Formulation Of Optimal Control Problem Research Articles

Bounding linear rainfall-runoff models with fractional derivatives applied to a barren catchment of the Jordan Rift Valley

A new concept is developed to mathematically understand the dynamics of the rainfall-runoff events in a barren catchment of the Jordan Rift Valley. Time series data of rainfall and runoff have been acquired at an observation point in the catchment. Due to the extreme arid environment, water current as the runoff from the catchment is ephemeral, and the rainfall-runoff events are clearly distinguishable from each other. Firstly, a pair of linear autoregressive models with exogenous input (ARX models) is identified to tightly bound each runoff time series using the simplex method of linear programming. The exogenous input part is compatible with the conventional unit hydrograph method, while the autoregressive part is regarded as a discretized differential operator of fractional orders. Then, a linear fractional differential equation is determined to approximate each linear ARX model, which restricts the perturbation of the actual causal relationship between rainfall intensity and runoff discharge. The resulting lower and upper bounding rainfall-runoff models with fractional derivatives are examined in the system-theoretic framework. Finally, a nominal model from which actual nonlinear and stochastic phenomena perturb is arranged to envelope the all upper bounding rainfall-runoff models in the frequency domain, leading to the formulation of a challenging fractional optimal control problem involving stochastic processes.

Comparison of different optimal control formulations for generating dynamically consistent crutch walking simulations using a torque-driven model

Walking impairment due to spinal cord injury can be improved using active orthoses, together with some type of external support such as crutches for balance. This study explores how optimal control problem formulation affects the ability to generate dynamically consistent crutch walking simulations with active orthosis assistance that closely reproduce experimental data. The investigation compares eight optimal tracking problem formulations using a torque-driven full-body skeletal model of a healthy subject walking with active knee-ankle-foot orthoses and forearm crutches. We have found that small adjustments to ground reactions are required to achieve convergence, tracking of joint instead of marker coordinates significantly improves convergence, and minimising joint jerks instead of joint accelerations has only a minor effect on the problem convergence. These results provide guidance for future work that seeks to develop predictive optimal control simulations that can identify orthosis control parameters that maximise improvement in walking function on an individual patient basis.

Prediction of three-dimensional crutch walking patterns using a torque-driven model

Computational prediction of 3D crutch-assisted walking patterns is a challenging problem that could be applied to study different biomechanical aspects of crutch walking in virtual subjects, to assist physiotherapists to choose the optimal crutch walking pattern for a specific subject, and to help in the design and control of exoskeletons, when crutches are needed for balance. The aim of this work is to generate a method to predict three-dimensional crutch-assisted walking motions following different patterns without tracking any experimental data. To reach this goal, we collected gait data from a healthy subject performing a four-point non-alternating crutch walking pattern, and developed a 3D torque-driven full-body model of the subject including the crutches and foot- and crutch-ground contact models. First, we developed a predictive (i.e., no tracking of experimental data) optimal control problem formulation to predict crutch walking cycles following the same pattern as the experimental data collected, using different cost functions. To reduce errors with respect to reference data, a cost function combining minimization terms of angular momentum, mechanical power, joint jerk and torque change was chosen. Then, the problem formulation was adapted to handle different foot- and crutch-ground conditions to make it capable of predicting three new crutch walking patterns, one of them at different speeds. A key aspect of our algorithm is that having ground reactions as additional controls allows one to define phases inside the cycle without the need of formulating a multiple-phase problem, thus facilitating the definition of different crutch walking patterns.

Stochastic optimal control — A concise introduction

<p style='text-indent:20px;'>This is a concise introduction to stochastic optimal control theory. We assume that the readers have basic knowledge of real analysis, functional analysis, elementary probability, ordinary differential equations and partial differential equations. We will present the following topics: (ⅰ) A brief presentation of relevant results on stochastic analysis; (ⅱ) Formulation of stochastic optimal control problems; (ⅲ) Variational method and Pontryagin's maximum principle, together with a brief introduction of backward stochastic differential equations; (ⅳ) Dynamic programming method and viscosity solutions to Hamilton-Jacobi-Bellman equation; (ⅴ) Linear-quadratic optimal control problems, including a careful discussion on open-loop optimal controls and closed-loop optimal strategies, linear forward-backward stochastic differential equations, and Riccati equations.</p>

Unified value-based feedback, optimization and risk management in complex electric energy systems

