Optimal Control Formulation Research Articles

Automating the design of tokamak experiment scenarios

The real-time control of plasma position, shape and current in a tokamak has to be ensured by a number of electrical circuits consisting of voltage suppliers and axisymmetric coils. Finding good target voltages/currents for the control systems is a very laborious, non-trivial task due to non-linear effects of plasma evolution. We introduce here an optimal control formulation to tackle this task and present in detail the main ingredients for finding numerical solutions: the finite element discretization, accurate linearizations and Sequential Quadratic Programming. Case studies for the tokamaks WEST and HL-2M highlight the flexibility and broad scope of the proposed optimal control formulation.

Lane Change Intention Awareness for Assisted and Automated Driving on Highways

Today the automotive industry faces a robust trend toward assisted and automated driving. The technology to accomplish this ambition has evolved rapidly over the last few years, and yet there are still a lot of algorithmical challenges left to make an automation of the driving task a safe and comfortable experience. One of the main remaining challenges is the comprehension of the current traffic situation and the anticipation of all traffic participants’ future driving behavior, which is needed for the technical system to obtain situation awareness: an indispensable foundation for successful decision-making. In this paper, a prediction framework is presented that is able to infer a driver's maneuver intention. This is achieved via a hybrid Bayesian network whose hidden layers represent a driver's lane contentedness. A pre-training of the network's parameters with simulated data provides for human interpretable parameters even after running the expectation maximization algorithm based on data gathered on German highways. Moreover, the future driving path of any traffic participant is predicted by solving an optimal control problem, whereby the parameters of the optimal control formulation are found via inverse reinforcement learning.

The piecewise parametric optimal control of uncertain linear quadratic models

ABSTRACTThe optimal control of linear quadratic model is given in a feedback form and determined by the solution of a Riccati equation. However, the control-related Riccati equation usually cannot be solved analytically such that the form of optimal control will become more complex. In this paper, we consider a piecewise parametric optimal control problem of uncertain linear quadratic model for simplifying the form of optimal control. By introducing a piecewise control parameter, a piecewise parametric optimal control model is established. Then we present a parametric optimisation method for solving the optimal piecewise control parameter. Finally, an uncertain inventory-promotion optimal control problem is discussed and a comparison is made to show the effectiveness of proposed piecewise parametric optimal control model.

Structured Decentralized Control of Positive Systems With Applications to Combination Drug Therapy and Leader Selection in Directed Networks

We study a class of structured optimal control problems in which the main diagonal of the dynamic matrix is a linear function of the design variable. While such problems are in general challenging and nonconvex, for positive systems we prove convexity of the $H_2$ and $H_\infty$ optimal control formulations which allow for arbitrary convex constraints and regularization of the control input. Moreover, we establish differentiability of the $H_\infty$ norm when the graph associated with the dynamical generator is weakly connected and develop a customized algorithm for computing the optimal solution even in the absence of differentiability. We apply our results to the problems of leader selection in directed consensus networks and combination drug therapy for HIV treatment. In the context of leader selection, we address the combinatorial challenge by deriving upper and lower bounds on optimal performance. For combination drug therapy, we develop a customized subgradient method for efficient treatment of diseases whose mutation patterns are not connected.

Inverse optimal control of a class of affine nonlinear systems

This paper proposes a systematic formulation of inverse optimal control (IOC) law based on a rather straightforward reduction of control Lyapunov function (CLF), applicable to a class of second-order nonlinear systems affine in the input. This method exploits the additional design degrees of freedom resulting from the non-uniqueness of the state dependent coefficient (SDC) formulation, which is widely used in pseudo-linear control techniques. The applicability of the proposed approach necessitates an apparently effortless SDC formulation satisfying an SDC matrix criterion in terms of the structure and characteristics of the state matrix, [Formula: see text]. Subsequently, a sufficient condition for the global asymptotic stability (g.a.s) of the closed-loop system is established. The SDC formulations conforming to the sufficient condition ensure the existence and determination of a smooth radially unbounded polynomial CLF of the form [Formula: see text], while offering a benevolent choice for the gain matrix [Formula: see text], in the CLF. The direct relationship between the gain matrix [Formula: see text] and state weighing matrix [Formula: see text] ensures optimization of an equivalent [Formula: see text]. This feature enables one to rightfully choose the gain matrix [Formula: see text] as per the performance requisites of the system. Finally, the application of the proposed methodology for the speed control of a permanent magnet synchronous motor validates the efficacy and design flexibility of the methodology.

