Singular Control Research Articles

Advanced Vertical and Lateral Coupling Dynamics Control for a Four In-wheel Motor Drive Electric Vehicle

This paper presents an advanced vertical and lateral coupling dynamics control for stabilizing the vertical and lateral motion of a four in-wheel motor drive electric vehicle. The novelty is to investigate the possibilities of enhancing the lateral stability of electric vehicle through the correct control of their suspension system, and further improve the suspension system performances using direct yaw control (DYC) without having a contradictory effect between the vertical and lateral controllers. There are two main objectives; the first is to improve ride comfort, road holding and vehicle lateral handling performance simultaneously by using a new combined full vehicle suspension system, and the second is to strengthen electric vehicle lateral handling performance by using a sliding mode technique for controlling the direct yaw moment. For vehicle suspension system control, we are seeking to design a skyhook control to control the suspension damper and a sliding mode control to stabilize the position of the sprung masses of the in-wheel damper vibration absorber (DVA). To solve the problems caused by the chattering phenomena and ensure the exponential stability of the sprung mass, the continuous singular terminal sliding-mode controller is selected to design the sliding mode control used as a DVA controller. Simulation results show that the coordinated control between DYC and combined full vehicle suspension system under several turns' maneuvers and under random road excitations, can stabilize electric vehicle lateral motion, enhance the vehicle ride comfort, road holding, enhance the rollover resistance performance and satisfy the main suspension performances simultaneously.

Approximation of <inline-formula><tex-math id="M1">$ N $</tex-math></inline-formula>-player stochastic games with singular controls by mean field games

This paper establishes that a class of $ N $-player stochastic games with singular controls, either of bounded velocity or of finite variation, can both be approximated by mean field games (MFGs) with singular controls of bounded velocity. More specifically, it shows (i) the optimal control to an MFG with singular controls of a bounded velocity $ \theta $ is shown to be an $ \epsilon_N $-NE to an $ N $-player game with singular controls of the bounded velocity, with $ \epsilon_N = O(\frac{1}{\sqrt{N}}) $, and (ii) the optimal control to this MFG is an $ (\epsilon_N + \epsilon_{\theta}) $-NE to an $ N $-player game with singular controls of finite variation, where $ \epsilon_{\theta} $ is an error term that depends on $ \theta $. This work generalizes the classical result on approximation $ N $-player games by MFGs, by allowing for discontinuous controls.

Singular optimal control for rendezvous problem for cooperative vehicle control

In this paper an unmanned aerial vehicle works cooperatively with a unmanned ground vehicle to aid in the automatic control of the ground vehicle. One challenge is when the aerial vehicle has to urgently rendezvous with the ground vehicle. An optimal control problem is formulated for such a case, the key contribution is the singular optimal control rendezvous problem formulation. Illustrative results are utilised to demonstrate this key formulation of singular optimal control solution as well as the optimal rendezvous trajectory analysis.

Irreversible Consumption Habit under Ambiguity: Singular Control and Optimal G-Stopping Time

Irreversible Consumption Habit under Ambiguity: Singular Control and Optimal G-Stopping Time

Time-Fuel-Optimal Navigation of a Commercial Aircraft in Cruise With Heading and Throttle Controls Using Pontryagin’s Maximum Principle

In this research, we consider the commercial aircraft trajectory optimization problem for a general cruise model with arbitrary spatial wind fields to be solved using the Pontryagin’s maximum principle. The model features two fundamental controls, namely “throttle setting” (appearing as a singular control) and “heading angle” (appearing as a regular control). For a problem with state-inequality constraints and minimum time-fuel objective, we show that the optimal “heading angle” can be described through the classic Zermelo’s navigation identity. We also demonstrate, by analyzing the switching function, that the singular “throttle setting” can be characterized through a feedback function that relies on both the optimal states and “heading angle”. The switching-point algorithm is employed to solve a case study where we inspect the optimality conditions and graph the optimal controls together with the optimal state and co-state variables.

