Solution Of Optimal Control Problem Research Articles

Fast Nonlinear Model Predictive Control Combining Online Trajectory Optimization and Value Function Regression

ABSTRACTModel predictive control (MPC) is a well‐developed method capable of handling complex control tasks. Implementation of an MPC requires solution of a deterministic finite horizon nonlinear optimal control (OC) problem. OC problems can be solved globally, explicitly expressing the optimal policy (and value function) as a function of the present state, or, locally, using online trajectory optimization, generating a solution that is only valid for the present state. For nonlinear problems however, neither is possible analytically, and the latter, finding a local solution online, is usually preferred. The difficulty with online trajectory optimization is that the solution must be available within a single sampling period. The main parameter affecting the computational demand of MPC is the length of the prediction horizon. The goal in this work is to reduce the length of the prediction horizon of a model predictive controller (MPC) to reduce computation time whilst preserving optimality guarantees. To this end, we propose approximation of the trajectory optimization problem for MPC by learning a finite horizon value function. The approximated value function is inserted into a truncated trajectory optimization problem so that the MPC can be attained with a reduced prediction horizon, ergo reduced computational load. By sampling the value function in a goal oriented way, we show that an effective approximate value function can be found by including both the function value and the gradients of the value function. The result is an accurate approximate MPC which leverages learning methodologies to reduce computational cost while still accounting for constraints.

Inverse demand tracking in transportation networks

AbstractThis paper deals with the reconstruction of the desired demand in an optimal control problem, stated over a tree-shaped transportation network which is governed by a linear hyperbolic conservation law. As desired demands typically undergo fluctuations due to seasonality or unexpected events making short-term adjustments necessary, such an approach can exemplary be used for forecasting from past data. We suggest to model this problem as a so-called inverse optimal control problem, i.e., a hierarchical optimization problem whose inner problem is the optimal control problem and whose outer problem is the reconstruction problem. In order to guarantee the existence of solutions in the function space framework, the hyperbolic conservation law is interpreted in weak sense allowing for control functions in Lebesgue spaces. For the computational treatment of the model, we transfer the hierarchical problem into a nonsmooth single-level one by plugging the uniquely determined solution of the inner optimal control problem into the outer reconstruction problem before applying techniques from nonsmooth optimization. Some numerical experiments are presented to visualize various features of the model including different types of noise in the demand and strategies of how to observe the network in order to obtain good reconstructions of the desired demand.

A Second-Order Numerical Method for a Class of Optimal Control Problems

The numerical solution of optimal control problems through second-order methods is examined in this paper. Controlled processes are described by a system of nonlinear ordinary differential equations. There are two specific characteristics of the class of control actions used. The first one is that controls are searched for in a given class of functions, which depend on unknown parameters to be found by minimizing an objective functional. The parameter values, in general, may be different at different time intervals. The second feature of the considered problem is that the boundaries of time intervals are also optimized with fixed values of the parameters of the control actions in each of the intervals. The special cases of the problem under study are relay control problems with optimized switching moments. In this work, formulas for the gradient and the Hessian matrix of the objective functional with respect to the optimized parameters are obtained. For this, the technique of fast differentiation is used. A comparison of numerical experiment results obtained with the use of first- and second-order optimization methods is presented.

An accurate wavelets‐collocation technique for neutral delay distributed‐order fractional optimal control problems

AbstractIn this study, a new class of optimal control problems called neutral delay distributed‐order fractional optimal control problems is introduced, this problem is solved based on an efficient computational scheme. To solve the problem, we derive an exact formula for the Riemann–Liouville fractional integral operator of Genocchi wavelets based on beta functions for the first time. By taking into account this operator, collocation method, and Gauss–Legendre integration formula, the solution of fractional optimal control problems (FOCPs) under consideration is converted to a nonlinear programming one to which existing well‐developed algorithms may be applied. The mentioned scheme is applied to both FOCPs with or without delay. Error analysis associated with the proposed idea is also investigated under several mild conditions. The effectiveness of the strategy is showed by several illustrative examples, furthermore, a comparison with the previous methods highlights the preference of this scheme.

