Nonlinear Optimal Control 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.

A nonlinear optimal control approach for dual-arm robotic manipulators

A nonlinear optimal control approach for dual-arm robotic manipulators

An optimal nonlinear fractional order controller for passive/active base isolation building equipped with friction-tuned mass dampers

This paper presents an optimal nonlinear fractional-order controller (ONFOC) designed to reduce the seismic responses of tall buildings equipped with a base-isolation (BI) system and friction-tuned mass dampers (FTMDs). The parameters for the BI and FTMD systems, as well as their combinations (BI-FTMD and active BI-FTMD or ABI-FTMD), were optimized separately using a multi-objective quantum-inspired seagull optimization algorithm (MOQSOA). The seismic performances of the BI, FTMD, BI-FTMD, and ABI-FTMD systems for a 15-storey building subjected to two far-field (Loma Prieta and Landers) and two near-fields (Tabas and Northridge) earthquakes were evaluated. The results indicated that structures with BI, FTMD, BI-FTMD, and ABI-FTMD systems outperformed the uncontrolled structure in reducing structural responses during the design earthquakes (Loma Prieta and Tabas). However, under validation earthquakes (Landers and Northridge), the peak acceleration of the building with the FTMD system was worse than that of the uncontrolled structure during the near-field Tabas earthquake. To address this issue, we proposed a combination of the active BI system and the FTMD system. Time history analysis results demonstrated that for the building equipped with the ABI-FTMD system, the peak displacement, peak acceleration, and peak inter-storey drift were reduced by approximately 60%, 64%, and 78%, respectively, as compared to the uncontrolled structure.

Adaptive Inverse Nonlinear Optimal Control Based on Finite-Time Concurrent Learning and Semidefinite Programming.

This article investigates the problem of inverse optimal control (IOC) for a class of nonlinear affine systems. An adaptive IOC approach is proposed to recover the cost functional using only the system state data, which integrates the finite-time concurrent learning (FTCL) technique and the semidefinite programming (SDP) technique. First, an identifier neural network (NN) is employed to approximate the unknown nonlinear control policy, and an FTCL-based update law is proposed to estimate the weights of the identifier NN online, which removes the traditional persistent excitation (PE) condition. Moreover, the finite-time convergence as well as the uniformly ultimately boundness (UUB) of estimation error of the identifier NN weights are analysed according to whether or not there exists the identifier NN approximation error. Then, with the help of a value NN for approximating the value function, an SDP problem with a quadratic objective function can be set up for determining the weighting matrices of the cost functional. Finally, simulation results are presented to validate the proposed method.

Heat-Equation-Based Smoothing Homotopy Method for Nonlinear Optimal Control Problems

This paper presents a new heat-equation-based smoothing homotopy method for solving nonlinear optimal control problems with the indirect method. The surrogates, derived from the heat equation solution, are first incorporated into the necessary conditions as replacement of the terminal state and costate variables. The homotopy process is then applied to the heat conduction time: a longer time results in a more uniform temperature distribution and greater stability against initial temperature variations, thereby making the corresponding homotopy problem much easier to solve; zero time implies no heat conduction and reverts to the original problem. Furthermore, capitalizing on the heat equation’s boundedness characteristic, an efficient approach for determining the hypersensitive parameters is proposed, thus obviating the necessity of manual tuning. Challenging numerical examples are provided to demonstrate the superior performance of the proposed method, indicating that its convergence and efficiency are significantly enhanced compared to the original smoothing homotopy methods.

A novel unified variational finite difference (VFD) solution method for optimal control problems

This article is aimed to propose a simple yet efficient unified numerical strategy for solving both linear and non-linear optimal control problems. To do so, the general form of quadratic performance index function and nonlinear state equations are considered first. Then, the idea of variational differential quadrature method is used to convert the integral/differential equations to the equivalent algebraic form. Since time is the only independent variable is this research, the finite difference method with an equally-spaced discretization scheme would be a more appropriate technique rather than the differential quadrature approach. So, the implemented numerical solution is called now as the variational finite difference method. The method of Lagrange multipliers is then utilized for minimization purpose and, as a result, the final set of nonlinear algebraic equations are obtained. Finally, the quadratic and triadic forms of non-linearity are considered and an explicit formulation is represented for the residual and Jacobian of the Newton-based iterative solution procedure. To demonstrate the accuracy and efficiency of the proposed approach in the quadratic optimal control area, several benchmark problems involving linear time-invariant, linear time-variant and nonlinear examples are successfully solved and the results are confirmed with those existed in the literature.

