Mixed-integer Optimal Control Problems Research Articles

A System of Systems approach for data centers optimization and integration into smart energy grids

This paper addresses the problem of proactive planning and optimizing the operation of a Systems of Systems (SoS) over a time horizon while considering the characteristics of each constituent system and complex interactions among them. We define a mathematical formalism for modeling complex systems composed of a mesh of sub-systems with linear and non-linear behaviors and abstractions like discrete time, atomic systems and interconnection of atomic system. The proposed modeling approach is simple enough to allow fast computations and simulations, and at the same time complex enough to capture the essential features of the real system thus allowing the mapping of proactive optimization problems to Mixed-Integer Optimal Control Problems. The proactive planning uses hierarchical optimization processes that compute predictions and optimization plans at various time granularities, each finer layer plan adjusting and refining the ones with higher granularity. To show case our approach we model a Data Center which is a well-known case of a large scale complex system aiming to plan and optimize its operation to use as much as possible the locally produced renewable energy and optimize its integration in smart grid advanced context. Simulation based results show a reduction of 5% of the carbon footprint and at the same time an increase in profit of more than 14% due to flexible energy shifting.

Time transformed mixed integer optimal control problems with impacts

AbstractThe solutions of mixed integer optimal control problems (MIOCPs) yield optimized trajectories for dynamical systems with instantly changing dynamical behavior. The instant change is caused by a changing value of the integer valued control function. For example, a changing integer value can cause a car to change the gear, or a mechanical system to close a contact. The direct discretization of a MIOCP leads to a mixed integer nonlinear program (MINLP) and can not be solved with gradient based optimization methods at once. We extend the work by Gerdts [1] and reformulate a MIOCP with integer dependent constraints as an ordinary optimal control problem (OCP). The discretized OCP can be solved using gradient based optimization methods. We show how the integer dependent constraints can be used to model systems with impact and present optimized trajectories of computational examples, namely of a lockable double pendulum and an acyclic telescope walker. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On the time transformation of mixed integer optimal control problems using a consistent fixed integer control function

Nonlinear control systems with instantly changing dynamical behavior can be modeled by introducing an additional control function that is integer valued in contrast to a control function that is allowed to have continuous values. The discretization of a mixed integer optimal control problem (MIOCP) leads to a non differentiable optimization problem and the non differentiability is caused by the integer values. The paper is about a time transformation method that is used to transform a MIOCP with integer dependent constraints into an ordinary optimal control problem. Differentiability is achieved by replacing a variable integer control function with a fixed integer control function and a variable time allows to change the sequence of active integer values. In contrast to other contributions, so called control consistent fixed integer control functions are taken into account here. It is shown that these control consistent fixed integer control functions allow a better accuracy in the resulting trajectories, in particular in the computed switching times. The method is verified on analytical and numerical examples.

Relaxing mixed integer optimal control problems using a time transformation

AbstractMixed integer control systems are used to model dynamical behavior that can change instantly, for example a driving car with different gears. Changing a gear corresponds to an instant change of the differential equation what is achieved in the model by changing the value of the integer control function. The optimal control of a mixed integer control system by a discretize‐then‐optimize approach leads to a mixed integer optimization problem that is not differentiable with respect to the integer variables, such that gradient based optimization methods can not be applied. In this work, differentiability with respect to all optimization variables is achieved by reformulating the mixed integer optimal control problem (MIOCP). A fixed integer control function and a time transformation are introduced. The combination of both allows to change the sequence of active differential equations by partially deactivating the fixed integer control function. In contrast to other works, here different fixed integer control functions are taken into account. Advantages of so called control consistent (CC) fixed integer control functions are discussed and confirmed on a numerical example. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Discrete mechanics and mixed integer optimal control of dynamical systems