A novel successive convexification framework with symplectic pseudospectral solutions for nonlinear optimal control problems of constrained pantograph delay systems

This paper proposes a novel symplectic pseudospectral iteration framework for the nonlinear optimal control problem (OCP) of a pantograph delay system with inequality constraints. Traditional numerical algorithms for OCPs with pantograph delay systems have predominantly focused on unconstrained scenarios, neglecting constrained problems. Therefore, we propose a novel symplectic pseudospectral iteration framework for constrained problems. First, we employ the successive convexification (SCvx) method to transform the original problem into a series of linear quadratic (LQ) problems, addressing nonlinearity. Then, leveraging time-scale transformations and parametric variational principles, we derive the first-order necessary conditions (FONCs) for these convexified problems, including a Hamiltonian two-point boundary value problem (HTBVP) with stretching terms and a linear complementarity problem. To handle pantograph time delays effectively, we employ a local Legendre-Gauss-Lobatto (LGL) pseudospectral method with proportional grid discretization. Finally, employing a symplectic indirect algorithm, we develop a symplectic pseudospectral method (SPM) for solving the transformed LQ problems, converting them into a system of sparse linear algebraic equations and a linear complementarity problem. Validation of the framework is performed using diverse illustrative examples, demonstrating strong agreement between SCvx and SPM in computational efficiency. Numerical experiments confirm the sparsity of the core matrix and robustness to initial guesses.

Numerical solution of optimal control problems for linear systems of ordinary differential equations

Abstract An original numerical matrix algorithm aimed at solving the optimal control problems for linear systems of ordinary differential equations with constant coefficients is proposed. The work of the algorithm is demonstrated with the problem, which consists in generating a given small disturbance of the Poiseuille flow in an infinite duct by blowing and suction through the walls. The costs of creating the leading modes and optimal disturbances are compared, which is of independent interest.

Adaptive dynamic programming for containment control with robustness analysis to iterative error: A global Nash equilibrium solution

Global Nash equilibrium is an optimal solution for each player in a graphical game. This paper proposes an iterative adaptive dynamic programming-based algorithm to solve the global Nash equilibrium solution for optimal containment control problem with robustness analysis to the iterative error. The containment control problem is transferred into the graphical game formulation. Sufficient conditions are given to decouple the Hamilton–Jacobi equations, which guarantee the solvability of the global Nash equilibrium solution. The iterative algorithm is designed to obtain the solution without any knowledge of system dynamics. Conditions of iterative error for global stability are given with rigorous proof. Compared with existing works, the design procedures of control gain and coupling strength are separated, which avoids trivial cases in the design procedure. The robustness analysis exactly quantifies the effect of the iterative error caused by various sources in engineering practice. The theoretical results are validated by two numerical examples with marginally stable and unstable dynamics of the leader.

Multiobjective optimal control problems. Stationary Navier–Stokes equations

This paper deals with the solution of some multi-objective optimal control problems for stationary Navier–Stokes equations. More precisely, we look for Pareto and Nash equilibria associated to standard cost functionals. First, we prove the existence of equilibria and we deduce appropriate optimality systems. Then, we analyse the existence and characterization of Pareto and Nash equilibria for the Navier–Stokes equations. Here, we use the formalism of Dubovitskii and Milyoutin., see [Girsanov FV. Lectures on mathematical theory of extremum problems. Berlin: Springer-Verlag; 1972. (Notes in economics and mathematical systems; vol. 67)]. Finally, we also present a finite element approximation of the bi-objective problem, we illustrate the techniques with several numerical experiments and we compare the Pareto and Nash equilibria.

Optimal Neumann control of the wave equation with L 1-control cost: the finite-time turnpike property

The finite-time turnpike property describes a situation where the optimal state reaches a steady state after finite time. The steady state is a solution of a static optimal control problem. We study an optimal control problem where the objective functional is the sum of an L 1 -norm control cost with a weight γ > 0 and a differentiable tracking term. We consider a vibrating string with homogeneous Dirichlet conditions at one end and Neumann control action at the other end. The tracking term is defined by the squared L 2 -norm of a non-collocated Neumann-observation. We show that the problem has a unique solution and that due to the non-smoothness of the L 1 -norm for sufficiently large T the optimal state reaches the desired state after a finite time that is equal to the minimal time where exact controllability holds. We also study the effect of smoothing of the control cost on the structure of the optimal control and show that the finite-time turnpike property also holds in the smoothing limit in a strong L 2 -sense.

