Optimal Control Formulation Research Articles

Optimal Control Approach to Helicopter Noise Abatement Trajectories in Nonstandard Atmospheric Conditions

This paper discusses the development of a software suite that aims to optimize helicopter trajectories with respect to the noise impact on the ground. The software suite has been developed around an advanced gradient-based optimization algorithm based on optimal control theory. The helicopter trajectories are modeled using an eight-degree-of-freedom flight mechanics model. To determine the noise impact on the ground, a helicopter noise model developed in this work consists of three modules, namely, a helicopter source noise model, a propagation model, and a noise impact model. To determine the noise levels on the ground, the source noise levels are determined from a database of aeroacoustically determined noise levels for varying flight conditions and projected on a hemisphere centered at the main rotor hub. The noise propagation model included in the suite is capable of determining the propagation loss between the source and the receiver in nonstandard atmospheric conditions, and it yields the total noise level in individual receiver locations. Finally, these noise levels can be quantified into a single noise impact criterion, which can be used as an optimization criterion in the optimal control formulation. To exemplify the capabilities of the suite, a hypothetical city center approach procedure is optimized for the noise impact in communities surrounding the helispot.

Combined Optimal Control and Combinatorial Optimization for Searching and Tracking Using an Unmanned Aerial Vehicle

Combined searching and tracking of objects using Unmanned Aerial Vehicles (UAVs) is an important task with many applications. One way to approach this task is to formulate path-planning as a continuous optimal control problem. However, such formulations will, in general, be complex and difficult to solve with global optimality. Therefore, we propose a two-layer framework, in which the first layer uses a Traveling-Salesman-type formulation implemented using combinatorial optimization to find a near-globally-optimal path. This path is refined in the second layer using a continuous optimal control formulation that takes UAV dynamics and constraints into consideration. Searching and tracking problems usually trade-off, often in a manual or ad-hoc manner, between searching unexplored areas and keeping track of already known objects. Instead, we derive a result that enables prioritization between searching and tracking based on the probability of finding a new object weighted against the probability of losing tracked objects. Based on this result, we construct a new algorithm for searching and tracking. This algorithm is validated in simulation, where it is compared to multiple base cases as well as a case utilizing perfect knowledge of the positions of the objects. The simulations demonstrate that the algorithm performs significantly better than the base cases, with an improvement of approximately 5-15%, while it is approximately 20-25% worse than the perfect case.

Approximation techniques for dynamic real-time optimization (DRTO) of distributed MPC systems

A dynamic real-time optimization (DRTO) strategy has recently been proposed for coordination of distributed MPC systems. The scheme utilizes a prediction model that accounts for the impact of the distributed controllers on the plant response. Two techniques are presented for approximating the closed-loop prediction within the DRTO formulation - a hybrid closed-loop and an input clipping formulation. The hybrid formulation generates closed-loop predictions for a limited number of time intervals along the DRTO prediction horizon, followed by an open-loop optimal control formulation for the rest of the horizon. The input clipping formulation utilizes an unconstrained MPC optimization formulation for each distributed MPC, coupled with an input saturation mechanism. The performance of the approximation techniques is evaluated through application to case studies based on linear and nonlinear dynamic plant models respectively. The approximation techniques are demonstrated to be more computationally efficient than the rigorous counterpart without significant loss in performance.

Model predictive control under forecast uncertainty for optimal operation of buildings with integrated solar systems

In this paper, we explore intelligent operation strategies, based on stochastic model predictive control (SMPC), for optimal utilization of solar energy in buildings with integrated solar systems. Our approach takes into account the uncertainty in solar irradiance forecast over a prediction horizon, using a new probabilistic time series autoregressive model, calibrated on the sky-cover forecast from a weather service provider. In the optimal control formulation, we model the effect of solar irradiance as non-Gaussian stochastic disturbance affecting the cost and constraints, and the nonconvex cost function is an expectation over the stochastic process. To solve this complex optimization problem, we introduce a new approximate dynamic programming methodology that represents the optimal cost-to-go functions using Gaussian process regression, and achieves good solution quality. In the final step, we use an emulator that couples physical system models in TRNSYS with the SMPC controller developed using Python and MATLAB to evaluate the closed-loop operation of a building-integrated system with a solar-assisted heat pump coupled with radiant floor heating. For the system and climate under consideration, the SMPC saves up to 44% of the electricity consumption for heating in a winter month, compared to a baseline well-tuned rule-based controller, and it is robust, imposing less uncertainty on thermal comfort violation.

