Mixed-integer Optimal Control Problems Research Articles

On Discrete Subproblems in Integer Optimal Control with Total Variation Regularization in Two Dimensions

We analyze integer linear programs that we obtain after discretizing two-dimensional subproblems arising from a trust-region algorithm for mixed integer optimal control problems with total variation regularization. We discuss NP-hardness of the discretized problems and the connection to graph-based problems. We show that the underlying polyhedron exhibits structural restrictions in its vertices with regard to which variables can attain fractional values at the same time. Based on this property, we derive cutting planes by employing a relation to shortest-path and minimum bisection problems. We propose a branching rule and a primal heuristic which improves previously found feasible points. We validate the proposed tools with a numerical benchmark in a standard integer programming solver. We observe a significant speedup for medium-sized problems. Our results give hints for scaling toward larger instances in the future. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: This work was supported by the Deutsche Forschungsgemeinschaft [Grant MA 10080/2-1]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2024.0680 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0680 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

A numerical study of transformed mixed-integer optimal control problems

AbstractTime transformation is a ubiquitous tool in theoretical sciences, especially in physics. It can also be used to transform switched optimal control problems into control problems with a fixed switching order and purely continuous decisions. This approach is known either as enhanced time transformation, time-scaling, or switching time optimization (STO) for mixed-integer optimal control. The approach is well understood and used widely due to its many favorable properties. Recently, several extensions and algorithmic improvements have been proposed. We use an alternative formulation, the partial outer convexification (POC), to study convergence properties of (STO). We introduce the open-source software package _ (Sager et al., czeile/ampl_mintoc: Math programming c release, 2024, https://doi.org/10.5281/zenodo.12520490). It is based on AMPL, designed for the formulation of mixed-integer optimal control problems, and allows to use almost identical implementations for (STO) and (POC). We discuss and explain our main numerical result: (STO) is likely to result in more local minima for each discretization grid than (POC), but the number of local minima is asymptotically identical for both approaches.

Clothoid-Based Path Planning for a Formation of Fixed-Wing UAVs

Unmanned aerial vehicles (UAVs) are playing an increasingly crucial role in many applications such as search and rescue, delivery services, and military operations. However, one of the significant challenges in this area is to plan efficient and safe trajectories for UAV formations. This paper presents an optimization procedure for trajectory planning for fixed-wing UAV formations using graph theory and clothoid curves. The proposed planning strategy consists of two main steps. Firstly, the geometric optimization of paths is carried out using graphs for each UAV, providing piece-wise linear paths whose smooth connections are made with clothoids. Secondly, the geometric paths are transformed into time-dependent trajectories, optimizing the assigned aircraft speeds to avoid collisions by solving a mixed-integer optimal control problem for each UAV of the flight formation. The proposed method is effective in achieving suboptimal paths while ensuring collision avoidance between aircraft. A sensitivity analysis of the main parameters of the algorithm was conducted in ideal conditions, highlighting the possibility of decreasing the length of the optimal path by about 4.19%, increasing the number of points used in the discretization and showing a maximum path length reduction of about 10% compared with the average solution obtained with a similar algorithm using a graph based on random directions. Furthermore, the use of clothoids, whose parameters depend on the UAV performance constraints, provides smoother connections, giving a significant improvement over traditional straight-line or circular trajectories in terms of flight dynamics compliance and trajectory tracking capabilities. The method can be applied to various UAV formation scenarios, making it a versatile and practical tool for mission planning.

Mixed-integer trajectory optimization with no-fly zone constraints for a hypersonic vehicle

This paper focuses on mixed-integer trajectory optimization of no-fly zones avoidance for a hypersonic vehicle. When there are multiple no-fly zones, the trajectory optimization problem has many local optimal solutions, and the performance of the solution depends on the selection of an initial guess. Essentially, the selection of the initial guess can be abstracted as a discrete path decision. To deal with this issue, this paper first integrates the discrete path decision and continuous trajectory optimization for no-fly zones avoidance and models these as a mixed-integer optimal control problem (MIOCP). As the corner stone, a creative directed graph is established to model the path decision problem, which is then formulated by integer constraints. Combined with the minimum effort optimal control problem (Problem 1), we form the MIOCP (Problem 2), thus integrating the path decision and trajectory optimization to improve global performance. Additionally, to solve the MIOCP, an effective iterative mixed-integer convex programming (IMICP) algorithm is customized by adding two ad-hoc techniques, relaxation and a proximity term, thus remedying the artificial infeasibility. Simulation results show that this approach is practically independent of the initial guess and has better performance than existing methods.

