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Active matter systems, composed of self-propelled agents that convert energy into directed motion, exhibit a wide range of emergent behaviors, such as motility-induced phase separation, flocking, and swarming. These phenomena, … Active matter systems, composed of self-propelled agents that convert energy into directed motion, exhibit a wide range of emergent behaviors, such as motility-induced phase separation, flocking, and swarming. These phenomena, observed across natural and engineered systems, hold immense potential for applications in programmable materials, directed assembly, and micro-robotics. However, precisely controlling their macroscopic continuum fields, e.g., density or flux, remains a significant challenge due to the complexity of multibody interactions and correlated particle dynamics. To address this challenge, we present a framework that combines physics-informed machine learning with Model Predictive Control. Our approach learns a closure model for complex particle interactions while incorporating known physical principles, resulting in an accurate predictive model suitable for real-time control. By integrating this model into a Model Predictive Control framework, we enable systematic optimization of control actions that can guide the system toward desired macroscopic behaviors. Through two illustrative examples, we showcase the versatility of the framework. First, we control the spatial distribution of particles by splitting them into two groups and dynamically juggling their densities. Second, we simultaneously control both the number density and the mean flux, guiding the latter to follow a prescribed sinusoidal profile. These results highlight the framework's potential to systematically control complex dynamics in active matter systems and provide a foundation for broader applications in programmable and adaptive materials.
We present the first general stability results for nonlinear offset-free model predictive control (MPC). Despite over twenty years of active research, the offset-free MPC literature has not shaken the assumption … We present the first general stability results for nonlinear offset-free model predictive control (MPC). Despite over twenty years of active research, the offset-free MPC literature has not shaken the assumption of closed-loop stability for establishing offset-free performance. In this paper, we present a nonlinear offset-free MPC design that is robustly stable with respect to the tracking errors, and thus achieves offset-free performance, despite plant-model mismatch and persistent disturbances. Key features and assumptions of this design include quadratic costs, differentiability of the plant and model functions, constraint backoffs at steady state, and a robustly stable state and disturbance estimator. We first establish nominal stability and offset-free performance. Then, robustness to state and disturbance estimate errors and setpoint and disturbance changes is demonstrated. Finally, the results are extended to sufficiently small plant-model mismatch. The results are illustrated by numerical examples.
We consider the asymptotic stability of MPC under plant-model mismatch, considering primarily models where the origin remains a steady state despite mismatch. Our results differ from prior results on the … We consider the asymptotic stability of MPC under plant-model mismatch, considering primarily models where the origin remains a steady state despite mismatch. Our results differ from prior results on the inherent robustness of MPC, which guarantee only convergence to a neighborhood of the origin, the size of which scales with the magnitude of the mismatch. For MPC with quadratic costs, continuous differentiability of the system dynamics is sufficient to demonstrate exponential stability of the closed-loop system despite mismatch. For MPC with general costs, a joint comparison function bound and scaling condition guarantee asymptotic stability despite mismatch. The results are illustrated in both algebraic and engineering examples. The tools developed to establish these results can address the stability of offset-free MPC, an open and interesting question in the MPC research literature.
The purpose of this note is to summarize the arguments required to derive the results appearing in robust minmax control of linear dynamical systems using a quadratic stage cost. The … The purpose of this note is to summarize the arguments required to derive the results appearing in robust minmax control of linear dynamical systems using a quadratic stage cost. The main results required in robust minmax control are Corollary 19 and Proposition 20. Moreover, the solution to the trust-region problem given in Proposition 15 and Lemma 16 may be of more general interest.
Maximum likelihood identification of linear time-invariant models is a difficult problem because it is, in general, a nonlinear semidefinite program, with semidefinite covariance matrix arguments and semidefinite filter stability constraints. … Maximum likelihood identification of linear time-invariant models is a difficult problem because it is, in general, a nonlinear semidefinite program, with semidefinite covariance matrix arguments and semidefinite filter stability constraints. To enforce filter stability, we establish a general theory of closed constraints on the system eigenvalues using LMI regions. To solve the identification problem, we employ a Cholesky factorization method that reduces the semidefinite program to a standard nonlinear program. Finally, we apply the identification algorithm to a class of linear plant and disturbance models commonly used in offset-free model predictive control applications. Specifically, we consider models that are structured with uncontrollable, integrating disturbance states. We solve this disturbance modeling problem, and validate the resulting controller and estimator performance, in two real-world case studies: first, a low-cost benchmark temperature control laboratory, and second, an industrial-scale chemical reactor at Eastman Chemical's Kingsport plant.
