Reduced order output feedback control design for PDE systems using proper orthogonal decomposition and nonlinear semidefinite programming

In this paper, an interior point trust region algorithm for the solution of a class of nonlinear semidefinite programming (SDP) problems is described and analyzed. Such nonlinear and nonconvex programs arise, e.g., in the design of optimal static or reduced order output feedback control laws and have the structure of abstract optimal control problems in a finite dimensional Hilbert space. The algorithm treats the abstract states and controls as independent variables. In particular, an algorithm for minimizing a nonlinear matrix objective functional subject to a nonlinear SDP-condition, a positive definiteness condition, and a nonlinear matrix equation is considered. The algorithm is designed to take advantage of the structure of the problem. It is an extension of an interior point trust region method to nonlinear and nonconvex SDPs, with a special structure which applies sequential quadratic programming techniques to a sequence of barrier problems and uses trust regions to ensure robustness of the iteration. Some convergence results are given, and, finally, several numerical examples demonstrate the applicability of the considered algorithm.

The method of Lyapunov functions is used to investigate the stability of systems with distributed and lumped parameters, described by linear partial and ordinary differential equations. The original high-order partial differential equations are represented by a first-order system of partial differential evolution equations and constraint equations; to do that, we introduce additional variables. Passing to the first-order system of partial differential equations and representing ordinary differential equations in the normal Cauchy form, we obtain a possibility to construct the Lyapunov function as a sum of integral and classical quadratic forms and develop general methods of the investigation of the stability of a broad class of systems with distributed and lumped parameters. For example, we consider the stability of the work of the wind-driven lift pump and take into account the elasticity of the shaft transmitting the torque from the wind engine to the pump.

Beyond Lack of Compactness and Lack of Stability of a Coupled Parabolic-Hyperbolic Fluid-Structure System.- A Continuous Adjoint Approach to Shape Optimization for Navier Stokes Flow.- Recent Advances in the Analysis of State-constrained Elliptic Optimal Control Problems.- Fast and Strongly Localized Observation for a perturbed Plate Equation.- Representations, Composition, and Decomposition of C 1,1-hypersurfaces.- On Some Nonlinear Optimal Control Problems with Vector-valued Affine Control Constraints.- Weak Solutions to a Model for Crystal Growth from the Melt in Changing Magnetic Fields.- Lavrentiev Prox-regularization Methods for Optimal Control Problems with Pointwise State Constraints.- Nonlinear Feedback Solutions for a Class of Quantum Control Problems.- Optimal Feedback Synthesis for Bolza Control Problem Arising in Linearized Fluid Structure Interaction.- Single-step One-shot Aerodynamic Shape Optimization.- Shape Differentiability of Drag Functional for Compressible Navier-Stokes Equations.- Null-controllability for a Coupled Heat-Finite-dimensional Beam System.- Feedback Modal Control of Partial Differential Equations.- Optimization Problems for Thin Elastic Structures.- A New Non-linear Semidefinite Programming Algorithm with an Application to Multidisciplinary Free Material Optimization.- How to Check Numerically the Sufficient Optimality Conditions for Infinite-dimensional Optimization Problems.- Hidden Boundary Shape Derivative for the Solution to Maxwell Equations and Non Cylindrical Wave Equations.

A design method of finite dimensional nonlinear disturbance observer based control (NLDOBC) is proposed for a class of nonlinear dissipative partial differential equation (PDE) systems, where the disturbance is modeled by an exosystem of nonlinear ordinary differential equations (ODEs). Motivated by the fact that the dominant structure of the dissipative PDE is usually characterized by a finite number of degrees of freedom, the modal decomposition technique is initially applied to the PDE system to give a slow subsystem of low dimensional nonlinear ODEs. By using the nonlinear slow subsystem and exosystem, a nonlinear disturbance observer (NLDO) is constructed to estimate the disturbance. Then, on the basis of NLDO, an NLDOBC method is proposed in terms of linear matrix inequality (LMI) to ensure the exponential stability of the closed-loop PDE system in the presence of the modeled disturbance. Finally, the simulation results on control of one dimensional Fisher diffusion-reaction system show the effectiveness of the proposed method.