Multi‐objective terminal trajectory optimization based on hybrid genetic algorithm pseudospectral method

AbstractDuring terminal guidance, the attack platform is provided with a high‐resolution image of the target area through the application of synthetic aperture radar. Additionally, the stealth trajectory with low observability can significantly impact mission success. This paper considers both the performance of missile‐borne synthetic aperture radar imaging and stealth performance as influencing factors for terminal trajectory optimization, which is modelled as a constrained multi‐objective optimization problem. The application of the pseudospectral method in the solution of optimal control problems has led to the proposal of the hybrid genetic algorithm pseudospectral optimization framework. The problem is decomposed into several single‐objective optimal control problems, which can generate a specific initial population for the genetic algorithm to obtain a set of Pareto‐optimal solutions. Finally, the numerical simulations demonstrate the effectiveness of the proposed optimization approach compared with the benchmark scheme.

Neural network approach to portfolio optimization with leverage constraints: a case study on high inflation investment

Motivated by the current global high inflation scenario, we aim to discover a dynamic multi-period allocation strategy to optimally outperform a passive benchmark while adhering to a bounded leverage limit. We formulate an optimal control problem to outperform a benchmark portfolio throughout the investment horizon. To obtain strategies under the bounded leverage constraint among other realistic constraints, we propose a novel leverage-feasible neural network (LFNN) to represent the control, which converts the original constrained optimization problem into an unconstrained optimization problem that is computationally feasible with gradient descent, without dynamic programming. We establish mathematically that the LFNN approximation can yield a solution that is arbitrarily close to the solution of the original optimal control problem with bounded leverage. We further validate the performance of the LFNN empirically by deriving a closed-form solution under jump-diffusion asset price models and show that a shallow LFNN model achieves comparable results on synthetic data. In the case study, we apply the LFNN approach to a four-asset investment scenario with bootstrap-resampled asset returns from the filtered high inflation regimes. The LFNN strategy is shown to consistently outperform the passive benchmark strategy by about 200 bps (median annualized return), with a greater than 90% probability of outperforming the benchmark at the end of the investment horizon.

Nonlinear optimal control for triangular tethered multi-satellite formations

Triangular satellite constellations are created by three satellites which are linked through tethers. Such formations serve distributed observation space missions or the creation of high-density satellite communication networks over high-frequency bands. In a tethered multi-satellite system consisting of three-satellites in triangular formation the associated dynamic model exhibits strong nonlinearities. Stabilization and precise positioning of the multi-satellite constellation is a nontrivial task and the solution of the associated nonlinear control problem is an elaborated procedure. In this article a novel nonlinear optimal control method is applied to the above-noted model the tethered multi-satellite system in triangular formation. First, the state-space model of the triangular tethered multi-satellite formation undergoes approximate linearization around a temporary operating point that is recomputed at each time-step of the control method. The linearization relies on Taylor series expansion and on the associated Jacobian matrices. For the linearized state-space model of the tethered satellites a stabilizing optimal (H-infinity) feedback controller is designed. This controller stands for the solution of the nonlinear optimal control problem under model uncertainty and external perturbations. To compute the controller’s feedback gains an algebraic Riccati equation is repetitively solved at each iteration of the control algorithm. The stability properties of the control method are proven through Lyapunov analysis. The proposed nonlinear optimal control approach achieves fast and accurate tracking of setpoints under moderate variations of the control inputs and a minimum dispersion of energy when changing the position of the satellites in their triangular tethered formation.

