Sparsity preserving optimal control of discretized PDE systems

Abstract

We focus on the problem of optimal control of large-scale systems whose models are obtained by discretization of partial differential equations using the Finite Element (FE) or Finite Difference (FD) methods. The motivation for studying this pressing problem originates from the fact that the classical numerical tools used to solve low-dimensional optimal control problems are computationally infeasible for large-scale systems. Furthermore, although the matrices of large-scale FE or FD models are usually sparse banded or highly structured, the optimal control solution computed using the classical methods is dense and unstructured. Consequently, it is not suitable for efficient centralized and distributed real-time implementations. We show that the a priori (sparsity) patterns of the exact solutions of the generalized Lyapunov equations for FE and FD models are banded matrices. The a priori pattern predicts the dominant non-zero entries of the exact solution. We furthermore show that for wellconditioned problems, the a priori patterns are not only banded but also sparse matrices. On the basis of these results, we develop two computationally efficient methods for computing sparse approximate solutions of generalized Lyapunov equations. Using these two methods and the inexact Newton method, we show that the solution of the generalized Riccati equation can be approximated by a banded matrix. This enables us to develop a novel computationally efficient optimal control approach that is able to preserve the sparsity of the control law. We perform extensive numerical experiments that demonstrate the effectiveness of our approach.

A novel method for adaptive control of deformable mirrors

“…The starting point for the research activities directed toward reducing the computational complexity can be approaches presented in. 8,[48][49][50][51][52][53][54][55]…”

A new hybrid method for two dimensional nonlinear variable order fractional optimal control problems

“…The algebraic system (46) can be solved by employing a suitable software such as Matlab or Maple. Finally, by computing A, B, K, S, R and L, numerical solution of the state function (x, t) and the control function (x, t) are approximated using Equations ( 34) and (35), respectively. The algorithm is described briefly in the sequel.…”

“…Singha and Nahak [32] designed an algorithm based on Laguerre polynomials for a class of FOCP. Interested readers can find other related articles in [33][34][35][36][37][38].…”

Subspace Identification of Temperature Dynamics

“…. , N estimate the state order n, system matrices A, B, C, and K, and the initial states of the state-space models ( 5)-( 6) and ( 7)- (8).…”

“…For example, heating and thermally induced deformations of lenses and mirrors in optical lithography machines used for integrated circuit manufacturing can seriously degrade the quality of manufactured circuits [1]. To develop methods and actuators to counteract these thermally induced deformations, it is of paramount importance that the temperature dynamics of optical elements is accurately modeled and controlled [2]- [8]. Besides this, modeling and control of temperature dynamics are important for temperature estimation and health monitoring of lithium-ion batteries [9], as well as for temperature control of greenhouses [10], [11], buildings [12]- [14], and for proper operation of a number of industrial processes and systems [15], [16].…”