Observers for PDEs are themselves PDEs. Therefore, producing real time estimates with such observers is computationally burdensome. For both finite-dimensional and ODE systems, moving-horizon estimators (MHE) are operators whose output … Observers for PDEs are themselves PDEs. Therefore, producing real time estimates with such observers is computationally burdensome. For both finite-dimensional and ODE systems, moving-horizon estimators (MHE) are operators whose output is the state estimate, while their inputs are the initial state estimate at the beginning of the horizon as well as the measured output and input signals over the moving time horizon. In this paper we introduce MHEs for PDEs which remove the need for a numerical solution of an observer PDE in real time. We accomplish this using the PDE backstepping method which, for certain classes of both hyperbolic and parabolic PDEs, produces moving-horizon state estimates explicitly. Precisely, to explicitly produce the state estimates, we employ a backstepping transformation of a hard-to-solve observer PDE into a target observer PDE, which is explicitly solvable. The MHEs we propose are not new observer designs but simply the explicit MHE realizations, over a moving horizon of arbitrary length, of the existing backstepping observers. Our PDE MHEs lack the optimality of the MHEs that arose as duals of MPC, but they are given explicitly, even for PDEs. In the paper we provide explicit formulae for MHEs for both hyperbolic and parabolic PDEs, as well as simulation results that illustrate theoretically guaranteed convergence of the MHEs.
Knowledge of the process, measurement, and cross noise covariance matrices (denoted Q, R, and S, respectively) is necessary for tasks such as state estimation and performance monitoring. Several different types … Knowledge of the process, measurement, and cross noise covariance matrices (denoted Q, R, and S, respectively) is necessary for tasks such as state estimation and performance monitoring. Several different types of algorithms have been developed to estimate these parameters from plant output data. Chief among them are the so-called correlation methods, such as autocovariance least squares (ALS). Despite the advances in covariance estimation algorithms, relatively little attention has been given to the topic of parameter identifiability. This paper discusses the limitations of when Q, R, and S are identifiable from output data. In particular, it is shown that for stable, linear time-invariant systems, the Kalman predictor gain, but not Q, R, and S, can be uniquely identified from the steady-state output autocovariance. Constrained ALS problems and the extension of the ALS problem to nonlinear and linear time-varying systems are also discussed.
We present an extension of the CasADi numerical optimization framework that allows arbitrary order NLP sensitivities to be calculated automatically and efficiently. The approach, which can be used together with … We present an extension of the CasADi numerical optimization framework that allows arbitrary order NLP sensitivities to be calculated automatically and efficiently. The approach, which can be used together with any NLP solver available in CasADi, is based on a sparse QR factorization and an implementation of a primal-dual active set method. The whole toolchain is freely available as open-source software and allows generation of thread-safe, self-contained C code with small memory footprint. We illustrate the toolchain using three examples; a sparse QP, an optimal control problem and a parameter estimation problem.
This paper presents an efficient computational method for solving the input-constrained, continuous time, infinite horizon, linear quadratic regulator problem to within a user specified tolerance. The infinite dimensional input trajectory … This paper presents an efficient computational method for solving the input-constrained, continuous time, infinite horizon, linear quadratic regulator problem to within a user specified tolerance. The infinite dimensional input trajectory is approximated with a piecewise linear function on a finite time discretization to ensure input constraint satisfaction. This approximate problem is then a standard finite dimensional quadratic program and is solved by conventional methods, generating an upper bound for the optimal value function. The finite time discretization is then refined by subdividing the intervals estimated to cause the largest decrease in the cost function. Convergence of the solution of this discretized problem towards the optimal continuous-time solution, as the discretization is refined, is proved. Exploiting the strict convexity of the original infinite dimensional problem, the gradient of the cost function with respect to the continuous-time input can be computed to generate a lower bound for the optimal cost. For computational efficiency, a lower bound for the solution of the discretized control at a very fine discretization can be used instead. The algorithm terminates when the difference between the upper and lower bounds meets a user supplied tolerance.
We propose in this note a method for computing the solution to the infinite horizon continuous-time constrained linear quadratic regulator. The method is based on two main ingredients: a multigrid … We propose in this note a method for computing the solution to the infinite horizon continuous-time constrained linear quadratic regulator. The method is based on two main ingredients: a multigrid method for placing a finite number of time intervals, and a piece-wise linear parameterization of the input within the intervals. The input values at the decision-time points and slopes within the time intervals are computed via quadratic programs (QPs). The grids are gradually refined to efficiently improve the accuracy of the solution, and the required matrices and vectors for all QPs are computed offline and stored to improve the online efficiency. Two examples are presented to show the main characteristics of the proposed method.