A Gauss–Newton-based decomposition algorithm for Nonlinear Mixed-Integer Optimal Control Problems

For the fast approximate solution of Mixed-Integer Non-Linear Programs (MINLPs) arising in the context of Mixed-Integer Optimal Control Problems (MIOCPs) a decomposition algorithm exists that solves a sequence of three comparatively less hard subproblems to determine an approximate MINLP solution. In this work, we propose a problem formulation for the second algorithm stage that is a convex approximation of the original MINLP and relies on the Gauss–Newton approximation. We analyze the algorithm in terms of approximation properties and establish a first-order consistency result. Then, we investigate the proposed approach considering a numerical case study of Mixed-Integer Optimal Control (MIOC) of a renewable energy system. The investigation shows that the proposed formulation can yield an improved integer solution regarding the objective of the original MINLP compared with the established Combinatorial Integral Approximation (CIA) algorithm.

Time-Domain Decomposition for Mixed-Integer Optimal Control Problems

We consider mixed-integer optimal control problems, whose optimality conditions involve global combinatorial optimization aspects for the corresponding Hamiltonian pointwise in time. We propose a time-domain decomposition, which makes this problem class accessible for mixed-integer programming using parallel-in-time direct discretizations. The approach is based on a decomposition of the optimality system and the interpretation of the resulting subproblems as suitably chosen mixed-integer optimal control problems on subintervals in time. An iterative procedure then ensures continuity of the states at the boundaries of the subintervals via co-state information encoded in virtual controls. We prove convergence of this iterative scheme for discrete-continuous linear-quadratic problems and present numerical results both for linear-quadratic as well as nonlinear problems.

Numerical Strategies for Mixed-Integer Optimization of Power-Split and Gear Selection in Hybrid Electric Vehicles

This paper presents numerical strategies for a computationally efficient energy management system that co-optimizes the power split and gear selection of a hybrid electric vehicle (HEV). We formulate a mixed-integer optimal control problem (MIOCP) that is transcribed using multiple-shooting into a mixed-integer nonlinear program (MINLP) and then solved by nonlinear model predictive control. We present two different numerical strategies, a Selective Relaxation Approach (SRA), which decomposes the MINLP into several subproblems, and a Round-n-Search Approach (RSA), which is an enhancement of the known ‘relax-n-round’ strategy. Subsequently, the resulting algorithmic performance and optimality of the solution of the proposed strategies are analyzed against two benchmark strategies; one using rule-based gear selection, which is typically used in production vehicles, and the other using dynamic programming (DP), which provides a global optimum of a quantized version of the MINLP. The results show that both SRA and RSA enable about 3.6% cost reduction compared to the rule-based strategy, while still being within 1% of the DP solution. Moreover, for the case studied RSA takes about 35% less mean computation time compared to SRA, while both SRA and RSA being about 99 times faster than DP. Furthermore, both SRA and RSA were able to overcome the infeasibilities encountered by a typical rounding strategy under different drive cycles. The results show the computational benefit of the proposed strategies, as well as the energy saving possibility of co-optimization strategies in which actuator dynamics are explicitly included.

Co-Optimization of Design and Control of Energy Efficient Hybrid Electric Vehicles Using Coordination Schemes

AbstractDesign and control co-optimization studies for hybrid vehicles have been proposed in the past. However, such works suffer from difficulties arising due to (a) diverse real- and integer-valued variables, (b) complex nonlinear powertrain dynamics and design interconnections, (c) conflicting objective functions with path constraints, and (d) high computational resources requirements. To meet these challenges, this study presents an efficient co-optimization framework for hybrid electric vehicles (HEVs) which is built using existing algorithms and coordination schemes. Particular emphasis is given to the simultaneous scheme and the decomposition-based scheme. The decomposition-based scheme with the problem decomposition proposed in this work can efficiently handle multitime scale state variables and both integer- and real-valued design and control optimization variables. This is demonstrated by solving the mixed-integer optimal design and control problem of a series hybrid vehicle over a 1-h long drive cycle with time discretization of 1 s. The problem complexity is elevated by using an increasing number of state variables (including battery state of charge, battery energy, and after-treatment system temperature), control variables (such as the engine power and engine on/off), and design parameters (such as the number of battery cells and the type and size of the engine). In addition, a multi-objective cost function is used to find a tradeoff solution between fuel consumption and emissions minimization. The results show that in terms of optimality of the solution, the decomposition-based scheme is comparable with the simultaneous but can give a 14% improvement in computational performance. The effectiveness of the proposed framework is demonstrated by comparing the co-optimization results against a baseline case in which only the optimal control problem is solved. The co-optimized solution yields up to 3.7% average genset efficiency improvement and a fuel consumption reduction to 1.6 kg from 2.5 kg, which is further reduced to 1.5 kg by adding the engine on-off control. Finally, a decision matrix is developed to provide guidance on the selection of the optimization algorithm and coordination scheme for any problem at hand.