AbstractMixed integer optimal control problems are a generalization of ordinary optimal control problems that include additional integer valued control functions. The integer control functions are used to switch instantaneously from one system to another. We use a time transformation (similar as in [1]) to get rid of the integer valued functions. This allows to apply gradient based optimization methods to approximate the mixed integer optimal control problem. The time transformation from [1] is adapted such that problems with distinct state domains for each system can be solved and it is combined with the direct discretization method DMOC [2,3] (Discrete Mechanics and Optimal Control) to approximate trajectories of the underlying optimal control problems. Our approach is illustrated with the help of a first example, the hybrid mass oscillator. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Mathematical Programs with Blocks of Vanishing Constraints Arising in Discretized Mixed-Integer Optimal Control Problems

A time transformation technique is used to discretize an optimal control problem with discrete-valued controls and mixed control-state constraints, which depend on the discrete-valued controls. The discretized optimal control problem leads to a mathematical program with blocks of vanishing constraints (MPBVC). The MPBVC is analyzed in terms of stationarity concepts and constraint qualifications extending previous results on mathematical programs with vanishing constraints to the case of blocks of vanishing constraints. A relaxation technique is employed to regularize the MPBVC and to solve it numerically. The relaxation method is validated for a flight path optimization problem taking into account a discrete number of positions for the flaps of the aircraft in a landing approach.

Flight trajectory design in the presence of contrails: Application of a multiphase mixed-integer optimal control approach

In this paper we study the 4D trajectory planning problem in a contrail sensitive environment. We identify the control inputs that steer the aircraft from the initial fix to the final fix following a horizontal route of waypoints while performing step climbs and descents, in order to minimize the overall flying cost of fuel consumption, CO2 emissions, passenger travel time, and persistent contrail formation. The optimal trajectory design problem is formulated as a multiphase mixed integer optimal control problem, which is converted into a mixed integer non-linear program by first making the unknown switching times part of the state, then applying a Hermite–Simpson direct collocation method, and finally introducing binary variables to model the decision making. We solve the mixed-integer nonlinear program using a branch-and-bound algorithm. The numerical results show the effectiveness of the approach.

Efficient upper and lower bounds for global mixed-integer optimal control

We present a control problem for an electrical vehicle. Its motor can be operated in two discrete modes, leading either to acceleration and energy consumption, or to a recharging of the battery. Mathematically, this leads to a mixed-integer optimal control problem (MIOCP) with a discrete feasible set for the controls taking into account the electrical and mechanical dynamic equations. The combination of nonlinear dynamics and discrete decisions poses a challenge to established optimization and control methods, especially if global optimality is an issue. Probably for the first time, we present a complete analysis of the optimal solution of such a MIOCP: solution of the integer-relaxed problem both with a direct and an indirect approach, determination of integer controls by means of the sum up rounding strategy, and calculation of global lower bounds by means of the method of moments. As we decrease the control discretization grid and increase the relaxation order, the obtained series of upper and lower bounds converge for the electrical car problem, proving the asymptotic global optimality of the calculated chattering behavior. We stress that these bounds hold for the optimal control problem in function space, and not on an a priori given (typically coarse) control discretization grid, as in other approaches from the literature. This approach is generic and is an alternative to global optimal control based on probabilistic or branch-and-bound based techniques. The main advantage is a drastic reduction of computational time. The disadvantage is that only local solutions and certified lower bounds are provided with no possibility to reduce these gaps. For the instances of the electrical car problem, though, these gaps are very small. The main contribution of the paper is a survey and new combination of state-of-the-art methods for global mixed-integer optimal control and the in-depth analysis of an important, prototypical control problem. Despite the comparatively low dimension of the problem, the optimal solution structure of the relaxed problem exhibits a series of bang-bang, path-constrained, and sensitivity-seeking arcs.

The Lagrangian relaxation for the combinatorial integral approximation problem

We are interested in methods to solve mixed-integer nonlinear optimal control problems constrained by ordinary differential equations and combinatorial constraints on some of the control functions. To solve these problems we use a first discretize, then optimize approach to get a specially structured mixed-integer nonlinear program (MINLP). We decompose this MINLP into a nonlinear program (NLP) and a mixed-integer linear program (MILP), which is called the combinatorial integral approximation problem (CIAP). Previous results guarantee an integer gap for the MINLP depending on the objective function value of the CIAP. The focus of this study is the analysis of the CIAP and of a tailored branch-and-bound method. We link the huge computational gains compared to state-of-the-art MILP solvers to an analysis of subproblems on the branching tree. To this end we study properties of the Lagrangian relaxation of the CIAP. Special focus is given to special ordered set constraints that are present due to an outer convexification of the control problem. Also subproblems that arise by the application of branch-and-bound schemes are of interest. We prove polynomial runtime of the algorithm for special cases and give numerical evidence for efficiency by means of a numerical benchmark problem.

