Optimal Control (OC) is the process of determining control and state trajectories for a dynamic system, over a period of time, in order to optimize a given performance index. With …
Optimal Control (OC) is the process of determining control and state trajectories for a dynamic system, over a period of time, in order to optimize a given performance index. With the increasing of variables and complexity, OC problems can no longer be solved analytically and, consequently, numerical methods are required. For this purpose, direct and indirect methods are used. Direct methods consist in the discretization of the OC problem, reducing it to a nonlinear constrained optimization problem. Indirect methods are based on the Pontryagin Maximum Principle, which in turn reduces to a boundary value problem. In order to have a more reliable solution, one can solve the same problem through different approaches. Here, as an illustrative example, an epidemiological application related to the rubella disease is solved using several software packages, such as the routine ode45 of Matlab, OC-ODE, DOTcvp toolbox, IPOPT and Snopt, showing the state of the art of numerical software for OC.
Dymos is a library for optimizing control schedules for dynamic systems -sometimes referred to as optimal control or trajectory optimization.There are a number of other optimal control libraries that tackle …
Dymos is a library for optimizing control schedules for dynamic systems -sometimes referred to as optimal control or trajectory optimization.There are a number of other optimal control libraries that tackle similar kinds of problems, such as OTIS4 (Paris et al., 2006), GPOPS-II (Patterson & Rao, 2014),and CASADI (Andersson et al., 2019).These tools all rely on gradient-based optimization to solve optimal control problems, though their methods of computing the gradients vary.Dymos is built on top of the OpenMDAO framework (Gray et al., 2019) and supports its modular derivative system which allows users to mix-and-match from finite-differencing, complex-step, hand-differentiated, and algorithmic differentiation.This flexibility allows Dymos to efficiently solve optimal control problems constructed with both ordinary differential equations (ODE) and differential-algebraic equations (DAE).Dymos can also help solve more general optimization problems where dynamics are only one part in a larger system-level model with additional -potentially computationally expensivecalculations that come before and after the dynamic calculations.These broader problems are commonly referred to as co-design, controls-co-design, and multidisciplinary design optimization.Dymos provides specific APIs and features that make it possible to integrate traditional optimal-control models into a co-design context, while still supporting analytic derivatives that are necessary for computational efficiency in these complex use cases.An example of a co-design problem that was solved with Dymos is the coupled trajectory-thermal design of an electric vertical takeoff and landing aircraft where the thermal management and propulsion systems were designed simultaneously with the flight trajectories to ensure no components overheated (Hendricks et al., 2020). Difference between optimal-control and co-designOptimal-control and co-design problems deal with dynamic systems.The evolution of the states over time is governed by an ordinary differential equation (ODE) or differential-algebraic equation (DAE):Here, x is a vector of time-varying state variables whose behavior is affected by time (t), a vector of dynamic controls (ū), and a vector of static design parameters ( d).To optimize a dynamic system we also need to account for the objective function (J):
We introduce the MATLAB-based software QuITO (Quasi-Interpolation based Trajectory Optimization) to numerically solve a wide class of constrained nonlinear optimal control problems (OCP). The solver is based on the QuITO …
We introduce the MATLAB-based software QuITO (Quasi-Interpolation based Trajectory Optimization) to numerically solve a wide class of constrained nonlinear optimal control problems (OCP). The solver is based on the QuITO (the same abbreviation) algorithm, which is a direct multiple shooting (DMS) technique that leverages a particular type of quasi-interpolation scheme for control trajectory parameterization. The software is equipped with several options for numerical integration, and optimization solvers along with a Graphical User Interface (GUI) to make the process of designing and solving the OCPs smooth and seamless for users with minimum coding experience. We demonstrate with two benchmark numerical examples the procedure to generate constrained state and control trajectories using QuITO.
This letter introduces the NOnSmooth Numerical Optimal Control (NOSNOC) open-source software package. It is a modular MATLAB tool based on CasADi and IPOPT for numerically solving Optimal Control Problems (OCP) …
This letter introduces the NOnSmooth Numerical Optimal Control (NOSNOC) open-source software package. It is a modular MATLAB tool based on CasADi and IPOPT for numerically solving Optimal Control Problems (OCP) with piecewise smooth systems (PSS). The tool supports: 1) automatic reformulation of systems with state jumps into PSS (via the time-freezing reformulation [Nurkanovi\'c et al., 2021]) and of PSS into computationally more convenient forms, 2) automatic discretization of the OCP via, e.g., the recently introduced Finite Elements with Switch Detection [Nurkanovi\'c et al., 2022] which enables high accuracy optimal control and simulation of PSS, 3) solution methods for the resulting discrete-time OCP. The nonsmooth discrete-time OCP are solved with techniques of continuous optimization in a homotopy procedure, without the use of integer variables. This enables the treatment of a broad class of nonsmooth systems in a unified way. Two tutorial examples are given. A benchmark shows that NOSNOC provides both faster and more accurate solutions than conventional approaches, including mixed-integer formulations.
We introduce the MATLAB-based software QuITO (Quasi-Interpolation based Trajectory Optimization) to numerically solve a wide class of constrained nonlinear optimal control problems (OCP). The solver is based on the QuITO …
We introduce the MATLAB-based software QuITO (Quasi-Interpolation based Trajectory Optimization) to numerically solve a wide class of constrained nonlinear optimal control problems (OCP). The solver is based on the QuITO (the same abbreviation) algorithm, which is a direct multiple shooting (DMS) technique that leverages a particular type of quasi-interpolation scheme for control trajectory parameterization. The software is equipped with several options for numerical integration, and optimization solvers along with a Graphical User Interface (GUI) to make the process of designing and solving the OCPs smooth and seamless for users with minimum coding experience. We demonstrate with two benchmark numerical examples the procedure to generate constrained state and control trajectories using QuITO.
The classical numerical methods for differential equations are a well-studied field. Nevertheless, these numerical methods are limited in their scope to certain classes of equations. Modern machine learning applications, such …
The classical numerical methods for differential equations are a well-studied field. Nevertheless, these numerical methods are limited in their scope to certain classes of equations. Modern machine learning applications, such as equation discovery, may benefit from having the solution to the discovered equations. The solution to an arbitrary equation typically requires either an expert system that chooses the proper method for a given equation, or a method with a wide range of equation types. Machine learning methods may provide the needed versatility. This article presents a method that uses an optimization algorithm for a parameterized approximation to find a solution to a given problem. We take an agnostic approach without dividing equations by their type or boundary conditions, which allows for fewer restrictions on the algorithm. The results may not be as precise as those of an expert; however, our method enables automated solutions for a wide range of equations without the algorithm’s parameters changing. In this paper, we provide examples of the Legendre equation, Painlevé transcendents, wave equation, heat equation, and Korteweg–de Vries equation, which are solved in a unified manner without significant changes to the algorithm’s parameters.
The numerical methods for differential equation solution allow obtaining a discrete field that converges towards the solution if the method is applied to the correct problem. Nevertheless, the numerical methods …
The numerical methods for differential equation solution allow obtaining a discrete field that converges towards the solution if the method is applied to the correct problem. Nevertheless, the numerical methods have the restricted class of the equations, on which the convergence with a given parameter set or range is proved. Only a few "cheap and dirty" numerical methods converge on a wide class of equations without parameter tuning with the lower approximation order price. The article presents a method that uses an optimization algorithm to obtain a solution using the parameterized approximation. The result may not be as precise as an expert one. However, it allows solving the wide class of equations in an automated manner without the algorithm's parameters change.