Discrete Mixed-Integer Shooting (DMIS): Algorithm and Application to Plug-In Hybrid Electric Vehicle Energy Management Accounting for Fuel Cranking and Actual Powertrain Efficiency Maps

This article presents an algorithm for mixed-integer nonlinear optimal control problems (OCPs) formulated directly in discrete time. In our current application, we address the energy management of a plug-in hybrid electric vehicle (PHEV) seeking to achieve minimum trip fuel consumption under a terminal state-of-charge (SOC) equality constraint while avoiding powertrain efficiency map approximations in order to maximize its fuel economy (FE). An additional engine cranking state with a nonsmooth state and control transition cost is then incorporated to prevent undesired engine on/off behaviors. We pursue a single shooting-based numerical algorithm, motivated by the success of the computationally efficient Pontryagin’s Minimum Principle (PMP)-based single shooting in achieving similar FE results as dynamic programming (DP) for the power-split optimization problem without the additional engine cranking state shown in prior work. The Discrete Mixed-Integer Shooting (DMIS) algorithm, with costate backward-in-time propagation, is presented here, avoiding unstable shooting iterations that are typically encountered in PMP-based single shooting. When applying DMIS to solve the power-split problem with cranking fuel consideration, we show through simulation that it leads to better FE results than the PMP-based single shooting without the cranking state, thus demonstrating the effectiveness of the DMIS algorithm. Finally, experimental dynamometer test results with the model predictive powertrain controller solved by DMIS for minimum trip fuel consumption are presented and discussed, demonstrating its real-time execution and the cranking fuel reduction capability.

Predictive energy management with engine switching control for hybrid electric vehicle via ADMM

This paper studies energy management (EM) of a power-split hybrid electric vehicle (HEV) equipped with planetary gear sets. We first formulate a mixed-integer global optimal control problem that includes a binary switching variable. Convex modeling, including the fuel model for a compound energy conversion unit, is then presented to reformulate the mixed-integer EM as a two-step program. For optimizing the engine switching and battery power decisions in the first step, we employ the alternating direction method of multipliers (ADMM) algorithm where the solution of the convex relaxation is used to initialize the non-convex problem. On the standard driving cycle, simulation results indicate that the ADMM based EM method saves 7.63% fuel compared to a heuristic method, and shows 99% optimality compared to a dynamic programming method, while saving three orders of magnitude in computing time. An ADMM combined model predictive control (ADMM-MPC) method is also developed that is suitable for receding horizon control scenarios. The ADMM-MPC method shows 5.28% fuel saving when implemented using a prediction horizon of 15 samples. Meanwhile, the mean computing time for MPC updates is 3.53ms. Our results demonstrate that the proposed ADMM is capable of real-time control.

A parallel algorithm based on quantum annealing and double-elite spiral search for mixed-integer optimal control problems in engineering

Many engineering processes can be summarized as mixed-integer optimal control problems (MIOCPs) owing to the needs for optimizing mixed-integer dynamic control policies. However, MIOCP is a challenging NP-hard problem with great computational complexity, resulting in slow convergence or premature convergence by most current heuristics. Accordingly, this study explores a new and effective hybrid Quantum Annealing-Double-Elite Spiral Search (QA-DESS) algorithm to address this issue. To be specific, QA algorithm specializes in solving integer optimization with high efficiency due to the unique quantum-tunnelling-based annealing mechanism. For the optimization of continuous decisions, a DESS algorithm which adopts adaptive Cauchy mutation and double-elite evolutionary mechanism to enhance global searching is designed. The hybrid QA-DESS algorithm integrates the strengths of such algorithms to better balance the exploration and exploitation abilities. The overall evolution performs to find the optimal mixed-integer decisions by interactive parallel computing of QA and DESS. Simulation results on benchmark functions and practical engineering MIOCPs verify that the proposed optimization algorithm is more excel at finding promising results than other 7 heuristics (including traditional and state-of-the-art algorithms).