Energy-Optimal Multi-Goal Motion Planning for Planar Robot Manipulators

In this work, the energy-optimal motion planning problem for planar robot manipulators with two revolute joints is studied, in which the end-effector of the robot manipulator is constrained to pass through a set of waypoints, whose sequence is not predefined. This multi-goal motion planning problem has been solved as a mixed-integer optimal control problem in which, given the dynamic model of the robot manipulator, the initial and final configurations of the robot, and a set of waypoints inside the workspace of the manipulator, one has to find the control inputs, the sequence of waypoints with the corresponding passage times, and the resulting trajectory of the robot that minimizes the energy consumption during the motion. The presence of the waypoint constraints makes this optimal control problem particularly difficult to solve. The mixed-integer optimal control problem has been converted into a mixed-integer nonlinear programming problem first making the unknown passage times through the waypoints part of the state, then introducing binary variables to enforce the constraint of passing once through each waypoint, and finally applying a fifth-degree Gauss---Lobatto direct collocation method to tackle the dynamic constraints. High-degree interpolation polynomials allow the number of variables of the problem to be reduced for a given numerical precision. The resulting mixed-integer nonlinear programming problem has been solved using a nonlinear programming-based branch-and-bound algorithm specifically tailored to the problem. The results of the numerical experiments have shown the effectiveness of the approach.

Mixed Integer Optimal Control in Minimum Time Multi-points Traversal Problem of Robotic Manipulators1

In this paper, the minimum time multi-points traversal problem is studied for robotic manipulators. It is shown that the problem can be formulated as a mixed integer optimal control problem. Cubic Hermite spline is applied to interpolate the desired path points and then a decomposition method is proposed to solve the problem. An outer level iteration and a series of inner level optimization processes are used in the decomposition method. The outer level iteration aims to search optimal permutation of the path points, which is realized by Genetic Algorithm (GA). And the inner level processes aim to cope with a series of path determined minimum time motion planning problems, which are solved by a direct transcription method. Minimum time multi-points traversal task of a 2-DOF robotic manipulator is used to demonstrate the effectiveness of the proposed approach.

Multiphase Mixed-Integer Optimal Control Approach to Aircraft Trajectory Optimization

In this paper, an approach to aircraft trajectory optimization is presented in which integer and continuous variables are considered. Integer variables model decision-making processes, and continuous variables describe the state of the aircraft, which evolves according to differential-algebraic equations. The problem is formulated as a multiphase mixed-integer optimal control problem. It is transcribed into a mixed-integer nonlinear programming problem by applying a fifth degree Gauss–Lobatto direct collocation method and is then solved using a nonlinear-programming-based branch-and-bound algorithm. The approach is applied to the following en route flight planning problem: Given an aircraft point mass model, a wind forecast, an airspace structure, and the relevant flying information regions with their associated overflying costs, find the control inputs that steer the aircraft from the initial fix to the final fix, following a route of waypoints while minimizing the fuel consumption and overflying costs during the flight. The decision-making process arises in determining the optimal sequence of waypoints. The optimal times at which the waypoints are to be overflown are also to be determined. Numerical results are presented and discussed, showing the effectiveness of the approach.