Optimal human navigation in steep terrain: a Hamilton–Jacobi–Bellman approach

Author(s): Parkinson, Christian; Arnold, David; Bertozzi, Andrea L; Chow, Yat Tin; Osher, Stanley | Abstract: We present a method for determining optimal walking paths in steep terrain using the level set method and an optimal control formulation. By viewing the walking direction as a control variable, we can determine the optimal control by solving a Hamilton-Jacobi-Bellman equation. We then calculate the optimal walking path by solving an ordinary differential equation. We demonstrate the effectiveness of our method by computing optimal paths which travel throughout mountainous regions of Yosemite National Park. We include details regarding the numerical implementation of our model and address a specific application of a law enforcement agency patrolling a nationally protected area.

Managing partially automated network traffic flow: Efficiency vs. stability

This paper analyzes how a central agent may bring a mixed traffic system including both human-driven and autonomous vehicles to an equilibrium that both maximizes the efficiency and is stable under the control. The evolution of the human drivers’ route choices, as well as the agent's control measures, is described using a joint day-to-day (DTD) dynamical model based on probability route choice. Within this setting, we show that (1) the fixed point of the proposed dynamical system coincides with the unique mixed equilibrium, and (2) the system is asymptotically stable in continuous time, namely it always converges to the mixed equilibrium from a given initial state. We then examine how alternative control policies may affect the transition trajectory leading to the mixed equilibrium. Two alternative control schemes are proposed and analyzed. The first, referred to as the stability-first control, aims to stabilize a given disequilibrium as soon as possible. The second seeks to minimize the total system cost accumulated over the transition period, hence called the efficiency-first control. We propose a continuous time optimal control formulation for both schemes and discuss how the formulation can be discretized and solved to local optimality using existing algorithms. Numerical experiments conducted on two illustrative examples highlight the differences among the three control schemes and how the share of autonomous vehicles affects the tradeoff between the efficiency and stability of the mixed traffic system.

Constrained Buckling of Spatial Elastica: Application of Optimal Control Method

A post-buckling analysis of a constant or variable length spatial elastica constrained by a cylindrical wall is performed for a first time by adopting an optimal control methodology. Its application in a constrained buckling analysis is shown to be superior when compared to other numerical techniques, as the inclusion of the unilateral constraints is feasible without the need of any special treatment or approximation. Furthermore, the formulation is simple and the optimal configurations of the spatial elastica can be also obtained by considering the minimization condition of the Hamiltonian. We first present the optimal control formulation for the constrained buckling problem of a constant length spatial elastica, including its associated necessary optimality conditions that constitute the Pontryagin's minimum principle. This fundamental constrained buckling problem is used to validate the proposed methodology. The general buckling problem of a variable length spatial elastica is then analyzed that consists of two parts; (1) the solution of the optimal control problem that involves the inserted elastica inside the conduit and (2) the derivation of the buckling load by taking into account the generation of the configurational or Eshelby-like force at the insertion point of the sliding sleeve. A variety of examples are accordingly presented, where the effects of factors, such as the presence of uniform pressure, the clearance of the wall, and the torsional rigidity, on the buckling response of the spatial elastica, are investigated.

Optimization of Fishing Strategies in Space and Time as a Non-convex Optimal Control Problem

The behavior of a fishing fleet and its impact onto the biomass of fish can be described by a nonlinear parabolic diffusion–reaction equation. Looking for an optimal fishing strategy leads to a non-convex optimal control problem with a bilinear control action. In this work, we present such an optimal control formulation, prove its well-posedness and derive first- and second-order optimality conditions. These results provide a basis for tailored finite element discretization as well as for Newton type optimization algorithms. First numerical test problems show typical features as so-called No-Take-Zones and maximal fishing quota (total allowable catches) as parts of an optimal fishing strategy.