Combinatorial Integral Approximation Decompositions for Mixed-Integer Optimal Control

Solving mixed-integer nonlinear programs (MINLPs) is hard from both a theoretical and practical perspective. Decomposing the nonlinear and the integer part is promising from a computational point of view. In general, however, no bounds on the objective value gap can be established and iterative procedures with potentially many subproblems are necessary. The situation is different for mixed-integer optimal control problems with binary variables that switch over time. Here, a priori bounds were derived for a decomposition into one continuous nonlinear control problem and one mixed-integer linear program, the combinatorial integral approximation (CIA) problem. In this article, we generalize and extend the decomposition idea. First, we derive different decompositions and analyze the implied a priori bounds. Second, we propose several strategies to recombine promising candidate solutions for the binary control functions in the original problem. We present the extensions for ordinary differential equations-constrained problems. These extensions are transferable in a straightforward way, though, to recently suggested variants for certain partial differential equations, for algebraic equations, for additional combinatorial constraints, and for discrete time problems. We implemented all algorithms and subproblems in AMPL for a proof-of-concept study. Numerical results show the improvement compared to the standard CIA decomposition with respect to objective function value and compared to general-purpose MINLP solvers with respect to runtime.

Switching Cost Aware Rounding for Relaxations of Mixed-Integer Optimal Control Problems: The 2-D Case

This article is concerned with a recently proposed switching cost aware rounding (SCARP) strategy in the combinatorial integral approximation for mixed-integer optimal control problems (MIOCPs). We consider the case of a control variable that is discrete-valued and distributed on a two-dimensional domain. While the theoretical results from the one-dimensional case directly apply to the multidimensional setting, the structure of the cost function in the graph-based rounding computation is significantly more involved in the two-dimensional case. We describe a set up of the computational graph and the traversal algorithm underlying the SCARP strategy that enable a transfer to the two-dimensional setting. We demonstrate the SCARP strategy in this two-dimensional setting using the example of a MIOCP from topology optimization. We compare the graph-based approach to a ground truth computation using an integer linear programming (ILP) solver. The graph-based approach becomes computationally intractable for medium grid sizes. We show that the one-dimensional SCARP algorithm can be employed on a serialization of the grid cells in these cases and still provides an efficient heuristic that yields superior performance compared with that of other rounding heuristics such as sum-up rounding (SUR).

Optimal feedback control for a class of fed-batch fermentation processes using switched dynamical system approach

<abstract><p>This paper considers an optimal feedback control problem for a class of fed-batch fermentation processes. Our main contributions are as follows. Firstly, a dynamic optimization problem for fed-batch fermentation processes is modeled as an optimal control problem of switched dynamical systems, and a general state-feedback controller is designed for this dynamic optimization problem. Unlike the existing switched dynamical system optimal control problem, the state-dependent switching method is applied to design the switching rule, and the structure of this state-feedback controller is not restricted to a particular form. Then, this problem is transformed into a mixed-integer optimal control problem by introducing a discrete-valued function. Furthermore, each of these discrete variables is represented by using a set of 0-1 variables. By using a quadratic constraint, these 0-1 variables are relaxed such that they are continuous on the closed interval $ [0, 1] $. Accordingly, the original mixed-integer optimal control problem is transformed intoa nonlinear parameter optimization problem. Unlike the existing works, the constraint introduced for these 0-1 variables are at most quadratic. Thus, it does not increase the number of locally optimal solutions of the original problem. Next, an improved gradient-based algorithm is developed based on a novel search approach, and a large number of numerical experiments show that this novel search approach can effectively improve the convergence speed of this algorithm, when an iteration is trapped to a curved narrow valley bottom of the objective function. Finally, numerical results illustrate the effectiveness of this method developed by this paper.</p></abstract>

Generic stochastic particle filter algorithm for predictive energy optimization of a Plug-in Hybrid Electric Vehicle extended by a battery temperature control and implemented on a Hardware-in-the-Loop system

This work presents an implementation and demonstration of a generic stochastic particle filter-based algorithm (GSPFA) to solve the energy management problem of a Plug-in Hybrid Electric Vehicle (PHEV, P2). The objective is to minimize energy consumption considering the driving mission-specific state of charge and the battery system’s desired temperature range. To achieve this objective, a two-level Model Predictive Control (MPC) approach is used for trajectory planning in the distance domain. The control signals are vehicle velocity, torque-split, gear shift, and the internal combustion engine on/off state. The mixed-integer nonlinear optimal control problem is solved by the second MPC. The PHEV model is executed on the real-time system of a Hardware-in-the-Loop test bench with a battery module as a real component to measure physical responses. A reference driving cycle (CADC URM130) without temperature control leads to a temperature increase of the battery of 2.1K. By allowing a temperature increase of only 1K under equal boundary conditions, the results show that this increase could be maintained with a temperature delta of 0.4K at the end of the cycle. By targeting a maximum temperature delta with identical boundary conditions, a temperature delta of 4K was achieved.

