Hybrid Optimal Control Research Articles

Discrete-time hybrid control in Borel spaces: Average cost optimality criterion

This paper addresses an optimal hybrid control problem in discrete-time with Borel state and action spaces. By hybrid we mean that the evolution of the state of the system may undergo deep changes according to structural modifications of the dynamic. Such modifications occur either by the position of the state or by means of the controller's actions. The optimality criterion is of a long-run ratio-average (or ratio-ergodic) type. We provide the existence of optimal average policies for this hybrid control problem by analyzing an associated dynamic programming equation. We also show that this problem can be translated into a standard (or non-hybrid) optimal control problem with cost constraints. Besides, we show that our model includes some special and important families of control problems, such as those with an impulsive or switching mode. Finally, to illustrate our results, we provide an example on a pollution-accumulation problem.

Hybrid Optimal Control under Mode Switching Constraints with Applications to Pesticide Scheduling

This paper concerns optimal mode-scheduling in autonomous switched-mode hybrid dynamical systems, where the objective is to minimize a cost-performance functional defined on the state trajectory as a function of the schedule of modes. The controlled variable, namely the modes’ schedule, consists of the sequence of modes and the switchover times between them. We propose a gradient-descent algorithm that adjusts a given mode-schedule by changing multiple modes over time-sets of positive Lebesgue measures, thereby avoiding the inefficiencies inherent in existing techniques that change the modes one at a time. The algorithm is based on steepest descent with Armijo step sizes along Gâteaux differentials of the performance functional with respect to schedule-variations, which yields effective descent at each iteration. Since the space of mode-schedules is infinite dimensional and incomplete, the algorithm’s convergence is proved in the sense of Polak’s framework of optimality functions and minimizing sequences. Simulation results are presented, and possible extensions to problems with dwell-time lower-bound constraints are discussed.

Occupation measure methods for modelling and analysis of biological hybrid systems

Mechanistic models in biology often involve numerous parameters about which we do not have direct experimental information. The traditional approach is to fit these parameters using extensive numerical simulations (e.g. by the Monte-Carlo method), and eventually revising the model if the predictions do not correspond to the actual measurements. In this work we propose a methodology for hybrid system model revision, when new types of functions are needed to capture time varying parameters. To this end, we formulate a hybrid optimal control problem with intermediate points as successive infinite-dimensional linear programs (LP) on occupation measures. Then, these infinite-dimensional LPs are solved using a hierarchy of semidefinite relaxations. The whole procedure is applied on a recent model for haemoglobin production in erythrocytes.

Hybrid optimal control problems for a class of semilinear parabolic equations

A class of optimal control problems of hybrid nature governed by semilinear parabolic equations is considered. These problems involve the optimization of switching times at which the dynamics, the integral cost, and the bounds on the control may change. First- and second-order optimality conditions are derived. The analysis is based on a reformulation involving a judiciously chosen transformation of the time domains. For autonomous systems and a time-independent integral cost, we prove that the Hamiltonian is constant in time when evaluated along the optimal controls and trajectories. A numerical example is provided.

A semi-Lagrangian algorithm in policy space for hybrid optimal control problems

The mathematical framework of hybrid system is a recent and general tool to treat control systems involving control action of heterogeneous nature. In this paper, we construct and test a semi-Lagrangian numerical scheme for solving the Dynamic Programming equation of an infinite horizon optimal control problem for hybrid systems. In order to speed up convergence, we also propose and analyze an acceleration technique based on policy iteration. Finally, we validate the approach via some numerical tests in low dimension.

On the Relation Between the Minimum Principle and Dynamic Programming for Classical and Hybrid Control Systems

Hybrid optimal control problems are studied for a general class of hybrid systems where autonomous and controlled state jumps are allowed at the switching instants and in addition to terminal and running costs switching between discrete states incurs costs. The statements of the Hybrid Minimum Principle and Hybrid Dynamic Programming are presented in this framework and it is shown that under certain assumptions the adjoint process in the Hybrid Minimum Principle and the gradient of the value function in Hybrid Dynamic Programming are governed by the same set of differential equations and have the same boundary conditions and hence are almost everywhere identical to each other along optimal trajectories. Analytic examples are provided to illustrate the results and, in particular, a Riccati formalism for linear quadratic hybrid tracking problems is presented.