<title>Abstract</title> Opytimal module is a Python/FEniCS framework to solve PDE-based optimization problems considering multiple controls. These problems can be defined in n-dimensional domains, for<bold> </bold><italic><bold>n </bold></italic><bold>∈ {1, 2, 3}</bold>, and …
<title>Abstract</title> Opytimal module is a Python/FEniCS framework to solve PDE-based optimization problems considering multiple controls. These problems can be defined in n-dimensional domains, for<bold> </bold><italic><bold>n </bold></italic><bold>∈ {1, 2, 3}</bold>, and complemented with mixed boundary conditions. The controls can be distributed or boundary type and may contribute to the cost function via the <bold>H</bold><sup><bold>1</bold></sup> norm. We show examples of the solution of control problems governed by the Poisson, Heat, and Stokes equations.
Combining direct and indirect methods to have the best of both worlds is an efficient method to solve numerically optimal control problems. A direct solver will typically provide information on …
Combining direct and indirect methods to have the best of both worlds is an efficient method to solve numerically optimal control problems. A direct solver will typically provide information on the structure of the optimal control, allowing an educated guess for indirect shooting. The control toolbox ct offers such possibilities and is presented on two examples. The first example has a bang-singular solution and is solved by chaining direct and indirect solvers. The second one consists in computing conjugate and cut loci on an ellipsoid of revolution, which is performed using a more advanced combination of indirect methods with differential continuation.
Overview of the FieldDifferential Algebraic Equations (DAEs) are mixed systems of differential and algebraic equations.It has been recognized for some time now that they have great potential both theoretically and …
Overview of the FieldDifferential Algebraic Equations (DAEs) are mixed systems of differential and algebraic equations.It has been recognized for some time now that they have great potential both theoretically and in applications.DAEs form one of the most elegant and simple ways to model a physical system because they allow for the creation of separate models for subcomponents that can then be pasted together via a network.As a consequence, this concept is used in many modern CAD/modeling systems like SIMULINK, Scicos and DYMOLA, although most software packages cannot fully exploit the full potential of DAE models.But this nice feature of DAEs for modeling has also a disadvantage, since it shifts all of the difficulties of a system onto the analysis and the numerical methods.For this reason in recent years much effort has been spent to analyze general DAEs and to derive suitable numerical methods either for general DAEs, see e.g.[9,19,18,20,29,38] or for special DAEs arising in applications, see e.g.[6,12,28,37].This analytical and numerical work so far has been primarily driven by the simulation community, where the desire was to simulate the behavior of a complex system which could be electrical, mechanical, chemical, or all three.Due to the ever growing complexity of models which pose new challenges, the field is developing rather rapidly including now also hybrid [2,3,4,11,21,31,39] and delay systems [16,17,25,26,27].Once a system can be modeled and simulated, there arises the need to control the process or optimize its performance.The control of physical processes is an important task in many applications and over the last two decades there have been tremendous advances in the theory and applications of control in almost all disciplines of science and technologies.Note that this includes not only the obvious applications such as designing a more efficient process, but also determining what are the control mechanisms inherent in complex biological systems and fitting models to data.
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 …
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.
Abstract COVID-19 pandemic response with non-pharmaceutical interventions is an intrinsic control problem. Governments weigh social distancing policies to avoid overload in the health system without significant economic impact. The mutability …
Abstract COVID-19 pandemic response with non-pharmaceutical interventions is an intrinsic control problem. Governments weigh social distancing policies to avoid overload in the health system without significant economic impact. The mutability of the SARS-CoV-2 virus, vaccination coverage, and mobility restriction measures change epidemic dynamics over time. A model-based control strategy requires reliable predictions to be efficient on a long-term basis. In this paper, a SEIR-based model is proposed considering dynamic feedback estimation. State and parameter estimations are performed on state estimators using augmented states. Three methods were implemented: constrained extended Kalman filter (CEKF), CEKF and smoother (CEKF & S), and moving horizon estimator (MHE). The parameters estimation was based on vaccine efficacy studies regarding transmissibility, severity of the disease, and lethality. Social distancing was assumed as a measured disturbance calculated using Google mobility data. Data from six federative units from Brazil were used to evaluate the proposed strategy. State and parameter estimations were performed from 1 October 2020 to 1 July 2021, during which Zeta and Gamma variants emerged. Simulation results showed that lethality increased between 11 and 30% for Zeta mutations and between 44 and 107% for Gamma mutations. In addition, transmissibility increased between 10 and 37% for the Zeta variant and between 43 and 119% for the Gamma variant. Furthermore, parameter estimation indicated temporal underreporting changes in hospitalized and deceased individuals. Overall, the estimation strategy showed to be suitable for dynamic feedback as simulation results presented an efficient detection and dynamic characterization of circulating variants.
Abstract Non-convex discrete-time optimal control problems in, e.g. , water or power systems, typically involve a large number of variables related through nonlinear equality constraints. The ideal goal is to …
Abstract Non-convex discrete-time optimal control problems in, e.g. , water or power systems, typically involve a large number of variables related through nonlinear equality constraints. The ideal goal is to find a globally optimal solution, and numerical experience indicates that algorithms aiming for Karush–Kuhn–Tucker points often find solutions that are indistinguishable from global optima. In our paper, we provide a theoretical underpinning for this phenomenon, showing that on a broad class of problems the objective can be shown to be an invariant convex function ( invex function) of the control decision variables when state variables are eliminated using implicit function theory. In this way, optimality guarantees can be obtained, the exact nature of which depends on the position of the solution within the feasible set. In a numerical example, we show how high-quality solutions are obtained with local search for a river control problem where invexity holds.
In recent years, the SUite of Nonlinear and DIfferential/ALgebraic equation Solvers (SUNDIALS) has been redesigned to better enable the use of application-specific and third-party algebraic solvers and data structures. Throughout …
In recent years, the SUite of Nonlinear and DIfferential/ALgebraic equation Solvers (SUNDIALS) has been redesigned to better enable the use of application-specific and third-party algebraic solvers and data structures. Throughout this work, we have adhered to specific guiding principles that minimized the impact to current users while providing maximum flexibility for later evolution of solvers and data structures. The redesign was done through the addition of new linear and nonlinear solvers classes, enhancements to the vector class, and the creation of modern Fortran interfaces. The vast majority of this work has been performed "behind-the-scenes," with minimal changes to the user interface and no reduction in solver capabilities or performance. These changes allow SUNDIALS users to more easily utilize external solver libraries and create highly customized solvers, enabling greater flexibility on extreme-scale, heterogeneous computational architectures.
We propose a learning-based model predictive control framework for mitigating the spread of epidemics. We capture the epidemic spreading process using a susceptible-infected-removed (SIR) epidemic model and consider testing for …
We propose a learning-based model predictive control framework for mitigating the spread of epidemics. We capture the epidemic spreading process using a susceptible-infected-removed (SIR) epidemic model and consider testing for isolation as the control strategy. In the framework, we use a daily testing strategy to remove (isolate) a portion of the infected population. Our goal is to keep the daily infected population below a certain level, while minimizing the total number of tests. Distinct from existing works on leveraging model predictive control in epidemic spreading, we learn the model parameters and compute the feedback control signal simultaneously. We illustrate the results by numerical simulation using COVID-19 data from India.