Experimental demonstration of grid-supportive scheduling of a polygeneration system using economic-MPC

Drawing off the technical flexibility of building polygeneration systems to support a rapidly expanding renewable electricity grid requires the application of advanced controllers like model predictive control (MPC) that can handle multiple inputs and outputs, uncertainties in forecast data, and plant constraints amongst other features. In this original work, an economic-MPC-based optimal scheduling of a real-world building energy system is demonstrated and its performance is evaluated against a conventional controller. The demonstration includes the steps to integrate an optimisation-based supervisory controller into a standard building automation and control system with off-the-shelf HVAC components and usage of state-of-art algorithms for solving complex nonlinear mixed integer optimal control problems. With the MPC, quantitative benefits in terms of 6–12% demand-cost savings and qualitative benefits in terms of better controller adaptability and hardware-friendly operation are identified. Further research potential for improving the MPC framework in terms of field-level stability, minimising constraint violations, and inter-system communication for its deployment in a prosumer-network is also identified.

Mixed-integer optimal control for multimodal chromatography

Multimodal chromatography is a powerful tool in the downstream processing of biopharmaceuticals. To fully benefit from this technology, an efficient process strategy must be determined beforehand. To facilitate this task, we employ a recent mechanistic model for multimodal chromatography, which takes salt concentration and pH into account, and we present a mathematical framework for the optimization of chromatographic processes. This framework also includes the use of discrete process controls in order to cover a wider range of chromatographic applications. We describe a procedure to numerically solve the resulting nonlinear mixed-integer optimal control problems. We discuss results of computational experiments, covering the cases where one wants to optimize the yield of the product or the batch-cycle time under specified purity requirements. The results indicate that a good separation can be achieved in a two-component system and that both salt concentration and discrete pH play an important role within the purification process.

Penalty alternating direction methods for mixed-integer optimal control with combinatorial constraints

We consider mixed-integer optimal control problems with combinatorial constraints that couple over time such as minimum dwell times. We analyze a lifting and decomposition approach into a mixed-integer optimal control problem without combinatorial constraints and a mixed-integer problem for the combinatorial constraints in the control space. Both problems can be solved very efficiently with existing methods such as outer convexification with sum-up-rounding strategies and mixed-integer linear programming techniques. The coupling is handled using a penalty-approach. We provide an exactness result for the penalty which yields a solution approach that convergences to partial minima. We compare the quality of these dedicated points with those of other heuristics amongst an academic example and also for the optimization of electric transmission lines with switching of the network topology for flow reallocation in order to satisfy demands.

A numerical method for interval multi-objective mixed-integer optimal control problems based on quantum heuristic algorithm

This article is concerned with the numerical solution for a class of interval multi-objective mixed-integer optimal control problems (IMOMIOCPs). The IMOMIOCPs under investigation are typical NP-hard problems involve unknown-but-bounded interval parameters, multiple objectives, and mixed-integer dynamic controls. Accordingly, a new numerical method based on quantum heuristic algorithm is designed, which has the following modules: (i) Control vector parameterization and the fourth order Runge–Kutta method for model discretization, (ii) interval programming based on interval credibility for addressing interval parameters, (iii) coevolution of Quantum Annealing and Quantum Krill Herd for searching the optimal mixed-integer decisions, and (iv) the multiple populations for multi-objective technology for establishing the Pareto optimal front. The analyses on convergence and computational complexity of the proposed optimization mechanism are given. Moreover, simulation results on benchmark functions and a practical engineering IMOMIOCP verify that the proposed numerical method is more excel at achieving promising results than some classic algorithms and state-of-the-art algorithms.

An optimal control approach to considering uncertainties in kinetic parameters in the maintenance scheduling and production of a process using decaying catalysts

This article presents a novel approach for optimising maintenance scheduling and production in a process using decaying catalysts while considering uncertainties in the kinetic parameters involved. The approach formulates this problem as a multistage mixed-integer optimal control problem (MSMIOCP) and uses a solution methodology that can offer a number of potential advantages over conventional methodologies. The solution methodology involves using a multiple scenario approach to consider parametric uncertainties and formulating a stochastic version of the MSMIOCP, which is solved as a standard nonlinear optimisation problem using a technique developed in a previous work. The proposed formulation and solution methodology are applied to identify the effects on the optimal process operation, of individual uncertainty of each parameter, of simultaneous uncertainty of all parameters and of the number of scenarios generated, as four case studies. The results obtained provide insights into these aspects and indicate the approach’s capability to solve this problem.