A Class of Linear Quadratic Gaussian Hybrid Optimal Control Problems with Realization–Independent Riccati Equations

A class of stochastic linear quadratic hybrid optimal control problems is presented for which the Hamiltonian boundary conditions appearing in the associated necessary optimality conditions of the Stochastic Hybrid Minimum Principle and Stochastic Hybrid Dynamic Programming are path-independent. Consequently, the linear quadratic Gaussian regulation problem associated with this class of stochastic hybrid optimal control problems can be solved via (stochastic) Riccati equations which are independent of the realization of stochastic diffusion terms. An analytic example of a scalar hybrid system is provided to illustrate the results, and the relation to the deterministic case is discussed.

On solving hybrid optimal control problems with higher index DAEs

The paper deals with hybrid optimal control problems described by higher index differential–algebraic equations (DAEs). We introduce a numerical procedure for solving these problems. The procedure has the following features: it is based on the appropriately defined adjoint equations formulated for the discretized equations being the result of the numerical integration of systems equations by an implicit Runge–Kutta method; the consistent initialization procedure is applied whenever control functions jumps, or state variables transition occurs. The procedure can cope with hybrid optimal control problems which are defined by DAEs with the index not exceeding three. Our approach does not require differentiation of some system equations in order to transform higher index DAEs to the underlying ordinary differential equations (ODEs). The presented numerical examples show that the proposed approach can be used to solve efficiently hybrid optimal control problems with higher index DAEs.

Optimal fractional order PI control applicability for enhanced dynamic behavior of on-grid solar PV systems

This paper aims at finding the optimal gain scheduling of the conventional Proportional Integral (PI) and the Fractional Order (FOPI) through the use of the advanced meta-heuristic optimization techniques in PV-grid connected systems. Therefore, the hybridization of the optimization techniques with both PI and FOPI controllers is proposed for attaining an optimal dynamic performance through a specific criterion. This latter is a function of different response characteristics such as the settling time, maximum overshoot, steady state error and execution time. Based on the proposed criterion, the meta-heuristic technique that provides the best enhanced dynamic performance can be simply identified. For this purpose, a comprehensive state of art of the on-grid PV system performance dilemma is initially discussed. The problem of designing current and voltage regulators for the PV-grid connected system is addressed, seeking for an adequate dynamic behavior. Then, different metaheuristic optimization techniques such as Artificial Bee Colony (ABC), Cuckoo Search (CS), Grey Wolf Optimizer (GWO), Mine Blast Algorithm (MBA), Whale Optimization Algorithm (WOA) and Moth Swarm Algorithm (MSA) are employed for determining the optimal PI/FOPI controller gains, forming a hybrid optimal FOPI or PI control. From the numerical simulation analysis of Matlab™/Simulink results, the significant dynamic behavior enhancement of the overall on-grid PV system is illustrated using PI/FOPI controllers optimally tuned with the proposed meta-heuristic algorithms considering solar irradiation variation conditions. The impact of the optimized PI and FOPI controllers on the overall system dynamic responses is explicitly analyzed while using different meta-heuristic algorithms through qualitative and quantitative comparisons.

Hybrid Optimal Tracking Control by Most Probable Trajectory Approach

In this paper, we study optimal tracking control problems for linear continuous-time hybrid systems. We adopt most probable trajectory approach to control system states and mode distributions simultaneously. We consider optimization problems for averaged systems and averaged performance indices throughout mode distributions.

Nonlinear hybrid control design of three-joint dual finger robots

The difficulty of finger robot grasping control designs is the inevitable coupling in the dynamics of finger joints and finger tips because this dynamics coupling effect lets the overall motion of the finger robots to be constrained by the object states when they are controlled to manipulate an object’s postures. Besides, from the practical implementation point of view, a stable and easy-to-implement control structure is also an important task for this topic. For solving these two design issues, an optimal hybrid control design which combines feedback linearization and nonlinear H2 control concepts for grasping design of three-joint dual finger robots is investigated in this study. This investigation makes two main contributions to the finger robot grasping control design: (1) an effective and acceptable control performance for finger robot grasping control designs under the effect of dynamics coupling is delivered and (2) a really simple nonlinear optimal control scheme is obtained.