Individual-based optimal weight adaptation for heterogeneous epidemic spreading networks

We study epidemic spreading on arbitrary weighted, directed, and heterogeneous complex networks. We propose an individual-based weight adaptation mechanism in which individuals’ contact strength is adaptable depending on the level of contagion spreading over the network. An optimal control formulation is presented to address the trade-off between the global infection level and the local weight adaptation cost corresponding to the topology of the underlying contact network. We prove the existence of a solution to the optimal control problem and obtain its explicit expression from a rigorous mathematical analysis. Additionally, we derive an approximation to the optimal solution and analyze its uniqueness with a specific cost function. In a series of numerical experiments, the forward-backward sweep method is adopted to verify the obtained theoretical results. Our findings uncover novel insights into the relationship between epidemic spreading on heterogeneous complex networks, the resulting dynamics of behavioral responses, and the associated control strategy costs, with important consequences for the future development of adaptive public health policies.

Optimal control formulation of pulse-based control using Koopman operator

In many applications, and in systems/synthetic biology, in particular, it is desirable to solve the switching problem, i.e., to compute control policies that force the trajectory of a bistable system from one equilibrium (the initial point) to another equilibrium (the target point). It was recently shown that for monotone bistable systems, this problem admits easy-to-implement open-loop solutions in terms of temporal pulses (i.e., step functions of fixed length and fixed magnitude). In this paper, we develop this idea further and formulate a problem of convergence to an equilibrium from an arbitrary initial point. We show that the convergence problem can be solved using a static optimization problem in the case of monotone systems. Changing the initial point to an arbitrary state allows building closed-loop, event-based or open-loop policies for the switching/convergence problems. In our derivations, we exploit the Koopman operator, which offers a linear infinite-dimensional representation of an autonomous nonlinear system and powerful computational tools for their analysis. Our solutions to the switching/convergence problems can serve as building blocks for other control problems and can potentially be applied to non-monotone systems. We illustrate this argument on the problem of synchronizing cardiac cells by defibrillation.

Effects of Structural Failure on the Safe Flight Envelope of Aircraft

The research presented in this paper focuses on the effects of structural failures on the safe flight envelope of an aircraft, which is the set of all the states in which safe maneuver of the aircraft can be assured. Nonlinear reachability analysis based on an optimal control formulation is performed to estimate the safe flight envelope using actual aircraft control surface inputs. This approach uses the physical model of an aircraft, where the aerodynamic stability and control derivatives are calculated using Digital Datcom. Symmetrical damages to a Cessna Citation II are considered with 25, 50, 75, and 100% spanwise vertical tail tip losses, leading to gradual shrinkage in the safe flight envelope. Based on the estimated safe flight envelopes, a discussion on the effects of structural damages and different flight conditions on the safe flight envelope is presented. In particular, the interpolatibility of the resulting safe flight envelopes is demonstrated. This property is essential for a novel database-driven flight envelope prediction method, where a database of safe flight envelopes is created offline to be accessed later in real time.

Optimal fuel consumption finite-thrust orbital hopping of aeroassisted spacecraft

In the paper, the problem of minimum-fuel aeroassisted spacecraft regional reconnaissance (orbital hopping) is considered. A new nonlinear constrained optimal control formulation is designed and constructed so as to describe this mission scenario. This formulation contains multiple exo-atmospheric and atmospheric flight phases and correspondingly, two sets of flight dynamics. The constructed continuous-time optimal control system is then discretized via a multi-phase global collocation technique. The resulting discrete-time system is optimized using a newly proposed gradient-based optimization algorithm. Several comparative simulations are carried out and the obtained optimal results indicate that it is effective and feasible to use the proposed multi-phase optimal control design for achieving the aeroassisted vehicle orbital hopping mission.