Innerization and Singular LQ Control Problem for Linear Discrete-Time Systems

In this paper, we discuss the LQ problem for discrete-time linear systems with singular weight. The interactor with all-pass property is used in order to make a given strictly proper system be proper. In our approach, the type of Riccati equation to be solved is different from the result for a square transfer function matrix case, where the input weight equals to zero. The relation between our approach and the singular weight case is discussed. Unlike the square case, the explicit solution to the problem cannot be obtained and a recursive method to solve the Riccati equation is shown. An application to the finite settling-time control is also discussed.

Optimality conditions of singular controls for systems with Caputo fractional derivatives

<p style='text-indent:20px;'>In this paper, we consider an optimal control problem in which a dynamical system is controlled by a nonlinear Caputo fractional state equation. The problem is investigated in the case when the Pontryagin maximum principle degenerates, that is, it is satisfied trivially. Then the second order optimality conditions are derived for the considered problem.</p>

Managing reputational risk in the decumulation phase of a pension fund

In this paper, we suggest strategies for a pension provider to avoid a loss of reputation due to possible pension reductions in the decumulation phase. In different settings, we determine optimal actions to keep the pension plan solvent, i.e. value of the assets always above the net present value of the pension liabilities. With this in mind, we solve suitable singular control problems. We show that, in expectation, the pension provider can cover the costs of the optimal action via sharing bonus payments with the policyholders.

Optimization Modeling for Advanced Syngas to Olefin Reactive Systems

Reactor designs with mixed catalysts play an important role in transforming a multiple reactor system to single-shot reactors. In addition to savings in capital and ease of implementation, single-shot reactors are useful to break equilibrium limitations, thereby increasing the yield and selectivity of desired product as shown in previous studies. However, the nonlinear and highly exothermic nature of mixed-catalyst systems makes it difficult for commercial process simulation and optimization tools to optimize these systems. This study describes the development and application of optimization strategies for mixed-catalyst, single-shot reactors for syngas to olefin (STO) processes. Finding the optimal catalyst distribution is challenging and requires advanced solution strategies for singular optimal control problems, which are poorly conditioned and often lead to flat response surfaces. The graded bed and partial-moving finite-element approaches are used to find the optimal catalyst distribution that maximizes the olefins yield. A 1.3% increase in the yield is observed from one zone to three zones. The yield further improves from three zones to the exact infinite dimensional solution by 0.2%. This improvement can be realized in practice by changing only the catalyst distribution, without any extra investment. Finally, the results suggest that a mixed-catalyst single shot reactor bed can be applied to other reaction mechanisms to increase reactor performance.

Lipschitz regularity of controls and inversion mapping for a class of smooth extremization problems

In the contest of optimal control problems, regularity results for optima are known when addressing fiber-strictly convex Lagrangian. For infinite time horizons, or for settings with infinite dimensional dynamics, the equivalence between minima/maxima and extremals could break down. Commonly, this is due to a loss of convexity/concavity of the cost functional or to a presence of state constraints, in which further controllability assumptions are needed. For many science applications, this trend is not required, as in energy saving problems. In the present paper, we deal with the set of a functional’s extremals subject to end-point restrictions. We consider an affine control system and a cost functional associated to an autonomous Lagrangian. The dynamics is smooth, satisfying the Lie bracket condition, and the functional is assumed merely Fréchet differentiable. Here we provide a regularity result for controls in the context of constrained extremization problems, under weaker conditions on Lagrangian than the not met classical ones. More precisely, we show a characterization for the Lipschitz regularity of controls associated with the extremal trajectories steering two fixed points, assuming the absence of singular controls. As main application, we construct a locally Lipschitz inversion mapping from the ambient space to the set of constrained extremals.

Singular versus combinatory glucose-sensitive signal control of metabolic sensor protein profiles in hypothalamic astrocyte cultures from each sex.