Nonlinear optimal frequency control for dynamic vibration absorber and its application

A principal limitation of passive dynamic vibration absorber (DVA) lies in its intrinsic design constraint, as it is tuned to a specific resonant frequency, rendering it less effective at mitigating vibrations over a variable frequency range. To ensure that the DVA retains its vibration reduction capability under loads with time-varying frequencies, this paper proposes a method for designing the optimal suspension frequency based on the external load frequency. The approach involves altering the stiffness of the DVA through semi-active or active control to achieve the optimal suspension frequency and the best vibration reduction performance. In this study, the ratio R between the optimal suspension frequency and the load frequency is defined, and the nonlinear characteristic of R, based on the lowest vibration transmission rate, is studied using a two degree-of-freedom DVA model. Furthermore, an optimal suspension frequency variation control strategy based on load frequency identification is presented. Combined with the typical load with time-varying frequency encountered in metro vehicles, a multi-body dynamics theoretical model and a rigid–flexible coupling dynamics model are established to verify the effectiveness of the nonlinear optimal frequency control. Finally, the full-size railway vehicle roller rig is used to verify the correctness of the principle of optimal frequency curve. The research results show that when the mass ratio is 0.1, compared with rigid suspension, the nonlinear optimal frequency control can reduce the root mean square (RMS) value of the carbody’s entire time period acceleration during variable speed operation by 62.2%, and the maximum RMS value is reduced by 83.6%. In nonlinear control, unlike passive DVA, the vibration is never higher than that of rigid suspension within a specific velocity range. Compared with linear control strategy, nonlinearity can further reduce the elastic vibration. The test results from roller rig indicate that the semi-active DVA can effectively reduce the elastic vibrations of the carbody, while also validating the correctness of the nonlinear optimal suspension frequency principle. Therefore, the nonlinear frequency control in this study can provide reference for structural vibration control under loads with time-varying frequencies.

Mainardi smoothing homotopy method for solving nonlinear optimal control problems

This paper proposes a new type of smoothing homotopy method for solving general nonlinear optimal control problems via the indirect method, leveraging the Mainardi kernel as the smoothing kernel. The Mainardi kernel is firstly derived from the fundamental solution of the time-fractional diffusion-wave equation, representing a generalized form of the Gaussian kernel. By altering the fractional derivative order, the kernel can seamlessly switch between non-Gaussian and Gaussian forms. Then, parts of the two-point boundary value problems are convolved with the smoothing kernel, and the resulting surrogates are incorporated into the necessary conditions, replacing the terminal state and costate variables. The homotopy process is used for the smoothing parameter: increasing this parameter enhances the smoothing effect, simplifying the homotopy problems, while a parameter value of zero implies no smoothing, reverting to the original problems. Additionally, explicit expressions for the Mainardi kernel at specific derivative orders are derived, avoiding the inefficiencies associated with fractional derivatives. Simulation examples demonstrate that the proposed method offers significant advantages in flexibility and convergence compared to the traditional Gaussian smoothing homotopy method.

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.