We provide an overview of a class of iterative convex approximation methods for nonlinear optimization problems with convex-over-nonlinear substructure. These problems are characterized by outer convexities on the one hand, …
We provide an overview of a class of iterative convex approximation methods for nonlinear optimization problems with convex-over-nonlinear substructure. These problems are characterized by outer convexities on the one hand, and nonlinear, generally nonconvex, but differentiable functions on the other hand. All methods from this class use only first order derivatives of the nonlinear functions and sequentially solve convex optimization problems. All of them are different generalizations of the classical Gauss-Newton (GN) method. We focus on the smooth constrained case and on three methods to address it: Sequential Convex Programming (SCP), Sequential Convex Quadratic Programming (SCQP), and Sequential Quadratically Constrained Quadratic Programming (SQCQP). While the first two methods were previously known, the last is newly proposed and investigated in this paper. We show under mild assumptions that SCP, SCQP and SQCQP have exactly the same local linear convergence – or divergence – rate. We then discuss the special case in which the solution is fully determined by the active constraints, and show that for this case the KKT conditions are sufficient for local optimality and that SCP, SCQP and SQCQP even converge quadratically. In the context of parameter estimation with symmetric convex loss functions, the possible divergence of the methods can in fact be an advantage that helps them to avoid some undesirable local minima: generalizing existing results, we show that the presented methods converge to a local minimum if and only if this local minimum is stable against a mirroring operation applied to the measurement data of the estimation problem. All results are illustrated by numerical experiments on a tutorial example.
The optimization of a controlled process in a simulation without access to the model itself is a common scenario and very relevant to many chemical engineering applications. A general approach …
The optimization of a controlled process in a simulation without access to the model itself is a common scenario and very relevant to many chemical engineering applications. A general approach is to apply a black-box optimization algorithm to a parameterized control scheme. The success then depends on the quality of the parametrization that should be low-dimensional though rich enough to express the salient features. This work proposes using solution snapshots to extract dominant modes of the temporal dynamics of a process and use them for low-dimensional parametrizations of control functions. The approach is called proper orthogonal decomposition in time (time-POD). We provide theoretical reasoning and illustrate the performance for the optimal control of a methanation reactor.
Differential-Algebraic Equations (DAEs) are the foundation of high-level equation-based languages for modeling physical dynamical systems. Simulating models in such languages requires a transformation known as index reduction that involves differentiating …
Differential-Algebraic Equations (DAEs) are the foundation of high-level equation-based languages for modeling physical dynamical systems. Simulating models in such languages requires a transformation known as index reduction that involves differentiating individual equations before numerical integration. Commercial and open-source implementations typically perform index reduction by symbolic differentiation (SD) and produce a Jacobian callback function with forward-mode automatic differentiation (AD). The former results in efficient runtime code, and the latter is asymptotically efficient in both runtime and code size. However, AD introduces runtime overhead caused by a non-standard representation of real numbers, and SD is not always applicable in models with general recursion. This work proposes a new approach that uses partial evaluation of AD in the context of numerical DAE solving to combine the strengths of the two differentiation methods while mitigating their weaknesses. Moreover, our approach selectively specializes partial derivatives of the Jacobian by exploiting structural knowledge while respecting a user-defined bound on the code size. Our evaluation shows that the new method both enables expressive modeling from AD and retains the efficiency of SD for many practical applications.
In this work we discuss the limits of direct methods for Optimal Control Problems (OCPs) for some classes of nonsmooth dynamic systems. We highlight the equivalence between Filippov systems and …
In this work we discuss the limits of direct methods for Optimal Control Problems (OCPs) for some classes of nonsmooth dynamic systems. We highlight the equivalence between Filippov systems and a subclass of Differential Complementarity Systems (DCSs). Direct methods for optimal control with DCSs yield Mathematical Programs with Complementarity Constraints (MPCC), which are often solved with relaxation or smoothing methods. Due to the equivalence, results from the first class transfer to the DCSs. Therefore, to get the right numerical sensitivities one has to have a step-size of h = o(σ), where σ is the relaxation or smoothing parameter in the MPCC. A possible consequence of wrong numerical sensitivities is the appearance of spurious local solutions of the discretized OCP. We demonstrate and highlight the limits of MPCC approaches in direct optimal control on a simple counterexample.
This paper discusses a bilevel optimization approach for free finite final time optimal control problems and addresses a numerical method for their approximate solution. The core idea is to decouple …
This paper discusses a bilevel optimization approach for free finite final time optimal control problems and addresses a numerical method for their approximate solution. The core idea is to decouple the final time optimization from the optimal control and state trajectory. This is rigorously formulated as an equivalent bilevel problem seeking, at the upper level, the optimal final time and optimal control and corresponding state at the lower level. Standard solvers for nonlinear optimal control can deal with the latter, while the former is a box-constrained optimization problem with one scalar decision variable. The interface between the two levels is based on the Hamilton function associated to the problem and its relationship with the cost function. A method for solving the upper level problem is developed, that combines a tailored fast first-order method with a robust and guaranteed root-finding algorithm. Finally, numerical results demonstrate the robustness of the method and show its limitations.
Multibang regularization and combinatorial integral approximation decompositions are two actively researched techniques for integer optimal control. We consider a class of polyhedral functions that arise particularly as convex lower envelopes …
Multibang regularization and combinatorial integral approximation decompositions are two actively researched techniques for integer optimal control. We consider a class of polyhedral functions that arise particularly as convex lower envelopes of multibang regularizers and show that they have beneficial properties with respect to regularization of relaxations of integer optimal control problems. We extend the algorithmic framework of the combinatorial integral approximation such that a subsequence of the computed discrete-valued controls converges to the infimum of the regularized integer control problem.
This paper considers the problem of computing Bayesian estimates of both states and model parameters for nonlinear state-space models. Generally, this problem does not have a tractable solution and approximations …
This paper considers the problem of computing Bayesian estimates of both states and model parameters for nonlinear state-space models. Generally, this problem does not have a tractable solution and approximations must be utilised. In this work, a variational approach is used to provide an assumed density which approximates the desired, intractable, distribution. The approach is deterministic and results in an optimisation problem of a standard form. Due to the parametrisation of the assumed density selected first and second order derivatives are readily available which allows for efficient solutions. The proposed method is compared against state-of-the-art Hamiltonian Monte Carlo in two numerical examples.
Solving fractional optimal control problems (FOCPs) with an approximate analytical method has been widely studied by many authors, but to guarantee the convergence of the series solution has been a …
Solving fractional optimal control problems (FOCPs) with an approximate analytical method has been widely studied by many authors, but to guarantee the convergence of the series solution has been a challenge. We solved this by integrating the Galerkin method of optimization technique into the whole region of the governing equations for accurate optimal values of control-convergence parameters <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"> <msubsup> <mi>C</mi> <mi>j</mi> <mi>s</mi> </msubsup> </math> . The arbitrary-order derivative is in the conformable fractional derivative sense. We use Euler–Lagrange equation form of necessary optimality conditions for FOCPs, and the arising fractional differential equations (FDEs) are solved by optimal homotopy asymptotic method (OHAM). The OHAM technique speedily provides the convergent approximate analytical solution as the arbitrary order derivative approaches 1. The convergence of the method is discussed, and its effectiveness is verified by some illustrative test examples.
In this article, by using the Brunovsky normal form, we provide a reformulation of the problem consisting in finding the actuator design which minimizes the controllability cost for finite-dimensional linear …
In this article, by using the Brunovsky normal form, we provide a reformulation of the problem consisting in finding the actuator design which minimizes the controllability cost for finite-dimensional linear systems with scalar controls. Such systems may be seen as spatially discretized linear partial differential equations with lumped controls. The change of coordinates induced by Brunovsky’s normal form allows us to remove the restriction of having to work with diagonalizable system dynamics and does not entail a randomization procedure as done in past literature on diffusion equations or waves. Instead, the optimization problem reduces to a minimization of the norm of the inverse of a change of basis matrix and allows for an easy deduction of existence of solutions, and for a clearer picture of some of the problem’s intrinsic symmetries. Numerical experiments help to visualize these properties, indicate further open problems, and also show a possible obstruction of using gradient-based algorithms—this is alleviated by using an evolutionary algorithm.