Event-triggered robust cooperative stabilization in nonlinearly interconnected multiagent systems

By way of the graph theoretic tools, we model closed-loop large-scale systems as two-layer multiagent systems with the agent layer describing nonlinearly interconnected dynamics and the control layer designed to stabilize the system by communications in agents’ neighborhoods. We consider a scenario in which the effect of dynamical interactions over the agent layer appears through some nonlinear functions which are in the range space of the control input matrix, yet the underlying agent layer interconnection topology and the associated nonlinearities are not known to the control layer designer. We develop a static linear cooperative algorithm and, having endowed each local controller with several buffers, we schedule the control layer information exchange events in a non-periodic non-synchronous manner. We combine graph theoretic and event triggering ideas with an optimal control formulation, and propose a design procedure to obtain the required robust control gain. We prove that all trajectories of the closed-loop multiagent system will exponentially converge to the origin in the presence of time-varying nonlinearly interconnected modeling uncertainties over agent layer. For the same cooperative stabilization algorithm, trading off with the exponential convergence to a neighborhood around the origin, we further show that a practical event triggering mechanism can be developed to better deal with the effect of modeling mismatch on the number of triggering events. We also prove the proposed event triggering mechanisms do not exhibit Zeno behavior to ensure the feasibility of the proposed ideas in the sense of reduced communication load and, thereby, energy consumption.

Geometric control with model predictive static programming on [formula omitted

This paper proposes a geometric attitude control approach with the model predictive static programming (MPSP) for a rigid body modeled on the Lie group SO(3). The modeling scheme can avoid the defects of the singularity associated with Euler angles and the unwinding phenomenon associated with quaternion, and the MPSP technique features a high computational efficiency and is capable of achieving hard terminal constraints. In this paper, the MPSP is extended to the geometric control on SO(3), using the variational principle, the left-invariance principle of SO(3) and the topology structure of the Lie algebra space so(3). The desired attitude is considered as the hard terminal constraint and the energy consumption is optimized in the optimal control formulation of the MPSP. Moreover, the system disturbance and the measurement error of the states are taken into account in the online application of the MPSP. Simulation results of a spacecraft attitude control illustrate the effectiveness of the proposed method.

Passive motion and active inference: Commentary on “Muscleless motor synergies and actions without movements: From motor neuroscience to cognitive robotics” by Vishwanathan Mohan, Ajaz Bhat and Pietro Morasso

Mohan et al offer an exhaustive and compelling survey of different paradigms for understanding an embodied exchange with the world. They describe a new way of looking at motor control that has emerged over the past years, – a view that resolves many issues with previous (optimal motor control) formulations. Pleasingly, this new way of looking at things picks up on historical themes that date back to the 19th century. For example, notions of synergy formation and the equilibrium point hypothesis, re-emerge with a new simplicity and explanatory power, when one takes a holistic and enactive view of embodied behaviour. Indeed, one might trace the ideas explored in Mohan et al back to foundational notions enshrined in ideomotor theory and Helmholtz's notion of unconscious inference. This new way of looking at things is centred on the notion of a forward or generative model that plays the functional role of a “plastic configurable internal representation of the body (body schema) as a critical link enabling the seamless continuum between motor control and imagery.” Mohan et al.

