Mixed-integer Optimal Control Problems Research Articles

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.

A Quantum Computing Based Numerical Method for Solving Mixed-Integer Optimal Control Problems

Mixed-integer optimal control problems (MIOCPs) usually play important roles in many real-world engineering applications. However, the MIOCP is a typical NP-hard problem with considerable computational complexity, resulting in slow convergence or premature convergence by most current heuristic optimization algorithms. Accordingly, this study proposes a new and effective hybrid algorithm based on quantum computing theory to solve the MIOCP. The algorithm consists of two parts: (i) Quantum Annealing (QA) specializes in solving integer optimization with high efficiency owing to the unique annealing process based on quantum tunneling, and (ii) Double-Elite Quantum Ant Colony Algorithm (DEQACA) which adopts double-elite coevolutionary mechanism to enhance global searching is developed for the optimization of continuous decisions. The hybrid QA/DEQACA algorithm integrates the strengths of such algorithms to better balance the exploration and exploitation abilities. The overall evolution performs to seek out the optimal mixed-integer decisions by interactive parallel computing of the QA and the DEQACA. Simulation results on benchmark functions and practical engineering optimization problems verify that the proposed numerical method is more excel at achieving promising results than other two state-of-the-art heuristics.

On mixed-integer optimal control with constrained total variation of the integer control

The combinatorial integral approximation (CIA) decomposition suggests solving mixed-integer optimal control problems by solving one continuous nonlinear control problem and one mixed-integer linear program (MILP). Unrealistic frequent switching can be avoided by adding a constraint on the total variation to the MILP. Within this work, we present a fast heuristic way to solve this CIA problem and investigate in which situations optimality of the constructed feasible solution is guaranteed. In the second part of this article, we show tight bounds on the integrality gap between a relaxed continuous control trajectory and an integer feasible one in the case of two controls. Finally, we present numerical experiments to highlight the proposed algorithm’s advantages in terms of run time and solution quality.

Extended multi-interval Legendre-Gauss-Radau pseudospectral method for mixed-integer optimal control problem in engineering

Many engineering optimisation problems can be summarised as mixed-integer optimal control problems (MIOCPs) owing to the needs for mixed-integer dynamic control decisions. However, the convergence theory of Legendre-Gauss-Radau (LGR) approximation fails to apply to such non-smooth and discontinuous optimal control problems. Therefore, this paper develops an extended multi-interval LGR pseudospectral method (EMLGR), which has the following features: (i) the mixed-integer controls at the end of each interval and the interval intersections are added as two new controls to avoid the unrestrained control and shorten the switching time of integer control, and (ii) a smart adaptive collocation monitor (SACM) is provided to optimise the polynomial order and interval structure for further reducing computational complexity and improving approximation precision. The detailed solution procedure of EMLGR is given in this study, and experimental studies including five challenging practical engineering MIOCPs are taken to verify the superiorities of the proposed EMLGR in efficiency, accuracy and stability.

Mixed-integer optimal control problems with switching costs: a shortest path approach

We investigate an extension of Mixed-Integer Optimal Control Problems by adding switching costs, which enables the penalization of chattering and extends current modeling capabilities. The decomposition approach, consisting of solving a partial outer convexification to obtain a relaxed solution and using rounding schemes to obtain a discrete-valued control can still be applied, but the rounding turns out to be difficult in the presence of switching costs or switching constraints as the underlying problem is an Integer Program. We therefore reformulate the rounding problem into a shortest path problem on a parameterized family of directed acyclic graphs (DAGs). Solving the shortest path problem then allows to minimize switching costs and still maintain approximability with respect to the tunable DAG parameter theta . We provide a proof of a runtime bound on equidistant rounding grids, where the bound is linear in time discretization granularity and polynomial in theta . The efficacy of our approach is demonstrated by a comparison with an integer programming approach on a benchmark problem.

Mixed-Integer Optimal Design and Energy Management of Hybrid Electric Vehicles With Automated Manual Transmissions

The energy efficiency improvement problems of hybrid electric vehicles (HEVs) with automated manual transmissions are investigated from both the design and control aspects for extended searching space in this work. To optimize the transmission, the gearshift commands and power distribution simultaneously, a combined mixed-integer optimal design and control problem (MIOCP) is formulated. The challenging MIOCP is then reformulated by introducing augmented terms in both objective function and the dynamic model to handle the integer control variables. The reformulated problem is solved with an optimal control software package developed by the authors, the performance of which is validated with dynamic programming and equivalent consumption minimization strategy. Optimal design and mixed-integer control results of the HEV are obtained for different cases, some existing confusions of transmission selection for the HEVs are clarified quantitatively. Results show that transmissions with fewer gear ratios can achieve competitive performances compared with the original one by using the proposed approach.