The ideas in this paper are motivated by an increased need for systematic data-enabled resource management of large-scale electric energy systems. The basic control objective is to manage uncertain disturbances, power imbalances in particular, by optimizing available power resources. To that end, we start with a centralized optimal control problem formulation of system-level performance objective subject to complex interconnection constraints and constraints representing highly heterogeneous internal dynamics of system components. To manage spatial complexity, an inherent multi-layered structure is utilized by modeling interconnection constraints in terms of unified power variables and their dynamics. Similarly, the internal dynamics of components and sub-systems (modules), including their primary automated feedback control, is modeled so that their input–output characterization is also expressed in terms of power variables. This representation is shown to be key to managing the multi-spatial complexity of the problem. In this unifying energy/power state space, the system constraints are all fundamentally convex, resulting in the convex dynamic optimization problem, for typically utilized quadratic cost functions. Based on this, an interactive multi-layered modeling and control method is introduced. While the approach is fundamentally based on the primal–dual decomposition of the centralized problem, this is formulated for the first time for the couple real-reactive power problem. It is also is proposed for the first time to utilize sensitivity functions of distributed agents for solving the primal distributed problem. Iterative communication complexity typically required for convergence of point-wise information exchange is replaced by the embedded distributed optimization by the modules when creating these functions. A theoretical proof of the convergence claim is given. Notably, the inherent multi-temporal complexity is managed by performing model predictive control (MPC)-based decision making when solving distributed primal problems. The formulation enables distributed decision-makers to value uncertainties and related risks according to their preferences. Ultimately, the distributed decision making results in creating a bid function to be used at the coordinating market-clearing level. The optimization approach in this paper provides a theoretical foundation for next-generation Supervisory Control and Data Acquisition (SCADA) in support of a Dynamic Monitoring and Decision Systems (DyMonDS) for a multi-layered interactive market implementation in which the grid users follow their sub-objectives and the higher layers coordinate interconnected sub-systems and the high-level system objectives. This forms a theoretically sound basis for designing IT-enabled protocols for secure operations, planning, and markets.

Multi-objective optimal control of a simple stochastic climate-economy model

Integrated assessment modelling of climate change aims to provide quantitative solutions to inform international climate policy by employing models where socio-economic and climatic systems are integrated. Among these models, DICE (Dynamic Integrated Climate-Economy), is used to perform cost-benefit analysis that returns as output the optimal emission reduction pathway. The model makes some important assumptions: future socioeconomic and climate system evolution is deterministic and economic damages of climate change are a quadratic function of the atmospheric temperature. In this study, propose a multi-objective stochastic optimal control problem formulation of the DICE model in order to account for stochastic disturbances and to align with physical targets posed by international agreements on climate change mitigation. The solutions are control policies that can handle stochastic disturbances outperforming the static inter-temporal optimization approach traditionally adopted. Moreover, such control policies are able to deal with multiple objectives making explicit the trade-offs between economic and environmental objectives.

Eco-Driving for Energy Efficient Cornering of Electric Vehicles in Urban Scenarios

In this paper, we propose a model for eco-driving that considers cornering effects. The proposed model purely relies on the geometric configuration of the vehicle and road. Consequently, we propose an eco-driving optimal control problem formulation that is suitable for both straight and curved trajectories in urban scenarios. Moreover, it can be applied for vehicles with front wheel drive (FWD) or rear wheel drive (RWD). We use a case study for an electric vehicle executing cornering maneuvers to validate the proposed approach with a high fidelity vehicle model. Results show an approximated improvement of 8% in energy savings with respect to traditional eco-driving strategies, especially in trajectories with large curvatures.

Data-Driven Optimal Control of Linear Time-Invariant Systems

We consider an optimal control problem for discrete-time linear time-invariant systems subject to unknown state-space model, constrained inputs and noisy output measurements. Since traditional model-based optimal control problem formulations and methods are not applicable to the problem under consideration, we propose a data-driven robust formulation based on the explicit model description derived from a single measured trajectory of the system. Then we propose an open-loop optimal feedback control scheme and show that its efficient implementation requires solution of a number of optimal estimation problems and a deterministic optimal control problem, all in data-driven formulations. The main contributions of this paper are the separation of the estimation and control processes in the data-driven context and the resulting robust feedback control scheme.

Fractional optimal control problem for ordinary differential equation in weighted Lebesgue spaces

In this paper, a necessary and sufficient condition, such as the Pontryagin’s maxi-mum principle for a fractional optimal control problem with concentrated parameters, is given by the ordinary fractional differential equation with a coefficient in weighted Lebesgue spaces. We discuss a formulation of fractional optimal control problems by a fractional differential equation in the sense of Caputo fractional derivative. The statement of the fractional optimal control problem is studied by using a new version of the increment method that essentially uses the concept of an adjoint equation of the integral form.

