Haar wavelet operational matrix method for solving constrained nonlinear quadratic optimal control problem

In this paper, an iterative computational method is proposed to handle constrained nonlinear quadratic optimal control problems. with terminal state constraints, saturation state and saturation control variables constraints. The method is based on replacing the original constrained nonlinear optimal control problem by a sequence of constrained linear time-varying quadratic optimal control problems. To this end, an iterative technique is used to replace the original constrained nonlinear dynamic system by a sequence of constrained linear time-varying systems. Then each of constrained linear time-varying quadratic optimal control problems is approximated by a quadratic programming problem by parameterizing each of the state variable by a finite length Legendre polynomials with unknown parameters. To show the effectiveness of the proposed method, simulation results of a constrained nonlinear optimal control problem are presented.

In this paper, we study an a priori error analysis for the quadratic optimal control problems governed by nonlinear elliptic partial differential equations using mixed finite element methods. The state and the costate are approximated by the lowest order Raviart-Thomas mixed finite element spaces, and the control is approximated by piecewise constant functions. A priori error estimates for the mixed finite element approximation of nonlinear optimal control problems are obtained. Some numerical examples are presented to confirm our theoretical results.

A EROSPACE engineering applications greatly stimulated the development of optimal control theory during the 1950s and 1960s, where the objective was to drive the system states in such a way that some defined cost was minimized. This turned out to have very useful applications in the design of regulators (where some steady state is to be maintained) and in tracking control strategies (where some predetermined state trajectory is to be followed). Among such applications was the problem of optimal flight trajectories for aircraft and space vehicles. Linear optimal control theory in particular has been very well documented and widely applied, where the plant that is controlled is assumed linear and the feedback controller is constrained to be linear with respect to its input. However, the availability of powerful low-cost microprocessors has spurred great advantages in the theory and applications of nonlinear control. The competitive era of rapid technological change, particularly in aerospace exploration, now demands stringent accuracy and cost requirements in nonlinear control systems. This has motivated the rapid development of nonlinear control theory for application to challenging, complex, dynamical real-world problems, particularly those that bear major practical significance in aerospace, marine, and defense industries. Infinite-time horizon nonlinear optimal control (ITHNOC) presents a viable option for synthesizing stabilizing controllers for nonlinear systems by making a state-input tradeoff, where the objective is to minimize the cost given by a performance index. The original theory of nonlinear optimal control dates from the 1960s. Various theoretical and practical aspects of the problem have been addressed in the literature over the decades since. In particular, the continuous-time nonlinear deterministic optimal control problem associated with autonomous (time-invariant) nonlinear regulator systems that are affine (linear) in the controls has been studied by many authors. The long-established theory of optimal control offers quite mature and well-documented techniques for solving this control-affine nonlinear optimization problem, based on dynamic programming or calculus of variations, but their application is generally a very tedious task. Bellman’s dynamic programming approach reduces to solving a nonlinear first-order partial differential equation (PDE), expressed by the Hamilton–Jacobi–Bellman (HJB) equation. The solution to the HJB equation gives the optimal performance/cost value (or storage) function and determines an optimal control in feedback form under some smoothness assumptions. Alternatively, in the classical calculus of variations, optimal control problems can be characterized locally in terms of the Hamiltonian dynamics arising from Pontryagin’s minimum principle. These are the characteristic equations of the HJB PDE, which result in a nonlinear, constrained two-point boundary value problem (TPBVP) that, in general, can only be solved by successive approximation of the optimal control input using iterative numerical techniques for each set of initial conditions. Numerically, even though the nonlinear TPBVP is somewhat easier to solve than the HJBPDE, control signals can only be determined offline and are thus best suited for feedforward control of plants for which the state trajectories are known a priori. Therefore, contrary to the dynamic programming approach, the resultant control law is not generally in feedback form. Open-loop control, however, is sensitive to random disturbances and requires that the initial state be on the optimal trajectory. In contrast, nonlinear optimal feedback has inherent robustness properties (inherent in the sense that it is obtained by ignoring uncertainty and disturbances). The potential difficulty with the HJB approach is that no efficient algorithm is available to solve the PDE when it is nonlinear and the problem dimension is high, making it impossible to derive exact expressions for optimal controls for most nontrivial problems of interest. The optimal can only be computed in special cases, such as linear dynamics and quadratic cost, or very low-dimensional systems. In particular, if the plant is linear time invariant (LTI) and the (infinite-time) performance index is quadratic, then the corresponding HJB equation for this infamous linear-quadratic regulator (LQR) problem reduces to an algebraic Riccati equation (ARE). Contrary to the well-developed and widely applied theory and computational tools for theRiccati equation (for example, see [1]), theHJB equation is difficult, if not impossible, to solve for most practical applications. The exact solution for the optimal control policies is very complex

A new fractional orthogonal basis and its application in nonlinear delay fractional optimal control problems

Optimal power split in a hybrid electric vehicle using direct transcription of an optimal control problem