This article studies the problem of prescribed‐time constrained tracking control for a class of hyperbolic partial differential equation (PDE) systems with actuator dynamics, which are described by a set of nonlinear ordinary differential equations (ODEs). Since the control input only appears in the ODE subsystem rather than directly on the boundary of PDE subsystem, the control task becomes quite challenging. The most important is that for the control of such ODE controlled PDE systems we mainly make the following two contributions: (1) the controlled output of the PDE system tracks the reference signal within the prescribed time; (2) the controlled output and all the actuator states are constrained. It is the first time that such a prescribed‐time constrained tracking control problem is addressed for the PDE‐ODE coupled system considered in this article. Through rigorous theoretical proof, it is demonstrated that all the system states and control signals are bounded and sufficiently continuous by configuring appropriate design parameters. Finally, the performance is investigated via numerical simulation.

Summary There is no standard framework for solving optimization problems for systems described by agent-based models (ABMs). We present a method for constructing individual-level controls that steer the population-level dynamics of an ABM towards a desired state. Our method uses a system of partial differential equations (PDEs) with control functions to approximate the dynamics of the ABM with control. An optimal control problem is formulated in terms of the PDE model to mimic the optimization goal of the ABM. Mathematical theory is used to derive optimal controls for the PDE model, which are numerically approximated and transformed for use in the ABM. We use the Sugarscape ABM, a prototype ABM that includes agent and environmental heterogeneity and accumulation of agent resources over time. We present a PDE model that approximates well the spatial, temporal, and resource dynamics of the Sugarscape ABM. In both models, control represents taxation of agent wealth with the goal to maximize total taxes collected while minimizing the impact of taxation on the population over a finite time. Solutions to the optimal control problem yield taxation rates specific to an agent's location and current wealth. The use of optimal controls (generated by the PDE model) within the ABM performed better than other controls we evaluated, even though some error was introduced between the ABM and PDE models. Our results demonstrate the feasibility of using a PDE to approximate an ABM for control purposes and illustrate challenges that can arise in applying this technique to sophisticated ABMs. Copyright © 2016 John Wiley & Sons, Ltd.

Classical theory for systems of the first order partial differential equations for a scalar function can be rephrased as the submanifold theory of contact manifolds (geometric first order jet spaces). In the same spirit, we will develop the geometric theory of systems of partial differential equations of second order for a scalar function as the Contact Geometry of Second Order, following E. Cartan.We will formulate the submanifold theory of second order jet spaces as the geometry of PD manifolds (R;D 1 ,D 2 ) of second order. Moreover we will establish the First Reduction Theorem for (R;D 1,D 2) admitting non-trivial Cauchy characteristic systems. By utilizing Parabolic Geometry, we will give, directly or combined with reduction theorems, several classes of systems of partial differential equations of second order, for which the model equation of each class admits the Lie algebra of infinitesimal contact transformations, which is finite dimensional and simple.

In this paper, we show direct connections between the conservation law (CL)-based method and the differential invariant (DI)-based method for obtaining nonlocally related systems and nonlocal symmetries for a given partial differential equation (PDE) system. For a PDE system with two independent variables, we show that the CL method is a special case for the DI method. For a PDE system with at least three independent variables, we show that the CL method, for a curl-type CL, is a special case for the DI method. We also consider the situation for a self-adjoint, i.e. variational, linear PDE system. Here, a solution of the linear PDE system yields a nonlocally related system for both approaches. In particular, the resulting nonlocally related systems need not be invertibly equivalent. Through an example, we show that three distinct nonlocally related systems can be obtained from an admitted point symmetry.

This article explores the observer-based feedback control problem for a nonlinear hyperbolic partial differential equations (PDEs) system. Initially, the polynomial fuzzy hyperbolic PDEs (PFHPDEs) model is established through the utilization of the fuzzy identification approach, derived from the nonlinear hyperbolic PDEs model. Various types of state estimation and controller design problems for the polynomial fuzzy PDEs system are discussed concerning the state estimation problem. To investigate the relaxed stability problem, Euler's homogeneous theorem, Lyapunov-Krasovskii functional with polynomial matrices (LKFPM), and the sum-of-squares (SOSs) approach are adopted. The exponential stabilization condition is formulated in terms of the spatial-derivative-SOSs (SD-SOSs). Additionally, a segmental algorithm is developed to find the feasible solution for the SD-SOS condition. Finally, a hyperbolic PDEs system and several numerical examples are provided to illustrate the validity and effectiveness of the proposed results.