Optimal Control of Multi-Source Electric Vehicles in Real Time Using Advisory Dynamic Programming

This paper presents a novel method, advisory dynamic programming (AD-DP), for power management of a fuel cell hybrid vehicle (FCHEV). The presented method embraces a new understanding of vehicle states as driver-dependent and -independent states in time domain to define a suitable state space to be used for dynamic programming. Driver-dependent states are defined in terms of multiple characteristic parameters for vehicle speed and power demand, corporately. Driver-independent states are defined in terms of discrete values for supercapacitor's state of charge $SoC_{sc}$ . Transition costs between all states in the state space are calculated offline and tabulated in look-up tables for online implementation. A state predictive model is developed based on the transition statistics of driver-dependent states for a suitable number of driving cycles. Backward calculation of the total transition cost for the predicted horizon in state space is used to define optimal power split strategy for the powertrain. The formulation of optimal control problem, in terms of situation-based solutions related to vehicle states, enables a significant reduction of computational steps and hence addresses the main challenge of real-time applications. The algorithm is adapted in terms of optimization horizon and number of discrete states for DIS to suit the real-time application. Experimental application of AD-DP, using an emulation test-rig, is conducted over different driving cycles. The obtained results reveal an improvement in energy efficiency up to 29% compared to the adaptive rule-based method. The contribution of this paper can be identified as: first, development of a corporate definition for vehicle states, that can be further implemented in optimization-based power management methods. Second, the formulation of an adaptive DP that requires lower computational steps and hence suits real-time applications in hybrid electric vehicles.

Explicit Solution for Constrained Scalar-State Stochastic Linear-Quadratic Control With Multiplicative Noise

We study in this paper, a class of constrained linear-quadratic (LQ) optimal control problem formulations for the scalar-state stochastic system with multiplicative noise, which has various applications, especially in the financial risk management. The linear constraint on both the control and state variables considered in our model destroys the elegant structure of the conventional LQ formulation and has blocked the derivation of an explicit control policy so far in the literature. We successfully derive in this paper, the analytical control policy for such a class of problems by utilizing the state separation property induced from its structure. We reveal that the optimal control policy is a piecewise affine function of the state and can be computed offline efficiently by solving two coupled Riccati equations. Under some mild conditions, we also obtain the stationary control policy for an infinite time horizon. We demonstrate the implementation of our method via some illustrative examples and show how to calibrate our model to solve dynamic constrained portfolio optimization problems.

Energy-Efficient Arrival with RTA Constraint for Multirotor eVTOL in Urban Air Mobility

The electric vertical takeoff and landing (eVTOL) aircraft can alleviate transportation congestion on the ground by using three-dimensional airspace efficiently. However, the endurance of lithium-ion polymer (Li-Po) batteries imposes critical constraints on the operational time span of an eVTOL aircraft on urban air mobility (UAM) passenger transportation mission. This research focuses on the formulation of fixed-final-time multiphase optimal control problem with energy consumption as the performance index for a multirotor eVTOL aircraft. The proposed multiphase optimal control problem formulation and the numerical solution enables a multirotor eVTOL aircraft to meet the assigned required time of arrival and achieve an energy-efficient arrival trajectory for a given concept of operation (CONOP), which is a critical enabler for the safe and efficient future eVTOL operations for passenger transportation and cargo delivery. The problem formulation is applied to a UAM passenger transportation use case with EHang 184 eVTOL aircraft, and an Uber-proposed vertiport for five different types of CONOPs. Finally, the energy consumed for all the CONOPs is compared to propose the most energy-efficient CONOP for a multirotor eVTOL on UAM passenger transportation mission. The proposed framework can also be used to address an energy-efficient cargo delivery application in a UAS traffic management context.

New aspects of time fractional optimal control problems within operators with nonsingular kernel

This paper deals with a new formulation of time fractional optimal control problems governed by Caputo-Fabrizio (CF) fractional derivative. The optimality system for this problem is derived, which contains the forward and backward fractional differential equations in the sense of CF. These equations are then expressed in terms of Volterra integrals and also solved by a new numerical scheme based on approximating the Volterra integrals. The linear rate of convergence for this method is also justified theoretically. We present three illustrative examples to show the performance of this method. These examples also test the contribution of using CF derivative for dynamical constraints and we observe the efficiency of this new approach compared to the classical version of fractional operators.

LP Formulations of Discrete Time Long-Run Average Optimal Control Problems: The NonErgodic Case

We formulate and study the infinite dimensional linear programming (LP) problem associated with the deterministic discrete time long-run average criterion optimal control problem. Along with its du...

