Accurate solution of differential-algebraic optimization problems

Decomposition strategies for large-scale dynamic optimization problems

In recent years direct methods have been developed for solving optimal control problems with ordinary differential equation (ODE) systems. Because of special problems arising from differential-algebraic equation (DAE) systems (e.g. computation of consistent initial values, higher index) these methods can not be applied to optimal control problems with DAE systems without advanced techniques. Therefore a new direct shooting approach for the numerical solution of optimal control problems with DAE systems even of higher index is presented. The method is used to simulate virtual test-drives of automobiles. Today detailed mathematical models of automobiles are generated automatically using mechanical multi-body simulation software. The resulting equations of motion are described by DAE systems of possibly higher index. By formulation of an optimal control problem subject to these equations of motion and additional constraints for the roadway a virtual test-driver is modeled.

The Kalman filter and its variants have been developed for state estimation in semi-explicit, index-1 DAE systems in current literature. In this work, we develop a method for state estimation in non-linear fully-implicit, index-1 differential algebraic equation (DAEs) systems. In order to extend the Kalman filtering techniques for fully-implicit index-1 DAE systems, in the correction step we convert the fully-implicit DAE into a system of ordinary differential equations (ODEs). This is achieved by the index reduction of DAE using the method of successive differentiation of algebraic equations. This is a challenging problem as the fully-implicit DAE system does not contain explicit algebraic states unlike in the semi-explicit case. In this work, we propose a linear transformation of the mass matrix which enables us to find candidate algebraic states for the system. This transformation on the mass matrix is relatively simpler than constructing the transformation matrices in the Weierstrass-Kronecker canonical form. We illustrate our proposed method with two examples, a linear and a non-linear fully-implicit index-1 DAE system.

Differential algebraic equation (DAE) systems of semi-explicit type arise naturally in the modeling of chemical engineering processes. The differential equations typically arise from dynamic conservation equations, while the algebraic constraints from constitutive equations, rate expressions, equilibrium relations, stoichiometric constraints, etc. Of particular interest are DAE systems of high index , i.e., those for which the algebraic constraints are singular and cannot be eliminated through appropriate substitutions. In this paper we provide an overview of generic classes of fast-rate chemical process models, which in the limit of infinitely fast rates, generate equilibrium-based models that are high-index DAE systems. These slow approximations of multi-time-scale systems can be obtained rigorously via singular perturbations. Two classes of nonstandard singularly perturbed systems leading to high-index DAEs are identified and analyzed. The first class arises in processes with fast rates of reaction or transport. We focus in particular on chemical reaction systems which often exhibit dynamics in multiple time-scales due to reaction rate constants that vary over widely different orders of magnitude. For such systems, we describe the sequential application of singular perturbations arguments for deriving nonlinear DAE models of the dynamics in the different time-scales. The second class arises in the modeling of tightly integrated process networks , i.e., those with large rates of recovery and recycle of material or energy. For such systems we describe a similar model reduction method for deriving DAE models of the slow network dynamics and discuss control-relevant considerations.

A class of parametric semi-explicit differential algebraic equation (DAE) systems up to index 2 is considered. It is well known that initial value problems with DAE systems do not have a solution for every initial value. The initial value has to be consistent. Therefore, a method for the calculation of consistent initial values for this class of systems is introduced. In addition, various applications need information about the dependency of the solution of an initial value problem with respect to given parameters. This question leads to a linear matrix DAE system, the sensitivity DAE system, for which consistent initial values have to be provided as well. An appropriate consistent initialization method based on the solution differentiability of parametric nonlinear optimization problems in combination with Newton's method is developed. An illustrative example shows the capability of the method.

Extended space method for parameter identifiability of DAE systems

This work presents a methodological framework, based on an indirect approach, for the automatic generation and numerical solution of Optimal Control Problems (OCP) for mechatronic systems, described by a system of Differential Algebraic Equations (DAEs). The equations of the necessary condition for optimality were derived exploiting the DAEs structure, according to the Calculus of Variation Theory. A collection of symbolic procedures was developed within general-purpose Computer Algebra Software. Those procedures are general and make it possible to generate both OCP equations and their jacobians, once any DAE mathematical model, objective function, boundary conditions and constraints are given. Particular attention has been given to the correct definition of the boundary conditions especially for models described with set of dependent coordinates. The non-linear symbolic equations, their jacobians with the sparsity patterns, generated by the procedures above mentioned, are translated into a C++ source code. A numerical code, based on a Newton Affine Invariant scheme, was also developed to solve the Boundary Value Problems (BVPs) generated by such procedures. The software and methodological framework here presented were successfully applied to the solution of the minimum-lap time problem of a racing motorcycle.

In this paper we discuss a couple of situations, where algebraic equations are to be attached to a system of one‐dimensional partial differential equations. Besides of models leading directly to algebraic equations because of the underlying practical background, for example in case of stationary equations, there are many others where the specific mathematical structure requires a certain reformulation leading to time‐independent equations. To be able to apply our approach to a large class of real‐life problems, we have to take into account flux formulations, constraints, switching points, different integration areas with transition conditions, and coupled ordinary differential algebraic equations (DAEs), for example. The system of partial differential algebraic equations (PDAEs) is discretized by the method of lines leading to a large system of differential algebraic equations which can be solved by any available implicit integration method. Standard difference formulas are applied to discretize first and second partial derivatives, and upwind formulae are used for transport equations. Proceeding from given experimental data, i.e., observation times and measurements, the minimum least squares distance of measured data from a fitting criterion is computed, which depends on the solution of the system of PDAEs. Parameters to be identified can be part of the differential equations, initial, transition, or boundary conditions, coupled DAEs, constraints, fitting criterion, etc. Also the switching points can become optimization variables. The resulting least squares problem is solved by an adapted sequential quadratic programming (SQP) algorithm which retains typical features of a classical Gauss‐Newton method by retaining robustness and fast convergence speed of SQP methods. The mathematical structure of the identification problems is outlined in detail, and we present a number of case studies to illustrate the different model classes which can be treated by our approach. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