Multiobjective Optimal Formulations for Bus Fleet Size of Public Transit under Headway-Based Holding Control

In recent years, with the development of advanced technologies for data collection, real-time bus control strategies have been implemented to improve the daily operation of transit systems, especially headway-based holding control which is a proven strategy to reduce bus bunching and improve service reliability for high-frequency bus routes, with the concept of regulating headways between successive buses. This hot topic has inspired the reconsideration of the traditional issue of fleet size optimization and the integrated bus holding control strategy. The traditional headway-based control method only focused on the regulation of bus headways, without considering the number of buses on the route. The number of buses is usually assumed as a given in advance and the task of the control method is to regulate the headways between successive buses. They did not consider the bus fleet size problem integrated with headway-based holding control method. Therefore, this work has presented a set of optimal control formulations to minimize the costs for the passengers and the bus company through calculating the optimal number of buses and the dynamic holding time, taking into account the randomness of passenger arrivals. A set of equations were formulated to obtain the operation of the buses with headway-based holding control or the schedule-based control method. The objective was to minimize the total cost for the passengers and the bus company in the system, and a Monte Carlo simulation based solution method was subsequently designed to solve the optimization model. The effects of this optimization method were tested under different operational settings. A comparison of the total costs was conducted between the headway-based holding control and the schedule-based holding control. It was found that the model was capable of reducing the costs of the bus company and passengers through utilizing headway-based bus holding control combined with optimization of the bus fleet size. The proposed optimization model could minimize the number of buses on the route for a guaranteed service level, alleviating the problem of redundant bus fleet sizes caused by bus bunching in the traditional schedule-based control method.

A Condition for Singular Optimal Control Formulation for a Tandem Pair of Gravity Fed Linear DC Machine Powered Mine Locomotives

Abstract The key focus in this study is on gravity fed linear DC machines used to drive mine locomotives. A case for optimal control singularity analysis for this tandem pair of gravity fed linear DC machines on a slope is thus presented. The key operational feature is for a generator locomotive to drive a motor locomotive with gravity providing the machine disparity for generator and motor action. Optimal control formulation is extended in this study, particularly optimal control singularity formulation. The optimal controller should be able to minimize the relative displacement between the machines. Mine locomotive dynamics results are also presented to show that the differential velocity and displacement are minimized. The performance index and Lagrangian are also minimized. The optimal control formulation is demonstrated via switching function differential calculus. The solution is shown to improve mine locomotive energy performance.

Solvability of optimization problem for the oscillation processes with optimal vector controls

The optimal control problem is investigated for oscillation processes, described by integrodifferential equations with the Fredholm operator when functions of external and boundary sources nonlinearly depend on components of optimal vector controls. Optimality conditions having specific properties in the case of vector controls were found. A sufficient condition is established for unique solvability of the nonlinear optimization problem and its complete solution is constructed in the form of optimal control, an optimal process, and a minimum value of the functional.

CLAIRE: A DISTRIBUTED-MEMORY SOLVER FOR CONSTRAINED LARGE DEFORMATION DIFFEOMORPHIC IMAGE REGISTRATION.

With this work we release CLAIRE, a distributed-memory implementation of an effective solver for constrained large deformation diifeomorphic image registration problems in three dimensions. We consider an optimal control formulation. We invert for a stationary velocity field that parameterizes the deformation map. Our solver is based on a globalized, preconditioned, inexact reduced space Gauss‒Newton‒Krylov scheme. We exploit state-of-the-art techniques in scientific computing to develop an eifective solver that scales to thousands of distributed memory nodes on high-end clusters. We present the formulation, discuss algorithmic features, describe the software package, and introduce an improved preconditioner for the reduced space Hessian to speed up the convergence of our solver. We test registration performance on synthetic and real data. We Demonstrate registration accuracy on several neuroimaging datasets. We compare the performance of our scheme against diiferent flavors of the Demons algorithm for diifeomorphic image registration. We study convergence of our preconditioner and our overall algorithm. We report scalability results on state-of-the-art supercomputing platforms. We Demonstrate that we can solve registration problems for clinically relevant data sizes in two to four minutes on a standard compute node with 20 cores, attaining excellent data fidelity. With the present work we achieve a speedup of (on average) 5× with a peak performance of up to 17× compared to our former work.