Linear Programming Formulation of Long-Run Average Optimal Control Problem

We formulate and study the infinite-dimensional linear programming problem associated with the deterministic long-run average cost control problem. Along with its dual, it allows one to characterize the optimal value of this control problem. The novelty of our approach is that we focus on the general case wherein the optimal value may depend on the initial condition of the system.

An optimal control approach for blood pressure regulation during head-up tilt

This paper presents an optimal control approach to modeling effects of cardiovascular regulation during head-up tilt (HUT). Many patients who suffer from dizziness or light-headedness are administered a head-up tilt test to explore potential deficits within the autonomic control system, which maintains the cardiovascular system at homeostasis. This system is complex and difficult to study in vivo, and thus we propose to use mathematical modeling to achieve a better understanding of cardiovascular regulation during HUT. In particular, we show the feasibility of using optimal control theory to compute physiological control variables, vascular resistance and cardiac contractility, quantities that cannot be measured directly, but which are useful to assess the state of the cardiovascular system. A non-pulsatile lumped parameter model together with pseudo- and clinical data are utilized in the optimal control problem formulation. Results show that the optimal control approach can predict time-varying quantities regulated by the cardiovascular control system. Our results compare favorable to our previous study using a piecewise linear spline approach, less a priori knowledge is needed, and results were obtained at a significantly lower computational cost.

Scalable Markov chain approximation for a safe intercept navigation in the presence of multiple vehicles

This paper studies a safe intercept navigation which accounts for the uncertainty of other vehicles’ trajectories, avoids collisions and any other positions in which vehicle safety is compromised. Since the number of vehicles can vary with time, it is important that the navigation strategy can quickly adjust to the current number of vehicles, i.e, that it scales well with the number of vehicles. The scalable strategy is based on a stochastic optimal control problem formulation of safe navigation in the presence of a single vehicle, denoted as the one-on-one vehicle problem. It is shown that safe navigation in the presence of multiple vehicles can be solved exactly as an auxiliary Markov decision problem. This allows us to approximate the solution based on the one-on-one vehicle optimal control solution and achieve scalable navigation. Our work is illustrated by a numerical example of safely navigating a vehicle in the presence of four other vehicles and by a robot experiment.

A decentralized energy-optimal control framework for connected automated vehicles at signal-free intersections

We address the problem of optimally controlling connected and automated vehicles (CAVs) crossing an urban intersection without any explicit traffic signaling, so as to minimize energy consumption subject to a throughput maximization requirement. We show that the solution of the throughput maximization problem depends only on the hard safety constraints imposed on CAVs and its structure enables a decentralized optimal control problem formulation for energy minimization. We present a complete analytical solution of these decentralized problems and derive conditions under which feasible solutions satisfying all safety constraints always exist. The effectiveness of the proposed solution is illustrated through simulation which shows substantial dual benefits of the proposed decentralized framework by allowing CAVs to conserve momentum and fuel while also improving travel time.

An Adaptive Economic Model Predictive Control Approach for Wind Turbines

Motivated by the reduction of overall wind power cost, considerable research effort has been focused on enhancing both efficiency and reliability of wind turbines. Maximizing wind energy capture while mitigating fatigue loads has been one of the main goals for control design. Recent developments in remote wind speed measurement systems (e.g., light detection and ranging (LIDAR)) have paved the way for implementing advanced control algorithms in the wind energy industry. In this paper, an LIDAR-assisted economic model predictive control (MPC) framework with a real-time adaptive approach is presented to achieve the aforementioned goal. First, the formulation of a convex optimal control problem is introduced, with linear dynamics and convex constraints that can be solved globally. Then, an adaptive approach is proposed to reject the effects of model-plant mismatches. The performance of the developed control algorithm is compared to that of a standard wind turbine controller, which is widely used as a benchmark for evaluating new control designs. Simulation results show that the developed controller can reduce the tower fatigue load with minimal impact on energy capture. For model-plant mismatches, the adaptive controller can drive the wind turbine to its optimal operating conditions while satisfying the optimal control objectives.

A New Formulation of the Fractional Optimal Control Problems Involving Mittag–Leffler Nonsingular Kernel

The aim of this paper is to propose a new formulation of the fractional optimal control problems involving Mittag–Leffler nonsingular kernel. By using the Lagrange multiplier within the calculus of variations and by applying the fractional integration by parts, the necessary optimality conditions are derived in terms of a nonlinear two-point fractional boundary value problem. Based on the convolution formula and generalized discrete Gronwall’s inequality, the numerical scheme for solving this problem is developed and its convergence is proved. Numerical simulations and comparative results show that the suggested technique is efficient and provides satisfactory results.