Approximate solution of nonlinear optimal control problems with scale delay function via a composite pseudospectral approach

Existing methods for nonlinear robust control often use scenario‐based approaches to formulate the control problem as large nonlinear optimization problems. The optimization problems are challenging to solve due to their size, especially if the control problems include time‐varying uncertainty. This paper draws from local reduction methods used in semi‐infinite optimization to solve robust optimal control problems with parametric and time‐varying uncertainty. By iteratively adding interim worst‐case scenarios to the problem, methods based on local reduction provide a way to manage the total number of scenarios. We show that the local reduction method for optimal control problems consists of solving a series of simplified optimal control problems to find worst‐case constraint violations. In particular, we present examples where local reduction methods find worst‐case scenarios that are not on the boundary of the uncertainty set. We also provide bounds on the error if local solvers are used. The proposed approach is illustrated with two case studies with parametric and additive time‐varying uncertainty. In the first case study, the number of scenarios obtained from local reduction is 101, smaller than in the case when all extreme scenarios are considered. In the second case study, the number of scenarios obtained from the local reduction is two compared to 512 extreme scenarios. Our approach was able to satisfy the constraints both for parametric uncertainty and time‐varying disturbances, whereas approaches from literature either violated the constraints or became computationally expensive.

Three algorithms for efficient solution of optimal control problems for high-dimensional systems are presented. Each bypasses the intermediate (and, unnecessary) step of open-loop model reduction. Each also bypasses the solution of the full Riccati equation corresponding to the LQR problem, which is numerically intractable for large n. Motivation for this effort comes from the field of model-based flow control, where open-loop model reduction often fails to capture the dynamics of interest (governed by the Navier–Stokes equation). Our minimum control energy method is a simplified expression for the well-known minimum-energy stabilizing control feedback that depends only on the left eigenvectors corresponding to the unstable eigenvalues of the system matrix A. Our Adjoint of the Direct-Adjoint method is based on the repeated iterative computation of the adjoint of a forward problem, itself defined to be the direct-adjoint vector pair associated with the LQR problem. Our oppositely-shifted subspace iteration (OSSI, the main new result of the present paper) method is based on our new subspace iteration method for computing the Schur vectors corresponding, notably, to the \(m\ll n\) central eigenvalues (near the imaginary axis) of the Hamiltonian matrix related to the Riccati equation of interest. Prototype OSSI implementations are tested on a low-order control problem to illustrate its behavior.

Mathematical formulation and numerical solutions of an optimal Dirichlet boundary control problem for the Boussinesq equations are considered. The solution of the optimal control problem is obtained by adjusting of the temperature on the boundary. We analyze finite element approximations. A gradient method for the solution of the discrete optimal control problem is presented and analyzed. Finally, the results of some computational experiments are presented.

We derive local error estimates for the discretization of optimal control problems governed by linear advection-diffusion partial differential equations (PDEs) using the streamline upwind/Petrov Galerkin (SUPG) stabilized finite element method. We show that if the SUPG method is used to solve optimization problems governed by an advection-dominated PDE, the convergence properties of the SUPG method is substantially different from the convergence properties of the SUPG method applied for the solution of an advection-dominated PDE. The reason is that the solution of the optimal control problem involves another advection-dominated PDE, the so-called adjoint equation, whose advection field is just the negative of the advection of the governing PDEs. For the solution of the optimal control problem, a coupled system involving both the original governing PDE as well as the adjoint PDE must be solved. We show that in the presence of a boundary layer, the local error between the solution of the SUPG discretized optimal control problem and the solution of the infinite dimensional problem is only of first order even if the error is computed locally in a region away from the boundary layer. In the presence of interior layers, we prove optimal convergence rates for the local error in a region away from the layer between the solution of the SUPG discretized optimal control problems and the solution of the infinite dimensional problem. Numerical examples are presented to illustrate some of the theoretical results.