We propose a method to design reduced-order output-feedback controllers for fluid flows with the use of data produced by full-order controllers. First, the full-order controller is obtained by combining an ensemble Kalman filter (EnKF) and a model predictive controller (MPC) that are designed based on the Navier–Stokes equations. The full-order controller has high computational complexity and, therefore, is not suitable for real-time implementation. Hence, we use the full-order controller in offline numerical simulations to generate data for data-driven design of the reduced-order controller with low computational complexity. We find a reduced-order subspace of a closed-loop system under the full-order control from the data. This subspace underlies the reduced-order output-feedback controller. The reduced-order state-feedback law is obtained by approximating the full-order MPC with the use of its input/output data. The reduced-order observer is designed for a reduced-order model that is derived by using the Gaussian process regression (GPR). The GPR enables us to design the reduced-order observer which can evaluate uncertainty due to state-dependent residuals of the reduced-order model. We demonstrate the proposed method for a control problem of a flow around a cylinder at the Reynolds number 100. Numerical simulations reveal that the reduced-order controller performs as almost well as the full-order controller for a set of initial states. In addition, robustness of the reduced-order controller to a temporal disturbance that is not considered in the control design is confirmed in the simulations.

For a class of nonlinear systems of partial differential equations of high order in a cylindrical domain the following boundary value problem is considered: Cauchy-type conditions are prescribed on the upper and lower bases of the cylinder, and a Robin-type condition is prescribed on the lateral part of the boundary. This boundary value problem is reduced in an equivalent way to a nonlinear functional equation in a certain subspace of the Sobolev space. Under certain assumptions about the nonlinear terms an a priori estimate is obtained for the solution of the problem in question and the existence of the solution is established, while in the case when these conditions fail, the absence of a solution is established. The question of uniqueness is also discussed for the solution. Bibliography: 18 titles.

A mathematical model for fluid transport in peritoneal dialysis is constructed. The model is based on a nonlinear system of two-dimensional partial differential equations with corresponding boundary and initial conditions. Using the classical Lie scheme, we establish that the base system of partial differential equations (under some restrictions on coefficients) is invariant under the infinite-dimensional Lie algebra, which enables us to construct families of exact solutions. Moreover, exact solutions with a more general structure are found using another (non-Lie) technique. Finally, it is shown that some of the solutions obtained describe the hydrostatic pressure and the glucose concentration in peritoneal dialysis.

The analysis of dynamic deformation processes is an important task in civil and in mechanical engineering. One main emphasis is set on the derivation of the objects transmission function, represented by systems of ordinary or partial differential equations (“white box”-models). The parametric identification of the initial and boundary conditions and relevant material parameters is one of the new challenges in the application of least-square algorithms. Embedding the investigation of the deformation process in a combination of a theoretical and an experimental system analysis suitable methods are provided with adaptive KALMAN-filtering as central identification tool. In this paper the approach of a full system analysis is shown quantifying a dynamic “white-box”-model for the calculation of thermal deformations of bar-shaped machine elements. The task was motivied from mechanical engineering searching new methods for the precise prediction and computational compensation of thermal influences in the heating and cooling phases of machine tools (i.e. robot arms, etc.). The quantification of thermal deformations under variable dynamic loads requires the modelling of the non-stationary spatial temperature distribution inside the object. Based upon FOURIERS law of heat flow the high-grade non-linear temperature gradient is represented by a system of partial differential equations within the framework of a dynamic Finite Element topology. It is shown that adaptive KALMAN-filtering is suitable to quantify relevant disturbance influences and to identify thermal parameters (i.e. thermal diffusivity) with a deviation of only 0,2%. As result an identified (and verified) parametric model for the realistic prediction respectively simu-lation of dynamic temperature processes is presented. Classifying the thermal bend as the main deformation quantity of bar-shaped machine tools, the temperature model is extended to a temperature deformation model. In lab tests with an aluminium column control measurements show that the identified model can be used to predict the columns bend with a mean deviation smaller than 10 mgon. Consequently the deformation model is a precise predictor and suitable for realistic simulations of thermal deformations under variable dynamic loads. In future our activities will be primarily focussed on applications in industrial manufacturing. 1. Parametric identification of dynamic deformation processes The creation of a realistic parametric dynamic deformation model is directly associated with the task to combine systems of differential equations (theoretical physical model) with empirical measurements. Consequently basic principles of system theory can be adapted to the geodetic deformation analysis. The parametric identification of a dynamic “white box”-model 3rd IAG / 12th FIG Symposium, Baden, May 22-24, 2006

A new family of exact solutions to the unsteady Navier–Stokes equations using canonical transformation with complex coefficients

Existence of series solutions for certain nonlinear systems of time fractional partial differential equations