The method of determining optimal control of the thermoelastic state of piece-homogeneous body using a stationary temperature field

This paper proposes a new highly effective method for determining the optimal control of the stress-strain state of spatially multi-connected composite bodies using a stationary temperature field. The proposed method is considered based on the example of a stationary axisymmetric thermoelastic problem for a space with a spherical inclusion and cavity. The proposed method is based on the generalized Fourier method and reduces the original problem to an equivalent problem of optimal control, in which the state of the object is determined by an infinite system of linear algebraic equations, the right-hand side of which parametrically depends on the control. At the same time, the functional of the cost of the initial problem is transformed into a quadratic functional, which depends on the state of the equivalent system and parametrically on the control. The limitation on the temperature distribution is replaced by the value of the control norm in the space of square summable sequences. In fact, this paper considers for the first time the problem of optimal control of an infinite system of linear algebraic equations and develops a method for its solution. The proposed method is based on presenting the solutions of infinite systems in a parametric form, which makes it possible to reduce equivalent problem to the problem of conditional extremum of a quadratic functional, which explicitly depends on the control. A further solution to this problem A further solution to this problem is found by the Lagrange method using the spectral decomposition of the quadratic functional matrix. found by the Lagrange method using the spectral decomposition of the quadratic functional matrix. The method developed in this paper is strictly justified. For all infinite systems, the Fredholm property of their operators is proved. As an important result necessary for substantiation, for the first time, an estimate from below of the module of the multi-parameter determinant of the resolving system of the boundary value problem of conjugation – space with a spherical inclusion – was obtained when solving it using the Fourier method. The theorem that establishes the conditions for the existence and uniqueness of the solution of equivalent problem or optimal control problem without restrictions in the space of square summable sequences is proved. The numerical algorithm is based on a reduction method for infinite systems of linear algebraic equations. Estimates of the practical accuracy of the numerical algorithm demonstrated the stability of the method and sufficiently high accuracy even with close location of the boundary surfaces. Graphs showing the optimal temperature distribution for various geometric parameters of the problem and their analysis are provided. The proposed method extends to boundary value problems with different geometries.

Optimal control for a variable-order diffusion-wave equation with a reaction term; A numerical study

In this paper, optimal control for a variable-order diffusion-wave equation with a reaction term is numerically studied, where the variable-order operator is defined in the sense of Caputo proportional constant. Necessary optimality conditions for the control problem are derived. Existence and uniqueness for the solutions of fractional optimal control problem are derived. The nonstandard weighted average finite difference method and the nonstandard leap-frog method are developed to study numerically the proposed problem. Moreover, the stability analysis of the methods is proved. Finally, in order to characterise the memory property of the proposed model, three test examples are given. It is found that the nonstandard weighted average finite difference method can be applied to study such variable-order fractional optimal control problems simply and effectively.

Bi-Basis Function Second Derivative Block Method for the Solution of Optimal Control Problems

This paper determines the state trajectory of a dynamic system over a period to optimize a given performance index. A one-step second derivative hybrid method based on the combination of Lucas and first Boubaker polynomials as basis functions is formulated for the solution of optimal control problems via the forward-backward sweep method. The technique is used to generate a set of hybrid schemes at selected grid and off-grid points with implementation in block form. The stability and convergence of the new method is discussed and the accuracy of the method is tested on some numerical examples. The results obtained correspond to a large extent with the results obtained by the exact solution. Bi-bases function, Block method, First Boubaker polynomials, Lucas polynomial, Optimal control problem

A direct transcription method for solving distributed-order fractional optimal control problems

This paper considers the numerical solution of distributed-order Fractional Optimal Control Problems (FOCPs). The common approaches for solving these problems involve approximating the problem with a multi-term FOCP and subsequently discretizing the resulting multi-term problem using either a direct or indirect method. However, this paper bypasses the need for approximation with a multi-term FOCP by introducing innovative approximation formulas for distributed-order fractional derivatives. These approximation formulas are directly derived from the conventional Grünwald–Letnikov, L1, and trapezoidal formulas. Building upon these formulas, we develop the relevant fractional-order distributed-order operational matrices. Utilizing the operational matrix, we easily discretize the distributed-order FOCP to a Non-Linear Programming (NLP), which can be solved by a suitable NLP-solver. Additionally, to enhance the efficiency of solving the resulting NLP, we determine the closed-forms of both the Jacobian of constraints and the gradient of the objective function. The presented method is characterized by its speed, simplicity, and ease of implementation. Moreover, it can be used to solve a wide range of distributed-order FOCPs, such as those involving nonlinear dynamics, free final times, free terminal conditions, and path constraints. By means of ample numerical tests, the accuracy and efficiency of the proposed method are assessed.