This article presents a unified approach to simultaneous solution of optimization and optimal control problems in batch distillation, operating under different modes of operation: variable, constant, or optimal reflux. The simplified, computationally efficient short‐cut method and a novel algorithm to solve the optimal control problems in batch distillation is the basis of this unified approach. The short‐cut method identifies the feasible region of operation essential for optimization and optimal control problems, and provides analytical partial derivatives of the model parameters crucial to the solution.The new algorithm for the solution of optimal combination of the maximum principle and NLP optimization techniques. It circumvents the problems associated with the maximum principle approach (iterative solution of a two‐point boundary value problem, undounded control variables, and inability to handle the simultaneous optimization and optimal control problem), and the coupled ODE discretization‐NLP optimization scheme for nonlinear models (higher system nonlinearities, multiplicity of solutions, sensitivity of convergence to initial guesses). This algorithm reduces the dimensionality of the problem, and the nature of the algorithm allows a common platform to optimal solutions of different operating conditions. This article also shows that different categories of the optimal control problems in batch distillation essentially involve the solution of the maximum distillate problem.

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 nonlinear programming problem which is implemented with adaptive critic neural network. The proposed simulation methods is illustrated by the optimal control problem of feeding adaptation of filter feeders of Daphnia. Results show that adaptive critic based systematic approach holds promise for obtaining the optimal control with control and state constraints.

1 A Survey on Computational Optimal Control.- Issues in the Direct Transcription of Optimal Control Problems to Sparse Nonlinear Programs.- Optimization in Control of Robots.- Large-scale SQP Methods and their Application in Trajectory Optimization.- Solving Optimal Control and Pursuit-Evasion Game Problems of High Complexity.- 2 Theoretical Aspects of Optimal Control and Nonlinear Programming.- Continuation Methods In Boundary Value Problems.- Second Order Optimality Conditions for Singular Extremals.- Synthesis of Adaptive Optimal Controls for Linear Dynamic Systems.- Control Applications of Reduced SQP Methods.- Time Optimal Control of Mechanical Systems.- 3 Algorithms for Optimal Control Calculations.- Second Order Algorithm for Time Optimal Control of a Linear System.- An SQP-type Solution Method for Constrained Discrete-Time Optimal Control Problems.- Numerical Methods for Solving Differential Games, Prospective Applications to Technical Problems.- Construction of the Optimal Feedback Controller for Constrained Optimal Control Problems with Unknown Disturbances.- Repetitive Optimization for Predictive Control of Dynamic Systems under Uncertainty.- Optimal Control of Multistage Systems Described by High-Index Differential-Algebraic Equations.- A New Class of a High Order Interior Point Method for the Solution of Convex Semiinfinite Optimization Problems.- A Structured Interior Point SQP Method for Nonlinear Optimal Control Problems.- 4 Software for Optimal Control Calculations.- Automated Approach for Optimizing Dynamic Systems.- ANDECS: A Computation Environment for Control Applications of Optimization.- Application of Automatic Differentiation to Optimal Control Problems.- OCCAL: A mixed symbolic-numeric Optimal Control CALculator.- 5 Applications of Optimal Control.- A Robotic Satellite with Simplified Design.- Nonlinear Control under Constraints of a Biological System.- An Object-Oriented Approach to Optimally Describe and Specify a SCADA System Applied to a Power Network.- Near-Optimal Flight Trajectories Generated by Neural Networks.- Performance of a Feedback Method with Respect to Changes in the Air-Density during the Ascent of a Two-Stage-To-Orbit Vehicle.- Linear Optimal Control for Reentry Flight.- Steady-State Modelling of Turbine Engine with Controllers.- Shortest Paths for Satellite Mounted Robot Manipulators.- Optimal Control of the Industrial Robot Manutec r3.

Receding horizon methods have become very popular in the last decades for the approximate solution of optimal control problems, model predictive control (MPC) being a popular choice. MPC is to a large extent a model-based feedforward technique and as such its performance is rather sensitive to the model quality. This has prompted much interest in the search of methods to make MPC more robust against deviations such as uncertain parameters. This article presents a way to incorporate additional sensitivity terms into the optimization problem in order to reduce the cost function's sensitivity to the model parameters. For tackling the problem Continuation/GMRES (C/GMRES), a method to solve receding horizon nonlinear optimal control problems in an efficient way, was chosen, however, the general formulation is not restricted to this particular method. The potential performance of the approach is shown by means of simulation examples.