Optimal Rendezvous Trajectory for Unmanned Aerial-Ground Vehicles

Fixed-wing unmanned aerial vehicles (UAVs) can be an essential tool for low-cost aerial surveillance and mapping applications in remote regions. There is, however, a key limitation, which is the fact that low-cost UAVs have limited fuel capacity and, hence, require periodic refueling to accomplish a mission. Moreover, the usual mechanism of commanding the UAV to return to a stationary base station for refueling can result in the fuel wastage and inefficient mission operation time. Alternatively, one strategy could be the use of an unmanned ground vehicle (UGV) as a mobile refueling unit, where the UAV will rendezvous with the UGV for refueling. In order to accurately perform this task in the presence of wind disturbances, we need to determine an optimal trajectory in three-dimensional taking UAV and UGV dynamics and kinematics into account. In this paper, we propose an optimal control formulation to generate a tunable UAV trajectory for rendezvous on a moving UGV that also addresses the possibility of the presence of wind disturbances. By a suitable choice of the value of an aggressiveness index that we introduce in our problem setting, we are able to control the UAV rendezvous behavior. Several numerical results are presented to illustrate the reliability and effectiveness of our approach.

Optimal Path Planning in Time-Varying Flows Using Adaptive Discretization

Autonomous marine vehicles (AMVs) are typically deployed for long periods of time in the ocean to monitor different physical, chemical, and biological processes. Given their limited energy budgets, it makes sense to consider motion plans that leverage the dynamics of the surrounding flow field so as to minimize energy usage for these vehicles. In this letter, we present a graph-search-based method to compute energy optimal paths for AMVs in 2-D time-varying flows. The novelty of the proposed algorithm lies in the use of an adaptive discretization scheme to construct the search graph. We demonstrate the proposed algorithm by computing optimal energy paths using an analytical time-varying flow model and using time-varying ocean flow data provided by the Regional Ocean Model System. We compare the output paths with those computed via an optimal control formulation and numerically demonstrate that the proposed method can overcome problems inherent in existing fixed discretization schemes.

Multi-criteria Optimization of PD Controllers for Plants including Integral Action

A simple design method for robust PD controllers is presented for systems including integral action. The design method is based on a multi-criteria optimal control formulation, which is easily solved by a few lines of MATLAB code. Most criteria are based on H∞ measures, but since the focus is on reference signal tracking and not load disturbance compensation, the settling time is also included as a relevant performance measure. Since a PD controller is equivalent to a lead filter, the optimal PD controller is compared with ordinary text book design rules for lead filters. More specifically, it is shown that the common recommendation to place the mid frequency of the lead filter at the desired gain crossover frequency often gives bad servo performance. The suggested optimal solution, still including robustness and control activity adjustments, is on the other hand a simple and flexible design method for arbitrary plants including integral action.

Adaptive-Critic-Based Robust Trajectory Tracking of Uncertain Dynamics and Its Application to a Spring–Mass–Damper System

In this paper, the robust trajectory tracking design of uncertain nonlinear systems is investigated by virtue of a self-learning optimal control formulation. The primary novelty lies in that an effective learning based robust tracking control strategy is developed for nonlinear systems under a general uncertain environment. The augmented system construction is performed by combining the tracking error with the reference trajectory. Then, an improved adaptive critic technique, which does not depend on the initial stabilizing controller, is employed to solve the Hamilton–Jacobi–Bellman (HJB) equation with respect to the nominal augmented system. Using the obtained control law, the closed-loop form of the augmented system is built with stability proof. Moreover, the robust trajectory tracking performance is guaranteed via Lyapunov approach in theory and then through simulation demonstration, where an application to a practical spring–mass–damper system is included.

A nonlinear model predictive control formulation for obstacle avoidance in high-speed autonomous ground vehicles in unstructured environments

ABSTRACTThis paper presents a nonlinear model predictive control (MPC) formulation for obstacle avoidance in high-speed, large-size autono-mous ground vehicles (AGVs) with high centre of gravity (CoG) that operate in unstructured environments, such as military vehicles. The term ‘unstructured’ in this context denotes that there are no lanes or traffic rules to follow. Existing MPC formulations for passenger vehicles in structured environments do not readily apply to this context. Thus, a new nonlinear MPC formulation is developed to navigate an AGV from its initial position to a target position at high-speed safely. First, a new cost function formulation is used that aims to find the shortest path to the target position, since no reference trajectory exists in unstructured environments. Second, a region partitioning approach is used in conjunction with a multi-phase optimal control formulation to accommodate the complicated forms the obstacle-free region can assume due to the presence of multiple obstacles in the prediction horizon in an unstructured environment. Third, the no-wheel-lift-off condition, which is the major dynamical safety concern for high-speed, high-CoG AGVs, is ensured by limiting the steering angle within a range obtained offline using a 14 degrees-of-freedom vehicle dynamics model. Thus, a safe, high-speed navigation is enabled in an unstructured environment. Simulations of an AGV approaching multiple obstacles are provided to demonstrate the effectiveness of the algorithm.