TACO: a toolkit for AMPL control optimization

We describe a set of extensions to the AMPL modeling language to conveniently model mixed-integer optimal control problems for ODE or DAE dynamic processes. These extensions are realized as AMPL user functions and suffixes and do not require intrusive changes to the AMPL language standard or implementation itself. We describe and provide TACO, a Toolkit for AMPL Control Optimization that reads AMPL stub.nl files and detects the structure of the optimal control problem. This toolkit is designed to facilitate the coupling of existing optimal control software packages to AMPL. We discuss requirements, capabilities, and the current implementation. Using the example of the multiple shooting code for optimal control MUSCOD-II, a direct and simultaneous method for DAE-constrained optimal control, we demonstrate how the problem information provided by the TACO toolkit is interfaced to the solver. In addition, we show how the MS-MINTOC algorithm for mixed-integer optimal control can be used to efficiently solve mixed-integer optimal control problems modeled in AMPL. We use the AMPL extensions to model three control problem examples and we discuss how those extensions affect the representation of optimal control problems. Solutions to these problems are obtained by using MUSCOD-II and MS-MINTOC inside the AMPL environment. A collection of further AMPL control models is provided on the web site http://mintoc.de . MUSCOD-II and MS-MINTOC have been made available on the NEOS Server for Optimization, using the TACO toolkit to enable input of AMPL models.

Modeling Mixed-Integer Constrained Optimal Control Problems in AMPL

AbstractModeling languages and systems for simulation and optimization of continuous ODE/DAE systems are commonly available. For the most part, they focus on convenience of user interaction, and are tightly coupled to one or a few selected numerical methods. Control problems with discrete and hybrid controls, called mixed-integer optimal control problems (MIOCPs), have recently gained increased attention as the potential for optimization is high. The mixed-integer optimization community however most often considers problems without dynamics, and relies on symbolic modeling languages such as AMPL. Access to many advances MI(N)LP codes is provided by the NEOS Server for Optimization through the AMPL modeling language. Addressing this gap, we describe a set of extensions to the AMPL modeling language to conveniently model mixed-integer optimal control problems for ODE or DAE dynamic processes. These extensions are easily realized and do not require intrusive changes to the AMPL language standard or implementation itself. An example of an optimal control problem solver interfaced with AMPL is the “multiple shooting code for optimal control” MUSCOD-II, a direct and simultaneous method for ODE/DAE-constrained optimal control, and its extension MS-MINTOC for mixed-integer optimal control. As an example, we use the described AMPL extensions to model a heavy duty truck control problem.

Combinatorial integral approximation

We are interested in structures and efficient methods for mixed-integer nonlinear programs (MINLP) that arise from a first discretize, then optimize approach to time-dependent mixed-integer optimal control problems (MIOCPs). In this study we focus on combinatorial constraints, in particular on restrictions on the number of switches on a fixed time grid. We propose a novel approach that is based on a decomposition of the MINLP into a NLP and a MILP. We discuss the relation of the MILP solution to the MINLP solution and formulate bounds for the gap between the two, depending on Lipschitz constants and the control discretization grid size. The MILP solution can also be used for an efficient initialization of the MINLP solution process. The speedup of the solution of the MILP compared to the MINLP solution is considerable already for general purpose MILP solvers. We analyze the structure of the MILP that takes switching constraints into account and propose a tailored Branch and Bound strategy that outperforms cplex on a numerical case study and hence further improves efficiency of our novel method.

The integer approximation error in mixed-integer optimal control

We extend recent work on nonlinear optimal control problems with integer restrictions on some of the control functions (mixed-integer optimal control problems, MIOCP). We improve a theorem (Sager et al. in Math Program 118(1): 109–149, 2009) that states that the solution of a relaxed and convexified problem can be approximated with arbitrary precision by a solution fulfilling the integer requirements. Unlike in previous publications the new proof avoids the usage of the Krein-Milman theorem, which is undesirable as it only states the existence of a solution that may switch infinitely often. We present a constructive way to obtain an integer solution with a guaranteed bound on the performance loss in polynomial time. We prove that this bound depends linearly on the control discretization grid. A numerical benchmark example illustrates the procedure. As a byproduct, we obtain an estimate of the Hausdorff distance between reachable sets. We improve the approximation order to linear grid size h instead of the previously known result with order $${\sqrt{h}}$$ (Hackl in Reachable sets, control sets and their computation, augsburger mathematisch-naturwissenschaftliche schriften. Dr. Bernd Wisner, Augsburg, 1996). We are able to include a Special Ordered Set condition which will allow for a transfer of the results to a more general, multi-dimensional and nonlinear case compared to the Theorems in Pietrus and Veliov in (Syst Control Lett 58:395–399, 2009).