Nonlinear Optimal Control of No-Insulation and Controlled-Insulation HTS Coils and Magnets

Nonlinear Optimal Control of No-Insulation and Controlled-Insulation HTS Coils and Magnets

An optimal midcourse guidance method for dual pulse air-to-air missiles using linear Gauss pseudospectral model predictive control method

This paper proposes an optimal midcourse guidance method for dual pulse air-to-air missiles, which is based on the framework of the linear Gauss pseudospectral model predictive control method. Firstly, a multistage optimal control problem with unspecified terminal time is formulated. Secondly, the control and terminal time update formulas are derived analytically. In contrast to previous work, the derivation process fully considers the Hamiltonian function corresponding to the unspecified terminal time, which is coupled with control, state, and costate. On the assumption of small perturbation, a special algebraic equation is provided to represent the equivalent optimal condition for the terminal time. Also, using Gauss pseudospectral collocation, error propagation dynamical equations involving the first-order correction term of the terminal time are transformed into a set of algebraic equations. Furthermore, analytical modification formulas can be derived by associating those equations and optimal conditions to eliminate terminal error and approach nonlinear optimal control. Even with their mathematical complexity, these formulas produce more accurate control and terminal time corrections and remove reliance on task-related parameters. Finally, several numerical simulations, comparisons with typical methods, and Monte Carlo simulations have been done to verify its optimality, high convergence rate, great stability and robustness.

Experimental backward integration for state-dependent differential Riccati equation (SDDRE): A case study on flapping-wing flying robot

Backward integration (BI) is a classical approach for solving optimal control arising from linear quadratic regulator (LQR) design in differential form. It proposes a two-round solution, the first one starting from a final boundary condition to the initial one to generate the optimal gain, and the next one, solving the control system in a forward loop. Implementation of the BI on the nonlinear optimal control, the state-dependent differential Riccati equation (SDDRE), is a challenge since in the backward loop, the state information is missing and is not straightforward like the LQR. Hence, a control for backward motion is required to regulate the system from the terminal to the initial desired states. While there have been some valuable works on the theoretical implementation of the BI in simulations for the SDDRE, this approach has not been reported in experiments for the best knowledge of the authors. Here in this work, the SDDRE is solved using the BI and two backward and forward solution steps, and it is experimentally applied for a flapping-wing flying robot (FWFR). The execution of the control system is done onboard using Raspberry Pi 4B as the processor which has limited computational capacity. The trajectory tracking of a line in a closed limited space is proposed for the FWFR flight. The objective is to position the robot bird at the corner of the rectangular limited space with minimum error in translation. The results have been presented for 21 flights to show the repeatability and presented the best-case minimum error of 10 (cm) at the end of the trajectory, in the YZ-plane. Considering approximately 12 (m) flight path, the error is found less than 1 percent of the travel distance. The results were compared with forward integration to confirm the correctness of the computation. The experimental flight dataset, MATLAB simulation codes, and experimental Python codes are available as supplementary material for this work in the online version of the paper.

A gradient-based optimization algorithm to solve optimal control problems with conformable fractional-order derivatives

In this paper, we solve numerically a class of fractional optimal control problems in the conformable sense. We first propose a nonlinear fractional optimal control problem subject to general equality and inequality constraints. Then, based on a novel numerical integration scheme, we present a discretization method for the governed fractional-order system as well as the cost and constraint functionals, which yields a discretized approximate problem. We further derive the gradients of the cost and constraint functionals in regard to the discretized controls. On the basis of this result, a numerical optimization approach to solve the approximate problem is developed. Finally, three non-trivial example problems are solved to illustrate the applicability and effectiveness of the developed approach.

Nonlinear Optimal Control Based on FBDEs and its Application to AGV.

This article focuses on solving a finite-horizon nonlinear optimal control problem by using the Pontryagin's maximum principle. In practical applications, linearization is a common approach for solving nonlinear dynamical systems. However, it is not universally applicable due to various reasons, such as instability and low accuracy. In contrast to linearization, the inherent challenge in directly solving the above nonlinear optimal control problem lies in addressing the highly coupled nonlinear forward and backward differential equations. In order to address this problem, an equivalent relationship is established between these equations and a new optimization problem. By exploiting the inherent relationship between supervised learning and an optimization problem from the view of a dynamical system, a deep neural network framework is constructed for describing the new optimization problem. Furthermore, a numerical algorithm for optimal control, which is very powerful for a large variety of nonlinear dynamical systems, is implemented by training a deep residual network. Finally, the effectiveness of the algorithm is demonstrated by solving a trajectory tracking control problem for automatic guided vehicle. The obtained results reveal that the proposed control scheme can achieve high-precision tracking.