Optimal Scheduling of GEO On-orbit Refuelling with Uncertain Object Satellites

The mission planning of Geostationary Earth Orbit (GEO) on-orbit refuelling with uncertain object satellites is studied in this paper. Three key problems should be resolved here: Object Satellite (OSc) selection, mission sequence planning and transfer trajectory optimization. The optimization goal is to select the OSc and find the optimal servicing sequence with the best optimization objective value. Three optimization objectives, i.e. fuel cost, the total number of the OScs and the sum of the OSc priority value are taken into account. Considering this mission as a hybrid optimal control problem, a mathematical model is proposed. Multi-objective particle swarm optimization is employed to address the model. Several mission scenarios are compared and discussed. Numerical experiments indicate that 1) the method developed in this paper could address the problem efficiently; and 2) OSc distribution and varieties in optimization objective emphasis have considerable influences on the optimization results.

Free chattering hybrid sliding mode control for a class of non‐linear systems: electric vehicles as a case study

In current study, in order to find the control of general uncertain nonlinear systems, a new optimal hybrid control approach called Optimal General Type II Fuzzy Sliding Mode (OGT2FSM) is presented. In order to estimate unknown nonlinear activities in monitoring dynamic uncertainties, the benefits of general type-2 fuzzy logic systems (GT2FLS) is utilized by the suggested controller. The suggested controller not only warranties the constancy and hardiness against uncertainties of the lumped resulted by external disturbances and un-modeled dynamics, but also considerably decreases the control chattering inherent in traditional sliding mode control. Moreover, the Modified Backtracking Search Algorithm (MBSA), which is a new heuristic algorithm, is used in the offered controller. A comparison is performed between the results of current study and the results of some of the most recent studies on the same topic, which are an Adaptive Interval Type-2 Fuzzy Logic Controller (AGT2FLC) and Conventional Sliding Mode Controller (CSMC), to assess the efficiency of the suggested controller. The suggested control scheme is finally used to the Electric Vehicles type as a case study. Results of simulation confirm the good performance the suggested controller.

Automated Solution of the Low-Thrust Interplanetary Trajectory Problem

Preliminary design of low-thrust interplanetary missions is a highly complex process. The mission designer must choose discrete parameters, such as the number of flybys, the bodies at which those flybys are performed, and in some cases, the final destination. In addition, a time history of control variables must be chosen that defines the trajectory. There are often many thousands, if not millions, of possible trajectories to be evaluated, which can be a very expensive process in terms of the number of human analyst hours required. An automated approach is therefore very desirable. This work presents such an approach by posing the mission design problem as a hybrid optimal control problem. The method is demonstrated on hypothetical missions to Mercury, the main asteroid belt, and Pluto.

Hybrid optimal control of an electric vehicle with a dual-planetary transmission

A hybrid systems framework is presented for the analysis and optimal control of an electric vehicle equipped with a seamless dual stage planetary transmission. A feature of special interest is that, due to the perpetual connectedness of the motor to the wheels via the seamless transmission, the mechanical degree of freedom changes during the transition period. These circumstances where autonomous and controlled state jumps at the switching instants are accompanied by changes in the dimension of the state space are reflected in the definition of hybrid systems and the corresponding statement of the Hybrid Minimum Principle (HMP). Furthermore, the state-dependent motor torque constraints which impose mixed input-state constraints are converted to state-independent input constraints via a change of variables and the introduction of auxiliary discrete states. Optimal control problems for the minimization of acceleration duration and the minimization of energy consumption for the acceleration task are formulated within the presented framework and simulation results are presented for the optimal control inputs and the optimal gear changing instants for reaching the speed of 100km/hr from the stationary initial condition. A phenomenon of note that appears in the dynamical evolution of the vehicle is the presence of power regeneration as a part of the acceleration task for the minimization of the energy consumption.

Real-Time Dynamic-Mode Scheduling Using Single-Integration Hybrid Optimization

This paper considers the problem of real-time mode scheduling in linear time-varying switched systems subject to a quadratic cost functional. The execution time of hybrid control algorithms is often prohibitive for real-time applications and typically may only be reduced at the expense of approximation accuracy. We address this trade-off by taking advantage of system linearity to formulate a projection-based approach so that no simulation is required during open-loop optimization. A numerical example shows how the proposed open-loop algorithm outperforms methods employing common numerical integration techniques. Additionally, we follow a receding-horizon scheme to apply real-time closed-loop hybrid control to a customized experimental setup, using the Robot Operating System (ROS). In particular, we demonstrate---both in Monte-Carlo simulation and in experiment---that optimal hybrid control efficiently regulates a cart and suspended mass system in real time.

