z_2$). The second one is that the initial conditions are given by an infinite number of points (or functions in the continuous case) which in turn creates several problems when dealing with stability. But that's not all, if you like surprises, then nD systems are for you as well. Goodman in a famous paper published in 1977 showed that contrary to the 1D case, the numerator can influence the stability of the system because of the existence of what he called nonessential singularity of the second kind (math stuff...). Have a look at the example he proposed:\\ $H_1(z_1,z_2)=\frac{(1-z_1)(1-z_2)}{2-z_1-z_2}$ is NOT BIBO stable.\\ $H_2(z_1,z_2)=\frac{(1-z_1)^8(1-z_2)^8}{2-z_1-z_2}$ is BIBO stable!\\ And finally, the underlying ring (functions of 2 or more variables) does not have a division algorithm and a lot of usual concept for 1D systems stems from this fact. Specific solutions are therefore needed for nD systems. The second point we would like to make is more about the physics. Many physical processes have a clear nD structures and it makes sense to model and study them as part of the multidimensional framework. Usual applications found in the literature are:\\ * Repetitive systems (such as metal rolling or long-wall coal cutting) * Iterative learning control (fishing everyday and getting better everyday) * Image processing and the use of nD digital filters. ======= What is the project about? ======= The MSDOS project questions the stability and stabilization of multidimensional systems. The project is naturally divided into 6 different tasks: * The first 3 ones (Tasks 1-2-3) will be devoted to the theory and focused on achieving the designated theoretical goals. More specifically, the task 1 concerns the stability of nD systems using nonlinear techniques mainly based on Lyapunov theory. Task 2 is mainly devoted to the study of repetitive systems and task 3 uses algebraic and geometric tools in order to analyze the problem of stabilization. Obviously these 3 tasks are interconnected. For instance a definition of stability given in task 1 will impact the work of task 2 or 3 and reciprocally. * The last 3 ones (Tasks 4-5-6) are designed to the practical side of the project and their designated goals. We will first try to apply the obtained theoretical results in order to study other infinite dimensional systems such as time delay systems (TDS) and partial differential equations (PDEs) using an nD approach (task 4). Task 5 will focus on the coding and implementation of packages which will help studying nD systems. Finally during task 6 we will try to aggregate all the results obtained during tasks 1-2-3-4-5 hence providing the basis for a complete course on the stability of multidimensional systems. The diagram given below summarizes the relations between the different tasks. {{ wiki:msdos.png }} ======= The research proposal ======= {{ biblio:msdos.pdf }}