Plasma medicine has emerged as a promising approach for treatment of biofilm-related and virus infections, assistance in cancer treatment, and treatment of wounds and skin diseases. Despite advances in learning-based …
Plasma medicine has emerged as a promising approach for treatment of biofilm-related and virus infections, assistance in cancer treatment, and treatment of wounds and skin diseases. Despite advances in learning-based and predictive control of plasma medical devices, there remain major challenges towards personalized and point-of-care plasma medicine. In particular, an important challenge arises from the need to adapt control policies after each treatment using (often limited) observations of therapeutic effects that can only be measured between treatments. Control policy adaptation is necessary to account for variable characteristics of plasma and target surfaces across different subjects and treatment scenarios, thus personalizing the plasma treatment to enhance its efficacy. To this end, this paper presents a data-efficient, "globally" optimal strategy to adapt deep learning-based controllers that can be readily embedded on resource-limited hardware for portable medical devices. The proposed strategy employs multi-objective Bayesian optimization to adapt parameters of a deep neural network (DNN)-based control law using observations of closed-loop performance measures. The proposed strategy for adaptive DNN-based control is demonstrated experimentally on a cold atmospheric plasma jet with prototypical applications in plasma medicine.
Efficient integrators with sensitivity propagation are an essential ingredient for the numerical solution of optimal control problems. This paper gives an overview on the acados integrators, their Python interface and …
Efficient integrators with sensitivity propagation are an essential ingredient for the numerical solution of optimal control problems. This paper gives an overview on the acados integrators, their Python interface and presents a workflow that allows using them with their sensitivities within a nonlinear programming (NLP) solver interfaced by CashDi. The implementation is discussed, demonstrated and provided as open-source software. The computation times of the proposed integrator and its sensitivity computation are compared to the native CasADi collocation integrator, CVODES and IDhS on different examples. A speedup of one order of magnitude for simulation and of up to three orders of magnitude for the forward sensitivity propagation is shown for an airborne wind energy system model.
Abstract This paper examines solution methods for mathematical programs with complementarity constraints (MPCC) obtained from the time-discretization of optimal control problems (OCPs) subject to nonsmooth dynamical systems. The MPCC theory …
Abstract This paper examines solution methods for mathematical programs with complementarity constraints (MPCC) obtained from the time-discretization of optimal control problems (OCPs) subject to nonsmooth dynamical systems. The MPCC theory and stationarity concepts are reviewed and summarized. The focus is on relaxation-based methods for MPCCs, which solve a (finite) sequence of more regular nonlinear programs (NLP), where a regularization/homotopy parameter is driven to zero. Such methods perform reasonably well on currently available benchmarks. However, these results do not always generalize to MPCCs obtained from nonsmooth OCPs. To provide a more complete picture, this paper introduces a novel benchmark collection of such problems, which we call . The problem set includes 603 different MPCCs and we split it into a few representative subsets to accelerate the testing. We compare different relaxation-based methods, NLP solvers, homotopy parameter update and relaxation parameter steering strategies. Moreover, we check whether the obtained stationary points allow first-order descent directions, which may be the case for some of the weaker MPCC stationarity concepts. In the best case, the Scholtes’ relaxation (SIAM J. Optim. 11 , 918–936, 2001) with (Math. Program. 106 , 25–57, 2006) as NLP solver manages to solve 73.8% of the problems. This highlights the need for further improvements in algorithms and software for MPCCs.
This paper introduces an open-source software for distributed and decentralized non-convex optimization named ALADIN-$\alpha$. ALADIN-$\alpha$ is a MATLAB implementation of the Augmented Lagrangian Alternating Direction Inexact Newton (ALADIN) algorithm, which …
This paper introduces an open-source software for distributed and decentralized non-convex optimization named ALADIN-$\alpha$. ALADIN-$\alpha$ is a MATLAB implementation of the Augmented Lagrangian Alternating Direction Inexact Newton (ALADIN) algorithm, which is tailored towards rapid prototyping for non-convex distributed optimization. An improved version of the recently proposed bi-level variant of ALADIN is included enabling decentralized non-convex optimization. A collection of application examples from different applications fields including chemical engineering, robotics, and power systems underpins the application potential of ALADIN-$\alpha$.
This letter introduces the NOnSmooth Numerical Optimal Control (NOSNOC) open-source software package. It is a modular MATLAB tool based on CasADi and IPOPT for numerically solving Optimal Control Problems (OCP) …
This letter introduces the NOnSmooth Numerical Optimal Control (NOSNOC) open-source software package. It is a modular MATLAB tool based on CasADi and IPOPT for numerically solving Optimal Control Problems (OCP) with piecewise smooth systems (PSS). The tool supports: 1) automatic reformulation of systems with state jumps into PSS (via the time-freezing reformulation [Nurkanovi\'c et al., 2021]) and of PSS into computationally more convenient forms, 2) automatic discretization of the OCP via, e.g., the recently introduced Finite Elements with Switch Detection [Nurkanovi\'c et al., 2022] which enables high accuracy optimal control and simulation of PSS, 3) solution methods for the resulting discrete-time OCP. The nonsmooth discrete-time OCP are solved with techniques of continuous optimization in a homotopy procedure, without the use of integer variables. This enables the treatment of a broad class of nonsmooth systems in a unified way. Two tutorial examples are given. A benchmark shows that NOSNOC provides both faster and more accurate solutions than conventional approaches, including mixed-integer formulations.
We present an efficient transcription method for highly oscillatory optimal control problems. For these problems, the optimal state trajectory consists of fast oscillations that change slowly over the time horizon. …
We present an efficient transcription method for highly oscillatory optimal control problems. For these problems, the optimal state trajectory consists of fast oscillations that change slowly over the time horizon. Out of a large number of oscillations, we only simulate a subset to approximate the slow change by constructing a semi-explicit differential-algebraic equation that can be integrated with integration steps much larger than one period. For the solution of optimal control problems with direct methods, we provide a way to parametrize the controls and regularize the state trajectory. Finally, we utilize the method to find a fuel-optimal orbit transfer of a low-thrust satellite. Using the novel method, we reduce the size of the resulting nonlinear program by more than one order of magnitude.
We present a novel reformulation of nonsmooth differential equations with state jumps which enables their easier simulation and use in optimal control problems without the need of using integer variables. …
We present a novel reformulation of nonsmooth differential equations with state jumps which enables their easier simulation and use in optimal control problems without the need of using integer variables. The main idea is to introduce an auxiliary differential equation to mimic the state jump map. Thereby, also a clock state is introduced which does not evolve during the runtime of the auxiliary system. The pieces of the trajectory that correspond to the parts when the clock state was evolving recover the solution of the original system with jumps. Our reformulation results in nonsmooth ordinary differential equations where the discontinuity is in the first time derivative of the trajectory, rather than in the trajectory itself. This class of systems is easier to handle both theoretically and numerically. We provide numerical examples demonstrating the ease of use of this reformulation in both simulation and optimal control. In the optimal control example a single call of a nonlinear programming (NLP) solver yields the same solution as a multi-stage formulation, without the need for exploring the optimal number of stages by enumeration or heuristics.