Pointwise Error Estimates of Numerical Solutions to Linear Quadratic Optimal Control Problems

An auxiliary optimal control problem is formulated that provides with its unique solution, a continuous representation of the global error of a numerical approximation to the solution of a linear quadratic optimal control problem. The resulting error functions are characterized as the unique solutions to an optimality system that is reformulated as a boundary value problem. With this formulation, reliable pointwise error estimates can be generated utilizing well-established techniques of defect control. A novel algorithm based on defect correction and defect control is presented that generates pointwise approximations to the global error of numerical optimal control solutions on a uniform grid. It is proven and numerically validated that this algorithm can generate pointwise estimates that approximate the true global error with a prescribed accuracy.

A Convergent Legendre Spectral Collocation Method for the Variable-Order Fractional-Functional Optimal Control Problems

In this paper, a numerical method is applied to approximate the solution of variable-order fractional-functional optimal control problems. The variable-order fractional derivative is described in the type III Caputo sense. The technique of approximating the optimal solution of the problem using Lagrange interpolating polynomials is employed by utilizing the shifted Legendre–Gauss–Lobatto collocation points. To obtain the coefficients of these interpolating polynomials, the problem is transformed into a nonlinear programming problem. The proposed method offers a significant advantage in that it does not require the approximation of singular integral. In addition, the matrix differentiation is calculated accurately and efficiently, overcoming the difficulties posed by variable-order fractional derivatives. The convergence of the proposed method is investigated, and to validate the effectiveness of our proposed method, some examples are presented. We achieved an excellent agreement between numerical and exact solutions for different variable orders, indicating our method’s good performance.

Non-Overlapping Domain Decomposition for 1D Optimal Control Problems Governed by Time-Fractional Diffusion Equations on Coupled Domains: Optimality System and Virtual Controls

We consider a non-overlapping domain decomposition method for optimal control problems of the tracking type governed by time-fractional diffusion equations in one space dimension, where the fractional time derivative is considered in the Caputo sense. We concentrate on a transmission problem defined on two adjacent intervals, where at the interface we introduce an iterative non-overlapping domain decomposition in the spirit of P.L. Lions for the corresponding first-order optimality system, such that the optimality system corresponding to the optimal control problem on the entire domain is iteratively decomposed into two systems on the respective sub-domains; this approach can be framed as first optimize, then decompose. We show that the iteration involving the states and adjoint states converges in the appropriate spaces. Moreover, we show that the decomposed systems on the sub-domain can in turn be interpreted as optimality systems of so-called virtual control problems on the sub-domains. Using this property, we are able to solve the original optimal control problem by an iterative solution of optimal control problems on the sub-domains. This approach can be framed as first decompose, then optimize. We provide a mathematical analysis of the problems as well as a numerical finite difference discretization using the L1-method with respect to the Caputo derivative, along with two examples in order to verify the method.

A new diagonalization based method for parallel-in-time solution of linear-quadratic optimal control problems

A new diagonalization technique for the parallel-in-time solution of linear-quadratic optimal control problems with time-invariant system matrices is introduced. The target problems are often derived from a semi-discretization of a Partial Differential Equation (PDE)-constrained optimization problem. The solution of large-scale time dependent optimal control problems is computationally challenging as the states, controls, and adjoints are coupled to each other throughout the whole time domain. This computational difficulty motivates the use of parallel-in-time methods. For time-periodic problems our diagonalization efficiently transforms the discretized optimality system into nt (=number of time steps) decoupled complex valued 2ny × 2ny systems, where ny is the dimension of the state space. These systems resemble optimality systems corresponding to a steady-state version of the optimal control problem and they can be solved in parallel across the time steps, but are complex valued. For optimal control problems with initial value state equations a direct solution via diagonalization is not possible, but an efficient preconditioner can be constructed from the corresponding time periodic optimal control problem. The preconditioner can be efficiently applied parallel-in-time using the diagonalization technique. The observed number of preconditioned GMRES iterations is small and insensitive to the size of the problem discretization.