Brain metabolic-sensory targets for modulatory glucose-sensitive endocrine and neurochemical signals remain unidentified. A hypothalamic astrocyte primary culture model was here used to investigate whether glucocorticoid receptor (GR) and noradrenergic signals regulate astrocyte glucose (glucose transporter-2 [GLUT2], glucokinase) and/or energy (5'-AMP-activated protein kinase [AMPK]) sensor reactivity to glucoprivation by sex. Glucose-supplied astrocytes of each sex showed increased GLUT2 expression after incubation with the GR agonist dexamethasone (DEX) or norepinephrine (NE); DEX plus NE (DEX/NE) augmented GLUT2 in the female, but not in male. Glucoprivation did not alter GLUT2 expression, but eliminated NE regulation of this protein in both sexes. Male and female astrocyte glucokinase profiles were refractory to all drug treatments, but were down-regulated by glucoprivation. Glucoprivation altered AMPK expression in male only, and caused divergent sex-specific changes in activated, i.e., phosphoAMPK (pAMPK) levels. DEX or DEX/NE inhibited (male) or stimulated (female) AMPK and pAMPK proteins in both glucose-supplied and -deprived astrocytes. In male, NE coincidently up-regulated AMPK and inhibited pAMPK profiles in glucose-supplied astrocytes; these effects were abolished by glucoprivation. In female, AMPK profiles were unaffected by NE irrespective of glucose status, whereas pAMPK expression was up-regulated by NE only during glucoprivation. Present outcomes document, for each sex, effects of glucose status on hypothalamic astrocyte glucokinase, AMPK, and pAMPK protein expression and on noradrenergic control of these profiles. Data also show that DEX and NE regulation of GLUT2 is sex-monomorphic, but both stimuli impose divergent sex-specific effects on AMPK and pAMPK. Further effort is warranted to characterize mechanisms responsible for sex-dimorphic GR and noradrenergic governance of hypothalamic astrocyte energy sensory function.

Enhanced moving finite element method based on error geometric estimation for simultaneous trajectory optimization

The direct method has been widely used in various fields of trajectory optimization due to the need to optimize the performance of a controlled dynamic system. An essential issue in the development of the direct method is the selection of the finite element mesh. Although there have been many pieces of research on adaptive mesh refinement, few works focused on finite element layout optimization before the refinement. This paper proposes an enhanced moving finite element method (EMFE), taking the length of each finite element as a variable, based on the geometric estimation of non-collocation-point error (NCPE) and generalized finite element length (GFEL). The features of EMFE include flexible problem reformulation, stable numerical calculation and the ability to locate breakpoints for singular control problems (the control variables appear linearly in the problems) and finding the local minimum numerical error. The theoretical basis of the ability to locate the breakpoints for singular control problems and the feature of NCPE estimate has also been investigated. Three numerical examples, two of which are singular control problems for lunar landing and the other is a multi-phase nonlinear problem for launch vehicles, demonstrated the performance and features of the proposed EMFE method.

A switched system approach for the direct solution of singular optimal control problems

Optimal control problems where the control variable appears linearly in the formulation lead to singular control arcs that introduce analytical and numerical complications during the solution process. In this work, the switching behavior (switching between singular and nonsingular control arcs) is exploited to solve singular optimal control problems using a direct method. The problem is reformulated as a switched system optimal control problem where each control arc defines a different subsystem, i.e., lower and upper limits and a singular arc. The switching between control arcs is handled using binary functions that depend on time. The switched system is embedded into a larger family by relaxing the binary functions to avoid integer variables. The variable switching times at which the control switches between arcs are mapped to fixed points in a new time domain using a time-scaling transformation. A direct transcription method along with control parametrization is applied to transform the embedded problem into a nonlinear programming formulation that contains few variables and constraints. A penalty term is introduced to obtain binary solutions for the optimal selection of control arcs. The proposed reformulation avoids the use of mesh refinement techniques and continuation methods, and allows to solve challenging problems in very short CPU times with high accuracy.