.Motivated by recent laboratory experiments, we study microbial populations with light-inducible genetic differentiation that generates a two-species microbial consortium relevant for bioproduction. First, we derive a hierarchy of models describing …
.Motivated by recent laboratory experiments, we study microbial populations with light-inducible genetic differentiation that generates a two-species microbial consortium relevant for bioproduction. First, we derive a hierarchy of models describing the evolution of the microbial populations, each with decreasing complexity. This sequential order reduction reveals the connections between several popular classes of models used in this context. Second, we demonstrate the analytical insight the order reduction provides by studying the optimal control of such a reduced-order system of nonlinear ordinary differential equations. Appealing to Pontryagin's maximum principle, we find different optimal control structures within different regions of the parameter space. Explicit solutions are obtained in a subset of parameter space, while, for the remainder of parameter space, closed-form solutions are obtained that depend on a scalar value that solves a particular transcendental equation. We show that a unique solution of the scalar equation exists and lies in a known compact interval, making its numerical approximation particularly easy. The analytical results are verified against direct numerical calculations.Keywordsmicrobial consortiabioengineeringoptimal controlmathematical modelingMSC codes92-1093C1534H0549K15
<p>Cancer is a complex group of diseases characterized by uncontrolled cell growth that can spread throughout the body, leading to serious health issues. Traditional treatments mainly include chemotherapy, surgery, and …
<p>Cancer is a complex group of diseases characterized by uncontrolled cell growth that can spread throughout the body, leading to serious health issues. Traditional treatments mainly include chemotherapy, surgery, and radiotherapy. Although combining different therapies is becoming more common, predicting how these treatments will interact and what side effects they may cause, such as gastrointestinal or neurological problems, can be challenging. This research applies optimal control theory (OCT) to create precise and personalized treatment plans for cancer patients. OCT helps identify the most effective doses of chemotherapy and immunotherapy by forecasting how various treatment combinations will impact tumor growth and the immune response over time. It optimizes the integration of chemotherapy with immunotherapy to minimize side effects while maximizing therapeutic benefits. The study proposes a model for managing malignant tumors using a mix of immunotherapy, vaccines, and chemotherapy. The aim is to develop the best treatment plan that reduces new tumor growth while keeping healthy cells stable. It also takes into account individual differences among patients, including variations in tumor biology and immune responses in both younger and older individuals. To do this, we compared different optimal control strategies: interior point optimization (IPOPT), an open-source tool for nonlinear optimization; state-dependent Riccati equation (SDRE), which adapts linear control methods for nonlinear situations; and approximate sequence Riccati equation (ASRE), a globally optimal feedback control approach for nonlinear systems. The optimization criterion showed that the proposed work achieved a cost value of 52.3573 for IPOPT, compared with 52.424 for both SDRE and ASRE. For $ \mathrm{C}\mathrm{D}{8}^{+} $ T cells, the proposed method maintained a consistent value of 1.6499 for continuous (C) and dosed (D) across all techniques. Tumor cell counts had a C value of 0.0007 for IPOPT, compared with 0.0006 for ISDRE and ASRE, with D values remaining at 0 across all methods. This comparison demonstrates the successful use of control theory techniques and highlights their potential for developing personalized and effective treatment strategies for complex cancer cases. By optimizing treatment schedules and dosages, OCT can help minimize the side effects of cancer therapies, thereby enhancing patients' overall quality of life.</p>
In order to anticipate rare and impactful events, we propose to quantify the worst-case risk under distributional ambiguity using a recent development in kernel methods -- the kernel mean embedding. …
In order to anticipate rare and impactful events, we propose to quantify the worst-case risk under distributional ambiguity using a recent development in kernel methods -- the kernel mean embedding. Specifically, we formulate the generalized moment problem whose ambiguity set (i.e., the moment constraint) is described by constraints in the associated reproducing kernel Hilbert space in a nonparametric manner. We then present the tractable approximation and its theoretical justification. As a concrete application, we numerically test the proposed method in characterizing the worst-case constraint violation probability in the context of a constrained stochastic control system.
We propose a fast optimization algorithm to find feasible solutions for a special class of switching structured problems. A type of continuous optimization problem that has switching structured constraints is …
We propose a fast optimization algorithm to find feasible solutions for a special class of switching structured problems. A type of continuous optimization problem that has switching structured constraints is called mathematical programs with switching constraints (MPSC). Relaxation methods are well known gradient descent-based approaches for solving MPSC. However, with the use of these conventional algorithms, we reveal that the sequence of solutions easily converges to an infeasible stationary point due to a special class of switching constraint whose feasible set is geometrically separated into mutually exclusive sets in variable space. We define this kind of switching constraint as the disjunctive allowable set constraint (DAS-constraint). To force the sequence of solutions to escape from such infeasible stationary points, we construct a new algorithm that introduces random sampling if convergence to an infeasible point is detected. Furthermore, to reduce the computation cost due to random sampling, we randomize only specific variables that are relevant to DAS-constraints. Numerical experiments show that feasible solutions can be found by using our proposed algorithm, even for large-scale optimization problems where no solutions are found within a practical time limit when using a conventional algorithm.
Atmospheric pressure plasma jets (APPJs) are increasingly used for biomedical applications. Reproducible and effective operation of APPJs hinges on controlling the nonlinear effects of plasma on a target substrate in …
Atmospheric pressure plasma jets (APPJs) are increasingly used for biomedical applications. Reproducible and effective operation of APPJs hinges on controlling the nonlinear effects of plasma on a target substrate in the face of intrinsic variabilities of the plasma as well as exogenous disturbances. This paper presents a low-memory fast approximate nonlinear model predictive control (NMPC) strategy for an APPJ with prototypical applications in plasma medicine. The NMPC objective is to regulate the delivery of the cumulative thermal effects of plasma to a substrate, while adhering to constraints pertaining to a patient's safety and comfort. Deep neural networks are used to approximate the implicit NMPC law with a cheap-to-evaluate explicit control law that has low memory requirements. Robust constraint satisfaction is guaranteed by projecting the output of the neural network onto a set that ensures the state stays within an appropriately defined invariant set. Closed-loop simulations and real-time control experiments indicate that the proposed approximate NMPC strategy is effective in handling nonlinear control costs at fast sampling times, while guaranteeing satisfaction of safety-critical system constraints. This work takes a crucial step toward fast NMPC of safety-critical plasma applications using resource-limited embedded control hardware.
Advection-Diffusion-Reaction (ADR) Partial Differential Equations (PDEs) appear in a wide spectrum of applications such as chemical reactors, concentration flows, and biological systems. A large number of these applications require the …
Advection-Diffusion-Reaction (ADR) Partial Differential Equations (PDEs) appear in a wide spectrum of applications such as chemical reactors, concentration flows, and biological systems. A large number of these applications require the solution of ADR equations involving time-varying coefficients, where analytical solutions are usually intractable. Numerical solutions on the other hand require fine discretization and are computationally very demanding. Consequently, the models are normally not suitable for real-time monitoring and control purposes. In this contribution, a reduced order modeling method for a general ADR system with time-varying coefficients is proposed. Optimality of the reduced order model regarding the reduction induced error is achieved by using an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathcal{H}_{2}$</tex> -norm reduction method. The efficacy of the method is demonstrated using two test cases. Namely, a case for an ADR with arbitrary dynamics varying coefficients and a second case including the modeling of an exemplary water quality distribution path with randomly generated demand. The reduced order models are evaluated against high fidelity simulations using MATLAB's finite element method PDE tool-box. It is shown that the reduction can achieve a significant computational speedup allowing for the usage of the model for real-time applications with sampling times in milliseconds range. Moreover, the constructed ROM is shown to achieve high prediction accuracy with the normalized mean square error below <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$2.3 \%$</tex> for a real-world water quality simulation test case.
