Discretized partial differential equations: Examples of control systems defined on modules

Patterned linear systems: Rings, chains, and trees

“…Brockett and Willems [3] showed that one way to find a near Toeplitz state feedback F such that u(t) = F x(t) achieves a desired trajectory x(t) is to model the Toeplitz system by a larger circulant system. A circulant solution can be easily reduced to a solution for the Toeplitz system.…”

Distributed Control: A Sequentially Semi-Separable Approach for Spatially Heterogeneous Linear Systems

“…1) The Structured Matrix Approach: In [15], finitely many spatially invariant subsystems connected in a certain topology are shown to generate system matrices with an easily diagonalizable structure. This allows for decoupling transformations and corresponding control design of computational complexity (where controllers, each of order are designed).…”

Distributed Control of Spatially Reversible Interconnected Systems with Boundary Conditions

“…Examples of such interconnected systems include formations of autonomous vehicles [11], [22], crossdirectional control in the pulp and paper and chemical process industry [13], [14], "smart structures" (large arrays of distributed micro electromechanical actuators and sensors) [2], and semidiscretized partial differential equations [3].…”

“…Early works [3], [17] showed that some systems, especially semidiscretized partial differential equations, can sometimes be treated as systems over modules. More recently, the papers [1], [5], [15] have considered so-called spatially invariant systems and used Fourier techniques or algebraic transformations to derive implementable and scalable optimal control algorithms, even in the limit of an infinite number of subunits.…”

“…It should be noted that an "embedding technique" similar to the method of images was already used in [3] to handle boundary conditions for the particular example of the semidiscretized heat equation on a finite interval. However, it was not emphasized that the possibility of using such a technique was due to symmetries of the problem.…”