Flatness Based Optimal Control for Induction Machine Drives

Aim of this study is to derive an efficient method and problem formulation for optimal control of induction machines in variable speed drives, as used for example in electric vehicles. Induction machine dynamics are thus represented as a differential flat system and voltage source inverter losses are considered as an average value model. Based on the canonical form of the equivalent flat system, a nonlinear discrete finite horizon optimal control problem with a predefined torque and speed profile is introduced. If it is required to find a global solution, this problem can be efficiently solved by discrete dynamic programming or, when time is of essence, by means of sequential quadratic programming. The former is applied to a practical electric vehicle load cycle with limiting voltage constraints. The solution demonstrates how optimal control methods may be beneficial to drive design.

Preconditioners and Tensor Product Solvers for Optimal Control Problems from Chemotaxis

In this paper, we consider the fast numerical solution of an optimal control formulation of the Keller--Segel model for bacterial chemotaxis. Upon discretization, this problem requires the solution of huge-scale saddle point systems to guarantee accurate solutions. We consider the derivation of effective preconditioners for these matrix systems, which may be embedded within suitable iterative methods to accelerate their convergence. We also construct low-rank tensor-train techniques which enable us to present efficient and feasible algorithms for problems that are finely discretized in the space and time variables. Numerical results demonstrate that the number of preconditioned GMRES iterations depends mildly on the model parameters. Moreover, the low-rank solver makes the computing time and memory costs sublinear in the original problem size.

Initiation versus Cessation Control Policies: Deriving Optimal Resource Allocation Strategies to Decrease Smoking Prevalence Under a Fixed Budget.

Background. Over several decades the tobacco control community has recommended and implemented smoking initiation and cessation interventions to reduce the smoking toll. It is necessary to study the combined effect of these interventions to allocate resources optimally. However, there is a paucity of studies that address the right combination of initiation and cessation policies over time to reduce smoking prevalence. Objective. To derive optimal trajectories of initiation and cessation interventions that minimize overall smoking prevalence over a specified period while satisfying a budget constraint. Methods. Using an established dynamic model of smoking prevalence, we employ an optimal control formulation to minimize overall smoking prevalence within a specified time period. The budget constraint is handled through an iterative application of a penalty function on above-budget expenditures. We further derive the optimal cost ratio of initiation versus cessation programs over time. To parameterize our model, we use results from two empirical interventions. The demographic data are from the National Health Interview Survey in the United States. Results. For our example, our results show that the optimal cost ratio (initiation over cessation) starts around 2.02 and gradually increases to 5.28 in 30 years. Smoking prevalence decreases significantly compared with the status quo, 8.54% in 30 years with no interventions versus the estimated 6.43% with interventions. In addition, the optimal units of initiation and cessation interventions increase over time. Conclusions. Our model provides a general framework to incorporate policy details in determining the optimal mix of smoking interventions.

Propagation of two independent sources of uncertainty in the electrocardiography imaging inverse solution

This study investigates the effects of the input parameter uncertainties (organ conductivities, boundary data, etc.) on the electrocardiography (ECG) imaging problem. These inputs are very important for the construction of the torso potential for the forward problem and for the non-invasive electrical potential on the heart surface in the case of the inverse problem. We propose a new stochastic formulation that allows us to combine both sources of errors. We formulate the forward and inverse stochastic problems by considering the input parameters as random fields and a stochastic optimal control formulation. In order to quantify multiple independent sources of uncertainties on the forward and inverse solutions, we attribute suitable probability density functions for each randomness source and apply stochastic finite elements based on generalized polynomial chaos method. The efficiency of this approach to solve the forward and inverse ECG problems and the usability to quantify the effect of organ conductivity and epicardial boundary data uncertainties in the torso are demonstrated by a number of numerical simulations on a two-dimensional computational mesh of a realistic torso geometry.