Numerical Computation of a Mixed-Integer Optimal Control Problem Based on Quantum Annealing

It is extremely challenging to solve the mixed-integer optimal control problems (MIOCPs) due to the complex computation in solving the integer decision variables. This paper presents a new method based on quantum annealing (QA) to solve MIOCP. The QA is a metaheuristic which applies quantum tunneling in the annealing process. It has a faster convergence speed in optimal-searching and is less likely to run into local minima. Hence, QA is applied to deal with this kind of optimization problems. First, MIOCP is transformed into a mixed-integer nonlinear programming (MINLP). Then, a method based on QA is adopted to solve the MINLP and acquire the optimal solution. At last, two benchmark examples including Lotka-Volterra type fishing problem and distillation column are presented and solved. The effectiveness of the methodology is verified by the acquired optimal schemes.

An optimal control approach to scheduling maintenance and production in parallel lines of reactors using decaying catalysts

In this work, an optimal control approach for scheduling maintenance and production in a process using decaying catalysts, originally developed in a previous study for a single reactor, is extended to parallel lines of reactors. Unlike traditional mixed-integer optimisation methods, this approach involves formulating this problem as a multistage mixed-integer optimal control problem (MSMIOCP) and using a methodology that enables solution as a standard nonlinear optimisation problem. The methodology’s features of a feasible path approach and scheduling catalyst changeovers without combinatorial optimisation techniques can be advantageous in providing reliable, robust and efficient solutions in comparison to mixed-integer methods. The MSMIOCP formulation is applied to the case study of an industrial process that operates a single feed over a set of 4 parallel reactors and produces a single product. The solutions obtained using the proposed methodology are of high quality and highlight the potential advantages of this approach over mixed-integer techniques.

Mixed-integer optimal control under minimum dwell time constraints

Tailored Mixed-Integer Optimal Control policies for real-world applications usually have to avoid very short successive changes of the active integer control. Minimum dwell time (MDT) constraints express this requirement and can be included into the combinatorial integral approximation decomposition, which solves mixed-integer optimal control problems (MIOCPs) to epsilon -optimality by solving one continuous nonlinear program and one mixed-integer linear program (MILP). Within this work, we analyze the integrality gap of MIOCPs under MDT constraints by providing tight upper bounds on the MILP subproblem. We suggest different rounding schemes for constructing MDT feasible control solutions, e.g., we propose a modification of Sum Up Rounding. A numerical study supplements the theoretical results and compares objective values of integer feasible and relaxed solutions.

Turnpike Properties in Discrete-Time Mixed-Integer Optimal Control

This note discusses properties of parametric discrete-time Mixed-Integer Optimal Control Problems (MIOCPs) as they often arise in model predictive control with discrete controls. We argue that, in want for a handle on similarity properties of parametric MIOCPs, the turnpike phenomenon known in optimal control is helpful. We provide sufficient turnpike conditions based on a dissipativity notion of MIOCPs, and we prove that the turnpike phenomenon allows specific and accurate guesses for the discrete controls. We also derive an easily checkable sufficient condition for dissipativity of linear-quadratic MIOCPs. Moreover, we show how the turnpike property can be used to derive efficient node-weighted branch-and-bound schemes tailored to parametric MIOCPs. We draw upon numerical examples to illustrate our findings.

A quantum computing-based numerical method of mixed-integer optimal control problems under uncertainty for alkali–surfactant–polymer flooding

ABSTRACT This article presents a numerical method based on quantum computing to solve two unresolved key issues for alkali–surfactant–polymer (ASP) flooding in oil exploitation: uncertainties affecting development planning and a switch control regarding injection oil wells. First, a fuzzy multi-objective mixed-integer optimal control model describing the mechanism of ASP flooding is established. Then, an improved possibilistic programming algorithm is presented to remove fuzziness and transform the model into a deterministic single-objective mixed-integer nonlinear programming (MINLP) model. On this basis, a bi-level quantum computing algorithm including quantum annealing and quantum ant colony algorithm is proposed to solve the MINLP and acquire the optimal mixed-integer control variables. The superiority of the proposed algorithm is verified by testing a number of benchmark examples. Finally, the numerical method based on quantum computing is adopted to solve successfully a mixed-integer optimal control problem for ASP flooding under uncertainty.