An infinite horizon controller that allows incorporation of input and state constraints in a receding horizon feedback strategy is developed. For both stable and unstable linear plants, feasibility of the … An infinite horizon controller that allows incorporation of input and state constraints in a receding horizon feedback strategy is developed. For both stable and unstable linear plants, feasibility of the contraints guarantees nominal closed-loop stability for all choices of the tuning parameters in the control law. The constraints' feasibility can be checked efficiently with a linear program. It is always possible to remove state constraints in the early portion of the infinite horizon to make them feasible. The controller's implementation requires only the solution of finite-dimensional quadratic programs.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>
This paper provides an overview of modelling, measurement, and control issues arising in systems modelled by population balances. The population balance is a partial differential equation describing the dynamics of … This paper provides an overview of modelling, measurement, and control issues arising in systems modelled by population balances. The population balance is a partial differential equation describing the dynamics of some general particle size distribution. The independent variables in the PDE are time and one or more internal particle coordinates, such as size, age, activity, etc., that fully characterize the state of the particle. Population balance models therefore can present a different set of issues than those arising in standard distributed parameter systems in which the independent variables are time and spatial location. The remaining process states, such as concentrations and temperature, are modelled - with integro-differential equations. The integro-differential equations and the population balance's nonlocal boundary conditions are the sources of interesting and problematic dynamic behavior in continuous processes. This behavior includes open-loop instability and long period oscillations. The solution of optimal control profiles for batch processes is also difficult and computationally expensive.

Commonly Cited References

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 overview paper reviews numerical methods for solution of optimal control problems in real-time, as they arise in nonlinear model predictive control (NMPC) as well as in moving horizon estimation … This overview paper reviews numerical methods for solution of optimal control problems in real-time, as they arise in nonlinear model predictive control (NMPC) as well as in moving horizon estimation (MHE). In the first part, we review numerical optimal control solution methods, focussing exclusively on a discrete time setting. We discuss several algorithmic ”building blocks” that can be combined to a multitude of algorithms. We start by discussing the sequential and simultaneous approaches, the first leading to smaller, the second to more structured optimization problems. The two big families of Newton type optimization methods, Sequential Quadratic Programming (SQP) and Interior Point (IP) methods, are presented, and we discuss how to exploit the optimal control structure in the solution of the linear-quadratic subproblems, where the two major alternatives are “condensing” and band structure exploiting approaches. The second part of the paper discusses how the algorithms can be adapted to the real-time challenge of NMPC and MHE. We recall an important sensitivity result from parametric optimization, and show that a tangential solution predictor for online data can easily be generated in Newton type algorithms. We point out one important difference between SQP and IP methods: while both methods are able to generate the tangential predictor for fixed active sets, the SQP predictor even works across active set changes. We then classify many proposed real-time optimization approaches from the literature into the developed categories.
We propose in this note a method for computing the solution to the infinite horizon continuous-time constrained linear quadratic regulator. The method is based on two main ingredients: a multigrid … We propose in this note a method for computing the solution to the infinite horizon continuous-time constrained linear quadratic regulator. The method is based on two main ingredients: a multigrid method for placing a finite number of time intervals, and a piece-wise linear parameterization of the input within the intervals. The input values at the decision-time points and slopes within the time intervals are computed via quadratic programs (QPs). The grids are gradually refined to efficiently improve the accuracy of the solution, and the required matrices and vectors for all QPs are computed offline and stored to improve the online efficiency. Two examples are presented to show the main characteristics of the proposed method.
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.
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).
This paper provides new conditions under which optimal controls are Lipschitz continuous for dynamic optimization problems with functional inequality constraints, a control constraint expressed in terms of a general closed … This paper provides new conditions under which optimal controls are Lipschitz continuous for dynamic optimization problems with functional inequality constraints, a control constraint expressed in terms of a general closed convex set and a coercive cost function. It is shown that the linear independence condition on active state constraints, present in the earlier literature, can be replaced by a less restrictive, positive linear independence condition that requires linear independence merely with respect to nonnegative weighting parameters. Smoothness conditions on the data, imposed in earlier work, are also relaxed. The new conditions for Lipschitz continuity of optimal controls are obtained by a detailed analysis of the implications of first order optimality conditions in the form of a nonsmooth maximum principle.