Stationary Discounted and Ergodic Mean Field Games with Singular Controls

We study stationary mean field games with singular controls in which the representative player interacts with a long-time weighted average of the population through a discounted and ergodic performance criteria. This class of games finds natural applications in the context of optimal productivity expansion in dynamic oligopolies. We prove the existence and uniqueness of the mean field equilibria, which are completely characterized through nonlinear equations. Furthermore, we relate the mean field equilibria for the discounted and ergodic games by showing the validity of an Abelian limit. The latter also allows us to approximate Nash equilibria of—so far unexplored—symmetric N-player ergodic singular control games through the mean field equilibrium of the discounted game. Numerical examples finally illustrate in a case study the dependency of the mean field equilibria with respect to the parameters of the games. Funding: The authors acknowledge financial support from the Alan Turing Institute and CMAP, École Polytechnique during this project. Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 317210226 – SFB 1283.

A Singular Perturbation Approach-Based Non-Cascade Sliding Mode Control for Surface-Mounted PMSMs

Motivated by the fact that electrical transients are rather fast compared with mechanical response, the traditional cascade control structure constituted by the inner current and outer speed loops is usually employed in the permanent magnet synchronous motors (PMSMs) servo control community. According to the above-mentioned time-scale characteristic of the PMSMs drive systems, this technique addresses the problems of the non-cascade sliding mode control (SMC) strategy for the surface-mounted PMSMs. Firstly, by appropriately introducing the singular perturbation theory, the corresponding mathematical equations are modeled as a singular perturbation system. Meanwhile, a composite sliding mode surface is constructed based on the Lyapunov equation, such that the system stability can be also guaranteed. Then, according to the exponential reaching law, a standard non-cascade SMC law is designed. Furthermore, an optimal nonlinear function-based tracking differentiator (TD) is presented to smooth the reference velocity value, while providing differential signals. As a result, a novel TD-based SMC strategy is synthesized by incorporating a nonlinear function, thus improving the inherent chattering phenomenon. Finally, a surface-mounted PMSM servo system is performed to illustrate the advantages and effectiveness of the proposed approaches. The main contribution of this paper is to present an alternative non-cascade SMC framework based on the singular perturbation approach, which provides a novel control structure for a PMSM speed regulation system.

Competition between populations: preventing domination of resistant population using optimal control

Emergence of pesticide-resistant strains of pests, antibiotic-resistant bacteria or chemotherapy-resistant tumor cells is a major obstacle standing in the way of effective external control of undesirable growth of these populations. Once the resistant subpopulation becomes dominant the means to control the population are effectively lost. Prevention of resistance may be therefore considered one of the optimal control goals to ensure long-term controllability.In this paper we formulate nonlinear optimal control problem motivated by the question of controlling the growth of two coexisting subpopulations representing two phenotypes of the same species. However, the model we consider is a standard competition model with populations following logistic growth, and therefore it is more general and can be used also when two competing populations are taken into account. We consider a situation when the control (e.g. pesticide or drug) has effect only on the specimen belonging to one (sensitive) population while having no effect on the other (resistant) population. We use a non-standard, nonlinear objective functional in which, aside from penalizing the overall size of both populations, we include an additional, nonlinear term to ensure that the control-sensitive population remains the dominant one.It is shown how the resistance penalty gives rise to locally-optimal singular controls which are key to achieving the balance between resistant and sensitive populations whenever the subpopulations are similar in terms of the model parameters. Analytic properties of the singular arcs and the structure of optimal control are investigated. Both theoretical and numerical results strongly suggest that the optimal control contains a singular interval.