Sequential Convex Programming for Nonlinear Optimal Control in UAV Trajectory Planning

In this paper, an algorithm is proposed to solve the non-convex optimization problem using sequential convex programming. An approximation method was used to solve the collision avoidance constraint. An iterative approach was utilized to estimate the non-convex constraints, replacing them with their linear approximations. Through the simulation, we can see that this method allows for quadcopters to take off from a given initial position and fly to the desired final position within a specified flight time. It is guaranteed that the quadcopters will not collide with each other in different scenarios.

A nonlinear optimal control approach for 3-DOF four-cable driven parallel robots

In this article, a nonlinear optimal control approach is proposed for the dynamic model of 3-DOF four-cable driven parallel robots (CDPR). To solve the associated nonlinear optimal control problem, the dynamic model of the 3-DOF cable-driven parallel robot 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 3-DOF cable-driven parallel robot a stabilizing optimal (H-infinity) feedback controller is designed. 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 reference setpoints under moderate variations of the control inputs and a minimum dispersion of energy.

Optimal Nonlinear Robust Sliding Mode Control of an Excitation System Based on Mixed $\mathcal{H}_{2}/\mathcal{H}_{\infty}$ Linear Matrix Inequalities

Optimal Nonlinear Robust Sliding Mode Control of an Excitation System Based on Mixed $\mathcal{H}_{2}/\mathcal{H}_{\infty}$ Linear Matrix Inequalities

Design of a reliable electrical photovoltaic generator based on power converters control

This work proposes a renewable energy generation system that operates with a similar behavior as a synchronous generator in terms of reliability in the power supply and inertial response properties against power changes and grid voltage disturbances. The main contributions of this work are: (1) the renewable energy variability managing by means of a battery energy storage strategy, whose charge and discharge are determined by a forecasted-based power adjustment algorithm, (2) the dynamical modeling of an inverter that includes physical inertia and its corresponding response to power and voltage variations, and (3) the synthesis of a nonlinear optimal control for the inertia-based inverter system where the reactive power injection is regulated. Simulation results for a photovoltaic-based generator are presented to demonstrate the effectiveness of the proposed methodologies.

Control strategy for trading off solar power and control input while rendezvous and docking

This paper presents a novel strategy for decoupling the attitude and orbit equations of a CubeSat for rendezvous and docking with an uncontrollable cooperative target. For computing a safe trajectory, the proposed control algorithm takes into account the solar energy received by the CubeSat. The CubeSat is equipped with a thruster for orbit maneuvers, magnetic coils for attitude control, and four fixed single-sided rectangular solar panels. The coupled dynamics arising from the dynamical model of the spacecraft is solved using a novel weighted vector-based approach. This paper presents a linearized nonlinear optimal control problem arising in small spacecrafts while trying to maximize solar energy input in the rendezvous and docking process. An intermediate orbit is defined and used to divide the problem into two different optimal control problems: rendezvous optimization and docking optimization problems. A geometrical approach based on the shape of the chaser and the target is contemplated for collision avoidance while docking. The proposed controller design is modified whenever the sunlight is obstructed by the Earth, and the maneuvering controls are redesigned accordingly for optimal rendezvous and docking. Numerical simulations have been carried out to show the efficacy of the proposed concept for steering the trade-off between solar energy and control input during the rendezvous and docking of CubeSat with a tumbling target.

Indirect method for solving non-linear optimal control of a non-rectilinear motion of a rocket with variable mass

In this paper, an optimal trajectory of the rocket angle with a variable mass will be calculated by considering the aerodynamic forces, the acceleration of gravity and moves with a non-rectilinear motion from a initial state to a final state with a known altitude. The aim is to optimize the lateral offset of the rocket. For this, we formulate an optimal control problem where the rocket angle is the control. In order to solve the problem, let applied Shooting method's based on the Pontryagin's maximum principle, and study the precision and a duration time. Finally, we validate the results by using MATLAB software.