Abstract Nearly all algorithms for linear model predictive control (MPC) either rely on the solution of convex quadratic programs (QPs) in real time, or on an explicit precalculation of this … Abstract Nearly all algorithms for linear model predictive control (MPC) either rely on the solution of convex quadratic programs (QPs) in real time, or on an explicit precalculation of this solution for all possible problem instances. In this paper, we present an online active set strategy for the fast solution of parametric QPs arising in MPC. This strategy exploits solution information of the previous QP under the assumption that the active set does not change much from one QP to the next. Furthermore, we present a modification where the CPU time is limited in order to make it suitable for strict real‐time applications. Its performance is demonstrated with a challenging test example comprising 240 variables and 1191 inequalities, which depends on 57 parameters and is prohibitive for explicit MPC approaches. In this example, our strategy allows CPU times of well below 100 ms per QP and was about one order of magnitude faster than a standard active set QP solver. Copyright © 2007 John Wiley &amp; Sons, Ltd.
Some implementations of interior-point algorithms obtain their search directions by solving symmetric indefinite systems of linear equations. The conditioning of the coefficient matrices in these so-called augmented systems deteriorates on … Some implementations of interior-point algorithms obtain their search directions by solving symmetric indefinite systems of linear equations. The conditioning of the coefficient matrices in these so-called augmented systems deteriorates on later iterations, as some of the diagonal elements grow without bound. Despite this apparent difficulty, the steps produced by standard factorization procedures are often accurate enough to allow the interior-point method to converge to high accuracy. When the underlying linear program is nondegenerate, we show that convergence to arbitrarily high accuracy occurs, at a rate that closely approximates the theory. We also explain and demonstrate what happens when the linear program is degenerate, where convergence to acceptable accuracy (but not arbitrarily high accuracy) is usually obtained.
An infinite horizon controller that allows incorporation of input and state constraints in a receding horizon feedback strategy is developed. For both stable and unstable linear plants, feasibility of the … An infinite horizon controller that allows incorporation of input and state constraints in a receding horizon feedback strategy is developed. For both stable and unstable linear plants, feasibility of the contraints guarantees nominal closed-loop stability for all choices of the tuning parameters in the control law. The constraints' feasibility can be checked efficiently with a linear program. It is always possible to remove state constraints in the early portion of the infinite horizon to make them feasible. The controller's implementation requires only the solution of finite-dimensional quadratic programs.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>
We solve the problem of stabilization of a class of linear first-order hyperbolic systems featuring n rightward convecting transport PDEs and one leftward convecting transport PDE. We design a controller, … We solve the problem of stabilization of a class of linear first-order hyperbolic systems featuring n rightward convecting transport PDEs and one leftward convecting transport PDE. We design a controller, which requires a single control input applied on the leftward convecting PDE's right boundary, and an observer, which employs a single sensor on the same PDE's left boundary. We prove exponential stability of the origin of the resulting plant-observer-controller system in the spatial L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> -sense.
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.
Summary A broadly applicable algorithm for computing maximum likelihood estimates from incomplete data is presented at various levels of generality. Theory showing the monotone behaviour of the likelihood and convergence … Summary A broadly applicable algorithm for computing maximum likelihood estimates from incomplete data is presented at various levels of generality. Theory showing the monotone behaviour of the likelihood and convergence of the algorithm is derived. Many examples are sketched, including missing value situations, applications to grouped, censored or truncated data, finite mixture models, variance component estimation, hyperparameter estimation, iteratively reweighted least squares and factor analysis.
A new approach for computing a sparsity pattern for a Hessian is presented: nonlinearity information is propagated through the function evaluation yielding the nonzero structure. A complexity analysis of the … A new approach for computing a sparsity pattern for a Hessian is presented: nonlinearity information is propagated through the function evaluation yielding the nonzero structure. A complexity analysis of the proposed algorithm is given. Once the sparsity pattern is available, coloring algorithms can be applied to compute a seed matrix. To evaluate the product of the Hessian and the seed matrix, a vector version for evaluating second order adjoints is analysed. New drivers of ADOL-C are provided implementing the presented algorithms. Runtime analyses are given for some problems of the CUTE collection.
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.