Here, a two-phase algorithm based on integrating particle swarm optimization (PSO) with modified hybrid genetic algorithm (MHGA) is proposed for solving the associated nonlinear programming problem of a nonlinear optimal control problem. In the first phase, PSO starts with a completely random initial swarm of particles, where each of them contains two random matrices in time nodes. After phase 1, to achieve more accurate solutions, the number of time nodes is increased. The values of the associated new control inputs are estimated by linear or spline interpolations using the curves computed in the phase 1. In addition, to maintain the diversity in the population, some additional individuals are added randomly. Next, in the second phase, MHGA, starts by the new population constructed by the above procedure and tries to improve the obtained solutions at the end of phase 1. MHGA combines a GA with a successive quadratic programming, SQP, as a local search. Finally, we implement the proposed algorithm on some well-known nonlinear optimal control problems. The numerical results show that the proposed algorithm can find almost better solution than other proposed algorithms. Keywords: Optimal control problem, Hybrid genetic algorithm, ?particle swarm optimization, Spline interpolation.

Existing methods for nonlinear robust control often use scenario-based approaches to formulate the control problem as nonlinear optimization problems. Increasing the number of scenarios improves robustness, while increasing the size of the optimization problems. Mitigating the size of the problem by reducing the number of scenarios requires knowledge about how the uncertainty affects the system. This paper draws from local reduction methods used in semi-infinite optimization to solve robust optimal control problems with parametric uncertainty. We show that nonlinear robust optimal control problems are equivalent to semi-infinite optimization problems and can be solved by local reduction. By iteratively adding interim globally worst-case scenarios to the problem, methods based on local reduction provide a way to manage the total number of scenarios. In particular, we show that local reduction methods find worst case scenarios that are not on the boundary of the uncertainty set. The proposed approach is illustrated with a case study with both parametric and additive time-varying uncertainty. The number of scenarios obtained from local reduction is 101, smaller than in the case when all 214+3x192 boundary scenarios are considered. A validation with randomly drawn scenarios shows that our proposed approach reduces the number of scenarios and ensures robustness even if local solvers are used.

An algorithm, based on quasi-linearization and state variable parametrization by using Chebyshev polynomials, is proposed to solve the nonlinear optimal control problem subject to terminal state constraints and control saturation constraints. By the virtue of the second method of quasi-linearization and state parametrization, the difficult nonlinear optimal control problem is converted into a sequence of standard quadratic programming problems. An example is given to show the effectiveness of the proposed algorithm.

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.

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.

Nonlinear optimal control problems are challenging to solve due to the prevalence of local minima that prevent convergence and/or optimality. This paper describes nearest-neighbors optimal control (NNOC), a data-driven framework for nonlinear optimal control using indirect methods. It determines initial guesses for new problems with the help of precomputed solutions to similar problems, retrieved using k-nearest neighbors. A sensitivity analysis technique is introduced to linearly approximate the variation of solutions between new and precomputed problems based on their variation of parameters. Experiments show that NNOC can obtain the global optimal solution orders of magnitude faster than standard random restart methods, and sensitivity analysis can further reduce the solving time almost by half. Examples are shown on two optimal control problems in vehicle control.

In this paper, the nonlinear optimal control problem is investigated for spacecraft formation flying with collision avoidance. Based on a nonlinear model of formation flying, two optimal tracking control laws are proposed to ensure the formation pattern converges to the predetermined configuration. The first optimal control law which combines Lyapunov optimizing control with a trajectory-following optimization technique is developed to solve the finite-time nonlinear optimal control problem. For the second controller, an extended [Formula: see text]-[Formula: see text] method is applied to design a closed-form feedback control scheme for nonlinear control problem of spacecraft formation flying with non-standard cost functions. By taking into account the flying safety, a repulsive control scheme is incorporated in the optimal controller to ensure collision avoidance and minimize the performance index. Finally, a numerical example is performed to demonstrate the effectiveness of the proposed approaches.

This paper investigates L∞-estimates for the general optimal control problems governed by two-dimensional nonlinear elliptic equations with pointwise control constraints using mixed finite element methods. The state and the co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. The authors derive L∞-estimates for the mixed finite element approximation of nonlinear optimal control problems. Finally, the numerical examples are given.

Multitask learning and nonlinear optimal control of the COVID-19 outbreak: A geometric programming approach

This work develops an optimization method based on a new class of basis function, namely the generalized Bernoulli polynomials (GBP), to solve a class of nonlinear 2-dim fractional optimal control problems. The problem is generated by nonlinear fractional dynamical systems with fractional derivative in the Caputo type and the Goursat–Darboux conditions. First, we use the GBP to approximate the state and control variables with unknown coefficients and parameters. Afterwards, we substitute the obtained values for the variables and parameters in the objective function, nonlinear fractional dynamical system and Goursat–Darboux conditions. The 2-dim Gauss–Legendre quadrature rule together with a fractional operational matrix construct a constrained problem, that is solved by the Lagrange multipliers method. The convergence of the GBP method is proved and its efficiency is demonstrated by several examples.