An optimal control approach to scheduling and production in a process using decaying catalysts

This article presents a novel approach to optimise scheduling and production planning to meet seasonal demand in an industrial process using decaying catalysts, based on its formulation as a multistage mixed-integer optimal control problem (MSMIOCP). Unlike existing methodologies, the MSMIOCP formulation allows to solve this problem as a standard nonlinear optimisation problem without combinatorial optimisation methods, which can be advantageous in providing reliable, robust and efficient solutions. Using this formulation, four case studies of this problem, differing in reaction or deactivation kinetics, are investigated. Two different solution implementations are used, each having their own relative advantages. The first implementation demonstrates a bang-bang behaviour for the linear scheduling controls, consistent with a theoretical analysis, but faces integration problems and does not always produce high quality solutions. The second implementation, while not demonstrating the bang-bang property, always produces high quality solutions and shows the advantages of the MSMIOCP formulation over existing methodologies.

MultiDimensional Sum-Up Rounding for Elliptic Control Systems

Partial outer convexification has been used to derive relaxations of mixed-integer optimal control problems (MIOCPs) that are constrained by time-dependent differential equations. The family of sum...

Approximation Properties and Tight Bounds for Constrained Mixed-Integer Optimal Control

We extend recent work on solving mixed-integer nonlinear optimal control problems (MIOCPs) to the case of integer control functions subject to constraints that involve a pointwise coupling of the state with the integer controls. We extend a theorem due to [S. Sager, H. Bock, and M. Diehl, Math. Program. Ser. A, 133 (2012), pp. 1--23] to the case of MIOCPs with constraints on the integer control and show that the integrality gap vanishes in function space when the coarseness of the rounding grid is driven to zero even after adding constraints of this type. For the time-discretized problem, we extend a sum-up rounding (SUR) scheme due to [S. Sager, C. Reinelt, and H. Bock, Math. Program. Ser. A, 118 (2009), pp. 109--149] to the new problem class. Our scheme permits one to constructively obtain an $\varepsilon$-feasible and $\varepsilon$-optimal binary feasible control. We derive new, tight upper bounds on the integer control approximation error made by SUR. For unconstrained binary controls on equidistant grids, we reduce the approximation error bound from $\mathcal{O}(|\Omega|)$ to $\mathcal{O}(\log |\Omega|)$ asymptotically for $|\Omega| \to \infty$ and a fixed coarseness of the rounding grid, where $|\Omega|$ is the number of binary controls. For constrained binary controls, we show that the approximation problem is more difficult, and we give a proof of an approximation error bound of complexity $\mathcal{O}(|\Omega|)$. A numerical example compares our approach to a state-of-the-art mixed-integer nonlinear programming solver and illustrates the applicability of our results when solving MIOCPs using the direct and simultaneous approach.

Economic Nonlinear Predictive Control for Real-Time Optimal Energy Management of Parallel Hybrid Electric Vehicles

This article presents an economic nonlinear hybrid model predictive control strategy for optimal energy management of parallel hybrid electric vehicles. Hybrid electric vehicles are controlled for operation in various driveline modes and the associated optimal control problem involves both continuous and discrete control variables. To solve the resultant mixed-integer nonlinear optimal control problem, we propose a hierarchical supervisory control architecture that consists of demand prediction, driveline mode determination, and real-time optimization. These three modules are designed independently and connected in series to perform computer-aided control. The demand prediction module uses a times series model to forecast the mechanical traction power requests of the driver over a prediction horizon based on vehicle speed, road grade, acceleration pedal scale, brake pedal scale, and past and current power demands. For a given forecasted power demand profile, the mode determination module decides a sequence of driveline modes that are presumed to be operated over the prediction horizon. The model-based real-time optimization corresponding to nonlinear model predictive control computes the optimal motor power over a prediction horizon, and the receding horizon scheme as feedback control is applied to repeat the processes of the three control modules. A dedicated case study with real driving data obtained from Hyundai IONIQ PHEV 2018 is presented to demonstrate the effectiveness in fuel economy and emission reduction offered by the proposed optimal energy management strategy. The proposed hierarchical real-time predictive optimization-based strategy is competitive with any exiting power management strategies such as dynamic programming and equivalent consumption minimization strategy in fuel economy and emission reduction while showing better charge-sustaining capability. This trade-off between fuel economy and charge-sustainability can be further improved by tuning the hyper-parameters in the proposed optimal control problem.