Active control of dielectric singularities in indium-tin-oxides hyperbolic metamaterials

Dielectric singularities (DSs) constitute one of the most exotic features occurring in the effective permittivity of artificial multilayers called hyperbolic metamaterials (HMMs). Associated to DSs, a rich phenomenology arises that justifies the ever-increasing interest profuse by the photonic community in achieving an active control of their properties. As an example, the possibility to “canalize” light down to the nanoscale as well as the capability of HMMs to interact with quantum emitters, placed in their proximity, enhancing their emission rate (Purcell effect), are worth mentioning. HMMs, however, suffer of an intrinsic lack of tunability of its DSs. Several architectures have been proposed to overcome this limit and, among them, the use of graphene outstands. Graphene-based HMMs recently shown outstanding canalization capabilities achieving λ/1660 light collimation. Despite the exceptional performances promised by these structures, stacking graphene/oxide multilayers is still an experimental challenge, especially envisioning electrical gating of all the graphene layers. In this paper, we propose a valid alternative in which indium-tin-oxide (ITO) is used as an electrically tunable metal. Here we have numerically designed and analyzed an ITO/SiO2 based HMM with a tunable canalization wavelength within the range between 1.57 and 2.74 μm. The structure feature light confinement of λ/8.8 (resolution of about 178 nm), self-focusing of the light down to 0.26 μm and Purcell factor of approximately 700. The proposed HMM nanoarchitecture could be potentially used in many applications, such as ultra-fast signal processing, high harmonic generation, lab-on-a-chip nanodevices, bulk plasmonic waveguides in integrated photonic circuits and laser diode collimators.

To the Problem of Motion Cueing Simulation on a Robotic Stand for Aircraft Flight

Various simulators with motion cueing simulation stands, which make it possible to create an acceleration environment for the pilot that is close to a real flight, are used for training aircraft pilots. The article considers the formulation of the motion cueing simulation on a stand based on an industrial manipulator. Motion cueing simulation algorithms include two phases: motion cueing simulation phase and phase of return to the working area center. During simulation phase the stand must implement such a movement that the angular accelerations acting on the person and the overload vector acting on the center of mass of the operator completely coincide with the real ones. If it is not possible then just the directions of these vectors should coincide. During the second phase the stand end point must return to the working area center with acceleration values below the threshold, but in the fastest way. This task can be presented as a generalization of the brachistochrone problem. The article considers the problem of the material point motion in a uniform gravity field along a curve located in a vertical plane, in the presence of restrictions on the trajectory curvature. It is necessary to choose the curve shape in such a way that the descent time is minimal. The problem solution is obtained by optimal control methods, the cases of regular and singular control realization are considered, the question of its conjugation. Also, the switching number between sections of regular and singular control is studied.

Mean-field games of finite-fuel capacity expansion with singular controls

We study Nash equilibria for a sequence of symmetric N-player stochastic games of finite-fuel capacity expansion with singular controls and their mean-field game (MFG) counterpart. We construct a solution of the MFG via a simple iterative scheme that produces an optimal control in terms of a Skorokhod reflection at a (state-dependent) surface that splits the state space into action and inaction regions. We then show that a solution of the MFG of capacity expansion induces approximate Nash equilibria for the N-player games with approximation error ε going to zero as N tends to infinity. Our analysis relies entirely on probabilistic methods and extends the well-known connection between singular stochastic control and optimal stopping to a mean-field framework.

A Note on Asymptotics Between Singular and Constrained Control Problems of One-Dimensional Diffusions

We study the asymptotic relations between certain singular and constrained control problems for one-dimensional diffusions with both discounted and ergodic objectives. In the constrained control problems the controlling is allowed only at independent Poisson arrival times. We show that when the underlying diffusion is recurrent, the solutions of the discounted problems converge in Abelian sense to those of their ergodic counterparts. Moreover, we show that the solutions of the constrained problems converge to those of their singular counterparts when the Poisson rate tends to infinity. We illustrate the results with drifted Brownian motion and Ornstein-Uhlenbeck process.