A Simulation-Based Traffic Signal Control for Congested Urban Traffic Networks

Traffic congestion in urban networks may lead to strong degradation in the utilization of the network infrastructure, which can be mitigated via suitable control strategies. This paper studies and analyzes the performance of an adaptive traffic-responsive strategy that controls the traffic light parameters in an urban network to reduce traffic congestion. A nearly optimal control formulation is adopted to avoid the curse of dimensionality occurring in the solution of the corresponding Hamilton–Jacobi–Bellman (HJB) optimal control problem. First, an (approximate) solution of the HJB is parametrized via an appropriate Lyapunov function; then, the solution is updated at each iteration in such a way to approach the nearly optimal solution, using a close-to-optimality index and information coming from the simulation model of the network (simulation-based design). Simulation results obtained using a traffic simulation model of the network Chania, Greece, an urban traffic network containing many varieties of junction staging, demonstrate the efficiency of the proposed approach, as compared with alternative traffic strategies based on a simplified linear model of the traffic network. It is shown that the proposed strategy can adapt to different traffic conditions and that low-complexity parametrizations of the optimal solution, a linear and a bimodal piecewise linear strategy, respectively, provide a satisfactory trade-off between computational complexity and network performance.

Relationships Between Maneuver Time and Energy for Reaction Wheel Attitude Control

The dichotomy between minimum time and minimum effort is well known. Minimum-time solutions are synonymous with large effort, whereas minimum effort solutions imply large time horizons. Shortest-time attitude maneuvers are minimum-time slews for agile reorientation of space vehicles. Intuition and experience would suggest that such maneuvers are expensive in terms of effort. This paper will show that this is not the case: Agile maneuvers exist within the energy budget associated with conventional attitude control systems. Moreover, even for conventional slew strategies (such as eigenaxis), energy requirements can be reduced. The energy savings are realized via a reallocation of the control effort by exploiting null motions within the control space, over the maneuver trajectory. A cost functional for minimum-energy slews is developed that is in line with true energy cost associated with reaction wheel-based attitude control systems. This energy metric is incorporated into a family of constrained nonlinear optimal control formulations whose solutions present a relationship between transfer time and energy. Both agile (off-eigenaxis) slews and conventional (eigenaxis) slews are studied. A trade space between transfer time and energy is identified, which can be exploited for mission operations, planning, and design.

Discrete-Time Formulation for Optimal Impact Control in Interaction Tasks

Lightweight manipulators are increasingly involved in industrial scenarios due to their intrinsic safety features allowing to share working space and to cooperate with humans/robots. In particular, interaction tasks are one of their main application. In fact, by imposing a compliant behavior (at software or hardware level) a target interaction can be tracked while ensuring safety during the whole task. Despite the wide range of control strategies developed to face the interaction control problem, the limited control frequency and the measurements noise (especially considering the estimation of end-effector wrenchs from joint side measurements/estimation) are the main limitation in order to achieve improved interaction tracking performance. This paper presents a discrete time formulation for impedance controlled tasks granting a free-overshoot contact force throughout the whole contact phase between the robot and a partially unknown environment, involving finite sampling and force measurements filtering. Moreover, since many applications require the manipulator to approach the not well-known positioned target environment, the proposed algorithm is capable to avoid any force overshoot during the initial contact phase, taking into account non-zero approaching velocities. The main control structure is used in both the free-motion and contact phases, without switching from different control laws, by properly optimizing the control gains solving the defined LQR optimal control problem. A probing task has been carried out in order to validate the control performance with particular attention to the smoothness of the response. Results show the avoidance of force overshoots and instabilities. Moreover, the method has been compared to a continuous time control algorithm, showing improved performance.