Dynamic optimization of dissipative PDE systems using nonlinear order reduction

On the n-Dimensional Phase Portraits

The phase portrait for dynamic systems is a tool used to graphically determine the instantaneous behavior of its trajectories for a set of initial conditions. Classic phase portraits are limited to two dimensions and occasionally snapshots of 3D phase portraits are presented; unfortunately, a single point of view of a third or higher order system usually implies information losses. To solve that limitation, some authors used an additional degree of freedom to represent phase portraits in three dimensions, for example color graphics. Other authors perform states combinations, empirically, to represent higher dimensions, but the question remains whether it is possible to extend the two-dimensional phase portraits to higher order and their mathematical basis. In this paper, it is reported that the combinations of states to generate a set of phase portraits is enough to determine without loss of information the complete behavior of the immediate system dynamics for a set of initial conditions in an n-dimensional state space. Further, new graphical tools are provided capable to represent methodically the phase portrait for higher order systems.

Data-Driven H∞ Control for Nonlinear Distributed Parameter Systems

The data-driven H∞ control problem of nonlinear distributed parameter systems is considered in this paper. An off-policy learning method is developed to learn the H∞ control policy from real system data rather than the mathematical model. First, Karhunen-Loève decomposition is used to compute the empirical eigenfunctions, which are then employed to derive a reduced-order model (ROM) of slow subsystem based on the singular perturbation theory. The H∞ control problem is reformulated based on the ROM, which can be transformed to solve the Hamilton-Jacobi-Isaacs (HJI) equation, theoretically. To learn the solution of the HJI equation from real system data, a data-driven off-policy learning approach is proposed based on the simultaneous policy update algorithm and its convergence is proved. For implementation purpose, a neural network (NN)- based action-critic structure is developed, where a critic NN and two action NNs are employed to approximate the value function, control, and disturbance policies, respectively. Subsequently, a least-square NN weight-tuning rule is derived with the method of weighted residuals. Finally, the developed data-driven off-policy learning approach is applied to a nonlinear diffusion-reaction process, and the obtained results demonstrate its effectiveness.

Adaptive optimal control of highly dissipative nonlinear spatially distributed processes with neuro-dynamic programming

Highly dissipative nonlinear partial differential equations (PDEs) are widely employed to describe the system dynamics of industrial spatially distributed processes (SDPs). In this paper, we consider the optimal control problem of the general highly dissipative SDPs, and propose an adaptive optimal control approach based on neuro-dynamic programming (NDP). Initially, Karhunen-Loève decomposition is employed to compute empirical eigenfunctions (EEFs) of the SDP based on the method of snapshots. These EEFs together with singular perturbation technique are then used to obtain a finite-dimensional slow subsystem of ordinary differential equations that accurately describes the dominant dynamics of the PDE system. Subsequently, the optimal control problem is reformulated on the basis of the slow subsystem, which is further converted to solve a Hamilton-Jacobi-Bellman (HJB) equation. HJB equation is a nonlinear PDE that has proven to be impossible to solve analytically. Thus, an adaptive optimal control method is developed via NDP that solves the HJB equation online using neural network (NN) for approximating the value function; and an online NN weight tuning law is proposed without requiring an initial stabilizing control policy. Moreover, by involving the NN estimation error, we prove that the original closed-loop PDE system with the adaptive optimal control policy is semiglobally uniformly ultimately bounded. Finally, the developed method is tested on a nonlinear diffusion-convection-reaction process and applied to a temperature cooling fin of high-speed aerospace vehicle, and the achieved results show its effectiveness.

Studies on Feedback Control of Cardiac Alternans

A beat-to-beat variation in the electric wave propagation morphology in myocardium is referred to as cardiac alternans and it has been linked to the onset of life threatening arrhythmias and sudden cardiac death. Experimental studies have demonstrated that alternans can be annihilated by the feedback modulation of the basic pacing interval in a small piece of cardiac tissue. In this work, we study the capability of feedback control to suppress alternans both spatially and temporally in an extracted rabbit heart and in a cable of cardiac cells. This work demonstrates real-time control of cardiac alternans in an extracted rabbit heart and provides an analysis of the control methodology applied in the case of a one-dimensional (1D) cable of cardiac cells. The real-time system control is realized through feedback by proportional perturbation of the basic pacing cycle length (PCL). The measurements of the electric wave propagation are obtained by optical mapping of fluorescent dye from the surface of the heart and are fed into a custom-designed software that provides the control action signal that perturbs the basic pacing cycle length. In addition, a novel pacing protocol that avoids conduction block is applied. A numerical analysis, complementary to the experimental study is also carried out, by the ionic model of a 1D cable of cardiac cells under a self-referencing feedback protocol, which is identical to the one applied in the experimental study. Further, the amplitude of alternans linear parabolic PDE that is associated with the 1D ionic cardiac cell cable model under full state feedback control is analyzed. We provide an analysis of the amplitude of alternans parabolic PDE which admits a standard evolutionary form in a well defined functional space. Standard modal decomposition techniques are used in the analysis and the controller synthesis is carried out through pole-placement. State and output feedback controller realizations are developed and the important issue of measurement noise in the controller implementation is addressed. The analysis of stabilization of the amplitude of alternans PDE is in agreement with the experimental results and numerical results produced by the ionic 1D cable of cardiac cells model. Finally, a discussion is provided in light of these results in order to use control to suppress alternans in the human myocardium.