We present Optimization Engine (OpEn): an open-source code generation framework for real-time embedded nonconvex optimization, which implements a novel numerical method. OpEn combines the proximal averaged Newton-type method for optimal …
We present Optimization Engine (OpEn): an open-source code generation framework for real-time embedded nonconvex optimization, which implements a novel numerical method. OpEn combines the proximal averaged Newton-type method for optimal control (PANOC) with the penalty and augmented Lagrangian methods to compute approximate stationary points of nonconvex problems. The proposed method involves very simple algebraic operations such as vector products, has a low memory footprint and exhibits very good convergence properties that allow the solution of nonconvex problems on embedded devices. OpEn's core solver is written is Rust — a modern, high-performance, memory-safe and thread-safe systems programming language — while users can call it from Python, MATLAB, C, C++, ROS or over a TCP socket.
1 Abstract Determining COVID-19 vaccination strategies presents many challenges in light of limited vaccination capacity and the heterogeneity of affected communities. Who should be prioritized for early vaccination when different …
1 Abstract Determining COVID-19 vaccination strategies presents many challenges in light of limited vaccination capacity and the heterogeneity of affected communities. Who should be prioritized for early vaccination when different groups manifest different levels of risks and contact rates? Answering such questions often becomes computationally intractable given that network size can exceed millions. We obtain a framework to compute the optimal vaccination strategy within seconds to minutes from among all strategies, including highly dynamic ones that adjust vaccine allocation as often as required, and even with modest computation resources. We then determine the optimal strategy for a large range of parameter values representative of various US states, countries, and case studies including retirement homes and prisons. The optimal is almost always one of a few candidate strategies, and, even when not, the suboptimality of the best among these candidates is minimal. Further, we find that many commonly deployed vaccination strategies, such as vaccinating the high risk group first, or administering second doses without delay, can often incur higher death rates, hospitalizations, and symptomatic counts. Our framework can be easily adapted to future variants or pandemics through appropriate choice of the compartments of the disease and parameters.
Prediction of the progression of an infectious disease outbreak in a population is an important task. Differential equations are often used to model an epidemic outbreak's behaviour but are challenging …
Prediction of the progression of an infectious disease outbreak in a population is an important task. Differential equations are often used to model an epidemic outbreak's behaviour but are challenging to parametrise. Furthermore, these models can suffer from misspecification, which biases the estimates. Stochastic models can help with misspecification but are even more expensive to simulate and perform inference with. Here, we develop an explicitly likelihood-based variation of the generalised profiling method as a tool for prediction and inference under model misspecification. Our approach allows us to carry out identifiability analysis and uncertainty quantification using profile likelihood-based methods without the need for marginalisation. We provide additional justification for this approach by introducing a new interpretation of the model approximation component as a stochastic constraint. This interpretation preserves the rationale for using profiling rather than integration to remove nuisance parameters while still providing a link back to stochastic models. We applied an initial version of this method during an outbreak of measles in Samoa in 2019-2020 and found that we achieved relatively fast, accurate predictions. Here present the most recent version of our method and its application to this measles outbreak, along with additional validation.
In this paper, a predictive-control-based approach is proposed for pandemic mitigation with multiple control inputs.Using previous results on the dynamical modeling of symptom-based testing, the testing intensity is introduced as …
In this paper, a predictive-control-based approach is proposed for pandemic mitigation with multiple control inputs.Using previous results on the dynamical modeling of symptom-based testing, the testing intensity is introduced as a new manipulable input to the control system model in addition to the stringency of non-pharmaceutical measures.The control objective is the minimization of the severity of interventions, while the main constraints are the bounds on the daily number of hospitalized people and on the total number of available tests.For the control design and simulation, a nonlinear dynamical model containing 14 compartments is used, where the effect of vaccination is also taken into consideration.The computation results clearly show that the optimization-based design of testing intensity significantly reduces the stringency of the measures to be introduced to reach the control goal and fulfill the prescribed constraints.
Prediction of the progression of an infectious disease outbreak is important for planning and coordinating a response. Differential equations are often used to model an epidemic outbreak's behaviour but are …
Prediction of the progression of an infectious disease outbreak is important for planning and coordinating a response. Differential equations are often used to model an epidemic outbreak's behaviour but are challenging to parameterise. Furthermore, these models can suffer from misspecification, which biases predictions and parameter estimates. Stochastic models can help with misspecification but are even more expensive to simulate and perform inference with. Here, we develop an explicitly likelihood-based variation of the generalised profiling method as a tool for prediction and inference under model misspecification. Our approach allows us to carry out identifiability analysis and uncertainty quantification using profile likelihood-based methods without the need for marginalisation. We provide justification for this approach by introducing a new interpretation of the model approximation component as a stochastic constraint. This preserves the rationale for using profiling rather than integration to remove nuisance parameters while also providing a link back to stochastic models. We applied an initial version of this method during an outbreak of measles in Samoa in 2019-2020 and found that it achieved relatively fast, accurate predictions. Here we present the most recent version of our method and its application to this measles outbreak, along with additional validation.
Iterative optimization algorithms depend on access to information about the objective function. In a differentiable programming framework, this information, such as gradients, can be automatically derived from the computational graph. …
Iterative optimization algorithms depend on access to information about the objective function. In a differentiable programming framework, this information, such as gradients, can be automatically derived from the computational graph. We explore how nonlinear control algorithms, often employing linear and/or quadratic approximations, can be effectively cast within this framework. Our approach illuminates shared components and differences between gradient descent, Gauss–Newton, Newton, and differential dynamic programming methods in the context of discrete time nonlinear control. Furthermore, we present line-search strategies and regularized variants of these algorithms, along with a comprehensive analysis of their computational complexities. We study the performance of the aforementioned algorithms on various nonlinear control benchmarks, including autonomous car racing simulations using a simplified car model. All implementations are publicly available in a package coded in a differentiable programming language.
Graph coloring has been employed since the 1980s to efficiently compute sparse Jacobian and Hessian matrices using either finite differences or automatic differentiation. Several coloring problems occur in this context, …
Graph coloring has been employed since the 1980s to efficiently compute sparse Jacobian and Hessian matrices using either finite differences or automatic differentiation. Several coloring problems occur in this context, depending on whether the matrix is a Jacobian or a Hessian, and on the specifics of the computational techniques employed. We consider eight variant vertex coloring problems here. This article begins with a gentle introduction to the problem of computing a sparse Jacobian, followed by an overview of the historical development of the research area. Then we present a unifying framework for the graph models of the variant matrix estimation problems. The framework is based upon the viewpoint that a partition of a matrix into structurally orthogonal groups of columns corresponds to distance-2 coloring an appropriate graph representation. The unified framework helps integrate earlier work and leads to fresh insights; enables the design of more efficient algorithms for many problems; leads to new algorithms for others; and eases the task of building graph models for new problems. We report computational results on two of the coloring problems to support our claims. Most of the methods for these problems treat a column or a row of a matrix as an atomic entity, and partition the columns or rows (or both). A brief review of methods that do not fit these criteria is provided. We also discuss results in discrete mathematics and theoretical computer science that intersect with the topics considered here.
