Accelerated disturbance damping of an unknown distributed system by nonlinear feedback

A Karhunen–Loève least-squares technique for optimization of geometry of a blunt body in supersonic flow

“…The KL decomposition has been used in conjunction with the Galerkin method of weighted residuals to develop approximate models of turbulent fluid mechanical phenomena, e.g., Aubry et al [18], and Sirovich and Park [19,20]. The first use of the KL method for a control application was by Chen and Chang [21], where it was used to control spatiotemporal patterns on a catalytic wafer using experimentally determined KL modes. Independently, Park and Cho [22] developed a KL Galerkin model of a nonlinear heat equation for control or parameter estimation [23][24][25][26][27][28].…”

Delay-independent sliding mode control for a class of quasi-linear parabolic distributed parameter systems with time-varying delay

“…Because parabolic PDE systems involve spatial differential operators, whose eigenspectrum can be partitioned into a finite dimensional slow one and an infinite dimensional stable fast one, one approximate method utilizes eigenfunction expansion techniques to obtain an approximate ordinary differential equations (ODE) representation of the original PDE system. It is then used to design the controller [7][8][9][10]. Although this approach has been very useful in designing control systems for parabolic PDEs, it may require keeping a large number of modes to derive an ODE model that yields the desired degree of approximation, leading to high dimensionality of the resulting controllers [11].…”

Using empirical Eigenfunctions and Galerkin method to two‐phase transport models

“…Since, a finite number of dominant modes of parabolic PDE systems practically determines the system dynamics [1], the dynamic behavior of such systems can be represented by finite-dimensional systems. For this reason, the Galerkin method has been used extensively to construct numerical solutions to parabolic PDEs [2]. The idea is to replace the given dynamics by an associated dynamics in which the PDE system is reduced to a set of ordinary differential equations that accurately describe the dominant dynamics of the PDE system.…”