Lossless Convexification of Optimal Control Problems with Semi-continuous Inputs

This paper presents a novel one-shot convex optimization method for finding globally optimal solutions of a class of mixed-integer non-convex optimal control problems. We consider problems with non-convex constraints that restrict the input norms to be either zero or lower-and upper-bounded. The non-convex problem is relaxed to a convex one whose optimal solution is proved to be optimal almost everywhere for the original problem, a procedure known as lossless convexification. The solution relies on second-order cone programming and demonstrates that a meaningful class of optimal control problems with binary variables can be solved reliably and in polynomial time. A rocket landing example with a coupled thrust-gimbal constraint corroborates the effectiveness of the approach.

Multiaircraft Optimal 4D Trajectory Planning Using Logical Constraints

This paper studies the trajectory planning problem for multiple aircraft with logical constraints in disjunctive form which arise in modeling passage through waypoints, distance-based and time-based separation constraints, decision-making processes, conflict resolution policies, no-fly zones, or obstacle or storm avoidance. Enforcing separation between aircraft, passage through waypoints, and obstacle avoidance is especially demanding in terms of modeling efforts. Indeed, in general, separation constraints require the introduction of auxiliary integer variables in the model; for passage constraints, a multiphase optimal control approach is used, and for obstacle avoidance constraints, geometric approximations of the obstacles are introduced. Multiple phases increase model complexity, and the presence of integer variables in the model has the drawback of combinatorial complexity of the corresponding mixed-integer optimal control problem. In this paper, an embedding approach is employed to transform logical constraints in disjunctive form into inequality and equality constraints which involve only continuous auxiliary variables. In this way, the optimal control problem with logical constraints is converted into a smooth optimal control problem which is solved using traditional techniques, thereby reducing the computational complexity of finding the solution. The effectiveness of the approach is demonstrated through several numerical experiments by computing the optimal trajectories of multiple aircraft in converging and intersecting arrival routes with time-based separation constraints, distance-based separation constraints, and operational constraints.

On the Mixed-Integer Linear-Quadratic Optimal Control With Switching Cost

This letter concerns the optimal control of a continuous-time dynamical system via continuous and discrete-valued control variables, where the objective functional also accounts for state-independent switching costs. The class of mixed-integer optimal control problems (OCPs) is interpreted as a bilevel problem, involving both switching times optimization, for a given sequence of modes, and purely continuous optimal control. Additionally, an original nonconvex formulation for the switching costs is introduced, in terms of cardinality, inspired by sparse optimization and compressed sensing techniques. We then adopt proximal algorithms to solve the resulting bilevel OCP with composite objective function. An efficient routine for evaluating the proximal operator is developed. Two examples are numerically solved via a proximal gradient method, discussed and compared with the literature. Although this letter focuses on switched linear time-varying dynamics and quadratic cost functionals with a specific formulation of the switching costs, the proposed approach may also apply to more general mixed-integer OCPs.

Optimisation of heat exchanger network cleaning schedules: Incorporating uncertainty in fouling and cleaning model parameters

The optimisation of the cleaning schedule in heat exchanger networks (HENs) subject to fouling is presented in which the impact of parametric uncertainty is considered. This work is based on the realisation that these HEN cleaning scheduling problems are multistage mixed-integer optimal control problems (MIOCPs) in which the controls, i.e. cleaning actions, exhibit bang-bang behaviour as they occur linearly in the system equations. A multiple scenario feasible path MIOCP approach is proposed, whereby many scenarios of the HEN multiperiod problems are stacked, sharing the same control actions. This is implemented on a 10 unit and a 25 unit HEN case studies representing crude oil refinery preheat trains (PHTs). Results show that there is a large difference in financial performance for the deterministic case versus the parametric uncertainty problem and hence it is vital that uncertainty be taken into account during the optimisation of schedules for HEN maintenance problems.

Efficient unmanned aerial vehicle formation rendezvous trajectory planning using Dubins path and sequential convex programming

ABSTRACTTrajectory planning of formation rendezvous of multiple unmanned aerial vehicles (UAVs) is formulated as a mixed-integer optimal control problem, and an efficient hierarchical planning approach based on the Dubins path and sequential convex programming is proposed. The proposed method includes the assignment of rendezvous points (high level) and generation of cooperative trajectories (low level). At the high level, the assignment of rendezvous points to UAVs is optimized to minimize the total length of Dubins-path-based approximate trajectories. The assignment results determine the geometric relations between the UAVs’ goals, which are used as equality constraints for generating trajectories. At the low level, trajectory generation is treated as a non-convex optimal control problem, which is transformed to a non-convex parameter optimization and then solved via sequentially performing convex optimization. Numerical experiments demonstrate that the proposed method can generate feasible trajectories and can outperform a typical nonlinear programming method in terms of efficiency.