A mean-field optimal control formulation of deep learning

Recent work linking deep neural networks and dynamical systems opened up new avenues to analyze deep learning. In particular, it is observed that new insights can be obtained by recasting deep learning as an optimal control problem on difference or differential equations. However, the mathematical aspects of such a formulation have not been systematically explored. This paper introduces the mathematical formulation of the population risk minimization problem in deep learning as a mean-field optimal control problem. Mirroring the development of classical optimal control, we state and prove optimality conditions of both the Hamilton-Jacobi-Bellman type and the Pontryagin type. These mean-field results reflect the probabilistic nature of the learning problem. In addition, by appealing to the mean-field Pontryagin's maximum principle, we establish some quantitative relationships between population and empirical learning problems. This serves to establish a mathematical foundation for investigating the algorithmic and theoretical connections between optimal control and deep learning.

A memory-distributed quasi-Newton solver for nonlinear programming problems with a small number of general constraints

We address the problem of parallelizing state-of-the-art nonlinear programming optimization algorithms. In particular, we focus on parallelizing quasi-Newton interior-point methods that use limited-memory secant Hessian approximations. Such interior-point methods are known to have better convergence properties and to be more effective on large-scale problems than gradient-based and derivative-free optimization algorithms. We target nonlinear and potentially nonconvex optimization problems with an arbitrary number of bound constraints and a small number of general equality and inequality constraints on the optimization variables. These problems occur for example in the form of optimal control, optimal design, and inverse problems governed by ordinary or partial differential equations, whenever they are expressed in a “reduced-space” optimization approach. We introduce and analyze the time and space complexity of a decomposition method for solving the quasi-Newton linear systems that leverages the fact that the quasi-Newton Hessian matrix has a small number of dense blocks that border a low-rank update of a diagonal matrix. This enables an efficient parallelization on memory-distributed computers of the iterations of the optimization algorithm, a state-of-the-art filter line-search interior-point algorithm by Wächter et. al. We illustrate the efficiency of the proposed method by solving structural topology optimization problems on up to 4608 cores on a parallel machine.

Formulation and interpretation of optimal braking and steering patterns towards autonomous safety-critical manoeuvres

ABSTRACTStability control of a vehicle in autonomous safety-critical at-the-limit manoeuvres is analysed from the perspective of lane keeping or lane changing, rather than that of yaw control as in traditional ESC systems. An optimal control formulation is developed, where the optimisation criterion is a linear combination of the initial and final velocity of the manoeuvre. Varying the interpolation parameter in this formulation turns out to result in an interesting family of optimal braking and steering patterns in stabilising manoeuvres. The two different strategies of optimal lane-keeping control and optimal yaw control are shown to be embedded in the formulation and result from the boundary values of the parameter. The results provide new insights and have the potential to be used for future safety systems that adapt the level of braking to the situation at hand, which is demonstrated through examples of how to exploit theresults.

Energy-maximising control of wave energy converters using a moment-domain representation

Wave Energy Converters (WECs) have to be controlled to ensure maximum energy extraction from waves while considering, at the same time, physical constraints on the motion of the real device and actuator characteristics. Since the control objective for WECs deviates significantly from the traditional reference “tracking” problem in classical control, the specification of an optimal control law, that optimises energy absorption under different sea-states, is non-trivial. Different approaches based on optimal control methodologies have been proposed for this energy-maximising objective, with considerable diversity on the optimisation formulation. Recently, a novel mathematical tool to compute the steady-state response of a system has been proposed: the moment-based phasor transform. This mathematical framework is inspired by the theory of model reduction by moment-matching and considers both continuous and discontinuous inputs, depicting an efficient and closed-form method to compute such a steady-state behaviour. This study approaches the design of an energy-maximising optimal controller for a single WEC device by employing the moment-based phasor transform, describing a pioneering application of this novel moment-matching mathematical scheme to an optimal control problem. Under this framework, the energy-maximising optimal control formulation is shown to be a strictly concave quadratic program, allowing the application of well-known efficient real-time algorithms.