Efficient multiple objective optimal control of dynamic systems with integer controls

In practical optimal control problems both integer control variables and multiple objectives can be present. The current paper proposes a generic and efficient solution strategy for these multiple objective mixed-integer optimal control problems (MO-MIOCPs) based on deterministic approaches. Hereto, alternative scalar multiple objective optimisation techniques as normal boundary intersection and normalised normal constraint are used to convert the original problem into a series of parametric single objective optimisation problems. These single objective mixed-integer optimal control problems are then efficiently solved through direct multiple shooting techniques which exploit convex relaxations of the original problem. Moreover, these relaxations enable to quickly approximate the final solution to any desired accuracy (without the need of solving integer problems). Consequently, the set of Pareto optimal solutions of the MO-MIOCP can be accurately obtained in highly competitive computation times. The proposed method is illustrated on (i) a testdrive case study with a complex car model which includes different gears and conflicting minimum time–minimum fuel consumption objectives, and (ii) a jacketed tubular reactor case study with conflicting conversion, heat recovery and installation costs.

Mixed-integer non-linear optimal control in systems biology and biotechnology: numerical methods and a software toolbox

Here we consider the problem of optimal manipulation of biological or biotechno-logical systems, formulated as a class of mixed-integer optimal control problems. We describe the current state of the art regarding the numerical solution of these problems, and a software implementation developed in our group (DOTcvp toolbox, based on Matlab).DOTcvp combines the control vector parameterization approach with a number of local deterministic and global stochastic and hybrid (mixed-integer) non-linear programming solvers and suitable dynamic process simulation methods so as to cover the solution of a wide class of problems. The performance of DOTcvp is illustrated considering representative set of benchmark problems, including the problem of drug displacement in a patient, the optimal operation of a fed-batch bioreactor, and the optimal control of intracellular calcium oscillations. The DOTcvp toolbox is freely available to academic users.

Direct methods with maximal lower bound for mixed-integer optimal control problems

Many practical optimal control problems include discrete decisions. These may be either time-independent parameters or time-dependent control functions as gears or valves that can only take discrete values at any given time. While great progress has been achieved in the solution of optimization problems involving integer variables, in particular mixed-integer linear programs, as well as in continuous optimal control problems, the combination of the two is yet an open field of research. We consider the question of lower bounds that can be obtained by a relaxation of the integer requirements. For general nonlinear mixed-integer programs such lower bounds typically suffer from a huge integer gap. We convexify (with respect to binary controls) and relax the original problem and prove that the optimal solution of this continuous control problem yields the best lower bound for the nonlinear integer problem. Building on this theoretical result we present a novel algorithm to solve mixed-integer optimal control problems, with a focus on discrete-valued control functions. Our algorithm is based on the direct multiple shooting method, an adaptive refinement of the underlying control discretization grid and tailored heuristic integer methods. Its applicability is shown by a challenging application, the energy optimal control of a subway train with discrete gears and velocity limits.

Solving mixed-integer optimal control problems by branch&bound: a case study from automobile test-driving with gear shift

The article discusses the application of the branch&bound method to a mixed integer non-linear optimization problem (MINLP) arising from a discretization of an optimal control problem with partly discrete control set. The optimal control problem has its origin in automobile test-driving, where the car model involves a discrete-valued control function for the gear shift. Since the number of variables in (MINLP) grows with the number of grid points used for discretization of the optimal control problem, the example from automobile test-driving may serve as a benchmark problem of scalable complexity. Reference solutions are computed numerically for two different problem sizes. A simple heuristic approach suitable for optimal control problems is suggested that reduces the computational amount considerably, though it cannot guarantee optimality anymore. Copyright © 2005 John Wiley & Sons, Ltd.