State Transition Tensors for Continuous-Thrust Control of Three-Body Relative Motion

A switched system approach for the direct solution of singular optimal control problems

Necessary conditions in terms of a local minimum principle are derived for optimal control problems subject to index-2 differential-algebraic equations, pure state constraints, and mixed control-state constraints. Differential-algebraic equations are composite systems of differential equations and algebraic equations, which arise frequently in practical applications. The local minimum principle is based on the necessary optimality conditions for general infinite optimization problems. The special structure of the optimal control problem under consideration is exploited and allows us to obtain more regular representations for the multipliers involved. An additional Mangasarian-Fromowitz-like constraint qualification for the optimal control problem ensures the regularity of a local minimum. An illustrative example completes the article.

Reachability analysis is a fundamental problem for safety verification and falsification of Cyber-Physical Systems (CPS) whose dynamics follow physical laws usually represented as differential equations. In the last two decades, numerous reachability analysis methods and tools have been proposed for a common class of dynamics in CPS known as ordinary differential equations (ODE). However, there is lack of methods dealing with differential algebraic equations (DAE), which is a more general class of dynamics that is widely used to describe a variety of problems from engineering and science, such as multibody mechanics, electrical circuit design, incompressible fluids, molecular dynamics, and chemical process control. Reachability analysis for DAE systems is more complex than ODE systems, especially for high-index DAEs because they contain both a differential part (i.e., ODE) and algebraic constraints (AC). In this paper, we propose a scalable reachability analysis for a class of high-index large linear DAEs. In our approach, a high-index linear DAE is first decoupled into one ODE and one or several AC subsystems based on the well-known Marz decoupling method utilizing admissible projectors. Then, the discrete reachable set of the DAE, represented as a list of star-sets, is computed using simulation. Unlike ODE reachability analysis where the initial condition is freely defined by a user, in DAE cases, the consistency of the initial condition is an essential requirement to guarantee a feasible solution. Therefore, a thorough check for the consistency is invoked before computing the discrete reachable set. Our approach successfully verifies (or falsifies) a wide range of practical, high-index linear DAE systems in which the number of state variables varies from several to thousands.

Practical Methods for Optimal Control using Nonlinear Programming

A method for the numerical solution of state-constrained optimal control problems subject to higher-index differential-algebraic equation (DAE) systems is introduced. For a broad and important class of DAE systems (semiexplicit systems with algebraic variables of different index), a direct multiple shooting method is developed. The multiple shooting method is based on the discretization of the optimal control problem and its transformation into a finite-dimensional nonlinear programming problem (NLP). Special attention is turned to the mandatory calculation of consistent initial values at the multiple shooting nodes within the iterative solution process of (NLP). Two different methods are proposed. The projection method guarantees consistency within each iteration, whereas the relaxation method achieves consistency only at an optimal solution. An illustrative example completes this article.

A neural network based optimal control synthesis is presented for solving optimal control problems with control and state constraints. The optimal control problem is transcribed into a nonlinear programming problem which is implemented with adaptive critic neural network. The proposed simulation method is illustrated by the optimal control problem of nitrogen transformation cycle model. Results show that adaptive critic based systematic approach holds promise for obtaining the optimal control with control and state constraints. INTRODUCTION Optimal control of nonlinear systems is one of the most active subjects in control theory. There is rarely an analytical solution although several numerical computation approaches have been proposed (for example, see (Polak, 1997), (Kirk, 1998)) for solving a optimal control problem. Most of the literature that deals with numerical methods for the solution of general optimal control problems focuses on the algorithms for solving discretized problems. The basic idea of these methods is to apply nonlinear programming techniques to the resulting finite dimensional optimization problem (Buskens at al., 2000). When Euler integration methods are used, the recursive structure of the resulting discrete time dynamic can be exploited in computing first-order necessary condition. In the recent years, the multi-layer feedforward neural networks have been used for obtaining numerical solutions to the optimal control problem. (Padhi at al., 2001), (Padhi et al., 2006). We have taken hyperbolic tangent sigmoid transfer function for the hidden layer and a linear transfer function for the output layer. The paper extends adaptive critic neural network architecture proposed by (Padhi at al., 2001) to the optimal control problems with control and state constraints. The paper is organized as follows. In Section 2, the optimal control problems with control and state constraints are introduced. We summarize necessary optimality conditions and give a short overview of basic result including the iterative numerical methods. Section 3 discusses discretization methods for the given optimal control problem. It also discusses a form of the resulting nonlinear programming problems. Section 4 presents a short description of adaptive critic neural network synthesis for optimal problem with state and control constraints. Section 5 consists of a nitrogen transformation model. In section 6, we apply the discussed methods to the nitrogen transformation cycle. The goal is to compare short-term and long-term strategies of assimilation of nitrogen compounds. Conclusions are presented in Section 7. OPTIMAL CONTROL PROBLEM We consider a nonlinear control problem subject to control and state constraints. Let x(t) ∈ R denote the state of a system and u(t) ∈ R the control in a given time interval [t0, tf ]. Optimal control problem is to minimize F (x, u) = g(x(tf )) + ∫ tf t0 f0(x(t), u(t))dt (1)