Sequential quadratic programming (SQP) methods have proved highly effective for solving constrained optimization problems with smooth nonlinear functions in the objective and constraints. Here we consider problems with general inequality …
Sequential quadratic programming (SQP) methods have proved highly effective for solving constrained optimization problems with smooth nonlinear functions in the objective and constraints. Here we consider problems with general inequality constraints (linear and nonlinear). We assume that first derivatives are available and that the constraint gradients are sparse. Second derivatives are assumed to be unavailable or too expensive to calculate. We discuss an SQP algorithm that uses a smooth augmented Lagrangian merit function and makes explicit provision for infeasibility in the original problem and the QP subproblems. The Hessian of the Lagrangian is approximated using a limited-memory quasi-Newton method. SNOPT is a particular implementation that uses a reduced-Hessian semidefinite QP solver (SQOPT) for the QP subproblems. It is designed for problems with many thousands of constraints and variables but is best suited for problems with a moderate number of degrees of freedom (say, up to 2000). Numerical results are given for most of the CUTEr and COPS test collections (about 1020 examples of all sizes up to 40000 constraints and variables, and up to 20000 degrees of freedom).
Receding horizon control requires the solution of an optimization problem at every sampling instant. We present efficient interior point methods tailored to convex multistage problems, a problem class which most …
Receding horizon control requires the solution of an optimization problem at every sampling instant. We present efficient interior point methods tailored to convex multistage problems, a problem class which most relevant MPC problems with linear dynamics can be cast in, and specify important algorithmic details required for a high speed implementation with superior numerical stability. In particular, the presented approach allows for quadratic constraints, which is not supported by existing fast MPC solvers. A categorization of widely used MPC problem formulations into classes of different complexity is given, and we show how the computational burden of certain quadratic or linear constraints can be decreased by a low rank matrix forward substitution scheme. Implementation details are provided that are crucial to obtain high speed solvers.We present extensive numerical studies for the proposed methods and compare our solver to three well-known solver packages, outperforming the fastest of these by a factor 2-5 in speed and 3-70 in code size. Moreover, our solver is shown to be very efficient for large problem sizes and for quadratically constrained QPs, extending the set of systems amenable to advanced MPC formulations on low-cost embedded hardware.
Abstract A near-optimal feedback control law for a distributed parameter system with a quadratic performance index is obtained by the method of trajectory approximation. The system equation is reduced to …
Abstract A near-optimal feedback control law for a distributed parameter system with a quadratic performance index is obtained by the method of trajectory approximation. The system equation is reduced to an approximate system of ordinary differential equations by the Galerkin—Kantorovich method, and then Pontryagin's maximum principle is applied to show that the feedback control law is linear and can be obtained as the solution of a matrix Riccati equation. Numerical computations performed using Chebyshev polynomials as the Galerkin weighting functions on the equation for a heat exchanger with wall flux forcing indicate that four thermocouples are enough to attain virtual optimality.
We present a new “lifting” approach for the solution of nonlinear optimization problems (NLPs) that have objective and constraint functions with intermediate variables. Introducing these as additional degrees of freedom …
We present a new “lifting” approach for the solution of nonlinear optimization problems (NLPs) that have objective and constraint functions with intermediate variables. Introducing these as additional degrees of freedom into the original problem, combined with adding suitable new constraints to ensure equivalence of the problems, we propose to solve this augmented system instead of the original system by a Newton-type method. This often offers advantages in terms of convergence rates and region of attraction. The main contribution of this article is an efficient algorithmic trick to generate the quantities needed for a Newton-type method on the augmented (“lifted”) system with (a) almost no additional computational cost per iteration compared to a nonlifted Newton method, and (b) with negligible programming burden. We derive lifted schemes for Newton's method, as well as for constrained Gauss–Newton and adjoint based sequential quadratic programming (SQP) methods, and show equivalence of the new efficiently lifted approaches with “full-space” lifted Newton iterations. We establish conditions on the intermediate functions that imply faster local quadratic convergence for lifted versus nonlifted Newton iterations, a phenomenon often observed in practice but not yet explained theoretically. Finally, we compare numerically the behavior of the lifted approach with the nonlifted approach on several test problems, including a large scale example with 27 million intermediate variables. The algorithms and examples are available as open-source code in the C++ package LiftOpt.
This paper deals with dynamic optimization of hybrid differential algebraic systems (Hybrid DAE) with explicit transitions, by means of direct collocation. Modes are described by binary variables that are treated …
This paper deals with dynamic optimization of hybrid differential algebraic systems (Hybrid DAE) with explicit transitions, by means of direct collocation. Modes are described by binary variables that are treated as continuous variables during the dynamic optimization, then a specific rounding procedure coupled with an adaptive mesh refinement is applied in an iterative way. The method is applied to a benchmark hybrid model of a Combined Heat and Power plant with heat storage.
The object-oriented software package OOQP for solving convex quadratic programming problems (QP) is described. The primal-dual interior point algorithms supplied by OOQP are implemented in a way that is largely …
The object-oriented software package OOQP for solving convex quadratic programming problems (QP) is described. The primal-dual interior point algorithms supplied by OOQP are implemented in a way that is largely independent of the problem structure. Users may exploit problem structure by supplying linear algebra, problem data, and variable classes that are customized to their particular applications. The OOQP distribution contains default implementations that solve several important QP problem types, including general sparse and dense QPs, bound-constrained QPs, and QPs arising from support vector machines and Huber regression. The implementations supplied with the OOQP distribution are based on such well known linear algebra packages as MA27/57, LAPACK, and PETSc. OOQP demonstrates the usefulness of object-oriented design in optimization software development, and establishes standards that can be followed in the design of software packages for other classes of optimization problems. A number of the classes in OOQP may also be reusable directly in other codes.
Time-optimal path following for robots considers the problem of moving along a predetermined Cartesian geometric path in minimum time. In practice this path need not be followed exactly, but within …
Time-optimal path following for robots considers the problem of moving along a predetermined Cartesian geometric path in minimum time. In practice this path need not be followed exactly, but within a certain tolerance; so that the motion may be executed faster. In this paper, we define this deviation as a tube around the given geometric path. This transforms the path following problem into a tube following problem. However, unlike the former, the latter is not convex.We propose a problem formulation that can still be solved efficiently, as illustrated by some numerical examples.
SUNDIALS is a suite of advanced computational codes for solving large-scale problems that can be modeled as a system of nonlinear algebraic equations, or as initial-value problems in ordinary differential …
SUNDIALS is a suite of advanced computational codes for solving large-scale problems that can be modeled as a system of nonlinear algebraic equations, or as initial-value problems in ordinary differential or differential-algebraic equations. The basic versions of these codes are called KINSOL, CVODE, and IDA, respectively. The codes are written in ANSI standard C and are suitable for either serial or parallel machine environments. Common and notable features of these codes include inexact Newton-Krylov methods for solving large-scale nonlinear systems; linear multistep methods for time-dependent problems; a highly modular structure to allow incorporation of different preconditioning and/or linear solver methods; and clear interfaces allowing for users to provide their own data structures underneath the solvers. We describe the current capabilities of the codes, along with some of the algorithms and heuristics used to achieve efficiency and robustness. We also describe how the codes stem from previous and widely used Fortran 77 solvers, and how the codes have been augmented with forward and adjoint methods for carrying out first-order sensitivity analysis with respect to model parameters or initial conditions.