This paper gives an approach to implementing a second-order primal-dual interior point method. It uses a Taylor polynomial of second order to approximate a primal-dual trajectory. The computations for the … This paper gives an approach to implementing a second-order primal-dual interior point method. It uses a Taylor polynomial of second order to approximate a primal-dual trajectory. The computations for the second derivative are combined with the computations for the centering direction. Computations in this approach do not require that primal and dual solutions be feasible. Expressions are given to compute all the higher-order derivatives of the trajectory of interest. The implementation ensures that a suitable potential function is reduced by a constant amount at each iteration. There are several salient features of this approach. An adaptive heuristic for estimating the centering parameter is given. The approach used to compute the step length is also adaptive. A new practical approach to compute the starting point is given. This approach treats primal and dual problems symmetrically. Computational results on a subset of problems available from netlib are given. On mutually tested problems the results show that the proposed method requires approximately 40 percent fewer iterations than the implementation proposed in Lustig, Marsten, and Shanno [Tech. Rep. TR J-89-11, Georgia Inst. of Technology, Atlanta, 1989]. It requires approximately 50 percent fewer iterations than the dual affine scaling method in Adler, Karmarkar, Resende, and Veiga [Math. Programming, 44 (1989), pp. 297–336], and 35 percent fewer iterations than the second-order dual affine scaling method in the same paper. The new approach for estimating the centering parameter and finding the step length and the starting point have contributed to the reduction in the number of iterations. However, the contribution due to the use of second derivative is most significant. On the tested problems, on the average the implementation shown was found to be approximately two times faster than OBl (version 02/90) described in Lustig, Marsten, and Shanno and 2.5 times faster than MINOS 5.3 described in Murtagh and Saunders [Tech. Rep. SOL 83-20, Dept. of Operations Research, Stanford Univ., Stanford, CA, 1983].
We investigate a modified Cholesky algorithm typical of those used in most interior-point codes for linear programming. Cholesky-based interior-point codes are popular for three reasons: their implementation requires only minimal … We investigate a modified Cholesky algorithm typical of those used in most interior-point codes for linear programming. Cholesky-based interior-point codes are popular for three reasons: their implementation requires only minimal changes to standard sparse Cholesky algorithms (allowing us to take full advantage of software written by specialists in that area); they tend to be more efficient than competing approaches that use alternative factorizations; and they perform robustly on most practical problems, yielding good interior-point steps even when the coefficient matrix of the main linear system to be solved for the step components is ill conditioned. We investigate this surprisingly robust performance by using analytical tools from matrix perturbation theory and error analysis, illustrating our results with computational experiments. Finally, we point out the potential limitations of this approach.
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.
In this paper, a problem of boundary stabilization of a class of linear parabolic partial integro-differential equations (P(I)DEs) in one dimension is considered using the method of backstepping, avoiding spatial … In this paper, a problem of boundary stabilization of a class of linear parabolic partial integro-differential equations (P(I)DEs) in one dimension is considered using the method of backstepping, avoiding spatial discretization required in previous efforts. The problem is formulated as a design of an integral operator whose kernel is required to satisfy a hyperbolic P(I)DE. The kernel P(I)DE is then converted into an equivalent integral equation and by applying the method of successive approximations, the equation's well posedness and the kernel's smoothness are established. It is shown how to extend this approach to design optimally stabilizing controllers. An adaptation mechanism is developed to reduce the conservativeness of the inverse optimal controller, and the performance bounds are derived. For a broad range of physically motivated special cases feedback laws are constructed explicitly and the closed-loop solutions are found in closed form. A numerical scheme for the kernel P(I)DE is proposed; its numerical effort compares favorably with that associated with operator Riccati equations.
Abstract The differential‐algebraic equation (DAE) optimization problem is transformed to a nonlinear programming problem by applying collocation on finite elements. The resulting problem is solved using a reduced space successive … Abstract The differential‐algebraic equation (DAE) optimization problem is transformed to a nonlinear programming problem by applying collocation on finite elements. The resulting problem is solved using a reduced space successive quadratic programming (rSQP) algorithm. Here, the variable space is partitioned into range and null spaces. Partitioning by choosing a pivot sequence for an LU factorization with partial pivoting allows us to detect unstable modes in the DAE system, which can now be stabilized without imposing new boundary conditions. As a result, the range space is decomposed in a single step by exploiting the overall sparsity of the collocation matrix; which performs better than the two‐step condensation method used in standard collocation solvers. To deal with ill‐conditioned constraints, we also extend the rSQP algorithm to include dogleg steps for the range space step that solves the collocation equations. The performance of this algorithm was tested on two well known unstable problems and on three chemical engineering examples including two reactive distillation columns and a plug‐flow reactor with free radicals. One of these is u batch column where an equilibrium reaction takes place. The second reactive distillation problem is the startup of a continuous column with competitive reactions. These optimization problems, which include more than 150 DAEs, ure solved in less than 7 CPU minutes on workstation class computers.
A new algorithm for computing integrals involving the matrix exponential is given. The method employs diagonal Padé approximation with scaling and squaring. Rigorous truncation error bounds are given and incorporated … A new algorithm for computing integrals involving the matrix exponential is given. The method employs diagonal Padé approximation with scaling and squaring. Rigorous truncation error bounds are given and incorporated in a Fortran subroutine. The computational aspects of this program are discussed and compared with existing techniques.
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.
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.
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.