A Hybrid Optimal Control Approach to Fuel-Efficient Aircraft Conflict Avoidance

We formulate fuel-optimal conflict-free aircraft trajectory planning as a hybrid optimal control problem. The discrete modes of the hybrid system capture the air traffic procedures for conflict resolution, e.g., speed and turn advisories. To solve problems of realistic dimensions arising from air traffic sector planning, we formulate a numerically tractable approach to solve the hybrid optimal control problem. The approach is based on introducing binary functions for each mode, relaxing the binary functions and including a penalty term on the relaxation. The transformed and discretized problem is a nonlinear program. We use the approach on a realistic case study with seven aircraft within an air traffic control sector, in which we find minimum-fuel conflict-free trajectories.

Optimal Calibration of Map-Based Energy Management for Plug-In Parallel Hybrid Configurations: A Hybrid Optimal Control Approach

In this paper, an optimization framework for the calibration of energy management in plug-in hybrid electric vehicles (PHEVs) is proposed. The framework is based on the modeling of hybrid vehicles as hybrid systems in the mathematical sense, i.e., as a system, whose input is composed of continuous and discrete variables. This allows for the flexible integration of discrete decisions, such as drive modes and gear selection. Hybrid optimal control problems are then formulated, which seek optimal continuous and discrete system inputs, and methods for the efficient solution are described. The framework also allows for the incorporation of losses that occur due to a change in a discrete variable. The results can then be used to automatically calculate lookup tables for optimal gear shifts, optimal torque split between the motor/generator and the internal combustion engine, and the determination of the drive mode (electric or hybrid mode). It is demonstrated that when switching cost is disregarded, the main challenge is still finding the initial costate value. Practical strategies for determining the costate value online are described, containing a rule-based method, a $\hbox{CO}_{2}$ optimal method, and predictive energy management. The most important implementation issues are discussed, and the results of real-world experiments of predictive energy management are shown.

Optimal Genetic-sliding Mode Control of VSC-HVDC Transmission Systems

This paper deals with the design of a hybrid optimal Genetic-Sliding Mode Control (GA-SMC) approach for VSC- HVDC transmission systems for improving the system's dynamic stability over a wide range of operating conditions considering different parameter variations and disturbances. For this purpose, a comprehensive state of the art of the VSC-HVDC stabilization dilemma is discussed. The nonlinear VSC-HVDC model is developed. The problem of de- signing a nonlinear feedback control scheme via two control strategies is addressed seeking a better performance. For ensuring robustness and chattering free behavior, the conventional SMC (C-SMC) scheme is realized using a boundary layer hyperbolic tangent function for the sliding surface. Then, the Genetic Algorithm (GA) is employed for determining the optimal gains for such SMC methodology forming a modified nonlinear GA-SMC control in order to conveniently stabilize the system end enhance its performance. The simulation results verify the enhanced performance of the VSC-HVDC transmission system controlled by SMC alone compared to the proposed optimal GA-SMC control. The comparative dynamic behavior analysis for both SMC and GA-SMC control schemes are presented.

Modeling air conditioning system with storage evaporator for vehicle energy management

Automotive Air Conditioning (A/C) system significantly affects fuel consumption and emission. Thus, Phase Change Material (PCM) is exploited in an innovative storage evaporator to improve the A/C system performance. Due to hybrid features introduced by mode switching when PCM changes its status between liquid and solid, the task of control-oriented modeling is particularly challenging. Upon the energy-based model built, an optimal control problem of an advanced A/C system with a storage evaporator is formulated as to find an optimal clutch command sequence balancing fuel consumption, cabin comfort and drivability constraints. In the scope of vehicle energy management, Dynamic Programming (DP) algorithm usually serves as a tool of obtaining benchmark optimal solution, against which results from other optimal algorithms are compared. However, a direct application of DP algorithm to the optimal control problem faces unexpected difficulty, because the discretization of state space is not feasible for an irregular multi-dimensional subspace formed by the multi-mode model. Alternatively, hybrid optimal control theory is pursued and a preliminary study is conducted to illustrate its promising application.