Representations of the SO(3) rotation group are crucial for airborne and aerospace applications. Euler angles is a popular representation in many applications, but yield models having singular dynamics. This issue …
Representations of the SO(3) rotation group are crucial for airborne and aerospace applications. Euler angles is a popular representation in many applications, but yield models having singular dynamics. This issue is addressed via non-singular representations, operating in dimensions higher than 3. Unit quaternions and the Direction Cosine Matrix are the best known non-singular representations, and favoured in challenging aeronautic and aerospace applications. All nonsingular representations yield invariants in the model dynamics, i.e. a set of nonlinear algebraic conditions that must be fulfilled by the model initial conditions, and that remain fulfilled over time. However, due to numerical integration errors, these conditions tend to become violated when using standard integrators, making the model inconsistent with the physical reality. This issue poses some challenges when non-singular representations are deployed in optimal control. In this paper, we propose a simple technique to address the issue for classical integration schemes, establish formally its properties, and illustrate it on the optimal control of a satellite.
Convex optimization problems arise frequently in many different fields. A comprehensive introduction to the subject, this book shows in detail how such problems can be solved numerically with great efficiency. …
Convex optimization problems arise frequently in many different fields. A comprehensive introduction to the subject, this book shows in detail how such problems can be solved numerically with great efficiency. The focus is on recognizing convex optimization problems and then finding the most appropriate technique for solving them. The text contains many worked examples and homework exercises and will appeal to students, researchers and practitioners in fields such as engineering, computer science, mathematics, statistics, finance, and economics.
My final lines go to my girlfriend Simone Evke.Thank you for your love, for your patience and support whenever my mind was entangled in upper
My final lines go to my girlfriend Simone Evke.Thank you for your love, for your patience and support whenever my mind was entangled in upper
Methods and software for derivative-based numerical optimization in general and simulation-based optimization in particular have seen a large rise in popularity over the past 30 years. Still, due to practical …
Methods and software for derivative-based numerical optimization in general and simulation-based optimization in particular have seen a large rise in popularity over the past 30 years. Still, due to practical difficulties in implementing many of the methods in a fast and reliable manner, it remains an underused technology both in academia and in industry. To address this, we present a set of methods and tools with the aim of making techniques for dynamic optimization more accessible. In particular, we present CasADi, an open-source software framework for numerical optimization and algorithmic differentiation (AD) that offers a level of abstraction which is lower than algebraic modeling languages, but higher than conventional AD tools. We also discuss several of the many application problems which have already been addressed with CasADi by researchers from diverse fields.
Direct optimal control methods first discretize a continuous-time Optimal Control Problem (OCP) and then solve the resulting Nonlinear Program (NLP). Sequential Quadratic Programming (SQP) is a popular family of algorithms …
Direct optimal control methods first discretize a continuous-time Optimal Control Problem (OCP) and then solve the resulting Nonlinear Program (NLP). Sequential Quadratic Programming (SQP) is a popular family of algorithms to solve this finite dimensional optimization problem. In the specific case of a least squares cost, the Generalized Gauss-Newton (GGN) method is a popular approach which works very well under some assumptions. This paper proposes a Sequential Convex Quadratic Programming (SCQP) scheme which exploits additional convexities in the NLP in order to generalize the GGN algorithm, possibly extend its applicability and improve its local convergence. These properties are studied in detail for the proposed SCQP algorithm, which will be compared to the classical GGN method using a numerical case study of the optimal control of an inverted pendulum.
This is the first entry-level book on algorithmic (also known as automatic) differentiation (AD), providing fundamental rules for the generation of first- and higher-order tangent-linear and adjoint code. The author …
This is the first entry-level book on algorithmic (also known as automatic) differentiation (AD), providing fundamental rules for the generation of first- and higher-order tangent-linear and adjoint code. The author covers the mathematical underpinnings as well as how to apply these observations to real-world numerical simulation programs. Readers will find many examples and exercises, including hints to solutions. Also included are the prototype AD tools dco and dcc for use with the examples and exercises. The derivative code compiler dcc provides first- and higher-order tangent-linear and adjoint modes for a limited subset of C/C++. In addition, readers will have access to a supplementary website containing sources of all software discussed in the book, additional exercises and comments on their solutions (growing over the coming years), links to other sites on AD, and errata. Audience: This book is intended for undergraduate and graduate students in computational science, engineering, and finance as well as applied mathematics and computer science. It will provide researchers and developers at all levels with an intuitive introduction to AD.
Convex optimization problems arise frequently in many different fields. This book provides a comprehensive introduction to the subject, and shows in detail how such problems can be solved numerically with …
Convex optimization problems arise frequently in many different fields. This book provides a comprehensive introduction to the subject, and shows in detail how such problems can be solved numerically with great efficiency. The book begins with the basic elements of convex sets and functions, and then describes various classes of convex optimization problems. Duality and approximation techniques are then covered, as are statistical estimation techniques. Various geometrical problems are then presented, and there is detailed discussion of unconstrained and constrained minimization problems, and interior-point methods. The focus of the book is on recognizing convex optimization problems and then finding the most appropriate technique for solving them. It contains many worked examples and homework exercises and will appeal to students, researchers and practitioners in fields such as engineering, computer science, mathematics, statistics, finance and economics.
This work is devoted to demonstration of the analysis on optimizing the output power harvested from vibration energy harvester.
This work is devoted to demonstration of the analysis on optimizing the output power harvested from vibration energy harvester.
This paper is about a real-time model predictive control (MPC) algorithm for a particular class of model based controllers, whose objective consists of a nominal tracking objective and an additional …
This paper is about a real-time model predictive control (MPC) algorithm for a particular class of model based controllers, whose objective consists of a nominal tracking objective and an additional learning objective. Here, the construction of the learning term is based on economic optimal experiment design criteria. It is added to the MPC objective in order to excite the system from time-to-time on purpose in order to improve the accuracy of the state and parameter estimates in the presence of incomplete or noise affected measurements. A particular focus of this paper is on so-called self-reflective model predictive control schemes, which have the property that the additional learning term can be interpreted as the expected loss of optimality of the controller in the presence of random measurement errors. The main contribution of this paper is a formulation-tailored algorithm, which exploits the particular structure of self-reflective MPC problems in order to speed-up the online computation. It is shown that, in contrast to generic state-of-the-art optimal control problem solvers, the proposed algorithm can solve the self-reflective optimization problems with reasonable additional computational effort and in real-time. The advantages of the proposed real-time scheme are illustrated by applying the algorithm to a nonlinear process control problem in the presence of measurement errors and process noise.
Airborne wind energy systems are capable of extracting energy from higher wind speeds at higher altitudes. The configuration considered in this paper is based on a tethered kite flown in …
Airborne wind energy systems are capable of extracting energy from higher wind speeds at higher altitudes. The configuration considered in this paper is based on a tethered kite flown in a pumping orbit. This pumping cycle generates energy by winching out at high tether forces and driving a generator while flying figures-of-eight, or lemniscates, as crosswind pattern. Then, the tether is reeled in while keeping the kite at a neutral position, thus leaving a net amount of generated energy. In order to achieve an economic operation, optimization of pumping cycles is of great interest. In this paper, first the principles of airborne wind energy will be briefly revisited. The first contribution is a singularity-free model for the tethered kite dynamics in quaternion representation, where the model is derived from first principles. The second contribution is an optimal control formulation and numerical results for complete pumping cycles. Based on the developed model, the setup of the optimal control problem (OCP) is described in detail along with its numerical solution based on the direct multiple shooting method in the CasADi optimization environment. Optimization results for a pumping cycle consisting of six lemniscates show that the approach is capable to find an optimal orbit in a few minutes of computation time. For this optimal orbit, the power output is increased by a factor of two compared to a sophisticated initial guess for the considered test scenario.