An enhanced scheme of transcribing the system dynamics for the numerical solution of optimal control problems is proposed. This new scheme is based on the standard method of direct collocation that converts an optimal control problem into a nonlinear programming problem via simultaneous state and control discretization. When compared with the standard method, the enhanced scheme has the advantage of higher solution accuracy with minimal additional computational effort. It is particularly suited for systems with states that are related to each other in a special form. For such systems, the ensuing nonlinear programming problem has the same number of constraints as those using the standard method. Numerical results on several optimal control problems using the enhanced scheme are presented, together with comparisons with the results obtained from the standard scheme.

The aim of this paper is to study an elliptic bilateral obstacle system (EBOS, for short) involving a nonlinear and nonhomogeneous partial differential operator and a multivalued term which is described by Clarke’s generalized gradient. First, we obtain the weak formulation of (EBOS) which is a variational-hemivariational inequality, and prove the unique solvability of the bilateral obstacle problem. Second, we consider a nonlinear optimal control problem governed by (EBOS) in which the control variable is the bilateral obstacle, and establish the existence of an optimal solution to the obstacle control problem under mild conditions. Then, employing the regularization technique and penalty approach, we introduce a family of approximating problems corresponding to the optimal control problem under consideration. Finally, a convergence result that any sequence of solutions for the approximating problems converges to an optimal solution of the original optimal control problem is delivered.

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 second-order algorithm is presented for the solution of continuous-time nonlinear optimal control problems. The algorithm is an adaptation of the trust region modifications of Newton's method and solves at each iteration a linear-quadratic control problem with an additional constraint. Under some assumptions, the proposed algorithm is shown to possess a global convergence property. A numerical example is presented to illustrate the method.

This paper describes a novel offline approach to estimate the parameters of a nonlinear system using the optimal control theory. In the conventional methods, only the output is considered in the identification process and the problem is stated as a nonlinear optimization problem. Nevertheless, the proposed method here is based on the least square error of both output and input. To do this, the problem is first stated as a tracking problem with some unknown parameters. The necessary and sufficiency conditions of optimality conditions are derived using the fundamental theorem of variational calculus. The problem is deployed as an effort optimal control problem which finally leads to a two point boundary value problem. To solve the resultant equations, a secondary optimal control problem is deployed to be solved using the homotopy algorithm. So the parameters are gradually converged to the real values. The proposed algorithm is analyzed for a one-link manipulator with the full measurements. Thereafter, the proposed method is applied on a real one-link manipulator and nonlinear model of permanent magnet dc motor. Application of the method for the identification of a few test beds at the AUT proves the validity and efficiency of the proposed approach.

Solution of nonlinear optimal control problems on analog parallel networks is proposed. Recurrent neural networks whose dynamic equations have a Lyapunov function are developed. Such circuits relax to an equilibrium which is the minimum of the Lyapunov function. Nonlinear optimal control problems are formulated in terms of a Lyapunov function and thus are solved using the recurrent networks. Convergence for linear and nonlinear classes of problems is considered. The method is demonstrated by developing and simulating a network to solve a nonlinear vibration problem. Simulation results demonstrate solution times are accurate and extremely fast. Solution times are shown to be independent of the size of the problem.

Due to the absence of an internal combustion engine and the corresponding waste heat, battery electric vehicles have a significantly reduced range in cold environments. Moreover, also at high ambient temperatures, the energy consumption of the air-conditioning system has a negative effect on the range. In this paper, a thermal management strategy for the passenger compartment of a battery electric vehicle is developed with the aim to reduce the power consumption of the heating, ventilation, and air-conditioning (HVAC) system. Simultaneously, the thermal comfort of the passengers has to be ensured. To address both objectives, a predictive, optimization-based approach for the thermal management system is developed. Models for the vehicle cabin and the HVAC system are derived and identified using measurement data. These models are used to formulate an optimal control problem, where also system limitations are considered explicitly. To avoid the computationally expensive solution of a nonlinear optimal control problem, it is approximated by a linear-quadratic model predictive control problem. This can be solved very efficiently and is suitable for real-time implementation on an automotive control unit. The proposed linear-quadratic strategy is compared with a baseline strategy and the nonlinear strategy to show the effectiveness of the proposed approach.